0$, then $(1 - K\\bar K)^{\\ell} = 2^{\\ell-1}(1 - K\\bar K)$, so the sum can be broken into the two sums given in the lemma statement. The two special cases follow directly.\n \\end{proof}\n\n \\begin{lemma}\\label{Drinfeld-calculations}\n Let $n$ be a positive integer. Then, the Drinfeld element $u\\in D\\Kn$ satisfies\n \\begin{align*}\n u &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor|w|/2\\rfloor}}{2}((-1)^{n+|w|}(1-K\\bar K)+K+\\bar K)w\\bar w,\\\\\n u^{-1} &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor(|w|+1)/2\\rfloor}}{2}((-1)^{n}(1-K\\bar K)+K+\\bar K)w\\bar w,\\\\\n S(u) &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor|w|/2\\rfloor}}{2}((-1)^{|w|}(1-K\\bar K)+K+\\bar K)w\\bar w,\\\\\n uS(u) &= \\frac{(-1)^{n}}{2}(1 - K\\bar K) + (1+K\\bar K)\\sum_{w\\in W}(-1)^{\\lfloor|w|/2\\rfloor}2^{|w|-1}w\\bar w,\\\\\n uS(u)^{-1} &= (K\\bar K)^n.\n \\end{align*}\n \\end{lemma}\n We see that $S(u)=u$ only when $n$ is even, wherein $uK$ and $u\\bar K$ are both ribbon elements in $D\\Kn$. When $n$ is odd, $K$ and $\\bar K$ still implement the antipode, but it is no longer the case that $uK$ and $u\\bar K$ are ribbon elements since $\\bar K^2\\neq uS(u)^{-1}\\neq K^2$.\n \\begin{proof}\n By definition,\n \\begin{align}\n u &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor|w|/2\\rfloor}}{2}( (1+\\bar K)(-\\bar K)^{|w|}\\bar w w + (1-\\bar K)(-\\bar K)^{|w|}\\bar w Kw)\\nonumber\\\\ \n &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor(|w|+1)/2\\rfloor}}{2}(1+K+\\bar K-K\\bar K)\\bar w w.\\label{u-barww}\n \\end{align}\n By applying Lemma \\ref{sum-by-length} to the functions $f(k) = \\frac{(-1)^{\\lfloor (k+1)/2\\rfloor}}{2}K(1+K\\bar K)$ and $f(k) = \\frac{(-1)^{\\lfloor (k+1)/2\\rfloor}}{2}(1 - K\\bar K)$, we get, by the binomial theorem,\n \\begin{align*}\n &=\\sum_{w\\in W} \\frac{(-1)^{\\lfloor|w|/2\\rfloor}}{2}(K+\\bar K)w\\bar w +\\sum_{w\\in W}(-1)^{\\lfloor (|w|+1)/2\\rfloor}\\sum_{\\ell=0}^{n-|w|} \\binom{n-|w|}{\\ell}(-1)^{\\ell+|w|}2^{\\ell-1}(1 - K\\bar K)w\\bar w,\\\\\n &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor|w|/2\\rfloor}}{2}((-1)^{n+|w|}(1-K\\bar K) + K + \\bar K)w\\bar w.\n \\end{align*}\n By Proposition \\ref{Drinfeld-props}, we see that\n \\begin{align*}\n u^{-1} &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor|w|/2\\rfloor}}{2}(\\bar w(1+\\bar K)S^2(w) + \\bar w (1-\\bar K)S^2(Kw)),\\\\\n &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor|w|/2\\rfloor}}{2}(((-1)^{|w|}(1-K\\bar K)+K+\\bar K)\\bar ww).\n \\intertext{Again, we apply the two special cases of Lemma \\ref{sum-by-length} to get}\n &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor (|w|+1)/2\\rfloor}}{2}\\left((K+\\bar K) + \\sum_{\\ell=0}^{n-|w|} \\binom{n-|w|}{\\ell}(-1)^{\\ell + |w|}2^\\ell (1-K\\bar K))\\right)w\\bar w,\\\\\n &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor (|w|+1)/2\\rfloor}}{2}\\left((K+\\bar K) + (-1)^{n} (1-K\\bar K))\\right)w\\bar w.\n \\end{align*}\n Now, we compute $S(u)$ using Equation \\eqref{u-barww}:\n \\begin{align*}\n S(u) &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor(|w|+1)/2\\rfloor}}{2}(1+K+\\bar K-K\\bar K)(-K)^{|w|}w(-\\bar K)^{|w|}\\bar w,\\\\\n &= \\sum_{w\\in W} \\frac{(-1)^{\\lfloor|w|/2\\rfloor}}{2}((-1)^{|w|}(1-K\\bar K)+K+\\bar K)w\\bar w.\n \\end{align*}\n Next, we compute the product $uS(u)$. First, let $\\hat w = \\prod_{i\\in\\mathbf{I}(w)} \\xi_i\\bar\\xi_i$. Note that $\\xi_i\\bar\\xi_i\\xi_j\\bar\\xi_j = \\xi_j\\bar\\xi_j\\xi_i\\bar\\xi_i$ for $i\\neq j$, so this product does not need a specified ordering. Then, we can write $u$ and $S(u)$ as\n \\begin{align*}\n u &= \\sum_{w\\in W} \\frac{1}{2}((-1)^{n+|w|}(1-K\\bar K)+K+\\bar K)\\hat w,\\\\\n S(u) &= \\sum_{w\\in W} \\frac{1}{2}((-1)^{|w|}(1-K\\bar K)+K+\\bar K)\\hat w.\n \\end{align*}\n We compute $uS(u)$ as follows:\n \\begin{align*}\n uS(u) &= \\sum_{w_1,w_2\\in W} \\frac{1}{4}((-1)^{|w_1|+n}(1-K\\bar K)+K+\\bar K)\\hat w_1((-1)^{|w_2|}(1-K\\bar K)+K+\\bar K)\\hat w_2,\\\\\n &= \\sum_{w_1,w_2\\in W} \\frac{1}{2}((-1)^{|w_1|+|w_2|+n}(1 - K\\bar K) + 1 + K\\bar K)\\prod_{i\\in \\mathbf{I}(w_1)\\cap \\mathbf{I}(w_2)}\\xi_i\\bar \\xi_i\\xi_i\\bar \\xi_i\\prod_{i\\in \\mathbf{I}(w_1)\\vartriangle \\mathbf{I}(w_2)} \\xi_i\\bar \\xi_i,\\\\\n &= \\sum_{w_1,w_2\\in W} \\frac{1}{2}((-1)^{|w_1|+|w_2|+n}(1 - K\\bar K) + 1 + K\\bar K)(1 - K\\bar K)^{|\\mathbf{I}(w_1)\\cap \\mathbf{I}(w_2)|}\\hat{\\mathbf{w}}(\\mathbf{I}(w_1)\\cup \\mathbf{I}(w_2)),\n \\end{align*}\n where $\\vartriangle$ denotes symmetric difference and, if $I\\subseteq \\{1,\\dots, n\\}$, then $\\hat{\\mathbf{w}}(I)=(\\xi_1\\bar\\xi_1)^{a_1}\\dots(\\xi_n\\bar\\xi_n)^{a_n}$, where $a_i = 1$ if $i\\in I$ and $a_i = 0$ otherwise. We can reindex the sum to take advantage of the $(1-K\\bar K)$ multiplier as follows:\n \\begin{align*}\n &= \\sum_{\\mathbf{I}(w_1)\\cap \\mathbf{I}(w_2)=\\varnothing} \\frac{1}{2}(K + \\bar K)\\hat w_1\\hat w_2 + \\sum_{w_1, w_2\\in W} 2^{|\\mathbf{I}(w_1)\\cap \\mathbf{I}(w_2)|-1}(-1)^{|w_1|+|w_2|+n}(1 - K\\bar K)\\hat{\\mathbf{w}}(\\mathbf{I}(w_1)\\cup \\mathbf{I}(w_2)).\n \\intertext{Given $w\\in W$, consider all summands of the form $(1-K\\bar K)\\hat w$. For each $\\ell_1=0,\\dots,|w|$, there are $\\binom{|w|}{\\ell_1}$ different $w_1\\in W$ such that $\\mathbf{I}(w_1)\\subseteq \\mathbf{I}(w)$ and $|w_1|=\\ell_1$. Fix such a $w_1$. Then, for each $\\ell_2=|w|-\\ell_1,\\dots,|w|$, there are $\\binom{\\ell_1}{\\ell_1+\\ell_2-|w|}$ different $w_2\\in W$ with $|w_2|=\\ell_2$ and $\\hat{\\mathbf{w}}(\\mathbf{I}(w_1)\\cup \\mathbf{I}(w_2)) = \\hat w$. In this case $|\\mathbf{I}(w_1)\\cap \\mathbf{I}(w_2)|=\\ell_1+\\ell_2-|w|$. Next, we consider all summands of the form $(1+K\\bar K)\\hat w$. For fixed $w_1\\subseteq \\mathbf{I}(w)$, there is exactly one $w_2$ such that $(K+\\bar K)\\hat w_1\\hat w_2=\\hat w$. There are $2^{|w|}$ such $w_1$. Moreover, the summand $\\frac{1}{2}(1 + K\\bar K)\\hat w_1\\hat w_2$ depends only on $w$. Indexing by $m=|w|-\\ell_2$, the above sum can be rewritten as}\n &= \\sum_{w\\in W}2^{|w|-1}(1 + K\\bar K)\\hat w + \\sum_{w\\in W}\\sum_{\\ell_1=0}^{|w|}\\sum_{m=0}^{\\ell_1}\\binom{|w|}{\\ell_1}\\binom{\\ell_1}{m}2^{\\ell_1-m-1}(-1)^{\\ell_1+|w|-m+n}(1 - K\\bar K)\\hat{w}\\\\\n &= \\sum_{w\\in W}2^{|w|-1}(1 + K\\bar K)\\hat w + \\sum_{w\\in W}\\sum_{\\ell_1=0}^{|w|}\\frac{\\binom{|w|}{\\ell_1}(-1)^{\\ell_1+|w|+n}}{2}(1 - K\\bar K)\\hat{w}\\\\\n &= \\sum_{w\\in W}2^{|w|-1}(1 + K\\bar K)\\hat w + \\frac{(-1)^{n}}{2}(1-K\\bar K).\n \\end{align*}\n The equation for $uS(u)$ follows. Finally, note that \n $$(K\\bar K)^n ((-1)^\\ell(1-K\\bar K) + K + \\bar K) = (-1)^{n+\\ell}(1-K\\bar K)+K+\\bar K,$$ \n so $u = (K\\bar K)^n S(u)$, and the last equation follows.\n \\end{proof}\n \n By Theorem \\ref{sommer}, for odd $n$, the ribbon extension $\\RKn$ is obtained by formally adding a grouplike $\\tilde k$ to $D\\Kn$ such that $\\tilde k^2 = K\\bar K$ and $S^2(x) = \\tilde kx\\tilde k^{-1}$. The latter is equivalent to requiring that $\\tilde k$ commutes with $K, \\bar K$ and anticommutes with each $\\xi_i, \\bar\\xi_i$. Note that since $\\bar K = K\\tilde k^2$, it need not be included in the generators. Thus, we can provide a simple presentation of $\\RKn$. \n \\begin{corollary}\n Let $n$ be odd. The Hopf algebra $\\RKn$ is generated (as an algebra) by the elements $K, \\tilde k, \\xi_1, \\dots, \\xi_n, \\bar\\xi_1,\\dots,\\bar\\xi_n$ which satisfy the following relations: for $i\\neq j$,\n \\begin{align*}\n K^2 &= 1, & \\xi_i^2 &= 0, & \\xi_i\\xi_j &= -\\xi_j\\xi_i, & K\\xi_i &= -\\xi_i K,\\\\\n \\tilde k^4 &= 1, & \\bar\\xi_i^2 &= 0, & \\bar\\xi_i\\bar\\xi_j &= -\\bar\\xi_j\\bar\\xi_i, & \\tilde k\\bar\\xi_i &= -\\bar\\xi_i \\tilde k,\\\\\n K\\tilde k &= \\tilde k K, & \\bar\\xi_i\\xi_i &= 1 - \\tilde k^2 - \\xi_i\\bar\\xi_i, & \\xi_i\\bar\\xi_j &= -\\bar\\xi_j\\xi_i, & K\\bar\\xi_i&=-\\bar\\xi_i K, & \\tilde k\\xi_i &= -\\xi_i \\tilde k.\n \\end{align*}\n \\end{corollary}\n\n\n\\subsection{Representations of $\\RKn$}\n For even $n$, by Corollary \\ref{ribbon-reps-factor}, $\\Rep(\\RKn)\\cong \\Rep(D\\Kn)\\boxtimes \\Vecm$. For odd $n$, the behavior is quite similar, though the category does not factor so nicely. Throughout this section, we assume $n$ is odd.\n\n In \\cite{modulardata2024}, it is shown that there are four simple $D\\Kn$-modules, two of which are their own projective covers. There are two one-dimensional simple $D\\Kn$-modules $V_1$ and $V_{K\\bar K}$. The projective covers $P_1$ and $P_{K\\bar K}$ of these modules are each of dimension $2^{2n}$. There are two $2^n$-dimensional projective simple $D\\Kn$-modules $V_K$ and $V_{\\bar K}$. It follows from Theorems \\ref{simples-double} and \\ref{indec-projectives-restrict} that all of these counts double. There are eight simple $\\RKn$-modules. There are four one-dimensional simple $\\RKn$-modules $V_1^\\pm$ and $V_{K\\bar K}^\\pm$. The projective covers $P_1^\\pm$ and $P_{K\\bar K}^\\pm$ of these modules are each of dimension $2^{2n}$. There are four $2^n$-dimensional projective simple $\\RKn$-modules $V_K^\\pm$ and $V_{\\bar K}^\\pm$. We now give explicit descriptions of these modules in terms of our presentation of $\\RKn$.\n \n Aligning with the notation established in \\cite{modulardata2024}, the one-dimensional simple $\\RKn$-modules $V_{1}^+, V_{1}^-, V_{K\\bar K}^+, V_{K\\bar K}^-$ are described by \n \\begin{alignat*}{5}\n \\forall \\lambda\\in V_{1}^+&:& \\hspace{1cm}K\\cdot\\lambda &=& \\lambda, \\hspace{1cm}\\tilde k\\cdot \\lambda &=& \\lambda, \\hspace{1cm}\\xi_i\\cdot\\lambda=\\bar\\xi_i\\cdot\\lambda &=&\\, 0,\\\\\n \\forall \\lambda\\in V_{1}^-&:&\\hspace{1cm}K\\cdot\\lambda &=& \\lambda, \\hspace{1cm}\\tilde k\\cdot \\lambda &=&\\, -\\lambda, \\hspace{1cm}\\xi_i\\cdot\\lambda=\\bar\\xi_i\\cdot\\lambda &=&\\, 0,\\\\\n \\forall \\lambda\\in V_{K\\bar K}^+&:&\\hspace{1cm}K\\cdot\\lambda &=&\\, -\\lambda, \\hspace{1cm}\\tilde k\\cdot \\lambda &=& \\lambda, \\hspace{1cm}\\xi_i\\cdot\\lambda=\\bar\\xi_i\\cdot\\lambda &=&\\, 0,\\\\\n \\forall \\lambda\\in V_{K\\bar K}^-&:&\\hspace{1cm}K\\cdot\\lambda &=&\\, -\\lambda, \\hspace{1cm}\\tilde k\\cdot \\lambda &=&\\, -\\lambda,\\hspace{1cm} \\xi_i\\cdot\\lambda=\\bar\\xi_i\\cdot\\lambda &=&\\, 0.\n \\end{alignat*} \n Observe that $V_1^{\\pm}|_{D\\Kn}=V_1$, $V_{K\\bar K}^{\\pm}|_{D\\Kn}=V_{K\\bar K}$.\n The projective covers $P_1^{\\pm}$ and $P_{K\\bar K}^\\pm$ are given by\n \\begin{align*}\n P_1^+ &= \\RKn(1 + K)(1 + \\tilde k + \\tilde k^2 + \\tilde k^3),\\\\\n P_1^- &= \\RKn(1 + K)(1 - \\tilde k + \\tilde k^2 - \\tilde k^3),\\\\\n P_{K\\bar K}^+ &= \\RKn(1 - K)(1 + \\tilde k + \\tilde k^2 + \\tilde k^3),\\\\\n P_{K\\bar K}^- &= \\RKn(1 - K)(1 - \\tilde k + \\tilde k^2 - \\tilde k^3).\n \\end{align*}\n They are indeed projective since $P_1^+ \\oplus P_1^-\\oplus P_{K\\bar K}^+\\oplus P_{K\\bar K}^-\\oplus \\RKn(1 - \\tilde k^2) = \\RKn$. The covering maps $P\\to V$ are all generated by $(1\\pm K)(1\\pm \\tilde k + \\tilde k^2 \\pm \\tilde k^3)\\mapsto 1$. The dimensions of the $P_{1}^\\pm$ and $P_{K\\bar K}^\\pm$ are $2^{2n}$.\n\n Set, as vector spaces, $V_{K}^{\\pm}=V_{\\bar K}^{\\pm}=(\\bC^2)^{\\otimes n}$. Define the matrices $\\Xi = \\begin{bmatrix}\n 0 & \\sqrt{2}\\\\\n 0 & 0\n \\end{bmatrix}$ and $\\sigma_Z = \\begin{bmatrix}\n 1 & 0\\\\\n 0 & -1\n \\end{bmatrix}$. Set $\\Xi_j = \\sigma_Z^{\\otimes j-1}\\otimes \\Xi\\otimes I_2^{\\otimes n-j}$ and $\\bar\\Xi_i = \\sigma_Z^{\\otimes j-1}\\otimes \\Xi^T\\otimes I_2^{\\otimes n-j}$. Then, we define the following $\\RKn$-module actions: for $j=1,\\dots, n$,\n \\begin{alignat*}{6}\n \\forall v\\in V_{K}^+&:& \\hspace{1cm}K\\cdot v &=& \\sigma_Z^{\\otimes n} v, \\hspace{1cm}\\tilde k\\cdot v &=& i\\sigma_Z^{\\otimes n} v, \\hspace{1cm}\\xi_j\\cdot v &=&\\, \\Xi_j v, \\hspace{1cm}\\bar\\xi_j\\cdot v &=&\\, \\bar\\Xi_j v,\\\\\n \\forall v\\in V_{K}^-&:& \\hspace{1cm}K\\cdot v &=& \\sigma_Z^{\\otimes n} v, \\hspace{1cm}\\tilde k\\cdot v &=& -i\\sigma_Z^{\\otimes n} v, \\hspace{1cm}\\xi_j\\cdot v &=&\\, \\Xi_j v, \\hspace{1cm}\\bar\\xi_j\\cdot v &=&\\, \\bar\\Xi_j v,\\\\\n \\forall v\\in V_{\\bar K}^+&:& \\hspace{1cm}K\\cdot v &=& -\\sigma_Z^{\\otimes n} v, \\hspace{1cm}\\tilde k\\cdot v &=& i\\sigma_Z^{\\otimes n} v, \\hspace{1cm}\\xi_j\\cdot v &=&\\, \\Xi_j v, \\hspace{1cm}\\bar\\xi_j\\cdot v &=&\\, \\bar\\Xi_j v,\\\\\n \\forall v\\in V_{\\bar K}^-&:& \\hspace{1cm}K\\cdot v &=& -\\sigma_Z^{\\otimes n} v, \\hspace{1cm}\\tilde k\\cdot v &=& -i\\sigma_Z^{\\otimes n} v, \\hspace{1cm}\\xi_j\\cdot v &=&\\, \\Xi_j v, \\hspace{1cm}\\bar\\xi_j\\cdot v &=&\\, \\bar\\Xi_j v.\n \\end{alignat*}\n Observe that again $V_K^{\\pm}|_{D\\Kn}=V_K$ and $V_{\\bar K}^{\\pm}|_{D\\Kn}=V_{\\bar K}$. Thus, these modules are necessarily simple and are, in particular, singly generated by $|0\\rangle^{\\otimes n}$. Let $f:V_K^+\\to V_K^-$ be a $\\RKn$-linear map. By considering the action of $\\frac{\\xi_1\\bar\\xi_1}{2}\\dots\\frac{\\xi_n\\bar\\xi_n}{2}$, we see that $f(|0\\rangle^{\\otimes n}) = c|0\\rangle^{\\otimes n}$ for some $c\\in \\bC$. Therefore, \n $$-if(|0\\rangle^{\\otimes n}) = \\tilde k\\cdot f(|0\\rangle^{\\otimes n}) = f(\\tilde k\\cdot|0\\rangle^{\\otimes n}) = i f(|0\\rangle^{\\otimes n}).$$\n It follows that $f$ is the zero map, so $V_K^+$ and $V_K^-$ must be distinct. The same argument distinguishes $V_{\\bar K}^+$ and $V_{\\bar K}^-$.\n \n \\begin{proposition}\\label{jordan-decomp}\n The Jordan decomposition of $P^{+}_{1}$ and $P^-_{K\\bar K}$ consists of $2^{2n-1}$ copies of $V_1^+$ and $2^{2n-1}$ copies of $V_{K\\bar K}^-$. The Jordan decomposition of $P^{-}_{1}$ and $P^+_{K\\bar K}$ consists of $2^{2n-1}$ copies of $V_1^-$ and $2^{2n-1}$ copies of $V_{K\\bar K}^+$. In particular, the Cartan matrix is\n $$\\begin{pmatrix}\n 2^{2n-1} & 0 & 0 & 2^{2n-1} & 0 & 0 & 0 & 0\\\\\n 0 & 2^{2n-1} & 2^{2n-1} & 0 & 0 & 0 & 0 & 0\\\\\n 0 & 2^{2n-1} & 2^{2n-1} & 0 & 0 & 0 & 0 & 0\\\\\n 2^{2n-1} & 0 & 0 & 2^{2n-1} & 0 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\n \\end{pmatrix}.$$\n \\end{proposition}\n \\begin{proof}\n This follows from the proof of \\cite[Thm. 7.2.4]{modulardata2024}, noting that $\\tilde k$ anticommutes with $\\xi_i$ for all $i$.\n \\end{proof}\n\n As expected, $uS(u)$ acts as $\\id_M$ on the $D\\Kn$-modules $M=V_1, V_{K\\bar K}$ and $-\\id_M$ on $M=V_K, V_{\\bar K}$. However, $uS(u)$ does not act as a scalar multiple of $\\id_M$ on $M=P_1$ or $M=P_{K\\bar K}$.\n\n \\begin{proposition}\\label{fusion-rules}\n Let $\\cdot:\\{+, -\\}^2\\to\\{+, -\\}$ be given by $\\pm\\cdot\\pm = +$ and $\\pm\\cdot\\mp = -$, and set $-(\\pm) = \\mp$. Then, for $s, t\\in \\{+, -\\}$, the following fusion rules hold.\n \\begin{align}\n V_{1}^s V_{1}^t \\cong V_{K\\bar K}^s V_{K\\bar K}^t&\\cong V_{1}^{s\\cdot t},& V_{K\\bar K}^s V_{1}^t &\\cong V_{K\\bar K}^{s\\cdot t},\\label{fusion:SS}\\\\\n V_K^s V_{1}^t&\\cong V_{K}^{s\\cdot t}, & V_{\\bar K}^s V_{1}^t&\\cong V_{\\bar K}^{s\\cdot t},\\label{fusion:SMa}\\\\ \n V_K^s V_{K\\bar K}^t&\\cong V_{\\bar K}^{s\\cdot t}, & V_K^s V_{K\\bar K}^t&\\cong V_{\\bar K}^{s\\cdot t},\\label{fusion:SMb}\\\\\n V_{1}^s P_{1}^t\\cong V_{K\\bar K}^s P_{K\\bar K}^t&\\cong P_{1}^{s\\cdot t}, & V_{K\\bar K}^s P_{1}^t\\cong V_{1}^s P_{K\\bar K}^t&\\cong P_{K\\bar K}^{s\\cdot t},\\label{fusion:SB}\\\\\n V_K^s V_K^t\\cong V_{\\bar K}^s V_{\\bar K}^t&\\cong P_{K\\bar K}^{s\\cdot t}, & V_K^s V_{\\bar K}^t&\\cong P_{1}^{s\\cdot t},\\label{fusion:MM}\n \\end{align}\n \\begin{align}\n V_{K}^{s} P_1^{t}\\cong V_{K}^{s} P_{K\\bar K}^{-t}\\cong V_{\\bar K}^{s} P_1^{-t}\\cong V_{\\bar K}^{s} P_{K\\bar K}^{t}&\\cong 2^{2n-1}V_K^{s\\cdot t}\\oplus 2^{2n-1}V_{\\bar K}^{-s\\cdot t},\\label{fusion:MB}\\\\\n P_1^s P_{1}^t \\cong P_{K\\bar K}^s P_{K\\bar K}^t\\cong P_1^s P_{K\\bar K}^{-t}&\\cong 2^{2n-1}P_1^{s\\cdot t}\\oplus 2^{2n-1}P_{K\\bar K}^{-s\\cdot t}.\\label{fusion:BB}\n \\end{align}\n \\end{proposition}\n \\begin{proof}\n Isomorphisms \\eqref{fusion:SS}-\\eqref{fusion:MB} almost follow from \\cite[Thm. 7.2.5]{modulardata2024} (and the comment after). However, care should be taken in verifying the sign of the modules. For example, in the fourth line, there is an isomorphism $P_{K\\bar K}^s\\to V_K^+V_K^+$ generated by $(1-K)(1+\\sigma\\tilde k+\\tilde k^2+\\sigma\\tilde k^3)\\mapsto |0\\rangle^{\\otimes n}\\otimes |1\\rangle^{\\otimes n}$ for some $\\sigma\\in\\{1, -1\\}$, where $s=\\sgn\\sigma$, but $s$ must be determined by the action of $\\tilde k$. In this case, since $\\tilde k$ is group-like, it acts as \n $$\\tilde k\\cdot (|0\\rangle^{\\otimes n}\\otimes |1\\rangle^{\\otimes n}) = (\\tilde k |0\\rangle^{\\otimes n}\\otimes \\tilde k |1\\rangle^{\\otimes n}) = (i|0\\rangle^{\\otimes n}\\otimes (-1)^n i|1\\rangle^{\\otimes n}) = |0\\rangle^{\\otimes n}\\otimes |1\\rangle^{\\otimes n}.$$ \n Thus, $\\tilde k$, by the isomorphism, must act by the identity on the generator $(1-K)(1+\\sigma\\tilde k+\\tilde k^2+\\sigma\\tilde k^3)$ of $P_{K\\bar K}^s$, meaning $\\sigma=1$ and $s=+$.\n\n Isomorphisms \\eqref{fusion:MB} follows from Proposition \\ref{jordan-decomp} and Isomorphisms \\eqref{fusion:SMa} and \\eqref{fusion:SMb}. Isomorphisms \\eqref{fusion:BB} follows from Isomorphisms \\eqref{fusion:MM} and \\eqref{fusion:MB}.\n \\end{proof}\n\n Propositions \\ref{jordan-decomp} and \\ref{fusion-rules} give some experimental evidence of the following general conjecture, which is also true when $H$ is ribbon. \n \\begin{conj}\\label{tensor-only-one}\n Let $H$ be a finite-dimensional quasitriangular Hopf algebra. Let $M_1, M_2$ be indecomposable $H$-modules and $\\tilde M_1, \\tilde M_2$ be $\\H$-modules for which $\\tilde M_1|_H\\cong M_1$ and $\\tilde M_2|_H\\cong M_2$ as $H$-modules. \n \\begin{enumerate}\n \\item Suppose the simple $H$-module $V$ is a composition factor of $M_1\\otimes M_2$. Then, there is exactly one $\\H$-module $\\tilde V$ (up to isomorphism) for which $\\tilde V|_H\\cong V$ and $\\tilde V$ is a composition factor of $\\tilde M_1\\otimes \\tilde M_2$.\n \\item Suppose the indecomposable $H$-module $M$ is a direct summand of $M_1\\otimes M_2$ as $H$-modules. Then, there is exactly one $\\H$-module $\\tilde M$ (up to isomorphism) for which $\\tilde M|_H\\cong M$ and $\\tilde M$ is a direct summand of $\\tilde M_1\\otimes \\tilde M_2$ as $\\H$-modules.\n \\end{enumerate}\n \\end{conj}\n\n As a special case of Conjecture \\ref{tensor-only-one}, we could set $M_1$ to be the trivial $H$-module in (1) to obtain the following; if the simple $H$-module $V$ is a composition factor of $M$ as $H$-modules, then there is exactly one $\\H$-module $\\tilde V$ (up to isomorphism) for which $\\tilde V|_H\\cong V$ and $\\tilde V$ is a direct summand of $\\tilde M$ as $\\H$-modules. This is again supported by Proposition \\ref{jordan-decomp}.\n \n\n\n \\begin{remark}\n Despite having modules which seem to pair up, neither $D\\Kn$ nor $\\Kn$ is of the form $\\H$ (as Hopf algebras) for some other quasitriangular Hopf algebra $H$. By factorizability, $D\\Kn$ for even $n$ cannot be written as $D\\Kn\\cong \\H$ for some $H$. If $\\Kn\\cong \\H$, then by Proposition \\ref{simples-double}, $\\Rep(H)$ would have precisely one simple object up to isomorphism. This implies $H$ is semisimple. By Corollary \\ref{semisimple-iff}, this would imply $\\H\\cong \\Kn$ is semisimple, which is a contradiction.\n \\end{remark} \n\n \\begin{proposition}\\label{not-deligne}\n For odd $n$, $\\Rep(\\RKn)\\not\\cong \\Rep(D\\Kn)\\boxtimes\\Vecm$ as monoidal categories.\n \\end{proposition}\n \\begin{proof}\n One can verify that such an equivalence cannot simultaneously preserve both Isomorphisms \\eqref{fusion:MM} and \\eqref{fusion:BB}.\n \\end{proof}\n \n\\section*{Acknowledgements}\nThe author is supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. 2139319. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. \n\nThe author thanks Y. Sommerh\\\"auser for a helpful discussion on cocycled crossed products. The author also thanks B. R. Jones for illustrating the proof of Lemma \\ref{sqrt-uSu} with the holomorphic functional calculus in the complex case. Finally, the author thanks K. Goodearl, Z. Wang, and Q. Zhang for many useful comments.\n\n\\bibliographystyle{abbrv}\n\\bibliography{zbib}\n\\end{document}\n\n"}}},{"rowIdx":285,"cells":{"id":{"kind":"string","value":"2412.20443v4"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20443v4"},"title":{"kind":"string","value":"Monogenic trinomials and class numbers of related quadratic fields"},"full_text":{"kind":"string","value":"\\documentclass[12]{amsart}\n\\usepackage{amsmath,amssymb,amsthm,color,enumerate,comment,centernot,enumitem,url,cite}\n\\usepackage{graphicx,relsize,bm}\n\\usepackage{mathtools}\n\\usepackage{array}\n\n\\makeatletter\n\\newcommand{\\tpmod}[1]{{\\@displayfalse\\pmod{#1}}}\n\\makeatother\n\n\n\n\\newcommand{\\ord}{\\operatorname{ord}}\n\n\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lemma}[thm]{Lemma}\n\\newtheorem{conj}[thm]{Conjecture}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{sch}[thm]{Scholium}\n\\newtheorem{guess}[thm]{Guess}\n\\newtheorem{ex}[thm]{Example}\n\\newtheorem{exs}[thm]{Examples}\n\\newtheorem{question}[thm]{Question}\n\\theoremstyle{remark}\n \\newtheorem*{remark}{{\\bf Remark}}\n \\newtheorem*{rems}{{\\bf Remarks}}\n\n \\newtheorem*{note}{Note}\n\\theoremstyle{definition}\n \\newtheorem{defn}[thm]{Definition}\n\\newtheorem{hyp}[thm]{Hypothesis}\n\\newtheorem{rem}[thm]{Remark}\n\n\\newtheorem{Con}{Conditions}[section]\n\n\\theoremstyle{THM}\n\\newtheorem*{THM}{Theorem}\n\n\\DeclareMathOperator{\\st}{\\left\\vert\\vphantom{\\frac{1}{1}}\\right.}\n\\DeclareMathOperator{\\lcm}{lcm}\n\n\\newcommand{\\abs}[1]{\\left|{#1}\\right|}\n\\def\\ds{\\displaystyle}\n\\def\\UU {{\\widehat{\\mathcal U}}}\n\\def\\g {{\\widehat{g}}}\n\\def\\uu {{\\widehat{u}}}\n\\def\\VV {{\\widehat{\\mathcal V}}}\n\\def\\vv {{\\widehat{v}}}\n\\def\\P {{\\mathcal P}}\n\\def\\O {{\\mathcal O}}\n\\def\\p {{\\mathfrak p}}\n\\def\\FF {{\\mathcal F}}\n\\def\\GG {{\\mathcal G}}\n\\def\\TT {{\\mathcal T}}\n\\def\\R {{\\mathbb R}}\n\\def\\Z {{\\mathbb Z}}\n\\def\\ZZ {{\\mathcal Z}}\n\\def\\NN {{\\mathcal N}}\n\\def\\QQ {{\\mathcal Q}}\n\\def\\N{{\\mathbb N}}\n\\def\\Q {{\\mathbb Q}}\n\\def\\T {{\\mathcal T}}\n\\def\\E {{\\mathcal E}}\n\\def\\B {{\\mathcal B}}\n\\def\\C {{\\mathcal C}}\n\\def\\S {{{\\mathcal S}}}\n\\def\\A {{\\mathcal A}}\n\\def\\D {{\\mathcal D}}\n\\def\\L {{\\mathcal L}}\n\\def\\Norm{{\\mathcal N}}\n\\def\\M {{\\mathcal M}}\n\\def\\h {{\\mathfrak h}}\n\\def\\X {{\\mathcal X}}\n\\def\\U {{\\mathcal U}}\n\\def\\V {{\\mathcal V}}\n\\def\\K{{\\mathcal K}}\n\\def\\PP {{\\widehat{\\mathcal P}}}\n\\def\\d {{\\rm det}}\n\\def\\PPP {{{\\mathfrak P}}}\n\\newcommand{\\im}{\\operatorname{im}}\n\n\\def\\k {k}\n\n\\def\\GG {{\\mathcal G}}\n\\def\\UU {{\\widehat{\\mathcal U}}}\n\\def\\uu {{\\widehat{u}}}\n\\def\\VV {{\\widehat{\\mathcal V}}}\n\\def\\vv {{\\widehat{v}}}\n\\def\\P {{\\mathcal P}}\n\\def\\F {{\\mathbb F}}\n\\def\\D {{\\mathcal D}}\n\\def\\Z {{\\mathbb Z}}\n\\def\\Q {{\\mathbb Q}}\n\\def\\C {{\\mathbb C}}\n\\def\\S {{{\\mathcal S}}}\n\\def\\U {{\\mathcal U}}\n\\def\\V {{\\mathcal V}}\n\\def\\W {{\\mathcal W}}\n\\def\\PP {{\\widehat{\\mathcal P}}}\n\\def\\RR {{{\\mathcal R}}}\n\\def\\PPP {{{\\mathfrak P}}}\n\\def\\CC {{\\mathcal C}}\n\\def\\Gal{{\\mbox{{\\rm{Gal}}}}}\n\n\\makeatletter\n\\@namedef{subjclassname@2020}{ \\textup{2020} Mathematics Subject Classification}\n\\makeatother\n\n\n\n\\def\\n{{\\mathbf n}}\n\\def\\tr{\\mathrm{tr}}\n\n\\def\\red#1 {\\textcolor{red}{#1 }}\n\\def\\blue#1 {\\textcolor{blue}{#1 }}\n\n\n\\numberwithin{equation}{section}\n\\def\\ds{\\displaystyle}\n\\def\\Z {{\\mathbb Z}}\n\n\\newcommand{\\Mod}[1]{\\ (\\mathrm{mod}\\enspace #1)}\n\\newcommand{\\mmod}[1]{\\ \\mathrm{mod}\\enspace #1}\n\n\\renewcommand{\\arraystretch}{1.25}\n\\begin{document}\n\n\n\\title[Monogenic trinomials and class numbers of related quadratic fields]{Monogenic trinomials and class numbers of related quadratic fields}\n\n\n\n \t\n\n\n\\author{Lenny Jones}\n\\address{Professor Emeritus, Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania 17257, USA}\n\\email[Lenny~Jones]{doctorlennyjones@gmail.com}\n\n\\date{\\today}\n\n\\begin{abstract}\nWe say that a monic polynomial $f(x)\\in \\Z[x]$ of degree $N\\ge 2$ is \\emph{monogenic} if $f(x)$ is irreducible over $\\Q$ and $\\{1,\\theta,\\theta^2,\\ldots ,\\theta^{N-1}\\}$ is a basis for the ring of integers of $\\Q(\\theta)$, where $f(\\theta)=0$. In this article, we investigate the divisibility of the class numbers of quadratic fields $\\Q(\\sqrt{\\delta})$ for certain families of monogenic trinomials $f(x)=x^N+Ax+B$, where $\\delta\\ne \\pm 1$ is a squarefree divisor of the discriminant of $f(x)$.\n\\end{abstract}\n\n\n\n\n\\subjclass[2020]{Primary 11R09, 11R29}\\keywords{monogenic, trinomial, class number, quadratic field}\n\n\\maketitle\n\n\n\n\\section{Introduction}\\label{Section:Intro}\nWe say that a monic polynomial $f(x)\\in \\Z[x]$ is \\emph{monogenic} if $f(x)$ is irreducible over $\\Q$ and $\\{1,\\theta,\\theta^2,\\ldots ,\\theta^{\\deg{f}-1}\\}$ is a basis for the ring of integers of $\\Q(\\theta)$, where $f(\\theta)=0$. Hence, $f(x)$ is monogenic if and only if $\\Z_K=\\Z[\\theta]$. For the minimal polynomial $f(x)$ of an algebraic integer $\\theta$ over $\\Q$, it is well known \\cite{Cohen} that\n\\begin{equation} \\label{Eq:Dis-Dis}\n\\Delta(f)=\\left[\\Z_K:\\Z[\\theta]\\right]^2\\Delta(K),\n\\end{equation}\nwhere $\\Delta(f)$ and $\\Delta(K)$ are the respective discriminants over $\\Q$ of $f(x)$ and the number field $K$.\nThus, from \\eqref{Eq:Dis-Dis}, $f(x)$ is monogenic if and only if $\\Delta(f)=\\Delta(K)$.\n\nMany authors have investigated divisibility properties of class numbers of quadratic fields. We present a few such results from the literature that are germane to our investigations in this paper.\nThe first theorem is from 1955 and is due to Ankeny and Chowla \\cite{AC}.\n\\begin{thm}\\label{Thm:AC}\n Let $n$ and $M$ be positive integers with $M\\ge 5$. If $\\delta:=M^{2n}+1$ is squarefree, then $n$ divides the class number $h_K$ of the real quadratic field $K=\\Q(\\sqrt{\\delta})$.\n\\end{thm}\nThe next theorem is due to M. Ram Murty \\cite{Murty}, which he published in 1998.\n\\begin{thm}\\label{Thm:Murty}\nSuppose that $\\delta:=1-M^n$ is squarefree with $M\\ge 5$ and odd. Then the class group of $\\Q(\\sqrt{\\delta})$ has an element of order $n$.\n\\end{thm}\n\\begin{rem}\\label{Rem:Murty}\n We point out that Murty's proof of Theorem \\ref{Thm:Murty} is also valid when $M=3$ and $n$ is odd.\n\\end{rem}\nIn 2000, Kishi and Miyake \\cite{KM} gave the following parameterization of the quadratic fields whose class numbers are divisible by 3:\n \\begin{thm}\\label{Thm:KM}\n Let $u,w\\in \\Z$ with $\\gcd(u,w)=1$, such that $d:=4uw^3-27u^2$ is not a square and one of the following sets of conditions holds:\n \\begin{align}\n\\label{C1} &3\\nmid w;\\\\\n\\label{C2} &3\\mid w, \\quad uw\\not \\equiv 3 \\pmod{9}, \\quad u\\equiv w \\pm 1 \\pmod{9};\\\\\n\\label{C3} &3\\mid w, \\quad uw \\equiv 3 \\pmod{9}, \\quad u\\equiv w\\pm 1 \\pmod{27}.\n \\end{align}\n Define\n \\begin{equation}\\label{Eq:f}\n f_{u,w}(x):=x^3-uwx-u^2.\n \\end{equation} If $f_{u,w}(x)$ is irreducible over $\\Q$, then the roots of $f_{u,w}(x)=0$ generate an unramified cyclic cubic extension of the quadratic field $\\Q(\\sqrt{d})$. Conversely, every quadratic field $K$ whose class number is divisible by {\\rm 3} and every unramified cyclic cubic extension of $K$ are given in this way by a suitable pair of integers $u$ and $w$.\n \\end{thm}\n\n\nThe next theorem is from 2009 and due to Kishi \\cite{K}.\n\\begin{thm}\\label{Thm:K}\nFor any positive integers $k$ and $n$ with $2^{2k}<3^n$ and $(k,n)\\ne (2,3)$, the class number of the imaginary quadratic field $\\Q(\\sqrt{2^{2k}-3^n})$ is divisible by $n$.\n\\end{thm}\n\n\n\n\n\n\nIn this paper, for certain monogenic trinomials $f(x)=x^N+Ax+B$, we use Theorems \\ref{Thm:Murty}, \\ref{Thm:KM} and \\ref{Thm:K} to investigate divisibility properties of the class numbers of quadratic fields $\\Q(\\sqrt{\\delta})$, where $\\delta\\ne \\pm 1$ is a divisor of $\\Delta(f)$. One motivation for such investigations is to determine whether there is some possible connection between the monogenicity of $f(x)$ and the class number of $\\Q(\\sqrt{\\delta})$. This notion is not completely outside the realm of possibility since there are well-known connections between $\\Delta(f)$ and the monogenicity of $f(x)$. For example, if $f(x)$ is monic and irreducible over $\\Q$ such that $\\Delta(f)$ is squarefree, then $f(x)$ is monogenic. A second motivation is to prove that there exist infinitely many distinct monogenic polynomials such that the class number of the related quadratic field can be arbitrarily large.\n\nWe summarize our approach to this study. Our somewhat natural starting point is to construct and examine families of polynomials $f(x)$ such that $\\Delta(f)$ has a divisor that is of the same form as the discriminants for the quadratic fields found in one of the known results: Theorem \\ref{Thm:Murty}, Theorem \\ref{Thm:KM} or Theorem \\ref{Thm:K}. For the construction of such polynomials, we focus on trinomials $f(x)=x^N+Ax+B$ because of the relative simplicity of their discriminants. Then, for each of these constructed families of trinomials, we provide necessary and sufficient conditions for the monogenicity of a trinomial in the family, and we show that no two such degree-$N$ monogenic trinomials generate the same extension over $\\Q$. Next, we confirm that these monogenic trinomials do satisfy the conditions of one of the Theorems \\ref{Thm:Murty}, \\ref{Thm:KM} and \\ref{Thm:K}, so that the divisibility conditions for the related quadratic field found in these known theorems do indeed hold. Finally, we examine the data for any obvious patterns.\n\nAlthough our investigations did not yield any apparent connections between the monogenicity of these trinomials and the divisibility of the class numbers of the related quadratic fields, we have been able to show that there exist infinitely many monogenic trinomials that fall within the context of each of the Theorems \\ref{Thm:Murty}, \\ref{Thm:KM} and \\ref{Thm:K}. Moreover, as mentioned above, we also show that no two degree-$N$ monogenic trinomials generate the same extension over $\\Q$. Consequently, in doing so, we have established our second motivation. \nIn the statement of each new theorem below, we indicate explicitly its association to the particular one of Theorems \\ref{Thm:Murty}, \\ref{Thm:KM} and \\ref{Thm:K}.\n More precisely, our main results are:\n\\begin{thm}(associated with Theorem \\ref{Thm:KM})\\label{Thm:Main1}\\\\\nLet $u$, $w$ and $d$ be integers that satisfy the conditions set forth in Theorem \\ref{Thm:KM}, and let $f_{u,w}(x)$ be as defined in \\eqref{Eq:f}. Suppose that $f_{u,w}(x)$ is irreducible over $\\Q$. Then\n\\begin{equation}\\label{Eq:Mono1}\nf_{u,w}(x)\\ \\mbox{is monogenic if and only if $u=1$ and $d$ is squarefree,}\n\\end{equation} the set\n\\begin{equation}\\label{FF1}\n\\FF_1=\\{f_{1,w}(x): \\mbox{$d$ is squarefree},\\}\n\\end{equation} is infinite, every element of $\\FF_1$\nis such that $\\Gal_{\\Q}(f_{a,b})\\simeq S_3$, the class number of the unique quadratic subfield $\\Q(\\sqrt{d})$ of the splitting field of $f_{a,b}(x)$ is divisible by $3$, and no two elements of $\\FF_1$ generate the same cubic extension over $\\Q$.\n \\end{thm}\n \\begin{thm}\\label{Thm:Main2}(associated with Theorem \\ref{Thm:KM})\n Let $a$ and $b$ be positive integers with $b\\ge 2$ and $(a,b)\\ne (1,2)$. Define\n \\[f_{a,b}(x):=x^3-2^{2a}x-3^{b-1} \\quad \\mbox{and} \\quad \\delta:=2^{6a+2}-3^{2b+1}.\\]\n Then\n \\begin{equation}\\label{Eq:Mono2}\nf_{a,b}(x)\\ \\mbox{is monogenic if and only if $\\delta$ is squarefree,}\n\\end{equation} and every element of the set\n\\begin{equation}\\label{FF2}\n\\FF_2=\\{f_{a,b}(x): \\mbox{$\\delta<0$ is squarefree}\\}\n\\end{equation} is such that $\\Gal_{\\Q}(f_{a,b})\\simeq S_3$, $\\FF_1\\cap \\FF_2=\\varnothing$, no two elements of $\\FF_2$ generate the same cubic extension of $\\Q$, and the class number of the unique imaginary quadratic subfield $\\Q(\\sqrt{\\delta})$ of the splitting field of $f_{a,b}(x)$ is divisible by $(6b+3)/\\gcd(3,2b+1)$.\n \\end{thm}\n\n\\begin{thm}\\label{Thm:Main3}(associated with Theorem \\ref{Thm:AC})\nLet $b,N\\in \\Z$ such that $b\\ge 1$, $N\\ge 6$ and $N\\equiv 2 \\pmod{4}$.\nDefine\n \\[f_{N,b}(x):=x^N-N(N-1)^bx-(N-1) \\quad \\mbox{and} \\quad \\delta:=(N-1)^{bN}+1.\\]\n Suppose that $f_{N,b}(x)$ is irreducible over $\\Q$. Then\n\\begin{equation}\\label{Eq:Mono3}\nf_{N,b}(x)\\ \\mbox{is monogenic if and only if $N(N-1)$ and $\\delta$ are squarefree.}\n\\end{equation} Furthermore, no two such monogenic trinomials $f_{N,b}(x)$ generate the same $N$th-degree extension of $\\Q$, and $bN/2$ divides the class number of the real quadratic field $\\Q(\\sqrt{\\delta})$.\n\\end{thm}\n\n\\begin{thm}\\label{Thm:Main4}(associated with Theorem \\ref{Thm:Murty})\nLet $b,N\\in \\Z$ with $b\\ge 1$, $N\\ge 3$ and $N\\equiv 3 \\pmod{4}$. Define\n\\[f_{N,b}(x)=x^N-x-(N-1)N^b\\quad \\mbox{and} \\quad \\delta:=1-N^{(b+1)N-b}.\\] Suppose that $f_{N,b}(x)$ is irreducible over $\\Q$. Then\n\\begin{equation}\\label{Eq:Mono4}\nf_{N,b}(x)\\ \\mbox{is monogenic if and only if $\\delta$ is squarefree.}\n\\end{equation} Furthermore, no two such monogenic trinomials $f_{N,b}(x)$ generate the same $N$th-degree extension of $\\Q$, and the ideal class group of the imaginary quadratic field $\\Q(\\sqrt{\\delta})$ contains an element of order $(b+1)N-b$.\n\\end{thm}\n\\section{Preliminaries}\\label{Section:Prelims}\nThe first theorem is a special case of a result due to Swan \\cite{Swan}.\n\\begin{thm}\\label{Thm:Swan}\nLet $f(x)=x^N+Ax+B\\in \\Q[x]$. Then\n\\[\n\\Delta(f)=(-1)^{N(N-1)/2}\\left(N^{N}B^{N-1}-(-1)^{N}(N-1)^{N-1}A^{N}\\right).\n\\]\n\\end{thm}\n\n\n\nThe theorem below is the special case of a theorem due to Jakhar, Khanduja and Sangwan \\cite[Theorem 1.1]{JKS2}, which is used to determine the monogenicity of an irreducible trinomial, when applied to trinomials of the form $f(x)=x^N+Ax+B$, where $N\\ge 3$.\n We use the notation $q^e\\mid \\mid n$ to mean that $q^e$ is the exact power of the prime $q$ that divides the integer $n$.\n\n\n\\begin{thm}{\\rm \\cite{JKS2}}\\label{Thm:JKS}\nLet $N\\ge 3$ be an integer.\nLet $K=\\Q(\\theta)$ be an algebraic number field with $\\theta\\in \\Z_K$, the ring of integers of $K$, having minimal polynomial $f(x)=x^{N}+Ax+B$ over $\\Q$. A prime factor $q$ of $\\Delta(f)$ does not divide $\\left[\\Z_K:\\Z[\\theta]\\right]$ if and only if $q$ satisfies one of the following conditions:\n\\begin{enumerate}[font=\\normalfont]\n \\item \\label{JKS:I1} when $q\\mid A$ and $q\\mid B$, then $q^2\\nmid B$;\n \\item \\label{JKS:I2} when $q\\mid A$ and $q\\nmid B$, then\n \\[\\mbox{either } \\quad q\\mid A_2 \\mbox{ and } q\\nmid B_1 \\quad \\mbox{ or } \\quad q\\nmid A_2\\left(-BA_2^{N}-\\left(-B_1\\right)^{N}\\right),\\]\n where $A_2=A/q$ and $B_1=\\frac{B+(-B)^{q^j}}{q}$ with $q^j\\mid\\mid N$;\n \\item \\label{JKS:I3} when $q\\nmid A$ and $q\\mid B$, then\n \\[\\mbox{either } \\quad q\\mid A_1 \\mbox{ and } q\\nmid B_2 \\quad \\mbox{ or } \\quad q\\nmid A_1\\left(-AA_1^{N-1}-\\left(-B_2\\right)^{N-1}\\right),\\]\n where $A_1=\\frac{A+(-A)^{q^\\ell}}{q}$ with $q^\\ell\\mid\\mid (N-1)$, and $B_2=B/q$;\n \\item \\label{JKS:I5} when $q\\nmid AB$, then $q^2\\nmid \\Delta(f)$.\n \\end{enumerate}\n\\end{thm}\n\n\n\\section{Proof of Theorem \\ref{Thm:Main1}}\n\\begin{proof}\nNote that replacing $u$ with $-u$ and $w$ with $-w$ in \\eqref{Eq:f} yields the same trinomial $f_{u,w}(x)=x^3-uwx-u^2$. Consequently, we can assume without loss of generality that $u>0$.\n A straightforward calculation shows that\n \\[\\Delta(f_{u,w})=4u^3w^3-27u^4.\\]\n\n We begin with the proof of \\eqref{Eq:Mono1}, and we assume first that $f_{u,w}(x)$ is monogenic. Let $q$ be a prime divisor of $\\Delta(f_{u,w})$, and suppose that $f_{u,w}(\\theta)=0$. If $u>1$, then condition \\eqref{JKS:I1} of Theorem \\ref{Thm:JKS} is not satisfied for every prime divisor $q$ of $u$; in which case, $q\\mid \\left[\\Z_K:\\Z[\\theta]\\right]$ and $f(x)$ is not monogenic. Thus, $u=1$ and we can focus on\n \\begin{equation}\\label{Eq:Newf}\n f_{1,w}(x)=x^3-wx-1\\quad \\mbox{and} \\quad \\Delta(f_{1,w})=d=4w^3-27.\n \\end{equation}\n Observe then from \\eqref{Eq:Newf} that $f_{u,w}(x)$ is irreducible over $\\Q$ if and only if $w\\not \\in \\{0,2\\}$ by the Rational Root Theorem.\n\nIf $q=3$, then $3\\mid w$ from \\eqref{Eq:Newf}.\n From condition \\eqref{JKS:I2} of Theorem \\ref{Thm:JKS}, we see that $B_1=0$, which implies that $3\\nmid A_2$ since $f(x)$ is monogenic. Hence, we have that $w\\equiv \\pm 3 \\pmod{9}$. However, with $u=1$, examining the two possibilities for $w$ reveals that neither condition \\eqref{C2} nor condition \\eqref{C3} of Theorem \\ref{Thm:KM} can be satisfied. For example, if $w\\equiv -3 \\pmod{9}$, then $uw\\equiv -3\\not \\equiv 3\\pmod{9}$, but $u-w\\equiv 4\\not \\equiv \\pm 1 \\pmod{9}$. Thus, $3\\nmid \\Delta(f_{1,w})$ and $3\\nmid w$, so that condition \\eqref{C1} of Theorem \\ref{Thm:KM} holds. As a consequence, we have shown that no cubic trinomial $f_{1,w}(x)$ in \\eqref{Eq:Newf} is monogenic when $3\\mid w$.\n\n Suppose next that $q$ is a prime divisor of $d$ such that $q^2\\mid d$. By the previous discussion, we have that $q\\ne 3$ and $q\\nmid w$. But then condition \\eqref{JKS:I5} of Theorem \\ref{Thm:JKS} is not satisfied, contradicting the fact that $f(x)$ is monogenic. Hence, $d$ is squarefree, which completes the proof in this direction.\n\n Conversely, suppose that $u=1$ and $d$ as in \\eqref{Eq:Newf} is squarefree. Let $q$ be a prime divisor of $\\Delta(f_{1,w})=d$. Then, from \\eqref{Eq:Newf}, conditions \\eqref{JKS:I1}, \\eqref{JKS:I3} and \\eqref{JKS:I5} of Theorem \\ref{Thm:JKS} are easily seen to be satisfied. If $q\\mid w$, then $q=3$ from \\eqref{Eq:Newf}, and $3^3\\mid d$, contradicting the fact that $d$ is squarefree, which completes the proof of \\eqref{Eq:Mono1}.\n\n Next, let $\\FF_1$ be the set as defined in \\eqref{FF1}. First note the fact that every element of $\\FF_1$ is such that $\\Gal_{\\Q}(f_{u,w})\\simeq S_3$, and that the class number of the unique quadratic subfield $\\Q(\\sqrt{d})$ of the splitting field of $f_{a,b}(x)$ is divisible by $3$, follows immediately from Theorem \\ref{Thm:KM}.\n\n Let $G(t):=4t^3-27\\in \\Z[t]$. It is easy to verify that $G(t)$ is irreducible over $\\Q$. Then, it follows from a result of Erd\\H{o}s \\cite{Erdos} that there exist infinitely many integers $z$ such that $G(z)$ is squarefree. (In fact, there exist infinitely many primes $p$ such that $G(p)$ is squarefree \\cite{Helfgott}.) Consequently, there exist infinitely many integers $w$ such that $d$ is squarefree, and hence, $\\FF_1$ is infinite.\n\n Finally, to see that no two elements of $\\FF_1$ generate the same cubic field, we assume, by way of contradiction, that $f_{1,w_1}(x)\\ne f_{1,w_2}(x)$ are elements of $\\FF_1$ such that $K_1=\\Q(\\theta_1)$ and $K_2=\\Q(\\theta_2)$ are equal, where $f_{1,w_i}(\\theta_i)=0$.\n Then, since $f_{1,w_i}(x)$ is monogenic by \\eqref{Eq:Mono1}, we have that $\\Delta(f_{1,w_i})=\\Delta(K_i)$, and so it follows from \\eqref{Eq:Dis-Dis} that\n \\begin{equation}\\label{Eq:Disequation}\n 4w_1^3-27=4w_2^3-27.\n \\end{equation} Since $w_1\\ne w_2$, it is easy to see that \\eqref{Eq:Disequation} has no integer solutions, which completes the proof of the theorem. \\end{proof}\n\n\n\n\\section{Proof of Theorem \\ref{Thm:Main2}}\n\\begin{proof}\nSince $f_{a,b}(x):=x^3-2^{2a}x-3^{b-1}$, we have from Theorem \\ref{Thm:Swan} that\n\\[\\Delta(f_{a,b})=\\delta=2^{6a+2}-3^{2b+1}.\\] We first determine the values of $(a,b)$ for which $f_{a,b}(x)$ is irreducible over $\\Q$. If $f_{a,b}(x)$ is reducible over $\\Q$, then $f_{a,b}(r)=0$ for some $r\\in \\Z$, so that $r=\\pm 3^c$ for some integer $c\\ge 0$ by the Rational Root Theorem. Suppose that $r=-3^c$. Then\n\\begin{align}\n \\nonumber f_{a,b}(-3^c)=0 \\Longrightarrow \\ & (-3^c)\\left((-3)^{2c}-2^{2a}\\right)=3^{b-1}\\\\\n \\nonumber\\Longrightarrow \\ & 3^{2c}-2^{2a}=-3^{b-1-c}\\\\\n \\nonumber\\Longrightarrow \\ & 3^{2c}+3^{b-1-c}=2^{2a}\\\\\n \\label{Root} \\Longrightarrow \\ & 3^{b-1-c}(3^{3c-b+1}+1)=2^{2a} \\quad \\mbox{or} \\quad 3^{2c}(1+3^{b-1-3c})=2^{2a}.\n\\end{align} From the first possibility in \\eqref{Root}, we deduce that $c=b-1$. Thus, we get the equation $2^{2a}-3^{2(b-1)}=1$, which has no solutions by Mih\\u{a}ilescu's theorem \\cite{M}. For the second possibility in \\eqref{Root}, we conclude that $c=0$. Hence, this possibility yields the equation $2^{2a}-3^{b-1}=1$, which has only the solution $(a,b)=(1,2)$, again by Mih\\u{a}ilescu's theorem \\cite{M}. Note then that $r=-1$ and\n\\[f_{1,2}(x)=(x+1)(x^2-x-3).\\] A similar argument shows that no solutions arise from assuming that $r=3^c$. Consequently, $f_{a,b}(x)$ is irreducible over $\\Q$ if and only if $(a,b)\\ne (1,2)$.\n\n We now prove \\eqref{Eq:Mono2}. With $A=-2^{2a}$, $B=-3^{b-1}$ and $q$ a prime divisor of $\\Delta(f_{a,b})=\\delta$, we see that only condition \\eqref{JKS:I5} is applicable in Theorem \\ref{Thm:JKS}. Hence, we conclude that $f_{a,b}(x)$ is monogenic if and only if $q^2\\nmid \\delta$ for all prime divisors $q$ of $\\delta$. That is, $f_{a,b}(x)$ is monogenic if and only if $\\delta$ is squarefree, which completes the proof of \\eqref{Eq:Mono2}.\n\nConsider next the set $\\FF_2$, as defined in \\eqref{FF2}. Observe that $\\FF_1\\cap \\FF_2=\\varnothing$ since $b\\ge 2$. To see that no two elements of $\\FF_2$ can generate the same cubic extension, we assume, by way of contradiction, that\n\\[K_1=\\Q(\\theta_1)=\\Q(\\theta_2)=K_2,\\]\nfor some $(a_1,b_1)\\ne (a_2,b_2)$, where $f_{a_i,b_i}(\\theta_i)=0$. Since each $f_{a_i,b_i}(x)$ is monogenic, then we have from \\eqref{Eq:Dis-Dis} that $\\delta_1=\\delta_2$, where\n\\[\\delta_i:=2^{6a_i+2}-3^{2b_i+1}.\\]\nWithout loss of generality, we can assume that $a_1>a_2$.\nThen, by rearranging the equation $\\delta_1=\\delta_2$, we see that\n\\[2^{6a_2+2}\\left(2^{6(a_1-a_2)}-1\\right)=3^{2b_2+1}\\left(3^{2(b_1-b_2)}-1\\right),\\]\nwhich implies that\n\\begin{equation}\\label{Eq:2equations}\n3^{2(b_1-b_2)}-2^{6a_2+1}=1 \\quad \\mbox{and} \\quad 2^{6(a_1-a_2)}-3^{2b_2+1}=1.\n\\end{equation} It is then easy to see that neither equation in \\eqref{Eq:2equations} has a solution by Mih\\u{a}ilescu's theorem \\cite{M}. Hence, no two trinomials in $\\FF_2$ generate the same cubic field.\n\n For each $f_{a,b}(x)\\in \\FF_2$, we define\n\\begin{gather}\n \\label{uwd} u:=3^{2b-2}, \\quad w:=2^{2a}, \\quad d:=4uw^3-27u^2=u\\delta\\\\\n\\nonumber \\mbox{and}\\\\\n \\label{g} g_{a,b}(x):=x^3-uwx-u^2=x^3-2^{2a}3^{2b-2}x-3^{4b-4}.\n\\end{gather}\nNote that\n\\begin{align*}\n3^{3b-3}f_{a,b}(x/3^{b-1})&=3^{3b-3}\\left((x/3^{b-1})^3-2^{2a}(x/3^{b-1})-3^{b-1}\\right)\\\\\n&=x^3-2^{2a}3^{2b-2}x-3^{4b-4}\\\\\n&=g_{a,b}(x),\n\\end{align*}\nand\n\\[\\Delta(g_{a,b})=3^{6b-6}\\left(2^{6a+2}-3^{2b+1}\\right)=u^2d=u^3\\delta=u^3\\Delta(f_{a,b}),\\] by Theorem \\ref{Thm:Swan}. Moreover, $g_{a,b}(x)$ is irreducible over $\\Q$ if and only if $(a,b)\\ne (1,2)$, $f_{a,b}(x)$ and $g_{a,b}(x)$ generate the same cubic field, but $g_{a,b}(x)$ is not monogenic since condition \\eqref{JKS:I1} of Theorem \\ref{Thm:JKS} is not satisfied with the prime $q=3$.\n We see from \\eqref{uwd} that $\\gcd(u,w)=1$, $d$ is not a square (since $\\delta<0$) and $3\\nmid w$. Hence, it follows from Theorem \\ref{Thm:KM} that $\\Gal_{\\Q}(f_{a,b})\\simeq \\Gal_{\\Q}(g_{a,b})\\simeq S_3$ and that $3\\mid h_K$, where $h_K$ is the class number of the imaginary quadratic field\n $K=\\Q(\\sqrt{d})=\\Q(\\sqrt{\\delta})$. Furthermore, since $\\delta=2^{2(3a+1)}-3^{2b+1}<0$ and $(3a+1,2b+1)\\ne (2,3)$, it follows from Theorem \\ref{Thm:K} that $(2b+1)\\mid h_K$. Consequently, we deduce that $(6b+3)/\\gcd(3,2b+1)$ divides $h_K$, which completes the proof of the theorem.\n\\end{proof}\n\\begin{rem}\nWe point out that in the proof of Theorem \\ref{Thm:Main2} a more elementary approach, such as in \\cite{H}, will suffice for every appeal\n to Mih\\u{a}ilescu's theorem \\cite{M}.\n\\end{rem}\n\\section{Proof of Theorem \\ref{Thm:Main3}}\n\\begin{proof}\nIn Theorem \\ref{Thm:Swan}, we have\n\\begin{equation}\\label{Eq:AB}\nA=-N(N-1)^{b} \\quad \\mbox{and}\\quad B=-(N-1),\n\\end{equation} so that\n\\[\\Delta(f_{N,b})=N^N(N-1)^{N-1}((N-1)^{bN}+1)=N^N(N-1)^{N-1}\\delta.\\]\n\n We first point out some important facts. Suppose that $q$ is a prime dividing $\\gcd(A,\\delta)$. Then, $q\\mid N$ so that\n\\[\\delta\\equiv (-1)^{bN}+1 \\equiv 2\\pmod{q},\\] since $2\\mid N$.\nBut $\\delta\\equiv 0 \\pmod{q}$, and therefore, $q=2$. Since $N\\equiv 2 \\pmod{4}$, we see that $\\delta\\equiv 2 \\pmod{4}$. Consequently,\n\\begin{equation}\\label{Eq:Fact}\n2\\mid\\mid \\delta\\quad \\mbox{and}\\quad \\gcd(A,\\delta)=2.\n\\end{equation}\n\nTo establish \\eqref{Eq:Mono3}, we let $q$ be a prime divisor of $\\Delta(f_{N,b})$, and we use \\eqref{Eq:AB} and Theorem \\ref{Thm:JKS}.\nObserve that $q\\mid A$ and $q\\mid B$ if and only if $q\\mid (N-1)$. Hence, condition \\eqref{JKS:I1} of Theorem \\ref{Thm:JKS} is satisfied for all prime divisors $q$ of $N-1$ if and only if $q^2\\nmid (N-1)$, or equivalently, $N-1$ is squarefree.\n\nNext, observe that $q\\mid A$ and $q\\nmid B$ if and only if $q\\mid N$. If $q=2$, then since $N\\equiv 2 \\pmod{4}$, we have $2\\mid \\mid N$, and in condition \\eqref{JKS:I2} of Theorem \\ref{JKS:I2}, we see that\n\\begin{gather*}\nA_2=\\frac{-N(N-1)^{b}}{2} \\equiv 1 \\pmod{2}\\\\\n\\mbox{and}\\\\\n B_1=\\frac{(N-1)^2-(N-1)}{2}=\\frac{(N-1)(N-2)}{2}\\equiv 0 \\pmod{2}.\n \\end{gather*} Hence,\n \\[A_2(-BA_2^N-(-B_1)^N)\\equiv 1 \\pmod{2},\\]\n so that condition \\eqref{JKS:I2} is satisfied for $q=2$.\n Suppose then that $q\\ne 2$ with $q^j\\mid \\mid N$ and write $N=q^jn$, where $q\\nmid n$. Then\n \\[A_2=\\frac{N(N-1)^b}{q}\\equiv (-1)^{b+1}q^{j-1}n \\pmod{q},\\]\n \\begin{align*}\n B_1&=\\frac{(q^jn-1)^{q^j}-(q^jn-1)}{q}\\\\\n &=\\frac{(-1)^{q^j}+q^{2j}n(-1)^{q^j-1}+\\sum_{k=2}^{q^j}\\binom{q^j}{k}q^{jk}n^k(-1)^{q^j-k}-(q^jn-1)}{q}\\\\\n &=q^{j-1}n(q^j-1)+\\sum_{k=2}^{q^j}\\binom{q^j}{k}q^{jk-1}n^k(-1)^{q^j-k}\\\\\n &\\equiv q^{j-1}n \\pmod{q},\n\\end{align*} and therefore,\n\\[A_2(-BA_2^N-(-B_1)^N)\\equiv (-1)^{b+2}2q^{(j-1)(N+1)}n^N \\not \\equiv 0 \\pmod{q}\\]\nif and only if $j=1$. Thus, condition \\eqref{JKS:I2} of Theorem \\ref{Thm:JKS} is satisfied for all odd prime divisors $q$ of $N$ if and only if $N$ is squarefree.\n\nSince $q\\nmid A$ and $q\\mid B$ is impossible, we see that condition \\eqref{JKS:I3} is not applicable.\n\nIf $q\\nmid AB$, then $q\\ne 2$ and $q\\mid \\delta$. Hence, condition \\eqref{JKS:I5} is satisfied for all prime divisors $q\\nmid AB$ if and only if $q^2\\nmid \\delta$, which is equivalent to $\\delta$ being squarefree by \\eqref{Eq:Fact}. Thus, we have established \\eqref{Eq:Mono3}.\n\nSuppose that the monogenic trinomials $f_{N,b_1}(x)$ and $f_{N,b_2}(x)$ generate the same $N$th-degree extension of $\\Q$. That is, $\\Q(\\alpha_1)=\\Q(\\alpha_2)$, where $f_{N,b_1}(\\alpha_1)=f_{N,b_2}(\\alpha_2)=0$. Then, by \\eqref{Eq:Dis-Dis}, we have that\n\\[(N-1)^{b_1N}+1=(N-1)^{b_2N}+1,\\] which implies that $b_1=b_2$ and $f_{N,b_1}(x)=f_{N,b_2}(x)$.\n\nFinally, it follows from Theorem \\ref{Thm:AC} that $bN/2$ divides the class number of the real quadratic field $\\Q(\\sqrt{\\delta})$.\n\\end{proof}\n\\begin{rem}\n We point out that if $N-1$ is squarefree with $N\\ge 6$, then $f_{N,b}(x)=x^N-N(N-1)^bx-(N-1)$ is $p$-Eisenstein for every prime divisor $p$ of $N-1$, and is hence irreducible over $\\Q$.\n\\end{rem}\n\\section{Proof of Theorem \\ref{Thm:Main4}}\n\\begin{proof}\nIn Theorem \\ref{Thm:Swan}, we have\n\\begin{equation}\\label{Eq:AB}\nA=-1 \\quad \\mbox{and}\\quad B=-(N-1)N^b,\n\\end{equation} so that\n\\[\\Delta(f_{N,b})=(N-1)^{N-1}(1-N^{(b+1)N-b})=(N-1)^{N-1}\\delta.\\]\n\nTo establish \\eqref{Eq:Mono4}, let $q$ be a prime divisor of $\\Delta(f_{N,b})$. Using \\eqref{Eq:AB}, we proceed through Theorem \\ref{Thm:JKS} to derive necessary and sufficient criteria to ensure the monogenicity of $f_{N,b}(x)$. Observe first that $q\\mid A$ is impossible so that conditions \\eqref{JKS:I1} and \\eqref{JKS:I2} of Theorem \\ref{Thm:JKS} are not applicable.\n\nIf $q\\nmid A$ and $q\\mid B$, then $q$ must divide $N-1$ since $q\\mid \\Delta(f_{N,b})$. We see in condition \\eqref{JKS:I3} of Theorem \\ref{Thm:JKS} that $A_1=0$ and $B_2=-(N-1)N^b/q$. Thus, $q\\nmid B_2$ if and only if $q^2\\nmid (N-1)$. Hence, condition \\eqref{JKS:I3} is satisfied for all prime divisors $q$ of $N-1$ if and only if $N-1$ is squarefree.\n\nIf $q\\nmid AB$, then $q\\mid \\delta$. Hence, condition \\eqref{JKS:I5} is satisfied for all prime divisors $q\\nmid AB$ if and only if $q^2\\nmid \\delta$, which is equivalent to $\\delta$ being squarefree. Since $\\delta$ being squarefree implies that $N-1$ is squarefree, we have established \\eqref{Eq:Mono4}.\n\nThe proof that no two monogenic trinomials $f_{N,b}(x)$ generate the same $N$-th-degree extension of $\\Q$ is similar to the corresponding proof in Theorem \\ref{Thm:Main3}, and we omit the details.\nFinally, it follows from Theorem \\ref{Thm:Murty} that the ideal class group of the imaginary quadratic field $\\Q(\\sqrt{\\delta})$ contains an element of order $(b+1)N-b$.\n\\end{proof}\n\\section{Examples to Illustrate the Theorems}\nAll computations were done using Sage.\n\n\\subsection{Theorem \\ref{Thm:Main1}} For $\\abs{w}\\le 10$, we give in Table \\ref{T:1} the values of $w$ and $d$ for trinomials $f_{1,w}(x)$ in $\\FF_1$, and the class numbers $h_K$ of the unique quadratic subfield $K=\\Q(\\sqrt{d})$ of the splitting field of $f_{1,w}(x)$.\n\n\\subsection{Theorem \\ref{Thm:Main2}}\nFor the seven smallest squarefree values of $\\abs{\\delta}$ with $1\\le a\\le 3$, $2\\le b \\le 6$ and $\\delta<0$, we give in Table \\ref{T:2} the values of $a$, $b$, $\\delta$, the class number $h$ of the imaginary quadratic field $\\Q(\\sqrt{\\delta})$, and the value of $n:=(6b+3)/\\gcd(3,2b+1)$.\n\n\\subsection{Theorem \\ref{Thm:Main3}}\nFor the two smallest squarefree values of $\\delta$, such that $N(N-1)$ is squarefree, we give in Table \\ref{T:3} the values of $N$, $b$, $\\delta$, the class number $h$ of the real quadratic field $\\Q(\\sqrt{\\delta})$, and $n:=bN/2$.\n\n\\subsection{Theorem \\ref{Thm:Main4}}\nFor the four smallest squarefree values of $\\abs{\\delta}$, we give in Table \\ref{T:4} the values of $N$, $b$, $\\delta$, the class number $h$ of the real quadratic field $\\Q(\\sqrt{\\delta})$, and $n:=(b+1)N-b$.\n\n\n\n\n\n \\begin{table}[h]\n \\begin{center}\n\\begin{tabular}{c|ccccccccccccc}\n $w$ & $-10$ & $-7$ & $-5$ & $-4$ & $-1$ & 1 & 4 & 5 & 7& 8 & 10 \\\\\n $d$ & $-4027$ & $-1399$ & $-527$ & $-283$ & $-31$ & $-23$ & 229 & 473 & 1345 & 2021 & 3973 \\\\\n $h_K$ & 9 & 27 & 18 & 3 & 3 & 3 & 3 & 3 & 6 & 3 & 6\n \\end{tabular}\n\\end{center}\n\\caption{Values of $w$ and $d$ for $f_{1,w}(x)\\in \\FF_1$ and class numbers of $\\Q(\\sqrt{d})$}\n \\label{T:1}\n\\end{table}\n\n\\begin{table}[h]\n \\begin{center}\n\\begin{tabular}{c|ccccccc}\n $a$ & 1 & 2 & 1 & 2 & 1 & 3 & 2\\\\\n $b$ & 3 & 4 & 4 & 5 & 5 & 6 & 6\\\\\n $\\delta$ & $-1931$ & $-3299$ & $-19427$ & $-160763$ & $-176891$ & $-545747$ & $-1577939$\\\\\n $h$ & 21 & 27 & 27 & 66 & 132 & 273 & 624\\\\\n $n$ & 21 & 9 & 9 & 33 & 33 & 39 & 39\n \\end{tabular}\n\\end{center}\n\\caption{Values of $a$, $b$, $\\delta$, class numbers $h$ of $\\Q(\\sqrt{\\delta})$ and $n$}\n \\label{T:2}\n\\end{table}\n\n\\begin{table}[h]\n \\begin{center}\n\\begin{tabular}{ccccc}\n $N$ & $b$ & $\\delta$ & $h$ & $n$\\\\ \\hline\n 6 & 1 & $15626$ & 24 & 3\\\\\n 6 & 2 & $244140626$ & 1248 & 6\n \\end{tabular}\n\\end{center}\n\\caption{Values of $N$, $b$, $\\delta$, class numbers $h$ of $\\Q(\\sqrt{\\delta})$ and $n$}\n \\label{T:3}\n\\end{table}\n\n \\begin{table}[h]\n \\begin{center}\n\\begin{tabular}{ccccc}\n $N$ & $b$ & $\\delta$ & $h$ & $n$\\\\ \\hline\n 3 & 2 & $-2186$ & 42 & 7\\\\\n 3 & 3 & $-19682$ & 108 & 9\\\\\n 3 & 4 & $-177146$ & 396 & 11\\\\\n 7 & 1 & $-96889010406$ & 196768 & 13\n \\end{tabular}\n\\end{center}\n\\caption{Values of $N$, $b$, $\\delta$, class numbers $h$ of $\\Q(\\sqrt{\\delta})$ and $n$}\n \\label{T:4}\n\\end{table}\n\n\n\\section*{Acknowledgement} \n\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{AC} N. C. Ankeny and S. Chowla, {\\em On the divisibility of the class number of quadratic fields}, Pacific J. Math. {\\bf 5} (1955), 321--324.\n\n\\bibitem{Cohen} H. Cohen, \\emph{A Course in Computational Algebraic Number Theory}, {Springer-Verlag}, 2000.\n\n\\bibitem{Erdos} P. Erd\\H{o}s, {\\em Arithmetical properties of polynomials}, J. London Math. Soc. {\\bf 28} (1953), 416--425.\n\n\\bibitem{Helfgott} H. Helfgott, {\\em Square-free values of $f(p)$, $f$ cubic}, Acta Math. {\\bf 213} (2014), no. 1, 107--135.\n\n\\bibitem{H} A. Herschfeld, {\\em The equation $2^x-3^y=d$}, Bull. Amer. Math. Soc. {\\bf 42} (1936), no. 4, 231--234.\n\n\\bibitem{JKS2} A. Jakhar, S. Khanduja and N. Sangwan, \\emph{Characterization of primes dividing the index of a trinomial}, Int. J. Number Theory {\\bf 13} (2017), no. 10, 2505--2514.\n\n\\bibitem{K} Y. Kishi, {\\em Note on the divisibility of the class number of certain imaginary quadratic\nfields}, Glasgow Math. J. {\\bf 51} (2009), 187--191; {\\em Corrigendum}, Glasgow Math. J. {\\bf 52}\n(2010), no. 1, 207--208.\n\n\\bibitem{KM} Y. Kishi and K. Miyake, {\\em Parametrization of the quadratic fields whose class numbers are divisible by three.} (English summary)\nJ. Number Theory {\\bf 80} (2000), no. 2, 209--217.\n\n\n\\bibitem{M} P. Mih\\u{a}ilescu, {\\em Primary cyclotomic units and a proof of Catalan's conjecture}, J. Reine Angew. Math. {\\bf 572} (2004), 167--195.\n\n\\bibitem{Murty} M. Ram Murty, {\\em The abc conjecture and exponents of class groups of quadratic fields}, Number theory (Tiruchirapalli, 1996), 85--95, Contemp. Math., {\\bf 210}, Amer. Math. Soc., Providence, RI, 1998.\n\n\\bibitem{Swan} R. Swan, \\emph{Factorization of polynomials over finite fields}, Pacific J. Math. {\\bf 12} (1962), 1099--1106.\n\n\\end{thebibliography}\n\n\n\n\\end{document}\n\n\n\\begin{table}[h]\n \\begin{center}\n\\begin{tabular}{ccccc}\n $a$ & $b$ & $\\delta$ & $h$ & $n$\\\\ \\hline\n 1 & 3 & $-1931$ & 21 & 21\\\\\n 1 & 4 & $-19427$ & 27 & 9\\\\\n 1 & 5 & $-176891$ & 132 & 33\\\\\n 2 & 4 & $-3299$ & 27 & 9\\\\\n 2 & 5 & $-160763$ & 66 & 33\\\\\n 3 & 6 & $-545747$ & 273 & 39\n \\end{tabular}\n\\end{center}\n\\caption{Values of $a$, $b$, $\\delta$, class numbers $h$ of $\\Q(\\sqrt{\\delta})$ and $n$}\n \\label{T:2A}\n\\end{table}\n\n\n\n\n\n"}}},{"rowIdx":286,"cells":{"id":{"kind":"string","value":"2412.20483v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20483v1"},"title":{"kind":"string","value":"Curvature, area and Gauss-Bonnet formula of the Moyal sphere"},"full_text":{"kind":"string","value":"\\documentclass[a4paper,12pt]{article}\n\\usepackage[left=3.17cm,right=3.17cm,top=2.54cm,\nheadheight=0.5cm,headsep=0.54cm,bottom=2.54cm,footskip=0.79cm\n]{geometry}\n\n\\usepackage{amsmath,amssymb,dsfont,cite}\n\n\\usepackage{graphicx,hyperref}\n\n\\usepackage{amsthm,float,subfigure}\n\n\\newtheorem{prop}{Property}\n\\newtheorem{thm}{Theorem}\n\n\n\\begin{document}\n\n\\title{Curvature, area and Gauss-Bonnet formula of the Moyal sphere}\n\n\\author{Han-Liang Chen$^{\\ast}$, Bing-Sheng Lin$^{\\dag}$\\\\\n\t\\small School of Mathematics, South China University of Technology,\n\t\\small Guangzhou 510641, China\\\\\n\t\\small $^{\\ast}$Email: mj16661905@163.com\\\\\n\t\\small $^{\\dag}$Email: sclbs@scut.edu.cn\\\\\n}\n\n\\date{\\today}\n\n\\maketitle\n\n\\begin{abstract}\n\tWe studied some geometric properties of the Moyal sphere. Using the conformal metric of the sphere in ordinary space and the matrix basis, we calculated the scalar curvature, total curvature integral and area of the Moyal sphere. We found that when the noncommutative parameter approaches to 0, the scalar curvature and area of the Moyal sphere return to those of the ordinary sphere. As the noncommutative parameter increases, the area of the Moyal sphere will decrease and eventually approach to 0. We found that the total curvature integral of the two-dimensional Moyal sphere still satisfies the usual Gauss-Bonnet formula and does not depend on the noncommutative parameter. We also calculated the approximate expression of the conformal metric with a constant curvature and obtained the corresponding correction function.\n\tIn addition, we also studied a type of generalized deformed Moyal sphere with two noncommutative parameters and obtained similar results. \n\\end{abstract}\n\n\n\n\\section{Introduction}\\label{sec1}\nThe ideas of noncommutative spacetime in physics started in 1947\\cite{Snyder}. In the 1980's, Connes formulated the mathematically rigorous framework of noncommutative geometry\\cite{Connes}. Using the language of noncommutative algebras, noncommutative geometry provides new perspectives on other branches of mathematics, such as operator algebras, differential geometry, algebraic geometry, K-theory, cyclic cohomology, number theory, measure theory, etc. It also has many novel and useful applications in physics\\cite{Seiberg, Douglas, Chaichian, Ettefaghi, lhc, Calmet, Couch, Muhuri, lh, Frob, Maris}. Many mathematicians have studied the generalizations of noncommutative analog of differential geometry from different perspectives\\cite{Madore, Kaygun, Tretkoff, Fathizadeh, Dabrowski, Brzezinski, Arnlind, Floricel, Sciandra}. For example, Connes studied the geometric properties of noncommutative manifolds by studying the noncommutative form of the so-called spectral triples corresponding to Riemann manifolds. Some authors defined calculus on noncommutative algebras and then use the classical formula to define and calculate the scalar curvature on some noncommutative spaces\\cite{Khalkhali,Wilson}.\n\nMoyal space is one of the simplest noncommutative spaces, and has many important applications in mathematics and physics\\cite{Gayral}. We will study some geometric properties of the Moyal sphere. As we all know, the calculations of geometric quantities on smooth manifolds depend on the operations of smooth functions on them, such as volume, curvature, vector field, differential form, tensor field, vector bundle, etc. The noncommutative geometric ideas based on this are described in detail in Ref.~\\cite{Connes}. In Ref.~\\cite{Eckstein}, the authors introduced the conformal metric $g=\\frac{1}{h^{2}}(dx^{1}dx^{1}+dx^{2}dx^{2})$ into the two-dimensional Moyal space. Using the Moyal matrix basis and the orthonormal system method, they obtained the scalar curvature formula, and proved that there is a family of constant curvature metrics. The authors called the Moyal space with such constant curvature metrics the Moyal sphere. They also derived the same Gauss-Bonnet formula with the metric in a specific case as that of the ordinary two-dimensional sphere.\n\nSince the classical sphere $\\mathbf{S}^{n}$ is homeomorphic to $\\mathbb{R}^{n}$ after removing a pole, and has a simple conformal metric $g=\\frac{4}{(1+r^{2})^{2}}(dx^{1}dx^{1}+dx^{2}dx^{2}+\\dots+dx^{n}dx^{n})$, one can use similar methods to generalize some results in Ref.~\\cite{Eckstein} to high dimensional cases.\nIn this paper, we will study the generalizations of high-dimensional spheres in Moyal space from two different perspectives. On the one hand, we will study the $2M$-dimensional Moyal space with spherical metric $g=\\frac{4A^{4}}{(r^{2}+A^{2})^{2}}(dx^{1}dx^{1}+dx^{2}dx^{2}+\\dots+dx^{2M}dx^{2M})$, and on the other hand, we will also study the Moyal space assigned a conformal metric with constant curvature similar to the classical sphere. These two noncommutative spaces can both return to the classical sphere when the noncommutative parameter approaches to 0, so they can be regarded as the generalizations of the ordinary sphere in Moyal space, and we also call them \\textit{Moyal spheres}.\n\nIn this paper we will calculate the area, curvature and total curvature integral of the Moyal spheres with certain dimensions, and study the conformal metric of the Moyal sphere with constant curvature.\nThis paper is organized as follows: In Sec.~\\ref{sec2}, we review the definitions and properties of Moyal star product and matrix basis. In Sec.~\\ref{sec3}, we derive the curvature expression of the Moyal space. In Sec.~\\ref{sec4}, we calculate the scalar curvature of the two-dimensional Moyal sphere. In Sec.~\\ref{sec5}, we calculate the Gauss-Bonnet formula of the two-dimensional Moyal sphere, and find that the result is the same as that of the ordinary two-dimensional sphere. This implies that the Moyal star product structure of the smooth function space does not change some topological properties of the two-dimensional sphere.\nIn Sec.~\\ref{sec6} we also calculate the scalar curvature expression of the four-dimensional Moyal sphere.\nIn Sec.~\\ref{sec7}, we calculate the area of any even-dimensional Moyal sphere and a type of deformed Moyal sphere. We find that the area of the Moyal sphere will decrease as the noncommutative parameter increases. When the noncommutative parameter approaches to infinity, the area of the Moyal sphere will approach to 0. In Sec.~\\ref{sec8}, we also study the conformal metric with constant curvature in the Moyal space, and gave an approximate expression of the factor. The last section are the conclusions.\n\n\n\\section{Moyal star product and matrix basis}\\label{sec2}\nThe Moyal $*$-product in the space of complex-valued functions on $\\mathbb{R}^{2M}$ (where $M$ is a positive integer) is defined as\\cite{Gayral}:\n\\begin{equation*}\n\t(f*g)(x):=(\\pi \\theta)^{-2M}\\int_{\\mathbb{R}^{2M}\\times \\mathbb{R}^{2M}}f(y)g(z)e^{\\frac{2\\mathbf{i}}{\\theta}(x-y)\\cdot\\Lambda(x-z)}d^{2M}yd^{2M}z,\n\\end{equation*}\nwhere the noncommutative parameter $\\theta$\\ is a real number, and the antisymmetric matrix\n$\\Lambda=I_{M}\\otimes \\varepsilon_2$, $I_{M}$\\ is the $M\\times M$ identity matrix, $\\varepsilon_2$ is the antisymmetric matrix\n$\\varepsilon_2\n=\\begin{pmatrix}\n\t0&1 \\\\\n\t-1 & 0\n\\end{pmatrix}$.\nThe smooth function space $C^{\\infty}(\\mathbb{R}^{2M})$ forms an algebra with respect to the Moyal star product, called the Moyal algebra. The corresponding noncommutative space is the so-called Moyal space, and its coordinate variables $\\{x_{1},x_{2},...,x_{2M}\\}$ satisfy the commutation relations:\n\\begin{equation}\n\\left[x_{2j-1},x_{2j}\\right]_*:=x_{2j-1}*x_{2j}-x_{2j}*x_{2j-1}\n=-\\left[x_{2j},x_{2j-1}\\right]_*=\\mathbf{i}\\theta,\\label{ms1}\n\\end{equation}\nwhere $j=1,\\dots,M$, and other commutators vanish.\n\nA Schwarz space is a non-unitary, associative, involutive Fr\\'echet algebra $(\\mathcal{S} (\\mathbb{R}^{2M})$,$*)$ with respect to the $*$-product, and it satisfies the following properties:\n\\begin{prop}\nLet $f,g\\in \\mathcal{S} (\\mathbb{R}^{2M},*) $, and $x=(x_{1},x_{2},...,x_{2M})$, there are\\cite{Gayral}\n\\begin{itemize}\n\t\\item[(i)]$(f*g)(x)\\in \\mathcal{S} (\\mathbb{R}^{2M})$.\n\t\\item[(ii)]When the variables of two functions are irrelevant with respect to the Moyal $*$-products, then their Moyal $*$-products return to the ordinary scalar products, for example: $f(x_{1},x_{2})*g(x_{3},x_{4})=f(x_{1},x_{2})g(x_{3},x_{4})$.\n\t\\item[(iii)]$\\partial^{j}(f*g)=\\partial^{j}f*g+f*\\partial^{j}g,(j=1,2,...,2M)$.\n\t\\item[(iv)]$\\int_{\\mathbb{R}^{2M}}(f*g)(x)d^{2M}x=\\int_{\\mathbb{R}^{2M}}f(x)g(x)d^{2M}x$.\n\\end{itemize}\n\\end{prop}\nNext, we introduce a set of functions on the Moyal space called the matrix basis $\\left \\{ f_{mn}\\right \\} _{m,n\\in \\mathbb{N}^{M}}$. First, we consider a function in the Schwarz space $\\mathcal{S}(\\mathbb{R}^{2M}) $:\n$f_{00}=2^{M}e^{-\\frac{1}{\\theta}\\sum_{j=1}^{2M}x_{j}^{2}}$. Let $m=(m_{1},m_{2},...,m_{M})$,\nand use multiple index notations: $ \\left | m\\right | =\\sum_{j=1}^{M}m_{j}$, $m!=\\prod_{j=1}^{M}m_{j} !$, $\\delta_{ij}=\\prod_{j=1}^{M}\\delta_{m_{j}n_{j}} $. The matrix basis $\\left \\{ f_{mn}\\right \\} _{m,n\\in \\mathbb{N}^{M}}$ can be defined as:\n\\begin{equation*}\nf_{mn}=\\frac{1}{\\sqrt{\\theta^{\\left | m\\right |+\\left | n\\right |}m!n!}}\\bar{a}^{m}*f_{00}*a^{n},\n\\end{equation*}\nwhere $a^{n}=a_{1}^{n_{1}}...a_{M}^{n_{M}}$, the annihilation and creation functions are $a_{l}=\\frac{1}{\\sqrt{2}}(x_{l}+\\mathbf{i}x_{l+M})$, $\\bar{a}_{l}=\\frac{1}{\\sqrt{2}}(x_{l}-\\mathbf{i}x_{l+M})$, respectively.\n\\begin{prop} The matrix basis $\\left \\{ f_{mn}\\right \\} _{m,n\\in \\mathbb{N}^{M}}$ satisfies the following properties:\n\\begin{itemize}\n\t\\item[(i)]$f_{mn}*f_{kl}=\\delta_{nk}f_{ml}$.\n\t\\item[(ii)]$\\bar{f}_{mn}=f_{nm}$.\n\t\\item[(iii)]$\\int_{\\mathbb{R}^{2M}}f_{mn} d^{2M}x=(2\\pi \\theta)^{M}\\delta_{mn}$.\n\\end{itemize}\n\\end{prop}\nIt is easy to see that $\\left \\{ f_{mn}\\right \\} _{m,n\\in \\mathbb{N}^{M}}$ is an orthogonal basis of the Schwarz space $\\mathcal{S} (\\mathbb{R}^{2M})$ under the $L^{2}$ norm. For the matrix basis on $\\mathcal{S} (\\mathbb{R}^{2})$, there are the following differential properties:\n\\begin{prop}\nConsider the differential operators on $\\mathcal{S} (\\mathbb{R}^{2})$: $\\partial=\\frac{1}{\\sqrt{2}}(\\partial_{x_{1}}-\\mathbf{i}\\partial_{x_{2}}),\\bar{\\partial}=\\frac{1}{\\sqrt{2}}(\\partial_{x_{1}}+\\mathbf{i}\\partial_{x_{2}})$, we have\\cite{Eckstein}\n\\begin{align*}\n\t\\partial f_{mn} &= \\sqrt{\\frac{n}{\\theta}}f_{m,n-1}-\\sqrt{\\frac{m+1}{\\theta}}f_{m+1,n}, \\\\\n\t\\bar{\\partial}f_{mn} &= \\sqrt{\\frac{m}{\\theta}}f_{m-1,n}-\\sqrt{\\frac{n+1}{\\theta}}f_{m,n+1}.\n\\end{align*}\n\\end{prop}\nThe above differential formulas are valid for $m,n \\ge 0$ (when a subscript is less than $0$, the term is set to be $0$). It is easy to obtain the following partial differential equations and the formula acted by the Laplace operator of the matrix basis:\n\\begin{align*}\n\\partial_{x_{1}}f_{mn} &=\\frac{1}{\\sqrt{2}}(\\partial+\\bar{\\partial})f_{mn}\\\\\n&=\\frac{1}{\\sqrt{2}}\\left( \\sqrt{\\frac{n}{\\theta}}f_{m,n-1}- \\sqrt{\\frac{m+1}{\\theta}}f_{m+1,n}+\\sqrt{\\frac{m}{\\theta}}f_{m-1,n}-\\sqrt{\\frac{n+1}{\\theta}}f_{m,n+1}\\right), \\\\\n\\partial_{x_{2}}f_{mn} &=\\frac{1}{\\sqrt{2}\\mathbf{i}}(\\bar{\\partial}-\\partial)f_{mn}\\\\\n&= \\frac{1}{\\sqrt{2}\\mathbf{i}}\\left( \\sqrt{\\frac{n}{\\theta}}f_{m,n-1}-\\sqrt{\\frac{m+1}{\\theta}}f_{m+1,n}-\\sqrt{\\frac{m}{\\theta}}f_{m-1,n}+\\sqrt{\\frac{n+1}{\\theta}}f_{m,n+1}\\right) , \\\\\n\\triangle f_{mn}\n&=2\\partial\\bar{\\partial}f_{mn}\\\\\n&=\\frac{2}{\\theta}\\left[-(m+n+1)f_{mn}+\\sqrt{(m+1)(n+1)}f_{m+1,n+1 }+\\sqrt{mn}f_{m-1,n-1}\\right].\n\\end{align*}\nIn addition, the matrix basis $\\{f_{mm}\\}_{m=0}^{\\infty}$ also satisfies the following very useful formulas:\n\\begin{equation*}\n\\sum_{m}f_{mm}=1 ,\\qquad \\sum_{m}mf_{mm}=\\frac{1}{2}\\left(\\frac{r^{2}}{\\theta}-1 \\right).\n\\end{equation*}\n\n\n\\section{Curvature of the Moyal space}\\label{sec3}\nFor an $n$-dimensional Riemannian manifold $M$, one can choose a set of local orthonormal bases $\\left\\{e_{i}\\right\\}$ in its local coordinate system. Using the properties of the Levi-Civita connection $\\nabla$, one can obtain the following expression of the curvature tensor:\n\\begin{align*}\n\tR(e_{i},e_{j},e_{i},e_{j})&=\\langle R(e_{i},e_{j})e_{i},e_{j}\\rangle\\\\\n\t&=\\langle\\nabla_{e_{j}}\\nabla_{e_{i}}e_{i}-\\nabla_{e_{i}}\\nabla_{e_{j}}e_{i}+\\nabla_{[ e_{i},e_{j}]}e_{i},e_{j}\\rangle \\\\\n\t&=e_{j}c_{jii}+e_{i}c_{ijj}-\\frac{1}{4}\\sum_{k}(c_{ kji}+c_{jik}+c_{kij})(c_{jik}+c_{ikj}+c_{jki})\\\\\n\t&\\qquad-\\sum_{k}c_{kii}c_{kjj}+ \\frac{1}{2}\\sum_{k}c_{ijk}(c_{jki}+c_{kij}+c_{jik}),\n\\end{align*}\nwhere $R(X,Y,Z,W)$ is the Riemann curvature tensor, $\\langle \\cdot ,\\cdot \\rangle$ is the Riemann metric of the manifold, $[e_{i},e_{j}]=\\displaystyle \\sum_{k=1}^{n} c_{ijk}e_{k}$, and $c_{ijk}$ are the structure constants of the commutators.\nThe scalar curvature $S$ is:\n\\begin{align*}\n\tS&=\\sum_{i,j=1}^{n}R(e_{i},e_{j},e_{i},e_{j})\\\\\n\t&=\\sum_{i,j=1}^{n}\\bigg[e_{j}c_{jii}+e_{i}c_{ijj}-\\sum_{k}c_{kii}c_{kjj }-\\frac{1}{4}\\sum_{k}(c_{kji}+c_{jik}+c_{kij})(c_{jik}+c_{ikj}+c_{jki})\\\\\n\t&\\qquad +\\frac{1}{2}\\sum_{k}c_{ijk}(c_{jki}+c_{kij}+c_{jik})\\bigg]\\\\\n\t&=2\\sum_{ij}^{n}e_{i}c_{ijj}-\\sum_{ijk}^{n}c_{kii}c_{kjj}-\\frac{1}{2}\\sum_{ijk}^{n}c_{ijk}c_{ikj}-\\frac{1}{4}\\sum_{ijk}^{n}c_{ijk}c_{ijk}.\n\\end{align*}\nConsidering the symmetry of the summation index, and using the Einstein summation convention, one can obtain the following scalar curvature formula:\n\\begin{equation*}\n\tS=2e_{i}c_{ijj}-c_{kii}c_{kjj}-\\frac{1}{4}c_{ijk}c_{ijk}-\\frac{1}{2}c_{ijk}c_{ikj}.\n\\end{equation*}\nIn the Moyal space, ordinary multiplications between functions will be replaced by Moyal $*$-products. Similar to the result in Ref.~\\cite{Eckstein}, there is the scalar curvature in the Moyal space:\n\\begin{equation}\\label{scf}\n\tS_\\theta=2e_{i}*c_{ijj}-c_{kii}*c_{kjj}-\\frac{1}{4}c_{ijk}*c_{ijk}-\\frac{1}{2}c_{ijk}*c_{ikj}.\n\\end{equation}\nDue to the symmetry of the sum, adding the $*$-product does not cause any confusion in the definition.\n\nConsider the conformal metric $g=h^{-2}(dx^{1}dx^{1}+dx^{2}dx^{2}+ \\dots dx^{n}dx^{n})$, where $h$ is some smooth positive function. We will calculate it using the set of local orthogonal basis $\\left\\{e_{i}=h\\frac{\\partial}{\\partial x^{i}}\\right\\}$. Note that the scalar curvature (\\ref{scf}) only contains the structure constants and their derivatives. So in order to obtain the scalar curvature, one only need to calculate the structure constants under the set of orthonormal basis :\n\\begin{align*}\n\t[e_{i},e_{j}]_*&=h*\\partial_{i}(h*\\partial_{j})-h*\\partial_{j}(h*\\partial_{i})\\\\\n\t&=h*\\partial_{i}(h)*h_*^{-1}*(h*\\partial_{j})-h*\\partial_{j}(h)*h_*^{-1}*(h*\\partial_{i})\\\\\n\t&=h*\\partial_{i}(h)*h_*^{-1}*e_{j}-h*\\partial_{j}(h)*h_*^{-1}*e_{i}\\\\\n\t&=\\sum_{k=1}^{n} c_{ijk}e_{k},\n\\end{align*}\nwhere $h_*^{-1}$ is the inverse of $h$ with respect to the Moyal $*$-product, $h*h_*^{-1}=h_*^{-1}*h=1$.\nNote that $c_{ijk}\\ne 0$ only when $i\\ne j$ and $k=i$ or $j$, and we have\n$$c_{iji}=-h*\\partial_{j}(h)*h_*^{-1}, \n\\qquad c_{ijj}=h*\\partial_{i}(h)*h_*^{-1},\\qquad(i\\ne j) .$$\nSubstituting the above expression into Eq.(\\ref{scf}), one can get the scalar curvature expression of the $2n$-dimensional Moyal space:\n\\begin{align}\\label{sss}\n\tS_\\theta&=2(n-1)h_*^{2}*\\partial_{ii}(h)*h_*^{-1}-(n^{2}-3n+2)h*\\partial_{i}(h)_*^{2}*h_*^{-1}\\nonumber\\\\\n\t&\\quad -2(n-1)h_*^{2}*\\partial_{i}(h)*h_*^{-1}*\\partial_{i}(h)*h_*^{-1}.\n\\end{align}\nHere we have used the Einstein summation convention, and the powers and inverses of functions in the above equation are in the sense of the Moyal $*$-product. It is easy to see that using the metric $g=h_*^{-2}(dx^{1}dx^{1}+dx^{2}dx^{2}+ ...+dx^{n}dx^{n}) $, the generalization of the classical scalar curvature formula (\\ref{scf}) is well defined (because the indices of the summation terms are symmetric, after commuting the star products of the two summation terms in the sum, the results are still the same).\n\n\n\\section{Scalar curvature of the two-dimensional Moyal sphere}\\label{sec4}\nLet us recall that in ordinary space, the metric of a sphere with a radius of $A$ in stereographic coordinates is:\n\\begin{equation*}\n\tg=\\frac{4A^4}{(A^2+r^{2})^{2}}(dx^{1}dx^{1}+dx^{2}dx^{2}).\n\\end{equation*}\nTherefore, the metric $g$ can be written as:\n\\begin{equation*}\n\tg=\\frac{1}{h^{2}}(dx^{1}dx^{1}+dx^{2}dx^{2}),\n\\end{equation*}\nwhere\n$$h=\\frac{A^2+r^{2}}{2A^2}=\\sum_{m}\\phi_{m}f_{mm},\\qquad \\phi_{m}=\\frac{2\\theta m+A^2+\\theta}{2A^2},\\quad(m=0,1,\\dots).$$\nAfter some straightforward calculations, one can obtain\n\\begin{equation*}\n\t\\triangle h=\\frac{2}{\\theta}\\sum_{m}[-(2m+1)\\phi_{m}+m\\phi_{m-1}+(m+1)\\phi_{m+1}]f_{mm},\n\\end{equation*}\n\\begin{equation*}\n\t\\partial_{i}(h)*h_*^{-1}*\\partial_{i}(h)\n\t=\\frac{1}{\\theta}\\sum_{m=0}\\left[ \\frac{m}{\\phi_{m-1}}(\\phi_{m}-\\phi_{m-1})^{2}+\\frac{m+1}{\\phi_{m+1}}(\\phi_{m+1}-\\phi_{m})^{2}\\right] f_{mm}.\n\\end{equation*}\nSubstituting the above results into Eq.(\\ref{sss}), one can get the scalar curvature of the two-dimensional Moyal sphere\n\\begin{eqnarray}\n\tS_\\theta&=&2[h_*^{2}*\\partial_{ii}(h)*h_*^{-1}-h_*^{2}*\\partial_{i}(h)*h_*^{-1}*\\partial_{i}(h)*h_*^{-1}]\\nonumber\\\\\n\t&=&\\frac{2}{\\theta}\\sum_{m}\\phi_{m}\\left[ \\frac{m+1}{\\phi_{m+1}}(\\phi_{m+1}^{2}-\\phi_{m}^{2})-\\frac{m}{\\phi_{m-1}}(\\phi_{m}^{2}-\\phi_{m-1}^{2})\\right] f_{mm}\\nonumber\\\\\n\t&=&\\sum_{m}\\left[ \\frac{2}{A^{2}}-\\frac{2\\theta(\\theta+A^{2})}{A^{4}(2\\theta m+A^{2}+3\\theta)}+\\frac{2\\theta(A^{2}-\\theta)}{A^{4}(2\\theta m+A^{2}-\\theta)}\\right] f_{mm}\\nonumber\\\\\n\t&=&\\frac{2}{A^{2}}-\\frac{2\\theta(A^{2}+\\theta)}{A^{4}}(r^{2}+A^{2}+2\\theta)_{*}^{-1}+\\frac{2\\theta(A^{2}-\\theta)}{A^{4}}(r^{2}+A^{2}-2\\theta)_{*}^{-1}.\\label{sth}\n\\end{eqnarray}\nNote that $S_\\theta=S_{-\\theta}$, without losing generality, one can be assumed that $\\theta>0$.\nIn order to avoid singularities, one can choose $\\theta\\in (0,\\frac{A^2}{2})$. Obviously, when $\\theta \\to 0$, there is $S_\\theta\\to \\frac{2}{A^2}$, which is the same as the scalar curvature of the sphere $\\mathbf{S}^{2}$ with a radius of $A$ in the classical case.\n\nWhen the noncommutative parameter $\\theta$ is very small, using the methods similar to those in Ref.~\\cite{Eckstein}, one can only consider terms that approximate to the $\\theta^2$ term. The inverse of a radial function $f$ with respect to the Moyal $*$-product has an approximate expansion:\n$$f_{*}^{-1}=f^{-1}+\\frac{\\theta^{2}}{4r}[(f')^{3}f^{-4}-f''f'f^{-3}].$$\nTherefore, the scalar curvature of the Moyal sphere (\\ref{sth}) can be approximately expressed as:\n\\begin{align*}\n\tS_\\theta&\\approx \\frac{2}{A^{2}}-\\frac{2\\theta(A^{2}+\\theta)}{A^{4}(r^{2}+A^{2}+2\\theta)}\n\t+\\frac{2\\theta(A^{2}-\\theta)}{A^{4}(r^{2}+A^{2}-2\\theta)}\\\\\n\t&= \\frac{2}{A^{2}}\\cdot \\eta_\\theta(\\lambda),\n\\end{align*}\nwhere $\\lambda=\\frac{A^{2}}{\\theta}$, and\n$$\n\\eta_\\theta(\\lambda)=1-\\frac{\\lambda+1}{\\lambda\\left( \\frac{r^{2}}{\\theta}+\\lambda+2\\right) }\n+\\frac{\\lambda-1}{\\lambda\\left( \\frac{r^{2}}{\\theta}+\\lambda-2\\right) }.\n$$\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=0.4]{2dscalar.png}\n\t\\caption{The curves of $\\eta_\\theta(\\lambda)$ with different values of $\\lambda$ ($\\theta=0.1$).}\\label{fig1}\n\\end{figure}\nIn Fig.~\\ref{fig1}, we show the curves of $\\eta_\\theta(\\lambda)$ with different values of $\\lambda$, where the noncommutative parameter $\\theta=0.1$.\nOne can see that for a given small $\\theta$, when $\\lambda \\to \\infty$, or equivalently, when the radius $A\\to \\infty$, the scalar curvature of the Moyal sphere will approach to that of the ordinary sphere. When $\\lambda \\to 0$, or equivalently, when the radius $A\\to 0$, the scalar curvature of the Moyal sphere is significantly different from that of the ordinary sphere for small $|r|$. But when $|r|$ is very large, the result also approaches to that of the ordinary sphere.\n\n\n\\section{Gauss-Bonnet formula for the Moyal sphere}\\label{sec5}\n\\begin{thm}[Gauss-Bonnet formula for compact surfaces]\n\tLet $M$ be a compact, oriented two-dimensional surface, there is\n\t\\begin{equation*}\n\t\t\\int_{M}Kd\\sigma=2\\pi \\chi(M),\n\t\\end{equation*}\n\twhere $K$ is the Gaussian curvature of $M$, $d\\sigma$ is the area element, and $\\chi(M)$ is the Euler characteristic number of $M$.\n\\end{thm}\nSince the scalar curvature $S=2K$ in two-dimensional surfaces, the Gauss-Bonnet formula can be written as:\n\\begin{equation*}\n\t\\int_{M}Sd\\sigma=4\\pi \\chi(M).\n\\end{equation*}\nIn particular, the Euler characteristic number of the two-dimensional sphere $\\mathbf{S}^{2}$ is $\\chi(\\mathbf{S}^{2})=2$, and $\\int_{\\mathbf{S}^{2}}Sd\\sigma=8\\pi$.\n\nNow let us calculate the Gauss-Bonnet formula for the two-dimensional Moyal sphere.\n\\begin{align*}\n\t\\int_{\\mathbf{S}^{2}}S_\\theta*d\\sigma\n\t&=\\int_{\\mathbb{R}^{2}}\\sum_{m}\\left[ \\frac{2}{A^{2}}-\\frac{2\\theta(\\theta+A^{2})}{A^{4}(2\\theta m+A^{2}+3\\theta)}+\\frac{2\\theta(A^{2}-\\theta)}{A^{4}(2\\theta m+A^{2}-\\theta)}\\right] f_{mm}*h_{*}^{-2} dxdy\\\\\n\t&=\\int_{\\mathbb{R}^{2}}\\left\\{ \\sum_{m}\\left[ \\frac{2}{A^{2}}-\\frac{2\\theta(\\theta+ A^{2})}{A^{4}(2\\theta m+A^{2}+3\\theta)}+\\frac{2\\theta(A^{2}-\\theta)}{A^{4}(2\\theta m+A^{2}-\\theta)}\\right] f_{mm}\\right. \\\\\n\t&\\qquad\\left. *\\sum_{m}\\frac{4A^{4}}{(2\\theta m +A^{2}+\\theta)^2}f_{mm}\\right\\} dxdy\\\\\n\t&=\\int_{\\mathbb{R}^{2}}\\sum_{m}\\frac{8(2\\theta A^{2}m+A^{4}-2\\theta^{2}+\\theta A^{2})}{(2\\theta m+A^{2}+\\theta)(2\\theta m+A^{2}-\\theta)(2\\theta m+A^{2}+3\\theta)}f_{mm}dxdy\\\\\n\t&=2\\pi\\theta\\sum_{m}\\frac{8(2\\theta A^{2}m+A^{4}-2\\theta^{2}+\\theta A^{2})} {(2\\theta m+A^{2}+\\theta)(2\\theta m+A^{2}-\\theta)(2\\theta m+A^{2}+3\\theta)}\\\\\n\t&=8\\pi.\n\\end{align*}\nThe above results show that the Gauss-Bonnet formula for the two-dimensional Moyal sphere is independent of the value of the noncommutative parameter $\\theta$. It is the same as that of the classical two-dimensional sphere $\\mathbf{S}^{2}$ with radius $A$.\n\nTherefore, the Gauss-Bonnet integrals for two-dimensional Moyal spheres of different radii are all equal to $8\\pi$.\nThis means that the integral of the scalar curvature on the surface in the noncommutative sense is only related to the topological invariant (Euler characteristic number) of the surface, and has nothing to do with the noncommutative parameter $\\theta$. This result is consistent with that in Ref.~\\cite{Eckstein}. But it should be noted that an appropriate $\\theta$ must be chosen here so that the inverse with respect to the Moyal star product of the matrix basis series exists.\n\n\n\n\\section{Scalar curvature of the four-dimensional Moyal sphere}\\label{sec6}\nConsidering the conformal metric $g=h^{-2}(dx^{1}dx^{1}+dx^{2}dx^{2}+dx^{3}dx^{3}+dx^{4}dx^{4})$ of the 4-dimensional sphere, one can similarly assume\n$$h=\\sum_{m}\\phi_{m}f_{mm}(x_1,x_2)+\\sum_{n}\\psi_{n}g_{nn}(x_3,x_4)=\\sum_{m,n=0}(\\phi_{m}+\\psi_{n})f_{mm}g_{nn}.\n$$\nAfter some straightforward calculations, one can obtain\n$$\n\th_*^{-1}*\\partial_{i}(h)^{2}=\\frac{1}{\\theta}\\sum_{m,n=0}\\frac{1}{\\phi_{m}+\\psi_{n}}[m(\\phi_{m}-\\phi_{m-1})^{2}+(m+1)(\\phi_{m+1}-\\phi_{m})^{2}]f_{mm}g_{nn}.\n$$\nThe scalar curvature of the four-dimensional Moyal space (\\ref{sss}) is\n\\begin{align*}\n\tS_\\theta&=6[h^{2}*\\partial_{ii}(h)*h^{-1}-h*\\partial_{i}(h)^{2}*h^{-1}-h^{2}*\\partial_{i}(h)*h^{-1}*\\partial_{i}(h)*h^{-1}]\\\\\n\t&=\\frac{6}{\\theta}\\sum_{m,n=0}(\\phi_{m}+\\psi_{n})\\Bigg\\{[-(4m+2)\\phi_{m}+2m\\phi_{m-1}+2(m+1)\\phi_{m+1}]\\\\\n\t&\\qquad-\\bigg[m(\\phi_{m}-\\phi_{m-1})^{2}\\bigg(\\frac{1}{\\phi_{m}+\\psi_{n}}+\\frac{1}{\\phi_{m-1}+\\psi_{n}}\\bigg)\\\\\n\t&\\quad\\qquad+(m+1)(\\phi_{m+1}-\\phi_{m})^{2}\\bigg(\\frac{1}{\\phi_{m}+\\psi_{n}}+\\frac{1}{\\phi_{m+1}+\\psi_{n}}\\bigg)\\bigg]\\Bigg\\}f_{mm}g_{nn}\\\\\n\t&\\quad+\\frac{6}{\\theta}\\sum_{m,n=0}(\\phi_{m}+\\psi_{n})\\Bigg\\{[-(4n+2)\\psi_{n} +2n\\psi_{n-1}+2(n+1)\\psi_{n+1}]\\\\\n\t&\\qquad-\\bigg[n(\\psi_{n}-\\psi_{n-1})^{2}\\bigg(\\frac{1}{\\phi_{m}+\\psi_{n}}+\\ frac{1}{\\psi_{n-1}+\\phi_{m}}\\bigg)\\\\\n\t&\\quad\\qquad+(n+1)(\\psi_{n+1}-\\psi_{n})^{2}\\bigg(\\frac{1}{\\phi_{m}+\\psi_{n}} +\\frac{1}{\\psi_{n+1}+\\phi_{m}}\\bigg)\\bigg]\\Bigg\\}f_{mm}g_{nn}.\n\\end{align*}\nFor the conformal metric of the four-dimensional unit sphere, we have\n$$\nh=\\frac{r^{2}+1}{2}=\\sum_{m,n}\\frac{2\\theta (m+n)+2\\theta+1}{2}f_{mm}g_ {nn}.\n$$\nTherefore, one can take $\\phi_{m}=\\frac{4\\theta m+2\\theta+1}{4} ,\\psi_{n}=\\frac{4\\theta n+2\\theta+1}{ 4}$. So we have\n\\begin{align*}\n\tS_\\theta&=\\sum_{m,n=0}\\left[12+12\\theta^{2}\\left( \\frac{m+n+2}{2\\theta(m+n)+4\\theta+1} -\\frac{m+n}{2\\theta(m+n)+1}\\right)\\right] f_{mm}g_{nn}\\\\\n\t&=12+12\\theta^{2}\\left[2\\theta+2(1+\\theta+\\sigma^{2})\\left(\\frac{\\rho^{2}}{\\theta}+1\\right)_{*}^{-1}\\right]_{*}^{-1}-12\\theta^{2}\\left[2\\theta+(1-2\\theta+\\sigma^{2})\\left(\\frac{\\rho^{2}}{\\theta}-1\\right)_{*}^{-1}\\right]_{*}^{-1}\\\\\n\t&\\quad+12\\theta^{2}\\left[2\\theta+2(1+\\theta+\\rho^{2})\\left(\\frac{\\sigma^{2}}{\\theta}+1\\right)_{*}^{-1}\\right]_{*}^{-1}-12\\theta^{2}\\left[2\\theta+(1-2\\theta+\\rho^{2})\\left(\\frac{\\sigma^{2}}{\\theta}-1\\right)_{*}^{-1}\\right]_{*}^{-1}\\\\\n\t&\\approx 12+12\\theta^{2}\\left\\{\\left[2\\theta+2(1+\\theta+\\sigma^{2})\\left(\\frac{\\rho^{2}}{\\theta}+1\\right)^{-1}\\right]^{-1}-\\left[2\\theta+(1-2\\theta+\\sigma^{2})\\left(\\frac{\\rho^{2}}{\\theta}-1\\right)^{-1}\\right]^{-1}\\right.\\\\\n\t&\\quad\\left.+\\left[2\\theta+2(1+\\theta+\\rho^{2})\\left(\\frac{\\sigma^{2}}{\\theta}+1\\right)^{-1}\\right]^{-1}-\\left[2\\theta+(1-2\\theta+\\rho^{2})\\left(\\frac{\\sigma^{2}}{\\theta}-1\\right)^{-1}\\right]^{-1}\\right\\},\n\\end{align*}\nwhere $\\rho^{2}=x_{1}^{2}+x_{2}^{2}, \\ \\sigma^{2}=x_{3}^{2}+x_{4}^{ 2}$.\t\nThe calculations of the above equation approximates to the second-order term of $\\theta$.\n\nIn order to avoid singularities, one can choose $0<\\theta <\\frac{1}{4}$. Obviously, when $\\theta \\to 0$, there is $S_\\theta\\to 12$. This is also the scalar curvature of the classical four-dimensional sphere.\nTaking $\\theta=0.01$, the values of the curvature $S_\\theta$ are shown in Fig.~\\ref{fig2}.\n\\begin{figure}[htbp]\n\t\\centering\n\t\\subfigure[]{\\includegraphics[scale=0.55]{curvaturesurface.png}}\\qquad\n\t\\subfigure[]{\\includegraphics[scale=0.4]{densityplot.png}}\n\t\\caption{The values of scalar curvature $S_\\theta$ with respect to the variables $\\rho$ and $\\sigma$ ($\\theta=0.01$).}\n\t\\label{fig2}\n\\end{figure}\nFrom these two figures, one can see that the values of the curvature is highly symmetrical with respect to $\\rho, \\sigma$. The curvature changes rapidly and is large near the origin and the coordinate axis, and it is small when $\\rho=\\sigma$ and $\\rho=-\\sigma$. However, they are all very close to 12, which is the curvature of the classic four-dimensional sphere. In addition, one can also find that in the sense of second-order approximation, when $\\rho^{2}+\\sigma^{2}=r^{2}$ is a constant, the curvature is not a constant.\n\n\n\\section{Area of the Moyal sphere}\\label{sec7}\nIn general, the conformal metric of the $2M$-dimensional Moyal sphere can be expressed as $g=h_{*}^{-2}(dx^{1}dx^{1}+\\dots+dx^{2M}dx^{2M})$,\nwhere\n$$\nh=\\frac{r^{2}+A^{2}}{2A^{2}}\n=\\sum_{m_{1},m_{2},\\dots,m_{M}}\\frac{2\\theta (m_{1}+m_{2}+\\dots m_{M})+A^{2}+M\\theta}{2A^{2}}f_{m_{1}m_{1}}f_{m_{2}m_{2}}\\dots f_{m_{M}m_{M}}.\n$$\nBy virtue of the formula for the classical sphere, one can calculate the area of the $2M$-dimensional Moyal sphere:\n\\begin{align}\\label{s2m}\n\t&\\int_{\\mathbf{S}^{2M}}dvol_{*}=\\int_{\\mathbb{R}^{2M}}h_{*}^{-2M}d^{2M}x\\nonumber\\\\\n\t&=\\int_{\\mathbb{R}^{2M}}\\sum_{m_{1},m_{2},\\dots,m_{M}}\\frac{(2A^{2})^{2M}}{\\left[2\\theta (m_{1}+m_{2}+\\dots m_{M})+A^{2}+M\\theta\\right]^{2M}}f_{m_{1}m_{1}}f_{m_{2}m_{2}}\\dots f_{m_{M}m_{M}}d^{2M}x\\nonumber\\\\\n\t&=\\left(2\\pi \\theta\\right)^{M}\\sum_{m_{1},m_{2},\\dots,m_{M}}\\frac{(2A^{2})^{2M}}{\\left[2\\theta (m_{1}+m_{2}+\\dots m_{M})+A^{2}+M\\theta\\right]^{2M}}\\nonumber\\\\\n\t&=\\frac{(2\\pi A^{4})^{M}}{\\theta^{M}}\\sum_{k=1}^{\\infty}\\frac{C_{k-2+M}^{M-1}}{\\left(k+\\frac{A^{2}}{2\\theta}+\\frac{M}{2}-1\\right) ^{2M}}\\nonumber\\\\\n\t&=\\frac{2\\pi^{M+\\frac{1}{2}}A^{2M}}{\\Gamma\\left(M+\\frac{1}{2}\\right) }\\cdot\\frac{2^{M-1}\\lambda^{M}}{\\sqrt{\\pi}}\\Gamma\\left(M+\\frac{1}{2}\\right)\\sum_{k=0}^{\\infty}\\frac{C_{k-1+M}^{M-1}}{\\left(k+\\frac{\\lambda}{2}+\\frac{M}{2}\\right)^{2M}}\\nonumber\\\\\n\t&=\\frac{2\\pi^{M+\\frac{1}{2}}A^{2M}}{\\Gamma\\left(M+\\frac{1}{2}\\right) }\\cdot\\gamma_{M}(\\lambda),\t\t\n\\end{align}\nwhere $\\Gamma(x)$ is the Gamma function, and\n\\[\n\\gamma_{M}(\\lambda)=\\frac{2^{M-1}\\lambda^{M}}{\\sqrt{\\pi}}\\Gamma\\left(M+\\frac{1}{2}\\right)\\sum_{k=0}^{\\infty}\\frac{C_{k-1+M}^{M-1}}{\\left( k+\\frac{\\lambda}{2}+\\frac{M}{2}\\right)^{2M}}.\n\\]\nFor a given $M$, $\\gamma_{M}(\\lambda)$ can also be expressed as a sum of polygamma functions. For example, for $M=1, 2$, we have\n\\[\n\\gamma_{1}(\\lambda)=\\frac{\\lambda}{2} \\Psi^{(1)}\\left(\\frac{\\lambda+1}{2}\\right),\\qquad\n\\gamma_{2}(\\lambda)=\\frac{\\lambda^{2}}{8}\\left[-6\\Psi^{(2)}\\left(1+\\frac{\\lambda}{2}\\right)-\\lambda\\Psi^{(3)}\\left(1+\\frac{\\lambda}{2}\\right)\\right],\n\\]\nwhere $\\Psi^{(n)}(x)$ is a polygamma function.\nTherefore, the areas of the 2-dimensional and 4-dimensional Moyal spheres are\n\\[\n\\int_{\\mathbf{S}^{2}}dvol_{*}=4\\pi A^{2}\\cdot \\gamma_{1}(\\lambda)\n=2\\pi A^{2}\\lambda \\Psi^{(1)}\\left(\\frac{\\lambda+1}{2}\\right),\n\\]\n\\[\n\\int_{\\mathbf{S}^{4}}dvol_{*}=\\frac{8\\pi^{2}A^{4}}{3}\\cdot\\gamma_{2}(\\lambda)\n=\\frac{\\pi^{2}A^{4}\\lambda^{2}}{3}\\left[-6\\Psi^{(2)}\\left(1+\\frac{\\lambda}{2}\\right)-\\lambda\\Psi^{(3)}\\left(1+\\frac{\\lambda}{2}\\right)\\right].\n\\]\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.35]{area.png}\n\\caption{$\\gamma_{M}(\\lambda)$ with different values of $M$}\\label{fig3}\n\\end{figure}\nFig.~\\ref{fig3} shows the graphs of $\\gamma_{M}(\\lambda)$ for $M=1,2,4,6$. It is easy to see that for any $M$ and a given radius $A$, when $\\theta \\to 0^{+}$, or equivalently, $\\lambda\\to \\infty$, there is $\\gamma_{M}(\\lambda)\\to 1$. The result (\\ref{s2m}) returns to the area of the ordinary $2M$-dimensional sphere\n$\\frac{2\\pi^{M+\\frac{1}{2}}A^{2M}}{\\Gamma\\left(M+\\frac{1}{2}\\right) }$. When the noncommutative parameter $\\theta \\to \\infty$, or equivalently, $\\lambda\\to 0^+$, there is $\\gamma_{M}(\\lambda)\\to 0$.\nThat means that the area of the $2M$-dimensional Moyal sphere will approach to 0.\n\nAnalogous to the case of ordinary spaces, if we consider the space enclosed by the sphere to be the volume of the corresponding higher-dimensional ball, then one can see that when the noncommutative parameter $\\theta \\to \\infty$, the volume enclosed by the Moyal sphere will also approach to 0.\n\nFurthermore, let us consider a type of deformed Moyal space. For the $4n$-dimensional case,\nthe deformed Moyal product is defined as\n\\[ {f \\star g = f \\cdot e^{\\frac{\\mathbf{i}}{2}\\sum_{i,j=1}^{4n} \\Theta_{ij} \\overset{\\leftarrow}{\\partial_i}\n\t\t\\overset{\\rightarrow}{\\partial_j}}} \\cdot g, \\]\nwhere the noncommutative parameter matrix $\\Theta$ is an $4 n \\times 4 n$ antisymmetric matrix\n$ \\Theta =\\theta I_{2n}\\otimes \\varepsilon_2+\\mu I_n\\otimes\n\\varepsilon_2\\otimes \\begin{pmatrix}\n\t1 & 0\\\\\n\t0 & -1\n\\end{pmatrix}, $\nand the parameter $\\mu$ is a real number.\nWith respect to the deformed Moyal product, the variables satisfy the following commutation relations:\n\\begin{eqnarray}\n&&\\left[x_{2j-1},x_{2j}\\right]_{\\star}=-\\left[x_{2j},x_{2j-1}\\right]_{\\star}=\\mathbf{i}\\theta,\\nonumber\\\\\n&&\\left[x_{4k},x_{4k-2}\\right]_{\\star}=-\\left[x_{4k-2},x_{4k}\\right]_{\\star}\n=\\left[x_{4k-3},x_{4k-1}\\right]_{\\star}=-\\left[x_{4k-1},x_{4k-3}\\right]_{\\star}=\\mathbf{i}\\mu,\\label{ms2}\n\\end{eqnarray}\nwhere $j=1,\\dots,2n$, $k=1,\\dots,n$, and other commutators vanish. Obviously, when $\\mu=0$, it returns to the ordinary $4n$-dimensional Moyal space (\\ref{ms1}).\n\nOne can consider the variable substitution $X'=DX$, where $X=(x_{1},x_{2},...,x_{4n})$, $X'=(x'_{1},x'_{2},...,x'_{4n})$, and the transformation matrix $D$ is\n\\[\nD=\\sqrt{\\frac{\\theta + \\sqrt{\\theta^2 + \\mu^2}}{2 \\sqrt{\\theta^2 +\\mu^2}}} \\left[I_{4n}+ \\frac{\\mu}{\\theta + \\sqrt{\\theta^2 + \\mu^2}} I_n\\otimes\\varepsilon_2\\otimes \\begin{pmatrix}\n\t0 & 1\\\\\n\t1 & 0\n\\end{pmatrix}\\right].\n\\]\nThe Moyal product now becomes\n\\[ {f \\ast' g = f\\cdot e^{\\frac{\\mathbf{i}}{2}\\sum_{i,j=1}^{4n} \\Theta'_{ij} \\overset{ \\leftarrow}{\\partial'_i}\n\t\t\\overset{\\rightarrow}{\\partial'_j}}} \\cdot g, \\]\nwhere $\\Theta' = \\theta' I_{2n}\\otimes \\varepsilon_2$.\nThe variables $\\{x'_{1},x'_{2},...,x'_{4n}\\}$ satisfy the commutative relations:\n\\begin{equation}\n\\left[x'_{2j-1},x'_{2j}\\right]_{*'}=-\\left[x'_{2j},x'_{2j-1}\\right]_ {*'}\n=\\mathbf{i} \\theta',\n\\qquad\\theta'=\\sqrt{\\theta^2 + \\mu^2},\n\\end{equation}\nwhere $j=1,\\dots,2n$, and other commutators vanish.\n\nTherefore, the area of the deformed $4n$-dimensional Moyal sphere (\\ref{ms2}) is\n\\begin{align*}\n\t\\int_{\\mathbf{S}^{4n}} dvol_{\\star} & = \\int_{\\mathbb{R}^{4n}}\n\th_{\\star}^{- 4n} d^{4n} x\\\\\n\t& = \\int_{{\\mathbb{R}'}^{4n}} h_{\\ast'}^{- 4n} |D^{-1}| d^{4n} x'\n\t= \\int_{{\\mathbb{R}'}^{4n}} h_{\\ast'}^{- 4n} d^{4n} x'\\\\\n\t&=\\frac{(2\\pi A^{4})^{2n}}{{\\theta'}^{2n}}\\sum_{k=0}^{\\infty}\\frac{C_{k+2n-1}^{2n-1}}{\\left( k+\\frac{A^{2}}{2\\theta'}+n\\right) ^{4n}}\\\\\n\t&=\\frac{2\\pi^{2n+\\frac{1}{2}}A^{4n}}{\\Gamma\\left(2n+\\frac{1}{2}\\right) }\\cdot\\gamma_{2n}(\\lambda'),\t\t\n\\end{align*}\nwhere $\\lambda'=\\frac{A^{2}}{\\theta'}$.\nOne can see that the area of the deformed $4n$-dimensional Moyal sphere (\\ref{ms2}) depends only on the radius $A$ and $\\theta'=\\sqrt{\\theta^2 + \\mu^2}$, and does not depend on $\\theta$ or $\\mu$ alone. Obviously, it has similar properties to the area of the normal $4n$-dimensional Moyal sphere (\\ref{s2m}).\n\n\n\\section{Constant curvature metrics of Moyal spaces}\\label{sec8}\nSimilar to Ref.~\\cite{Eckstein}, here we will study the conformal metric with constant curvature of the 4-dimensional Moyal space, and calculate the expression of its factor $h$.\nIn this case, the Moyal $*$-product is defined as:\n$$f(x_{1},x_{2},x_{3},x_{4})*g(x_{1},x_{2},x_{3},x_{4})=f\\cdot e^{\\frac{\\mathbf{i}\\theta}{2}(\\overleftarrow{\\partial_{1}}\\overrightarrow{\\partial_{2}}-\\overleftarrow{\\partial_{2}}\\overrightarrow{\\partial_{1}}+\\overleftarrow{\\partial_{3}}\\overrightarrow{\\partial_{4}}-\\overleftarrow{\\partial_{4}}\\overrightarrow{\\partial_{3}})}\\cdot g.$$\nHere we only consider the case where $f, g$ are both radial functions. Similarly, $f*g$ can be expanded to the second-order term of $\\theta$,\n$$\nf*g =fg-\\frac{\\theta^{2}}{8r}\\left( f''g'+f'g''+\\frac{2}{r}f'g'\\right) + O(\\theta^{2}).\n$$\nIf $f$ is a radial function, then $f_{*}^{-1}$ is also a radial function. Ignoring the higher-order terms of $\\theta^{2}$, one can get\n\\begin{equation*}\n\tf_{*}^{-1}=f^{-1}+\\frac{\\theta^{2}}{4r}\\left[f^{-4}(f')^{3}-f^ {-3}f'f''-\\frac{1}{r}(f')^{2}f^{-3}\\right].\n\\end{equation*}\nConsider the case where the scalar curvature of the 4-dimensional Moyal sphere is constant, that is\n$$\nS_\\theta=6h_*^{2}*[\\partial_{ii}(h)-h_*^{-1}\\partial_{i}(h)^{2}-\\partial_{i}(h)* h_*^{-1}*\\partial_{i}(h)]*h_*^{-1}=C,\n$$\nwhere $C$ is a constant. Then there is the constant curvature equation\n\\begin{equation}\\label{hh}\n\th*[\\partial_{ii}(h)-h_*^{-1}\\partial_{i}(h)^{2}-\\partial_{i}(h)*h_*^{-1}*\\partial_{i}(h)]=\\frac{C}{6},\n\\end{equation}\nApproximating to the $\\theta^{2}$ term, we assume that $h=\\frac{A^{2}+r^{2}}{2A^{2}}+\\theta^{2}\\epsilon(r)=h_0+\\theta^{2}\\epsilon(r)$, where $h_0=\\frac{A^{2}+r^{2}}{2A^{2}}$, then $h_{*}^{-1}=h^{-1}-\\frac{4\\theta^{2}A^{4}}{(A^{2}+r^{2})^{4}} $. Substituting these into the constant curvature equation (\\ref{hh}), after some straightforward calculations, one can obtain\n$$\n\\frac{2}{A^{2}}+\\theta^{2}\\left(\\frac{A^{2}+r^{2}}{2A^{2}}\\epsilon''(r)+\\frac{3A^{2}-5r^{2}}{2A^ {2}r}\\epsilon'(r)+\\frac{4}{A^{2}}\\epsilon(r)+\\frac{4}{A^{2}(A^{2}+r^{2})^{2}}\\right)=\\frac{C}{6}.\n$$\n\nIf the curvature of the Moyal sphere is required to be equal to the curvature of the classic 4-dimensional sphere $C=\\frac{12}{A^{2}}$, then there is the equation:\n$$\n\\epsilon''(r)+\\frac{3A^{2}-5r^{2}}{r(A^{2}+r^{2})}\\epsilon'(r)+\\frac{8}{A^{2}+r^{2}}\\epsilon(r)+\\frac{8}{(A^{2}+r^{2})^{3}}=0.\n$$\nOne can see that the error function $\\epsilon(r)$ does not depend on the noncommutative parameter $\\theta$. It is easy to find the general solution of the equation:\n\\begin{align*}\n\t\\epsilon(r)=&C_{1}(r^{2}-A^{2})\n\t+\\frac{C_{2}}{2r^{2}}\\left(r^{6}-A^{2}r^{4}-17A^{4}r^{2}+A^{6}+12(r^{2}-A^{2})A^{2}r^{2}\\ln r\\right)\\\\\n\t&-\\frac{1}{30A^{6}r^{2}(A^{2}+r^{2})}\\left(A^{6}-3A^{4}r^{2}+6A^{2}r^{4}+6(r^{4}-A^{4})r^{2}\\ln\\frac{r^2}{A^{2}+r^{2}}\\right),\n\\end{align*}\nwhere $C_{1}$, $C_{2}$ are integration constants. When $C_{2}=\\frac{1}{15A^{8}}$, $\\epsilon(r)$ is well defined on $[ 0,\\infty)$.\nFurthermore, if $C_{1}=-\\frac{13+12\\ln A}{30A^6}$, there is $\\epsilon(0)=0$.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[scale=0.33]{constanthh.png}\n\t\\caption{$\\frac{h^{-2}(r)}{h_0^{-2}}$ with different radii $A$ ($\\theta=0.1$)}\\label{fig4}\n\\end{figure}\nFig.~\\ref{fig4} shows the ratio of two functions $\\frac{h^{-2}(r)}{h_0^{-2}}$ for different radii $A$ when $\\theta=0.1$, $C=\\frac{12}{A^{2}}$, $C_{1}=-\\frac{13+12\\ln A}{30A^6}$, $C_{2}=\\frac{1}{15A^{8}}$.\nOne can see that for a given sufficiently small $\\theta$, the smaller the radius $A$ of the sphere, the greater the deviation of the conformal metric of constant curvature of Moyal space from that of the classical case. Conversely, the larger the radius $A$ of the sphere, the closer the conformal metric of constant curvature is to the result of the classical sphere.\n\nSimilarly, one can also consider high-dimensional cases.\nFor any $2m$-dimensional Moyal space (\\ref{ms1}), ignoring the higher-order terms of $\\theta^{2}$, the approximate expansion for the Moyal product of radial functions $f, g$ is:\n\\begin{equation*}\n\tf*g=fg-\\frac{\\theta^{2}}{8r}\\left[f''g'+f'g''+\\frac{2(m-1)}{r}f'g'\\right].\n\\end{equation*}\nSo the corresponding inverse with respect to the star product is:\n\\begin{equation*}\n\tf_{*}^{-1}=f^{-1}+\\frac{\\theta^{2}}{4r}\\left[f^{-4}(f')^{3}-f^{-3}f'f''-\\frac{m-1}{r}(f')^{2}f^{-3}\\right].\n\\end{equation*}\nSimilar to the previous approach, one can assume $h=\\frac{A^{2}+r^{2}}{2A^{2}}+\\theta^{2}\\epsilon(r)$. Substituting it into the corresponding constant curvature equation, it is not difficult to obtain the differential equation for the error function $\\epsilon(r)$:\n\\begin{equation*}\n\t\\epsilon''(r)+\\frac{A^{2}(2m-1)-(2m+1)r^{2}}{r(A^{2}+r^{2})}\\epsilon'(r)+\\frac{4m}{A^{2}+ r^{2}}\\epsilon(r)+\\frac{8(2m-1)\\left[A^{2}m+(m-2)r^{2}\\right]}{A^{2} (A^{2}+r^{2})^{3}}=0.\n\\end{equation*}\n\nFurthermore, one can also consider the constant curvature metric of the deformed $4n$-dimensional Moyal sphere (\\ref{ms2}).\nThe approximate expansion of the deformed Moyal product of the radial function $f,g$ is as follow:\n\\begin{equation*}\n\tf\\star g=fg-\\frac{\\theta^{2}+\\mu^{2}}{8r}\\left[f''g'+f'g''+\\frac{2(2n-1 )}{r}f'g'\\right],\n\\end{equation*}\nand\n\\begin{equation*}\n\tf_{\\star}^{-1}=f^{-1}+\\frac{\\theta^{2}+\\mu^{2}}{4r}\\left[f^{-4}(f' )^{3}-f^{-3}f'f''-\\frac{2n-1}{r}(f')^{2}f^{-3}\\right].\n\\end{equation*}\nIn this case, the approximate expression of the conformal metric factor $h$ can be assumed to be $h=\\frac{A^{2}+r^{2}}{2A^{2}}+(\\theta^{2}+\\mu^2)\\epsilon(r)$. Similarly, one can obtain the differential equation of the error function $\\epsilon(r)$:\n\\begin{equation*}\n\t\\epsilon''(r)+\\frac{A^{2}(4n-1)-(4n+1)r^{2}}{r(A^{2}+r^{2})}\\epsilon'(r)+\\frac{8n}{A^{2}+ r^{2}}\\epsilon(r)+\\frac{16(4n-1)\\left[A^{2}n+(n-1)r^{2}\\right]}{A^{2}(A^{2}+r^{2})^{3}}=0.\n\\end{equation*}\n\nFrom the formulas of the error functions above, we can see that this type of deformed $4n$-dimensional Moyal space (\\ref{ms2}) and the usual $4n$-dimensional Moyal space (\\ref{ms1}) have the same error function $\\epsilon(r)$.\nSo for given noncommutative parameters $\\theta, \\mu$, their constant curvature conformal metrics have similar properties and are also similar to those of the 4-dimensional Moyal sphere. Similarly, for given sufficiently small noncommutative parameters $\\theta, \\mu$, the smaller the radius $A$ of the Moyal sphere, the greater the deviation of its constant curvature conformal metric from that of the classical case; The larger the radius $A$ of the Moyal sphere, the closer its constant curvature conformal metric is to the result of the classical sphere.\n\n\n\\section{Conclusions}\\label{sec9}\nIntroducing the Moyal product into the smooth function space $C^{\\infty}(\\mathbb{R}^{n})$ on the Euclidean space $\\mathbb{R}^{n}$, one can obtain the generalized function space $(C^{\\infty}(\\mathbb{R}^{n}),\\ast)$ with noncommutative products, and the corresponding space is the noncommutative Moyal space. If we introduce into the Moyal space a metric similar to the spherical metric of ordinary space, we will get a so-called Moyal sphere.\n\nIn this paper, we study the Moyal sphere from two different aspects.\nFirst, with the help of the conformal metric of the sphere in ordinary space, we introduced the Moyal product in the conformal factor and obtained the corresponding spherical metric in the noncommutative sense.\nWe calculated the scalar curvature and area of the Moyal sphere in this case. We found that when the noncommutative parameter approaches to 0, the Moyal sphere returns to an ordinary one. As the noncommutative parameter increases, the area of the Moyal sphere will decrease. When the noncommutative parameter approaches to infinity, the area of the Moyal sphere will approach to 0. \nAnalogous to the case of ordinary space, if we consider the space enclosed by the sphere to be the volume of the corresponding higher-dimensional ball, then one can see that when the noncommutative parameter approaches to infinity, the volume enclosed by the Moyal sphere will also approach to 0. In addition, we found that the total curvature integral of the two-dimensional Moyal sphere exactly coincides with the Gauss-Bonnet formula of the two-dimensional Euclidean space, and the result does not depend on the noncommutative parameter $\\theta$. This implies that the noncommutative algebraic structure of the smooth function space does not change some topological properties of the two-dimensional Moyal sphere.\n\nIn the classical case, a sphere corresponds to a space of constant curvature. Therefore, we also study the conformal metric of constant curvature of the Moyal space. We obtained the approximate expression of the conformal metric of the Moyal space with constant curvature in the sense of the second order approximation.\nWe found that for given sufficiently small noncommutative parameters, the smaller the radius $A$ of the Moyal sphere, the greater the deviation of its constant curvature conformal metric from that of the classical case; The larger the radius $A$ of the Moyal sphere, the closer its constant curvature conformal metric is to the result of the classical sphere.\n\nIn addition, we also calculated the area and constant curvature metric expression of a generalized deformed Moyal sphere with two noncommutative parameters, and obtained similar results. When these noncommutative parameters approach to 0, the Moyal star product will return to the ordinary product, and these curvatures and areas will return to the results of the ordinary sphere.\n\nThese results are significant for the studies of mathematical structures and physical properties of noncommutative spaces. To our knowledge, these results have\nnot been reported in the literatures.\nFurther, we will study the Gauss-Bonnet-Chern formula for the 4-dimensional Moyal sphere, and work on this direction is in progress. \n\n\n\\section*{Acknowledgments}\nThis work is partly supported by the National Natural Science Foundation of China (Grant No. 11911530750), the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2019A1515011703).\n\n\n\\begin{thebibliography}{100}\n\\bibitem{Snyder} H. S. Snyder, ``Quantized space-time,'' {\\it Phys. Rev.} {\\bf 71}, 38 (1947).\n\n\\bibitem{Connes} A. 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Phys.} {\\bf 2024}, 119 (2024).\n\n\\bibitem{Madore} J. Madore, ``An introduction to noncommutative differential geometry and its physical applications'' (Cambridge University Press, Cambridge, 1999).\n\n\\bibitem{Kaygun} A. Kaygun and M. Khalkhali, ``Hopf modules and noncommutative differential geometry,''\n{\\it Lett Math Phys} {\\bf 76}, 77 (2006).\n\n\\bibitem{Tretkoff} A. Connes and P. Tretkoff, ``The Gauss-Bonnet theorem for the noncommutative two torus,'' in {\\it Noncommutative geometry, arithmetic, and related topics,} edited by C. Consani and A. Connes (Johns Hopkins University Press, Maryland, 2012), pp. 141-158.\n\n\\bibitem{Fathizadeh} F. Fathizadeh and M. Khalkhali, ``The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure,'' {\\it J. Noncommut. Geom.} {\\bf 6}, 457 (2012).\n\n\\bibitem{Dabrowski} L. Dabrowski and A. Sitarz, ``Curved noncommutative torus and Gauss–Bonnet,'' {\\it J. Math. Phys.} {\\bf 54}, 013518 (2013).\n\n\\bibitem{Brzezinski} T Brzezi\\'{n}ski, ``Noncommutative differential geometry of generalized Weyl algebras,''\n{\\it SIGMA} {\\bf 12}, 059 (2016).\n\n\\bibitem{Arnlind} J. Arnlind and M. Wilson, ``Riemannian curvature of the noncommutative 3-sphere,'' {\\it J. Noncommut. Geom.} {\\bf 11}, 507 (2017).\n\n\\bibitem{Floricel} R. Floricel, A. Ghorbanpour, and M. Khalkhali, ``The Ricci curvature in noncommutative geometry,'' {\\it J. Noncommut. Geom.} {\\bf 13}, 269 (2019).\n\n\\bibitem{Sciandra} A. Sciandra and T. Weber, ``Noncommutative differential geometry on crossed product algebras,'' {\\it Journal of Algebra} {\\bf 664}, 129 (2025).\n\n\\bibitem{Khalkhali} F. Fathizadeh and M. Khalkhali, ``Scalar curvature for the noncommutative two torus,'' {\\it J. Noncommut. Geom.} {\\bf 7}, 1145 (2013).\n\n\\bibitem{Wilson} M. Wilson, ``Gauss-Bonnet-Chern type theorem for the noncommutative four-sphere,''\n{\\it Doctoral dissertation}, The University of Western Ontario (Canada) (2016).\n\n\\bibitem{Gayral} V. Gayral, J. M. Gracia-Bond\\'{i}a, B. Iochum, T. Sch\\\"{u}cker, and J. C. V\\'{a}rilly, ``Moyal planes are spectral triple,'' {\\it Commun. Math. Phys.} {\\bf 246}, 569 (2004).\n\n\\bibitem{Eckstein} M. Eckstein, A. Sitarz, and R. Wulkenhaar, ``The Moyal sphere,'' {\\it J. Math. Phys.} {\\bf 57}, 112301 (2016).\n\\end{thebibliography}\n\\end{document}\n\n\n"}}},{"rowIdx":287,"cells":{"id":{"kind":"string","value":"2501.01383v2"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2501.01383v2"},"title":{"kind":"string","value":"Electrical networks and data analysis in phylogenetics"},"full_text":{"kind":"string","value":"\\documentclass[11pt,reqno]{amsart}\n\\usepackage[utf8]{inputenc}\n\n\\usepackage{tikz-cd}\n\\usepackage{pdfpages}\n\\usepackage{graphicx}\n\\usepackage{amssymb,amsmath, amscd, latexsym,amsfonts,bbm, amsthm,stmaryrd, enumerate,phaistos}\n\\usepackage{yhmath}\n\\usepackage{comment}\\usepackage{todonotes}\\usepackage[all]{xy}\n\\usepackage[vcentermath]{youngtab}\n\\usepackage{float}\n\\renewcommand{\\familydefault}{cmss}\n\\usepackage{tabularx}\n\\usepackage{amsmath, amssymb, latexsym, enumerate, graphicx,tikz, stmaryrd,pifont,ifsym}\n\\usetikzlibrary{arrows,decorations,decorations.pathmorphing,positioning}\n\\usepackage{mdframed}\n\\usepackage{mdwlist}\n\\usepackage[pdftex]{hyperref}\n\\usepackage{amscd,mathrsfs,epic,empheq,float}\n\\usepackage{bbm}\n\\usepackage{cases}\n\\usepackage[all]{xy}\n \n\\def\\theequation{\\arabic{section}.\\arabic{equation}}\n\\usepackage{tikz}\n\\setcounter{tocdepth}{1}\n\n\\usepackage{tikz-cd}\n\\newenvironment{defn}{\\vspace{0.3cm}\\par\\noindent\\refstepcounter{theorem}\\begin{exafont}Definition \\thetheorem.\\end{exafont}\\hspace{\\labelsep}}{\\vspace{0.3cm}\\par}\n\n\\theoremstyle{definition}\n\n\n\n\n\\renewcommand{\\familydefault}{cmss}\n\n\\newcommand{\\dist} {3mm}\n\\newcommand{\\Xn}{{\\op{X}}_n}\n\\newcommand{\\cS}{\\mathcal{S}}\n\\newcommand{\\mV}{\\mathbb{V}}\n\\newcommand{\\bt}{\\mathbf t} \n\\newcommand{\\bb}{\\mathbf b} \n\\newcommand{\\bd}{\\mathbf d} \n\\newcommand{\\oS}{\\op{S}} \n\\newcommand{\\Func}{\\op{Func}_n}\n\\newcommand{\\eu}{\\op{eu}} \n\\newcommand{\\inc}{\\op{inc}} \n\\newcommand{\\Supp}{\\op{Supp}} \n\\newcommand{\\bw}{\\op{w}} \n\n\\newcommand{\\ii}{\\mathbf i} \n\\newcommand{\\RR}{\\mathbb R} \n\\newcommand{\\CC}{\\mathbb C} \n\\newcommand{\\QQ}{\\mathbb Q}\n\\newcommand{\\ZZ}{\\mathbb Z}\n\n\n\\usepackage{pict2e}\n\\def\\StrangeCross\n\n\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amscd}\n\\usepackage{amsmath}\n\\def\\bea{\\begin{eqnarray}}\n\\def\\eea{\\end{eqnarray}}\n\\def\\nn{\\nonumber}\n\n\\def\\AA{\\mathcal{A}}\n\\def\\ZZ{\\mathbb{Z}}\n\\def\\CC{\\mathbb{C}}\n\\def\\PP{\\mathbb{P}}\n\\def\\RR{\\mathbb{R}}\n\\def\\QQ{\\mathbb{Q}}\n\\def\\GG{\\mathbb{G}}\n\\def\\TT{\\mathbb{T}}\n\n\\def\\kk{\\Bbbk}\n\n\\def\\gg{\\mathfrak{g}} \n\n\\usepackage{pict2e}\n\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{question}[theorem]{Question}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{problem}[theorem]{Problem}\n\\newenvironment{exafont}{\\begin{bf}}{\\end{bf}}\n\n\n\n\\newtheorem{corollary}[theorem]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{conv}[theorem]{Conventions}\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\newcommand{\\op}{\\operatorname}\n\\newcommand{\\Mat}{\\operatorname{Mat}}\n\\newcommand{\\mC}{\\mathbb{C}}\n\\newcommand{\\ov}{\\overline{\\otimes}}\n\\newcommand{\\la}{\\lambda}\n\\newcommand{\\bH}{{\\bf H}}\n\\newcommand{\\bP}{{\\bf P}}\n\\newcommand{\\End}{\\op{End}}\n\\newcommand{\\cU}{\\mathcal{U}}\n\\newcommand{\\cZ}{\\mathcal{Z}}\n\\newcommand{\\mg}{\\mathfrak{gl}_2}\n\\newcommand{\\modH}{{$\\op{mod}$-$\\bH$}}\n\\begin{document}\n\n\n\\author[V. Gorbounov]{V.~Gorbounov}\n\\address{V.~G.: Faculty of Mathematics, National Research University Higher School of Economics, Usacheva 6, 119048 Moscow, Russia}\n\\email{vgorb10@gmail.com }\n\n\\author[A. Kazakov]{A.~Kazakov}\n\\address{A.~K.: Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Russia, 119991, Moscow, GSP-1, 1 Leninskiye Gory, Main Building; Centre of Integrable Systems, P. G. Demidov Yaroslavl State University, Sovetskaya 14, 150003, Yaroslavl, Russia;\nCenter of Pure Mathematics, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russian Federation;\nKazan Federal University, N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan, 420008, Russia}\n\\email{anton.kazakov.4@mail.ru}\n\\title{Electrical networks and data analysis in phylogenetics}\n\n\n\\maketitle\n\n\\begin{abstract}\n\nA classic problem in data analysis is studying the systems of subsets defined by either a similarity or a dissimilarity function on $X$ which is either observed directly or derived from a data set.\n\nFor an electrical network there are two functions on the set of the nodes defined by the resistance matrix and the response matrix either of which defines the network completely.\nWe argue that these functions should be viewed as a similarity and a dissimilarity function on the set of the nodes moreover they are related via the covariance mapping also known as the Farris transform or the Gromov product. We will explore the properties of electrical networks from this point of view.\n\nIt has been known for a while that the resistance matrix defines a metric on the nodes of the electrical networks. Moreover for a circular electrical network this metric obeys the Kalmanson property as it was shown recently. We will call such a metric an electrical Kalmanson metric. \n The main results of this paper is a new description of the electrical Kalmanson metrics in the set of all Kalmanson metrics in terms of the geometry of the positive Isotropic Grassmannian whose connection to the theory of electrical networks was discovered earlier. \n\nOne important area of applications where Kalmanson metrics are actively used is the theory of phylogenetic networks which are a generalization of phylogenetic trees. Our results allow us to use in phylogenetics the powerful methods of reconstruction of the minimal graphs of electrical networks and possibly open the door into data analysis for the methods of the theory of cluster algebras.\n\\end{abstract}\n\\setcounter{tocdepth}{3}\n\\tableofcontents\nMSC2020: 14M15, 82B20, 05E10, 05C50, 05C10, 92D15, 94C15, 90C05, 90C59, 05C12.\n\nKey words: Electrical networks, circular split metrics, Kalmanson metrics\n\n\\section{Introduction}\nThe theory of electrical networks goes back to the work of Gustav Kirchhoff around mid 1800 and since then it has been a source of remarkable achievements in combinatorics, algebra, geometry, mathematical physics, and electrical engineering. \n\nAn electrical network is a graph with positive weights, conductances attached to the edges, and a chosen subset of the set of vertices which are called the boundary vertices or nodes. An important characteristic of an electrical network with only two nodes $i$ and $j$ is the effective resistance $R_{ij}$, that is the voltage at node $i$ which, when node $j$ is held at zero volts, causes a unit current to flow through the circuit from node $i$ to node $j$. The effective resistance defines a metric on the set of nodes that is widely used in chemistry, for example, \\cite{Kl}. \nFor convenience, we will organize the effective resistance $R_{ij}$ in a matrix $R$ setting $R_{ii}=0$ for all $i$.\n\nIn the case where there are more than two nodes, there is an important generalization of $R_{ij}$. Given an electrical network with $n$ nodes, there is a linear endomorphism of the vector space of functions defined on the nodes, constructed as follows: for each such a function, there is a unique extension of that function to all the vertices which satisfies Kirchhoff's current law at each interior vertex. This function then gives the current $I$ in the network at the boundary vertices defining a linear map which is called the Dirichlet-to-Neumann map or the network response. The matrix of this map is called the response matrix. It plays a key role in the theory and applications of the electrical network \\cite{CIM}.\n\nThe above two matrices define each other and moreover it is possible to reconstruct a planar circular electrical network if these matrices are known \\cite{CIM}.\n\nA classic problem in data analysis is studying the systems of subsets defined by either a similarity or a dissimilarity function on $X$ which is either observed directly or derived from a data set.\nWhile the latter makes use of splits and split metrics, the key ingredients of the former are systems of clusters, subsets of $X$, and elementary similarity functions. One can interpret splits as distinctive features and clusters as common features, see \\cite{GandC} and \\cite{MS} for an introduction to these ideas.\n\nWe argue that the \"inverse\" of the response and the resistance matrices should be viewed as a similarity and a dissimilarity function on the set of nodes of an electrical network and as such they are related via the covariance mapping also known as the Farris transform or the Gromov product, see Remark \\ref{inverse} for more details. \n\nWe will explore the properties of electrical networks from this point of view.\nIn this paper we will work with the resistance matrix. The connection of the response matrix to data analysis will be presented in a future publication.\n\nIn computational biology species are naturally assigned collections of symbols from a given set, called characters, and one can construct a distance between two species (such as Hamming distance) which records the proportion of characters where the two species differ. Such a record can be encoded by a real symmetric, nonnegative matrix called a dissimilarity matrix $M$. An important problem in phylogenetics is to reconstruct a weighted tree $T$ with the set of leaves equal to the set of of species and the matrix of the tree metric defined by $T$ on the set of leaves is equal to $M$, see \\cite{MS} for a nice introduction to these ideas.\n In most cases, a tree structure is too constraining. The notion of a split network is a generalization of a tree in which a certain type of cycles is allowed. A geometric space of such networks was introduced \\cite{DP}, forming a natural extension of the work by Billera, Holmes, and Vogtmann on tree space. \nIt has been studied from different points of view since it is related to a number of objects in mathematics: the compactification of the real moduli space of curves, to the Balanced Minimal Evolution polytopes and Symmetric Traveling Salesman polytopes to name a few.\n\nThe appropriate metric on the set of species defined by a split network has a very special property called the Kalmanson property which distinguishes it completely in the set of all metrics \\cite{BD1}, \\cite{MS}.\n\nStefan Forcey has discovered recently that the resistance metric defined by a circular planar electric network obeys the Kalmanson property \\cite{F}. We will call such a split metric an {\\it electrical Kalmanson metric}. This important result puts a new light on the theory of electrical networks connecting it with phylogenetics and metric geometry.\n\nThe purpose of this paper is twofold: firstly we want to collect in one place the relevant facts from the research areas mentioned above and indicate interesting connections between them, secondly we provide a new description of the set of the electrical Kalmanson metrics inside the set of all Kalmanson metrics given in Theorem \\ref{th: dual}. For the later we will exploit the connection between the space of circular planar electrical networks and the non-negative Isotropic Grassmannian $\\mathrm{IG}_{\\geq 0}(n-1, 2n)$ which uses the effective resistance matrix \\cite{BGGK}, Theorem $5.6$ as opposed to the response matrix. This description is a modification of the original construction given by Thomas Lam \\cite{L} and developed further in \\cite{BGKT}, \\cite{CGS}.\n\nIt turns out that the Kalmanson property itself is a consequence of this connection and the description of the electrical Kalmanson metrics we provide is given entirely in terms of the geometry of the non-negative part of this projective variety. In our description \nchecking whether a given Kalmanson metric is an electrical Kalmanson metric amounts to checking that the Plücker coordinates of an explicitly defined point in the above Grassmannian are non-negative. \n\nMoreover, using the results from \\cite{L} we propose an algorithm for reconstructing a minimal circular planar electrical network out of a given resistance matrix which might be useful for possible applications in phylogenetics.\n\n\n\nIt is remarkable that the theory of positivity might play a role in studying phylogenetic networks since it would allow us to apply the powerful machinery of the theory of cluster algebras developed for describing positivity in mathematical objects \\cite{Z}.\n\n\nThis story should be extended to the compactifications of the respective spaces, taking cactus networks, the known strata in the compactification of circular planar electrical networks, to the newly defined compactified split systems as it was already started in \\cite{DF}. In this picture the cactus networks should correspond to the pseudometrics playing the role of dissimilarity functions.\n\nThe connection of the tropical geometry of the Grassmannians and the space of trees and tree metrics found in \\cite{SS} is another interesting direction for developing our work.\n\nFinally building of an example from \\cite{F} we show in Remark \\ref{nk} that the planarity condition on the network is far from being necessary to guarantee the Kalmanson condition for the resistance metric. It leaves the description of the set of electrical Kalmanson metrics as an interesting question.\n\n{\\bf Acknowledgments.} For V. G. this article is an output of a research project implemented as part of the Basic Research Program at the National Research University Higher School of Economics (HSE University). Working on this project V. G. also visited the Max Plank Institute of Mathematics in the Sciences in Leipzig, Germany in the Fall of 2024 and BIMSA in Beijing, China in the Summer 2024.\nResearch of A.~K. on Sections \\ref{sec:kalman} was supported by the Russian Science Foundation project No. 20-71-10110 (https://rscf.ru/en/project/23-71-50012) which finances the work of A.K. at P. G. Demidov Yaroslavl State University.\nResearch of A.K. on Section \\ref{sec:rec} was supported by the state assignment of MIPT (project FSMG-2023-0013).\n\nThe authors are grateful to Borya Shapiro, Misha Shapiro, Anton Petrunin, Lazar Guterman and especially to Stefan Forcey for useful discussions and helpful suggestions. The authors are grateful to the referees for carefully reading the paper and suggesting a number of useful comments to make it better.\n\n\\section{The space of electrical networks and the positive Isotropic Grassmannian}\n\\subsection{The space of electrical networks}\n\n\\begin{definition}\n A electrical network $\\mathcal E$ is a graph $\\Gamma$ with positive $\\omega$ weights (conductances) attached to the edges and a chosen subset of the set of vertices which are called the boundary vertices or the nodes. \n\\end{definition}\nIn this paper we will denote by $n$ the number of the nodes.\n\nAs explained in the introduction the key result in the theory of electrical networks says the boundary voltages and the currents are related to each other linearly via a matrix $M_R(\\mathcal E)=(x_{ij})$ called the {\\it response matrix} of a network. $M_R(\\mathcal E)$ has the following properties:\n\\begin{itemize}\n \\item $M_R(\\mathcal E)$ is a $n\\times n$ symmetric matrix;\n \\item All the non-diagonal entries $x_{ij}$ of $M_R(\\mathcal E)$ are non-positive;\n \\item For each row the sum of all its entries is equal to $0$.\n\\end{itemize}\n\nGiven an electrical network $\\mathcal E$ the response matrix $M_R(\\mathcal E)$ can be calculated as the Schur complement of a submatrix in the Laplacian matrix of the graph of $\\mathcal E$ \\cite{CIM}. \nSuppose that $M$ is a square matrix and $D$ is a non-singular square lower right-hand corner submatrix of $M$, so that $M$ has the block structure. \n\\[\n\\begin{pmatrix}\nA&B\\\\\nC&D\n\\end{pmatrix}\n\\]\nThe Schur complement of $D$ in $M$ is the matrix $M/D = A - BD^{-1} C$. The Schur complement satisfies the following identity\n\\[\\text{det} M = \\text{det} (M/D)\\text{det} D\\]\nLabeling the vertices starting from the nodes we get the Laplacian matrix of the graph representing $\\mathcal E$ in a two by two block form as above. The submatrix $D$ corresponds to the connections between the internal vertices and is known to be non degenerate. Then $M_R(\\mathcal E)=L/D$.\n\nThere are many electrical networks which have the same response matrix, we will describe them now. The following five local network transformations given below are called the {\\it electrical transformations}. \nTwo electrical networks are said to be equivalent if they can be obtained from each other by a sequence of the electrical transformations. This is an equivalence relation of course, so the set of electrical networks is partitioned in the equivalence classes.\n\\begin{theorem} \\cite{CIM} \\label{gen_el_1}\nThe electrical transformations preserve the response matrix of an electrical network.\n\\end{theorem}\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.9\\textwidth]{eltranseng.jpg}\n \\hspace{-1.5cm} \\caption{Electrical transformations }\n \\label{fig:el_trans}\n\\end{figure}\n\nIn this paper we will deal with a particular type of connected electrical networks, the {\\it connected circular planar electrical networks}, unless otherwise stated. For these we require in addition the graph of $\\mathcal E$ to be planar and such that the nodes are located on a circle and enumerated clockwise while the rest of the vertices are situated inside of this circle.\n\nWe will denote by $E_n$ the set of equivalence classes of the circular electrical networks in this paper. Note that the notation for this set of equivalence classes used in \\cite{CIM} and in \\cite{CIW} is $\\Omega_n$. \nThe set $E_n$ allows the following elegant description.\n\\begin{definition}\nLet $P=(p_1,\\ldots,p_k)$ and $Q=(q_1,\\ldots,q_k)$ be disjoint ordered subsets of the nodes arranged on a circle, then $(P;Q)=(p_1,\\ldots,p_k;q_1,\\ldots,q_k)$ is a {\\it circular pair} if $p_1,\\ldots, p_k,q_k,\\ldots,q_1$ are in non-overlapping circular order around the boundary circle. Let $(P;Q)$ be a circular pair then the determinant of the submatrix $ M(P;Q)$ whose rows are labeled by $(p_1,\\ldots,p_k)$ and the columns labeled by $(q_1,\\ldots,q_k)$ is called the {\\it circular minor} associated with a circular pair $(P;Q)$.\n\\end{definition}\n\n\\begin{theorem} \\cite{CIM}, \\cite{CdV} \\label{Set of response matrices all network}\n The set of response matrices of the the elements of $E_n$ is precisely the set of the matrices $M$ such that\n \\begin{itemize}\n \\item $M$ is a symmetric matrix;\n \\item All the non-diagonal entries of $M$ are non-positive;\n \\item For each row the sum of all its entries is equal to $0$.\n \\item For any $k\\times k$ circular minor $(-1)^k\\det M(P;Q) \\geq 0$.\n \\item The kernel of $M$ is generated by the vector $(1,1,\\dots,1)$.\n\\end{itemize}\nMoreover, any two circular electrical networks which have the same response matrix are equivalent.\n\\end{theorem}\nWe will need the following notion later. \n\\begin{definition}\nLet $\\mathcal E$ be a connected circular electrical network on a graph $\\Gamma$. The {\\it dual electrical network} $\\mathcal E^*$ is defined in the following way\n\\begin{itemize}\n\\item the graph $\\Gamma^*$ of $\\mathcal E^*$ is the dual graph to $\\Gamma$; \n\\item for any pair of dual edges of $\\Gamma$ and $\\Gamma^*$ their conductances are reciprocal to each other;\n\\item the labeling of the nodes of $\\mathcal E^{*}$ is determined by the requirement that the first node of $\\mathcal E^{*}$ lies between the first and second node of $\\mathcal E$.\n\\end{itemize}\n\\end{definition}\n\\subsection{Cactus electrical networks}\n\nSetting an edge conductance to zero or infinity in $\\mathcal E$ makes sense. According to the Ohm law it means that we delete or contract this edge. Doing it we either get isolated nodes or some nodes get glued together. We will consider the resulting network as a network with $n$ nodes, remembering how nodes get identified. Such a network is called {\\it a cactus electrical network} with $n$ nodes. One can think of it as a collection of ordinary circular electrical networks with the total number of the nodes equal to $n$, glued along some of these nodes. Note, that the graph of such a network is planar, but it does not have to be connected.\n \n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=1.0\\textwidth]{cactuswithout.jpg}\n \\caption{A cactus electrical network with 4 nodes }\n \\label{fig:cactus}\n\\end{figure}\n\nThe electrical transformations can be applied to the cactus electrical networks. We denote by $\\overline{E}_n$ the set of equivalence classes with respect to the electrical transformations of the cactus electrical networks with $n$ nodes.\n\nThe definition of a cactus network was introduced in \\cite{L} where it was proved that the set $\\overline{E}_n$ is a compactification of $E_n$ in the appropriate sense. Note that the space of cactus networks in denoted by $E_n$ in \\cite{L}. We apologize for this unfortunate clash of notations.\n\n\\subsection{Lam embedding} \\label{sec: lam emb} \n\nRecall that the real Grassmannian $\\mathrm{Gr}(k, n)$\nis a differentiable manifold that parameterizes the set of all \n$k$-dimensional linear subspaces of a vector space $\\mathbb{R}^n$. In fact it has a structure of a projective algebraic variety. The Plücker embedding is an embedding of the Grassmannian $\\mathrm{Gr}(k, n)$ into the projectivization of the $k$-th exterior power of the vector space $\\mathbb{R}^n$\n\\[\\iota :\\mathrm {Gr} (k,n)\\rightarrow \\mathrm {P}(\\Lambda ^{k}\\mathbb{R}^n)\\]\nSuppose $W\\subset \\mathbb{R}^n$ is a $k$-dimensional subspace. To define \n$\\iota (W)$, choose a basis\n$(w_{1},\\cdots ,w_{k})$ for \n$W$, and let \n$\\iota (W)$ be the projectivization of the wedge product of these basis elements:\n$\\iota (W)=[w_{1}\\wedge \\cdots \\wedge w_{k}]$, where \n$ [\\,\\cdot \\,]$ denotes the projective equivalence class. \n\nFor practical calculations one can view the matrix whose rows are the coordinates of the basis vectors $(w_{1},\\cdots ,w_{k})$ as a representative of this equivalence class.\nFor any ordered sequence $I$ of $k$ positive integers\n$1\\leq i_{1}<\\cdots <i_{k}\\leq n$ denote by $\\Delta_I$ the determinant of a $k\\times k$ submatrix of the above matrix with columns labeled by the numbers from $I$. The numbers $\\Delta_I$ are called the Plücker coordinates of the point $W$ of $\\mathrm Gr(k,n)$. They are defined up to a common non zero factor.\n\n\n\\begin{definition}\nThe totally non-negative Grassmannian $\\mathrm{Gr}_{\\geq 0}(k, m)$ is the subset of the points of the Grassmannian $\\mathrm{Gr}(k, n)$ whose Plücker coordinates $\\Delta_I$ have the same sign or equal to zero.\n\\end{definition}\n\nThe following theorem of T. Lam \\cite{L} is one of the key results about the space of electrical networks.\n\\begin{theorem} \\label{th: main_gr}\nThere is a bijection \n\\[\\overline {E}_n \\cong \\mathrm{Gr}_{\\geq 0}(n-1,2n)\\cap \\mathbb{P}H\\]\nwhere $H$ is a certain subspace of $\\bigwedge^{n-1}\\mathbb{R}^{2n}$ of dimension equal to the Catalan number $C_n$. \n\nMoreover, the image of the set $E_n$ under this bijection is exactly the set of points with the Plücker coordinates $\\Delta_{24\\dots 2n-2}$ and $\\Delta_{13\\dots 2n-3}$ not equal to zero.\n\\end{theorem}\n\n\n\nWe will recall the explicit construction of the embedding of ${E}_n$ obtained in \\cite{BGKT}, which is induced by the above bijection.\nLet $\\mathcal E$ be a circular electrical network with the response matrix $M_R(\\mathcal E)=(x_{ij}).$ The following $n\\times 2n$ matrix\n\\begin{equation} \\label{omega_eq}\n\\Omega(\\mathcal E)=\\left(\n\\begin{array}{cccccccc}\nx_{11} & 1 & -x_{12} & 0 & x_{13} & 0 & \\cdots & (-1)^n\\\\\n-x_{21} & 1 & x_{22} & 1 & -x_{23} & 0 & \\cdots & 0 \\\\\nx_{31} & 0 & -x_{32} & 1 & x_{33} & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \n\\end{array}\n\\right) \n\\end{equation}\ngives the point in $\\mathrm{Gr}_{\\geq 0}(n-1,2n)\\cap \\mathbb{P}H$ which corresponds to $\\mathcal E$ under the Lam bijection. \n\nSince the sum of the entries in each row of $M_R(\\mathcal E)$ is equal to $0$ (see Theorem \\ref{Set of response matrices all network}), we have that $\\Omega(\\mathcal E)$ has the rank equal to $n-1$. Therefore, the dimension of the row space of $\\Omega(\\mathcal E)$ is equal to $n-1$, hence it defines a point in $\\mathrm{Gr}(n-1,2n)$. The Plücker coordinates of the point associated with $\\Omega(\\mathcal E)$ can be calculated as the maximal size minors of the matrix $\\Omega'(\\mathcal E)$ obtained from $\\Omega(\\mathcal E)$ by deleting, for example, the first row.\n\\begin{theorem} \\cite{BGK} \\label{about sur}\nLet $A=(a_{ij})$ be a matrix which satisfies the first three conditions of Theorem \\ref{Set of response matrices all network} and $\\Omega(A)$ be a matrix constructed according to the formula \\eqref{omega_eq}. \nIf $\\Omega(A)$ defines a point in $Gr_{\\geq 0}(n-1, 2n)$ and the Plücker coordinate $\\Delta_{13\\dots 2n-3}\\bigl(\\Omega(A)\\bigr)$ is not equal to zero, then there is a connected electrical network $\\mathcal E \\in E_n$ such that \n$A=M_R(\\mathcal E).$\n\\end{theorem}\n\n\\begin{example}\n For the network $\\mathcal E \\in E_4$ seen in the Figure \\ref{treedaul} the matrix $\\Omega(\\mathcal E)$ has the following form:\n \\begin{equation*}\n\\Omega(\\mathcal E) = \\left(\n\\begin{array}{cccccccc}\n\\dfrac{5}{8}& 1 & \\dfrac{1}{8} & 0 & -\\dfrac{1}{8} & 0 & \\dfrac{3}{8} & 1 \\\\\n& & & & & & & \\\\\n\\dfrac{1}{8}& 1 & \\dfrac{5}{8} & 1 & \\dfrac{3}{8} & 0 & -\\dfrac{1}{8} & 0 \\\\\n& & & & & & & \\\\\n-\\dfrac{1}{8}& 0 &\\dfrac{3}{8} & 1 & \\dfrac{5}{8} & 1 & \\dfrac{1}{8} & 0 \\\\\n& & & & & & & \\\\\n\\dfrac{3}{8}& 0 & -\\dfrac{1}{8} & 0 & \\dfrac{1}{8} & 1 & \\dfrac{5}{8} & 1 \\\\\n\\end{array}\n\\right).\n\\end{equation*}\n\\end{example}\n\nIn fact, the row space of $\\Omega(\\mathcal E)$ is isotropic with respect to a particular symplectic form. This refines the above embedding to a submanifold \n$$\\mathrm{IG}_{\\geq 0}(n-1, 2n)\\subset \\mathrm{Gr}_{\\geq 0}(n-1, 2n)$$\nmade out of isotropic subspaces of $\\mathbb{R}^{2n}$ \\cite{BGKT}, \\cite{CGS}.\n\n\\section{Characterization of resistance distance} \\label{sec:kalman}\n\\subsection{Resistance metric}\nNow we assume that an electrical network is just connected.\n\\begin{definition} \\label{def:eff-resist}\n Let $\\Gamma, \\omega$ be a connected graph with a weight function $\\omega$ on the edges and $i,\\, j\\in \\Gamma$ be two of its vertices. Consider it as an electrical network $\\mathcal E_{ij}(\\Gamma, \\omega)$ on a graph $\\Gamma$ with a conductivity function $\\omega$ by declaring the vertices $i$ and $j$ to be the nodes while the remaining vertices to be internal vertices.\n \nApply the voltages $U=(U_i, U_j)$ to the nodes of $\\mathcal E_{ij}(\\Gamma, \\omega)$ such that they induce the boundary currents $I=(1, -1)$. Then the {\\it effective resistance} between the nodes $i$ and $j$ is defined as \n $$R_{ij}=|U_i-U_j|$$\nObviously $R_{ij}=R_{ji}$. \n\\end{definition}\n\nThis definition can be stated which is due to Gustav Kirchhoff in a beautiful way using the combinatorics of the graph. Two pieces of notation are required: \nrecall the generating function of spanning-trees of $\\Gamma$ is defined as follows\n\\[T(\\Gamma,\\omega) = \\sum_{T\\in T(\\Gamma)} \\omega(T),\\]\nwhere the sum is taken over the set $T(\\Gamma)$ of all spanning trees of $\\Gamma$ and $\\omega(T)$ is defined as the product of the weights of the edges of $T$. \nFor a graph $\\Gamma, \\omega$ as above and vertices $i,j \\in V$, we let $\\Gamma/ij$ denote the graph obtained by merging the two vertices $i$ and $j$ together into a single vertex. \n\\begin{theorem}\\label{KF} (Kirchhoff’s Formula \\cite{MTT}, \\cite{Wa}). Let $(\\Gamma,\\omega)$ be an electrical network, and let $i,j \\in V$. The effective resistance between $i$ and $j$ in $\\Gamma$ is \n\n\\[R_{ij} =\\frac{T(\\Gamma/ij;\\omega)} {T(\\Gamma;\\omega)} \\]\n\\end{theorem}\nThe following lemma relates the effective resistances and the response matrix entries is well known. \n\\begin{lemma} \\cite{KW 2011} \\label{lem:eff-resist}\nLet $\\mathcal E$ be a connected electrical network with $n$ nodes\\\\ $\\{1, \\dots, i, \\dots, j, \\dots, n\\} $, and let the boundary voltages $U = (U_1, \\dots , U_n)$ be such that\n\\begin{equation} \\label{eq-resist}\n M_R(\\mathcal E)U = -e_i + e_j,\n\\end{equation}\n where $e_k, \\ k \\in \\{1, \\dots , n\\}$ is the standard basis of $\\mathbb{R}^n$.\nThen $$|U_i - U_j|=R_{ij}.$$\n\\end{lemma}\n\\begin{proof}\n Indeed, if the boundary voltages $U$ are such as in \\eqref{eq-resist} we can consider all the vertices except $i$ and $j$ as the inner vertices, and then $|U_i - U_j|$ is precisely as in Definition \\ref{def:eff-resist}. \n\\end{proof}\nFor convenience, we will organize the effective resistances $R_{ij}$ between the nodes of $\\mathcal E$ in a symmetric matrix $R_{\\mathcal E}$ setting $R_{ii}=0$ for all $i$.\nWe call this matrix the {\\it resistance matrix} of $\\mathcal E$ and denote it by $R_{\\mathcal E}$. \n\nFrom Lemma \\ref{lem:eff-resist} it follows that \n$$R_{ij}=(-e_i+e_j)^t\\bigl(M_R(\\mathcal E)\\bigr)^{-1}(-e_i+e_j),$$\nwhere $\\bigl(M_R(\\mathcal E)\\bigr)^{-1}(-e_i+e_j)$ means a vector $U$ which satisfies \\eqref{eq-resist}. Notice that such a vector always exists. \n\n\\begin{proposition}\\cite{KW 2011} \\label{th: about inverse resp} \nLet $\\mathcal E$ be a connected electrical network. Denote by $M'_R(\\mathcal E)$ the matrix obtained from $M_R(\\mathcal E)$ by deleting the last row and the last column, then $M'_R(\\mathcal E)$ is invertible. The matrix elements of its inverse are given by the formula\n \\begin{equation*}\nM'_R(\\mathcal E)^{-1}_{ij}=\\begin{cases}\n R_{in},\\, \\text{if}\\,\\, i=j \\\\\n \\frac{1}{2}(R_{in}+R_{jn}-R_{ij}),\\, \\text{if}\\,\\, i\\not = j,\\\\\n \n \\end{cases}\n \\end{equation*} \n\n\\end{proposition}\n\nProposition \\ref{th: about inverse resp} and Lemma \\ref{eq-resist} show that the resistance and the response matrices define each other. Therefore, Theorem \\ref{gen_el_1} would hold if we replace the response matrix with the resistance matrix in its statement.\n\n\\begin{remark}\\label{inverse} The formula from Proposition \\ref{th: about inverse resp} is well known in different areas of mathematics. It appeared in the literature under the names the Gromov product, the Farris transform, the Covariance mapping between the Cut and the Covariance cones, see \\cite{GandC} for more information. This allows us to view $M'_R(\\mathcal E)^{-1}$, the \"inverse\" of the response matrix, as a similarity function corresponding under the Covariance mapping to a dissimilarity function represented by the resistance metric $R_{\\mathcal E}$. We are planning to explore the properties of the resistance matrix from these points of view in the future publications.\n\\end{remark}\nProposition \\ref{th: about inverse resp} provides a simple proof that the effective resistances $R_{ij}$ satisfy the triangle inequality.\n\\begin{theorem} \\label{th:about metric}\n Let $\\mathcal E$ be an electrical network on a connected network $\\Gamma$ then for any of three nodes $k_1, k_2$ and $k_3$ the triangle inequality holds:\n $$R_{k_1k_3}+R_{k_2k_3}-R_{k_1k_2} \\geq 0.$$\n Hence the set of all $R_{vw}$ defines a metric on the nodes of $\\Gamma$.\n \n\\end{theorem}\n\\begin{proof} \n Let $\\mathcal E_{k_1k_2k_3}$ be a connected electrical network obtained from $\\mathcal E$ by declaring the vertices $k_1, k_2, k_3$ to be the boundary nodes, while remaining vertices are declared to be inner and $M_R(\\mathcal E_{k_1k_2k_3})$ be its response matrix, then according to Proposition \\ref{th: about inverse resp} we have that:\n $$M'_R(\\mathcal E_{k_1k_2k_3})^{-1}_{k_1k_2}=\\frac{1}{2}(R_{k_1k_3}+R_{k_2k_3}-R_{k_1k_2}),$$\n therefore to get the statement it is enough to verify that $M'_R(\\mathcal E_{k_1k_2k_3})^{-1}_{k_1k_2} \\geq 0.$ \nIndeed, the matrix $M'_R(\\mathcal E_{k_1k_2k_3})$ has the following form:\n \\begin{equation*} \nM'_R(\\mathcal E_{k_1k_2k_3})=\\left(\\begin{matrix}\nx_{k_1k_1} & x_{k_1k_2} \\\\\n x_{k_1k_2} & x_{k_2k_2} \n\\end{matrix} \\right) = \\left(\\begin{matrix}\n-x_{k_1k_2}-x_{k_1k_3} & x_{k_1k_2} \\\\\n x_{k_1k_2} & -x_{k_1k_2}-x_{k_2k_3} \n\\end{matrix} \\right).\n\\end{equation*}\n By the direct computation we obtain that \n \\begin{equation*}\n M'_R(\\mathcal E_{k_1k_2k_3})^{-1}_{k_1k_2}=\\frac{-x_{k_1k_2}}{\\det M'_R(\\mathcal E_{k_1k_2k_3}) }=\\frac{-x_{k_1k_2}}{x_{k_1k_2}x_{k_2k_3}+x_{k_1k_3}x_{k_1k_2}+x_{k_1k_3}x_{k_2k_3}},\n \\end{equation*}\n By the definition of the response matrix all $x_{k_ik_j} \\leq 0, i\\neq j$ which implies the statement of the theorem. \n\\end{proof}\n\n\nTo describe the properties of the resistance metric associated with a connected circular electrical network, we will provide a formula for the embedding of $E_n$ into $\\mathrm{Gr}_{\\geq 0}(n-1, 2n)$ described in Theorem \\ref{th: main_gr} which uses the effective resistances matrix instead of the response matrix.\n\nLet $\\mathcal E$ be a connected network with the resistance matrix $R_{\\mathcal E}$, define a point in $Gr(n-1,2n)$ associated to it as the row space of the matrix:\n\\begin{equation} \\label{eq:omega_n,r}\n \\Omega_{R}(\\mathcal E)=\\left(\\begin{matrix}\n1 & m_{11} & 1 & -m_{12} & 0 & m_{13} & 0 & \\ldots \\\\\n0 & -m_{21} & 1 & m_{22} & 1 & -m_{23} & 0 & \\ldots \\\\\n0 & m_{31} & 0 & -m_{32} & 1 & m_{33} & 1 & \\ldots \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \n\\end{matrix}\\right),\n\\end{equation}\nwhere \n$$m_{ij}= -\\frac{1}{2}(R_{i,j}+R_{i+1,j+1}-R_{i,j+1}-R_{i+1,j}).$$\nNotice that the matrix $M(R_{\\mathcal E})=(m_{ij})$ is symmetric and the sum of the matrix entries in each row is zero. In other words, it looks like a response matrix of an electrical network.\nThere is a reason for this as it was discovered by R. Kenyon and D. Wilson.\n\\begin{theorem} \\cite{KW 2011} \\label{ken-wen}\n Let $\\mathcal E$ be a connected circular electrical network and $\\mathcal E^{*}$ be its dual, then the following holds:\n \\begin{equation} \\label{form:xij}\n x^{*}_{ij}=-\\frac{1}{2}(R_{i,j}+R_{i+1,j+1}-R_{i,j+1}-R_{i+1,j})\n \\end{equation}\n \\begin{equation} \\label{form:rij}\n R^{*}_{ij}=-\\sum \\limits_{i'<j': \\ D_{S_{ij}}(i', j') \\neq 0}x_{i'j'},\n \\end{equation}\nwhere $x^*_{ij}$ are the matrix elements of the response matrix of the dual network $\\mathcal E^*$. \n\\end{theorem}\n\nSince $M(R_{\\mathcal E})$ is a degenerate matrix, the Plücker coordinates of a point $Gr(n-1, 2n)$ associated with $\\Omega_{R}(\\mathcal E)$ can be calculated as the maximal size minors of the matrix $\\Omega'_{ R}(\\mathcal E)$ obtained from $\\Omega_{R}(\\mathcal E)$ by deleting, for example, the last row.\n\n\\begin{theorem} \\cite{BGGK} \\label{th:aboout omeganr}\nThe row space of $\\Omega_{R}(\\mathcal E)$ defines the same point in the $Gr_{\\geq 0}(n-1,2n)$ as the point $\\Omega(\\mathcal E)$ defined by the Lam embedding, see Theorem \\ref{th: main_gr}. In particular, the Plücker coordinate $\\Delta_{24\\dots 2n-2}(\\Omega_{R}(\\mathcal E))$ is not 0 if and only if $\\mathcal E$ is a connected circular electrical network.\n\nPutting it together\n\\[ \n\\Omega_{R}(\\mathcal E)=\\Omega(\\mathcal E^{*})s=\\Omega(\\mathcal E)\\] \nwhere $s \\in \\Mat_{2n \\times 2n}$ is the shift operator\n\\[s=\\left(\\begin{matrix}\n0 & 1 & 0 & 0 & \\cdots & 0 \\\\\n0 & 0 & 1 & 0 & \\cdots & 0 \\\\\n0 & 0 & 0 & 1 & \\cdots & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n0 & 0 & 0 & 0 & \\cdots & 1 \\\\\n(-1)^{n} & 0 & 0 & 0 & \\cdots & 0\n\\end{matrix}\\right),\\]\\label{cyclic operator s}\nSee the proof of \\cite[Theorem 5.6]{BGGK} for more details.\n\\end{theorem}\n\n\nMany interesting inequalities involving $R_{ij}$ follow from the positivity of the Plücker coordinates of the point represented by $\\Omega_{R}(\\mathcal E)$. For some of them we have found an explicit combinatorial meaning, others are still waiting to be interpreted.\n\nBelow we will deduce the Kalmanson property for the metric $R_{\\mathcal E}$ as a consequence of the positivity described above. This fact was established in \\cite{F} earlier using different methods. \n\n\\begin{theorem} \\label{charkalm}\n Let $\\mathcal E$ be a connected circular electrical network and let $i_1, i_2, i_3, i_4$ be any four nodes in the circular order as it is shown in the Figure \\ref{kalmanson}. Then the Kalmanson inequalities hold:\n \\begin{equation} \\label{kal_1}\n R_{i_1i_3}+R_{i_2i_4}\\geq R_{i_2i_3}+R_{i_1i_4}, \n \\end{equation}\n \\begin{equation} \\label{kal_2}\n R_{i_1i_3}+R_{i_2i_4}\\geq R_{i_1i_2}+R_{i_3i_4}. \n \\end{equation}\n\\end{theorem}\n\n\n\\begin{proof}\n Let $\\mathcal E_{i_1i_2i_3i_4}$ be an electrical network obtained from $\\mathcal E$ by declaring the vertices $i_1, i_2, i_3, i_4$ the boundary nodes, and the rest of the vertices are internal. Obviously $\\mathcal E_{i_1i_2i_3i_4}$ is a connected circular electrical network in $E_4$ and let $\\Omega'_{R}(\\mathcal E_{i_1i_2i_3i_4})$ be its matrix defined by the formula \\ref{eq:omega_n,r}.\n By direct computations we conclude that\n\n $$\\Delta_{567}\\bigl(\\Omega'_{ R}(\\mathcal E_{i_1i_2i_3i_4})\\bigr)=\\frac{1}{2}(R_{i_1i_3}+R_{i_2i_4} -R_{i_1i_2}-R_{i_3i_4}),$$\n $$\\Delta_{123}\\bigl(\\Omega'_{R}(\\mathcal E_{i_1i_2i_3i_4})\\bigr)=\\frac{1}{2}(R_{i_1i_3}+R_{i_2i_4}- R_{i_2i_3}-R_{i_1i_4}).$$\n Taking into account that all the minors $\\Delta_{I}\\bigl(\\Omega'_{ R}(\\mathcal E_{i_1i_2i_3i_4})\\bigr)$ are non-negative we obtain that the Kalmanson inequalities hold for the metric defined by the resistances.\n\\end{proof}\n \\begin{figure}[h!]\n\\center\n\\includegraphics[width=60mm]{kalmanson.jpg}\n\\caption{$4$-point Kalmanson property}\n\\label{kalmanson}\n\\end{figure}\n\\begin{remark}\nNote that the equations \\ref{kal_1}, \\ref{kal_2} together are equivalent to the tropical inequality:\n\\[\n R_{i_{1}i_{3}}+R_{i_{2}i_{4}}\\geq\\max(R_{i_{2}i_{3}}+R_{i_{1}i_{4}},\\,R_{i_{1}i_{2}}+R_{i_{3}i_{4}})\n \\]\n\\end{remark}\n\\begin{definition} Let $X$ be a finite set and $D$ is a metric on it. If there is a circular order on $X$ such that the inequalities from the theorem above hold for any four points of $X$ in this circular order, we call this metric the {\\it Kalmanson metric}.\n\\end{definition}\nTherefore the resistance metric $R_{\\mathcal E}$ defined by a circular electrical network $\\mathcal E$ is a Kalmanson metric.\n\n\\subsection{Circular split systems and electrical networks} \\label{sec:splitmetr}\n\\begin{definition}\n A {\\it split} $S$ of a set $X=\\{1, \\dots, n\\}$ is a partition of $X$ into two non-empty, disjoint subsets $A$ and $B$, $ A \\sqcup B = X$. A split is called trivial if either $A$ or $B$ has cardinality $1.$\n A collection of splits $\\mathcal{S}$ is called a {\\it split system }.\n\\end{definition}\n\nThe pseudo metric associated with a split $S$ is defined by the following matrix $D:$\n \\begin{equation*}\n D_{A|B}(k,l)=\\begin{cases}\n 1, \\ \\text{if} \\ |A \\cap \\{k,l\\}|=1, \\\\\n 0, \\ \\text{otherwise.}\n \\end{cases}\n \\end{equation*}\n\n \nA circular order of $X$ can be drawn as a polygon with the elements of $X$ labeling the sides. A {\\it circular split system} is a split system for which a circular order exists such that all the splits can be simultaneously drawn as sides or diagonals of the labeled polygon. Trivial splits are sides of the polygon, separating the label of that side from the rest, while a non-trivial split $A|B$ is a diagonal separating the sides labeled by $A$ and $B$. For any circular split system we can visualize it by such a polygonal representation, or instead choose a visual representation using sets of parallel edges for each split; these representations are called {\\it circular split networks}. A set of parallel edges displays a split $A|B$ if the removal of those edges leaves two connected components with respective sets of terminals $A$ and $B$, see the Figure \\ref{splex}. \n\\begin{definition} We call a {\\it weighted circular split system} a circular split system with a positive weight attached to each set of parallel edges, which display the splits. Such a weighted circular split system defines a pseudo metric on the set $X$ by the formula\n\\begin{equation} \\label{decom1}\n D_S=\\sum \\limits_{A|B \\in S } \\alpha_{A|B}D_{A|B},\n \\end{equation}\nwhere $\\alpha_{A|B}$ is the weight of the set of parallel edges which defines the split $A|B$.\n\\end{definition}\nThe split $(\\{1,2,3,4\\}|\\{5,6,7,8\\})$ of the set of the sides of the polygon on the left defined by the yellow diagonal in the Figure \\ref{splex} corresponds to the set of parallel edges with the weight $\\alpha_1$ in the graph on the left.\n\\begin{figure}[h!]\n\\center\n\\includegraphics[width=90mm]{splyweight.jpeg}\n\\caption{Circular split system and their polygon representations. Removing any set of parallel edges defines a split of the set of leaves of the graph on the right which coincides with the split of the set of the sides of a polygon on the left obtained by removing an appropriate diagonal.}\n\\label{splex}\n\\end{figure}\n\nThe following theorem gives a beautiful characterization of the set of Kalmanson metrics \\cite{BD}.\n\\begin{theorem} \\label{BD} A metric $d$ is a Kalmanson metric with respect to a circular order $c$ if and only if $d = D_S$ for a unique weighted circular split system $S$, (not necessarily containing all trivial splits) with each split $A|B$ of $S$ having both parts contiguous in that circular order $c$.\n\\end{theorem}\n\nSince the resistance metric defined by a planar circular electrical network satisfies the Kalmanson condition, it corresponds to a unique weighted circular split system, which we call an {\\it electrical circular split system} following \\cite{F} or, given Theorem \\ref{charkalm}, an {\\it electrical Kalmanson metric}.\n\nWe will introduce a slightly different way of labeling the splits, it will be useful for us later.\n\\begin{definition} \\label{circ-system}\nLet $X$ be a set of nodes on a circle labeled clockwise by the symbols $1$ to $n$. Define the dual set $X^d$ consisting of nodes labeled by the symbols $\\overline{1}$ to $\\overline{n}$ in such a way that each $\\overline{j}$ lies between $j$ and $j+1.$ Then each chord connecting $\\overline{i}$ and $\\overline{j}$ defines a split of $X$ which we will denote by $S_{ij}$.\n\nIt is obvious that the set $S_{ij}$ forms a circular split system which we will denote by $\\mathcal{S}_c$. \n\n\\end{definition}\nThe following theorem gives the formula for the weights of a weighted circular split systems \\ref{BD}. \n\\begin{theorem} \\cite{GandC}, \\cite{HKP} \\label{th:decomp}\n Let $X$ be a set as in Definition \\ref{circ-system} and let $D=(d_{ij})$ be a Kalmanson metric defined on $X$. Then the following split decomposition holds:\n \\begin{equation} \\label{decom}\n D=\\sum \\limits_{S_{ij} \\in \\mathcal{S}_c } \\omega_{ij}D_{S_{ij}},\n \\end{equation}\n where the coefficients $\\omega_{ij}$ are defined by the formula $$\\omega_{ij}=\\frac{1}{2}(d_{i,j}+d_{i+1,j+1}-d_{i,j+1}-d_{i+1,j}).$$\n\\end{theorem}\n\\begin{remark}\\label{KW} The formula \\eqref{decom} for $R_{ij}$ can be obtained from Theorem \\ref{ken-wen} since taking the dual electrical network is an involution up to a shift: $\\mathcal E^{**}=\\mathcal E'$, where $\\mathcal E'$ is obtained from $\\mathcal E$ by shifting the numeration of the nodes of $\\mathcal E$ by $1$ clockwise:\n\n \\begin{align} \\label{formdual}\n R_{ij}=R^{**}_{i-1j-1}=-\\sum \\limits_{\\bar{i}'<\\bar{j}': \\ D_{S_{i-1j-1}}(\\bar{i}', \\bar{j}') \\neq 0}x^{*}_{\\bar{i}'\\bar{j}'}=-\\sum \\limits_{i'<j': \\ D_{S_{ij}}(i', j') \\neq 0}x^{*}_{i'j'}= \\\\ \\nonumber\n \\hspace{-40mm} {\\sum \\limits_{i'<j': \\ D_{S_{ij}}(i', j') \\neq 0}\\frac{1}{2}(R_{i,j}+R_{i+1,j+1}-R_{i,j+1}-R_{i+1,j})},\n\\end{align} \nwhere $\\{1, \\dots, n\\}$ and $\\{\\overline1, \\dots, \\overline n\\}$ are the labels of the nodes of $\\mathcal E$ and $\\mathcal E^{*}$ respectively. Thus Theorem \\ref{ken-wen} in fact provides an alternative way to observe that the effective resistance distance is the Kalmanson metric. \n\\end{remark}\n\\begin{figure}[H]\n\\center\n\\includegraphics[width=80mm]{toformualdual.jpg}\n\\caption{Labeling in the formula \\eqref{formdual} }\n\\label{fig:treedaul}\n\\end{figure}\n\\begin{definition}\nDefine a matrix $M(D)$\n\\begin{align}\n M(D)_{ij} = \\begin{dcases*}\n \\sum_{k \\not= i}\\omega_{ik}, & \\text{if } i = j,\\\\\n -{\\omega_{ij}}, & \\text{if } $i \\not= j,$ \n \\end{dcases*}\n\\end{align} \nNotice that $\\sum_{k \\not= i}\\omega_{ik}=-\\omega_{ii}=d_{ii+1}.$\n\\end{definition}\n\n\nGiven a Kalmanson metric as a matrix \\( D \\), one can decide whether it is an electrical Kalmanson metric in a straightforward but computationally heavy way. First one must reorder the matrix columns and rows to ensure its entries satisfy the Kalmanson inequalities, then find the unique response matrix $M$ that corresponds to a given Kalmanson metric matrix using the pseudoinverse formula, see \\cite{F} (Lemma 3.1), and then check all the circular minors of that $M$ for non-negativity. \n\nHere is our main result, it gives a new characterization of the Kalmanson metrics, which are the resistance metrics of circular electrical networks without using inversion of matrices.\n\\begin{theorem} \\label{th: dual}\n Let $D$ be a Kalmanson metric matrix with the correct order of rows and columns, then $D$ is the effective resistance matrix of a connected circular electrical network $\\mathcal E$ if and only if the matrix $\\Omega_{D}$ constructed from $D$ according to the formula \\eqref{eq:omega_n,r} defines a point $X$ in $\\mathrm{Gr}_{\\geq 0}(n-1, 2n)$ and the Plücker coordinate $\\Delta_{24\\dots 2n-2}(X)$ does not vanish.\n\\end{theorem}\n\\begin{proof}\n\nNecessity follows from Theorem \\ref{th:aboout omeganr}.\n\nTo prove sufficiency, assume that $\\Omega_{D}$ defines a point $X$ in $\\mathrm{Gr}_{\\geq 0}(n-1, 2n)$ and $\\Delta_{24\\dots 2n-2}(\\Omega'_{D}) \\neq 0$. By direct computations we conclude that the matrix $\\Omega_{D}s^{-1}$ has the form \\eqref{omega_eq} and $\\Delta_{13\\dots 2n-3}\\bigr((\\Omega_{D}s^{-1})'\\bigl)\\neq 0$. Since the action of $s$ preserves the non-negativity according to \\cite{L3}, the matrix $\\Omega_{D}s^{-1}$ defines a point of $\\mathrm{Gr}_{\\geq 0}(n-1, 2n)$. Using the surjectivity of Lam's embedding (see Theorem \\ref{about sur}) and Theorem \\ref{th:aboout omeganr}, we obtain that both $\\Omega_{D}$ and $\\Omega_{D}s^{-1} $ are associated with connected networks. \n\nFinally, due to Theorem \\ref{th:aboout omeganr} and Theorem \\ref{ken-wen} we have that the matrix $M(D)=(m_{ij}),$ where \n\\[m_{ij}=-\\dfrac{1}{2}(d_{i,j}+d_{i+1,j+1}-d_{i,j+1}-d_{i+1,j})\\]\ncan be identified with the response matrix of a network $\\mathcal E^{*},$ associated with $\\Omega_{D}s^{-1}.$\n\n\n It remains to prove that $d_{ij}$ are equal to the effective resistances $R_{ij}$ of the network $\\mathcal E,$ associated with $\\Omega_{D}.$ Since the metrics $d_{ij}$ and $R_{ij}$ are both Kalmanson and the weights in their split decomposition are equal, $\\omega_{ij}=-m_{ij}$ (see the formula \\eqref{decom}) , we conclude that:\n $$d_{ij}=-\\sum \\limits_{i'(i, j)j'(i, j)} m_{i'j'}= R_{ij}.$$\n\\end{proof}\nThe following theorem is also a convenient test for detecting electrical Kalmanson metrics. It literally falls in our lap due to Theorems \\ref{ken-wen} and \\ref{th:decomp}.\n\n\n\\begin{theorem} \\label{th dual2}\nLet $D$ be a Kalmanson metric on $X$,\nthen $D$ is an Electric Kalmanson metric of a connected circular electrical network $\\mathcal E$ if and only if the rank of $M(D)$ is equal to $n-1$ and the circular minors of the matrix $M(D)$ are non-negative after multiplying by $(-1)^k$ where $k$ is the size of the minor.\n\nGiven this, the matrix $M(D)$ is the response matrix of the dual to the electrical network $\\mathcal E$. \n\\end{theorem}\n\n\\begin{proof}\n\n\nIf any $k \\times k$ circular minors of the matrix $M(D)$ is positive after multiplying by $(-1)^k$ then $M(D)$ defines a response matrix of a network $\\mathcal E'$ due to Theorem \\ref{Set of response matrices all network}.\nHence we conclude that $\\mathcal E'$ is connected (see Theorem \\ref{Set of response matrices all network}) given the condition on the rank of $M(D)$.\n\nThe matrix $\\Omega( \\mathcal E')$ defines a point in $Gr_{\\geq 0}(n-1, 2n)$ hence $\\Omega(\\mathcal E')s$ does as well. According to Theorem \\ref{eq:omega_n,r} \n$$\\Omega(\\mathcal E')s=\\Omega(\\mathcal E)=\\Omega_{R}(\\mathcal E)$$ \nfor a connected network $\\mathcal E$ with the resistance matrix $R_{\\mathcal E}=M(D)$, moreover $\\mathcal E'=\\mathcal E^{*} $. It remains to prove that $R_{ij}=d_{ij}, $ it can be done as it has been explained in the proof of Theorem \\ref{th: dual}.\n\nSuppose now that $D$ comes from a circular electrical split system associated with a connected circular electrical network $\\mathcal E$, then due to Theorem \\ref{eq:omega_n,r} we conclude that $\\Omega_{R}(\\mathcal E)s^{-1}$ defines a point in $Gr_{\\geq 0}(n-1, 2n)$ associated with a connected network $\\mathcal E^{*}$ and $M(D)=M_R(\\mathcal E^*).$ The properties of $M(D)$ in the statement of the theorem follow from Theorem \\ref{Set of response matrices all network}.\n\\end{proof}\nA few remarks are in order.\n\\begin{remark}\nMany statements in this paper can be extended to the connected cactus networks. In particular, one can obtain that the effective resistances of a connected cactus network give rise to a Kalmanson pseudometric and a Kalmanson pseudometric matrix $D$ can be identified with the resistance matrix of a connected cactus networks if and only if the matrix $\\Omega_{D}$ constructed from $D$ according to the formula \\eqref{eq:omega_n,r} defines a point $X$ in $\\mathrm{Gr}_{\\geq 0}(n-1, 2n)$.\n\\end{remark}\n\\begin{remark}\\label{nk}\nIn Remark 4.3, [F] S. Forcey gives an example of a non-planar network whose resistance metric is nevertheless Kalmanson. It is easy to generalise this example in the following way. Let's start with a circular planar electrical network, whose resistance metric therefore is Kalmanson, such that all the inequalities required by the Kalmanson condition are strict. This means the resistance metric is represented by a full circular planar split system. Add an edge with a variable conductance $x$ to it in such a way that it is not a circular planar anymore, as in the Figure \\ref{nkal}.\nIt is clear from the Kirchhoff formula \\ref{KF} that each effective resistance $R_{ij}$ changes as a function of $x$ according to the formula\n$$R^{\\text new}_{ij}=\\frac{R_{ij}+r_{ij}x}{1+Tx}$$\nwhere $r_{ij}$ and $T$ are non-negative numbers and $T$ does not depend on $i,\\,j$. Therefore, for small $x$ the Kalmanson condition for the resistance metric will hold for the new network, although it is not any longer circular planar. This means that the planar circular Kalmanson metrics are sitting far away from the boundary of the space of all Kalmanson metrics.\n\\end{remark}\n\\begin{figure}\n\\center\n\\includegraphics[width=50mm]{non-plan-kalma.jpg}\n\\caption{A non-planar electrical network with nodes \\{1,\\dots,5\\} }\n\\label{nkal}\n\\end{figure}\n\\begin{remark}\\label{plueclerformetric}\n Let $X$ and $D$ be as in Theorem \\ref{th:decomp}, then the matrix $M(D)$ defined above is the response matrix of a connected electrical network, not necessarily circular planar. Namely, it is symmetric, all non-diagonal elements are negative and the sum of the elements in any row is zero.\n The connection of the resistance matrix of this network to the original metric $D$ is an interesting question.\n \n Moreover, such a matrix defines a point \n $$\\mathrm{IG}(n-1, 2n)\\subset \\mathrm{Gr}(n-1, 2n)$$\n according to Theorem \\ref{th:aboout omeganr}. Its Plücker coordinates are interesting invariants of the metric $D$. \nWe are planning to address these questions in a future publication.\n\\end{remark}\n\\begin{remark} The resistance metric has the following important property: its\nsquare root $\\sqrt {D(i,j)}$ is $L_2$ embeddable \\cite{K}, hence the electric Kalmanson metrics have that property. There is a well known condition for $L_2$ embeddability stated in terms of the minors of the Caley-Menger matrix \\cite{GandC}. It is interesting to see if this condition also describes the electric Kalmanson metrics in the set of all Kalmanson metrics.\n\\end{remark}\n\\begin{figure}[h!]\n\\center\n\\includegraphics[width=70mm]{treedual.jpg}\n\\caption{A tree network $\\mathcal E$ and its dual network $\\mathcal E^{*},$ \\\\ all the conductances are equal to $1$}\n\\label{treedaul}\n\\end{figure}\nWe will provide an example to illustrate the theorem above.\n\\begin{example} \nLet $T$ be a tree with four leaves as in the Figure \\ref{treedaul} with the weights of edges all equal to one. The resistance metric matrix of the tree is \n\\[D=\n\\begin{pmatrix}\n0& 3 & 3& 2 \\\\\n3& 0& 2& 3\\\\\n3& 2& 0& 3 \\\\\n2&3&3&0\n\\end{pmatrix}\n\\]\nIt coincides with the matrix of the geodesic distance for trees. Its split decomposition is as follows as one sees easily\n\\[D=D_{S_{12}}+D_{S_{13}}+D_{S_{14}}+D_{S_{23}}+D_{S_{34}}+D_{S_{43}}=D_{2|134}+D_{23|14}+D_{1|234}+D_{3|124}+D_{4|123}\\]\nThe response matrix of the dual network multiplied by $-1$ is\n\\[\n\\begin{pmatrix}\n-3& 1& 1& 1 \\\\\n1& -2& 1& 0\\\\\n1& 1& -3 & 1 \\\\\n1&0&1&-2\n\\end{pmatrix}\n\\]\nIt coincides with the incidence matrix of the dual graph since there are no internal vertices.\nOur correspondence between the splits and pairs of numbers is given in the table\n\\\\\n\\[\\begin{tabular}{ |c|c|c|c|c |}\n $(\\bar1\\bar2)$& $(\\bar1\\bar3)$& $(\\bar1\\bar4)$& $(\\bar2\\bar3)$&$ (\\bar3\\bar4)$ \\\\\n \\hline\n $(2|134)$ & $(14|23)$&$ (1|234)$&$(3|124)$&$(4|123)$ \\\\\n\\end{tabular}\\]\n\\\\\nmatching the coefficients in the split decomposition of $D$ and the coefficients of the response matrix of the dual network.\n\n\\end{example}\n\n\n\n\\section{Reconstruction of network topology} \\label{sec:rec}\nAs we mentioned in the introduction one of the important problems in applied mathematics can be formulated as follows:\n\\begin{problem} \\label{black-box}\n Suppose we are given a matrix $D=(d_{ij})$, whose entries are the distances between $n$ terminal nodes of an unknown weighted graph $G$. It is required to recover the graph $G$ and the edge weights which are consistent with the given matrix. \n\\end{problem}\nIf the terminal nodes are the boundary vertices of a circular electrical networks $\\mathcal E $ and $D=R_{\\mathcal E}$, then due to Proposition \\ref{th: about inverse resp} we conclude that Problem \\ref{black-box} can be identified with the {\\it black box problem}, see \\cite{CIW}.\nIt is also known as the discrete Calderon problem or the discrete inverse electro impedance tomography problem.\n\nIf a graph $G$ is an unknown tree $T$ and the entries of $D_T=(d_{ij})$ are equal to the weights of the paths between vertices of $T$, then Problem \\ref{black-box} can be identified with the minimal tree reconstruction problem, which plays an important role in phylogenetics \\cite{HRS}.\n\\begin{definition}\n We will call a matrix $D=(d_{ij})$ a {\\it tree realizable}, if there is a tree $T$ such that \n \\begin{itemize}\n \\item $D=D_T$ i.e. $d_{ij}$ are equal to the weights of the paths between terminal nodes of $T$;\n \\item a set of terminal nodes contains all leafs of $T;$\n \\item there are no vertices of degree $2.$ \n \\end{itemize}\n \n\\end{definition}\n\n\\begin{theorem} \\cite{CR}, \\cite{HY} \\label{minimaltree}\n If a matrix $D$ is tree realizable, then there is an unique tree $T$ such that $ D=D_T.$\n\n\\end{theorem}\n\n\nIt is not difficult to see that if the graph of a circular electrical network $\\mathcal E $ is a tree $T$ and its boundary nodes contain all tree leafs, then $R_{\\mathcal E}=D_{\\bar T}$ where $\\bar T$ and $T$ are identical as unweighted trees and the weights of the corresponded edges are reciprocal. This observation allows us to use in phylogenetics the algorithm for reconstruction of electrical networks \\cite{L} from a given resistance matrix. Our reconstruction method is different from the methods suggested in \\cite{F}, \\cite{FS}. \n\n\\begin{definition}\n The median graph of a circular network $\\mathcal E $ with the graph $\\Gamma$ is the graph $\\Gamma_M$ whose internal vertices are the midpoints of the edges of $\\Gamma$ and two internal vertices are connected by an edge if the edges of the original graph $\\Gamma$ are adjacent in one face. The boundary vertices of $\\Gamma_M$ are defined as the intersection of the natural extensions of the edges of $\\Gamma_M$ with the boundary circle. Since the interior vertices of the median graph have degree four, we can define the strands of the median graph as the paths which always go straight through any degree four vertex. \n\nThe strands naturally define a permutation $\\tau(\\mathcal E)$ on the set of points $\\{1, \\dots, 2n\\}.$\n\\end{definition}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.4\\textwidth]{strad.jpg}\n \\caption{Star-shape network, its median graph and the strand permutation $\\tau(\\mathcal E)=(14)(36)(25)$ }\n \\label{fig:triangle}\n\\end{figure}\n\\begin{theorem} \\cite{CIW} \\label{strand}\n A circular electrical network is defined uniquely up to electrical transformations by its strand permutation.\n\\end{theorem}\n Denote by $A_i$ the columns of the matrix $\\Omega_{ R}(\\mathcal E)$ and define the column permutation $g(\\mathcal E)$ as follows:\n $g(\\mathcal E)(i)=j,$ if $j$ is the minimal number such that $A_i \\in \\ \\mathrm{span}(A_{i+1}, \\dots, A_{j} ),$ where the indices are taken modulo $2n$.\n\\begin{theorem}\\label{permutationj} \\label{th:aboutstradsper}\n The following holds $$g(\\mathcal E)+1=\\tau(\\mathcal E).$$ \n\\end{theorem}\n \\begin{proof}\n This result follows from the fact that $\\Omega_{ R}(\\mathcal E)$ is an explicit parametrization of the Lam embedding, see Proposition $5.17$ \\cite{L}. \n \\end{proof}\n\\begin{definition}\nA circular electrical network is called minimal if the strands of its median graph do not have self-intersections; any two strands intersect at most one point and the median graph has no loops or lenses, see the Figure \\ref{fig:loop}.\n\\end{definition}\n\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.3\\textwidth]{loop.jpg}\n \\caption{A lens obtained by the intersection of strands $\n\\alpha_1$ and $\n\\alpha_2$ }\n \\label{fig:loop}\n\\end{figure}\n\nBased on Theorem \\ref{permutationj} we suggest the following reconstruction algorithm (see Example \\ref{ex:rec-ex}):\n\\begin{itemize}\n \\item For a given matrix $R_{\\mathcal E}$ construct the matrix $\\Omega_{ R}(\\mathcal E);$\n \\item Using $\\Omega_{ R}(\\mathcal E)$ calculate a strand permutation $\\tau(\\mathcal E)$;\n \\item The permutation $\\tau(\\mathcal E)$ defines a strand diagram, which can be transformed to a median graph of a \\textit{minimal} circular electrical network $\\mathcal E$ using the procedure described in \\cite{CIW}. \n \\item From the median graph we recover the network $\\mathcal E$ as in \\cite{CIW} or \\cite{F}.\n\\end{itemize}\n\n\\begin{theorem} \\cite{CIW}\\label{electrominim}\nAny circular electrical network is equivalent to a minimal network. \n\nAny two minimal circular electrical networks which share \nthe same response matrix, and hence the same effective resistance matrix, can be converted to each other only by the star-triangle transformation. As a consequence we obtain that any two equivalent minimal circular electrical networks have the same number of edges which is less or equal than $\\frac{n(n-1)}{2}$.\n\nIf two minimal circular electrical networks are equivalent, they have \nthe same strand permutation.\n\\end{theorem}\nInformally, the theorem says that removing all the loops, the internal vertices of degree $1$ and reduction of all the parallels and the series any network can be converted into a minimal network.\n \nLet us compare the last result with Theorem \\ref{minimaltree}.\n\\begin{theorem} \\label{tomin}\nLet $\\mathcal E$ be a minimal electrical network whose graph is a tree $T$, then such a tree is unique.\n \n Let $\\mathcal{E}(T, \\omega)$ be a circular planar electrical network on a tree $T$ such that all its leaves are nodes and there are no inner vertices of degree $2$, then $\\mathcal{E}(T, \\omega)$ is minimal.\n\\end{theorem}\n\\begin{proof}\nIndeed, it is easy to see that $R_T$ is a tree realizable by $\\bar T$. Therefore, $T$ is unique by Theorem \\ref{minimaltree}. This proves the first statement.\n\nTo prove the second statement, it is enough to verify that the median graph $T_M$ of $\\mathcal{E}(T, \\omega)$ has no lenses. Indeed, each strand of $T_M$ corresponds to a simple path between nodes in the original graph, see the Figure \\ref{fig:treepath}. Since $T$ has no inner vertices of degree $2$ these paths can intersect at most once. Therefore, $T_M$ has no lenses and $\\mathcal{E}(T, \\omega)$ is minimal.\n\\end{proof}\n \\begin{figure}[h!]\n\\center\n\\includegraphics[width=100mm]{pathtree.jpg}\n\\caption{On the left: A path which corresponds to a strand; On the right: the stands around a node of the degree $2$}\n\\label{fig:treepath}\n\\end{figure}\nWe propose the following algorithm for reconstruction of the minimal tree for a given tree metric $D_T$ based on Theorem \\ref{tomin}:\n\\begin{itemize}\n \\item Do all steps of the algorithm described above to obtain a minimal network $\\mathcal E$ such that $R_{\\mathcal E}=D_T$;\n \\item Transform $\\mathcal E$ to a minimal tree by a sequence of the star-triangle transformation. \n\\end{itemize}\nHeuristically, by (\\cite{Bu}, Theorem $1$), we assume that the second step can be performed monotonically, converting all triangles into stars.\n\\begin{remark}\n A more advanced technique called the chamber ansatz when applied to $\\Omega_{ R}(\\mathcal E)$ gives an algorithm for recovering not only the topology of the network but the weights of the edges as well. This method is described in \\cite{Ka}. \n\\end{remark}\n\nWe will illustrate our algorithm by an example.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=1.2\\textwidth]{rectop.jpg}\n \\caption{ An example of a reconstruction of a network topology}\n \\label{fig:recon}\n\\end{figure}\n\\begin{example} \\label{ex:rec-ex}\n Consider a dissimilarity matrix\n \\[D=\n\\begin{pmatrix}\n0& 3 & 3& 2 \\\\\n3& 0& 2& 3\\\\\n3& 2& 0& 3 \\\\\n2&3&3&0\n\\end{pmatrix}\n\\]\nThen the matrix $\\Omega'_{ D}$ has the form\n\\[\\Omega'_D=\n\\begin{pmatrix}\n1& 3 & 1& 1 & 0 & -1& 0 &1 \\\\\n0& 1 & 1& 2 & 1 & 1& 0 &0\\\\\n0& -1 & 0& 1 & 1 & 3& 1 &1 \n\\end{pmatrix}\n\\]\n\\end{example}\nand satisfies the Theorem \\ref{th: dual}, therefore $\\Omega'_{ D}=\\Omega'_R(\\mathcal E)$ for a network $\\mathcal E \\in E_4.$\n\n\nBy direct computations we verify that $g(\\mathcal E)=\n[4\\,6\\,5\\,7\\,8\\,2\\,1\\,3]$ in the one window notation as it is used in \\cite{L}, therefore the strand permutation $\\tau(\\mathcal E)$ is $\\tau(\\mathcal E)=(15)(27)(36)(48).$ This strand permutation defines a minimal network as it is shown in the Figure \\ref{fig:recon}, which can be transformed to a tree by applying one star-triangle transformation. \n\\\n\n\n\n\\begin{thebibliography}{9999999}\n\\bibitem{BD} H-J. 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George and D. E. Speyer, Electrical networks and Lagrangian Grassmannians, https://arxiv.org/abs/2106.15418 (2021).\n\\bibitem{CR} J. C. Culberson, P. Rudnicki, A fast algorithm for constructing trees from distance matrices. Information Processing Letters, Vol. 30, No. 4, pp. 215-220 (1989).\n\\bibitem{CIM} E. B. Curtis, D. Ingerman, J. A. Morrow, Circular planar graphs and resistor networks. Linear algebra and its applications, Vol.283, No. 1-3, pp.115-150 (1998).\n\\bibitem{CIW} B. Curtis, J. A. Morrow, Inverse problems for electrical networks. World Scientific. Vol.13 (2000).\n\n\\bibitem{GandC} M. Deza, M. Laurent, Geometry of Cuts and Metrics, Springer Berlin, DOI https://doi.org/10.1007/978-3-642-04295-9.\n\\bibitem{CdV} Colin de Verdi`ere, Yves réseaux électriques planaires. I. Comment. Math. Helv. 69 (1994), no. 3, 351374.\n\\bibitem{DF} S. Devadoss, S. 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Particularly, we obtain that effective resistances of connected cactus network give rise to Kalmanson pseudometrics. Kalmanson pseudometric $D$ can be identified with effective resistances of a connected cactus networks if and only if:\n\\begin{itemize}\n \\item $\\Omega_D \\in Gr_{\\geq 0}(n-1, 2n);$\n \\item the circular minors of the matrix $M(D)$ are non-negative after multiplying by $(-1)^k$.\n\\end{itemize}\nAnalyzing it we conclude that for Kalmanson metrics the non-degenerative conditions in Theorem \\ref{th: dual} and Theorem \\ref{th dual2} automatically satisfy due to $d_{lm}>0 \\ \\forall l\\neq m. $ \n\\end{remark}\n\n\n\n"}}},{"rowIdx":288,"cells":{"id":{"kind":"string","value":"2412.20562v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20562v1"},"title":{"kind":"string","value":"Some Necessary and Sufficient Conditions for Diophantine Graphs"},"full_text":{"kind":"string","value":"\\documentclass{article}\n\n\n\\usepackage{amssymb,latexsym,amsmath,amsthm,amsfonts,graphics}\n\\usepackage{graphicx}\n\\graphicspath{ {Figures/} }\n\n\\usepackage{caption}\n\\usepackage{subcaption}\n\\usepackage[rightcaption]{sidecap}\n\n\\usepackage{color}\n\\usepackage{lineno}\n\\usepackage{multirow}\n\\usepackage{epstopdf}\n\\usepackage{rotating}\n\\usepackage{cite}\n\n\n\\usepackage[a4paper, total={6.8in, 9in}]{geometry}\n\\usepackage{hyperref}\n\\usepackage{tikz}\n\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{cor}{Corollary}[section]\n\\newtheorem{lem}{Lemma}[section]\n\\newtheorem{prop}{Proposition}[section]\n\\newtheorem{dfn}{Definition}[section]\n\\newtheorem{ex}{Example}[section]\n\\newtheorem{conj}{Conjecture}[section]\n\\newtheorem{rem}{Remark}[section]\n\\setcounter{MaxMatrixCols}{10}\n\n\\newcommand{\\marginlabel}[1]{\\mbox{}\\marginpar{\\raggedleft\\hspace{0pt}#1}}\n\\newcommand{\\h}{\\mbox{$\\cal H$}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\Complex}{\\mathbb{C}}\n\\newcommand{\\Field}{\\mathbb{F}}\n\\newcommand{\\RPlus}{\\Real^{+}}\n\\captionsetup[figure]{name={Figure},labelsep=period}\n\\captionsetup[table]{name={Table},labelsep=period}\n\n\\makeatletter\n\\def\\ps@pprintTitle{ \\let\\@oddhead\\@empty\n \\let\\@evenhead\\@empty\n \\def\\@oddfoot{\\centerline{\\thepage}} \\let\\@evenfoot\\@oddfoot}\n\\makeatother\n\n\\begin{document}\n\n\n\\begin{center}\n{\\bf {\\Large Some Necessary and Sufficient Conditions for Diophantine Graphs}}\\\\\n\\end{center}\n\n\n\\begin{center}\n{ \\bf M. A. Seoud*$^3$, \\ A. Elsonbaty*$^2$, \\ A. Nasr*$^1$, \\ M. Anwar*$^4$}\n\\vspace{3mm}\\\\\n *Department of Mathematics, Faculty of Science, Ain Shams University, 11566, Abbassia, Cairo, Egypt.\n\\vspace{3mm}\\\\\ne-mails: $^1$ \\ \\href{mailto:amr_fatouh@sci.asu.edu.eg}{\\url{amr_fatouh@sci.asu.edu.eg}}, $^2$ \\ \\href{mailto:ahmedelsonbaty@sci.asu.edu.eg}{\\url{ahmedelsonbaty@sci.asu.edu.eg}},\\\\\n \\hspace{0.9cm}$^3$ \\ \\href{mailto:m.a.seoud@sci.asu.edu.eg}{\\url{m.a.seoud@sci.asu.edu.eg}},\\hspace{0.2cm} $^4$ \\ \\href{mailto:mohamedanwar@sci.asu.edu.eg}{\\url{mohamedanwar@sci.asu.edu.eg}},\n\\end{center}\n\n\\begin{center}\n MSC code: 05A10, 05C07, 05C78, 11A05, 11A25, 11B75, 11D04, 11D88.\n\\end{center}\n\n\n\n\n\n\\begin{abstract}\nA linear Diophantine equation $ax+by=n$ is solvable if and only if $\\gcd(a,b)$ divides $n$. A graph $G$ of order $n$ is called Diophantine if there exists a labeling function $f$ of vertices such that $\\gcd(f(u),f(v))$ divides $n$ for every two adjacent vertices $u,v$ in $G$. In this work, maximal Diophantine graphs on $n$ vertices, $D_n$, are defined, studied and generalized. The independence number, the number of vertices with full degree and the clique number of $D_n$ are computed. Each of these quantities is the basis of a necessary condition for the existence of such a labeling.\n\\end{abstract}\n\n\n\\begin{flushleft}\n\\textbf{Keywords}: Diophantine graph, Maximal Diophantine graph, labeling isomorphism, $\\gamma$-labeled graph.\n\\end{flushleft}\n\n\\section{Introduction}\n\n\\hspace{0.5cm} Assuming that a graph $G=(V, E)$ is a finite simple undirected graph with $|V|$ vertices and $|E|$ edges, where $V=V(G)$ is the vertex set, $E=E(G)$ is the edge set, $|V|$ is called the order of the graph $G$ and $|E|$ is called the size of the graph $G$. In general, $|X|$ denotes the cardinality of a set $X$. $\\delta(G)$ denotes the minimum degree of the vertices in a graph $G$. A set of vertices $S$ of a graph $G$ is said to be an independent set or a free set if for all $u,v\\in S$, $u,v$ are nonadjacent in $G$. The independence number, denoted by $\\alpha(G)$, is the maximum order of an independent set of vertices of a graph $G$. The operation of adding an edge $e=uv$ to a graph $G$ joining the vertices $u,v$ yields a new graph with the same vertex set $V(G)$ and edge set $E(G)\\cup\\{uv\\}$, which is denoted $G+\\{uv\\}$. The operation of deleting an edge $e=uv$ from a graph $G$ removes only that edge, the resulting graph is denoted $G-\\{uv\\}$. A spanning subgraph of a graph $G$ is a subgraph of $G$ obtained by deleting edges only, adding edges to a graph $G$ yields a spanning supergraph of $G$. The join of two graphs $G$ and $H$ is denoted by $G+H$, it has the following vertex set $V(G+H)= V(G)\\cup V(H)$ and edge set $E(G+H)=E(G)\\cup E(H)\\cup\\{uv: u\\in V(G) \\ \\mbox{and} \\ v\\in V(H)\\}$. $K_n,\\overline{K_n}$ and $C_n$ denote the complete graph, the null graph and the cycle graph of order $n$ respectively. We follow terminology and notations in graph theory as in A. Bickle \\cite{Bickle}, J. L. Gross; J. Yellen; P. Zhang \\cite{G-Y-Z}, F. Harary \\cite{Harary} and K. H. Rosen \\cite{Rosen2}.\n\nThe concept of prime labeling was introduced by R. Entringer and was discussed in a paper by A. Tout \\cite{Tout}. A graph $G$ is called a prime graph if there exists a bijective map $f:V\\rightarrow \\{1, 2, \\dots, n\\}$ such that for all $uv\\in E$, $(f(u),f(v))=1$. Some authors investigated algorithms for prime labeling in \\cite{sonbaty} and necessary and sufficient conditions are studied in \\cite{Seoud1}, \\cite{Seoud-Y}. The notion of Diophantine labeling is an extension of that of prime labeling. In this paper, we give a brief summary of some definitions and some results pertaining to Diophantine graphs. A generalization encompassing prime graphs, Diophantine graphs and another type of graph labeling is introduced and discussed. In maximal Diophantine graphs, an arithmetic function is established to calculate the number of vertices with full degree and the order of the maximal clique or the maximal complete subgraph, the independence number is computed and necessary and sufficient conditions are provided with these bounds. Moreover, an explicit formula for a vertex with minimum degree and minimum label is proved. Furthermore, a new perspective on degree sequences for establishing necessary conditions is presented. Relevant definitions and notations from number theory are mentioned. We follow the basic definitions and notations of number theory as in T. M. Apostol \\cite{Apostol} and D. Burton \\cite{Burton}.\n\n\n\nThis manuscript is structured as follows. Section 2 provides some results of $\\gamma$-labelings. Section 3 is partitioned into three subsections, each presents some results related to maximal Diophantine graphs. Subsection 3.1 discusses some basic bounds and necessary and sufficient conditions for maximal Diophantine graphs. Subsection 3.2 and 3.3 provided some necessary conditions and explore properties of the minimum degree and the degree sequence in maximal Diophantine graphs. Section 4 includes some examples of non-Diophantine graphs to explain the relation among these necessary conditions.\n\n\\begin{dfn}\\label{dfn2}\\cite{Nasr}\n Let $G$ be a graph with $n$ vertices. The graph $G$ is called a Diophantine graph if there exists a bijective map $f:V\\rightarrow \\{1, 2, \\dots, n\\}$ such that for all $uv\\in E$, $(f(u),f(v))\\mid n$. Such a map $f$ is called a Diophantine labeling of $G$. A maximal Diophantine graph with $n$ vertices, denoted by $(D_n,f)$, is a Diophantine graph such that adding any new edge yields a non-Diophantine graph. If there is no ambiguity, we drop $f$ from $(D_n,f)$ and write it simply $D_n$.\n\\end{dfn}\nClearly, if a graph $G$ is Diophantine, then $|E(G)|\\leq|E(D_n)|$. A formula that computes the number of edges of $D_n$ can be found in \\cite{Nasr}. Some maximal Diophantine graphs are given in the next example.\n\n\n\\begin{ex}\nThe following three graphs are examples of maximal Diophantine graphs.\n\\begin{figure*}[h!]\n\\centering\n\\begin{subfigure}{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}\n [scale=.6,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v9) at (0,4) {$9$};\n \\node (v1) at (3,2.5) {$1$};\n \\node (v7) at (3.7,0) {$7$};\n \\node (v5) at (-3,2.5) {$5$};\n \\node (v3) at (-3.7,0) {$3$};\n\n \\node (v2)[circle,fill=red!20] at (-3,-2.5) {$2$};\n \\node (v4)[circle,fill=red!20] at (-1,-3) {$4$};\n \\node (v6)[circle,fill=red!20] at (1,-3) {$6$};\n \\node (v8)[circle,fill=red!20] at (3,-2.5) {$8$};\n \\draw (v1) -- (v2);\n \\draw (v1) -- (v3);\n \\draw (v1) -- (v4);\n \\draw (v1) -- (v5);\n \\draw (v1) -- (v6);\n \\draw (v1) -- (v7);\n \\draw (v1) -- (v8);\n \\draw (v1) -- (v9);\n\n \\draw (v3) -- (v2);\n \\draw (v3) -- (v4);\n \\draw (v3) -- (v5);\n \\draw (v3) -- (v6);\n \\draw (v3) -- (v7);\n \\draw (v3) -- (v8);\n \\draw (v3) -- (v9);\n\n \\draw (v5) -- (v2);\n \\draw (v5) -- (v4);\n \\draw (v5) -- (v6);\n \\draw (v5) -- (v7);\n \\draw (v5) -- (v8);\n \\draw (v5) -- (v9);\n\n \\draw (v7) -- (v2);\n \\draw (v7) -- (v4);\n \\draw (v7) -- (v6);\n \\draw (v7) -- (v8);\n \\draw (v7) -- (v9);\n\n \\draw (v9) -- (v2);\n \\draw (v9) -- (v4);\n \\draw (v9) -- (v6);\n \\draw (v9) -- (v8);\n \\end{tikzpicture}\\caption{Graph $D_9$}\n \\end{subfigure}\n~~~\n\\begin{subfigure}{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}\n [scale=.6,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v4) at (3.5,0) {$4$};\n\n \\node (v1) at (3.7,2) {$1$};\n \\node (v2) at (2.5,4) {$2$};\n \\node (v10) at (0,4.9) {$10$};\n \\node (v7) at (-2.5,4) {$7$};\n \\node (v5) at (-3.7,2) {$5$};\n\n \\node (v8) at (-3.5,0) {$8$};\n\n \\node (v3)[circle,fill=red!20] at (0,-2.5) {$3$};\n \\node (v6)[circle,fill=red!20] at (-2,-2) {$6$};\n \\node (v9)[circle,fill=red!20] at (2,-2) {$9$};\n \\draw (v1) -- (v2);\n \\draw (v1) -- (v3);\n \\draw (v1) -- (v4);\n \\draw (v1) -- (v5);\n \\draw (v1) -- (v6);\n \\draw (v1) -- (v7);\n \\draw (v1) -- (v8);\n \\draw (v1) -- (v9);\n \\draw (v1) -- (v10);\n\n \\draw (v5) -- (v2);\n \\draw (v5) -- (v3);\n \\draw (v5) -- (v4);\n \\draw (v5) -- (v6);\n \\draw (v5) -- (v7);\n \\draw (v5) -- (v8);\n \\draw (v5) -- (v9);\n \\draw (v5) -- (v10);\n\n \\draw (v7) -- (v2);\n \\draw (v7) -- (v3);\n \\draw (v7) -- (v4);\n \\draw (v7) -- (v6);\n \\draw (v7) -- (v8);\n \\draw (v7) -- (v9);\n \\draw (v7) -- (v10);\n\n \\draw (v2) -- (v3);\n \\draw (v2) -- (v4);\n \\draw (v2) -- (v6);\n \\draw (v2) -- (v8);\n \\draw (v2) -- (v9);\n \\draw (v2) -- (v10);\n\n \\draw (v10) -- (v3);\n \\draw (v10) -- (v4);\n \\draw (v10) -- (v6);\n \\draw (v10) -- (v8);\n \\draw (v10) -- (v9);\n\n \\draw (v4) -- (v3);\n \\draw (v4) -- (v6);\n \\draw (v4) -- (v9);\n\n \\draw (v8) -- (v3);\n \\draw (v8) -- (v6);\n \\draw (v8) -- (v9);\n \\end{tikzpicture}\\caption{Graph $D_{10}$}\n \\end{subfigure}\n~~\n \\begin{subfigure}{0.25\\textwidth}\n \\centering\n \\begin{tikzpicture}\n [scale=.6,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v9) at (3.7,0) {$9$};\n \\node (v1) at (3,2.5) {$1$};\n \\node (v11) at (1.5,4) {$11$};\n \\node (v7) at (-1.5,4) {$7$};\n \\node (v5) at (-3,2.5) {$5$};\n \\node (v3) at (-3.7,0) {$3$};\n\n \\node (v2)[circle,fill=red!20] at (-3,-2.5) {$2$};\n \\node (v4)[circle,fill=red!20] at (-1.5,-3) {$4$};\n \\node (v6)[circle,fill=red!20] at (0,-3.5) {$6$};\n \\node (v8)[circle,fill=red!20] at (1.5,-3) {$8$};\n \\node (v10)[circle,fill=red!20] at (3,-2.5) {$10$};\n \\draw (v1) -- (v2);\n \\draw (v1) -- (v3);\n \\draw (v1) -- (v4);\n \\draw (v1) -- (v5);\n \\draw (v1) -- (v6);\n \\draw (v1) -- (v7);\n \\draw (v1) -- (v8);\n \\draw (v1) -- (v9);\n \\draw (v1) -- (v10);\n \\draw (v1) -- (v11);\n\n \\draw (v11) -- (v2);\n \\draw (v11) -- (v3);\n \\draw (v11) -- (v4);\n \\draw (v11) -- (v5);\n \\draw (v11) -- (v6);\n \\draw (v11) -- (v7);\n \\draw (v11) -- (v8);\n \\draw (v11) -- (v9);\n \\draw (v11) -- (v10);\n\n \\draw (v7) -- (v2);\n \\draw (v7) -- (v3);\n \\draw (v7) -- (v4);\n \\draw (v7) -- (v5);\n \\draw (v7) -- (v6);\n \\draw (v7) -- (v8);\n \\draw (v7) -- (v9);\n \\draw (v7) -- (v10);\n\n \\draw (v5) -- (v2);\n \\draw (v5) -- (v3);\n \\draw (v5) -- (v4);\n \\draw (v5) -- (v6);\n \\draw (v5) -- (v8);\n \\draw (v5) -- (v9);\n\n \\draw (v3) -- (v2);\n \\draw (v3) -- (v4);\n \\draw (v3) -- (v8);\n \\draw (v3) -- (v10);\n\n \\draw (v9) -- (v2);\n \\draw (v9) -- (v4);\n \\draw (v9) -- (v8);\n \\draw (v9) -- (v10);\n \\end{tikzpicture} \\caption{Graph $D_{11}$}\n \\end{subfigure}\\caption{Some maximal Diophantine graphs $D_9$, $D_{10}$ and $D_{11}$}\\label{figure0}\n\\end{figure*}\n\\end{ex}\n\n\\begin{dfn}\\cite{Nasr}\n For a given an integer $n\\in \\Z^+$ and a prime $p\\in \\mathbb{P}$, the successor of the $p$-adic valuation is denoted by $\\acute{v}_p(n):=v_p(n)+1$, where $v_p(n)$ is the $p$-adic valuation, $ \\Z^+$ is set of positive integers and $\\mathbb{P}$ is the set of prime numbers. The number $p^{\\acute{v}_p(n)}$ is called the critical prime power number with respect to $p,n$.\n\\end{dfn}\n\nIn the rest of this paper, the following arithmetic functions $\\pi,\\omega$ and $\\tau$ will be used, (see \\cite{Apostol}, \\cite{Burton}): Let $n\\in \\Z^+$.\n\\begin{equation*}\n \\pi(n):=\\big|\\{p\\in\\mathbb{P}: 2\\leq p\\leq n\\}\\big|, \\quad \\omega(n):=\\big|\\{p\\in\\mathbb{P}: p\\mid n, \\ 2\\leq p\\leq n\\}\\big|,\n \\quad\\tau(n):=\\big|\\{d\\in \\Z^+ : d\\mid n\\}\\big|.\n\\end{equation*}\n\n\\begin{lem}\\label{lem1}\\cite{Nasr}\n Suppose $D_n$ is a maximal Diophantine graph of order $n$. For every $u,v\\in V(D_n)$, $uv\\notin E(D_n)$ if and only if there exists $p\\in\\mathbb{P}$ such that \n $$f(u), f(v)\\in M_{p^{\\acute{v}_{p}(n)}}:=\\left\\{kp^{\\acute{v}_{p}(n)}: \\ k=1,2,\\dots,\\left\\lfloor\\frac{n}{p^{\\acute{v}_{p}(n)}}\\right\\rfloor\\right\\}.$$\n\\end{lem}\n\n\\begin{thm}\\label{lem2}\\cite{Nasr}\n Suppose $D_n$ is a maximal Diophantine graph of order $n$. For every $u\\in V(D_n)$,\n $$\\deg(u)=n-1\\quad\\mbox{if and only if}\\quad f(u)\\mid n\\quad\\mbox{\\textbf{or}}\\quad \\frac{n}{2}<f(u)=p^{\\acute{v}_p(n)}<n,$$\n where $p\\in\\mathbb{P}$ and the exclusive \\textbf{or} will be typed in bold while the inclusive or is as usual.\n\\end{thm}\n\n\nThe reduced label $f^*(u)$ of a vertex $u$ in a labeled graph $G$ with $n$ vertices is defined as $f^*(u):=\\frac{f(u)}{(f(u), n)}.$\n\n \n\n\\begin{lem}\\label{lem3}\\cite{Nasr}\n Suppose $D_n$ is a maximal Diophantine graph of order $n$ and $u,v\\in V(D_n)$. If $f(u)\\mid f(v)$, then $N(u)\\supseteq N(v)$, where $N(s)$ defines the neighborhood of $s$ as the set of all vertices in $D_n$ that join the vertex $s$.\n\\end{lem}\n\n\\begin{thm}\\label{thm_eq-deq2}\\cite{Nasr}\n Suppose $D_n$ is a maximal Diophantine graph of order $n$. Let $u,v\\in V(D_n)$ such that $f(u)\\mid f(v)$, $f(v)$ is not a prime power number and $f^*(u)>1$. If $\\deg(u)=\\deg(v)$, then $f^*(u),f^*(v)$ have the same prime factors.\n\\end{thm}\n\n\\begin{cor}\\label{cor1}\\cite{Nasr}\n Suppose $D_n$ is a maximal Diophantine graph of order $n$ and $u,v\\in V(D_n)$ such that $f(v)=tf(u)$ for some $t\\geq1$. If $t\\mid n$ and $(t, f(u))=1$, then $\\deg(u)=\\deg(v)$.\n\\end{cor}\n\n\n\\section{$\\gamma$-Labelings of Graphs }\n\n\\hspace{0.cm}The following definition is a generalization of Definition \\ref{dfn2}.\n\\begin{dfn}\\label{dfn3}\n Let $G$ be a graph with $n$ vertices. The graph $G$ is called an $\\gamma$-labeled graph if there exists a bijective map $f:V\\rightarrow \\{x_1, x_2, \\dots, x_n\\}$ such that $f(u),f(v)$ satisfy some conditions, where $\\{x_1, x_2, \\dots, x_n\\}$ is any set of $n$ elements. Such a map $f$ is called an $\\gamma$-labeling. A maximal $\\gamma$-labeled graph with $n$ vertices, denoted by $(\\Gamma_n,f)$, is a $\\gamma$-labeled graph in which for all $uv\\notin E(\\Gamma_n)$, $\\Gamma_n+\\{uv\\}$ is not a $\\gamma$-labeled graph.\n\\end{dfn}\n\nThe reader should not be confused the notion of $\\gamma$-labeling as provided in Definition \\ref{dfn3} with the concept of $\\alpha$-valuation that presented in the seminal work of A. Rosa \\cite{Rosa}.\n\n\n\\begin{dfn}\\cite{S-C-L}\n Let $(G_1,f_1),(G_2,f_2)$ be two labeled graphs, where $f_1:V(G_1)\\rightarrow \\{x_1, x_2, \\dots, x_n\\}$ and $f_2:V(G_2)\\rightarrow \\{x_1, x_2, \\dots, x_n\\}$ are two bijective maps. The labeled graphs $(G_1,f_1),(G_2,f_2)$ are said to be labeling isomorphic, denoted by $(G_1,f_1)\\cong_l (G_2,f_2)$, if there exists a bijective map $\\varphi:V(G_1)\\rightarrow V(G_2)$ such that for all $u,v\\in V(G_1)$, $uv\\in E(G_1)$ if and only if $\\varphi(u)\\varphi(v)\\in E(G_2)$ and $f_1(u)=\\big(f_2\\circ\\varphi\\big)(u).$\n\\end{dfn}\n\n\\begin{thm}\\label{thm-equivalance}\n A maximal $\\gamma$-labeled graph $\\Gamma_n$ is unique up to labeling isomorphism.\n\\end{thm}\n\\begin{proof}\n Suppose $(\\Gamma_n,f_1)$ and $(\\acute{\\Gamma}_n,f_2)$ are two maximal $\\gamma$-labeled graphs of order $n$, where the two maps \n $$f_1:V(\\Gamma_n)\\rightarrow \\{x_1, x_2, \\dots, x_n\\}\\quad \\mbox{and}\\quad f_2:V(\\acute{\\Gamma}_n)\\rightarrow \\{x_1, x_2, \\dots, x_n\\}$$\n are $\\gamma$-labelings of $\\Gamma_n$ and $\\acute{\\Gamma}_n$ satisfying certain conditions, say condition $C$. Define a map \n $$\\varphi:V(\\Gamma_n)\\rightarrow V(\\acute{\\Gamma}_n)\\quad \\mbox{by}\\quad \\varphi(u)=f_2^{-1}(f_1(u)).$$ \n Therefore, $\\varphi$ is one to one (for let $u,v\\in V(\\Gamma_n)$, $\\varphi(u)=\\varphi(v)$. Then we obtain $f_2^{-1}(f_1(u))=f_2^{-1}(f_1(v))$; accordingly, $f_1(u)=f_1(v)$. Consequently, $u=v$), $\\varphi$ is onto (since $\\varphi$ is one to one and $|V(\\Gamma_n)|=|V(\\acute{\\Gamma}_n)|=n$), $\\varphi$ is preserving the adjacency and non-adjacency of $\\Gamma_n$ and $\\acute{\\Gamma}_n$ (for the reason that let $u,v\\in V(\\Gamma_n)$ such that $uv\\in E(\\Gamma_n)$. Then we have the two labels $f_1(u),f_1(v)$ satisfy $C$. Since, $f_1(u)=f_2(\\varphi(u))$ and $f_1(v)=f_2(\\varphi(v))$ (see Figure \\ref{fig.}), we get $f_2(\\varphi(u)),f_2(\\varphi(v))$ satisfy $C$. Consequently, $\\varphi(u)\\varphi(v)\\in E(\\acute{\\Gamma}_n)$ and the converse is similar) and let $u\\in V(\\Gamma_n)$, $\\varphi(u)=f_2^{-1}(f_1(u))$. Therefore, $f_1(u)=f_2(\\varphi(u))=(f_2\\circ\\varphi)(u)$. Hence, the two graphs $(\\Gamma_n,f_1)$ and $(\\acute{\\Gamma}_n,f_2)$ are labeling isomorphic.\n\\end{proof}\n\n\\begin{figure*}[h!]\n\\centering\n \\begin{tikzpicture}\n [scale=.8,auto=center]\n \\node (v) at (0,1.33) {$\\equiv$};\n \\node (v1) at (0,0) {$\\{x_1, x_2, \\dots, x_n\\}$};\n \\node (v2) at (-2,2) {$V(\\Gamma_n)$};\n \\node (v3) at (2,2) {$V(\\acute{\\Gamma}_n)$};\n \\path[->] (v2)edge [align=left, below] node {$f_1$} (v1);\n \\path[->] (v3)edge [align=left, below] node {$f_2$} (v1);\n \\path[->] (v2)edge [align=left, above] node {$\\varphi$} (v3);\n \\end{tikzpicture}\n \\caption{$(\\Gamma_n,f_1)\\cong_l (\\acute{\\Gamma}_n,f_2)$}\\label{fig.}\n\\end{figure*}\n\n\\begin{cor}\\label{thm-equivalance1}\n The graphs $D_n$ are unique up to labeling isomorphism.\n\\end{cor}\n\n\\begin{thm}\n Suppose $G$ is a graph with order $n$ and $\\Gamma_n$ is the maximal $\\gamma$-labeled graph with order $n$. $G$ is an $\\gamma$-labeled graph if and only if $G$ is labeling isomorphic to a spanning subgraph of $\\Gamma_n$.\n\\end{thm}\n\\begin{proof}\n Suppose $\\Gamma_n$ is the maximal $\\gamma$-labeled graph with order $n$ and a graph $G$ is a $\\gamma$-labeled graph with order $n$. Then there exists $f:V(G)\\rightarrow \\{x_1, x_2, \\dots, x_n\\}$ is a bijective map such that $f(u),f(v)$ satisfy certain conditions, say condition $C$ and define \n $$T:=\\{uv:uv\\notin E(G) \\ \\mbox{and} \\ f(u),f(v) \\ \\mbox{satisfy} \\ C\\}.$$\n Consequently, the spanning supergraph $G+T$ of $G$ is a $\\gamma$-labeled graph of order $n$ and the set $E(G)\\cup T$ is set of all edges such that $f(u),f(v)$ satisfy $C$. Let $\\acute{u}\\acute{v}\\notin E(G)\\cup T$. Then we have that the two labels $f(\\acute{u}),f(\\acute{v})$ do not satisfy $C$. Therefore, the spanning supergraph $G+(T\\cup\\{\\acute{u}\\acute{v}\\})$ of $G$ is not a $\\gamma$-labeled graph with a $\\gamma$-labeling satisfy $C$. Consequently, $G+T$ is the maximal $\\gamma$-labeled graph of order $n$. Thus, using Theorem \\ref{thm-equivalance}, we have that $G+T$ is labeling isomorphic to $\\Gamma_n$. Hence, the graph $G$ is labeling isomorphic to a spanning subgraph of the maximal $\\gamma$-labeled graph $\\Gamma_n$.\\\\\n\n Conversely, suppose $\\Gamma_n$ is the maximal $\\gamma$-labeled graph with order $n$ and a graph $G$ is labeling isomorphic to a spanning subgraph of the maximal $\\gamma$-labeled graph $\\Gamma_n$. Let $T$ be the set of deleted edges of $\\Gamma_n$ such that the graph $G$ is labeling isomorphic to $\\Gamma_n-T$. Then we have\n $$|V(G)|=|V(\\Gamma_n-T)|=|V(\\Gamma_n)| \\quad \\mbox{and} \\quad V(\\Gamma_n)=V(\\Gamma_n-T).$$\n Therefore, using the same $\\gamma$-labeling of $\\Gamma_n$, we have $\\Gamma_n-T$ is a $\\gamma$-labeled graph. Since the graph $G$ is labeling isomorphic to $\\Gamma_n-T$, hence the graph $G$ is a $\\gamma$-labeled graph.\n\\end{proof}\n\n\\begin{cor}\\label{spanning-thm}\n A graph $G$ of order $n$ is Diophantine if and only if $G$ is labeling isomorphic to a spanning subgraph of $D_n$.\n\\end{cor}\n\n\n\n\n\\section{Basic Bounds of the Maximal Diophantine Graphs $D_n$}\n\n\\subsection{Some Necessary and Sufficient Conditions for $D_n$ }\n\n\\hspace{0.5cm} In what follows, let $(D_n,f)$ denote the maximal Diophantine graph of order $n$, with Diophantine labeling $f$ and $F(G)$ denote the number of full degree vertices of a graph $G$. The next two theorems present two different methods that compute the quantity $F(D_n)$. \n\\begin{thm}\\label{fulldegree2}\n If $p_i^{\\acute{v}_{p_i}(n)}<\\frac{n}{2}$, $i=1, 2, \\dots, r$, then the number of full degree vertices in $D_n$ is given by\n\\begin{equation*}\n F(D_n) =n-\\sum_{1\\leq i\\leq r}\\left\\lfloor\\frac{n}{p_i^{\\acute{v}_{p_i}(n)}}\\right\\rfloor\n +\\sum_{1\\leq i<j\\leq r}\\left\\lfloor\\frac{n}{p_i^{\\acute{v}_{p_i}(n)}p_j^{\\acute{v}_{p_j}(n)}}\\right\\rfloor\n -\\dots +(-1)^{r}\\left\\lfloor\\frac{n}{\\prod\\limits_{1\\leq i\\leq r}p_i^{\\acute{v}_{p_i}(n)}}\\right\\rfloor,\n\\end{equation*}\n where $p_1, p_2, \\dots, p_r$ are distinct prime numbers.\n\\end{thm}\nThe proof of Theorem \\ref{fulldegree2} is straightforward by applying Lemma \\ref{lem1}, Theorem \\ref{lem2} and the inclusion-exclusion principle (see \\cite{Rosen2}). For a very large $n\\in \\Z^+$, the above formula does not provide efficient upper and lower bounds for the quantity $F(D_n)$. There is an alternative approach to determine the quantity $F(D_n)$ by using the following arithmetic function\n$$\\gamma_x(n):=\\left|\\left\\{p^{\\acute{v}_p(n)}: p\\mid n, \\ x<p^{\\acute{v}_p(n)}<n, \\ p\\in\\mathbb{P}\\right\\}\\right|,$$\nwhere $n\\in \\Z^+$ and a positive real number $x<n$. This function is utilized for computing not only the number of vertices with full degree in $D_n$ but also the order of the maximal clique of $D_n$ as follows in Theorems \\ref{fulldegree}, \\ref{complete_subgraph}. Obviously, for every $n\\in \\Z^+$, $\\gamma_1(n)\\leq\\omega(n)$, for every $p\\in\\mathbb{P}$, $k\\in \\Z^+$ and a positive real number $x<n$, $\\gamma_x\\left(p^k\\right)=0$ and also, for every $n,m\\in\\Z^+$ with $m<n$, $\\gamma_m(n)=\\gamma_1(n)-\\gamma_1(m)$. \n\n\\begin{thm} \\label{fulldegree}\n The number of vertices with full degree in $D_n$ is given by\n\\begin{equation*}\nF(D_n)=\\tau(n) + \\pi(n-1)-\\pi\\left(\\frac{n}{2}\\right) + \\gamma_{\\frac{n}{2}}(n).\n\\end{equation*}\nIn particular, if $n$ is a prime number, we have \n$$F(D_n)=\\pi(n)-\\pi\\left(\\frac{n}{2}\\right) +1.$$\n\\end{thm}\n\\begin{proof}\n Let $D_n$ be the maximal Diophantine graph with order $n$. Define the following three sets\n\\begin{equation*}\n S_1:=\\{d\\in \\Z^+ : d\\mid n\\}, \\quad\n S_2:=\\left\\{p\\in\\mathbb{P}: \\frac{n}{2} < p < n\\right\\}, \\quad\n S_3:=\\left\\{ p^{\\acute{v}_p(n)} : p\\mid n, \\ \\frac{n}{2}< p^{\\acute{v}_p(n)} < n, \\ p\\in\\mathbb{P} \\right\\}.\n\\end{equation*}\nConsequently, using Theorem \\ref{lem2}, one can see that $ S_1\\cup S_2\\cup S_3$ is the set of labels of the full degree vertices in $D_n.$ Clearly, $S_1,S_2$ and $S_3$ are mutually disjoint sets and\n\n $$|S_1|=\\tau(n),\\quad |S_2|=\\pi(n-1)-\\pi\\left(\\frac{n}{2}\\right)\\quad \\mbox{and}\\quad |S_3|=\\gamma_{\\frac{n}{2}}(n),$$\nand hence \n$$F(D_n)= \\tau(n) + \\pi(n-1)-\\pi\\left(\\frac{n}{2}\\right) + \\gamma_{\\frac{n}{2}}(n).$$\nIn case of $n$ is a prime number, we have $F(D_n)= \\pi(n)-\\pi\\left(\\frac{n}{2}\\right)+1$.\n\\end{proof}\n\n\n\\begin{cor}\\label{corVI2}\n Let $G$ be a graph with order $n$. If the graph $G$ is Diophantine, then $F(G)\\leq F(D_n)$.\n\\end{cor}\n\n\n\nThe clique number, denoted by $Cl(G)$, is the order of the maximal clique of a graph $G$. Although $\\omega(G)$ is the standard notation of the clique number, we have chosen $Cl(G)$ in this study to prevent confusion with the arithmetic function $\\omega(n)$. The following theorem gives the order of the maximal clique in $D_n$.\n\n\\begin{thm}\\label{complete_subgraph} \nThe clique number of $D_n$ is given by \n$$Cl(D_n)= \\tau(n) + \\pi(n) - \\omega(n) + \\gamma_1(n).$$ \nIn particular, if $n$ is a prime number, we have\n$$Cl(D_n)=\\pi(n)+1.$$\n\\end{thm}\n\\begin{proof}\n Let $D_n$ be the maximal Diophantine graph with order $n$. Define the following three sets\n\\begin{equation*}\n S_1:=\\{d\\in \\Z^+ : d\\mid n\\}, \\quad\n S_2:=\\{p\\in\\mathbb{P}: p\\nmid n, \\ 1 < p < n\\}, \\quad\n S_3:=\\left\\{p^{\\acute{v}_p(n)}: p\\mid n, \\ 1<p^{\\acute{v}_p(n)}<n, \\ p\\in\\mathbb{P}\\right\\}.\n\\end{equation*}\n Therefore, any two vertices in $V(D_n)$ that is labeled by integers from the set $S_1\\cup S_2\\cup S_3$ are adjacent, since for any two distinct labels $\\ell_1,\\ell_2$, we have\n \\begin{equation*}\n \\begin{cases}\n (\\ell_1, \\ell_2)=1, & \\mbox{if} \\ \\ell_1, \\ell_2\\in S_2\\cup S_3\\\\ \n &\\\\\n (\\ell_1, \\ell_2)\\mid n, & \\mbox{if} \\ \\ell_1\\in S_1. \\\\\n \\end{cases}\n \\end{equation*}\n Consequently, one can see that $ S_1\\cup S_2\\cup S_3$ is the set of labels of vertices that are in the maximal clique of $D_n.$\n Suppose contrary that $u\\in V(D_n)$ is a vertex $u$ of the maximal clique in $D_n$ such that $f(u)\\notin S_1\\cup S_2\\cup S_3.$ Then we have $f(u)\\nmid n$. Therefore, there exists a prime number $p_0$ such that $p_0^{\\acute{v}_{p_0}(n)}\\mid f(u)$; otherwise, for every a prime number $p$, $p^{\\acute{v}_p(n)}\\nmid f(u)$, so we get $v_p(f(u))<\\acute{v}_p(n)=v_p(n)+1$. Consequently, $v_p(f(u))\\leq v_p(n)$ which is a contradiction of $f(u)\\nmid n$. Let $\\ell=p_0^{\\acute{v}_{p_0}(n)}$ be a certain label. Then we have $\\ell\\in S_2\\cup S_3$, $\\ell\\mid f(u)$ and $\\ell\\neq f(u)$. So, $(f(u),\\ell)=\\ell\\nmid n,$ which contradicts the completeness of the maximal clique in $D_n$. Therefore, the set $S_1\\cup S_2\\cup S_3$ has all labels of vertices in the maximal clique of $D_n$. Obviously, $S_1, S_2$ and $S_3$ are mutually disjoint sets and\n $$|S_1|=\\tau(n),\\quad |S_2|=\\pi(n)-\\omega(n)\\quad \\mbox{and}\\quad |S_3|=\\gamma_1(n),$$\n we obtain\n $$Cl(D_n)=\\tau(n) + \\pi(n) - \\omega(n) + \\gamma_1(n).$$\n If $n$ is a prime number, then $Cl(D_n)=\\pi(n)+1.$\n\\end{proof}\n\n\\begin{cor} \\label{corVI3}\n Let $G$ be a graph with order $n$. If the graph $G$ is Diophantine, then $Cl(G)\\leq Cl(D_n)$.\n\\end{cor}\n\n\\begin{rem} Let $D_n$ be the maximal Diophantine graph of order $n$. Then\n\\begin{itemize}\n \\item[1.] $|E(D_n)|\\geq\\frac{1}{2}Cl(D_n)\\big(Cl(D_n)-1\\big)\\geq \\frac{1}{2}F(D_n)\\big(F(D_n)-1\\big),$\n \\item[2.] if $D_n$ is not a complete graph, then $F(D_n)\\leq\\delta(D_n)$,\n \\item[3.] for every $n\\in \\Z^+$, $F(D_n)\\leq Cl(D_n)\\leq n$.\n\\end{itemize}\n\\end{rem}\n\n\\begin{lem}\n For every a prime number $p\\leq\\frac{n}{2}$, $p\\mid n$ and $p^{\\acute{v}_p(n)}>\\frac{n}{2}$ if and only if $D_n$ is a complete graph.\n\\end{lem}\n\\begin{proof}\n Assume $p\\leq\\frac{n}{2}$ is prime number such that $p\\mid n$ and $p^{\\acute{v}_p(n)}>\\frac{n}{2}$. Suppose contrary that the maximal Diophantine graph $D_n$ is not a complete graph. Then there exist $u,v\\in V(D_n)$ such that $uv\\notin E(D_n)$. Therefore, using lemma \\ref{lem1}, there exists a prime number $p$ such that $f(u),f(v)\\in M_{p^{\\acute{v}_p(n)}}$. Let $f(u)=tp^{\\acute{v}_p(n)}$ and $f(v)=s p^{\\acute{v}_p(n)}$ for some $t,s\\geq1$ and $t<s$. Then, $p^{\\acute{v}_p(n)}<\\frac{n}{s}\\leq\\frac{n}{2},$\n this contradicts the assumption. Hence, $D_n$ is a complete graph.\\\\\n\n Conversely, let $D_n$ be a complete graph and consider contrary that there exists a prime number $p\\leq\\frac{n}{2}$ such that $p\\nmid n$ or $p^{\\acute{v}_p(n)}<\\frac{n}{2}$, otherwise, if $p^{\\acute{v}_p(n)}=\\frac{n}{2}$, then $p^{\\acute{v}_p(n)}\\mid n$ that is a contradiction. Then we have the following two cases. In case of $p\\leq\\frac{n}{2}$ and $p\\nmid n$, we obtain $2p<n$. Then we get $(p, 2p)=p\\nmid n$. Therefore, $F(D_n)<n$. In the other case of $p^{\\acute{v}_p(n)}<\\frac{n}{2}$, we have $(p^{\\acute{v}_p(n)}, 2p^{\\acute{v}_p(n)})= p^{\\acute{v}_p(n)}\\nmid n$. Therefore, $F(D_n)<n$. Consequently, from the two cases, $D_n$ is not a complete graph, this contradicts the hypothesis.\n\\end{proof}\n\n\\begin{thm}\n The independence number of $D_n$ is given by\n $$\\alpha(D_n)=\\max\\limits_{2\\leq p\\leq n}\\left\\lfloor\\frac{n}{p^{\\acute{v}_p(n)}}\\right\\rfloor,$$\n where $p\\in\\mathbb{P}$. In particular, if $n$ is odd, we have \n $$\\alpha(D_n)=\\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{thm}\n\\begin{proof}\n Let $f(u),f(v)$ be two labels, where $u,v\\in V(D_n)$. Then, using Lemma \\ref{lem1}, $uv\\notin E(D_n)$ if and only if there exists $p\\in\\mathbb{P}$ such that $f(u), f(v)\\in M_{p^{\\acute{v}_{p}(n)}}.$\n Therefore, the set of vertices of $D_n$ with labels in $M_{p^{\\acute{v}_p(n)}}$\n is an independent set.\n Hence, \n $$\\alpha(D_n)=\\max\\limits_{2\\leq p\\leq n}\\left|M_{p^{\\acute{v}_p(n)}}\\right|=\\max\\limits_{2\\leq p\\leq n}\\left\\lfloor\\frac{n}{p^{\\acute{v}_p(n)}}\\right\\rfloor.$$\n If $n$ is an odd, then the set of nonadjacent vertices in $D_n$ with labels in $M_2=\\left\\{2, 4, 6, \\dots, 2\\left\\lfloor\\frac{n}{2}\\right\\rfloor\\right\\}$ is a maximal independent set. \n Hence, $\\alpha(D_n)=\\left\\lfloor\\frac{n}{2}\\right\\rfloor$.\n\\end{proof}\n\n\\begin{cor} \\label{corVI4}\n Let $G$ be a graph with order $n$. If the graph $G$ is Diophantine, then $\\alpha(G)\\geq\\alpha(D_n)$.\n\\end{cor}\n\n\\begin{thm}(\\textbf{sufficient condition})\\label{thm_sufficient}\n Let $G$ be a graph with order $n$. If $\\alpha(G)\\geq n-F(D_n)$, then $G$ is a Diophantine graph.\n\\end{thm}\n\\begin{proof}\n Let $G$ be a graph with $n$ vertices such that $\\alpha(G)\\geq n-F(D_n)$. Suppose $S$ is a subgraph of $G$ with number of vertices less than or equal to $F(D_n)$ vertices. Then, using Theorem \\ref{lem2}, the number that is either a divisor of $n$ or is of the form $p^{\\acute{v}_{p}(n)}$, where $\\frac{n}{2}<p^{\\acute{v}_{p}(n)}<n$ can be used as labels in the vertices of the subgraph $S$ of $G$. Therefore, the other labels can be assigned in the remaining independent vertices of order $\\alpha(G)$ from the graph $G$. Hence, $G$ is a Diophantine graph.\n\\end{proof}\n\n\\begin{ex}\n The graph $G=K_3+\\overline{K_4}$ satisfies the sufficient condition in Theorem \\ref{thm_sufficient} as $\\alpha(G)\\geq 7-F(D_7)$; accordingly, the graph $G$ is a Diophantine graph as seen in Figure \\ref{figure1} part (a).\n\\begin{figure*}[h!]\n\\centering\n \\begin{subfigure}{0.4\\textwidth}\n \\centering\n \\begin{tikzpicture}\n [scale=.5,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v1) at (0,3) {$1$};\n \\node (v5) at (3,4) {$5$};\n \\node (v7) at (-3,4) {$7$};\n \\node (v3)[circle,fill=red!20] at (3,0) {$3$};\n \\node (v2)[circle,fill=red!20] at (-3,0) {$2$};\n \\node (v6)[circle,fill=red!20] at (1,0) {$6$};\n \\node (v4)[circle,fill=red!20] at (-1,0) {$4$};\n \\draw (v1) -- (v2);\n \\draw (v1) -- (v3);\n \\draw (v1) -- (v4);\n \\draw (v1) -- (v5);\n \\draw (v1) -- (v6);\n \\draw (v1) -- (v7);\n\n \\draw (v5) -- (v2);\n \\draw (v5) -- (v3);\n \\draw (v5) -- (v4);\n \\draw (v5) -- (v6);\n \\draw (v5) -- (v7);\n\n \\draw (v7) -- (v2);\n \\draw (v7) -- (v3);\n \\draw (v7) -- (v4);\n \\draw (v7) -- (v6);\n \\end{tikzpicture}\\caption{The graph $G=K_3+\\overline{K_4}$} \n \\end{subfigure}\n ~~~~~~~\n \\begin{subfigure}{0.4\\textwidth}\n \\centering\n \\begin{tikzpicture}\n [scale=.5,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v1) at (1,6) {$1$};\n \\node (v5) at (-1,6) {$5$};\n \\node (v7) at (-3,4) {$7$};\n \\node (v3) at (3,4) {$3$};\n \\node (v2)[circle,fill=red!20] at (-2,2) {$2$};\n \\node (v6)[circle,fill=red!20] at (0,1.5) {$6$};\n \\node (v4)[circle,fill=red!20] at (2,2) {$4$};\n \\draw (v1) -- (v2);\n \\draw (v1) -- (v3);\n \\draw (v1) -- (v4);\n \\draw (v1) -- (v5);\n \\draw (v1) -- (v6);\n \\draw (v1) -- (v7);\n\n \\draw (v5) -- (v2);\n \\draw (v5) -- (v3);\n \\draw (v5) -- (v4);\n \\draw (v5) -- (v6);\n \\draw (v5) -- (v7);\n\n \\draw (v7) -- (v2);\n \\draw (v7) -- (v3);\n \\draw (v7) -- (v4);\n \\draw (v7) -- (v6);\n\n \\draw (v3) -- (v2);\n \\draw (v3) -- (v4);\n \\end{tikzpicture}\\caption{The maximal Diophantine graph $D_7$}\n \\end{subfigure}\\caption{}\\label{figure1} \n\\end{figure*}\n\\end{ex}\nThis sufficient condition is not a necessary condition. For instance, $D_7$ is a Diophantine graph. However, $D_7$ does not meet the sufficient condition since $\\alpha(D_7)<7-F(D_7)$.\n\\subsection{The Minimum Degree Vertex with Minimum Label}\n\n \\hspace{0.5cm} Let $V_{\\delta}(G)$ denote the set of vertices with minimum degree in $G$, i.e., $V_{\\delta}(G):=\\{u\\in V(G):\\deg(u)=\\delta(G)\\}$. So, we denote the following $f(u_0):=\\min\\{f(u): u\\in V_{\\delta}(D_n)\\},$\n in which the vertex $u_0$ is the vertex in $V_{\\delta}(D_n)$ with minimum label and $f(u_0)$ is the smallest label of a vertex in $V_{\\delta}(D_n)$.\n\n\\begin{rem} Let $u_0$ be the vertex in $V_{\\delta}(D_n)$ with minimum label.\n \\begin{itemize}\n \\item[1.] $f(u_0)=1$ if and only if $D_n$ is a complete graph.\n \\item [2.] For every $n>3$, $\\delta(D_n)\\geq 3$.\n \\end{itemize}\n\\end{rem}\n\n\\begin{thm}\n Suppose the maximal Diophantine graph $D_n$ is not a complete graph and the finite sequence \n $$p_1^{\\acute{v}_{p_{_1}}(n)}< p_2^{\\acute{v}_{p_{_2}}(n)}<\\dots< p_s^{\\acute{v}_{p_s}(n)}<\\frac{n}{2},$$ \n where $p_i$, $i = 1, 2, \\dots, s$ are distinct prime numbers. If the vertex $u_0\\in V_{\\delta}(D_n)$ has minimum label, then there exits an integer $r<s$ such that $$f(u_0)=\\prod\\limits_{i=1}^{r}p_i^{\\acute{v}_{p_i}(n)}<n\\quad \\mbox{and}\\quad p_{r+1}^{\\acute{v}_{p_{r+1}}(n)}f(u_0)>n.$$\n\\end{thm}\n\\begin{proof}\n Suppose the maximal Diophantine graph $D_n$ of order $n$ is not a complete graph. Let $u_0$ be the vertex in $V_{\\delta}(D_n)$ with minimum label. Then\n $$1<f(u_0)=\\min\\{f(u): u\\in V_{\\delta}(D_n)\\}<n.$$ Therefore, we have \n \\begin{equation}\\label{eqn8}\n f(u_0)=\\prod\\limits_{j=1}^{r_1}p_j^{\\alpha_j} \\prod\\limits_{i=1}^{r_2}q_i^{\\beta_i}<n. \n \\end{equation}\n where $p_j,q_i$ are distinct prime numbers and $\\alpha_j,\\beta_i$ are two non-negative integers such that for every $j=1,2,\\dots r_1$, $i=1,2,\\dots r_2$\n \\begin{equation}\\label{eqn12}\n \\alpha_j\\geq \\acute{v}_{p_j}(n)\\quad\\mbox{and}\\quad 0\\leq \\beta_i<\\acute{v}_{h_i}(n).\n \\end{equation}\n Clearly, the two terms of equation \\eqref{eqn8} are relatively primes and $\\prod\\limits_{i=1}^{r_2}q_i^{\\beta_i}\\mid n$. Otherwise, if $\\prod\\limits_{i=1}^{r_2}q_i^{\\beta_i}\\nmid n$, then there exists a prime number $q_i\\mid f(u_0)$ such that $q_i^{\\beta_i}\\nmid n$. Therefore, there exists $i=1,2,\\dots r_2$ such that $\\beta_i\\geq\\acute{v}_{q_i}(n)$ which contradicts equation \\eqref{eqn12}. Let $u\\in V(D_n)$ such that $f(u)=\\prod\\limits_{j=1}^{r_1}p_j^{\\alpha_j}$. Then, using equation \\eqref{eqn8}, we get\n \\begin{equation}\\label{eqn10}\n f(u_0)=f(u)\\prod\\limits_{i=1}^{r_2}q_i^{\\beta_i}. \n \\end{equation}\n Therefore, using Corollary \\ref{cor1} and equation \\eqref{eqn10},\n $$f(u)\\leq f(u_o)\\quad \\mbox{and}\\quad \\deg(u)=\\deg(u_0),$$\n This contradicts the hypothesise of the minimal label of $u_0$ unless $\\beta_i=0$ in equation \\eqref{eqn10}. Thus,\n \\begin{equation}\\label{eqn11}\n f(u_0)=f(u)=\\prod\\limits_{j=1}^{r_1}p_j^{\\alpha_j},\n \\end{equation}\n where $\\alpha_j\\geq \\acute{v}_{p_j}(n)$. Then $\\alpha_j=\\acute{v}_{p_j}(n)+k_j$ for some $k_j\\geq 0$. Consequently, using equation \\eqref{eqn11}, we get\n \\begin{equation}\\label{eqn9}\n f(u_0)=\\prod\\limits_{j=1}^{r_1}p_j^{v_{p_j}(n)+k_j}= \\prod\\limits_{j=1}^{r_1}p_j^{\\acute{v}_{q_j}(n)} \\prod\\limits_{j=1}^{r_1}p_j^{k_j}.\n \\end{equation}\n Let $v\\in V(D_n)$ such that $f(v)=\\prod\\limits_{j=1}^{r_1}p_j^{\\acute{v}_{p_j}(n)}.$ Then, using equation \\eqref{eqn9}, we obtain\n \\begin{equation}\\label{eqn1}\n f(u_0)= \\prod\\limits_{j=1}^{r_1}p_j^{k_j} f(v).\n \\end{equation}\n Thus, using Lemma \\ref{lem3}, $\\deg(v)\\leq \\deg(u_o)$. Since $u_o$ has minimum degree, we have $\\deg(v)=\\deg(u_o)$.\n Therefore,\n $$f(v)\\leq f(u_o)\\quad \\mbox{and}\\quad \\deg(v)=\\deg(u_o).$$\n Since $f(u_o)$ is the minimum label and using equation \\eqref{eqn1}, we get $k_j=0$. Consequently we have\n \\begin{equation}\\label{eqn14}\n f(u_0)=f(v)=\\prod\\limits_{j=1}^{r_1}p_j^{\\acute{v}_{p_j}(n)}<n.\n \\end{equation}\n Given a finite sequence $1<p_1^{\\acute{v}_{p_{_1}}(n)}<p_2^{\\acute{v}_{p_{_2}}(n)}<\\dots<p_s^{\\acute{v}_{p_s}(n)}<\\frac{n}{2},$ where $p_i$, $i = 1, 2, \\dots, s$ are distinct primes. Then we have\n \\begin{equation*}\n \\left|M_{p_1^{\\acute{v}_{p_{_1}}(n)}}\\right| \\geq \\left|M_{p_2^{\\acute{v}_{p_{_2}}(n)}}\\right| \\geq \\dots \\geq \\left|M_{p_s^{\\acute{v}_{p_s}(n)}}\\right|.\n \\end{equation*}\n Since $p_i^{\\acute{v}_{p_i}(n)}\\mid f(u_0)$ for some $i=1, 2, \\dots s$ and $u_0$ is the vertex in $V_{\\delta}(D_n)$ with minimum label, we have the following cases:\n \\\\\n Case i: If the prime factors of $f(u_0)$ are $i$ primes, then $f(u_0)\\in M_{p_1^{\\acute{v}_{p_{_1}}(n)}}, \\ f(u_0)\\in M_{p_2^{\\acute{v}_{p_{_2}}(n)}}, \\ \\dots, \\ f(u_0)\\in M_{p_i^{\\acute{v}_{p_i}(n)}},$\n where $i=1,2,\\dots, r$ for some $r<s$. Then, using equation \\eqref{eqn14}, we have the following formula \n $$f(u_0)=\\prod\\limits_{j=1}^{i}p_j^{\\acute{v}_{p_j}(n)}<n.$$\n where $i=1,2,\\dots, r$ and $r<s$. Suppose contrarily that for every $i=1,2,\\dots, r$ and $r<s$ such that\n $p_{i+1}^{\\acute{v}_{p_{i+1}}(n)}f(u_0)<n.$\n Let $w\\in V(D_n)$ such that $f(w)=p_{i+1}^{\\acute{v}_{p_{i+1}}(n)}f(u_0).$\n Then, using Lemma \\ref{lem3}, $\\deg(w)\\leq \\deg(u_o)$. Since the degree of $u_0$ is minimum, we get $\\deg(w)=\\deg(u_0)$. However, since\n $$f^*(u_0)=\\frac{f(u_o)}{(f(u_o),n)}=\\prod\\limits_{j=1}^{i}p_j\\quad\\mbox{and}\\quad f^*(w)=\\frac{f(w)}{(f(w),n)}=\\prod\\limits_{j=1}^{i+1}p_j,$$\n therefore the reduced labels $f^*(u_0),f^*(u)$ do not have the same prime factors which contradicts Theorem \\ref{thm_eq-deq2}. Hence, the proof follows.\n\\end{proof}\n\nClearly, one can see that $\\delta(D_n)=\\deg(u_0)$ and the degree of every vertex in $D_n$ is provided by \n\\begin{thm}\\cite{Nasr}.\\label{deg(u)}\n If $f^*(u)=\\prod\\limits_{i=1}^{r}p_i^{k_i}$, where $u\\in V(D_n)$, $p_i$, $i=1,2,\\dots,r$ are distinct prime numbers and $k_i\\geq0$, $i=1,2,\\dots,r$, then\n\\begin{equation*}\n\\deg(u)=\\left\\{\\begin{array}{ll}\n n-1, & \\hbox{$f^*(u)=1.$} \\\\\n n-\\sum\\limits_{1\\leq i\\leq r}\\bigg\\lfloor\\frac{n}{p_i^{\\acute{v}_{p_i}(n)}}\\bigg\\rfloor\n +\\sum\\limits_{1\\leq i.j\\leq r}\\bigg\\lfloor\\frac{n}{p_i^{\\acute{v}_{p_i}(n)}p_j^{\\acute{v}_{p_j}(n)}}\\bigg\\rfloor\n -\\dots +(-1)^{r}\\Bigg\\lfloor\\frac{n}{\\prod\\limits_{1\\leq i\\leq r}p_i^{\\acute{v}_{p_i}(n)}}\\Bigg\\rfloor, & \\hbox{$f^*(u)>1.$}\n \\end{array}\n \\right.\n\\end{equation*}\n\\end{thm}\n\n\\begin{cor}\\label{thmVI1}\n Let $G$ be a graph of order $n$. If the graph $G$ is Diophantine, then $\\delta(G)\\leq \\delta(D_n)$.\n\\end{cor}\n\\begin{proof}\n Let $G$ be a Diophantine graph of order $n$. Then, using Corollary \\ref{spanning-thm}, the graph $G$ is labeling isomorphic to a spanning subgraph (say $\\acute{G}$) of $D_n$, i.e., $G\\cong_l\\acute{G}$. Hence, $\\delta(G)=\\delta(\\acute{G})\\leq\\delta(D_n)$\n\\end{proof}\n\n\\begin{thm}\\label{lem5}\n There exists a vertex $w\\in V_{\\delta}(D_n)$ such that $\\frac{n}{2}< f(w)< n$.\n\\end{thm}\n\\begin{proof}\nlet $u\\in V_{\\delta}(D_n)$. Then the two cases follow, in the case of $\\frac{n}{2}< f(u)< n$, we have nothing to prove. In the other case of $1<f(u)<\\frac{n}{2}$, we get $\\frac{n}{2}< 2^t f(u)< n$ for some $t>0$. Let $f(w)=2^tf(u)$, where $w\\in V(D_n)$, $t>0$. Then we have $\\frac{n}{2}<f(w)< n$. Using Lemma \\ref{lem3}, we get $N(u)\\supseteq N(w)$. Therefore, $\\deg(w)\\leq \\deg(u)$. Since $u\\in V_{\\delta}(D_n)$, therefore $\\deg(w)=\\deg(u)=\\delta(D_n)$. Hence, form the two cases, There exists a vertex $w\\in V_{\\delta}(D_n)$ such that $\\frac{n}{2}< f(w)< n$.\n\\end{proof}\n\\subsection{The Degree Sequences of graphs}\n\n\n\n\\begin{dfn}\n Let $G$ be a graph of order $n$. A finite sequence $S_G=\\{g_i\\}_{i=0}^{n-1}=(g_0, g_1, \\dots, g_{n-1})$, where $g_i:=|\\{v\\in V(G):\\deg(v)=i\\}|$, $i=0,1, \\dots, n-1$ is called the vertex-degree sequence of $G$.\n\\end{dfn}\n\nThe reader notices that the standard notion of degree sequence is different in this literature. Obviously, we obtain the following two equations $\\sum\\limits_{i=0}^{n-1}g_i=|V(G)|$ and $\\sum\\limits_{i=0}^{n-1} ig_i=2|E(G)|$. \n\n\\begin{ex} \n $S_{D_7}=(0, 0, 0, 1, 2, 1, 3)$, and\n $S_{D_{11}}=(0,0,0,0,1,1,3,0,2,1,3)$. \\end{ex}\n\n\n\\begin{thm}\\label{thmVI2}\n Let $G$ be a graph of order $n$ with $S_G=\\{g_i\\}_{i=0}^{n-1}$ and $D_n$ be the maximal Diophantine graph of order $n$ with $S_{D_n}=\\{d_i\\}_{i=0}^{n-1}$. If the graph $G$ is Diophantine then for each $k\\in\\{0,1,\\dots,n-1\\}$, \n $$\\sum\\limits_{i=0}^{k} g_{i}\\geq \\sum\\limits_{i=0}^{k} d_{i}.$$\n\\end{thm}\n\\begin{proof}\n Assume $(D_n,f_1)$ is the maximal Diophantine graph of order $n$ with $S_{D_n}=\\{d_i\\}_{i=0}^{n-1}$ and $(G,f_2)$ is a Diophantine graph of order $n$ with $S_G=\\{g_i\\}_{i=0}^{n-1}$, where \n $$f_1:V(D_n)\\rightarrow \\{1,2,\\dots,n\\}, \\quad f_2:V(G)\\rightarrow \\{1,2,\\dots,n\\}.$$\n are Diophantine labelings of $D_n,G$ respectively. Suppose contrarily that there exists $k_0 \\in\\{0,1,\\dots,n-1\\}$ such that\n \\begin{equation}\\label{eqn5}\n \\sum\\limits_{i=0}^{k_0} g_{i}< \\sum\\limits_{i=0}^{k_0} d_{i}.\n \\end{equation}\n Using Corollary \\ref{spanning-thm}, we get that $(G,f_2)$ is labeling isomorphic to a spanning subgraph of $(D_n,f_1)$. Let $(\\acute{G},f_1)$ be a spanning subgraph of $(D_n,f_1)$ such that \n $(G,f_2)\\cong_l(\\acute{G},f_1).$ Then there is a bijective map $\\varphi:V(G)\\rightarrow V(\\acute{G})$ such that for all $u,\\acute{u}\\in V(G)$, \n $u\\acute{u}\\in E(G)$ if and only if $\\varphi(u)\\varphi(\\acute{u})\\in E(\\acute{G})$ and $f_1(u)=f_2(\\varphi(u)).$\n Then for every $u\\in V(G)$,\n \\begin{equation}\\label{eqn2}\n \\deg(u)=\\deg(\\varphi(u)).\n \\end{equation}\n Define a map $\\acute{\\varphi}:V(G)\\rightarrow V(D_n)$ such that for every $u\\in V(G)$, $\\acute{\\varphi}(u):=\\varphi(u).$ \n Then the map $\\acute{\\varphi}$ is bijective and for every $u\\in V(G)$, $f_2(u)=f_1(\\varphi(u))=f_1(\\acute{\\varphi}(u))$.\n Since $(\\acute{G},f_1)$ is a spanning subgraph of $(D_n,f_1)$ and using equation \\eqref{eqn2}, one can see that for every $u\\in V(G)$,\n \\begin{equation}\\label{eqn3}\n \\deg(u)\\leq \\deg(\\acute{\\varphi}(u)),\n \\end{equation} \n Define the following two sets \n $$A_{D_n}(k):=\\{v\\in V(D_n):\\deg(v)\\leq k\\}\\quad\\mbox{and} \\quad A_G(k):=\\{u\\in V(G):\\deg(u)\\leq k\\},$$ \n where $k=0,1,\\dots,n-1$. Therefore, using equation \\eqref{eqn5}, there exists $k_0\\in\\{ 0,1,\\dots,n-1\\}$ such that\n \\begin{equation*}\n |A_{G}(k_0)|=\\sum\\limits_{i=0}^{k_0} g_{i}< \\sum\\limits_{i=0}^{k_0} d_{i}=|A_{D_n}(k_0)|.\n \\end{equation*}\n Consequently, there exists a vertex $v\\in A_{D_n}(k_0)\\subseteq V(D_n)$ such that $u=\\acute{\\varphi}^{-1}(v)\\notin A_{G}(k_0).$ Then,\n \\begin{equation*}\n \\deg(u)> k_0\\geq\\deg(v)=\\deg(\\acute{\\varphi}(u)),\n \\end{equation*}\n which contradicts equation \\eqref{eqn3}. Hence, for every $k\\in \\{0,1,\\dots,n-1\\}$,\n \\begin{equation*}\n \\sum\\limits_{i=0}^{k} g_{i}\\geq \\sum\\limits_{i=0}^{k} d_{i},\n \\end{equation*}\n which completes the proof\n\\end{proof}\n\n\\begin{rem}\\label{cor2}\n Let a graph $G$ of order $n$ have $S_G=\\{g_i\\}_{i=0}^{n-1}$ and the maximal Diophantine graph $D_n$ have $S_{D_n}=\\{d_i\\}_{i=0}^{n-1}$. If $\\sum\\limits_{i=0}^{k} g_{i}\\geq\\sum\\limits_{i=0}^{k} d_{i}$ holds for every $k=0,1,\\dots,n-1$, then $$|E(G)|\\leq|E(D_n)|\\quad \\mbox{or}\\quad F(G)\\leq F(D_n)\\quad\\mbox{or}\\quad \\delta(G)\\leq\\delta(D_n).$$\n\\end{rem}\n\nThe proof of Remark \\ref{cor2} closely resembles the corollaries in \\cite{Seoud1}. The following table presents the quantities $|E(G)|$, $F(G)$, $Cl(G)$, $\\alpha(G)$, $\\delta(G)$ and $S_{G}$, where $G=D_n$, $n=4,\\dots,20$. \n\\begin{table}[h]\n\\centering\n\\begin{tabular}{c c c c c c c c}\n\\hline\n$n$ & $|E(D_n)|$ & $F(D_n)$ & $Cl(D_n)$ & $\\alpha(D_n)$ & $\\delta(D_n)$ & $S_{D_n}=(d_i)$ \\\\\n\\hline\\hline\n4 & 6 & 4 & 4 & 1 & 3 & $(0,0,0,4)$ \\\\\n5 & 9 & 3 & 4 & 2 & 3 & $(0,0,0,2,3)$ \\\\\n6 & 15 & 6 & 6 & 1 & 5 & $(0,0,0,0,0,6)$ \\\\\n7 & 17 & 3 & 5 & 3 & 3 & $(0,0,0,1,2,1,3)$ \\\\\n8 & 27 & 6 & 7 & 2 & 6 & $(0,0,0,0,0,0,2,6)$ \\\\\n9 & 30 & 5 & 6 & 4 & 5 & $(0,0,0,0,0,4,0,0,5)$ \\\\\n10 & 41 & 5 & 7 & 3 & 7 & $(0,0,0,0,0,0,0,3,2,5)$ \\\\\n11 & 41 & 3 & 6 & 5 & 4 & $(0,0,0,0,1,1,3,0,2,1,3)$ \\\\\n12 & 65 & 10 & 11 & 2 & 10 & $(0,0,0,0,0,0,0,0,0,0,2,10)$ \\\\\n13 & 57 & 4 & 7 & 6 & 5 & $(0,0,0,0,0,2,1,3,0,2,0,1,4)$ \\\\\n14 & 81 & 6 & 8 & 4 & 8 & $(0,0,0,0,0,0,0,0,1,0,3,2,2,6)$ \\\\\n15 & 83 & 7 & 9 & 7 & 7 & $(0,0,0,0,0,0,0,1,6,0,0,0,0,1,7)$ \\\\\n16 & 106 & 7 & 10 & 5 & 9 & $(0,0,0,0,0,0,0,0,0,1,0,4,0,2,2,7)$ \\\\\n17 & 95 & 4 & 8 & 8 & 6 & $(0,0,0,0,0,0,2,1,1,4,1,0,2,0,1,1,4)$ \\\\\n18 & 143 & 9 & 11 & 4 & 14 & $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,3,2,9)$ \\\\\n19 & 119 & 5 & 9 & 9 & 7 & $(0,0,0,0,0,0,0,3,1,1,4,1,0,2,0,0,1,1,5)$ \\\\\n20 & 173 & 10 & 13 & 6 & 14 & $(0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,4,10)$ \\\\\n\\hline\n\\end{tabular}\n\\caption{Basic Bounds of $D_n$, $n=4,\\dots,20$}\n\\label{table1}\n\\end{table}\n\\section{Necessary Conditions for Diophantine Graphs}\n \\hspace{0.5cm} According to Definition \\ref{dfn2}, Corollaries \\ref{corVI2}, \\ref{corVI3}, \\ref{corVI4}, \\ref{thmVI1} and Theorem \\ref{thmVI2}, each of the following six conditions $\\textbf{C1}, \\dots,\\textbf{C6}$ listed below constitutes a necessary condition for the existence of a Diophantine labeling for a given graph $G$ of order $n$:\\\\\n\n \\textbf{C1}. $|E(G)|\\leq|E(D_n)|$.\\quad \\textbf{C2}. $F(G)\\leq F(D_n)$.$\\quad$ \\textbf{C3}. $Cl(G)\\leq Cl(D_n)$.$\\quad$ \\textbf{C4}. $\\alpha(G)\\geq\\alpha(D_n)$.\\quad\n\n \\textbf{C5}. $\\delta(G)\\leq\\delta(D_n)$.\\hspace{0.8cm} \\textbf{C6}. There exists $k\\in\\{0, 1, \\dots, n-1\\}$ such that $\\sum\\limits_{i=0}^{k} g_{i}\\geq\\sum\\limits_{i=0}^{k} d_{i}$.\\\\\n \n \nNotice that, based on Remark \\ref{cor2} and Example \\ref{example1}, \\textbf{C6} is stronger than each of the following three conditions \\textbf{Ci}, i = 1, 2, 5. Conditions \\textbf{Ci}, i = 1, 2, 3, 4, 5 are mutually independent while conditions \\textbf{Ci}, i = 3, 4, 6 are also mutually independent. Additionally, Examples \\ref{example1} and \\ref{example2} illustrate some relations among these conditions.\n\n\\begin{ex}\\label{example1}\n In this example, the previous six necessary conditions are studied in the following six graphs $G_i$, $i=1,\\dots,6$ as shown in Figure \\ref{figure2}. The basic bounds for these graphs are given in Table \\ref{table2} and the corresponding maximal Diophantine graphs $D_7$ and $D_{11}$. The graph $G_1$ does not satisfy \\textbf{C4}, the graph $G_2=C_4+\\overline{K_3}$ does not satisfy \\textbf{C5}, \\textbf{C6} $\\left(\\mbox{for} \\ \\sum\\limits_{i=0}^{3}g_{i}<\\sum\\limits_{i=0}^{3}d_{i}\\right)$, the graph $G_3$ does not satisfy \\textbf{C3}, the graph $G_4=K_4+\\overline{K_7}$ does not satisfy \\textbf{C2}, \\textbf{C6} $\\left(\\mbox{for} \\ \\sum\\limits_{i=0}^{9} g_{i}<\\sum\\limits_{i=0}^{9} d_{i} \\right)$, the graph $G_5$ does not satisfy \\textbf{C1}, \\textbf{C6} $\\left(\\mbox{for} \\ \\sum\\limits_{i=0}^{5} g_{i}<\\sum\\limits_{i=0}^{5} d_{i} \\right)$ and the graph $G_6$ does not satisfy \\textbf{C6} $\\left(\\mbox{for} \\ \\sum\\limits_{i=0}^{8}g_{i}<\\sum\\limits_{i=0}^{8}d_{i} \\right)$. Therefore, the graphs $G_i$, $i=1,\\dots,6$ are not Diophantine. However, the other necessary conditions are satisfied for the graphs $G_i$, $i=1,2,\\dots,6$.\n\\end{ex}\n\n\n\\begin{figure*}[h!]\n\\centering\n \\begin{subfigure}{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}\n[scale=.7,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v1)[circle,fill=red!20] at (2,0) {$4$};\n \\node (v2) at (0,2) {$2$};\n \\node (v3) at (-2,0) {$3$};\n \\node (v4) at (0,-2) {$1$};\n \\node (v5) at (2,-2) {$5$};\n \\node (v6)[circle,fill=red!20] at (4,0) {$6$};\n \\node (v7) at (2,2) {$7$};\n\\draw (v1) -- (v2);\n\\draw (v1) -- (v3);\n\\draw (v1) -- (v4);\n\\draw (v1) -- (v5);\n\\draw (v1) -- (v7);\n\n\\draw (v2) -- (v3);\n\\draw (v2) -- (v4);\n\\draw (v2) -- (v5);\n\\draw (v2) -- (v7);\n\n\\draw (v3) -- (v4);\n\\draw (v3) -- (v5);\n\\draw (v3) -- (v7);\n\n\\draw (v4) -- (v5);\n\n\\draw (v5) -- (v6);\n\n\\draw (v6) -- (v7);\n \\end{tikzpicture}\\caption{Graph $G_1$}\n \\end{subfigure}\n~\n \\begin{subfigure}{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}\n[scale=.7,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v1) at (1.8,2) {$1$};\n \\node (v2) at (1.8,0.6) {$2$};\n \\node (v3) at (0,-2) {$3$};\n \\node (v4) at (0,-0.6) {$4$};\n\n \\node (v5)[circle,fill=red!20] at (6,1.5) {$5$};\n \\node (v6)[circle,fill=red!20] at (6,0) {$6$};\n \\node (v7)[circle,fill=red!20] at (6,-1.5) {$7$};\n\\draw (v1) -- (v2);\n\\draw (v1) -- (v4);\n\\draw (v1) -- (v5);\n\\draw (v1) -- (v6);\n\\draw (v1) -- (v7);\n\n\\draw (v2) -- (v3);\n\\draw (v2) -- (v5);\n\\draw (v2) -- (v6);\n\\draw (v2) -- (v7);\n\n\\draw (v3) -- (v4);\n\\draw (v3) -- (v5);\n\\draw (v3) -- (v6);\n\\draw (v3) -- (v7);\n\n\\draw (v4) -- (v5);\n\\draw (v4) -- (v6);\n\\draw (v4) -- (v7);\n \\end{tikzpicture}\\caption{Graph $G_2$}\n \\end{subfigure}\n~\n \\begin{subfigure}{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}\n[scale=.6,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v1)[circle,fill=red!20] at (2,2) {$1$};\n \\node (v2)[circle,fill=red!20] at (2,0.8) {$2$};\n \\node (v3)[circle,fill=red!20] at (2,-0.8) {$3$};\n \\node (v4)[circle,fill=red!20] at (2,-2) {$4$};\n\n \\node (v5) at (-2,-2) {$6$};\n \\node (v6) at (-1,0) {$7$};\n \\node (v7) at (-2,2) {$8$};\n \\node (v8)[circle,fill=red!20] at (-4,2.5) {$9$};\n \\node (v9) at (-6,1.3) {$10$};\n \\node (v10) at (-6,-1.3) {$11$};\n \\node (v11) at (-4,-2.5) {$5$};\n\\draw (v1) -- (v5);\n\\draw (v1) -- (v6);\n\\draw (v1) -- (v7);\n\n\\draw (v2) -- (v5);\n\\draw (v2) -- (v6);\n\\draw (v2) -- (v7);\n\n\\draw (v3) -- (v5);\n\\draw (v3) -- (v6);\n\\draw (v3) -- (v7);\n\n\\draw (v4) -- (v5);\n\\draw (v4) -- (v6);\n\\draw (v4) -- (v7);\n\n\\draw (v5) -- (v6);\n\\draw (v5) -- (v7);\n\\draw (v5) -- (v8);\n\\draw (v5) -- (v9);\n\\draw (v5) -- (v10);\n\\draw (v5) -- (v11);\n\n\\draw (v6) -- (v7);\n\\draw (v6) -- (v8);\n\\draw (v6) -- (v9);\n\\draw (v6) -- (v10);\n\\draw (v6) -- (v11);\n\n\\draw (v7) -- (v8);\n\\draw (v7) -- (v9);\n\\draw (v7) -- (v10);\n\\draw (v7) -- (v11);\n\n\\draw (v8) -- (v9);\n\\draw (v8) -- (v10);\n\\draw (v8) -- (v11);\n\n\\draw (v9) -- (v10);\n\\draw (v9) -- (v11);\n\n\\draw (v10) -- (v11);\n \\end{tikzpicture}\\caption{Graph $G_3$}\n \\end{subfigure}\n~\n \\begin{subfigure}{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}\n[scale=.6,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v1) at (3,1) {$1$};\n \\node (v2) at (1,3) {$2$};\n \\node (v3) at (-1,3) {$3$};\n \\node (v4) at (-3,1) {$4$};\n\n \\node (v5)[circle,fill=red!20] at (-3.5,-1) {$10$};\n \\node (v6)[circle,fill=red!20] at (-2.5,-2) {$5$};\n \\node (v7)[circle,fill=red!20] at (-1.3,-2.5) {$6$};\n \\node (v8)[circle,fill=red!20] at (0,-3) {$7$};\n \\node (v9)[circle,fill=red!20] at (1.3,-2.5) {$8$};\n \\node (v10)[circle,fill=red!20] at (2.5,-2) {$9$};\n \\node (v11)[circle,fill=red!20] at (3.5,-1) {$11$};\n\\draw (v1) -- (v2);\n\\draw (v1) -- (v3);\n\\draw (v1) -- (v4);\n\\draw (v1) -- (v5);\n\\draw (v1) -- (v6);\n\\draw (v1) -- (v7);\n\\draw (v1) -- (v8);\n\\draw (v1) -- (v9);\n\\draw (v1) -- (v10);\n\\draw (v1) -- (v11);\n\n\\draw (v2) -- (v3);\n\\draw (v2) -- (v4);\n\\draw (v2) -- (v5);\n\\draw (v2) -- (v6);\n\\draw (v2) -- (v7);\n\\draw (v2) -- (v8);\n\\draw (v2) -- (v9);\n\\draw (v2) -- (v10);\n\\draw (v2) -- (v11);\n\n\\draw (v3) -- (v4);\n\\draw (v3) -- (v5);\n\\draw (v3) -- (v6);\n\\draw (v3) -- (v7);\n\\draw (v3) -- (v8);\n\\draw (v3) -- (v9);\n\\draw (v3) -- (v10);\n\\draw (v3) -- (v11);\n\n\\draw (v4) -- (v5);\n\\draw (v4) -- (v6);\n\\draw (v4) -- (v7);\n\\draw (v4) -- (v8);\n\\draw (v4) -- (v9);\n\\draw (v4) -- (v10);\n\\draw (v4) -- (v11);\n \\end{tikzpicture}\\caption{Graph $G_4$}\n \\end{subfigure}\n~\n \\begin{subfigure}{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}\n[scale=.6,auto=center,every node/.style={circle,fill=blue!20}]\n \\node (v1) at (3.7,0) {$1$};\n \\node (v2) at (3,2.5) {$2$};\n \\node (v3) at (1.5,4) {$3$};\n \\node (v4) at (-1.5,4) {$4$};\n \\node (v5) at (-3,2.5) {$5$};\n \\node (v6) at (-3.7,0) {$6$};\n\n \\node (v7)[circle,fill=red!20] at (-3,-2.5) {$10$};\n \\node (v8)[circle,fill=red!20] at (-1.5,-3) {$7$};\n \\node (v9)[circle,fill=red!20] at (0,-3) {$8$};\n \\node (v10)[circle,fill=red!20] at (1.5,-3) {$9$};\n \\node (v11)[circle,fill=red!20] at (3,-2.5) {$11$};\n\\draw (v1) -- (v2);\n\\draw (v1) -- (v3);\n\\draw (v1) -- (v4);\n\\draw (v1) -- (v5);\n\\draw (v1) -- (v6);\n\\draw (v1) -- (v7);\n\\draw (v1) -- (v8);\n\\draw (v1) -- (v9);\n\\draw (v1) -- (v10);\n\\draw (v1) -- (v11);\n\n\\draw (v2) -- (v3);\n\\draw (v2) -- (v4);\n\\draw (v2) -- (v5);\n\\draw (v2) -- (v7);\n\\draw (v2) -- (v8);\n\\draw (v2) -- (v9);\n\\draw (v2) -- (v10);\n\\draw (v2) -- (v11);\n\n\\draw (v3) -- (v4);\n\\draw (v3) -- (v5);\n\\draw (v3) -- (v6);\n\\draw (v3) -- (v7);\n\\draw (v3) -- (v8);\n\\draw (v3) -- (v9);\n\\draw (v3) -- (v10);\n\\draw (v3) -- (v11);\n\n\\draw (v4) -- (v5);\n\\draw (v4) -- (v6);\n\\draw (v4) -- (v8);\n\\draw (v4) -- (v9);\n\\draw (v4) -- (v10);\n\\draw (v4) -- (v11);\n\n\\draw (v5) -- (v6);\n\\draw (v5) -- (v8);\n\\draw (v5) -- (v9);\n\\draw (v5) -- (v10);\n\\draw (v5) -- (v11);\n\n\\draw (v6) -- (v7);\n\\draw (v6) -- (v8);\n\\draw (v6) -- (v9);\n\\draw (v6) -- (v10);\n\\draw (v6) -- (v11);\n \\end{tikzpicture}\\caption{Graph $G_5$}\n \\end{subfigure}\n~\n \\begin{subfigure}{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}\n[scale=.6,auto=center,every node/.style={circle,fill=blue!20}]\n \\node[circle,fill=blue!20](v1) at (4,1) {$1$};\n \\node[circle,fill=blue!20](v2) at (2.5,3) {$2$};\n \\node[circle,fill=blue!20](v3) at (0,4) {$3$};\n \\node[circle,fill=blue!20](v4) at (-2.5,3) {$4$};\n \\node[circle,fill=blue!20](v5) at (-4,1) {$5$};\n\n \\node[circle,fill=red!20](v6) at (-4,-1) {$6$};\n \\node[circle,fill=red!20](v7) at (-2.5,-1.5) {$7$};\n \\node[circle,fill=red!20](v8) at (0,-2) {$8$};\n \\node[circle,fill=red!20](v9) at (2.5,-1.5) {$9$};\n \\node[circle,fill=red!20](v10) at (4,-1) {$10$};\n\n \\node[circle,fill=red!20](v11) at (0,5.5) {$11$};\n\n\\draw (v2) -- (v1);\n\\draw (v2) -- (v3);\n\\draw (v2) -- (v4);\n\\draw (v2) -- (v5);\n\\draw (v2) -- (v6);\n\\draw (v2) -- (v7);\n\\draw (v2) -- (v8);\n\\draw (v2) -- (v9);\n\\draw (v2) -- (v10);\n\\draw (v2) -- (v11);\n\n\\draw (v3) -- (v1);\n\\draw (v3) -- (v4);\n\\draw (v3) -- (v5);\n\\draw (v3) -- (v6);\n\\draw (v3) -- (v7);\n\\draw (v3) -- (v8);\n\\draw (v3) -- (v9);\n\\draw (v3) -- (v10);\n\\draw (v3) -- (v11);\n\n\\draw (v4) -- (v1);\n\\draw (v4) -- (v5);\n\\draw (v4) -- (v6);\n\\draw (v4) -- (v7);\n\\draw (v4) -- (v8);\n\\draw (v4) -- (v9);\n\\draw (v4) -- (v10);\n\\draw (v4) -- (v11);\n\n\\draw (v1) -- (v5);\n\\draw (v1) -- (v6);\n\\draw (v1) -- (v7);\n\\draw (v1) -- (v8);\n\\draw (v1) -- (v9);\n\\draw (v1) -- (v10);\n\n\\draw (v5) -- (v6);\n\\draw (v5) -- (v7);\n\\draw (v5) -- (v8);\n\\draw (v5) -- (v9);\n\\draw (v5) -- (v10);\n \\end{tikzpicture}\\caption{Graph $G_6$}\n \\end{subfigure}\\caption{The graphs $G_1,G_2,G_3,G_4,G_5$ and $G_6$ are non-Diophantine}\\label{figure2}\n\\end{figure*}\n\n\\begin{table}[h!]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n Graph $G$ & $|E(G)|$ & $F(G)$ & $Cl(G)$ & $\\alpha(G)$ & $\\delta(G)$ & $S_{G}$ \\\\\n \\hline\n $D_7$ & 17 & 3 & 5 & 3 & 3 & $(0,0,0,1,2,1,3)$ \\\\\n \\hline\n $G_1$ & 15 & 0 & 5 & \\fbox{2} & 2 & $(0,0,1,0,2,4,0)$ \\\\\n $G_2$ & 16 & 0 & 3 & 3 & \\fbox{4} & $\\fbox{(0,0,0,0,3,4,0)}$ \\\\\n \\hline\n $D_{11}$ & 41 & 3 & 6 & 5 & 4 & $(0,0,0,0,1,1,3,0,2,1,3)$ \\\\\n \\hline\n $G_3$ & 33 & 3 & \\fbox{7} & 5 & 3 & $(0,0,0,4,0,0,4,0,0,0,3)$ \\\\\n $G_4$ & 34 & \\fbox{4} & 5 & 7 & 4 & $\\fbox{(0,0,0,0,7,0,0,0,0,0,4)}$ \\\\\n $G_5$ & \\fbox{42} & 2 & 6 & 5 & 5 & $\\fbox{(0,0,0,0,1,0,4,0,2,2,2)}$ \\\\\n $G_6$ & 38 & 3 & 6 & 6 & 3 & $\\fbox{(0,0,0,1,0,5,0,0,0,2,3)}$ \\\\\n \\hline\n \\end{tabular}\n \\caption{Some Basic Bounds of $G_i$, $i=1,\\dots,6$ and The Corresponding Bounds of $D_7$ and $D_{11}$}\\label{table2}\n\\end{table}\n\n\\begin{ex}\\label{example2}\n The following graph $G$ in Figure \\ref{figure3} is not Diophantine (see \\cite{sonbaty}). However, condition \\textbf{C1} is satisfied for $|E(G)|=37<41=|E(D_{11})|$, condition \\textbf{C2} is satisfied for $F(G)=3=F(D_{11})$, condition \\textbf{C3} is satisfied for $Cl(G)=6=Cl(D_{11})$, condition \\textbf{C4} is satisfied for $\\alpha(G)=6>5=\\alpha(D_{11})$, condition \\textbf{C5} is satisfied for $\\delta(G)=3<4=\\delta(D_{11})$ and condition \\textbf{C6} is satisfied for the two sequences \n $$S_G=\\{g_i\\}_{i=0}^{10}=(0,0,0,1,1,4,0,0,1,1,3)\\quad\\mbox{and}\\quad S_{D_{11}}=\\{d_i\\}_{i=0}^{10}=(0,0,0,0,1,1,3,0,2,1,3)$$\n hold the following condition, for every $k=0,1,\\dots,10$\n $$\\sum\\limits_{i=0}^{k}g_{i}\\geq\\sum\\limits_{i=0}^{k}d_{i}.$$ \n Hence, all six conditions \\textbf{Ci}, i=1, \\dots, 6 are necessary and insufficient conditions.\n \\end{ex}\n \n\\begin{figure*}[h!]\n \\centering\n \\begin{tikzpicture}\n [scale=0.6,auto=center]\n \\node[circle,fill=blue!20](v1) at (4,1) {$5$};\n \\node[circle,fill=blue!20](v2) at (2.5,3) {$1$};\n \\node[circle,fill=blue!20](v3) at (0,4) {$11$};\n \\node[circle,fill=blue!20](v4) at (-2.5,3) {$7$};\n \\node[circle,fill=blue!20](v5) at (-4,1) {$3$};\n\n \\node[circle,fill=red!20](v8) at (-0.8,-1.5) {$2$};\n \\node[circle,fill=red!20](v6) at (-2.5,-1.5) {$6$};\n \\node[circle,fill=red!20](v9) at (0.8,-1.5) {$8$};\n \\node[circle,fill=red!20](v7) at (2.5,-1.5) {$4$};\n \\node[circle,fill=red!20](v11) at (-4,-1) {$9$};\n \\node[circle,fill=red!20](v10) at (4,-1) {$10$};\n \\draw (v2) -- (v1);\n \\draw (v2) -- (v3);\n \\draw (v2) -- (v4);\n \\draw (v2) -- (v5);\n \\draw (v2) -- (v6);\n \\draw (v2) -- (v7);\n \\draw (v2) -- (v8);\n \\draw (v2) -- (v9);\n \\draw (v2) -- (v10);\n \\draw (v2) -- (v11);\n\n \\draw (v3) -- (v1);\n \\draw (v3) -- (v4);\n \\draw (v3) -- (v5);\n \\draw (v3) -- (v6);\n \\draw (v3) -- (v7);\n \\draw (v3) -- (v8);\n \\draw (v3) -- (v9);\n \\draw (v3) -- (v10);\n \\draw (v3) -- (v11);\n\n \\draw (v4) -- (v1);\n \\draw (v4) -- (v5);\n \\draw (v4) -- (v6);\n \\draw (v4) -- (v7);\n \\draw (v4) -- (v8);\n \\draw (v4) -- (v9);\n \\draw (v4) -- (v10);\n \\draw (v4) -- (v11);\n\n \\draw (v1) -- (v5);\n \\draw (v1) -- (v6);\n \\draw (v1) -- (v7);\n \\draw (v1) -- (v8);\n \\draw (v1) -- (v9);\n \\draw (v1) -- (v10);\n\n \\draw (v5) -- (v7);\n \\draw (v5) -- (v8);\n \\draw (v5) -- (v9);\n \\draw (v5) -- (v10);\n \\end{tikzpicture}\n\\caption{A graph $G$ does not satisfy the six necessary conditions thought $G$ is not Diophantine}\n\\label{figure3}\n\\end{figure*}\n\\newpage\n\\begin{flushleft}\n\\Large\\textbf{Declarations}\n\\end{flushleft}\n\\begin{flushleft}\n\\textbf{Conflict of interest}: The authors declare that they have no conflict of interests.\n\\end{flushleft}\n\n\\begin{thebibliography}{9}\n\n\n\n\\bibitem{Apostol} {\\bf Apostol, T.M.},\\newblock{\\it~Introduction to Analytic Number Theory}. Springer Scince+Business Media, New York, 1976.\n\n\n\\bibitem{Bickle} {\\bf Bickle, A.},\\newblock{\\it~Fundamentals of Graph Theory}, American Mathematical Society, Providence, Rhode Island, USA, 2020.\n\n\\bibitem{Burton} {\\bf Burton, D.M.},\\newblock{\\it~Elementary number theory,} 7th ed., a business unit of The McGraw-Hill Companies, New York, 2011.\n\n\\bibitem{sonbaty} {\\bf Elsonbaty, A.; Al-harbi, E.},\\newblock{\\it~Iterative independence numbers for prime graphs.}, Ars Combinatorai, Vol. 151, 2020.\n\n\\bibitem{G-Y-Z} {\\bf Gross, J.L.; Yellen, J.; Zhang, P.},\\newblock{\\it~Handbook of Graph Theory}, 2nd ed., CRC Press, Taylor \\& Francis Group, USA, 2014.\n\n\\bibitem{Harary} {\\bf Harary, F.},\\newblock{\\it~Graph Theory}, Addison-Wesley Publishing Company, Reading, Massachusetts, 1969.\n\n\\bibitem{S-C-L} {\\bf Hsieh, S.-M.; Hsu, C.-C.; Hsu, L.-F.},\\newblock{\\it~Efficient Method to Perform Isomorphism Testing of Labeled Graphs}, Computational Science and Its Applications - ICCSA 2006, part V, pp 422-431.\n\n\\bibitem{Nasr} {\\bf Nasr, A.; Elsonbaty, A.; Seoud, M.A.; Anwar, M.}, \\newblock{\\it~Diophantine graphs}, submittd for publication.\n\n\\bibitem{Robert} {\\bf Robert, A.M.},\\newblock{\\it~A Course in $p$-adic Analysis.} Springer Science+Business Media, LLC, New York, 2000.\n\n\\bibitem{Rosa} {\\bf Rosa, A.},\\newblock{\\it~On certain valuations of the vertices of a graph}, Main Library-University of Lows, pp 349-355, 1967.\n\n\n\\bibitem{Rosen2} {\\bf Rosen, K.H.}, \\newblock{\\it~Discrete Mathematics and Its Applications,} 7th ed., a business unit of The McGraw-Hill Companies, New York, 2012.\n\n\\bibitem{Seoud1} {\\bf Seoud, M.A.; Elsonbaty, A.; Mahran, A.E.A.}, \\newblock{\\it~On prime graphs}, Ars Combinatorai, Vol. 104, pp 241-260, 2012.\n\n\\bibitem{Seoud-Y} {\\bf Seoud, M.A.; Youssef, M.Z.}, \\newblock{\\it~On prime labelings of graphs.} Congressus Numerantium, Vol. 141, pp 203-215, 1999.\n\n\\bibitem{Tout} {\\bf Tout, A.; Dabboucy, A.N.; Howalla, K.}, \\newblock{\\it~Prime labeling of graphs}. National Academy Science Letters, Vol. 11, pp 365-368, 1982.\n\n\n\n\n\\end{thebibliography}\n\n\n\\end{document}\n\n\n"}}},{"rowIdx":289,"cells":{"id":{"kind":"string","value":"2412.20589v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20589v1"},"title":{"kind":"string","value":"Countable models of weakly quasi-o-minimal theories I"},"full_text":{"kind":"string","value":"\\documentclass{amsart}\n\\usepackage{amssymb,amsmath,amsthm}\n\\usepackage{enumerate}\n\\usepackage{eucal}\n\\usepackage{mathabx}\n\\usepackage{xcolor}\n\\usepackage{tasks}\n\\usepackage{hyperref}\n\n\\hypersetup{\n final,\n colorlinks,\n linkcolor={red!50!black},\n citecolor={red!50!black},\n urlcolor={red!50!black}\n}\n \n\\theoremstyle{definition}\n\\newtheorem{Theorem}{Theorem}[section]\n\\newtheorem{Proposition}[Theorem]{Proposition}\n\\newtheorem{Lemma}[Theorem]{Lemma}\n\\newtheorem{Corollary}[Theorem]{Corollary}\n\\newtheorem{Claim}{Claim}\n\\newtheorem*{Claim*}{Claim}\n \n\\theoremstyle{definition}\n\\newtheorem{Definition}[Theorem]{Definition}\n\\newtheorem{Fact}[Theorem]{Fact}\n\\newtheorem{Conjecture}[Theorem]{Conjecture}\n\\newtheorem{Example}[Theorem]{Example}\n\\newtheorem{Remark}[Theorem]{Remark}\n\\newtheorem{Observation}[Theorem]{Observation}\n\\newtheorem{Question}{Question}\n\\newtheorem{Notation}[Theorem]{Notation}\n\\newtheorem{Theorem1}{Theorem}\n\\newtheorem{Corollary1}{Corollary}\n\n\\usepackage{etoolbox}\\usepackage{orcidlink}\n\n\\def\\forces{\\vdash}\n\\def\\strok{\\restriction}\n\\def\\wom{\\textit{wom}}\n\n\\def\\Ind#1#2{#1\\setbox0=\\hbox{$#1x$}\\kern\\wd0\\hbox to 0pt{\\hss$#1\\mid$\\hss}\n\\lower.9\\ht0\\hbox to 0pt{\\hss$#1\\smile$\\hss}\\kern\\wd0}\n\n\\def\\ind{\\mathop{\\mathpalette\\Ind{}}}\n\n\\def\\notind#1#2{#1\\setbox0=\\hbox{$#1x$}\\kern\\wd0\n\\hbox to 0pt{\\mathchardef\\nn=12854\\hss$#1\\nn$\\kern1.4\\wd0\\hss}\n\\hbox to 0pt{\\hss$#1\\mid$\\hss}\\lower.9\\ht0 \\hbox to 0pt{\\hss$#1\\smile$\\hss}\\kern\\wd0}\n\n\\def\\dep{\\mathop{\\mathpalette\\notind{}}}\n\n\\DeclareMathOperator{\\wor}{\\perp\\hspace*{-0.4em}^{\\mathit w}}\n\\DeclareMathOperator{\\nwor}{\\not\\perp\\hspace*{-0.4em}^{\\mathit w}}\n\\DeclareMathOperator{\\fwor}{\\perp\\hspace*{-0.4em}^{\\mathit f}}\n\\DeclareMathOperator{\\nfor}{\\not\\perp\\hspace*{-0.4em}^{\\mathit f}}\n\\DeclareMathOperator{\\Mon}{\\mathfrak C}\n\\DeclareMathOperator{\\tp}{tp}\n\\DeclareMathOperator{\\itp}{itp}\n\\DeclareMathOperator{\\dcl}{dcl}\n\\DeclareMathOperator{\\conv}{conv}\n\\DeclareMathOperator{\\Inv}{Inv}\n\\DeclareMathOperator{\\Th}{Th}\n\\DeclareMathOperator{\\Aut}{Aut}\n\n\\usepackage[margin=30mm]{geometry}\n\n\\title[Trivial types and shifts]{Countable models of weakly quasi-o-minimal theories I} \n\\author[S.\\ Moconja]{Slavko Moconja\\ \\orcidlink{0000-0003-4095-8830}}\n\\address[S.\\ Moconja]{University of Belgrade, Faculty of mathematics, Belgrade, Serbia}\n\\email{slavko@matf.bg.ac.rs}\n\n\\author[P.\\ Tanovi\\'c]{Predrag Tanovi\\'c\\ \\orcidlink{0000-0003-0307-7508}}\n\\address[P.\\ Tanovi\\'c]{Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia}\n\\email{tane@mi.sanu.ac.rs}\n\\thanks{The authors are supported by the Science Fund of the Republic of Serbia,\ngrant 7750027--SMART}\n\n\\begin{document} \n\n\\begin{abstract} \nWe introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global extensions of a weakly o-minimal type, in which case we say that the type is trivial. In the o-minimal case, we prove that every definable complete 1-type over a model is trivial. We prove that the triviality has several favorable properties; in particular, it is preserved in nonforking extensions of a weakly o-minimal type and under weak nonorthogonality of weakly o-minimal types. We introduce the notion of a shift in a linearly ordered structure that generalizes the successor function. Then we apply the techniques developed to prove that every weakly quasi-o-minimal theory that admits a definable shift has $2^{\\aleph_0}$ countable models.\n\\end{abstract}\n\n \\maketitle\n \n\\section{Introduction}\n\n\n\nIn this paper, we continue the work on the classification theory for countable models of weakly quasi-o-minimal theories; recall that a complete first-order theory is weakly quasi-o-minimal if for some (equivalently, all by \\cite[Theorem 1(ii)]{MTwmon}) $0$-definable linear order $<$ every definable\\footnote{Throughout the paper, definable means with parameters.}\nsubset of $\\Mon$ is a Boolean combination of unary $0$-definable sets and convex sets.\nIn \\cite{MT}, the classification for binary weakly quasi-o-minimal $T$ was completed; recall that $T$ is binary if every formula is equivalent to a Boolean combination of formulae in two free variables. Here, by classification, we mean: first, listing the relevant properties of $T$ that imply $I(T,\\aleph_0)=2^{\\aleph_0}$, and then, assuming that all these properties fail for $T$, finding a reasonable system of invariants for isomorphism types of countable models of $T$. By reasonable invariants, structures that can be easily counted are usually meant, so that Vaught's conjecture for $T$ can be confirmed. In \\cite[Theorem 1]{MT}, four conditions are listed, each of which guarantees $I(T,\\aleph_0)=2^{\\aleph_0}$ for binary weakly quasi-o-minimal $T$:\n\\begin{enumerate}[\\hspace{10pt}(C1)]\n \\item $T$ is not small;\n \\item There is a non-convex type $p\\in S_1(T)$;\n \\item There is a non-simple type $p\\in S_1(T)$;\n \\item There is a nonisolated forking extension of some $p\\in S_1(T)$ over a 1-element domain; \n\\end{enumerate}Then, in \\cite[Theorem 2]{MT}, assuming that none of (C1)-(C4) holds, the invariants of countable models are described as certain sequences of dense order-types (including $\\mathbf 0$ and $\\mathbf 1$), and in \\cite[Theorem 8.10]{MT} the exact value of $I(T,\\aleph_0)$ was computed. \nOutside the binary context, the above list is inadequate; we know that there are weakly o-minimal (non-binary) theories with three countable models that satisfy conditions (C3) and (C4). As for the other conditions, (C1) trivially works for general $T$, while in the subsequent part of this article \\cite{MTclass2}, we will prove that (C2) works too. The principal theorem of this paper introduces a novel significant condition.\n\\begin{Theorem1}\\label{Theorem1_shift}\nIf a weakly quasi-o-minimal theory has a definable shift on $(\\Mon,<)$, then $I(T,\\aleph_0)=2^{\\aleph_0}$.\n\\end{Theorem1}\nThe non-existence of shifts, together with the failure of (C1) and (C2), will suffice to confirm Vaught's and Martin's conjecture for a wide subclass of the weakly quasi-o-minimal theories that includes all quasi-o-minimal and all almost $\\aleph_0$-categorical theories (\\cite{MTclass2}); however, for the whole class, it is unlikely that the three conditions suffice. \n\nThe motivation for Theorem \\ref{Theorem1_shift} we found in the work of Alibek, Baizhanov and Zambarnaya \\cite{Alibek}, where they introduce the notion of quasi-successors as a generalization of the successor function; our notion of shifts is a minor modification of this notion. \nShifts are defined as follows. Let $S(x,y)$ be a formula, $\\mathcal S=(S(a,\\Mon)\\mid a\\in \\Mon)$, and let $c\\in \\Mon$. Then $S(a,\\Mon)$ is an $\\mathcal S$-shift if the following two conditions hold:\n\\begin{enumerate}[\\hspace{10pt}(1)] \n\\item $S(a,\\Mon)$ is a convex subset of $\\Mon$ with minimum $a$ for all $a\\in\\Mon$; and\n\\item $(S^{(n)}(c,\\Mon)\\mid n\\in\\mathbb N)$ is a strictly $\\subset$-increasing family of sets, where $S^{(1)}(x,y):=S(x,y)$ and $S^{(n+1)}(x,y):=(\\exists t)(S^{(n)}(x,t)\\land S(t,y))$.\n\\end{enumerate} \nShifts generalize the successor functions in infinite discrete orders in the following sense. Let $S(x)$ denote the successor of $x$ if it exists. Consider the formula $R(x,y):= (y=x \\vee y=S(x))$ and the family $\\mathcal R=(R(a,\\Mon)\\mid a\\in \\Mon)$. Then $R(c,\\Mon)$ is a $\\mathcal R$-shift, provided that $S^{(n)}(c)$ is defined for all $n\\in \\mathbb N$. \nIn general, it is known that the existence of an infinite definable discrete order in a model of a countable complete first-order theory $T$ implies $I(T,\\aleph_0)=2^{\\aleph_0}$, see \\cite{Tane}. Therefore, it is interesting to know whether Theorem \\ref{Theorem1_shift} holds outside the weakly quasi-o-minimal context. \n\n\\begin{Question}\nIf $T=\\Th(M,<,f,c,\\ldots)$ is countable, where $<$ is a linear order, $f:M\\to M$, and the sequence $(f^{(n)}(c)\\mid n\\in \\mathbb N)$ strictly increases, must $T$ have $2^{\\aleph_0}$ countable models? \n\\end{Question}\n\nThroughout the paper, first-order model theory techniques will be used exclusively. We will rely on results from our recent paper \\cite{MTwom}, in which the notion of weak o-minimality for complete types of an arbitrary first-order theory is introduced and several favorable model-theoretic properties of weakly o-minimal types are proved; these are reviewed in Section \\ref{Section_preliminaries}. \nThe key new notion in this work is that of trivial weakly o-minimal types. They play a central role in the proof of Theorem \\ref{Theorem1_shift}, making the proof not generalize outside the weakly quasi-o-minimal context. We believe also that trivial types will play a decisive role in the proof of Vaught's conjecture for weakly quasi-o-minimal theories. \n\nThe paper is organized as follows. Section \\ref{Section_preliminaries} contains preliminaries. We give a thorough overview of the results from \\cite{MTwom} on weakly o-minimal types that will be used in this and subsequent articles. We also prove some properties of convex weakly o-minimal types ($p\\in S(A)$ is convex if its locus is a convex subset of some $A$-definable linear order).\n\nIn Section \\ref{Section_trivial}, motivated by the notion of triviality for types in stable theories, that was introduced and studied by Baldwin and Harrington in \\cite{BH}, \nwe introduce the notions of triviality and order-triviality for invariant types in an arbitrary theory. The idea behind the triviality of the type in the stable case is that every pairwise independent set of realizations is independent (as a set). Instead of sets of realizations, we consider sequences of realizations of $\\mathfrak p_{\\restriction A}$, where $\\mathfrak p$ is an $A$-invariant global type, and consider them independent if they are Morley in $\\mathfrak p$ over $A$; thus $\\mathfrak p$ is trivial over $A$ if $(a_i,a_j)\\models \\mathfrak p^2_{\\restriction A}$ (for all $i<j<\\omega$) implies that $(a_i\\mid i\\in\\omega)$ is Morley in $\\mathfrak p$ over $A$.\nFor order triviality, we require that $(a_i,a_{i+1})\\models \\mathfrak p^2_{\\restriction A}$ (for all $i\\in\\omega$) implies that $(a_i\\mid i\\in\\omega)$ is Morley; this cannot occur in the stable case. In the NIP case, we show that the triviality of $\\mathfrak p$ over $A$ transfers to any $B\\supseteq A$. We also prove some nice properties of weak orthogonality ($\\wor$) for trivial and order-trivial types. \n\nIn Section \\ref{Section_trivial_wom} we introduce and study triviality in the context of weakly o-minimal types. \nIf the type $p=\\mathfrak p_{\\restriction A}$ is weakly o-minimal, we show that the triviality and order-triviality of $\\mathfrak p$ agree; we also prove that the triviality of one $A$-invariant extension of $p$ implies triviality of the other; in that case, we say that $p$ is trivial. Surprisingly, trivial types are common in o-minimal theories; we show that all definable types $p\\in S_1(M)$ of an o-minimal structure $M$ are trivial. \nWe prove in Theorem \\ref{Theorem_nwor preserves triviality}: the triviality of $p$ transfers to its nonforking extensions, that triviality is preserved under $\\nwor$ of weakly o-minimal types, and that $\\wor$ of trivial types transfers to their nonforking extensions. We also prove that every trivial type in a theory with few countable models is both convex and simple. \n\nIn Section \\ref{Section_shifts}, we introduce shifts and prove Theorem \\ref{Theorem1_shift}. \n\n\n\n\n\n\\section{Preliminaries}\\label{Section_preliminaries} \n\nThroughout the paper, we use standard model-theoretical terminology and notation. We work in $\\Mon$, a large, saturated (monster) model of a complete, first-order (possibly multi-sorted) theory $T$ in a language $L$. \nThe singletons and tuples of elements from $\\Mon$ are denoted by $a,b,c,\\dots$. The letters $A,B,A', B',\\dots$ are reserved for small (of cardinality $<|\\Mon|$) subsets of the monster, while $C,D,C',D',\\dots$ are used to denote arbitrary sets of tuples. By a formula $\\phi(x)$, we mean one with parameters. For a formula $\\phi(x)$ we denote $\\phi(C)= \\{c\\in C\\mid \\models \\phi(c)\\}$, and similarly for partial types. A set $D$ is definable ($A$-definable) if $D=\\phi(\\Mon)$ for some formula $\\phi(x)$ ($\\phi(x)\\in L_A$); similarly for type-definable sets. \n$S_n(C)$ denotes the set of all complete $n$-types over $C$ and $S(C)=\\bigcup_{n\\in\\mathbb N}S_n(C)$. \nTypes from $S(\\Mon)$ are called global types.\nA global type $\\mathfrak p(x)$ is {\\em $A$-invariant} if $(\\phi(x,a_1)\\leftrightarrow \\phi(x,a_2))\\in\\mathfrak p$ for all $L_A$-formulae $\\phi(x,y)$ and all tuples $a_1,a_2$ with $a_1\\equiv a_2\\,(A)$; $\\mathfrak p$ is {\\em invariant} if it is $A$-invariant for some $A$. Let $\\mathfrak p$ be $A$-invariant and let $(I,<)$ be a linear order; we say that the sequence of tuples $(a_i\\mid i\\in I)$ is a {\\em Morley sequence} in $\\mathfrak p$ over $A$ if $a_i\\models \\mathfrak p_{\\restriction A\\,a_{<i}}$ holds for all $i\\in I$. Note that we allow Morley sequences to have arbitrary (even finite) order-type. \n\nDividing and forking have usual meaning. Complete types over $A$, $p(x)$ and $q(y)$, are weakly orthogonal, denoted by $p\\wor q$, if $p(x)\\cup q(y)$ determines a complete type; $p$ is forking orthogonal to $q$, denoted by $p\\fwor q$, if $\\tp(a/bA)$ does not fork over $A$ for all $a\\models p$ and $b\\models q$. \n\nA partial type $p(x)$ over $A$ is {\\em NIP} if there are no formula $\\varphi(x,y)$, an $A$-indiscernible sequence $(a_n\\mid n<\\omega)$ in $p$ and a tuple $b$, such that $\\models\\varphi(a_n,b)$ iff $n$ is even. A partial type $p(x)$ over $A$ is {\\em dp-minimal} if there are no formula $\\varphi( x, y)$, an $A$-indiscernible sequence $I$ of tuples of length $|y|$ and $ a\\models p$ such that $\\varphi( a, y)$ has at least four alternations on $I$. (This characterization of dp-minimal types is due to Kaplan, Onshuus, and Usvyatsov; see \\cite[Proposition 2.8]{KOU}). Recall that every dp-minimal types is NIP.\n\n\n\nThe notation related to linear orders is mainly standard. Let $(X,<)$ be a linear order; $<$ is defined on the power set of $X$ by $Y<Z$ iff $y<z$ for all $y\\in Y$ and $z\\in Z$.\n$D\\subseteq X$ is an initial part of $X$ if $a\\in D$ and $b<a$ imply $b\\in D$; similarly, the final parts are defined. $D$ is convex if $a<c<b$ and $a,b\\in D$ imply $c\\in D$; \n$a\\in X$ is an upper (lower) bound of $D$ if $D<a$ ($a<D$) holds. \nWe emphasize the following notation for $D_1,D_2\\subseteq X$. \n\\begin{itemize} \n \\item $\\sup D_1\\leqslant \\sup D_2$ \\ if and only if \\ $D_2<x$ implies $D_1<x$ for all $x\\in X$; \n \\item $\\sup D_1<\\sup D_2$ if $\\sup D_1\\leqslant \\sup D_2$ but $\\sup D_2\\not\\leqslant \\sup D_1$. \n\\end{itemize}\nThis differs a bit from the standard notion of $\\sup$, which will not be used. \n\n\n\n\\subsection{Relative definability} \nHere, we recall some facts and conventions from \\cite{MTwom}. \nLet $p(x)$ be a partial type over $A$. A subset $X\\subseteq p(\\Mon)$ is {\\em relatively $B$-definable within $p(\\Mon)$} if $X=\\phi(\\Mon)\\cap p(\\Mon)$ for some $L_B$-formula $\\phi(x)$ (usually we will have $A\\subseteq B$); here, $\\phi(x)$ is called a relative definition of $X$ within $p(\\Mon)$, and we will also say that $X$ is relatively defined by $\\phi(x)$ within $p(\\Mon)$. \n\n\n\n\nThe family of all relatively $B$-definable subsets of a type-definable set is closed for finite Boolean combinations. Also,\nif $X_i$ is relatively $B$-definable within $P_i$ ($1\\leqslant i\\leqslant n$), then $\\prod_{1\\leqslant i\\leqslant n}X_i$ is relatively $B$-definable within $\\prod_{1\\leqslant i\\leqslant n}P_i$.\nIn particular, the relative definability of (finitary) relations within $P$ is well defined. We will say that a structure $(P,R_i)_{i\\in I}$ is relatively definable if each relation $R_i$ is such within $P$. For example, if a relation $R\\subseteq P^2$ is relatively defined by a formula $\\phi(x,y)$ and if $(P,R)$ is a linear order, then we say that $\\phi(x,y)$ relatively defines a linear order on $P$. Also, if $P$ and $Q$ are type-definable sets, then the relative definability of (graphs of) functions $f:P\\to Q$ is well defined. \n \n\nThe universal properties of relatively definable structures are expressed by certain $L_{\\infty,\\omega}$-sentences, which we will call {\\em $\\tp$-universal sentences} and informally denote by $(\\forall x_1\\models p_1)\\dots(\\forall x_n\\models p_n)\\ \\psi(x_1,\\dots,x_n)$, where $p_1(x_1),\\dots,p_n(x_n)$ are partial types and $\\psi(x_1,\\dots,x_n)$ is a first-order formula; formally, $(\\forall x\\models p)\\ \\phi(x)$ is $(\\forall x)(\\bigwedge_{\\theta\\in p}\\theta(x)\\rightarrow \\phi(x))$. \nThe properties of relatively definable relations (and their defining formulae) expressed by these sentences are called {\\em $\\tp$-universal properties}. For example, ``the formula $x\\leq y$ relatively defines a preorder on $p(\\Mon)$'' and ``$\\leq$ is a relatively definable preorder on $p(\\Mon)$'' are $\\tp$-universal properties.\nThe following is a version of the compactness that will be applied when dealing with $\\tp$-universal properties. \n\n\\begin{Fact}\\label{Fact_L_P_sentence}\nSuppose that $p_1(x_1),\\dots,p_n(x_n)$ are partial types that are closed for finite conjunctions and let $\\phi(x_1,\\dots,x_n)$ be an $L_{\\Mon}$-formula such that $\\Mon\\models (\\forall x_1\\models p_1)\\dots(\\forall x_n\\models p_n)\\ \\phi(x_1,\\dots,x_n)$. Then there are $\\theta_i(x_i)\\in p_i$ ($1\\leqslant i\\leqslant n$) such that: \n$$\\Mon\\models (\\forall x_1\\dots x_n)\\left(\\bigwedge_{1\\leqslant i\\leqslant n}\\theta_i'(x_i)\\rightarrow \\phi(x_1,\\dots,x_n)\\right)$$\nfor all formulae $\\theta_i'(x_i)$ such that $\\theta_i'(\\Mon)\\subseteq \\theta_i(\\Mon)$ ($1\\leqslant i\\leqslant n$). \n\\end{Fact} \n\n\n\n\\noindent {\\bf Convention.} Whenever $<$ is a relatively definable order on $p(\\Mon)$, we assume the formula $x<y$ is its relative definition; similarly, $E(x,y)$ is a relative definition of an equivalence relation $E$. \n\n\n\\begin{itemize}\n \\item Let $<$ be a relatively definable linear order on $p(\\Mon)$. This is a tp-universal property, so by compactness, there is a formula $\\theta(x)\\in p$ such that the formula $x<y$ relatively defines a linear order on $\\theta(\\Mon)$, that is, $(\\theta(\\Mon),<)$ is a linear order. Orders of this form are called {\\it definable extensions} of $(p(\\Mon),<)$.\n \n \\item Consider the structure $\\mathcal P=(p(\\Mon),<,E)$ where $<$ is a relatively definable linear order and $E$ a relatively definable convex equivalence relation on $p(\\Mon)$. It is not hard to see that a finite conjunction of tp-universal properties is also tp-universal, so by compactness, there is a formula $\\theta(x)\\in p$ such that $(\\theta(\\Mon),<,E)$ is a definable extension of $\\mathcal P$: $x<y$ defines a linear order and $E(x,y)$ a convex equivalence relation on $\\theta(\\Mon)$. \n\\end{itemize} \n\n\n\\subsection{Weakly o-minimal types}\n\nIn this subsection, we survey basic properties of weakly o-minimal types. A complete type $p\\in S(A)$ is {\\it weakly o-minimal} if there is a relatively $A$-definable linear order $<$ on $p(\\Mon)$ such that each relatively definable subset of $(p(\\Mon),<)$ consists of a finite number of convex components; in that case, $(p,<)$ is called a {\\it weakly o-minimal pair over $A$}. Weak o-minimality of the type does not depend on the particular choice of the order: By Corollary 3.3. from \\cite{MTwom}, if $(p,<)$ is a weakly o-minimal pair over $A$, then so is $(p,<')$ for each relatively $A$-definable linear order $<'$ on $p(\\Mon)$. \n\nWe introduced so-types\\footnote{Here, {\\em so} stands for {\\em stationarily ordered}.} and so-pairs in \\cite{MT}. It is easy to see that weakly o-minimal types (pairs) are so-types (so-pairs), that is, if $\\mathbf p=(p,<)$ is a weakly o-minimal pair over $A$, then for each relatively definable subset $D\\subseteq p(\\Mon)$ one of the sets $D$ and $p(\\Mon)\\smallsetminus D$ is right-eventual in $(p(\\Mon),<)$ (contains a final part of $(p(\\Mon),<)$) and one of them is left-eventual. In particular, we can define the left ($\\mathbf p_l$) and the right ($\\mathbf p_r$) globalization of $\\mathbf p$:\n\\begin{itemize} \n\\item $\\mathbf p_l(x):=\\{\\phi(x)\\in L_{\\Mon}\\mid \\phi(\\Mon)\\cap p(\\Mon) \\mbox{ is left-eventual in } (p(\\Mon),<)\\}$ \\ and\n\n\\item $\\mathbf p_r(x):=\\{\\phi(x)\\in L_{\\Mon}\\mid \\phi(\\Mon)\\cap p(\\Mon) \\mbox{ is right-eventual in } (p(\\Mon),<)\\}$.\n\\end{itemize}\nIt is easy to see that $\\mathbf p_l$ and $\\mathbf p_r$ are global $A$-invariant extensions of $p$; by \\cite[Fact 1.12]{MTwom}, they are the only such extensions. Moreover, by \\cite[Corollary 4.2]{MTwom}, they are the only global nonforking extensions of $p$. Forking extensions of $p$ have a simple description: By \\cite[Corollary 4.2]{MTwom}, for any formula $\\phi(x)$ consistent with $p(x)$ we have: $p(x)\\cup \\{\\phi(x)\\}$ forks over $A$ iff $p(x)\\cup \\{\\phi(x)\\}$ divides over $A$ iff\n$\\phi(x)$ relatively defines a bounded (from both sides) subset in $(p(\\Mon),<)$. Such formulas are called {\\it relatively $p$-bounded}. Note that the first two conditions do not refer to any specific order, so these formulae relatively define subsets that are bounded with respect to any relatively $A$-definable linear order. \n\n\\begin{Fact}\\label{Fact_basic_S_p_wom}\nLet $\\mathbf p= (p,<_p)$ be a weakly o-minimal pair over $A$ and let $B\\supseteq A$. \nConsider the set $S_p(B)=\\{q(x)\\in S_n(B)\\mid p(x)\\subseteq q(x)\\}$.\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item $q(\\Mon)$ is a convex subset of $p(\\Mon)$ and $(q,<_p)$ is a weakly o-minimal pair over $B$ for each $q\\in S_p(B)$.\n\\item The set $S_p(B)$ is linearly ordered by: $q<_pq'$ iff $q(\\Mon)<_pq'(\\Mon)$; $(\\mathbf p_{l})_{\\restriction B}=\\min S_p(B)$ and $(\\mathbf p_{r})_{\\restriction B}=\\max S_p(B)$.\n\\item For all $q\\in S_p(B)$: $q$ forks over $A$ if and only if the locus $q(\\Mon)$ is bounded (from both sides) in $(p(\\Mon),<_p)$ if and only if $q$ contains a relatively $p$-bounded formula.\n\\item $(\\mathbf p_{l})_{\\restriction B}$ and $(\\mathbf p_{r})_{\\restriction B}$ are the only nonforking extensions of $p$ in $S_p(B)$.\n\\end{enumerate}\n\\end{Fact}\n\\begin{proof}The proofs for all the statements can be found in \\cite{MTwom}: part (a) corresponds to Lemma 2.6(ii), part (b) to Corollary 2.7, and parts (c) and (d) are derived from Corollary 4.2. \n\\end{proof} \n\nThe key notion in studying the forking independence of weakly o-minimal types in \\cite{MTwom} was the right (and left) $\\mathbf p$-genericity. \nLet $\\mathbf p=(p,<)$ be a weakly o-minimal pair over $A$ and let $B\\supseteq A$. By Fact \\ref{Fact_basic_S_p_wom}(a), the sets $(q(\\Mon)\\mid q\\in S_p(B))$ form a convex partition of $(p(\\Mon),<_p)$. The maximal, or the rightmost element of this partition, is $(\\mathbf p_{r})_{\\restriction B}(\\Mon)$; this motivates the definition of the right $\\mathbf p$-genericity.\n\n \n\n\\begin{Definition}Let $\\mathbf p=(p,<_p)$ be a weakly o-minimal pair over $A$ and let $B$ be {\\it any} small set.\n\\begin{enumerate}[\\hspace{10pt}(a)]\n \\item $a$ is {\\em right $\\mathbf p$-generic} over $B$, denoted by $B\\triangleleft^{\\mathbf p} a$, if $a\\models (\\mathbf p_{r})_{\\restriction AB}$; \n \\item $a$ is {\\em left $\\mathbf p$-generic} over $B$ if $a\\models (\\mathbf p_{l})_{\\restriction AB}$;\n \\item $\\mathcal D_p(B):=\\{a\\in p(\\Mon)\\mid a\\dep_A B\\}$;\n \\item $\\mathcal D_p:=\\{(x,y)\\in p(\\Mon)^2\\mid x\\dep_A y\\}$.\n\\end{enumerate}\n\\end{Definition}\n\\begin{Remark}\\label{Remark_pgeneric_first}\nLet $\\mathbf p=(p,<_p)$ be a weakly o-minimal pair over $A$. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item To avoid possible confusion, no special notation for the left $\\mathbf p$-genericity was chosen. However, $B\\triangleleft^{\\mathbf p^*}a$ means that $a$ is left $\\mathbf p$-generic over $B$, where $\\mathbf p^*=(p,>_p)$ is the reverse of $\\mathbf p$.\n\\item By Fact \\ref{Fact_basic_S_p_wom}(d), the only nonforking extensions of $p$ in $S(AB)$ are $(\\mathbf p_{l})_{\\restriction AB}$ and $(\\mathbf p_{r})_{\\restriction AB}$. Therefore, $a\\models p$ is (left or right) $\\mathbf p$-generic over $B$ if and only if $a\\ind_A B$; that is:\n\n($\\triangleleft^{\\mathbf p}$-comparability) \\ \\ \\ $a\\ind_A B$ \\ \\ if and only if \\ \\ $B\\triangleleft^{\\mathbf p}a$ or $B\\triangleleft^{\\mathbf p^*}a$. \n\n\\item By Fact \\ref{Fact_basic_S_p_wom}(b) we have $(\\mathbf p_{l})_{\\restriction AB}=\\min S_p(B)$, so the locus $(\\mathbf p_{l})_{\\restriction AB}(\\Mon)$, which is the set of all left $\\mathbf p$-generic elements over $B$, is an initial part of $p(\\Mon)$. Similarly, the set of all right $\\mathbf p$-generic elements over $B$, is a final part of $p(\\Mon)$.\n\n\\item By Fact \\ref{Fact_basic_S_p_wom}(c), the set $\\mathcal D_p(B)$ consists of all non-$\\mathbf p$-generic elements. \nSince the left $\\mathbf p$-generic elements form an initial part and the right $\\mathbf p$-generic a final part of $(p(\\Mon),<_p)$, the set $\\mathcal D_p(B)$, if nonempty, is a convex subset of $p(\\Mon)$; in that case, by part (c), $a<_p\\mathcal D_p(B)<_p b$ holds whenever $a$ is left $\\mathbf p$-generic and $b$ right $\\mathbf p$-generic over $B$. \nIn particular, $\\mathcal D_p(a)$ is a convex, bounded subset of $p(\\Mon)$ for all $a\\models p$.\n\n\\item The set $\\mathcal D_p(B)$, if nonempty, is relatively $\\bigvee$-definable over $AB$ within $p(\\Mon)$, by which we mean that $\\mathcal D_p(B)=p(\\Mon)\\cap\\bigcup_{\\phi\\in\\Phi}\\phi(\\Mon)$ holds for some family $\\Phi$ of $L_{AB}$-formulas in variable $x$. Here, $\\Phi$ can be taken to be the set of all relatively $p$-bounded $L_{AB}$-formulae, since by Fact \\ref{Fact_basic_S_p_wom}(c) we have $a\\in \\mathcal D_p(B)$ if and only if $p(x)\\cup \\{\\phi(x)\\}\\subseteq \\tp(a/AB)$ for some relatively $p$-bounded formula $\\phi(x)$. Therefore, $\\mathcal D_p(B)$ is a convex relatively $\\bigvee$-definable subset of $p(\\Mon)$. \n\n\\item $p\\wor\\tp(B/A)$ if and only if $(\\mathbf p_{l})_{\\restriction AB}=(\\mathbf p_{r})_{\\restriction AB}$. \n\\end{enumerate}\n\\end{Remark}\n\n \n \nThe following theorem gathers key properties of right (left) $\\mathbf p$-genericity, viewed as a binary relation on $p(\\Mon)$; these are expressed as properties of the structure\n$(p(\\Mon), \\triangleleft^{\\mathbf p},<_p,\\mathcal D_p)$.\n\n\\begin{Theorem}\\label{Theorem4} Let $\\mathbf p=(p,<_p)$ be a weakly o-minimal pair over $A$. \n\\begin{enumerate}[\\hspace{10pt}(a)] \n\\item \\ $b\\models p$ is right $\\mathbf p$-generic over $a\\models p$ if and only if $a$ is left $\\mathbf p$-generic over $b$ \\ ($a\\triangleleft^{\\mathbf p} b\\Leftrightarrow b\\triangleleft^{\\mathbf p^*} a$). \n\\item $(p(\\Mon), \\triangleleft^{\\mathbf{p}})$ is a strict partial order and $(p(\\Mon), <_p)$ its linear extension. \n\\item The relations $\\mathcal D_{p}$ and the $\\triangleleft^{\\mathbf p}$-incomparability are the same, $<_p$-convex equivalence relation on $p(\\Mon)$. In particular, for all $a,b\\models p$, $\\mathcal D_p(a)$ is a convex subset of $(p(\\Mon),<_p)$ and:\n\\begin{center}$a\\in\\mathcal D_p(b)\\Leftrightarrow a\\dep_A b\\Leftrightarrow \\lnot(a\\triangleleft^{\\mathbf p} b\\vee b\\triangleleft^{\\mathbf p} a)\\Leftrightarrow b\\dep_A a\\Leftrightarrow b\\in\\mathcal D_p(a)$.\n\\end{center}\n\\item The quotient $(p(\\Mon) / \\mathcal{D}_p, <_p)$ is a dense linear order.\n\\end{enumerate} \n\\end{Theorem}\n\\begin{proof}\n(a) is \\cite[Lemma 4.8(ii)]{MTwom}, (b)-(d) are from \\cite[Theorem 4]{MTwom}. \n\\end{proof}\n\n \n\n\\begin{Remark}\\label{Remark_trianglep_binary_basicprops}\nLet $\\mathbf p=(p,<_p)$ be a weakly o-minimal pair over $A$ and let $a,b\\models p$. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item Theorem \\ref{Theorem4}(c) implies the following version of $\\triangleleft^{\\mathbf p}$-comparability: $a\\ind_A b\\Leftrightarrow (a\\triangleleft^{\\mathbf p} b\\vee b\\triangleleft^{\\mathbf p} a)$.\n\n\\item By \\cite[Remark 4.14]{MTwom}, each of the following conditions is equivalent to $a\\triangleleft^{\\mathbf p} b$:\n\n\\ $(a,b)\\models (\\mathbf p_r^2)_{\\restriction A}$; \\ $(b,a)\\models (\\mathbf p_l^2)_{\\restriction A}$; \\ $a<_pb\\land a\\ind_A b$; \\\n $a<_p \\mathcal D_p(b)$; \\ $\\mathcal D_p(a)<_p b$; \\ $\\mathcal D_p(a)<_p\\mathcal D_p(b)$.\n \\end{enumerate}\n\\end{Remark}\n\n\n\n \n\n\\begin{Definition}\nTwo weakly o-minimal pairs over $A$, $\\mathbf p=(p,<_p)$ and $\\mathbf q=(q,<_q)$, are said to be {\\it directly non-orthogonal}, denoted by $\\delta_A(\\mathbf p,\\mathbf q)$, if $p\\nwor q$ and whenever $a\\models p$ is right $\\mathbf p$-generic over $b\\models q$, then $b$ is left $\\mathbf q$-generic over $a$, and vice versa, whenever $b$ is right $\\mathbf q$-generic over $a$, then $a$ is left $\\mathbf p$-generic over $b$. \n\\end{Definition}\n\n\n\nIntuitively, $\\delta_A(\\mathbf p,\\mathbf q)$ describes that orders $<_p$ and $<_q$ ``have the same direction''. This is justified by the next theorem, where we prove that $\\delta_A$ is an equivalence relation and that whenever $\\mathbf p\\nwor \\mathbf q$ (meaning $p\\nwor q$), the pair $\\mathbf p$ has the same direction with exactly one of the pairs $\\mathbf q$ and $\\mathbf q^*$; that is, $\\delta_A(\\mathbf p,\\mathbf q)$ or $\\delta_A(\\mathbf p,\\mathbf q^*)$. \n\n\n\n\\begin{Theorem}\\label{Theorem_nonorthogonality}\\cite[Theorem 5.9]{MTwom}\nLet $\\mathcal{W}_A$ denote the set of all weakly o-minimal types over $A$ and $\\mathcal P_A$ the set of all such pairs over $A$. Let $\\mathcal{W}_A(\\Mon)$ be the set of all realizations of types from $\\mathcal{W}_A$.\n\\begin{enumerate}[\\hspace{10pt}(a)]\n \\item $\\nwor$ is an equivalence relation on both $\\mathcal{W}_A$ and $\\mathcal P_A$. \\\n \\item $\\delta_A$ is an equivalence relation on $\\mathcal P_A$; $\\delta_A$ refines $\\nwor$ by splitting each class into two classes, each of them consisting of the reverses of pairs from the other class.\n \\item $x\\dep_A y$ is an equivalence relation on $\\mathcal{W}_A(\\Mon)$. \n \\item $\\nfor$ is an equivalence relation on $\\mathcal{W}_A$.\n\\end{enumerate}\n\\end{Theorem}\n\n\n \\begin{Fact}\\label{Fact_notdelta}\nSuppose that $\\mathbf p=(p,<_p)$ and $\\mathbf q=(q,<_q)$ are weakly o-minimal pairs over $A$ and $\\mathbf p\\nwor\\mathbf q$. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item Exactly one of $\\delta_A(\\mathbf p,\\mathbf q)$ and $\\delta_A(\\mathbf p,\\mathbf q^*)$ holds. \n\n\\item $\\lnot\\delta_A(\\mathbf p,\\mathbf q)$ if and only if $\\delta_A(\\mathbf p,\\mathbf q^*)$ if and only if $a\\triangleleft^{\\mathbf q} b\\triangleleft^{\\mathbf p}a$ holds for some $a\\models p$ and $b\\models q$. \n\n\\item If $p=q$, then $\\delta_A(\\mathbf p,\\mathbf q)$ if and only if $\\mathbf p_r=\\mathbf q_r$ if and only if $\\triangleleft^{\\mathbf p}=\\triangleleft^{\\mathbf q}$ on $p(\\Mon)$.\n\\end{enumerate}\n\\end{Fact}\n\\begin{proof}\n(a) follows from Theorem \\ref{Theorem_nonorthogonality}(b), (b) from \\cite[Remark 5.3]{MTwom}, and (c) from \\cite[Lemma 5.4]{MTwom}.\n\\end{proof}\n\nAs a consequence of Theorem \\ref{Theorem_nonorthogonality} and Fact \\ref{Fact_notdelta}(c), we have the following definition.\n\n\\begin{Definition}\nLet $\\mathcal F$ be a part of a $\\delta_A$-class of weakly o-minimal pairs and let $\\mathcal F(\\Mon)$ be the set of all realizations of types from $\\mathcal F$. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item For $a,b\\in \\mathcal F(\\Mon)$ \ndefine \\ $a\\triangleleft^{\\mathcal F} b$ \\ if and only if \\ $a\\triangleleft ^{\\mathbf q}b$ holds for some (or equivalently, {\\it any} according to Fact \\ref{Fact_notdelta}(c)) weakly o-minimal pair $\\mathbf q=(q,<_q)\\in\\mathcal F$ such that $b\\models q$. \n\n\\item $\\mathcal D_{\\mathcal F}:=\\{(x,y)\\in \\mathcal F(\\Mon)^2\\mid x\\dep_A y\\}$.\n\\end{enumerate}\n\\end{Definition}\n\n\n\\begin{Theorem}[{\\cite[Theorem 5]{MTwom}}]\\label{Theorem5}\nLet $\\mathcal F$ be a part of a $\\delta_A$-class of non-algebraic weakly o-minimal pairs over $A$. Then there is an $A$-invariant order $<^{\\mathcal F}$ on $\\mathcal F(\\Mon)$ such that the structure \\ \n$(\\mathcal F(\\Mon),\\triangleleft^{\\mathcal F}, <^{\\mathcal F},\\mathcal D_{\\mathcal F})$ satisfies the following conditions: \n\\begin{enumerate}[\\hspace{10pt} (a)]\n\\item $(\\mathcal{F}(\\Mon), \\triangleleft^{\\mathcal F})$ is a strict partial order and $(\\mathcal{F}(\\Mon), <^{\\mathcal F})$ is its linear extension; \n\n\\item Relations $\\mathcal D_{\\mathcal F}$ and the $\\triangleleft^{\\mathcal F}$-incomparability are the same, $<^{\\mathcal F}$-convex equivalence relation on $\\mathcal{F}(\\Mon)$. The quotient $(\\mathcal{F}(\\Mon)/\\mathcal D_{\\mathcal F},<^{\\mathcal F})$ is a dense linear order.\n\\item For each $p\\in S(A)$ represented in $\\mathcal F$, the order $<_p=(<^{\\mathcal F})_{\\restriction p(\\Mon)}$ is relatively $A$-definable and the pair $(p,<_p)$ is directly non-orthogonal to pairs from $\\mathcal F$. $(p(\\Mon), \\triangleleft^{\\mathbf p},<_p,\\mathcal D_p)$ is a substructure of $(\\mathcal F(\\Mon),\\triangleleft^{\\mathcal F}, <^{\\mathcal F},\\mathcal D_{\\mathcal F})$. \n\\end{enumerate}\nMoreover, if $(\\mathbf p_i=(p_i,<_i)\\mid i\\in I)$ is a family of pairs from $\\mathcal F$, with the types $(p_i\\mid i\\in I)$ mutually distinct, then the order $<_{\\mathcal F}$ can be chosen to extend each $<_i$ for $i\\in I$. \n\\end{Theorem}\n\n\\begin{Theorem}\\label{Theorem_triangle_mathcal F}\nLet $\\mathcal F$ be part of a $\\delta_A$-class of weakly o-minimal pairs over $A$ and let $<_{\\mathcal F}$ be given by Theorem \\ref{Theorem5}. Then for all small $B$, $\\mathbf p=(p,<_p)\\in\\mathcal F$, and $a,b,c\\in \\mathcal{F}(\\Mon)$ the following hold true: \n\\begin{enumerate}[\\hspace{10pt} (a)]\n\\item (Existence) \\ There exists a $c\\models p$ such that $B\\triangleleft^{\\mathcal F}c$;\n\\item (Density) \\ If $B\\triangleleft^{\\mathcal F} b$, then there is a $c$ such that $B\\triangleleft^{\\mathbf p} c\\triangleleft^{\\mathcal F} b$; \n\\item (Symmetry) \\ $a\\ind_A b$ if and only if $b\\ind_A a$; \n\\item \\ \n$a \\triangleleft^{\\mathcal F} b\\Leftrightarrow b\\triangleleft^{\\mathcal F^*} a$ \\ ($b$ is right $\\mathcal F$-generic over $a$ if and only if $a$ is left $\\mathcal F$-generic over $b$.) \n\\item ($\\triangleleft^{\\mathcal F}$-comparability) \\ $a\\ind_A B\\Leftrightarrow (B \\triangleleft^{\\mathcal F} a\\vee B\\triangleleft^{\\mathcal F^*} a)$\n\\ and \\ \n$a\\ind_A b\\Leftrightarrow (a \\triangleleft^{\\mathcal F} b\\vee b\\triangleleft^{\\mathcal F} a)$;\n\\item (Transitivity) \\ $B\\triangleleft^{\\mathcal F} a \\triangleleft^{\\mathcal F} b$ \\ implies \\ $B\\triangleleft^{\\mathcal F} b$. \n \\item $B\\triangleleft^{\\mathcal F} a \\land a\\dep_A b$ implies $B\\triangleleft^{\\mathcal F} b\\land a\\dep_{B}b$;\n\\item $B\\triangleleft^{\\mathcal F} a<^{\\mathcal F}b$ implies $B\\triangleleft^{\\mathcal F} b$; \n\\item $B\\triangleleft^{\\mathcal F} a<^{\\mathcal F}b\\land \\, a\\ind_B b$ implies $Ba\\triangleleft^{\\mathcal F} b$; \n\\item $B\\triangleleft^{\\mathcal F}a<^{\\mathcal F}b<^{\\mathcal F}c\\land a\\dep_{B}c$ implies $b\\dep_{B} a \\land b\\dep_{B}c$. \n\\end{enumerate}\n\\end{Theorem}\n\\begin{proof}\n(a)-(f) are Theorem 5.11(i)-(vi) of \\cite{MTwom} and (g)-(j) are Proposition 5.12(iii)-(vi) of \\cite{MTwom}. \n\\end{proof} \n\nMost applications of Theorem \\ref{Theorem_triangle_mathcal F} will be in the case where $\\mathcal F=\\{\\mathbf p,\\mathbf q\\}$ (with $\\delta_A(\\mathbf p,\\mathbf q)$ assumed). This is summarized in the next corollary.\n\n\\begin{Corollary}\\label{Cor_triangle_mathbf_pq_properties}\n $\\delta_A(\\mathbf p,\\mathbf q)$ implies that for all $B$, $a\\models p$ and $b\\models q$:\n\\begin{enumerate}[\\hspace{10pt} (a)]\n\\item (Existence) \\ There exists a $a'\\models p$ such that $B\\triangleleft^{\\mathbf p}a'$;\n\\item (Density) \\ If $B\\triangleleft^{\\mathbf q} b$, then there is a $a'\\models p$ such that $B\\triangleleft^{\\mathbf p} a'\\triangleleft^{\\mathbf q} b$; \n\\item (Symmetry) \\ $a\\ind_A b$ if and only if $b\\ind_A a$; \n\\item \\ \n$a \\triangleleft^{\\mathbf q} b\\Leftrightarrow b\\triangleleft^{\\mathbf p^*} a$ \\ ($b$ is right $\\mathbf q$-generic over $a$ if and only if $a$ is left $\\mathbf p$-generic over $b$.) \n\\item ($\\triangleleft^{\\mathbf p}$-comparability) \\ $a\\ind_A B\\Leftrightarrow (B \\triangleleft^{\\mathbf p} a\\vee B\\triangleleft^{\\mathbf p^*} a)$\n\\ and \\ \n$a\\ind_A b\\Leftrightarrow (a \\triangleleft^{\\mathbf q} b\\vee b\\triangleleft^{\\mathbf p} a)$;\n\\item (Transitivity) \\ $B\\triangleleft^{\\mathbf p} a \\triangleleft^{\\mathbf q} b$ \\ implies \\ $B\\triangleleft^{\\mathbf q} b$; \n \\item $B\\triangleleft^{\\mathbf p} a \\land a\\dep_A b$ implies $B\\triangleleft^{\\mathbf q} b\\land a\\dep_{B}b$.\n\\end{enumerate}\n\\end{Corollary}\n\n\\begin{Corollary}\\label{Corolary_a_1_equiv_a_2_AB}\nSuppose that $p\\in S(A)$ is a weakly o-minimal type and $a_1,a_2\\models p$. Then for all $B$: \n\\begin{center}\n $a_1\\dep_A a_2 \\ \\land \\ a_1\\ind_A B$ \\ implies \\ $a_1\\equiv a_2 \\,(AB)$. \n\\end{center}\n\\end{Corollary}\n\\begin{proof}\nAssume $a_1\\dep_A a_2 \\ \\land \\ a_1\\ind_A B$. Choose $\\mathbf p=(p,<)$, a weakly o-minimal pair over $A$, such that $B\\triangleleft^{\\mathbf p} a_1$. Then by Corolary \\ref {Cor_triangle_mathbf_pq_properties}(g), applied to $\\mathbf p=\\mathbf q$, $B\\triangleleft^{\\mathbf p} a_1$ and $a_1\\dep_A a_2$ together imply $B\\triangleleft^{\\mathbf p} a_2$. Therefore, $a_1\\equiv a_2\\,(AB)$. \n\\end{proof}\n \n\n\n\\begin{Fact}[{\\cite[Corollary 5.14]{MTwom}}]\\label{Fact_delta_implies_deltaB_fworB}\nSuppose that $\\mathbf p,\\mathbf q$ are weakly o-minimal pairs over $A$ with $\\delta_A(\\mathbf p,\\mathbf q)$. Let $B\\supseteq A$. Define $p_B=(\\mathbf p_r)_{\\restriction B}$, $\\mathbf p_B=(p_B,<_p)$ and analogously define $q_B$ and $\\mathbf q_B$. \n \\begin{enumerate}[\\hspace{10pt} (a)]\n\\item $\\delta_B(\\mathbf p_B,\\mathbf q_B)$ holds; in particular, $p_B\\nwor q_B$ and $\\mathbf p_r\\nwor \\mathbf q_r$;\n\\item If $p\\nfor q$, then $p_B\\nfor q_B$ and, in particular, $\\mathbf p_r\\nfor\\mathbf q_r$. \n\\end{enumerate}\n\\end{Fact}\n\n\n\\begin{Fact}[{\\cite[Proposition 6.4]{MTwom}}]\n Every weakly o-minimal type is dp-minimal, so NIP. \n\\end{Fact}\n\n\\subsection{Interval types and convex types}\n\nLet $(D,<)$ be an $A$-definable linear order. A formula with parameters $\\phi(x)$ is said to be {\\em convex} if it implies $x\\in D$ and defines a $<$-convex subset of $D$. For a convex formula $\\phi(x)$, by $\\phi^-(x)$ and $\\phi^+(x)$ we will denote formulae $x\\in D\\land x<\\phi(\\Mon)$ and $x\\in D\\land x>\\phi(\\Mon)$ respectively. Note that both $\\phi^-(x)$ and $\\phi^+(x)$ are convex formulae given over the same parameters as $\\phi(x)$, and that $\\phi^-(\\Mon)<\\phi(\\Mon)<\\phi^+(\\Mon)$ is a convex partition of $D$.\n\nBy an {\\em interval type in $(D,<)$ over $B\\supseteq A$} we mean a maximal partial type $\\Pi(x)$ consisting of convex $L_B$-formulae. Since for each convex $L_B$-formula $\\phi(x)$ exactly one of $\\phi^-(x)$, $\\phi(x)$ and $\\phi^+(x)$ belongs to an interval type in $(D,<)$ over $B$, an interval type in $(D,<)$ over $B$ may be described as a consistent set of convex $L_B$-formulae $\\Pi(x)$ such that for each convex $L_B$-formula $\\phi(x)$ either $\\phi(x)\\in \\Pi(x)$ or $\\phi(x)$ is inconsistent with $\\Pi(x)$. \n\nThe set of all interval types in $(D,<)$ over $B\\supseteq A$ is denoted by $IT(B)$. Note that, although we do not emphasize it, $IT(B)$ depends on $A$, $D$ and $<$. However, these will always be clear from the context, so we feel that easing the notation is in order. The set $IT(B)$ is endowed with a compact Hausdorff topology in the usual way. \n\nFor $a\\in D$, {\\em the interval type of $a$ in $(D,<)$ over $B\\supseteq A$}, denoted by $\\itp(a/B)$ (we again suppress $A$, $D$ and $<$), is the set of all convex $L_B$-formulas that are satisfied by $a$. It is easy to see that $\\itp(a/B)\\in IT(B)$. For a type $p\\in S_x(B)$ implying $x\\in D$, by $p^{conv}$ we denote the subtype of all convex $L_B$-formulae from $p$. Again, it is easy to see that $p^{conv}\\in IT(B)$.\n\nIn the following lemma, we list some of the basic facts about interval types.\n\n\\begin{Lemma}\\label{Lemma_interval_types}\nLet $B\\supseteq A$.\n\\begin{enumerate}[\\hspace{10pt} (a)]\n \\item For all $\\Pi\\in IT(B)$, $\\Pi(\\Mon)$ is a convex subset of $D$.\n \\item For distinct $\\Pi_1,\\Pi_2\\in IT(B)$, $\\Pi_1(\\Mon)$ and $\\Pi_2(\\Mon)$ are disjoint and $<$-comparable.\n \\item The space $IT(B)$ is naturally linearly ordered by $<$.\n \\item For all $p\\in S_x(B)$ implying $x\\in D$, $p^{conv}(\\Mon)$ is the convex hull of $p(\\Mon)$.\n \\item If $\\Pi\\in IT(B)$ and if $p\\in S_x(B)$ is a completion of $\\Pi$, then $\\Pi(\\Mon)$ is the convex hull of $p(\\Mon)$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n(a) Clearly, $\\Pi(\\Mon)$ is convex as an intersection of convex sets.\nFor (b), if $\\phi(x)\\in\\Pi_1\\smallsetminus\\Pi_2$, then $\\phi(x)$ is inconsistent with $\\Pi_2$, so $\\Pi_1(\\Mon)\\cap\\Pi_2(\\Mon)=\\emptyset$.\nNow, (c) follows directly from (a) and (b).\n\n(d) It is clear that $p^{conv}(\\Mon)$ contains the convex hull of $p(\\Mon)$. If $a\\in D$ is not in the convex hull of $p(\\Mon)$, then $a<p(\\Mon)$ or $a>p(\\Mon)$. Without loss, suppose that the former holds. By compactness, there is a $\\theta(x)\\in p(x)$ such that $a<\\theta(\\Mon)$. Denote by $\\theta^{conv}(x)$ the formula that describes the convex hull of $\\theta(\\Mon)\\cap D$. Then $a<\\theta^{conv}(\\Mon)$ and $\\theta^{conv}(x)\\in p^{conv}$, so $a\\notin p^{conv}(\\Mon)$.\n\n(e) Clearly, $\\Pi\\subseteq p^{conv}$ holds, so $\\Pi=p^{conv}$ follows by the maximality of $\\Pi$, and the conclusion follows by (d).\n\\end{proof}\n\nIn the literature, the term ``convex type\" is commonly used to denote an interval type, but here we will follow \\cite{MT} and assign it a different meaning.\n\n\\begin{Definition}\\label{Definition_convex}\nA complete type $p\\in S(A)$ is {\\it convex} if there exists an $A$-definable linear order such that $p(\\Mon)$ is its convex subset.\n\\end{Definition}\n\nLet $p\\in S(A)$ be a convex type and let $<$ be a relatively $A$-definable linear order on $p(\\Mon)$. A priori, there is no reason why there should be an $A$-definable extension of $(p(\\Mon),<)$ in which $p(\\Mon)$ is convex. However, if $p$ is also weakly o-minimal, then we will prove in Proposition \\ref{Proposition_convex_type_witness} below that such extensions exist. We need the following lemma. \n\n\n\n\n\\begin{Lemma}\\label{Lemma_D_p_rel_def_convex}\nLet $\\mathbf p=(p,<)$ be a weakly o-minimal pair over $A$ and let $(D,<)$ be an $A$-definable extension of $(p(\\Mon),<)$. Then there is a family $\\Phi(x,y)=\\{\\phi_i(x,y)\\mid i\\in I\\}$ of $L_{A}$-formulae such that for all $a\\models p$, the disjunction $\\bigvee_{i\\in I}\\phi_i(x,a)$ relatively defines $\\mathcal D_p(a)$ within $p(\\Mon)$ and for all $i\\in I$ the set $\\phi_i(\\Mon,a)$ is a convex subset of $D$. \n\\end{Lemma}\n\\begin{proof}\nLet $\\Phi_0(x,a)$ denote the set of all relatively $p$-bounded $L_{Aa}$-formulae in variable $x$. \nBy Remark \\ref{Remark_pgeneric_first}(e), the disjunction $\\bigvee \\Phi_0(x,a)$ relatively defines $\\mathcal D_p(a)$ within $p(\\Mon)$. Define: \\\n\\begin{center}\n$\\Phi(x,a)=\\{\\phi(x,a)\\in \\Phi_0\\mid \\mbox{ $\\phi(\\Mon,a)$ is a convex subset of $D$}\\}$.\n\\end{center}\nWe claim that the family $\\Phi(x,y)$ satisfies the conclusion of the lemma: $\\bigvee \\Phi(x,a)$ relatively defines $\\mathcal D_p(a)$ within $p(\\Mon)$. To prove this, it suffices to show that for each formula $\\phi(x,a)\\in \\Phi_0(x,a)$ there exists a $\\phi^*(x,a)\\in \\Phi(x,a)$ such that $\\phi(p(\\Mon))\\subseteq \\phi^*(p(\\Mon))$. Let $\\phi(x,a)\\in \\Phi_0$ and let $b_1,b_2\\models p$ satisfy $b_1\\triangleleft^{\\mathbf p}a\\triangleleft^{\\mathbf p}b_2$. By Remark \\ref{Remark_trianglep_binary_basicprops} we have $b_1<\\mathcal D_p(a)<b_2$. \nSince the set $\\phi(p(\\Mon),a)$ is a subset of $\\mathcal D_p(a)$ we have: $p(x)\\cup\\{\\phi(x,a)\\}\\vdash b_1<x<b_2$. By compactness, there is a formula $\\theta(x)\\in p$, without loss implying $x\\in D$, such that $\\models \\theta(x)\\land \\phi(x,a)\\rightarrow b_1<x<b_2$. Let $\\phi^*(x,a)$ be a formula that defines the convex hull of $\\phi(\\Mon,a)\\cap \\theta(\\Mon)$ in $(D,<)$. Clearly, $b_1<\\phi^*(\\Mon,a)<b_2$, so $\\phi^*(x,a)$ is a relatively $p$-bounded formula and $\\phi(x,a)\\in \\Phi(x,a)$. As $\\theta(x)\\in p$, the formula $\\theta(x)\\land \\phi(x,a)$ relatively defines the same subset of $p(\\Mon)$ as $\\phi(x,a)$ does; clearly, this implies the desired conclusion $\\phi^*(p(\\Mon),a) \\supseteq \\phi(p(\\Mon),a)$ and completes the proof of the lemma. \n\\end{proof}\n\n\n\n\\begin{Proposition}\\label{Proposition_convex_type_witness} \nSuppose that $\\mathbf p=(p,<)$ is a weakly o-minimal pair over $A$ and that $p$ is convex. Then there is a formula $\\theta(x)\\in p$ such that $(\\theta(\\Mon),<)$ is an $A$-definable extension of $(p(\\Mon),<)$ and $p(\\Mon)$ is a convex subset of $\\theta(\\Mon)$; we will say that $\\theta(x)$ witnesses the convexity of $\\mathbf p$. \n\\end{Proposition}\n\\begin{proof}\nLet $(D,<)$ be an $A$-definable extension of $(p(\\Mon),<)$. Since $p$ is convex, there is an $A$-definable linear order $(D_p,<_p)$ that contains $p(\\Mon)$ as a convex subset; by replacing the domains $D$ and $D_p$ by their intersection, we may assume $D=D_p$. Denote $\\mathbf p=(p,<)$ and $\\mathfrak p=(p,<_p)$. Reversing the order $<_p$ if necessary, we may assume that $<$ and $<_p$ have the same orientation; that is, $\\mathbf p_r=\\mathfrak p_r$. \n\nLet $a\\models p$ and let $\\Phi(x,y)$ be the set of formulas given by Lemma \\ref{Lemma_D_p_rel_def_convex}: $\\bigvee\\Phi(x,a)$ relatively defines $\\mathcal D_p(a)$ within $p(\\Mon)$ and for all $\\phi(x,a)\\in \\Phi(x,a)$ the set $\\phi_i(\\Mon,a)$ is a convex subset of $D$. Then $\\{\\phi(\\Mon,a)<x\\mid \\phi(x,a)\\in \\Phi(x,a)\\}$ is a type, denote it by $\\bigvee\\Phi(D,a)<x$. Similarly, define the type $x<\\bigvee\\Phi(D,a)$. \nSince $\\bigvee\\Phi(x,a)$ relatively defines the set $\\mathcal D_p(a)$ within $p(\\Mon)$, we have $p(x)\\cup (\\bigvee\\Phi(D,a)<x)\\vdash (\\mathbf p_r)_{\\restriction Aa}(x)$ and, since $\\mathbf p_r=\\mathfrak p_r$, $p(x)\\cup (\\bigvee\\Phi(D,a)<x)\\vdash a<_px$. Similarly, for all $b\\models p$ we have $p(x)\\cup (x<\\bigvee\\Phi(D,b))\\vdash x<_p b$. So\n\\begin{equation*}p(x)\\cup \\left(\\bigvee\\Phi(D,a)< x\\right)\\vdash a<_p x \\ \\mbox{ and } \\ p(x)\\cup \\left(x<\\bigvee\\Phi(D,b)\\right)\\vdash x<_p b .\n\\end{equation*}\nBy compactness, there is a formula $\\theta(x)\\in p(x)$ implying $x\\in D$ such that \n\\begin{equation}\\{\\theta(x)\\}\\cup \\left(\\bigvee\\Phi(D,a)< x\\right)\\vdash a<_p x \\ \\mbox{ and } \\ \\{\\theta(x)\\}\\cup \\left(x<\\bigvee\\Phi(D,b)\\right)\\vdash x<_p b .\n\\end{equation}\n\\setcounter{equation}{0} \nWe claim that $\\theta(x)$ satisfies the conclusion of the lemma: $p(\\Mon)$ is convex in $(\\theta(\\Mon),<)$. To prove it, assume $a',b'\\models p$ and $\\models a'<c<b'\\land \\theta(c)$; we need to prove $c\\models p$. Choose $a,b\\models p$ satisfying $a\\triangleleft^{\\mathbf p}a'$ and $b'\\triangleleft^{\\mathbf p}b$. Since each formula from $\\Phi(x,a)$ defines a convex subset of $D$, the locus of the type $\\bigvee\\Phi(D,a)<x$ is a final part of $(D,<)$; clearly, it contains $a'$, so $a'<c\\in D$ implies $c\\models (\\bigvee\\Phi(D,a)<x)$, which together with $\\models \\theta(c)$, by (1), implies $a<_pc$. Similarly, $c<_pb$. Since $p(\\Mon)$ is convex in $(\\theta(\\Mon),<_p)$ and $a<_p c<_p b$, we conclude $c\\models p$, as desired. \n\\end{proof}\n\n\\setcounter{equation}{0} \n\n \n\\begin{Corollary}\\label{Corollary_convex_iff_isolated_in_IT}\nLet $(D,<)$ be an $A$-definable linear order, let $\\Pi\\in IT(A)$, and let $S_{\\Pi}=\\{p\\in S_1(A)\\mid \\Pi(x)\\subseteq p(x)\\}$. A weakly o-minimal type $p\\in S_{\\Pi}$ is convex if and only if it is an isolated point of $S_{\\Pi}$. \n\\end{Corollary}\n\\begin{proof}\nFirst, suppose that $p\\in S_\\Pi$ is convex. By Lemma \\ref{Proposition_convex_type_witness} there is a formula $\\theta(x)\\in p$ such that $p(\\Mon)$ is convex in $(\\theta(\\Mon),<)$. Replacing $\\theta(x)$ by $\\theta(x)\\land x\\in D$, we obtain $\\theta(\\Mon)\\subseteq D$, so $\\Pi(x)\\cup \\{\\theta(x)\\}\\vdash p(x)$; $p$ is an isolated point of $S_{\\Pi}$. \n\nTo prove the other implication, assume that $\\theta(x)\\in p$ (implies $x\\in D$ and) isolates $p$ within $S_\\Pi$; then $p(\\Mon)=\\Pi(\\Mon)\\cap\\theta(\\Mon)$. As $\\Pi(\\Mon)$ is convex in $(D,<)$, it follows that $p(\\Mon)$ is convex in $(\\theta(\\Mon),<)$. Therefore, $p$ is a convex type. \\end{proof}\n\n\n \\begin{Lemma}\\label{Lemma_wqom_interval_type}\nSuppose that $\\Th(\\Mon,<,\\dots)$ is weakly quasi-o-minimal.\n\\begin{enumerate}[\\hspace{10pt}(a)] \n\\item $\\delta_A(\\mathbf p,\\mathbf q)$ holds for all $p,q\\in S_1(A)$ extending the same interval type ($\\mathbf p=(p,<),\\mathbf q=(q,<)$). \n\\item If $\\mathcal F$ is a $\\delta_A$ class, then the order $<^{\\mathcal F}$ in Theorem \\ref{Theorem5} can be chosen so that for each $\\Pi\\in IT(A)$ with $\\Pi(\\Mon)\\cap\\mathcal F(\\Mon)\\neq\\emptyset$ one of $(<^{\\mathcal F})_{\\restriction \\Pi(\\Mon)}=\\ <$ and $(<^{\\mathcal F})_{\\restriction \\Pi}(\\Mon)=\\ >$ holds.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n(a) Let $\\Pi\\in IT(A)$ and let $p,q\\in S_{\\Pi}$ be distinct completions of $\\Pi$. First, we show $p\\nwor q$. We know $p^{conv}=q^{conv}=\\Pi$, so by Lemma \\ref{Lemma_interval_types}(d) the sets $p(\\Mon)$ and $q(\\Mon)$ have the same convex hull in $(\\Mon,<)$. Thus, for any $a\\models p$ there are $b_1,b_2\\models q$ such that $b_1<a<b_2$; clearly, $\\tp(ab_1)\\neq \\tp(ab_2)$ witnesses $p\\nwor q$. \nNext, we claim that $a\\triangleleft^{\\mathbf q} b$ implies $a<b$. \nAssume that $b$ is right $\\mathbf q$-generic over $a$. By Remark \\ref{Remark_pgeneric_first}(c), the set of all right $\\mathbf q$-generic elements over $a$, which is the locus of $\\tp(b/Aa)$, is a final part of $(q(\\Mon),<)$; since $q(\\Mon)$ is included in the convex hull of $p(\\Mon)$, some $\\mathbf q$-generic element, say $b'$, satisfies $a<b'$; $a<b$ follows, proving the claim. Similarly, we see that $b\\triangleleft^{\\mathbf p}a$ implies $b<a$.\nTherefore, $a\\triangleleft^{\\mathbf q}b\\triangleleft^{\\mathbf p}a$ is impossible. By Fact \\ref{Fact_notdelta} this implies $\\delta_A(\\mathbf p,\\mathbf q)$. \n\n(b) Follows from part (a) and the ``moreover\" part of Theorem \\ref{Theorem5}.\n\\end{proof}\n\n \\noindent\n{\\bf Convention.} Whenever the theory $T=\\Th(\\Mon,<,\\dots)$ is weakly quasi-o-minimal, $\\mathcal F$ is part of a $\\delta_A$-class, and $<^{\\mathcal F}$ satisfies the conclusion of Theorem \\ref{Theorem5}, we will assume that $<^{\\mathcal F}$ also satisfies the conclusion of the previous lemma.\n\n\n\\begin{Lemma}\\label{Lemma_forking_on_interval_type}\nSuppose that $\\Th(\\Mon,<,\\dots)$ is weakly quasi-o-minimal.\nFor each $\\Pi\\in IT(A)$ define: \n\\begin{center}$S_{\\Pi}=\\{p\\in S_1(A)\\mid \\Pi(x)\\subseteq p(x)\\} \\ \\ \\ \\mbox{ and } \\ \\ \\ \\mathcal D_{\\Pi}=\\{(a,b)\\in \\Pi(\\Mon)^2\\mid a\\dep_A b\\}.$\n\\end{center}\n\\begin{enumerate}[\\hspace{10pt}(a)] \n\\item $(a,b)\\in\\mathcal D_{\\Pi}$ \\ if and only if \\ $a,b\\in \\Pi(\\Mon)$ and $\\tp(a/Ab)$ contains a $\\Pi$-bounded formula.\n\\item $\\mathcal D_{\\Pi}$ is a convex equivalence relation on $\\Pi(\\Mon)$. \n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof} \n(a) Let $(a,b)\\in \\Pi(\\Mon)$ and let $p=\\tp(a/A)$. First, suppose that $(a,b)\\in\\mathcal D_{\\Pi}$. Then $a\\dep_A b$, so there is a relatively $p$-bounded formula $\\phi(x)\\in\\tp(a/Ab)$ in $(p(\\Mon),<)$; that is, there are $a_1,a_2\\models p$ such that $p(x)\\cup \\{\\phi(x)\\}\\vdash a_1<x<a_2$. \nBy compactness, there is $\\theta(x)\\in p$ such that \n$\\theta(x)\\land\\phi(x)\\vdash a_1<x<a_2$. Clearly, $\\theta(x)\\land \\phi(x)\\in \\tp(a/Ab)$ is a $\\Pi$-bounded formula, proving the left-to-right implication. To prove the other, it suffices to note that, since $\\Pi(\\Mon)$ is the convex hull of $p(\\Mon)$, every $\\Pi$-bounded formula $\\phi(x)\\in\\tp(a/Ab)$ is also $p$-bounded, so $\\phi(x)$ witnesses $a\\dep_Ab$. \n\n(b) Let $<^{\\mathcal F}$ be given by Lemma \\ref{Lemma_wqom_interval_type}(b). By Theorem \\ref{Theorem5}(b), $\\mathcal D_{\\mathcal F}=\\{(a,b)\\in \\mathcal F(\\Mon)^2\\mid a\\dep_A b\\}$ is a $<^{\\mathcal F}$-convex equivalence relation on $\\mathcal F(\\Mon)$. Since $\\mathcal D_{\\mathcal F\\restriction \\Pi(\\Mon)}=\\mathcal D_{\\Pi}$ and $(<^{\\mathcal F})$ is equal to $<$ or $>$ on $\\Pi(\\Mon)$, we conclude that $\\mathcal D_{\\Pi}$ is $<$-convex on $\\Pi(\\Mon)$. \n\\end{proof}\n\n\n\n\n\n\\section{Order-trivial types}\\label{Section_trivial}\n\n\n\nThe notion of triviality for types in stable theories was introduced and studied by Baldwin and Harrington in \\cite{BH}: a stationary type $p\\in S_x(A)$ is trivial if, for all $B\\supseteq A$, any pairwise independent triple of realizations of the nonforking extension of $p$ is independent (as a set) over $B$. They showed that triviality of the type $p$ is preserved in all nonforking extensions and restrictions, suggesting that triviality is a property of the global nonforking extension of $p$, which is also a unique $A$-invariant, global extension of $p$. Therefore, a global type $\\mathfrak p$ is trivial if for some (equivalently all) sets $A$ over which $\\mathfrak p$ is invariant, any pairwise independent over $A$ set of realizations of $\\mathfrak p_{\\restriction A}$ is independent over $A$. Now, if we want to consider an $A$-invariant type $\\mathfrak p$ in an arbitrary first-order theory, then it is natural to consider Morley sequences in $\\mathfrak p$ over $A$ as independent. \n\n\\begin{Definition}\\label{Def_trivial_global_type}\nA global type $\\mathfrak p\\in S_x(\\Mon)$ is {\\em trivial over $A$} if it is non-algebraic, $A$-invariant and whenever the members of the sequence $I=(a_i\\mid i\\in \\omega)$ are such that $(a_i,a_j)$ is a Morley sequence in $\\mathfrak p$ over $A$ for all $i<j\\in\\omega$, then $I$ is a Morley sequence in $\\mathfrak p$ over $A$. \n\\end{Definition}\n\n \n\\begin{Remark}\\phantomsection\\label{Remark_basic_trivial}\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item Let $T$ be stable and let $p\\in S_x(A)$ be stationary and non-algebraic. Denote by $\\mathfrak p$ the global nonforking extension of $p$. Then $p$ is trivial in the Baldwin-Harrington sense if and only if $\\mathfrak p$ is trivial over $A$ in the sense of Definition \\ref{Def_trivial_global_type}.\n\n\\item If $T$ is a binary theory, then any non-algebraic, $A$-invariant global type is trivial over $A$. Examples of binary theories are: the theory of random graphs, theories of colored orders, etc.\n\n\\item If a global type $\\mathfrak p$ is trivial over $A$, then so is every finite power $\\mathfrak p^n$.\n\\end{enumerate}\n\\end{Remark}\n\n\\begin{Remark}\\label{Remark_trivial_basic2}\nLet $\\mathfrak p\\in S_x(\\Mon)$ be a non-algebraic, $A$-invariant type.\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item An equivalent way to state that $\\mathfrak p$ is trivial over $A$ is: for every $n\\in\\mathbb N$, the type \\ $\\bigcup_{i<j< n}(\\mathfrak p^2)_{\\restriction A}(x_i,x_j)$ \\ implies a unique complete type over $A$ in variables $x_0,\\dots,x_{n-1}$ (in which case this type must be $(\\mathfrak p^n)_{\\restriction A}(x_0,\\dots,x_{n-1})$).\n\n\\item Similarly as in \\cite{BH}, by a {\\em $\\mathfrak p$-triangle} over $A$ we will mean a triplet $(a_0,a_1,a_2)$ of realizations of $\\mathfrak p_{\\restriction A}$ which is not a Morley sequence in $\\mathfrak p$ over $A$, but each pair $(a_i,a_j)$, $i<j<3$, is. When $\\mathfrak p$ is not trivial over $A$, we can find a finite (possibly empty) Morley sequence $\\bar a$ in $\\mathfrak p$ over $A$ such that $\\mathfrak p$ is not trivial over $A\\bar a$, as witnessed by a triangle. To do this, choose minimal $n$ such that some sequence of size $n+3$, say $(a_0,a_1,\\ldots, a_{n+2})$, realizes the type $\\bigcup_{i<j< n+3}(\\mathfrak p^2)_{\\restriction A}(x_i,x_j)$ but does not realize $(\\mathfrak p^n)_{\\restriction A}(x_0,\\dots,x_{n+2})$. Then put $\\bar a=(a_0,\\ldots,a_{n-1})$ and observe that $(a_n,a_{n+1},a_{n+2})$ is a $\\mathfrak p$-triangle over $\\bar aA$. \n\\end{enumerate}\n\\end{Remark}\n\n Poizat in \\cite{Goode} introduced triviality in the context of stable theories. A stable theory $T$ is trivial if, for all parameter sets $A$, any family of pairwise independent tuples over $A$ is independent over $A$. \n It is easy to see that every global non-algebraic type in a trivial, stable theory is trivial over any parameter set over which it is based. \nThe following fact is worth noting, but since we will not use it later, we leave the proof to the reader.\n\n \n\\begin{Fact} \nA complete theory $T$ is stable and trivial if and only if every global non-algebraic type is trivial (in the sense of Definition \\ref{Def_trivial_global_type}) over some small set of parameters. \n\\end{Fact}\n \n\nBelow, we will introduce order-triviality, a strong form of triviality for global types. The motivation for this comes from the weakly o-minimal case: we will prove in Lemma \\ref{Lemma wom left trivial iff right trivial} that for every weakly o-minimal pair $(p,<)$ over $A$, the type $\\mathbf p_r$ is trivial over $A$ if and only if it is order-trivial over $A$. \n\n\n\n\n \n\n\\begin{Definition}\nA global type $\\mathfrak p\\in S_x(\\Mon)$ is {\\em order-trivial over $A$} if it is non-algebraic, $A$-invariant and whenever the sequence $I=(a_i\\mid i\\in \\omega)$ is such that $(a_i,a_{i+1})$ is a Morley sequence in $\\mathfrak p$ over $A$ for all $i\\in\\omega$, then $I$ is a Morley sequence in $\\mathfrak p$ over $A$. \n\\end{Definition}\n\n\n\n\\begin{Remark}\\phantomsection\\label{Remark_basic_ordertrivial}\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item A global $A$-invariant type $\\mathfrak p\\in S_x(\\Mon)$ is order-trivial over $A$ if and only if \nfor every $n\\in\\mathbb N$, the type $\\bigcup_{i< n}(\\mathfrak p^2)_{\\restriction A}(x_i,x_{i+1})$ implies a unique complete type over $A$ in variables $x_0,\\dots,x_{n-1}$ (in which case this type must be $(\\mathfrak p^{n+1})_{\\restriction A}(x_0,\\dots,x_{n})$).\n\n\\item If $\\mathfrak p$ is order-trivial over $A$, then $\\mathfrak p$ is trivial over $A$ and every power $\\mathfrak p^n$ is order-trivial over $A$.\n\n\\item A non-algebraic symmetric global invariant type $\\mathfrak p$ is {\\it not} order-trivial over a small set of parameters (where $\\mathfrak p$ is symmetric if $\\mathfrak p^2(x,y)=\\mathfrak p^2(y,x)$). \nIndeed, if $\\mathfrak p$ is symmetric and $A$-invariant, take $(a_0,a_1)\\models \\mathfrak p^2_{\\restriction A}$ and consider the sequence $a_0,a_1,a_0,a_1,\\dots$; clearly, this is not a Morley sequence in $\\mathfrak p$ over $A$, although each consecutive pair is. In particular, there are no order-trivial types in stable theories nor in the theory of random graphs. \n\\end{enumerate}\n\\end{Remark}\n\n\nAll examples of indiscernible sequences of order-trivial types that we know of are strictly increasing with respect to some definable partial order; yet, it remains uncertain if this is always the case.\n\n\\begin{Question}\nIf $\\mathfrak p(x)\\in S_x(\\Mon)$ is order-trivial over $A$, must there exist a $B\\supseteq A$ and a $B$-definable partial order such that all Morley sequences in $\\mathfrak p$ over $B$ are strictly increasing? \n\\end{Question}\n\n \n\\begin{Proposition}\\label{Prop_basic_order_trivial}\nSuppose that $\\mathfrak p\\in S(\\Mon)$ is order-trivial over $A$ and that the type $p=\\mathfrak p_{\\restriction A}$ is NIP. Let $I=(a_i\\mid i\\in \\omega)$ be a Morley sequence in $\\mathfrak p$ over $A$ and let $B\\supseteq A$. Then $I$ is a Morley sequence in $\\mathfrak p$ over $B$ if and only if $a_0\\models \\mathfrak p_{\\restriction B}$.\n\\end{Proposition}\n\\begin{proof} The left-to-right implication is clear. For the converse, suppose on the contrary that $a_0\\models \\mathfrak p_{\\restriction B}$ but there is $n\\in\\omega$ such that $\\bar a=(a_0,\\dots,a_{n-1})$ is not Morley in $\\mathfrak p$ over $B$. Choose a formula $\\phi(\\bar x,b)\\in \\tp(\\bar a/B)$ such that $\\phi(\\bar x,b)\\notin \\mathfrak p^n$. Define the sequence $J=(\\bar c^i\\mid i\\in\\omega)$, where $\\bar c^i=(c_0^i,\\dots,c_{n-1}^i)\\models (\\mathfrak p^n)_{\\restriction A}$, in the following way: Let $\\bar c^0=\\bar a$. Suppose that $J_{<k}=(\\bar c^i\\mid i<k)$ has already been defined. First choose $c_0^k\\models \\mathfrak p_{\\restriction Bc_{n-1}^{k-1}}$ and then $\\bar c^k=(c_0^k,\\dots,c_{n-1}^k)$ such that:\n\\begin{equation}\\tag{$\\dagger$}\\label{eq prop 2.9}\n\\bar c^{k}\\equiv \\bar a\\,(B)\\mbox{ if $k$ is even \\ \\ and \\ \\ }\\bar c^{k}\\models \\mathfrak p^n_{\\restriction B}\\mbox{ if $k$ is odd}.\n\\end{equation}\nNote that this is possible as $c_0^k,a_0\\models\\mathfrak p_{\\restriction B}$.\nConsider the sequence: $$c_0^0,\\dots,c_{n-1}^0,c_0^1,\\dots,c_{n-1}^1,\\dots, c_0^k,\\dots,c_{n-1}^k,\\dots.$$\nNote that, by our choice of $c_0^k$ and $\\bar c^k$, any pair of consecutive elements of this sequence\nrealizes the type $(\\mathfrak p^2)_{\\restriction A}$, so, by order-triviality, the sequence is Morley in $\\mathfrak p$ over $A$. It follows that $J$ is a Morley sequence in $\\mathfrak p^n$ over $A$. \nBy our choice of $\\phi(\\bar x,b)$, condition (\\ref{eq prop 2.9}) implies: $\\models \\phi(\\bar c^k,b)$ if and only if $k$ is even. Therefore, the indiscernible sequence $J$ and the formula $\\phi(\\bar x,b)$ show that the type $(\\mathfrak p^n)_{\\restriction A}$ is not NIP. Contradiction!\n\\end{proof}\n\n\\begin{Corollary}\\label{Cor_otrivialoverA_otrivialoverB}\nSuppose that $\\mathfrak p\\in S(\\Mon)$ is order-trivial over $A$ and that the type $p=\\mathfrak p_{\\restriction A}$ is NIP. Then $\\mathfrak p$ is order-trivial over any $B\\supseteq A$.\n\\end{Corollary}\n\\begin{proof}\nLet $I=(a_i\\mid i\\in\\omega)$ be a sequence of realizations of $\\mathfrak p_{\\restriction B}$ such that $(a_i,a_{i+1})\\models (\\mathfrak p^2)_{\\restriction B}$ for all $i\\in\\omega$. We need to show that $I$ is a Morley sequence in $\\mathfrak p$ over $B$. Note that $(a_i,a_{i+1})\\models(\\mathfrak p^2)_{\\restriction A}$ for each $i\\in\\omega$, so by order-triviality of $\\mathfrak p$ over $A$, $I$ is a Morley sequence in $\\mathfrak p$ over $A$. Since $a_0$ realizes $\\mathfrak p_{\\restriction B}$, by Proposition \\ref{Prop_basic_order_trivial}, $I$ is a Morley sequence in $\\mathfrak p$ over $B$. Therefore, $\\mathfrak p$ is order-trivial over $B$. \n\\end{proof}\n\n\\begin{Corollary}\\label{Cor wom lifts to omega for o trivial}\nLet $\\mathfrak p,\\mathfrak q\\in S(\\Mon)$ be order-trivial over $A$ such that $p=\\mathfrak p_{\\restriction A}$ and $q=\\mathfrak q_{\\restriction A}$ are NIP types. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item If $r\\in S(A)$ and $p\\wor r$, then $\\mathfrak (\\mathfrak p^{\\omega})_{\\restriction A}\\wor r$.\n\\item If $p\\wor q$ then $(\\mathfrak p^{\\omega})_{\\restriction A}\\wor (\\mathfrak q^{\\omega})_{\\restriction A}$.\n\\end{enumerate}\n\\end{Corollary}\n\\begin{proof}\n(a) Let $c\\models r$, $\\bar a=(a_0,a_1,\\dots)\\models(\\mathfrak p^\\omega)_{\\restriction Ac}$ and $\\bar a'=(a_0',a_1',\\dots)\\models(\\mathfrak p^\\omega)_{\\restriction A}$; we need to prove $\\bar a\\equiv\\bar a'\\,(Ac)$. Since $p\\wor r$, $a_0\\equiv a_0'\\,(Ac)$, so $a_0'\\models \\mathfrak p_{\\restriction Ac}$. By Proposition \\ref{Prop_basic_order_trivial}, $\\bar a'\\models(\\mathfrak p^\\omega)_{\\restriction Ac}$, and we are done.\n\n(b) By (a), $(\\mathfrak p^\\omega)_{\\restriction A}\\wor q$, so, again by (a), $(\\mathfrak p^\\omega)_{\\restriction A}\\wor (\\mathfrak q^\\omega)_{\\restriction A}$.\n\\end{proof}\n\n\n\n\n\n\\section{Trivial weakly o-minimal types}\\label{Section_trivial_wom}\n \n\n \n\n\n\\begin{Lemma}\\label{Lemma wom left trivial iff right trivial}\nLet $\\mathbf p=(p,<)$ be a weakly o-minimal pair over $A$. The following are equivalent: \n\\begin{enumerate}[\\hspace{10pt}(1)]\n\\item $\\mathbf p_r$ is trivial over $A$; \n\\item Every $\\triangleleft^{\\mathbf p}$-increasing sequence of realizations of $p$ is Morley in $\\mathbf p_r$ over $A$; \n\\item $\\mathbf p_r$ is order-trivial over $A$;\n\\item Every $\\triangleleft^{\\mathbf p}$-decreasing sequence of realizations of $p$ is Morley in $\\mathbf p_l$ over $A$; \n\\item $\\mathbf p_l$ is order-trivial over $A$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nRecall that $(p(\\Mon), \\triangleleft^{\\mathbf p})$ is a strict partial order and that, by Remark \\ref{Remark_trianglep_binary_basicprops}(b), $a\\triangleleft^{\\mathbf p}b$, $(a,b)\\models (\\mathbf p_r^2)_{\\restriction A}$, and $(b,a)\\models (\\mathbf p_l^2)_{\\restriction A}$ are mutually equivalent for all $a,b\\models p$; in particular, $(\\mathbf p_l^2\\restriction A)(x,y)=(\\mathbf p_r^2\\restriction A)(y,x)$. \n\n\n(1)$\\Rightarrow$(2) Suppose that $\\mathbf p_r$ is trivial over $A$ and let $I=(a_i\\mid i\\in \\omega)$ be a $\\triangleleft^{\\mathbf p}$-increasing sequence. Then for all $i<j<\\omega$ we have $a_i\\triangleleft^{\\mathbf p} a_j$; hence, $(a_i,a_j)\\models (\\mathbf p_r^2)_{\\restriction A}$. Due to the triviality of $\\mathbf p_r$, $I$ is a Morley sequence in $\\mathbf p_r$ over $A$. \n\n (2)$\\Leftrightarrow$(3) Every $\\triangleleft^\\mathbf p$-increasing sequence is Morley if and only if:\n\\begin{enumerate}\n\\item[(r)] For all $n\\in \\mathbb N$ the type $\\bigcup_{i<n}(\\mathbf p_r^2)_{\\restriction A}(x_i,x_{i+1})$ has a unique completion over $A$.\n\\end{enumerate}\nBy Remark \\ref{Remark_basic_ordertrivial}, this is equivalent to the order-triviality of $\\mathbf p_r$ over $A$. \n\nAs (3)$\\Rightarrow$(1) is immediate from the definitions, we conclude (1)$\\Leftrightarrow$(2)$\\Leftrightarrow$(3). Similarly, (4)$\\Leftrightarrow$(5). \n\n (2)$\\Leftrightarrow$(4) Every $\\triangleleft^{\\mathbf p}$-increasing sequence of realizations of $p$ is Morley in $\\mathbf p_r$ over $A$ if and only if condition (r) holds. \nAs $(\\mathbf p_l^2)_{\\restriction A}(x,y)=(\\mathbf p_r^2)_{\\restriction A}(y,x)$, condition (r) is equivalent with:\n\\begin{enumerate}\n\\item[(l)] For all $n\\in \\mathbb N$ the type $\\bigcup_{i<n}(\\mathbf p_l^2)_{\\restriction A}(x_i,x_{i+1})$ has a unique completion over $A$,\n\\end{enumerate}\nwhich holds if and only if every $\\triangleleft^{\\mathbf p}$-decreasing sequence of realizations of $p$ is Morley in $\\mathbf p_l$ over $A$.\n\\end{proof}\n\n\n\\begin{Definition}\nA weakly o-minimal type $p\\in S(A)$ is {\\em trivial} if one (equivalently, both) of its $A$-invariant globalizations is (order-) trivial over $A$.\n\\end{Definition}\n\n\\noindent{\\bf Convention.} \\ For $\\mathbf p=(p,<)$ a weakly o-minimal pair over $A$, by $x_1\\triangleleft^{\\mathbf p} x_2\\triangleleft^{\\mathbf p} \\dots \\triangleleft^{\\mathbf p}x_{n}$ we will denote the partial type $\\bigcup_{1\\leqslant i< n} (\\mathbf p_r^2)_{\\restriction A}(x_i,x_{i+1})$.\n\n \n\\begin{Remark}\\label{Remark trivial wom expressing via triangleleft}\nLet $\\mathbf p=(p,<)$ be a weakly o-minimal pair over $A$. As a consequence of Lemma \\ref{Lemma wom left trivial iff right trivial} we have the following: \n\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item $p$ is trivial if and only if $x_1\\triangleleft^{\\mathbf p} x_2\\triangleleft^{\\mathbf p} \\dots \\triangleleft^{\\mathbf p}x_{n}$ determines a complete type for all $n\\in\\mathbb N$. \n\n\\item If $p$ is trivial, then a sequence $I=(a_i\\mid i< \\omega)$ of realizations of $p$ is Morley in a nonforking globalization of $p$ over $A$ if and only if $I$ is strictly $\\triangleleft^{\\mathbf p}$-monotone for some (all) relatively $A$-definable order $<$; equivalently, if the sequence $(\\mathcal D_p(a_i)\\mid i<\\omega)$ is strictly $<$-monotone. \n\\end{enumerate}\n\\end{Remark}\n\n\n\\begin{Lemma}\\label{Lemma_triv_preserved_in_nonforking}\nLet $\\mathbf p=(p,<)$ be a weakly o-minimal pair over $A$. Suppose that $p$ is trivial.\n\\begin{enumerate}[\\hspace{10pt}(a)]\n \\item Every nonforking extension of $p$ is trivial.\n \\item The sequence $I=(a_i\\mid i< \\omega)$ of realizations of $p$ is a Morley sequence in $\\mathbf p_r$ over $B\\supseteq A$ if and only if \\ $B\\triangleleft^{\\mathbf p} a_0\\triangleleft^{\\mathbf p}a_1\\triangleleft^{\\mathbf p} a_2\\triangleleft^{\\mathbf p}\\dots$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}(a) Let $q\\in S(B)$ be a nonforking extension of $p$. \nThen $q=(\\mathbf p_r)_{\\restriction B}$ or $q=(\\mathbf p_l)_{\\restriction B}$. Since $p$ is trivial, according to Lemma \\ref{Lemma wom left trivial iff right trivial} both $\\mathbf p_r$ and $\\mathbf p_l$ are order-trivial over $A$, so by Corollary \\ref{Cor_otrivialoverA_otrivialoverB}, they are order-trivial over $B$. In particular, $q$ is trivial. \n\n(b) First, assume \\ $B\\triangleleft^{\\mathbf p} a_0\\triangleleft^{\\mathbf p}a_1\\triangleleft^{\\mathbf p} a_2\\triangleleft^{\\mathbf p}\\dots$. Since $I$ is strictly $\\triangleleft^{\\mathbf p}$-increasing, by Remark \\ref{Remark trivial wom expressing via triangleleft}, it is a Morley sequence in $\\mathbf p_r$ over $A$. Since $a_0\\models (\\mathbf p_r)_{\\restriction B}$, Proposition \\ref{Prop_basic_order_trivial} applies: $I$ is a Morley sequence in $\\mathbf p_r$ over $B$. This proves one direction of the equivalence. To prove the other, assume $I$ is Morley in $\\mathbf p_r$ over $B$. Then $a_0\\models (\\mathbf p_r)_{\\restriction B}$ implies $B\\triangleleft^{\\mathbf p} a_0$. Since $I$ is Morley over $A$, $a_0\\triangleleft^{\\mathbf p}a_1\\triangleleft^{\\mathbf p} a_2\\triangleleft^{\\mathbf p}\\dots$ follows. Therefore, \\ $B\\triangleleft^{\\mathbf p}a_0\\triangleleft^{\\mathbf p}a_1\\triangleleft^{\\mathbf p} a_2\\triangleleft^{\\mathbf p}\\dots$ \n\\end{proof}\n\n In the following lemma, we establish an important property of trivial types, which will be referred to as the strong transitivity property later in the text. \n \n\\begin{Lemma}[Strong transitivity]\\label{Lemma_trivial_strong_trans}\nSuppose that $\\mathbf p$ and $\\mathbf q$ are directly nonorthogonal weakly o-minimal pairs over $A$ and that $p$ is trivial. Then $B\\triangleleft^{\\mathbf p}a\\triangleleft^{\\mathbf q} b$ implies $Ba\\triangleleft^{\\mathbf q} b$ for all parameters $B$. \n\\end{Lemma}\n\\begin{proof}\nAssume $B\\triangleleft^{\\mathbf p}a\\triangleleft^{\\mathbf q}b$. Choose $a'$ with $a\\triangleleft^{\\mathbf p}a'\\triangleleft^{\\mathbf q}b$; that is possible by the density property (Corollary \\ref {Cor_triangle_mathbf_pq_properties}(b)). By Lemma \\ref{Lemma_triv_preserved_in_nonforking}(b), $B\\triangleleft^{\\mathbf p}a\\triangleleft^{\\mathbf p}a'$ implies that $(a,a')$ is a Morley sequence in $\\mathbf p_r$ over $B$, so $Ba\\triangleleft^{\\mathbf p} a'$ holds. By the transitivity property (Corollary \\ref {Cor_triangle_mathbf_pq_properties}(6)) $Ba\\triangleleft^{\\mathbf p}a'\\triangleleft^{\\mathbf q}b$ implies $Ba\\triangleleft^{\\mathbf q}b$.\n\\end{proof}\n\n\n\\begin{Lemma}\\label{Lemma_trivial_iff_Dp=Dq}\nA weakly o-minimal type $p\\in S(A)$ is trivial if and only if $\\mathcal D_q(a)=\\mathcal D_p(a)$ for all nonforking extensions $q$ of $p$ and all $a\\models q$.\n\\end{Lemma}\n\\begin{proof} Let $\\mathbf p=(p,<)$ be a weakly o-minimal pair over $A$. \n\n($\\Rightarrow$) Assume that $p$ is trivial. Let $q\\in S(B)$ be a nonforking extension of $p$ and let $a\\models q$; we need to prove $\\mathcal D_p(a)=\\mathcal D_q(a)$. As $q$ is a nonforking extension of $p$, $a$ is left or right $\\mathbf p$-generic over $B$; without loss, assume $B\\triangleleft^{\\mathbf p} a$. \nBy Theorem \\ref{Theorem_triangle_mathcal F}(g) we know that for all $b\\models p$, $B\\triangleleft^{\\mathbf p}a\\land a\\dep_A b$ implies $B\\triangleleft^{\\mathbf p}b\\land a\\dep_Bb$; hence,\n$\\mathcal D_p(a)\\subseteq \\mathcal D_q(a)$. If this inclusion were proper, then there would be $a'\\in\\mathcal D_q(a)$ with $a\\ind_A a'$; in particular, $a'\\models q$ and thus $B\\triangleleft^{\\mathbf p}a'$. By the $\\triangleleft^{\\mathbf p}$-comparability property, Corollary \\ref {Cor_triangle_mathbf_pq_properties}(5), $a\\ind_A a'$ implies $a\\triangleleft^{\\mathbf p}a'$ or $a'\\triangleleft^{\\mathbf p}a$. In the first case, we have $B\\triangleleft^{\\mathbf p}a\\triangleleft^{\\mathbf p}a'$, which,\nby Lemma \\ref{Lemma_triv_preserved_in_nonforking}(b), implies that $(a,a')$ is Morley over $B$ and, in particular, $a'\\ind_B a$; this contradicts $a'\\in \\mathcal D_q(a)$. The second case is dealt with similarly. Therefore, $\\mathcal D_p(a)=\\mathcal D_q(a)$, as desired. \n\n($\\Leftarrow$) Suppose that $\\mathcal D_p(a)=\\mathcal D_q(a)$ holds for all nonforking extensions $q$ and all $a\\models q$. \nWe will prove that every $\\triangleleft^{\\mathbf p}$-increasing sequence $I=(a_i\\mid i\\in \\omega)$ of realizations of $p$ is Morley in $\\mathbf p_r$ over $A$; by Lemma \\ref{Lemma wom left trivial iff right trivial}, this implies that $p$ is trivial.\nBy induction on $n>0$, we prove that $(a_0,a_1,\\dots,a_n)$ is Morley (in $\\mathbf p_r$ over $A$). The case $n=1$ is clear, so assume that $(a_0,a_1,\\dots,a_n)$ is Morley for some $n>0$; in particular, $a_{<n}\\triangleleft^{\\mathbf p} a_n\\triangleleft^{\\mathbf p}a_{n+1}\\triangleleft^{\\mathbf p}\\ldots$.\nPut $q=(\\mathbf p_r)_{\\restriction Aa_{<n}}$ and $\\mathbf q=(q,<)$. Then $q$ is a nonforking extension of $p$, $\\mathbf p_r=\\mathbf q_r$, and $a_n,a_{n+1}\\models q$. By the above assumption, we have $\\mathcal D_p(a_n)=\\mathcal D_q(a_{n})$ and $\\mathcal D_p(a_{n+1})=\\mathcal D_q(a_{n+1})$. As $I$ is $\\triangleleft^{\\mathbf p}$-increasing, we get $\\mathcal D_p(a_n)<\\mathcal D_p(a_{n+1})$; hence $\\mathcal D_q(a_n)<\\mathcal D_q(a_{n+1})$; the latter implies that $(a_n,a_{n+1})$ is Morley in $\\mathbf q_r$ over $Aa_{<n}$, so $(a_0,\\ldots,a_n,a_{n+1})$ is Morley in $\\mathbf p_r$ over $A$, finishing the proof. \n\\end{proof}\n\n \n\n\\begin{Lemma}\\label{Lemma trivial wom morley in nonforking is left or right morley}\nLet $\\mathbf p=(p,<)$ be a weakly o-minimal pair over $A$ and let $q\\in S(B)$ be a nonforking extension of $p$. Suppose that $\\mathfrak q$ is a global nonforking extension of $q$. Then every\nMorley sequence in $\\mathfrak q$ over $B$ is a Morley sequence in $\\mathbf p_r$ or $\\mathbf p_l$ over $A$.\n \\end{Lemma}\n\\begin{proof}By Lemma \\ref{Lemma_triv_preserved_in_nonforking}, the triviality of $p$ is preserved in nonforking extensions, so $q$ is trivial. \nSuppose that $I=(a_i\\mid i\\in\\omega)$ is a Morley sequence in $\\mathfrak q$ over $B$. By Remark \\ref{Remark trivial wom expressing via triangleleft}, the sequence $(\\mathcal D_q(a_i)\\mid i\\in\\omega)$ is strictly $<$ monotone, so by\nLemma \\ref{Lemma_trivial_iff_Dp=Dq} the sequence $(\\mathcal D_p(a_i)\\mid i\\in\\omega)$\n is also strictly $<$-monotone; by Remark \\ref{Remark trivial wom expressing via triangleleft} again, $I$ is a Morley sequence in $\\mathbf p_r$ or $\\mathbf p_l$ over $A$.\n\\end{proof}\n\n\n\n\\begin{Remark}\\label{Remark_notrivial_wom_witness}Let $\\mathbf p=(p,<)$ be a weakly o-minimal pair over $A$. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item Recall that a sequence $(a_1,a_2,a_3)$ of realizations of $p$ is a $\\mathbf p_r$-triangle over $A$ if it is not Morley in $\\mathbf p_r$ over $A$, but every pair $(a_i,a_j) \\ (i<j)$ is; that is, $a_1 \\triangleleft^{\\mathbf p}a_2\\triangleleft^{\\mathbf p} a_3$ and $a_1a_2a_3\\not\\models (\\mathbf p_r^3)_{\\restriction A}(x,y,z)$. \nIn other words, $(a_1,a_2,a_3)$ is a $\\mathbf p_r$-triangle over $A$ if and only if the type $\\tp_{xyz}(a_1a_2a_3/A)$ is a complete extension of the partial type $x\\triangleleft^{\\mathbf p} y\\triangleleft^{\\mathbf p}z$ that is different from $(\\mathbf p_r^3)_{\\restriction A}(x,y,z)$. In particular, the type $x\\triangleleft^{\\mathbf p} y\\triangleleft^{\\mathbf p}z$ is incomplete if and only if $\\mathbf p_r$-triangles over $A$ exist.\n\\item Similarly as in (a), we see that the types of $\\mathbf p_l$-triangles over $A$ correspond to completions of $x\\triangleleft^{\\mathbf p} y\\triangleleft^{\\mathbf p}z$ that are different from $(\\mathbf p_l^3)_{\\restriction A}(z,y,x)$. In particular, $\\mathbf p_r$-triangles over $A$ exist if and only if $\\mathbf p_l$-triangles over $A$ exist. \n\\item $(a_1,a_2,a_3)$ is a $\\mathbf p_r$-triangle over $A$ if $a_1 \\triangleleft^{\\mathbf p}a_2\\triangleleft^{\\mathbf p} a_3$ and $a_1a_2\\not\\triangleleft^{\\mathbf p} a_3$. \nSince by Theorem \\ref{Theorem_triangle_mathcal F}, $a_1 \\triangleleft^{\\mathbf p}a_2\\triangleleft^{\\mathbf p} a_3$ and $a_2\\ind _{Aa_1} a_3$ together imply $a_1a_2\\triangleleft^{\\mathbf p}a_3$, $(a_1,a_2,a_3)$ is a $\\mathbf p_r$-triangle over $A$ if and only if \\ $a_1 \\triangleleft^{\\mathbf p}a_2\\triangleleft^{\\mathbf p} a_3 \\ \\ \\mbox{and} \\ \\ a_2\\dep_{Aa_1} a_3.$ \n \\item \nBy Remark \\ref{Remark_trivial_basic2} the nontriviality of $\\mathbf p_r$ over $A$ can be witnessed by a finite (possibly empty) Morley sequence $\\bar b$ in $\\mathbf p_r$ over $A$ and elements $a_1,a_2,a_3$ that form a $\\mathbf p_r$-triangle over $A\\bar b$. \nThis can be expressed by: \\ $\\bar b\\triangleleft^{\\mathbf p} a_1 \\triangleleft^{\\mathbf p_{\\bar b}}a_2\\triangleleft^{\\mathbf p_{\\bar b}} a_3 \\ \\ \\ \\mbox{and} \\ \\ \\ a_2\\dep_{A\\bar ba_1} a_3$ \\ (where $\\mathbf p_{\\bar b}=((\\mathbf p_r)_{\\restriction A\\bar b},<)$).\n\\end{enumerate}\n\\end{Remark} \n\n \n\n\\begin{Lemma}\nThe following are equivalent for a weakly o-minimal pair $\\mathbf p=(p,<)$ over $A$. \n\\begin{enumerate}[\\hspace{10pt} (1)]\n \\item There exists a $\\mathbf p_r$-triangle over $A$; \n \\item There exists a $\\mathbf p_l$-triangle over $A$;\n \\item $(\\mathbf p_l)_{\\restriction Aa}\\nwor (\\mathbf p_r)_{\\restriction Aa}$ \\ for all $a\\models p$. \n \\item The type $x\\triangleleft^{\\mathbf p}y\\triangleleft^{\\mathbf p}z$ has at least two completions in $S_{xyz}(A)$;\n\\item $(\\mathbf p_r^3)_{\\restriction A}(x,y,z)\\neq (\\mathbf p_l^3)_{\\restriction A}(z,y,x)$;\n \\item If $(a,b,c)\\models (\\mathbf p_r^3)_{\\restriction A}(x,y,z)$, then $(c,b,a)$ is a $\\mathbf p_l$-triangle over $A$ (that is, $a\\dep_{Ac}b$); \n \\item If $(a,b,c)\\models (\\mathbf p_l^3)_{\\restriction A}(x,y,z)$, then $(c,b,a)$ is a $\\mathbf p_r$-triangle over $A$ (that is, $a\\dep_{Ac}b$).\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n(1)$\\Leftrightarrow$(2)$\\Leftrightarrow$(4) and (5)$\\Leftrightarrow$(6)$\\Leftrightarrow$(7) follow by Remark \\ref{Remark_notrivial_wom_witness}. \n\n(3)$\\Leftrightarrow$(4) Let $a\\models p$. The type $x\\triangleleft^{\\mathbf p}y\\triangleleft^{\\mathbf p}z$ has at least two completions in $S_{xyz}(A)$ if and only if the type $x\\triangleleft^{\\mathbf p}a\\triangleleft^{\\mathbf p}z$ has at least two completions in $S_{xz}(Aa)$. The latter is equivalent to $(\\mathbf p_l)_{\\restriction Aa}\\nwor (\\mathbf p_r)_{\\restriction Aa}$, because $p(x)\\cup \\{x\\triangleleft^{\\mathbf p}a\\}$ is equivalent to $(\\mathbf p_l)_{\\restriction Aa}(x)$ and $p(z)\\cup \\{a\\triangleleft^{\\mathbf p}z\\}$ to $(\\mathbf p_r)_{\\restriction Aa}(z)$. \n\n(5)$\\Rightarrow$(4) is immediate, so it remains to prove (4)$\\Rightarrow$(5). Assume (4). Let $(a,b,c)$ be a Morley sequence in $\\mathbf p_r$ over $A$ and let $(c,b',a)$ be a Morley sequence in $\\mathbf p_l$ over $A$. We will prove $b\\nequiv b'\\,(Aac)$; this implies (5). Denote $p_a=(\\mathbf p_r)_{\\restriction Aa}$ and $\\mathbf p_a=(p_a,<)$, and consider $\\Pi(x)=a\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}c$ as a partial type over $Aac$. By Remark \\ref{Remark_pgeneric_first}(c), the set $p_a(\\Mon)$, which is defined by $a\\triangleleft^{\\mathbf p}x$, is a final part of $(p(\\Mon),<)$. Similarly, $x\\triangleleft^{\\mathbf p}c$ defines an initial part of $(p(\\Mon),<)$, so $\\Pi(\\Mon)$ is a convex subset of $(p(\\Mon),<)$. If $q\\in S_x(Aac)$ is a completion of $\\Pi(x)$, then $q$ is an extension of $p$, so by the weak o-minimality of $p$, $q(\\Mon)$ is convex in $(p(\\Mon),<)$ and hence in $(\\Pi(\\Mon),<)$; by (4), $q(\\Mon)$ is a proper subset of $\\Pi(\\Mon)$.\nNow, we claim that the locus of $\\tp(b/Aac)$, call it $P$, is an initial part of $\\Pi(\\Mon)$. \nTo prove this, note that the sequence $(b,c)$ is Morley in $\\mathbf p_r$ over $Aa$, so $b\\triangleleft^{\\mathbf p_a}c$, that is, $b$ is left $\\mathbf p_a$-generic over $c$. We conclude that $P$ consists of all left $(\\mathbf p_a)_l$-generic elements over $c$; by Remark \\ref{Remark_pgeneric_first}(c), $P$ is an initial part of $p_a(\\Mon)$. Since $\\Pi(\\Mon)$ is an initial part of $p_a(\\Mon)$ and $P\\subseteq \\Pi(\\Mon)$, we conclude that $P$ is an initial part of $\\Pi(\\Mon)$.\nAnalogously, we see that the locus of $\\tp(b'/Aac)$ is a proper final part of $\\Pi(\\Mon)$. Therefore, $b\\not\\equiv b'\\,(Aac)$.\n\\end{proof}\n \n\n\\begin{Theorem}\\label{Theorem_nwor preserves triviality}\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item Nonforking extensions of a trivial weakly o-minimal type are all trivial, but the triviality of some nonforking extension does not guarantee triviality of the type. \n\n\\item Triviality is preserved under $\\nwor$ of weakly o-minimal types over the same domain.\n\n\\item Weak orthogonality of trivial types transfers to their nonforking extensions. \n\\end{enumerate}\n\\end{Theorem}\n\\begin{proof}(a) The first clause is Lemma \\ref{Lemma_triv_preserved_in_nonforking}(a); the second will be justified in Example \\ref{Example_trivover0}. \n\n(b) Let $p$ and $q$ be weakly o-minimal types over $A$ with $p\\nwor q$. Assuming that $p$ is nontrivial, we will prove that $q$ is also nontrivial. Choose relatively $A$-definable orders $<_p$ and $<_q$ such that the weakly o-minimal pairs $\\mathbf p=(p,<_p)$ and $\\mathbf q=(q,<_q)$ are directly nonorthogonal. \nBy Remark \\ref{Remark_notrivial_wom_witness}(d) there are $D\\supseteq A$ and $a_1,a_2,a_3$ such that $D\\triangleleft^{\\mathbf p}a_1$, $Da_1\\triangleleft^{\\mathbf p}a_2\\triangleleft^{\\mathbf p} a_3$ and $a_2\\dep_{Da_1} a_3$. \nBy the density property, Corollary \\ref {Cor_triangle_mathbf_pq_properties}(b), there are $b,b'$ satisfying $a_2\\triangleleft^{\\mathbf q}b\\triangleleft^{\\mathbf q}b'\\triangleleft^{\\mathbf p}a_3$. By Theorem \\ref{Theorem_triangle_mathcal F},\n$Da_1\\triangleleft^{\\mathbf p}a_2\\triangleleft^{\\mathbf q} b\\triangleleft^{\\mathbf q}b'\\triangleleft^{\\mathbf p} a_3$ and $a_2\\dep_{Da_1} a_3$ together imply\n$b\\dep_{Da_1} b'$. \nIf $q$ were trivial, then by the strong transitivity property, Lemma \\ref{Lemma_trivial_strong_trans}, $Da_1\\triangleleft^{\\mathbf q}b\\triangleleft^{\\mathbf q}b'$ would imply $Da_1b\\triangleleft^{\\mathbf q}b'$, which contradicts $b\\dep_{Da_1} b'$. Therefore, $q$ is nontrivial, proving (b).\n\n(c) Suppose that $p,q\\in S(A)$ are trivial weakly o-minimal types with $p\\wor q$ and that $p_1,q_1\\in S(B)$ are their nonforking extensions. We need to prove $p_1\\wor q_1$. \nSuppose on the contrary that $p_1\\nwor q_1$. By part (a) of the theorem, both $p_1$ and $q_1$ are trivial, since $p$ and $q$ are. \nChoose relatively $A$-definable orders $<_p$ and $<_q$ such that the weakly o-minimal pairs $\\mathbf p_1=(p_1,<_p)$ and $\\mathbf q_1=(q_1,<_q)$ are directly nonorthogonal. Let $\\mathbf p=(p,<_p)$ and $\\mathbf q=(q,<_q)$.\nChoose $a_0,a_1\\models p_1$ and $b_0,b_1\\models q_1$ such that \\ $a_0\\triangleleft^{\\mathbf q_1}b_0\\triangleleft^{\\mathbf q_1}b_1\\triangleleft^{\\mathbf p_1}a_1$. Then \\ $\\tp(a_0b_0/B)\\neq \\tp(a_1b_1/B)$. To see this, first observe that $p_1\\nwor q_1$ implies $\\tp(a_0b_0/B)\\neq \\tp(a_1b_0/B)$ as $a_0$ is left- and $a_1$ is right $\\mathbf p_1$-generic over $b_0$; also observe that \n$b_0\\triangleleft^{\\mathbf q_1}b_1\\triangleleft^{\\mathbf p_1}a_1$ implies $\\tp(a_1b_0/B)=\\tp(a_1b_1/B)$; therefore, $\\tp(a_0b_0/B)\\neq \\tp(a_1b_1/B)$. Witness this by a formula $\\phi(x,y,c)\\in \\tp(a_0b_0/B)$ such that $\\phi(x,y,B)\\notin \\tp(a_1b_1/B)$. \nDefine sequences $I=(a_n\\mid n<\\omega)$ and $J=(b_n\\mid n<\\omega)$ such that the following holds for all $n\\geqslant 1$:\n$$a_{2n-1}\\triangleleft^{\\mathbf p_1}a_{2n}\\ \\ \\ \\ \\mbox{ and }\\ \\ \\ \\ a_0b_0a_1b_1\\equiv a_{2n}b_{2n}a_{2n+1}b_{2n+1}\\, (B).$$\nNote that for all $n<\\omega$, $a_{2n}\\triangleleft^{\\mathbf q_1}b_{2n}\\triangleleft^{\\mathbf q_1}b_{2n+1}\\triangleleft^{\\mathbf p_1}a_{2n+1}\\triangleleft^{\\mathbf p_1}a_{2n+2}$ holds.\nHence, $I$ is strictly $\\triangleleft^{\\mathbf p_1}$-increasing, so, by Remark \\ref{Remark trivial wom expressing via triangleleft}, $I$ is a Morley sequence in $(\\mathbf p_1)_r$ over $B$. Similarly, $J$ is a Morley sequence in $(\\mathbf q_1)_r$ over $B$. By Lemma \\ref{Lemma trivial wom morley in nonforking is left or right morley}(b), $I$ is a Morley sequence in a nonforking globalization of $p$ over $A$; as it is $<_p$-increasing, $I$ is a Morley sequence in $\\mathbf p_r$ over $A$. Similarly, $J$ is a Morley sequence in $\\mathbf q_r$ over $A$. By Corollary \\ref{Cor wom lifts to omega for o trivial}(b), $\\tp(I/A)\\wor \\tp(J/A)$ as $p\\wor q$. In particular, $I$ and $J$ are mutually indiscernible over $A$ and the sequence $(a_nb_n\\mid n<\\omega)$ is indiscernible over $A$. Since $a_0b_0a_1b_1\\equiv a_{2n}b_{2n}a_{2n+1}b_{2n+1}\\,(B)$ implies $\\models\\phi(a_{2n},b_{2n},c)\\land\\lnot \\phi(a_{2n+1},b_{2n+1},c)$, the type $\\tp(a_0b_0/A)$ is not NIP. Contradiction.\n\\end{proof}\n\nFor directly non-orthogonal weakly o-minimal pairs we can strengthen part (c) of the theorem: the transfer of weak and forking non-orthogonality occurs in both directions, from and towards non-forking extensions.\n\n\\begin{Corollary}\nSuppose that $\\mathbf p=(p,<_p)$ and $\\mathbf q=(q,<_q)$ are weakly o-minimal pairs over $A$. Let $B\\supseteq A$, $p_B=(\\mathbf p_r)_{\\restriction B}$, and $q_B=(\\mathbf q_r)_{\\restriction B}$. Suppose that $p$ and $q$ are trivial types. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item $p\\wor q \\Rightarrow p_B\\wor q_B$ \\ and \\ $\\delta_A(\\mathbf p,\\mathbf q) \\Rightarrow p_B\\nwor q_B$.\n\n\\item $\\delta_A(\\mathbf p,\\mathbf q)$ implies \\ $p\\fwor q \\Leftrightarrow p_B\\fwor q_B$. \n\\end{enumerate}\n\\end{Corollary}\n\\begin{proof}\n(a) The first claim is Theorem \\ref{Theorem_nwor preserves triviality}(c) and the second follows by Fact \\ref{Fact_delta_implies_deltaB_fworB}(a). \n\n(b) Assume $\\delta_A(\\mathbf p,\\mathbf q)$. By Fact \\ref{Fact_delta_implies_deltaB_fworB}(b) we have $p\\nfor q \\Rightarrow p_B\\nfor q_B$. \nTo prove the other direction, assume $p\\fwor q$ and let $a\\models p_B$ and $b\\models q_B$; we need to prove $a\\ind_B b$. $p\\fwor q$ implies $a\\ind_A b$, \nso $a\\triangleleft^{\\mathbf q} b$ or $b\\triangleleft^{\\mathbf p} a$. Without loss, assume $a\\triangleleft^{\\mathbf q} b$. By the strong transitivity property, $B\\triangleleft^{\\mathbf p}a\\triangleleft^{\\mathbf q}b$ implies $Ba\\triangleleft^{\\mathbf q}b$; hence, $b\\ind_B a$, as desired. \n\\end{proof}\n\n\nWe have already shown that every $\\triangleleft^{\\mathbf p}$-increasing sequence of realizations of a trivial type is Morley in $\\mathbf p_r$. This generalizes to realizations of trivial types from the same $\\delta_A$-class in the following way: \n\n\\begin{Definition}\nLet $\\mathcal F$ be a $\\delta_A$-class of weakly o-minimal pairs over $A$ and let $(I,<_I)$ be a linear order. A sequence $(a_i\\mid i\\in I)$ of realizations of types from $\\mathcal F$ is {\\em $\\mathcal F$-independent over $A$} if $(a_{j}\\mid j<i)\\triangleleft^{\\mathcal F} a_i$ holds for all $i\\in I$. \n\\end{Definition}\n\n\\begin{Proposition}\nLet $\\mathcal F$ be a $\\delta_A$-class such that the types appearing in $\\mathcal F$ are trivial. Then every $\\triangleleft^{\\mathcal F}$-increasing sequence of realizations of types from $\\mathcal F$ is $\\mathcal F$-independent over $A$. \n\\end{Proposition} \n\\begin{proof}\nIt suffices to prove the claim for finite sequences. We proceed by induction on the length of a sequence. The claim is trivial for sequences of length $\\leq 2$, so suppose that we have a $\\triangleleft^{\\mathcal F}$-increasing sequence $(a_1,\\dots,a_n,a_{n+1})$ for $n\\geqslant 2$. By the induction hypothesis, $(a_j\\mid j<i)\\triangleleft^{\\mathcal F}a_i$ holds for all $i\\leqslant n$. In particular, we have $(a_j\\mid j<n)\\triangleleft^{\\mathcal{F}} a_n\\triangleleft^{\\mathcal{F}}a_{n+1}$. By strong transitivity, Lemma \\ref{Lemma_trivial_strong_trans}, we conclude $(a_j\\mid j\\leqslant n)\\triangleleft^{\\mathcal F}a_{n+1}$, and we are done.\n\\end{proof}\n\n\n\n\n\\subsection{Trivial 1-types in o-minimal theories}\n\nThroughout this subsection, $T$ denotes an o-minimal theory (with respect to some dense linear order $<$). Recall that an element $a$ of an o-minimal structure is said to be non-trivial if there is an open interval $I$ containing $a$ and a definable function $f:I\\times I\\to \\Mon$ that is strictly increasing in both coordinates; otherwise, $a$ is trivial. By the Trichotomy Theorem (\\cite{PeS}), $a$ is non-trivial if and only if there is a definable group structure on some open, convex neighborhood of $a$. \nOn the other hand, as\nshown in \\cite{MRS}, all elements of $\\Mon$ are trivial if and only if $(\\Mon,\\dcl)$ is a degenerate pregeometry if and only if $T$ is binary. In fact, \nif $a$ is trivial, then it follows from Lemmas 2.1 and 2.2 of \\cite{MRS} that $\\tp(a)$ is trivial. In general, the converse is not true, as illustrated by the following example. \n\\begin{Example}\\phantomsection\\label{Example_trivial_real_additive}\nConsider the ordered group of reals $(\\mathbb R,+,<,0)$; clearly, it is an o-minimal structure. There is a unique type $p\\in S_1(\\emptyset)$ that contains $0<x$ and the pair $\\mathbf p=(p,<)$ is weakly o-minimal. \nFor all $A$, the type $(\\mathbf p_{r})_{\\restriction A}$ is determined by $Gp(A)<x$, where $Gp(A)$ is the subgroup generated by $A$. Using this, it is easy to see that $\\mathbf p_r$ is order-trivial over $\\emptyset$:\nan increasing sequence of realizations of $p$, $(a_i\\mid i\\in\\omega)$, is Morley in $\\mathbf p_r$ over $\\emptyset$ if and only if $Gp(a_i)<a_{i+1}$ holds for all $i\\in \\omega$, or equivalently, if $a_i\\triangleleft^{\\mathbf p} a_{i+1}$ (for all $i\\in\\omega$). Therefore, $\\mathbf p_r$ is trivial over $\\emptyset$, but the $\\mathrm{dcl}$-pregeometry on $p(\\Mon)$ is not degenerate.\n\\end{Example}\nTherefore, the triviality of the type is not equivalent to the triviality of its realizations. In the proof of Proposition \\ref{Proposition_Ramakrishnan}, we will see that the triviality of $\\tp(+\\infty)$ is related to the existence of ``uniform bounds on growth\" of functions, studied by Friedman and Miller in \\cite{FM} and Ramakrishnan in \\cite{Ramakrishnan}; we adapt the existence of uniform bounds to the context of arbitrary types as follows:\n\n\\begin{Definition}\\label{Definition_p_germs_bdd}\nLet $\\mathfrak p\\in S_1(\\Mon)$. We say that {\\it $\\mathfrak p$-germs are $A$-bounded} if $\\mathfrak p$ is $A$-invariant and for all $B\\supseteq A$ and all relatively $B$-definable functions $f:\\mathfrak p_{\\restriction B}(\\Mon)\\to \\mathfrak p_{\\restriction B}(\\Mon)$, there exists a relatively $A$-definable function $g:p(\\Mon)\\to p(\\Mon)$ such that $f(x)\\leqslant g(x)$ for all $x\\in \\mathfrak p_{\\restriction B}(\\Mon)$. \n\\end{Definition} \n\nEquivalently, $\\mathfrak p$-germs are $A$-bounded if every relatively $\\Mon$-definable function mapping the locus of $\\mathfrak p$ in a larger monster into itself is bounded by a relatively $A$-definable such function.\n\n\\begin{Remark}\\label{Remark_boundedgerms}\nIf $\\mathfrak p$-germs are $A$-bounded and $B,f$ are as in Definition \\ref{Definition_p_germs_bdd}, then there are relatively $A$-definable functions $g,g' :\\mathfrak p_{\\restriction B}(\\Mon)\\to \\mathfrak p_{\\restriction B}(\\Mon)$ with $g'(x)\\leqslant f(x)\\leqslant g(x)$. By the Monotonicity Theorem, $f$ is strictly increasing, so there is a relatively $A$-definable $g_1$ with $f(x)^{-1}\\leqslant g_1(x)$. Since $f$ and $g_1$ are strictly increasing, $g_1^{-1}(x)\\leqslant f(x)$ follows, so $g'=g_1^{-1}(x)$ works. \n\\end{Remark}\n \nLet $p\\in S_1(A)$ and let $a\\models p$. Recall that the set $\\mathcal D_p(a)$ is the union of all relatively $Aa$-definable subsets of $p(\\Mon)$ that are bounded in $(p(\\Mon),<)$; since $p(\\Mon)$ is a convex subset of the monster, each of these bounded subsets is included in a segment with endpoints in $\\dcl(Aa)\\cap p(\\Mon)$. Therefore, the set $\\mathcal D_p(a)$ equals the union of all segments $[f(a),g(a)]$, where $f,g:p(\\Mon)\\to p(\\Mon)$ are relatively $A$-definable functions; $\\mathcal D_p(a)$ can also be described as the convex hull of $\\dcl(Aa)\\cap p(\\Mon)$. \n\n\\begin{Lemma}\\label{Lemma_ominimal}\nA type $p\\in S_1(A)$ (in an o-minimal theory) is trivial if and only if $\\mathbf p_r$-germs are $A$-bounded if and only if $\\mathbf p_l$-germs are $A$-bounded.\n\\end{Lemma} \n\\begin{proof}We will prove that $p\\in S_1(A)$ is trivial if and only if $\\mathbf p_r$-germs are $A$-bounded; a similar proof establishes the other equivalence.\n\n\nFirst, suppose that $p$ is trivial and $B\\supseteq A$. Denote $q=(\\mathbf p_r)_{\\restriction B}$ and let $f:q(\\Mon)\\to q(\\Mon)$ be relatively $B$-definable. By Lemma \\ref{Lemma_trivial_iff_Dp=Dq}, for any $a\\models q$ we have $\\mathcal D_p(a)=\\mathcal D_q(a)$. Hence, $f(a)\\in \\mathcal D_p(a)$, so there exists $b\\in \\mathcal D_p(a)\\cap\\dcl(Aa)$ such that $f(a)\\leqslant b$; clearly, $b=g(a)$ for some relatively $A$-definable function $g$. \n\nTo prove the other implication, suppose that $\\mathbf p_r$-germs are $A$-bounded. First, we {\\it claim} that $\\mathcal D_{(\\mathbf p_r)_{\\restriction B}}(a)=\\mathcal D_p(a)$ holds for all $B\\supseteq A$ and all $a\\models (\\mathbf p_r)_{\\restriction B}$. To prove it, let $[f_1(a),f_2(a)]\\subseteq \\mathcal D_{(\\mathbf p_r)_{\\restriction B}}(a)$, where $f_1(a),f_2(a)\\in \\dcl(Ba)$. By Remark \\ref{Remark_boundedgerms}, there are \n$g_1(a),g_2(a)\\in \\dcl(Aa)\\cap p(\\Mon)$ such that $g_1(a)\\leqslant f_1(a)<f_2(a)\\leqslant g_2(a)$. Then $[f_1(a),f_2(a)]\\subseteq [g_1(a),g_2(a)]\\subseteq \\mathcal D_p(a)$; the claim follows. \nNow, we will prove that $\\mathbf p_r$ is trivial over $A$; that is, assuming that $I=(a_i\\mid i\\in \\omega)$ is a sequence of realizations of $p$ such that $\\mathcal D_p(a_0)<\\mathcal D_p(a_1)< \\dots$, we will show that $I$ is Morley in $\\mathbf p_r$ over $A$. Put $p_0=(\\mathbf p_r)_{\\restriction Aa_0}$. Clearly, $a_i\\models p_0$ for all $i\\geqslant 1$, so $\\mathcal D_{p_0}(a_i)=\\mathcal D_p(a_i)$ holds by the above claim. Then $\\mathcal D_{p_0}(a_1)<\\mathcal D_{p_0}(a_2)<\\dots$ implies that $(a_1,a_2)$ is Morley in $\\mathbf p_r$ over $Aa_0$, so $(a_0,a_1,a_2)$ is Morley in $\\mathbf p_r$ over $A$. Continuing in this way (induction), we see that $I$ is Morley in $\\mathbf p_r$ over $A$. Therefore, $\\mathbf p_r$ is trivial over $A$ and $p$ is trivial.\n\\end{proof}\n \n\\begin{Proposition}\\label{Proposition_Ramakrishnan} \nIf $(M,<, \\dots)$ is a densely ordered o-minimal structure, then every definable type $p\\in S_1(M)$ is trivial. \n\\end{Proposition}\n\\begin{proof} Suppose that $p$ is definable and nonalgebraic. \nAfter possibly modifying the order $<$, while preserving the o-minimality, we may assume that $p=\\tp(+\\infty/M)$. \nWe will show that $\\mathbf p_r$-germs are $M$-bounded; consequently, by Lemma \\ref{Lemma_ominimal}, $p$ is trivial. Let\n$B\\supseteq M$ and let $f:(\\mathbf p_r)_{\\restriction B}(\\Mon)\\to (\\mathbf p_r)_{\\restriction B}(\\Mon)$ be a relatively $b$-definable function ($b\\in B^n$); this is also expressed by (the formula) $\\lim_{x\\to +\\infty}f(x)=+\\infty$. By compactness, there are a $M$-definable set $D\\subseteq \\Mon^n$ and a $M$-definable function $F:D\\times \\Mon\\to \\Mon$ such that $b\\in D$, $F(b,x)=f(x)$, and\n$\\lim_{x\\to +\\infty}F(d,x)=+\\infty$ (for all $d\\in D$). By \\cite[Theorem 1.2]{Ramakrishnan}, there exists a $M$-definable function $g:\\Mon\\to\\Mon$ such that for all $d\\in D$: $F(d,x)\\leqslant g(x)$ holds for all sufficiently large $x\\in M$. In particular, $f(x)=F(b,x)\\leqslant g(x)$ holds for some (equivalently, all) $x\\in (\\mathbf p_r)_{\\restriction B}(\\Mon)$, as desired.\n \\end{proof}\n \n\\begin{Example} \nConsider the ordered group of rationales $(\\mathbb Q,+,<,0)$, elementarily embedded in the group of reals. By Proposition \\ref{Proposition_Ramakrishnan} we know that all definable types belonging to $S_1(\\mathbb Q)$ are trivial; we will show that these are the only trivial types there. Let $p\\in S_1(\\mathbb Q)$ be non-definable. Then there is an irrational number $\\alpha$ such that $p=\\tp(\\alpha/\\mathbb Q)$. Since $\\alpha$ is the unique realization of $p$ in $\\mathbb R$, for each $a\\in p(\\Mon)$ we have $\\dcl(\\mathbb Qa)\\cap p(\\Mon)=\\{a\\}$. This implies that there are exactly three extensions of $p(x)\\cup p(y)$ in $S_2(a)$, determined by $x<y$, $x=y$, and $y<x$, respectively. In particular, every two distinct realizations of $p$ are independent over $\\emptyset$. It follows that if $a<b$ are distinct realizations of $p$, then $(a,\\frac{a+b}{2},b)$ is a triangle of realizations of $p$, so $p$ is non-trivial. \n\\end{Example}\n\n\\begin{Example}[$\\emptyset$-invariant type which is trivial over $\\{0\\}$, but not over $\\emptyset$]\\label{Example_trivover0} \nConsider the affine reduct of the additive ordered group of reals $(\\mathbb R,m,<)$, where $m(x,y,z):=x-y+z$. This structure is o-minimal with a unique type $p\\in S_1(\\emptyset)$ and exactly three complete 2-types determined by $x<y$, $y<x$, and $x=y$, respectively; for example, if $a<b$, then $x\\longmapsto \\frac{x-a}{b-a}$ is an automorphism mapping $(a,b)$ to $(0,1)$, so $\\tp(a,b)=\\tp(0,1)$. In particular, every pair of distinct elements is independent. However, $p$ is non-trivial, since the triple $(0,1,2)$ is not a Morley sequence in $\\mathbf p_r$ due to $m(1,0,1)=2$. \nLet $p_0(x)$ be the complete type over $\\{0\\}$ determined by $0<x$; clearly, $p_0(x)=(\\mathbf p_r)_{\\restriction \\{0\\}}$. Similarly as in Example \\ref{Example_trivial_real_additive}, we may conclude that $p_0$ is trivial; this is because the structures\n$(\\mathbb R,m,<,0)$ and $(\\mathbb R,+,<,0)$ are interdefinable. Therefore, the type $\\mathbf p_r$ is trivial over $\\{0\\}$ but not over $\\emptyset$. \n\\end{Example}\n\n\n\\subsection{Omitting trivial weakly o-minimal types} \n\n\n\nIn this subsection, we will prove that in a countable theory with few countable models, every trivial weakly o-minimal type over a finite domain is both convex and simple.\nRecall that a weakly o-minimal type $p\\in S(A)$ is {\\it simple} if $x\\dep_Ay$ is a relatively definable (equivalence) relation on $p(\\Mon)$ (in which case, by $\\Aut_{A}(\\Mon)$-invariance, it is relatively $A$-definable); that is, $\\mathcal D_p=\\{(x,y)\\in p(\\Mon)^2\\mid x\\dep_A y\\}$ is relatively definable within $p(\\Mon)^2$ or, equivalently, $\\mathcal D_p(a)$ is relatively definable for some/all $a\\models p$. \nIn the next proposition, we characterize weakly o-minimal types that are both convex and simple; there (and later) by $a\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}b$ we denote the partial type $p(x)\\cup(\\mathcal D_p(a)<x)\\cup (x<\\mathcal D_p(b))$ (where $\\mathbf p=(p,<)$ is a weakly o-minimal pair over $A$). \n\n\n\\begin{Proposition}\\label{Prop middle definable iff simple and convex}\nSuppose that $\\mathbf p=(p,<)$ is a weakly o-minimal pair over $A$ and $A\\triangleleft^{\\mathbf p}a\\triangleleft^{\\mathbf p}b$. Then $p$ is both simple and convex if and only if the type $a\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}b$ has a definable locus.\n\\end{Proposition}\n\\begin{proof}Let $D$ denote the locus of $a\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}b$. Notice that $D$ is a $\\mathcal D_p$-closed subset of $p(\\Mon)$: if $c\\in D$ then $\\mathcal D_p(c)\\subseteq D$; this holds since by Remark \\ref{Remark_trianglep_binary_basicprops}(b) $a\\triangleleft^{\\mathbf p}c\\triangleleft^{\\mathbf p}b$ is equivalent with $a<\\mathcal D_p(c)<b$.\n\n\n$(\\Rightarrow)$ Suppose that $D$ is definable. Then $D$ is $Aab$-definable, since it is clearly $Aab$-invariant. Let $\\theta(y,x,z)$ be an $L_A$-formula such that $\\theta(a,x,b)$ defines $D$. First, we prove that $p$ is simple, that is, $\\mathcal D_p(e)$ is relatively definable within $p(\\Mon)$ for some/all $e\\models p$. In fact, we prove more: $\\mathcal D_p(e)$ is definable. Without loss, assume $e\\in D$. \nWe {\\it claim} that $\\mathcal D_p(e)$ is defined by the following formula:\n$$\\sigma(x)=\\theta(a,x,b)\\land\\lnot \\theta(a,x,e)\\land\\lnot\\theta(e,x,b).$$\nIndeed, note that $\\sigma(x)$ is equivalent with $a\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}b\\land x\\ntriangleleft^{\\mathbf p}e \\land e\\ntriangleleft^{\\mathbf p}x$. For all $x\\models p$, $x\\ntriangleleft^{\\mathbf p}e \\land e\\ntriangleleft^{\\mathbf p}x$ expresses the $<$-incompatibility of the classes $\\mathcal D_p(x)$ and $\\mathcal D_p(e)$, so by Theorem \\ref{Theorem4}(c) is equivalent with $x\\in \\mathcal D_p(e)$. As $\\mathcal D_p(e)\\subset D$, we conclude that $\\sigma(x)$ is equivalent with $x\\in \\mathcal D_p(e)$. This proves the claim, so $p$ is simple. \nIt remains to prove that $p$ is convex. Note that:\n$$\\{\\theta(a,x_1,b),\\theta(a,x_2,b),x_1<x<x_2\\}\\cup p(x)\\vdash \\theta(a,x,b);$$\nIndeed, if $e_1,e,e_2$ satisfy the left-hand side (for $x_1,x,x_2$ respectively), then $e_1,e,e_2\\models p$ and $a\\triangleleft^{\\mathbf p}e_1<e<e_2\\triangleleft^{\\mathbf p}b$, so $a\\triangleleft^{\\mathbf p}e\\triangleleft^{\\mathbf p}b$, and therefore $\\models\\theta(a,e,b)$. By compactness, find $\\pi(x)\\in p(x)$ such that:\n\\begin{equation}\\label{eq convex}\\tag{$\\dagger$}\n (\\forall x_1,x,x_2)(\\theta(y,x_1,z)\\land\\theta(y,x_2,z)\\land x_1<x<x_2\\land\\pi(x)\\rightarrow \\theta(y,x,z))\\in\\tp_{y,z}(a,b/A).\n\\end{equation}\nMoreover, we can assume that $\\pi(\\Mon)\\subseteq D$.\nWe now {\\em claim} that $\\pi(x)$ witnesses the convexity of $p$: $p(\\Mon)$ is a $<$-convex subset of $(\\pi(\\Mon),<)$. Take $e_1,e_2\\models p$ and $d\\in\\pi(\\Mon)$ such that $e_1<d<e_2$; we need to show $d\\in p(\\Mon)$. Clearly, $e_1,e_2\\in p(\\Mon)$. \nChoose $a',b'\\models p$ so that $a'\\triangleleft^{\\mathbf p}e_1$ and $e_2\\triangleleft^{\\mathbf p}b'$; then $a'\\triangleleft^{\\mathbf p}e_1<d<e_2\\triangleleft^{\\mathbf p}b'$. This implies $a'\\triangleleft^{\\mathbf p}e_1\\triangleleft^{\\mathbf p}b'$ and $a'\\triangleleft^{\\mathbf p}e_2\\triangleleft^{\\mathbf p}b'$, so $\\models\\theta(a',e_1,b')\\land\\theta(a',e_2,b')$ is valid. Also $a'\\triangleleft^{\\mathbf p}b'$, so $a'b'\\equiv ab\\ (A)$, hence from (\\ref{eq convex}) we have:\n$$\\models (\\theta(a',e_1,b')\\land\\theta(a',e_2,b')\\land e_1<d<e_2\\land\\pi(d))\\rightarrow \\theta(a',d,b')\\,.$$\nThe left side of this implication is valid, so $\\models \\theta(a',d,b')$. In particular, $d\\models p$, and we are done.\n\n$(\\Leftarrow)$ Suppose now that $p$ is simple and convex. By Lemma \\ref{Proposition_convex_type_witness}, there is $\\pi(x)\\in p(x)$ witnessing the convexity: $(\\pi(\\Mon),<)$ is a definable extension of $(p(\\Mon),<)$ and $p(\\Mon)$ is convex in $\\pi(\\Mon)$. \nBy the simplicity of $p$, for $e\\models p$, $\\mathcal D_p(e)$ is a relatively $Ae$-definable subset of $p(\\Mon)$, defined by $\\phi(x,e)$ say.\nThen the locus of $a\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}b$ is clearly defined by $\\pi(x)\\land \\phi(\\Mon,a)<x<\\phi(\\Mon,b)$. \n\\end{proof}\n\n\n \n\n\\begin{Theorem}\\label{Theorem trivial implies simple and convex}\nSuppose that $T$ is countable and $I(\\aleph_0,T)<2^{\\aleph_0}$. Then every trivial weakly o-minimal type over a finite domain is simple and convex. \n\\end{Theorem}\n\\begin{proof}\nAssuming that $p$ is a trivial weakly o-minimal type over a finite domain that is not both convex and simple, we will prove $I(\\aleph_0,T)=2^{\\aleph_0}$. We may assume that $\\mathbf p=(p,<)$ is a weakly o-minimal pair over $\\emptyset$ (as for finite $A$, $I(\\aleph_0,T_A)=2^{\\aleph_0}$ implies $I(\\aleph_0,T)=2^{\\aleph_0}$). \nFor every countable endless linear order $\\mathbb I=(I,<_I)$ we will find a countable $M_{\\mathbb I}\\models T$ such that $(p(M_{\\mathbb I})/\\mathcal D_p,<)\\cong \\mathbb I$; clearly, for nonisomorphic orders $\\mathbb I$ and $\\mathbb J$ the corresponding models $M_{\\mathbb I}$ and $M_{\\mathbb J}$ will be nonisomorphic. As there are $2^{\\aleph_0}$ many pairwise nonisomorphic such orders, the desired conclusion follows. \n\nFix $\\mathbb I=(I,<_I)$ and let $A_I=(a_i\\mid i\\in I)$ be a $\\triangleleft^{\\mathbf p}$-increasing sequence of realizations of $p$; since $p$ is trivial, $A_I$ is a Morley sequence in $\\mathbf p_r$ over $\\emptyset$. Put \\ $\\Pi_I(x):=p(x)\\cup\\bigcup_{i\\in I}x\\notin\\mathcal D_p(a_i)$. \n \n\\begin{Claim*} \nThere exists a countable model $M_{\\mathbb I}\\supseteq A_I$ that omits $\\Pi_I(x)$.\n\\end{Claim*}\n\n\\noindent{\\em Proof of Claim.} Suppose not. Then, by the Omitting Types Theorem there is an $L$-formula $\\theta(x,\\bar y)$ and a finite subsequence of $A_I$, $\\bar a=(a_{i_1},\\dots,a_{i_n})$, such that $\\theta(x,\\bar a)$ is consistent and $\\theta(x,\\bar a)\\vdash \\Pi_I(x)$. \nMoreover, we can assume that $D=\\theta(\\Mon,\\bar a)$ is a convex subset of $(p(\\Mon),<)$; indeed, $\\theta(x,\\bar a)\\vdash p(x)$ holds, so according to the weak o-minimality of $p$, $\\theta(\\Mon,\\bar a)\\subseteq p(\\Mon) $ has finitely many convex components in $(p(\\Mon),<)$, and we can replace $\\theta(x,\\bar a)$ with a formula that defines one of them so that $\\theta(x,\\bar a)\\vdash \\Pi_I(x) $ still holds. \nTherefore, $D:=\\theta(\\Mon,\\bar a)$ is a convex set that is disjoint from each of the (convex) sets $(\\mathcal D_p(a_i)\\mid i\\in I)$. This implies that for each $i\\in I$ either $\\mathcal D_p(a_i)<D$ or $D<\\mathcal D_p(a_i)$ holds, so every element $e\\in D$ is $\\triangleleft^{\\mathbf p}$-comparable to $a_i$ and we can arrange $A_{\\mathbb I}\\cup \\{e\\}$ into a $\\triangleleft^{\\mathbf p}$-increasing sequence. \nThen the triviality of $p$ implies that this sequence is Morley in $\\mathbf p_r$ over $\\emptyset$, and that it remains such after replacing $e$ with any other $e'\\in D$; in particular, $e\\equiv e'\\,(A_{\\mathbb I})$ holds, so $\\theta(x,\\bar a)$ isolates a complete type $q\\in S(A_{\\mathbb I})$. Since $\\theta(x,\\bar a)$ uses only parameters $\\bar a$, the type $q_{\\restriction \\bar a}\\in S(\\bar a)$ is isolated by $\\theta(x,\\bar a)$ ($q_{\\restriction \\bar a}(\\Mon)=D$), and $q_{\\restriction \\bar a}\\wor\\tp(A_{\\mathbb I}/\\bar a)$ holds. Now we prove:\n\\begin{equation}\n \\mbox{ $\\mathcal D_p(a_{i_k})<D<\\mathcal D_p(a_{i_{k+1}})$ \\ holds for some $k<n$. }\n\\end{equation}\\setcounter{equation}{0}\nWe have a sequence of convex sets $\\mathcal D_p(a_{i_0})<\\mathcal D_p(a_{i_1})<\\dots <\\mathcal D_p(a_{i_n})$ that are disjoint from $D$ (which is also convex). Therefore, to prove (1), it suffices to exclude $D<\\mathcal D_p(a_{i_0})$ and $\\mathcal D_p(a_{i_n})<D$. \nSince $\\mathbb I$ is endless, the type $\\mathbf p_{r\\restriction \\bar a}$ is realized in $A_{\\mathbb I}$, so $q_{\\restriction \\bar a}\\wor\\mathbf p_{r\\restriction \\bar a}$. \nWe conclude that the locus $q_{\\restriction \\bar a}(\\Mon)=D$ does not contain any elements above $\\mathcal D_p(a_{i_n})$, as all realize $\\mathbf p_{r\\restriction \\bar a}$. This excludes $\\mathcal D_p(a_{i_n})<D$; similarly, $D<\\mathcal D_p(a_{i_0})$ is ruled out, proving (1). \n\nNote that (1) implies $\\theta(x,\\bar a)\\vdash a_{i_k}\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}a_{i_{k+1}}$. \nAlso note that $a_{i_k}\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}a_{i_{k+1}}$ determines a complete type in $S(\\bar a)$, as $a_{i_k}\\triangleleft^{\\mathbf p}d\\triangleleft^{\\mathbf p}a_{i_{k+1}}$ implies that $(a_{i_0},\\ldots,a_{i_k},d,a_{i_{k+1}},\\ldots,a_{i_n})$ is a Morley sequence in $\\mathbf p_r$ over $\\emptyset$; this type is $q_{\\restriction \\bar a}(x)$, which is isolated. In particular, the locus of $a_{i_k}\\triangleleft^{\\mathbf p}x\\triangleleft^{\\mathbf p}a_{i_{k+1}}$ is definable, so by Proposition \\ref{Prop middle definable iff simple and convex}, $p$ is convex and simple. This contradicts the initial assumptions and completes the proof of Claim.\\hfill$\\blacksquare$\n\n\\smallskip\nLet the model $M_{\\mathbb I}$ be given by the claim. To complete the proof of the theorem, it suffices to note that since $M_{\\mathbb I}$ omits $\\Pi_{\\mathbb I}(x)$ and since $A_I\\subseteq M_{\\mathbb I}$, the classes $(\\mathcal D_p(a_i)\\mid i\\in I)$ are the only $\\mathcal D_p$-classes represented in $p(M_{\\mathbb I})$, so $(p(M_{\\mathbb I})/\\mathcal D_p,<)\\cong \\mathbb I$.\n\\end{proof} \n\n\n \n\\section{Semiintervals and shifts}\\label{Section_shifts}\n \nIn this section, we first introduce the notions of semiintervals and shifts and then prove Theorem \\ref{Theorem1_shift}. \n\nLet $(D,<)$ be any linear order. We say that $S\\subseteq D$ {\\it is a semi-interval of $(D,<)$} if $S$ is convex in $(D,<)$ and $a=\\min S$ exists; in that case, we also say that $S$ is a semiinterval with minimum $a$, and denote it by $S_a$ to emphasize $a=\\min S_a$.\n\n\n\\begin{Definition}\\label{Definition_family_semiintervals}\nLet $(D,<)$ be a linear order and let $\\mathcal S=(S_x\\mid x\\in D)$ be a family of semi-intervals of $D$. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item $\\mathcal S$ is an {\\it $A$-definable family of semiintervals of $(D,<)$} if $(D,<)$ is $A$-definable and there exists an $L_A$-formula $S(x,y)$ such that $S_a=S(a,\\Mon)$ holds for all $a\\in D$.\n\\item The family $\\mathcal S$ is {\\it monotone} if $\\sup S_x\\leqslant \\sup S_y$ for all elements $x\\leqslant y$ of $D$. \n\\end{enumerate}\n\\end{Definition} \nRecall that by $\\sup S_x\\leqslant \\sup S_y$ we mean that any upper bound of $S_y$ is also an upper bound of $S_x$.\n\n\nSuppose that $\\mathcal S=(S_x\\mid x\\in D)$ is an $A$-definable family of semiintervals of $(D,<)$, defined by $S(x,y)$. Recursively, define: \n\\begin{center}\n $S^{(1)}(x,y):=S(x,y)$ \\ \\ and \\ \\ $S^{(n+1)}(x,y):=(\\exists t)(S^{(n)}(x,t)\\land S(t,y))$ \\ ($n\\in\\mathbb N$). \n\\end{center}\nFurther in the text, we will use the following notation:\n\\begin{center}$\\mathcal S^{n}:=(S^{(n)}(x,D)\\mid x\\in D)$, \\ \\ $S_x^n:=S^{(n)}(x,D)$ \\ \\ and \\ \\ $S^{\\omega}_x:=\\bigcup_{n\\in\\mathbb N} S_x^n$. \n\\end{center}\n \n\\begin{Observation}\\label{Lemma_semiint_verybasic1}\nSuppose that $\\mathcal S=(S_x\\mid x\\in D)$ is an $A$-definable family of semiintervals of $(D,<)$. \n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item $\\mathcal S^{n}$ is an $A$-definable family of semiintervals of $(D,<)$; if $\\mathcal S$ is monotone, then so is $\\mathcal S^n$ for all $n\\in \\mathbb N$. \n\\item $(S_x^n\\mid n\\in\\mathbb N)$ is a $\\subseteq$-increasing sequence for all $x\\in D$. If $S_x^n=S_x^{n+1}$ for some $n\\in \\mathbb N$, then $S_x^n=S_x^m$ for all $m\\geqslant n$, so $S_x^n=S_x^\\omega$. Hence, the sequence is strictly increasing or eventually constant.\n\\item $(S^{\\omega}_x\\mid x\\in D)$ is a family of semiintervals of $(D,<)$; it is monotone if $\\mathcal S$ is monotone.\n\\item $\\models S^{(n+m)}(x,y)\\leftrightarrow (\\exists t)(S^{(n)}(x,t)\\land S^{(m)}(t,y))$, so \n$S^{n+m}_a=\\bigcup_{t\\in S_a^n}S_t^m$ for all $m\\in\\mathbb N$.\n\\end{enumerate}\n\\end{Observation}\n \n\n \n\\begin{Definition}\\label{Def_shift}\\label{Definition_shift} \nLet $\\mathcal S=(S_x\\mid x\\in D)$ be a family of semiintervals of $(D,<)$. We say that the semiinterval $S_a\\in\\mathcal S$ is an {\\em $\\mathcal S$-shift} if the sequence $(S_a^n\\mid n\\in\\mathbb N)$ is strictly $\\subseteq$-increasing. \n\\end{Definition}\n\n\n\n\\begin{Remark}\\label{Remark_shift} \nLet $\\mathcal S$ be a family of semiintervals of $(D,<)$ and let $S_a\\in \\mathcal S$. By Observation \\ref{Lemma_semiint_verybasic1}(b) we have two options: either $S_a$ is an $\\mathcal S$-shift, or $S_a^n=S^{\\omega}_a$ holds for some $n\\in\\mathbb N$. \n\\end{Remark}\n\n\n\n\\begin{Example}\n(1) \\ Consider the theory $T=\\Th(\\mathbb Q,<,S)$ where $<$ is the usual order on the rationales and $S$ is defined by: \\ $(x,y)\\in S$ if and only if $x\\leqslant y<x+\\sqrt{2}$. Here, the formula $S(x,y)$ defines a monotone family of semiintervals with $S_0^n=\\{y\\mid 0\\leqslant y<n\\sqrt 2\\}$, so $S_0$ is a shift. Since $T$ is weakly o-minimal, $I(T,\\aleph_0)=2^{\\aleph_0}$ follows by Theorem \\ref{Theorem1_shift}; it also follows by Theorem \\ref{Theorem trivial implies simple and convex}, since the unique type $p\\in S_1(\\emptyset)$ is trivial and non-simple. \n\n \n(2) \\ A typical example of a monotone family without shifts is as follows. Let $(\\Mon,<,E)$ be a linear order with a convex equivalence relation $E$. Define $S(x,y):=x\\leqslant y\\land E(x,y)$; $S_x$ is a final part of the class $[x]_E$ that begins at $x$. Here $S_x^n=S_x$ holds for all $n$, so $S_x$ is not a shift. \n\\end{Example}\n\n\n\n\\begin{Lemma}\\label{Lemma_shift_properties}\nSuppose that $\\mathcal S$ is a monotone $A$-definable family of semiintervals of $(D,<)$, $a\\in D$, and $b\\in S^{\\omega}_a$. Then $S^{\\omega}_b$ is a final part of $S^{\\omega}_a$, and if $S_a$ is an $\\mathcal S$-shift, then so is $S_b$. \n\\end{Lemma}\n\\begin{proof}\nChoose $n\\in\\mathbb N$ such that $b\\in S^n_a$. As we observed in \\ref{Lemma_semiint_verybasic1}(d), $S^{n+m}_a=\\bigcup_{x\\in S_a^n}S_x^m$ holds for all $m\\in\\mathbb N$. This, together with $b\\in S_a^n$ implies $S^{m+n}_a\\supseteq S^m_b$, so $S_a^\\omega\\supseteq S_b^\\omega$ holds. \nSince the family $\\mathcal S^m$ is monotone for all $m\\in\\mathbb N$, $a\\leqslant b$ implies $\\sup S_a^m\\leqslant \\sup S_b^m$ and thus $\\sup S_a^\\omega\\leqslant \\sup S_b^\\omega$. This, together with $S_a^\\omega\\supseteq S_b^\\omega$ implies that $S_b^\\omega$ is a final part of $S_a^\\omega$ and proves the first claim. \nTo prove the second, assume that $S_b$ is not an $\\mathcal S$-shift; we prove that $S_a$ is also not an $\\mathcal S$-shift. By Remark \\ref{Remark_shift}, there is $k\\in\\mathbb N$ with $S_b^k=S_b^\\omega$. By the above proved, we have $S^k_b\\subseteq S_a^{k+n}$; since $S_b^\\omega=S_b^k$ is a final part of $S_a^\\omega$, we conclude that the set $S_a^{k+n}$ contains a final part of $S_a^\\omega$. Clearly, this implies $S_a^{k+n}=S_a^\\omega$, so $S_a$ is not an $\\mathcal S$-shift by Remark \\ref{Remark_shift}. This proves the second claim. \n\\end{proof}\n\n\\begin{Lemma}\\label{Lemma_exists_monotonefamily} \nIf some $A$-definable set $D$ admits an $A$-definable family of semiintervals (with respect to some linear order) such that $S_a$ is its shift, then it also admits an $Aa$-definable order $<$ with $a=\\min D$, and an $Aa$-definable monotone family of semiintervals of $(D,<)$ that contains $S_a$ as a shift. \n\\end{Lemma}\n\\begin{proof}\nLet $\\mathcal S=(S_x\\mid x\\in D)$ be an $A$-definable family of semiintervals such that $S_a$ is a $\\mathcal S$-shift. Modify $<$ and $\\mathcal S$ as follows.\nUsing $a$ as a parameter, first modify the family $\\mathcal S$ by setting $S_x=\\{x\\}$ for $x\\in (-\\infty,a)$, and then modify the order $<$ by\nmoving the segment $(-\\infty,a)$ to the right of $[a,+\\infty)$. Notice that the modified family is $Aa$-definable, that $S_{a}$ is still an $\\mathcal S$-shift, and that $a=\\min D$.\n\nFor $x\\in D$ define: $R_x=\\bigcup_{t\\leqslant x}S_t\\cap [x,+\\infty)$. It is easy to see that $\\mathcal R=(R_x\\mid x\\in D)$ is a monotone $Aa$-definable family of semiintervals of $(D,<)$ such that $R_a=S_a$, $S_x\\subseteq R_x$ for all $x\\in D$, and $S_a^k\\subseteq R_a^k$ for all $k\\in\\mathbb N$. We will prove that $R_a$ is a $\\mathcal R$-shift. \n \n\\noindent{\\em Claim.} \\ \\ $b\\in S_a^n$ \\ implies \\ $R_b^k\\subseteq S_a^{k(n+1)}$.\n\nBy induction on $k$. For $b\\in S_a^n$ we have \\ $R_b\\subseteq \\bigcup_{x\\leqslant b}S_x\\subseteq \\bigcup_{x\\in S_a^n}S_x=S_a^{n+1}$; here, the first inclusion follows by the definition of $R_b$, the second by $b\\in S_a^n$, and the equality by \\ref{Lemma_semiint_verybasic1}(d). This proves the case $k=1$. Assume that the claim holds for all $x\\in S_a^n$ and some $k\\in \\mathbb N$: $R_x^k\\subseteq S_a^{k(n+1)}$. We have:\n\\begin{center}\n $R_b^{k+1}=\\bigcup_{x\\in R_b^k}R_x\\subseteq \\bigcup_{x\\in S_a^{k(n+1)}}R_x \\subseteq \\bigcup_{x\\in S_a^{k(n+1)}} S_x^{n+1}= S_a^{(k+1)(n+1)}$; \n\\end{center}\nHere, the equalities follow by \\ref{Lemma_semiint_verybasic1}(d), the first inclusion by the induction hypothesis, and the second by the case $k=1$. This proves the claim. \n\nTherefore, $S_a^k\\subseteq R_a^k\\subseteq S_a^{(k+1)(n+1)}$ holds for all $k\\in\\mathbb N$. This, together with the fact that the family $(S_a^n\\mid n\\in\\mathbb N)$ is strictly increasing, implies that the family $(R_a^k\\mid k\\in\\mathbb N)$ must strictly increase, too; $R_a$ is a $\\mathcal R$-shift.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{Theorem1_shift}}\n\n\nIn this subsection, we will prove Theorem \\ref{Theorem1_shift}. \nThroughout, assume that $T$ is a countable weakly quasi-o-minimal theory with respect to $<$, $\\mathcal S=(S_x\\mid x\\in \\Mon)$ is a definable family of semiintervals of $(\\Mon,<)$, $c_0\\in \\Mon$, and $S_{c_0}$ is an $\\mathcal S$-shift. We need to prove $I(T,\\aleph_0)=2^{\\aleph_0}$. \n\n\nAbsorb in the language the parameter $c_0$ and a finite set of parameters needed to define the family $\\mathcal S$; note that this does not affect the desired conclusion $I(T,\\aleph_0)=2^{\\aleph_0}$.\nBy Lemma \\ref{Lemma_exists_monotonefamily}, after possibly modifying $<$ and $\\mathcal S$, we can also assume that $\\mathcal S$ is monotone with $c_0=\\min \\Mon$. \nTherefore, for the rest of this section, we will assume: \\begin{itemize} \n\\item $T$ is small; \n\\item $\\mathcal S=(S_x\\mid x\\in\\Mon)$ \\ is a $0$-definable, monotone family of semiintervals of $(\\Mon,<)$;\n\\item $c_0=\\min \\Mon$ \\ and \\ $S_{c_0}$ is an $\\mathcal S$-shift; $S^{1}_{c_0}\\subset S^{2}_{c_0}\\subset S^{3}_{c_0}\\subset\\dots$ is a family of initial parts of $\\Mon$. \n\\end{itemize}\nThe following notation is convenient:\n\\begin{itemize}\n \\item $\\mathcal C:=S^{\\omega}_{c_0} =\\bigcup_{n\\in\\mathbb N} S^{n}_{c_0}$; \n \\item $\\Pi_A(x):=\\{\\phi(x)\\in L_A\\mid \\mbox{$\\phi(\\Mon)$ is convex and contains a final part of $\\mathcal C$}\\}$; $\\Pi(x):=\\Pi_{\\emptyset}(x)$; \\ \\ \n \\item $S_{\\Pi}(A):=\\{p\\in S_1(A)\\mid \\Pi_A(x)\\subseteq p(x)\\}$. \n\\end{itemize}\n\n \n\n\\begin{Remark}\n\\begin{enumerate}[\\hspace{10pt}(a)] \n\\item $\\mathcal C$ is an initial part of $(\\Mon,<)$, and $\\mathcal C<x$ is $\\emptyset$-type-definable. For all $n\\in \\mathbb N$, the formula $S^n_{c_0}<x$ belongs to $\\Pi_A(x)$.\n \\item Lemma \\ref{Lemma_shift_properties} implies that $S_c$ is an $\\mathcal S$-shift and $S^\\omega_c$ is a final part of $\\mathcal C$ for all $c\\in \\mathcal C$. \n \\item $\\Pi_A(x)$ is a partial type, $(\\mathcal C<x)\\subseteq \\Pi_A(x)$, and $\\Pi_A(x)$ is finitely satisfiable in $\\mathcal C$.\n\\end{enumerate}\n\\end{Remark}\n \n\nWe will say that $p\\in S_1(A)$ is a {\\it $\\mathcal C$-type} if $p$ is finitely satisfiable but not realized in $\\mathcal C$. $\\mathcal C$-types have extension property: if $p\\in S_1(A)$ is a $\\mathcal C$-type and $A\\subseteq B$, then there is a $\\mathcal C$-type $q\\in S_1(B)$ with $p\\subseteq q$.\nMost of our efforts are aimed at proving Corollary \\ref{Corollary_Ctype_trivial}, which states that every $\\mathcal C$-type over a finite domain is trivial. Then, using the description of the $\\mathcal C$-types proved in the next lemma, we will be able to quickly complete the proof of Theorem \\ref{Theorem1_shift}. \n\n\\begin{Lemma}\\label{Lemma_Pi_basic}\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item For all $p\\in S_1(A)$ the following conditions are equivalent:\n\\begin{tasks}(3)\n \\task[(1)] \\ $p\\in S_{\\Pi}(A)$; \n \\task[(2)] \\ $p$ is a $\\mathcal C$-type;\n \\task[(3)] \\ $\\mathcal C=\\{x\\in\\Mon\\mid x< p(\\Mon)\\}$. \n\\end{tasks}\n\\item $\\mathcal C\\cup\\Pi_A(\\Mon)$ is an initial part of $\\Mon$.\n\\item $\\Pi_A(x)$ is the minimal interval type from $IT(A)$ consistent with $\\mathcal C<x$. \n\\item $S_{\\Pi}(A)=\\{(\\mathbf p_l)_{\\restriction A}\\mid p\\in S_{\\Pi}(\\emptyset)\\}$. \n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof} (a) (1)$\\Rightarrow$(2) Assume $p\\in S_\\Pi(A)$. Then $(\\mathcal C<x)\\subseteq \\Pi_A(x)\\subseteq p(x)$, so $p$ is not realized in $\\mathcal C$. To prove that $p$ is a $\\mathcal C$-type, it remains to show that $p(x)$ is finitely satisfiable in $\\mathcal C$. Suppose that an $L_A$-formula $\\phi(x)$ is not realized in $\\mathcal C$. Then the set of realizations of (a formula) $x<\\phi(\\Mon)$ is convex in $\\Mon$ and contains $\\mathcal C$; hence $(x<\\phi(\\Mon))\\in \\Pi_A(x)\\subseteq p(x)$ and thus $p(\\Mon)<\\phi(\\Mon)$. In particular, $\\phi(x)\\notin p$. Therefore, $p$ contains only formulae that are satisfied in $\\mathcal C$; $p$ is a $\\mathcal C$-type. \n\n(2)$\\Rightarrow$(3) Suppose that $p$ is a $\\mathcal C$-type. We need to prove that $\\mathcal C$ is the set of all lower bounds of $p(\\Mon)$. Since $p$ is not realized in $\\mathcal C$, which is an initial part of $\\Mon$, we conclude $\\mathcal C< p(\\Mon)$, so any $c\\in\\mathcal C$ is a lower bound of $p(\\Mon)$. To prove (3), it remains to show that there are no other lower bounds; assuming $\\mathcal C<a$ we will prove $a\\nless p(\\Mon)$. Since $p$ is a $\\mathcal C$-type, by the extension property there is a $b\\in p(\\Mon)$ such that $\\tp(b/aA)$ is a $\\mathcal C$-type. Then, since $(x<a)\\in \\tp(c/aA)$ for all $c\\in \\mathcal C$, we have $(x<a)\\in \\tp(b/aA)$. \nNow, $b<a$ and $b\\in p(\\Mon)$ together imply $a\\nless p(\\Mon)$, so $a$ is not a lower bound of $p(\\Mon)$. \n\n\n(3)$\\Rightarrow$(1) Suppose that $\\mathcal C$ is the set of all lower bounds of $p(\\Mon)$. To prove $p\\in S_{\\Pi}(A)$, i.e. $\\Pi_A(x)\\subseteq p(x)$, assume $\\phi(x)\\in\\Pi_A(x)$; we need to prove $\\phi(x)\\in p$.\nSince the set $\\phi(\\Mon)$ contains a final part of $\\mathcal C$, \nthere is a $n\\in\\mathbb N$ such that $\\mathcal C\\subseteq S^n_{c_0}\\cup \\phi(\\Mon)$. \nPut $\\psi(x):=(\\forall y)(y\\leqslant x\\rightarrow y\\in S^n_{c_0}\\cup \\phi(\\Mon))$. Clearly, \n$\\mathcal C\\subseteq \\psi(\\Mon)$, so by compactness $\\{\\psi(x)\\land\\ S^n_{c_0}<x\\mid n\\in\\mathbb N\\}$ is consistent; let $b$ realize it. Then $\\mathcal C<b$, so $b$ is not a lower bound of $p(\\Mon)$. Hence, there is $a\\models p$ such that $a<b$. \nThen $\\models \\psi(b)$ implies $\\models \\phi(a)$, so $\\phi(x)\\in p$. Therefore, $\\Pi_A(x)\\subseteq p(x)$.\n\n(b) Assuming $b<a$ and $a\\in \\mathcal C\\cup \\Pi_A(\\Mon)$ we need to prove $b\\in \\mathcal C\\cup \\Pi_A(\\Mon)$. This is immediate if $a\\in \\mathcal C$ or $b\\in\\mathcal C$. So assume $a,b\\notin\\mathcal C$ and let $p=\\tp(a/A), q=\\tp(b/A)$. Then $p\\in S_{\\Pi}(A)$ as $a\\in \\Pi_A(\\Mon)$. By (1)$\\Leftrightarrow$(3) from part (a) of the lemma, $\\mathcal C$ is the set of all lower bounds of $p(\\Mon)$. In particular, $b$ is not a lower bound of $p(\\Mon)$, so $a'<b<a$ holds for some $a'\\models p$. It follows that $p$ and $q$ extend the same interval type from $IT(A)$ and that $p(\\Mon)$ and $q(\\Mon)$ have the same set of lower bounds, so $q\\in S_{\\Pi}(A)$ follows by (1)$\\Leftrightarrow$(3) from part (a) of the lemma. Therefore, $b\\in\\Pi_A(\\Mon)$, as desired. \n\n(c) We prove that $\\Pi_A(x)$ is an interval type, i.e. a maximal consistent set of convex $L_A$-formulae. Let $\\phi(x)$ be a convex $L_A$-formula such that $\\phi(x)\\notin \\Pi_A(x)$; we show that $\\phi(x)$ is inconsistent with $\\Pi_A(x)$. As $\\phi(x)$ is convex and $\\phi(x)\\notin \\Pi_A(x)$, the set $\\phi(\\Mon)$ does not contain a final part of $\\mathcal C$. Then one of the convex components of $\\lnot\\phi(\\Mon)$, say $D$, contains a final part of $\\mathcal C$. This implies that the formula expressing $x\\in D$ belongs to $\\Pi_A(x)$. Since $x\\in D$ is inconsistent with $\\phi(x)$, it follows that $\\phi(x)$ is inconsistent with $\\Pi_A(x)$, as desired. Therefore, $\\Pi_A(x)$ is an interval type. The minimality of $\\Pi_A(x)$ follows from (b).\n\n(d) Let $p\\in S_{\\Pi}(\\emptyset)$. By (1)$\\Leftrightarrow$(3) from part (a), $\\mathcal C$ is the set of all the lower bounds of $p(\\Mon)$. Let $p_A=(\\mathbf p_l)_{\\restriction A}$. \nBy the definition of $\\mathbf p_l$, the set $p_A(\\Mon)$ is an initial part of $p(\\Mon)$, so $p(\\Mon)$ and $p_A(\\Mon)$ share the same set, $\\mathcal C$, of lower bounds; $p_A\\in S_{\\Pi}(A)$ follows by part (a) of the lemma. Therefore, $\\{(\\mathbf p_l)_{\\restriction A}\\mid p\\in S_{\\Pi}(\\emptyset)\\}\\subseteq S_{\\Pi}(A)$. \nTo prove the converse, suppose that $q_A\\in S_{\\Pi}(A)$ and let $q=(q_A)_{\\restriction \\emptyset}$; clearly, $q\\in S_{\\Pi}(\\Mon)$. Now, part (a) of the lemma applies to both $q$ and $q_A$, so the sets $q(\\Mon)$ and $q_A(\\Mon)$ have the same set, $\\mathcal C$, of lower bounds. This, together with the fact that $q_A(\\Mon)$ is a convex subset of $q(\\Mon)$, implies that $q_A(\\Mon)$ is an initial part of $q(\\Mon)$, so $q_A=(\\mathbf q_l)_{\\restriction A}$, proving the converse. Therefore, $S_{\\Pi}(A)=\\{(\\mathbf p_l)_{\\restriction A}\\mid p\\in S_{\\Pi}(\\emptyset)\\}$.\n\\end{proof}\n\nNext, we consider forking as a binary relation on $\\Pi_A(\\Mon)$: \n\\begin{itemize}\n \\item $\\mathcal D_{\\Pi_A}:=\\{(x,y)\\in\\Pi_A(\\Mon)^2\\mid x\\dep_A y\\}$; \\ \\ \\ \\ $\\mathcal D_{\\Pi_A}(a):=\\{x\\in \\Pi_A(\\Mon)\\mid (x,a)\\in\\mathcal D_{\\Pi_{A}}\\}$.\n\\end{itemize}\nRecall that a set $D\\subseteq \\Mon$ is $\\Pi_A$-bounded if there are $a_1,a_2\\in \\Pi_A(\\Mon)$ such that $a_1<D<a_2$. \n\n\n \n\\begin{Lemma}\\label{Lemma_forking_on_D_Pi}\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item $(a,b)\\in \\mathcal D_{\\Pi_{A}}$ \\ if and only if \\ $\\tp(a/Ab)$ contains a $\\Pi_A$-bounded formula; \n\n\\item $\\mathcal D_{\\Pi_A}$ is a convex equivalence relation on $\\Pi_A(\\Mon)$.\n\n\\item $\\mathcal S^{\\omega}_a\\subseteq \\mathcal D_{\\Pi_A}(a)$ \\ holds for all $a\\in\\Pi_A(\\Mon)$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n(a) and (b) follow from Lemma \\ref{Lemma_forking_on_interval_type}. To prove (c), let $a\\in \\Pi_A(\\Mon)$ and we prove that $S_a$ is $\\Pi_A$-bounded. \nPut $p=\\tp(a/A)$. Choose $b\\in p(\\Mon)$ such that $a$ is left $\\mathbf p$-generic over $b$. By Lemma \\ref{Lemma_Pi_basic}(a), $\\tp(a/bA)$ is a $\\mathcal C$-type. Note that for all $c\\in\\mathcal C$ we have $S_c<b$, so the formula (expressing) $S_x<b$ belongs to $\\tp_x(a/bA)$. In particular, $S_a<b$ and thus $a\\leqslant S_a<b$. Therefore, $x\\in S_a$ is a $\\Pi_A$-bounded formula, so $S_a\\subseteq \\mathcal D_{\\Pi_A}(a)$; $\\mathcal S^{n}_a\\subseteq \\mathcal D_{\\Pi_A}(a)$ follows by induction, hence $\\mathcal S^{\\omega}_a\\subseteq \\mathcal D_{\\Pi_A}(a)$.\n\\end{proof}\n \n\n\\begin{Definition}\nLet $D\\subseteq \\Mon$ and $a\\in D$. We say that $a$ is an {\\it $\\mathcal S$-minimal element of $D$} if $a=c_0$ or if there exists $b< D$ such that $a\\in S_b$.\n\\end{Definition}\nUsually, when we say that $a\\in D$ is $\\mathcal S$-minimal, the minimality is with respect to $D$. Note that if $D$ is definable, then ``$x$ is an $\\mathcal S$-minimal element of $D$'' is also definable. \n\n\\begin{Lemma}\\label{Lemma_S_minimal_exist}\n\\begin{enumerate}[\\hspace{10pt}(a)]\n\\item Every definable set $D$ that intersects $\\mathcal C\\cup \\Pi(\\Mon)$ contains an $\\mathcal S$-minimal element. Moreover, if $D$ is $\\bar b$-definable, then there is an $\\mathcal S$-minimal element $a\\in D$ such that $\\tp(a/\\bar b)$ is isolated. \n\n\\item Suppose that $D\\subseteq \\Pi(\\Mon)$ is the locus of an isolated type (over some parameters). Then every element of $D$ is $\\mathcal S$-minimal and, whenever $D$ intersects some $\\mathcal D_{\\Pi_A}$-class for some $A$, then it is completely contained within that class. \n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof} \n(a) First, consider the case $D\\cap \\mathcal C\\neq 0$. If $c_0\\in D$, then $c_0$ is $\\mathcal S$-minimal in $D$. Assume $c_0\\notin D$ and let $n\\in\\mathbb N$ be minimal with $S^n_{c_0}\\cap D\\neq 0$. We claim that every element $a\\in S^n_{c_0}\\cap D$ is $\\mathcal S$-minimal in $D$. If $n=1$ then $c_0$ witnesses the $\\mathcal S$-minimality of $a$ in $D$, as $c_0<D$ and $a\\in S_{c_0}$ hold. If $n>1$, then every element $a\\in D\\cap S^n_{c_0}$ is $\\mathcal S$-minimal in $D$: $a\\in S^n_{c_0}$ implies that $a\\in S_d$ holds for some $d\\in S^{n-1}_{c_0}$; the assumed minimality of $n$ implies $d<D$, so $d$ witnesses the $\\mathcal S$-minimality of $a$ in $D$. \n\nThe second case is $D\\cap\\Pi(\\Mon)\\neq 0$. Let $\\phi(x,\\bar b)$ be a formula that defines $D$ and let $d\\in D\\cap \\Pi(\\Mon)$. \nLet $\\sigma(y,\\bar z)$ be an $L$-formula that says ``$y$ is an $\\mathcal S$-minimal element of $\\phi(\\Mon,\\bar z)$''. \nBy the first case, for each $\\bar b'\\in \\Mon^{|\\bar b|}$ the following holds: if the set $\\phi(\\Mon,\\bar b')$\nmeets $\\mathcal C$ then $\\phi(\\Mon,\\bar b')$ has an $\\mathcal S$-minimal element. Therefore, \nfor all $c\\in\\mathcal C$ \nwe have: \\ $\\models (\\forall \\bar z)(\\phi(c,\\bar z) \\rightarrow (\\exists y)\\, \\sigma(y,\\bar z))$. Since $\\tp(b)$ is finitely satisfiable in $\\mathcal C$, the same holds with $c$ replaced by $d$; $D$ contains an $\\mathcal S$-minimal element. To prove the ``moreover\" part, assume $D$ is $\\bar b$-definable. Choose an $L_{\\bar b}$-formula $\\phi(x)$ that expresses ``$x$ is an $\\mathcal S$-minimal element of $D$''; it is consistent by part (a), so, since $T$ is small, $\\phi(x)$ belongs to some isolated type of $S_1(\\bar b)$. \n\n\n(b) Let $\\phi(x,\\bar b)$ define $D$ and isolate a complete type in $S_1(\\bar b)$. By (a), we know that some element of $x\\in D$ satisfies the formula ``$x$ is an $\\mathcal S$-minimal element of $\\phi(\\Mon,\\bar b)$''. Since $x\\in D$ isolates a complete type, all elements of $D$ satisfy that formula. This proves the first assertion. To prove the second, assume that $a\\in D\\cap \\Pi_A(\\Mon)$; we need to show $D\\subseteq \\mathcal D_{\\Pi_A}(a)$. By the first claim, $a$ is $\\mathcal S$-minimal in $D$, so there is a $b\\in \\Pi(\\Mon)$ witnessing that: $b<D$ and $a\\in S_b$. By Lemma \\ref{Lemma_forking_on_D_Pi}(c), we have $\\mathcal S^{\\omega}_b\\subseteq \\mathcal D_{\\Pi_A}(b)$, so $a\\in S_b$ implies $\\mathcal D_{\\Pi_A}(a)=\\mathcal D_{\\Pi_A}(b)$. Therefore, to complete the proof, it suffices to prove $D\\subseteq S_a\\cup S_b$. \n\nSuppose on the contrary that $d\\in D\\smallsetminus (S_a\\cup S_b)$. \nNote that $S_a\\cup S_b$ is a convex set, so $b\\in S_b$ and $b<D$ together imply $(S_b\\cup S_a)<d$. By the above proved, $d\\in D$ is $\\mathcal S$-minimal; let $b'\\in \\Pi_A(\\Mon)$ be a witness for that: $b'<D$ and $d\\in S_{b'}$. Then $b'<a$ as $b'<D\\ni a$, so the monotonicity of $\\mathcal S$ implies $\\sup S_{b'}\\leqslant \\sup S_a$. On the other hand, $S_a<d$ and $d\\in S_{b'}$ imply $\\sup S_{a}<\\sup S_{b'}$. Contradiction.\n\\end{proof}\n\n\n\\begin{Lemma}\\label{Lemma_every_convexshift_istrivial} \nEvery convex $\\mathcal C$-type over a finite domain is trivial. \n\\end{Lemma}\n\\begin{proof}\nToward contradiction assume that $A_0$ is finite and $r\\in S_1(A_0)$ is a non-trivial convex $\\mathcal C$-type. \nBy Remark \\ref{Remark_notrivial_wom_witness}, there is a finite Morley sequence, $\\bar b$, in $\\mathbf{r}_l$ over $A_0$ such that the type $p=(\\mathbf{r}_l)_{\\restriction A_0\\bar b}$ is non-trivial, as witnessed by a triangle; put $A=A_0\\bar b$. Note that Lemma \\ref{Lemma_Pi_basic}(a) guarantees:\n\n(1) \\ $p\\in S_1(A)$ is a $\\mathcal C$-type and $p\\in S_{\\Pi}(A)$.\n\n\\noindent\nAlso note that $p$ is convex, as a complete extension of a convex weakly o-minimal type $r$. Since $\\Pi_A(x)$ is an interval type and since $p\\in S_{\\Pi}(A)$ is convex, $p$ is an isolated point of $S_{\\Pi}(A)$. Then there is a formula $\\theta(x)$ such that: \n\n(2) \\ $\\theta(x)\\in p$ \\ and \\ $\\Pi_A(x)\\cup\\{\\theta(x)\\}\\vdash p(x)$.\n\n\\noindent \nNote that this guarantees that $p(\\Mon)$ is convex in $(\\theta(\\Mon),<)$. \nLet $(a_3,a_2,a_1)$ be a $\\mathbf p_l$-triangle over $A$: \n\n(3) \\ $\\mathcal D_p(a_1)<\\mathcal D_p(a_2) <\\mathcal D_p(a_3)$, \\ and \\ $a_1\\dep_{Aa_3} a_2$.\n\n \n\\noindent \nBy Lemma \\ref{Lemma_forking_on_D_Pi}, the classes $\\mathcal D_{\\Pi_A}(a_1),\\mathcal D_{\\Pi_A}(a_2)$, and $\\mathcal D_{\\Pi_A}(a_3)$ are convex; they are disjoint because $a_1,a_2$, and $a_3$ are pairwise independent over $A$. Therefore, $\\mathcal D_{\\Pi_A}(a_1)<\\mathcal D_{\\Pi_A}(a_2) <\\mathcal D_{\\Pi_A}(a_3)$.\nNow, we claim that a triple $(a_1,a_2,a_3)$ can be found to also satisfy the following:\n\n(4) \\ $\\tp(a_1/a_2a_3A)$ is isolated.\n\n\\noindent\nTo prove (4), start with a triple $(a_1',a_2,a_3)$ satisfying $\\mathcal D_{\\Pi_A}(a_1')<\\mathcal D_{\\Pi_A}(a_2) <\\mathcal D_{\\Pi_A}(a_3)$ and $a_1'\\dep_{Aa_3} a_2$. Put $q=(\\mathbf p_l)_{\\restriction a_3A}$; clearly, $q(\\Mon)$ is an initial part of $p(\\Mon)$ and $a_1',a_2\\in q(\\Mon)$. Note that $q(\\Mon)$ is a convex subset of $(\\theta(\\Mon),<)$, as $p(\\Mon)$ is. Choose a formula $\\phi(x,y)\\in\\tp(a_1'a_2/a_3A)$ that implies $x<y\\land \\theta(x)\\land \\theta(y)$ and witnesses $a_1'\\dep_{Aa_3} a_2$: \n$\\phi(\\Mon,a_2)$ is a $q$-bounded subset of $\\Mon$. Then $\\phi(\\Mon,a_2)\\subseteq \\theta(\\Mon)$ implies that $\\phi(\\Mon,a_2)$ is a $q$-bounded subset of $(\\theta(\\Mon), <)$, so since $q(\\Mon)$ is a convex subset of $(\\theta(\\Mon),<)$, we derive $\\phi(\\Mon,a_2)\\subset q(\\Mon)$.\n\n\n Let $a_1$ be an $\\mathcal S$-minimal element of $\\phi(\\Mon,a_2)$ with $\\tp(a_1/a_2a_3A)$ isolated; it exists by Lemma \\ref{Lemma_S_minimal_exist}. Clearly, $a_1\\models q$ and the triple $(a_1,a_2,a_3)$ satisfy condition (4). We will prove that (3) is also satisfied. \n Choose $b\\in\\Pi_A(\\Mon)$ to witness the $\\mathcal S$-minimality of $a_1$ in $\\phi(\\Mon,a_2)$: $b< \\phi(\\Mon,a_2)$ and $a_1\\in S_b$.\n By Lemma \\ref{Lemma_forking_on_D_Pi}(b), $a_1\\in S_b$ implies $\\mathcal D_{\\Pi_A}(a_1)=\\mathcal D_{\\Pi_A}(b)$. In addition, $b<\\phi(\\Mon,a_2)$ and $\\models \\phi(a_1',a_2)$ imply $b\\leqslant a_1'$, which together with $\\mathcal D_{\\Pi_A}(a_1')<\\mathcal D_{\\Pi_A}(a_2)$ implies $\\mathcal D_{\\Pi_A}(b)<\\mathcal D_{\\Pi_A}(a_2)$. Hence $\\mathcal D_{\\Pi_A}(a_1)<\\mathcal D_{\\Pi_A}(a_2)<\\mathcal D_{\\Pi_A}(a_3)$. As $\\mathcal D_p(a_i)\\subseteq \\mathcal D_{\\Pi_A}(a_i)$ clearly holds for $i=1,2,3$, we conclude $\\mathcal D_{\\Pi_A}(a_1)<\\mathcal D_{\\Pi_A}(a_2)<\\mathcal D_{\\Pi_A}(a_3)$. Since $a_1\\in\\phi(\\Mon,a_2)$ and since the formula $\\phi(x,a_2)$ has been chosen to witness $a_1'\\dep_{a_3A} a_2$, we conclude $a_1\\dep_{a_3A} a_2$. Therefore, the triple $(a_1,a_2,a_3)$ satisfies conditions (3) and (4).\n\n\\smallskip\nFrom now on assume that $(a_1,a_2,a_3)$ satisfies conditions (3) and (4). Before stating the next claim, recall from the proof of (4) that the type $q=\\tp(a_1/a_3A)=\\tp(a_2/a_3A)$ is convex and that $q(\\Mon)$ is convex in $(\\theta(\\Mon), <)$. We claim:\n\n(5) \\ $\\tp(a_2/a_1a_3A)$ is isolated.\n\n\\noindent \nWe know that $a_2\\dep_{Aa_3} a_1$ holds and that the type $\\tp(a_1/a_2a_3A)$ is isolated, so there is a formula $\\phi'(x,y)\\in\\tp(a_1,a_2/a_3A)$ implying $x<y\\land \\theta(x)\\land \\theta(y)$ such that $\\phi'(a_1,\\Mon)$ is a $q$-bounded subset of $q(\\Mon)$ and $\\phi'(x,a_2)$ isolates $\\tp(a_1/a_2a_3A)$. \nAs $\\phi'(a_1,\\Mon)$ is a $q$-bounded subset of $\\theta(\\Mon)$ and $q(\\Mon)$ is convex in $(\\theta(\\Mon),<)$, we conclude $\\phi'(a_1,\\Mon)\\subset q(\\Mon)$. \nChoose $a_2'\\in\\phi'(a_1,\\Mon)$ such that $\\tp(a_2'/a_1a_3A)$ is isolated; then $a_2\\in q(\\Mon)$. Now, both $a_2$ and $a_2'$ realize the type $q\\in S_1(a_3A)$, so $a_2\\equiv a_2'\\,(a_3A)$. Since the formula $\\phi'(x,a_2)$ isolates a complete type in $S_1(a_2a_3A)$, the formula $\\phi'(x,a_2')$ isolates the $a_3A$-conjugate of that type in $S_1(a_2'a_3A)$. Then $\\models \\phi'(a_1,a_2)\\land \\phi(a_1,a_2')$ implies $a_1a_2\\equiv a_1a_2'\\,(a_3A)$. Since $\\tp(a_2'/a_1a_3A)$ is isolated, so is $\\tp(a_2/a_1a_3A)$; this proves (5).\n\n\n\\smallskip \nLet $\\psi(a_1,y,a_3)$ be a formula that isolates $\\tp(a_2/a_1a_3A)$ and let $M$ be a countable model prime over $a_1a_3A$. We claim:\n\n(6) \\ The set $\\{\\mathcal D_p(x)\\mid x\\in p(M)\\}$ is densely ordered by $<$.\n\n\\noindent\nWe need to prove that for all $b_1,b_3\\in p(M)$ satisfying $\\mathcal D_p(b_1)<\\mathcal D_p(b_3)$ there exists a $b_2\\in M$ with $\\mathcal D_p(b_1)<\\mathcal D_p(b_2)<\\mathcal D_p(b_3)$. Note that $b_1b_3\\equiv a_1a_3\\,(A)$, so the formula $\\psi(b_1,y,b_3)$ isolates a complete type in $S_1(b_1b_3A)$; that type is realized by some $b_2\\in M$. Then $a_1a_2a_3\\equiv b_1b_2b_3\\, (A)$ and, in particular, $\\mathcal D_p(b_1)<\\mathcal D_p(b_2)<\\mathcal D_p(b_3)$. This proves (6). \n\n\\smallskip \nLet $(b_i\\mid i\\in\\mathbb Q)$ be an increasing sequence of elements of $p(M)$ with $\\mathcal D_p(b_i)<\\mathcal D_p(b_j)$ for all rational numbers $i<j$. As $M$ is prime over $a_1a_3A$, for each $i\\in\\mathbb Q$ the type $\\tp(b_i/a_1a_3A)$ is isolated, by $\\phi_i(x)$ say. By Lemma \\ref{Lemma_S_minimal_exist} we know $\\phi_i(\\Mon)\\subseteq \\mathcal D_p(b_i)$. This, together with $\\mathcal D_p(b_i)<\\mathcal D_p(b_j)$ implies $\\phi_i(\\Mon)<\\phi_j(\\Mon)$. It is easy to see that for each irrational number $r$ the following set:\n$$p_r(x):=\\{\\phi_i(\\Mon)<x\\mid i\\in\\mathbb Q \\land i<r\\}\\cup \\{x<\\phi_j(\\Mon)\\mid j\\in\\mathbb Q\\land j<r\\}$$\nis consistent. Our final claim in this proof is:\n\n(7) \\ $p_r(x)\\cup p_s(x)$ is inconsistent for all irrational numbers $r\\neq s$.\n\n\\noindent\nTo prove the claim, suppose that $r<s$ are irrational. Choose $i\\in \\mathbb Q$ with $r<i<s$. Then $(x<\\phi_i(\\Mon))\\in p_r$ and $(\\phi_i(\\Mon)<x)\\in p_s$, so $p_r(x)\\cup p_s(x)$ is inconsistent.\nWe have found $2^{\\aleph_0}$ pairwise inconsistent partial types over $a_1a_3A$, so $T$ is not small. Contradiction.\n \\end{proof}\n\n\\begin{Corollary}\\label{Corollary_Ctype_trivial} \nEvery $\\mathcal C$-type over a finite domain is trivial.\n\\end{Corollary}\n\\begin{proof}\nSuppose that $A$ is finite and that $p\\in S_1(A)$ is a $\\mathcal C$-type. Since $T$ is small, there is a convex type $q$ that extends the same interval type as $p$; clearly, $p\\nwor q$. By Lemma \\ref{Lemma_every_convexshift_istrivial}, $q$ is trivial, so by Theorem \\ref{Theorem_nwor preserves triviality}(a), $p\\nwor q$ implies that $p$ is also trivial. \n\\end{proof}\n\n\\begin{Lemma}[$I(\\aleph_0,T)<2^{\\aleph_0}$]\\label{Lema_S_Pi_isfinite} \nEvery type in $S_{\\Pi}(\\emptyset)$ is both convex and simple. $S_{\\Pi}(\\emptyset)$ is a finite set.\n\\end{Lemma}\n\\begin{proof}\nBy Lemma \\ref{Lemma_Pi_basic} the set $S_{\\Pi}(\\emptyset)$ consists of all completions of the interval type $\\Pi(x)$, so $S_{\\Pi}(\\emptyset)$ is a closed subspace of $S_1(\\emptyset)$. Corollary \\ref{Corollary_Ctype_trivial} guarantees that each $p\\in S_{\\Pi}(\\emptyset)$ is trivial, and then Theorem \\ref{Theorem trivial implies simple and convex} implies that such $p$ is both convex and simple. \nBy Corollary \\ref{Corollary_convex_iff_isolated_in_IT}, convex types are isolated points of $S_{\\Pi}(\\emptyset)$, so $S_{\\Pi}(\\emptyset)$ is a discrete closed subspace of $S_1(\\emptyset)$; $S_{\\Pi}(\\emptyset)$ is finite. \n\\end{proof}\n\n\n\n\\begin{Lemma}[$I(\\aleph_0,T)<2^{\\aleph_0}$]\\label{Lema_forking_reldef} \n$\\mathcal D_{\\Pi}$ is a relatively $\\emptyset$-definable equivalence relation on $\\Pi(\\Mon)$. \n\\end{Lemma}\n\\begin{proof}\nWe need to prove that the set $\\mathcal D_{\\Pi}(a)=\\{x\\in\\Pi(\\Mon)\\mid x\\dep a\\}$ is $a$-definable for all $a\\in \\Pi(\\Mon)$.\nBy Lemma \\ref{Lema_S_Pi_isfinite} every $p\\in S_{\\Pi}(\\emptyset)$ is simple, so there exists a definable convex equivalence relation $E_p$ on $\\Mon$ such that $E_{p\\restriction p(\\Mon)}=\\mathcal D_p$. For each $q\\in S_{\\Pi}(\\emptyset)$ with $q\\nfor \\tp(a)$ choose an element $a_q\\in q(\\Mon)$ with $a_q\\dep a$. Note that the class $[a_q]_{E_q}$ contains exactly those realizations of $q$ that fork with $a$, so $\\mathcal D_{\\Pi}(a)=\\bigcup \\{[a_q]_{E_q}\\mid q\\nfor \\tp(a), q\\in S_{\\Pi}(\\emptyset)\\}$; given that $S_{\\Pi}(\\emptyset)$ is finite, as proved in Lemma \\ref{Lema_S_Pi_isfinite}, it follows that $D_{\\Pi}(a)$ is definable; since it is $\\Aut_a(\\Mon)$-invariant, it is $a$-definable. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{Theorem1_shift}]\nWe need to prove $I(\\aleph_0,T)=2^{\\aleph_0}$. By way of contradiction, assume $I(\\aleph_0,T)<2^{\\aleph_0}$. Then Lemma \\ref{Lema_forking_reldef} applies: $\\mathcal D_{\\Pi}$ is a convex relatively $\\emptyset$-definable equivalence relation on $\\Pi(\\Mon)$. \nSince $\\Pi(\\Mon)$ is a convex subset of $\\Mon$, there is a $\\emptyset$-definable convex equivalence relation $E$ on $\\Mon$ such that $E_{\\restriction \\Pi(\\Mon)}=\\mathcal D_{\\Pi}$. Let $a\\in \\Pi(\\Mon)$. Then $[a]_E=\\mathcal D_{\\Pi}(a)$ because $\\mathcal D_{\\Pi}(a)$ is a $\\Pi$-bounded subset of $\\Pi(\\Mon)$.\nBy Lemma \\ref{Lemma_forking_on_D_Pi}, we know that\n$S_x\\subset \\mathcal D_{\\Pi}(x)$ holds for all $x\\in [a]_E$, so \\ $\\models \\forall x(x\\in [a]_E \\rightarrow S_x \\subseteq [a]_E)$. \n\nSince $\\tp(a)$ is a $\\mathcal C$-type, \\ $\\models (\\forall x)(x\\in [c]_E \\rightarrow S_x \\subseteq [c]_E)$ holds for some $c\\in\\mathcal C$. \nThen $S^{\\omega}_c\\subseteq [c]_E$. By Lemma \\ref{Lemma_shift_properties}, $S_c$ is an $\\mathcal S$-shift, so the sequence $(S_c^n\\mid n\\in\\mathbb N\\}$ is strictly increasing. By compactness there exists an $b\\in[c]_E$ such that $S_c^n<b$ holds for all $n\\in\\mathbb N$, that is, $S^{\\omega}_c<b$. By Lemma \\ref{Lemma_shift_properties} the set $S^{\\omega}_c$ is a final part of $\\mathcal C$, so we conclude $\\mathcal C<b$. Then, since the class $[c]_E$ is convex, there exists $b'<b$ with $b'\\in[c]_E\\cap \\Pi(\\Mon)$. By our choice of $E$, the class $[b']_E=\\mathcal D_\\Pi(b')$ is disjoint from $\\mathcal C$. Hence $c\\notin[b']_E$.\nContradiction. The proof of Theorem \\ref{Theorem1_shift} is now complete. \n\\end{proof}\n\n\\subsection{A generalization}\n\n \nNow, let $T$ be any complete countable theory with infinite models. \nA ``local\" context of weak quasi-o-minimality was introduced in \\cite{MTwom}: A partial type $\\Pi(x)$ is weakly quasi-o-minimal over $A$ if $\\Pi(x)$ is over $A$ and every type $p\\in S_x(A)$ is weakly o-minimal. In that case, we also say that the set $\\Pi(\\Mon)$ is weakly quasi-o-minimal over $A$. \nThis definition was motivated (and justified) by the next proposition, whose corollary is that the theory $T$ is weakly quasi-o-minimal if and only if the formula $x=x$ is weakly quasi-o-minimal. \n\n\\begin{Proposition}[{\\cite[Proposition 3.9]{MTwom}}] \nLet $P$ be type-definable over $A$. Then $P$ is weakly quasi-o-minimal over $A$ if and only if there exists a relatively $A$-definable linear order $<$ on $P$ such that every relatively definable subset of $P$ is a Boolean combination of $<$-convex and relatively $A$-definable sets. In that case, the latter is true for any relatively $A$-definable linear order $<$ on $P$.\n\\end{Proposition}\n\n\nNow, it is routine to generalize Theorem \\ref{Theorem1_shift} to the local definable weakly quasi-o-minimal context.\n\n\\begin{Theorem}Suppose that for some finite set of parameters $A$, there is a definable weakly quasi-o-minimal order over $A$ possessing a monotone $A$-definable family of semiintervals which includes a shift. Then $I(T,\\aleph_0)=2^{\\aleph_0}$. \n\\end{Theorem}\n\n \n\n\n\n\\begin{thebibliography}{9}\n\\bibitem{Alibek} Alibek, A. A., B. S. Baizhanov, and T. S. Zambarnaya. \"Discrete order on a definable set and the number of models.\" Mathematical Journal 14.3 (2014): 5-13.\n\\bibitem{BH} Baldwin, John T., and Leo Harrington. \"Trivial pursuit: remarks on the main gap.\" Annals of Pure and Applied Logic 34.3 (1987): 209-230.\n\\bibitem{Belegradek} Belegradek, O., Peterzil, Y., Wagner, F. (2000). Quasi-O-Minimal Structures. The Journal of Symbolic Logic, 65(3), 1115–1132.\n\\bibitem{Goode} Goode, John B. \"Some trivial considerations.\" The Journal of symbolic logic 56.2 (1991): 624-631. \n\n\\bibitem{KOU} Kaplan Itay, Alf Onshuus, and Alexander Usvyatsov. \"Additivity of the dp-rank.\" Transactions of the American Mathematical Society 365.11 (2013): 5783-5804.\n\n\n\\bibitem{FM} Friedman, Harvey, and Chris Miller. \"Expansions of o-minimal structures by fast sequences.\" The Journal of Symbolic Logic 70.2 (2005): 410-418.\n\\bibitem{MRS} Mekler Alan, Matatyahu Rubin, and Charles Steinhorn. \"Dedekind completeness and the algebraic complexity of o-minimal structures.\" Canadian journal of mathematics 44.4 (1992): 843-855.\n\\bibitem{MT} Moconja, Slavko, and Predrag Tanovi\\'c. \"Stationarily ordered types and the number of countable models.\" Annals of Pure and Applied Logic 171.3 (2020): 102765.\n\\bibitem{MTwmon} Moconja, Slavko, and Predrag Tanovi\\'c. \"Does weak quasi-o-minimality behave better than weak o-minimality?.\" Archive for Mathematical Logic 61.1 (2022): 81-103.\n\\bibitem{MTwom} Moconja, Slavko, and Predrag Tanovi\\'c. \"Weakly o-minimal types.\" arXiv preprint arXiv:2404.08260 (2024).\n\\bibitem{MTclass2} Moconja, Slavko, and Predrag Tanovi\\'c. Countable models of weakly quasi-o-minimal theories II. (in preparation)\n\\bibitem{PeS}Peterzil, Ya'acov, and Sergei Starchenko. \"A trichotomy theorem for o-minimal structures.\" Proceedings of the London Mathematical Society 77.3 (1998): 481-523.\n\\bibitem{Ramakrishnan}Ramakrishnan, Janak. \"Uniform bounds on growth in o‐minimal structures.\" Mathematical Logic Quarterly 56.4 (2010): 406-408.\n\\bibitem{Tane} Tanovi\\'c, Predrag. \"Vaught’s Conjecture for Theories of Discretely Ordered Structures.\" Notre Dame Journal of Formal Logic 65.3 (2024): 247-257.\n\n\\end{thebibliography}\n\n\n\n\\end{document}\n\n\n\n"}}},{"rowIdx":290,"cells":{"id":{"kind":"string","value":"2412.20652v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20652v1"},"title":{"kind":"string","value":"Hyperbolic knots with arbitrarily large torsion order in knot Floer homology"},"full_text":{"kind":"string","value":"\n\n\n\\pdfoutput=1 \n\n\n\n\\documentclass{amsart}\n\\usepackage{amssymb}\n\n\\usepackage{graphicx}\n\n\n\\usepackage{caption} \n\\captionsetup[table]{skip=10pt}\n\n\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{lemma}[theorem]{Lemma}\n\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{claim}[theorem]{Claim}\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{question}[theorem]{Question}\n\n\\theoremstyle{definition}\n\\newtheorem{example}[theorem]{Example}\n\n\n\\theoremstyle{remark}\n\\newtheorem{remark}[theorem]{Remark}\n\n\\numberwithin{equation}{section}\n\n\\newcommand{\\FL}{{\\rm FL}}\n\n\n\n\n\n\\begin{document}\n\n\\title[Hyperbolic knots with large torsion order]{Hyperbolic knots with arbitrarily large torsion order\nin knot Floer homology}\n\n\n\n\\author[K. Himeno]{Keisuke Himeno}\n\\address{Graduate School of Advanced Science and Engineering, Hiroshima University,\n1-3-1 Kagamiyama, Higashi-hiroshima, 7398526, Japan}\n\\email{himeno-keisuke@hiroshima-u.ac.jp}\n\\thanks{The first author was supported by JST SPRING, Grant Number JPMJSP2132. }\n\\author[M. Teragaito]{Masakazu Teragaito}\n\\address{Department of Mathematics and Mathematics Education, Hiroshima University,\n1-1-1 Kagamiyama, Higashi-hiroshima 7398524, Japan.}\n\\email{teragai@hiroshima-u.ac.jp}\n\\thanks{The second author has been partially supported by JSPS KAKENHI Grant Number JP20K03587.}\n\n\n\n\\subjclass[2020]{Primary 57K10; Secondary 57K18}\n\n\\date{\\today}\n\n\n\\commby{}\n\n\\begin{abstract}\nIn knot Floer homology, there are two types of torsion order. \nOne is the minimal power of the action of the variable $U$ to annihilate the\n$\\mathbb{F}_2[U]$-torsion submodule of the minus version of knot Floer homology $\\mathrm{HFK}^-(K)$.\nThis is introduced by Juh\\'{a}sz, Miller and Zemke, and denoted by $\\mathrm{Ord}(K)$.\nThe other, $\\mathrm{Ord}'(K)$, introduced by Gong and Marengon, is similarly defined for the $\\mathbb{F}_2[U]$-torsion submodule of the unoriented knot Floer homology $\\mathrm{HFK}'(K)$.\n\nFor both torsion orders, it is known that arbitrarily large values are realized by torus knots.\nIn this paper, we prove that they can be realized by hyperbolic knots, most of which are twisted torus knots.\nTwo torsion orders are argued in a unified way by using the Upsilon torsion function introduced by Allen and Livingston.\nWe also give the first infinite family of hyperbolic knots which shares a common Upsilon torsion function.\n\n\\end{abstract}\n\n\\keywords{twisted torus knot, torsion order, Upsilon torsion function, knot Floer homology}\n\n\\maketitle\n\n\n\\section{Introduction}\\label{sec:intro}\n\n\nThere are two types of torsion order in knot Floer homology.\nThe first one is introduced by Juh\\'{a}sz, Miller and Zemke \\cite{JMZ}.\nRecall that the minus version of knot Floer homology $\\mathrm{HKF}^-(K)$ is a finitely generated module over\nthe polynomial ring $\\mathbb{F}_2[U]$.\nLet us denote $\\mathrm{Tor}(\\mathrm{HFK}^-(K))$ its $\\mathbb{F}_2[U]$-torsion submodule.\nThen the torsion order of a knot $K$ is defined as\n\\[\n\\mathrm{Ord}(K)=\\min \\{ k\\ge 0 \\mid U^k\\cdot \\mathrm{Tor}(\\mathrm{HFK}^-(K))=0 \\} \\in \\mathbb{N}\\cup \\{0\\}.\n\\]\nOf course, for the unknot $O$, $\\mathrm{Ord}(O)=0$.\nSince knot Floer homology detects the unknot \\cite{OS0}, $\\mathrm{Ord}(K)\\ge 1$ when $K$ is non-trivial.\nFor example, for the torus knot $T(p,q)$ with $1<p<q$, $\\mathrm{Ord}(T(p,q))=p-1$ \\cite{JMZ}.\nHence arbitrarily large values of torsion order can be realized by torus knots.\nThere are several applications for knot cobordisms. See also \\cite{HKP}.\n\n\n\n\nThe second is similarly defined in \\cite{GM} by using the torsion submodule of Ozsv\\'{a}th, Stipsicz and Szab\\'{o}'s unoriented knot Floer homology $\\mathrm{HFK}'(K)$, \nwhich is also a module over $\\mathbb{F}_2[U]$ (\\cite{OSS}),\ninstead of $\\mathrm{HFK}^-(K)$.\nHence\n\\[\n\\mathrm{Ord}'(K)=\\min \\{ k\\ge 0 \\mid U^k\\cdot \\mathrm{Tor}(\\mathrm{HFK}'(K))=0 \\} \\in \\mathbb{N}\\cup \\{0\\}.\n\\]\nAgain, $\\mathrm{Ord}'(K)=0$ if and only if $K$ is trivial.\n(For, $\\mathrm{HFK}'(O)=\\mathbb{F}_2[U]$, which is torsion-free \\cite[Corollary 2.15]{OSS}.\nConversely,\nif $\\mathrm{HFK}'(K)$ is torsion-free, then $\\mathrm{HFK}'(K)=\\mathbb{F}_2[U]=\n\\mathrm{HFK}'(O)$ \\cite[Proposition 3.5]{OSS}.\nSo, the unoriented knot Floer complexes $\\mathrm{CFK}'(K)$ and $\\mathrm{CFK}'(O)$ \nshare the same homology, which\nimplies chain homotopy equivalence between them \\cite[Proposition A.8.1]{OSS2}.\nSince setting $U=0$ reduces the complex into the hat version of knot Floer complex\n\\cite[Proposition 2.4]{OSS}, we have\n$\\widehat{\\mathrm{HFK}}(K)\\cong \\widehat{\\mathrm{HFK}}(O)$ by \\cite[Proposition A.3.5]{OSS2}.\nThis implies $K=O$.)\n\nGong and Marengon \\cite[Lemma 7.1]{GM} verify $\\mathrm{Ord}'(T(p,p+1))=\\lfloor \\frac{p}{2} \\rfloor$.\nHence arbitrarily large values of this torsion order can be realized by torus knots, again.\n\n\n\nAs shown in \\cite{AL}, two types of torsion order can be unified\nin terms of the Upsilon torsion function $\\Upsilon^{\\mathrm{Tor}}_K(t)$,\n which is a piecewise linear continuous function defined on the interval $[0,2]$.\nThe derivative of $\\Upsilon^{\\mathrm{Tor}}_K(t)$ near $0$ equals to $\\mathrm{Ord}(K)$, and\n$\\Upsilon^{\\mathrm{Tor}}_K(1)=\\mathrm{Ord}'(K)$.\nWe remark that the Upsilon torsion function and two types of torsion order are not concordance invariats.\n\n\n\nThe main purpose of this paper is to confirm that arbitrarily large values of \nthese two types of torsion order can be realized by hyperbolic knots.\nExcept a few small values, we make use of twisted torus knots.\n\n\n\\begin{theorem}\\label{thm:main}\nLet $K$ be a twisted torus knot $T(p,kp+1;2,1)$ with $k\\ge 1$.\n\\begin{itemize}\n\\item[(1)] If $p\\ge 2$, then $\\mathrm{Ord}(K)=p-1$.\n\\item[(2)] If $p\\ge 4$, then $\\mathrm{Ord}'(K)=\\lfloor\\frac{p-2}{2}\\rfloor$.\n\\end{itemize}\n\\end{theorem}\n\n\n\nUnfortunately, a twisted torus knot $T(p,kp+1;2,1)$ is not hyperbolic when $p<5$ (see Proposition \\ref{prop:hyp}).\nHowever, an additional argument gives the following.\n\n\\begin{corollary}\\label{cor:main}\nLet $N\\ge 1$ be a positive integer.\nThen there exist infinitely many hyperbolic knots $K_1$ and $K_2$ with $\\mathrm{Ord}(K_1)=N$ and $\\mathrm{Ord}'(K_2)=N$.\n\\end{corollary}\n\n\n\n\\begin{corollary}\\label{cor:main2}\nThere exist infinitely many hyperbolic knots that share the same Upsilon torsion function.\n\\end{corollary}\n\nWe pose a simple question.\n\n\\begin{question}\nLet $M$ and $N$ be positive integers.\nDoes there exist a knot $K$ with $(\\mathrm{Ord}(K), \\mathrm{Ord}'(K))=(M,N)$?\n\\end{question}\n\n\n\n\n\n\n\\begin{remark}\nFor two types of torsion order, \nthe original symbols are $\\mathrm{Ord}_v(K)$ and $\\mathrm{Ord}_U(K)$ (see \\cite{JMZ,GM}).\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Twisted torus knots}\n\nA twisted torus knot is obtained from a torus knot of type $(p,q)$ by twisting $r$ adjacent strands by $s$ full twists.\nThe resulting knot is denoted by $T(p,q;r,s)$ as in literatures \\cite{L0,L1,L2,L3}.\n\nThroughout this section, let $K$ be the twisted torus knot $T(p,kp+1;2,1)$ with $p\\ge 2, k\\ge 1$.\nClearly, if $p=2$, then $T(2,2k+1;2,1)=T(2,2k+3)$.\nAlso, Lee \\cite{L2,L3} shows that $T(3,3k+1;2,1)=T(3,3k+2)$, and\n$T(4,4k+1;2,1)$ is the $(2,8k+3)$-cable of $T(2,2k+1)$.\nWe will show later that $T(p,kp+1;2,1)$ is hyperbolic if $p\\ge 5$ (Proposition \\ref{prop:hyp}).\nSince these knots are the closure of a positive braid, it is fibered by \\cite{S}.\nIn particular, the Seifert algorithm on a positive braided diagram gives a fiber, which is a minimal genus Seifert surface.\nThus we know that it has genus $(kp^2-kp+2)/2$.\nHence $K$ is non-trivial.\n\n\n\n\n\\begin{lemma}\\label{lem:tunnel}\n$K$ is an L--space knot.\\end{lemma}\n\n\\begin{proof}\nThis follows from \\cite{V}.\\end{proof}\n\n\n\n\n\n\\begin{lemma}\\label{lem:alex}\nThe Alexander polynomial $\\Delta_K(t)$ of $K$ is given by\n\\[\n\\Delta_K(t)=\n\\begin{cases}\n\\begin{aligned}\n1&+\\sum_{i=1}^k(-t +t^{p})t^{(i-1)p}\\\\\n&+\\sum_{i=1}^{p-3} \\sum_{j=1}^{k}( -t^{ikp+1}+t^{ikp+2}-t^{ikp+2+i}+t^{(ik+1)p} ) t^{(j-1)p}\\\\\n&+\\sum_{i=1}^{k+1}( -t^{kp(p-2)+1} + t^{kp(p-2)+2} )t^{(i-1)p},\n\\end{aligned}\n & \\text{ if $p\\ge 3$},\\\\\n1-t+t^2-\\dots +t^{2k+2}, &\\text{if $p=2$.}\n\\end{cases}\n\\]\n\\end{lemma}\n\n\\begin{proof}\nWhen $p=2$, it is well known that\n\\[\n\\Delta_K(t)=\\frac{ (1-t) (1-t^{2(2k+3)}) } { (1-t^2) (1-t^{2k+3} ) }=1-t+t^2-\\dots +t^{2k+2},\n\\]\nsince $K=T(2,2k+3)$ as mentioned before.\n\nAssume $p\\ge 3$.\nThe conclusion essentially follows from \\cite{Mo}.\nIn his notation, our knot $K$ is $\\Delta(p,kp+1,2)$ with $r=p-1$.\nHence\n\\begin{align*}\n\\Delta_K(t) &=\\frac{1-t} {(1-t^p)(1-t^{kp+1}) }\\cdot \\\\\n& \\qquad ( 1- (1-t) ( t^{(p-1)(kp+1)+1} + t^{kp+1} ) -t^{p(kp+1)+2} ).\n\\end{align*}\n\nThe second factor is changed as\n\\[\n\\begin{split}\n 1 &- (1-t) ( t^{(p-1)(kp+1)+1} + t^{kp+1} ) -t^{p(kp+1)+2} =\n 1- t^{(p-1)(kp+1)+1}- t^{kp+1}\\\\\n & + t^{(p-1)(kp+1)+2} + t^{kp+2} -t^{p(kp+1)+2} = (1-t^{kp+1})+\\\\\n &+t^{kp+2}(1-t^{ (kp+1)(p-2)})+ t^{(p-1)(kp+1)+2}(1-t^{kp+1}).\n\\end{split}\n\\]\nThus\n\\[\n\\Delta_K(t)=\\frac{1-t}{1-t^p}\\cdot ( 1 +t^{kp+2} \\sum_{i=0}^{p-3} t^{i(kp+1)}+t^{(p-1)(kp+1)+2} ) .\n\\]\n\nWe set \n\\begin{align*}\nA&=\\sum_{i=1}^k(-t +t^{p})t^{(i-1)p}, \\\\\nB&=\\sum_{i=1}^{p-3} \\sum_{j=1}^{k}( -t^{ikp+1}+t^{ikp+2}-t^{ikp+2+i}+t^{(ik+1)p} ) t^{(j-1)p}, \\\\\nC&=\\sum_{i=1}^{k+1}( -t^{kp(p-2)+1} + t^{kp(p-2)+2} )t^{(i-1)p}.\n\\end{align*}\nThen it is straightforward to calculate\n\\begin{align*}\n(1-t^p)A&= -t+t^p+t^{kp+1}-t^{(k+1)p},\\\\\n(1-t^p)B&= -t^{kp+1}+t^{(k+1)p}+(1-t) \\sum_{i=0}^{p-3} t^{(i+1)kp+i+2}+t^{kp(p-2)+1}-t^{kp(p-2)+2}, \\\\\n(1-t^p)C&= -t^{kp(p-2)+1} + t^{kp(p-2)+2} + t^{kp(p-2)+1+(k+1)p} - t^{kp(p-2)+2+(k+1)p}.\n\\end{align*}\nHence\n\\[\n\\begin{split}\n(1-t^p)(1+A+B+C)&=1-t +(1-t) \\sum_{i=0}^{p-3} t^{(i+1)kp+i+2} \\\\\n&\\qquad +(1-t) t^{kp(p-2)+1+(k+1)p}\\\\\n&=(1-t)( 1+ t^{kp+2}\\sum_{i=0}^{p-3}t^{i(kp+1)}+t^{ (p-1)(kp+1)+2 }).\n\\end{split}\n\\]\nThis shows that $\\Delta_K(t)=1+A+B+C$ as desired.\n\\end{proof}\n\n\n\\begin{corollary}\\label{cor:gap}\nThe gaps of the exponents of the Alexander polynomial of $K$ are\n\\[\n(1,p-1)^k,(1,1,1,p-3)^k,(1,1,2,p-4)^k,\\dots, (1,1,p-3,1)^k,1,1,(p-1,1)^k\\]\nif $p\\ge 3$, and\n$1^{2k+2}$ if $p=2$.\nHere, the power indicates the repetition.\n(We remark that the above sequence is $(1,2)^k,1,1,(2,1)^k$ when $p=3$.)\n\\end{corollary}\n\n\n\n\n\n\nTo prove that our twisted torus knot $K=T(p,kp+1;2,1)$ is hyperbolic when $p\\ge 5$,\nwe give a more general result by using \\cite{I}.\nA knot $k$ is called a \\textit{fully positive braid knot\\/} \nif it is the closure of a positive braid which contains at least one full twist.\n\n\\begin{proposition}\\label{prop:full}\nLet $k$ be a fully positive braid knot.\nIf $k$ is a tunnel number one, satellite knot, then $k$ is a cable knot.\n\\end{proposition}\n\n\\begin{proof}\nBy \\cite{MS}, $k$ has a torus knot $T(r,s)$ as a companion. We may assume that $1<r<s$.\nThen Theorem 1.2 of \\cite{I} claims that\nthe pattern $P$ is represented by a positive braid in a solid torus.\n\n\nLet us recall the construction of \\cite{MS}.\nStarting from a $2$-bridge link $K_1\\cup K_2$,\nconsider the solid torus $E(K_2)$ containing $K_1$.\nRemark that $K_1$ and $K_2$ are unknotted.\nFor the companion knot $T(r,s)$,\nconsider a homeomorphism from $E(K_2)$ to the tubular neighborhood of $T(r,s)$,\nwhich sends the preferred longitude of $E(K_2)$ to the regular fiber of the Seifert fibration in the exterior of $T(r,s)$.\nHence our pattern knot $P$, which is defined under preserving preferred longitudes, \n is obtained from $K_1$ by adding $rs$ positive full twists to $E(K_2)$.\nSince $K_1$ is unknotted, we can set the pattern $P$ as the closure of a positive braid\n\\[\n\\sigma_{i(1)} \\sigma_{i(2)} \\dots \\sigma_{i(n-1)} (\\sigma_1 \\sigma_2 \\dots \\sigma_{n-1})^{nrs}\n\\]\nfor some $n\\ge 2$, where $\\{i(1), i(2), \\dots, i(n-1)\\}=\\{1,2,\\dots, n-1\\}$.\n(If the initial part before $rs$ full twists contains more than $n-1$ generators,\nthen the Seifert algorithm gives a fiber surface of the closure $K_1$, which has positive genus.)\n\n\n\nFor two braids $\\beta_1$ and $\\beta_2$, we write $\\beta_1\\sim \\beta_2$ if they are conjugate or equivalent.\n\n\\begin{claim}\\label{cl:braid}\n$\\sigma_{i(1)} \\sigma_{i(2)} \\dots \\sigma_{i(n-1)} (\\sigma_1 \\sigma_2 \\dots \\sigma_{n-1})^{nrs}\n\\sim (\\sigma_1\\sigma_2 \\dots \\sigma_{n-1})^{nrs+1}$.\n\\end{claim}\n\n\\begin{proof}[Proof of Claim \\ref{cl:braid}]\nPut $F=(\\sigma_1 \\sigma_2 \\dots \\sigma_{n-1})^{nrs}$, which is central in the braid group.\nFirst, write $\\sigma_{i(1)} \\sigma_{i(2)} \\dots \\sigma_{i(n-1)} F=U_1 \\sigma_1 U_2 F$, where\n$U_i$ is a word without $\\sigma_1$, which is possibly empty.\nThen $U_1\\sigma_1 U_2 F \\sim \\sigma_1 U_2F U_1\\sim \\sigma_1 U_2U_1F$.\nNext, set $U_2U_1=V_1\\sigma_2 V_2$, where $V_i$ is a (possibly, empty) word without $\\sigma_1, \\sigma_2$.\nNote that $\\sigma_1$ and $V_1$ commute.\nThen \n\\[\n\\sigma_1 U_2U_1F =\\sigma_1 V_1\\sigma_2 V_2 F \\sim V_1\\sigma_1 \\sigma_2 V_2F\\sim \\sigma_1\\sigma_2V_2FV_1 \\sim \\sigma_1\\sigma_2 V_2V_1F.\n\\]\nRepeating this procedure, we have the conclusion.\n\\end{proof}\n\nThus the pattern $P$ is the closure of a braid $(\\sigma_1\\sigma_2\\dots \\sigma_{n-1})^{nrs+1}$.\nThis means that $k$ is a cable knot.\n\\end{proof}\n\n\n\\begin{remark}\nLee \\cite[Question 1.2]{L3} asks whether $T(p,q;r, s)$ is a cable knot, if it is a satellite knot under a condition that\n$1<p<q$, $r\\ne q$, $r$ is not a multiple of $p$, $1<r\\le p+q$ and $s>0$.\nProposition \\ref{prop:full}\ngives a positive answer if the knot has tunnel number one, which is known to be true when $r\\in \\{2,3\\}$ (see \\cite{JL}).\n\\end{remark}\n\n\n\n\\begin{proposition}\\label{prop:hyp}\nIf $p\\ge 5$, then $K$ is hyperbolic.\n\\end{proposition}\n\n\\begin{proof}\nFirst, $K=T(p,kp+1;2,1)$ is a torus knot if and only if\n$p=2,3$ by \\cite[Theorem 1.1]{L2}.\nHence we know that our knot is not a torus knot.\n\nAssume that $K$ is a satellite knot for a contradiction.\nWe remark that $K$ has tunnel number one.\n(A short arc at the extra full twist gives an unknotting tunnel.)\nProposition \\ref{prop:full} shows\nthat $K$ is the $(n,nrs+1)$-cable of $T(r,s)$.\nThen $K=T(4,4m+1; 2,1)$ for some $m\\ge 1$ by \\cite{L3}.\nThis is a contradiction, because of Lemma \\ref{lem:alex} and $p\\ge 5$.\nThus we have shown that $K$ is neither a torus knot nor a satellite knot, so\n$K$ is hyperbolic.\n\\end{proof}\n\n\n\n\\section{Upsilon torsion function}\\label{sec:upsilon-torsion}\n\nIn this section, we determine the Upsilon torsion function $\\Upsilon^{\\mathrm{Tor}}(K)$ of $K=T(p,kp+1;2,1)$.\nSince $K$ is an L--space knot (Lemma \\ref{lem:tunnel}), the full knot Floer complex $\\mathrm{CFK}^\\infty(K)$ is determined\nby the Alexander polynomial (\\cite{OS}). \nIt has the form of staircase diagram described by the gaps of Alexander polynomial.\nIf the gaps are given as a sequence $a_1,a_2,\\dots, a_n$, then\nthe terms give the length of horizontal and vertical steps.\nMore precisely, let $g$ be the genus of $K$.\nStart at the vertex $(0,g)$ on the coordinate plane.\nGo right $a_1$ steps, and down $a_2$ steps, and so on. Finally, we reach $(g,0)$.\nBy the symmetry of the Alexander polynomial, the staircase inherits the symmetry along the line $y=x$.\n\n\n\nWe follow the process in \\cite[Appendix]{AL}.\nHowever, we assign a modified filtration level $\\FL$ to each generator of the complex.\nIf a generator $x$ has the coordinate $(p,q)$, then $\\FL(x)=tq+(2-t)p$.\nIn fact, for any $t\\in [0,2]$,\n$\\FL$ defines a real-valued function on $\\mathrm{CFK}^\\infty(K)$.\nThen, for all $s\\in \\mathbb{R}$, $\\mathcal{F}_s$ is spanned by all vectors\n$x \\in \\mathrm{CFK}^\\infty(K)$ such that $\\FL(x)\\le s$.\nThe collection $\\{\\mathcal{F}_s\\}$ gives a filtration on $\\mathrm{CFK}^\\infty(K)$. See \\cite{L}.\n(Remark that this filtration level is just the twice of that used in \\cite{AL}.)\nSince $\\mathcal{F}_s\\subset \\mathcal{F}_u$ if $s\\le u$, \na generator $x_i\\in \\mathcal{F}_u$ can be added by $x_j\\in \\mathcal{F}_s$, without any change of the filtration level.\nThat, $\\FL(x_i)=\\FL(x_i+x_j)$.\n\n\n\nFor the staircase complex,\nrepeating a change of basis gradually splits the complex into a single isolated generator\nand separated arrows.\nThen the value of the Upsilon torsion function is given as the maximum difference between filtration levels among the arrows.\n\nSince the Upsilon torsion function, defined on $[0,2]$, is symmetric along $t=1$, it suffices to consider the domain $[0,1]$.\n\nAs the simplest case, we demonstrate the process when $p=2$.\n\n\\begin{example}\\label{ex:p=2}\nLet $p=2$.\nThen $K=T(2,2k+3)$ as mentioned before, and we show that\nits Upsilon torsion function $\\Upsilon^{\\mathrm{Tor}}_K(t)=t \\ (0\\le t\\le 1)$, independent of $k$.\n\nBy Corollary \\ref{cor:gap}, the gaps of the exponents of the Alexander polynomial is $1,1,\\dots, 1$ (repeated $2k+2$ times).\nHence the staircase diagram has the form as shown in Figure \\ref{fig:p=2}, where\n $A_i$ has Maslov grading $0$, but $B_i$ has grading $1$, and each arrow has length one.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.7]{p=2.pdf} \n\\caption{Left: The staircase diagram when $p=2$. The generator $A_i$ has grading $0$, but $B_i$ has grading $1$.\nEach arrow has length one.\nRight: By adding $A_0$ to $A_1,\\dots, A_{k+1}$, the generator $A_0$ is isolated from the complex.}\\label{fig:p=2}\n\\end{figure}\n\nEach generator is assigned the filtration level $\\FL$.\nThe difference between filtration levels among the generators is important.\nWe have $\\FL(B_{i+1})-\\FL(A_i)=2-t$ and $\\FL(B_i)-\\FL(A_i)=t$, because each arrow has length one.\nThus we have\n\\[\n\\FL(A_0)\\le \\FL(A_1)\\le \\dots \\le \\FL(A_{k+1}),\n\\]\nwhere each equality occurs only when $t=1$.\nHence $A_0$ has the lowest filtration level among the generators with grading $0$.\nAdd $A_0$ to $A_1,\\dots, A_{k+1}$. Then the generator $A_0$ is isolated from the complex as shown in Figure \\ref{fig:p=2}.\n(Recall that we use $\\mathbb{F}_2$ coefficients.)\nIn the remaining part of the complex, $A_1+A_0$ is the lowest, since $\\FL(A_i+A_0)=\\FL(A_i)$ for $i=1,2,\\dots,k+1$. \nTo simplify the notation, we keep the same symbol $A_i$, instead of $A_i+A_0$, after this, if no confusion can arise.\n\nAdd $A_1$ to the other generators with grading $0$, except $A_0$.\nThen the arrow $B_1\\rightarrow A_1$ is split off from the complex.\nRepeating this process leads to the decomposition of the original staircase into one isolated generator $A_0$ and \n$k+1$ vertical arrows.\nFor each arrow, the difference of filtration levels is equal to $t$, so the maximum difference is $t$ among the arrows.\nThis shows $\\Upsilon^{\\mathrm{Tor}}_K(t)=t$.\n\\end{example}\n\n\n\n\\begin{theorem}\\label{thm:upsilon-torsion}\nLet $p\\ge 4$.\nThe Upsilon torsion function $\\Upsilon^{\\mathrm{Tor}}_K(t)$ is given as\n\\[\n\\Upsilon_K^{{\\rm Tor}}(t)=\n\\begin{cases}\n(p-1)t & (0\\le t \\le \\frac{2}{p})\\\\\n2-t & (\\frac{2}{p}\\le t \\le \\frac{2}{p-2})\\\\\n(p-3)t & (\\frac{2}{p-2}\\le t \\le \\frac{4}{p})\\\\\n2m+(-m-1)t & (\\frac{2m}{p}\\le t \\le \\frac{2m}{p-1},\\ m=2,\\dots, \\lfloor\\frac{p-1}{2}\\rfloor)\\\\\n(p-2-m)t & (\\frac{2m}{p-1}\\le t\\le \\frac{2(m+1)}{p},\\ m=2,\\dots,\\lfloor\\frac{p}{2}\\rfloor-1).\n\\end{cases}\n\\] \nIn particular, $\\Upsilon^{\\mathrm{Tor}}_K(1)=\\lfloor \\frac{p-2}{2}\\rfloor$.\n\\end{theorem}\n\n\n\n\\begin{proof}\nRecall that the gaps are\n\\[\n(1,p-1)^k,(1,1,1,p-3)^k,(1,1,2,p-4)^k,\\dots, (1,1,p-3,1)^k,1,1,(p-1,1)^k\n\\]\nby Corollary \\ref{cor:gap}.\nWe name the generators of the staircase as in Figures \\ref{Fig1}, \\ref{Fig2} and \\ref{Fig3}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.3]{Fig1.pdf} \n\\caption{The first part corresponds to $(1,p-1)^k$. The generators $A_i$ have Maslov grading $0$, but\n$B_i$ have $1$.\n The number $p-1$ next to each vertical arrow indicates the length.\nEach horizontal arrow has length one.\nHere, $C^1_0=A_k$. }\\label{Fig1}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.3]{Fig2.pdf} \n\\caption{Left: The second part corresponds to $(1,1,j,p-2-j)^k\\ (j=1,\\ldots,p-3)$.\nThe generators $C_*^j$ and $E_*^j$ have Maslov grading $0$, but the others have $1$.\nRight: This is a connecting part between $(1,1,j,p-2-j)$ and $(1,1,j+1, p-2-(j+1))$.}\\label{Fig2}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.3]{Fig3.pdf} \n\\caption{The last part corresponds to $1,1,(p-1,1)^k$.\nThe generators $G$ and $A_*'$ have Maslov grading $0$, but the others have $1$.\n}\\label{Fig3}\n\\end{figure}\n\nIn particular, we have \n the difference between filtration levels of certain generators with Maslov grading 0\n as in Table \\ref{table:filtration}.\n The argument is divided into 4 cases.\n\n\n\n\n\\begin{table}[htbp]\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{l |l}\n\\hline\nDifference & Indices\\\\\n\\hline\n$\\FL(A_i)-\\FL(A_{i-1})=2-pt$ & $i=0,\\ldots,k$\\\\\n$\\FL(E^j_i)-\\FL(C^j_{i-1})=2-2t\\ge 0$ & $i=1,\\ldots,k;\\ j=1,\\ldots,p-3$\\\\\n$\\FL(C^j_i)-\\FL(E^j_i)=(2-p)t+2j $& $i=1,\\ldots,k;\\ j=1,\\ldots,p-3$\\\\\n$\\FL(C^j_i)-\\FL(C^j_{i-1})=-pt+2(j+1)$ & $i=1,\\ldots,k;\\ j=1,\\ldots,p-3$\\\\\n$\\FL(C^j_0)-\\FL(C^{j-1}_0)=(-pt+2j)k$ & $j=2,\\ldots,p-3$ \\\\\n$\\FL(A'_0)-\\FL(G)=2-2t\\ge 0$ & \\\\\n$\\FL(A'_i)-\\FL(A'_{i-1})=-pt+2p-2>0$ & $i=1,\\ldots,k$\\\\\n\\hline\n\\end{tabular}\n\\caption{Difference between filtration levels of the generators\nwith Maslov grading $0$.}\n\\label{table:filtration}\n\\end{table}\n\n\\medskip\n\\textbf{Case 1. $0\\le t \\le \\frac{2}{p}$.}\nThen any difference in Table \\ref{table:filtration} is at least $0$.\nHence $A_0$ has the lowest filtration level among the generators with grading $0$, whose filtration levels\nincrease when we go to the right.\n\nExactly as in Example \\ref{ex:p=2}, \nthe staircase complex is decomposed into a single isolated generator $A_0$\nand separated vertical arrows $B_i\\rightarrow A_i\\ (i=1,2,\\dots,k)$.\nHence the maximum difference of filtration levels on the arrows is $(p-1)t$.\nThis gives $\\Upsilon^{\\mathrm{Tor}}_K(t)=(p-1)t$ for $0\\le t \\le 2/p$.\n\n\\medskip\n\\textbf{Case 2. $\\frac{2}{p}\\le t \\le \\frac{4}{p}$.}\nThen $\\FL(A_0)\\ge \\FL(A_1)\\ge \\dots \\ge \\FL(A_k)$.\nAfter $A_k$, the filtration levels increase among the generators with grading $0$, so $A_k$ is the lowest.\nAdd $A_k$ to the other generators with grading $0$.\nThen $A_k$ will be isolated, and the complex splits into two parts.\nWe say that the first part, which starts at $A_0$ and ends at $B_k$, is \\textit{N-shaped}, but the second, which starts at $D_1^1$ and ends at $A_k'$, is \\textit{mirror N-shaped}.\nIn general,\nif a ``zigzag'' complex starts and ends at horizontal arrows, then it is N-shaped.\nIf it starts and ends at vertical arrows, then it is mirror N-shaped.\n\nFor the first part, add $A_{k-1}$ to the others with grading $0$, which splits the arrow $A_{k-1} \\leftarrow B_k$ off.\nRepeat this as in Case 1. Then the N-shaped complex is decomposed into separated horizontal arrows $A_{i-1}\\leftarrow B_i\\ (i=1,2,\\dots,k)$, each of which has difference $2-t$.\nThe mirror N-shaped complex is also decomposed into vertical arrows similarly.\nThus the maximum difference among them is $(p-3)t$.\n\nCompare $2-t$ and $(p-3)t$.\nIf $\\frac{2}{p}\\le t \\le \\frac{2}{p-2}$, then $2-t \\ge (p-3)t$.\nIf $\\frac{2}{p-2}\\le t \\le \\frac{4}{p}$, then $2-t\\le (p-3)t$.\nHence $\\Upsilon^{\\mathrm{Tor}}_K(t)=2-t$ for $\\frac{2}{p}\\le t\\le \\frac{2}{p-2}$, and $(p-3)t$ for $\\frac{2}{p-2}\\le t \\le\\frac{4}{p}$.\n\n\n\\medskip\n\\textbf{Case 3. $\\frac{2m}{p}\\le t \\le \\frac{2m}{p-1}\\ (m=2,\\dots,\\lfloor (p-1)/2\\rfloor)$.}\n\nFrom Table \\ref{table:filtration}, we see that $C_k^{m-1}=C_0^m$ is the lowest among the generators with grading $0$.\nAdding this to the others with grading $0$ decomposes the complex into one isolated generator $C_0^m$,\nthe N-shaped one between $A_0$ and $F_k^{m-1}$ and the mirror N-shaped one between $D_1^m$ and $A_k'$.\n\nAs before, the mirror N-shaped complex can be decomposed into vertical arrows.\nThe longest arrows has length $(p-2-m)t$.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.4]{3zigzag.pdf} \n\\caption{The N-shaped complex between $A_0$ and $F_k^{m-1}$ after isolating the lowest vertex $C_0^m$, where $k=2$.\nThe height indicates the filtration level of each generator.\nAs before, we keep the same notation for generators after a change of basis.\n}\\label{fig:case3}\n\\end{figure}\n\nThe N-shaped complex is described in Figure \\ref{fig:case3}.\nWe have\n\\[\n\\begin{split}\n\\FL(A_0)\\ge \\FL(A_1)\\ge \\cdots &\\ge \\FL(A_k=C_0^1)\\ge \\FL(C_1^1)\\ge \\dots \\ge \\FL(C_k^1=C_0^2)\\\\\n&\\ge \\FL(C_1^2)\\ge \\dots \\ge \\FL(C_{k-1}^{m-1})\n\\end{split}\n\\]\nand $\\FL(E_i^j)\\ge \\FL(C_{i-1}^j)\\ (i=1,2,\\dots,k; j=1, 2\\dots, m-1 )$.\n\nHence $C_{k-1}^{m-1}$ is the lowest.\nAdding this to the others with grading $0$ on the left splits an N-shaped complex \n$C_{k-1}^{m-1} \\leftarrow D_k^{m-1} \\rightarrow E_k^{m-1} \\leftarrow F_k^{m-1}$\noff.\nFor the remaining part, the lowest is $C_{k-2}^{m-1}$.\nAgain, adding this to the others with grading $0$ on the left splits an N-shaped complex\n$C_{k-2}^{m-1} \\leftarrow D_{k-1}^{m-1} \\rightarrow E_{k-1}^{m-1} \\leftarrow F_{k-1}^{m-1}$\noff.\nRepeat this, then we obtain an N-shaped complex between $A_0$ and $B_k$, and\nN-shaped complexes \n$C_{i-1}^j \\leftarrow D_{i}^{j} \\rightarrow E_{i}^{j} \\leftarrow F_{i}^{j} \\ (i=1,\\dots, k; j=1,\\dots,m-1)$.\n\nFor the former, the process as in Case 2 yields separated horizontal arrows, each of which has difference $2-t$.\nLet us consider the latter N-shaped ones.\nSince $\\FL(F_i^j)-\\FL(D_i^j)=-t+(2-t)j\\ge 0$, add $D_i^j$ to $F_i^j$.\nAfter that, add $C_{i-1}^j$ to $E_i^j$.\nAs shown in Figure \\ref{fig:case3N}, \nthis change of basis decomposes the complex into a pair of arrows.\nOne has difference $2-t$, and the other has difference $\\FL(F_i^j)-\\FL(C_{i^-1}^j)=2(j+1)+(-j-2)t$.\nNote $2-t\\le 2(j+1)+(-j-2)t$.\nFurthermore, for $1\\le j\\le m-1$, $j=m-1$ attains the maximum value, $2m+(-m-1)t$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.8]{3Nnew.pdf} \n\\caption{A change of basis for an N-shaped complex.\nAdd $D_i^j$ to $F_i^j$, and $C_{i-1}^j$ to $E_i^j$.}\\label{fig:case3N}\n\\end{figure}\n\nHence we need to compare the values $(p-2-m)t$ and $2m+(-m-1)t$.\nSince $2m+(-m-1)t \\ge (p-2-m)t$,\nwe have $\\Upsilon^{\\mathrm{Tor}}_K(t)=2m+(-m-1)t$ for this case.\n\n\\medskip\n\\textbf{Case 4. $\\frac{2m}{p-1}\\le t \\le \\frac{2(m+1)}{p}\\ (m=2,\\dots,\\lfloor p/2\\rfloor-1)$.}\n\nAs in Case 3, $C_k^{m-1}=C_0^m$ is the lowest.\nSo, adding this to the others with grading $0$ decomposes the complex into one isolated generator $C_0^m$,\nthe N-shaped one and the mirror N-shaped one, again.\n\nFor the N-shaped complex, the situation is the same as in Case 3.\nThus we have an arrow with maximum difference $2m+(-m-1)t$ from this N-shaped complex.\n\nHowever, we need to handle the mirror N-shaped complex differently now.\n\nFirst, consider the case where $\\frac{2m}{p-1}\\le t\\le \\frac{2m}{p-2}$.\nThen the filtration levels of the generators with grading $0$ increase as going to the right.\nSo, as in Case 3, this part can be decomposed into vertical arrows, and the longest has length $(p-2-m)t$.\n\nSecond, consider the case where $\\frac{2m}{p-2}\\le t\\le \\frac{2(m+1)}{p}$.\nThen $\\FL(E_i^m)\\ge \\FL(C_i^m)\\ge \\FL(C_{i-1}^m)\\ (i=1,2,\\dots,k)$, but\nthe filtration levels of\nthe remaining generators with grading $0$, $C_0^{m+1},E_1^{m+1},C_1^{m+1},\\dots,G,A_0',\\dots A_k'$,\nincrease as going to the right.\nSee Figure \\ref{fig:case4}.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.35]{4zigzag.pdf} \n\\caption{The mirror N-shaped complex when $\\frac{2m}{p-2}\\le t\\le \\frac{2(m+1)}{p}$, where $k=2$.\nThe generator $C_1^m$ is the lowest.}\\label{fig:case4}\n\\end{figure}\n\nHere, $C_1^m$ is the lowest.\nAdding this to the others with grading $0$ on the right splits a mirror N-shaped complex \n$D_1^m\\rightarrow E_1^m \\leftarrow F_1^m \\rightarrow C_1^m$ off.\nThen $C_2^m$ is the lowest in the remaining part.\nRepeating this yields mirror N-shaped complexes $D_i^m\\rightarrow E_i^m \\leftarrow F_i^m \\rightarrow C_i^m\\ (i=1,2,\\dots,k)$,\nand one more mirror N-shaped one between $D_1^{m+1}$ and $A_k'$.\nFor the last one, the previous process gives vertical arrows.\nFor each mirror N-complex $D_i^m\\rightarrow E_i^m \\leftarrow F_i^m \\rightarrow C_i^m$,\nwe remark $\\FL(F_i^m)-\\FL(D_i^m)=(-m-1)t+2m\\ge 0$.\nHence adding $D_i^m$ to $F_i^m$ yields a pair of vertical arrows as shown in Figure \\ref{fig:case4N}.\nThus we have only vertical arrows, whose longest length is $(p-2-m)t$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.8]{4Nnew.pdf} \n\\caption{A change of basis for a mirror N-shaped complex.\nAdding $D_i^m$ to $F_i^m$ yields a pair of vertical arrows.}\\label{fig:case4N}\n\\end{figure}\n\n\n\nFinally, compare $2m+(-m-1)t$ and $(p-2-m)t$.\nSince $\\frac{2m}{p-2}\\le t\\le \\frac{2(m+1)}{p}$, the latter is bigger.\nThen $\\Upsilon^{\\mathrm{Tor}}_K(t)=(p-2-m)t$ for this case.\n\\end{proof}\n\n\\begin{example}\nWhen $p=6$,\n\\[\n\\Upsilon^{\\mathrm{Tor}}_K(t)=\n\\begin{cases}\n5t & (0\\le t\\le \\frac{1}{3}) \\\\\n2-t & (\\frac{1}{3}\\le t\\le \\frac{1}{2}) \\\\\n3t & (\\frac{1}{2}\\le t\\le \\frac{2}{3}) \\\\\n4-3t & (\\frac{2}{3}\\le t\\le \\frac{4}{5}) \\\\\n2t & (\\frac{4}{5}\\le t\\le 1).\n\\end{cases}\n\\]\nSee Figure \\ref{fig:p=6}.\n\\end{example}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics*[scale=0.7]{p=6.pdf} \n\\caption{The Upsilon torsion function $\\Upsilon^{\\mathrm{Tor}}_K(t)$ \nof $K=T(6,6k+1;2;1)$.\nThen $\\Upsilon^{\\mathrm{Tor}}_K(1)=2$.}\\label{fig:p=6}\n\\end{figure}\n\n\n\n\\section{Torsion order}\n\nWe are ready to prove Theorem \\ref{thm:main}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\nBy \\cite{AL}, $\\Upsilon'^{\\mathrm{Tor}}_K(0)=\\mathrm{Ord}(K)$ and\n$\\Upsilon^{\\mathrm{Tor}}_K(1)=\\mathrm{Ord}'(K)$.\nThus Theorem \\ref{thm:upsilon-torsion} immediately gives\n$\\mathrm{Ord}(K)=p-1$ and $\\mathrm{Ord}'(K)=\\lfloor (p-2)/2 \\rfloor$ when $p\\ge 4$.\n\nWhen $p\\in \\{2,3\\}$, $K$ is a torus knot, and\n$\\mathrm{Ord}(K)$ is equal to the longest gap in the exponents of the Alexander\npolynomial by \\cite[Lemma 5.1]{JMZ}.\nHence it is $p-1$ by Corollary \\ref{cor:gap}.\n(Indeed, the latter argument proves $\\mathrm{Ord}(K)=p-1$ for any $p\\ge 2$.)\n\\end{proof}\n\n\n\\begin{proof}[Proof of Corollary \\ref{cor:main}]\nBy Proposition \\ref{prop:hyp}, the twisted torus knot $K=T(p,kp+1;2,1)$ is hyperbolic if $p\\ge 5$.\nSince $K$ has genus $(kp^2-kp+2)/2$,\ndistinct choices of $k$, with a fixed $p$, give distinct knots.\n\nSet $K_2=K$ with $p=2N+3\\ge 5$. Then $K_2$ is hyperbolic\nand $\\mathrm{Ord}'(K_2)=\\lfloor (p-2)/2\\rfloor=N$.\n\nIf $N\\ge 4$, then set $K_1=K$ with $p=N+1$.\nThen $K_1$ is hyperbolic and $\\mathrm{Ord}(K_1)=p-1=N$.\n\nTo complete the proof,\nwe need infinitely many hyperbolic knots $K_1$ whose\n$\\mathrm{Ord}(K_1)$ takes each of the values $1,2, 3$.\n\n\n\\begin{enumerate}\n\\item\nBy \\cite[Corollary 1.8]{JMZ}, $\\mathrm{Ord}(L)\\le b(L)-1$ for any knot $L$.\nHence if $K_1$ is a hyperbolic $2$-bridge knot, then $\\mathrm{Ord}(K_1)=1$.\n\\item\nLet $K_1=T(3,4;2,s)$ with $s\\ge 2$.\nThen $K_1$ is an L--space knot (\\cite{V}), and twist positive in the sense of \\cite{KM}.\nIn the proof of \\cite[Theorem 1.3]{KM}, they show\nthat $\\mathrm{Ord}(K_1)=2$.\nBy \\cite{L1,L2}, $K_1$ is hyperbolic.\nSince $K_1$ has genus $s+3$, distinct choices of $s$ give distinct knots.\n\\item\nFinally, there are infinitely many hyperbolic L--space knots $\\{k_n\\}$, defined in \\cite[Section 2]{BK}, \nwith $\\mathrm{Ord}(k_n)=3$. (See \\cite[Proposition 5.1]{KM}.)\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\n\n\\begin{proof}[Proof of Corollary \\ref{cor:main2}]\nLet $K=T(p,kp+1;2,1)$ with $p\\ge 5$.\nThen $K$ is hyperbolic by Proposition \\ref{prop:hyp}.\nAfter fixing $p$,\nTheorem \\ref{thm:upsilon-torsion} shows that\nthe Upsilon torsion function does not depend on $k$.\n\\end{proof}\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n\n\\begin{thebibliography}{32}\n\n\\bibitem{AL}\nS. Allen and C. Livingston,\n\\textit{An Upsilon torsion function for knot Floer homology},\npreprint. \\texttt{arXiv:2208.04768}.\n\n\\bibitem{BK}\nK. Baker and M. Kegel,\n\\textit{Census L--space knots are braid positive, except for one that is not},\nAlgebr. Geom. Topol. \\textbf{24} (2024), no. 1, 569--586.\n\n\n\n\n\n\n\n\n\\bibitem{GM}\nS. Gong and M. Marengon,\n\\textit{Nonorientable link cobordisms and torsion order in Floer homologies},\nAlgebr. Geom. Topol. \\textbf{23} (2023), no. 6, 2627--2672.\n\n\\bibitem{HKP}\nJ. Hom, S. Kang and J. Park,\n\\textit{Ribbon knots, cabling, and handle decompositions},\nMath. Res. Lett. \\textbf{28} (2021), no. 5, 1441--1457.\n\n\\bibitem{I}\nT. Ito,\n\\textit{Satellite fully positive braid links are braided satellite of\nfully positive braid links}, preprint.\n\\texttt{arXiv:2402.01129}.\n\n\n\\bibitem{JMZ}\nA. Juh\\'{a}sz, M. Miller and I. Zemke,\n\\textit{Knot cobordisms, bridge index, and torsion in Floer homology},\n J. Topology \\textbf{13} (2020), no.4, 1701--1724.\n\n\n\n\\bibitem{KM}\nS. Krinshna and H. Morton,\n\\textit{Twist positivity, L--space knots, and concordance},\npreprint. \\texttt{arXiv:2211.17109}.\n\n\\bibitem{JL}\nJ.~H. Lee, \n\\textit{Twisted torus knots $T(p,q;3,s)$ are tunnel number one},\nJ. Knot Theory Ramifications {\\bf 20} (2011), no.~6, 807--811.\n\n\n\n\\bibitem{L0}\nS. Lee,\n\\textit{Knot types of twisted torus knots},\nJ. Knot Theory Ramifications \\textbf{26} (2017), no. 12, 1750074 (7 pages). \n\n\\bibitem{L1}\nS. Lee,\n\\textit{Satellite knots obtained by twisting torus knots: hyperbolicity of twisted torus knots},\nInt. Math. Res. Not. IMRN (2018), no. 3, 785--815. \n\n\\bibitem{L2}\nS. Lee,\n\\textit{Positively twisted torus knots which are torus knots},\nJ. Knot Theory Ramifications \\textbf{28} (2019), no. 3, 1950023, 13 pp. \n\n\\bibitem{L3}\nS. Lee,\n\\textit{Cable knots obtained by positively twisting torus knots},\nJ. Knot Theory Ramifications \\textbf{32} (2023), no. 3, Paper No. 2350018, 15 pp.\n\n\n\\bibitem{L}\nC. Livingston, \n\\textit{Notes on the knot concordance invariant upsilon},\nAlgebr. Geom. Topol. \\textbf{17} (2017), no. 1, 111--130.\n\n\\bibitem{MS}\nK. Morimoto and M. Sakuma, \n\\textit{On unknotting tunnels for knots},\nMath. Ann. \\textbf{289} (1991), no. 1, 143--167.\n\n\n\\bibitem{Mo}\nH. Morton, \n\\textit{The Alexander polynomial of a torus knot with twists},\nJ. Knot Theory Ramifications \\textbf{15} (2006), no.8, 1037--1047.\n\n\n\n\\bibitem{OSS}\nP. Ozsv\\'{a}th, A. Stipsicz and Z. Szab\\'{o},\n\\textit{Unoriented knot Floer homology and the unoriented four-ball genus},\n Int. Math. Res. Not. IMRN (2017), no.17, 5137--5181.\n\n\\bibitem{OSS2}\nP. Ozsv\\'{a}th, A. Stipsicz and Z. Szab\\'{o},\n\\textit{Grid homology for knots and links},\nMathematical Surveys and Monographs, vol. 208, \nAmerican Mathematical Society, Providence, RI, 2015.\n\n\n\\bibitem{OS0}\nP. Ozsv\\'{a}th and Z. Szab\\'{o},\n\\textit{Holomorphic disks and genus bounds},\nGeom. Topol. \\textbf{8} (2004), 311--334.\n\n\n\n\\bibitem{OS}\nP. Ozsv\\'{a}th and Z. Szab\\'{o},\n\\textit{On knot Floer homology and lens space surgeries},\nTopology \\textbf{44} (2005), 1281--1300.\n\n\n\n\\bibitem{S}\nJ. R. Stallings,\n\\textit{ Constructions of fibred knots and links},\nProc. Sympos. Pure Math., XXXII\nAmerican Mathematical Society, Providence, RI, 1978, pp. 55--60.\n\n\n\\bibitem{V}\nF. Vafaee,\n\\textit{On the knot Floer homology of twisted torus knots},\nInt. Math. Res. Not. IMRN (2015), no. 15, 6516--6537.\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n"}}},{"rowIdx":291,"cells":{"id":{"kind":"string","value":"2412.20642v2"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20642v2"},"title":{"kind":"string","value":"Direct and inverse spectral problems for the Schrodinger operator with double generalized Regge boundary conditions"},"full_text":{"kind":"string","value":"\n\\documentclass[reqno,12pt,centertags]{article}\n\\usepackage{amsmath,amsthm,amscd,amssymb,latexsym,upref,stmaryrd}\n\\usepackage[cp1251]{inputenc}\n\\usepackage[english]{babel}\n\\usepackage{mathtext}\n\\usepackage{amsfonts}\n\\usepackage{graphicx}\n\\usepackage[numbers,sort&compress]{natbib}\n\\usepackage{amsmath}\n\\newcommand{\\highlight}[1]{\\textbf{#1}}\n\\numberwithin{equation}{section}\\numberwithin{figure}{section}\n\\usepackage{hyperref}\n\\newcommand*{\\mailto}[1]{\\href{mailto:#1}{\\nolinkurl{#1}}}\n\\textwidth=17 cm \\textheight=22cm \\oddsidemargin=-5mm\n\\evensidemargin=-5mm\n\\mathsurround=2pt \\topmargin=0cm\n\n\n\\usepackage{pgfplots}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\newtheorem{corollary}{Corollary}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{remark}{Remark}\n\\theoremstyle{proposition}\n\\newtheorem{proposition}{Proposition}\n\n\n\\begin{document}\n\\thispagestyle{empty}\n\n{\\center \\large\\bf Direct and inverse spectral problems for the Schr\\\"{o}dinger operator with double generalized Regge boundary conditions}\n\\\\[0.5cm]\n\\noindent {\\bf Xiao-Chuan Xu}\\footnote{School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu,\nPeople's Republic of China, {\\it Email:\nxcxu@nuist.edu.cn}}\n\\noindent {\\bf and Yu-Ting Huang}\\footnote{School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu,\nPeople's Republic of China, {\\it Email: hyt00142024@163.com}}\n\\\\\n\n\\noindent{\\bf Abstract.}\n{In this paper, we study the direct and inverse spectral problems for the Schr\\\"{o}dinger operator with two generalized Regge boundary conditions. For the direct problem, we give the properties of the spectrum, including the asymptotic distribution of the eigenvalues. For the inverse problems, we prove several uniqueness theorems, including the cases: even potential, two-spectra, as well as the general partial inverse problem.}\n\n\n\\medskip\n\\noindent {\\it Keywords:} Schr\\\"{o}dinger operator, generalized Regge boundary condition, spectral asymptotics, inverse spectral problem\n\\medskip\n\n\\noindent {\\it 2020 Mathematics Subject Classification:} 34A55; 34L20; 34L40; 34B07\n\\section{Introduction}\n\nConsider the following generalized Regge problem $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$\n\\begin{equation}\\label{k1}\n \\ell y:=-y''+q(x)y=\\lambda^2 y ,\\quad x\\in(0,a),\\\\\n \\end{equation}\n \\begin{equation}\\label{k2}\n \\quad y'\\left(0\\right)-\\left(i\\alpha_{0}\\lambda + \\beta_{0}\\right)y\\left(0\\right)=0,\\\\\n \\end{equation}\n \\begin{equation}\\label{k3}\n \\quad y'\\left(a\\right)+\\left(i\\alpha\\lambda +\\beta\\right)y\\left(a\\right)=0,\\\\\n\\end{equation}\nwhere the complex-valued potential function$q\\in L^2(0,a)$, $\\alpha_{0}\\ge0,\\alpha>0$ and $\\beta_{0},\\beta\\in\\mathbb{C}$.\nThe problem $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$ arises in various mathematical and physical models. For example, the Liouville transformation can transform the problem of smooth inhomogeneous string vibration with viscous damping at both ends of the string into the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$ with $\\alpha,\\alpha_0>0$. When $\\alpha_{0}=0$, it describes the problem of an inhomogeneous string vibration with no damping at the left end but with various damping at the right end \\cite{MV, mv}. The problem $L\\left(q,1,0,1,0\\right)$ is the resonance scattering problem on the real line \\cite{EK,ST,XF,Zo1}.\n\nIn the direct and inverse spectral theories, the Schr\\\"{o}dinger operator with boundary conditions independent of spectral parameters, i.e., the problem $L\\left(q,0,\\beta_{0},0,\\beta\\right)$, has the comparatively perfect researches (see the monographs \\cite{GV,Bb,V}). For the inverse spectral problem of $L\\left(q,0,\\beta_{0},0,\\beta\\right)$, in general, in order to recover the potential and the parameters in the boundary conditions, one needs to specify two spectra. Whereas, in some cases, one spectrum is enough, such as, (i) the even inverse problem \\cite{GV}: $q(x)=q(a-x)$ and $\\beta=\\beta_0$, and (ii) the half-inverse problem \\cite{HB}: $\\beta$ is given and $q(x)$ is known a priori on $(\\frac{a}{2},a)$. After the half-inverse problem, many authors also continued to study the general partial inverse problems, i.e., the potential is given on a subinterval $(b,a)$ with arbitrary $b\\in (0,a)$. The uniqueness theorems of the general partial inverse problems were proved in \\cite{DGS,FB,M,MH} and other works. One can refer to the review \\cite{BP} for the general summaries of partial inverse problems.\n\nWhen the boundary conditions contain the spectral parameter $\\lambda$, the results of the problem will change. There are many works related Schr\\\"{o}dinger operator with the boundary condition dependent on $\\lambda^2$ (see, e.g., \\cite{BN1,BN2,FY0,G,G1,WW} and the references therein).\nThe dependency form of spectral parameters in the boundary condition \\eqref{k2} or \\eqref{k3} is different from the works mentioned above. The boundary condition \\eqref{k2} or \\eqref{k3} is called the generalized Regge boundary condition. This kind of boundary condition has attracted a lot of attentions from many scholars (see, e.g., \\cite{BY,BS,WY,mv,RS,VC,XP,x,X,Y}). When there is only one generalized Regge boundary condition, one spectrum uniquely recovers all the unknown coefficients \\cite{mv,VC,XP,EK0,RS,x,X,Y}. When there are two generalized Regge boundary conditions, the situation becomes different, in particular, one spectrum is not enough to uniquely recover all the unknown coefficients. As far as we know, few ones considered two generalized Regge boundary conditions \\eqref{k2} and \\eqref{k3} directly for the Schr\\\"{o}dinger operator. Whereas, these two boundary conditions were studied by some scholars for the differential pencil (see \\cite{BY,BS,WY})\n \\begin{equation}\\label{hx9}\n -y''+(2\\lambda p(x)+q(x))y=\\lambda^2 y.\n \\end{equation}\nIn \\cite{BY,WY}, the two-spectra theorem is proved for the problem \\eqref{k2}-\\eqref{hx9}: the spectrum of the problem \\eqref{k2}-\\eqref{hx9} and the spectrum of the problem \\eqref{k3}, \\eqref{hx9} and $y(0)=0$ uniquely determine $p$ and $q$ as well as all the unknown parameters in \\eqref{k2} and \\eqref{k3}. The uniqueness theorem of the half inverse problem was proved in \\cite{BS}. Although \\eqref{k1} is a particular case of \\eqref{hx9}, the two-spectra theorem in \\cite{BY,WY} is not applicable for the problem $L\\left(q,\\alpha_0,\\beta_{0},\\alpha,\\beta\\right)$ considered here. Because the spectrum of the problem \\eqref{k3}, \\eqref{hx9} and $y(0)=0$ with $p=0$ is enough to uniquely determine all the unknowns \\cite{VC,X}. Hence, the two-spectra theorem for the problem $L\\left(q,\\alpha_0,\\beta_{0},\\alpha,\\beta\\right)$ should be reformulated and proved. Moreover, there is no result for the even inverse problem or the general partial inverse problem of the problem $L\\left(q,\\alpha_0,\\beta_{0},\\alpha,\\beta\\right)$ with $(1-\\alpha)(\\alpha_0-1)\\ne0$, which are also interesting and important issues and should be investigated.\n\n\n\n\n\n\n\n\n\n\nIn this paper, we study the direct and inverse spectral problems for the boundary value problem $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$. For the direct problem, we give some properties of the eigenvalues, in particular, give the asymptotic behavior. For the inverse spectral problem, we prove four uniqueness theorems, including the two-spectra theorem, even inverse problem as well as the general partial inverse problem.\n\n\n\n\nThis paper is structured as follows. In the second section, we introduce three characteristic functions. In the third section, we investigate the asymptotic distribution of eigenvalues and the fundamental properties of eigenvalues in the lower plane. In the last section, we consider the inverse problems and prove the uniqueness theorems.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Characteristic functions}\nIn this section, we introduce three characteristic functions.\n Let $s\\left(\\lambda,x\\right)$ and $c\\left(\\lambda,x\\right)$ be the solutions of the initial value problems for \\eqref{k1} with the initial conditions $s\\left(\\lambda,0\\right)=0$, $s'\\left(\\lambda,0\\right)=1$ and $c\\left(\\lambda,0\\right)=1$, $c'\\left(\\lambda,0\\right)=\\beta_{0}$, respectively.\nIt was shown in \\cite[p.9]{V} that\n\\begin{equation}\\label{k4}\n s\\left(\\lambda,x\\right)=\\frac{\\sin{\\lambda x}}{\\lambda}+\\int_{0}^{x}K_{1}\\left(x,t\\right)\\frac{\\sin{\\lambda t}}{\\lambda}dt,\n\\end{equation}\n\\begin{equation}\\label{k5}\n c\\left(\\lambda,x\\right)=\\cos{\\lambda x}+\\int_{0}^{x}G\\left(x,t,\\beta_{0}\\right)\\cos{\\lambda t}dt,\n\\end{equation}\nwhere\n\\begin{equation}\\label{k6}\n K_{1}\\left(x,t\\right)=K\\left(x,t\\right)-K\\left(x,-t\\right),\n\\end{equation}\n\\begin{equation}\\label{k6}\n G\\left(x,t,\\beta_{0}\\right)=\\beta_{0}+K\\left(x,t\\right)+K\\left(x,-t\\right)+\\beta_{0}\\int_{t}^{x}[K\\left(x,\\xi\\right)-K\\left(x,-\\xi\\right)]d\\xi,\n\\end{equation}\nand $K\\left(x,t\\right)$ is the unique solution of the integral equation\n\n\\begin{equation}\n\\notag\n K\\left(x,t\\right)=\\frac{1}{2}\\int_{0}^{\\frac{x+t}{2}}q(s)ds+\n \\int_{0}^{\\frac{x+t}{2}}\\int_{0}^{\\frac{x-t}{2}}q\\left(s+v\\right)K\\left(s+v,s-v\\right)dv ds,\n\\end{equation}\nin the region $\\left\\{\\left(x,t\\right)\\in (0,a)\\times(0,a):\\lvert t \\rvert\\leq x \\right\\}$. Moreover,\n\\begin{equation}\\label{mza}\n G\\left(x,x,\\beta_0\\right)=\\beta_0+K_{1}\\left(x,x\\right),\\quad K_{1}\\left(x,x\\right)=K\\left(x,x\\right)=\\frac{1}{2}\\int_{0}^{x}q\\left(t\\right)dt.\n\\end{equation}\nLet\n\\begin{equation}\\label{hx4}\n y\\left(\\lambda,x\\right)=c\\left(\\lambda,x\\right)+i\\alpha_{0}\\lambda s\\left(\\lambda,x\\right).\n\\end{equation}\nThen\n\\begin{equation}\\label{r2}\n y\\left(\\lambda,x\\right)=\\cos{\\lambda x}+i\\alpha_{0}\\sin{\\lambda x}+G(x,x,\\beta_{0})\\frac{\\sin{\\lambda x}}{\\lambda}-i\\alpha_{0}K_{1}\\left(x,x\\right)\\frac{\\cos{\\lambda x}}{\\lambda}+\\frac{\\Psi_{1}\\left(\\lambda,x\\right)}{\\lambda},\n\\end{equation}\nand\n\\begin{equation}\\label{k8}\n y'\\left(\\lambda,x\\right)=-\\lambda \\sin{\\lambda x}+\\left[G(x,x,\\beta_{0})+i\\alpha_{0}\\lambda\\right]\\cos{\\lambda x}+i\\alpha_{0}K_{1}\\left(x,x\\right)\\sin{\\lambda x}+\\Psi_{2}\\left(\\lambda,x\\right),\n\\end{equation}\nwhere\n\\begin{equation}\\label{hx}\n \\Psi_{1}\\left(\\lambda,x\\right)=-\\int_{0}^{x}G_{t}\\left(x,t,\\beta_{0}\\right)\\sin{\\lambda t}dt+i\\alpha_{0}\\int_{0}^{x}K_{1t}\\left(x,t\\right)\\cos{\\lambda t}dt,\n\\end{equation}\nand\n\\begin{equation}\n \\Psi_{2}\\left(\\lambda,x\\right)=\\int_{0}^{x}G_{x}\\left(x,t,\\beta_{0}\\right)\\cos{\\lambda t}dt+i\\alpha_{0}\\int_{0}^{x}K_{1x}\\left(x,t\\right)\\sin{\\lambda t}dt.\n\\end{equation}\n The spectrum of problem $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$ coincides with the sets of zeros of the entire function\n\\begin{equation}\\label{k9}\n \\Delta_+\\left(\\lambda\\right)=y'\\left(\\lambda,a\\right)+\\left(i\\alpha\\lambda+\\beta\\right)y\\left(\\lambda,a\\right),\n\\end{equation}\n which is called the \\emph{characteristic function} of the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right).$ Denote $\\left<y,z\\right>=yz'-y'z$. It is easy to show that if $y$ and $z$ are solutions of \\eqref{k1}, then $\\left<y,z\\right>$ is independent of $x$. Let $\\phi\\left(\\lambda,x\\right)$ be the solution of the problem for \\eqref{k1} with initial conditions $\\phi\\left(\\lambda,a\\right)=1$ and $\\phi'\\left(\\lambda,a\\right)=-\\left(i\\alpha\\lambda+\\beta\\right).$ Then we have\n \\begin{equation}\\label{e10}\n \\Delta_+\\left(\\lambda\\right)=\\left<\\phi\\left(\\lambda,x\\right),y\\left(\\lambda,x\\right)\\right>.\n \\end{equation}\n Define\n \\begin{equation}\\label{e15}\n \\Delta_{-}\\left(\\lambda\\right)=y'\\left(\\lambda,a\\right)-\\left(i\\alpha\\lambda-\\beta\\right)y\\left(\\lambda,a\\right).\n \\end{equation}\n Obviously, $\\Delta_{-}\\left(\\lambda\\right)$ is the \\emph{characteristic function} of the problem $L\\left(q,\\alpha_{0},\\beta_{0},-\\alpha,\\beta\\right)$, i.e., the equation \\eqref{k1} with the boundary conditions \\eqref{k2} and $$y'\\left(\\lambda,a\\right)-\\left(i\\alpha\\lambda-\\beta\\right)y\\left(\\lambda,a\\right)=0.$$\n Using \\eqref{k9} and \\eqref{e15}, it is easy to verify\n \\begin{equation}\\label{e7}\n \\Delta_+\\left(\\lambda\\right)\\Delta_+\\left(-\\lambda\\right)-\\Delta_{-}\\left(\\lambda\\right)\\Delta_{-}\\left(-\\lambda\\right)=4\\alpha\\alpha_{0}\\lambda^2.\n \\end{equation}\n We also consider the function\n \\begin{equation}\\label{o30}\n \\Delta_{0}(\\lambda)=y\\left(\\lambda,a\\right).\n \\end{equation}\nIt is easy to show that the function $\\Delta_{0}(\\lambda)$ is the \\emph{characteristic function} of the following problem\n\\begin{equation}\\label{hx12}\n -y''+q_{1}(x)y=\\lambda^2 y ,\\quad x\\in(0,a),\n\\end{equation}\n\\begin{equation}\\label{hx13}\n y(0)=0,\n\\end{equation}\n\\begin{equation}\\label{hx14}\n y'(a)+(i\\alpha_{0}\\lambda+\\beta_{0})y(a)=0,\n\\end{equation}\nwhere $q_{1}(x)=q(a-x)$.\n\n\n\n\n\n \\section{Properties of eigenvalues}\n In this section, we investigate the properties of the spectrum of the problems $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$. Let's first give the asymptotic distribution of the eigenvalues.\n\\begin{theorem}\\label{th1}\n $\\left(1\\right)$ If $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)>0$, then the eigenvalues $\\{\\lambda_{k}^\\pm\\}_{k\\in\\mathbb{Z}}$ of the problems $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$, respectively, have the following asymptotic behavior:\n \\begin{equation}\\label{k11}\n \\begin{aligned}\n \\lambda_{k}^\\pm=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-\\frac{1}{2}\\right){\\rm sgn} k+\\frac{i P_0^\\pm}{2a}+\\frac{P}{k}+\\frac{\\beta_{k}}{k}, \\quad \\lvert k \\rvert \\to \\infty,\n \\end{aligned}\n \\end{equation}\n where $\\{\\beta_{k}\\}_{k=-\\infty,k\\neq0}^{\\infty}\\in l_{2}$ and where\n \\begin{equation}\\label{k10}\n \\begin{aligned}\n P_0^\\pm=\\ln{\\left(\\frac{|\\alpha_{0}\\pm\\alpha+1\\pm\\alpha\\alpha_{0}|}{|\\alpha_{0}\\pm\\alpha-(1\\pm\\alpha\\alpha_{0})|}\\right)},\\quad P=\\frac{\\beta_0}{\\pi\\left(1-\\alpha_{0}^2\\right)}+\\frac{\\beta}{\\pi\\left(1-\\alpha^2\\right)}+\\frac{K_{1}\\left(a,a\\right)}{\\pi}.\n \\end{aligned}\n \\end{equation}\n $\\left(2\\right)$ If $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)<0$, then the eigenvalues $\\{\\lambda_{k}^\\pm\\}_{k\\in\\mathbb{Z}_{0}}$, $\\mathbb{Z}_{0}=\\mathbb{Z}\\setminus\\{0\\}$, of the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$, respectively, have the following asymptotic behavior:\n \\begin{equation}\\label{k12}\n \\begin{aligned}\n \\lambda_{k}^\\pm=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-1\\right){\\rm sgn} k+\\frac{iP_0^\\pm}{2a}+\\frac{P}{k}+\\frac{\\beta_{k}}{k}, \\quad \\lvert k \\rvert \\to \\infty,\n \\end{aligned}\n \\end{equation}\n where $P_0^\\pm$ and $P$ are given \\eqref{k10} and $\\{\\beta_{k}\\}_{k=-\\infty,k\\neq0}^{\\infty}\\in l_{2}$.\\\\\n $\\left(3\\right)$ If $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)=0$, then the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$ may have only finitely many eigenvalues. If there exist infinitely many eigenvalues $\\{\\lambda_{k}^+\\}$, then $Im \\lambda_{k}^+ \\to \\infty$ as $\\lvert k\\rvert\\to \\infty$.\n\\end{theorem}\n\\begin{proof}\n Using \\eqref{r2}-\\eqref{k8} we obtain \\begin{align}\\label{c16}\n \\Delta_\\pm\\left(\\lambda\\right)=&\\lambda[i\\left(\\alpha_{0}\\pm\\alpha\\right)\\cos{\\lambda a}-\\left(1\\pm\\alpha\\alpha_{0}\\right)\\sin{\\lambda a}]+i[\\alpha_{0}\\beta\\pm\\alpha \\omega+\\alpha_{0} K_{1}\\left(a,a\\right)]\\sin{\\lambda a}\\\\\n \\notag &+[\\omega+\\beta\\pm\\alpha_{0}\\alpha K_{1}\\left(a,a\\right)]\\cos{\\lambda a}+\\psi_\\pm\\left(\\lambda\\right).\n \\end{align}\n where $\\omega=G(a,a,\\beta_{0})$ and\n \\begin{equation}\\label{o80}\n \\psi_\\pm\\left(\\lambda\\right)=\\omega\\beta\\frac{\\sin{\\lambda a}}{\\lambda}-i\\beta\\alpha_{0}K_{1}(a,a)\\frac{\\cos{\\lambda a}}{\\lambda}+(\\beta \\pm i\\alpha\\lambda)\\frac{\\Psi_{1}\\left(\\lambda,a\\right)}{\\lambda}+\\Psi_{2}\\left(\\lambda,a\\right).\n \\end{equation}\nIt is obvious that the function $\\psi_\\pm(\\lambda)$ are entire functions of exponential type $\\le a$ and belong to $L^2(-\\infty,\\infty)$.\nRewrite \\eqref{c16} as\n \\begin{align}\\label{k13}\n \\notag -i\\Delta_\\pm\\left(\\lambda\\right)=&\\lambda[\\left(\\alpha_{0}\\pm\\alpha\\right)\\cos{\\lambda a}+i\\left(1\\pm\\alpha\\alpha_{0}\\right)\\sin{\\lambda a}]+[\\alpha_{0}\\beta\\pm\\alpha \\omega+\\alpha_{0} K_{1}\\left(a,a\\right)]\\sin{\\lambda a}\\\\\n &-i[\\omega+\\beta\\pm\\alpha_{0}\\alpha K_{1}\\left(a,a\\right)]\\cos{\\lambda a}-i\\psi_\\pm\\left(\\lambda\\right).\n \\end{align}\nThe right hand side of \\eqref{k13} is of the form \\eqref{A.7} in Lemma \\ref{LA2} with $M^\\pm=\\alpha_{0}\\beta\\pm\\alpha \\omega+\\alpha_{0} K_{1}\\left(a,a\\right)$ and $N^\\pm=\\omega+\\beta\\pm\\alpha_{0}\\alpha K_{1}\\left(a,a\\right)$, and the number $P$ defined in \\eqref{A.9} becomes\n \\begin{equation}\n P^\\pm=\\frac{\\left(1\\pm\\alpha_{0}\\alpha\\right)N^\\pm-\\left(\\alpha_{0}\\pm\\alpha\\right)M^\\pm}{\\pi[\\left(1\\pm\\alpha_{0}\\alpha\\right)^2-\\left(\\alpha_{0}\\pm\\alpha\\right)^2]}\n =\\frac{(1-\\alpha^2)\\omega+(1-\\alpha_0^2)\\beta+(\\alpha^2-1)\\alpha_0^2K_1(a,a)}{\\pi(1-\\alpha_0^2)(1-\\alpha^2)}.\n \\end{equation}\n Using Lemma \\ref{LA2} and noting $\\omega=\\beta_0+K_1(a,a)$, we obtain \\eqref{k11}-\\eqref{k12}.\n\n When $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)=0$, we have $\\alpha_0=1$ or $\\alpha=1$. In the case $\\alpha_{0}=1$, we have\n \\begin{equation}\\label{o26}\n \\Delta_\\pm(\\lambda)\\!=\\!\\left[\\!(1\\pm\\alpha)\\!\\left(i\\lambda+\\frac{\\omega}{2}+\\frac{K_{1}(a,a)}{2}\\!\\right)\\!+\\beta\\!\\right]e^{i\\lambda a}+\\frac{1\\mp\\alpha}{e^{i\\lambda a}}\\!\\left[\\frac{\\omega}{2}-\\frac{K_{1}(a,a)}{2}\\right]\\!+\\psi_\\pm(\\lambda).\n \\end{equation}\n When $q=0$ and $\\beta_{0}=0$ in \\eqref{o26}, we have\n \\begin{equation}\n \\Delta_\\pm(\\lambda)=[i\\lambda(1\\pm\\alpha)+\\beta]e^{i\\lambda a}.\n \\end{equation}\n It implies that the unique zero of $\\Delta_+(\\lambda)$ is $\\frac{i\\beta}{1+\\alpha}$ and $\\Delta_-(\\lambda)$ also has the only zero $\\frac{i\\beta}{1-\\alpha}$ if $\\alpha\\ne1$. If $\\Delta_+(\\lambda)$ has infinitely many eigenvalues $\\{\\lambda_{k}^+\\}$, then $\\lvert\\lambda_{k}^+\\rvert\\to\\infty$ as $\\lvert k\\rvert\\to\\infty.$ Substituting $\\lambda=\\lambda_{k}^+$ into \\eqref{o26}, respectively, and noting that $\\psi(\\lambda_k^+)=o(e^{|{\\rm Im}\\lambda_k^+|a})$, then we have\n \\begin{equation}\\label{o27}\n (1+\\alpha)\\!\\!\\left[i\\lambda_{k}^++\\frac{\\omega}{2}+\\frac{K_{1}(a,a)}{2}\\right]+\\beta\\!=\\!\\frac{1-\\alpha}{e^{2i\\lambda_{k}^+ a}}\n \\!\\left[\\!\\frac{K_{1}(a,a)}{2}-\\frac{\\omega}{2}\\!\\right]\\!\\!+o(1)e^{\\left(\\lvert {\\rm Im}\\lambda_{k}^+\\rvert+{\\rm Im}\\lambda_{k}^+\\right)a}.\n \\end{equation}\n If $ {\\rm Im}\\lambda_{k}^+$ is bounded or goes to $-\\infty$ as $\\lvert k\\rvert\\to\\infty$, then the left hand side of \\eqref{o27} is unbounded and the right hand is bounded, which is a contradiction. It shows that $\\lim_{k\\to\\infty}\\sup {\\rm Im}\\lambda_{k}^+=\\infty.$ The left hand side of \\eqref{o27} tending to $\\infty$ implies for the right hand side that $2{\\rm Im}\\lambda_{k}^+\\to\\infty$ or $\\lvert {\\rm Im}\\lambda_{k}^+\\rvert+{\\rm Im}\\lambda_{k}^+\\to\\infty$ as $k\\to\\infty$, which gives ${\\rm Im}\\lambda_{k}^+\\to\\infty$. Similarly, we can also obtain the same results in the case $\\alpha=1$.\nThe proof is complete.\n\\end{proof}\n\\begin{theorem}\\label{th2x}\n (1) There are at most a finite number of the eigenvalues of $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$ lying in the closed lower half-plane. If $(\\alpha_0-1)(1-\\alpha)\\ne0$ and $(\\alpha_0+1)|\\alpha-1|>(\\alpha+1)|\\alpha_0-1|$ ($(\\alpha_0+1)|\\alpha-1|<(\\alpha+1)|\\alpha_0-1|$), then there are at most a finite number of the eigenvalues of $L\\left(q,\\alpha_{0},\\beta_{0},-\\alpha,\\beta\\right)$ lying in the closed lower (upper) half-plane. \\\\\n (2) If $q(x)$ is a real-valued function and $\\beta,\\beta_{0}\\in\\mathbb{R}$, then the eigenvalues of the problems $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$ have the following properties:\\\\\n (i) If $0$ is an eigenvalue of the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$, then it is simple.\\\\\n (ii) If $\\lambda$ is an eigenvalue of the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$, then $-\\overline{\\lambda}$ is also an eigenvalue, where $\\overline{\\lambda}$ is the complex conjugate of $\\lambda$.\\\\\n (iii) All nonzero eigenvalues of the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$ in the lower half-plane lie on the negative imaginary semiaxis $i\\mathbb{R}_-$ and are simple.\\\\\n (iv) Assume that there are $\\kappa$ eigenvalues of $L\\left(q,\\alpha_{0},\\beta_{0},\\alpha,\\beta\\right)$ on $i\\mathbb{R}_-$, denoted by $\\lambda_{-j}=-i\\lvert \\lambda_{-j} \\rvert$,$j=1,\\ldots,\\kappa$, satisfying $\\lvert \\lambda_{-j} \\rvert<\\lvert \\lambda_{-(j+1)}\\rvert$. Then $$i\\dot{\\Delta}_+\\left(-i\\lvert\\lambda_{-j}\\rvert\\right)(-1)^{\\kappa-j}<0,\\quad\n \\Delta_{0}\\left(-i\\lvert\\lambda_{-j}\\rvert\\right)(-1)^{\\kappa-j}>0,\\quad j=1,\\ldots,\\kappa,$$ where $\\dot{\\Delta}_+(\\lambda)=\\frac{d {\\Delta_+}(\\lambda)}{d\\lambda}$. Moreover, the zeros of $\\Delta_{0}(\\lambda)$ and $\\Delta_+(\\lambda)$ on $i\\mathbb{R}_-$ interlace each other.\n\\end{theorem}\n\\begin{proof}\n (1) From Theorem \\ref{th1}, it is easy to see that there are at most a finite number of the eigenvalues $\\{\\lambda_n^+\\}$ lying in the closed lower half-plane. Assume that $(\\alpha_0-1)(1-\\alpha)\\ne0$, (i) if $(\\alpha_0+1)|\\alpha-1|>(\\alpha+1)|\\alpha_0-1|$, then $\\frac{|\\alpha_{0}-\\alpha+1-\\alpha\\alpha_{0}|}{|\\alpha_{0}-\\alpha-(1-\\alpha\\alpha_{0})|}>1$, which implies $P_0^->0$; (ii) if $(\\alpha_0+1)|\\alpha-1|<(\\alpha+1)|\\alpha_0-1|$, then $\\frac{|\\alpha_{0}-\\alpha+1-\\alpha\\alpha_{0}|}{|\\alpha_{0}-\\alpha-(1-\\alpha\\alpha_{0})|}\\in (0,1)$, which implies $P_0^-<0$.\n\n (2) (i)\n Due to \\eqref{k1}, then the differentiation of \\eqref{k1} with respect to $\\lambda$ gives\n \\begin{equation}\\label{e12}\n -\\dot{y}''\\left(\\lambda,x\\right)+q\\left(x\\right)\\dot{y}\\left(\\lambda,x\\right)=2\\lambda y\\left(\\lambda,x\\right)+\\lambda^2\\dot{y}\\left(\\lambda,x\\right).\n \\end{equation}\n Multiply \\eqref{k1} and \\eqref{e12} by $\\dot{y}$ and $y$, respectively, take the difference, and get\n \\begin{equation}\\label{e5s}\n 2\\lambda y^2\\left(\\lambda,x\\right) =\\left[y'\\left(\\lambda,x\\right)\\dot{y}\\left(\\lambda,x\\right)-\\dot{y}'\\left(\\lambda,x\\right)y\\left(\\lambda,x\\right)\\right]'.\n \\end{equation}\n Let us integrate both side of \\eqref{e5s}, and using \\eqref{k2}, then we have\n \\begin{align}\n \\notag\n 2\\lambda\\int_{0}^{a}y^2\\left(\\lambda,x\\right)dx&=\\int_{0}^{a}\\left[y'\\left(\\lambda,x\\right)\\dot{y}\\left(\\lambda,x\\right)-\\dot{y}'\\left(\\lambda,x\\right)y\\left(\\lambda,x\\right)\\right]'dx\\\\\n &\\notag=y'\\left(\\lambda,a\\right)\\dot{\\Delta}_{0}\\left(\\lambda\\right)-\\dot{y}'\\left(\\lambda,a\\right)\\Delta_{0}\\left(\\lambda\\right)+i\\alpha_{0}\\\\\n &\\label{o1}=\\Delta_+\\left(\\lambda\\right)\\dot{\\Delta}_{0}\\left(\\lambda\\right)-\\dot{\\Delta}_+\\left(\\lambda\\right)\\Delta_{0}\\left(\\lambda\\right)+i\\alpha \\Delta_{0}^{2}(\\lambda)+i\\alpha_{0}.\n \\end{align}\n Letting $\\lambda=0$ in \\eqref{o1}, due to $\\Delta_+(0)=0$, we have\n \\begin{equation}\n \\dot{\\Delta}_+\\left(0\\right)\\Delta_{0}\\left(0\\right)=i\\alpha \\Delta_{0}^{2}(0)+i\\alpha_{0},\n \\end{equation}\n which implies\n \\begin{equation}\n -i\\dot{\\Delta}_+\\left(0\\right)\\Delta_{0}\\left(0\\right)=\\alpha \\Delta_{0}^{2}(0)+\\alpha_{0}>0.\n \\end{equation}\n Hence, if $0$ is an eigenvalue, then it is simple.\n\n (ii) Note that\n \\begin{equation*}\n \\overline{ c^{(\\nu)}\\left(\\lambda,x\\right)}=c^{(\\nu)}\\left(-\\bar{\\lambda},x\\right), \\quad \\overline{ s^{(\\nu)}\\left(\\lambda,x\\right)}=s^{(\\nu)}\\left(-\\bar{\\lambda},x\\right),\\quad \\nu=0,1.\n \\end{equation*}\n It follows from \\eqref{hx4} that\n \\begin{align}\\label{k16}\n y^{(\\nu)}\\left(-\\overline{\\lambda},x\\right)= \\overline{y^{(\\nu)}\\left(\\lambda,x\\right)},\\quad \\nu=0,1,\n \\end{align}\nwhich implies from \\eqref{k9} and \\eqref{e15} that\n \\begin{equation}\\label{m1s}\n \\overline{\\Delta_\\pm \\left(\\lambda\\right)}=\\Delta_\\pm\\left(-\\overline{\\lambda}\\right).\n \\end{equation}\nThe assertion in (i) follows from \\eqref{m1s}.\n\n (iii) Assume that there is an eigenvalue $\\lambda_{0}=\\sigma-i\\tau\\left(\\tau>0\\right)$, i.e., $\\Delta_+\\left(\\lambda_{0}\\right)=0$. Using the initial condition of $y(\\lambda_0,x)$ and integration by the parts, we calculate\n \\begin{align}\\label{e1}\n \\notag &\\lambda_{0}^2\\int_{0}^{a}\\lvert y\\left(\\lambda_{0},x\\right)\\rvert^2dx=\n \\int_{0}^{a}\\ell y(\\lambda_0,x)y\\left(-\\overline{\\lambda}_{0},x\\right)dx\\\\\n \\notag= &i\\alpha\\left(\\lambda_{0}+\\overline{\\lambda}_{0}\\right)\\lvert y\\left(\\lambda_{0},a\\right)\\rvert^2+i\\alpha_0\\left(\\lambda_{0}+\\overline{\\lambda}_{0}\\right)+ \\int_{0}^{a} y(\\lambda_0,x)\\ell y\\left(-\\overline{\\lambda}_{0},x\\right)dx\\\\\n =& 2i\\sigma\\left[\\alpha\\lvert y(\\lambda_{0},a)\\rvert^2+\\alpha_{0}\\right]+\n \\notag \\overline{\\lambda}_{0}^2\\int_{0}^{a}\\lvert y\\left(\\lambda_{0},x\\right)\\rvert^2dx,\n \\end{align}\nwhich implies\n \\begin{equation}\\label{e3}\n - 2\\sigma\\tau\\int_{0}^{a}\\lvert y\\left(\\lambda_{0},x\\right)\\rvert^2dx=\\alpha\\sigma\\lvert y(\\lambda_{0},a)\\rvert^2+\\sigma\\alpha_{0}.\n \\end{equation}\n By observing \\eqref{e3}, we have that $\\tau<0$ if $\\sigma\\neq0$, which contradicts the fact that $\\tau>0$. Thus $\\sigma=0$.\n\n Next, we verify that the eigenvalues $\\lambda_{0}=-i\\tau \\left(\\tau>0\\right)$ is simple. Letting $\\lambda=\\lambda_{0}=-i\\tau$ in \\eqref{o1}, due to $\\Delta_+\\left(-i\\tau\\right)=0$, we have\n \\begin{align}\n \\dot{\\Delta}_+\\left(-i\\tau\\right)\\Delta_{0}\\left(-i\\tau\\right)=2i\\tau\\int_{0}^{a}y^2\\left(-i\\tau ,x\\right)dx+i\\alpha \\Delta_{0}^{2}(-i\\tau)+i\\alpha_{0},\n \\end{align}\nwhich implies\n\\begin{equation}\\label{o2}\n -i\\dot{\\Delta}_+\\left(-i\\tau\\right)\\Delta_{0}\\left(-i\\tau\\right)>0.\n\\end{equation}\nTherefore, if $\\lambda_{0}$ is a zero of $\\Delta_+(\\lambda)$ on the negative imaginary semiaxis, then it is simple.\n\n (iv) Note that\n \\begin{equation}\\label{m1}\n \\cos({-i\\tau a})=\\frac{e^{-\\tau a}+e^{\\tau a}}{2}, \\sin({-i\\tau a})=\\frac{-e^{-\\tau a}+e^{\\tau a}}{2i}.\n \\end{equation}\n Substituting \\eqref{m1} into \\eqref{c16}, then we have\n \\begin{align}\n \\Delta_+(-i\\tau)\\notag=&\\left[\\left(\\alpha+\\alpha_{0}\\right)\\frac{e^{-\\tau a}+e^{\\tau a}}{2}+\\left(1+\\alpha\\alpha_{0}\\right)\\frac{-e^{-\\tau a}+e^{\\tau a}}{2}\\right]\\tau\\\\&\\notag+\\left[\\omega+\\beta+\\alpha\\alpha_{0}K_{1}\\left(a,a\\right)\\right]\\frac{e^{-\\tau a}+e^{\\tau a}}{2}\\\\\n \\notag &+\\left[\\alpha_{0}\\beta+\\alpha\\omega+\\alpha_{0}K_{1}\\left(a,a\\right)\\right]\\frac{-e^{-\\tau a}+e^{\\tau a}}{2}+O\\left(\\frac{e^{\\tau a}}{\\tau}\\right).\n \\end{align}\n Then\n \\begin{equation}\\label{q9}\n \\Delta_+(-i\\tau)\\to +\\infty, \\quad \\tau\\to +\\infty.\n \\end{equation}\n Note that $\\Delta_+(\\lambda)$ has only simple zeros on on the negative imaginary semiaxis. It follows from \\eqref{q9} that $-i\\dot{\\Delta}_+\\left(-i\\lvert\\lambda_{-\\kappa}\\rvert\\right)>0$. Consequently, we have\n \\begin{equation}\n \\notag\n -i\\dot{\\Delta}_+\\left(-i\\lvert\\lambda_{-j}\\rvert\\right)(-1)^{\\kappa-j}>0.\n \\end{equation}\n Together with \\eqref{o2}, we have $ \\Delta_{0}\\left(-i\\lvert\\lambda_{-j}\\rvert\\right)(-1)^{\\kappa-j}>0.$\n\n\n Substituting $\\lambda=-i\\tau$ into \\eqref{o1}, we obtain\n\\begin{align}\n\\notag\n & \\Delta_+\\left(-i\\tau \\right)\\frac{\\mathrm{d}}{\\mathrm{d}(-i\\tau)}\\Delta_{0}\\left(-i\\tau\\right)-\\Delta_{0}\\left(-i\\tau\\right)\\frac{\\mathrm{d}}{\\mathrm{d} (-i\\tau)}\\Delta_+\\left(-i\\tau\\right)\\\\&=-2i\\tau\\int_{0}^{a}\\lvert y\\left( -i\\tau,x\\right)\\rvert^2dx-i\\alpha \\Delta_{0}^{2}(-i\\tau)-i\\alpha_{0}.\n\\end{align}\nConsequently,\n\\begin{align}\n\\notag\n \\frac{\\mathrm{d}}{\\mathrm{d}\\tau }\\left(\\frac{\\Delta_{0}\\left(-i\\tau\\right)}{\\Delta_+\\left(-i\\tau\\right)}\\right)&=\\frac{1}{\\Delta_+^2\\left(-i\\tau\\right)}\\left[2\\tau\\int_{0}^{a}\\lvert y\\left(-i\\tau,x\\right)\\rvert^2dx+\\alpha\\Delta_{0}^{2}(-i\\tau)+\\alpha_{0}\\right]>0,\n\\end{align}\nthus the function $\\frac{\\Delta_{0}\\left(-i\\tau\\right)}{\\Delta_+\\left(-i\\tau\\right)}$ monotonically increasing for $\\tau\\in\\mathbb{R}^{+}\\verb|\\|\\left\\{ \\lvert \\lambda_{-j} \\rvert|j=1,\\ldots,\\kappa\\right\\}$\nwith\n\\begin{align*}\n &\\lim_{\\tau\\to \\lvert \\lambda_{-j} \\rvert^\\pm}\\frac{\\Delta_{0}\\left(-i\\tau\\right)}{\\Delta_+\\left(-i\\tau\\right)}=\\mp\\infty,\\quad j=2,\\dots,\\kappa-1\\\\\n &\\label{e5}\\lim_{\\tau\\to \\lvert \\lambda_{-1} \\rvert^{+}}\\frac{\\Delta_{0}\\left(-i\\tau\\right)}{\\Delta_+\\left(-i\\tau\\right)}=-\\infty,\\quad\n \\notag\\lim_{\\tau\\to \\lvert \\lambda_{-\\kappa} \\rvert^{-}}\\frac{\\Delta_{0}\\left(-i\\tau\\right)}{\\Delta_+\\left(-i\\tau\\right)}=+\\infty.\n\\end{align*}\nHence, $\\Delta_{0}\\left(-i\\lvert \\lambda_{-j} \\rvert\\right)\\Delta_{0}\\left(-i\\lvert \\lambda_{-(j+1)} \\rvert\\right)<0$. It follows that the function $\\Delta_{0}(\\lambda)$ has one zero in the interval $\\left(-i\\lvert \\lambda_{-(j+1)} \\rvert,-i\\lvert \\lambda_{-j} \\rvert\\right)$, then we have that the zeros of $\\Delta_+(\\lambda)$ and $\\Delta_{0}(\\lambda)$ interlace on the negative imaginary semiaxis.\nThe proof is complete.\n\\end{proof}\n\\section{Inverse problems}\nIn this section, we study the inverse spectral problems. We will consider the even inverse problem, two-spectra theorem and the general partial inverse problem, and prove four uniqueness theorems.\n\n\\begin{lemma}\\label{hx11}\nLet $f(z)$ be an entire function of exponential type, and has the asymptotics\n\\begin{equation}\\label{hx10}\n f(z)=z[c_1\\cos z +c_2\\sin z ]+O(e^{|{\\rm Im}z|}),\\quad |z|\\to \\infty,\n\\end{equation}\nwhere $c_1$ and $c_2$ are constants. If $|c_1|^2+|c_2|^2\\ne0$ and there exists one non-zero $c_0\\in \\{c_1,c_2,c_1+c_2,c_1-c_2\\}$ and $c_0$ is known, then $f(z)$ is uniquely determined by all its zeros (including multiplicity).\n\\end{lemma}\n\\begin{proof}\nLet $\\{z_n\\}$ (counted with multiplicities) be the non-zero zeros of the function $f(z)$. The Hadamard's factorization theorem implies\n\\begin{equation}\\label{3.0}\nf(z)=ce^{bz}E(z),\\quad E(z):=z^{s}\\prod_{z_n\\ne0}\\left(1-\\frac{z}{z_n}\\right)e^{\\frac{z}{z_n}},\n\\end{equation}\nwhere $b,c$ are the constants to be determined, and $s\\ge 0$. Without loss of generality, assume $c_1\\ne 0$ and is known. Then from (\\ref{hx10}) we have\n\\begin{equation*}\n \\frac{f(2n\\pi)}{2n\\pi}=c_1(1+o(1)),\\quad n\\to\\infty,\n\\end{equation*}\nwhich implies from (\\ref{3.0}) that\n\\begin{equation}\\label{3.0s}\n b=-\\lim_{n\\to\\infty} \\frac{\\ln E(2n\\pi)}{2n\\pi},\n\\quad\n c= c_1\\left[\\lim_{n\\to\\infty}\\frac{e^{2n\\pi b}E(2n \\pi)}{2n\\pi}\\right]^{-1}.\n\\end{equation}\nIt follows from (\\ref{3.0}) and (\\ref{3.0s}) that all zeros of $f(z)$ uniquely determine $f(z)$. The proof is complete.\n\\end{proof}\n\n\n\nIn the following theorem, under the assumption that $0$ is not an eigenvalue, we prove a uniqueness result for the even inverse problem of the problem $L(q,\\alpha,\\beta,\\alpha,\\beta)$.\n\\begin{theorem}\n Assume that $\\Delta_+\\left(0\\right)\\neq0$, $q\\left(x\\right)=q\\left(a-x\\right)$, $\\alpha=\\alpha_{0}$ and $\\beta=\\beta_{0}$. Then all the eigenvalues $\\left\\{\\lambda_{k}^+\\right\\}$ (including multiplicity) uniquely determine $\\alpha$, $\\beta$ and $q(x)$ a.e. on $(0,a)$.\n\\end{theorem}\n\\begin{proof}\n Using Theorem \\ref{th1} and $\\alpha=\\alpha_{0}$, we know that $\\frac{2\\alpha}{1+\\alpha^2}$ is uniquely determined. Indeed, we can first determine whether $\\alpha=1$ or not, by observing the number and the imaginary parts of the eigenvalues $\\left\\{\\lambda_{k}^+\\right\\}$. If $\\alpha\\ne1$, then we can use the asymptotics of the eigenvalues to recover $\\frac{2\\alpha}{1+\\alpha^2}$. Using Lemma \\ref{hx11} and \\eqref{c16}, whether $\\alpha=1$ or not, we can obtain that the function $\\frac{\\Delta_+(\\lambda)}{1+\\alpha^2}$ is uniquely determined from its zeros. Using \\eqref{k9}, \\eqref{e15} and \\eqref{o30}, we have\n \\begin{equation}\\label{o31}\n \\frac{\\Delta_+(\\lambda)}{1+\\alpha^2}-\\frac{\\Delta_{-}(\\lambda)}{1+\\alpha^2}=\\frac{2i\\alpha\\lambda\\Delta_{0}(\\lambda)}{1+\\alpha^2}.\n \\end{equation}\nIt is known \\cite{X} that all zeros of $\\Delta_{0}(\\lambda)$ uniquely determine $q(x)$, $\\alpha$ and $\\beta$ under the assumption $\\alpha\\ne 1$. When $\\alpha=1$, one can use a similar argument in \\cite[Theorem 3.1]{XP} to prove the uniqueness theorem. Thus, it is enough to prove $\\frac{\\Delta_+(\\lambda)}{1+\\alpha^2}$ uniquely determines $\\frac{\\Delta_{-}(\\lambda)}{1+\\alpha^2}$.\n Define\n \\begin{equation*}\n y_{a}\\left(\\lambda,x\\right):=c\\left(\\lambda,a-x\\right)-i\\alpha\\lambda s\\left(\\lambda,a-x\\right).\n \\end{equation*}\n Obviously, it satisfies\n \\begin{align}\n \\notag\n y_{a}\\left(\\lambda,x\\right)= y(-\\lambda,a-x),\\quad y_{a}\\left(\\pm \\lambda,a\\right)=1,\\quad y'_{a}\\left(\\pm \\lambda,a\\right)=\\pm i\\alpha\\lambda-\\beta.\n \\end{align}\nSince $q(x)=q(a-x)$, $\\alpha=\\alpha_0$ and $\\beta=\\beta_0$, using \\eqref{e15}, we have\n \\begin{equation}\\label{e8}\n \\Delta_{-}\\left(\\lambda\\right)=-\\left<y\\left(\\lambda,x\\right),y_{a}\\left(\\lambda,x\\right)\\right>.\n \\end{equation}\n Letting $x=\\frac{a}{2}$ in \\eqref{e8}, then we have\n \\begin{align}\n \\notag\n \\Delta_{-}\\left(\\lambda\\right)&=-y\\left(\\lambda,\\frac{a}{2}\\right)y'_{a}\\left(\\lambda,\\frac{a}{2}\\right)+y_{a}\\left(\\lambda,\\frac{a}{2}\\right)y'\\left(\\lambda,\\frac{a}{2}\\right)\\\\\n &\\notag=-y_{a}\\left(-\\lambda,\\frac{a}{2}\\right)y'_{a}\\left(\\lambda,\\frac{a}{2}\\right)-y_{a}\\left(\\lambda,\\frac{a}{2}\\right)y_{a}'\\left(-\\lambda,\\frac{a}{2}\\right).\n \\end{align}\n It follows that $\\Delta_{-}\\left(-\\lambda\\right)=\\Delta_{-}\\left(\\lambda\\right)$. Using \\eqref{e7}, we have\n \\begin{equation}\n \\frac{\\Delta_{-}(\\lambda)}{1+\\alpha^2}=\\pm\\sqrt{\\frac{\\Delta_+(\\lambda)\\Delta_+(-\\lambda)}{(1+\\alpha^2)^2}-\\frac{4\\alpha^2\\lambda^2}{(1+\\alpha^2)^2}}\n \\end{equation}\n To determine which branch is the right choose, we note that $\\frac{\\Delta_{-}(0)}{1+\\alpha^2}=\\frac{\\Delta_+(0)}{1+\\alpha^2}\\neq0$ that follows from \\eqref{k9} and \\eqref{e15}. Thus, $\\frac{\\Delta_{-}(\\lambda)}{1+\\alpha^2}$ is uniquely determined by $\\frac{\\Delta_+(\\lambda)}{1+\\alpha^2}$.\n The proof is complete.\n\\end{proof}\n\\begin{remark}\nIf $\\Delta_+(0)=0$, then the uniqueness may not hold. It was known \\cite{EK,Zo1} that for the problem $L(q,1,0,1,0)$, in the case $\\Delta_+(0)=0$, there exist two even potentials corresponding to the same set of eigenvalues.\n\\end{remark}\n\n\nNow, let us prove the so-called two-spectra theorem. Note that $\\Delta_+(\\lambda)$ and $\\Delta_-(\\lambda)$ have no common non-zero zeros.\n\n\\begin{theorem}\\label{tht}\nIf $\\alpha_0\\ne1$ (or $\\alpha\\ne1$) and is known a priori, then all zeros of $\\Delta_+(\\lambda)$ and all non-zero zeros of $\\Delta_-(\\lambda)$ (including multiplicity) uniquely determine $\\alpha$ ($\\alpha_0$), $\\beta_0$, $\\beta$ and $q(x)$ a.e. on $(0,a)$.\n\\end{theorem}\n\\begin{proof}\nSince $\\alpha_0\\ne1$ (or $\\alpha\\ne1$) and is known a priori, using Theorem \\ref{th1}, we can determine the sign of $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)$. Indeed, if there are only finitely many eigenvalues $\\{\\lambda_n^+\\}$ or infinitely many eigenvalues $\\{\\lambda_n^+\\}$ with unbounded imaginary parts, then $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)=0$. If there are infinitely many eigenvalues $\\{\\lambda_n^+\\}$ with bounded imaginary parts, then $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)\\ne0$. In the latter case, if $|\\sin (a{\\rm Re}\\lambda_n^+)|\\to 1$ as $n\\to\\infty $ then $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)>0$; if $|\\sin (a{\\rm Re} \\lambda_n^+)|\\to 0$ as $n\\to\\infty $ then $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)<0$. In particular, we can determine whether $\\alpha=1$ ($\\alpha_0=1$) or not. If $\\alpha\\neq1$ ($\\alpha_0\\neq1$), then we can recover the term $\\frac{\\alpha+\\alpha_0}{1+\\alpha\\alpha_0}$ from the the asymptotics of the eigenvalues, and so $\\alpha$ ($\\alpha_0$) is uniquely determined. Using Lemma \\ref{hx11} and \\eqref{c16}, we can obtain that the function $\\Delta_+(\\lambda)$ is uniquely determined from its zeros. If $\\alpha=1$, since $\\alpha_0$ is known, by using Lemma \\ref{hx11} and \\eqref{c16}, we can still obtain that the function $\\Delta_+(\\lambda)$ is uniquely determined from its zeros. Using \\eqref{e7}, it is easy to get whether $\\lambda=0$ is a zero of $\\Delta_-(\\lambda)$. If $\\Delta_-(0)=0$, then the multiplicity of $\\lambda=0$ is also uniquely determined by using \\eqref{e7}. Then, together with the condition of this theorem, we know that all zeros of $\\Delta_-(\\lambda)$ are known. Then using Lemma \\ref{hx11} and \\eqref{c16}, we know that $\\Delta_-(\\lambda)$ is uniquely determined by its zeros.\nUsing \\eqref{k9}, \\eqref{e15} and \\eqref{o30}, we have\n \\begin{equation}\\label{o31s}\n {\\Delta_+(\\lambda)}+{\\Delta_{-}(\\lambda)}=y'\\left(\\lambda,a\\right)+\\beta y\\left(\\lambda,a\\right).\n \\end{equation}\n The right-hand side of \\eqref{o31s} is the characteristic function of the problem \\eqref{hx12}, \\eqref{hx14} and $y'(0)-\\beta y(0)=0$.\n Using Theorem 3.1 in \\cite{XP}, we complete the proof.\n\\end{proof}\n\n\\begin{remark}(1) If $q(x)$ is real valued and $\\beta,\\beta_0\\in \\mathbb{R}$, then we can use the signs of imaginary parts of zeros of $\\Delta_-(\\lambda)$ instead of the zeros of $\\Delta_-(\\lambda)$. Indeed, after obtaining $\\Delta_+(\\lambda)$ and $\\alpha\\alpha_0$, the function $g(\\lambda):=\\Delta_-(\\lambda)\\Delta_-(-\\lambda)$ is determined (cf.\\eqref{e7}). Let $\\{\\xi_n\\}$ (counted with multiplicities) be the zeros of $g(\\lambda)$ in $\\overline{\\mathbb{C}_+}:=\\{\\lambda: {\\rm Im}\\lambda\\ge0\\}$. Then using Theorem \\ref{th2x} (2)(ii), we know that $\\xi_n$ or $\\bar{\\xi}_n$ is a zero of $\\Delta_-(\\lambda)$. Define\n\\begin{equation*}\n\\sigma_n={\\rm sgn } ({\\rm Im} \\lambda_n^-)=\\left\\{ \\begin{split}\n +1,\\quad &\\text{if} \\quad \\lambda_n^-= \\xi_n\\in \\mathbb{C}_+, \\\\\n -1,\\quad &\\text{if} \\quad \\lambda_n^-=\\bar{\\xi}_n\\in \\mathbb{C}_-,\\\\\n 0,\\quad &\\text{if} \\quad \\lambda_n^-={\\xi}_n\\in \\mathbb{R}.\n \\end{split}\\right.\n\\end{equation*}\nUsing the set of signs $\\{\\sigma_n\\}$ and the zeros of $g(\\lambda)$ in $\\overline{\\mathbb{C}_+}$, we can uniquely recovered all zeros of $\\Delta_-(\\lambda)$. We admit that this property was first observed by Korotyaev \\cite{EK} in studying the inverse resonance problem $L(q,1,0,1,0)$. By contrast, when $\\alpha\\ne1$ and $\\alpha_0\\ne1$, it is possible that all zeros of $\\Delta_+(\\lambda)$ together with at most a finite number of the signs $\\{\\sigma_n\\}$ uniquely recover all zeros of $\\Delta_-(\\lambda)$. For example, if $\\alpha$ and $\\alpha_0$ are determined such that $P_0^->0$, then from the asymptotics of $\\lambda_n^-$ we know that there exist only finitely many $\\lambda_n^-$s\nin $\\overline{\\mathbb{C}_-}$. So, in this case, there exist at most finitely many elements in $\\{\\xi_n\\}$ belonging to the set of zeros of $\\Delta_-(-\\lambda)$ and the other elements are zeros of $\\Delta_-(\\lambda)$. That is to say, in this case, we only need at most a finite number of signs to distinguish the zeros of $\\Delta_-(-\\lambda)$ and the zeros of $\\Delta_-(\\lambda)$ from all zeros of $g(\\lambda)$ in $\\overline{\\mathbb{C}_+}$, and so all zeros of $\\Delta_-(\\lambda)$ are determined.\n\n\n(2) If the known $\\alpha_0$ is equal to $1$, it is unclear that if $\\alpha$ can be determined. If $\\alpha\\ne1$ and is given, then it is included in Theorem \\ref{tht}. If $\\alpha_0=1=\\alpha$, then $\\Delta_+(\\lambda)$ is still uniquely determined from its zeros by using Lemma \\ref{hx11} and \\eqref{c16}. Whereas, $\\Delta_-(\\lambda)$ is uniquely determined from its zeros provided that there is a non-zero $b_0\\in\\{\\beta,\\beta_0,\\beta+\\beta_0,\\beta-\\beta_0\\}$ and $b_0$ is known. If $\\beta=\\beta_0=0$, i.e., in the case of the resonance problem $L(q,1,0,1,0)$, all zeros of $\\Delta_-(\\lambda)$ cannot uniquely determine $\\Delta_-(\\lambda)$ unless an additional sign is given (see \\cite{EK}).\n\n(3) If $(\\alpha_0-1)(\\alpha-1)\\ne0$ and the sign of either $\\alpha_0-1$ or $\\alpha-1$ is known a priori, then $\\alpha$ and $\\alpha_0$ can all be uniquely recovered from the asymptotics of the eigenvalues $\\{\\lambda_k^\\pm\\}$. Indeed, we can first recover two numbers $P_0^\\pm$ and then it follows from \\eqref{k10} that\n\\begin{equation*}\ne^{P_0^++P_0^-}=\\frac{(\\alpha_0+1)^2}{(\\alpha_0-1)^2},\\quad e^{P_0^+-P_0^-}=\\frac{(\\alpha+1)^2}{(\\alpha-1)^2}.\n\\end{equation*}\nIf the sign of $\\alpha_0-1$ (or $\\alpha-1$) is known, then $\\alpha_0$ ($\\alpha$) is obtained uniquely. Substituting $\\alpha_0$ (or $\\alpha$) into $P_0^+$, we get $\\alpha$ ($\\alpha_0$).\n\\end{remark}\n\n\n\nNext, let us give two uniqueness theorems for the general partial inverse problem, namely, the potential function is known in a subinterval $(b,a)$ with arbitrary $b\\in (0,a)$.\n\n\\begin{lemma}[\\textup{see \\cite[p.173]{B}}]\\label{le1}\n For any entire function $f(z)\\not\\equiv0$ of exponential type, the following inequality holds true:\n \\begin{equation}\n \\mathop {\\varliminf }\\limits_{r \\to \\infty} \\frac{n_f(r)}{r}\\leq\\frac{1}{2\\pi}\\int_0^{2\\pi}h_f(\\theta)d\\theta,\n \\end{equation}\n where $n_f(r)$ denotes the number of zeros of $f(z)$ in the disk $|z|\\le r$, and $h_f(\\theta)$ is the indicator function of $f(z)$ that is defined by\n \\begin{equation*}\n h_f(\\theta):=\\mathop {\\varlimsup }\\limits_{r \\to \\infty }\\frac{\\ln |f(re^{i\\theta})|}{r}.\n \\end{equation*}\n\\end{lemma}\n\nFrom Theorem \\ref{th1}, we see that\n\\begin{equation*}\n n_{\\Delta_\\pm}(r)=\\frac{2tr}{\\pi}(1+o(1)),\\quad r\\to\\infty,\n\\end{equation*}\nwhere $t= a$ if $(\\alpha_0-1)(1-\\alpha)\\ne0$ and $t\\le a$ if $(\\alpha_0-1)(1-\\alpha)=0$.\n Let $\\Omega_\\pm$ be the sets of the zeros of the functions $\\Delta_\\pm(\\lambda)$, respectively.\nDenote $\\Omega_0:=\\Omega_+\\cup\\Omega_-$. Let $ n_{\\Omega_0}(r)$ be the number of the points in $\\Omega_0\\cap \\{\\lambda:|\\lambda|\\le r\\}.$ \n\n\n\n\n\n\\begin{theorem}\\label{th6}\n Assume that $q(x)$ is known a priori a.e. on $\\left(b,a\\right)$ with $b\\in (0,a)$ and $\\alpha$ and $\\beta$ are given. If the subset $\\Omega_0$ satisfies\n \\begin{equation}\n n_{\\Omega_0}(r)=\\frac{2mr}{\\pi}[1+o(1)],\\quad r \\to \\infty,\\quad m>2b,\n \\end{equation}\n then $\\Omega_0$ uniquely determines $\\alpha_{0}$ and $\\beta_{0}$ and $q(x)$ a.e. on $\\left(0,b\\right)$.\n\\end{theorem}\n\\begin{proof}\n Suppose that there are two problems $L(q,\\alpha_0,\\beta_0,\\alpha,\\beta)$ and $L(\\tilde{q},\\tilde{\\alpha}_0,\\tilde{\\beta}_0,\\tilde{\\alpha},\\tilde{\\beta})$, which satisfy that $q(x)=\\tilde{q}(x)$ a.e. on $\\left(b,a\\right)$ and $\\alpha=\\tilde{\\alpha}$ and $\\beta=\\tilde{\\beta}$. Let us prove $q(x)=\\tilde{q}(x)$ on $\\left(0,b\\right)$ and $\\alpha_{0}=\\tilde{\\alpha}_{0}$ and $\\beta_{0}=\\tilde{\\beta}_{0}$ if $\\Omega_0=\\tilde{\\Omega}_0$. Define\n \\begin{equation}\\label{p1}\n F(\\lambda)=y\\left(\\lambda,b\\right)\\tilde{y'}\\left(\\lambda,b\\right)-\\tilde{y}\\left(\\lambda,b\\right)y'\\left(\\lambda,b\\right)\n \\end{equation}\n Combing \\eqref{r2} and \\eqref{k8}, then we have\n \\begin{equation}\\label{p9}\n \\lvert F(\\lambda)\\rvert\\leq C|\\lambda|e^{2b\\lvert {\\rm Im}\\lambda\\rvert}, \\quad \\lvert\\lambda\\rvert\\to\\infty, \\quad C>0.\n \\end{equation}\n Let $\\lambda=re^{i\\theta}$, so $\\lvert {\\rm Im}\\lambda\\rvert=r\\lvert\\sin\\theta\\rvert.$ Then using \\eqref{p9}, we have\n \\begin{equation}\n h_{F}(\\theta):=\\varlimsup\\limits_{r \\to \\infty}\\frac{\\ln|F(re^{i\\theta})|}{r} \\le 2b|\\sin \\theta|,\n \\end{equation}\n which implies\n \\begin{equation}\n \\int_{0}^{2\\pi}h_{F}(\\theta)d\\theta \\le\n 2b\\int_{0}^{2\\pi}|\\sin \\theta|d\\theta =\n 4b\\int_{0}^{\\pi}\\sin \\theta d\\theta = 8b.\n \\end{equation}\n Since $q(x)=\\tilde{q}(x)$ a.e. on $\\left(b,a\\right)$ and $\\alpha=\\tilde{\\alpha}$ and $\\beta=\\tilde{\\beta}$, then $\\phi(\\lambda,b)=\\tilde{\\phi}(\\lambda,b)$ for all $\\lambda\\in \\mathbb{C}$.\n\n\n Note that $\\phi(\\lambda,x)$ meets the conditions $\\phi(\\lambda,a)=1$ and $\\phi'(\\lambda,a)=-(i\\alpha\\lambda+\\beta)$, it is easy to get the function $\\phi(-\\lambda,x)$ to satisfy the conditions $\\phi(-\\lambda,a)=1$ and $\\phi'(-\\lambda,a)=(i\\alpha\\lambda-\\beta)$. Then we can rewrite \\eqref{e15} as\n\\begin{equation}\\label{o64}\n \\Delta_{-}(\\lambda)=\\left<\\phi(-\\lambda,x),y(\\lambda,x)\\right>.\n\\end{equation}\n Letting $x=b$ in \\eqref{e10} and \\eqref{o64}, we get\n \\begin{align}\\label{w1}\n \\Delta_\\pm\\left(\\lambda\\right)=y'\\left(\\lambda,b\\right)\\phi\\left(\\pm\\lambda,b\\right)-\\phi'\\left(\\pm\\lambda,b\\right)y\\left(\\lambda,b\\right),\n \\end{align}\n \\begin{equation}\\label{w2}\n \\tilde{\\Delta}_\\pm\\left(\\lambda\\right)=\\tilde{y}'\\left(\\lambda,b\\right)\\phi\\left(\\pm\\lambda,b\\right)-\\phi'\\left(\\pm\\lambda,b\\right)\\tilde{y}\\left(\\lambda,b\\right).\n \\end{equation}\n Multiply \\eqref{w1} and \\eqref{w2} by $\\tilde{y}'\\left(\\lambda,b\\right)$ and $y'\\left(\\lambda,b\\right)$, respectively, take the difference and get\n \\begin{equation}\\label{w3}\n F\\left(\\lambda\\right)=\\frac{y'\\left(\\lambda,b\\right)\\tilde{\\Delta}_\\pm \\left(\\lambda\\right)-\\tilde{y}'\\left(\\lambda,b\\right)\\Delta_\\pm \\left(\\lambda\\right)}{\\phi'\\left(\\pm \\lambda,b\\right)}.\n \\end{equation}\n Multiply \\eqref{w1} and \\eqref{w2} by $\\tilde{y}\\left(\\lambda,b\\right)$ and $y\\left(\\lambda,b\\right)$, respectively, take the difference and get\n \\begin{equation}\\label{w4}\n F\\left(\\lambda\\right)=\\frac{y\\left(\\lambda,b\\right)\\tilde{\\Delta}_\\pm \\left(\\lambda\\right)-\\tilde{y}\\left(\\lambda,b\\right)\\Delta_\\pm \\left(\\lambda\\right)}{\\phi\\left(\\pm \\lambda,b\\right)}.\n \\end{equation}\n Considering \\eqref{w3} and \\eqref{w4}, as $\\phi\\left(\\pm \\lambda,b\\right)$ and $\\phi'\\left(\\pm\\lambda,b\\right)$ cannot vanish simultaneously, then we conclude that all common zeros (including multiplicity) of $\\Delta_\\pm\\left(\\lambda\\right)$ and $\\tilde{\\Delta}_\\pm\\left(\\lambda\\right)$ are zeros of $F\\left(\\lambda\\right)$.\n Thus, we have\n \\begin{equation}\n \\varliminf\\limits_{r \\to \\infty}\\frac{n_{F}(r)}{r} \\ge\n \\lim_{r \\to \\infty}\\frac{n_{\\Omega_0}(r)}{r} =\n \\frac{2m}{\\pi}.\n \\end{equation}\n Using Lemma \\ref{le1}, if the entire function $F(\\lambda)\\not\\equiv0$, then\n \\begin{equation}\n \\frac{2m}{\\pi}\\le \\varliminf\\limits_{r \\to \\infty}\\frac{n_{F}}{r} \\le \\frac{1}{2\\pi}\\int_0^{2\\pi}h_{F}(\\theta)d\\theta \\le\n \\frac{4b}{\\pi},\n \\end{equation}\n which implies $m\\leq 2b$. This contradicts $m>2b$. Hence, $F(\\lambda)\\equiv0$, namely, $\\frac{y'\\left(\\lambda,b\\right)}{y\\left(\\lambda,b\\right)}=\\frac{\\tilde{y}'\\left(\\lambda,b\\right)}{\\tilde{y}\\left(\\lambda,b\\right)}$. Since $y'\\left(\\lambda,b\\right)$ and $y\\left(\\lambda,b\\right)$ do not have common zeros, the zeros of ${y\\left(\\lambda,b\\right)}$ are uniquely determined. Using Theorem 3.2 in \\cite{X}, we obtain that $q(x)=\\tilde{q}(x)$ a.e. on $\\left(0,b\\right)$, $\\alpha_{0}=\\tilde{\\alpha}_{0}$ and $\\beta_{0}=\\tilde{\\beta}_{0}$. The proof is complete.\n\\end{proof}\n\n\n\\begin{corollary}\\label{th7}\n If $q(x)$ is known a priori a.e. on $\\left(b,a\\right)$ with $b<\\frac{a}{2}$ and $\\alpha$ and $\\beta$ are known, then the subset $\\Omega:=\\Omega_+$ (or $\\Omega:=\\Omega_-$) satisfying\n \\begin{equation}\n n_{\\Omega}(r)=\\frac{2mr}{\\pi}[1+o(1)],\\quad r \\to \\infty,\\quad 2b<m\\le a,\n \\end{equation}\n uniquely determines $\\alpha_{0}$, $\\beta_{0}$ and $q(x)$ a.e. on $\\left(0,b\\right)$.\n\\end{corollary}\n\n\nIn the above theorem, it needs $m$ to be greater than $2b$. In the following theorem, we consider the critical case $m=2b$. In this case, we assume $(\\alpha_0-1)(1- \\alpha)\\ne0$.\nDenote\n\\begin{equation}\\label{xk1}\n g_\\pm (\\lambda):=\\lambda[i\\left(\\alpha_{0}\\pm \\alpha\\right)\\cos{\\lambda a}-\\left(1\\pm\\alpha\\alpha_{0}\\right)\\sin{\\lambda a}].\n\\end{equation}\nLet $\\{\\mu_k^\\pm\\}$ be zeros of $g_\\pm(\\lambda)$, respectively. It is obvious that\n\\begin{equation}\\label{o28}\n \\begin{cases}& \\!\\!\\!\\!\\mu_{0}^\\pm=0 ,\\;\n \\mu_{k}^\\pm=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-\\frac{1}{2}\\right){\\rm sgn}k+\\frac{iP_0^\\pm}{2a},\\; k\\in\\mathbb{Z}_0,\\ \\text{if}\\ \\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)>0, \\\\\n & \\!\\! \\!\\!\\!\\! \\mu_{-1}^\\pm=0,\\;\\mu_{k}^\\pm=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-1\\right){\\rm sgn}k+\\frac{iP_0^\\pm}{2a}, k\\in\\mathbb{Z}_{0}\\setminus\\{-1\\} ,\\ \\text{if}\\ \\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)<0.\n \\end{cases}\n\\end{equation}\nFrom Theorem \\ref{th1}, we see that\n\\begin{equation*}\n \\lambda_k^\\pm =\\mu_k^\\pm +O(1/k),\\quad k\\to\\infty.\n\\end{equation*}\n\\begin{lemma}[\\textup{\\cite[Proposition B.6]{FB}}]\\label{lem2}\n Assume that $E_0(\\rho)$ is an entire function of order less than one. If $\\lim_{\\lvert t\\rvert\\to\\infty, t\\in\\mathbb{R}}E_0(it)=0$, then $E_0(\\rho)\\equiv0$ on the whole complex plane.\n\\end{lemma}\n\n\\begin{theorem}\\label{thxx}\n Assume that $(\\alpha_0-1)(1-\\alpha)\\ne0$, and $q(x)$ is known a priori a.e. on $(b,a)$ with $b\\in (0,a)$ and $\\alpha$ and $\\beta$ are known. If the subset $\\Omega_{1}:=\\{\\lambda_{k_{j}}^+\\}_{j\\in \\mathbb{Z}_1}\\cup \\{\\lambda_{k_{j}}^-\\}_{j\\in \\mathbb{Z}_1}$ satisfies\n \\begin{equation}\\label{i1}\n \\sum_{j\\in \\mathbb{Z}_1}\\frac{\\lvert\\lambda_{k_{j}}^+-a\\mu_{j}^+/b_+\\rvert}{\\lvert j\\rvert+1}+\\sum_{j\\in \\mathbb{Z}_1}\\frac{\\lvert\\lambda_{k_{j}}^--a\\mu_{j}^-/b_-\\rvert}{\\lvert j\\rvert+1}<\\infty,\\quad b_++b_-=2b,\n \\end{equation}\n where $\\mathbb{Z}_1=\\mathbb{Z}_0$ if $(\\alpha_0-1)(1- \\alpha)<0$ and $\\mathbb{Z}_1=\\mathbb{Z}$ if $(\\alpha_0-1)(1- \\alpha)>0$, then $\\Omega_{1}$ uniquely determines $\\alpha_{0},\\beta_{0}$ and $q(x)$ a.e. on $(0,b)$.\n\\end{theorem}\n\\begin{proof}\n We only deal with the case $(\\alpha_0-1)(1- \\alpha)>0$, i.e., $\\mathbb{Z}_1=\\mathbb{Z}$. Let $\\rho=\\lambda^{2}$. Define\n \\begin{equation}\\label{p2}\n G(\\rho)=F(\\lambda)F(-\\lambda),\\quad\n \\Phi(\\rho)=\\prod_{j\\in \\mathbb{Z}}\\left(1-\\frac{\\rho}{(\\lambda_{k_{j}}^+)^{2}}\\right)\\left(1-\\frac{\\rho}{(\\lambda_{k_{j}}^-)^{2}}\\right),\n \\quad\n E_0(\\rho)=\\frac{G(\\rho)}{\\Phi(\\rho)},\n \\end{equation}\n where $F$ is defined in \\eqref{p1}. Since $F(\\lambda)$ is an entire function of order $\\le1$ and $F(\\lambda)F(-\\lambda)$ is an even function of $\\lambda$, then $G(\\rho)$ is an entire function of $\\rho $ of order $\\le\\frac{1}{2}$. Let us show that $E_0(\\rho)$ is also an entire function of $\\rho$ of order $\\le\\frac{1}{2}$. Since all zeros of $\\Phi(\\rho) $ are zeros of $G(\\rho)$, it is enough to show that $\\Phi(\\rho) $ is an entire function of $\\rho $ of order $\\le\\frac{1}{2}$.\n By virtue of \\eqref{k11} and \\eqref{k12}, we have\n \\begin{equation}\\label{4.26}\n \\frac{1}{(\\lambda_{k_{j}}^\\pm)^{2}}=O\\left(\\frac{1}{j^2}\\right), \\quad j\\to\\infty.\n \\end{equation}\n It follows that the series\n \\begin{equation}\n \\notag\n \\sum_{j\\in \\mathbb{Z}}\\left(\\left\\lvert \\frac{\\rho}{(\\lambda_{k_{j}}^+)^{2}} \\right\\rvert+\\left\\lvert \\frac{\\rho}{(\\lambda_{k_{j}}^-)^{2}}\\right\\rvert\\right)\n \\end{equation}\n converges uniformly on bounded subsets of $\\mathbb{C}$. Consequently, the infinite product $\\Phi(\\rho)$ in \\eqref{p2} converges to an entire function of $\\rho$, with its roots being exactly $(\\lambda_{k_{j}}^+)^{2}$ and $(\\lambda_{k_{j}}^-)^{2}$, $j\\in \\mathbb{Z}$. Denote\n \\begin{equation}\n \\Xi_\\Phi:=\\inf \\left\\{p: \\sum_{j\\in \\mathbb{Z}}\\left(\\frac{1}{\\left \\lvert(\\lambda_{k_{j}}^+)^{2p}\\right\\rvert} +\\frac{1}{\\left\\lvert (\\lambda_{k_{j}}^-)^{2p}\\right\\rvert}\\right)<\\infty\\right\\},\n \\end{equation}\n which is called \\emph{convergence exponent of zeros} of the canonical product of $\\Phi(\\rho)$ in \\eqref{p2}. Obviously, the estimate \\eqref{4.26} implies $\\Xi_\\Phi\\le\\frac{1}{2}$. Note that the order of canonical product of an entire function coincides with its convergence exponent of zeros (see \\cite[p.16]{B}). It follows that the order of canonical\nproduct of $\\Phi(\\rho)$\n is less than or equal to $1/2$. Using Hadamard's factorization theorem, we know that the infinite product in \\eqref{p2} is\nthe canonical product of the function\n$\\Phi(\\rho)$, and so the order of $\\Phi(\\rho)$\n is at most $1/2$.\n\n Using the Lemma \\ref{lem2} and the end part of proof of Theorem \\ref{th6}, to prove Theorem \\ref{thxx}, it is enough to prove\n \\begin{equation}\n \\lim_{\\lvert t\\rvert\\to\\infty,t\\in\\mathbb{R}}E_0(it)=0.\n \\end{equation}\n Define\n \\begin{equation}\n \\Phi_{0}\\left(\\rho\\right)=g_+\\left(\\frac{b_+\\lambda}{a}\\right)g_+\\left(-\\frac{b_+\\lambda}{a}\\right)g_-\\left(\\frac{b_-\\lambda}{a}\\right)\n g_-\\left(-\\frac{b_-\\lambda}{a}\\right),\n \\end{equation}\n where the functions $g_\\pm$ are defined in \\eqref{xk1}.\n Using the condition $d_++d_-=2b$, we have that for large $|t|$, there holds\n \\begin{equation}\\label{xk4}\n \\lvert\\Phi_{0}\\left(it\\right)\\rvert\\geq C\\lvert t\\rvert^{2}e^{4b\\lvert {\\rm Im} \\sqrt{t}\\rvert},\\quad C>0.\n \\end{equation}\n Using the Hadamard's factorization theorem, we can also have\n \\begin{equation}\n \\Phi_{0}\\left(\\rho\\right)=C_1\\rho^2\\prod_{j\\in \\mathbb{Z}_0}\\left(1-\\frac{\\rho}{(\\zeta_{j}^+)^{2}}\\right)\\left(1-\\frac{\\rho}{(\\zeta_{j}^-)^{2}}\\right),\n \\end{equation}\n where $\\zeta_{j}^\\pm=\\frac{a}{b_\\pm}\\mu_{j}^\\pm$ and $C_1$ is a constant. Then\n \\begin{align*}\n \\left |\\frac{\\Phi_{0}\\left(it\\right)}{\\Phi\\left(it\\right)}\\right|\\notag&\\le C\\prod_{j\\in \\mathbb{Z}_0}\\frac{\\left(1-\\frac{it}{(\\zeta_{j}^+)^{2}}\\right)\\left(1-\\frac{it}{(\\zeta_{j}^-)^{2}}\\right)}{\\left(1-\\frac{it}{(\\lambda_{k_{j}}^+)^{2}}\\right)\\left(1-\\frac{it}{(\\lambda_{k_{j}}^-)^{2}}\\right)}\\\\\n &\\leq C\\!\\!\\prod_{j\\in \\mathbb{Z}_0}\\!\\!\\left[1+\\left|\\frac{(\\zeta_{j}^+)^{2}-(\\lambda_{k_{j}}^+)^{2}}{(\\lambda_{k_{j}}^+)^{2}-it}\\right|\\right]\\!\\!\\left[1+\\left|\\frac{(\\zeta_{j}^-)^{2}-(\\lambda_{k_{j}}^-)^{2}}{(\\lambda_{k_{j}}^-)^{2}-it}\\right|\\right]\\prod_{j\\in \\mathbb{Z}_0}\\left|\\frac{(\\lambda_{k_{j}}^+)^{2}}{(\\zeta_{j}^+)^{2}}\\right|\\left|\\frac{(\\lambda_{k_{j}}^-)^{2}}{(\\zeta_{j}^-)^{2}}\\right|,\n \\end{align*}\n which implies from \\eqref{i1} that\n\\begin{equation}\\label{xk6}\n \\left |\\frac{\\Phi_{0}\\left(it\\right)}{\\Phi\\left(it\\right)}\\right|=O(1), \\quad\\lvert t\\rvert\\to \\infty.\n\\end{equation}\nUsing \\eqref{p9}, \\eqref{p2}, \\eqref{xk4} and \\eqref{xk6}, we get\n\\begin{equation}\n \\lvert E_0\\left(it\\right)\\rvert=\\left|\\frac{G(it)\\Phi_{0}(it)}{\\Phi_{0}(it)\\Phi(it)}\\right|=O\\left(\\frac{1}{\\lvert t\\rvert}\\right), \\quad\\lvert t\\rvert\\to \\infty.\n\\end{equation}.\\\\\nThe proof is complete.\n\\end{proof}\n\\begin{corollary}\\label{cor2}\n Assume that $(\\alpha_0-1)(1-\\alpha)\\ne0$. If $q(x)$ is known a priori a.e. on $(b,a)$ with $b<\\frac{a}{2}$, and $\\alpha$,$\\beta$ are known, then the subset $\\{\\lambda_{k_{j}}^+\\}_{j\\in \\mathbb{Z}_1}$ (or $ \\{\\lambda_{k_{j}}^-\\}_{j\\in \\mathbb{Z}_1}$) satisfying\n \\begin{equation*}\n \\sum_{j\\in \\mathbb{Z}_1}\\frac{\\lvert\\lambda_{k_{j}}^+-\\frac{a\\mu_{j}^+}{2b}\\rvert}{\\lvert j\\rvert+1}<\\infty,\\quad \\left(\\text{or}\\quad \\sum_{j\\in \\mathbb{Z}_1}\\frac{\\lvert\\lambda_{k_{j}}^--\\frac{a\\mu_{j}^-}{2b}\\rvert}{\\lvert j\\rvert+1}<\\infty,\\right)\n \\end{equation*}\n uniquely determines $\\alpha_{0}$, $\\beta_{0}$ and $q(x)$ a.e. on $\\left(0,b\\right)$.\n\\end{corollary}\n\n\n\\noindent\\textbf{\\large Appendix}\n\\\\[-2mm]\n\n\\appendix\n\\setcounter{equation}{0}\n\\renewcommand\\theequation{A.\\arabic{equation}}\n\\setcounter{lemma}{0}\n\\renewcommand\\thelemma{A.\\arabic{lemma}}\n\nIn Appendix, we give the asymptotics of zeros of the entire function\n \\begin{equation}\\label{A.7}\n \\varphi(\\lambda)=\\phi^{(1)}(\\lambda)+M \\sin{\\lambda a}-iN \\cos{\\lambda a}+\\psi(\\lambda),\n \\end{equation}\nwhere $M,\\ N\\in\\mathbb{C}$, $\\psi$ is an entire function of exponential type $\\leq a$ and belongs to $L^2(-\\infty,\\infty)$ and\n \\begin{equation}\\label{A.1}\n \\phi^{(1)}(\\lambda)=\\lambda(\\sigma_1\\cos{\\lambda a}+i\\sigma_2\\sin{\\lambda a}),\n \\end{equation}\nwith $\\sigma_1,\\sigma_2\\in \\mathbb{R}$. In \\cite[p.178]{MV}, the asymptotics of zeros of $ \\varphi(\\lambda)$ is given for the condition $\\sigma_1\\ne\\sigma_2$ and $\\sigma_1,\\sigma_2\\ge0$. Indeed, this condition can be weaker. Let us first prove the following lemma.\n\n\\begin{lemma}\\label{LA1}\n Let $\\sigma_1$, $\\sigma_2$ be real numbers with $\\sigma_1\\ne\\pm\\sigma_2$. Then the zeros of the entire function $\\phi^{(1)}(\\lambda)$ defined in \\eqref{A.1}\n have the following asymptotic representations.\\\\\n $\\left(1\\right)$ If $\\left(\\sigma_2-\\sigma_1\\right)\\left(\\sigma_2+\\sigma_1\\right)<0$, then the zeros $\\{\\lambda_{k}^{(1)}\\}_{k\\in\\mathbb{Z}}$ of $\\phi^{(1)}(\\lambda)$ are $\\lambda_{0}^{(1)}=0$,\n \\begin{equation}\\label{XA1}\n \\begin{aligned}\n \\lambda_{k}^{(1)}=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-\\frac{1}{2}\\right){\\rm sgn} k+\\frac{i}{2a}\\ln{\\left(\\frac{|\\sigma_2+\\sigma_1|}{|\\sigma_2-\\sigma_1|}\\right)}, \\quad k\\in\\mathbb{Z}_0,\\quad \\mathbb{Z}_0=\\mathbb{Z}\\setminus\\{0\\},\n \\end{aligned}\n \\end{equation}\n $\\left(2\\right)$ If $\\left(\\sigma_2-\\sigma_1\\right)\\left(\\sigma_2+\\sigma_1\\right)>0$, then the zeros $\\{\\lambda_{k}^{(1)}\\}_{k\\in \\mathbb{Z}_0}$ of $\\phi^{(1)}(\\lambda)$ are $\\lambda_{-1}^{(1)}=0$:\n \\begin{equation}\\label{XA2}\n \\begin{aligned}\n \\lambda_{k}^{(1)}=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-1\\right){\\rm sgn} k+\\frac{i}{2a}\\ln{\\left(\\frac{|\\sigma_2+\\sigma_1|}{|\\sigma_2-\\sigma_1|}\\right)}, \\quad k\\in \\mathbb{Z}_0\\setminus\\{-1\\}.\n \\end{aligned}\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\nNote that\n\\begin{equation}\\label{A7}\n \\cos{\\lambda a}=\\frac{e^{i\\lambda a}+e^{-i\\lambda a}}{2}, \\quad \\sin{\\lambda a}=\\frac{e^{i\\lambda a}-e^{-i\\lambda a}}{2i}.\n\\end{equation}\nThen $\\frac{1}{\\lambda}\\phi^{(1)}(\\lambda)=0$ is equivalent to\n\\begin{equation}\\label{A.5}\n \\left(\\sigma_2+\\sigma_1\\right)e^{i\\lambda a}=\\left(\\sigma_2-\\sigma_1\\right)e^{-i\\lambda a}.\n\\end{equation}\nSince $\\sigma_1\\ne\\pm \\sigma_2$, it follows that\n\\begin{equation}\\label{A11}\n e^{2i\\lambda a}=\\frac{\\sigma_2-\\sigma_1}{\\sigma_2+\\sigma_1}.\n\\end{equation}\nTaking the logarithm on both sides of \\eqref{A11}, we have\n\\begin{equation*}\n 2i\\lambda_k^{(1)} a=\\ln{\\left(\\frac{\\sigma_2-\\sigma_1}{\\sigma_2+\\sigma_1}\\right)}+2k\\pi i,\\quad k\\in \\mathbb{Z}.\n\\end{equation*}\nNote that $\\ln z=\\ln |z|+i\\arg z$ with $\\arg z\\in (-\\pi,\\pi]$. It follows that\n\\begin{equation}\n \\lambda_{k}^{(1)}=\\begin{cases}& \\!\\!\\!\\!\n \\frac{k\\pi}{a}+\\frac{i}{2a}\\ln{\\left(\\frac{\\sigma_2+\\sigma_1}{\\sigma_2-\\sigma_1}\\right)},\\; k\\in\\mathbb{Z},\\ \\text{if}\\ \\frac{\\sigma_2+\\sigma_1}{\\sigma_2-\\sigma_1}>0, \\\\\n & \\!\\! \\!\\!\\!\\! \\frac{\\pi}{a}\\left(k-\\frac{1}{2}\\right)+\\frac{i}{2a}\\ln{\\left(\\frac{|\\sigma_2+\\sigma_1|}{|\\sigma_2-\\sigma_1|}\\right)},\\; k\\in\\mathbb{Z},\\ \\text{if}\\ \\frac{\\sigma_2+\\sigma_1}{\\sigma_2-\\sigma_1}<0,\n \\end{cases}\n\\end{equation}\nwhich is equivalent to \\eqref{XA1} and \\eqref{XA2}. The proof is complete.\n\\end{proof}\nUsing Lemma \\ref{LA1} and a similar argument to the proof of Lemma 7.1.3 in \\cite{MV}, one can get the following lemma.\n\\begin{lemma}\\label{LA2}\nLet $\\sigma_1$ and $\\sigma_2$ be real numbers with $\\sigma_1\\ne\\pm\\sigma_2$, and the entire function $ \\varphi(\\lambda)$ is defined in \\eqref{A.7}.\\\\\n$\\left(1\\right)$ If $\\left(\\sigma_2-\\sigma_1\\right)\\left(\\sigma_2+\\sigma_1\\right)<0$, then the zeros $\\{\\lambda_{k}\\}_{k\\in\\mathbb{Z}}$ of $\\varphi(\\lambda)$ have the following asymptotic behavior:\n \\begin{equation}\n \\begin{aligned}\n \\lambda_{k}=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-\\frac{1}{2}\\right){\\rm sgn} k+\\frac{i}{2a}\\ln{\\left(\\frac{|\\sigma_2+\\sigma_1|}{|\\sigma_2-\\sigma_1|}\\right)}+\\frac{P}{k}+\\frac{\\gamma_k}{k}, \\quad \\lvert k\\rvert\\to\\infty,\n \\end{aligned}\n \\end{equation}\n where $\\{\\gamma_{k}\\}_{k=-\\infty,k\\neq0}^{\\infty}\\in l_{2}$ and\n \\begin{equation}\\label{A.9}\n P=\\frac{\\sigma_2 N-\\sigma_1 M}{\\pi(\\sigma_2^2-\\sigma_1^2)}.\n \\end{equation}\n $\\left(2\\right)$ If $\\left(\\sigma_2-\\sigma_1\\right)\\left(\\sigma_2+\\sigma_1\\right)>0$, then the zeros $\\{\\lambda_{k}\\}_{k\\in\\mathbb{Z}_0}$ of $\\varphi(\\lambda)$ have the following asymptotic behavior:\n \\begin{equation}\n \\begin{aligned}\n \\lambda_{k}=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-1\\right){\\rm sgn} k+\\frac{i}{2a}\\ln{\\left(\\frac{|\\sigma_2+\\sigma_1|}{|\\sigma_2-\\sigma_1|}\\right)}+\\frac{P}{k}+\\frac{\\gamma_k}{k}, \\quad \\lvert k\\rvert\\to\\infty.\n \\end{aligned}\n \\end{equation}\n where $P$ is given \\eqref{A.9} and $\\{\\gamma_{k}\\}_{k=-\\infty,k\\neq0}^{\\infty}\\in l_{2}$.\n\\end{lemma}\n\\begin{proof}\nThe proof is similar to the proof of Lemma 7.1.3 in \\cite{MV}. But for the convenience of readers, we just give the main idea of the proof.\n\nDenote $G_\\delta:=\\{\\lambda\\in\\mathbb{C}: \\lvert \\lambda-\\lambda_{k}^{(1)}\\rvert\\geq\\delta>0, k\\in\\mathbb{Z}_0\\}$.\nClearly, there exist positive constants $C_\\delta$ and $K$ such that\n\\begin{equation}\\label{A10}\n \\lvert\\phi^{(1)}(\\lambda)\\rvert\\geq C_\\delta\\lvert \\lambda \\rvert e^{a\\lvert{\\rm Im}\\lambda \\rvert}, \\quad \\lambda \\in G_\\delta,\\quad \\lvert \\lambda \\rvert\\geq \\lambda^* ,\n\\end{equation}\nfor sufficiently large $\\lambda^*=\\lambda^*(\\delta)$, and\n\\begin{equation}\\label{A.12}\n \\lvert \\varphi(\\lambda)-\\phi^{(1)}(\\lambda)\\rvert\\le Ke^{a\\lvert{\\rm Im}\\lambda \\rvert}.\n\\end{equation}\nIt follows that\n\\begin{equation}\n \\lvert\\phi^{(1)}(\\lambda)\\rvert>\\lvert \\varphi(\\lambda)-\\phi^{(1)}(\\lambda)\\rvert,\\quad \\lambda\\in G_\\delta\\cap \\{\\lambda:|\\lambda|\\ge \\lambda^*\\},\n\\end{equation}\n for sufficiently large $\\lambda^*$. Therefore, there exist sufficient large numbers $R_k$ and sufficiently small numbers $\\delta_k>0$ such that\n \\begin{equation}\n \\varphi(\\lambda)\\ne0, \\quad \\lvert \\phi^{(1)}(\\lambda)\\rvert>\\lvert \\varphi(\\lambda)-\\phi^{(1)}(\\lambda)\\rvert,\\quad \\lambda\\in\\Gamma_{R_k}\\cup \\gamma_{\\delta_k},\n\\end{equation}\nwhere\n\\begin{equation}\n \\Gamma_{R_k}:=\\{\\lambda\\in\\mathbb{C}: \\lvert \\lambda \\rvert=R_k\\},\\quad \\gamma_{\\delta_k}:=\\{\\lambda\\in\\mathbb{C}: \\lvert \\lambda-\\lambda_{k}^{(1)}\\rvert=\\delta_k\\}.\n\\end{equation}\nUsing Rouch\\`{e}'s theorem, we conclude that $\\varphi(\\lambda)$ and $\\phi^{(1)}(\\lambda)$ have the same number of zeros, counted with multiplicity, inside $\\Gamma_{R_k}$ and $\\gamma_{\\delta_k}$. In particular, in $\\gamma_{\\delta_k}$ there is exactly one zero of $\\varphi(\\lambda)$. It follows\nthat\n\\begin{equation}\\label{A12}\n \\lambda_k=\\lambda_{k}^{(1)}+\\varepsilon_k,\\quad \\varepsilon_k=o(1),\\quad |k|\\to\\infty.\n\\end{equation}\nSubstituting \\eqref{A12} into \\eqref{A.7}, we get for large $|k|$ that\n \\begin{align}\\label{XC0}\n \\notag\n 0=&\\varphi(\\lambda_k)=\\phi^{(1)}(\\lambda_{k}^{(1)}+\\varepsilon_k)+M \\sin{(\\lambda_{k}^{(1)}+\\varepsilon_k) a}-iN \\cos{(\\lambda_{k}^{(1)}+\\varepsilon_k) a}+\\psi(\\lambda_k)\\\\\n \\notag=& (\\sigma_1\\lambda_{k}-iN)\\cos{(\\lambda_{k}^{(1)}+\\varepsilon_k) a}+(i\\sigma_2\\lambda_{k}+M)\\sin{(\\lambda_{k}^{(1)}+\\varepsilon_k) a}+\\psi(\\lambda_k)\\\\\n \\notag=&(\\sigma_1\\lambda_{k}-iN)\\left[\\cos{(\\lambda_{k}^{(1)} a)}\\cos{(\\varepsilon_k a)}-\\sin{(\\lambda_{k}^{(1)} a)}\\sin{(\\varepsilon_k a)}\\right]\\\\\n \\notag &+(i\\sigma_2\\lambda_{k}+M)\\left[\\sin{(\\lambda_{k}^{(1)} a)}\\cos{(\\varepsilon_k a)}+\\cos{(\\lambda_{k}^{(1)} a)}\\sin{(\\varepsilon_k a)}\\right]+\\psi(\\lambda_k).\\\\\n \\notag=&\\psi(\\lambda_k)+\\frac{\\lambda_k}{\\lambda_k^{(1)}}\\phi^{(1)}(\\lambda_k^{(1)})\\cos (\\varepsilon_ka)+\\lambda_k\\left[-\\sigma_1\\sin{(\\lambda_{k}^{(1)} a)}+i\\sigma_2\\cos{(\\lambda_{k}^{(1)} a)}\\right]\\sin{(\\varepsilon_k a)}\\\\\n &+\\left[M\\sin{(\\lambda_{k}^{(1)} a)}-iN\\cos{(\\lambda_{k}^{(1)} a)}\\right]\\cos{(\\varepsilon_k a)}+\\left[M\\cos{(\\lambda_{k}^{(1)} a)}+iN\\sin{(\\lambda_{k}^{(1)} a)}\\right]\\sin{(\\varepsilon_k a)}.\n \\end{align}\n Using \\eqref{A11} and \\eqref{A7}, we have for large $|k|$ that\n \\begin{equation}\\label{XC}\n -\\sigma_1\\sin{(\\lambda_{k}^{(1)} a)}+i\\sigma_2\\cos{(\\lambda_{k}^{(1)} a)}=ie^{-i\\lambda_k^{(1)}a}(\\sigma_2-\\sigma_1),\n \\end{equation}\n \\begin{equation}\\label{XC1}\n M\\sin{(\\lambda_{k}^{(1)} a)}-iN\\cos{(\\lambda_{k}^{(1)} a)}=ie^{-i\\lambda_k^{(1)}a}\\frac{M\\sigma_1-N\\sigma_2}{\\sigma_2+\\sigma_1},\n \\end{equation}\n \\begin{equation}\\label{XC2}\n M\\cos{(\\lambda_{k}^{(1)} a)}+iN\\sin{(\\lambda_{k}^{(1)} a)}=e^{-i\\lambda_k^{(1)}a}\\frac{M\\sigma_2-N\\sigma_1}{\\sigma_2+\\sigma_1}.\n \\end{equation}\nSubstituting \\eqref{XC}--\\eqref{XC2} into \\eqref{XC0}, and noting $ \\phi^{(1)}(\\lambda_{k}^{(1)})=0$, we have\n\\begin{equation}\\label{XC3}\n \\sin{(\\varepsilon_k a)}=\\frac{i\\left(N\\sigma_2-M\\sigma_1\\right)\\cos(\\varepsilon_ka)-\\psi(\\lambda_k)(\\sigma_2+\\sigma_1)e^{i\\lambda_k^{(1)}a}}{i\\lambda_k(\\sigma_2^2-\\sigma_1^2)+\\left(M\\sigma_2-N\\sigma_1\\right)} .\n\\end{equation}\nNote that\n\\begin{equation}\\label{XC4}\n \\sin{\\varepsilon_k a}=\\varepsilon_k a+O(\\lvert \\varepsilon_k\\rvert^{-3}), \\ \\cos{\\varepsilon_k a}=1+O(\\lvert \\varepsilon_k\\rvert^{-2}), \\ \\lambda_k=\\frac{k\\pi }{a}[1+O(1/k)], \\ |k|\\to \\infty.\n\\end{equation}\nSubstituting \\eqref{XC4} into \\eqref{XC3}, and noting $\\{\\psi(\\lambda_k)\\}\\in l_2$, we get\n\\begin{equation}\n \\varepsilon_k=\\frac{P}{k}+\\frac{\\gamma_k}{k}.\n\\end{equation}\nThe proof is complete.\n\\end{proof}\n\n\n\n\n\\begin{thebibliography}{99}\n\\bibitem{BN1} N.P. Bondarenko, Inverse Sturm-Liouville problem with analytical functions in the boundary condition, \\emph{Open Mathematics} \\textbf{18} (2020), 512-528.\n\\bibitem{BN2} N.P. Bondarenko, Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the\nboundary condition. \\emph{Math. Methods Appl. Sci.} \\textbf{43} (2020) 7009-7021.\n\\bibitem{BP} N.P. Bondarenko, Partial inverse Sturm-Liouville problems. \\emph{Mathematics} \\textbf{11} (2023), 2408. (44pp).\n\\bibitem{BY} S. A. Buterin, V. A. Yurko Inverse spectral problem for pencils of differential operators on a finite interval, \\emph{Vestnik Bashkirskogo Universiteta} \\textbf{4} (2006), 8-12.\n\\bibitem{BS} S. A. Buterin, On half inverse problem for differential pencils with the spectral parameter in boundary conditions, \\emph{Tamkang Journal of Mathematics} \\textbf{42} (2011), 355-364.\n\n\\bibitem{DGS} R. del Rio, F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential, III. Updating boundary conditions, \\emph{Internat. Math. Res. Notices} \\textbf{15} (1997), 751-758.\n\\bibitem{GV} G. Freiling, V. Yurko, Inverse Sturm-Liouville Problems and their Applications, NOVA Science Publishers, New York, 2001.\n\\bibitem{FY0} G. Freiling, V. Yurko, Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, \\emph{Inverse Problems} \\textbf{26} (2010), 055003 (17pp).\n\\bibitem{FB} F. Gesztesy, B. Simon, Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum, \\emph{Trans. Amer. Math. Soc.} \\emph{352} (2000), 2765-2787.\n\\bibitem{G} N. J. Guliyev, On two-spectra inverse problems. \\emph{Proc. Amer. Math. Soc.}, \\textbf{148} (2020), 4491-4502.\n\\bibitem{G1} N. J. Guliyev, Essentially isospectral transformations and their applications. \\emph{Ann. Mat. Pura Appl.} \\textbf{199} (2020), 1621-1648.\n\\bibitem{HB} H. Hochstadt, B. Lieberman, An inverse Sturm-Liouville problem with mixed given data, \\emph{SIAM J. Appl. Math.} \\textbf{34} (1978), 676-680.\n\\bibitem{MH} M. Horv\\'{a}th, Inverse spectral problems and closed exponential systems. \\emph{Ann. Math.} {\\bf162} (2005) 885-918.\n\\bibitem{M} M. Horv\\'{a}th, On the inverse spectral theory of Schr\\\"{o}dinger and Dirac operators, \\emph{Trans. Amer. Math. Soc.} \\textbf{353} (2001), 4155-4171.\n\\bibitem{EK} E. Korotyaev, Inverse resonance scattering on the real line, \\emph{Inverse Problems} {\\bf 21} (2005), 325-341.\n\\bibitem{EK0} E. Korotyaev, Inverse resonance scattering for Schr\\\"{o}dinger operator on the half line. \\emph{Asymptotic Anal.} \\textbf{37} (2004), 215-226.\n\n\n\\bibitem{Bb} B. M. Levitan, Inverse Sturm-Liouville Problems, Nauka, Moscow, 1987.\n\\bibitem{B} B. Levin, Distribution of zeros of entire functions, AMS Transl. vol.5, Providence RI 1964.\n\\bibitem{MV} M. M\\\"{o}ller, V. Pivovarchik, Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications, OT 246, Birkh\\\"{a}user, Cham, 2015.\n\\bibitem{mv} M. M\\\"{o}ller, V. Pivovarchik, Direct and inverse Robin-Regge problems, \\emph{Electron. J. Differential Equations} \\textbf{2017} (2017), 1-18.\n\\bibitem{V} V. A. Marchenko, Sturm-Liouville Operators and Applications, Oper. Theory Adv. Appl. 22, Birkh\\\"{a}user, Basel, 1986.\n\\bibitem{VC} V. Pivovarchik, C. van der Mee, The inverse generalized Regge problem, \\emph{Inverse Problems} \\textbf{17} (2001), 1831-1845.\n\\bibitem{RS} W. Rundell, P. Sacks, Numerical technique for the inverse resonance problem, \\emph{J. Computational and Applied Mathematics} {\\bf170} (2004), 337-347.\n\\bibitem{ST} S. A. Stepin, A. G. Tarasov, Asymptotic distribution of resonances for one-dimensional Schr\\\"{o}dinger operators with compactly supported potential, \\emph{Mat. Sb.} \\textbf{198} (2007), 1787-1804; \\bibitem{WY} Y. P. Wang, V.A. Yurko, Inverse spectral problems for differential pencils with boundary conditions dependent on the spectral parameter, \\emph{Math. Methods Appl. Sci.} \\textbf{40} (2017), 3190-3196.\n \\bibitem{WW} Z. Wei, G. Wei, On the uniqueness of inverse spectral problems associated with incomplete spectral data, \\emph{J. Math. Anal. Appl.} \\textbf{462} (2018), 697-711.\n\\bibitem{XP} X.-C. Xu, Inverse spectral problems for the generalized Robin-Regge problem with complex coefffcients, \\emph{J. Geom. Phys.} \\textbf{159} (2021),\nPaper No. 103936.\n\\bibitem{X} X.-C. Xu, N. P. Bondarenko, On the local solvability and stability for the inverse spectral problem of the generalized\nDirichlet-Regge problem, \\emph{Acta Math. Sin. (Engl. Ser.)} \\textbf{38} (2022), no. 7, 1229-1240.\n\\bibitem{x} X.-C. Xu, N.P. Bondarenko, Local solvability and stability of the generalized inverse Robin-Regge problem with complex coefficients. \\emph{Journal of Inverse and Ill-posed Problems} \\textbf{31} (2023), 711-721.\n\\bibitem{XF} X.-C. Xu, C.-F. Yang, Inverse resonance problems for the Schr\\\"{o}dinger operator on the real line with mixed given data, \\emph{Lett. Math. Phys.} \\textbf{108} (2018), 213-223.\n\\bibitem{Y} V.A. Yurko, On boundary value problems with a parameter in the boundary conditions, \\emph{Soviet J. Contemporary Math. Anal.} \\textbf{19} (1984), 62-73.\n\\bibitem{Zo1} M. Zworski, A remark on isopolar potentials. \\emph{SIAM J. Math. Anal.} \\textbf{32} (2001), 1324-1326.\n\\end{thebibliography}\n\n\n\n\n\n\n\n\n\n\\end{document}\nLet's prove the theorem of Ambarzumian's type.\n\\begin{theorem}\n If the eigenvalues of the problem $L(q,\\alpha_{0},0,\\alpha,0)$ (or $L(q,\\alpha_{0},0,-\\alpha,0)$) with real $q$ are \\eqref{o28}, then $q(x)=0$ a.e. on $(0,a)$.\n\\end{theorem}\n\\begin{proof}\n From \\eqref{k10} and the asymptotics of the eigenvalues, we see that $K_{1}(a,a)=0$. Let $y_{1}$ be an eigenfunction for the eigenvalue $\\lambda=0$, then we have\n \\begin{equation*}\n -y''_{1}(x)+q(x)y_{1}(x)=0, \\quad y'_{1}(0)=y'_{1}(a)=0.\n \\end{equation*}\nThus, $y_1$ is an eigenfunction corresponding to the first eigenvalue of the problem $L(q,0,0,0,0)$.\n Following the same argument as the proof of Theorem 1.4.1 in \\cite{GV}, we can prove this theorem.\n\\end{proof}\n\n\n\n\\begin{theorem}\\label{th1}\n $\\left(1\\right)$ If $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)>0$, then the eigenvalues $\\{\\lambda_{k}^\\pm\\}_{k\\in\\mathbb{Z}}$ of the problems $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$, respectively, have the following asymptotic behavior:\n \\begin{equation}\\label{k11}\n \\begin{aligned}\n \\lambda_{k}^\\pm=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-\\frac{1}{2}\\right){\\rm sgn} k+\\frac{i P_0^\\pm}{2a}+\\frac{P}{k}+\\frac{\\beta_{k}}{k}, \\quad \\lvert k \\rvert \\to \\infty,\n \\end{aligned}\n \\end{equation}\n where $\\{\\beta_{k}\\}_{k=-\\infty,k\\neq0}^{\\infty}\\in l_{2}$ and where\n \\begin{equation}\\label{k10}\n \\begin{aligned}\n P_0^\\pm=\\ln{\\left(\\frac{|\\alpha_{0}\\pm\\alpha+1\\pm\\alpha\\alpha_{0}|}{|\\alpha_{0}\\pm\\alpha-(1\\pm\\alpha\\alpha_{0})|}\\right)},\\quad P=\\frac{\\beta_0}{\\pi\\left(1-\\alpha_{0}^2\\right)}+\\frac{\\beta}{\\pi\\left(1-\\alpha^2\\right)}+\\frac{K_{1}\\left(a,a\\right)}{\\pi}.\n \\end{aligned}\n \\end{equation}\n $\\left(2\\right)$ If $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)<0$, then the eigenvalues $\\{\\lambda_{k}\\}_{k\\in\\mathbb{Z}_{0}}$, $\\mathbb{Z}_{0}=\\mathbb{Z}\\setminus\\{0\\}$, of the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$, respectively, have the following asymptotic behavior:\n \\begin{equation}\\label{k12}\n \\begin{aligned}\n \\lambda_{k}^\\pm=\\frac{\\pi}{a}\\left(\\lvert k\\rvert-1\\right){\\rm sgn} k+\\frac{iP_0^\\pm}{2a}+\\frac{P}{k}+\\frac{\\beta_{k}}{k}, \\quad \\lvert k \\rvert \\to \\infty,\n \\end{aligned}\n \\end{equation}\n where $P_0^\\pm$ and $P$ are given \\eqref{k10} and $\\{\\beta_{k}\\}_{k=-\\infty,k\\neq0}^{\\infty}\\in l_{2}$.\\\\\n $\\left(3\\right)$ If $\\left(\\alpha_{0}-1\\right)\\left(1-\\alpha\\right)=0$, then the problem $L\\left(q,\\alpha_{0},\\beta_{0},\\pm\\alpha,\\beta\\right)$ may have only finitely many eigenvalues. If there exist infinitely many eigenvalues $\\{\\lambda_{k}^+\\}$, then $Im \\lambda_{k}^+ \\to \\infty$ as $\\lvert k\\rvert\\to \\infty$.\n\\end{theorem}\n\nor equivalently,\n\\begin{equation}\n\\notag\n \\lambda_k^0=\\begin{cases}& \\!\\!\\!\\!\n \\frac{\\pi}{a}\\left(\\lvert k\\rvert-1\\right){\\rm sgn} k+\\frac{iP_0^-}{2a},\\; k\\in\\mathbb{Z}_{0},\\ \\text{if}\\ \\left(1-\\alpha_{0}\\right)\\left(1-\\alpha\\right)>0, \\\\\n & \\!\\! \\!\\!\\!\\! \\frac{\\pi}{a}\\left(\\lvert k\\rvert-\\frac{1}{2}\\right){\\rm sgn} k+\\frac{i P_0^-}{2a}, k\\in\\mathbb{Z},\\ \\text{if}\\ \\left(1-\\alpha_{0}\\right)\\left(1-\\alpha\\right)<0,\n \\end{cases}\n\\end{equation}\n\n\n\n\n"}}},{"rowIdx":292,"cells":{"id":{"kind":"string","value":"2412.20703v2"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20703v2"},"title":{"kind":"string","value":"The Restricted Inverse Optimal Value Problem under Weighted Bottle-neck Hamming distance on trees"},"full_text":{"kind":"string","value":"\n\\documentclass[a4paper,fleqn]{cas-dc}\n\n\n\n\n\\def\\tsc#1{\\csdef{#1}{\\textsc{\\lowercase{#1}}\\xspace}}\n\\tsc{WGM}\n\\tsc{QE}\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{amsmath}\n\n\\usepackage{makeidx} \\usepackage{graphicx}\n\\usepackage{amsfonts, amssymb, amsmath, cite,bm}\n\\usepackage{algorithm}\n\\usepackage{algorithmic}\n\\usepackage{color}\n\\usepackage{float}\n\\usepackage{orcidlink}\n\\usepackage{multirow}\n\\usepackage{multicol}\n\\renewcommand{\\algorithmicrequire}{ \\textbf{Input:}} \\renewcommand{\\algorithmicensure}{ \\textbf{Output:}} \\newtheorem{thm}{Theorem}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem{alg}[thm]{Algorithm}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{defn}{Definition}\n\\newcommand{\\subsubsubsection}[1]{\\paragraph{#1}\\mbox{}\\\\} \n\\newcommand{\\eq}{\\begin{equation}}\n\\newcommand{\\nq}{\\end{equation}}\n\\newcommand\\eqa{\\begin{eqnarray}}\\newcommand\\nqa{\\end{eqnarray}}\n\\newcommand{\\lbl}[1]{\\label{#1}}\n\\newcommand\\bess{\\begin{eqnarray*}}\\newcommand\\eess{\\end{eqnarray*}}\n\\newcommand{\\nn}{\\nonumber}\n\\newcommand{\\mb}{\\mbox{} \\hspace}\n\\newcommand{\\hb}{\\hfill $\\Box$}\n\\newcommand{\\ALG}{}\n\\newenvironment{breakablealgorithm}\n{\t\\begin{center}\n\t\t\\refstepcounter{algorithm}\t\t\\hrule height.8pt depth0pt \\kern2pt\t\t\\renewcommand{\\caption}[2][\\relax]{\t\t\t{\\raggedright\\textbf{\\ALG Algorithm~\\thealgorithm} ##2\\par}\t\t\t\\ifx\\relax##1\\relax \t\t\t\\addcontentsline{loa}{algorithm}{\\protect\\numberline{\\thealgorithm}##2}\t\t\t\\else \t\t\t\\addcontentsline{loa}{algorithm}{\\protect\\numberline{\\thealgorithm}##1}\n\t\t\t\\kern2pt\\hrule\\kern2pt\n\t\t}\n\t}{\t\t\\kern2pt\\hrule\\relax\t\\end{center}\n}\n\n\\begin{document}\n\\let\\WriteBookmarks\\relax\n\\def\\floatpagepagefraction{1}\n\\def\\textpagefraction{.001}\n\n\\shorttitle{The Restricted Inverse Optimal Value Problem} \n\n\\shortauthors{Q. Zhang et~al.} \n\n\\title [mode = title]{The Restricted Inverse Optimal Value Problem on Shortest Path on Trees under Weighted Bottleneck Hamming distance } \n\n\\tnotemark[1] \n\n\\tnotetext[1]{The work of Q. Zhang was supported by National Natural Science Foundation of China (1230012046).} \n\n\\author[1]{Qiao Zhang \\orcidlink{0009-0004-4723-2017} }\\cormark[1]\\ead{qiaozhang@cczu.edu.cn}\n\\cortext[1]{Corresponding author}\n\\author[2]{Xiao Li \\orcidlink{0009-0003-4222-6577}}\\ead{xli2576@wisc.edu}\n\\author[3]{Xiucui Guan \\orcidlink{0000-0002-2653-1868}}\\ead{xcguan@163.com}\n\n\\address[1]{Aliyun School of Big Data, Changzhou University, Changzhou 213164, Jiangsu, China }\n\\address[2]{Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA }\n\\address[3]{School of Mathematics, Southeast University, Nanjing 210096, Jiangsu, China}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{abstract}\nWe consider the Restricted Inverse Optimal Value Problem on Shortest Path on Trees (RIOVSPT$_{BH}$) under weighted bottleneck Hamming distance.\nThe problem aims to minimize the total cost under weighted bottleneck Hamming distance such that the length of the shortest root-leaf path of the tree is lower-bounded by a given value by adjusting the lengths of some edges. Additionally, the specified lower bound must correspond to the length of a particular root-leaf path.\nThrough careful analysis of the problem's structural properties, we develop an algorithm with $O(n\\log n)$ time complexity to solve (RIOVSPT$_{BH}$). Furthermore, by removing the path-length constraint, we derive the Minimum Cost Shortest Path Interdiction Problem on Trees, for which we present an $O(n\\log n)$ time algorithm that operates under weighted bottleneck Hamming distance. Extensive computational experiments demonstrate the efficiency and effectiveness of both algorithms.\n\\end{abstract}\n\n\n\n\n\n\\begin{keywords}\n Inverse optimal value problem \\sep Interdiction problem\\sep Shortest path \\sep Bottleneck Hamming distance \\sep Tree\n\\end{keywords}\n\n\\maketitle\n\n\\section{Introduction}\n\nThe inverse shortest path problem \\textbf{(ISP)} has been a foundational topic in inverse optimization since its introduction by Burton and Toint\\cite{BT-MP-ISP-1992}. \nConsider a weighted graph $G=(V,E,w)$ with a set of $s$-$t$ paths $P_j(j=0,1,\\ldots,r-1)$, where $V$ and $E$ denote the sets of nodes and edges, respectively. The \\textbf{(ISP)} seeks to minimally adjust the weight vector such that a prescribed path $P_0$ becomes the shortest path under some measurement criterion. For a vector $f$ and a path $P_j$, we define $f(P_j):=\\sum_{e\\in P_{j}}f(e)$.\nThen the mathematical model for the problem \\textbf{(ISP)} \\cite{BT-MP-ISP-1992} can be stated as follows.\n\\begin{eqnarray}\n&\\min_{\\bar w}&\\|\\bar w-w\\| \\nonumber\\\\\n\\textbf{(ISP)} &s.t.& \\bar w(P_0)\\leq \\bar w(P_j),j=1,2,\\cdots, r-1,\\nn\\\\\n&&\\bar w(e)\\geq 0, e\\in E.\\nonumber\n\\end{eqnarray}\n\n\n\nA more challenging variant, the inverse optimal value problem on shortest paths \\textbf{(IOVSP)}, has emerged as an important research direction, though it has received comparatively less attention than \\textbf{(ISP)}. In \\textbf{(IOVSP)}, rather than providing a desired solution a priori, the objective is to achieve a specified optimal value. Given a weighted graph $G=(V,E,w)$, the \\textbf{(IOVSP)} problem seeks to adjust the weight vector while minimizing the total adjustment cost, such that the shortest path length equals a prescribed value $D$. Next is the mathematical formulation of \\textbf{(IOVSP)}~\\cite{CH-N-ISPL}:\n\\begin{eqnarray}\n&\\min_{\\tilde w}&\\|\\tilde w-w\\| \\nonumber\\\\\n\\textbf{(IOVSP)} &\\text{s.t.}& \\min\\limits_{0\\leq j\\leq r-1} \\tilde w(P_j)= D, \\nonumber\\\\\n&&\\tilde w(e)\\geq 0, \\quad e\\in E.\\nonumber\n\\end{eqnarray}\n\n\n\nNow we consider a restricted inverse optimal value problem on shortest path on\ntrees (denoted by \\textbf{(RIOVSPT)}). Let $T=(V, E, w)$ be an edge-weighted tree rooted at $v_1$, where $V:=\\{v_1, v_2, \\cdots, v_n\\}$ and $E:=\\{e_2, e_3, \\cdots, e_{n}\\}$ are the sets of nodes and edges, respectively. Let $Y:=\\{t_0,t_1, \\cdots$ $,t_{r-1}\\}$ be the set of leaves. Let $w(e)$, $l(e)$ and $u(e)$ be the original weight, the lower and upper bounds of the adjusted weight for an edge $e\\in E$, respectively, where $l(e)\\leq w(e)\\leq u(e)$. Let $c(e)>0$ be a cost to ajust the weight of edge $e\\in E$. Denoted by $P_{v_i, v_j}$ the unique path from $v_i$ to $v_j$ on the tree $T$ and record $P_k:=P_{v_1, t_k}$ as the ``root-leaf path\" of the leaf node $t_k(k=0,1,2,\\cdots,r-1)$ in brief.\nThe problem \\textbf{(RIOVSPT)} aims at adjusting some weights of edges to minimize the total cost under some measurement such that the length of the shortest path of the tree is lower-bounded by a given value $D$, which is just the restriction of the length of the given path $P_0$. The mathematical model can be stated below.\n\\begin{eqnarray}\n&\\min_{\\tilde w}&\\|\\tilde w-w\\| \\nonumber\\\\\n\\textbf{ (RIOVSPT) }&s.t.&\\tilde w(P_k)\\geq D, t_k\\in Y, \\label{eq-RIOVSPT}\\\\\n&& \\tilde w(P_0)=D,\\nonumber\\\\\n&& l(e)\\leq \\tilde w(e)\\leq u(e), e\\in E.\\nonumber\n\\end{eqnarray}\n\nThe feasible region of \\textbf{(RIOVSPT)} can be characterized as the intersection of the feasible regions of \\textbf{(ISP)} and \\textbf{(IOVSP)} when restricted to tree structures. This structural relationship provides important insights into the problem's properties and solution space.\n\nZhang et al.\\cite{ZhangQ22} proposed an $O(n^2)$ time algorithm to solve the problem \\textbf{(RIOVSPT$_1$)} under weighted $l_1$ norm. Meanwhile, they also solved the problem \\textbf{(RIOVSPT$_1$)} under unit $l_1$ norm in linear time. \n\nIn a seminal contribution to the field, Zhang et al.~\\cite{ZGP-LBIOVMST-JGO,ZGZ-IOVMST-OL} introduced the inverse optimal value problem on minimum spanning trees \\textbf{(RIOVMST)}. This problem incorporates dual constraints: ensuring a specified spanning tree $T^0$ maintains its minimum spanning tree status while simultaneously achieving a prescribed weight value $K$. Their research under the unit $l_{\\infty}$ norm for measuring adjustment costs yielded two strongly polynomial-time algorithms, both achieving $O(mn)$ time complexity---one for the unbounded case~\\cite{ZGZ-IOVMST-OL} and another for the lower-bounded case~\\cite{ZGP-LBIOVMST-JGO}, where $|V|:=n, |E|:=m$.\n\nBuilding upon this foundation, Jia et al.~\\cite{Jia2023} made several significant contributions. For the $l_\\infty$ norm variant, they developed an efficient binary search method achieving $O(m^2 \\log n)$ time complexity, while also proposing a more efficient $O(mn)$ algorithm specifically for the unit $l_\\infty$ norm case. Their subsequent work~\\cite{Jia2024} addressed the more challenging weighted $l_1$ norm variant, presenting an algorithm with time complexity $O(m^2 n^2 \\log n \\log(n c_{\\max}))$, where $c_{\\max}$ denotes the maximum element in the cost vector $c$.\nComplementing these advances, Wang et al.~\\cite{WGZ-CIOVMST-JOCO} introduced novel algorithmic techniques for the bounded \\textbf{RIOVMST} problem under bottleneck Hamming distance, achieving an $O(mn \\log m)$ time complexity.\n\n\n\n\nThe practical significance of \\textbf{(RIOVSPT)} is particularly evident in telecommunication bandwidth pricing, as highlighted by Paleologo and Takriti~\\cite{PT-ANV-BT}. Consider a bandwidth provider pricing city-to-city links on a specialized bandwidth tree network (excluding node $v_0$) in \\textbf{Figure.\\ref{Fig1}}\\cite{ZhangQ22}. Given an observed traded price $D$ for a bandwidth link between cities $v_1$ and $v_0$, the sum of prices along the cheapest path between these cities must equal $D$ to prevent arbitrage opportunities. Without this equilibrium, traders could exploit price differentials by purchasing the cheaper option (either the direct traded link or the sequence of links along the cheapest path) and selling the more expensive alternative.\nIn this context, suppose path $P_{11}$ through node $v_{11}$ represents the optimal route for emergency response scenarios, while alternative paths may be suboptimal due to construction budget constraints. To ensure reliable emergency communication between $v_1$ and $v_0$, the total price along path $P_{11}$ must equal $D$.\n\\begin{figure}\n \\centering\n\\includegraphics[width=0.7\\linewidth]{Fig-background.png}\n \\caption{\\cite{ZhangQ22}A special bandwidth network between the cities $v_1$ and\n$v_0$ with prices on the links.}\n \\label{Fig1}\n\\end{figure}\n\nZhang, Guan et al. \\cite{ZhangQ20, ZhangQ21} studied the minimum cost shortest path interdiction problem by upgrading edges on trees (\\textbf{MCSPIT}). In their work, they did not consider the restriction on the path \\( P_0 \\) as outlined in problem (\\textbf{RIOVSPT}). The \\textbf{(MCSPIT)} problem seeks to minimize the total cost of upgrading selected edges under a given measurement, ensuring that the length of the shortest path on the tree is bounded below by a specified value \\( D \\). The mathematical model for this problem can be formulated as follows.\n\\begin{eqnarray}\n&\\min_{\\bar w}& \\|\\bar w-w\\|\\nonumber\\\\\n\\textbf{(MCSPIT)}&s.t.&\\bar w(P_k)\\geq D, t_k\\in Y, \\label{eq-MCSPIT}\\\\\n&& w(e)\\leq \\bar w(e)\\leq u(e), e\\in E.\\nonumber\n\\end{eqnarray}\n\nIt is obvious that the optimal solution of the problem (\\textbf{RIOVSPT}) is a feasible solution of the problem \\textbf{(MCSPIT)}. \n\nUnder the weighted $ l_1 $ norm, Zhang, Guan et al.\\cite{ZhangQ20} proposed a primal-dual algorithms in $ O(n^2) $ time for the problems \\textbf{(MCSPIT$_1$)}. Under the weighted Hamming distance, they\\cite{ZhangQ21} demonstrated that the problem are $\\mathcal{NP}$-hard. While for the unit sum-Hamming distance, they developed an $ O(n^4\\log n) $ time algorithm for the problem \\textbf{(MCSPIT$_{\\bm{USH}}$)} by employing a combination of dynamic programming . Subsequently, with some local optimization, Yi et al. \\cite{YiL22} improved upon this algorithm, achieving two reduced time complexity of $ O(n^3 \\log n) $ for this problem. \n\n\n\nwhen the weighted bottleneck Hamming distance is applied to $\\|\\cdot\\|$ in models (\\ref{eq-RIOVSPT})-(\\ref{eq-MCSPIT}), the problems \\textbf{(RIOVSPT$\\bm{_{BH}}$)} and \\textbf{(MCSPIT$\\bm{_{BH}}$)} under weighted bottleneck Hamming distance can be formulated as follows. \n \\begin{eqnarray}\n &\\min_{\\tilde w}& C(\\tilde w):=\\max_{e\\in E}c(e)H(\\tilde w(e),w(e)) \\nonumber\\\\\n \\textbf{(RIOVSPT$\\bm{_{BH}}$)} &s.t.&\\tilde w(P_k)\\geq D, t_k\\in Y, \\nonumber\\\\\n && \\tilde w(P_0)=D,\\label{eq-RIOVSPTBH}\\\\\n && l(e)\\leq \\tilde w(e)\\leq u(e), e\\in E.\\nonumber\n \\end{eqnarray}\n\n\\begin{eqnarray}\n&\\min_{\\bar w}& M(\\bar w):=\\max_{e\\in E}c(e)H(\\bar w(e),w(e)) \\nonumber\\\\\n\\textbf{(MCSPIT$\\bm{_{BH}}$)}&s.t.&\\bar w(P_k)\\geq D, t_k\\in Y, \\label{eq-MCSPITBH}\\\\\n&& w(e)\\leq \\bar w(e)\\leq u(e), e\\in E.\\nonumber\n\\end{eqnarray}\nwhere Hamming distance $H(\\bar w(e),w(e))$ between $\\bar w(e)$ and $w(e)$ is defined as $$H(\\bar w(e),w(e))=\\left\\{\\begin{array}{ll}1,&\\bar w(e)\\neq w(e),\\\\0,&\\bar w(e)= w(e).\\end{array}\\right.$$\n\n \nThis paper is structured as follows: Section 2 presents our solution methodology for the Restricted Inverse Optimal Value Problem on Trees under weighted bottleneck Hamming distance (\\textbf{RIOVSPT}$_{BH}$). Section 3 develops an algorithmic approach for the Minimum Cost Shortest Path Interdiction Problem (\\textbf{MCSPIT}$_{BH}$). Section 4 evaluates the computational performance of both algorithms through extensive numerical experiments. Finally, Section 5 concludes the paper with a summary of our findings and directions for future research.\n\n\n\n\\section{Solve the problems (RIOVSPT$_{BH}$).}\n\n\nFor problem (RIOVSPT$_{BH}$) in the model (\\ref{eq-RIOVSPTBH}), the optimal value must be one of the costs of edges. Notice that once an edge $e$ is adjusted, no matter how or in what ajusting amount, the relevant Hamming distance of $e$ is 1 and its adjusting cost is $c(e)$. Hence, suppose $\\tilde c$ is an objective value. Then the edges with costs less than $\\tilde c$ can be adjusted to maintain the objective value.\n\nFor convenience, we first sort the edges $e$ by their values of $c(e)$ \n in non-decreasing order and removing any duplicates to produce the sequence $ \\{\\alpha_{n^*}\\} $: \n\\begin{equation} \\label{eq:Sequence}\n c(e_{\\alpha_1}) < c(e_{\\alpha_2}) < \\cdots < c(e_{\\alpha_{n^*}}).\n\\end{equation}\n\n\t\t Let $C_k:=c(e_{\\alpha_k})$, $k=1,2,\\cdots,{n^*}$. \n\nThen for any one cost $C_k$, the feasibility of the problem \\textbf{(RIOVSPT$\\bm{_{BH}}$)} can be judged in the following theorem, where $E_k := \\{e \\in E \\mid c(e) \\leq C_k\\}$. Let $\\mathcal{C} := \\{C_k \\mid k=1,2,\\cdots,{n^*}\\}$ be the set of costs of edges in the sequence $ \\{\\alpha_{n^*}\\} $.\n\n\\begin{thm}\\label{thm-fea=RIOVSPTBH}\nThe problem \\textbf{(RIOVSPT$\\bm{_{BH}}$)} has a feasible solution with objective value $C_k \\in \\mathcal{C}$ if and only if:\n\\begin{enumerate}\n \\item $l(P_0 \\cap E_k) + w(P_0 \\setminus E_k) \\leq D \\leq u(P_i \\cap E_k) + w(P_i \\setminus E_k)$ for all $t_i \\in Y$, and\n \\item $l(P_0 \\cap E_k) + w(P_0 \\setminus E_k) \\leq l(P_i \\cap P_0 \\cap E_k) \\\\\n + u(P_i \\setminus P_0 \\cap E_k) + w(P_i \\setminus E_k)$ for all $t_i \\in Y$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n \\textbf{Necessity.} If $\\tilde{w}$ is a feasible solution of the problem \\textbf{(RIOVSPT$\\bm{_{BH}}$)}, then we have $l(e) \\leq \\tilde{w}(e) \\leq u(e)$ for all $e \\in E$. Moreover,\n$$\nu(P_i \\cap E_k) + w(P_i \\setminus E_k) \\geq \\tilde{w}(P_i) \\geq D, \\quad \\forall t_i \\in Y \\setminus \\{t_0\\},\n$$\nand\n$$\\begin{aligned}\nu(P_0 \\cap E_k) + w(P_0 \\setminus E_k) &\\geq \\tilde{w}(P_0) = D \\\\&\\geq l(P_0 \\cap E_k) + w(P_0 \\setminus E_k).\n\\end{aligned}\n$$\nHence, condition 1 holds.\n\nNext, suppose that\n$$\\begin{aligned}\n&l(P_0 \\cap E_k) + w(P_0 \\setminus E_k)\\\\& > l(P_{t^*} \\cap P_0 \\cap E_k) + u(P_{t^*} \\setminus P_0 \\cap E_k) + w(P_{t^*} \\setminus E_k),\\end{aligned}\n$$\nwhere\n$$\nt^* := \\arg\\min_{t_i \\in Y} \\Big(l(P_i \\cap P_0 \\cap E_k) + u(P_i \\setminus P_0 \\cap E_k) + w(P_i \\setminus E_k)\\Big).\n$$\n\n\\textbf{Case (i):} If $P_{t^*} \\cap P_0 = \\emptyset$, then\n$$\\begin{aligned}\nD &\\geq l(P_0 \\cap E_k) + w(P_0 \\setminus E_k) \\\\&> u(P_{t^*} \\cap E_k) + w(P_{t^*} \\setminus E_k) \\geq \\bar{w}(P_{t^*}),\\end{aligned}\n$$\nwhich contradicts with the feasibility of $\\bar{w}$.\n\n\\textbf{Case (ii):} If $P_{t^*} \\cap P_0 \\neq \\emptyset$, we have\n\\begin{eqnarray*}\n && u(P_{t^*} \\setminus P_0 \\cap E_k) + w(P_{t^*} \\setminus P_0 \\setminus E_k) < l(P_0 \\cap E_k) \\\\&&+ w(P_0 \\setminus E_k) - l(P_{t^*} \\cap P_0 \\cap E_k) - w(P_{t^*} \\cap P_0 \\setminus E_k)\\\\&=&l(P_0 \\setminus P_{t^*} \\cap E_k) + w(P_0 \\setminus P_{t^*} \\setminus E_k),\n\\end{eqnarray*}\nand thus\n$$\\begin{aligned}\nD &\\leq \\tilde{w}(P_{t^*}) = \\tilde{w}(P_{t^*} \\cap P_0) + \\tilde{w}(P_{t^*} \\setminus P_0) \\\\\n&\\leq \\tilde{w}(P_{t^*} \\cap P_0) + u(P_{t^*} \\setminus P_0 \\cap E_k) + w(P_{t^*} \\setminus P_0 \\setminus E_k) \\\\\n&<\\tilde{w}(P_{t^*} \\cap P_0) + l(P_0 \\setminus P_{t^*} \\cap E_k) + w(P_0 \\setminus P_{t^*} \\setminus E_k) \\\\\n&\\leq \\tilde{w}(P_0),\\end{aligned}\n$$\nwhich contradicts with $\\tilde{w}(P_0) = D$. Hence, condition 2 holds.\n\n \n\n\\textbf{Sufficiency.} Suppose the conditions 1-2 hold. Then we have \n\\begin{eqnarray*}\n&& l(P_0 \\cap E_k) + w(P_0 \\setminus E_k) \\leq D\\\\&\\leq&\\min_{t_i \\in Y}\\big(u(P_i \\cap E_k) + w(P_i \\setminus E_k)\\big) \\\\&\\leq& u(P_0 \\cap E_k) + w(P_0 \\setminus E_k). \n\\end{eqnarray*}\n\n\nLet $ e_j = (v_i, v_j) \\in P_0 \\cap E_k $ be the first edge on $ P_0 \\cap E_k $ from $ v_1 $ satisfying $$ \n\\begin{aligned}\n & u(P_{v_1, v_i} \\cap E_k) + w(P_{v_1, v_i} \\setminus E_k) + l(P_{v_i, t_0} \\cap E_k)\\\\ &+ w(P_{v_i, t_0} \\setminus E_k) \n \\leq D\\leq u(P_{v_1, v_j} \\cap E_k) + w(P_{v_1, v_j} \\setminus E_k)\\\\& + l(P_{v_j, t_0} \\cap E_k) + w(P_{v_j, t_0} \\setminus E_k).\n\\end{aligned}\n$$\nNotice that the special cases when $ v_i = v_1 $ with \n$$\\begin{aligned}\n&~~~~~u(P_{v_1, v_i} \\cap E_k) + w(P_{v_1, v_i} \\setminus E_k) \\\\&= u(P_{v_1, v_1} \\cap E_k) + w(P_{v_1, v_1} \\setminus E_k) = 0\\end{aligned}\n$$\nand when $ v_j = t_0 $ with \n$$\\begin{aligned}\n&~~~~~l(P_{v_j, t_0} \\cap E_k) + w(P_{v_j, t_0} \\setminus E_k) \\\\&= l(P_{t_0, t_0} \\cap E_k) + w(P_{t_0, t_0} \\setminus E_k) = 0\\end{aligned}\n$$\nhave been considered. \n\nThen we can construct\n\\begin{equation} \\label{fs-RIOVSPTBH}\n \\tilde{w}(e) := \n\\begin{cases} \n u(e), & e \\in E \\setminus P_{v_i, t_0} \\cap E_k, \\\\\n l(e), & e \\in P_{v_j, t_0} \\cap E_k, \\\\\n \\delta, & e = e_j = (v_i, v_j), \\\\\n w(e), & \\text{otherwise},\n\\end{cases}\n\\end{equation}\nwhere \n$ \n\\delta = D - u(P_{v_1, v_i} \\cap E_k) - w(P_{v_1, v_i} \\setminus E_k) $ \n\n$~~~~~~~~~~~$ $ - l(P_{v_j, t_0} \\cap E_k) - w(P_{v_j, t_0} \\setminus E_k).\n$\n\nNext, we show that $ \\bar{w} $ is a feasible solution of the problem \\textbf{(RIOVSPT$\\bm{_{BH}}$)}.\n\n\\textbf{ (i)} For $ e \\in E \\setminus \\{e_j\\} $, we have\n$l(e) \\leq \\tilde{w}(e) \\leq u(e). $ \n\n\nFor $ e_j = (v_i, v_j) \\in E_k $, we have \n$$\n\\begin{aligned}\n & u(P_{v_1, v_i} \\cap E_k) + w(P_{v_1, v_i} \\setminus E_k) + l(v_i, v_j) + l(P_{v_j, t_0} \\cap E_k)\\\\&~~~~ + w(P_{v_j, t_0} \\setminus E_k) = u(P_{v_1, v_i} \\cap E_k) + w(P_{v_1, v_i} \\setminus E_k) \\\\ &~~~~+ l(P_{v_i, t_0} \\cap E_k) + w(P_{v_i, t_0} \\setminus E_k)\n \\leq D \\leq u(P_{v_1, v_j} \\cap E_k) \\\\&~~~~+ w(P_{v_1, v_j} \\setminus E_k) +l(P_{v_j, t_0} \\cap E_k) + w(P_{v_j, t_0} \\setminus E_k) \n \\\\& = u(P_{v_1, v_i} \\cap E_k)+ w(P_{v_1, v_i} \\setminus E_k) + u(v_i, v_j) \\\\&~~~~+ l(P_{v_j, t_0} \\cap E_k) + w(P_{v_j, t_0} \\setminus E_k).\n\\end{aligned}\n$$\nThus, we have \n$ \nl(e_j) \\leq \\delta = \\tilde{w}(e_j) \\leq u(e_j).\n$\n\n\\textbf{ (ii)} Let $Y^1:=\\{t'\\in Y|E(P_{t'}\\cap P_0)=\\emptyset\\}$ be the set of the leaf nodes whose root-leaf paths have no intersecting edges with the path $P_0$. \n \n For ${t'}\\in Y^1,$ $\\bar w(P_{t'})= u(P_{t'}\\cap E_k)+w(P_{t'}\\backslash E_k)\\geq D$.\n\t$$\\begin{aligned}\n\t\t&\\tilde{w}(P_0)= \\tilde w(P_{v_1,v_i})+ \\tilde w(P_{v_j,t_0})+ \\tilde w(v_i,v_j)\\\\ &=u(P_{v_1,v_i}\\cap E_k)+w(P_{v_1,v_i}\\backslash E_k) +l(P_{v_j,t_0}\\cap E_k)\\\\&~~~~+w(P_{v_j,t_0}\\backslash E_k)+\\Big(D-u(P_{v_1,v_i}\\cap E_k)-w(P_{v_1,v_i}\\backslash E_k)\\\\&~~~~-l(P_{v_j,t_0}\\cap E_k)-w(P_{v_j,t_0}\\backslash E_k)\\Big)=D.\n\t\\end{aligned}$$\n \n\tFor ${t'}\\in Y\\backslash (Y^1\\cup \\{t_0\\})$, let $v^*_t\\in P_{t'}\\cap P_0$ be the last intersection node on $P_{t'}\\cap P_0$. \n \n\t\\textbf{(a)} If $v^*_t\\in P_{v_1,v_i}$, then\n\t$\\bar w(P_{t'})=u(P_{t'}\\cap E_k)+w(P_{t'}\\backslash E_k)\\geq D.$\n \n\t\\textbf{(b)} If $v^*_t=v_j$, we have$$ \\begin{aligned}\n\t &l(P_{v_1,v_j}\\cap E_k )+w(P_{v_1,v_j}\\backslash E_k)+l(P_{v_j,t_0}\\cap E_k)\\\\&~~~~+w(P_{v_j,t_0}\\backslash E_k)=l(P_0\\cap E_k)+w(P_0\\backslash E_k)\\\\&\\leq \\min_{t'\\in Y}(l(P_{t'}\\cap P_0\\cap E_k)+u(P_{t'}\\backslash P_0\\cap E_k)+w(P_{t'}\\backslash E_k))\\\\&\\leq l(P_{v_1,v^*_t}\\cap E_k)+u(P_{v^*_t,{t'}}\\cap E_k)+w(P_{t'}\\backslash E_k)\\\\&=l(P_{v_1,v_j}\\cap E_k)+w(P_{v_1,v_j}\\backslash E_k)+u(P_{v_j,{t'}}\\cap E_k)\\\\&~~~~+w(P_{v_j,{t'}}\\backslash E_k)\n\t\\end{aligned}$$ \n and thus \n $$l(P_{v_j,t_0}\\cap E_k)+w(P_{v_j,t_0}\\backslash E_k)\\leq u(P_{v_j,{t'}}\\cap E_k)+w(P_{v_j,{t'}}\\backslash E_k).$$ Then\n\t$$\\begin{aligned}\n\t\t&\\tilde w(P_{t'})= \\tilde w(P_{v_1,v_i})+ \\tilde w(P_{v_j,{t'}})+\\tilde w(v_i,v_j)\\\\&=u(P_{v_1,v_i}\\cap E_k)+w(P_{v_1,v_i}\\backslash E_k)+u(P_{v_j,{t'}}\\cap E_k)\\\\&~~~~+w(P_{v_j,{t'}}\\backslash E_k)+\\Big(D-u(P_{v_1,v_i}\\cap E_k)-w(P_{v_1,v_i}\\backslash E_k)\\\\&~~~~-l(P_{v_j,t_0}\\cap E_k)-w(P_{v_j,t_0}\\backslash E_k)\\Big)\\geq D.\n\t\\end{aligned}$$\n\t\\textbf{(c)} If $v^*_t\\in P_{v_j,t_0}\\backslash \\{v_j\\}$, we have $$\\begin{aligned}\n\t &l(P_{v_1,v_j}\\cap E_k)+w(P_{v_1,v_j}\\backslash E_k)+l(P_{v_j,t_0}\\cap E_k)\\\\&~~~~+w(P_{v_j,t_0}\\backslash E_k)=l(P_0\\cap E_k)+w(P_0\\backslash E_k)\\\\&\\leq \\min_{t'\\in Y}(l(P_{t'}\\cap P_0\\cap E_k)+u(P_{t'}\\backslash P_0\\cap E_k)+w(P_{t'}\\backslash E_k))\\\\&\\leq l(P_{v_1,v_j}\\cap E_k)+w(P_{v_1,v_j}\\backslash E_k)+l(P_{v_j,v^*_t}\\cap E_k)\\\\&~~~~+u(P_{v^*_t,{t'}}\\cap E_k)+w(P_{v_j,{t'}}\\backslash E_k)\n\t\\end{aligned} $$\n and thus $l(P_{v_j,t_0}\\cap E_k)+w(P_{v_j,t_0}\\backslash E_k)\\leq l(P_{v_j,v^*_t})\\\\\n +u(P_{v^*_t,{t'}}\\cap E_k)+w(P_{v_j,{t'}}\\backslash E_k).$ Then we have\n\t$$\\begin{aligned}\n\t\t&\\tilde w(P_{t'})= \\tilde w(P_{v_1,v_i})+\\tilde w(P_{v_j,v^*_t})+ \\tilde w(P_{v^*_t,{t'}})+\\tilde w(v_i,v_j)\\\\\n\t\t&=u(P_{v_1,v_i}\\cap E_k)+w(P_{v_1,v_i}\\backslash E_k)+l(P_{v_j,v^*_t}\\cap E_k)\\\\&~~~~+u(P_{v^*_t,{t'}}\\cap E_k)+w(P_{v_j,{t'}}\\backslash E_k)+\\Big(D-u(P_{v_1,v_i}\\cap E_k)\\\\&~~~~-w(P_{v_1,v_i}\\backslash E_k)-l(P_{v_j,t_0}\\cap E_k)-w(P_{v_j,t_0}\\backslash E_k)\\Big)\\geq D.\n\t\\end{aligned}$$\n\tTherefore, the problem \\textbf{(RIOVSPT$\\bm{_{BH}}$)} is feasible. \n\\end{proof}\n\nFor convenience, we intoduce the following definition.\n\\begin{defn}\n If a cost $\\tilde c\\in \\mathcal{C}$ satisfies the conditions 1-2 in \\textbf{Theorem \\ref{thm-fea=RIOVSPTBH}}, we call $\\tilde c$ feasible. Otherwise, we call it infeasible.\n\\end{defn}\n \nThen we can solve the problem \\textbf{ (RIOVSPT$\\bm{_{BH}}$)} by finding the minimum feasible cost in the sequence $ \\{\\alpha_{n^*}\\} $. We can obviously have the following lemma.\n\\begin{lem}\n Suppose $\\tilde c,\\hat c\\in \\mathcal{C}$ and $\\hat c\\geq \\tilde c $. If $\\tilde c$ is feasible, so is $\\hat c$. If $\\hat c$ is infeasible, so is $\\tilde c$. \n\\end{lem}\n \n\n\n If the maximum cost $C_{n^*}$ in $\\mathcal{C}$ is infeasible, we have the following theorem. \n\\begin{thm}\n If $C_{n^*}$ is infeasible, then the problem \\textbf{ (RIOVSP-T$\\bm{_{BH}}$)} is infeasible. \n\\end{thm}\n\nHence, for the feasible problem, we can search for the minimum feasible cost by a binary search method based on sorting the costs on $\\mathcal{C}$ ascendingly.Then a relevant optimal solution can be constructed as formula (\\ref{fs-RIOVSPTBH}). \n\nBased on the analysis above, we propose \\textbf{Algorithm \\ref{alg-RIOVSPTBH}} to solve the problem \\textbf{ (RIOVSPT$\\bm{_{BH}}$)} and followed by the corresponding time complexity analysis. \n\n\\begin{breakablealgorithm}\n\t\\renewcommand{\\thealgorithm}{1}\n\t\\caption{$[\\tilde w, C(\\tilde w)]$:=\\textbf{RIOVSPT}$_{BH}(T, w,l, u, c, D,t_0)$ }\n\t\\label{alg-RIOVSPTBH}\n\t\\begin{algorithmic}[1]\n\t\t\n\t\t\\REQUIRE The tree $T(V,E)$ rooted at a root node $v_1$ with the set $Y$ of leaf nodes; the cost vector $c$, three weight vectors $w,l,u$ and the given leaf node $t_0$ and a given value $D$.\n\t\t\\ENSURE An optimal solution $\\tilde w$ and its relevant objective value $C(\\tilde w)$.\n \n\\IF {$\\max_{e\\in E}c(e)$ is infeasible}\n\\RETURN `` The problem is infeasible!\"\n\\ELSE\n\t\t\\STATE Sort the values of $c(e)$ \n in non-decreasing order and removing any duplicates to produce the sequence $ \\{\\alpha_{n^*}\\} $: \n\\begin{equation*} c(e_{\\alpha_1}) < c(e_{\\alpha_2}) < \\cdots < c(e_{\\alpha_{n^*}}).\n\\end{equation*}\n\n\t\t\\STATE Let $C_j:=c(e_{\\alpha_j})$, $j=1,2,\\cdots,{n^*}$. \n \n \\STATE Initialization: $a := 1, b := n^*$, $k^* := 0, k := \\lfloor\\frac{a+b}2\\rfloor$.\n \n \\WHILE{$k^* =0$} \n \\IF {$a=b$ or $b=a+1$}\n \\STATE $k^* := b$, \\textbf{break.}\n \n \\ENDIF\n \\IF {$C_k$ is feasible}\n \n \\STATE Update $b := k, k := \\lfloor\\frac{a+b}2\\rfloor$.\n \\ELSE \\STATE Update $a := k, k := \\lfloor\\frac{a+b}2\\rfloor$.\n \n \n \\ENDIF\n \\ENDWHILE\n\t\t\\RETURN $\\tilde w$ as formula (\\ref{fs-RIOVSPTBH}), $C(\\tilde w):=C_{k^*}$.\n\\ENDIF\t\t\n\t\\end{algorithmic}\n\\end{breakablealgorithm}\n\\begin{thm}\nThe problem \\textbf{ (RIOVSPT$\\bm{_{BH}}$)} can be solved by \\textbf{Algorithm 1} in $O(n\\log n)$ time.\n\\end{thm}\n\\begin{proof}\n The main calculations are in \\textbf{Lines 4-18} in \\textbf{Algorithm \\ref{alg-RIOVSPTBH}}. Sorting in \\textbf{Line 4} costs $O(n\\log n)$ time and there are $O(\\log n )$ iterations in the binary search method in \\textbf{Line 7-16}, where it takes $O(n)$ time to check the feasibility of $C_k$ in \\textbf{Line 11} in each iteration. Hence, the theorem holds.\n\\end{proof}\n\n\n\n\\section{Solve the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)}}\n\n\n\nWe next solve the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)} based on solving the problem \\textbf{(MSPIT$\\bm{_{BH}}$)}, which aims to maximize the root-leaf shortest path of a tree while ensuring that the total cost of upgrading selected edges under weighted bottle-neck Hamming distance does not exceed a given value $M$. The mathematical model for this problem can be formulated as follows.\n \\begin{eqnarray}\n&\\max_{\\hat w}& D(\\hat w):=\\min_{t_k\\in Y}\\hat w(P_k) \\label{eq-MSPITBH}\\\\\n\\textbf{(MSPIT$\\bm{_{BH}}$)}&s.t.&\\max_{e\\in E}c(e)H(\\hat w(e),w(e)) \\leq M, \\nonumber\\\\\n&& w(e)\\leq \\hat w(e)\\leq u(e), e\\in E.\\nonumber\n\\end{eqnarray}\n \nHere is the property of the optimal solutions of the two problems.\n\n\\begin{thm}[\\cite{ZhangQ21},Theorem 4]\nIn model (\\ref{eq-MCSPITBH})-(\\ref{eq-MSPITBH}), for $e\\in E$, if the edge length vector $\\tilde w$ is optimal, so is the edge length vector $w^*$ defined as follows.\n$$w^*(e)=\\left\\{\n\\begin{array}{ll}\nw(e), &\\text{if }\\tilde {w}(e)=w(e)\\\\\nu(e), &\\text{if }\\tilde {w}(e)\\neq{w(e)}.\n\\end{array}\\right.$$\n\\end{thm}\n\nHence, in the problems \\textbf{(MSPIT$\\bm{_{BH}}$)} and \\textbf{(MCSPIT$\\bm{_{BH}}$)}, if one edge is upgraded,then we can upgrade it to its upper bound. \n\n\n\n\n\\begin{thm}\n\tThe solution \\begin{eqnarray}\\hat w(e)=\\left\\{\\begin{array}{ll}u(e),&c(e)\\leq M\\\\w(e),&c(e)>M\\end{array}\\right.\\label{os-MSPITBH}\\end{eqnarray} is an optimal solution of the problem \\textbf{(MSPIT$\\bm{_{BH}}$)}.\n\\end{thm}\n\\begin{proof}\n\\textbf{(1)} For any edge $ e \\in E $ with $ c(e) \\leq M $, we have $ w(e) \\leq \\hat{w}(e) = u(e) $, and $ c(e)H(\\hat{w}(e), w(e)) = c(e) \\leq M $. For any edge $ e \\in E $ with $ c(e) > M $, we have $ \\hat{w}(e) = w(e) \\leq u(e) $ and $ c(e)H(\\hat{w}(e), w(e)) = 0 \\leq M $. Therefore, \n$$\n\\max_{e \\in E} c(e)H(\\hat{w}(e), w(e)) \\leq M.\n$$\nThis implies that $ \\hat{w} $ is a feasible solution of the problem \\textbf{(MSPIT$\\bm{_{BH}}$)}.\n\n\\textbf{(2)} Suppose $ \\hat{w} $ is not optimal, and let $ w^* $ be an optimal solution. Then, \n$\nD(w^*) = w^*(P_{t^*}) > D(\\hat{w}) = \\hat{w}(P_{\\hat {t}}).\n$\n\n\\textbf{Case (i):} If $ t^* = \\hat{t} $, then there exists an edge $ \\hat{e} \\in P_{t^*} $ such that $ u(\\hat{e}) \\geq w^*(\\hat{e}) > \\hat{w}(\\hat{e}) = w(\\hat{e}) $ as $ w^*(P_{t^*}) > \\hat{w}(P_{t^*}) $. In this case, \n$\nc(\\hat{e})H(w^*(\\hat{e}), w(\\hat{e})) = c(\\hat{e}) > M,\n$\nwhich contradicts with the feasibility of $ w^* $.\n\n\\textbf{Case (ii):} If $ t^* \\neq \\hat{t} $, then $ w^*(P_{\\hat{t}}) \\geq w^*(P_{t^*}) > \\hat{w}(P_{\\hat{t}}) \\leq \\hat{w}(P_{t^*}) $. Consequently, there exists an edge $ \\hat{e} \\in P_{\\hat{t}} $ such that $ u(\\hat{e}) \\geq w^*(\\hat{e}) > \\hat{w}(\\hat{e}) = w(\\hat{e}) $. Similarly, \n$$\nc(\\hat{e})H(w^*(\\hat{e}), w(\\hat{e})) = c(\\hat{e}) > M,\n$$\nwhich again contradicts with the feasibility of $ w^* $.\n\nThus, in both cases, $ \\hat{w} $ must be an optimal solution to the problem \\textbf{(MSPIT$\\bm{_{BH}}$)}.\n\\end{proof}\n\n\\begin{thm}\nThe problem \\textbf{(MSPIT$\\bm{_{BH}}$)} can be solved by formla (\\ref{os-MSPITBH}) in $ O(n) $ time.\n\\end{thm}\n\nWe next address the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)}. From model (\\ref{eq-MCSPITBH}), the objective value is one of the costs of edges. For any one cost $\\tilde c\\in \\mathcal{C} := \\{C_k \\mid k=1,2,\\cdots,{n^*}\\}$, we consider the length of shortest path based on this cost $\\tilde c$, which is just the problem \\textbf{(MSPIT$\\bm{_{BH}}$)} with $M=\\tilde c.$\n\\begin{lem}\n If $\\hat c,\\tilde c\\in \\mathcal{C}$ and $\\hat c>\\tilde c$, we can obtain $ \\hat{w}$ with $\\hat c$ and $\\tilde{w}$ with $\\tilde c$ from formula (\\ref{os-MSPITBH}). \n Then $D(\\hat{w})\\geq D(\\tilde{w})$, where $D(\\hat{w}):=\\min_{t_k\\in Y}\\hat w(P_k)$. \n\\end{lem}\n\nBy a binary search method based on the sequence $\\{\\alpha_{n^*}\\}$ in (\\ref{eq:Sequence}), we can determine the minimum cost on $\\mathcal{C}$ whose corresponding length of shortest path is no less than $D$. Then the optimal objective value of the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)} can be obtained, so does its optimal solution.\n\nThe process begins by the ascending sequence $\\{\\alpha_{n^*}\\} $ in (\\ref{eq:Sequence}). \nFor any $ C_j= c(e_{\\alpha_j}) $, we can obtain an optimal solution $ \\hat{w}^j $ of the problem \\textbf{(MSPIT$\\bm{_{BH}}$)} by replacing $M$ with $C_j$ in (\\ref{os-MSPITBH}) as well as its corresponding shortest root-leaf distance $D_j:=D(\\hat{w}^j ):=\\min_{t_i\\in Y}\\hat{w}^j(P_i)$. \nSince $\\{ C_j \\}$ is increasing, the output $ D_j $ obviously is non-decreasing.\n\nSpecifically, for $ C_{n^*} $ and $ C_1 $, from formula (\\ref{os-MSPITBH}), we can achieve the relevant optimal solutions $\\hat{w}^{n^*}$ with its shortest root-leaf distance $ D_{n^*}:=D(\\hat{w}^{n^*})$ and $\\hat{w}^1$ with $D_1:=D(\\hat{w}^1)$,respectively. \n\n Based on above analysis, we derive the following straightforward lemmas:\n\n\\begin{lem}\nThe optimal objective value of the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)} lies within the range $ [0, C_{n^*}] $.\n\\end{lem}\n\n\\begin{lem}\\label{lem-MCSPITinfty-feasible}\nIf $ \\bar {w} $ is an optimal solution to the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)}, then the relevant shortest root-leaf distance $ D(\\bar{w}) $ satisfies $ D(w) \\leq D(\\bar{w}) \\leq D_{n^*} $.\n\\end{lem}\n\nBy \\textbf{Lemma \\ref{lem-MCSPITinfty-feasible}}, the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)} is infeasible for $D > D_{n^*}$, as all the weights of edges have been upgraded to their maximum capacity. Conversely, for $D \\le D(w)$, the optimal solution is $w$ with an optimal value $0$, since the weight vector $w$ already satisfies the feasibility conditions. Consequently, we can state the following theorems.\n\n\\begin{thm}\n\tIf $D> D_{n^*}$, the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)} is infeasible.\n\\end{thm}\n\n\\begin{thm}\n If $D \\le D(w)$, then $w$ is an optimal solution of the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)} and the optimal value is $0$.\n\\end{thm}\n\nThen we consider the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)} when $D(w)<D\\leq D_{n^*}$. \n\nTo be specific, based on the ascending sequence in \\eqref{eq:Sequence} and a binary search method, we can determine the index $k$ satisfying $D_k < D \\leq D_{k+1}$ and then the corresponding $C_{k+1}$ is just the minimum cost to satisfy the constraints of the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)}. \n\n\n\n\n\nBased on the analysis above, we have the following \\textbf{Algorithm \\ref{alg-MCSPITBH}} to solve the problem \\textbf{(MCSPIT$\\bm{_{BH}}$)}.\n\n\\begin{breakablealgorithm}\n\t\\renewcommand{\\thealgorithm}{2}\n\t\\caption{$[\\bar w, M^*]:=\\textbf{MCSPIT}_{\\bm{BH}}(T, w, u, c, D)$}\n\t\\label{alg-MCSPITBH}\n\t\\begin{algorithmic}[1]\n\t\t\\REQUIRE The tree $T(V,E)$ rooted at a root node $v_1$ with the set $Y$ of leaf nodes; the cost vector $c$, two weight vectors $w,u$ and a given value $D$.\n\t\t\\ENSURE An optimal solution $\\bar w$ and its relevant objective value $M^*$.\n\t\t\\STATE Sort the edges $e$ by their values of $c(e)$ in ascending order(remove the repeating elements and suppose there are $n^*$ different values). Renumbered as follows: \\\\\n\t\t$c(e_{\\alpha_1})<c(e_{\\alpha_2})< \\cdots< c(e_{\\alpha_{n^*}}).$ \n\t\t\\STATE Let $C_j:=c(e_{\\alpha_j})$, $j=1,2,\\cdots,{n^*}$.\n \\STATE$D(w):=\\min_{t_k\\in Y}w(P_k)$.\\\\\n Obtain $\\hat w^{n^*}$ with $C_{n^*}$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $D_{n^*}:=D(\\hat w^{n^*})$.\n\t\t\n Obtain $\\hat w^1$ with $C_1$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $D_1:=D(\\hat w^1)$.\n \\IF{$D\\leq D(w)$}\n \\RETURN ($ w$, $0$).\n\t\t\\ELSIF {$D> D_{n^*}$}\n\t\t\\RETURN ``the problem is infeasible!\"\n\t\t\\ELSE\n \\STATE Initialization: $a := 1, b := n^*$, $k^* := 0, k := \\lfloor\\frac{a+b}2\\rfloor$.\n \n \\WHILE{$k^* = 0$}\n \\IF{$a = b$}\n \\STATE $k^* := a$, \\textbf{break.}\n \\ENDIF\n \\STATE Obtain $\\hat w^k$ with $C_k$ and its relevant length of shortest root-leaf distance $D_k :=D(\\hat w^k)$.\\\\\n Obtain $\\hat w^{k+1}$ with $ C_{k+1}$ and its relevant length of shortest root-leaf distance $D_{k+1} := D(\\hat w^{k+1})$.\n \\IF{$ D_{k+1} < D$}\n \\STATE Update $a := k, k := \\lfloor\\frac{a+b}2\\rfloor$.\n \\ELSIF{$D_k > D$}\n \\STATE Update $b := k, k := \\lfloor\\frac{a+b}2\\rfloor$.\n \\ELSIF{$D_{k}<D \\le D_{k+1}$}\n \\STATE $k^* := k+1$.\n \\ENDIF\n \\ENDWHILE\n\t\t\\RETURN $\\bar w:=w^{k^*}, M^*:=C_{k^*}$.\n\t\t\\ENDIF\n\t\\end{algorithmic}\n\\end{breakablealgorithm}\n The main calculations in \\textbf{Algorithm \\ref{alg-MCSPITBH}} are the cycles of the binary search method, where two \\textbf{(MSPIT$\\bm{_{BH}}$)} problems are solved in each iteration. Hence, \n\\begin{thm}\nThe problem \\textbf{(MCSPIT$\\bm{_{BH}}$)} can be solved by \\textbf{Algorithm \\ref{alg-MCSPITBH}} in $O(n\\log n)$ time.\n\\end{thm}\n\n\n\\section{Numerical experiments}\n\n\\subsection{An example to show the process of algorithms}\nFor the better understanding of \\textbf{Algorithm \\ref{alg-RIOVSPTBH}} and \\textbf{Algorithm \\ref{alg-MCSPITBH}}, Example 1 is given to show the detailed computing process.\n\n\\noindent\\textbf{Example 1.} As is illustrated in \\textbf{Figure.\\ref{Fig1}}, a special bandwidth network between the cities $v_1$ and $v_0$ is given with prices on the links, which is a tree network(excluding the node $v_0$) with an observed traded price $D:=39$. Let $V:=\\{v_1,v_2,\\cdots,v_{17}\\}$, $E:=\\{e_2,e_3,\\cdots,e_{17}\\}$. Let the initial,lower and upper bounds vectors of the costs are $$\\begin{aligned}\n &w:=(7, 12, 8, 6, 1, 12, 14, 19, 11, 11, 17, 9, 38, 10, 14, 17),\\\\&\n l: =(3 , 7 , 4, 3, 0 , 6 , 4 , 16 , 10 , 3, 13 , 5 , 37, 9, 8 , 9),\\\\&\n u: =(9 , 18, 12, 9, 7, 14 , 16 , 22, 20 , 14, 26, 12 , 48, 14, 16 , 20),\n\\end{aligned}$$ \\text{respectively}.The penalty vector of adjusting the costs between cities is $$c:=(7,2,4,14,3,13,9,8, 2,7,2,1,15,12,13,14).$$\n\n\\textbf{(1)} Suppose path $P_{11}$ through node $v_{11}$ is the optimal route for emergency response scenarios. Try to determine a pricing scheme to prevent arbitrage opportunities and also to ensure reliable emergency communication.\n\n\n\n\\textbf{(2)} Try to determine a pricing scheme just to prevent arbitrage opportunities. \n\nWe first show the detailed process to solve the problem \\textbf{(1)} by \\textbf{Algorithm \\ref{alg-RIOVSPTBH}}.\n\n\\textbf{In Lines 1-2:} $\\max_{e\\in E}c(e)=15,$ $E'=\\{e\\in E|c(e)\\leq 15\\}=E$. Then in \\textbf{Theorem \\ref{thm-fea=RIOVSPTBH}}, on the one hand, $l(P_0 \\cap E') + w(P_0 \\setminus E') =l(P_0) =29\\leq D=39$ and $ \\min_{t_k\\in Y}u(P_k)=u(P_{v_{17}})=50\\geq D:=40$, then for all $t_k \\in Y, u(P_k \\cap E') + w(P_k \\setminus E')=P_k)\\geq D:=39.$ On the other hand, \n$l(P_0 \\cap E') + w(P_0 \\setminus E') =l(P_0 )=29\\leq \\min_{t_k\\in Y }\\Big(l(P_k \\cap P_0\\cap E')+ u(P_k \\setminus P_0 \\cap E') + w(P_k \\setminus E')\\Big)=\\min_{t_k\\in Y }\\Big(l(P_k \\cap P_0)+ u(P_k \\setminus P_0 ) \\Big)=l(P_0)=29\\leq l(P_k \\cap P_0 \\cap E')+ u(P_k \\setminus P_0 \\cap E') + w(P_k \\setminus E')$ for all $t_k \\in Y$. Hence, $\\max_{e\\in E}c(e)=15$ is feasible. \n\n\\textbf{In Lines 4-5:} Construct the sequence $\\{\\alpha_{n^*}\\}: C_1=c(e_{13})=1<C_2=c(e_{10})=2<C_3=c(e_{6})=3<C_4=c(e_{4})=4<C_5=c(e_{2})=7<C_6=c(e_{9})=8<C_7=c(e_{8})=9<C_8=c(e_{15})=12<C_9=c(e_{7})=13<C_{10}=c(e_{5})=14<C_{11}=c(e_{14})=15. $\n\n\\textbf{In Lines 7-16:a binary search method.} \\\\\nInitialization: $a:=1,b:=n^*=11,k^*:=0,k:=6$.\n\n\\textbf{In the first iteration}, $C_6=8, E_6=\\{e_2,e_3,e_4,e_6,e_9,$ $e_{10},e_{11},e_{12},e_{13}\\}$. $l(P_0 \\cap E_6) + w(P_0 \\setminus E_6)= l(P_0 \\cap E_6)=29\\leq D:=39\\leq \\min_{t_k\\in Y}\\big( u(P_k \\cap E_6) + w(P_k \\setminus E_6)\\big)=u(P_{v_{17}} \\cap E_6) + w(P_{v_{17}} \\setminus E_6)=41,$ $l(P_0 \\cap E_6) + w(P_0 \\setminus E_6)=29 \\leq \\min_{t_k\\in Y}\\big(l(P_k \\cap P_0 \\cap E_6)+ u(P_k \\setminus P_0 \\cap E_6) + w(P_k \\setminus E_6)\\big)=l(P_0)=29$.Hence, $C_6=8$ is feasible. Then update $b:=6,k:=3.$\n\n\\textbf{In the second iteration}, $C_3=3,E_3=\\{e_3,e_6,e_{10},e_{12}\\}.$ $l(P_0 \\cap E_3) + w(P_0 \\setminus E_3)=40>D:=40$. Hence, $C_3=3$ is infeasible. Then update $a:=3,k:=4.$\n\n\\textbf{In the third iteration}, $C_4=4,E_4=\\{e_3,e_4,e_6,e_{10},e_{12},$ $e_{13}\\}.$ $l(P_0 \\cap E_4) + w(P_0 \\setminus E_4)=40> D:=39$. Hence, $C_4=4$ is infeasible. Then update $a:=4,k:=5.$ \n\n\\textbf{In the fourth iteration}, $C_5=7, E_5=\\{e_2,e_3,e_4,e_6,e_{10},$ $e_{11},e_{12},e_{13}\\}$. $l(P_0 \\cap E_5) + w(P_0 \\setminus E_5)= l(P_0 \\cap E_6)=32\\leq D:=39\\leq \\min_{t_k\\in Y}\\big( u(P_k \\cap E_5) + w(P_k \\setminus E_5)\\big)=u(P_{v_{17}} \\cap E_5) + w(P_{v_{17}} \\setminus E_6)=41,$ $l(P_0 \\cap E_5) + w(P_0 \\setminus E_5)=32 \\leq \\min_{t_k\\in Y}\\big(l(P_k \\cap P_0 \\cap E_5)+ u(P_k \\setminus P_0 \\cap E_5) + w(P_k \\setminus E_5)\\big)=l(P_0\\cap E_5) + w(P_0 \\setminus E_5)=32$.Hence, $C_5=7$ is feasible. Then update $b:=5,k:=4.$ Then $b=a+1,k^*:=b=5$. \n\n\\textbf{In Line 17: determine an optimal solution and its corresponding objective value.} \nFrom (\\ref{fs-RIOVSPTBH}), \\\\\n$e_j=e_{10}=(v_9,v_{10})$. \nWe can obtain an optimal solution $$\\tilde w:=(\\textbf{9}, \\textbf{18}, \\textbf{12}, 6, \\textbf{7}, 12, 14, 19, \\textbf{ 17}, \\textbf{3}, 17, \\textbf{12}, 38, 10, 14, 17)$$ and its corresponding objective value is $C(\\tilde w):=C_{k^*}=C_5=7.$ Then $\\bar w$ is a pricing scheme to prevent arbitrage opportunities and also to ensure reliable emergency communication.\n\n\n\nWe next show the detailed process to solve the problem \\textbf{(2)} by \\textbf{Algorithm \\ref{alg-MCSPITBH}}.\n\n\\textbf{In Lines 1-2:} Construct the sequence $\\{\\alpha_{n^*}\\}$ as above. \n\n\\textbf{In Line 3:} Calculate $D(w):=\\min_{t_k\\in Y}w(P_k)=w(P_{v_6})=34$, Obtain $\\hat w^{11}$ with $C_{11}=15$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $D_{11}:=D(\\hat w^{11})=u(P_{v_{17}})=50$. Obtain $\\hat w^1$ with $C_1=1$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $D_1:=D(\\hat w^1)=\\hat w^1(P_{v_{6}})=34$.\n\n\\textbf{In Lines 4-7:} judging special cases. $D_{11}=50>D:=39>D(w)=34$. Then the problem is feasible. \n\n\\textbf{In Lines 9-22: a binary search method.} \\\\\nInitialization: $a:=1,b:=n^*=11,k^*:=0,k:=6$.\n\n\\textbf{In the first iteration}, obtain $\\hat w^6$ with $C_6=8$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $D_6 :=D(\\hat w^6)=\\hat w^6(P_{v_{17}})=41$.\nObtain $\\hat w^{7}$ with $ C_{7}=9$ and its relevant length of shortest root-leaf distance $D_{7} := D(\\hat w^{7})=\\hat w^7(P_{v_{17}})=41$. Then $D_6>D:=39$. Update $b:=6, k:=3$.\n\n\n\\textbf{In the second iteration}, obtain $\\hat w^3$ with $C_3=3$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $D_3 :=D(\\hat w^3)=\\hat w^3(P_{v_{17}})=41$.\nObtain $\\hat w^{4}$ with $ C_{4}=4$ and its relevant length of shortest root-leaf distance $D_{4} := D(\\hat w^{4})=\\hat w^4(P_{v_{17}})=41$. Then $D_3>D:=39$. Update $b:=3, k:=2$.\n\n\\textbf{In the third iteration}, obtain $\\hat w^2$ with $C_2=2$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $D_2 :=D(\\hat w^2)=\\hat w^2(P_{v_{6}})=40$.\nObtain $\\hat w^{3}$ with $ C_{3}=3$ and its relevant length of shortest root-leaf distance $D_{3} := D(\\hat w^{3})=\\hat w^3(P_{v_{17}})=41$. Then $D_2>D:=39$. Update $b:=2, k:=1$.\n\n\\textbf{In the fourth iteration}, obtain $\\hat w^1$ with $C_1=1$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $D_1 :=D(\\hat w^1)=\\hat w^1P_{v_{6}})=40$.\nObtain $\\hat w^{2}$ with $ C_{2}=2$ and its relevant length of shortest root-leaf distance $D_{2} := D(\\hat w^{2})=\\hat w^2(P_{v_{6}})=40$. Then $D_1>D:=39$. Update $b:=1, k:=1$. Then $k^*:=a=1$ as $a=b=1$. \n\n\\textbf{In Line 23: determine an optimal solution and its corresponding objective value.} \nObtain $$\\hat w^1=(7, 12, 8, 6, \\textbf{7}, 12, 14, 19, 11, 11, 17, 9, 38, 10, 14, 17)$$ with $C_1=1$ from formula (\\ref{os-MSPITBH}) and its relevant length of shortest root-leaf distance $$D_1 :=D(\\hat w^1)=\\hat w^1(P_{v_{6}})=40\\geq D:=39.$$ Then $\\bar w:=\\hat w^1 $ is a pricing scheme just to prevent arbitrage opportunities.\n\n\n\n\\subsection{Numerical experiments}\nWe present the numerical experimental results for Algorithms \\ref{alg-RIOVSPTBH} and \\ref{alg-MCSPITBH} in Table \\ref{table:ex}. We tested these algorithms on six randomly generated trees with vertex numbers ranging from 1000 to 5000.\nWe randomly generated the vectors $u$, $c$ and $w$ such that $0\\leq w \\leq u$ and $c > 0$. We generated $D$ for each tree based on $n$ with randomness. In this table the average, maximum and minimum CPU time are denoted by $T_i$, $T_i^{max}$ and $T_i^{min}$, $i=1,2$, respectively.\n\nThe experimental results demonstrate that both Algorithms \\ref{alg-RIOVSPTBH} and \\ref{alg-MCSPITBH} perform efficiently on large-scale trees, with Algorithm \\( T_1 \\) outperforming Algorithm \\( T_2 \\) in terms of CPU time since the sorting algorithm used in Algorithm \\ref{alg-RIOVSPTBH} is more stable than the binary search in Algorithm \\ref{alg-MCSPITBH}. \n\n\\begin{table}[htbp!]\n \\centering\n \\caption{Performance of Algorithms \\ref{alg-RIOVSPTBH} and \\ref{alg-MCSPITBH}.}\n \\label{table:ex} \n \\begin{tabular}{ccccc}\\toprule \n \\textbf{Complexity} & \\textbf{\\( n \\)} & \\textbf{1000} & \\textbf{3000} & \\textbf{5000} \\\\ \n \\midrule\n \\( O(n \\log n) \\) & \\( T_1 \\) & 0.3052 & 1.0427 & 1.8561 \\\\\n & \\( T_1^{\\text{max}} \\) & 0.5428 & 1.2613 & 2.1547 \\\\\n & \\( T_1^{\\text{min}} \\) & 0.1871 & 0.9701 & 1.4467 \\\\\n \\midrule \n \\( O(n \\log n) \\) & \\( T_2 \\) & 0.5147 & 1.4928 & 2.5431 \\\\\n & \\( T_2^{\\text{max}} \\) & 0.6742 & 1.7624 & 2.9127 \\\\\n & \\( T_2^{\\text{min}} \\) & 0.3392 & 1.2691 & 2.1103 \\\\ \n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\section{Conclusion and further research}\nThis paper addresses the restricted inverse optimal value problem for shortest paths under weighted bottleneck Hamming distance on trees \\textbf{(RIOVSPT\\textsubscript{BH})}. This optimization problem has significant applications in transportation and communication networks. Through rigorous analysis of its mathematical properties, we develop an efficient $O(n \\log n)$ algorithm based on binary search methodology to solve \\textbf{(RIOVSPT\\textsubscript{BH})}. Additionally, we present a solution to the related \\textbf{(MCSPIT\\textsubscript{BH})} problem, achieving the same time complexity of $O(n \\log n)$.\n\nOur findings establish a foundation for future research directions. Potential extensions include investigating the restricted inverse optimal value problem on shortest paths for more complex graph structures, particularly series-parallel graphs and block graphs. 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Journal of Global Optimization. 2023; 86(1): 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}\n\\def\\testc{\\testa\\testa\\testa\\testa}\n\\def\\testb{\\testc\\testc\\testc\\testc\\testc}\n\\long\\def\\test{\\testb\\par\\testb\\par\\testb\\par}\n\n\\pinclude{\\copy\\contbox\\printSq{\\LastPage}}\n\n\\title{Documentation for Elsevier's CAS \\LaTeX\\ template}\n\n\\author{Elsevier Ltd}\n\\contact{elsarticle@stmdocs.in}\n\n\\version{2.3}\n\\date{\\today}\n\\maketitle\n\n\\section{Introduction}\nThis bundle provides two classfiles, namely \\verb+cas-sc.cls+ and\n\\verb+cas-dc.cls+ and corresponding template files for typesetting\njournal articles supposed to go through Elsevier's updated workflow.\n\\verb+cas-sc.cls+ is meant for one-column, the other \n\\verb+cas-dc.cls+ for two-column layout. These are now accepted for\nsubmitting articles both in Elsevier's electronic submission system and\nelsewhere.\n\n\\subsection{Usage}\n\\begin{enumerate}\n\\item \\verb+cas-sc.cls+ for single column journals. \n\n\\begin{vquote}\n \\documentclass[<options>]{cas-sc}\n\\end{vquote}\n\\item \\verb+cas-dc.cls+ for single column journals. \n\n\\begin{vquote}\n \\documentclass[<options>]{cas-dc}\n\\end{vquote}\n\\end{enumerate}\nand have an option \\verb+longmktitle+ to handle long front matter. \n\n\\section{Front matter}\n\n\\begin{vquote}\n\\title [mode = title]{This is a specimen $a_b$ title} \n\\tnotemark[1,2]\n\n\\tnotetext[1]{This document is the results of the research\n project funded by the National Science Foundation.}\n\n\\tnotetext[2]{The second title footnote which is a longer text \n matter to fill through the whole text width and overflow \n into another line in the footnotes area of the first page.}\n\n\\end{vquote}\n\n\\begin{vquote}\n\n\\author[1,3]{J.K. Krishnan}[type=editor,\n auid=000,bioid=1,\n prefix=Sir,\n role=Researcher,\n orcid=0000-0001-0000-0000]\n\\cormark[1]\n\\fnmark[1]\n\\ead{jkk@example.in}\n\\ead[url]{www.jkkrishnan.in}\n\n\\credit{Conceptualization of this study, \n Methodology, Software}\n\n\\affiliation[1]{organization={Department of Physics, \n J.K. Institute of Science},\n addressline={Jawahar Nagar}, \n city={Trivandrum},\n postcode={695013}, \n state={Kerala},\n country={India}}\n\n\\author[2,4]{Han Thane}[style=chinese]\n\n\\author[2,3]{William {J. Hansen}}[ role=Co-ordinator,\n suffix=Jr,\n ]\n\\fnmark[2]\n\\ead{wjh@example.org}\n\\ead[URL]{https://www.university.org}\n\n\\credit{Data curation, Writing - Original draft preparation}\n\n\\end{vquote}\n\n\\begin{vquote}\n\n\\affiliation[2]{organization={World Scientific University},\n addressline={Street 29}, \n postcode={1011 NX}, \n postcodesep={}, \n city={Amsterdam},\n country={The Netherlands}}\n\n\\author[1,3]{T. Rafeeq}\n\\cormark[2]\n\\fnmark[1,3]\n\\ead{t.rafeeq@example.in}\n\\ead[URL]{www.campus.in}\n\n\\affiliation[3]{organization={University of Intelligent \n Studies},\n addressline={Street 15}, \n city={Jabaldesh},\n postcode={825001}, \n state={Orissa}, \n country={India}}\n\n\\cortext[cor1]{Corresponding author}\n\\cortext[cor2]{Principal corresponding author}\n\\fntext[fn1]{This is the first author footnote, but is common \n to third author as well.}\n\\fntext[fn2]{Another author footnote, this is a very long \n footnote and it should be a really long footnote. But this \n footnote is not yet sufficiently long enough to make two \n lines of footnote text.}\n\n\\nonumnote{This note has no numbers. In this work we \n demonstrate $a_b$ the formation Y\\_1 of a new type of \n polariton on the interface between a cuprous oxide slab \n and a polystyrene micro-sphere placed on the slab.\n }\n\\end{vquote}\n\n\\begin{vquote}\n\n\\begin{abstract}[S U M M A R Y]\nThis template helps you to create a properly formatted \n \\LaTeX\\ manuscript.\n\n\\begin{abstract} ... \\end{abstract} and \\begin{keyword} \n... \\end{keyword} which contain the abstract and keywords \nrespectively. Each keyword shall be separated by \na \\sep command.\n\\end{abstract}\n\n\\begin{keywords}\nquadrupole exciton \\sep polariton \\sep WGM \\sep BEC\n\\end{keywords}\n\n\\maketitle\n\\end{vquote}\n\n\\subsection{Title}\n\n\\verb+\\title+ command have the below options:\n\\begin{enumerate}\n\\item \\verb+title:+ Document title\n\\item \\verb+alt:+ Alternate title\n\\item \\verb+sub:+ Sub title\n\\item \\verb+trans:+ Translated title\n\\item \\verb+transsub:+ Translated sub title\n\\end{enumerate}\n\n\\begin{vquote}\n \\title[mode=title]{This is a title}\n \\title[mode=alt]{This is a alternate title}\n \\title[mode=sub]{This is a sub title}\n \\title[mode=trans]{This is a translated title}\n \\title[mode=transsub]{This is a translated sub title}\n\\end{vquote}\n\n\n\\subsection{Author}\n\\verb+\\author+ command have the below options: \n\n\\begin{enumerate}\n\\item \\verb+auid:+ Author id\n\\item \\verb+bioid:+ Biography id\n\\item \\verb+alt:+ Alternate author\n\\item \\verb+style:+ Style of author name, eg.\\ chinese\n\\item \\verb+prefix:+ Prefix, eg.\\ Sir\n\\item \\verb+suffix:+ Suffix\n\\item \\verb+degree:+ Degree\n\\item \\verb+role:+ Role\n\\item \\verb+orcid:+ ORCID\n\\item \\verb+collab:+ Collaboration\n\\item \\verb+anon:+ Anonymous author\n\\item \\verb+deceased:+ Deceased author\n\\item \\verb+twitter:+ Twitter account\n\\item \\verb+facebook:+ Facebook account\n\\item \\verb+linkedin:+ LinkedIn account\n\\item \\verb+plus:+ Google plus account\n\\item \\verb+gplus:+ Google plus account\n\\end{enumerate}\n\n\\begin{vquote}\n\\author[1,3]{Author Name}[type=editor,\n auid=000,bioid=1,\n prefix=Sir,\n role=Researcher,\n orcid=0000-0001-0000-0000,\n facebook=<facebook id>,\n twitter=<twitter id>,\n linkedin=<linkedin id>,\n gplus=<gplus id>]\n\\end{vquote}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{sc-sample.pdf}\n\\caption{Single column output (classfile: cas-sc.cls).}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{dc-sample.pdf}\n\\caption{Double column output (classfile: cas-dc.cls).}\n\\end{figure}\n\n\\subsection{Various Marks in the Front Matter}\n\nThe front matter becomes complicated due to various kinds\nof notes and marks to the title and author names. Marks in\nthe title will be denoted by a star ($\\star$) mark;\nfootnotes are denoted by super scripted Arabic numerals,\ncorresponding author by an Conformal asterisk (*) mark.\n\n\\subsubsection{Title marks}\n\nTitle mark can be entered by the command, \\verb+\\tnotemark[<num>]+\nand the corresponding text can be entered with the command\n\\verb+\\tnotetext[<num>]+ \\verb+{<text>}+. An example will be:\n\n\\begin{vquote}\n\\title[mode=title]{Leveraging social media news to predict\n stock index movement using RNN-boost}\n\n\\tnotemark[1,2]\n\n\\tnotetext[1]{This document is the results of the research\n project funded by the National Science Foundation.}\n\n\\tnotetext[2]{The second title footnote which is a longer \n text matter to fill through the whole text width and \n overflow into another line in the footnotes area of \n the first page.}\n\\end{vquote}\n\n\\verb+\\tnotemark+ and \\verb+\\tnotetext+ can be anywhere in\nthe front matter, but should be before \\verb+\\maketitle+ command.\n\n\\subsubsection{Author marks}\n\nAuthor names can have many kinds of marks and notes:\n\n\\begin{vquote}\n footnote mark : \\fnmark[<num>]\n footnote text : \\fntext[<num>]{<text>}\n affiliation mark : \\author[<num>]\n email : \\ead{<emailid>}\n url : \\ead[url]{<url>}\n corresponding author mark : \\cormark[<num>]\n corresponding author text : \\cortext[<num>]{<text>}\n\\end{vquote}\n\n\\subsubsection{Other marks}\n\nAt times, authors want footnotes which leave no marks in\nthe author names. The note text shall be listed as part of\nthe front matter notes. Class files provides\n\\verb+\\nonumnote+ for this purpose. The usage\n\n\\begin{vquote}\n\\nonumnote{<text>}\n\\end{vquote}\n\n\\noindent and should be entered anywhere before the \\verb+\\maketitle+\ncommand for this to take effect. \n\n\\subsection{Abstract and Keywords}\n\nAbstract shall be entered in an environment that starts\nwith\\break \\verb+\\begin{abstract}+ and ends with\n\\verb+\\end{abstract}+. Longer abstracts spanning more than\none page is also possible in slass file even in double\ncolumn mode. We need to invoke \\verb+longmktitle+ option in the\nclass loading line for this to happen smoothly.\n\nThe key words are enclosed in a \\verb+{keyword}+\nenvironment.\n\n\\begin{vquote}\n\\begin{abstract}\n This is an abstract. \\lipsum[3]\n\\end{abstract}\n\n\\begin{keywords}\n First keyword \\sep Second keyword \\sep Third \n keyword \\sep Fourth keyword\n\\end{keywords}\n\\end{vquote}\n\n\\section{Main Matter}\nMain matter contains sections, paragraphs, equations and floats like\ntables, figures, textboxes etc.\n\\subsection{Tables}\n\\subsubsection{Normal tables}\n\n\\begin{vquote}\n\\begin{table}\n \\caption{This is a test caption.}\n \\begin{tabular*}{\\tblwidth}{@{} LLLL@{} }\n \\toprule\n Col 1 & Col 2\\\\\n \\midrule\n 12345 & 12345\\\\\n 12345 & 12345\\\\\n 12345 & 12345\\\\\n 12345 & 12345\\\\\n 12345 & 12345\\\\\n 12345 & 12345\\\\\n \\bottomrule\n \\end{tabular*}\n\\end{table}\n\\end{vquote}\n\n\\subsubsection{Span tables}\n\n\\begin{vquote}\n\\begin{table*}[width=.9\\textwidth,cols=4,pos=h]\n \\caption{This is a test caption.}\n \\begin{tabular*}{\\tblwidth}{@{} LLLLLL@{} }\n \\toprule\n Col 1 & Col 2 & Col 3 & Col4 & Col5 & Col6 & Col7\\\\\n \\midrule\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n 12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n \\bottomrule\n \\end{tabular*}\n\\end{table*}\n\\end{vquote}\n\n\\subsection{Figures}\n\\subsubsection{Normal figures}\n\\begin{vquote}\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[scale=.75]{Fig1.pdf}\n\t\\caption{The evanescent light - $1S$ quadrupole coupling. \n See also Fig. \\protect\\ref{FIG:2}).}\n\t\\label{FIG:1}\n\\end{figure}\n\\end{vquote}\n\n\\subsubsection{Span figures}\n\n\\begin{vquote}\n\\begin{figure*}\n\t\\centering\n\t \\includegraphics[width=\\textwidth,height=2in]{Fig2.pdf}\n\t\\caption{Schematic of formation of the evanescent polariton on\n\tlinear chain of \\PMS.}\n \\label{FIG:2}\n\\end{figure*}\\end{vquote}\n\n\\subsection{Theorem and theorem like environments}\n\nCAS class file provides a few hooks to format theorems and\ntheorem like environments with ease. All commands the\noptions that are used with \\verb+\\newtheorem+ command will work\nexactly in the same manner. Class file provides three\ncommands to format theorem or theorem like environments:\n\n\\begin{enumerate}\n\\item \\verb+\\newtheorem+ command formats a theorem in\n\\LaTeX's default style with italicized font for theorem\nstatement, bold weight for theorem heading and theorem\nnumber typeset at the right of theorem heading. It also\noptionally accepts an argument which will be printed as an\nextra heading in parentheses. Here is an example coding and\noutput:\n\n\\begin{vquote}\n\\newtheorem{theorem}{Theorem}\n\\begin{theorem}\\label{thm}\n The \\WGM evanescent field penetration depth into the \n cuprous oxide adjacent crystal is much larger than the \n \\QE radius: \n \\begin{equation*}\n \\lambda_{1S}/2 \\pi \\left({\\epsilon_{Cu2O}-1}\n \\right)^{1/2} = 414 \\mbox{ \\AA} \\gg a_B = 4.6 \n \\mbox{ \\AA} \n \\end{equation*}\n\\end{theorem}\n\\end{vquote}\n\n\\item \\verb+\\newdefinition+ command does exactly the same\nthing as with except that the body font is up-shape instead\nof italic. See the example below:\n\n\\begin{vquote}\n\\newdefinition{definition}{Definition}\n\\begin{definition}\n The bulk and evanescent polaritons in cuprous oxide\n are formed through the quadrupole part of the light-matter\n interaction:\n \\begin{equation*}\n H_{int} = \\frac{i e }{m \\omega_{1S}} {\\bf E}_{i,s} \n \\cdot {\\bf p}\n \\end{equation*}\n\\end{definition}\n\\end{vquote}\n\n\\item \\verb+\\newproof+ command helps to define proof and\ncustom proof environments without counters as provided in\nthe example code. Given below is an example of proof of\ntheorem kind.\n\n\\begin{vquote}\n\\newproof{pot}{Proof of Theorem \\ref{thm}}\n\\begin{pot}\n The photon part of the polariton trapped inside the \\PMS\n moves as it would move in a micro-cavity of the effective\n modal volume $V \\ll 4 \\pi r_{0}^{3} /3$. Consequently, it\n can escape through the evanescent field. This evanescent\n field essentially has a quantum origin and is due to\n tunneling through the potential caused by dielectric\n mismatch on the \\PMS surface. Therefore, we define the\n \\emph{evanescent} polariton (\\EP) as an evanescent light -\n \\QE coherent superposition.\n\\end{pot}\n\\end{vquote}\n\n\\end{enumerate}\n\n\\subsection{Enumerated and Itemized Lists}\n\nCAS class files provides an extended list processing macros\nwhich makes the usage a bit more user friendly than the\ndefault LaTeX list macros. With an optional argument to the\n\\verb+\\begin{enumerate}+ command, you can change the list\ncounter type and its attributes. You can see the coding and\ntypeset copy. \n\n\\begin{vquote}\n\\begin{enumerate}[1.]\n \\item The enumerate environment starts with an optional\n argument `1.' so that the item counter will be suffixed\n by a period as in the optional argument.\n \\item If you provide a closing parenthesis to the number in the\n optional argument, the output will have closing \n parenthesis for all the item counters.\n \\item You can use `(a)' for alphabetical counter and `(i)' for\n roman counter.\n \\begin{enumerate}[a)]\n \\item Another level of list with alphabetical counter.\n \\item One more item before we start another.\n \\begin{enumerate}[(i)]\n \\item This item has roman numeral counter.\n\\end{vquote}\n\n\\begin{vquote}\n \\item Another one before we close the third level.\n \\end{enumerate}\n \\item Third item in second level.\n \\end{enumerate}\n \\item All list items conclude with this step.\n\\end{enumerate}\n\n\\section{Biography}\n\n\\verb+\\bio+ command have the below options:\n\\begin{enumerate}\n \\item \\verb+width:+ Width of the author photo (default is 1in).\n \\item \\verb+pos:+ Position of author photo.\n\\end{enumerate}\n\n\\begin{vquote}\n\\bio[width=10mm,pos=l]{tuglogo.jpg}\n \\textbf{Another Biography:}\n Recent experimental \\cite{HARA:2005} and theoretical\n \\cite{DEYCH:2006} studies have shown that the \\WGM can travel\n along the chain as \"heavy photons\". Therefore the \\WGM \n acquires the spatial dispersion, and the evanescent \n quadrupole polariton has the form (See Fig.\\ref{FIG:3}):\n\\endbio\n\\end{vquote}\n\n\\section[CRediT...]{CRediT authorship contribution statement}\n\nGive the authorship contribution after each author as \n\n\\begin{vquote}\n \\credit{Conceptualization of this study, Methodology, \n Software}\n\\end{vquote}\n\nTo print the details use \\verb+\\printcredits+ \n\n\\begin{vquote}\n\\author[1,3]{J.K. Krishnan}[type=editor,\n auid=000,bioid=1,\n prefix=Sir,\n role=Researcher,\n orcid=0000-0001-0000-0000]\n\\end{vquote}\n\n\\begin{vquote}\n\\cormark[1]\n\\fnmark[1]\n\\ead{jkk@example.in}\n\\ead[url]{www.jkkrishnan.in}\n\n\\credit{Conceptualization of this study, Methodology, Software}\n\n\\affiliation[1]{organization={Department of Physics, \n J.K. Institute of Science},\n addressline={Jawahar Nagar}, \n city={Trivandrum},\n postcode={695013}, \n state={Kerala},\n country={India}}\n\n\\author[2,4]{Han Thane}[style=chinese]\n\n\\author[2,3]{William {J. Hansen}}[ role=Co-ordinator,\n suffix=Jr,\n ]\n\\fnmark[2]\n\\ead{wjh@example.org}\n\\ead[URL]{https://www.university.org}\n\n\\credit{Data curation, Writing - Original draft preparation}\n\n . . .\n . . .\n . . .\n \\printcredits\n\\end{vquote}\n\n\\section{Bibliography}\n\nFor CAS categories, two reference models are recommended.\nle{cas-model2-names.bst}.\nFormer will format the reference list and their citations according to\nnumbered scheme whereas the latter will format according name-date or\nauthor-year style. Authors are requested to choose any one of these\naccording to the journal style. You may download these from \n\nThe above bsts are available in the following location for you to\ndownload:\n\n\\url{https://support.stmdocs.in/wiki/index.php?title=Model-wise_bibliographic_style_files} \n\\hfill $\\Box$\n\n\\end{document}\n\n\n\n"}}},{"rowIdx":293,"cells":{"id":{"kind":"string","value":"2412.20700v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20700v1"},"title":{"kind":"string","value":"Optimal rolling of fair dice using fair coins"},"full_text":{"kind":"string","value":"\\documentclass[12pt]{article}\n\n\\usepackage[backend=biber,style=numeric,]{biblatex}\n\\addbibresource{optimalDDG.bib}\n\n\\usepackage{amsfonts, amsmath}\n\n\\usepackage{amsthm}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}{Lemma}\n\n\\usepackage{booktabs}\n\n\\usepackage{etoolbox}\n\\preto\\tabular{\\setcounter{alglinecounter}{0}}\n\\newcounter{alglinecounter}\n\\newcommand{\\gk}{\\stepcounter{alglinecounter}\\arabic{alglinecounter})}\n\n\\usepackage{tikz}\n\n\\newcommand{\\unifdist}{\\textsf{Unif}}\n\\newcommand{\\berndist}{\\textsf{Bern}}\n\\newcommand{\\prob}{\\mathbb{P}}\n\\newcommand{\\mean}{\\mathbb{E}}\n\\newcommand{\\ind}{\\mathbb{I}}\n\n\\begin{document}\n\n\\title{Optimal rolling of fair dice using fair coins}\n\n\\author{Mark Huber and Danny Vargas}\n\n\\maketitle\n\n\\begin{abstract} In 1976, Knuth and Yao presented an algorithm for sampling from a finite distribution using flips of a fair coin that on average used the optimal number of flips. Here we show how to easily run their algorithm for the special case of rolling a fair die that uses memory linear in the input. Analysis of this algorithm yields a bound on the average number of coin flips needed that is slightly better than the original Knuth-Yao bound. This can then be extended to discrete distributions in a near optimal number of flips again using memory linear in the input.\n\\end{abstract}\n\n\\section{Introduction}\n\nA \\emph{probabilisitic Turing machine} is a Turing machine that has access to an independent, identically distributed (iid) sequence of flips of a fair coin. Specifically, say that \\( B \\) is the flip of a fair coin if \\( B \\) is equally likely to be 1 or 0. This means that \\( B \\) is uniform over \\( \\{0, 1\\} \\), written \\( B \\sim \\unifdist(\\{0, 1\\}) \\). Alternatively, say that \\( B \\) has the Bernoulli distribution with parameter \\( 1 / 2 \\), and write \\( B \\sim \\berndist(1 / 2) \\).\n\nA natural question to ask is how many flips of a fair coin would it take (on average) to generate a draw from a roll of a fair die? To be precise, if \\( \\prob(X = i) = 1 / i \\) for \\( i \\in \\{1, \\ldots, n \\} \\) say that \\( X \\) is the roll of a fair \\( n \\)-sided die. Write \\( X \\sim \\unifdist(\\{1, \\ldots, n\\}) \\) or \\( X \\sim \\text{d}n \\) for short.\n\nIn 1976, Knuth and Yao~\\cite{knuth1976complexity} developed the optimal algorithm for generating from any discrete probability distribution using a number of fair coin flips. Various authors (such as~\\cite{baidya2024efficient, saad2020fast, sinha2013high}) have looked at how to implement Knuth-Yao as efficiently as possible in different use cases.\n\nThe purpose of this work is to show how to implement their algorithm for generating from a fair die in a very simple way using a \\emph{randomness recycler}~\\cite{fill2000randomness, huber2016perfect} approach.\n\nThe memory used by the algorithm consists of storing the number \\( n \\), the original input, storing \\( m \\) a positive integer which is at most \\( 2n - 1 \\), storing \\( X \\) a positive integer at most \\( m \\), and one bit. Altogether, this uses \\( 3 \\lceil \\log_2(n) \\rceil + 3 \\) bits of memory when the input uses \\( \\lceil \\log_2(n) \\rceil \\) bits of memory.\n\nThe algorithm operates as follows. The idea is to always have a state that is an ordered pair \\( (X, m) \\) of two random integers. At any time during the algorithm's run, if the computation conducted by the algorithm used fair coin flips, then \\( [X | m] \\sim \\text{d}m \\). If \\( m = n \\), then the algorithm is done, since \\( X \\sim \\text{d}n \\) as desired.\n\nIf \\( m < n \\), the next step is to flip a fair coin \\( B \\) and change the state as follows.\n\\[\n(X, m) \\mapsto (X + Bm, 2m).\n\\]\nThe new state has \\( [X + Bm \\mid 2m] \\sim \\text{d}(2m) \\). This allows us to increase the number of sides of the die that \\( X \\) is a draw from until the number of sides reaches at least \\( n \\).\n\nAt this point, if \\( m \\geq n \\) and \\( X \\leq n \\) then \n\\[\n(X, m) \\mapsto (X, n).\n\\]\nThat is to say, accept \\( X \\) as a draw from \\( \\text{d}n \\). However, if \\( X > n \\), do not throw away the state and start over. Instead, note that \\( [X - m \\mid m - n] \\sim \\text{d}(m - n) \\). In other words, what is left over after subtracting the die roll is still a roll of a fair die, just one with a smaller number \\( m - n \\) of sides. That is,\n\\[\n(X, m) \\mapsto (X - n, m - n).\n\\]\n\n\nReturn to the doubling procedure and repeat until acceptance occurs.\n\nTo illustrate, consider trying to generate a roll from a 5-sided die. Begin in the state \\( (1, 1) \\). Since a 1-sided die only has output 1, \\( X = 1 \\) is a valid draw from a \\( \\text{d}1 \\).\n\nNow let \\( B_1, B_2, B_3, \\ldots \\) be the iid flips of the fair coin used by the algorithm. First, the number of sides of the die is doubled to 2, and the current draw from the die is \\( 1 + 1 \\cdot B_1 \\). So the new state is \\( (1 + B_1, 2) \\). Since \\( 2 < 5 \\), another coin is flipped to get to state \\( (1 + B_1 + 2 B_2, 4) \\). Since \\( 4 < 5 \\), flip another coin to get to state \\( (1 + B_1 + 2 B_2 + 4 B_3, 8) \\).\n\nNote that \\( B_1 + 2B_2 + 4B_3 \\) is uniform over \\( \\{0, 1, 2, \\ldots, 7\\} \\), so \\( X = 1 + B_1 + 2 B_2 + 4B_3 \\sim \\text{d}8 \\). At this point, if \\( X \\leq 5 \\), then this number is uniform over \\( \\{1, 2, 3, 4, 5\\} \\) and the algorithm terminates. If \\( X > 5 \\), then \\( X \\) is uniform over \\( \\{6, 7, 8\\} \\). Therefore \\( X - 5 \\) is uniform over \\( \\{1, 2, 3\\} \\), so \\( X - 5 \\) is a 3-sided die roll. Therefore the next state is\n\\[\n(X \\ind(X \\leq 5) + (X - 5) \\ind(X > 5), 5 \\ind(X \\leq 5) + (8 - 5) \\ind(X > 5))\n\\]\nwhere \\( \\ind(r) \\) is the indicator function that evaluates to 1 if expression \\( r \\) is true, and 0 otherwise.\n\nAt this point, if the second component is \\( 5 \\) the algorithm terminates, otherwise it continues by doubling the number of sides on the die from 3 to 6. Then it accepts if the state is at most \\( 6 \\), otherwise returning to state \\( (1, 1) \\). The evolution of the state for \\( n = 5 \\) is given in Figure~\\ref{FIG:example}. At a given state, when there is one child, take that path, when there are two, choose the child uniformly from the two choices.\n\n\\begin{figure}[h!]\n \\begin{center}\n \\begin{tikzpicture}[xscale=0.8]\n \\draw (7,0) node (A) {\\( (1, 1) \\)};\n \\draw (3, -1) node (B) {\\( (1, 2) \\)};\n \\draw (1, -2) node (D) {\\( (1, 4) \\)};\n \\draw (5, -2) node (E) {\\( (3, 4) \\)};\n \\draw (11, -1) node (C) {\\( (2, 2) \\)};\n \\draw (9, -2) node (F) {\\( (2, 4) \\)};\n \\draw (13, -2) node (G) {\\( (4, 4) \\)};\n\n\n \\draw (0, -3) node (H) {\\( (1, 8) \\)};\n \\draw (2, -3) node (I) {\\( (5, 8) \\)};\n\n \\draw (4, -3) node (J) {\\( (3, 8) \\)};\n \\draw (6, -3) node (K) {\\( (7, 8) \\)};\n\n \\draw (8, -3) node (L) {\\( (2, 8) \\)};\n \\draw (10, -3) node (M) {\\( (6, 8) \\)};\n\n \\draw (12, -3) node (N) {\\( (4, 8) \\)};\n \\draw (14, -3) node (O) {\\( (8, 8) \\)};\n\n \\draw (0, -4) node (P) {\\( (1, 5) \\)};\n \\draw (2, -4) node (Q) {\\( (5, 5) \\)};\n\n \\draw (4, -4) node (R) {\\( (3, 5) \\)};\n \\draw (6, -4) node (S) {\\( (2, 3) \\)};\n \\draw(5, -5) node (X) {\\( (2, 6) \\)};\n \\draw(5,-6) node(DD) {\\( (2, 5) \\)};\n \\draw(7, -5) node (Y) {\\( (5, 6) \\)};\n \\draw(7, -6) node (EE) {\\( (5, 5) \\)};\n\n \\draw (8, -4) node (T) {\\( (2, 5) \\)};\n \\draw (10, -4) node (U) {\\( (1, 3) \\)};\n \\draw(9, -5) node (Z) {\\( (1, 6) \\)};\n \\draw(11, -5) node (AA) {\\( (4, 6) \\)};\n \\draw(9, -6) node (FF) {\\( (1, 5) \\)};\n \\draw(11, -6) node (GG) {\\( (4, 5) \\)};\n\n \\draw (12, -4) node (V) {\\( (4, 5) \\)};\n \\draw (14, -4) node (W) {\\( (3, 3) \\)};\n \\draw(13, -5) node (BB) {\\( (3, 6) \\)};\n \\draw(15, -5) node (CC) {\\( (6, 6) \\)};\n \\draw(13, -6) node (HH) {\\( (3, 5) \\)};\n \\draw(15, -6) node (II) {\\( (1, 1) \\)};\n\n \n \n \\draw[->] (A) to (B);\n \\draw[->] (A) to (C);\n \\draw[->] (B) to (D);\n \\draw[->] (B) to (E);\n \\draw[->] (C) to (F);\n \\draw[->] (C) to (G);\n \\draw[->] (D) to (H);\n \\draw[->] (D) to (I);\n \\draw[->] (E) to (J);\n \\draw[->] (E) to (K);\n \\draw[->] (F) to (L);\n \\draw[->] (F) to (M);\n \\draw[->] (G) to (N);\n \\draw[->] (G) to (O);\n \\draw[->] (H) to (P);\n \\draw[->] (I) to (Q);\n \\draw[->] (J) to (R);\n \\draw[->] (K) to (S);\n \\draw[->] (L) to (T);\n \\draw[->] (M) to (U);\n \\draw[->] (N) to (V);\n \\draw[->] (O) to (W);\n \\draw[->] (S) to (X);\n \\draw[->] (S) to (Y);\n \\draw[->] (U) to (Z);\n \\draw[->] (U) to (AA);\n \\draw[->] (W) to (BB);\n \\draw[->] (W) to (CC);\n \\draw[->] (X) to (DD);\n \\draw[->] (Y) to (EE);\n \\draw[->] (Z) to (FF);\n \\draw[->] (AA) to (GG);\n \\draw[->] (BB) to (HH);\n \\draw[->] (CC) to (II);\n \n \\end{tikzpicture}\n \\end{center}\n \\caption{From each state, each child is equally likely to be chosen. When a state of the form \\( (i, 5) \\) is reached, output \\( i \\). When state \\( (1, 1) \\) is reached, begin again at the root.}\n \\label{FIG:example}\n\\end{figure}\n \nIn general, the pseudocode for this algorithm is as follows.\n\n\\vspace*{12pt}\n\\begin{tabular}{rl}\n \\toprule\n \\multicolumn{2}{l}{\\textbf{Algorithm} \\textsf{optimal\\_uniform}(n) } \\\\\n \\midrule\n \\gk & \\hspace{0em} \\( m \\leftarrow 1,\\ X \\leftarrow 1 \\) \\\\\n \\gk & \\hspace*{0em} while \\( m < n \\) \\\\\n \\gk & \\hspace*{1em} draw \\( B \\) uniform over \\( \\{0, 1\\} \\) \\\\\n \\gk & \\hspace*{1em} \\( X \\leftarrow X + Bm \\), \\( m \\leftarrow 2m \\) \\\\\n \\gk & \\hspace*{1em} if \\( m \\geq n \\) and \\( X \\leq n \\) then \\( m \\leftarrow n \\) \\\\\n \\gk & \\hspace*{1em} if \\( m \\geq n \\) and \\( X > n \\) then \\( X \\leftarrow X - n \\), \\( m \\leftarrow m - n \\) \\\\ \n \\gk & \\hspace*{0em} return \\( X \\) \\\\ \n \\bottomrule\n\\end{tabular}\n\\vspace*{12pt}\n\nThis type of algorithm, where either acceptance occurs, or rejection occurs and as much of the randomness leftover is used in the next state is called the \\emph{randomness recycler} protocol~\\cite{fill2000randomness, huber2016perfect}.\n\nThis method works best with a fair die. The \\emph{Alias method}~\\cite{walker1977efficient} gives a close to efficient method of generating from a loaded die, that is, an arbitrary distribution over a finite number of states. The new method presented here can be extended to handle this situation as well, but it is less simple to implement.\n\nThe rest of the work is organized as follows. The next section contains a proof that the algorithm is correct and that it uses the optimal number of coin flips. Section~\\ref{sec:three} shows an upper bound on the expected number of flips needed that is slightly better than that of~\\cite{knuth1976complexity}. \nThe last section then shows how this technique can be extended to loaded dice.\n\n\\section{Proof of correctness}\n\n\\begin{theorem}\n\\label{thm:one}\n The output of \\( \\textsf{optimal\\_uniform}(n) \\) is uniform over \\( \\{1, \\ldots, n \\} \\).\n\\end{theorem}\n\nIt helps to have the following fact in place.\n\n\\begin{lemma}\n At the end of each line of the algorithm, for state \\( (X, m) \\) it holds that \\( [X \\mid m] \\sim \\text{d}m \\).\n\\end{lemma}\n\n\\begin{proof}\n The state \\( (X, m) \\) is changed by lines 1, 4, 5, and 6. When \\( (X, m) = (1, 1) \\), it is trivially true that \\( X \\sim \\text{d}1 \\), so line 1 maintains this invariant.\n\n Suppose \\( (X, m) \\) has \\( X \\sim \\text{d}m \\) before running line 4, 5, or 6. For line 4, if \\( X \\sim \\unifdist(\\{1, \\ldots, m\\} \\) then \\( X + m \\sim \\unifdist(\\{m + 1, \\ldots, 2m\\} \\). Since \\( B \\) is equally likely to be \\( 0 \\) or \\( 1 \\), \\( X + Bm \\) is equally likely to be any element of \\( \\{1, \\ldots, 2m \\} \\) and the invariant is maintained.\n\n For line 5, it is a well known fact that for \\( X \\sim \\text{d}m \\) where \\( m \\geq n \\), \\( [X | X \\leq n] \\sim \\text{d}n \\). Since \\( X \\leq n \\) leads to \\( m = n \\), \\( [X | m = n] \\sim \\text{d}n \\), Similarly for line 6, if \\( X \\sim \\text{d}m \\) where \\( m > n \\), \\( [X \\mid X > n] \\sim \\unifdist(\\{n + 1, \\ldots, m \\} \\). So \\( [X - n \\mid X > n] \\sim \\unifdist(\\{1, \\ldots, m - n\\} \\).\n \n An induction then completes the proof.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:one}]\nThe theory of the \\emph{randomness recycler} (see for instance~\\cite{fill2000randomness,huber2016perfect}) says that two conditions are necessary and sufficient to establish correctness. The first is that \\( [X | m] \\sim \\text{d}m \\) throughout the running of the algorithm, which is true from the previous lemma. The second is that the algorithm terminates with probability 1. After at most \\( \\lceil \\log_2(n) \\rceil \\) steps there will have been at least one opportunity for \\( m \\) to grow to be at least \\( n \\). When \\( m \\in \\{n, n + 1, \\ldots, 2n - 1\\} \\), there is at least a \\( 1 / m \\geq 1 / (2n) \\) chance of termination. Hence the probability that the algorithm terminates approaches 1 as the number of steps goes to infinity.\n\\end{proof}\n\n\\section{The optimal DDG}\n\nIn fact, this simple algorithm is not only correct, it is optimal in the following sense. \n\nConsider any algorithm that utilizes random bits to reach an outcome. This can be represented using a \\emph{discrete distribution generating} tree, called a DDG for short. This is a rooted binary tree where each leaf of the tree is associated with an outcome. The algorithm works by starting with the current node equal to the root of the tree. A coin is flipped, if the result is 0 move to the left child of the current node, otherwise move to the right. If the resulting node is a leaf, output the outcome associated with that leaf. Otherwise make this the new current node, and repeat.\n\nFigure~\\ref{fig:DDG} shows two DDGs that output a random variable with probabilities \\( (3 / 8, 1 / 2, 1 / 8) \\) over outcomes \\( (1, 2, 3) \\). The tree on the right will be preferred in practice because the average depth (and hence average number of coin flips used by the algorithm) is smaller.\n\n\\begin{figure}[ht!]\n\\centering\n \\begin{tikzpicture}\n \\draw (0, 0) to (-1, -1);\n \\draw (0, 0) to (1, -1);\n \\draw (-1, -1) to (-1.5, -2);\n \\draw (-1, -1) to (-0.5, -2);\n \\draw (1, -1) to (1.5, -2);\n \\draw (1, -1) to (0.5, -2);\n \\draw (-1.5, -2) to (-1.75, -3) node[below] {1};\n \\draw (-1.5, -2) to (-1.25, -3) node[below] {1};\n \\draw (-0.5, -2) to (-0.25, -3) node[below] {1};\n \\draw (-0.5, -2) to (-0.75, -3) node[below] {2};\n \\draw (1.5, -2) to (1.75, -3) node[below] {2};\n \\draw (1.5, -2) to (1.25, -3) node[below] {2};\n \\draw (0.5, -2) to (0.25, -3) node[below] {2};\n \\draw (0.5, -2) to (0.75, -3) node[below] {3};\n \\end{tikzpicture} \n \\hspace*{12pt}\n \\begin{tikzpicture}\n \\draw (0, 0) to (-1, -1) node[below] {2};\n \\draw (0, 0) to (1, -1);\n \\draw (1, -1) to (1.5, -2);\n \\draw (1, -1) to (0.5, -2) node[below] {1};\n \\draw (1.5, -2) to (1.75, -3) node[below] {1};\n \\draw (1.5, -2) to (1.25, -3) node[below] {3};\n \\end{tikzpicture} \n \\caption{Two DDGs for \\( (3 / 8, 1 / 2, 1 / 8) \\). The DDG on the left will always use 3 flips to determine the outcome, while the one on the right uses 1 flip with probability \\( 1 / 2 \\), 2 flips with probability \\( 1 / 4 \\), or 3 flips with probability \\( 1 / 4 \\).}\n \\label{fig:DDG}\n\\end{figure}\n\nKnuth and Yao defined an optimal DDG for a probability distribution \\( p = (p_1, \\ldots, p_n) \\) as follows. Let \\( N(T) \\) denote the random number of bits used by DDG \\( T \\). For \\( p \\), say that \\( T \\) is \\emph{optimal} if for any other DDG \\( T' \\) with output \\( p \\), for all \\( i \\), \\( \\prob(N(T) > i) \\leq \\prob(N(T') > i) \\). \n\nAnother way to say this is that the number of flips used by the optimal tree is stochastically dominated by the number of flips used by any other tree.\n\nAs usual, define the \\emph{level} of a node to be the number of edges in the path from the root to the node. Knuth and Yao showed the following result~\\cite{knuth1976complexity}.\n\\begin{theorem}\n For probability vector \\( (p_1, \\ldots, p_n) \\), a DDG is optimal if and only if outcome \\( i \\) appears on level \\( j \\) \n a number of times equal to \n \\[\n \\lfloor 2^j p_i \\rfloor - 2 \\lfloor 2^{j - 1} p_i \\rfloor\n \\] (this is the \\( j \\)th bit in the binary expansion of \\( p_i \\), using the finite expansion when possible.)\n\\end{theorem}\n\nThis is equivalent to the following.\n\\begin{lemma}\n For probability vector \\( (p_1, \\ldots, p_n) \\), a DDG is optimal if and only if outcome \\( i \\) appears on any given level either 0 or 1 times, and there is no \\( k \\) such that outcome \\( i \\) appears at level \\( k, k + 1, k + 2, \\ldots \\). \n\\end{lemma}\n\n\\begin{proof}\n If outcome \\( i \\) appears once at level \\( j \\), this leaf contributes \\( 1 / 2^j \\) to the probability of outcome \\( i \\). Since \\( p_i \\) has a unique binary expansion where there are not an infinite number of trailing 1's, this DDG must be equivalent to \\( i \\) appearing on level \\( i \\) if and only if the (finite if possible) binary expansion of \\( p_i \\) has a 1 at level \\( i \\). \n\\end{proof}\n\nThis characterization can show that \\( \\textsf{optimal\\_uniform}(n) \\) is optimal.\n\n\\begin{theorem}\n The algorithm \\( \\textsf{optimal\\_uniform}(n) \\) yields an optimal DDG.\n\\end{theorem}\n\n\\begin{proof}\n\n Consider a state \\( (X, m) \\) at a level greater than 1 in the tree. Then there is exactly one state \\( (X', m') \\) and coin flip \\( B' \\) that moves to this state.\n\n If the previous level did not have any leaves, then \\( m' = m / 2 \\), \\( B' = \\ind(X > m') \\), and \n \\[\n X' = X - m' \\ind(X > m')\n \\]\n is the only state and flip that leads to \\( (X, m) \\).\n\n If the previous level did have leaves, then the state that moves at line 6 to \\( (X, m) \\) is \\( (X + n, m + n) \\). Therefore, the state at the previous level that moves to this state is \\( m' = (m + n) / 2 \\), \\( B' = \\ind(X > m') \\), and \n \\[\n X' = X + n - B m'\n \\]\n is the only option for the previous state and flip that leads to the current state.\n\n But since the initial state is fixed at \\( (1, 1) \\), this forms the basis of an induction proof that any state \\( (X, m) \\) at a later level is determined by a unique set of flips. In particular, if \\( (X, m, B) \\) moves to a leaf at a particular level, then there is a unique set of flips that led to that state. That means that for any outcome at a particular level, there is exactly one leaf at that level that results in that state. Therefore each outcome appears on at most one leaf at each level.\n\n Now consider how many levels in a row the DDG can have a particular outcome as a leaf. A leaf only occurs when \\( m \\geq n \\) and \\( X \\leq n \\) at line 6. If \\( X > n \\), then at the next state \\( m \\leftarrow m - n \\). \n\n Therefore, if there are \\( r \\) values for \\( X \\) that reject at that level, and at the next level there are at most \\( 2r - n \\) states of \\( X \\) that can reject at line 6.\n\n At the next level there will be at most \\( 4r - 2n - n = 4r - 3n \\) values of \\( X \\) that reject, then \\( 8r - 7n \\), then \\( 16r - 15n \\) and so on. After \\( k \\) levels there will be \\( 2^k r - (2^k - 1)n = 2^k(r - (1 - (1 / 2^k)n \\) states of \\( X \\) that lead to rejection. Because \\( r < n \\), this either reaches 0 or goes negative, indicating that rejection does not occur. If rejection does not occur at a particular level then there cannot be a leaf at that same level.\n\n Therefore, it cannot be the case that the DDG for this algorithm has an infinite sequence of consecutive levels with leaves. Therefore, by the previous lemma this DDG must be optimal.\n \n\\end{proof}\n\n\\section{Upper bounding the expected number of flips}\n\\label{sec:three}\n\nIt is straightforward to calculate exactly the average number of flips used by \\( \\textsf{optimal\\_sampling}(n) \\) for any particular \\( n \\).\n\nFor instance, if \\( n = 5 \\) it takes three flips to make \\( m = 8 \\), at which point there is a \\( 5 / 8 \\) chance of accepting the result and terminating the algorithm. If it does not accept then \\( m = 8 - 5 = 3 \\), and one more flip makes \\( m = 6 \\). At this point either the algorithm accepts with probability \\( 5 / 6 \\) or rejects and sets \\( m = 6 - 5 = 1 \\), so the algorithm starts from the beginning. Hence\n\\[\n\\mean[N_{n = 5}] = 3 + \\frac{3}{8}\\left[1 + \\frac{1}{6} \\mean[N_{n = 5}] \\right],\n\\]\nso \\( \\mean[N_{n = 5}] = 3.6 \\).\n\nNote that any DDG for a fair die will need at least \\( \\lceil \\log_2(n) \\rceil \\) bits on average. Results in~\\cite{knuth1976complexity} give that \\( \\log_2(n) + 2 \\) bits suffice on average. The following result strengthens this slightly.\n\n\\begin{theorem}\n The expected number of flips used by \\( \\textsf{optimal\\_sampling}(n) \\) is at most \\( \\lceil \\log_2(n) \\rceil + 1 \\).\n\\end{theorem}\n\n\\begin{proof}\n Start the algorithm by flipping coins until the initial state \\( m_1 \\) has doubled to be at least \\( n \\). This will require \\( f_1 = \\lceil \\log_2(n / m) \\rceil \\) flips. After these flips, say that the state is \\( (X_2, m_2) \\) where \\( m_2 \\in \\{n, \\ldots, 2n - 1\\} \\). To be specific,\n \\[\n m_2 = 2^{\\lceil \\log_2(n / m) \\rceil}.\n \\]\n\n The chance of accepting is \\( n / m_2 \\), so the chance that further work is needed is \\( 1 - n / m_2 \\). If further work is needed, then \\( f_2 \\) more flips are needed to double \\( m_2 \\) back up to be at least \\( n \\), and so on. Altogether the expected number of flips \\( t \\) is \n \\[\n t = f_1 + \\left(1 - \\frac{n}{m_2} \\right) \\left[ f_2 + \\left(1 - \\frac{n}{m_3}\\right)\\left[f_3 + \\left( 1 - \\frac{n}{m_4} \\right) \\cdots \\right.\\right.\n \\]\n where \\( f_k = \\lceil \\log_2(n / (n - m_{k})) \\rceil \\) for \\( k \\geq 2 \\).\n \n Suppose this is thought of as a function of \\( m_2, m_3, \\ldots \\) chosen independently of each other in \\( \\{n, \\ldots, 2n - 1\\} \\). Then once \\( m_2 \\) is chosen to maximize the result, \\( m_3 \\) should receive the same value because of the self-similarity of the expression. \n Hence there is \\( m^* \\in \\{n, \\ldots, 2n - 1\\} \\) such that \n \\[\n t \\leq f_1 + \\left(1 - \\frac{n}{m^*} \\right) \\left[ f_{m^*} + \\left(1 - \\frac{n}{m^*}\\right)\\left[f_{m^*} + \\left( 1 - \\frac{n}{m^*} \\right) \\cdots \\right.\\right.\n \\]\n\n Setting \n \\[\n \\alpha = \\left(1 - \\frac{n}{m^*} \\right) \\left[ f_{m^*} + \\left(1 - \\frac{n}{m^*}\\right)\\left[f_{m^*} + \\left( 1 - \\frac{n}{m^*} \\right) \\cdots \\right.\\right.\n \\]\n gives\n \\[\n \\alpha = \\left(1 - \\frac{n}{m^*} \\right) \\left[ f_{m^*} + \\alpha \\right],\n \\]\n so\n \\[\n \\alpha = \\left( \\frac{m^*}{n} - 1 \\right) \\left(\\lceil \\log_2\\left(\\frac{n}{m^* - n} \\right) \\right\\rceil.\n \\]\n\n Let \\( \\beta = (m^* - n) / n \\). If \\( \\beta \\in (1 / 2, 1] \\), then \\( \\alpha = 1 \\cdot 1 = 1 \\). If \\( \\beta \\in (1 / 4, 1/ 2] \\), then \\( \\alpha = (1 / 2) \\cdot 2 = 1 \\). If \\( \\beta \\in (1 / 8, 1 / 4] \\), \\( \\alpha = (1 / 4) \\cdot 3 = 3 / 4 \\), and for smaller \\( \\beta \\) the value of \\( \\alpha \\) only shrinks further.\n\n Hence \\( t \\leq \\lceil \\log_2(n) \\rceil + 1 \\).\n \n\\end{proof}\n\n\\section{Extending to discrete distributions}\n\\label{sec:four}\n\nFor the general algorithm, the idea is to also keep state \\( (X, m) \\), but use \\( X \\) to draw uniformly from the set of outcomes that have a \\( i \\) in their bit description at the bit equal to the number of flips of the coin taken. \n\n\\vspace*{12pt}\n\\begin{tabular}{rl}\n \\toprule\n \\multicolumn{2}{l}{\\textbf{Algorithm} \\( \\textsf{optimal\\_discrete}(p = (p_1, p_2, \\ldots)) \\) } \\\\\n \\midrule\n \\gk & \\hspace{0em} \\( m \\leftarrow 1,\\ X \\leftarrow 1, Y \\leftarrow 0 \\) \\\\\n \\gk & \\hspace*{0em} while \\( Y = 0 \\) \\\\\n \\gk & \\hspace*{1em} let \\( A \\leftarrow\\{i:2p_i \\geq 1\\} \\),\n \\( n \\leftarrow \\#(A) \\) \\\\\n \\gk & \\hspace*{1em} for all \\( i \\), let \\( p_i \\leftarrow 2p_i - \\ind(2p_i \\geq 1)\\) \\\\\n \\gk & \\hspace*{1em} draw \\( B \\) uniform over \\( \\{0, 1\\} \\) \\\\\n \\gk & \\hspace*{1em} \\( X \\leftarrow X + Bm \\), \\( m \\leftarrow 2m \\) \\\\\n \\gk & \\hspace*{1em} if \\( m \\geq n \\) and \\( X \\leq n \\) then \\( m \\leftarrow n \\), let \\( Y \\) be the \\( X \\)th element of \\( A \\). \\\\\n \\gk & \\hspace*{1em} if \\( m \\geq n \\) and \\( X > n \\) then \\( X \\leftarrow X - n \\), \\( m \\leftarrow m - n \\) \\\\ \n \\gk & \\hspace*{0em} return \\( Y \\) \\\\ \n \\bottomrule\n\\end{tabular}\n\\vspace*{12pt}\n\n\\begin{theorem}\n Algorithm \\( \\textsf{optimal\\_discrete}(p) \\) is optimal.\n\\end{theorem}\n\n\\begin{proof}\n Although the value of \\( n \\) changes from flip to flip, the arguments used in the uniform case remain the same. Here there is a unique set of flips that results in a particular state \\( (X, m) \\) and hence any particular leaf outcome appears only once at a level. \n\n Here \\( A \\) is chosen after \\( d \\) flips so that \\( i \\in A \\) if and only if \\( \\lceil 2^d A_i \\rceil = 1 \\), so the DDG must be optimal.\n\\end{proof}\n\nOf course this algorithm is a bit more abstract than the first, given that to actually implement line 3 requires an analysis of the probability distribution being used to determine which of the outcomes have a 1 bit in the position equal to the number of flips. \n\n\n\\section{Conclusion}\n\nThe Knuth-Yao optimal sampler can be implemented using a randomness recycler type algorithm. This allows for a very simple implementation of the algorithm for generating the roll of a fair die and can be extended to handle general discrete distributions given the ability to draw out the \\( i \\)th bit in the binary expansion of the probabilities.\n\n\\printbibliography\n\n\\end{document}\n\nFor distributions that are finite with a finite probability vector \\( (p_1, \\ldots, p_n) \\), this requires at most \\( \\Theta(N n) \\) steps of work, where \\( N \\) is the number of bits used by the optimal DDG.\n\n\nA more concrete way of doing things is to use the \\emph{Alias} method of Walker~\\cite{walker1977efficient}.\n\nGiven input \\( (p_1, \\ldots, p_n) \\), Walker showed how to create in linear time (and memory) a table \\( f:\\{0, 1\\} \\rightarrow \\{1, \\ldots, n\\} \\) and a new probability vector \\( (a_1, \\ldots, a_n) \\) such that if \\( U \\sim \\text{d}n \\) and \\( [B \\mid U] \\sim \\berndist(a_U) \\), then \\( f(B, U) \\sim (p_1, \\ldots, p_n) \\).\n\nNote that flipping a \\( \\berndist(p) \\) random variable can be done for any \\( p \\in (0, 1) \\) using on average 2 flips with Knuth-Yao sampling.\n\n\\vspace*{12pt}\n\\begin{tabular}{rl}\n \\toprule\n \\multicolumn{2}{l}{\\textbf{Algorithm} \\( \\textsf{optimal\\_bern}(p) \\) } \\\\\n \\midrule\n \\gk & \\hspace*{0em} repeat \\\\\n \\gk & \\hspace*{1em} draw \\( B \\) uniform over \\( \\{0, 1\\} \\) \\\\\n \\gk & \\hspace*{1em} if \\( B = 1 \\) and \\( p \\geq 1 / 2 \\) then return(1) \\\\\n \\gk & \\hspace*{1em} if \\( B = 1 \\) and \\( p < 1 / 2 \\) then return(0) \\\\\n \\gk & \\hspace*{1em} if \\( B = 0 \\) and \\(p \\geq 1 / 2\\) then \\( p \\leftarrow 2p - 1 \\) \\\\\n \\gk & \\hspace*{1em} if \\( B = 0 \\) and \\(p < 1 / 2\\) then \\( p \\leftarrow 2p \\) \\\\\n \\bottomrule\n\\end{tabular}\n\\vspace*{12pt}\n\nTherefore, the number of flips needed to generate from \\( (p_1, \\ldots, p_n) \\) (after linear time \npreprocessing) \n\n\n"}}},{"rowIdx":294,"cells":{"id":{"kind":"string","value":"2412.20782v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20782v1"},"title":{"kind":"string","value":"A randomisation method for mean-field control problems with common noise"},"full_text":{"kind":"string","value":"\\documentclass[\n 10pt,\n a4paper\n]{article}\n\\usepackage{amsmath}\n\\usepackage{color} \n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{accents}\n\\usepackage[utf8]{inputenc}\n\\usepackage[english]{babel}\n\\usepackage{bbm}\n\\usepackage{chngcntr}\n\\usepackage{enumitem}\n\\usepackage{listings}\n\\usepackage{lmodern} \\usepackage{mathtools}\n\\usepackage[unicode=true,bookmarks=false,hypertexnames=false]{hyperref}\n\\usepackage{graphicx,tikz,tabularx} \\usepackage{csquotes}\n\\usepackage[textwidth=16cm,textheight=23.5cm]{geometry}\n\\usepackage[capitalise,noabbrev,nameinlink,sort]{cleveref}\n\\let\\etoolboxforlistloop\\forlistloop \\usepackage{autonum}\n\\let\\forlistloop\\etoolboxforlistloop \\cslet{blx@noerroretextools}\\empty \n\n \n\\counterwithin{equation}{section}\n\n\\newcommand{\\ubar}[1]{\\underaccent{\\bar}{#1}}\n\n\\DeclarePairedDelimiter\\abs{\\lvert}{\\rvert}\\DeclarePairedDelimiter\\norm{\\lVert}{\\rVert}\n\\DeclareMathOperator*{\\esssup}{ess\\,sup}\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\DeclareMathOperator*{\\argmin}{arg\\,min}\n\n\\newcommand{\\<}{\\langle}\n\\renewcommand{\\>}{\\rangle}\n\n\\let\\oldc\\c \\renewcommand{\\c}[1]{\\mathcal{#1}}\n\n\\renewcommand{\\b}[1]{\\mathbb{#1}}\n\\renewcommand{\\bf}[1]{\\mathbf{#1}}\n\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\renewcommand{\\P}{\\mathbb{P}}\n\\DeclareMathOperator{\\E}{\\mathbb{E}}\n\\DeclareMathOperator{\\Var}{Var}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\n\\newcommand{\\Pc}{\\mathcal{P}}\n\n\\newcommand{\\1}{\\mathbbm{1}}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{definition}[theorem]{Definition}\n\n\n\n\\crefalias{enumi}{part}\n\\newtheorem{assumption}{Assumption}\n\\crefname{assumption}{Assumption}{Assumptions}\n\n\\renewcommand{\\theassumption}{\\Alph{assumption}}\n\\newlist{assumptionenum}{enumerate}{1} \\setlist[assumptionenum]{label=(\\theassumption\\arabic*)}\\crefalias{assumptionenumi}{assumption}\n\n\n\\let\\eqref\\labelcref\n\\makeatletter\n\\autonum@generatePatchedReferenceCSL{eqref}\n\\makeatother\n\n\n\n\\def\\blu#1{{\\color{blue}#1}}\n\\def\\rd#1{{\\color{red}#1}}\n\\def\\gr#1{{\\color{green}#1}}\n\\def\\red#1{{\\color{red}#1}}\n\\def\\ora#1{{\\color{orange}#1}}\n\\def\\cy#1{{\\color{cyan}#1}}\n\\def\\teal#1{{\\color{teal}#1}}\n\n\n\\begin{document}\n\n\\title{A randomisation method for mean-field control problems with common noise}\n\\author{Robert Denkert\\thanks{Humboldt University of Berlin, Department of Mathematics, rob at denkert.eu; This author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -- Project-ID 410208580 -- IRTG2544 (``Stochastic Analysis in Interaction'').}\n\\quad Idris Kharroubi\\thanks{\nLPSM, UMR CNRS 8001, Sorbonne Universit\\'e\n and Universit\\'e Paris Cit\\'e, idris.kharroubi at sorbonne-universite.fr. Research of the author partially supported by Agence Nationale de la Recherche (ReLISCoP grant ANR-21-CE40-0001).\n}\n\\quad Huy\\^en Pham\\thanks{Ecole Polytechnique, CMAP, huyen.pham at polytechnique.edu; This author is supported by the BNP-PAR Chair ``Futures of Quantitative Finance\", and by FiME, Laboratoire de Finance des March\\'es de l'Energie, and the ``Finance and Sustainable Development'' EDF - CACIB Chair}\n}\n\\maketitle\n\n\\begin{abstract}\n We study mean-field control (MFC) problems with common noise using the control randomisation framework, where we substitute the control process with an independent Poisson point process, controlling its intensity instead. To address the challenges posed by the mean-field interactions in this randomisation approach, we reformulate the admissible control as $L^0$-valued processes adapted only to the common noise. We then construct the randomised control problem from this reformulated control process, and show its equivalence to the original MFC problem. Thanks to this equivalence, we can represent the value function as the minimal solution to a backward stochastic differential equation (BSDE) with constrained jumps. Finally, using this probabilistic representation, we derive a randomised dynamic programming principle (DPP) for the value function, expressed as a supremum over equivalent probability measures.\n\\end{abstract}\n\n\\vspace{5mm}\n\n\\noindent \\textbf{ MSC Classification}: 60H30; 60K35; 60K37; 93E20. \n\n\n\\vspace{3mm}\n\n\n\\noindent \\textbf{ Keywords:}{ Mean-field control with common noise; control randomisation; decomposition of processes;\n randomised dynamic programming principle; backward stochastic differential equations. \n \n\n\n\n\\section{Introduction}\n\nWe will consider mean-field control problems with the following state dynamics\n\\begin{align}\n dX_s^{t,\\xi,\\alpha}\n &= b(s, \\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\P}_s},X^{t,\\xi,\\alpha}_s,\\alpha_s)ds + \\sigma(s,\\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\P}_s},X^{t,\\xi,\\alpha}_s,\\alpha_s) dW_s\\\\\n &\\quad + \\sigma^0(s,\\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\P}_s},X^{t,\\xi,\\alpha}_s,\\alpha_s) dB_s,\\qquad X^{t,\\xi,\\alpha}_t = \\xi,\n\\end{align}\nwith two independent Brownian motions $W,B$ on some probability space $(\\Omega,\\c F,\\P)$ and an independent given starting value $\\xi$ and a control process $\\alpha$ in some action space $A$.\nHere $(\\P^{\\c F^{B,\\P}_s}_{X^{t,\\xi,\\alpha}_s})_{s\\in [t,T]}$ denotes the $\\b F^{B,\\P}$-predictable and $\\P$-a.s.\\@ continuous version of $(\\c L(X^{t,\\xi,\\alpha}_s|\\c F^{B,\\P}_s))_{s\\in [t,T]}$.Our goal is to maximise the reward functional\n\\begin{align}\n J(t,\\xi,\\alpha) \\coloneqq \\E\\Big[ g(X^{t,\\xi,\\alpha}_T, \\P_{X^{t,\\xi,\\alpha}_T}^{{\\c F}^{B,\\P}_T}) + \\int_t^T f(s,X^{t,\\xi,\\alpha}_s,\\P_{X^{t,\\xi,\\alpha}_s}^{{\\c F}^{B,\\P}_s},\\alpha_s) ds \\Big].\n\\end{align}\nThere are generally two main approaches to solving mean-field (or McKean-Vlasov) control problems in the literature. The first is the stochastic maximum principle, which characterises the optimal control by an adjoint backward stochastic differential equation (BSDE). In the context of mean-field control problems, the stochastic maximum principle was first studied by \\cite{buckdahn_general_2011,andersson_maximum_2011} for cases where the coefficients only depend on the law through its moments. Building on the notion of differentiability introduced by \\cite{lions2007theorie} in his course at Coll\\`ege de France, \\cite{carmona_forwardbackward_2015} extended this approach to general MFC problems; see also the standard work \\cite{carmona_probabilistic_1_2018} for a broader overview of the related literature.\nThe second approach revolves around the dynamic programming principle (DPP), which breaks the original optimisation problem down into several local optimisation problems. Combined with an Itô formula for McKean-Vlasov diffusions, this method leads to a Hamilton-Jacobi-Bellman (HJB) equation on the Wasserstein space that characterises the value function. \\cite{pham_bellman_2018} were the first to develop such a DPP in the general setting without common noise, while \\cite{pham_dynamic_2017} derived a DPP for MFC problems with common noise, where the admissible controls are solely adapted to the common noise. To our knowledge, the most comprehensive results on mean-field control problems involving common noise can be found in works of \\cite{djete_mckeanvlasov_2022,djete_mckeanvlasov_2022-1}.\n\nIn this paper, we establish a randomised dynamic programming principle based on a randomised formulation of the control problem. The idea is to replace the control process by an independent Poisson point process, and then optimising over a set of equivalent probability measures, effectively making the intensity of the Poisson point process the new control.\nThe randomisation method was introduced by \\cite{bouchard_stochastic_2009} to deal with optimal switching problems, with further works for optimal switching problems found in \\cite{elie_bsde_2014,elie_adding_2014,elie_probabilistic_2010} and with \\cite{fuhrman_optimal_2020} addressing the infinite dimensional case. Additionally, it has been applied to impulse control problems in \\cite{kharroubi_backward_2010} and to optimal stopping in non-Markovian settings in \\cite{fuhrman_representation_2016}. \\cite{fuhrman_randomized_2015} applied the randomisation method to non-Markovian regular control problems. \nIn addition, \\cite{kharroubi_feynmankac_2015} studied the constrained BSDE for the randomised value function across a wide range of control problems, establishing a general connection between a class of constrained BSDEs and HJB equations.\nThe randomisation framework has also found applications beyond establishing a randomised DPP. Specifically, \n\\cite{kharroubi_discrete_2015,kharroubi_numerical_2014} proposed a numerical scheme for fully non-linear HJB equations based on the constrained BSDEs arising from the control randomisation framework, leading to various applications including portfolio optimisation \\cite{zhang_dynamic_2019} and algorithmic trading \\cite{abergel_algorithmic_2020}.\nAdditionally, \\cite{denkert_control_2024} introduced a policy gradient method for continuous-time reinforcement learning based on the equivalence between the original problem and its randomised formulation.\n\nIn a mean-field setting, the randomisation method as a tool has previously only been considered by \\cite{bayraktar_randomized_2018} based on a decoupled MKV representation introduced by \\cite{buckdahn_mean-field_2017} in the uncontrolled case.\n\\cite{bayraktar_randomized_2018} provided a randomisation method for mean-field control problems without common noise and used this to obtain a Feynman-Kac formulation and also a randomised dynamic programming principle. \nTheir results apply to MFC problems with \\emph{deterministic} initial laws of the form $\\pi = \\delta_x$, $x\\in\\R^d$. This result does not extend to general initial laws as the following example illustrates.\n\n\\begin{example}\nWe consider the following controlled, decoupled dynamics\n\\begin{align}\n dX^{\\xi,\\alpha}_s &= (\\alpha_s + \\E[X^{\\xi,\\alpha}_s]) ds,\\qquad X^{\\xi,\\alpha}_0 = \\xi \\sim \\frac 1 2 \\delta_{-1} + \\frac 1 2 \\delta_1,\\\\\n dX^{\\xi,x,\\alpha}_s &= (\\alpha_s + \\E[X^{\\xi,\\alpha}_s]) ds,\\qquad X^{\\xi,x,\\alpha}_0 = x,\n\\end{align}\nwhere the control process $\\alpha$ should be $A=[-1,1]$-valued, $x\\in\\R$ and $\\xi$ a square-integrable random variable with distribution $\\xi\\sim \\frac 1 2 \\delta_{-1} + \\frac 1 2 \\delta_1$. Further we assume that $T=1$ and that the reward functional is given by\n\\[\nJ(\\xi,\\alpha) = \\E\\big[(X^{\\xi,\\alpha}_1 - 1)_+ + (X^{\\xi,\\alpha}_1 - \\tfrac 5 2)_-\\big].\n\\]\nThen the value function is given by\n\\[\nV(\\xi) = \\sup_\\alpha J(\\xi,\\alpha),\n\\]\nand solving for the optimal control, we find $\\alpha^*_s \\equiv 1$ for all $s\\in [0,1]$, which results in $V(\\xi) = J(\\xi,\\alpha^*) % = \\frac 1 2 \\cdot (e-1) + \\frac 1 2 \\cdot 0\n= \\frac 1 2 (e-1)$.\n\n\nAt the same time, the decoupled value function in \\cite{bayraktar_randomized_2018} is defined by\n\\[\nV(\\xi,x) = \\sup_\\alpha \\E\\big[(X^{\\xi,x,\\alpha}_1 - 1)_+ + (X^{\\xi,x,\\alpha}_1 - \\tfrac 5 2)_-\\big].\n\\]\nFor the starting value $x=-1$, the optimal control is $\\alpha^*_s \\equiv -1$ for $s\\in [0,1]$, resulting in $V(\\xi,-1) = e - \\frac 5 2$. For $x=1$, the optimal control is $\\alpha^*_s \\equiv 1$ for all $s\\in [0,1]$, leading to $V(\\xi,1) = e-1$.\n\n\nAccording to \\cite[Proposition 2.2]{bayraktar_randomized_2018}, we would have $V(\\xi) = \\E_{\\xi'\\sim\\xi}[V(\\xi,\\xi')] = \\frac 1 2 V(\\xi,-1) + \\frac 1 2 V(\\xi,1)$, which is not the case.\n\\end{example}\n\nIn this paper, we improve on the randomisation approach in \\cite{bayraktar_randomized_2018} to handle general mean-field control problems and to allow for common noise.\nUnlike the previous approach that relies on the decoupled SDEs, we directly work with the mean-field SDE. This approach presents a challenge because we must ensure that the mean-field interaction is preserved during the control randomisation.\n\n\nOur approach is based on a novel reformulation of the admissible control set as a set of $L^0$-valued processes, adapted solely to the common noise, where we require that at each time point $s\\in [t,T]$, the reformulated admissible control takes values in $L^0(\\Omega,\\c G\\lor\\c F^W_s,\\P;A)$. This decomposition of the original controls is motivated by the observation that for the McKean-Vlasov dynamics, by viewing not each particle but instead the whole current conditional distribution as the state, the idiosyncratic noise will become part of the \"deterministic\" state dynamics, and only the common noise remains as \"stochastic noise\" for this system. This viewpoint has the advantage that it leads to a very natural way to apply the control randomisation method, as randomising over such $L^0$-valued processes adapted solely to the common noise keeps the mean-field interaction, in form of the conditional distribution of state process given the common noise, intact.\n\nWe establish an isomorphism between the original and the reformulated admissible control set, up to modification of processes. A key challenge lies in the way back from a reformulated control to an original control, as it presents a subtle measurability issue. Specifically, given two filtrations $\\b F = (\\c F_t)_t$, $\\b G = (\\c G_t)_t$ and an $\\b F$-progressive process $\\hat\\alpha:[0,T]\\times\\Omega\\to L^0(\\Omega',\\c G_{T},\\P;A)$ satisfying $\\hat\\alpha_t(\\omega) \\in L^0(\\Omega',\\c G_t,\\P;A)$ for all $(t,\\omega)\\in [0,T]\\times\\Omega$, such as the reformulated control, we need to find an $\\b F\\otimes\\b G$-progressive process $\\alpha:[0,T]\\times\\Omega\\times\\Omega'\\to A$ that is a modification of $\\hat\\alpha$, meaning that\n\\[\n\\alpha_t(\\omega,\\cdot) = \\hat\\alpha_t(\\omega)\\text{ in }L^0(\\Omega',\\c G_{T},\\P;A)\\qquad\\text{for all }(t,\\omega)\\in [0,T]\\times\\Omega,\n\\]\nas detailed in \\cref{proposition selecting a measurable process}.\n\nWe introduce a randomised control problem based on the reformulated admissible control set, by replacing the reformulated control by an independent Poisson process taking values in $L^0$. By showing that we can again find a corresponding progressive $A$-valued control process, we are able to define the randomised state dynamics. The optimisation takes now place over a set of equivalent probability measures that vary the intensity of the Poisson point process. Crucially, the prior decomposition of the control now ensures that the conditional law of state given the common noise remains unchanged under different randomised controls. This allows us to apply the standard tools from the randomisation framework, although adapted to account for the additional layer of abstraction between the original and reformulated control. Our first main result shows that the randomised value function coincides with the value function of the original problem.\n\nSubsequently we represent the value function as the minimal solution to a backward stochastic differential equation (BSDE) with constrained jumps. This result is obtained by studying a class of penalised BSDEs corresponding to value functions for a restricted set of randomised controls.\n\nFinally, with this probabilistic representation of the value function, we directly derive a randomised dynamic programming principle (DPP), expressed as a supremum over equivalent probability measures. In the benchmark case where the value function is sufficiently regular, we recover the standard (non-randomised) DPP for mean-field control problems, which has been established in the comprehensive works of \\cite{djete_mckeanvlasov_2022,djete_mckeanvlasov_2022-1}.\n\nThe remainder of this paper is organised as follows. In \\cref{section original problem formulation} we formulate the mean-field control problem. In \\cref{section problem l2 formulation}, we introduce the reformulated admissible control set and shows its equivalence to the original problem. \\cref{section randomised problem} constructs the randomised control problem and gives our first main result \\cref{theorem equivalence original and randomised value function} on the equivalence of the non-randomised and the randomised problem. Finally, \\cref{section bsde characterisation and randomised dpp} characterises the value function as the minimal solution to a constrained BSDE, enabling us to subsequently derive the randomised DPP.\n\n\n\\paragraph{Notation} Given a random variable $X$, we will denote the (non-augmented) filtration generated by $X$ by $\\b F^X$. Further, for a filtration $\\b F$, we will denote the $\\P$-augmented filtration by $\\b F^{\\P}$ and the predictable $\\sigma$-algebra w.r.t. $\\b F$ by $Pred(\\b F)$. For a path $w:[0,T]\\to A$, we denote its stopped path by $[w]_t \\coloneqq (w_{s\\land t})_{s\\in [0,T]}$.\n\nFor some integer $\\ell$, we denote by $|\\cdot|$, and $\\cdot$ the Euclidean norm and the inner (or scalar) product on $\\R^\\ell$. \nWe denote by $\\c{P}(\\R^d)$ the space of Borel probability measures on $\\R^d$ and by $\\c{P}_2(\\R^d)$ the set of elements of $\\c{P}(\\R^d)$ having a finite second order moment. We equip $\\c{P}_2(\\R^d)$ with the 2-Wasserstein distance $\\c{W}_2$ so that it is a metric space itself and even a Polish space. We recall that the 2-Wasserstein distance on $\\c{P}_2(\\R^d)$ is defined by \n\\begin{equation}\\label{eq definition w_2 distance}\n \\c{W}_2(\\mu, \\nu) = \\Big( \\inf_{X\\sim\\mu, Y\\sim\\nu} \\E[|X-Y|^2] \\Big)^{\\frac 1 2} \n\\end{equation}\nfor $\\mu, \\nu \\in \\c{P}_2(\\R^d)$. We also denote by $C^\\ell$ the set of continuous functions from $[0,T]$ to $\\R^\\ell$ and by $\\mathbf{C}^\\ell=(\\c C ^\\ell_t)_{t \\in [0,T]}$ its canonical filtration. Further we denote the Wiener measure on $C^\\ell$ by $\\mu^\\ell_W$. Finally, given a Polish space $E$, we write $D([0,T];E)$ for the space of $E$-valued càdlàg functions on $[0,T]$ equipped with the usual Skorokhod topology.\n\n\n\n\\section{Setting}\\label{section original problem formulation}\n\nLet $(\\Omega,\\c F,\\P)$ be a complete probability space supporting an $m$-dimensional Brownian motion $W$ and an independent $n$-dimensional Brownian motion $B$. Further we suppose that there exists a separable $\\sigma$-algebra $\\c G\\subseteq \\c F$ independent of $\\b F^{W,B}$ sufficiently large such that $\\{\\P_\\xi \\,|\\, \\xi \\in L^2(\\Omega,\\c G,\\P;\\R^d) \\} = \\c P_2(\\R^d)$.\nWe define the set of admissible controls $\\c A$ as all $\\b F^{W,B}\\lor \\c G$-predictable\\footnote{We could also allow all $(\\b F^{W,B,\\P}\\lor\\c G)$-predictable processes, since by \\cite[Lemma 2.17(b)]{jacod_limit_2003},\nevery $(\\b F^{W,B,\\P}\\lor\\c G)$-predictable process is indistinguishable from an $(\\b F^{W,B}\\lor \\c G)$-predictable process. We choose $(\\b F^{W,B}\\lor\\c G)$-predictable processes only to slightly simplify the presentation in \\cref{section problem l2 formulation}.\n}\nprocesses taking values in some Polish space $A$ equipped with some metric $\\rho < 1$ (otherwise we replace it by $\\frac{\\rho}{1+\\rho}$). Let $b,\\sigma,\\sigma^0:~[0,T]\\times \\R^d\\times \\c{P}_2(\\R^d)\\times {A}\\rightarrow \\R^d,\\R^{d\\times m},\\R^{d\\times n}$ be three measurable functions that we suppose to satisfy the following assumption.\n\\begin{assumption}\\label{assumptions sde coefficients}\n\\begin{assumptionenum}\n\\item The functions $b,\\sigma$ and $\\sigma^0$ are continuous.\n\\item There exists a constant $L$ such that\n\\begin{align}\n&|b(t,x,\\mu,a)-b(t,y,\\nu,a)|+|\\sigma(t,x,\\mu,a)-\\sigma(t,y,\\nu,a)|+|\\sigma^0(t,x,\\mu,a)-\\sigma^0(t,y,\\nu,a)|\\\\\n&\\leq L\\Big(|x-y|+\\c{W}_2(\\mu,\\nu)\\Big),\n\\end{align}\nfor all $x,y\\in\\R^d$, $\\mu,\\nu\\in \\c{P}_2(\\R^d)$, $t\\in[0,T]$ and $a\\in A$.\n\\item There exists a constant $M$ such that\n\\begin{align}\n|b(t,0,\\delta_0,a)|+|\\sigma(t,0,\\delta_0,a)|+|\\sigma^0(t,0,\\delta_0,a)| & \\leq M,\n\\end{align}\nfor all $t\\in[0,T]$ and $a\\in A$.\n\\end{assumptionenum}\n\\end{assumption}\nGiven an initial time $t\\in[0,T]$, an initial condition $\\xi\\in L^2(\\Omega,\\c G\\lor \\c F^W_t,\\P;\\R^d)$ and an admissible control $\\alpha\\in \\c A$, we consider the $\\b F^{W,B,\\P}\\lor\\c G$-progressive state process $(X^{t,\\xi,\\alpha}_s)_{s\\in [t,T]}$ defined by\n\\begin{align}\n X_r^{t,\\xi,\\alpha}\n &= \\xi+\\int_t^rb(s, X^{t,\\xi,\\alpha}_s,\\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\P}_s},\\alpha_s)ds + \\int_t^r \\sigma(s,X^{t,\\xi,\\alpha}_s,\\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\P}_s},\\alpha_s) dW_s\\\\\n &\\quad +\\int_t^r \\sigma^0(s,X^{t,\\xi,\\alpha}_s,\\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\P}_s},\\alpha_s) dB_s,\\qquad r\\in[t,T]\\;.\n \\label{eq mfc state dynamics}\n\\end{align}\nHere we denote by $(\\P^{\\c F^{B,\\P}_s}_{X^{t,\\xi,\\alpha}_s})_{s\\in [t,T]}$ the $\\b F^{B,\\P}$-predictable and $\\P$-a.s.\\@ continuous version of $(\\c L(X^{t,\\xi,\\alpha}_s|\\c F^{B,\\P}_s))_{s\\in [t,T]}$, that is $\\P^{\\c F^{B,\\P}_s}_{X^{t,\\xi,\\alpha}_s} = \\c L(X^{t,\\xi,\\alpha}_s|\\c F^{B,\\P}_s)$, $\\P$-a.s., for all $s\\in [t,T]$. The existence of such a version is ensured by \\cite[Lemma A.1]{djete_mckeanvlasov_2022-1}, since the sets of $\\b F^{B,\\P}$-optional and $\\b F^{B,\\P}$-predictable processes coincide by \\cite[Chapter V, Corollary 3.3]{revuz_continuous_1999}.\nWe note that under \\cref{assumptions sde coefficients}, we have existence and uniqueness of $(X^{t,\\xi,\\alpha}_s)_{s\\in [t,T]}$ and\n\\begin{align}\n\\sup_{\\alpha\\in \\c A}\\E\\Big[\\sup_{s\\in[t,T]}|X^{t,\\xi,\\alpha}_s|^2\\Big] & < \\infty\\;.\n\\end{align} \n\nLet $f:~[0,T]\\times \\R^d\\times \\c{P}_2(\\R^d)\\times {A}\\rightarrow \\R$ and $g:~ \\R^d\\times \\c{P}_2(\\R^d)\\rightarrow \\R$ be two Borel-measurable functions on which we make the following assumption.\n\\begin{assumption}\\label{assumptions reward functional} There exists a constant $M>0$ such that\n\\begin{align}\n|f(t,x,\\mu,a)|+|g(x,\\mu)| &\\leq M\\Big( 1+|x|+ \\Big(\\int_{\\R^d}|y|^2\\mu(dy)\\Big)^{\\frac 1 2} \\Big)\n\\end{align}\nfor all $(t,x,\\mu,a)\\in[0,T]\\times \\R^d\\times \\c{P}_2(\\R^d)\\times A$.\n\\end{assumption}\n\\begin{remark}\nIn certain results, such as the equivalence between the original and the randomised problem in \\cref{theorem equivalence original and randomised value function}, the linear growth assumption on $f$ and $g$ can be relax to quadratic growth. Nevertheless, for the law invariance result of the value function by \\cite{djete_mckeanvlasov_2022-1}, and to establish the (randomised) DPP in \\cref{section randomised dpp} and the BSDE characterisation of the value function in \\cref{section bsde characterisation}, we need the more restrictive linear growth assumption.\n\\end{remark}\nWe next define the function $J$ by\n\\begin{align}\n J(t,\\xi,\\alpha) \\coloneqq \\E\\Big[ g(X^{t,\\xi,\\alpha}_T, \\P_{X^{t,\\xi,\\alpha}_T}^{{\\c F}^{B,\\P}_T}) + \\int_t^T f(s,X^{t,\\xi,\\alpha}_s,\\P_{X^{t,\\xi,\\alpha}_s}^{{\\c F}^{B,\\P}_s},\\alpha_s) ds \\Big]\n\\end{align}\nfor $t\\in[0,T]$, $\\xi\\in L^2(\\Omega,\\c G\\lor \\c F^W_t,\\P;\\R^d)$ and $\\alpha\\in \\c A$. Under \\cref{assumptions sde coefficients,assumptions reward functional}, the function $J$ is well defined. We consider the optimal value of $J$ \nover the set of admissible controls $\\c A$. Consequently, we define the value function as\n\\[\nV(t,\\xi) \\coloneqq \\sup_{\\alpha\\in \\c A} J(t,\\xi,\\alpha) < \\infty.\n\\]\n\nWe note that \\cite{djete_mckeanvlasov_2022,djete_mckeanvlasov_2022-1} show that the value function does not depend on the choice of the probability space, and that it depends on $\\xi$ only through its law $\\P_\\xi$.\\footnote{They furthermore show that the strong and the weak formulation of the problem both lead to the same value function.}\n\n\\begin{proposition}[{\\cite[Proposition 2.4]{djete_mckeanvlasov_2022-1}}]\n\\label{proposition value function law invariant and independent of probabilistic setting}\nThe value function is \\emph{law invariant}, that is, it depends on $\\xi$ only through its law $\\P_\\xi$. \nFurthermore, the value function does not depend on the specific choice of $(\\Omega,\\c F,\\P)$ and $\\c G$, as long as $\\c G$ is rich enough.\n\\end{proposition}\n\nTherefore, with slight abuse of notation, we will also write\n\\[\nV(t,\\P_\\xi) = V(t,\\xi).\n\\]\n\n\n\\section{Reformulating the set of admissible controls}\\label{section problem l2 formulation}\n\nIn this section, we derive an equivalent reformulation of the original mean-field control problem, that will form the basis for developing the randomised formulation in \\cref{section randomised problem}. The key idea behind this reformulation is that in McKean-Vlasov dynamics, by treating the entire current distribution as the state space rather than focusing on individual particles, the idiosyncratic noise becomes part of the deterministic state dynamics, leaving the common noise as the only \"true noise\" affecting the system. This leads us to reframe the control set $\\c A$ of $\\b F^{W,B}\\lor\\c G$-predictable processes by connecting it to a suitable set $\\hat{\\c A}$ of only $\\b F^B$-predictable processes. This approach aligns the control problem thus aligning the control set more closely with the traditional (non-mean-field) framework, where the control is adapted to the external noise.\n\n\nIn this spirit, let us consider the sets\n\\[\n\\c A_s \\coloneq L^0(\\Omega,\\c G\\lor\\c F^W_s,\\P;A),\\qquad s\\in [0,T],\n\\]\nof $\\P$-equivalence classes of $\\c G\\lor\\c F^W_s$-$\\c B(A)$-measurable functions $\\phi:\\Omega\\to A$ equipped with the metric\n\\[\n\\hat\\rho(\\phi^1,\\phi^2) \\coloneqq \\E^\\P[\\rho(\\phi^1,\\phi^2)] \n< 1\n,\\qquad\\text{for all } \\phi^1,\\phi^2\\in \\c A_s.\n\\]\nThey will now take the role of the action space, each at its corresponding time $s\\in [0,T]$. We remark that $\\c A_u\\subseteq\\c A_v$, for $u\\leq v$. Further, since $A$ is a Polish space and $\\c G$ and $\\c F^W_s$ are separable, the spaces $\\c A_s$ are also Polish, $s\\in [0,T]$.\nFinally, we define the set $\\hat{\\c A}$ of all $\\b F^B$-predictable processes $\\hat\\alpha:[0,T]\\times\\Omega\\to \\c A_T$, such that additionally $\\hat\\alpha_s\\in\\c A_s$ for $s\\in [0,T]$. \n\nWe will spend the rest of this section on the relation between the sets $\\c A$ and $\\hat{\\c A}$. More precisely, in \\cref{subsection connecting A and L0 A processes} we will introduce in which sense we can identify such process $\\hat\\alpha:[0,T]\\times\\Omega\\to \\c A_T$ with the original control processes of the form $\\alpha:[0,T]\\times\\Omega\\to A$ in a slightly more general setting, which we will need in \\cref{section randomised problem}. Afterwards, in \\cref{subsection equivalent action set}, we will use this concept to establish an equivalence between the action sets $\\c A$ and $\\hat{\\c A}$, which will aid in the upcoming construction of the randomised control problem in \\cref{section randomised problem}.\n\n\n\\subsection{Identifying decomposed controls with original controls}\\label{subsection connecting A and L0 A processes}\n\nSince in \\cref{section randomised problem} we are going to extend our probability space for the randomised formulation to allow for an additional Poisson point process $\\mu$, we will in this section consider more generally processes defined on an extension $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ of our original probability space $(\\Omega,\\c F,\\P)$. This means that $\\hat \\Omega = \\Omega\\times \\tilde \\Omega$, $\\tilde \\Omega \\times \\c F\\subseteq \\hat{\\c F} $ and $\\hat \\P(\\tilde\\Omega\\times \\cdot)=\\P$. We denote by $\\pi:\\hat\\Omega\\to\\Omega$ the canonical projection, which by definition is probability preserving. We next identify $\\sigma$-algebras on $\\Omega$ as $\\sigma$-algebras on $\\hat\\Omega$ by replacing them with their reciprocal images by $\\pi$. We also extend the definition of random variables on $\\Omega$ to $\\hat \\Omega$ by composing by $\\pi$.\n\nFor a suitably general class of processes $\\hat\\alpha:[0,T]\\times\\hat\\Omega\\to \\c A_T$ and $\\alpha:[0,T]\\times\\hat\\Omega\\to A$, we will define the identification relation as follows.\n\n\\begin{definition}\nLet $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ be an extension of $(\\Omega,\\c F,\\P)$ and $\\b F$ a filtration on the extension independent of $\\c G,W$. Then we say that an $\\b F^W\\lor\\c G\\lor \\b F$-progressive process $\\alpha:[0,T]\\times\\hat\\Omega\\to A$ $\\b F$-identifies to a $\\b F$-progressive process $\\hat\\alpha:[0,T]\\times\\hat\\Omega\\to \\c A_T = L^0(\\Omega,\\c G\\lor\\c F^W_T,\\P;A)$ if there exists an $(\\c G\\lor\\b F^W)\\otimes\\b F$-progressive process $\\Phi:[0,T]\\times\\Omega\\times\\hat\\Omega\\to A$ such that\n\\[\n\\alpha_s(\\hat\\omega) = \\Phi_s(\\pi(\\hat\\omega),\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega,\n\\]\nand, viewed as equality in $L^0(\\Omega,\\c G\\lor\\c F^W_T,\\P;A)$,\n\\[\n\\hat\\alpha_s(\\hat\\omega) = \\Phi_s(\\cdot,\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega.\n\\]\n\\end{definition}\n\nWe remark that just by the definition it is not clear whether given an $\\alpha$ also an $\\hat\\alpha$ identifying with $\\alpha$ exists (and vice versa). However, we can show that, if it then exists, $\\hat\\alpha$ given $\\alpha$ (and similarly $\\alpha$ given $\\hat\\alpha$) is unique up to modification, as the following \\cref{lemma alpha bar alpha uniqueness} shows.\n\n\\begin{lemma}\\label{lemma alpha bar alpha uniqueness}\n Let $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ be an extension of $(\\Omega,\\c F,\\P)$ and for $i\\in\\{1,2\\}$ let $\\alpha^i$ be a $\\b F^W\\lor\\c G\\lor\\b F^i$-progressive process $\\b F^i$-identifying to a $\\b F^i$-progressive process $\\hat\\alpha^i$. Then for all $s\\in [0,T]$,\n \\[\n \\E^{\\hat\\P}[\\rho(\\alpha^1_s,\\alpha^2_s)] = \\E^{\\hat\\P}[\\hat\\rho(\\hat\\alpha^1_s,\\hat\\alpha^2_s)].\n \\]\n In particular, $\\alpha^1$ is a modification of $\\alpha^2$ iff $\\hat\\alpha^1$ is a modification of $\\hat\\alpha^2$.\n\\end{lemma}\n\\begin{proof}\n For $i\\in\\{1,2\\}$, by definition there exists $(\\c G\\lor\\b F^W)\\otimes\\b F^i$-progressive processes $\\Phi^i:[0,T]\\times\\Omega\\times\\hat\\Omega\\to A$ such that\n \\[\n \\alpha^i_s(\\hat\\omega) = \\Phi^i_s(\\pi(\\hat\\omega),\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega,\n \\]\n and, viewed as equality in $L^0(\\Omega,\\c G\\lor\\c F^W_T,\\P;A)$,\n \\[\n \\hat\\alpha^i_s(\\hat\\omega) = \\Phi^i_s(\\cdot,\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega.\n \\]\n Defining the filtration $\\b F\\coloneqq \\b F^1\\lor\\b F^2$, we note that $\\b F$ is also independent of $\\c G,W$ and $\\Phi^1,\\Phi^2$ are also $(\\c G\\lor\\b F^W)\\otimes\\b F$-progressive. Thus, for all $s\\in [0,T]$,\n \\begin{align}\n \\E^{\\hat\\P}[\\rho(\\alpha^1_s,\\alpha^2_s)] &= \\int_{\\hat\\Omega} \\rho(\\Phi^1_s(\\pi(\\hat\\omega),\\hat\\omega),\\Phi^2_s(\\pi(\\hat\\omega),\\hat\\omega)) \\hat\\P(d\\hat\\omega)\\\\\n &= \\int_{\\hat\\Omega} \\int_{\\hat\\Omega} \\rho(\\Phi^1_s(\\pi(\\hat\\omega^1),\\hat\\omega^2),\\Phi^2_s(\\pi(\\hat\\omega^1),\\hat\\omega^2)) \\hat\\P(d\\hat\\omega^1)\\hat\\P(d\\hat\\omega^2)\n = \\E^{\\hat\\P}[\\hat\\rho(\\hat\\alpha^1_s,\\hat\\alpha^2_s)],\n \\end{align}\n where the second equality comes from the independence of $\\c G$, $W$ and $\\b F$.\n\\end{proof}\n\nMoreover, in some cases, e.g. if the filtration $\\b F$ is generated by a càdlàg process, we can even show that uniqueness in joint law holds for this the identification, see the following \\cref{lemma joint law N bar N map}, which will be useful in the case of Poisson processes in \\cref{section randomised problem}. \n\\begin{lemma}\\label{lemma joint law N bar N map}\n Let $(\\hat\\Omega^i,\\hat{\\c F}^i,\\hat\\P^i)$, $i=\\{1,2\\}$ be two extensions of $(\\Omega,\\c F,\\P)$. Denotes by $\\c G^i,W^i$ be the canonical extensions of $\\c G,W$, and by $\\pi^i:\\hat\\Omega^i\\to\\Omega$ be the canonical projection, $i=\\{1,2\\}$. Further let on each $(\\hat\\Omega^i,\\hat{\\c F}^i,\\hat\\P^i)$ be a càdlàg,\n $\\b F^{N^i}$-progressive process $N^i$ independent of $\\c G^i$ and $W^i$ taking values in a Polish space $E$, an $\\b F^{W^i,N^i}\\lor\\c G^i$-progressive map $\\alpha^i:[0,T]\\times\\hat\\Omega^i\\to A$ and an $\\b F^{N^i}$-progressive $\\hat\\alpha^i:[0,T]\\times\\hat\\Omega^i\\to\\c A_T$ which $\\b F^{N^i}$-identifies to $\\alpha^i$, $i=\\{1,2\\}$. Then the following statements (of equality in finite dimensional distributions) are equivalent\n \\begin{enumerate}[label=(\\roman*)]\n \\item\\label{lemma 4.2 assumption i} $\\c L^{\\hat\\P^1}(\\pi^1,N^1,\\alpha^1) = \\c L^{\\hat\\P^2}(\\pi^2,N^2,\\alpha^2)$,\n \\item\\label{lemma 4.2 assumption ii} $\\c L^{\\hat\\P^1}(\\pi^1,N^1,\\hat\\alpha^1) = \\c L^{\\hat\\P^2}(\\pi^2,N^2,\\hat\\alpha^2)$,\n \\item\\label{lemma 4.2 assumption iii} $\\c L^{\\hat\\P^1}(\\pi^1,N^1,\\alpha^1,\\hat\\alpha^1) = \\c L^{\\hat\\P^2}(\\pi^2,N^2,\\alpha^2,\\hat\\alpha^2)$.\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nBy definition there exists $(\\c G\\lor\\b F^W)\\otimes\\b F^{N^i}$-progressive processes $\\Phi^i:[0,T]\\times\\Omega\\times\\hat\\Omega^i\\to A$ such that\n\\[\n\\alpha^i_s(\\hat\\omega^i) = \\Phi^i_s(\\pi^i(\\hat\\omega^i),\\hat\\omega^i),\\qquad\\text{for all }(s,\\hat\\omega^i)\\in [0,T]\\times\\hat\\Omega^i,\n\\]\nand, viewed as equality in $L^0(\\Omega,\\c G\\lor\\c F^W_T,\\P;A)$,\n\\[\n\\hat\\alpha^i_s(\\hat\\omega^i) = \\Phi^i_s(\\cdot,\\hat\\omega^i),\\qquad\\text{for all }(s,\\hat\\omega^i)\\in [0,T]\\times\\hat\\Omega^i.\n\\]\nThen the Doob-Dynkin Lemma implies that there exists a $\\c B([0,T])\\otimes(\\c G\\lor\\c F^W_T)\\otimes\\c B(D([0,T];E))$-measurable $\\Psi^i:[0,T]\\times\\Omega\\times D([0,T];E)\\to A$ such that\n\\[\n\\Phi^i_s(\\omega,\\hat\\omega^i) = \\Psi^i_s(\\omega,N^i(\\hat\\omega^i)),\\qquad\\text{for all }(s,\\omega,\\hat\\omega^i)\\in[0,T]\\times\\Omega\\times\\hat\\Omega^i.\n\\]\nNow we define ${\\alpha^2}':[0,T]\\times\\hat\\Omega^2\\to A$ and ${\\hat\\alpha^2}':[0,T]\\times\\hat\\Omega^2\\to\\c A_T$ via\n\\[\n{\\alpha^2_s}'(\\hat\\omega^2) \\coloneqq \\Psi^1_s(\\pi^2(\\hat\\omega^2),N^2(\\hat\\omega^2)),\\quad\\text{for all }(s,\\hat\\omega^2)\\in [0,T]\\times\\hat\\Omega^2,\n\\]\nand\n\\[\n{\\hat\\alpha^2_s}'(\\hat\\omega^2) = \\Psi^1_s(\\cdot,N^2(\\hat\\omega^2)),\\qquad\\text{for all }(s,\\hat\\omega^2)\\in [0,T]\\times\\hat\\Omega^2.\n\\]\nBy construction, ${\\hat\\alpha^2}'$ is $\\c B([0,T])\\otimes\\c F^{N^2}_T$-$\\c B(A)$-measurable and ${\\alpha^2}'$ is $\\c B([0,T]\\otimes(\\c F^{W^2,N^2}_T\\lor\\c G)$-measurable.\n\nNow let us assume that \\labelcref{lemma 4.2 assumption i} holds, which means that $\\c L^{\\hat\\P^1}(\\pi^1,N^1,\\alpha^1) = \\c L^{\\hat\\P^2}(\\pi^2,N^2,\\alpha^2)$. Then by construction\n\\begin{align}\n\\c L^{\\hat\\P^1}(\\pi^1,N^1,\\alpha^1,\\alpha^1,\\hat\\alpha^1) = \\c L^{\\hat\\P^2}(\\pi^2,N^2,\\alpha^2,{\\alpha^2}',{\\hat\\alpha^2}'),\n\\label{eq lemma 4.2 joint law equality}\n\\end{align}\nwhich shows that $\\alpha^2_s = {\\alpha^2_s}'$, $\\hat\\P^2$-a.s., for all $s\\in [0,T]$, which means that $\\alpha^2$ is a modification of ${\\alpha^2_s}'$. Therefore, using that $\\Psi^2(\\pi^2(\\cdot),N^2(\\cdot))$ is $\\c B([0,T])\\otimes (\\c G^2\\lor\\c F^{W^2}_T)\\otimes \\c F^{N^2}_T$-measurable and that $\\c G^2\\lor\\c F^{W^2}_T$ and $N^2$ are independent, we obtain\n\\begin{align}\n\\E^{\\hat\\P^2}[\\rho(\\alpha^2_s,{\\alpha^2_s}')] = \\E^{\\hat\\P^2}[\\rho(\\Psi^2_s(\\pi^2,N^2),\\Psi^1_s(\\pi^2,N^2))] = \\E^{\\hat\\P^2}[\\hat\\rho(\\hat\\alpha^2_s,{\\hat\\alpha^2_s}')] = 0,\n\\end{align}\nand hence $\\hat\\alpha^2_s = {\\hat\\alpha^2_s}'$, $\\hat\\P^2$-a.s., for all $s\\in [0,T]$, which together with \\eqref{eq lemma 4.2 joint law equality} implies \\labelcref{lemma 4.2 assumption iii}. The direction \\labelcref{lemma 4.2 assumption ii} $\\Rightarrow$ \\labelcref{lemma 4.2 assumption iii} can be proven similarly.\n\\end{proof}\n\n\\begin{remark}\n It is not strictly necessary for there to be a càdlàg process $N$ generating the filtration $\\b F=\\b F^N$. The result \\cref{lemma joint law N bar N map} holds more generally as long as the filtration $\\b F$ is countably generated. However, in this paper, we focus on filtrations $\\b F=\\b F^N$ generated by càdlàg, piece-wise constant processes $N$, such as Poisson processes, as these will be relevant to the randomisation framework outlined in \\cref{section randomised problem}.\n\\end{remark}\n\n\n\n\n\\subsection{Equivalence of the action sets $\\c A$ and $\\hat{\\c A}$}\\label{subsection equivalent action set}\n\nIn this section we will show that for each $\\alpha\\in\\c A$ there exists an $\\hat\\alpha\\in\\hat{\\c A}$ such that $\\alpha$ identifies with $\\hat\\alpha$ and vice versa. We equip $\\c A$ and $\\hat{\\c A}$ with the following pseudometrics\n\\[\nd_{\\c A}^{\\hat\\P}(\\alpha^1,\\alpha^2) = \\E^{\\hat\\P}\\Big[\\int_0^T \\rho(\\alpha^1_s,\\alpha^2_s)ds \\Big],\\qquad d_{\\hat{\\c A}}^{\\hat\\P}(\\hat\\alpha^1,\\hat\\alpha^2) \\coloneqq \\E^{\\hat\\P}\\Big[\\int_0^T \\hat\\rho(\\hat\\alpha^1_s,\\hat\\alpha^2_s) ds\\Big].\n\\]\nBy \\cref{lemma alpha bar alpha uniqueness}, this will then define an isomorphism between the induced quotient spaces $\\c A/_\\sim$ and $\\hat{\\c A}/_\\sim$, where $\\alpha^1\\sim\\alpha^1$ (resp. $\\hat\\alpha^1\\sim\\hat\\alpha^2$) iff $d_{\\c A}^{\\hat\\P}(\\alpha^1,\\alpha^2) = 0$ (resp. $d_{\\hat{\\c A}}^{\\hat\\P}(\\hat\\alpha^1,\\hat\\alpha^2) = 0$).\n\nIn particular, this justifies considering the optimal control problem over the action set $\\hat{\\c A}$ rather than $\\c A$, which will be crucial for \\cref{section randomised problem}, as it provides a more natural starting point for the construction of the randomised control problem.\n\n\n\\subsubsection{From $\\c A$ to $\\hat{\\c A}$}\nFor the construction of an $\\hat\\alpha\\in\\hat{\\c A}$, which $\\b F^B$-identifies to a given $\\alpha\\in\\c A$, we first prove the following general results about decomposing predictable processes.\n\n\\begin{proposition}\\label{proposition decompose filtrations of predictable processes}\n Let $(\\Omega,\\c F)$ be a measurable space with two filtrations $\\b F$ and $\\b G$ and let $A$ be a Polish space.\n Then a process $X:[0,T]\\times\\Omega\\to A$ is $\\b F\\lor\\b G$-predictable if and only if there exists a $\\b F\\otimes\\b G$-predictable process $Y:[0,T]\\times\\Omega\\times\\Omega\\to A$ defined on the space $(\\Omega\\times \\Omega,\\c F\\otimes \\c F)$ such that\n \\[\n X_t(\\omega) = Y(t,\\omega,\\omega),\\qquad\\text{for all }(t,\\omega)\\in [0,T]\\times\\Omega.\n \\]\n\\end{proposition}\n\\begin{proof}\n Given $Y$, the $\\b F\\lor\\b G$-predictability of $X$ is obvious. So let us focus on the other direction.\n We assume that the Polish space $A$ is isomorphic $\\R$, and thus w.l.o.g. assume that $A = \\R$.\n \n We will prove this result using the functional monotone class theorem. To this end, we denote by $\\c H$ the space of $\\b F\\lor\\b G$-predictable functions $X:[0,T]\\times\\Omega\\to\\R$ such that a corresponding $Y^X$ exists. We note that if $X^1,X^2\\in\\c H$ and $c\\in\\R$, then also $X^1 X^2,X^1+c X^2\\in\\c H$ since we can choose $Y^{X^1 X^2} \\coloneqq Y^{X^1}Y^{X^2}$ and $Y^{X^1+c X^2} \\coloneqq Y^{X^1}+c Y^{X^2}$. Furthermore, if $(X^n)_n\\subseteq \\c H$ and $0\\leq X^n\\uparrow X$ to a bounded process $X$, then we can choose $Y^X\\coloneqq \\limsup_n Y^{X^n}$ to conclude that also $X\\in\\c H$.\n\n Thus, recalling the generator of $Pred(\\b F\\lor\\b G)$, we only need to show that (i) $\\1_{(t_1,t_2]\\times B\\cap C}\\in\\c H$ for all $0\\leq t_1<t_2\\leq T$ and $B\\in \\c F_{t_1},C\\in\\c G_{t_1}$ (ii) $\\1_{\\{0\\}\\times B\\cap C}\\in\\c H$ for all $B\\in \\c F_{0-},C\\in\\c G_{0-}$. By defining $Y^{\\1_{(t_1,t_2]\\times B\\cap C}}\\coloneqq \\1_{(t_1,t_2]\\times B\\times C}$ (respectively $Y^{\\1_{\\{0\\}\\times B\\cap C}}\\coloneqq \\1_{\\{0\\}\\times B\\times C}$) this is clear and the result follows.\n\\end{proof}\n\n\\begin{proposition}\\label{proposition l0 control is predictable}\nLet $(\\Omega,\\c F,\\b F)$ a filtered space, $(\\Omega',\\c G,\\P)$ a probability space and $X:[0,T]\\times\\Omega\\times\\Omega'\\to A$ an $\\b F\\otimes\\c G$-predictable process taking values in some Polish space $A$. Then the process $Y:[0,T]\\times\\Omega\\to L^0(\\Omega',\\c G,\\P;A)$, where $L^0$ is meant w.r.t. $\\P$-equivalence classes, defined by\n\\[\nY_t(\\omega) \\coloneqq X_t(\\omega,\\cdot) \\in L^0(\\Omega',\\c G,\\P;A),\\qquad (t,\\omega)\\in [0,T]\\times\\Omega,\n\\]\nis $\\b F$-predictable.\n\\end{proposition}\n\n\\begin{proof}\nNote that it suffices to show the claim for $A=\\R$ since every Polish space is isomorphic to a Borel subset of $\\R$. Therefore we will only consider the case $A=\\R$, and carry out the proof using the functional monotone class theorem. To this end, let $\\c H$ be the set of such processes $X$ for which $Y^X$ is indeed $\\b F$-predictable. Note that if $X\\equiv 1$, then $Y^X\\equiv 1$ is trivially $\\b F$-predictable. Similarly, if $X^1,X^2\\in \\c H$ and $c\\in\\R$, then also $X^1 X^2, X^1+c X^2\\in\\c H$ since $Y^{X^1 X^2} = Y^{X^1} Y^{X^2}$ and $Y^{X^1 + c X^2} = Y^{X^1} + c Y^{X^2}$. Furthermore, if $(X^n)_n\\subseteq \\c H$ and $0\\leq X^n\\uparrow X$ to a bounded process $X$, then $Y^X = \\lim_n Y^{X^n}$ is also $\\b F$-predictable, and thus $X\\in\\c H$.\n\nFinally, we need to verify that $\\c H$ contains a generating set for $Pred(\\b F)$. For this we note that both (i) $\\1_{(t_1,t_2]\\times B\\times C}$ for $0 \\leq t_1 < t_2\\leq T$ and $B\\in \\c F_{t_1},C\\in\\c G$ and (ii) $\\1_{\\{0\\}\\times B\\times C}$ for $B\\in\\c F_{0-},C\\in\\c G$ are in $\\c H$ since the corresponding processes $Y$ are clearly $\\b F$-predictable and thus $\\c H$ indeed contains all $\\b F\\otimes\\c H$-predictable processes $X$.\n\\end{proof}\n\nPutting these results together, we can now prove the following \\cref{proposition bar alpha given alpha well defined}.\n\n\\begin{proposition}\\label{proposition bar alpha given alpha well defined}\n For every $\\alpha\\in\\c A$, there exists an $\\hat\\alpha\\in\\hat{\\c A}$ that $\\b F^B$-identifies to $\\hat\\alpha$.\n\\end{proposition}\n\\begin{proof}\nWe recall that $\\alpha\\in\\c A$ is $\\b F^{W,B}\\lor\\c G$-predictable, and thus by \\cref{proposition decompose filtrations of predictable processes} there exists a $(\\c G\\lor\\b F^W)\\otimes\\b F^B$-predictable process $\\Psi:[0,T]\\times\\Omega\\times\\Omega \\to A$ such that\n\\[\n\\Psi_s(\\omega,\\omega) = \\alpha_s(\\omega),\\qquad\\text{for all }(s,\\omega)\\in [0,T]\\times\\Omega.\n\\]\nNext we note that we can canonically extend $\\Psi$ to $\\Phi:[0,T]\\times\\Omega\\times\\hat\\Omega\\to A$, that means it satisfies $\\Phi = \\Psi(\\cdot,\\pi(\\cdot))$. \nIn particular $\\Phi:[0,T]\\times\\Omega\\times\\hat\\Omega\\to A$ is $(\\c G\\lor\\b F^W)\\otimes\\b F^B$-predictable, and thus by \\cref{proposition l0 control is predictable}, the process $\\hat\\alpha:[0,T]\\times\\hat\\Omega\\to L^0(\\Omega,\\c G\\lor\\c F^W_T,\\P;A) = \\c A_T$ defined by\n\\[\n\\hat\\alpha_s(\\hat\\omega) \\coloneqq \\Phi_s(\\cdot,\\hat\\omega),\n\\]\nis $\\b F^B$-predictable and satisfies $\\hat\\alpha_s(\\hat\\omega)\\in L^0(\\Omega,\\c G\\lor\\c F^W_s,\\P;A) = \\c A_s$ for all $s\\in [0,T]$, $\\hat\\omega\\in\\hat\\Omega$. Therefore $\\hat\\alpha\\in\\hat{\\c A}$ and $\\alpha$ is $\\b F^B$-identifying to $\\hat\\alpha$.\n\\end{proof}\n\n\n\n\\subsubsection{From $\\hat{\\c A}$ to $\\c A$}\n\nOur goal in this section is constructing an $\\alpha\\in\\c A$ which $\\b F^{\\hat\\alpha}$-identifies to a given $\\hat\\alpha\\in\\hat{\\c A}$. For this, we will need several auxiliary result. The first key tool is the following measurable selection result.\n\n\\begin{proposition}\\label{proposition selecting a measurable process}\n Let $(\\Omega,\\c F,\\b F)$ be a filtered space and $(\\Omega',\\c G,\\b G,\\P)$ be a filtered probability space, such that $\\c G$ is separable. Let $A$ be a Polish space and $X:[0,T]\\times\\Omega\\to L^0(\\Omega',\\c G,\\P;A)$ be a $(\\c B([0,T])\\otimes\\c F)$-$\\c B(L^0(\\Omega',\\c G,\\P;A))$-measurable and $\\b F$-adapted process. \n Further assume that $X_t(\\omega) \\in L^0(\\Omega',\\c G_t,\\P;A)\\subseteq L^0(\\Omega',\\c G,\\P;A)$ for all $(t,\\omega)\\in [0,T]\\times\\Omega$.\n\n Then there exists a $(\\c B([0,T])\\otimes\\c F\\otimes\\c G)$-$\\c B(A)$-measurable and $\\b F\\otimes\\b G$-adapted process $Y:[0,T]\\times\\Omega\\times\\Omega'\\to A$ such that\n \\[\n Y_t(\\omega,\\cdot) = X_t(\\omega)\\text{ in }L^0(\\Omega',\\c G,\\P;A),\\qquad\\text{for all }(t,\\omega)\\in [0,T]\\times\\Omega.\n \\]\n Further, if $X$ is $\\b F$-progressive, then $Y$ can also be chosen $\\b F\\otimes\\b G$-progressive.\n\\end{proposition}\n\n\\begin{proof}\n\\begin{enumerate}[wide,label=(\\roman*)]\n \\item \\label{proof proposition selecting a measurable process case A unit interval}\nLet us first consider the case that $A = [0,1]$. \nThe following construction and proof is based on \\cite[Theorem IV.30]{dellacherie_probabilities_1978} and its detailed version in \\cite{kaden_progressive_2004}. Since $[0,1]$ is bounded and thus the notions of convergence in probability and convergence in $L^1$ coincide (and therefore induce the same topology), we will in the following consider $L^1$ instead of $L^0$.\nLet $X:[0,T]\\times\\Omega\\to L^1(\\Omega',\\c G,\\P;[0,1])$ be a $(\\c B([0,T])\\otimes\\c F)$-$\\c B(L^1(\\Omega',\\c G,\\P;[0,1]))$-measurable and $\\b F$-adapted process. Then, since $\\c G$ and thus also $L^1(\\Omega',\\c G,\\P;[0,1])$ is separable, we can approximate $X$ uniformly by countable sums of simple predictable processes as follows:\n\nLet $n\\in\\N$, and let $(A^n_k)_k\\subset\\c B(L^1(\\Omega',\\c G,\\P;[0,1]))$ be a Borel sets of diameter $d^n_k \\coloneqq \\sup_{x,y\\in A^n_k} \\norm{x-y}_{L^1(\\Omega',\\c G,\\P;[0,1])} \\leq 2^{-n}$ covering $L^1(\\Omega',\\c G,\\P;[0,1])$, that is $\\bigcup_k A^n_k = L^1(\\Omega',\\c G,\\P;[0,1])$. Further let $(B^n_k)_k\\subseteq \\c B([0,T])\\otimes \\c F$ be the corresponding preimages under $X$, that is $B^n_k \\coloneqq X^{-1}(A^n_k)$ for all $k\\in\\N$. We note that if $X$ is $\\b F$-progressive, then $B^n_k\\in Prog(\\b F)$.\nNow for each $k\\in\\N$, we choose a $\\c G$-measurable $\\alpha^n_k:\\Omega'\\to [0,1]$ such that $\\alpha^n_k\\in A^n_k$ (more precise: the $\\P$-equivalence class of $\\alpha^n_k$ is in $A^n_k$). Then we can define\n\\[\n\\Phi^n_s(\\omega,\\omega') \\coloneqq \\sum_{k=1}^\\infty \\alpha^n_k(\\omega') \\1_{B^n_k}(s,\\omega),\\qquad (s,\\omega,\\omega')\\in [0,T]\\times\\Omega\\times\\Omega'.\n\\]\nThus we observe that $\\norm{\\Phi^n_s(\\omega,\\cdot) - X_s(\\omega)}_{L^1(\\Omega',\\c G,\\P;[0,1])}\\leq 2^{-n}\\to 0$ uniformly in $(s,\\omega) \\in [0,T]\\times\\Omega$. Moreover, the $\\Phi^n$ are $\\c B([0,T])\\otimes \\c F\\otimes \\c G$-measurable, but not yet adapted to $\\b F\\otimes\\b G$, since we only have $\\alpha^n_k\\in L^1(\\Omega',\\c G,\\P;[0,1])$.\n\nTo fix this, we define $t^n_k\\coloneqq \\inf \\{t \\geq 0 \\,|\\, (t,\\omega)\\in B^n_k\\text{ for some }\\omega\\in\\Omega\\}$ and distinguish two cases: If the infimum is attained, i.e. $t^n_k \\in B^n_k(\\omega)$ for some $\\omega\\in\\Omega$, we note that $Z^n_k \\coloneqq X_{t^n_k} \\in L^1(\\Omega',\\c G_{t^n_k},\\P;[0,1])$. In particular there exists a $\\c G_{t^n_k}$-measurable $\\hat Z^n_k:\\Omega'\\to [0,1]$ such that $\\hat Z^n_k(\\cdot) = Z^n_k$ in $L^1(\\Omega',\\c G,\\P;[0,1])$. Furthermore, by construction $\\norm{\\hat Z^n_k(\\cdot) - \\alpha^n_k(\\cdot)}_{L^1(\\Omega',\\c G,\\P;[0,1])}\\leq 2^{-n}$.\n\nIn the case that $t^n_k \\not\\in B^n_k(\\omega)$ for all $\\omega\\in\\Omega$, then we can instead find $(s^n_{k,l},\\omega^n_{k,l})_l\\subseteq B^n_k$ such that $s^n_{k,l}\\downarrow t^n_k$.\nWe now consider the sequence $(Z^n_{k,l})_l \\coloneqq (X_{s^n_{k,l}}(\\omega^n_{k,l}))_l\\subseteq L^1(\\Omega',\\c G,\\P;[0,1])$. Note that by construction $Z^n_{k,l} \\in L^1(\\Omega',\\c G_{s^n_{k,l}},\\P;[0,1])$ for all $l\\in\\N$.\nNow since $(Z^n_{k,l})_l$ are uniformly bounded, by \\cite[Theorem II.25]{dellacherie_probabilities_1978}, the sequence is relatively compact in the weak $\\sigma(L^1,L^\\infty)$ topology. Thus, \\cite[Theorem II.24]{dellacherie_probabilities_1978} implies that the sequence $(Z^n_{k,l})_l$ converges (along a subsequence) in the weak $\\sigma(L^1,L^\\infty)$ topology to some $Z^n_k\\in L^1(\\Omega',\\c G,\\P;[0,1])$. Further, since $Z^n_{k,l}\\in L^1(\\Omega',\\c G_{s^n_{k,l}},\\P;[0,1])$ and $s^n_{k,l}\\downarrow t^n_k$, we conclude that $Z^n_k \\in L^1(\\Omega',\\c G_{t^n_k+},\\P;[0,1])$,\nwhich means that we can find a $\\c G_{t^n_k+}$-measurable $\\hat Z^n_k:\\Omega'\\to [0,1]$ such that $\\hat Z^n_k(\\cdot) = Z^n_k$ in $L^1(\\Omega',\\c G,\\P;[0,1])$.\nAt the same time, by construction $\\norm{\\hat Z^n_k(\\cdot) -\\alpha^n_k(\\cdot)}_{L^1(\\Omega',\\c G,\\P;[0,1])} \\leq \\liminf_{l\\to\\infty} \\norm{Z^n_{k,l} - \\alpha^n_k(\\cdot)}_{L^1(\\Omega',\\c G,\\P;[0,1])} \\leq 2^{-n}$ since the $L^1$-norm is l.s.c.\\@ w.r.t.\\@ weak $\\sigma(L^1,L^\\infty)$ convergence.\n\nTherefore, by defining\n\\[\nY^n_s(\\omega,\\omega') \\coloneqq \\sum_{k=1}^\\infty \\hat Z^n_k(\\omega') \\1_{B^n_k}(s,\\omega),\\qquad (s,\\omega,\\omega')\\in [0,T]\\times\\Omega\\times\\Omega',\n\\]\nwe obtain $(\\c B([0,T])\\otimes\\c F\\otimes\\c G)$-$\\c B([0,1])$-measurable and $\\b F\\otimes\\b G$-adapted processes $Y^n$ for $n\\in\\N$, which by construction satisfy for all $(s,\\omega)\\in [0,T]\\times \\Omega$,\n\\begin{align}\n\\norm{Y^n_s(\\omega,\\cdot) - \\Phi^n_s(\\omega,\\cdot)}_{L^1(\\Omega',\\c G,\\P;[0,1])} &\\leq \\sup_k \\norm{\\hat Z^n_k - \\alpha^n_k}_{L^1(\\Omega',\\c G,\\P;[0,1])}\n= \\sup_k \\norm{Z^n_k - \\alpha^n_k}_{L^1(\\Omega',\\c G,\\P;[0,1])} \\leq 2^{-n},\n\\end{align}\nand thus\n\\begin{align}\n&\\norm{Y^n_s(\\omega,\\cdot) - X_s(\\omega)}_{L^1(\\Omega',\\c G,\\P;[0,1])}\\\\\n&\\leq \\norm{Y^n_s(\\omega,\\cdot) - \\Phi^n_s(\\omega,\\cdot)}_{L^1(\\Omega',\\c G,\\P;[0,1])}\n+ \\norm{\\Phi^n_s(\\omega,\\cdot) - X_s(\\omega)}_{L^1(\\Omega',\\c G,\\P;[0,1])}\n\\leq 2^{-n+1} \\to 0,\n\\end{align}\nuniformly in $(s,\\omega)\\in [0,T]\\times\\Omega$.\nFinally, by defining the point-wise limit\n\\[\nY_s(\\omega,\\omega') \\coloneqq \\liminf_{n\\to\\infty} Y^n_s(\\omega,\\omega'),\\qquad (s,\\omega,\\omega') \\in [0,T]\\times\\Omega\\times\\Omega',\n\\]\nwe obtain the desired $(\\c B([0,T])\\otimes\\c F\\otimes\\c G)$-$\\c B([0,1])$-measurable and $\\b F\\otimes\\b G$-adapted process, which satisfies by construction $X_s(\\omega) = Y_s(\\omega,\\cdot)$ in $L^1(\\Omega',\\c G,\\P;[0,1])$ for all $(s,\\omega)\\in [0,T]\\times\\Omega$.\n\n\nFurthermore, if $X$ is $\\b F$-progressive, then $B^n_k\\in Prog(\\b F)$, for $n,k\\in\\N$ and thus, using a similar argument as \\cite[Lemma 2.7]{kaden_progressive_2004}, we see that the constructed processes $\\hat Z^n_k \\1_{B^n_k}$ will be $\\b F\\otimes\\b G$-progressive. This in turn implies that then also $Y^n = \\sum_k \\hat Z^n_k \\1_{B^n_k}$ and $Y = \\liminf Y^n$ are $\\b F\\otimes\\b G$-progressive.\n\n\n \\item \nLet us now consider the case where $A$ is a general Polish space. By a standard result, there exists a Borel-subset $I \\subseteq [0,1]$ and a Borel-isomorphism $\\phi:A\\to I\\subseteq [0,1]$ between $A$ and $I$.\\footnote{If $A$ is uncountable, one may choose $I=[0,1]$.}\nWe define a left-inverse $\\psi:[0,1]\\to A$ to $\\phi$ by $\\psi = \\1_I \\phi^{-1} + \\1_{I^c} a_0$, where $a_0\\in A$ is a fixed but arbitrary element of the Polish space $A$. Clearly, $\\psi$ is Borel-measurable and $\\psi\\circ\\phi = \\operatorname{id}_A$.\n\nGiven this identification of $A$ with $I\\subseteq [0,1]$, every process $X:[0,T]\\times\\Omega\\to L^0(\\Omega',\\c G,\\P;A)$ can be viewed as a process taking values in $L^0(\\Omega',\\c G,\\P;[0,1])$. Specifically, we define $\\bar X \\coloneqq \\phi(X):[0,T]\\times\\Omega\\to L^0(\\Omega',\\c G,\\P;I) \\subseteq L^0(\\Omega',\\c G,\\P;[0,1])$. Then $\\bar X$ is \na $(\\c B([0,T])\\otimes \\c F)$-$(\\c B(L^0(\\Omega',\\c G,\\P;[0,1])))$-measurable and $\\b F$-adapted process. Furthermore, it holds that \n$\\bar X_t(\\omega) \\in L^0(\\Omega',\\c G_t,\\P;[0,1])\\subseteq L^0(\\Omega',\\c G,\\P;[0,1])$ for all $(t,\\omega)\\in [0,T]\\times\\Omega$.\n\n\nBy \\cref{proof proposition selecting a measurable process case A unit interval}, there exists a corresponding $(\\c B([0,T])\\otimes \\c F\\otimes\\c G)$-$\\c B([0,1])$-measurable and $\\b F\\otimes\\b G$-adapted process $\\bar Y:[0,T]\\times\\Omega\\times\\Omega'\\to [0,1]$, such that\n\\[\n \\bar Y_t(\\omega,\\cdot) = \\bar X_t(\\omega)\\text{ in }L^0(\\Omega',\\c G,\\P;[0,1]),\\qquad\\text{for all }(t,\\omega)\\in [0,T]\\times\\Omega.\n\\]\nIf $\\bar X$ is $\\b F$-progressive, then $\\bar Y$ can also be chosen to be $\\b F\\otimes\\b G$-progressive.\n\n\n\nNext, we map this process back to $A$ by defining $Y\\coloneqq \\psi(\\bar Y):[0,T]\\times\\Omega\\times\\Omega'\\to A$. By construction, $Y$ is $(\\c B([0,T])\\otimes \\c F\\otimes\\c G)$-$\\c B(A)$-measurable and $\\b F\\otimes\\b G$-adapted. Furthermore, if $X$, and thus $\\bar X$, is $\\b F$-progressive, then $\\bar Y$, and thus $Y$, can also be chosen to be $\\b F\\otimes\\b G$-progressive. Finally we have\n\\[\nY_t = \\psi(\\bar Y_t(\\omega,\\cdot)) = \\psi(\\bar X_t(\\omega)) = \\psi(\\phi(X_t(\\omega))) = X_t(\\omega)\\text{ in }L^0(\\Omega',\\c G,\\P;A),\\qquad\\text{for all }(t,\\omega)\\in [0,T]\\times\\Omega,\n\\]\nwhich shows that $Y$ is the desired process.\n\n\\end{enumerate}\n\\end{proof}\n\nWhile the preceding measurable selection result \\cref{proposition selecting a measurable process} allows us to select a progressive process for each $\\hat\\alpha\\in\\hat{\\c A}$, the definition of the action set $\\c A$ requires predictability. To obtain such a predictable process, we will leverage the fact that a large class of predictable processes can be mapped to the canonical space $(C^d, \\c C^d_T, \\mathbf C^d, \\mu^d_W)$, see \\cref{proposition canonical representation of predictable processes}. W.r.t.\\@ this canonical filtration, the concepts of progressiveness and predictability then coincide, see also the upcoming \\cref{lemma canonical space predictable adapted measurable equivalence}. Consequently, we will begin by proving these two auxiliary results, which are extensions of \\cite[Propositions 10 and 9]{claisse_pseudo-markov_2016}.\n\n\\begin{proposition}\\label{proposition canonical representation of predictable processes}\nLet $(\\Omega,\\c F,\\b F)$ be a filtered space, let $(\\Omega',\\c G)$ be a measurable space with a continuous process $X:[0,T]\\times\\Omega'\\to\\R^d$ and its generated filtration $\\b G^X$ and let $A$ be a Polish space.\nThen a process $Y:[0,T]\\times\\Omega\\times\\Omega'\\to A$ is $\\b F\\otimes \\b G^X$-predictable if and only if there exists a $\\b F\\otimes \\mathbf C^d$-predictable process $\\phi:[0,T]\\times\\Omega\\times C^d\\to A$ defined on the space $(\\Omega\\times C^d,\\c F\\otimes \\c C^d_T)$, where $\\mathbf C^d$ is the canonical filtration on $C^d$, such that\n\\[\nY_t(\\omega,\\omega') = \\phi(t,\\omega,[X(\\omega')]_t) = \\phi(t,\\omega,X(\\omega')),\\qquad\\text{for all }(t,\\omega,\\omega')\\in [0,T]\\times\\Omega\\times\\Omega',\n\\]\nwhere $[w]_t \\coloneqq (w_{s\\land t})_{s\\in [0,T]}$.\n\\end{proposition}\n\\begin{proof}\nWe first prove that $\\psi:(t,\\omega,\\omega')\\mapsto (t,\\omega,[X(\\omega')]_t)$ is $Pred(\\b F\\otimes\\b G^X)$-$Pred(\\b F\\otimes\\mathbf C^d)$-measurable. To this end, we recall that $Pred(\\b F\\otimes\\mathbf C^d)$ is generated by all sets of the form (i) $(t_1,t_2]\\times C\\times D$, where $0\\leq t_1<t_2\\leq T$ and $C\\in\\c F_{t_1}$, $D\\in\\c C^d_{t_1}$, and (ii) $\\{0\\}\\times C\\times D$ where $C\\in \\c F_{0-}$, $D\\in\\{\\emptyset,C^d\\}$. Thus, the measurability directly follows from noticing that $\\psi^{-1}((t_1,t_2]\\times C\\times D) = (t_1,t_2]\\times C\\times X^{-1}(D)\\in Pred(\\b F\\otimes\\b G^X)$, since for $s\\geq t_1$, $X^{-1}_{t_1}(D) = ([X]_s)^{-1}_{t_1}(D)$, respectively from $\\psi(\\{0\\}\\times C\\times D) = \\{0\\}\\times C\\times D'\\in Pred(\\b F\\otimes\\b G^X)$, where $D' = \\emptyset$ if $D=\\emptyset$ and $D' = \\Omega'$ if $D=C^d$.\n\nAt the same time, we observe that $\\psi%:(t,\\omega,\\omega')\\mapsto(t,\\omega,[X(\\omega')]_t)\n$ also generates the $\\sigma$-algebra $Pred(\\b F\\otimes\\b G^X)$, that is $Pred(\\b F\\otimes\\b G^X) = \\sigma(\\psi)$. Recall that the measurability of $\\psi$ shows that $\\sigma(\\psi)\\subseteq Pred(\\b F\\otimes\\b G^X)$. For the other direction, we note that since $\\c G^X_t = \\sigma(X_s^{-1}(C)|s\\in [0,t],C\\in\\c B(\\R^d))$, the $\\sigma$-algebra algebra $Pred(\\b F\\otimes\\b G^X)$ is generated by all sets of the form (i) $(t_2,t_3]\\times B\\times X_{t_1}^{-1}(C)$, where $0\\leq t_1\\leq t_2 < t_3\\leq T$ and $B\\in\\c F_{t_2}$, $C\\in\\c B(\\R^d)$, and (ii) $\\{0\\}\\times B\\times D$, where $B\\in\\c F_{0-}$ and $D\\in \\{\\emptyset,\\Omega'\\}$.\nFor (i), noting that for $s\\geq t_1$, $X_{t_1}^{-1}(C) = ([X]_s)_{t_1}^{-1}(C) = \\{w^d_s\\in C\\}\\in\\c C^d_s$, we see that $(t_2,t_3]\\times B\\times X_{t_1}^{-1}(C) = \\psi^{-1}((t_2,t_3]\\times B\\times \\{w_{t_1}\\in C\\})\\in\\psi^{-1}(Pred(\\b F\\otimes\\mathbf C^d))$.\nFor (ii), we note that $\\{0\\}\\times B\\times D = \\psi^{-1}(\\{0\\}\\times B\\times D')$, where $D' = \\emptyset$ if $D=\\emptyset$ and $D' = C^d$ if $D=\\Omega'$, which implies that $\\{0\\}\\times B\\times D\\in \\psi^{-1}(Pred(\\b F\\otimes\\b C^d))$.\nTherefore also $Pred(\\b F\\otimes\\b G^X)\\subseteq \\sigma(\\psi)$, and we conclude that $Pred(\\b F\\otimes\\b G^X) = \\sigma(\\psi)$.\n\n\nUsing this auxiliary result, we can now prove both directions separately as follows.\n\\begin{enumerate}[wide,label=(\\roman*)]\n \\item Let $Y:[0,T]\\times\\Omega\\times\\Omega'\\to A$ be $\\b F\\otimes\\b G^X$-predictable. Using that $\\psi%:(t,\\omega,\\omega')\\mapsto (t,\\omega,[X(\\omega')]_t)\n $ is $Pred(\\b F\\otimes\\mathbf C^d)$-$Pred(\\b F\\otimes\\b G^X)$-measurable, and that $Pred(\\b F\\otimes\\mathbf C^d) = \\sigma(\\psi)$, an application of the Doob-Dynkin Lemma gives us the desired $\\b F\\otimes\\mathbf C^d$-predictable process $\\phi:[0,T]\\times\\Omega\\times C^d\\to A$ with $Y(t,\\omega,\\omega') =(\\phi\\circ\\psi)(t,\\omega,\\omega') = \\phi(t,\\omega,[X(\\omega')]_t)$ for all $(t,\\omega,\\omega')\\in [0,T]\\times\\Omega\\times\\Omega'$. Finally, since $[\\cdot]_t\\circ[\\cdot]_t = [\\cdot]_t$, we conclude that $\\phi(t,\\omega,X(\\omega')) = Y_t(\\omega,\\omega') = \\phi(t,\\omega,[X(\\omega')]_t)$ for all $(t,\\omega,\\omega')\\in [0,T]\\times\\Omega\\times\\Omega'$.\n \\item For the other direction, let $\\phi:[0,T]\\times\\Omega\\times C^d\\to A$ be such an $\\b F\\otimes\\mathbf C^d$-predictable process. Then since $\\psi%:(t,\\omega,\\omega')\\mapsto (t,\\omega,[X(\\omega')]_t)\n $ is $Pred(\\b F\\otimes\\b G^X)$-$Pred(\\b F\\otimes\\mathbf C^d)$-measurable, we directly obtain that $Y:[0,T]\\times\\Omega\\times\\Omega'\\to A$, which satisfies $Y_t(\\omega,\\omega') = \\phi(t,\\omega,[X(\\omega')]_t) = (\\phi \\circ \\psi)(t,\\omega,\\omega')$ for all $(t,\\omega,\\omega')\\in [0,T]\\times\\Omega\\times\\Omega'$, is $\\b F\\otimes\\b G^X$-predictable.\n\\end{enumerate}\n\\end{proof}\n\n\nThe following generalisation of \\cite[Proposition 9]{claisse_pseudo-markov_2016} is proven the same way as in \\cite{claisse_pseudo-markov_2016}, but since the proof is quite short, we will include it for the reader's convenience.\n\n\\begin{lemma}\\label{lemma canonical space predictable adapted measurable equivalence}\nLet $(\\Omega,\\c F)$ be a measurable space and $(C^d,\\c C^d_T,\\mathbf C^d,\\mu^d_W)$ the canonical filtered probability space of continuous $\\R^d$-valued paths. Let $E$ be a Polish space equipped with its Borel $\\sigma$-algebra $\\c B(E)$. Then for a process $X:[0,T]\\times\\Omega\\times C^d\\to E$ the following statements are equivalent:\n\\begin{enumerate}[label=(\\roman*)]\n \\item $X$ is $\\c F\\otimes \\mathbf C^d$-predictable,\n \\item $X$ is $\\c F\\otimes \\mathbf C^d$-optional,\n \\item $X$ is $\\c F\\otimes \\mathbf C^d$-progressive,\n \\item $X$ is $\\c B([0,T])\\otimes\\c F\\otimes\\c C^d_T$-$\\c B(E)$-measurable and $\\c F\\otimes \\mathbf C^d$-adapted,\n \\item $X$ is $\\c B([0,T])\\otimes\\c F\\otimes\\c C^d_T$-$\\c B(E)$-measurable and satisfies\n \\[\n X_s(\\omega,w^d) = X_s(\\omega,[w^d]_s),\\qquad\\text{for all }(s,\\omega,w^d) \\in [0,T]\\times\\Omega\\times C^d.\n \\]\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\\begin{enumerate}[wide,label=(\\roman*)]\n \\item It is known that (i) $\\Rightarrow$ (ii) $\\Rightarrow$ (iii) $\\Rightarrow$ (iv).\n \\item Regarding (iv) $\\Rightarrow$ (v), we first note that $\\c F\\otimes\\c C^d_s = \\sigma((\\omega,w^d)\\mapsto (\\omega,[w^d]_s))$ for all $s\\in [0,T]$. Thus using that $E$ is Polish together with $X_s$ being $\\c F\\otimes\\c C^d_s$-measurable, we can apply the Doob-Dynkin Lemma to obtain an $\\c F\\otimes\\c C^d_s$-measurable $Y_s:\\Omega\\times C^d\\to E$ such that $X_s(\\omega,w^d) = Y_s(\\omega,[w^d]_s)$ for all $(\\omega,w^d)\\in \\Omega\\times C^d$. Now using that $[\\cdot]_s\\circ [\\cdot]_s = [\\cdot]_s$ for all $s\\in [0,T]$, we conclude that $X_s(\\omega,[w^d]_s) = Y_s(\\omega,[w^d]_s) = X_s(\\omega,w^d)$ for all $(s,\\omega,w^d)\\in [0,T]\\times\\Omega\\times C^d$.\n \\item Regarding (v) $\\Rightarrow$ (i), we note that $\\psi:(s,\\omega,w^d)\\mapsto (s,\\omega,[w^d]_s)$ is $\\c F\\otimes\\b C^d$-predictable. To this end, we recall since $\\c C^d_T = \\sigma(\\{w^d_t\\in B\\} | t\\in [0,T], B\\in\\c B(\\R^d))$ that $\\c B([0,T])\\otimes\\c F\\otimes\\c C^d_T$ is generated by all sets of the form $D = A\\times B\\times \\{w^d_t\\in C\\}$, where $A\\in\\c B([0,T])$, $B\\in\\c F$ and $t\\in [0,T]$, $C\\in\\c B(\\R^d)$. For such sets $D$, we have $\\psi^{-1}(D) = \\{(s,\\omega,w^d)\\in A\\times B\\times C^d \\,|\\, ([w^d]_s)_t \\in C\\} = \\{(s,\\omega,w^d)\\in A\\times B\\times C^d \\,|\\, w^d_{s\\land t}\\in C\\} \\in Pred(\\c F\\otimes\\b C^d)$ since $(s,\\omega,w^d)\\mapsto w^d_{s\\land t}$ is continuous and $\\b C^d$-adapted and hence $\\b C^d$-predictable. Therefore $\\psi$ is $\\c F\\otimes\\b C^d$-predictable, and we can conclude that $X = X\\circ \\psi$ is also $\\c F\\otimes\\b C^d$-predictable.\n\\end{enumerate}\n\\end{proof}\n\n\nWith these tools in hand, we are now ready to go back to the action sets $\\c A$ and $\\hat{\\c A}$ and prove the following \\cref{corollary alpha given alpha bar well defined}. The key idea is noting that $\\c A_s = L^0(\\Omega,\\c G\\lor\\c F^W_s,\\P;A)$ and $L^0(\\Omega\\times C^m,\\c G\\otimes\\c C^m_s,\\P\\otimes\\mu^m_W;A)$ are isomorphic: by decomposing $\\alpha\\in\\c A_s$ similar to the preceding \\cref{proposition decompose filtrations of predictable processes} and together with the Doob-Dynkin Lemma, we can find for each $\\alpha\\in\\c A_s$ an $\\tilde\\alpha\\in L^0(\\Omega\\times C^m,\\c G\\otimes\\c C^m_s,\\P\\otimes\\mu^m_W;A)$ such that $\\alpha(\\omega) = \\tilde\\alpha(\\omega,W(\\omega))$ for all $\\omega\\in\\Omega$, and vice versa. Furthermore, since $\\c G$ and $W$ are independent, we also see that $\\norm{\\alpha}_{\\c A_s} = \\norm{\\tilde\\alpha}_{L^0(%\\Omega\\times C^m,\\c G\\otimes\\c C^m_s,\n\\P\\otimes\\mu^m_W;A\n)}$, that means that their respective norms also coincide. Using this canonical probability space has now the advantage that we can use \\cref{lemma canonical space predictable adapted measurable equivalence} to show the predictability of the process, which allows us to prove the following \\cref{corollary alpha given alpha bar well defined}.\n\n\\begin{proposition}\\label{corollary alpha given alpha bar well defined}\n For every $\\hat\\alpha\\in\\hat{\\c A}$, there exists an $\\alpha\\in\\c A$ that $\\b F^B$-identifies to $\\hat\\alpha$.\n\\end{proposition}\n\\begin{proof}\nStarting from $\\hat\\alpha\\in\\hat{\\c A}$, recalling that $\\hat\\alpha$ is $\\b F^B$-predictable, we can by \\cite[Proposition 9]{claisse_pseudo-markov_2016} (or \\cref{proposition canonical representation of predictable processes} with $\\Omega$ being the trivial probability space) find a $\\mathbf C^n$-predictable $\\hat\\Psi:[0,T]\\times C^n \\to \\c A_T% = L^0(\\Omega,\\c G\\lor\\c F^W_T,\\P;A)\n$ with $\\hat\\Psi(s,w)\\in \\c A_s$ for all $(s,w)\\in [0,T]\\times C^n$ such that\n\\[\n\\hat\\alpha_s(\\hat\\omega) = \\hat\\Psi_s([B(\\hat\\omega)]_s) = \\hat\\Psi_s(B(\\hat\\omega)),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega.\n\\]\nFurthermore as indicated before, we use that $\\c A_s$ is isomorphic to $L^0(\\Omega\\times C^m,\\c G\\otimes\\c C^m_s,\\P\\otimes\\mu^m_W;A)$ for all $s\\in [0,T]$, and we therefore can find a $\\mathbf C^n$-predictable map $\\tilde\\Psi:[0,T]\\times C^n\\to L^0(\\Omega\\times C^m,\\c G\\otimes\\c C^m_T,\\P\\otimes\\mu^m_W;A)$ such that (viewed as equality in $\\c A_T$)\n\\[\n\\tilde\\Psi_s(B(\\hat\\omega))(\\cdot,W(\\cdot)) = \\hat\\Psi_s(B(\\hat\\omega))(\\cdot),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega\\times\\Omega.\n\\]\nNote that since $\\hat\\Psi_s(\\hat\\omega)\\in\\c A_s$, also $\\tilde\\Psi_s(\\hat\\omega)\\in L^0(\\Omega\\times C^m,\\c G\\otimes\\c C^m_s,\\P\\otimes\\mu^m_W;A)$.\nNext we apply \\cref{proposition selecting a measurable process}, which enables us to select a $\\c G\\otimes\\mathbf C^m\\otimes\\mathbf C^n$-progressive process $\\Psi:[0,T]\\times\\Omega\\times C^m\\times C^n\\to A$ from the equivalence classes from $\\tilde\\Psi$ in the sense that (as equality in $L^0(\\Omega\\times C^m,\\c G\\otimes \\c C^m_T,\\P\\otimes\\mu^m_W;A)$)\n\\[\n\\tilde\\Psi_s(w) = \\Psi_s(\\cdot,\\cdot,w),\\qquad\\text{for all }(s,w)\\in [0,T]\\times C^n.\n\\]\nNow $\\Psi$ is defined on the canonical space and hence \\cref{lemma canonical space predictable adapted measurable equivalence} shows that $\\Psi$ is also $\\c G\\otimes\\mathbf C^m\\otimes\\mathbf C^n$-predictable. Finally, we define the process $\\Phi:[0,T]\\times\\Omega\\times\\hat\\Omega\\to A$ by\n\\[\n\\Phi_s(\\omega,\\hat\\omega) \\coloneqq \\Psi_s(\\omega,W(\\omega),B(\\hat\\omega)) ,\\qquad\\text{for all }(s,\\omega,\\hat\\omega)\\in [0,T]\\times\\Omega\\times\\hat\\Omega,\n\\]\nwhich by \\cref{proposition canonical representation of predictable processes,proposition decompose filtrations of predictable processes} is now $(\\c G\\lor\\b F^W)\\otimes\\b F^B$-predictable, and satisfies by construction (viewed as equality in $\\c A_T$)\n\\[\n\\hat\\alpha_s(\\hat\\omega) = \\Phi_s(\\cdot,\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in[0,T]\\times\\hat\\Omega.\n\\]\nNow it only remains to define the process $\\alpha:[0,T]\\times\\hat\\Omega\\to A$ by\n\\[\n\\alpha_s(\\hat\\omega) \\coloneqq \\Phi_s(\\pi(\\hat\\omega),\\hat\\omega),\\qquad (s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega,\n\\]\nand we obtain the desired $\\b F^{W,B}\\lor\\c G$-predictable process $\\alpha\\in\\c A$ which $\\b F^B$-identifies to $\\hat\\alpha$.\n\\end{proof}\n\n\nFinally, we conclude this section with the following \\cref{lemma existence N bar N}, which establishes the existence of an $\\alpha$ identifying to a given $\\hat\\alpha$ for a more general class of $\\hat\\alpha$. This result is particularly important for \\cref{section randomised problem}, where we introduce the randomisation framework, as the Poisson process involved will neither be predictable nor adapted to $\\b \\c G\\lor\\b F^{W,B}$ and thus will not fit into the setting of \\cref{corollary alpha given alpha bar well defined}.\n\n\\begin{corollary}\\label{lemma existence N bar N}\n Let $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ be an extension of $(\\Omega,\\c F,\\P)$ and $\\b F$ be a filtration independent of $\\c G,W$. Further let $\\hat\\alpha:[0,T]\\times\\hat\\Omega\\to\\c A_T$ be an $\\b F$-progressive process such that $\\hat\\alpha_s(\\hat\\omega) \\in \\c A_s$ for all $(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega$.\n Then there exists an $\\b F^W\\lor\\c G\\lor\\b F$-progressive and $\\b F^W\\lor\\c G\\lor\\c F_T$-predictable process $\\alpha$ which $\\b F$-identifies to $\\hat\\alpha$.\n\\end{corollary}\n\\begin{proof}\n Similarly to the construction in the proof of \\cref{corollary alpha given alpha bar well defined}, we start by using that $\\c A_s$ is isomorphic to $L^0(\\Omega\\times C^m,\\c G\\otimes\\c C^m_s,\\P\\otimes\\mu^m_W;A)$ for all $s\\in [0,T]$, and we therefore can find a $\\b F$-progressive process $\\tilde\\alpha:[0,T]\\times\\hat\\Omega\\to L^0(\\Omega\\times C^m,\\c G\\otimes\\c C^m_T,\\P\\otimes\\mu^m_W;A)$ such that (viewed as equality in $\\c A_T$)\n \\[\n \\tilde\\alpha_s(\\hat\\omega)(\\cdot,W(\\cdot)) = \\hat\\alpha_s(\\hat\\omega)(\\cdot),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega,\n \\]\n and $\\tilde\\alpha_s(\\hat\\omega) \\in L^0(\\Omega\\times C^m,\\c G\\otimes\\c C^m_s,\\P\\otimes\\mu^m_W;A)$ for all $(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega$.\n Now using \\cref{proposition selecting a measurable process}, we can select a $\\c G\\otimes\\mathbf C^m\\otimes\\b F$-progressive process $\\Psi:[0,T]\\times\\Omega\\times C^m\\times\\hat\\Omega\\to A$ such that, viewed as equality in $\\c A_T$,\n \\[\n \\Psi_s(\\cdot,\\cdot,\\hat\\omega) = \\tilde\\alpha_s(\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega.\n \\]\n Since $\\Psi$ is defined on the canonical space, \\cref{lemma canonical space predictable adapted measurable equivalence} shows that $\\Psi$ is also $\\c G\\otimes\\mathbf C^m\\otimes\\c F_T$-predictable. Finally we define the process $\\Phi:[0,T]\\times\\Omega\\times\\hat\\Omega\\to A$ by\n \\[\n \\Phi_s(\\omega,\\hat\\omega) \\coloneqq \\Psi_s(\\omega,W(\\omega),\\hat\\omega)\n ,\\qquad\\text{for all }(s,\\omega,\\hat\\omega)\\in [0,T]\\times\\Omega\\times\\hat\\Omega,\n \\]\n which is then $(\\c G\\lor\\b F^W)\\otimes\\b F$-progressive and by \\cref{proposition canonical representation of predictable processes,proposition decompose filtrations of predictable processes} also $(\\c G\\lor\\b F^W)\\otimes\\c F_T$-predictable. Now we only need to define the process $\\alpha:[0,T]\\times\\hat\\Omega\\to A$ by\n \\[\n \\alpha_s(\\hat\\omega) \\coloneqq \\Phi_s(\\pi(\\hat\\omega),\\hat\\omega),\\qquad (s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega,\n \\]\n which is then the desired $\\b F^W\\lor\\c G\\lor\\b F$-progressive and $\\b F^W\\lor\\c G\\lor\\c F_T$-predictable process which $\\b F$-identifies to $\\hat\\alpha$.\n\\end{proof}\n\n\n\n\\section{Randomised problem}\\label{section randomised problem}\n\nIn this section, we derive a randomised formulation for mean-field control problems with common noise.\nThe main idea of the randomisation method introduced in \\cite{bouchard_stochastic_2009} is to replace the control process by an independent Poisson point process taking values in the action space, and then controlling its intensity instead. While in the non-mean-field case, the set $A$ would be the natural choice for the action space, in which the aforementioned Poisson process should take its values, in the mean-field case this choice runs into problem concerning the additional mean-field dependence of the state dynamics when changing the intensity of the Poisson point process. In fact, in \\cite{bayraktar_randomized_2018} (without common noise), the authors instead introduced an additional space $\\c A_{\\text{step}}$ of all $\\b F^W$-progressive step processes taking values in $A$, which they then took as action space for the randomising Poisson point process. The motivation behind this approach is that the conditional law process $(\\P^{\\c F^{B,\\P}_s}_{X_s})_s$ is $\\b F^{B,\\P}$-progressive and as such well-behaved when tilting the Poisson point process $\\mu$ w.r.t.\\@ $\\b F^{B,\\mu}$-predictable intensities.\n\nIn this paper, we take a similar approach. On the one hand, we simplify the idea of \\cite{bayraktar_randomized_2018} by considering the action spaces $\\c A_s = L^0(\\Omega,\\c G\\lor\\c F^W_s,\\P;A)$ defined in \\cref{section problem l2 formulation}, which only represent the (conditional) distribution of $\\alpha$ at time $s$, instead of considering a whole piece-wise constant process in $\\c A_{\\text{step}}$ as action at a given time point.\\footnote{The idea behind $\\c A_{\\text{step}}$ could be interesting when further extending this setting to include path-dependence. However for now, while it seems to ease the construction of the randomising Poisson point process, it later complicates the proofs and the resulting BSDE formulation. Further it seems to be hard to get rid of the dependence on the specific sequences used to construct $\\c A_{\\text{step}}$.}\nOn the other hand, we extend the setting to additionally consider problems with common noise, which is aided by viewing $\\hat{\\c A}$ as the set of admissible controls instead of $\\c A$.\n\nOur first step will be defining an $\\c A_s$-valued Poisson point process. Now defining a Poisson point process with marks taking values on different spaces $\\c A_s$ depending on the current time $s\\in [0,T]$ is difficult, but we are going to utilise that $\\c A_0\\subseteq\\c A_s\\subseteq\\c A_T$ for all $s\\in [0,T]$.\n\n\\begin{assumption}\\label{assumptions lambda family}\nWe assume that $(\\lambda_s)_{s\\in [0,T]}$ is a family of finite measures on $\\c A_T$ such that\n\\begin{assumptionenum}[label=(C\\arabic*)]\n \\item the topological support of each $\\lambda_s$ is given by $\\c A_s$ for all $s\\in [0,T]$,\n \\item $\\lambda_s\\ll \\lambda_r$ for all $0\\leq s\\leq r\\leq T$,\n \\item $\\sup_{s\\in [0,T]} \\lambda_s(\\c A_s) < \\infty$.\n\\end{assumptionenum}\n\\end{assumption}\n\nSince it may not be obvious why such a family exists, we will provide a possible construction in \\cref{remark construction lambda family}. Note that since $\\lambda_s$ topological support is given by $\\c A_s$, we will view $\\lambda_s$, depending on the context, as measure on $\\c A_T$ or $\\c A_s$ respectively.\n\n\\begin{lemma}\\label{remark construction lambda family}\nSuch a family $(\\lambda_s)_{s\\in [0,T]}$ satisfying \\cref{assumptions lambda family} exists.\n\\end{lemma}\n\\begin{proof}\n Since each $\\c A_s$ is Polish,\\footnote{Recall that every uncountable Polish space is Borel-isomorphic to $[0,1]$, which we can equip with the Lebesgue measure.} we can construct a family of probability measures $(\\kappa_s)_{s\\in [0,T]\\cap\\Q}$ on $\\c A_T$ such that the topological support of each $\\kappa_s$ is given by $\\c A_s$ for all $s\\in [0,T]\\cap\\Q$, and by normalisation, we can further assume that $\\sup_{s\\in [0,T]\\cap\\Q} \\kappa_s(\\c A_s) < \\infty$.\n Note that this is a countable family and let $(s_n)_n = [0,T]\\cap\\Q$ be an enumeration of it. Now we define $\\lambda_r \\coloneqq \\sum_{s_n\\leq r} 2^{-n} \\kappa_{s_n}$ for all $r\\in [0,T]$. We see that by construction $\\lambda_s\\ll\\lambda_r$ for all $t\\leq s\\leq r\\leq T$. Further, we recall that the filtration $\\c G\\lor\\b F^{W,\\P}$ is left-continuous, see \\cite[Chapter 2, Problem 7.6]{karatzas_brownian_1998}, and thus we can approximate every indicator function $\\1_A \\in\\c A_r$, $A\\in \\c G\\lor\\c F^{W,\\P}_r$ by a series of indicator functions $\\1_{A_n}\\in\\c A_{q_n}$, $A_n\\in\\c G\\lor\\c F^{W,\\P}_{q_n}$ with $q_n\\leq r$,$q_n\\in\\Q$. Since such indicator functions generate $\\c A_s = L^0(\\Omega,\\c G\\lor\\c F^W,\\P;A)$, this shows that $\\bigcup_{q\\leq r, q\\in\\Q} \\c A_q\\subseteq\\c A_r$ is dense in $\\c A_r$, and thus the topological support of each $\\lambda_r$ is really given by $\\c A_r$.\n\\end{proof}\n\nTo construct a randomised setting, let us suppose we are given such a family $(\\lambda_s)_{s\\in [0,T]}$, and $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ is (if needed) an extension of the original space $(\\Omega,\\c F,\\P)$ such that there exists a Poisson random measure $\\mu$ on $(0,T]\\times\\c A_T$, independent of $\\c G,W,B$, with intensity $\\frac{d\\lambda_s}{d\\lambda_T}(\\alpha) \\lambda_T(d\\alpha)ds = \\lambda_s(d\\alpha) ds$. Note that with slight abuse of notation, we again denote the canonical extensions of the $\\sigma$-algebra $\\c G$, the (non-augmented) filtrations $\\b F^W,\\b F^{B}$, the random variable $\\xi$ and the processes $W,B$ with the same symbols as their original versions. Further, we denote the canonical projection from $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ to $(\\Omega,\\c F,\\P)$ by $\\pi:\\hat\\Omega\\to\\Omega$, and recall that by definition it is probability preserving, that is $\\hat\\P(\\pi^{-1}(A)) = \\P(A)$, for all $A\\in\\c F$.\n\nGiven an initial time $t\\in [0,T]$, we define the processes $B^t,\\mu^t$ via\n\\[\nB^t\\coloneqq (B_{\\cdot\\lor t} - B_t),\\quad \\mu^t\\coloneqq \\mu(\\cdot\\cap (t,T]\\times\\c A_T)).\n\\]\nNote that by construction $B^t,\\mu^t$ are $\\hat\\P$-independent of $\\c F^{B,\\mu,\\hat\\P}_t$.\nNext, we construct the piece-wise constant $\\b F^{\\mu^t}$-progressive process $\\hat I^{t,\\alpha_t}:[t,T]\\times\\hat\\Omega\\to \\c A_T$ as follows\n\\begin{align}\n \\hat I^{t,\\alpha_t}_s \\coloneqq \\alpha_t + \\int_{(t,s]}\\int_{\\c A_r} (\\alpha - \\hat{I}^{t,\\alpha_t}_{r-}) \\mu(dr,d\\alpha),\\qquad s\\in [t,T],\n \\label{eq randomised poisson point process}\n\\end{align}\nwhere $\\alpha_t\\in\\c A_t\\subseteq\\c A_T$ is a fixed chosen initial action.\nNote that we will later show that both the particular choices of $\\alpha_t\\in\\c A_t$ and the extension $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ of $(\\Omega,\\c F,\\P)$ have no influence on the resulting value function. Furthermore, we observe in the following \\cref{corollary I given bar I is well defined} that there exists a $\\b F^{W,\\mu^t}\\lor\\c G$-progressive process $I^{t,\\alpha_t}$ which $\\b F^{\\hat I^{t,\\alpha_t}}$-identifies to $\\hat I^{t,\\alpha_t}$, and that this choice is even unique in joint law, independent of the chosen extension $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ of $(\\Omega,\\c F,\\P)$. In the following we will use the notation $\\pi|_{\\c G}:(\\hat\\Omega,\\c G)\\to (\\Omega,\\c G)$ to denote the $\\c G$-measurable projection from $\\hat\\Omega$ to $\\Omega$. \n\n\\begin{corollary}\\label{corollary I given bar I is well defined}\n Given $\\hat I^{t,\\alpha_t}$, we can choose a càdlàg, $\\b F^W\\lor\\c G\\lor\\c F^{\\hat I^{t,\\alpha_t}}_T$-predictable and $\\b F^{W,\\hat I^{t,\\alpha_t}}\\lor\\c G$-progressive $I^{t,\\alpha_t}$ which $\\b F^{\\hat I^{t,\\alpha_t}}$-identifies to $\\hat I^{t,\\alpha_t}$, unique up to indistinguishability.\n In particular, $I^{t,\\alpha_t}$ is also $\\b F^W\\lor\\c G\\lor\\c F^{\\mu^t}_T$-predictable and $\\b F^{W,\\mu^t}\\lor\\c G$-progressive.\n\n Further, this choice is even unique in the joint law $(\\pi|_{\\c G},\\xi,W,B,\\mu,\\hat I^{t,\\alpha_t},I^{t,\\alpha_t})$, by which we mean that if $(\\hat\\Omega',\\hat{\\c F}',\\hat\\P')$ is a another extension of $(\\Omega,\\c F,\\P)$ on which suitable $\\mu,\\hat I^{t,\\alpha_t},I^{t,\\alpha_t}$ are defined, then also\n \\[\n \\c L^{\\hat\\P}(\\pi|_{\\c G},\\xi,W,B,\\mu,\\hat I^{t,\\alpha_t},I^{t,\\alpha_t}) = \\c L^{\\hat\\P'}(\\pi|_{\\c G},\\xi,W,B,\\mu,\\hat I^{t,\\alpha_t},I^{t,\\alpha_t}).\n \\]\n\\end{corollary}\n\\begin{proof}\n From \\cref{lemma existence N bar N}\\footnote{To match the setting described in \\cref{section problem l2 formulation}, we can define $\\c G'\\coloneqq\\c G\\lor\\c F^W_t$ and consider $\\hat I'\\coloneqq \\hat I^{t,\\alpha_t}_{\\cdot - t}:[0,T-t]\\times\\hat\\Omega\\to\\c A_T$ and $W' \\coloneqq W^t_{\\cdot - t}$. Note that, for example, $\\hat I'$ is then $\\c G'\\lor\\b F^{W'}$-progressive since $\\c G'\\lor\\b F^{W'} = (\\c G\\lor\\c F^W_{t+s})_{s\\in [0,T-t]}$.}\n we obtain an $\\b F^W\\lor\\c G\\lor\\c F^{\\mu^t}_T$-predictable and $\\b F^{W,\\mu^t}\\lor\\c G$-progressive process $I:[t,T]\\times\\Omega\\to A$ that $\\b F^{\\hat I^{t,\\alpha_t}}$-identifies to $\\hat I^{t,\\alpha_t}$. Next we define\n \\begin{align}\n I^{t,\\alpha_t}_s\\coloneqq I_t + \\sum_{r\\in(t,s],\\hat I^{t,\\alpha_t}_{r-}\\not= \\hat I^{t,\\alpha_t}_r} (I_r - I_{r-}),\\qquad s\\in [t,T],\n \\label{eq corollary 4.2 construction I t alpha_t}\n \\end{align}\n which by construction is piece-wise constant and in particular càdlàg. Furthermore, just like $I$, the process $I^{t,\\alpha_t}$ is also $\\b F^W\\lor\\c G\\lor\\c F^{\\mu^t}_T$-predictable and $\\b F^{W,\\mu^t}\\lor\\c G$-progressive. Finally, since $I$ is $\\b F^{\\hat I^{t,\\alpha_t}}$-identifying with $\\hat I^{t,\\alpha_t}$, there exists a $(\\c G\\lor\\b F^W)\\otimes\\b F^{\\hat I^{t,\\alpha_t}}$-progressive process $\\Psi:[t,T]\\times\\Omega\\times\\hat\\Omega\\to A$ such that\n \\begin{align}\n I_s(\\hat\\omega) = \\Psi_s(\\pi(\\hat\\omega),\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega,\n \\label{eq corollary 4.2 I Psi}\n \\end{align}\n and, viewed as equality in $\\c A_T$,\n \\begin{align}\n \\hat I^{t,\\alpha_t}_s(\\hat\\omega) = \\Psi_s(\\cdot,\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega.\n \\label{eq corollary 4.2 hat I Psi}\n \\end{align}\n Now defining $\\Phi:[t,T]\\times\\Omega\\times\\hat\\Omega\\to A$ via\n \\[\n \\Phi_s\\coloneqq \\Psi_t + \\sum_{r\\in(t,s],\\hat I^{t,\\alpha_t}_{r-}\\not= \\hat I^{t,\\alpha_t}_r} (\\Psi_r - \\Psi_{r-}) ,\\qquad s\\in [t,T],\n \\]\n we obtain an $(\\c G\\lor\\c F^W)\\otimes\\b F^{\\hat I^{t,\\alpha_t}}$-progressive process, which due to \\eqref{eq randomised poisson point process,eq corollary 4.2 construction I t alpha_t,eq corollary 4.2 hat I Psi,eq corollary 4.2 I Psi} now also satisfies\n \\[\n I^{t,\\alpha_t}_s(\\hat\\omega) = \\Phi_s(\\pi(\\hat\\omega),\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega,\n \\]\n and, viewed as equality in $\\c A_T$,\n \\[\n \\hat I^{t,\\alpha_t}_s(\\hat\\omega) = \\Phi_s(\\cdot,\\hat\\omega),\\qquad\\text{for all }(s,\\hat\\omega)\\in [0,T]\\times\\hat\\Omega.\n \\]\n Hence, $I^{t,\\alpha_t}$ also $\\b F^{\\hat I^{t,\\alpha_t}}$-identifies to $\\hat I^{t,\\alpha_t}$.\n\n Finally the uniqueness result follows from \\cref{lemma existence N bar N,lemma joint law N bar N map}, noting that $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ and $(\\hat\\Omega',\\hat{\\c F}',\\hat\\P')$ are both extensions of $(\\Omega,\\c G\\lor\\c F^{W,B}_T,\\P)$.\n\\end{proof}\n\nHaving a well-defined corresponding control process $I^{t,\\alpha_t}$, we can now define the $\\b F^{W,B^t,\\mu^t,\\hat\\P}\\lor\\c G$-progressive state process $X^{t,\\xi,\\alpha_t}$ as the unique solution to the following uncontrolled dynamics\n\\begin{align}\n dX^{t,\\xi,\\alpha_t}_s &= b(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds + \\sigma(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) dW_s\\\\\n &\\qquad+ \\sigma^0(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) dB_s,\\qquad X^{t,\\xi,\\alpha_t}_t = \\xi.\n \\label{eq sde randomised problem}\n\\end{align}\nAgain $(\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s})_{s\\in [t,T]}$ shall denote the $\\b F^{B^t,\\mu^t,\\hat\\P}$-optional and $\\hat\\P$-a.s.\\@ continuous version of the conditional law $(\\c L(X^{t,\\xi,\\alpha_t}_s|\\c F^{B,\\mu,\\hat\\P}_s))_{s\\in [t,T]}$, that is $\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s} = \\c L(X^{t,\\xi,\\alpha_t}_s|\\c F^{B,\\mu,\\hat\\P}_s)$, $\\hat\\P$-a.s., for all $s\\in [t,T]$. We recall that the existence of such a version is ensured by \\cite[Lemma A.1]{djete_mckeanvlasov_2022-1}. Further, \\cref{assumptions sde coefficients} ensures by standard arguments the existence and uniqueness of $X^{t,\\xi,\\alpha_t}$.\n\nAs admissible controls to our randomised problem, we consider the set $\\c V$ of all $Pred(\\b F^{B,\\mu})\\otimes \\c B(\\c A_T)$-measurable processes which are strictly bounded away from $0$ and $\\infty$, that is $0 < \\inf_{[0,T]\\times\\hat\\Omega\\times \\c A_T} \\nu \\leq \\sup_{[0,T]\\times\\hat\\Omega\\times\\c A_T} \\nu < \\infty$. Further, we will also introduce the subset $\\c V_t\\subseteq\\c V$ of $Pred(\\b F^{B^t,\\mu^t})\\otimes \\c B(\\c A_T)$-measurable processes.\nNote that by Girsanov's Theorem, see \\cite[Proposition 14.4.I]{daley_introduction_2008},\nwe can define for each admissible control $\\nu\\in \\c V$ a probability measure using the Doléans-Dade exponential $L^\\nu$ as follows\n\\begin{align}\\label{eq randomised control girsanov formula}\n \\frac{d\\hat\\P^\\nu}{d\\hat\\P}\\Big|_{\\c F^{B,\\mu,\\hat\\P}_s} \\coloneqq L^\\nu_s \\coloneqq \\exp\\Big(\\int_{(0,s]} \\int_{\\c A_r} \\log \\nu_r(\\alpha) \\mu(dr,d\\alpha) - \\int_0^s \\int_{\\c A_r} (\\nu_r(\\alpha) - 1) \\lambda_r(d\\alpha)dr \\Big),\\qquad s\\in [0,T],\n\\end{align}\nsuch that under $\\hat\\P^\\nu$ the Poisson random measure $\\mu$ has the intensity $\\nu_s(\\alpha) ds\\lambda_s(d\\alpha)$, (recall that $\\mu((0,s],du)$ is supported on $\\c A_s\\subseteq\\c A_T$). By the following \\cref{lemma d P nu d P is square integrable}, which can be proved in the same way as \\cite[Lemma 2.4]{kharroubi_feynmankac_2015}, we see that $\\hat\\P^\\nu$ is indeed well-defined.\n\n\\begin{lemma}\\label{lemma d P nu d P is square integrable}\n For every $\\nu\\in\\c V$ the process $L^\\nu$ defined in \\eqref{eq randomised control girsanov formula} is a uniform integrable martingale and $L_T \\in L^2(\\hat\\Omega,\\c F^{B,\\mu}_T,\\hat\\P;\\R)$.\n\\end{lemma}\n\nWe note that by standard arguments, since $\\hat\\P_\\xi = \\hat\\P^\\nu_\\xi$ for all $\\nu\\in\\c V$, from \\cref{assumptions sde coefficients} we obtain the following uniform estimate\n\\begin{align}\\label{eq randomised state dynamics basic estimate}\n \\sup_{\\alpha_t\\in\\c A_t} \\sup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[\\sup_{s\\in [t,T]} |X^{t,\\xi,\\alpha_t}_s|^2 \\Big] < \\infty.\n\\end{align}\nOur goal is now to maximise the following randomised reward functional\n\\begin{align}\n J^{\\c R}(t,\\xi,\\alpha_t,\\nu) \\coloneqq \\E^{\\hat\\P^\\nu}\\Big[ g(\\hat\\P_{X^{t,\\xi,\\alpha_t}_T}^{\\c F^{B,\\mu,\\hat\\P}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P_{X^{t,\\xi,\\alpha_t}_s}^{\\c F^{B,\\mu,\\hat\\P}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds \\Big],\n\\end{align}\nwhere $\\b E^{\\hat\\P^\\nu}$ denotes the expectation under the probability measure $\\hat\\P^\\nu$ corresponding to $\\nu\\in\\c V$.\nFinally, we also define the randomised value function via\n\\[\nV^{\\c R}(t,\\xi,\\alpha_t) \\coloneqq \\sup_{\\nu\\in\\c V} J^{\\c R}(t,\\xi,\\alpha_t,\\nu) < \\infty,\n\\]\nwhich we know by \\cref{assumptions reward functional} and \\eqref{eq randomised state dynamics basic estimate} is well-defined. We note in the following \\cref{proposition equivalence v v_t randomised value function}, that this is equivalent to taking only optimising over the controls in $\\c V_t$.\n\n\\begin{proposition}\\label{proposition equivalence v v_t randomised value function}\nLet $t\\in [0,T]$ and $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$. Defining the randomised problem over the restricted control set $\\c V_t\\subseteq\\c V$ leads to the same value function, that is\n\\[\nV^{\\c R}(t,\\xi,\\alpha_t) = \\sup_{\\nu\\in\\c V_t} J^{\\c R}(t,\\xi,\\alpha_t),\n\\]\nfor all $\\alpha_t\\in\\c A_t\\subseteq\\c A_T$.\n\\end{proposition}\n\\begin{proof}\n We first note that $\\sup_{\\nu\\in\\c V_t} J^{\\c R}(t,\\xi,\\alpha_t,\\nu) \\leq \\sup_{\\nu\\in\\c V} J^{\\c R}(t,\\xi,\\alpha_t,\\nu) \\eqqcolon V^{\\c R}(t,\\xi,\\alpha_t)$ follows directly from $\\c V_t\\subseteq\\c V$.\n For the other direction, let us fix $\\nu\\in\\c V$.\n Since $\\nu$ is $Pred(\\b F^{B,\\mu})\\otimes\\c B(\\c A_T)$-measurable, by \\cref{proposition decompose filtrations of predictable processes},\\footnote{Note that $\\b F^{B,\\mu} = \\b F^{B,\\mu}_{t\\land \\cdot}\\lor \\b F^{B^t,\\mu^t} \\subseteq \\c F^{B,\\mu}_t\\lor\\b F^{B^t,\\mu^t}$.}\n we can decompose $\\nu$ into an $\\c F^{B,\\mu}_t\\otimes Pred(\\b F^{B^t,\\mu^t})\\otimes\\c B(\\c A_T)$-measurable process $\\Upsilon:[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\to\\c A_T$ with $0<\\inf_{[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\times\\c A_T} \\Upsilon\\leq \\sup_{[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\times\\c A_T} \\Upsilon < \\infty$ such that $\\nu(\\hat\\omega) = \\Upsilon(\\hat\\omega,\\hat\\omega)$, for all $\\hat\\omega\\in\\hat\\Omega$. Note that in particular $\\Upsilon(\\hat\\omega,\\cdot)\\in\\c V_t$ for all $\\hat\\omega\\in\\hat\\Omega$. Now noting that $X^{t,\\xi,\\alpha_t}$ and $I^{t,\\alpha_t}$ are $\\hat\\P$-independent of $\\c F^{B,\\mu,\\hat\\P}_t$, we see that\n using freezing lemma, see e.g.\\@ \\cite[Lemma 4.1]{baldi_stochastic_2017}, that\n \\begin{align}\n J^{\\c R}(t,\\xi,\\alpha_t,\\nu)\n &= \\E^{\\hat\\P}\\Big[L^{\\nu}_T g(\\hat\\P_{X^{t,\\xi,\\alpha_t}_T}^{\\c F^{B,\\mu,\\hat\\P}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P_{X^{t,\\xi,\\alpha_t}_s}^{\\c F^{B,\\mu,\\hat\\P}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds \\Big]\\\\\n &= \\E^{\\hat\\P}\\Big[L^\\nu_t \\E^{\\hat\\P}\\Big[\\frac{L^\\nu_T}{L^\\nu_t} \\Big(g(\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big)\\Big| \\c F^{B,\\mu,\\hat\\P}_t\\Big]\\Big]\\\\\n &= \\int_{\\hat\\Omega} L^{\\nu(\\hat\\omega)}_t \\E^{\\hat\\P}\\Big[\\frac{L^{\\Upsilon(\\hat\\omega,\\cdot)}_T}{L^{\\Upsilon(\\hat\\omega,\\cdot)}_t} \\Big(g(\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big)\\Big] \\hat\\P(d\\hat\\omega)\\\\\n &= \\int_{\\hat\\Omega} L^{\\nu(\\hat\\omega)}_t \\E^{\\hat\\P}\\Big[L^{\\Upsilon(\\hat\\omega,\\cdot)}_T \\Big(g(\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big)\\Big] \\hat\\P(d\\hat\\omega)\\\\\n &= \\int_{\\hat\\Omega} L^{\\nu(\\hat\\omega)}_t J^{\\c R}(t,\\xi,\\alpha_t,\\Upsilon(\\hat\\omega,\\cdot)) \\hat\\P(d\\hat\\omega)\\\\\n &\\leq \\int_{\\hat\\Omega} L^{\\nu(\\hat\\omega)}_t \\sup_{\\upsilon\\in\\c V_t} J^{\\c R}(t,\\xi,\\alpha_t,\\upsilon) \\ \\hat\\P(d\\hat\\omega) = \\sup_{\\upsilon\\in\\c V_t} J^{\\c R}(t,\\xi,\\alpha_t,\\upsilon),\n \\end{align}\n using that for every $\\hat\\omega\\in\\hat\\Omega$ we have $\\Upsilon(\\hat\\omega,\\cdot)\\in\\c V_t\\subseteq\\c V$ and thus $\\frac{L^{\\Upsilon(\\hat\\omega,\\cdot)}_T}{L^{\\Upsilon(\\hat\\omega,\\cdot)}_t}$ is $\\hat\\P$-independent of $\\c F^{B,\\mu,\\hat\\P}_t$, together with $\\E^{\\hat\\P}[L^{\\upsilon}_t] = 1$ and $L^\\upsilon_t$ being $\\c F^{B,\\mu,\\hat\\P}_t$-measurable, for all $\\upsilon\\in\\c V$.\n\\end{proof}\n\n\\begin{remark}\\label{remark V > 0 vs V geq 0}\n We could also relax the requirement $\\inf_{[0,T]\\times\\hat\\Omega\\times \\c A_T} \\nu > 0$ to $\\nu\\geq 0$. This leads to the same value function $V^{\\c R}$, as also noted in \\cite[Remark 3.1]{bandini_backward_2018}. However, to simplify our arguments, especially later in the proof of \\cref{theorem equivalence original and randomised value function}, we will consider only $\\inf_{[0,T]\\times\\hat\\Omega\\times \\c A_T}\\nu > 0$ as this condition ensures that $\\hat\\P\\sim\\hat\\P^\\nu$.\n\\end{remark}\n\n\\begin{remark}\\label{remark conditional distribution unchanged under measure change}\n Given $\\nu\\in\\c V$, since $\\nu$ is $\\b F^{B,\\mu}$-predictable and $\\hat\\P^\\nu\\sim\\hat\\P$, the process $\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s}$ is also $\\b F^{B^t,\\mu^t,\\hat\\P^\\nu}$-optional and for all $s\\in [t,T]$ satisfies $\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s} = \\c L^{\\hat\\P^\\nu}(X^{t,\\xi,\\alpha_t}_s | \\c F^{B,\\mu,\\hat\\P^\\nu}_s)$, $\\hat\\P^\\nu$-a.s., which hence justifies writing $\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s} = \\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s}$.\n\\end{remark}\n\n\\begin{lemma}\\label{lemma J R continuous}\nThe reward functional $J^{\\c R}(t,\\cdot,\\cdot,\\cdot):L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)\\times\\c A_t\\times\\c V\\to \\R$ is continuous for each $t\\in [0,T]$, where we equip $\\c V$ with the $L^\\infty$-norm.\n\\end{lemma}\n\\begin{proof}\nLet $(\\xi^n,\\alpha_t^n,\\nu^n)_n\\subseteq L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)\\times\\c A_t\\times\\c V$ with $(\\xi^n,\\alpha_t^n,\\nu^n) \\to (\\xi^\\infty,\\alpha_t^\\infty,\\nu^\\infty) \\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)\\times\\c A_t\\times\\c V$ in $L^2(\\P)\\times L^0(\\hat\\P)\\times L^\\infty(ds\\times\\hat\\P)$. Further let us define $\\bar\\nu \\coloneqq \\sup_{k\\in\\N\\cup\\{\\infty\\}} \\nu^k$. Note that since $\\nu^n\\to\\nu^\\infty$ in $L^\\infty(ds\\times\\hat\\P)$, by construction $\\bar\\nu\\in\\c V$ and thus $\\hat\\P^{\\bar\\nu} \\sim \\hat\\P$. By \\cref{lemma alpha bar alpha uniqueness}, and using that $\\rho < 1$ we see that also $d_{\\c A}^{\\hat\\P^{\\bar\\nu}}(I^{t,\\alpha_t^n},I^{t,\\alpha_t^\\infty}) = d_{\\hat{\\c A}}^{\\hat\\P^{\\bar\\nu}}(\\hat I^{t,\\alpha_t^n},\\hat I^{t,\\alpha_t^\\infty}) \\to 0$. Furthermore, since $\\hat\\P|_{\\c G\\lor\\c F^W_t} = \\hat\\P^{\\bar\\nu}|_{\\c G\\lor\\c F^W_t}$, we also have $\\xi^n\\to\\xi^\\infty$ in $L^2(\\hat\\P^{\\bar\\nu})$.\nSince $X^{t,\\xi^n,\\alpha_t^n}$ and $X^{t,\\xi^\\infty,\\alpha_t^\\infty}$ are $\\c G\\lor\\b F^{W,B,\\mu,\\hat\\P^{\\bar\\nu}}$-progressive, and since $\\c G\\lor\\b F^W$ is independent of $\\b F^{B,\\mu}$ under $\\hat\\P^{\\bar\\nu}$, we see that, for all $s\\in [t,T]$, $\\hat\\P^{\\bar\\nu}$-a.s.,\n\\[\n\\c W_2(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^n,\\alpha_t^n}_s},\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s})^2\n\\leq \\E^{\\hat\\P^{\\bar\\nu}}[|X^{t,\\xi^n,\\alpha_t^n}_s - X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s|^2 | \\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s]\n= \\E^{\\hat\\P^{\\bar\\nu}}[|X^{t,\\xi^n,\\alpha_t^n}_s - X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s|^2 | \\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_T],\n\\]\nwhich implies due to the $\\hat\\P^{\\bar\\nu}$-a.s.\\@ continuity of $(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^n,\\alpha_t^n}_s})_s$ and $(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s})_s$ that\n\\[\n\\sup_{s\\in [t,T]} \\c W_2(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^n,\\alpha_t^n}_s},\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s})^2\n\\leq \\E^{\\hat\\P^{\\bar\\nu}}\\Big[ \\sup_{s\\in [t,T]} |X^{t,\\xi^n,\\alpha_t^n}_s - X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s|^2 \\,\\Big|\\,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_T\\Big].\n\\]\nThus from \\cref{assumptions sde coefficients}, standard arguments using the BDG-inequality and Gronwall's Lemma show that\n\\begin{align}\n \\E^{\\hat\\P^{\\bar\\nu}}\\Big[ \\sup_{s\\in [t,T]} \\Big(|X^{t,\\xi^n,\\alpha_t^n}_s - X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s|^2 + \\c W_2(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^n,\\alpha_t^n}_s},\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s})^2\\Big)\\Big] \\to 0.\n \\label{eq lemma J R continuous convergence of X}\n\\end{align}\nFinally, we see that for all $n\\in\\N$,\n\\begin{align}\n &|J^{\\c R}(t,\\xi^n,\\alpha_t^n,\\nu^n) - J^{\\c R}(t,\\xi^\\infty,\\alpha_t^\\infty,\\nu^\\infty)|\\\\\n &\\leq \\E^{\\hat\\P^{\\bar\\nu}}\\Big[\\Big| \\frac{L^{\\nu^n}_T}{L^{\\bar\\nu}_T} \\Big(g(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_T}_{X^{t,\\xi^n,\\alpha_t^n}_T},X^{t,\\xi^n,\\alpha_t^n}_T) + \\int_t^T f(s,\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^n,\\alpha_t^n}_s},X^{t,\\xi^n,\\alpha_t^n}_s,I^{t,\\alpha_t^n}_s) ds \\Big)\\\\\n &\\qquad - \\frac{L^{\\nu^\\infty}_T}{L^{\\bar\\nu}_T} \\Big(g(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_T}_{X^{t,\\xi^\\infty,\\alpha_t^\\infty}_T},X^{t,\\xi^\\infty,\\alpha_t^\\infty}_T) + \\int_t^T f(s,\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s},X^{t,\\xi^\\infty,\\alpha_t^\\infty}_s,I^{t,\\alpha_t^\\infty}_s) ds \\Big) \\Big| \\Big].\n\\end{align}\nThus, together with, for all $k\\in\\N\\cup\\{\\infty\\}$,\n\\begin{align}\n 0\\leq \\frac{L^{\\nu^k}_T}{L^{\\bar\\nu}_T}\n &= \\exp\\Big(\\int_{[0,T]}\\int_{\\c A_s} \\log \\frac{\\nu^k_s(\\alpha)}{\\bar\\nu_s(\\alpha)} \\mu(ds,d\\alpha) - \\int_0^T \\int_{\\c A_s} (\\nu^k_s(\\alpha) - \\bar\\nu_s(\\alpha)) \\lambda_s(d\\alpha)ds \\Big)\\\\\n &\\leq \\exp\\Big(\\int_0^T \\int_{\\c A_s} \\lambda_s(d\\alpha)ds\\Big)^{\\norm{\\bar\\nu}_\\infty} < \\infty,\n\\end{align}\nand \\eqref{eq lemma J R continuous convergence of X}, \\eqref{eq randomised state dynamics basic estimate} and \\cref{assumptions reward functional}, we obtain the desired result by dominated convergence.\n\\end{proof}\n\nOur first main result is that this randomised problem is in some sense equivalent to the original problem. Note that this then also allows us to transfer the properties of $V$ from \\cref{proposition value function law invariant and independent of probabilistic setting} to $V^{\\c R}$.\nWe give the proof in \\cref{section proof of equivalence randomised non randomised formulation}.\n\n\\begin{theorem}\\label{theorem equivalence original and randomised value function}\n The randomised problem is equivalent to the original MFC problem in the sense that for all $t\\in [0,T]$ and all $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$ and $\\alpha_t\\in\\c A_t\\subseteq\\c A_T$,\n \\[\n V(t,\\xi) = V^{\\c R}(t,\\xi,\\alpha_t).\n \\]\n In particular, $V^{\\c R}$ depends neither on the initial action $\\alpha_t\\in\\c A_t$, the family $(\\lambda_s)_{s\\in [0,T]}$ nor the extension $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ of $(\\Omega,\\c F,\\P)$.\n\\end{theorem}\n\n\\begin{remark}\n By \\cref{proposition value function law invariant and independent of probabilistic setting}, $V$ depends on $\\xi$ only through its law $\\P_\\xi$ and is independent of the specific choice of $(\\Omega,\\c F,\\P)$ and $\\c G$. The equivalence result above thus implies that $V^{\\c R}$ also depends only on $\\P_\\xi$ and remains the same for all spaces $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ for which suitable $\\c G,W,B,\\mu$ exist, since every space is a trivial extension of itself.\n\\end{remark}\n\n\n\n\n\n\\section{Randomised dynamic programming}\\label{section bsde characterisation and randomised dpp}\n\nIn this section, we establish a randomised dynamic programming principle to characterise the value function of our mean-field control problem. Following the randomisation approach, we will first derive a BSDE representation for the (randomised) value function through penalisation techniques. This representation, combined with the equivalence between the original and randomised formulations established in \\cref{theorem equivalence original and randomised value function}, subsequently enables us to derive the (randomised) dynamic programming principle. \n\n\\subsection{BSDE representation}\\label{section bsde characterisation}\n\nWe start by deriving a BSDE representation for the value function $V$ using its randomised setting from \\cref{theorem equivalence original and randomised value function}. \nFor this, we first define the following spaces\n\\begin{itemize}\n \\item $\\c S^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})$, the set of all càdlàg $\\b F^{B,\\mu,\\hat\\P}$-adapted processes $Y:[t,T]\\times\\hat\\Omega\\to\\R$ processes $Y$ such that $\\E^{\\hat\\P}[\\int_t^T |Y_u|^2 du] < \\infty$,\n \\item $L^2_{W,[t,T]}(\\b F^{B,\\mu,\\hat\\P})$, the set of all $\\b F^{B,\\mu,\\hat\\P}$-predictable processes $Z:[t,T]\\times\\hat\\Omega\\to\\R^n$ that satisfy $\\E^{\\hat\\P}[\\int_t^T |Z_u|^2 du] < \\infty$,\n \\item $L^2_{\\lambda,[t,T]}(\\b F^{B,\\mu,\\hat\\P})$, the set of all $Pred(\\b F^{B,\\mu,\\hat\\P})\\otimes\\c B(A)$-measurable processes $U:[t,T]\\times\\hat\\Omega\\times\\c A_T\\to\\R$ such that $\\E^{\\hat\\P}[\\int_t^T \\int_{\\c A_u} |U_u(\\alpha)|^2 \\lambda_u(d\\alpha)du] < \\infty$,\n \\item $\\c K^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})\\subseteq\\c S^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})$ the subset of all processes $K:[t,T]\\times\\hat\\Omega\\to\\R$ that are additionally nondecreasing, $\\b F^{B,\\mu,\\hat\\P}$-predictable and satisfy $K_t = 0$.\n\\end{itemize}\nAs it is standard for the randomisation approach, we will characterise the value function as the minimal solution to a certain constrained BSDE. For this, our main tool will be \\cite[Theorem 2.1]{kharroubi_feynmankac_2015} since it replaces the problem of characterising such a minimal solution by studying the limit of solutions to a class of penalised BSDEs. Hence, we will first study in the following \\cref{lemma penalised bsde minimal solution existence} the penalised BSDE \\eqref{eq penalised bsde value function} and show that it corresponds to the optimisation over the following restricted randomised control set $\\c V^n\\subseteq\\c V$ of all $Pred(\\b F^{B,\\mu})\\otimes\\c B(\\c A_T)$-measurable processes $\\nu:[0,T]\\times\\hat\\Omega\\times\\c A_T\\to (0,n]$ with $\\inf_{[0,T]\\times\\hat\\Omega\\times\\c A_T} \\nu > 0$, based on the methodology by \\cite{kharroubi_feynmankac_2015,bayraktar_randomized_2018,bandini_backward_2018}. The proof is given in \\cref{appendix proof lemma penalised bsde minimal solution existence}. \n\n\\begin{lemma}\\label{lemma penalised bsde minimal solution existence}\n For $n\\in\\N$, $t\\in [0,T]$, $\\alpha_t\\in\\c A_t\\subseteq\\c A_T$ and $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$, there exists a unique solution\n \\[\n (Y^{n,t,\\xi,\\alpha_t},Z^{n,t,\\xi,\\alpha_t},U^{n,t,\\xi,\\alpha_t})\\in \\c S^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})\\times L^2_{W,[t,T]}(\\b F^{B,\\mu,\\hat\\P})\\times L^2_{\\lambda,[t,T]}(\\b F^{B,\\mu,\\hat\\P})\n \\]\n to following penalised BSDE,\n \\begin{align}\\label{eq penalised bsde value function}\n Y^{n,t,\\xi,\\alpha_t}_s &= \\E^{\\hat\\P}[g(X^{t,\\xi,\\alpha_t}_T,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T}) | \\c F^{B,\\mu,\\hat\\P}_T] + \\int_s^T \\E^{\\hat\\P}[f(r,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_r}_{X^{t,\\xi,\\alpha_t}_r},X^{t,\\xi,\\alpha_t}_r,I^{t,\\alpha_t}_r) | \\c F^{B,\\mu,\\hat\\P}_r] dr\\\\\n &\\qquad - \\int_s^T Z^{n,t,\\xi,\\alpha_t}_r dB_r + n \\int_s^T \\int_{\\c A_r} (U^{n,t,\\xi,\\alpha_t}_r(\\alpha))_+ \\lambda_r(d\\alpha) dr - \\int_s^T \\int_{\\c A_r} U^{n,t,\\xi,\\alpha_t}_r(\\alpha) \\mu(dr,d\\alpha).\n \\end{align}\n Further, for any $s\\in [t,T]$ and $r\\in [s,T]$, we also have the following representation,\n \\begin{align}\\label{eq Y n t xi snell envelope formula}\n Y^{n,t,\\xi,\\alpha_t}_s\n &= \\esssup_{\\nu\\in\\c V^n} \\E^{\\hat\\P^\\nu}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_s^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big],\\qquad\\hat\\P\\text{-a.s.},\n \\end{align}\n and moreover for any $r\\in [t,T]$, we have the following estimate\n \\begin{align}\\label{eq Y n t xi snell envelope upper bound at s=t}\n \\E^{\\hat\\P}[Y^{n,t,\\xi,\\alpha_t}_t] \\leq \\sup_{\\nu\\in\\c V^n} \\E^{\\hat\\P^\\nu}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_t^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\Big].\n \\end{align}\n\\end{lemma}\n\nUsing the previous result on penalised BSDEs, we can now show the following BSDE formulation for our randomised control problem. Our approach is similar to \\cite{bayraktar_randomized_2018,bandini_backward_2018} and based on \\cite[Theorem 2.1]{kharroubi_feynmankac_2015} which connects penalised BSDEs like \\eqref{eq penalised bsde value function} with constrained BSDEs of the form \\eqref{eq constrained bsde value function}. The proof is given in \\cref{appendix proof lemma bsde minimal solution existence}. \n\n\\begin{theorem}\\label{lemma bsde minimal solution existence}\n Let $(t,m)\\in [0,T]\\times\\c P_2(\\R^d)$. For any $\\alpha_t\\in\\c A_t\\subseteq\\c A_T$ and $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$ such that $\\P_\\xi = m$, there exists a unique minimal solution\n \\[\n (Y^{t,\\xi,\\alpha_t},Z^{t,\\xi,\\alpha_t},U^{t,\\xi,\\alpha_t},K^{t,\\xi,\\alpha_t})\\in \\c S^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})\\times L^2_{W,[t,T]}(\\b F^{B,\\mu,\\hat\\P})\\times L^2_{\\lambda,[t,T]}(\\b F^{B,\\mu,\\hat\\P})\\times \\c K^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P}),\n \\]\n in the sense that for any other solution $(\\tilde Y^{t,\\xi,\\alpha_t},\\tilde Z^{t,\\xi,\\alpha_t},\\tilde U^{t,\\xi,\\alpha_t},\\tilde K^{t,\\xi,\\alpha_t})$ we have $Y^{t,\\xi,\\alpha_t}\\leq \\tilde Y^{t,\\xi,\\alpha_t}$, to the following constrained BSDE\n \\begin{align}\\label{eq constrained bsde value function}\n Y^{t,\\xi,\\alpha_t}_s &= \\E^{\\hat\\P}[g(X^{t,\\xi,\\alpha_t}_T,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T}) | \\c F^{B,\\mu,\\hat\\P}_T] + \\int_s^T \\E^{\\hat\\P}[f(r,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_r}_{X^{t,\\xi,\\alpha_t}_r},X^{t,\\xi,\\alpha_t}_r,I^{t,\\alpha_t}_r) | \\c F^{B,\\mu,\\hat\\P}_r] dr\\\\\n &\\qquad - \\int_s^T Z^{t,\\xi,\\alpha_t}_r dB_r + K^{t,\\xi,\\alpha_t}_T - K^{t,\\xi,\\alpha_t}_s - \\int_s^T \\int_{\\c A_r} U^{t,\\xi,\\alpha_t}_r(\\alpha) \\mu(dr,d\\alpha),\\\\\n U^{t,\\xi,\\alpha_t}_s(\\alpha)&\\leq 0,\\qquad \\text{for }\\lambda_s(d\\alpha)\\text{-a.e. }\\alpha \\in \\c A_s,\\text{ for }ds\\text{-a.e. }s\\in [t,T].\n \\end{align}\n Further for any $s\\in [t,T]$ and $r\\in [s,T]$ we also have the following representation,\n \\begin{align}\\label{eq recursive snell envelope formula}\n Y^{t,\\xi,\\alpha_t}_s\n &= \\esssup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[Y^{t,\\xi,\\alpha_t}_r + \\int_s^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big],\\qquad \\hat\\P\\text{-a.s}.\n \\end{align}\n Finally, $Y^{t,\\xi,\\alpha_t}_t$ is $\\hat\\P$-a.s. constant, and satisfies for any $r\\in [t,T]$ also\n \\begin{align}\\label{eq recursive snell envelope formula at s=t}\n Y^{t,\\xi,\\alpha_t}_t\n &= \\sup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[Y^{t,\\xi,\\alpha_t}_r + \\int_t^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\Big],\\qquad\\hat\\P\\text{-a.s.},\n \\end{align}\n and thus $Y^{t,\\xi,\\alpha_t}_t = V(t,m)$, $\\hat\\P$-a.s.\n\\end{theorem}\n\n\n\n\n\\subsection{Randomised dynamic programming principle}\\label{section randomised dpp}\n\nRecalling that in \\cref{section randomised problem} we have established an equivalence between the value function of the original and the randomised problem, we note that $V^{\\c R} = V$ only depends on time and initial law of $X$. Thus the question about a dynamic programming principle is well-posed and in this section we want to use the randomised formulation to establish such a DPP for $V^{\\c R}$ and thus also for $V$.\n\nWe start by studying the flow property of the randomised state dynamics \\eqref{eq sde randomised problem}. For this, we will need to define solutions $X^{t,\\zeta,\\gamma_t}$ to \\eqref{eq sde randomised problem} for more general initial conditions $\\zeta\\in L^2(\\hat\\Omega,\\c G\\lor\\c F^{W,B,\\mu}_t,\\hat\\P;\\R^d)$ and $\\gamma_t \\in L^0(\\hat\\Omega,\\c G\\lor\\c F^{W,B,\\mu}_t,\\hat\\P;A)$. We note that as in \\cref{section randomised problem} the existence and uniqueness of $X^{t,\\zeta,\\gamma_t}$ is standard since both the existence and uniqueness of $I^{t,\\gamma_t}$ given $\\hat I^{t,\\gamma_t}$ by \\cref{corollary I given bar I is well defined} also extend to such more general initial conditions. If moreover $\\sup_{\\nu\\in\\c V} \\E^{\\hat\\P}[|\\zeta|^2] < \\infty$ holds, then similarly to the estimate in \\eqref{eq randomised state dynamics basic estimate}, we obtain\n\\begin{align}\n\\sup_{\\gamma_t}\\sup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[\\sup_{s\\in [t,T]} |X^{t,\\zeta,\\gamma_t}_s|^2 \\Big] < \\infty.\n\\label{eq extended randomised state dynamics basic estimate}\n\\end{align}\nThis now allows us to write down the following flow property for \\eqref{eq sde randomised problem}.\n\n\\begin{lemma}\\label{lemma flow property}\n Let $t\\in [0,T]$ and $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$. Then for all $s\\in [t,T]$, the following flow property holds\n \\[\n \\big(X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_u,I^{s,I^{t,\\alpha_t}_s}_u,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_u}_{X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_u}\\big)_{u\\in [s,T]} = \\big(X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_u}_{X^{t,\\xi,\\alpha_t}_u}\\big)_{u\\in [s,T]},\\qquad\\hat\\P\\text{-a.s}.\n \\]\n\\end{lemma}\n\n\\begin{proof}\nWe note that by \\cref{corollary I given bar I is well defined}, $I^{t,\\alpha_t}$ and $I^{s,I^{t,\\alpha_t}}$ are indistinguishable on $[s,T]$. Thus $X^{t,\\xi,\\alpha_t}$ and $X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}$ both solve \\eqref{eq sde randomised problem} on $[s,T]$, and hence the flow property follows from the strong uniqueness of solutions to \\eqref{eq sde randomised problem}.\n\\end{proof}\n\nNext let us take a closer look at such randomised settings with a randomised strategy $\\nu\\in\\c V$ and general initial conditions $\\zeta\\in L^2(\\hat\\Omega,\\c G\\lor\\c F^{W,B,\\mu}_t,\\hat\\P;\\R^d)$, $\\gamma_t\\in L^0(\\hat\\Omega,\\c G\\lor\\c F^{W,B,\\mu}_t,\\hat\\P;A)$. We want to decompose both $\\zeta,\\gamma_t$ and $\\nu$ along the $\\sigma$-algebra $\\c F^{B,\\mu,\\hat\\P}_t$, to be able to relate this to our original setting with $\\xi\\in L^2(\\hat\\Omega,\\c G\\lor\\c F^W_t,\\hat\\P;\\R^d)$ and $\\alpha_t\\in\\c A_t$, which we studied in the previous sections. For this, we recall that we introduced $B^t\\coloneqq (B_{\\cdot\\lor t} - B_t)$ and $\\mu^t\\coloneqq \\mu(\\cdot\\cap ((t,T]\\times\\c A_T))$ in \\cref{section randomised problem}.\nWe then note that by \\cref{proposition decompose filtrations of predictable processes}, we can decompose both $\\zeta,\\gamma_t$ and $\\nu$ into unique $\\Xi\\in L^2(\\hat\\Omega\\times\\hat\\Omega,\\c F^{B,\\mu}_t\\otimes(\\c G\\lor\\c F^W_t),\\hat\\P\\otimes\\hat\\P;\\R^d)$, $C_t\\in L^0(\\hat\\Omega\\times\\hat\\Omega,\\c F^{B,\\mu}_t\\otimes(\\c G\\lor\\c F^W_t),\\hat\\P\\otimes\\hat\\P;A)$ and an $\\c F^{B,\\mu}_t\\otimes Pred(\\b F^{B^t,\\mu^t})\\otimes\\c B(\\c A_T)$-measurable process $\\Upsilon$ with $0<\\inf_{[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\times\\c A_T} \\Upsilon\\leq \\sup_{[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\times\\c A_T} \\Upsilon < \\infty$ such that $(\\zeta,\\gamma_t,\\nu)(\\hat\\omega) = (\\Xi,C_t,\\Upsilon)(\\hat\\omega,\\hat\\omega)$, for $\\hat\\P$-a.a.\\@ $\\hat\\omega\\in\\hat\\Omega$. This decomposition is the motivation for the following lemma, which studies how this decomposition reflects in the reward functional.\n\n\\begin{lemma}\\label{lemma generalised randomised payoff decomposition} \n Let $t\\in [0,T]$, $\\zeta\\in L^2(\\hat\\Omega,\\c G\\lor\\c F^{W,B,\\mu}_t,\\hat\\P;\\R^d)$, $\\gamma_t\\in L^0(\\hat\\Omega,\\c G\\lor\\c F^{W,B,\\mu}_t,\\hat\\P;A)$ and $\\nu\\in\\c V$,\n and let $\\Xi\\in L^2(\\hat\\Omega\\times\\hat\\Omega,\\c F^{B,\\mu}_t\\otimes(\\c G\\lor\\c F^W_t),\\hat\\P\\otimes\\hat\\P;\\R^d)$, $C_t\\in L^0(\\hat\\Omega\\times\\hat\\Omega,\\c F^{B,\\mu}_t\\otimes(\\c G\\lor\\c F^W_t),\\hat\\P\\otimes\\hat\\P;A)$ and $\\Upsilon$ be an $\\c F^{B,\\mu}_t\\otimes Pred(\\b F^{B^t,\\mu^t})\\otimes\\c B(\\c A_T)$-measurable process such that $0<\\inf_{[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\times\\c A_T} \\Upsilon\\leq \\sup_{[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\times\\c A_T} \\Upsilon < \\infty$ (in particular, $\\Upsilon(\\hat\\omega,\\cdot)\\in\\c V_t$ for every $\\hat\\omega\\in\\hat\\Omega$), such that $(\\zeta,\\gamma_t,\\nu)(\\hat\\omega) = (\\Xi,C_t,\\Upsilon)(\\hat\\omega,\\hat\\omega)$, for $\\hat\\P$-a.a.\\@ $\\hat\\omega\\in\\hat\\Omega$. Assuming that $\\sup_{\\upsilon\\in\\c V} \\E^{\\hat\\P^\\upsilon}[|\\zeta|^2]<\\infty$, then it holds that\n \\begin{align}\n &\\E^{\\hat\\P^\\nu}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_T}_{X^{t,\\zeta,\\gamma_t}_T},X^{t,\\zeta,\\gamma_t}_T) + \\int_t^T f(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\zeta,\\gamma_t}_s},X^{t,\\zeta,\\gamma_t}_s,I^{t,\\gamma_t}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_t\\Big](\\hat\\omega)\\\\\n &= \\E^{\\hat\\P^\\upsilon}\\Big[g(\\hat\\P^{\\upsilon,\\c F^{B,\\mu,\\hat\\P^\\upsilon}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\upsilon,\\c F^{B,\\mu,\\hat\\P^\\upsilon}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds \\Big]\\bigg|_{\\xi = \\Xi(\\hat\\omega,\\cdot),\\alpha_t = C_t(\\hat\\omega,\\cdot),\\upsilon = \\Upsilon(\\hat\\omega,\\cdot)}\\\\\n &= J^{\\c R}(t,\\xi,\\alpha_t,\\upsilon)\\Big|_{\\xi = \\Xi(\\hat\\omega,\\cdot),\\alpha_t = C_t(\\hat\\omega,\\cdot), \\upsilon = \\Upsilon(\\hat\\omega,\\cdot)},\\qquad\\text{ for }\\hat\\P\\text{-a.a.\\@ }\\hat\\omega\\in\\hat\\Omega.\n \\label{eq generalised randomised payoff decomposition}\n \\end{align}\n\\end{lemma}\n\\begin{proof}\nTo prove the formula \\eqref{eq generalised randomised payoff decomposition}, we will first consider the case where $\\hat\\omega\\to (\\Xi(\\hat\\omega,\\cdot),C_t(\\hat\\omega,\\cdot),\\Upsilon(\\hat\\omega,\\cdot))$ takes finitely many values, and then use continuity of $J$ in $\\xi,\\alpha_t,\\nu$ and of $(X,\\P^{\\c F^{B,\\mu,\\P}}_X)$ in $\\Xi,C_t,\\Upsilon$ to obtain the result for general $\\Xi,C_t,\\Upsilon$. \\begin{enumerate}[wide,label=(\\roman*)]\n\\item\\label{lemma 7.2 step i} Let us assume that $\\hat\\omega\\mapsto (\\Xi(\\hat\\omega,\\cdot),C_t(\\hat\\omega,\\cdot),\\Upsilon(\\hat\\omega,\\cdot))$ only takes finitely many values (in the spaces $L^2(\\hat\\Omega,\\c G\\lor\\c F^W_t,\\hat\\P;\\R^d)$, $\\c A_t$ and $\\c V_t$), and thus are of the form\n\\begin{align}\n&\\Xi(\\hat\\omega^1,\\hat\\omega^2) = \\sum_{k=1}^K \\xi^k(\\pi(\\hat\\omega^2))\\1_{Z^k}(\\hat\\omega^1),\\quad C_t(\\hat\\omega^1,\\hat\\omega^2) = \\sum_{l=1}^L \\alpha_t^l(\\hat\\omega^2)\\1_{H^l}(\\hat\\omega^1),\\\\\n&\\Upsilon(\\hat\\omega^1,\\hat\\omega^2) = \\sum_{m=1}^M \\upsilon^m(\\hat\\omega^2)\\1_{E^m}(\\hat\\omega^1) ,\\quad\\text{for }(\\hat\\omega^1,\\hat\\omega^2)\\in\\hat\\Omega\\times\\hat\\Omega,\n\\end{align}\nwhere $(Z^k)_k,(H^l)_l,(E^m)_m$ are each disjoint partitions of $\\hat\\Omega$, and $\\xi^k\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$, $\\alpha^l\\in\\c A_t$ and $\\upsilon^m\\in\\c V_t$. Let us now consider the processes $\\hat I^{t,\\alpha^l_t}$ given by \\eqref{eq randomised poisson point process}, and the corresponding $I^{t,\\alpha^l_t}$ which $\\b F^{\\hat I^{t,\\alpha^l_t}}$-identifies to $\\hat I^{t,\\alpha^l_t}$ from \\cref{corollary I given bar I is well defined}. We see that then $\\sum_{l=1}^L I^{t,\\alpha^l_t}\\1_{H^l}$ by construction $\\b F^{\\hat I^{t,\\gamma_t}}$-identifies to \n$\\hat I^{t,\\gamma_t}$ and thus again by \\cref{corollary I given bar I is well defined} is indistinguishable from $I^{t,\\gamma_t}$ under $\\hat\\P\\sim\\hat\\P^\\nu$.\nSimilarly, denoting the corresponding unique solutions to \\eqref{eq sde randomised problem} with starting value $\\xi^k$ by $X^{s,\\xi^k,\\alpha^l_t}$, we note that by strong uniqueness of solutions to \\eqref{eq sde randomised problem} the processes $X^{s,\\zeta,\\gamma_t}$ and $\\sum_{k=1}^K \\sum_{l=1}^L X^{t,\\xi^k,\\alpha^l_t} \\1_{Z^k\\cap H^l}$ are indistinguishable under $\\hat\\P\\sim\\hat\\P^\\nu$. Finally, we also define the tilted probability measures $\\hat\\P^{\\upsilon^m}\\sim\\hat\\P$ belonging to $\\upsilon^m$ by $\\frac{d\\hat\\P^{\\upsilon^m}}{d\\hat\\P}\\big|_{\\c F^{B,\\mu,\\hat\\P}_u} \\coloneqq L^{\\upsilon^m}_u$ for $u\\in [0,T]$, see \\eqref{eq randomised control girsanov formula}, and note that by construction $L^\\nu = \\sum_{m=1}^M L^{\\upsilon^m} \\1_{E^m}$. All together, this allows us to decompose the LHS of \\eqref{eq generalised randomised payoff decomposition} as follows\n\\begin{align}\n &\\E^{\\hat\\P^\\nu}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_T}_{X^{t,\\zeta,\\gamma_t}_T},X^{t,\\zeta,\\gamma_t}_T) + \\int_t^T f(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\zeta,\\gamma_t}_s},X^{t,\\zeta,\\gamma_t}_s,I^{t,\\gamma_t}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_t\\Big]\\\\\n &= \\sum_{k=1}^K \\sum_{l=1}^L \\sum_{m=1}^M \\1_{Z^k\\cap H^l\\cap E^m} \\E^{\\hat\\P^{\\upsilon^m}}\\Big[g(\\hat\\P^{\\upsilon^m,\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_T}_{X^{t,\\xi^k,\\alpha_t^l}_T},X^{t,\\xi^k,\\alpha_t^l}_T)\\\\\n &\\hspace{145pt} + \\int_t^T f(s,\\hat\\P^{\\upsilon^m,\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_s}_{X^{t,\\xi^k,\\alpha_t^l}_s},X^{t,\\xi^k,\\alpha_t^l}_s,I^{t,\\alpha_t^l}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_t\\Big].\n \\label{eq lemma 7.2 decompose randomised payoff for simple functions}\n\\end{align}\n\nFor each summand, we note that since $\\upsilon^m\\in\\c V_t$, by construction $\\xi^k,\\alpha^l_t,W,B^t,\\mu^t$ are $\\hat\\P^{\\upsilon^m}$ independent of $\\c F_t^{B,\\mu,\\hat\\P^{\\upsilon^m}}$.\nThus from the strong uniqueness of solutions to \\eqref{eq sde randomised problem} we can deduce that $X^{t,\\xi^k,\\alpha_t^l}$ and $(\\hat\\P^{\\upsilon^m,\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_s}_{X^{t,\\xi^k,\\alpha_t^l}_s})_{s\\in [t,T]}$ are $\\hat\\P^{\\upsilon^m}$-independent of $\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_t$. For this, one can e.g.\\@ note that the standard construction of a solution to \\eqref{eq sde randomised problem} using a Picard iteration leads to a process $\\hat\\P^{\\upsilon^m}$-independent of $\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_t$ in every step and thus also in the limit. Therefore\n\\begin{align}\n &\\E^{\\hat\\P^{\\upsilon^m}}\\Big[g(\\hat\\P^{\\upsilon^m,\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_T}_{X^{t,\\xi^k,\\alpha_t^l}_T},X^{t,\\xi^k,\\alpha_t^l}_T) + \\int_t^T f(s,\\hat\\P^{\\upsilon^m,\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_s}_{X^{t,\\xi^k,\\alpha_t^l}_s},X^{t,\\xi^k,\\alpha_t^l}_s,I^{t,\\alpha_t^l}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_t\\Big]\\\\\n &= \\E^{\\hat\\P^{\\upsilon^m}}\\Big[g(\\hat\\P^{\\upsilon^m,\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_T}_{X^{t,\\xi^k,\\alpha_t^l}_T},X^{t,\\xi^k,\\alpha_t^l}_T) + \\int_t^T f(s,\\hat\\P^{\\upsilon^m,\\c F^{B,\\mu,\\hat\\P^{\\upsilon^m}}_s}_{X^{t,\\xi^k,\\alpha_t^l}_s},X^{t,\\xi^k,\\alpha_t^l}_s,I^{t,\\alpha_t^l}_s) ds\\Big]\\\\\n &= J^{\\c R}(s,\\xi^k,\\alpha_t^l,\\upsilon^m),\\quad \\hat\\P\\text{-a.s.},\n\\end{align}\nwhich together with \\eqref{eq lemma 7.2 decompose randomised payoff for simple functions}, shows that the claim \\eqref{eq generalised randomised payoff decomposition} holds for $\\hat\\omega\\mapsto (\\Xi(\\hat\\omega,\\cdot),C_t(\\hat\\omega,\\cdot),\\Upsilon(\\hat\\omega,\\cdot))$ only taking finitely many values.\n\\item Let us now consider general $(\\Xi,C_t,\\Upsilon)$. We first note that due to $\\rho < 1$, and since $\\Upsilon$ is essentially bounded, we can find a sequence $(\\Xi^n,C_t^n,\\Upsilon^n)_n$ such that $(\\Xi^n,C_t^n,\\Upsilon^n) \\to (\\Xi,C_t,\\Upsilon)$ in $L^2\\times L^\\infty\\times L^\\infty$ and a.s., such that $\\sup_{\\upsilon\\in\\c V}\\E^{\\hat\\P^\\upsilon\\otimes\\hat\\P^\\upsilon}[|\\Xi^n|^2] < \\infty$ and each $\\hat\\omega\\mapsto (\\Xi^n(\\hat\\omega,\\cdot),C_t^n(\\hat\\omega,\\cdot),\\Upsilon^n(\\hat\\omega,\\cdot))$ only takes finitely many values. \nCorrespondingly we define $\\zeta^n,\\gamma_t^n,\\nu^n$ by $(\\zeta^n,\\gamma_t^n,\\nu^n)(\\hat\\omega)\\coloneqq (\\Xi^n,C_t^n,\\Upsilon^n)(\\hat\\omega,\\hat\\omega)$, which by construction then satisfy $(\\zeta^n,\\gamma_t^n,\\nu^n)\\to (\\zeta,\\gamma_t,\\nu)$ in $L^2\\times L^\\infty\\times L^\\infty$ and a.s., and $\\sup_{\\upsilon\\in\\c V}\\E^{\\hat\\P^\\upsilon}[|\\zeta^n|^2] < \\infty$. \nThus by \\cref{lemma 7.2 step i}, we already know that \\eqref{eq generalised randomised payoff decomposition} holds for each $(\\Xi^n,C_t^n,\\Upsilon^n)$ and $(\\zeta^n,\\gamma_t^n,\\nu^n)$, and our task now will be showing that \\eqref{eq generalised randomised payoff decomposition} still holds in the limit, for which it is sufficient to show the convergence of the left- and right-hand sides as $n\\to\\infty$.\n\nTo this end, for the right-hand side of \\eqref{eq generalised randomised payoff decomposition}, we note that $J^{\\c R}:[0,T]\\times L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)\\times\\c A_t\\times\\c V\\to \\R$ is continuous in $\\xi,\\alpha_t,\\nu$ by \\cref{lemma J R continuous}, and thus\n\\[\n\\lim_{n\\to\\infty} J^{\\c R}(t,\\Xi^n(\\hat\\omega,\\cdot),C_t^n(\\hat\\omega,\\cdot),\\Upsilon^n(\\hat\\omega,\\cdot))\n= J^{\\c R}(t,\\Xi(\\hat\\omega,\\cdot),C_t(\\hat\\omega,\\cdot),\\Upsilon(\\hat\\omega,\\cdot)),\n\\qquad\\text{for $\\hat\\P$-a.a.\\@ }\\hat\\omega\\in\\hat\\Omega.\n\\]\nFor the left-hand side of \\eqref{eq generalised randomised payoff decomposition}, we introduce $\\bar\\nu\\coloneqq \\nu\\lor\\sup_n \\nu^n \\in\\c V$. Then using the conditional Jensen's inequality, we obtain that\n\\begin{align}\n&\\E^{\\hat\\P^{\\bar\\nu}}\\bigg[\\bigg|\\E^{\\hat\\P^{\\nu^n}}\\Big[g(\\hat\\P^{\\nu^n,\\c F^{B,\\mu,\\hat\\P^{\\nu^n}}_T}_{X^{t,\\zeta^n,\\gamma_t^n}_T},X^{t,\\zeta^n,\\gamma_t^n}_T) + \\int_t^T f(s,\\hat\\P^{\\nu^n,\\c F^{B,\\mu,\\hat\\P^{\\nu^n}}_s}_{X^{t,\\zeta^n,\\gamma_t^n}_s},X^{t,\\zeta^n,\\gamma_t^n}_s,I^{t,\\gamma_t^n}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^{\\nu^n}}_t\\Big]\\\\\n&\\qquad - \\E^{\\hat\\P^{\\nu}}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^{\\nu}}_T}_{X^{t,\\zeta,\\gamma_t}_T},X^{t,\\zeta,\\gamma_t}_T) + \\int_t^T f(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^{\\nu}}_s}_{X^{t,\\zeta,\\gamma_t}_s},X^{t,\\zeta,\\gamma_t}_s,I^{t,\\gamma_t}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^{\\nu}}_t\\Big]\\bigg| \\bigg]\\\\\n&\\leq \\E^{\\hat\\P^{\\bar\\nu}}\\bigg[\\bigg|\\frac{L^{\\nu^n}_T}{L^{\\bar\\nu}_T} \\Big(g(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_T}_{X^{t,\\zeta^n,\\gamma_t^n}_T},X^{t,\\zeta^n,\\gamma_t^n}_T) + \\int_t^T f(s,\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\zeta^n,\\gamma_t^n}_s},X^{t,\\zeta^n,\\gamma_t^n}_s,I^{t,\\gamma_t^n}_s) ds\\Big)\\\\\n&\\qquad - \\frac{L^\\nu_T}{L^{\\bar\\nu}_T} \\Big(g(\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_T}_{X^{t,\\zeta,\\gamma_t}_T},X^{t,\\zeta,\\gamma_t}_T) + \\int_t^T f(s,\\hat\\P^{\\bar\\nu,\\c F^{B,\\mu,\\hat\\P^{\\bar\\nu}}_s}_{X^{t,\\zeta,\\gamma_t}_s},X^{t,\\zeta,\\gamma_t}_s,I^{t,\\gamma_t}_s) ds\\Big)\\bigg|\\bigg],\n\\end{align}\nand thus repeating the arguments in the proof of \\cref{lemma J R continuous} using the estimate \\eqref{eq extended randomised state dynamics basic estimate}, we conclude that\n\\begin{align}\n&\\E^{\\hat\\P^{\\nu^n}}\\Big[g(\\hat\\P^{\\nu^n,\\c F^{B,\\mu,\\hat\\P^{\\nu^n}}_T}_{X^{t,\\zeta^n,\\gamma_t^n}_T},X^{t,\\zeta^n,\\gamma_t^n}_T) + \\int_t^T f(s,\\hat\\P^{\\nu^n,\\c F^{B,\\mu,\\hat\\P^{\\nu^n}}_s}_{X^{t,\\zeta^n,\\gamma_t^n}_s},X^{t,\\zeta^n,\\gamma_t^n}_s,I^{t,\\gamma_t^n}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^{\\nu^n}}_t\\Big]\\\\\n&\\to \\E^{\\hat\\P^{\\nu}}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^{\\nu}}_T}_{X^{t,\\zeta,\\gamma_t}_T},X^{t,\\zeta,\\gamma_t}_T) + \\int_t^T f(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^{\\nu}}_s}_{X^{t,\\zeta,\\gamma_t}_s},X^{t,\\zeta,\\gamma_t}_s,I^{t,\\gamma_t}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^{\\nu}}_t\\Big]\\text{ in }L^1(\\hat\\P^{\\bar\\nu}).\n\\end{align}\nThus from the uniqueness of limits, we conclude that \\eqref{eq generalised randomised payoff decomposition} also holds for general $(\\Xi,C_t,\\Upsilon)$, which completes the proof.\n\\end{enumerate}\n\\end{proof}\n\nThis allows us now to obtain the following more general connection between the mean-field control problem and the BSDE in \\cref{theorem value function bsde characterisation}.\n \n\\begin{theorem}\\label{theorem value function bsde characterisation}\n Let $t\\in [0,T]$ and $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$. Further let $\\alpha_t\\in\\c A_t\\subseteq \\c A_T$ and let $(Y^{t,\\xi,\\alpha_t},Z^{t,\\xi,\\alpha_t},U^{t,\\xi,\\alpha_t},K^{t,\\xi,\\alpha_t})$ be the unique minimal solution to \\eqref{eq constrained bsde value function} from \\cref{lemma bsde minimal solution existence}. Then for any $s\\in [t,T]$, it holds that\n \\[\n Y^{t,\\xi,\\alpha_t}_s = V(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s}),\\qquad \\hat\\P\\text{-a.s.}\n \\]\n\\end{theorem}\n\n\\begin{proof}\nWe start by decomposing $(X_s^{t,\\xi,\\alpha_t},I_s^{t,\\alpha_t})$. Since $X_s^{t,\\xi,\\alpha_t}$ and $I_s^{t,\\alpha_t}$ are both $\\c G\\lor\\c F^{W,B,\\mu,\\hat\\P}_s$-measurable, by e.g.\\@ \\cref{proposition decompose filtrations of predictable processes} there exist $\\c F^{B,\\mu}_s\\otimes (\\c G\\lor\\c F^W_s)$-measurable $\\zeta:\\hat\\Omega\\times\\hat\\Omega\\to\\R^d$ and $C_s:\\hat\\Omega\\times\\hat\\Omega\\to A$ such that\n\\[\n(X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s)(\\hat\\omega) = (\\zeta,C_s)(\\hat\\omega,\\hat\\omega), \\quad \\text{for $\\hat\\P$-a.a.\\@ }\\hat\\omega\\in\\hat\\Omega.\n\\]\nNote that by construction $\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s}(\\hat\\omega) = \\hat\\P_{\\zeta(\\hat\\omega,\\cdot)}$, for $\\hat\\P$-a.a.\\@ $\\hat\\omega\\in\\hat\\Omega$, since $\\c F^{B,\\mu,\\hat\\P}_t$ and $\\c G\\lor\\c F^{W,\\hat\\P}_t$ are independent under $\\hat\\P$.\nHence, from \\cref{proposition equivalence v v_t randomised value function,theorem equivalence original and randomised value function}, we now see that\n\\[\nV(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s}(\\hat\\omega))\n= V^{\\c R}(s,\\zeta(\\hat\\omega,\\cdot),C_s(\\hat\\omega,\\cdot))\n= \\sup_{\\nu\\in\\c V_s} J^{\\c R} (s,\\zeta(\\hat\\omega,\\cdot),C_s(\\hat\\omega,\\cdot),\\nu),\n\\ \\text{for $\\hat\\P$-a.a.\\@ }\\hat\\omega\\in\\hat\\Omega,\n\\]\n\nAt the same time, we note that \\cref{lemma bsde minimal solution existence} (for $r=T$) together with the flow property in \\cref{lemma flow property} implies that $\\hat\\P$-a.s.\n\\begin{align}\n Y^{t,\\xi,\\alpha_t}_s &= \\esssup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_s^T f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big]\\\\\n &= \\esssup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_T}_{X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_T},X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_T)\\\\\n &\\hspace{65pt}+ \\int_s^T f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_u},X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_u,I^{t,\\alpha_t}_u) du\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big].\n\\end{align}\nIn particular, the claim now follows if we can show that $\\hat\\P$-a.s.\n\\begin{align}\n&\\sup_{\\nu\\in\\c V_s} J^{\\c R} (s,\\zeta(\\hat\\omega,\\cdot),C_s(\\hat\\omega,\\cdot),\\nu)\\\\\n&= \\esssup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_T}_{X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_T},X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_T)\\\\\n&\\hspace{65pt}+ \\int_s^T f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_u},X^{s,X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s}_u,I^{t,\\alpha_t}_u) du\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big].\n\\end{align}\n\nTo this end, we note that by the estimate \\eqref{eq randomised state dynamics basic estimate}, we have $\\sup_{\\nu\\in\\c V}\\E^{\\hat\\P^\\nu}[|\\zeta|^2] = \\sup_{\\nu\\in\\c V}\\E^{\\hat\\P^\\nu}[|X^{t,\\xi,\\alpha_t}_s|^2] < \\infty$.\nHence \"$\\leq$\" follows from \\cref{lemma generalised randomised payoff decomposition} since $\\c V_s\\subseteq\\c V$. On the other hand, for \"$\\geq$\" it suffices to notice that we can decompose every $\\nu\\in\\c V$ by \\cref{proposition decompose filtrations of predictable processes} into an $\\c F^{B,\\mu}_s\\otimes Pred(\\b F^{B^s,\\mu^s})\\otimes\\c B(\\c A_T)$-measurable process $\\Upsilon$ such that $\\nu_r(\\hat\\omega,\\alpha) = \\Upsilon_r(\\hat\\omega,\\hat\\omega,\\alpha)$ for all $\\hat\\omega\\in\\hat\\Omega,\\alpha\\in\\c A_T,r\\in [0,T]$, as this direction now follows from \\cref{lemma generalised randomised payoff decomposition} since $\\Upsilon(\\hat\\omega,\\cdot)\\in\\c V_s$ for $\\hat\\P$-a.a.\\@ $\\hat\\omega\\in\\hat\\Omega$.\n\\end{proof}\n\nFinally, we can now conclude the following randomised dynamic programming principle (DPP).\n\n\\begin{theorem}\\label{theorem randomised dpp}\n Let $(t,m) \\in [0,T]\\times \\c P_2(\\R^d)$. Then for any choice of $\\alpha_t\\in\\c A_t\\subseteq\\c A_T$ and $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$ with $\\hat\\P_\\xi = m$, it holds for $s\\in [t,T]$ that\n \\begin{align}\n V(t,m) = \\sup_{\\nu\\in \\c V} \\E^{\\hat\\P^\\nu}\\Big[ V(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s}) + \\int_t^s f(r,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_r}_{X^{t,\\xi,\\alpha_t}_r},X^{t,\\xi,\\alpha_t}_r,I^{t,\\alpha_t}_r) dr \\Big].\n \\label{eq randomised dpp}\n \\end{align}\n\\end{theorem}\n\\begin{proof}\nThis follows directly from \\cref{lemma bsde minimal solution existence,theorem value function bsde characterisation}.\n\\end{proof}\n\n\\begin{remark}\nAssuming that the value function $V$ is sufficiently regular, this approach leads to the standard (non-randomised) DPP for mean-field control problems, which has been well-studied in the literature, with \\cite{djete_mckeanvlasov_2022,djete_mckeanvlasov_2022-1} providing one of the most comprehensive studies. Specifically, we can interpret the right-hand side in \\eqref{eq randomised dpp} as a new control problem with running reward $f$ and terminal reward $V$. Thus, as long as $V$ satisfies the assumptions on $g$ in \\cref{assumptions reward functional}, we can apply the same theory developed in this paper again to this new problem. Consequently, by \\cref{theorem equivalence original and randomised value function}, we then obtain\n\\begin{align}\nV(t,m)\n&= \\sup_{\\nu\\in \\c V} \\E^{\\hat\\P^\\nu}\\Big[ V(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s}) + \\int_t^s f(r,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_r}_{X^{t,\\xi,\\alpha_t}_r},X^{t,\\xi,\\alpha_t}_r,I^{t,\\alpha_t}_r) dr \\Big]\\\\\n&= \\sup_{\\alpha\\in\\c A} \\E^\\P\\Big[ V(s,\\P^{\\c F^{B,\\P}_s}_{X^{t,\\xi,\\alpha}_s}) + \\int_t^s f(r,\\P^{\\c F^{B,\\P}_r}_{X^{t,\\xi,\\alpha}_r},X^{t,\\xi,\\alpha}_r,\\alpha_r) dr \\Big].\n\\end{align}\n\\end{remark}\n\n\n\\appendix\n\n\n\\section{Proof of \\cref{theorem equivalence original and randomised value function}}\n\\label{section proof of equivalence randomised non randomised formulation}\n\nOur goal in this section is proving that $V = V^{\\c R}$. For this let us fix $t\\in [0,T]$ and $\\alpha_t\\in\\c A_t$. Furthermore, let us fix also the family $(\\lambda_s)_{s\\in [0,T]}$ satisfying \\cref{assumptions lambda family}. As already outlined in \\cref{theorem equivalence original and randomised value function}, we will obtain as a side result that $V^{\\c R}$ is independent of specific choice of $\\alpha_t\\in\\c A_t$, which we will come into play later again in the upcoming dynamic programming result in \\cref{section randomised dpp}. In what follows, we will prove the two direction $V\\leq V^{\\c R}$ and $V^{\\c R}\\leq V$ separately. The general structure of the proofs is similar to \\cite{bandini_backward_2018,bayraktar_randomized_2018}, albeit the detailed arguments are naturally different in our mean-field setting due to the presence of the mean-field interaction and the common noise.\n\nA crucial tool will be the following \\cref{proposition randomised setting is independent of the extension}, which shows that the randomised value function $V^{\\c R}$ is independent of the specific choice of the extension of the probability space $(\\Omega,\\c F,\\P)$. This will allow us to simplify the upcoming proofs by considering suitable extensions of $(\\Omega,\\c F,\\P)$ as needed for each direction. \n\n\\begin{lemma}\\label{proposition randomised setting is independent of the extension}\n The randomised value function $V^{\\c R}$ does not depend on the specific extension $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ of $(\\Omega,\\c F,\\P)$.\n\\end{lemma}\n\\begin{proof}\nLet $t\\in [0,T]$, $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$ and $\\alpha_t\\in\\c A_t$. Further let $(\\lambda_s)_{s\\in [0,T]}$ satisfy \\cref{assumptions lambda family}, and let $(\\hat\\Omega^i,\\hat{\\c F}^i,\\hat\\P^i)$, $i\\in\\{1,2\\}$ be both extensions of $(\\Omega,\\c F,\\P)$, each carrying a Poisson point measure $\\mu^i$ with intensity $\\lambda_s(d\\alpha)ds$ independent of $\\c G^i,W^i,B^i$.\nWe denote the respective canonical projections by $\\pi^i:(\\hat\\Omega^i,\\hat{\\c F}^i)\\to(\\Omega,\\c F)$. By the extension property, we have then that $\\c L^{\\hat\\P^1}(\\pi^1) = \\c L^{\\hat\\P^2}(\\pi^2)$. Therefore, by introducing $\\pi^i|_{\\c G^i}:(\\hat\\Omega^i,\\hat{\\c G}^i)\\to (\\Omega,\\c G)$, $i\\in \\{1,2\\}$, we observe that also $\\c L^{\\hat\\P^1}(\\pi^1|_{\\c G^1},W^1,B^1) = \\c L^{\\hat\\P^2}(\\pi^2|_{\\c G^2},W^2,B^2)$.\nFinally, using the independence of $\\c G^i,W^i,B^i$ from $\\mu^i$ and thus also from the corresponding $\\hat I^{t,\\alpha_t,i}$ defined by \\eqref{eq randomised poisson point process}, together with $\\c L^{\\hat\\P^1}(\\mu^1) = \\c L^{\\hat\\P^2}(\\mu^2)$ this shows that\n\\[\n\\c L^{\\hat\\P^1}(\\pi^1|_{\\c G^1},W^1,B^1,\\hat I^{t,\\alpha_t,1},\\mu^1) = \\c L^{\\hat\\P^2}(\\pi^1|_{\\c G^2},W^2,B^2,\\hat I^{t,\\alpha_t,2},\\mu^2).\n\\]\nThen \\cref{corollary I given bar I is well defined} shows that the corresponding càdlàg, $\\b F^{W^i}\\lor\\c G^i\\lor\\c F^{\\mu^{t,i}}_T$-predictable and $\\b F^{W^i,\\mu^{t,i}}\\lor\\c G^i$-progressive $I^{t,\\alpha_t,i}$ which $\\b F^{\\hat I^{t,\\alpha_t,i}}$-identify to $\\hat I^{t,\\alpha_t,i}$, $i\\in\\{1,2\\}$ are unique (up to indistinguishability) and moreover satisfy\n\\[\n\\c L^{\\hat\\P^1}(\\pi^1|_{\\c G^1},W^1,B^1,\\hat I^{t,\\alpha_t,1},I^{t,\\alpha_t,1},\\mu^1) = \\c L^{\\hat\\P^2}(\\pi^2|_{\\c G^2},W^2,B^2,\\hat I^{t,\\alpha_t,2},I^{t,\\alpha_t,2},\\mu^2).\n\\]\nUsing the weak uniqueness of solutions to the SDE \\eqref{eq sde randomised problem}, we hence obtain for the corresponding unique solutions $X^{t,\\xi^i,\\alpha_t,i}$ to \\eqref{eq sde randomised problem}, $i\\in\\{1,2\\}$, that\n\\begin{align}\n\\c L^{\\hat\\P^1}(X^{t,\\xi^1,\\alpha_t,1},\\mu^1,I^{t,\\alpha_t,1},(\\hat\\P^{1,\\c F^{B^1,\\mu^1,\\hat\\P^1}_s}_{X^{t,\\xi^1,\\alpha_t,1}_s})_s) = \\c L^{\\hat\\P^2}(X^{t,\\xi^2,\\alpha_t,2},\\mu^2,I^{t,\\alpha_t,2},(\\hat\\P^{2,\\c F^{B^2,\\mu^2,\\hat\\P^2}_s}_{X^{t,\\xi^2,\\alpha_t,2}_s})_s).\n\\label{eq proposition 4.9 law equivalence of state processes}\n\\end{align}\n\nNow let $\\nu^1\\in\\c V^1$ be a randomised control and $\\hat\\P^{1,\\nu^1}\\sim\\hat\\P^1$ be the corresponding tilted probability measure. Then using \\cite[Lemma 4.2]{bandini_backward_2018} and \\cref{proposition canonical representation of predictable processes} we can represent it in the following form, for all $(s,\\hat\\omega^1,a)\\in [0,T]\\times\\hat\\Omega^1\\times\\c A_T$,\n\\begin{align}\n\\label{eq proposition 4.9 representation nu 1}\n \\nu^1_s(\\hat\\omega^1,a) &= \\nu^{(0)}_s(B^1(\\hat\\omega^1),a) \\ \\1_{[0,\\tau^1_1(\\hat\\omega^1)]}(s)\\\\\n &\\quad+ \\sum_{k\\geq 1} \\nu^{(k)}_s(B^1(\\hat\\omega^1),(\\tau^1_1(\\hat\\omega^1),\\alpha^1_1(\\hat\\omega^1)),\\dots,(\\tau^1_k(\\hat\\omega^1),\\alpha^1_k(\\hat\\omega^1)),a) \\ \\1_{(\\tau^1_k(\\hat\\omega^1), \\tau^1_{k+1}(\\hat\\omega^1)]}(s),\n\\end{align}\nfor $Pred(\\mathbf C^n)\\otimes \\c B(([0,\\infty)\\times \\c A_T)^k)\\otimes\\c B(\\c A_T)$-$\\c B((0,\\infty))$-measurable processes $\\nu^{(k)}:C^n\\times ([0,\\infty)\\times \\c A_T)^k\\times \\c A_T\\to (0,\\infty)$, for all $k\\geq 0$, where $(\\tau^1_k,\\alpha^1_k)_{k\\geq 1}$ are the events of $\\mu^1$. This representation now allows us to transfer the control $\\nu^1$ to $(\\hat\\Omega^2,\\hat{\\c F}^2,\\hat\\P^2)$ by defining, for all $(s,\\hat\\omega^1,a)\\in [0,T]\\times\\hat\\Omega^1\\times\\c A_T$,\n\\begin{align}\n\\label{eq proposition 4.9 representation nu 2}\n \\nu^2_s(\\hat\\omega^2,a) &= \\nu^{(0)}_s(B^2(\\hat\\omega^2),a) \\ \\1_{[0,\\tau^2_1(\\hat\\omega^2)]}(s)\\\\\n &\\quad + \\sum_{k\\geq 1} \\nu^{(k)}_s(B^2(\\hat\\omega^2),(\\tau^2_1(\\hat\\omega^2),\\alpha^2_1(\\hat\\omega^2)),\\dots,(\\tau^2_k(\\hat\\omega^2),\\alpha^2_k(\\hat\\omega^2)),a) \\ \\1_{(\\tau^2_k(\\hat\\omega^1), \\tau^2_{k+1}(\\hat\\omega^2)]}(s),\n\\end{align}\nwhere $(\\tau^2_k,\\alpha^2_k)_{k\\geq 1}$ are the events of $\\mu^2$. We note that by \\cref{proposition canonical representation of predictable processes}, $\\nu^2$ is $Pred(\\b F^{B^2,\\mu^2})\\otimes\\c B(\\c A_T)$-measurable and thus $\\nu^2\\in\\c V^2$. Furthermore, the representations \\eqref{eq proposition 4.9 representation nu 1,eq proposition 4.9 representation nu 2} together with \\eqref{eq proposition 4.9 law equivalence of state processes} show that\n\\begin{align}\n \\c L^{\\hat\\P^1}(X^{t,\\xi^1,\\alpha_t,1},\\mu^1,I^{t,\\alpha_t,1},(\\hat\\P^{1,\\c F^{B^1,\\mu^1,\\hat\\P^1}_s}_{X^{t,\\xi^1,\\alpha_t,1}_s})_s,\\nu^1) = \\c L^{\\hat\\P^2}(X^{t,\\xi^2,\\alpha_t,2},\\mu^2,I^{t,\\alpha_t,2},(\\hat\\P^{2,\\c F^{B^2,\\mu^2,\\hat\\P^2}_s}_{X^{t,\\xi^2,\\alpha_t,2}_s})_s,\\nu^2).\n\\end{align}\nFinally, this allows us to conclude that\n\\begin{align}\n J^{\\c R,1}(t,\\xi,\\alpha_t,\\nu^1) &= \\E^{\\hat\\P^1}\\Big[L^{\\nu^1}_T g(\\hat\\P_{1,X^{t,\\xi^1,\\alpha_t,1}_T}^{\\c F^{B^1,\\mu^1,\\hat\\P^1}_T},X^{t,\\xi^1,\\alpha_t,1}_T) + \\int_t^T f(s,\\hat\\P_{1,X^{t,\\xi^1,\\alpha_t,1}_s}^{\\c F^{B^1,\\mu^1,\\hat\\P^1}_s},X^{t,\\xi,\\alpha_t,1}_s,I^{t,\\alpha_t,1}_s) ds \\Big]\\\\\n &= \\E^{\\hat\\P^2}\\Big[L^{\\nu^2}_T g(\\hat\\P_{2,X^{t,\\xi^1,\\alpha_t,2}_T}^{\\c F^{B^2,\\mu^2,\\hat\\P^2}_T},X^{t,\\xi^2,\\alpha_t,2}_T) + \\int_t^T f(s,\\hat\\P_{2,X^{t,\\xi^2,\\alpha_t,2}_s}^{\\c F^{B^2,\\mu^2,\\hat\\P^2}_s},X^{t,\\xi,\\alpha_t,2}_s,I^{t,\\alpha_t,2}_s) ds \\Big]\\\\\n &= J^{\\c R,2}(t,\\xi,\\alpha_t,\\nu^2),\n\\end{align}\nwhere $L^{\\nu^i}_T$ is given by \\eqref{eq randomised control girsanov formula}.\nSince $\\nu^1\\in\\c V^1$ was arbitrary, this shows that\n\\[\nV^{\\c R,1}(t,\\xi,\\alpha_t) = \\sup_{\\nu^1\\in{\\c V}^1} J^{\\c R,1}(t,\\xi,\\alpha_t,\\nu^1) \\leq \\sup_{\\nu^2\\in{\\c V}^2} J^{\\c R,2}(t,\\xi,\\alpha_t,\\nu^2) = V^{\\c R,2}(t,\\xi,\\alpha_t).\n\\]\nBy the same arguments we can also show that $V^{\\c R,2}(t,\\xi,\\alpha_t)\\leq V^{\\c R,1}(t,\\xi,\\alpha_t)$, and thus we conclude that $V^{\\c R,1} \\equiv V^{\\c R,2}$.\n\\end{proof}\n\n\\subsection{Proof of $V\\leq V^{\\c R}$}\\label{subsection V leq V^R}\n\nThis direction relies on the result in \\cite[Proposition A.1]{bandini_randomisation_2016_extended_arxiv_version}\\footnote{\\cite{bandini_randomisation_2016_extended_arxiv_version} is the extended version of the paper \\cite{bandini_backward_2018}, and \\cite[Proposition A.1]{bandini_randomisation_2016_extended_arxiv_version} is a more general version of \\cite[Proposition 4.1]{bandini_backward_2018}.} which allows us to approximate general progressive controls with suitable point processes having both full support and a bounded intensity. Note that this requires taking the view point of the admissible control set as the set of $\\b F^B$-predictable processes taking values in $\\c A_T$ as described in \\cref{section problem l2 formulation}, since this allows us to view the mean-field dynamics as a standard (albeit infinite-dimensional) control problem, for which we are now able to use the above approximation result.\n\nLet us fix $t\\in [0,T]$, $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$ and some $\\alpha_t\\in\\c A_t$. Our goal is now to show that for any control $\\alpha\\in\\c A$, it holds that $J(t,\\xi,\\alpha) \\leq V^{\\c R}(t,\\xi,\\alpha_0)$. To this end, let us fix some arbitrary $\\alpha\\in\\c A$, then by \\cref{proposition bar alpha given alpha well defined} there exists a $\\hat\\alpha\\in\\hat{\\c A}$ which is $\\b F^B$-identifying to $\\alpha$, and furthermore by \\cref{lemma alpha bar alpha uniqueness} this $\\hat\\alpha\\in\\hat{\\c A}$ is unique up to modification.\n\nOur first step will be constructing approximating $\\c A_T$-valued Poisson point processes for the $\\c A_T$-valued control process $\\hat\\alpha\\in\\hat{\\c A}$.\nWe recall that these processes are supposed to take the role of $\\hat I^{t,\\alpha_t}$ in the randomised setting which we are going to construct, and thus we want them to start in $\\alpha_t$ at time $t$. For this reason, we will first establish an approximation on $[t,T]$, and afterwards extend the underlying Poisson random measure to $(0,T]$.\nWe note that $(\\hat\\alpha_s)_{s\\in [t,T]}$ is a $\\b F^B$-progressive (in fact by the construction in \\cref{proposition bar alpha given alpha well defined} it is even $\\b F^B$-predictable), and that $\\hat\\alpha_s\\in\\c A_s\\subseteq\\c A_T$ for all $s\\in [t,T]$. Thus, by \\cite[Proposition A.1]{bandini_randomisation_2016_extended_arxiv_version}, there exists a suitable extension $(\\Omega\\times\\Omega',(\\c F\\otimes\\c F')^{\\P\\otimes\\P'},\\P\\otimes\\P')$ of our probability space $(\\Omega,\\c F,\\P)$, \nsuch that for any $\\delta > 0$ there exists a marked point process $(\\tau^\\delta_k,\\alpha^\\delta_k)_{k\\geq 1}$ with the corresponding Poisson random measure $\\mu^\\delta = \\sum_{k\\geq 1} \\delta_{(\\tau^\\delta_k,\\alpha^\\delta_k)}$ on $(t,T]\\times\\c A_T$ such that\n\\begin{enumerate}[label=(\\roman*)]\n \\item $\\c F^{\\mu^\\delta,\\P\\otimes\\P'}_T\\subseteq \\c F^{B,\\P\\otimes\\P'}_T\\lor\\c F'$,\\footnote{With slight abuse of notation, we identify $\\c F'$ with its canonical extension to $\\Omega\\times\\Omega'$.}\n so in particular $\\mu^\\delta$ is independent of $\\c G$ and $W$ under $\\P\\otimes\\P'$,\\footnote{The independence follows from the explicit construction in \\cite[Proposition A.1]{bandini_randomisation_2016_extended_arxiv_version} which uses only $\\b F^{B,\\P\\otimes\\P'}$ together with a fixed family of random variables defined on $(\\Omega',\\c F',\\P')$, which is independent of $\\delta$.}\n \\item the process $\\hat I^{t,\\alpha_t,\\delta}$ defined by\n \\[\n \\hat I^{t,\\alpha_t,\\delta}_s \\coloneqq \\alpha_t \\1_{[t,\\tau^\\delta_1)} + \\sum_{k\\geq 1} \\alpha^\\delta_k \\1_{[\\tau^\\delta_k,\\tau^\\delta_{k+1})} = \\alpha_t + \\int_{(t,s]}\\int_{\\c A_T} (\\alpha - I^{t,\\alpha_t,\\delta}_{r-}) \\mu^\\delta(dr,d\\alpha),\\qquad s\\in [t,T],\n \\]\n satisfies $d_{\\hat{\\c A},[t,T]}^{\\P\\otimes\\P'}(\\hat\\alpha,\\hat I^{t,\\alpha_t,\\delta}) < \\delta$,\n \\item the $\\b F^{B,\\mu^\\delta,\\P\\otimes\\P'}$-compensator of $\\mu^\\delta$ is absolutely continuous with respect to $\\lambda_s(d\\alpha)ds$ and can be written as\n \\[\n \\nu^\\delta_s(\\alpha)\\lambda_s(d\\alpha)ds,\n \\]\n where $\\nu^\\delta$ is $Pred(\\b F^{B,\\mu^\\delta})\\otimes\\c B(\\c A_T)$-measurable and satisfies\n \\[\n 0 < \\inf_{[t,T]\\times\\Omega\\times\\Omega'\\times\\c A_T} \\nu^\\delta \\leq \\sup_{[t,T]\\times\\Omega\\times\\Omega'\\times\\c A_T} \\nu^\\delta < \\infty,\n \\]\n \\item $\\alpha^\\delta_k\\in\\c A_{\\tau^\\delta_k}$, for all $k\\geq 1$.\n\\end{enumerate}\nNote that this statement is slightly different than \\cite[Proposition A.1]{bandini_randomisation_2016_extended_arxiv_version}. At first glance, property (i) may seem new, but it directly follows from the explicit construction in \\cite[Proposition A.1]{bandini_randomisation_2016_extended_arxiv_version} constructing $\\mu^\\delta$ from $\\b F^{B,\\P\\otimes\\P'}$ and a family of random variables defined on $(\\Omega',\\c F',\\P')$, which is independent of $\\delta$.\nThe more crucial difference, however, is that while the action set $A$ in \\cite{bandini_randomisation_2016_extended_arxiv_version} is constant over time, our action sets $\\c A_s$ vary with $s\\in [t,T]$, which now leads to the adapted form of the intensity process in (iii) and the additional property (iv), necessary for consistency with the control randomisation introduced in \\cref{section randomised problem}.\nFortunately, we can extend its proof to cover our use case with minimal changes. Our admissible controls $\\hat\\alpha\\in\\hat{\\c A}$ satisfy additionally $\\hat\\alpha_s\\in\\c A_s$ for all $s\\in [0,T]$, which allows us to choose the step function approximation $\\bar\\alpha = \\sum_{n=0}^\\infty \\alpha_n \\1_{[t_n,t_{n+1})}$ on \\cite[pp.\\@ 41-42]{bandini_randomisation_2016_extended_arxiv_version} such that each $\\hat\\alpha_n\\in\\c A_{t_n}\\subseteq\\c A_T$. We then replace the kernel $q^m$ on \\cite[p.\\@ 42]{bandini_randomisation_2016_extended_arxiv_version} with a family of kernels $(q^m_{s})_{s\\in [0,T]}$, where each $q^m_{s}$ is constructed from $\\lambda_{s}$ instead of $\\lambda$ and thus supported on $\\c A_s$ instead of $\\c A_T$. The definition of $\\beta^m_n$ on \\cite[p.\\@ 42]{bandini_randomisation_2016_extended_arxiv_version} is also modified to include the time, so $\\beta^m_n\\coloneqq q^m_{R^m_n}(\\alpha_n,U^m_n)$. The estimates on \\cite[pp.\\@ 42-43]{bandini_randomisation_2016_extended_arxiv_version} remain valid, and we obtain a marked point process $\\kappa^m$ with the compensator $\\tilde\\kappa^m(dt,d\\alpha) = \\sum_{n\\geq 1} \\1_{(S^m_n,R^m_n]}(t) q^m_t(\\alpha_n,d\\alpha)\\lambda_{nm} dt$, replacing $q^m$ by $q^m_t$ in \\cite[Lemma A.1]{bandini_randomisation_2016_extended_arxiv_version} and its proof. Finally, by replacing $\\lambda(d\\alpha)dt$ by $\\lambda_t(d\\alpha)dt$ in the intensity of the additional \"small\" intensity Poisson process $\\pi^k$ on \\cite[p.\\@ 45]{bandini_randomisation_2016_extended_arxiv_version}, we can obtain the desired point measure $\\mu^\\delta = \\kappa^m + \\pi^k$ without further modifications.\n\nTo extend the Poisson random measure $\\mu^\\delta$ to $[0,T]\\times\\c A_T$, we will extend the probability space again by a probability space $(\\Omega'',\\c F'',\\P'')$ on which a Poisson random measure $\\mu$ on $(0,t]\\times\\c A_T$ with $\\b F^{\\mu^\\delta}$-intensity $\\lambda_s(d\\alpha)ds$ lives. Hence we obtain $(\\hat\\Omega,\\hat{\\c F},\\hat\\Q) \\coloneqq (\\Omega\\times\\Omega'\\times\\Omega'',(\\c F\\otimes\\c F'\\otimes\\c F'')^{\\P\\otimes\\P'\\otimes\\P''},\\P\\otimes\\P'\\otimes\\P'')$, such that $\\mu^\\delta \\coloneqq \\mu(\\cdot \\cap ((0,t]\\times\\c A_T)) + \\mu^\\delta(\\cdot\\cap((t,T]\\times\\c A_T))$ has the $\\b F^{B,\\mu^\\delta}$-intensity $\\nu^\\delta_s(\\alpha)\\lambda_s(d\\alpha)ds$, where we extend $\\nu^\\delta$ to $[0,t]\\times\\c A_T$ with $\\nu^\\delta|_{[0,t]\\times\\c A_T}\\equiv 1$.\nAs before, with slight abuse of notation, we identify each sigma algebra $\\c F$ with its canonical extension to the product space. \nNote that property (i) also extends to $\\mu^\\delta$ on $[0,T]$, that is, $\\c F^{\\mu^\\delta,\\hat\\Q}_T \\subseteq \\c F^{B,\\hat\\Q}_T\\lor\\c F'\\otimes\\c F''$, and in particular $\\mu^\\delta$ on $[0,T]$ is again independent of $\\c G$ and $W$ under $\\hat\\Q$.\n\nHaving defined approximations $(\\hat I^{t,\\alpha_t,\\delta})_\\delta$ to $\\hat\\alpha$, we will now also show that the corresponding state processes converge. We first note that by \\cref{corollary I given bar I is well defined} for every $\\delta > 0$, there exist an $\\b F^{W,\\mu^\\delta}\\lor\\c G$-progressive, càdlàg $I^{t,\\alpha_t,\\delta}$ that $\\b F^{I^{t,\\alpha_t,\\delta}}$-identifies to $\\hat I^{t,\\alpha_t,\\delta}$. Moreover, by \\cref{lemma alpha bar alpha uniqueness}, we have\n\\begin{align}\n d_{\\c A,[t,T]}^{\\hat\\Q}(\\alpha,I^{t,\\alpha_t,\\delta}) = d_{\\hat{\\c A},[t,T]}^{\\hat\\Q}(\\hat\\alpha,\\hat I^{t,\\alpha_t,\\delta}) = d_{\\hat{\\c A},[t,T]}^{\\P\\otimes\\P'}(\\hat\\alpha,\\hat I^{t,\\alpha_t,\\delta}) < \\delta,\\quad\\text{for all }\\delta>0.\n \\label{eq proof V leq V^R alpha I delta close}\n\\end{align}\nCorrespondingly, we now define $X^{t,\\xi,\\alpha_t,\\delta}$ as the unique solution to the SDE\n\\begin{align}\ndX^{t,\\xi,\\alpha_t,\\delta}_s\n&= b(s,\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s},X^{t,\\xi,\\alpha_t,\\delta}_s,I^{t,\\alpha_t,\\delta}_s) ds + \\sigma(s,\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s},X^{t,\\xi,\\alpha_t,\\delta}_s,I^{t,\\alpha_t,\\delta}_s) dW_s\\\\\n&\\qquad+ \\sigma^0(s,\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s},X^{t,\\xi,\\alpha_t,\\delta}_s,I^{t,\\alpha_t,\\delta}_s) dB_s,\\qquad X^{t,\\xi,\\alpha_t,\\delta}_t = \\xi.\n\\label{eq randomised state dynamics X t xi alpha_t delta}\n\\end{align}\nWe next show that $X^{t,\\xi,\\alpha_t,\\delta}$ converges to the state process $X^{t,\\xi,\\alpha}$ for our control $\\alpha\\in\\c A$ in the original control problem, as $\\delta\\to 0$. To this end, we note that we can rewrite the conditional distributions occurring in \\eqref{eq mfc state dynamics,eq randomised state dynamics X t xi alpha_t delta} as follows: Since $X^{t,\\xi,\\alpha}$ is $\\c G\\lor\\b F^{W,B,\\hat\\Q}$-progressive and by construction $\\c G,W,B$ and $\\c F'\\otimes\\c F''$ are independent under $\\hat\\Q$, we see that $\\hat\\Q^{\\c F^{B,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha}_s} = \\hat\\Q^{\\c F^{B,\\hat\\Q}_T\\lor\\c F'\\otimes\\c F''}_{X^{t,\\xi,\\alpha}_s}$, $\\hat\\Q$-a.s., for all $s\\in [t,T]$. For the $\\c G\\lor\\b F^{W,B,\\mu^\\delta,\\hat\\Q}$-progressive process $X^{t,\\xi,\\alpha_t,\\delta}$, we note that by property (i) of $\\mu^\\delta$, we have $\\c F^{\\mu^\\delta,\\hat\\Q}_T\\subseteq\\c F^{B,\\hat\\Q}_T\\lor\\c F'\\otimes\\c F''$, which together with the independence of $\\c G,W$ from $B,\\c F'\\otimes\\c F''$ under $\\hat\\Q$ shows that $\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s} = \\hat\\Q^{\\c F^{B,\\hat\\Q}_T\\lor\\c F'\\otimes\\c F''}_{X^{t,\\xi,\\alpha_t,\\delta}_s}$, $\\hat\\Q$-a.s., for all $s\\in [t,T]$. \nTherefore we can estimate, for all $s\\in [t,T]$, $\\hat\\Q$-a.s.,\n\\[\n\\c W_2(\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s}, \\hat\\Q^{\\c F^{B,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha}_s})^2 = \\c W_2(\\hat\\Q^{\\c F^{B,\\hat\\Q}_T\\lor\\c F'\\otimes\\c F''}_{X^{t,\\xi,\\alpha_t,\\delta}_s},\\hat\\Q^{\\c F^{B,\\hat\\Q}_T\\lor\\c F'\\otimes\\c F''}_{X^{t,\\xi,\\alpha}_s})^2 \\leq \\E^{\\hat\\Q}\\big[|X^{t,\\xi,\\alpha_t,\\delta}_s - X^{t,\\xi,\\alpha}_s|^2 \\,\\big|\\,\\c F^{B,\\hat\\Q}_T\\lor\\c F'\\otimes\\c F''\\big],\n\\]\nwhich implies that\n\\[\n\\sup_{s\\in[t,T]} \\c W_2(\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s},\\hat\\Q^{\\c F^{B,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha}_s})^2\n\\leq \\E^{\\hat\\Q}\\Big[\\sup_{s\\in[t,T]} |X^{t,\\xi,\\alpha_t,\\delta}_s - X^{t,\\xi,\\alpha}_s|^2 \\,\\Big|\\,\\c F^{B,\\hat\\Q}_T\\lor\\c F'\\otimes\\c F'' \\Big].\n\\]\nNow, from \\eqref{eq proof V leq V^R alpha I delta close}, and using \\cref{assumptions sde coefficients}, standard arguments involving the BDG-inequality and Gronwall's Lemma lead to the following estimate\n\\[\n\\E^{\\hat\\Q}\\Big[\\sup_{s\\in[t,T]} \\Big(|X^{t,\\xi,\\alpha_t,\\delta}_s - X^{t,\\xi,\\alpha}_s|^2 + \\c W_2(\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s},\\hat\\Q^{\\c F^{B,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha}_s})^2 \\Big) \\Big] \\to 0,\n\\]\nand thus by \\cref{assumptions reward functional} that\n\\begin{align}\n&\\E^{\\hat\\Q}\\Big[g(\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_T}_{X^{t,\\xi,\\alpha_t,\\delta}_T},X^{t,\\xi,\\alpha_t,\\delta}_T) + \\int_t^T f(s,\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s},X^{t,\\xi,\\alpha_t,\\delta}_s,I^{t,\\alpha_t,\\delta}_s) ds \\Big]\\\\\n&\\to \\E^{\\hat\\Q}\\Big[g(\\hat\\Q^{\\c F^{B,\\hat\\Q}_T}_{X^{t,\\xi,\\alpha}_T},X^{t,\\xi,\\alpha}_T) + \\int_t^T f(s,\\hat\\Q^{\\c F^{B,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha}_s},X^{t,\\xi,\\alpha}_s,\\alpha_s) ds \\Big] = J(t,\\xi,\\alpha).\n\\end{align}\n\nTherefore, the claim follows if we prove that\n\\[\n\\E^{\\hat\\Q}\\Big[g(\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_T}_{X^{t,\\xi,\\alpha_t,\\delta}_T},X^{t,\\xi,\\alpha_t,\\delta}_T) + \\int_t^T f(s,\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s},X^{t,\\xi,\\alpha_t,\\delta}_s,I^{t,\\alpha_t,\\delta}_s) ds \\Big] \\leq V^{\\c R}(t,\\xi,\\alpha_t),\\qquad\\text{for all }\\delta > 0.\n\\]\nFor this, we will express the left-hand side as the pay-off of a randomised control, using that by \\cref{proposition randomised setting is independent of the extension} we can freely choose our randomised probability setting. We will define a new probability measure $\\hat\\P^{(\\delta)}\\sim\\hat\\Q$ on $(\\hat\\Omega,\\hat{\\c F})$ via\n\\[\n\\frac{d\\hat\\P^{(\\delta)}}{d\\hat\\Q}\\Big|_{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s} \\coloneqq L^{\\frac 1 {\\nu^\\delta}}_s = \\exp\\Big(\\int_{(0,s]} \\int_{\\c A_r} \\log \\frac 1 {\\nu^\\delta_r(\\alpha)} \\ \\mu^\\delta(dr,d\\alpha) - \\int_0^s \\int_{\\c A_r} \\Big(\\frac 1 {\\nu^\\delta_r(\\alpha)} - 1\\Big) \\lambda_r(d\\alpha)dr \\Big),\\quad s\\in [0,T].\n\\]\nBy \\cref{lemma d P nu d P is square integrable} and Girsanov's Theorem, see \\cite[Proposition 14.4.I]{daley_introduction_2008}, we note that $\\mu^\\delta$ has under $\\hat\\P^{(\\delta)}$ the intensity $\\lambda_s(d\\alpha)ds$, since by property (iii) the process $\\nu^\\delta$ and thus also $\\frac 1 {\\nu^\\delta}$ are strictly bounded away from $0$ and $\\infty$. Further, following the same arguments as in \\cite[page 2155]{fuhrman_randomized_2015} shows that $\\mu^\\delta$ is independent of $W,B$ and also $B$ resp. $W$ are still $\\b F^{B,\\mu^\\delta,\\hat\\P^{(\\delta)}}$- resp. $\\b F^{W,\\mu^\\delta,\\hat\\P^{(\\delta)}}$-Brownian motions under $\\hat\\P^{(\\delta)}$. Finally, since $\\nu^\\delta$ is $Pred(\\b F^{B,\\mu^\\delta})\\otimes\\c B(\\c A_T)$-measurable and thus independent of $\\c G$ under $\\hat\\Q$, this shows that $\\mu^\\delta$ stays independent of $\\c G$ under $\\hat\\P^{(\\delta)}$.\n\nNext, we observe that the Radon-Nikodym derivative given by $\\frac{d\\hat\\P^{(\\delta)}}{d\\hat\\Q}|_{\\c F^{B,\\mu^\\delta,\\hat\\Q}_\\cdot} = L^{\\frac 1 {\\nu^\\delta}}$ is by construction $\\b F^{B,\\mu^\\delta}$-progressive and that $\\E^{\\P'\\otimes\\P''}[\\frac{d\\hat\\P^{(\\delta)}}{d\\hat\\Q}|_{\\c F^{B,\\mu^\\delta,\\hat\\Q}_T}(\\omega,\\cdot)] = \\E^{\\P'\\otimes\\P''}[L^{\\frac 1 {\\nu^\\delta}}_T(\\omega,\\cdot)] \\equiv 1$, for $\\P$-a.e.\\@ $\\omega\\in\\Omega$, which shows that $(\\hat\\Omega,\\hat{\\c F},\\hat\\P^{(\\delta)})$ is an extension of $(\\Omega,\\c F,\\P)$.\nFinally, as outlined in \\cref{remark conditional distribution unchanged under measure change}, we see that $X^{t,\\xi,\\alpha_t,\\delta}$ satisfies the (uncontrolled) randomised state equation \\eqref{eq sde randomised problem} under $\\hat\\P^{(\\delta)}$, and thus $(\\hat\\Omega,\\hat{\\c F},\\hat\\P^{(\\delta)})$ forms a randomised setting as defined in \\cref{section randomised problem}. \n\nFrom the properties (iii) and (iv) of $\\mu^\\delta$, we note that $\\nu^\\delta\\in\\c V$ is an admissible randomised control to this randomised setting.\nFurther, by construction $\\hat\\Q = (\\P^{(\\delta)})^{\\nu^\\delta}$ is the probability measure belonging to the randomised control $\\nu^\\delta\\in\\c V$, and thus we conclude by \\cref{proposition randomised setting is independent of the extension} that\n\\[\n\\E^{\\hat\\Q}\\Big[g(\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_T}_{X^{t,\\xi,\\alpha_t,\\delta}_T},X^{t,\\xi,\\alpha_t,\\delta}_T) + \\int_t^T f(s,\\hat\\Q^{\\c F^{B,\\mu^\\delta,\\hat\\Q}_s}_{X^{t,\\xi,\\alpha_t,\\delta}_s},X^{t,\\xi,\\alpha_t,\\delta}_s,I^{t,\\alpha_t,\\delta}_s) ds \\Big] = J^{\\c R}_{(\\hat\\Omega,\\hat{\\c F},\\hat\\P^{(\\delta)})}(t,\\xi,\\alpha_t,\\nu^\\delta) \\leq V^{\\c R}(t,\\xi,\\alpha_t).\n\\]\nFinally, since $\\alpha\\in\\c A$ was arbitrary, we thus conclude that $V(t,\\xi) = \\sup_{\\alpha\\in\\c A} J(t,\\xi,\\alpha) \\leq V^{\\c R}(t,\\xi,\\alpha_t)$ for all $\\alpha_t\\in\\c A_t$.\n\n\n\n\\subsection{Proof of $V^{\\c R}\\leq V$}\n\nBased on the approach in \\cite{bandini_backward_2018,bayraktar_randomized_2018}, we will consider a canonical extension (in some sense) $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ of $(\\Omega,\\c F,\\P)$, using that by \\cref{proposition randomised setting is independent of the extension} we are free to choose the extension we want to work on. Together with $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$, we will in \\cref{subsection extended auxiliary problem} define an auxiliary problem, which essentially extends the original control problem to this extended space. Denoting its value function by $V^{\\c E}$, our goal for this section will be proving that $V^{\\c R}\\leq V^{\\c E}\\leq V$.\n\n\n\\subsubsection{The canonical extended space and an auxiliary problem}\\label{subsection extended auxiliary problem}\n\nWe will start by construction the space $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$, which we will need for the later proofs in this section. We recall from \\cref{section problem l2 formulation} that $(\\lambda_s)_{s\\in [0,T]}$ is a family of measures on $\\c A_T$ such that each $\\lambda_s$ is fully supported on $\\c A_s$, and $\\lambda_s\\ll\\lambda_r$ for $0\\leq s\\leq r\\leq T$. Now following essentially the construction in \\cite{bandini_backward_2018}, we can find a surjective, measurable map $\\pi_{\\c A_T}:\\R\\to\\c A_T$ and a family of measures $(\\lambda'_s)_{s\\in [0,T]}$ on $\\R$ such that $\\lambda_s = \\lambda'_s\\circ\\pi_{\\c A_T}^{-1}$ and $\\lambda'_s\\ll\\lambda'_r$ for $0\\leq s\\leq r\\leq T$. This allows us to consider the canonical space $(\\Omega',\\c F',\\P')$ of a nonexplosive Poisson point process on $(0,\\infty)\\times\\R$ with intensity $\\lambda'_s(dr)ds$, which is constructed as follows: Let $\\Omega'$ be the set of all sequences $\\omega' = (\\tau_k,r_k)_{k\\geq 1}\\subseteq (0,\\infty)\\times\\R$ with $\\tau_k < \\tau_{k+1} < \\infty$ and $\\tau_k\\to\\infty$. Next we denote the canonical process by $(T_k,R_k)_{k\\geq 1}$ and let $\\mu' \\coloneqq \\sum_{k\\geq 1} \\delta_{(T_k,R_k)}$ be the corresponding point measure. Letting $\\P'$ be the (unique) probability measure, under which $\\mu'$ is a Poisson random measure with intensity $\\lambda'_s(dr)ds$, (recall that $\\lambda_s$ and thus $\\lambda'_s$ are bounded), we define $\\c F'\\coloneqq \\sigma(T_k,R_k|k\\geq 1)\\lor\\c N^{\\P'}$, where $\\c N^{\\P'}$ denotes all $\\P'$-null sets. We note that by construction $\\c F' = \\c F^{\\mu,\\P'}_\\infty$.\nFinally, we define $A_k\\coloneqq \\pi_{\\c A_T}(R_k)\\in\\c A_T$ for each $k\\geq 1$ and note that by construction $A_k\\in\\c A_{T_k}$, $\\P'$-a.s. Therefore we define the random point measure $\\mu \\coloneqq \\sum_{k\\geq 1} \\delta_{(T_k,A_k)}$, which by construction has intensity $\\lambda_s(d\\alpha)ds$ under $\\P'$.\nIn the following we will denote by $(\\hat\\Omega,\\hat{\\c F},\\hat\\P) \\coloneqq (\\Omega\\times\\Omega',(\\c F\\otimes\\c F')^{\\P\\otimes\\P'},\\P\\otimes\\P')$ the extension of $\\Omega$ with $\\Omega'$. As usual, we denote the canonical extension of $\\c G,\\xi,W,B,\\c F'$ to $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ with the same symbol.\n\nNext we will define an auxiliary problem on this extended space. For this, we define $\\c A^{\\c E}$ as the set of all $\\b F^{W,B}\\lor\\c G\\lor\\c F'$-predictable processes $\\alpha$. Then for each $t\\in [0,T]$, $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^n)$ and $\\alpha\\in\\c A^{\\c E}$, we denote by $X^{t,\\xi,\\alpha}$ the unique $\\b F^{W,B,\\hat\\P}\\lor\\c G\\lor\\c F'$-progressive process solving the state dynamics\n\\begin{align}\n dX_s^{t,\\xi,\\alpha}\n &= b(s, \\hat\\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\hat\\P}_s\\lor\\c F'},X^{t,\\xi,\\alpha}_s,\\alpha_s)ds + \\sigma(s,\\hat\\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\hat\\P}_s\\lor\\c F'},X^{t,\\xi,\\alpha}_s,\\alpha_s) dW_s\\\\\n &\\quad + \\sigma^0(s,\\hat\\P_{X^{t,\\xi,\\alpha}_s}^{\\c F^{B,\\hat\\P}_s\\lor\\c F'},X^{t,\\xi,\\alpha}_s,\\alpha_s) dB_s,\\qquad X^{t,\\xi,\\alpha}_t = \\xi.\n \\label{eq extended problem state dynamics}\n\\end{align}\nAs before, here $(\\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha}_s})_{s\\in [t,T]}$ shall denote the $\\b F^{B,\\hat\\P}\\lor\\c F'$-predictable and $\\hat\\P$-a.s.\\@ continuous version of $(\\c L(X^{t,\\xi,\\alpha}_s|\\c F^{B,\\hat\\P}_s\\lor\\c F'))_{s\\in [t,T]}$, that is $\\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha}_s} = \\c L(X^{t,\\xi,\\alpha}_s|\\c F^{B,\\hat\\P}_s\\lor\\c F')$, $\\hat\\P$-a.s., for all $s\\in [t,T]$. We recall that the existence of such a version is ensured by \\cite[Lemma A.1]{djete_mckeanvlasov_2022-1}, since $(t,\\omega,\\omega')\\mapsto (B_t(\\omega),\\omega')$ is a continuous Hunt process and thus by \\cite[Chapter 3, Section 2.3, pp. 30–32]{chung_introduction_1990}, $O(\\b F^{B,\\hat\\P}\\lor\\c F') = Pred(\\b F^{B,\\hat\\P}\\lor\\c F')$.\nAgain we note that by standard results, the existence and uniqueness of $X^{t,\\xi,\\alpha}$ follows from \\cref{assumptions sde coefficients}, as well as the basic estimate\n\\begin{align}\n \\sup_{\\alpha\\in\\c A^{\\c E}} \\E^{\\hat\\P}\\Big[\\sup_{s\\in [t,T]} |X^{t,\\xi,\\alpha}_s|^2 \\Big] < \\infty.\n\\end{align}\nFinally, by \\cref{assumptions reward functional} the following reward functional\n\\[\nJ^{\\c E}(t,\\xi,\\alpha) \\coloneqq \\E^{\\hat\\P}\\Big[ g(\\hat\\P_{X^{t,\\xi,\\alpha}_T}^{{\\c F}^{B,\\hat\\P}_T\\lor\\c F'},X^{t,\\xi,\\alpha}_T) + \\int_t^T f(s,\\hat\\P_{X^{t,\\xi,\\alpha}_s}^{{\\c F}^{B,\\hat\\P}_s\\lor\\c F'},X^{t,\\xi,\\alpha}_s,\\alpha_s) ds \\Big]\n\\]\nis again well-defined, and we can introduce the corresponding value function as\n\\[\nV^{\\c E}(t,\\xi) \\coloneqq \\sup_{\\alpha\\in\\c A^{\\c E}} J^{\\c E}(t,\\xi,\\alpha) < \\infty.\n\\]\n\n\n\\subsubsection{Proof of $V^{\\c E}\\leq V$}\\label{subsection V^E leq V}\n\nLet $(\\hat\\Omega,\\hat{\\c F},\\hat\\P)$ be the canonical extended space constructed in \\cref{subsection extended auxiliary problem}, and \nlet us fix $t\\in [0,T]$ and $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$. \nFurther, let $\\alpha\\in\\c A^{\\c E}$ be an admissible control to the extended problem and $X^{t,\\xi,\\alpha}$ the corresponding state process satisfying \\eqref{eq extended problem state dynamics}. Since by definition $\\alpha:[0,T]\\times\\Omega\\times\\Omega'\\to A$ is $(\\b F^{W,B}\\lor\\c G)\\otimes\\c F'$-predictable, we see that for all $\\omega'\\in\\Omega'$, the process $\\alpha^{\\omega'}\\coloneqq \\alpha(\\cdot,\\omega')$ is $\\b F^{W,B}\\lor\\c G$-predictable and in particular $\\alpha^{\\omega'}\\in\\c A$. This motivates us to consider for each $\\omega'\\in\\Omega'$ the corresponding state process $X^{t,\\xi,\\alpha^{\\omega'}}$ to $\\alpha^{\\omega'}$ satisfying the state dynamics \\eqref{eq mfc state dynamics}, and our next goal is showing that\n\\[\nX^{t,\\xi,\\alpha}(\\cdot,\\omega') = X^{t,\\xi,\\alpha^{\\omega'}},\\ \\P\\text{-a.s.},\\quad \\text{for }\\P'\\text{-a.a. }\\omega'\\in\\Omega'.\n\\]\nTo this end, we note that since $(\\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha}_s})_{s\\in [t,T]}$ is $\\b F^{B,\\hat\\P}\\lor\\c F'$-predictable and for our canonical extended space $\\b F^{B,\\hat\\P}\\lor\\c F' = (\\b F^B\\otimes\\c F')^{\\hat\\P}$, it follows that for $\\P'$-a.a.\\@ $\\omega'\\in\\Omega'$, the process $(\\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha}_s}(\\cdot,\\omega'))_{s\\in [t,T]}$ is $\\b F^{B,\\P}$-predictable. At the same time, we see that since $\\hat\\P = \\P\\otimes\\P'$, for all $s\\in [t,T]$ and for $\\P'$-a.a.\\@ $\\omega'\\in\\Omega'$,\n\\[\n\\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha}_s}(\\cdot,\\omega') = \\c L^{\\hat\\P}(X^{t,\\xi,\\alpha}_s | (\\c F^B_s\\otimes\\c F')^{\\hat\\P})(\\cdot,\\omega') = \\c L^{\\P}(X^{t,\\xi,\\alpha}_s(\\cdot,\\omega') | \\c F^{B,\\P}_s),\\quad\\P\\text{-a.s},\n\\]\nand thus $(\\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha}_s}(\\cdot,\\omega'))_{s\\in [t,T]}$ is a suitable $\\P$-a.s.\\@ continuous and $\\b F^{B,\\P}$-predictable version of the conditional law process $(\\c L^{\\P}(X^{t,\\xi,\\alpha}_s(\\cdot,\\omega') | \\c F^{B,\\P}_s))_{s\\in [t,T]}$, for $\\P'$-a.a.\\@ $\\omega'\\in\\Omega'$. Finally, this shows that for $\\P'$-a.a.\\@ $\\omega'\\in\\Omega'$, the process $X^{t,\\xi,\\alpha}(\\cdot,\\omega')$ also satisfies the SDE \\eqref{eq mfc state dynamics} with the control $\\alpha^{\\omega'} = \\alpha(\\cdot,\\omega')$, and thus by the uniqueness of strong solutions to \\eqref{eq mfc state dynamics}, we deduce that $X^{t,\\xi,\\alpha}(\\cdot,\\omega') = X^{t,\\xi,\\alpha^{\\omega'}}$, $\\P$-a.s.\n\nThis now allows us to rewrite the extended reward functional $J^{\\c E}$ in terms of $J$ as follows,\n\\begin{align}\nJ^{\\c E}(t,\\xi,\\alpha)\n&= \\int_{\\Omega'} \\E\\Big[g(\\hat\\P^{\\c F^{B,\\hat\\P}_T\\lor\\c F'}_{X^{t,\\xi,\\alpha}_T}(\\cdot,\\omega'),X^{t,\\xi,\\alpha}_T(\\cdot,\\omega'))\\\\\n&\\qquad + \\int_t^T f(s, \\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha}_s}(\\cdot,\\omega'),X^{t,\\xi,\\alpha}_s(\\cdot,\\omega'),\\alpha_s(\\cdot,\\omega')) ds\\Big] \\P'(d\\omega')\\\\\n&= \\int_{\\Omega'} J(t,\\xi,\\alpha^{\\omega'}) \\P'(d\\omega')\n\\leq V(t,\\xi),\n\\end{align}\nand since $\\alpha\\in\\c A^{\\c E}$ was chosen arbitrary, this shows that $V^{\\c E}(t,\\xi) = \\sup_{\\alpha\\in\\c A^{\\c E}} J^{\\c E}(t,\\xi,\\alpha)\\leq V(t,\\xi)$.\n\n\n\\subsubsection{Proof of $V^{\\c R}\\leq V^{\\c E}$}\\label{subsection V^R leq V^E}\n\nRecalling that by \\cref{proposition randomised setting is independent of the extension} we are free to choose our randomised probability space, we will consider the canonical extended space $(\\hat\\Omega,\\hat{\\c F},\\hat\\P) = (\\Omega \\times \\Omega',(\\c F\\otimes\\c F')^{\\P\\otimes\\P'},\\P\\otimes\\P')$ from \\cref{subsection extended auxiliary problem}. Let us further fix $t\\in [0,T]$, $\\xi\\in L^2(\\Omega,\\c G\\lor\\c F^W_t,\\P;\\R^d)$ and $\\alpha_t\\in\\c A_t$. We now need to show that for any $\\nu\\in\\c V$, we have $J^{\\c R}(t,\\xi,\\alpha_t,\\nu) \\leq V^{\\c E}(t,\\xi)$.\n\nTo this end, let us fix $\\nu\\in\\c V$ and let $\\hat\\P^\\nu\\sim\\hat\\P$ be the corresponding tilted probability measure under which $\\mu$ has the intensity $\\nu_s(\\alpha)\\lambda_s(d\\alpha)ds$, defined in \\eqref{eq randomised control girsanov formula}.\nFollowing the approach by \\cite{bandini_backward_2018}, we will construct a piece-wise constant process $\\hat\\alpha^\\nu\\in\\c A^{\\c E}$ that has under $\\hat\\P$ the same distribution as the Poisson point process $\\hat I^{t,\\alpha_t}$ under $\\hat\\P^\\nu$ using \\cite[Lemma 4.3]{bandini_backward_2018}.\nTo set this up, we define for each $\\omega\\in\\Omega$ the process $\\nu^\\omega:[0,T]\\times\\Omega'\\times\\c A_T\\to (0,\\infty)$ by $\\nu^\\omega_s(\\omega',\\alpha)\\coloneqq \\nu_s(\\omega,\\omega',\\alpha)$. This $\\nu^\\omega$ is then $Pred(\\b F^\\mu)\\otimes\\c B(\\c A_T)$-measurable and satisfies $0\\leq \\inf_{[0,T]\\times\\Omega'\\times\\c A_T} \\nu^\\omega\\leq \\sup_{[0,T]\\times\\Omega'\\times\\c A_T} \\nu^\\omega < \\infty$, see also \\cite[Lemma 4.2]{bandini_backward_2018}. Correspondingly we define probability measures ${\\P'}^{\\nu^\\omega}\\sim\\P'$ on $(\\Omega',{\\c F}')$ via $\\frac{d{\\P'}^{\\nu^\\omega}}{d\\P'}|_{\\c F^\\mu_T} = L^{\\nu^\\omega}_T$, where $L^{\\nu^\\omega}$ is given by \\eqref{eq randomised control girsanov formula}. Using an analogous result to \\cref{lemma d P nu d P is square integrable}, we note that ${\\P'}^{\\nu^\\omega}$ is indeed well-defined, and moreover we observe that by construction $\\hat\\P^\\nu(d\\omega,d\\omega') = {\\P'}^{\\nu^\\omega}(d\\omega')\\P(d\\omega)$.\n\nNext we apply \\cite[Lemma 4.3]{bandini_backward_2018} and obtain a process\n\\[\n\\hat\\alpha'^\\nu = \\alpha_t \\1_{[t,\\tau'^\\nu_1)} + \\sum_{k\\geq 1} \\alpha'^\\nu_k \\1_{[\\tau'^\\nu_k,\\tau'^\\nu_{k+1})},\n\\]\nwhere $\\tau'^\\nu_k\\geq t$ are $\\b F^{B,\\hat\\P}\\lor\\c F'$-stopping times and $\\alpha'^\\nu_k$ are $\\c F^{B,\\hat\\P}_{\\tau^\\nu_k}\\lor\\c F'$-measurable, $k\\geq 1$, such that\n\\[\n\\c L^{\\P'}(\\hat\\alpha'^\\nu(\\omega,\\cdot)) = \\c L^{{\\P'}^{\\nu^\\omega}}(\\hat I^{t,\\alpha_t}),\\qquad\\text{for }\\P\\text{-a.a. }\\omega\\in\\Omega.\n\\]\nNow we note that $(t,\\omega,\\omega')\\mapsto (B_t(\\omega),\\omega')$ is a continuous Hunt process and thus by \\cite[Chapter 3, Section 2.3, pp. 30–32]{chung_introduction_1990} every $\\b F^{B,\\hat\\P}\\lor\\c F'$-optional time is also $\\b F^{B,\\hat\\P}\\lor\\c F'$-predictable. This allows us to use the arguments in \\cite[Lemma 6]{claisse_pseudo-markov_2016} to conclude that there exist $\\b F^B\\lor\\c F'$-predictable stopping times $\\tau^\\nu_k$ and $\\c F^B_{\\tau^\\nu_k}\\lor\\c F'$-measurable $\\alpha^\\nu_k$ such that $\\tau^\\nu_k = \\tau'^\\nu_k$ and $\\alpha^\\nu_k = \\alpha'^\\nu_k$, $\\hat\\P$-a.s. Therefore, the process\n\\[\n\\hat\\alpha^\\nu \\coloneqq \\alpha_t \\1_{[t,\\tau^\\nu_1)} + \\sum_{k\\geq 1} \\alpha^\\nu_k \\1_{[\\tau^\\nu_k,\\tau^\\nu_{k+1})}\n\\]\nis $\\b F^B\\lor\\c F'$-predictable and indistinguishable from $\\hat\\alpha'^\\nu$, and thus we have\n\\[\n\\c L^{\\P'}(\\hat\\alpha^\\nu(\\omega,\\cdot)) = \\c L^{\\P'}(\\hat\\alpha'^\\nu(\\omega,\\cdot)) = \\c L^{{\\P'}^{\\nu^\\omega}}(\\hat I^{t,\\alpha_t}),\\qquad\\text{for }\\P\\text{-a.a. }\\omega\\in\\Omega.\n\\]\n\nWe note that this implies that\n\\begin{align}\n\\c L^{\\hat\\P}(\\pi|_{\\c G},\\xi,W,B,\\hat\\alpha^\\nu) = \\c L^{\\hat\\P^\\nu}(\\pi|_{\\c G},\\xi,W,B,\\hat I^{t,\\alpha_t}),\n\\label{eq proof V R <= V E law equality}\n\\end{align}\nwhere $\\pi|_{\\c G}:(\\hat\\Omega,\\c G)\\to (\\Omega,\\c G)$ is the canonical $\\c G$-measurable projection, since for all bounded, measurable functions $\\phi:(\\Omega\\times\\R^d\\times\\R^m\\times\\R^n,\\c G\\times\\c B(\\R^d\\times\\R^m\\times\\R^n))\\to\\R$ and $\\psi:(\\c A_T,\\c B(\\c A_T))\\to\\R$ we have\n\\begin{align}\n &\\E^{\\hat\\P^\\nu}[\\phi(\\pi|_{\\c G},\\xi,W,B) \\psi(\\hat I^{t,\\alpha_t})] \\\\\n &= \\int_{\\Omega\\times\\Omega'} \\phi(\\omega,\\xi(\\omega),W(\\omega),B(\\omega)) \\psi(\\hat I^{t,\\alpha_t}(\\omega,\\omega')) \\hat\\P^\\nu(d\\omega,d\\omega')\\\\\n &= \\int_\\Omega \\phi(\\omega,\\xi(\\omega),W(\\omega),B(\\omega)) \\int_{\\Omega'} \\psi(\\hat I^{t,\\alpha_t}(\\omega,\\omega')) {\\P'}^{\\nu^\\omega}(d\\omega') \\P(d\\omega)\\\\\n &= \\int_\\Omega \\phi(\\omega,\\xi(\\omega),W(\\omega),B(\\omega)) \\int_{\\Omega'} \\psi(\\hat\\alpha^\\nu(\\omega,\\omega')) \\P'(d\\omega') \\P(d\\omega)\\\\\n &= \\E^{\\hat\\P}[\\phi(\\pi|_{\\c G},\\xi,W,B) \\psi(\\hat\\alpha^\\nu)].\n\\end{align}\n\nAs a next step, we recall that by \\cref{corollary I given bar I is well defined} there is an unique (up to indistinguishability) càdlàg, $\\b F^{W}\\lor\\c G\\lor\\c F^{\\hat I^{t,\\alpha_t}}_T$-predictable and $\\b F^{W,\\hat I^{t,\\alpha_t}}\\lor\\c G$-progressive process $I^{t,\\alpha_t}$ $\\b F^{\\hat I^{t,\\alpha_t}}$-identifying to $\\hat I^{t,\\alpha_t}$. Similarly, \\cref{lemma existence N bar N,lemma alpha bar alpha uniqueness} shows that there is an unique (up to modification) $\\b F^{W}\\lor\\c G\\lor\\c F^{\\hat\\alpha^\\nu}_T$-predictable and $\\b F^{W,\\hat\\alpha^\\nu}\\lor\\c G$-progressive process $\\alpha^\\nu$ which $\\b F^{\\hat\\alpha^\\nu}$-identifies to $\\hat\\alpha^\\nu$, and moreover together with \\eqref{eq proof V R <= V E law equality}, we obtain by \\cref{lemma joint law N bar N map} that\n\\begin{align}\n\\c L^{\\hat\\P}(\\pi|_{\\c G},\\xi,W,B,\\hat\\alpha^\\nu,\\alpha^\\nu) = \\c L^{\\hat\\P^\\nu}(\\pi|_{\\c G},\\xi,W,B,\\hat I^{t,\\alpha_t},I^{t,\\alpha_t}).\n\\label{eq proof V R <= V E second law equality}\n\\end{align}\nWe note that $\\alpha^\\nu$ is $\\b F^{W,B}\\lor\\c G\\lor\\c F'$-predictable and hence $\\alpha^\\nu\\in\\c A^{\\c E}$.\n\nSince $I^{t,\\alpha_t}$ is $\\b F^{W,\\hat I^{t,\\alpha_t}}\\lor\\c G$-progressive, there exists a unique $\\b F^{W,B^t,\\hat I^{t,\\alpha_t},\\hat\\P}\\lor\\c G$-progressive process $X^{t,\\xi,\\alpha_t}$ solving the randomised state dynamics \\eqref{eq sde randomised problem}, and similar since $\\alpha^\\nu$ is $\\b F^{W,\\hat\\alpha^\\nu}\\lor\\c G$-progressive, there exists a unique $\\b F^{W,B,\\hat\\P}\\lor\\c G\\lor\\c F'$-progressive $X^{t,\\xi,\\alpha^\\nu}$ process solving \\eqref{eq extended problem state dynamics}. This implies that the corresponding conditional law process $(\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s})_{s\\in [t,T]}$ is $\\b F^{B,\\hat I^{t,\\alpha_t},\\hat\\P^\\nu}$-progressive, since $\\c G,W$ are $\\hat\\P^\\nu$-independent of $B,\\mu$ and similar that $(\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha^\\nu}_s})_{s\\in [t,T]}$ is $\\b F^{B,\\hat\\alpha^\\nu,\\hat\\P}$-progressive, since $\\c G,W$ are $\\hat\\P$-independent of $B,\\c F'$. This implies that\n\\[\n\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s} = \\hat\\P^{\\nu,\\c F^{B,\\hat I^{t,\\alpha_t},\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s},\n\\qquad \\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha^\\nu}_s} = \\hat\\P^{\\c F^{B,\\hat\\alpha^\\nu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha^\\nu}_s},\\qquad\\hat\\P\\text{-a.s.},\n\\]\nand therefore both $X^{t,\\xi,\\alpha_t}$ and $X^{t,\\xi,\\alpha^\\nu}$ satisfy the following equation\n\\begin{align}\n dX_s &= b(s,\\hat\\P^{\\c F^{B,\\hat\\alpha,\\hat\\P}_s}_{X_s},X_s,\\alpha_s) ds + \\sigma(s,\\hat\\P^{\\c F^{B,\\hat\\alpha,\\hat\\P}_s}_{X_s},X_s,\\alpha_s) dW_s + \\sigma^0(s,\\hat\\P^{\\c F^{B,\\hat\\alpha,\\hat\\P}_s}_{X_s},X_s,\\alpha_s) dB_s,\\ X_t = \\xi,\n \\label{eq proof V R <= V E auxiliary SDE}\n\\end{align}\nwith the corresponding control $\\hat\\alpha = \\hat I^{t,\\alpha_t}$ respectively $\\hat\\alpha = \\hat\\alpha^\\nu$. Consequently, together with \\eqref{eq proof V R <= V E second law equality} the weak uniqueness of solutions to \\eqref{eq proof V R <= V E auxiliary SDE} now shows that\n\\[\n\\c L^{\\hat\\P}((\\hat\\P^{\\c F^{B,\\hat\\P}_s\\lor\\c F'}_{X^{t,\\xi,\\alpha^\\nu}_s})_{s\\in [t,T]}, X^{t,\\xi,\\alpha^\\nu},\\alpha^\\nu)\n= \\c L^{\\hat\\P^\\nu}((\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s})_{s\\in [t,T]}, X^{t,\\xi,\\alpha_t},I^{t,\\alpha_t}).\n\\]\n\nFinally, this implies that\n\\begin{align}\n J^{\\c R}(t,\\xi,\\alpha_t,\\nu)\n &= \\E^{\\hat\\P^\\nu}\\Big[ g(\\hat\\P_{X^{t,\\xi,\\alpha_t}_T}^{\\c F^{B,\\mu,\\hat\\P}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P_{X^{t,\\xi,\\alpha_t}_s}^{\\c F^{B,\\mu,\\hat\\P}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds \\Big]\\\\\n &= \\E^{\\hat\\P}\\Big[g(X^{t,\\xi,\\alpha^\\nu}_T, \\hat\\P_{X^{t,\\xi,\\alpha^\\nu}_T}^{{\\c F}^{B,\\hat\\P}_T\\lor\\c F'}) + \\int_t^T f(s,X^{t,\\xi,\\alpha^\\nu}_s,\\hat\\P_{X^{t,\\xi,\\alpha^\\nu}_s}^{{\\c F}^{B,\\hat\\P}_s\\lor\\c F'},\\alpha^\\nu_s) ds \\Big]\n = J^{\\c E}(t,\\xi,\\alpha),\n\\end{align}\nand since $\\nu\\in\\c V$ was arbitrary, this shows that $V^{\\c R}(t,\\xi,\\alpha_t) = \\sup_{\\nu\\in\\c V} J^{\\c R}(t,\\xi,\\alpha_t,\\nu) \\leq V^{\\c E}(t,\\xi)$.\n\n\n\n\\section{Proof of \\cref{lemma penalised bsde minimal solution existence}}\n\\label{appendix proof lemma penalised bsde minimal solution existence}\n\n\\begin{proof}\n\\begin{enumerate}[wide,label=(\\roman*)]\n\\item \nWe first note that we can view the penalised BSDE \\eqref{eq penalised bsde value function} as BSDE on $[0,T]$ with the driver\n\\[\nF(s,\\hat\\omega,u) \\coloneqq \\1_{[t,T]}(s) \\Big(H_{s-}(\\hat\\omega) + n \\int_{\\c A_s} (u(\\alpha))_+ \\lambda_s(d\\alpha)\\Big),\n\\]\nwhere $H_r \\coloneqq \\E^{\\hat\\P}[f(r,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_r}_{X^{t,\\xi,\\alpha_t}_r},X^{t,\\xi,\\alpha_t}_r,I^{t,\\alpha_t}_r) | \\c F^{B,\\mu,\\hat\\P}_r]$, and terminal value\n\\[\nG \\coloneqq \\E^{\\hat\\P}[g(X^{t,\\xi,\\alpha_t}_T,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T}) | \\c F^{B,\\mu,\\hat\\P}_T].\n\\]\nThen by \\cite[Lemma 2.4]{tang_necessary_1994}, there exists a unique solution $(Y^{n,t,\\xi,\\alpha_t},Z^{n,t,\\xi,\\alpha_t},U^{n,t,\\xi,\\alpha_t}) \\in \\c S^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})\\times L^2_{W,[t,T]}(\\b F^{B,\\mu,\\hat\\P})\\times L^2_{\\lambda,[t,T]}(\\b F^{B,\\mu,\\hat\\P})$ to \\eqref{eq penalised bsde value function}, noting that while boundedness of the driver $F$ in \\cite[(2.15)]{tang_necessary_1994} is not fulfilled by our driver $F$, this assumption is only needed to show that for each $\\bar U\\in L^2_{\\lambda,[t,T]}(\\b F^{B,\\mu,\\hat\\P})$,\\begin{align}\n\\E^{\\hat\\P}\\Big[\\Big|H + \\int_0^T F(s,\\bar U_s) ds \\Big|^2\\Big] < \\infty,\n\\end{align}\nwhich in our setting holds since by \\cref{assumptions reward functional} together with \\eqref{eq randomised state dynamics basic estimate}, we have\n\\begin{align}\n&\\E^{\\hat\\P}\\Big[\\Big|H + \\int_0^T F(s,\\bar U_s) ds \\Big|^2\\Big]\\\\\n&\\leq 2\\E^{\\hat\\P}\\Big[|\\E^{\\hat\\P}[g(X^{t,\\xi,\\alpha_t}_T,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T}) | \\c F^{B,\\mu,\\hat\\P}_T]|^2\\\\\n&\\qquad + T \\int_t^T \\Big|\\E^{\\hat\\P}[f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) | \\c F^{B,\\mu,\\hat\\P}_s] + n \\int_{\\c A_s} (\\bar U_s(\\alpha))_+ \\lambda_s(d\\alpha) \\Big|^2 ds\\Big]\\\\\n&\\leq 2\\E^{\\hat\\P}\\Big[|g(X^{t,\\xi,\\alpha_t}_T,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T})|^2 + T \\int_t^T |f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s)|^2 ds\\\\\n&\\qquad + n T \\sup_{s\\in [0,T]} \\lambda_s(\\c A_s) \\int_t^T \\int_{\\c A_s} |\\bar U_s(\\alpha)|^2 \\lambda_s(d\\alpha) ds \\Big] < \\infty.\n\\end{align}\n\n\\item\\label{proof lemma 6.1 step 2}\nNext we will prove the representation \\eqref{eq Y n t xi snell envelope formula}. From the BSDE \\eqref{eq penalised bsde value function}, for any $\\nu\\in\\c V$, by taking the conditional expectation under $\\hat\\P^\\nu$ given $\\c F^{B,\\mu,\\hat\\P^\\nu}_s$, we obtain for all $s\\in [t,T]$ and $r\\in [s,T]$,\n\\begin{align}\n Y^{n,t,\\xi,\\alpha_t}_s\n &= \\E^{\\hat\\P^\\nu}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_s^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\\\\n &\\qquad + \\int_s^r\\int_{\\c A_u} \\big(n (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_+ - \\nu_u(\\alpha) U^{n,t,\\xi,\\alpha_t}_u(\\alpha)\\big) \\lambda_u(d\\alpha) du\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big].\n \\label{eq Y n t xi expectation}\n\\end{align}\nNoting that for $\\nu\\in\\c V^n$ we have $0 < \\nu\\leq n$ and thus $n(U^{n,t,\\xi,\\alpha_t})_+ - \\nu U^{n,t,\\xi,\\alpha_t} \\geq 0$, we see that \\eqref{eq Y n t xi expectation} shows that for all $\\nu\\in\\c V^n$, $s\\in [t,T]$ and $r\\in [s,T]$,\n\\[\nY^{n,t,\\xi,\\alpha_t}_s\n\\geq \\E^{\\hat\\P^\\nu}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_s^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du \\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big],\n\\]\nand hence\n\\[\nY^{n,t,\\xi,\\alpha_t}_s\n\\geq \\esssup_{\\nu\\in\\c V^n} \\E^{\\hat\\P^\\nu}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_s^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du \\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big].\n\\]\n\nFor the other direction, we define for each $\\varepsilon\\in (0,1]$ the randomised control $\\nu^{n,\\varepsilon}\\in\\c V^n$ via\n\\begin{align}\n\\nu^{n,\\varepsilon}_s(\\alpha) \\coloneqq \\begin{cases*}\nn\\1_{\\{U^{n,t,\\xi,\\alpha_t}_s(\\alpha) \\geq 0\\}} + \\varepsilon\\1_{\\{U^{n,t,\\xi,\\alpha_t}_s(\\alpha) < 0\\}}&, on $[t,T]\\times\\c A_T$\\\\\n1&, on $[0,t)\\times\\c A_T$.\n\\end{cases*}\n\\label{eq Y n t xi definition of nu n varepsilon}\n\\end{align}\nCorrespondingly, we define the probability measures $\\hat\\P^{\\nu^{n,\\varepsilon}}\\sim\\hat\\P$ via $\\frac{d\\hat\\P^{\\nu^{n,\\varepsilon}}}{d\\hat\\P}\\big|_{\\c F^{B,\\mu,\\hat\\P}_u} \\coloneqq L^{n,\\varepsilon}_u$ for all $u\\in [0,T]$, see \\eqref{eq randomised control girsanov formula}. Thus from \\eqref{eq Y n t xi expectation} we obtain for all $s\\in [t,T]$ and $r\\in [s,T]$,\n\\begin{align}\nY^{n,t,\\xi,\\alpha_t}_s\n&\\leq \\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_s^r f(u,\\hat\\P^{\\nu^{n,\\varepsilon},\\c F^{B,\\mu,\\hat\\P^{\\nu^{n,\\varepsilon}}}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du \\Big| \\c F^{B,\\mu,\\hat\\P^{\\nu^{n,\\varepsilon}}}_s\\Big]\\\\\n&\\qquad + \\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}\\Big[\\varepsilon\\int_s^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big|\\c F^{B,\\mu,\\hat\\P^{\\nu^{n,\\varepsilon}}}_s \\Big]\\\\\n&\\leq \\esssup_{\\nu\\in\\c V^n} \\E^{\\hat\\P^\\nu}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_s^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du \\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_s\\Big]\\\\\n&\\qquad + \\varepsilon \\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}\\Big[\\int_s^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big|\\c F^{B,\\mu,\\hat\\P^{\\nu^{n,\\varepsilon}}}_s \\Big].\n\\label{eq Y n t xi upper bound}\n\\end{align}\nTherefore the desired result follows by letting $\\varepsilon\\downarrow 0$, if we can show that the last term vanishes in the limit. \nTo estimate the last term, we observe that since by construction $0 < \\nu^{n,\\varepsilon} \\leq \\nu^{n,1}$ for all $\\varepsilon\\in (0,1]$, we have that for all $s\\in [t,T]$,\n\\begin{align}\n0\\leq \\frac{L^{\\nu^{n,\\varepsilon}}_T}{L^{\\nu^{n,\\varepsilon}}_s} \\frac{L^{\\nu^{n,1}}_s}{L^{\\nu^{n,1}}_T}\n&= \\exp\\Big(\\int_{[s,T]}\\int_{\\c A_r} \\log \\frac{\\nu^{n,\\varepsilon}_r(\\alpha)}{\\nu^{n,1}_r(\\alpha)} \\mu(dr,d\\alpha) - \\int_0^T \\int_{\\c A_r} (\\nu^{n,\\varepsilon}_r(\\alpha) - \\nu^{n,1}_r(\\alpha)) \\lambda_r(d\\alpha)dr \\Big)\\\\\n&\\leq \\exp\\Big(\\int_0^T \\int_{\\c A_r} \\lambda_r(d\\alpha)dr\\Big)^{\\norm{\\nu^{n,1}}_\\infty} \\eqqcolon C < \\infty.\n\\end{align}\nThis allows us to bound the last term in \\eqref{eq Y n t xi upper bound} as follows\n\\begin{align}\n&\\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}\\Big[\\int_s^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big|\\c F^{B,\\mu,\\hat\\P^{\\nu^{n,\\varepsilon}}}_s \\Big]\\\\\n&= \\E^{\\hat\\P}\\Big[\\frac{L^{\\nu^{n,\\varepsilon}}_T}{L^{\\nu^{n,\\varepsilon}}_s} \\int_s^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big|\\c F^{B,\\mu,\\hat\\P}_s \\Big]\\\\\n&\\leq C \\E^{\\hat\\P}\\Big[\\frac{L^{\\nu^{n,1}}_T}{L^{\\nu^{n,1}}_s} \\int_s^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big|\\c F^{B,\\mu,\\hat\\P}_s \\Big]\\\\\n&= C \\E^{\\hat\\P^{\\nu^{n,1}}}\\Big[ \\int_s^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big|\\c F^{B,\\mu,\\hat\\P^{\\nu^{n,1}}}_s \\Big].\n\\end{align}\nThis term is now $L^1(\\hat\\P^{\\nu^{n,1}})$-integrable, and due to $\\hat\\P^{\\nu^{n,1}}\\sim\\hat\\P$ also $\\hat\\P$-a.s.\\@ finite, since\n\\begin{align}\n&\\E^{\\hat\\P^{\\nu^{n,1}}}\\Big[ \\E^{\\hat\\P^{\\nu^{n,1}}}\\Big[ \\int_s^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big|\\c F^{B,\\mu,\\hat\\P^{\\nu^{n,1}}}_s \\Big] \\Big]\\\\\n&\\leq \\bigg(\\E^{\\hat\\P}[ |L^{\\nu^{n,1}}_T|^2 ] \\E^{\\hat\\P}\\Big[\\Big(\\int_s^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du\\Big)^2\\Big] \\bigg)^{\\frac 1 2}\\\\\n&\\leq \\bigg(\\E^{\\hat\\P}[|L^{\\nu^{n,1}}_T|^2 ] \\E^{\\hat\\P}\\Big[\\int_s^r \\int_{\\c A_u} |U^{n,t,\\xi,\\alpha_t}_u(\\alpha)|^2 \\lambda_u(d\\alpha) du \\Big] \\int_s^r \\lambda_u(\\c A_u) du \\bigg)^{\\frac 1 2} < \\infty,\n\\end{align}\nsince $U^{n,t,\\xi,\\alpha_t}\\in L^2_{\\lambda,[t,T]}(\\b F^{B,\\mu,\\hat\\P})$ and $L^{\\nu^{n,1}}_T \\in L^2(\\hat\\Omega,\\c F^{B,\\mu}_T,\\hat\\P;\\R)$ by \\cref{lemma d P nu d P is square integrable}.\n\n\\item Finally, let us turn to \\eqref{eq Y n t xi snell envelope upper bound at s=t}. By taking the expectation with respect to $\\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}$ in \\eqref{eq Y n t xi upper bound} for $s=t$, we obtain hat for all $r\\in [t,T]$ and $\\varepsilon\\in (0,1]$,\n\\begin{align}\n\\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}[Y^{n,t,\\xi,\\alpha_t}_t]\n&\\leq \\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_t^r f(u,\\hat\\P^{\\nu^{n,\\varepsilon},\\c F^{B,\\mu,\\hat\\P^{\\nu^{n,\\varepsilon}}}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du \\Big]\\\\\n&\\qquad + \\varepsilon \\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}\\Big[\\int_t^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big].\n\\end{align}\nSince by construction $\\frac{d\\hat\\P^{\\nu^{n,\\varepsilon}}}{d\\hat\\P}|_{\\c F^{B,\\mu,\\hat\\P}_t} = L^{\\nu^{n,\\varepsilon}}_t \\equiv 1$, we see that $\\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}[Y^{n,t,\\xi,\\alpha_t}_t] = \\E^{\\hat\\P}[Y^{n,t,\\xi,\\alpha_t}_t]$, and thus\n\\begin{align}\n\\E^{\\hat\\P}[Y^{n,t,\\xi,\\alpha_t}_t] = \\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}[Y^{n,t,\\xi,\\alpha_t}_t]\n&\\leq \\sup_{\\nu\\in\\c V^n} \\E^{\\hat\\P^{\\nu}}\\Big[Y^{n,t,\\xi,\\alpha_t}_r + \\int_t^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^{\\nu}}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du \\Big]\\\\\n&\\qquad + \\varepsilon \\E^{\\hat\\P^{\\nu^{n,\\varepsilon}}}\\Big[\\int_t^r \\int_{\\c A_u} (U^{n,t,\\xi,\\alpha_t}_u(\\alpha))_- \\lambda_u(d\\alpha) du \\Big].\n\\end{align}\nFinally, by letting $\\varepsilon\\downarrow 0$, we obtain \\eqref{eq Y n t xi snell envelope upper bound at s=t}, recalling that we have already shown in \\cref{proof lemma 6.1 step 2} that the last term will vanish.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\\section{Proof of \\cref{lemma bsde minimal solution existence}}\n\\label{appendix proof lemma bsde minimal solution existence}\n\n\\begin{proof}\n\\begin{enumerate}[wide,label=(\\roman*)]\n\\item\\label{proof theorem 6.2 part i} We will start by proving the existence and uniqueness of such a minimal solution to \\eqref{eq constrained bsde value function} using \\cite[Theorem 2.1]{kharroubi_feynmankac_2015}, which tells us that this unique minimal solution to \\eqref{eq constrained bsde value function} should be the increasing limit of the solutions to the penalised BSDEs \\eqref{eq penalised bsde value function} obtained from \\cref{lemma penalised bsde minimal solution existence}. For this we view \\eqref{eq constrained bsde value function} and the penalised versions \\eqref{eq penalised bsde value function} as a BSDEs on $[0,T]$ with the driver $F(s,\\hat\\omega) \\coloneqq \\1_{[t,T]}(s) H_{s-}(\\hat\\omega)$, where $H_r \\coloneqq \\E^{\\hat\\P}[f(r,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_r}_{X^{t,\\xi,\\alpha_t}_r},X^{t,\\xi,\\alpha_t}_r,I^{t,\\alpha_t}_r) | \\c F^{B,\\mu,\\hat\\P}_r]$, and the terminal value $G \\coloneqq \\E^{\\hat\\P}[g(X^{t,\\xi,\\alpha_t}_T,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T}) | \\c F^{B,\\mu,\\hat\\P}_T]$. Then we note that \\cite[Assumption (H0)]{kharroubi_feynmankac_2015} holds: (ii) and (iii) are naturally satisfied as the generator $F$ does not depend on $Y,Z,U$, and (i) holds since $\\xi\\in L^2(\\hat\\Omega,\\c G\\lor\\c F^W_t,\\hat\\P;\\R^d)$ together with\\cref{assumptions reward functional} and \\eqref{eq randomised state dynamics basic estimate} show that\n\\begin{align}\n\\E^{\\hat\\P}\\Big[|G|^2 + \\int_0^T |F(s)|^2 ds\\Big] &\\leq \\E^{\\hat\\P}\\Big[|g(X^{t,\\xi,\\alpha_t}_T,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T})|^2 + \\int_t^T |f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s)|^2 ds\\Big] < \\infty.\n\\end{align}\nWe remark that \\cite[Assumption (H1)]{kharroubi_feynmankac_2015} is only needed to show the following estimate\n\\begin{align}\n\\sup_n \\E^{\\hat\\P}\\Big[\\sup_{s\\in [t,T]} |Y^{n,t,\\xi,\\alpha_t}_s|^2\\Big] < \\infty.\n\\label{eq theorem 6.2 H1 estimate}\n\\end{align}\nThus, instead of verifying \\cite[Assumption (H1)]{kharroubi_feynmankac_2015}, we will directly show that \\eqref{eq theorem 6.2 H1 estimate} holds. For this, we note that following the same arguments as \\cite[Lemma 4.9]{fuhrman_randomized_2015}, from the representation \\eqref{eq Y n t xi snell envelope formula} in \\cref{lemma penalised bsde minimal solution existence} together with \\cref{assumptions sde coefficients,assumptions reward functional}, we can show that\n\\[\n\\sup_{s\\in [t,T]} |Y^{n,t,\\xi,\\alpha_t}_s|^2 \\leq C\\Big( 1 + \\E^{\\hat\\P}\\Big[ \\sup_{s\\in [t,T]} |X^{t,\\xi,\\alpha_t}_s|^2 \\Big|\\c F^{B,\\mu,\\hat\\P}_T\\Big]\\Big),\\quad\\hat\\P\\text{-a.s.},\n\\]\nfor a constant $C$ independent of $n$, which together with \\eqref{eq randomised state dynamics basic estimate} implies \\eqref{eq theorem 6.2 H1 estimate}.\nThus by \\cite[Theorem 2.1]{kharroubi_feynmankac_2015}, the BSDE \\eqref{eq constrained bsde value function} has a unique minimal solution\n\\[\n(Y^{t,\\xi,\\alpha_t},Z^{t,\\xi,\\alpha_t},U^{t,\\xi,\\alpha_t},K^{t,\\xi,\\alpha_t})\\in \\c S^2_{[0,T]}(\\b F^{B,\\mu,\\hat\\P})\\times L^2_{W,[0,T]}(\\b F^{B,\\mu,\\hat\\P})\\times L^2_{\\lambda,[0,T]}(\\b F^{B,\\mu,\\hat\\P})\\times \\c K^2_{[0,T]}(\\b F^{B,\\mu,\\hat\\P}),\n\\]\nwhen viewed as BSDE on the whole time interval $[0,T]$, and which satisfies $Y^{n,t,\\xi,\\alpha_t}_s\\uparrow Y^{t,\\xi,\\alpha_t}_s$, for all $s\\in [t,T]$, $\\hat\\P$-a.s.\\@ and in $L^2(ds\\otimes\\hat\\P)$. Therefore, as a last step we need to show that also $K^{t,\\xi,\\alpha_t}_t = 0$ and thus $K^{t,\\xi,\\alpha_t}\\in \\c K^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})$, which we will postpone to \\cref{proof theorem 6.2 part iii} of the proof as it will follow from showing that $Y^{t,\\xi,\\alpha_t}_t$ is $\\hat\\P$-a.s.\\@ constant.\n\n\n\\item Regarding the representation \\eqref{eq recursive snell envelope formula}, we first note that by \\cref{lemma penalised bsde minimal solution existence}, we have the formula \\eqref{eq Y n t xi snell envelope formula} for $Y^{n,t,\\xi,\\alpha_t}$ for all $n\\in\\N$. Thus, using that $\\c V_n\\subseteq\\c V_{n+1}\\subseteq\\c V$ and $\\bigcup_{n\\in\\N} \\c V_n = \\c V$, we conclude from $Y^{n,t,\\xi,\\alpha_t}_s\\uparrow Y^{t,\\xi,\\alpha_t}_s$, for all $s\\in [t,T]$, $\\hat\\P$-a.s., by monotone convergence that \\eqref{eq recursive snell envelope formula} holds for $Y^{t,\\xi,\\alpha_t}$.\n\n\n\\item\\label{proof theorem 6.2 part iii} Next we will prove that $Y^{t,\\xi,\\alpha_t}_t$ is $\\hat\\P$-a.s.\\@ constant, and with it that $K^{t,\\xi,\\alpha_t}_t = 0$ and hence $K^{t,\\xi,\\alpha_t}\\in\\c K^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})$, which also completes the construction in \\cref{proof theorem 6.2 part i}. For this, we will show that $Y^{t,\\xi,\\alpha_t}_t = V(t,m)$, $\\hat\\P$-a.s.\n\nFirst, we note that \\eqref{eq recursive snell envelope formula} together with \\eqref{eq constrained bsde value function} implies that\n\\begin{align}\n Y^{t,\\xi,\\alpha_t}_t\n &= \\esssup_{\\nu\\in\\c V} \n \\E^{\\hat\\P^\\nu}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_t\\Big],\\ \\hat\\P\\text{-a.s.}\n \\label{eq Y t xi reward representation}\n\\end{align}\nNext, we observe that by \\cref{proposition decompose filtrations of predictable processes} we can decompose each $\\nu\\in\\c V$ into an $\\c F^{B,\\mu}_t\\otimes Pred(\\b F^{B^t,\\mu^t})\\otimes\\c B(\\c A_T)$-measurable process $\\Upsilon$ with $0<\\inf_{[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\times\\c A_T} \\Upsilon\\leq \\sup_{[0,T]\\times\\hat\\Omega\\times\\hat\\Omega\\times\\c A_T} \\Upsilon < \\infty$ such that $\\nu(\\hat\\omega) = \\Upsilon(\\hat\\omega,\\hat\\omega)$, for all $\\hat\\omega\\in\\hat\\Omega$. \nIn particular, we have $\\Upsilon(\\hat\\omega,\\cdot)\\in\\c V_t$ for every $\\hat\\omega\\in\\hat\\Omega$.\nNow recalling that $X^{t,\\xi,\\alpha_t}$ and $I^{t,\\alpha_t}$ are $\\hat\\P$-independent of $\\c F^{B,\\mu,\\hat\\P}_t$, we can apply the freezing lemma, see e.g.\\@ \\cite[Lemma 4.1]{baldi_stochastic_2017}, to rewrite the conditional expectation in \\eqref{eq Y t xi reward representation} as follows,\n\\begin{align}\n &\\E^{\\hat\\P^\\nu}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_t\\Big]\\\\\n &= \\E^{\\hat\\P}\\Big[\\frac{L^\\nu_T}{L^\\nu_t} \\Big(g(\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big)\\Big| \\c F^{B,\\mu,\\hat\\P}_t\\Big]\\\\\n &= \\E^{\\hat\\P}\\Big[\\frac{L^\\upsilon_T}{L^\\upsilon_t} \\Big(g(\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big)\\Big]\\bigg|_{\\upsilon = \\Upsilon(\\hat\\omega,\\cdot)}\\\\\n &= \\E^{\\hat\\P}\\Big[L^\\upsilon_T \\Big(g(\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\c F^{B,\\mu,\\hat\\P}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big)\\Big]\\bigg|_{\\upsilon = \\Upsilon(\\hat\\omega,\\cdot)}\\\\\n &= J^{\\c R}(t,\\xi,\\alpha_t,\\upsilon)\\big|_{\\upsilon = \\Upsilon(\\hat\\omega,\\cdot)},\\qquad\\hat\\P\\text{-a.s.},\n\\end{align}\nsince for every $\\hat\\omega\\in\\hat\\Omega$ we have $\\upsilon = \\Upsilon(\\hat\\omega,\\cdot)\\in\\c V_t$ and thus $\\frac{L^\\upsilon_T}{L^\\upsilon_t}$ is $\\hat\\P$-independent of $\\c F^{B,\\mu,\\hat\\P}_t$ together with $\\E^{\\hat\\P}[L^{\\upsilon}_t] = 1$ and $L^\\upsilon_t$ being $\\c F^{B,\\mu,\\hat\\P}_t$-measurable. Therefore by using \\eqref{eq Y t xi reward representation} together with \\cref{proposition equivalence v v_t randomised value function,theorem equivalence original and randomised value function} we obtain one the one hand\n\\begin{align}\n V(t,m) &= \\sup_{\\nu\\in\\c V_t} J^{\\c R}(t,\\xi,\\alpha_t,\\nu)\\\\\n &= \\sup_{\\nu\\in\\c V_t} \\E^{\\hat\\P^\\nu}\\Big[g(\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_T}_{X^{t,\\xi,\\alpha_t}_T},X^{t,\\xi,\\alpha_t}_T) + \\int_t^T f(s,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_s}_{X^{t,\\xi,\\alpha_t}_s},X^{t,\\xi,\\alpha_t}_s,I^{t,\\alpha_t}_s) ds\\Big| \\c F^{B,\\mu,\\hat\\P^\\nu}_t\\Big]\n \\leq \n Y^{t,\\xi,\\alpha_t}_t,\n\\end{align}\nand on the other hand also\n\\begin{align}\n Y^{t,\\xi,\\alpha_t}_t &= \\sup_{\\nu\\in\\c V} J^{\\c R}(t,\\xi,\\alpha_t,\\upsilon)\\big|_{\\upsilon = \\Upsilon(\\hat\\omega,\\cdot)} \\leq \\sup_{\\upsilon\\in\\c V_t} J^{\\c R}(t,\\xi,\\alpha_t,\\upsilon) = V(t,m).\n\\end{align}\nThis shows that $Y^{t,\\xi,\\alpha_t}_t = V(t,m)$, $\\hat\\P$-a.s., and in particular that $Y^{t,\\xi,\\alpha_t}_t$ is $\\hat\\P$-a.s.\\@ constant. Finally, we note that this also implies $K^{t,\\xi,\\alpha_t}_t = 0$, which shows that $K^{t,\\xi,\\alpha_t}\\in\\c K^2_{[t,T]}(\\b F^{B,\\mu,\\hat\\P})$.\n\n\n\\item\nFinally, let us prove the representation \\eqref{eq recursive snell envelope formula at s=t}.\nFrom the BSDE \\eqref{eq constrained bsde value function}, by taking the conditional expectation under $\\hat\\P^\\nu$, we obtain for all $r\\in [t,T]$ and $\\nu\\in\\c V$,\n\\[\nY^{t,\\xi,\\alpha_t}_t \\geq \\E^{\\hat\\P^\\nu}[Y^{t,\\xi,\\alpha_t}_t] \\geq \\E^{\\hat\\P^\\nu}\\Big[Y^{t,\\xi,\\alpha_t}_r + \\int_t^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\Big],\n\\]\nsince $U^{t,\\xi,\\alpha_t}$ is nonnegative and $K^{t,\\xi,\\alpha_t}$ is nondecreasing, and $Y^{t,\\xi,\\alpha_t}_t$ is $\\hat\\P$-a.s.\\@ constant. Taking the supremum over all $\\nu\\in\\c V$ then shows that\n\\[\nY^{t,\\xi,\\alpha_t}_t \\geq \\sup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[Y^{t,\\xi,\\alpha_t}_r + \\int_t^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\Big].\n\\]\n\nFor the other direction, we recall that by \\cref{lemma penalised bsde minimal solution existence}, we have the upper estimate \\eqref{eq Y n t xi snell envelope upper bound at s=t} for $Y^{n,t,\\xi,\\alpha_t}$, for all $n\\in\\N$. Thus, using that $\\c V_n\\subseteq\\c V_{n+1}\\subseteq\\c V$ and $\\bigcup_{n\\in\\N} \\c V_n = \\c V$, we deduce from $Y^{n,t,\\xi,\\alpha_t}_s\\uparrow Y^{t,\\xi,\\alpha_t}_s$, for all $s\\in [t,T]$, $\\hat\\P$-a.s., by monotone convergence that, for all $r\\in [t,T]$,\n\\[\nY^{t,\\xi,\\alpha_t}_t = \\E^{\\hat\\P}[Y^{t,\\xi,\\alpha_t}_t]\n\\leq \\sup_{\\nu\\in\\c V} \\E^{\\hat\\P^\\nu}\\Big[Y^{t,\\xi,\\alpha_t}_r + \\int_t^r f(u,\\hat\\P^{\\nu,\\c F^{B,\\mu,\\hat\\P^\\nu}_u}_{X^{t,\\xi,\\alpha_t}_u},X^{t,\\xi,\\alpha_t}_u,I^{t,\\alpha_t}_u) du\\Big].\n\\]\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\\let\\c\\oldc \n\\bibliographystyle{plain}\n\\bibliography{bibliography}\n\n\\end{document}\n\n\n\n\n\n"}}},{"rowIdx":295,"cells":{"id":{"kind":"string","value":"2412.20778v2"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20778v2"},"title":{"kind":"string","value":"An accurate approach to determining the spatiotemporal vehicle load on bridges based on measured boundary slopes"},"full_text":{"kind":"string","value":"\\documentclass[preprint,12pt]{elsarticle}\n\n\\usepackage{amssymb}\n\\usepackage{amsmath,amssymb,amsfonts}\n\\usepackage{tikz}\n\n\\usetikzlibrary{decorations.pathmorphing, patterns, decorations.markings}\n\n\\newtheorem{thm}{Theorem}\n\\newtheorem{Cor}{Corollary}\n\\newtheorem{Lem}{Lemma}\n\\newdefinition{rmk}{Remark}\n\\newproof{proof}{Proof}\n\\hoffset=-.6in\n\\voffset=-1.1in\n\\setlength{\\textheight}{9.3in}\n\\setlength{\\textwidth}{6.8in}\n\\journal{Applied Mathematical Modelling}\n\n\\begin{document}\n\n\\begin{frontmatter}\n\n\\title{An accurate approach to determining the spatiotemporal vehicle load on bridges based on\nmeasured boundary slopes}\n\n\\author[1]{Alemdar Hasanov\\fnref{fn1}}\n\\ead{alemdar.hasanoglu@gmail.com}\n\\author[2]{Onur Baysal\\corref{cor1}\n\\fnref{fn2}}\n\\ead{onur.baysal@um.edu.mt}\n\n\n\n\\cortext[cor1]{Corresponding author}\n\\fntext[fn1]{Emeritus Professor. \\c{S}ehit Ekrem Dsitrict, Altun\\c{s}ehir Str., Ayazma Villalari, No: 22. Bah\\c{c}ecik - Ba\\c{s}iskele, Kocaeli, 41030}\n\\fntext[fn2]{Senior Lecturer, Department of Mathematics, University of Malta, Malta}\n\n\\address[1]{Kocaeli University, 41001, \\.{I}zmit/Kocaeli, T\\\"{u}rkiye}\n\\address[2]{University of Malta, Msida, Malta}\n\n\\begin{abstract}\nIn this paper, a novel mathematical model is developed to evaluate the spatiotemporal vehicle loads on long bridges from slope measurements made at the ends of a bridge based on Euler-Bernoulli beam model with internal and external damping. The mathematical modelling of this phenomena leads to the inverse source problem of determining the spatiotemporal vehicle load $F(x,t)$ in the variable coefficient Euler-Bernoulli equation $\\rho_A(x)u_{tt}+\\mu(x) u_{t}+(r(x)u_{xx})_{xx}+(\\kappa(x)u_{xxt})_{xx}=F(x,t)$,\n$(x,t)\\in \\Omega_T:=(0,\\ell)\\times (0,T)$ subject to the \"simply supported\" boundary conditions $u(0,t)=(r(x)u_{xx}+(\\kappa(x)u_{xxt})_{x=0}=0$, $u(\\ell,t)=(r(x)u_{xx}+(\\kappa(x)u_{xxt})_{x=\\ell}=0$, from the both measured outputs: $\\theta_1(t):=u_x(0,t)$ and $\\theta_2(t):=u_x(\\ell,t)$, that is, the measured boundary slopes. It is shown that the input-output maps $(\\Phi F)(t):=u_x(0,t;F)$, $(\\Psi F)(t):=u_x(\\ell,t;F)$, $F \\in \\mathcal{F}\\subset L^2(\\Omega_T)$, corresponding to the inverse problem, are compact and Lipschitz continuous. Then Tikhonov functional $J(F)=\\Vert \\Phi F-\\theta_1 \\Vert_{L^2(0,T)}^2+\\Vert \\Psi F-\\theta_2 \\Vert_{L^2(0,T)}^2$ is introduced to prove the existence of a quasi-solution to the inverse problem. An explicit gradient formula for the Fr\\'{e}chet derivative of the Tikhonov functional is derived. The Lipschitz continuity of the Fr\\'{e}chet gradient, which guarantees the monotonicity of iterations in gradient methods, has been proven.\n\\end{abstract}\n\\begin{keyword}\nSpatiotemporal vehicle loads identification, damped Euler-Bernoulli beam, inverse source problem, solvability of the inverse problem, Fr\\'{e}chet gradient.\n\\end{keyword}\n\n\\end{frontmatter}\n\n\n\\section{Introduction}\n\nIncorrectly assessing vehicle-induced loads can lead to fatigue cracking or even bridge collapse. Furthermore, establishing relationships between these loads using experimentally measurable data is an important basis for bridge design, safety assessment, maintenance, and reinforcement. The statistical data obtained from these relationships can help us better understand the behavior of bridges under vehicle loads. Therefore, accurate and reliable estimation of vehicle loads is very important.\n\nThe dynamic effects of moving loads on bridges were not recognized until the mid-19th century. The analysis and history of moving load problems are described by Timoshenko in his book \\cite{Timoshenko:1953}. The monograph \\cite{Fryba:1972} was the first to give the analysis of numerous basic moving load problems and their analytical solutions. Later, these problems extensively were reviewed in the book \\cite{Yang:2004}. An overview, analysis and history of dynamic problems caused by moving loads are given in article \\cite{Ouyang:2011}. An isogeometric approach to dynamic analysis of beam constructions subjected to moving vehicles was proposed in \\cite{VanDo:2017}, based on the Euler-Bernoulli model.\n\nThe primary live loads on bridges are vehicle loads, which are critical parameters for bridge health monitoring. However, traditional Weigh-in-Motion (WIM) systems \\cite{Lydon:2016} require the installation of weighing devices embedded in the road surface, which requires traffic interruptions during installation. However, this process is time-consuming and costly, which prevents the method from being widely implemented. As a sequel, Bridge Weigh-in-Motion (BWIM) systems \\cite{Moses:1979}, developed in the 1970s, offer relatively easy installation and somewhat lower installation costs compared to traditional WIM systems. Over the last 40 years, BWIM systems have achieved a high level of research maturity, established technical methodologies, and high measurement accuracy. However, they still require special instrumentation and have complex installation requirements. Furthermore, the specificity of BWIM systems, where each bridge requires a special system, results in high initial installation and subsequent maintenance costs. This limits the deployment of dynamic weighing systems to only a few specific bridges.\n\nIn recent years, major developments in computer vision and image processing technologies have attracted a lot of attention in the field of civil engineering. In this context, the study of using computer vision to identify vehicle loads on bridges has also been proposed by various papers. In \\cite{Ojio:2016} the authors propose a method that uses a controller to simultaneously control the cameras on the bridge deck and underneath the bridge. The camera on the bridge deck can measure the\ndynamic displacement response data of the bridge data and receive the vehicle's axle information. They proved the applicability of employing computer vision to determine vehicle weight by identifying vehicle loads through the analysis of both sets of data. In \\cite{Martini:2022}, a computer vision-based approach to determine the tire loads of moving cars, the load places on the bridge, and the displacement response of the bridge was proposed. A bridge influence line model was then built in accordance with these findings. This technique made use of multiple cameras cooperating. By identifying tire types and getting tire pressure information from a database, the on-bridge cameras calculated vehicle loads. Vehicle load positions and the bridge displacement reaction were picked up by the under-bridge cameras. They created a bridge influence line model by integrating these data, offering a fresh method for computer vision-based bridge structure monitoring.\n\nThe use of computer vision for non-contact, target-free displacement measurements was discussed in detail in \\cite{Khuc:2018}. Here, an iterative approximation algorithm to construct the displacement unit influence surface (UIS) was proposed. This algorithm allows to estimate the equivalent moving loads on the bridge under multiple vehicle loads using camera data and computer vision algorithms.\n\nTo solve for vehicle load, a spatiotemporal connection model of structural displacement, vehicle load, and load distribution was developed in \\cite{Tang:2024}. To confirm the viability of the proposed method, engineering practice experiments and model bridge testing under varied loading scenarios are carried out. According to the results of the model bridge tests, the structural displacement determined by traffic video measurement can accurately represent the displacement characteristics of the structure.\n\nAs an alternative to the above methods and approaches, in this paper we propose a new mathematical model of vehicle-bridge interaction based on the damped Euler–Bernoulli beam, which takes into account all the main physical parameters, and a new solution algorithm based on weak solution theory for the forward problem and a quasi-solution approach combined with the adjoint method for the inverse problem. In this model, the angles at the ends of the bridge are used as measurable data. The advantage of this approach is that it allows for almost zero-cost load estimation on bridges equipped with traffic surveillance, meets more practical needs with extremely low monitoring costs, and promotes the widespread application of vehicle load estimation. In addition, unlike known methods, the error margin of this method in load estimation is below $10\\%$.\n\nThe rest of the paper is organized as follows. In Section 2, we describe the mathematical model of bridge-vehicle load interaction, and formulate the identification problem. Analysis of the weak solution of the corresponding forward problem with a'pripori estimates is discussed in Section 3. In Section 4, we introduce the input-output operator and prove some properties of this operator.\nThe corresponding Tikhonov functional is introduced in Section 5. The Fr\\'echet derivative of this functional through a suitable adjoint problem is derived in Section 6. In Section 7, the Lipschitz continuity of the Fr\\'{e}chet gradient is proved. \n\n\n\\section{The mathematical model of bridge-vehicle load interaction}\n\nWithin the Euler-Bernoulli damped beam model, the vibration of a simply supported long bridge under the spatiotemporal load $F(x,t)$ is described by the following mathematical model:\n\\begin{eqnarray}\\label{1}\n\\left\\{ \\begin{array}{ll}\n\\rho_A(x) u_{tt}+\\mu(x)u_{t}-(T(x)u_{x})_{x}+ (r(x)u_{xx}+\\kappa(x)u_{xxt})_{xx} =F(x,t),\\\\\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (x,t)\\in \\Omega_{T}:=(0,\\ell)\\times (0,T);\\\\ [2pt]\nu(x,0)=u_{t}(x,0)=0, ~x \\in (0,\\ell); \\\\ [2pt]\nu(0,t)=\\left(r(x)u_{xx}+\\kappa(x)u_{xxt}\\right)_{x=0}=0; \\\\\n\\qquad \\qquad \\qquad \\qquad u(\\ell,t)=\\left (r(x)u_{xx}+\\kappa(x)u_{xxt}\\right)_{x=\\ell}=0,~t \\in [0,T].\n\\end{array} \\right.\n\\end{eqnarray}\n\nHere and below, $u(x,t)$ is the transverse deflection in position $x\\in (0,\\ell)$ and at the time $t\\in \\times [0,T]$, while $T>0$ is the final time instance, which may be small enough, and $\\ell>0$ is the length of the beam. Further, $\\rho_A(x):=\\rho(x)A_s(x)$, while $\\rho(x)>0$ and $A_s(x)>0$ are the mass density and the cross-sectional area, $r(x):=E(x)I(x)>0$ is the flexural rigidity (or bending stiffness) of a nonhomogeneous beam while $E(x)>0$ is the elasticity modulus and $I(x)>0$ is the moment of inertia. $T_r(x)\\ge 0$ is the axial tensile force.\n\nThe external and internal damping mechanisms are given by the terms $\\mu(x)u_t$ and $(\\kappa(x)u_{xxt})_{xx}$, respectively. The coefficients $\\mu(x)\\ge 0$ and $\\kappa(x)>0$ are called the viscous (internal) damping and the strain-rate or Kelvin-Voigt damping coefficients, respectively. The coefficient $\\kappa(x):=c_d(x)I(x)$ represents energy dissipated by friction internal to the beam, while $c_d>0$ is the strain-rate damping coefficient.\n\nThe function $F(x,t)$ is the spatiotemporal load expressing the effect of the moving car on the bridge.\n\nFigure 1 represents the geometry of the problem (\\ref{1}).\n\nSuppose that the slopes $\\theta_0(t)$ and $\\theta_\\ell(t)$ at the ends $x=0$ and $x=\\ell$ of the bridge are given as measurable data:\n\\begin{eqnarray}\\label{2}\n\\theta_0(t):=u_x(0,t),~\\theta_\\ell(t):=u_x(\\ell,t),~ t \\in [0,T].\n\\end{eqnarray}\n\nWithin the framework of the mathematical model (\\ref{1})-(\\ref{2}), the problem of determining the unknown spatiotemporal load $F(x,t)$ based on these data, is defined as follows:\\\\\n\\emph{Find the unknown spatiotemporal load $F(x,t)$ in (\\ref{1}) from the measured slopes $\\theta_0(t)$ and $\\theta_\\ell(t)$ introduced in (\\ref{2}).}\n\nThe problem (\\ref{1})-(\\ref{2}) is defined as a spatiotemporal load identification problem or an inverse source problem with two Dirichlet measured outputs $\\theta_0(t)$ and $\\theta_\\ell(t)$, according to generally accepted terminology.\n\nFor a given function $F(x,t)$ from some class of admissible loads, the initial boundary value problem (\\ref{1}) will be referred as the \\emph{forward problem}.\n\n\n\\begin{figure}\n \\centering\n \\hspace*{0.6cm} \\begin{tikzpicture}[scale=.85]\n \\draw[color=black,ultra thick] (0.0,0.0) -- (-0.2,-0.2);\n \\draw[color=black,ultra thick] (0.0,0.0) -- (0.2,-0.2);\n \\draw[color=black,ultra thick] (-0.50,-0.20) -- (0.55,-0.20);\n \\draw[color=gray] (-0.50,-0.45) -- (-0.25,-0.20);\n \\draw[color=gray] (-0.25,-0.45) -- (0,-0.20);\n \\draw[color=gray] (0,-0.45) -- (0.25,-0.20);\n \\draw[color=black] (0.25,-0.45) -- (0.50,-0.20);\n \\draw [line width=0.2mm, black] (0.0,0.0) -- (3.2,0.8);\n \\draw [line width=0.2mm, black] (8.5,0.75) -- (12.7,0.0);\n \\node[label=right:{\\scriptsize $ \\theta_0(t):=u_x(0,t)$}] at (2.4,0.18) {};\n \\node[label=right:{\\scriptsize $ \\theta_\\ell(t):=u_x(\\ell,t)$}] at (8.0,0.18) {};\n \\draw[thin, ->] (2.7,0) arc (0:85:0.5);\n \\draw[thin, <-] (9.1,0.63) arc (0:100:-0.6);\n \\draw[dashed] (0,0) -- (4*3.14,0);\n \\draw[thick, ->] (5.0,1.1)-- (5.5,1.1);\n \\draw[thick, <-] (5.0,0.9)-- (5.5,0.9);\n \\draw[thick, ->] (7.8,1.1)-- (8.3,1.1);\n \\draw[thick, <-] (7.8,0.9)-- (8.3,0.9);\n \\node[label=right:{\\small$ F(x,t)$}] at (5.6,1.4) {};\n \\draw[thick, ->] (6.2,1.1)-- (6.2,0.78);\n \\draw[thick, ->] (6.5,1.1)-- (6.5,0.78);\n \\draw[thick, ->] (6.8,1.1)-- (6.8,0.78);\n \\draw[thick,->] (4*3.14,0) -- (14,0) node [right]{$x$};\n \\draw[thick,->] (0,-1.5) -- (0,1.75) node [left]{$u$};\n \\draw[color=black,ultra thick,smooth,domain=0:{4*3.14}] plot (\\x,{sin(0.25*deg(\\x))/1.5});\n \\node[label=right:{\\scriptsize$ u(0,t)=0$}] at (-.3,-0.7) {};\n \\node[label=right:{\\scriptsize$(r(x)u_{xx}+(\\kappa(x)u_{xxt})_{x=0}=0$}] at (-.3,-1.1) {};\n \\node[label=left:{\\scriptsize$u(\\ell,t)=0$}] at (12,-0.7) {};\n \\node[label=left:{\\scriptsize $(r(x)u_{xx}+(\\kappa(x)u_{xxt})_{x=\\ell}=0$}] at (14.5,-1.1) {};\n \\draw[color=black,ultra thick] (12.5,0.0) -- (-0.2+12.5,-0.2);\n \\draw[color=black,ultra thick] (12.5,0.0) -- (0.2+12.5,-0.2);\n \\draw[color=black,ultra thick] (12.0,-0.20) -- (13,-0.20);\n \\draw[color=gray] (12.5-0.50,-0.45) -- (12.5-0.25,-0.20);\n \\draw[color=gray] (12.5-0.25,-0.45) -- (12.5,-0.20);\n \\draw[color=gray] (12.5,-0.45) -- (12.5+0.25,-0.20);\n \\draw[color=black] (12.5+0.25,-0.45) -- (12.5+0.50,-0.20);\n \\end{tikzpicture}\n \\caption{Geometry of the spatiotemporal vehicle load on a long bridge based on the damped Euler-Bernoulli beam model}\n\\label{Fig-1}\n \\end{figure}\n\n\n\\section{Analysis of the forward problem and a'pripori estimates}\n\nWe assume that the inputs and outputs in (\\ref{1}) and (\\ref{2}) satisfy the following physically justified basic conditions with a minimum requirement of smoothness:\n\\begin{eqnarray} \\label{3}\n\\left \\{ \\begin{array}{ll}\n\\rho_A, \\mu, T_r, r, \\kappa \\in L^{\\infty}(0,\\ell),~F \\in L^2(\\Omega_T),\\\\ [1pt]\n\\theta_0,\\theta_\\ell \\in L^2(0,T),\\\\ [1pt]\n0 < \\rho_0 \\le \\rho_A (x)\\le \\rho_1,~0\\le T_{r0} \\le T_r(x) \\le T_{r1},~0< r_0 \\le r(x) \\le r_1, \\\\ [1pt]\n0 \\le \\mu_0\\le \\mu(x)\\le \\mu_1,~0 < \\kappa_0 \\le \\kappa(x)\\le \\kappa_1,~x \\in (0,\\ell).\n\\end{array} \\right.\n\\end{eqnarray}\n\nWe use the weak solution theory developed in \\cite{Baysal:2019} and \\cite{Hasanov:Romanov:2021} to derive some a'priori estimates for the weak solution Euler-Bernoulli beam equation subject to clamped boundary conditions. For cantilever beams with Kelvin-Voigt damping, similar estimates have been derived in \\cite{Sakthivel:2024} .\n\n\\begin{thm}\\label{Theorem-1}\nAssume that the basic conditions (\\ref{3}) hold. Then for the weak solution\n$u\\in L^2(0,T;\\mathcal{V}^2(0,\\ell))$ with $u_t\\in L^2(0,T;L^2(0,\\ell))$, $u_{tt}\\in L^2(0,T;H^{-2}(0,\\ell))$ of the forward problem (\\ref{1}), the following estimates are satisfied:\n\\begin{eqnarray}\\label{4}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert u_t \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{1}{\\rho_0}\\,C_e^2\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}, \\\\ [10pt]\n\\Vert u_t \\Vert^2_{L^2(0,T;L^2(0,\\ell))} \\leq (C_e^2 -1) \\,\\Vert F \\Vert^2_{L^2(\\Omega_T)},\\\\ [12pt]\n\\displaystyle \\Vert u_{xx} \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{1}{r_0}\\,C_e^2\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}, \\\\ [10pt]\n\\displaystyle \\Vert u_{xx} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\rho_0}{r_0}\\,(C_e^2 -1)\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}, \\\\ [13pt]\n\\displaystyle \\Vert u_{xxt} \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{1}{\\kappa_0}\\,C_e^2\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}, \\\\ [10pt]\n\\displaystyle \\Vert u_{xxt} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\rho_0}{\\kappa_0}\\,(C_e^2 -1)\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)},\n\\end{array} \\right.\n\\end{eqnarray}\nwhere $\\mathcal{V}^2(0,\\ell):=\\{v\\in H^2(0,\\ell):~ v(0)=v(\\ell)=0\\}$,\n\\begin{eqnarray}\\label{5}\nC_e^2=\\exp(T/\\rho_0),\n\\end{eqnarray}\nand $\\rho_0,r_0>0$ are the constants introduced in (\\ref{3}).\n\\end{thm}\n{\\bf Proof.} Utilize the following identities\n\\begin{eqnarray}\\label{6}\n\\left. \\begin{array}{ll}\n\\displaystyle 2\\int_0^t \\int_0^\\ell \\left (r(x)u_{xx}\\right )_{xx} u_{\\tau} dx d\\tau \\\\ [6pt]\n\\qquad = \\displaystyle 2\\int_0^t \\int_0^\\ell \\left \\{\\left [(r(x)u_{xx})_x u_{\\tau}-r(x)u_{xx} u_{x\\tau}\\right ]_x+\\frac{1}{2}\\,\\left (r(x)u_{xx}^2\\right )_{\\tau} \\right \\} dx d\\tau,\\\\ [16pt]\n\\displaystyle 2\\int_0^t \\int_0^\\ell \\left (\\kappa(x)u_{xx\\tau}\\right)_{xx} u_{\\tau} dx d\\tau \\\\ [6pt]\n\\qquad = \\displaystyle 2\\int_0^t \\int_0^\\ell \\left \\{ \\left [(\\kappa (x)u_{xx\\tau})_x u_{\\tau}\n-\\kappa(x)u_{xx\\tau} u_{x\\tau}\\right ]_x+\\kappa (x)u^2_{xx\\tau} \\right \\}dx d\\tau,\n\\end{array} \\right.\n\\end{eqnarray}\nafter multiplying both sides of equation (\\ref{1}) by $2u_t(x,t)$, integrating the result over $\\Omega_t:=(0,\\ell)\\times (0,t)$, $t\\in (0,T)$. In view of the homogeneous initial and boundary conditions, next we get the following \\emph{energy identity}:\n\\begin{eqnarray}\\label{7}\n\\int_0^\\ell \\left [\\rho_A(x) u_t^2 +r(x)u_{xx}^2+\\kappa(x)u_{xx\\tau}^2 +T_r(x)u_x^2\\right ]dx \\qquad \\qquad \\qquad \\quad\n \\nonumber \\\\ [1pt]\n\\qquad \\qquad +2 \\int_0^t \\int_0^\\ell \\mu (x)u_{\\tau}^2 dx d \\tau = 2\\int_0^t \\int_0^\\ell F(x,\\tau) u_{\\tau}dx d \\tau,~t\\in[0,T].\n\\end{eqnarray}\nThe following main integral inequality is derived from the identity (\\ref{7}) under the basic conditions (3):\n\\begin{eqnarray}\\label{8}\n\\rho_0 \\int_0^\\ell u_t^2dx +r_0 \\int_0^\\ell u_{xx}^2dx+\\kappa_0 \\int_0^\\ell u_{xx\\tau}^2 dx\n+T_{r0} \\int_0^\\ell u_x^2dx \\qquad \\qquad \\nonumber \\\\ [1pt]\n\\qquad +2 \\int_0^t \\int_0^\\ell \\mu (x) u_{\\tau}^2 dx d \\tau \\le\n \\int_0^t \\int_0^\\ell u^2_{\\tau}dx d \\tau +\\int_0^t \\int_0^\\ell F^2(x,\\tau)dx d \\tau,~\n\\end{eqnarray}\nfor all $t\\in [0,T]$.\n\nThe first consequence of (\\ref{8}) is the following inequality:\n\\begin{eqnarray*}\n\\rho_0 \\int_0^\\ell u_t^2dx \\le \\int_0^t \\int_0^\\ell u^2_{\\tau}dx d \\tau +\\int_0^t \\int_0^\\ell F^2(x,\\tau)dx d \\tau,~t\\in [0,T].\n\\end{eqnarray*}\nHere the Gr\\\"onwall-Bellman inequality is used to get:\n\\begin{eqnarray}\\label{9}\n\\displaystyle \\int_0^\\ell u_t^2dx \\le \\frac{1}{\\rho_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}\n\\exp(t/\\rho_0),~t\\in [0,T].\n\\end{eqnarray}\nThe first two estimates in (\\ref{4}) are derived from this inequality.\n\nAs the second consequence of the main integral inequality (\\ref{8}), we deduce that\n\\begin{eqnarray*}\nr_0 \\int_0^\\ell u_{xx}^2dx \\le\n \\int_0^t \\int_0^\\ell u^2_{\\tau}dx d \\tau +\\int_0^t \\int_0^\\ell F^2(x,\\tau)dx d \\tau,~\n\\end{eqnarray*}\nWith the estimate\n\\begin{eqnarray}\\label{10}\n\\displaystyle \\int_0^\\tau \\int_0^\\ell u_t^2dx \\le \\Vert F \\Vert^2_{L^2(\\Omega_T)}\n\\left [\\exp(t/\\rho_0)-1\\right ],~t\\in [0,T].\n\\end{eqnarray}\nwhich is obtained by integration of the inequality (\\ref{9}), this inequality yields:\n\\begin{eqnarray*}\nr_0 \\int_0^\\ell u_{xx}^2dx \\le \\Vert F \\Vert^2_{L^2(\\Omega_T)}\\exp(t/\\rho_0),~t\\in [0,T].\n\\end{eqnarray*}\nThis inequality readily yields the third and fourth estimations in (\\ref{4}).\n\nFinally, we use estimate (\\ref{10}) with the third consequence\n\\begin{eqnarray*}\n\\kappa_0 \\int_0^\\ell u_{xx\\tau}^2 dx \\le\n \\int_0^t \\int_0^\\ell u^2_{\\tau}dx d \\tau +\\int_0^t \\int_0^\\ell F^2(x,\\tau)dx d \\tau\n\\end{eqnarray*}\nof the main integral inequality to obtain the following estimate:\n\\begin{eqnarray*}\n\\kappa_0 \\int_0^\\ell u_{xx\\tau}^2 dx \\le\n \\Vert F \\Vert^2_{L^2(\\Omega_T)}\\exp(t/\\rho_0),~t\\in [0,T].\n\\end{eqnarray*}\nThe fifth and sixth estimates in (\\ref{4}) are easily obtained from this inequality.\n \\hfill$\\Box$\n\n\\begin{rmk}\\label{Remark-1}\nThis theorem, which obviously generalizes Theorem 11.1.2 of \\cite{Hasanov:Romanov:2021} for the case when the Kelvin-Voigt damping coefficient $\\kappa(x)$ is in the equation, allows us to obtain important trace estimates. In the case when $\\kappa(x)$, the fifth and sixth estimates in (\\ref{4}) can be obtained only for the regular weak solution of the forward problem (\\ref{1}). Therefore, these considerations can also be interpreted as the Kelvin-Voigt damping coefficient increasing the regularity of the weak solution \\cite{Sakthivel:2024}.\n\\end{rmk}\n\n\\begin{Cor}\\label{Corollary-1}\nIf conditions of Theorem \\ref{Theorem-1} are satisfied, then the following trace estimates are true:\n\\begin{eqnarray}\\label{11}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert u_{x}(0,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{r_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}, \\\\ [10pt]\n\\displaystyle \\Vert u_{xt}(0,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{\\kappa_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)},\\\\ [12pt]\n\\displaystyle \\Vert u_{x}(\\ell,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{r_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}, \\\\ [10pt]\n\\displaystyle \\Vert u_{xt}(\\ell,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{\\kappa_0}\\,\\Vert F \\Vert^2_{L^2(\\Omega_T)},\n\\end{array} \\right.\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{12}\n\\displaystyle C_1^2=\\frac{5\\ell \\rho_0}{3}\\,(C_e^2 -1),\n\\end{eqnarray}\nand $C_e>0$ is the constant introduced in (\\ref{5}).\n\\end{Cor}\n{\\bf Proof.} With the aid of Rolle's theorem, we have the following inequality:\n\\begin{eqnarray}\\label{13}\n\\displaystyle \\int_0^\\ell v_x^2(x) dx \\le \\frac{\\ell^2}{2} \\int_0^\\ell v_{xx}^2(x) dx,\n\\end{eqnarray}\nfor a function $v \\in H^2(0,\\ell)$ satisfying the conditions $v(0)=v(l)=0$.\n\nOn the other hand, the identity\n\\begin{eqnarray*}\n\\displaystyle u_x(0,t)=- \\frac{1}{\\ell} \\int_0^\\ell \\left (\\ell-x)u_x(x,t) \\right )_{x}dx\n\\end{eqnarray*}\nfor the weak solution of the forward problem (\\ref{1}) implies:\n\\begin{eqnarray*}\n\\displaystyle u_x(0,t) \\le \\frac{2}{\\ell^2} \\left (\\int_0^\\ell u_x(x,t) dx\\right )^2+\n\\frac{2}{\\ell^2} \\left (\\int_0^\\ell (\\ell-x)u_{xx}(x,t)dx\\right )^2 \\\\ [8pt]\n\\displaystyle \\qquad \\qquad \\le \\frac{2}{\\ell} \\int_0^\\ell u^2_x(x,t) dx+\n\\frac{2\\ell}{3} \\int_0^\\ell u^2_{xx}(x,t)dx.\n\\end{eqnarray*}\nHence,\n\\begin{eqnarray*}\n\\displaystyle \\int_0^T u^2_x(0,t) dx \\le \\frac{2}{\\ell} \\int_0^T \\int_0^\\ell u^2_x(x,t) dx dt+\n\\frac{2\\ell}{3} \\int_0^T \\int_0^\\ell u^2_{xx}(x,t)dx dt.\n\\end{eqnarray*}\nIn view of (\\ref{13}), applied to the weak solution of the forward problem (\\ref{1}), this yields:\n\\begin{eqnarray}\\label{14}\n\\displaystyle \\int_0^T u^2_x(0,t) dx \\le \\frac{5 \\ell}{3} \\int_0^T \\int_0^\\ell u^2_{xx}(x,t)dx.\n\\end{eqnarray}\nIn the same way, we deduce that\n\\begin{eqnarray}\\label{15}\n\\displaystyle \\int_0^T u^2_{xt}(0,t) dx \\le \\frac{5 \\ell}{3} \\int_0^T \\int_0^\\ell u^2_{xxt}(x,t)dx.\n\\end{eqnarray}\n\nIt is evident that the similar to (\\ref{14}) and (\\ref{15}) estimations are valid also for the norms $\\Vert u_{x}(\\ell,\\,\\cdot) \\Vert^2_{L^2(0,T)}$ and $\\Vert u_{xt}(\\ell,\\,\\cdot) \\Vert^2_{L^2(0,T)}$, respectively. \n\nThe desired estimations (\\ref{11}) are obtained if we take into account the third, fourth, fifth and sixth estimates in (\\ref{4}) in inequalities (\\ref{14}) and (\\ref{15}) above. \\hfill$\\Box$\n\n\\section{The input-output operators}\n\nWe define the \\emph{set of admissible loads (inputs)} as follows:\n\\begin{eqnarray}\\label{16}\n\\mathcal{F}:=\\{F \\in L^2(\\Omega_T):\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)} \\leq C_F\\}.\n\\end{eqnarray}\nwhere $C_F>0$ independent on $F(x,t)$ constant.\n\nLet $F \\in \\mathcal{F}$ be given. Denote by $u(x,t;F)$ the unique weak solution of the\n the forward problem (\\ref{1}) corresponding to this input. Then $u_x(0,t;F)$ and $u_x(\\ell,t;F)$ are the outputs, and we introduce the input-output operators as follows:\n\\begin{eqnarray}\\label{17}\n\\left. \\begin{array}{ll}\n\\Phi_0(F)(t):=u_x(0,t;F),~F \\in \\mathcal{F},\\, t\\in [0,T], \\\\ [6pt]\n\\Phi_0: \\mathcal{F} \\subset L^2(\\Omega_T) \\mapsto L^2(0,T); \\\\[10pt]\n\\Phi_\\ell(F)(t):=u_x(\\ell,t;F),~F \\in \\mathcal{F},\\, t\\in [0,T], \\\\ [6pt]\n\\Phi_\\ell : \\mathcal{F} \\subset L^2(\\Omega_T) \\mapsto L^2(0,T).\n\\end{array} \\right.\n\\end{eqnarray}\nIn view of these operators the inverse source problem (\\ref{1})-(\\ref{2}) can be reformulated as the following system of operator equations:\n \\begin{eqnarray}\\label{18}\n\\left \\{ \\begin{array}{ll}\n\\Phi_0(F)(t)=\\theta_0(t),~\\theta_0 \\in L^2(0,T), \\\\ [6pt]\n\\Phi_\\ell(F)(t)=\\theta_\\ell(t),~\\theta_\\ell \\in L^2(0,T),~F \\in \\mathcal{F},\\, t\\in [0,T].\n\\end{array} \\right.\n\\end{eqnarray}\n\nThe following lemma will make it possible for us to assert the ill-posedness of the considered inverse problem.\n\n\\begin{Lem}\\label{Lemma-1}\nAssume that conditions of Theorem \\ref{Theorem-1} are satisfied. Then the input-output operators introduced in (\\ref{17}) are compact.\n\\end{Lem}\n{\\bf Proof.} Let $\\{F^{(m)} \\}\\subset \\mathcal{F}$, $m=1,2,3,\\, ...$, be a sequence of admissible loads. Denote by $\\{\\Phi_0(F^{(m)}) \\}, \\{\\Phi_\\ell(F^{(m)}) \\}\\subset \\mathcal{F}$ the sequences of outputs: $\\Phi_0(F^{(m)})(t)=u_x(0,t;F^{(m)})$, $\\Phi_\\ell(F^{(m)})(t)=u_x(\\ell,t;F^{(m)})$.\nFrom the estimates (\\ref{11}), it follows that the norms are bounded in the norm of the Sobolev space $H^1(0,T)$, and hence compact in $L^2(0,T)$. This means that the input-output operators transform a bounded in $L^2(0,T)$ set $\\{F^{(m)} \\}$, $m=1,2,3,\\, ...$, to the compact in $L^2(0,T)$ sets $\\{\\Phi_0(F^{(m)}) \\}$, and $\\{\\Phi_\\ell(F^{(m)}) \\}$. Hence, these operators are compact. \\hfill$\\Box$\n\n\\begin{rmk}\\label{Remark-2}\nHere, unlike Lemma 11.2.1 of \\cite{Hasanov:Romanov:2021}, the compactness property is obtained without assuming the existence of a \\emph{regular }weak solution, thanks to the presence of the Kelving-Voigh damping term in the mathematical model (\\ref{1}). This feature was first found in the study \\cite{Sakthivel:2024}.\n\\end{rmk}\n\nFrom Lemma \\ref{Lemma-1}, it follows, in particular, that the \\emph{inverse problem (\\ref{1})-(\\ref{2}) is ill-posed}.\n\n\\begin{Lem}\\label{Lemma-2}\nUnder the conditions of Theorem \\ref{Theorem-1}, the input-output operators (\\ref{17}) are Lipschitz continuous, that is,\n\\begin{eqnarray}\\label{19}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert \\Phi_0(F_1)-\\Phi_0(F_2) \\Vert_{L^2(0,T)}\n\\leq C_L \\Vert F_1-F_2 \\Vert_{L^2(\\Omega_T)}, \\\\ [8pt]\n\\displaystyle \\Vert \\Phi_\\ell(F_1)-\\Phi_\\ell(F_2) \\Vert_{L^2(0,T)}\n\\leq C_L \\Vert F_1-F_2 \\Vert_{L^2(\\Omega_T)}, ~\\forall F_1,F_2 \\in \\mathcal{F},\n\\end{array} \\right.\n\\end{eqnarray}\nwhere $C_L=C_1/\\sqrt{r_0} >0$ is the Lipschitz constant, and $C_1>0$ is the constant introduced in (\\ref{12}).\n\\end{Lem}\n{\\bf Proof.} Denote by $\\delta u(x,t):=u(x,t;F_1) -u(x,t;F_2)$ the weak solution of the forward problem (\\ref{1}) with $F(x,t)$ replaced by $\\delta F(x,t):=F_1(x,t)-F_2(x,t)$, $F_1,F_2 \\in \\mathcal{F}$. Then, from estimates (\\ref{11}) it follows that\n\\begin{eqnarray*}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert \\delta u_{x}(0,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{r_0}\\, \\Vert \\delta F \\Vert^2_{L^2(\\Omega_T)}, \\\\ [10pt]\n\\displaystyle \\Vert \\delta u_{x}(\\ell,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{r_0}\\, \\Vert \\delta F \\Vert^2_{L^2(\\Omega_T)}.\n\\end{array} \\right.\n\\end{eqnarray*}\nBy definition of the input-output operators, $\\delta u_{x}(0,t)=\\Phi_0(F_1)-\\Phi_0(F_2)$, $\\delta u_{x}(\\ell,t)=\\Phi_\\ell (F_1)-\\Phi_\\ell (F_2)$, and\nthese estimates lead to the required statements (\\ref{19}). \\hfill$\\Box$\n\nWe will show that the property of the input-output operators yields the Lipschitz continuity of the Tikhonov functional, which will be introduced below.\n\n\\section{The Tikhonov functional and existence of a quasi-solution}\n\nDue to random noise in the measured outputs $\\theta_0(t)$ and $\\theta_\\ell(t)$, it is evident that exact equalities in the system of equations (\\ref{18}) are not achievable in practice. As a consequence, one needs to introduce the Tikhonov functional\n\\begin{eqnarray}\\label{20}\n\\displaystyle J(F):=\\frac{1}{2} \\int_0^T \\left [\\Phi_0(F)(t) -\\theta_0(t) \\right ]^2dt\n+ \\frac{1}{2} \\int_0^T \\left [\\Phi_\\ell(F)(t) -\\theta_\\ell(t) \\right ]^2dt \\quad \\nonumber \\\\ [6pt]\n\\qquad \\equiv \\frac{1}{2} \\int_0^T \\left [u_x(0,t;F) -\\theta_0(t) \\right ]^2dt\n+ \\frac{1}{2} \\int_0^T \\left [u_x(\\ell,t;F) -\\theta_\\ell(t) \\right ]^2dt,\n\\end{eqnarray}\n$F \\in \\mathcal{F}$, and consider the following minimization problem for this functional:\n \\begin{eqnarray}\\label{21}\nJ(F_*)=\\inf_{F \\in \\mathcal{F}} J(F).\n\\end{eqnarray}\n\nA solution of the minimization problem (\\ref{21}) is called a quasi-solution of the inverse source problem (\\ref{1})-(\\ref{2}).\n\n\\begin{Lem}\\label{Lemma-3}\nLet conditions of Theorem \\ref{Theorem-1} hold. Then the Tikhonov functional (\\ref{20}) is Lipschitz continuous:\n\\begin{eqnarray}\\label{22}\n\\displaystyle \\vert J(F_1)-J(F_2) \\vert_{L^2(0,T)}\n\\leq C_J \\Vert F_1-F_2 \\Vert_{L^2(\\Omega_T)}, ~\\forall F_1,F_2 \\in \\mathcal{F},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n\\displaystyle C_J=\\left [\\frac{2C_1 C_F}{\\sqrt{r_0}} +\\Vert \\theta_0 \\Vert_{L^2(0,T)}+\\Vert \\theta_\\ell \\Vert_{L^2(0,T)}\\right ] C_L\\,,\n\\end{eqnarray*}\nis the Lipschitz constant, $C_1>0$ and $C_F>0$ are the constants introduced in (\\ref{12}) and (\\ref{16}), respectively.\n\\end{Lem}\n{\\bf Proof.} We can easily prove that\n\\begin{eqnarray*}\n\\displaystyle \\vert J(F_1)-J(F_2) \\vert_{L^2(0,T)}\\quad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\\\ [8pt]\n\\qquad \\leq \\frac{1}{2} \\left [ \\Vert \\Phi_0(F_1) \\Vert_{L^2(0,T)}\n+\\Vert \\Phi_0(F_2) \\Vert_{L^2(0,T)} +2\\Vert \\theta_0 \\Vert_{L^2(0,T)}\\right ]\\Vert \\delta \\Phi_0(F)\\Vert_{L^2(0,T)} \\\\ [8pt]\n \\qquad +\\displaystyle \\frac{1}{2} \\left [ \\Vert \\Phi_\\ell(F_1) \\Vert_{L^2(0,T)}\n+\\Vert \\Phi_\\ell(F_2) \\Vert_{L^2(0,T)} +2\\Vert \\theta_\\ell \\Vert_{L^2(0,T)}\\right ] \\Vert \\delta \\Phi_\\ell(F)\\Vert_{L^2(0,T)},\n\\end{eqnarray*}\nfor all $F_1,F_2 \\in \\mathcal{F}$, where $\\Vert \\delta \\Phi_0(F)\\Vert_{L^2(0,T)}= \\Vert \\Phi_0(F_1)-\\Phi_0(F_2) \\Vert_{L^2(0,T)}$ and $\\Vert \\delta \\Phi_\\ell(F)\\Vert_{L^2(0,T)}= \\Vert \\Phi_\\ell(F_1)-\\Phi_\\ell(F_2) \\Vert_{L^2(0,T)}$, respectively. Using here (\\ref{11}) and (\\ref{16}) we obtain:\n\\begin{eqnarray*}\n\\displaystyle \\vert J(F_1)-J(F_2) \\vert_{L^2(0,T)}\\quad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\\\ [8pt]\n\\qquad \\leq \\left [\\frac{C_1 C_F}{\\sqrt{r_0}}+ \\Vert \\theta_0\\Vert_{L^2(0,T)}\\right ]\\Vert \\delta \\Phi_0(F)\\Vert_{L^2(0,T)} \\\\ [8pt]\n \\qquad +\\displaystyle \\left [\\frac{C_1 C_F}{\\sqrt{r_0}}+\n \\Vert \\theta_\\ell \\Vert_{L^2(0,T)}\\right ] \\Vert \\delta \\Phi_\\ell(F)\\Vert_{L^2(0,T)},\n\\end{eqnarray*}\n\nWith the continuity property of the input-output operators, this leads to the required result (\\ref{22}). \\hfill$\\Box$\n\n\\begin{thm}\\label{Theorem-2}\nAssume that the basic conditions (\\ref{3}) hold, and $F \\in \\mathcal{F}$, where $\\mathcal{F}$ is the set of admissible loads. Then the minimization problem (\\ref{21}) for the Tikhonov functional has at least one solution in $\\mathcal{F}$, that is the inverse source problem -(\\ref{1})-(\\ref{2}) has a quasi-solution.\n\\end{thm}\n\nThe proof of this theorem follows the same procedure as that of Theorem 10.1.11 of (\\ref{21}) \\cite{Hasanov:Romanov:2021}.\n\n\\section{Fr\\'echet derivative of the Tikhonov functional}\n\nLet $F, F+\\delta F \\in \\mathcal{F}$. Denote by $\\delta J(F):= J(F+\\delta F )-J(F)$ the increment of the Tikhonov functional introduced in (\\ref{20}). Then,\n\\begin{eqnarray}\\label{23}\n\\displaystyle \\delta J(F)= \\int_0^T \\left [u_x(0,t;F) -\\theta_0(t) \\right ]\\delta u_x(0,t;F) dt +\\frac{1}{2} \\int_0^T \\left (\\delta u_x(0,t;F) \\right )^2 dt \\nonumber \\\\ [8pt]\n\\quad \\displaystyle + \\int_0^T \\left [u_x(\\ell,t;F) -\\theta_\\ell(t) \\right ] \\delta u_x(\\ell,t;F) dt +\\frac{1}{2} \\int_0^T \\left (\\delta u_x(\\ell,t;F) \\right )^2 dt,~~\n\\end{eqnarray}\n\n\\begin{Lem}\\label{Lemma-4}\nLet conditions the basic conditions (\\ref{3}) hold. Then the following integral relationship\nholds:\n\\begin{eqnarray}\\label{24}\n\\displaystyle \\int_0^T p(t)\\delta u_x(0,t)dt +\\int_0^T q(t)\\delta u_x(\\ell,t)dt\n= \\int_0^T \\int_0^\\ell \\delta F(x,t) \\varphi (x,t) dx dt,\n\\end{eqnarray}\nfor all $p,q \\in L^2(0,T)$, where the function $\\varphi (x,t)$ is the weak solution of the following backward problem:\n\\begin{eqnarray}\\label{25}\n\\left\\{ \\begin{array}{ll}\n\\rho_A(x) \\varphi_{tt}-\\mu(x)\\varphi_{t}-(T(x)\\varphi_{x})_{x}+ (r(x)\\varphi_{xx}-\\kappa(x)\\varphi_{xxt})_{xx} =0,\\\\ [2pt]\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (x,t)\\in \\Omega_{T}:=(0,\\ell)\\times (0,T);\\\\ [4pt]\n\\varphi(x,T)=\\varphi_{t}(x,T)=0, ~x \\in (0,\\ell); \\\\ [4pt]\n\\varphi(0,t)=0,~-\\left(r(x)\\varphi_{xx}-\\kappa(x)\\varphi_{xxt}\\right)_{x=0}=p(t); \\\\ [2pt]\n\\qquad \\qquad \\varphi(\\ell,t)=0,~\\left (r(x)\\varphi_{xx}-\\kappa(x)\\varphi_{xxt}\\right)_{x=\\ell}=q(t),~t \\in [0,T],\n\\end{array} \\right.\n\\end{eqnarray}\nwith the inputs $p,q \\in L^2(0,T)$, and $\\delta u(x,t)$ is the weak solution of the forward problem (\\ref{1}) with $F(x,t)$ replaced by $\\delta F(x,t)$.\n\\end{Lem}\n{\\bf Proof.} Apply the integration by parts formula several times after multiplying both sides of equation (\\ref{1}) for $ \\delta u (x,t)$ by an arbitrary function $\\varphi(x,t)$ and integrating over $(0,T)$. Next, we get:\n\\begin{eqnarray*}\n\\int_0^T \\int_0^{\\ell} \\left [\\rho_A(x)\\varphi_{tt}-\\mu(x)\\varphi_{t}+\\left (r(x)\\varphi_{xx}- \\kappa (x)\\varphi_{xxt} \\right)_{xx}- \\left (T_r(x)\\varphi_{x}\\right)_{x}\\right] \\delta u \\,dx dt ~\\nonumber \\\\ [2pt]\n\\quad + \\int_0^\\ell \\left [\\rho_A(x) \\delta u_t \\varphi-\\rho_A(x) \\delta u \\varphi_t +\\mu(x) \\delta u \\varphi - \\delta u \\left(\\kappa(x)\\varphi_{xx}\\right)_{xx}\\right ]_{t=0}^{t=T} \\,dx \\nonumber \\\\ [2pt]\n~~ + \\int_0^T \\left [\\left(r(x) \\delta u_{xx}\\right)_{x}\\varphi-r(x)\\delta u_{xx} \\varphi_{x}+\nr(x)\\delta u_{x} \\varphi_{xx}-\\delta u \\left (r(x)\\varphi_{xx}\\right)_{x} \\right ]_{x=0}^{x=\\ell} \\,dt \\nonumber \\\\ [2pt]\n~+ \\int_0^T \\left [\\left(\\kappa(x) \\delta u_{xxt}\\right)_{x}\\varphi-\\kappa(x)\\delta u_{xxt} \\varphi_{x}-\n\\kappa(x)\\delta u_{x} \\varphi_{xxt}+\\delta u \\left (\\kappa(x)\\varphi_{xxt}\\right)_{x} \\right ]_{x=0}^{x=\\ell} \\,dt \\nonumber \\\\ [2pt]\n\\quad + \\int_0^T \\left [T_r(x) \\varphi_x \\delta u-T_r(x)\\varphi \\delta u_{x} \\right ]_{x=0}^{x=\\ell} \\,dt = \\int_0^T \\int_0^\\ell \\delta F(x,t) \\varphi (x,t) dx dt.\n\\end{eqnarray*}\nHere, we assume that the function $\\varphi(x,t)$ solves the backward problem (\\ref{25}).\nIn view of the homogeneous boundary, initial, and final conditions in (\\ref{1}) and (\\ref{25}), the required integral relationship (\\ref{24}) is then obtained. \\hfill$\\Box$\n\nConsidering the difference between the outputs $u_x(0,t;F)$ and $u_x(\\ell,t;F)$, corresponding to the admissible $F \\in \\mathcal{F}$, and the measured outputs $\\theta_0(t)$ and $\\theta_\\ell(t)$ in the Tikhonov functional (\\ref{20}), we choose the inputs to the backward problem (\\ref{25}) based on these differences, as follows:\n\\begin{eqnarray}\\label{26}\n\\left\\{ \\begin{array}{ll}\np(t)=u_x(0,t;F)-\\theta_0(t),\\\\ [2pt]\nq(t)=u_x(\\ell,t;F)-\\theta_\\ell(t),~t \\in [0,T],\\, F \\in \\mathcal{F}.\n\\end{array} \\right.\n\\end{eqnarray}\n\nThe backward problem (\\ref{25}) with the inputs (\\ref{26}) is defined as the \\emph{adjoint problem, corresponding to the inverse problem (\\ref{1})-(\\ref{2}):}\n\\begin{eqnarray}\\label{27}\n\\left\\{ \\begin{array}{ll}\n\\rho_A(x) \\varphi_{tt}-\\mu(x)\\varphi_{t}-(T(x)\\varphi_{x})_{x}+ (r(x)\\varphi_{xx}-\\kappa(x)\\varphi_{xxt})_{xx} =0,\\\\ [2pt]\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (x,t)\\in \\Omega_{T}:=(0,\\ell)\\times (0,T);\\\\ [4pt]\n\\varphi(x,T)=\\varphi_{t}(x,T)=0, \\, x \\in (0,\\ell); \\\\ [4pt]\n\\varphi(0,t)=0,\\,\\left(r(x)\\varphi_{xx}-\\kappa(x)\\varphi_{xxt}\\right)_{x=0}=u_x(0,t;F)-\\theta_0(t); \\\\ [2pt]\n\\quad \\varphi(\\ell,t)=0,\\,\\left (r(x)\\varphi_{xx}-\\kappa(x)\\varphi_{xxt}\\right)_{x=\\ell}=u_x(\\ell,t;F)-\\theta_\\ell(t),\\,t \\in [0,T].\n\\end{array} \\right.\n\\end{eqnarray}\n\nThe adjoint problem (\\ref{27}), as well as the backward problem (\\ref{25}), are well-posed problems, as the change of the variable $\\tau=T-t$ shows.\n\nFurthermore, the integral relationship (\\ref{24}) with the inputs introduced in (\\ref{26}) is defined as the \\emph{input-output relationship}:\nholds:\n\\begin{eqnarray}\\label{28}\n\\displaystyle \\int_0^T \\left [u_x(0,t;F)-\\theta_0(t) \\right ]\\delta u_x(0,t)dt\n+\\int_0^T \\left [u_x(\\ell,t;F)-\\theta_\\ell(t) \\right ]\\delta u_x(\\ell,t)dt \\nonumber \\\\ [2pt]\n\\qquad \\qquad \\qquad = \\int_0^T \\int_0^\\ell \\delta F(x,t) \\varphi (x,t;F) dx dt.\n\\end{eqnarray}\nThis integral identity expresses, as its name suggests, a relationship between the input $F(x,t)$, the outputs $u_x(0,t;F)$, $u_x(\\ell,t;F)$, and the measured outputs $\\theta_0(t)$, $\\theta_\\ell(t)$ of the inverse problem through the solution $\\varphi (x,t;F)$ of the adjoint problem (\\ref{27}).\n\nAs a consequence of the increment formula (\\ref{23}) for the Tikhonov functional and the input-output relationship (\\ref{28}), we obtain the following formula\n\\begin{eqnarray}\\label{29}\n\\displaystyle \\delta J(F)= \\int_0^T \\int_0^\\ell \\varphi (x,t;F)\\,\\delta F(x,t)dx dt \\qquad \\qquad \\qquad \\qquad \\qquad ~ \\nonumber \\\\ [2pt]\n~\\qquad \\qquad +\\frac{1}{2} \\int_0^T \\left (\\delta u_x(0,t;F) \\right )^2 dt +\\frac{1}{2} \\int_0^T \\left (\\delta u_x(\\ell,t;F) \\right )^2 dt,~~\n\\end{eqnarray}\nwhich is implies \\emph{the formal gradient formula}:\n\\begin{eqnarray}\\label{30}\n\\displaystyle J'(F)= \\varphi (x,t;F),\\, \\mbox {a.e.} ~(x,t)\\in \\Omega_T,\\, F \\in \\mathcal{F}.\n\\end{eqnarray}\n\nTo justify the gradient formula (\\ref{30}), we need to examine the adjoint problem (\\ref{27}) in more detail, obtaining also the necessary estimates for the weak solution. As we will see below, this formula may not be valid for every measured output data $\\theta_0, \\theta_\\ell \\in L^2(0,T)$, which are, at the same time, inputs for the adjoıint problem (\\ref{27}).\n\nWe employ the change of variable $\\tau=T-t$, $t\\in [0,T]$, to transform the backward problem (\\ref{25}) for $\\varphi (x,t) \\equiv \\varphi (x,t;F)$ to the following initial boundary value problem\n\\begin{eqnarray}\\label{31}\n\\left\\{ \\begin{array}{ll}\n\\rho_A(x) \\phi_{\\tau \\tau}-\\mu(x)\\phi_{\\tau}-(T(x)\\phi_{x})_{x}+ (r(x)\\phi_{xx}-\\kappa(x)\\phi_{xx\\tau})_{xx} =0,\\\\ [2pt]\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (x,\\tau)\\in \\Omega_{T}:=(0,\\ell)\\times (0,T);\\\\ [4pt]\n\\phi(x,0)=\\phi_{\\tau}(x,0)=0, ~x \\in (0,\\ell); \\\\ [4pt]\n\\phi(0,\\tau)=0,~-\\left(r(x)\\phi_{xx}-\\kappa(x)\\phi_{xx\\tau}\\right)_{x=0}=\\widetilde{p}(\\tau); \\\\ [2pt]\n\\qquad \\qquad \\varphi(\\ell,\\tau)=0,~\\left (r(x)\\phi_{xx}-\\kappa(x)\\phi_{xx\\tau}\\right)_{x=\\ell}=\\widetilde{q}(\\tau),~\\tau \\in [0,T],\n\\end{array} \\right.\n\\end{eqnarray}\nfor the function $\\phi (x,\\tau) = \\varphi (x,t)$, with the inputs $\\widetilde{p}(\\tau)=p(t)$ and $\\widetilde{q}(\\tau)=q(t)$.\n\n\\begin{Lem}\\label{Lemma-5}\nAssume that in addition to the basic conditions (\\ref{3}), the inputs $\\widetilde{p}(\\tau)$ and $\\widetilde{q}(\\tau)$ in (\\ref{31}) satify the following regularity conditions:\n\\begin{eqnarray}\\label{32}\n\\left. \\begin{array}{ll}\n\\widetilde{p},\\widetilde{q} \\in H^1(0,T).\n\\end{array} \\right.\n\\end{eqnarray}\nThen for the weak solution $\\phi\\in L^2(0,T;\\mathcal{V}^2(0,\\ell))$ with $\\phi_\\tau\\in L^2(0,T;L^2(0,\\ell))$, $\\phi_{\\tau\\tau}\\in L^2(0,T;H^{-2}(0,\\ell))$ of the transformed problem (\\ref{31}), the following estimates hold:\n\\begin{eqnarray}\\label{33}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert \\phi_{xx} \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\exp (T)\\, C_0^2 \\, \\Vert \\widetilde{Q}\\,' \\Vert^2_{L^2(0,T)}, \\\\ [8pt]\n\\displaystyle \\Vert \\phi_{xx} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\left (\\exp (T)-1\\right ) C_0^2 \\, \\Vert \\widetilde{Q}\\,' \\Vert^2_{L^2(0,T)}, \\\\ [10pt]\n\\displaystyle \\Vert \\phi_\\tau \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\exp (T)r_0}{2\\rho_0}\\,C_0^2 \\, \\Vert \\widetilde{Q}\\,' \\Vert^2_{L^2(0,T)}, \\\\ [12pt]\n\\displaystyle \\Vert \\phi_\\tau \\Vert^2_{L^2(0,T;L^2(0,\\ell))} \\leq\n\\frac{\\left (\\exp (T)-1\\right )r_0}{2\\rho_0}\\,C_0^2 \\, \\Vert \\widetilde{Q}\\,' \\Vert^2_{L^2(0,T)}, \\\\ [16pt]\n\\displaystyle \\Vert \\phi_{xx\\tau} \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\exp (T)r_0}{2\\kappa_0}\\,C_0^2 \\, \\Vert \\widetilde{Q}\\,' \\Vert^2_{L^2(0,T)}, \\\\ [12pt]\n\\displaystyle \\Vert \\phi_{xx\\tau} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\left (\\exp (T)-1\\right )r_0 }{2\\kappa_0}\\,C_0^2 \\, \\Vert \\widetilde{Q}\\,' \\Vert^2_{L^2(0,T)},\n\\end{array} \\right.\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\\label{34}\n\\left. \\begin{array}{ll}\n\\displaystyle C_0^2= \\frac{20 \\ell \\,C_T}{3r^2_0},~ C_T=\\max (2/T,\\, 1+3T/3), \\\\ [13pt]\n\\Vert \\widetilde{Q} \\Vert^2_{L^2(0,T)}= \\Vert \\widetilde{p} \\Vert^2_{L^2(0,T)}+\\Vert \\widetilde{q} \\Vert^2_{L^2(0,T)}.\n\\end{array} \\right.\n\\end{eqnarray}\n\\end{Lem}\n{\\bf Proof.}\nMultiply both sides of equation (\\ref{31}) by $2\\phi_t(x,\\tau)$, integrate it over $\\Omega_\\tau:=(0,\\ell)\\times (0,\\tau)$, $\\tau \\in [0,T]$, and use the identities (\\ref{6}). Applying the integration by parts formula, using the initial and boundary conditions, after elementary transformations we obtain:\n\\begin{eqnarray}\\label{35}\n\\int_0^\\ell \\left [\\rho_A(x) \\phi_\\tau^2 +r(x)\\phi_{xx}^2+\\kappa(x)\\phi_{xx\\tau}^2 +T_r(x)\\phi_x^2\\right ]dx+2 \\int_0^\\tau \\int_0^\\ell \\mu (x)\\phi_{\\eta}^2 dx d \\eta\n \\nonumber \\\\ [1pt]\n =2 \\int_0^\\tau \\widetilde{p}(\\eta)\\phi_{x\\eta}(0,\\eta) d\\eta+ 2 \\int_0^\\tau \\widetilde{q}(\\eta)\\phi_{x\\eta}(\\ell,\\eta) d\\eta,~\\tau \\in[0,T].~\n\\end{eqnarray}\nUsing the $\\varepsilon$-inequality and the consequence $\\phi_{x}(0,0)=\\phi_{x}(\\ell,0)=0$ of the homogeneous initial conditions, we estimate the right-hand side integrals, as follows:\n\\begin{eqnarray*}\n\\left. \\begin{array}{ll}\n\\displaystyle 2 \\int_0^\\tau \\widetilde{p}(\\eta)\\phi_{x\\eta}(0,\\eta) d\\eta=\n-2 \\int_0^\\tau \\widetilde{p}\\,'(\\eta)\\phi_{x}(0,\\eta) d\\eta+2 \\widetilde{p}(\\tau)\\phi_{x}(0,\\tau) \\qquad \\qquad\\\\ [8pt]\n\\displaystyle \\qquad \\le \\varepsilon \\left [\\int_0^\\tau \\phi^2_{x}(0,\\eta) d\\eta +\\phi^2_{x}(0,\\tau)\\right ]+ \\frac{1}{\\varepsilon}\\, \\left [\\int_0^T \\left (\\widetilde{p}\\,'(\\tau)\\right)^2 d\\tau + \\left (\\widetilde{p}(\\tau)\\right)^2\\right ],~\\\\ [14pt]\n\\displaystyle 2 \\int_0^\\tau \\widetilde{q}(\\eta)\\phi_{x\\eta}(\\ell,\\eta) d\\eta=-2 \\int_0^\\tau \\widetilde{q}\\,'(\\eta)\\phi_{x}(\\ell,\\eta) d\\eta+2 \\widetilde{q}(\\tau)\\phi_{x}(\\ell,\\tau) \\qquad \\qquad \\\\ [8pt]\n\\displaystyle \\qquad \\le \\varepsilon \\left [\\int_0^\\tau \\phi^2_{x}(\\ell,\\eta) d\\eta +\\phi^2_{x}(\\ell,\\tau)\\right ]+ \\frac{1}{\\varepsilon}\\, \\left [\\int_0^T \\left (\\widetilde{q}\\,'(\\tau)\\right)^2 d\\tau + \\left (\\widetilde{q}(\\tau)\\right)^2\\right ],\n \\end{array} \\right.\n\\end{eqnarray*}\nfor all $\\tau \\in[0,T]$. Here we use the identities\n\\begin{eqnarray*}\n\\displaystyle \\widetilde{p}(\\tau)=\\frac {1}{\\tau}\\int_0^\\tau \\left (\\eta \\widetilde{p}(\\eta)\\right)' d\\eta,~\n\\displaystyle \\displaystyle \\widetilde{q}(\\tau)=\\frac {1}{\\tau}\\int_0^\\tau \\left (\\eta \\widetilde{q}(\\eta)\\right)' d\\eta,~\\tau \\in(0,T]\n\\end{eqnarray*}\nto deduce that\n\\begin{eqnarray*}\n\\displaystyle \\left (\\eta \\widetilde{p}(\\eta)\\right)^2\n\\le \\frac {2}{\\tau} \\int_0^\\tau \\left (\\widetilde{p}(\\eta)\\right)^2 d\\eta+\n\\frac {2\\tau}{3} \\int_0^\\tau \\left (\\widetilde{p}\\,'(\\eta)\\right)^2 d\\eta,\\\\ [1pt]\n\\displaystyle \\left (\\eta \\widetilde{q}(\\eta)\\right)^2\n\\le \\frac {2}{\\tau} \\int_0^\\tau \\left (\\widetilde{q}(\\eta)\\right)^2 d\\eta+\n\\frac {2\\tau}{3} \\int_0^\\tau \\left (\\widetilde{q}\\,'(\\eta)\\right)^2 d\\eta,\n\\end{eqnarray*}\nHence,\n\\begin{eqnarray}\\label{36}\n\\left. \\begin{array}{ll}\n\\displaystyle 2 \\int_0^\\tau \\widetilde{p}(\\eta)\\phi_{x\\eta}(0,\\eta) d\\eta\n\\le \\varepsilon \\left [\\int_0^\\tau \\phi^2_{x}(0,\\eta) d\\eta +\\phi^2_{x}(0,\\tau)\\right ]\\\\ [8pt]\n\\displaystyle \\qquad \\qquad + \\frac{1}{\\varepsilon}\\, \\left [\\frac {2}{\\tau} \\int_0^\\tau \\left (\\widetilde{p}(\\eta)\\right)^2 d\\eta+\n\\left (1+\\frac {2\\tau}{3}\\right ) \\int_0^\\tau \\left (\\widetilde{p}\\,'(\\eta)\\right)^2 d\\eta\\right ],~\\\\ [14pt]\n\\displaystyle 2 \\int_0^\\tau \\widetilde{q}(\\eta)\\phi_{x\\eta}(\\ell,\\eta) d\\eta \\le \\varepsilon \\left [\\int_0^\\tau \\phi^2_{x}(\\ell,\\eta) d\\eta +\\phi^2_{x}(\\ell,\\tau)\\right ] \\\\ [8pt]\n\\displaystyle \\qquad \\qquad + \\frac{1}{\\varepsilon}\\, \\left [\\frac {2}{\\tau} \\int_0^\\tau \\left (\\widetilde{q}(\\eta)\\right)^2 d\\eta+\n\\left (1+\\frac {2\\tau}{3}\\right ) \\int_0^\\tau \\left (\\widetilde{q}\\,'(\\eta)\\right)^2 d\\eta\\right ].\n \\end{array} \\right.\n\\end{eqnarray}\nFor the terms in the first right-hand side square brackets, we use the trace estimate (\\ref{14}) and its analogue for $x=\\ell$. Then we get:\n\\begin{eqnarray}\\label{37}\n\\left. \\begin{array}{ll}\n\\displaystyle \\int_0^\\tau \\phi^2_{x}(0,\\eta) d\\eta +\\phi^2_{x}(0,\\tau) \\le \\frac{5\\ell}{3}\\left [ \\int_0^\\tau \\int_0^\\ell \\phi^2_{x}(x,\\eta) d\\eta dx +\\int_0^\\ell \\phi^2_{x}(x,\\tau) dx \\right ],\\\\ [14pt]\n\\displaystyle \\int_0^\\tau \\phi^2_{x}(\\ell,\\eta) d\\eta +\\phi^2_{x}(\\ell,\\tau) \\le \\frac{5\\ell}{3}\\left [ \\int_0^\\tau \\int_0^\\ell \\phi^2_{x}(x,\\eta) d\\eta dx +\\int_0^\\ell \\phi^2_{x}(x,\\tau) dx\\right ],\n \\end{array} \\right.\n\\end{eqnarray}\nfor all $\\tau \\in[0,T]$. Further, we estimate the terms in the second right-hand side square brackets of (\\ref{36}) as follows:\n\\begin{eqnarray}\\label{38}\n\\left. \\begin{array}{ll}\n\\displaystyle \\frac {2}{\\tau} \\int_0^\\tau \\left (\\widetilde{p}(\\eta)\\right)^2 d\\eta+\n\\left (1+\\frac {2\\tau}{3}\\right ) \\int_0^\\tau \\left (\\widetilde{p}\\,'(\\eta)\\right)^2 d\\eta\\le C_T\\int_0^T \\left (\\widetilde{p}\\,'(\\eta)\\right)^2 d\\eta,~\\\\ [10pt]\n\\displaystyle \\frac {2}{\\tau} \\int_0^\\tau \\left (\\widetilde{q}(\\eta)\\right)^2 d\\eta+\n\\left (1+\\frac {2\\tau}{3}\\right ) \\int_0^\\tau \\left (\\widetilde{q}\\,'(\\eta)\\right)^2 d\\eta \\le C_T \\int_0^T \\left (\\widetilde{q}\\,'(\\eta)\\right)^2 d\\eta,\n \\end{array} \\right.\n\\end{eqnarray}\n$ \\tau \\in[0,T]$, with $C_T>0$ introduced in (\\ref{34}). Substituting (\\ref{36}) with (\\ref{37}) and (\\ref{38}) in (\\ref{35}), we deduce that \n\\begin{eqnarray}\\label{39}\n\\displaystyle \\int_0^\\ell \\rho_A(x) \\phi_\\tau^2dx +\\left (r_0-\\frac{5\\ell \\varepsilon}{3}\\right)\\int_0^\\ell \\phi_{xx}^2 dx+\\kappa_0 \\int_0^\\ell \\phi_{xx\\tau}^2 dx\n\\qquad \\qquad \\quad \\nonumber \\\\ [1pt]\n\\qquad +\\int_0^\\ell T_r(x)\\phi_x^2dx +2 \\int_0^\\tau \\int_0^\\ell \\mu (x)\\phi_{\\eta}^2 dx d \\eta \\qquad \\qquad \\qquad \\qquad \\qquad \\nonumber \\\\ [1pt]\n\\qquad \\qquad \\le \\frac{5\\ell\\varepsilon}{3} \\int_0^\\tau \\int_0^\\ell\\phi_{xx}^2 dx d \\eta\n + \\frac{C_T}{\\varepsilon} \\left [\\Vert \\widetilde{p}\\,' \\Vert^2_{L^2(0,T)}+\\Vert \\widetilde{q}\\,' \\Vert^2_{L^2(0,T)} \\right ],\n\\end{eqnarray}\nfor all $\\tau \\in[0,T]$. We choose the arbitrary parameter $\\varepsilon>0$ from the condition\n$r_0-5\\ell \\varepsilon/3>0$, as follows:\n\\begin{eqnarray*}\n\\varepsilon= \\frac{3r_0}{10\\ell}.\n\\end{eqnarray*}\nThen (\\ref{39}) yields:\n\\begin{eqnarray}\\label{40}\n\\displaystyle \\int_0^\\ell \\rho_A(x) \\phi_\\tau^2dx +\\frac{r_0}{2}\\int_0^\\ell \\phi_{xx}^2 dx+\\kappa_0 \\int_0^\\ell \\phi_{xx\\tau}^2 dx\n\\qquad \\qquad \\qquad \\qquad \\qquad \\nonumber \\\\ [1pt]\n +\\int_0^\\ell T_r(x)\\phi_x^2dx +2 \\int_0^\\tau \\int_0^\\ell \\mu (x)\\phi_{\\eta}^2 dx d \\eta \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\nonumber \\\\ [1pt]\n \\qquad \\le \\frac{r_0}{2} \\int_0^\\tau \\int_0^\\ell \\phi_{xx}^2 dx d \\eta\n + \\frac{10\\ell C_T}{3r_0} \\left [\\Vert \\widetilde{p}\\,' \\Vert^2_{L^2(0,T)}+\\Vert \\widetilde{q}\\,' \\Vert^2_{L^2(0,T)} \\right ],\n\\end{eqnarray}\nfor all $\\tau \\in[0,T]$.\n\nThe required estimates (\\ref{33}) are derived from the main inequality (\\ref{40}), as in the proof of Theorem \\ref{Theorem-1}. \\hfill$\\Box$\n\nIn view of (\\ref{26}),\n\\begin{eqnarray*}\n\\left. \\begin{array}{ll}\n\\Vert \\widetilde{p}\\,' \\Vert^2_{L^2(0,T)}=\\Vert p' \\Vert^2_{L^2(0,T)}=\n\\Vert u_{xt}(0,\\,\\cdot) -\\theta'_0\\Vert^2_{L^2(0,T)}, \\\\ [8pt]\n\\Vert \\widetilde{q}\\,' \\Vert^2_{L^2(0,T)}=\\Vert q' \\Vert^2_{L^2(0,T)}=\n\\Vert u_{xt}(\\ell,\\,\\cdot) -\\theta'_\\ell \\Vert^2_{L^2(0,T)}.\n\\end{array} \\right.\n\\end{eqnarray*}\nWith the inequalities\n\\begin{eqnarray*}\n\\left. \\begin{array}{ll}\n\\Vert u_{xt}(0,\\,\\cdot) -\\theta'_0\\Vert^2_{L^2(0,T)}\\le\n2\\Vert u_{xt}(0,\\,\\cdot) \\Vert^2_{L^2(0,T)}+ 2\\Vert \\theta'_0\\Vert^2_{L^2(0,T)}, \\\\ [8pt]\n\\Vert u_{xt}(\\ell,\\,\\cdot) -\\theta'_\\ell \\Vert^2_{L^2(0,T)}\\le\n2\\Vert u_{xt}(\\ell,\\,\\cdot)\\Vert^2_{L^2(0,T)}+2\\Vert \\theta'_\\ell \\Vert^2_{L^2(0,T)},\n\\end{array} \\right.\n\\end{eqnarray*}\nand the trace estimates in (\\ref{11}) this leads to the following estimates:\n\\begin{eqnarray}\\label{41}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert p' \\Vert^2_{L^2(0,T)}\\le \\frac{2C_1^2}{\\kappa_0} \\Vert F\\Vert^2_{L^2(\\Omega_T)}+ 2\\Vert \\theta'_0\\Vert^2_{L^2(0,T)}, \\\\ [10pt]\n\\displaystyle \\Vert q' \\Vert^2_{L^2(0,T)}\\le \\frac{2C_1^2}{\\kappa_0} \\Vert F\\Vert^2_{L^2(\\Omega_T)}+ 2\\Vert \\theta'_\\ell \\Vert^2_{L^2(0,T)}.\n\\end{array} \\right.\n\\end{eqnarray}\n\nThe right-hand side norms $\\Vert \\theta'_0\\Vert_{L^2(0,T)}$ and $\\Vert \\theta'_\\ell\\Vert_{L^2(0,T)}$ in estimates (\\ref{41}) provide insight into the necessary conditions for existence of the weak solution to the adjoint problem (\\ref{27}). Namely, the measured outputs $\\theta_0(t)$ and $\\theta_\\ell(t)$ must not be from the space $L^2(0,T)$, but\nfrom the space of smoother functions $H^1(0,T)$.\n\n\\begin{thm}\\label{Theorem-3}\nAssume that the basic conditions (\\ref{3}) are satisfied. Suppose, in addition, that the measured outputs satisfy the regularity conditions (\\ref{32}). Then for there exists a weak solution $\\varphi \\in L^2(0,T;\\mathcal{V}^2(0,\\ell))$ with $\\varphi_t\\in L^2(0,T;L^2(0,\\ell))$, $\\varphi_{tt}\\in L^2(0,T;H^{-2}(0,\\ell))$ of the adjoint problem (\\ref{27}), and the following a'piori estimates hold:\n\\begin{eqnarray}\\label{42}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert \\varphi_{xx}\\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq 2\\exp(T) C_0^2\\left [\\frac{2C_1^2}{\\kappa_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}\n+ \\Vert \\Theta \\Vert^2_{L^2(\\Omega_T)} \\right ], \\\\ [10pt]\n\\displaystyle \\Vert \\varphi_{xx} \\Vert^2_{L^2(0,T;L^2(0,\\ell))} \\leq 2\\left (\\exp(T)-1\\right) C_0^2 \\left [\\frac{2C_1^2}{\\kappa_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)}\n+ \\Vert \\Theta \\Vert^2_{L^2(\\Omega_T)} \\right ],\\\\ [12pt]\n\\displaystyle \\Vert \\varphi_{t} \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\exp(T)r_0}{\\rho_0}\\,C_0^2 \\left [\\frac{2C_1^2}{\\kappa_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)} + \\Vert \\Theta \\Vert^2_{L^2(\\Omega_T)} \\right ], \\\\ [10pt]\n\\displaystyle \\Vert \\varphi_{t} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\left (\\exp(T)-1\\right )r_0}{\\rho_0}\\,C_0^2 \\left [\\frac{2C_1^2}{\\kappa_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)} + \\Vert \\Theta \\Vert^2_{L^2(\\Omega_T)} \\right ], \\\\ [13pt]\n\\displaystyle \\Vert \\varphi_{xxt} \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\exp(T)r_0}{\\kappa_0}\\,C_0^2 \\left [\\frac{2C_1^2}{\\kappa_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)} + \\Vert \\Theta \\Vert^2_{L^2(\\Omega_T)} \\right ], \\\\ [10pt]\n\\displaystyle \\Vert \\varphi_{xxt} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\left (\\exp(T)-1\\right )r_0}{\\kappa_0}\\,C_0^2 \\left [\\frac{2C_1^2}{\\kappa_0}\\, \\Vert F \\Vert^2_{L^2(\\Omega_T)} + \\Vert \\Theta \\Vert^2_{L^2(\\Omega_T)} \\right ],\n\\end{array} \\right.\n\\end{eqnarray}\nwhere $\\mathcal{V}^2(0,\\ell)$ is the subspace of the Sobolev space $H^2(0,\\ell)$ introduced in Theorem \\ref{Theorem-1},\n\\begin{eqnarray}\\label{43}\n\\Vert \\Theta \\Vert^2_{L^2(\\Omega_T)}= \\Vert \\theta_0 \\Vert^2_{L^2(\\Omega_T)}+ \\Vert \\theta_\\ell \\Vert^2_{L^2(\\Omega_T)},\n\\end{eqnarray}\nand $C_0,C_1>0$ are the constants defined in (\\ref{34}) and (\\ref{12}), respectively.\n\\end{thm}\n{\\bf Proof.} The proof of the existence of the weak solution can be done in a similar way to the proof of the related theorems in \\cite{Baysal:2019} and \\cite{Sakthivel:2024}.\n\nEstimates (\\ref{33}) clearly hold for the corresponding norms of the weak solution $\\varphi(x,t)$ of the adjoint problem (\\ref{27}), with $\\Vert \\widetilde {Q}\\,' \\Vert^2_{L^2(0,T)}$ replaced by\n\\begin{eqnarray*}\n\\displaystyle \\Vert Q' \\Vert^2_{L^2(0,T)}= \\Vert p' \\Vert^2_{L^2(0,T)}+ \\Vert q' \\Vert^2_{L^2(0,T)}\\,.\n\\end{eqnarray*}\nNamely,\n\\begin{eqnarray*}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert \\varphi_{xx} \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\exp (T)\\, C_0^2 \\, \\Vert Q' \\Vert^2_{L^2(0,T)}, \\\\ [8pt]\n\\displaystyle \\Vert \\varphi_{xx} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\left (\\exp (T)-1\\right ) C_0^2 \\, \\Vert Q' \\Vert^2_{L^2(0,T)}, \\\\ [10pt]\n\\displaystyle \\Vert \\varphi_\\tau \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\exp (T)r_0}{2\\rho_0}\\,C_0^2 \\, \\Vert Q' \\Vert^2_{L^2(0,T)}, \\\\ [8pt]\n\\displaystyle \\Vert \\varphi_\\tau \\Vert^2_{L^2(0,T;L^2(0,\\ell))} \\leq\n\\frac{\\left (\\exp (T)-1\\right )r_0}{2\\rho_0}\\,C_0^2 \\, \\Vert Q' \\Vert^2_{L^2(0,T)}, \\\\ [10pt]\n\\displaystyle \\Vert \\varphi_{xx\\tau} \\Vert^2_{L^{\\infty}(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\exp (T)r_0}{2\\kappa_0}\\,C_0^2 \\, \\Vert Q' \\Vert^2_{L^2(0,T)}, \\\\ [8pt]\n\\displaystyle \\Vert \\varphi_{xx\\tau} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\left (\\exp (T)-1\\right )r_0}{2\\kappa_0}\\,C_0^2 \\, \\Vert Q' \\Vert^2_{L^2(0,T)}.\n\\end{array} \\right.\n\\end{eqnarray*}\nFurther, as a consequence of (\\ref{41}), we deduce that\n\\begin{eqnarray*}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert Q' \\Vert^2_{L^2(0,T)} \\le \\frac{4C_1^2}{\\kappa_0} \\Vert F\\Vert^2_{L^2(\\Omega_T)}+ 2\\left [\\Vert \\theta'_0\\Vert^2_{L^2(0,T)}+\\Vert \\theta'_\\ell \\Vert^2_{L^2(0,T)}\\right ].\n\\end{array} \\right.\n\\end{eqnarray*}\nWe obtain the necessary estimates (\\ref{41}) by substituting this in the aforementioned inequalities. \\hfill$\\Box$\n\nWe can now use Theorem \\ref{Theorem-3} and the estimates (\\ref{11}) to justify the formal gradient formula (\\ref{30}).\n\n\\begin{thm}\\label{Theorem-4}\nSuppose that conditions of Theorem \\ref{Theorem-3} are satisfied. Then the Tikhonov functional introduced in (\\ref{20}) is Fr\\'{e}chet differentiable, and for the Fr\\'{e}chet gradient of this functional, the gradient formula (\\ref{30}) is valid.\n\\end{thm}\n{\\bf Proof.} Denote by $\\delta u(x,t)$ the solution of problem (\\ref{1}) with $F(x,t)$ replaced by $\\delta F(x,t)$. Then, it follows from the estimates (\\ref{11}) applied to the solution $\\delta u(x,t)$ that\n\\begin{eqnarray*}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert \\delta u_{x}(0,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{r_0}\\, \\Vert \\delta F \\Vert^2_{L^2(\\Omega_T)}, \\\\ [10pt]\n\\displaystyle \\Vert \\delta u_{x}(\\ell,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{r_0}\\, \\Vert \\delta F \\Vert^2_{L^2(\\Omega_T)}.\n\\end{array} \\right.\n\\end{eqnarray*}\nHence, the last two right-hand side integrals in (\\ref{29}) are of the order\n $\\mathcal{O}\\left ( \\Vert \\delta F \\Vert_{L^2(\\Omega_T)}^2\\right )$, which implies that\n the Tikhonov functional is Fr\\'{e}chet differentiable and the formula (\\ref{30}) is well-defined.\n This implies the proof. \\hfill$\\Box$\n\n\\section{The Lipschitz continuity of the Fr\\'{e}chet gradient and monotonicity of iterations}\n\n It is a well-known fact that important properties, such as the monotonicity of iterations and the rate of convergence in gradient methods, are results of the Lipschitz continuity of the Fr\\'{e}chet gradient $J'(F)$, $F \\in \\mathcal{F}$, of the Tikhonov functional \\cite{Hasanov:2007}. Thus, for example, in the Landweber iteration algorithm $F^{n+1}=F^{n}-\\omega_n J'(F^{n})$, applied to the inverse problem (\\ref{1})-(\\ref{2}), the relaxation parameter $\\omega_n>0$ can be estimated through the Lipschitz constant \\cite{Hasanov:Romanov:2021, Hasanov:2007}.\n\n\\begin{thm}\\label{Theorem-5}\nAssume that conditions of Theorem \\ref{Theorem-3} are satisfied. Then the Fr\\'{e}chet gradient of the Tikhonov functional, defined (\\ref{30}), is Lipschitz continuous,\n\\begin{eqnarray}\\label{44}\n\\Vert J'(F+\\delta F)- J'(F)\\Vert_{L^2(\\Omega_T)}\n\\leq L_G \\,\\Vert \\delta F \\Vert_{L^2(\\Omega_T)},\n\\end{eqnarray}\nwith the Lipschitz constant\n\\begin{eqnarray}\\label{45}\n \\displaystyle L_G = \\sqrt {\\frac{\\exp (T)-1}{2\\kappa_0}} \\,\\ell^2 \\,C_0 \\,C_1,\n\\end{eqnarray}\nwhere the constants $C_1>0$ and $C_0>0$ are defined in (\\ref{12}) and (\\ref{34}), respectively.\n\\end{thm}\n{\\bf Proof.} By the definition,\n\\begin{eqnarray}\\label{46}\n\\Vert J'(F+\\delta F)- J'(F)\\Vert^2_{L^2(\\Omega_T)}= \\Vert \\delta \\varphi \\Vert^2_{L^2(0,T;L^2(0,\\ell))},\n\\end{eqnarray}\nwhere $\\delta \\varphi (x,t):= \\varphi (x,t;F+\\delta F)-\\varphi (x,t;F)$. Here, $\\varphi (x,t;F+\\delta F)$ and $\\varphi (x,t;F)$ are the weak solutions of the adjoint problem (\\ref{27}) for given $F +\\delta F \\in \\mathcal{F}$ and $F \\in \\mathcal{F}$, respectively, and $\\delta \\varphi (x,t)$ solves the following problem:\n\\begin{eqnarray}\\label{47}\n\\left\\{ \\begin{array}{ll}\n\\rho_A(x) \\delta \\varphi_{tt}-\\mu(x)\\delta \\varphi_{t}-(T(x)\\delta \\varphi_{x})_{x}+ (r(x)\\delta \\varphi_{xx}-\\kappa(x)\\delta \\varphi_{xxt})_{xx} =0,\\\\ [2pt]\n\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad (x,t)\\in \\Omega_{T}:=(0,\\ell)\\times (0,T);\\\\ [4pt]\n\\delta \\varphi(x,T)=\\delta \\varphi_{t}(x,T)=0, \\, x \\in (0,\\ell); \\\\ [4pt]\n\\delta \\varphi(0,t)=0,\\,\n\\left(r(x)\\delta \\varphi_{xx}-\\kappa(x)\\varphi_{xxt}\\right)_{x=0}=\\delta u_x(0,t); \\\\ [2pt]\n\\quad \\varphi(\\ell,t)=0,\\,\\left (r(x)\\varphi_{xx}-\\kappa(x)\\varphi_{xxt}\\right)_{x=\\ell}=\n\\delta u_x(\\ell,t;F),\\,t \\in [0,T],\n\\end{array} \\right.\n\\end{eqnarray}\nwith the inputs $\\delta u_x(0,t):=u_x(0,t;F+\\delta F)-u_x(0,t;F)$ and\n$\\delta u_x(\\ell,t):=u_x(\\ell,t;F+\\delta F)-u_x(\\ell,t;F)$. From the second estimate\nof (\\ref{33}) applied to the solution $\\delta \\varphi (x,t)$ of the problem (\\ref{47})\nwe deduce that\n\\begin{eqnarray}\\label{48}\n\\displaystyle \\Vert \\delta \\varphi_{xx} \\Vert^2_{L^2(0,T;L^2(0,\\ell))} \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\nonumber \\\\ [1pt]\n\\leq \\left (\\exp (T)-1\\right ) C_0^2 \\,\n\\left [\\Vert \\delta u_{xt}(0,\\cdot) \\Vert^2_{L^2(0,T)}+\\Vert \\delta u_{xt}(0,\\cdot) \\Vert^2_{L^2(0,T)} \\right ].\n\\end{eqnarray}\nWe employ the second and fourth trace estimates in (\\ref{11}) for the weak solution $\\delta u(x,t):=u(x,t;F+\\delta F) -u(x,t;F)$ of the problem (\\ref{1}) with $F(x,t)$ replaced by $\\delta F(x,t)$. Then, we have\n\\begin{eqnarray*}\n\\left. \\begin{array}{ll}\n\\displaystyle \\Vert \\delta u_{xt}(0,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{\\kappa_0}\\, \\Vert \\delta F \\Vert^2_{L^2(\\Omega_T)},\\\\ [10pt]\n\\displaystyle \\Vert \\delta u_{xt}(\\ell,\\,\\cdot) \\Vert^2_{L^2(0,T)}\n\\leq \\frac{C_1^2}{\\kappa_0}\\,\\Vert \\delta F \\Vert^2_{L^2(\\Omega_T)}.\n\\end{array} \\right.\n\\end{eqnarray*}\nHence, by (\\ref{48}),\n\\begin{eqnarray}\\label{49}\n\\displaystyle \\Vert \\delta \\phi_{xx} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{2\\left (\\exp (T)-1\\right )}{\\kappa_0} \\,C_0^2 C_1^2 \\,\n\\Vert \\delta F \\Vert^2_{L^2(\\Omega_T)}.\n\\end{eqnarray}\n\nOn the other hand, by the inequality (\\ref{13}), applied to the solution of (\\ref{47}),\n\\begin{eqnarray*}\n\\displaystyle \\Vert \\delta \\varphi_{x} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\ell^2}{2}\\, \\Vert \\delta \\varphi_{xx} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}.\n\\end{eqnarray*}\nIf we consider here the inequality\n\\begin{eqnarray*}\n\\displaystyle \\Vert \\delta \\varphi \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\ell^2}{2}\\, \\Vert \\delta \\varphi_{x} \\Vert^2_{L^2(0,T;L^2(0,\\ell))},\n\\end{eqnarray*}\nwhich can be easily proven for the function $\\delta \\varphi (x,t)$ satisfying the condition $\\delta \\varphi (0,t)=0$, then we obtain\n\\begin{eqnarray*}\n\\displaystyle \\Vert \\delta \\varphi \\Vert^2_{L^2(0,T;L^2(0,\\ell))}\n\\leq \\frac{\\ell^4}{4}\\, \\Vert \\delta \\varphi_{xx} \\Vert^2_{L^2(0,T;L^2(0,\\ell))}.\n\\end{eqnarray*}\nTaking into account this inequality in the estimate (\\ref{49}), we arrive at the requested result.\n\\hfill$\\Box$\n\n\n\n\n\\begin{thebibliography}{6}\n\n\\bibitem{Timoshenko:1953}\nS.P. Timoshenko, History of Strength of Materials: With a Brief Account of the History of Theory of Elasticity and Theory of Structures, McGraw-Hill, New York, 1953.\n\\bibitem{Fryba:1972}\nL. Fryba, Vibration of Solids and Structures under Moving Loads, Noordhoff, Groningen, 1972.\n\\bibitem{Yang:2004}\nB.Y. Yang, J.D. Yau, Y.S. Wu, Vehicle-Bridge Interaction Dynamics, World Scientific Publishing Co., 2004.\n\\bibitem{Ouyang:2011}\nH. Ouyang, Moving-load dynamic problems: A tutorial (with a brief overview), Mechanical Systems and Signal Processing, 25 (2011) 2039–2060.\n\\bibitem{VanDo:2017}\nVan Do, Vuong Nguyen, Thanh Hai Ong, and Chien H. Thai, Dynamic responses of Euler–Bernoulli beam subjected to moving vehicles using isogeometric approach, Applied Mathematical Modelling 51 (2017): 405-428.\n\\bibitem{Lydon:2016}\nM. Lydon, S. E. Taylor, D. Robinson, A. Mufti, and E.J.O. Brien, Recent developments in bridge weigh in motion (B-WIM), Journal of Civil Structural Health Monitoring 6 (2016) 69-81.\n\\bibitem{Moses:1979}\nF. Moses, Weigh-in-motion system using instrumented bridges, Transportation Engineering Journal of ASCE, 105(3) (1979) 233-249.\n\\bibitem{Ojio:2016}\nT. Ojio, C.H. Carey, E.J. OBrien, C. Doherty, and S.E. Taylor, Contactless bridge weigh-in-motion. Journal of Bridge Engineering, 21(7) (2016) 04016032.\n\\bibitem{Martini:2022}\nA. Martini, E.M. Tronci, M.Q. Feng, and R.Y. Leung, A computer vision-based method for bridge model updating using displacement influence lines, Engineering Structures 259 (2016) 114129.\n\\bibitem{Khuc:2018}\nT. Khuc and F.N. Catbas, Structural identification using computer vision–based bridge health monitoring, Journal of Structural Engineering 144(2)2 (2018) 04017202.\n\\bibitem{Tang:2024} L. Tang, X.B. Liu, Y.J. Liu, et al. A bridge dynamic response analysis and load recognition method using traffic imaging, Scientific Reports 14 (2024) 18742.\n\\bibitem{Hasanov:Romanov:2021} A. Hasanov and A. G. Romanov, Introduction to Inverse Problems for Differential Equations, 2nd ed., Springer, New York, 2021.\n\\bibitem{Baysal:2019}\nO. Baysal and A. Hasanov, Solvability of the clamped Euler–Bernoulli beam equation, Applied Mathematics Letters 93 (2019) 85-90.\n\\bibitem{Sakthivel:2024}\nK. Sakthivel, A. Hasanov and A. Dileep, Inverse problems of identifying the unknown transverse shear force in the Euler-Bernoulli beam with Kelvin-Voigt damping, Journal of Inverse and Ill-Posed Problems 32(1) (2024) 75-106.\n\\bibitem{Hasanov:2007}\nA. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from\nthe final overdetermination: Weak solution approach, Journal of Mathematical Analysis and Applications, 330 (2007),766-779.\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n\n\n\n"}}},{"rowIdx":296,"cells":{"id":{"kind":"string","value":"2412.20775v6"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20775v6"},"title":{"kind":"string","value":"On Spectral Graph Determination"},"full_text":{"kind":"string","value":"\\documentclass[11pt,twoside,reqno]{amsart}\n\\linespread{1.05}\n\\usepackage[colorlinks=true,citecolor=blue]{hyperref}\n\n\\numberwithin{equation}{section}\n\\DeclareMathOperator*{\\essinf}{ess\\,inf}\n\\makeatletter\n\\makeatother\n\n\\newcommand{\\ep}{\\varepsilon}\n\\newcommand{\\eps}[1]{{#1}_{\\varepsilon}}\n\n\\usepackage{amsmath, amssymb, amsthm, amsfonts, cite, dsfont, enumerate, epsfig, float, geometry, doi,\n\tinfwarerr, mathrsfs, mathtools, mathrsfs, 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\\newcommand{\\dgr}[1]{\\Degree_{#1}} \\newcommand{\\trace}[1]{\\text{Tr}{(#1)}}\n\\newcommand{\\Gmats}{\\{\\A,\\LM,\\Q, {\\bf{\\mathcal{L}}}, \\overline{\\A}, \\overline{\\LM}, \\overline{\\Q}, \\overline{{\\bf{\\mathcal{L}}}}\\} }\n\\newcommand{\\AM}[1]{\\text{AM}{(#1)}}\n\\newcommand{\\GM}[1]{\\text{GM}{(#1)}}\n\\newcommand{\\diag}[1]{\\operatorname{diag}\\bigl(#1\\bigr)}\n\\newcommand\\qfrac[3][1pt]{\\frac{\t\t\\ThisStyle{\\addstackgap[#1]{\\SavedStyle#2}}}{\t\t\\ThisStyle{\\addstackgap[#1]{\\SavedStyle#3}}}}\n\n\\MHInternalSyntaxOn\n\\renewcommand{\\dcases}\n{\n\t\\MT_start_cases:nnnn\n\t{\\quad}\n\t{$\\m@th\\displaystyle##$\\hfil}\n\t{$\\m@th\\displaystyle##$\\hfil}\n\t{\\lbrace}\n}\n\\MHInternalSyntaxOff\n\n\n\\geometry{left=1in, right=1in, top=1in, bottom=1in}\n\\makeatletter\\c@MaxMatrixCols=15\\makeatother\n\n\\begin{document}\n\\setlength{\\baselineskip}{1.15\\baselineskip}\n\n\\title{On Spectral Graph Determination}\n\n\\author{Igal Sason \\and Noam Krupnik \\and Suleiman Hamud \\and Abraham Berman}\n\n\\maketitle\n\\thispagestyle{empty}\n\n\\vspace*{-0.8cm}\n\\begin{center}\n{\\em Technion - Israel Institute of Technology, Technion City, Haifa 3200003, Israel}\n\\end{center}\n\n\\vskip 4mm {\\noindent {\\bf Abstract.}\nThe study of spectral graph determination is a fascinating area of research in spectral graph\ntheory and algebraic combinatorics. This field focuses on examining the spectral characterization\nof various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic\ngraphs, and analyzing the conditions under which a graph's spectrum uniquely determines its structure.\nThis paper presents an overview of both classical and recent advancements in these topics, along\nwith newly obtained proofs of some existing results, which offer additional insights.\n\n\\vspace*{0.2cm}\n\\noindent {\\bf Keywords.}\nSpectral graph theory, spectral graph determination, cospectral nonisomorphic graphs,\nHaemers' conjecture, Tur\\'{a}n graphs, graph operations.\n\n\\vspace*{0.2cm}\n\\noindent {\\bf 2020 Mathematics Subject Classification.} 05C50, 05C75, 05C76.\n\n\\vspace*{0.2cm}\n\\noindent {\\bf Correspondence}: Igal Sason, Technion - Israel Institute of Technology, Technion City,\nHaifa 3200003, Israel. Email: eeigal@technion.ac.il; Tel: +97248294699.\n\n\\tableofcontents{}\n\n\\section{Introduction}\n\\label{section: Introduction}\n\nSpectral graph theory lies at the intersection of combinatorics and matrix theory, exploring the structural\nand combinatorial properties of graphs through the analysis of the eigenvalues and eigenvectors of matrices\nassociated with these graphs \\cite{BrouwerH2011,Chung1997,CvetkovicDS1995,CvetkovicRS2010,GodsilR2001}.\nSpectral properties of graphs offer powerful insights into a variety of useful graph characteristics, enabling\nthe determination or estimation of features such as the independence number, clique number, chromatic number,\nand the Shannon capacity of graphs, which are notoriously NP-hard to compute.\n\nA particularly intriguing topic in spectral graph theory is the study of cospectral graphs, i.e., graphs that\nshare identical multisets of eigenvalues with respect to one or more matrix representations. While isomorphic\ngraphs are always cospectral, non-isomorphic graphs may also share spectra, leading to the study of non-isomorphic\ncospectral (NICS) graphs. This phenomenon raises profound questions about the extent to which a graph’s spectrum\nencodes its structural properties. Conversely, graphs determined by their spectrum (DS graphs) are uniquely\nidentifiable, up to isomorphism, by their eigenvalues. In other words, a graph is DS if and only if no other\nnon-isomorphic graph shares the same spectrum.\n\nThe problem of spectral graph determination and the characterization of DS graphs dates back to the pioneering\n1956 paper by G\\\"{u}nthard and Primas \\cite{GunthardP56}, which explored the interplay between graph theory and\nchemistry. This paper posed the question of whether graphs can be uniquely determined by their spectra with respect\nto their adjacency matrix $\\A$.\n\nWhile every graph can be determined by its adjacency matrix, which enables the determination\nof every graph by its eigenvalues and a basis of corresponding eigenvectors, the characterization of graphs\nfor which eigenvalues alone suffice for identification forms a fertile area of research in spectral graph theory.\nThis research holds both theoretical interest and practical implications.\n\nSubsequent studies have broadened the scope of this question to include determination by the spectra of other\nsignificant matrices, such as the Laplacian matrix ($\\LM$), signless Laplacian matrix ($\\Q$), and normalized\nLaplacian matrix (${\\bf{\\mathcal{L}}}$), among many other matrices associated with graphs.\nThe study of cospectral and DS graphs with respect to these matrices has become a cornerstone of spectral graph\ntheory. This line of research has far-reaching applications in diverse fields, including chemistry and molecular\nstructure analysis, physics and quantum computing, network communication theory, machine learning, and data science.\n\nOne of the most prominent conjectures in this area is Haemers' conjecture \\cite{Haemers2016,Haemers2024}, which\nposits that most graphs are determined by the spectrum of their adjacency matrices ($\\A$-DS). Despite many efforts\nin proving this open conjecture, some theoretical and experimental progress on the theme of this conjecture has been\nrecently presented in \\cite{KovalK2024,WangW2024}, while also graphs or graph families that are not DS continue to be\ndiscovered. Haemers’ conjecture has spurred significant interest in classifying DS graphs and understanding the factors\nthat influence spectral determination, particularly among special families of graphs such as regular graphs, strongly\nregular graphs, trees, graphs of pyramids, as well as the construction of NICS graphs by a variety of graph operations.\nStudies in these directions of research have been covered in the seminal works by Schwenk \\cite{Schwenk1973}, and by van\nDam and Haemers \\cite{vanDamH03,vanDamH09}, as well as in more recent studies (in part by the authors) such as\n\\cite{AbdianBTKO21,AbiadH2012,AbiadBBCGV2022,Butler2010,ButlerJ2011,BermanCCLZ2018,Butler2016,ButlerH2016,BuZ2012,BuZ2012b,\nCamaraH14,DasP2013,DuttaA20,GodsilM1982,HamidzadeK2010,HamudB24,JinZ2014,KannanPW22,KoolenHI2016,KoolenHI2016b,KovalK2024,KrupnikB2024,\nLinLX2019,LiuZG2008,MaRen2010,OboudiAAB2021,OmidiT2007,OmidiV2010,Sason2024,YeLS2025,ZhangLY09,ZhangLZY09,ZhouBu2012}, and references therein.\nSpecific contributions of these papers to the problem of the spectral determination of graphs are addressed in the continuation of this article.\n\nThis paper surveys both classical and recent results on spectral graph determination, also presenting newly obtained\nproofs of some existing results, which offer additional insights.\n\nThe paper emphasizes the significance of adjacency spectra ($\\A$-spectra), and it provides conditions for $\\A$-cospectrality, $\\A$-NICS, and\n$\\A$-DS graphs, offering examples that support or refute Haemers’ conjecture.\nWe furthermore address the cospectrality of graphs with respect to the Laplacian, signless Laplacian, and normalized Laplacian matrices. For regular\ngraphs, cospectrality with respect to any one of these matrices (or the adjacency matrix) implies cospectrality with respect to all the others,\nenabling a unified framework for studying DS and NICS graphs across different matrix representations. However, for irregular graphs, cospectrality\nwith respect to one matrix does not necessarily imply cospectrality with respect to another. This distinction underscores the complexity of\nanalyzing spectral properties in irregular graphs, where the interplay among different matrix representations becomes more intricate and often\nnecessitates distinct techniques for characterization and comparison.\n\nThe structure of the paper is as follows: Section~\\ref{section: preliminaries} provides preliminary material in matrix theory, graph theory,\nand graph-associated matrices. Section~\\ref{section: DS graphs} focuses on graphs determined by their spectra (with respect to one or\nmultiple matrices). Section~\\ref{section: special families of graphs} examines special families of graphs and their determination by\nadjacency spectra. Section~\\ref{section: graph operations} analyzes unitary and binary graph operations, emphasizing their impact on\nspectral determination and construction of NICS graphs. Finally, Section~\\ref{section: summary and outlook} concludes the paper with\nopen questions and an outlook on spectral graph determination, highlighting areas for further research.\n\n\n\\section{Preliminaries}\n\\label{section: preliminaries}\nThe present section provides preliminary material and notation in matrix theory, graph theory,\nand graph-associated matrices, which serves for the presentation of this paper.\n\n\\subsection{Matrix Theory Preliminaries}\n\\label{subsection: Matrix Theory Preliminaries}\n\nThe following standard notation in matrix theory is used in this paper:\n\\begin{itemize}\n\t\\item $\\Reals^{n\\times m}$ denotes the set of all $n \\times m$ matrices with real entries,\n\t\\item $\\Reals^{n} \\triangleq \\Reals^{n\\times 1}$ denotes the set of all $n$-dimensional column vectors with real entries,\n\t\\item $\\I{n}\\in\\Reals^{n\\times n}$ denotes the $n \\times n$ identity matrix,\n \\item $\\mathbf{0}_{k,m} \\in\\Reals^{k\\times m}$ denotes the $k \\times m$ all-zero matrix,\n \\item $\\J{k,m}\\in\\Reals^{k\\times m}$ denotes the $k \\times m$ all-ones matrix,\n\t\\item $\\mathbf{1}_n \\triangleq \\J{n,1} \\in \\Reals^n$ denotes the $n$-dimensional column vector of ones.\n\\end{itemize}\n\nThroughout this paper, we deal with real matrices.\n\nThe concepts of \\emph{Schur complement} and \\emph{interlacing of eigenvalues} are useful in papers on\nspectral graph determination and cospectral graphs, and are also addressed in this paper.\n\n\\begin{definition}\n\\label{definition: Schur complement}\nLet $\\mathbf{M}$ be a block matrix\n\\begin{align}\n\\mathbf{M}=\n\\begin{pmatrix}\n\\mathbf{A} & \\mathbf{B}\\\\\n\\mathbf{C} & \\mathbf{D}\n\\end{pmatrix},\n\\end{align}\nwhere the block $\\mathbf{D}$ is invertible. The \\emph{Schur complement of $D$ in $M$} is\n\\begin{align}\n\\label{definition: eq - Schur complement}\n\\mathbf{M/D}= \\mathbf{A}-\\mathbf{BD}^{-1}\\mathbf{C}.\n\\end{align}\n\\end{definition}\n\nSchur proved the following remarkable theorem:\n\\begin{theorem}[Theorem on the Schur complement \\cite{Schur1917}]\n\\label{theorem: Schur complement}\nIf $D$ is invertible, then\n\\begin{align}\n\\label{eq: Schur's formula}\n\\det{\\mathbf{M}} & =\\det(\\mathbf{M/D}) \\, \\det{\\mathbf{D}}.\n\\end{align}\n\\end{theorem}\n\n\\begin{theorem}[Cauchy Interlacing Theorem \\cite{ParlettB1998}]\n\\label{thm:interlacing}\nLet $\\lambda_{1} \\ge \\ldots \\ge \\lambda_{n}$ be the eigenvalues of a symmetric matrix $\\mathbf{M}$\nand let $\\mu_{1}\\ge\\ldots\\ge\\mu_{m}$ be the eigenvalues of a \\emph{principal $m \\times m$ submatrix\nof $\\mathbf{M}$} (i.e., a submatrix that is obtained by deleting the same set of rows and columns\nfrom $M$) then $\\lambda_{i}\\ge\\mu_{i}\\ge\\lambda_{n-m+i}$ for $i=1,\\ldots,m$.\n\\end{theorem}\n\n\\begin{definition}[Completely Positive Matrices]\n\\label{definition: completely positive matrix}\nA matrix $\\A \\in \\Reals^{n \\times n}$ is called {\\em completely positive} if there exists a matrix\n${\\mathbf{B}} \\in \\Reals^{n \\times m}$ whose all entries are nonnegative such that $\\A = {\\mathbf{B}} {\\mathbf{B}}^\\mathrm{T}$.\n\\end{definition}\nA completely positive matrix is therefore symmetric and all its entries are nonnegative. The interested reader\nis referred to the textbook \\cite{ShakedBbook19} on completely positive matrices, also addressing their connections\nto graph theory.\n\n\\begin{definition}[Positive Semidefinite Matrices]\n\\label{definition: positive semidefinite matrix}\nA matrix $\\A \\in \\Reals^{n \\times n}$ is called {\\em positive semidefinite} if $\\A$ is symmetric, and\nthe inequality $\\underline{x}^{\\mathrm{T}} \\A \\underline{x} \\geq 0$ holds for every column vector $\\underline{x} \\in \\Reals^n$.\n\\end{definition}\n\\begin{proposition}\n\\label{proposition: positive semidefinite matrix}\nA symmetric matrix is positive semidefinite if and only if one of the following conditions hold:\n\\begin{enumerate}\n\\item All its eigenvalues are nonnegative (real) numbers.\n\\item There exists a matrix ${\\mathbf{B}} \\in \\Reals^{n \\times m}$ such that $\\A = {\\mathbf{B}} {\\mathbf{B}}^\\mathrm{T}$.\n\\end{enumerate}\n\\end{proposition}\n\nThe next result readily follows.\n\\begin{corollary}\n\\label{corollary: c.p. yields p.s.}\nA completely positive matrix is positive semidefinite.\n\\end{corollary}\n\\begin{remark}\n\\label{remark: matrix of order 5}\nRegarding Corollary~\\ref{corollary: c.p. yields p.s.}, it is natural to ask whether, under certain conditions, a positive\nsemidefinite matrix whose all entries are nonnegative is also completely positive. By \\cite[Theorem~3.35]{ShakedBbook19}, this\nholds for all square matrices of order $n \\leq 4$. Moreover, \\cite[Example~3.45]{ShakedBbook19} also presents an explicit example\nof a matrix of order~5 that is positive semidefinite with all nonnegative entries but is not completely positive.\n\\end{remark}\n\n\\subsection{Graph Theory Preliminaries}\n\\label{subsection: Graph Theory Preliminaries}\n\nA graph $\\Gr{G} = (\\V{\\Gr{G}}, \\E{\\Gr{G}})$ forms a pair\nwhere $\\V{\\Gr{G}}$ is a set of vertices and $\\E{\\Gr{G}}\\subseteq \\V{\\Gr{G}} \\times \\V{\\Gr{G}}$\nis a set of edges.\n\nIn this paper all the graphs are assumed to be\n\\begin{itemize}\n\t\\item {\\em finite} - $\\bigcard{\\V{\\Gr{G}}}<\\infty$,\n\t\\item {\\em simple} - $\\Gr{G}$ has no parallel edges and no self loops,\n\t\\item {\\em undirected} - the edges in $\\Gr{G}$ are undirected.\n\\end{itemize}\n\nWe use the following terminology:\n\\begin{itemize}\n\\item The {\\em degree}, $d(v)$, of a vertex $v\\in \\V{\\Gr{G}}$ is the number of vertices in $\\Gr{G}$ that are adjacent to $v$.\n\\item A {\\em walk} in a graph $\\Gr{G}$ is a sequence of vertices in $\\Gr{G}$, where every two consecutive vertices in the sequence\nare adjacent in $\\Gr{G}$.\n\\item A {\\em path} in a graph is a walk with no repeated vertices.\n\\item A {\\em cycle} $\\Cycle$ is a closed walk, obtained by adding an edge to a path in $\\Gr{G}$.\n\\item The {\\em length of a path or a cycle} is equal to its number of edges. A {\\em triangle} is a cycle of length~3.\n\\item A {\\em connected graph} is a graph in which every pair of distinct vertices is connected by a path.\n\\item The {\\em distance} between two vertices in a connected graph is the length of a shortest path that connects them.\n\\item The {\\em diameter} of a connected graph is the maximum distance between any two vertices in the graph, and the diameter\nof a disconnected graph is set to be infinity.\n\\item The {\\em connected component} of a vertex $v \\in \\V{\\Gr{G}}$ is the subgraph whose vertex set $\\set{U} \\subseteq \\V{\\Gr{G}}$\nconsists of all the vertices that are connected to $v$ by any path (including the vertex $v$ itself), and its edge set consists of\nall the edges in $\\E{\\Gr{G}}$ whose two endpoints are contained in the vertex set $\\set{U}$.\n\\item A {\\em tree} is a connected graph that has no cycles (i.e., it is a connected and {\\em acyclic} graph).\n\\item A {\\em spanning tree} of a connected graph $\\Gr{G}$ is a tree with the vertex set $\\V{\\Gr{G}}$ and some of the edges of~$\\Gr{G}$.\n\\item A graph is {\\em regular} if all its vertices have the same degree.\n\\item A {\\em $d$-regular} graph is a regular graph whose all vertices have degree $d$.\n\\item A {\\em bipartite graph} is a graph $\\Gr{G}$ whose vertex set is a disjoint union of two subsets such that no two vertices in\nthe same subset are adjacent.\n\\item A {\\em complete bipartite graph} is a bipartite graph where every vertex in each of the two partite sets is adjacent to all\nthe vertices in the other partite set.\n\\end{itemize}\n\n\\begin{definition}[Complement of a graph]\nThe \\emph{complement} of a graph $\\Gr{G}$, denoted by $\\CGr{G}$, is a graph whose vertex set is $\\V{\\Gr{G}}$, and its edge set\nis the complement set $\\CGr{\\E{\\Gr{G}}}$.\nEvery vertex in $\\V{\\Gr{G}}$ is nonadjacent to itself in $\\Gr{G}$ and $\\CGr{G}$, so $\\{i,j\\} \\in \\E{\\CGr{G}}$ if and only if\n$\\{i, j\\} \\notin \\E{\\Gr{G}}$ with $i \\neq j$.\n\\end{definition}\n\n\\begin{definition}[Disjoint union of graphs]\n\\label{def:disjoint_union_graphs}\nLet $\\Gr{G}_1, \\ldots, \\Gr{G}_k$ be graphs.\nIf the vertex sets in these graphs are not pairwise disjoint,\nlet $\\Gr{G}'_2, \\ldots, \\Gr{G}'_k$ be isomorphic copies of\n$\\Gr{G}_2, \\ldots, \\Gr{G}_k$, respectively, such that none\nof the graphs $\\Gr{G}_1, \\Gr{G}'_2, \\ldots \\Gr{G}'_k$ have\na vertex in common. The disjoint union of these graphs,\ndenoted by $\\Gr{G} = \\Gr{G}_1 \\DU \\ldots \\DU \\Gr{G}_k$,\nis a graph whose vertex and edge sets are equal to the\ndisjoint unions of the vertex and edge sets of\n$\\Gr{G}_1, \\Gr{G}'_2, \\ldots, \\Gr{G}'_k$\n($\\Gr{G}$ is defined up to an isomorphism).\n\\end{definition}\n\n\\begin{definition}\nLet $k\\in \\naturals$ and let $\\Gr{G}$ be a graph. Define\n$k \\Gr{G} = \\Gr{G} \\DU \\Gr{G} \\DU \\ldots \\DU \\Gr{G}$\nto be the disjoint union of $k$ copies of $\\Gr{G}$.\n\\end{definition}\n\n\n\\begin{definition}[Join of graphs]\n\\label{definition: join of graphs}\nLet $\\Gr{G}$ and $\\Gr{H}$ be two graphs with disjoint vertex sets.\nThe join of $\\Gr{G}$ and $\\Gr{H}$ is defined to be their disjoint union,\ntogether with all the edges that connect the vertices in $\\Gr{G}$ with\nthe vertices in $\\Gr{H}$. It is denoted by $\\Gr{G} \\vee \\Gr{H}$.\n\\end{definition}\n\n\n\\begin{definition}[Induced subgraphs]\n\\label{definition: Induced subgraphs}\nLet $\\Gr{G}=(\\Vertex,\\Edge)$ be a graph, and let $\\set{U} \\subseteq \\Vertex$. The \\emph{subgraph of $\\Gr{G}$ induced\nby $\\set{U}$} is the graph obtained by the vertices in $\\set{U}$ and the edges in $\\Gr{G}$ that has both ends on $\\set{U}$.\nWe say that $\\Gr{H}$ is an \\emph{induced subgraph of $\\Gr{G}$}, if it is induced by some $\\set{U} \\subseteq \\Vertex$.\n\\end{definition}\n\n\\begin{definition}[Strongly regular graphs]\n\\label{definition: strongly regular graphs}\nA regular graph $\\Gr{G}$ that is neither complete nor empty is called a\n{\\em strongly regular} graph with parameters $(n,d,\\lambda,\\mu)$,\nwhere $\\lambda$ and $\\mu$ are nonnegative integers, if the following conditions hold:\n\\begin{enumerate}[(1)]\n\\item \\label{Item 1 - definition of SRG}\n$\\Gr{G}$ is a $d$-regular graph on $n$ vertices.\n\\item \\label{Item 2 - definition of SRG}\nEvery two adjacent vertices in $\\Gr{G}$ have exactly $\\lambda$ common neighbors.\n\\item \\label{Item 3 - definition of SRG}\nEvery two distinct and nonadjacent vertices in $\\Gr{G}$ have exactly $\\mu$ common neighbors.\n\\end{enumerate}\nThe family of strongly regular graphs with these four specified parameters is denoted by $\\srg{n}{d}{\\lambda}{\\mu}$.\nIt is important to note that a family of the form $\\srg{n}{d}{\\lambda}{\\mu}$ may contain multiple nonisomorphic strongly\nregular graphs. Throughout this work, we refer to a strongly regular graph as $\\srg{n}{d}{\\lambda}{\\mu}$ if it belongs to\nthis family.\n\\end{definition}\n\n\\begin{proposition}[Feasible parameter vectors of strongly regular graphs]\n\\label{proposition: necessary condition for the parameter vector of SRGs}\nThe four parameters of a strongly regular graph $\\srg{n}{d}{\\lambda}{\\mu}$ satisfy the equality\n\\begin{align}\n\\label{eq: necessary condition for the parameter vector of SRGs}\n(n-d-1)\\mu = d(d-\\lambda-1).\n\\end{align}\n\\end{proposition}\n\\begin{remark}\n\\label{remark: necessary condition for the parameter vector of SRGs}\nEquality~\\eqref{eq: necessary condition for the parameter vector of SRGs} provides a necessary, but\nnot sufficient, condition for the existence of a strongly regular graph $\\srg{n}{d}{\\lambda}{\\mu}$.\nFor example, as shown in \\cite{Haemers93}, no $(76,21,2,7)$ strongly regular graph exists, even though\nthe condition $(n-d-1)\\mu = 378 = d(d-\\lambda-1)$ is satisfied in this case.\n\\end{remark}\n\n\\begin{notation}[Classes of graphs]\n\\noindent\n\n\\begin{itemize}\n\\item $\\CoG{n}$ is the complete graph on $n$ vertices.\n\\item $\\PathG{n}$ is the path graph on $n$ vertices.\n\\item $\\CoBG{\\ell}{r}$ is the complete bipartite graph whose degrees of partite sets are $\\ell$ and $r$ (with possible equality between $\\ell$ and $r$).\n\\item $\\SG{n}$ is the star graph on $n$ vertices $\\SG{n} = \\CoBG{1}{n-1}$.\n\\end{itemize}\n\\end{notation}\n\n\\begin{definition}[Integer-valued functions of a graph]\n\\noindent\n\\begin{itemize}\n\\item Let $k \\in \\naturals $. A \\emph{proper} $k$-\\emph{coloring} of a graph $\\Gr{G}$\nis a function $c \\colon \\V{\\Gr{G}} \\to \\{1,2,...,k\\}$, where $c(v) \\ne c(u)$ for every\n$\\{u,v\\}\\in \\E{\\Gr{G}}$. The \\emph{chromatic number} of $\\Gr{G}$, denoted by $\\chrnum{\\Gr{G}}$,\nis the smallest $k$ for which there exists a proper $k$-coloring of $\\Gr{G}$.\n\\item A \\emph{clique} in a graph $\\Gr{G}$ is a subset of vertices $U\\subseteq \\V{\\Gr{G}}$\nwhere the subgraph induced by $U$ is a complete graph. The \\emph{clique number} of $\\Gr{G}$,\ndenoted by $\\omega(\\Gr{G})$, is the largest size of a clique in $\\Gr{G}$; i.e., it is the\nlargest order of an induced complete subgraph in $\\Gr{G}$.\n\\item An \\emph{independent set} in a graph $\\Gr{G}$ is a subset of vertices $U\\subseteq \\V{\\Gr{G}}$,\nwhere $\\{u,v\\} \\notin \\E{\\Gr{G}}$ for every $u,v \\in U$. The \\emph{independence number} of $\\Gr{G}$,\ndenoted by $\\indnum{\\Gr{G}}$, is the largest size of an independent set in $\\Gr{G}$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{definition}[Orthogonal and orthonormal representations of a graph]\n\\label{def: orthogonal representation}\nLet $\\Gr{G}$ be a finite, simple, and undirected graph, and let $d \\in \\naturals$.\n\\begin{itemize}\n\\item\nAn {\\em orthogonal representation} of the graph $\\Gr{G}$ in the $d$-dimensional\nEuclidean space $\\Reals^d$ assigns to each vertex $i \\in \\V{\\Gr{G}}$\na nonzero vector ${\\bf{u}}_i \\in \\Reals^d$ such that\n${\\bf{u}}_i^{\\mathrm{T}} {\\bf{u}}_j = 0$ for every $\\{i, j\\} \\notin \\E{\\Gr{G}}$\nwith $i \\neq j$. In other words, for every two distinct and nonadjacent vertices in the graph,\ntheir assigned nonzero vectors should be orthogonal in $\\Reals^d$.\n\\item\nAn {\\em orthonormal representation} of $\\Gr{G}$ is additionally represented by unit vectors,\ni.e., $\\| {\\bf{u}}_i \\| = 1$ for all $i \\in \\V{\\Gr{G}}$.\n\\item\nIn an orthogonal (orthonormal) representation of $\\Gr{G}$, every two nonadjacent vertices in $\\Gr{G}$\nare mapped (by definition) into orthogonal (orthonormal) vectors, but adjacent vertices may not necessarily\nbe mapped into nonorthogonal vectors. If ${\\bf{u}}_i^{\\mathrm{T}} {\\bf{u}}_j \\neq 0$ for all\n$\\{i, j\\} \\in \\E{\\Gr{G}}$, then such a representation of $\\Gr{G}$ is called {\\em faithful}.\n\\end{itemize}\n\\end{definition}\n\n\\begin{definition}[Lov\\'{a}sz $\\vartheta$-function \\cite{Lovasz79_IT}]\n\\label{definition: Lovasz theta function}\nLet $\\Gr{G}$ be a finite, simple, and undirected graph. Then, the {\\em Lov\\'{a}sz\n$\\vartheta$-function of $\\Gr{G}$} is defined as\n\\begin{eqnarray}\n\\label{eq: Lovasz theta function}\n\\vartheta(\\Gr{G}) \\triangleq \\min_{{\\bf{c}}, \\{{\\bf{u}}_i\\}} \\, \\max_{i \\in \\V{\\Gr{G}}} \\,\n\\frac1{\\bigl( {\\bf{c}}^{\\mathrm{T}} {\\bf{u}}_i \\bigr)^2} \\, ,\n\\end{eqnarray}\nwhere the minimum on the right-hand side of \\eqref{eq: Lovasz theta function} is taken over all\nunit vectors ${\\bf{c}}$ and all orthonormal representations $\\{{\\bf{u}}_i: i \\in \\V{\\Gr{G}} \\}$\nof $\\Gr{G}$. In \\eqref{eq: Lovasz theta function}, it suffices to consider orthonormal representations\nin a space of dimension at most $n = \\card{\\V{\\Gr{G}}}$.\n\\end{definition}\n\nThe Lov\\'{a}sz $\\vartheta$-function of a graph $\\Gr{G}$ can be calculated by solving (numerically) a convex optimization problem.\nLet ${\\bf{A}} = (A_{i,j})$ be the $n \\times n$ adjacency matrix of $\\Gr{G}$ with\n$n \\triangleq \\card{\\V{\\Gr{G}}}$. The Lov\\'{a}sz\n$\\vartheta$-function $\\vartheta(\\Gr{G})$ can be expressed as the solution of the following semidefinite programming (SDP) problem:\n\\vspace*{0.2cm}\n\\begin{eqnarray}\n\\label{eq: SDP problem - Lovasz theta-function}\n\\mbox{\\fbox{$\n\\begin{array}{l}\n\\text{maximize} \\; \\; \\mathrm{Tr}({\\bf{B}} \\J{n}) \\\\\n\\text{subject to} \\\\\n\\begin{cases}\n{\\bf{B}} \\succeq 0, \\\\\n\\mathrm{Tr}({\\bf{B}}) = 1, \\\\\nA_{i,j} = 1 \\; \\Rightarrow \\; B_{i,j} = 0, \\quad i,j \\in \\OneTo{n}.\n\\end{cases}\n\\end{array}$}}\n\\end{eqnarray}\n\n\\vspace*{0.1cm}\nThere exist efficient convex optimization algorithms (e.g., interior-point methods) to compute\n$\\vartheta(\\Gr{G})$, for every graph $\\Gr{G}$, with a precision of $r$ decimal digits, and a\ncomputational complexity that is polynomial in $n$ and $r$. The reader is referred to\nSection~2.5 of \\cite{Sason2024} for an account of the various interesting properties of\nthe Lov\\'{a}sz $\\vartheta$-function. Among these properties, the sandwich theorem states that\nfor every graph $\\Gr{G}$, the following inequalities hold:\n\\begin{align}\n\\label{eq1: sandwich}\n\\indnum{\\Gr{G}} \\leq \\vartheta(\\Gr{G}) \\leq \\chrnum{\\CGr{G}}, \\\\\n\\label{eq2: sandwich}\n\\clnum{\\Gr{G}} \\leq \\vartheta(\\CGr{G}) \\leq \\chrnum{\\Gr{G}}.\n\\end{align}\nThe usefulness of \\eqref{eq1: sandwich} and \\eqref{eq2: sandwich} lies in the fact that while the\nindependence, clique, and chromatic numbers of a graph are NP-hard to compute, the Lov\\'{a}sz\n$\\vartheta$-function can be efficiently computed as a bound in these inequalities by solving the\nconvex optimization problem in \\eqref{eq: SDP problem - Lovasz theta-function}.\n\n\\bigskip\n\\subsection{Matrices associated with a graph}\n\\label{subsection: Matrices associated with a graph}\n\n\\subsubsection{Four matrices associated with a graph}\n\n\\noindent\n\n\\vspace*{0.1cm}\nLet $\\Gr{G}=(\\Vertex,\\Edge)$ be a graph with vertices $\\left\\{ v_{1},...,v_{n}\\right\\} $.\nThere are several matrices associated with $\\Gr{G}$. In this survey, we consider four of them,\nall are symmetric matrices in $\\mathbb{R}^{n\\times n}$: the \\emph{adjacency matrix} ($\\A$),\n\\emph{Laplacian matrix} ($LM$), \\emph{signless Laplacian matrix} ($\\Q$), and the\n\\emph{normialized Laplacian matrix} (${\\bf{\\mathcal{L}}}$).\n\n\\begin{enumerate}\n\\item The adjacency matrix of a graph $\\Gr{G}$, denoted by $\\A = \\A(\\Gr{G})$, has the binary-valued entries\n\\begin{align}\n\\label{eq: adjacency matrix}\n(\\A(\\Gr{G}))_{i,j}=\n\\begin{cases}\n1 & \\mbox{if} \\, \\{v_i,v_j\\} \\in \\E{\\Gr{G}}, \\\\\n0 & \\mbox{if} \\, \\{v_i,v_j\\} \\notin \\E{\\Gr{G}}.\n\\end{cases}\n\\end{align}\n\\item The Laplacian matrix of a graph $\\Gr{G}$, denoted by $\\LM = \\LM(\\Gr{G})$, is given by\n\\begin{align}\n\\LM(\\Gr{G}) = \\D(\\Gr{G})-\\A(\\Gr{G}),\n\\end{align}\nwhere\n\\begin{align}\n\\D(\\Gr{G}) = \\diag{d(v_1), d(v_2), \\ldots ,d(v_n)}\n\\end{align}\nis the {\\em diagonal matrix} whose entries in the principal diagonal are the degrees of the $n$ vertices of $\\Gr{G}$.\n\\item The signless Laplacian martix of a graph $\\Gr{G}$, denoted by $\\Q = \\Q(\\Gr{G})$, is given by\n\\begin{align}\n\\label{eq: signless Laplacian martix}\n\\Q(\\Gr{G}) = \\D(\\Gr{G})+\\A(\\Gr{G}).\n\\end{align}\n\\item The normalized Laplacian matrix of a graph $\\Gr{G}$, denoted by $\\mathcal{L}(\\Gr{G})$, is given by\n\\begin{align}\n\\label{eq: normalized Laplacian matrix}\n\\mathcal{L}(\\Gr{G}) = \\D^{-\\frac12}(\\Gr{G}) \\, \\LM(\\Gr{G}) \\, \\D^{-\\frac12}(\\Gr{G}),\n\\end{align}\nwhere\n\\begin{align}\n\\D^{-\\frac12}(\\Gr{G}) = \\diag{d^{-\\frac12}(v_1), d^{-\\frac12}(v_2), \\ldots, d^{-\\frac12}(v_n)},\n\\end{align}\nwith the convention that if $v \\in \\V{\\Gr{G}}$ is an isolated vertex in $\\Gr{G}$ (i.e., $d(v)=0$), then\n$d^{-\\frac12}(v) = 0$.\t\nThe entries of ${\\bf{\\mathcal{L}}} = (\\mathcal{L}_{i,j})$ are given by\n\\begin{align}\n\\mathcal{L}_{i,j} =\n\\begin{dcases}\n\\begin{array}{cl}\n1, \\quad & \\mbox{if $i=j$ and $d(v_i) \\neq 0$,} \\\\[0.2cm]\n-\\dfrac{1}{\\sqrt{d(v_i) \\, d(v_j)}}, \\quad & \\mbox{if $i \\neq j$ and $\\{v_i,v_j\\} \\in \\E{\\Gr{G}}$}, \\\\[0.5cm]\n0, \\quad & \\mbox{otherwise}.\n\\end{array}\n\\end{dcases}\n\\end{align}\t\n\\end{enumerate}\n\nIn the continuation of this section, we also occasionally refer to two other matrices that are associated with undirected graphs.\n\\begin{definition}\n\\label{definition: incidence matrix}\nLet $\\Gr{G}$ be a graph with $n$ vertices and $m$ edges. The {\\em incidence matrix} of $\\Gr{G}$, denoted by ${\\mathbf{B}} = {\\mathbf{B}}(\\Gr{G})$\nis an $n \\times m$ matrix with binary entries, defined as follows:\n\\begin{align}\nB_{i,j} =\n\\begin{cases}\n1 & \\text{if vertex \\(v_i \\in \\V{\\Gr{G}}\\) is incident to edge \\(e_j \\in \\E{\\Gr{G}}\\)}, \\\\\n0 & \\text{if vertex \\(v_i \\in \\V{\\Gr{G}}\\) is not incident to edge \\(e_j \\in \\E{\\Gr{G}}\\)}.\n\\end{cases}\n\\end{align}\nFor an undirected graph, each edge $e_j$ connects two vertices $v_i$ and $v_k$, and the corresponding column in\n$\\mathbf{B}$ has exactly two $1$'s, one for each vertex.\n\\end{definition}\n\n\\begin{definition}\n\\label{definition: oriented incidence matrix}\nLet $\\Gr{G}$ be a graph with $n$ vertices and $m$ edges. An {\\em oriented incidence matrix} of $\\Gr{G}$, denoted by ${\\mathbf{N}} = {\\mathbf{N}}(\\Gr{G})$\nis an $n \\times m$ matrix with ternary entries from $\\{-1, 0, 1\\}$, defined as follows. One first selects an arbitrary orientation to each edge in $\\Gr{G}$,\nand then define\n\\begin{align}\nN_{i,j} =\n\\begin{cases}\n-1 & \\text{if vertex \\(v_i \\in \\V{\\Gr{G}}\\) is the tail (starting vertex) of edge \\(e_j \\in \\E{\\Gr{G}}\\)}, \\\\\n+1 & \\text{if vertex \\(v_i \\in \\V{\\Gr{G}}\\) is the head (ending vertex) of edge \\(e_j \\in \\E{\\Gr{G}}\\)}, \\\\\n\\hspace*{0.2cm} 0 & \\text{if vertex \\(v_i \\in \\V{\\Gr{G}}\\) is not incident to edge \\(e_j \\in \\E{\\Gr{G}}\\)}.\n\\end{cases}\n\\end{align}\nConsequently, each column of $\\mathbf{N}$ contains exactly one entry equal to 1 and one entry equal to $-1$, representing the head and tail of the corresponding\noriented edge in the graph, respectively, with all other entries in the column being zeros.\n\\end{definition}\n\nFor $X\\in \\left\\{ A,L,Q,\\mathcal{L} \\right\\}$, the \\emph{$X$-spectrum} of a graph $\\Gr{G}$, $\\sigma_X(G)$, is the multiset of the eigenvalues\nof $X(G)$. We denote the elements of the multiset of eigenvalues of $\\{\\A, \\LM, \\Q, \\mathcal{L}\\}$, respectively, by\n\\begin{align}\n\\label{eq2:26.09.23}\n& \\Eigval{1}{\\Gr{G}} \\geq \\Eigval{2}{\\Gr{G}} \\geq \\ldots \\geq \\Eigval{n}{\\Gr{G}}, \\\\\n\\label{eq3:26.09.23}\n& \\mu_1(\\Gr{G}) \\leq \\mu_2(\\Gr{G}) \\leq \\ldots \\leq \\mu_n(\\Gr{G}), \\\\\n\\label{eq4:26.09.23}\n& \\nu_1(\\Gr{G}) \\geq \\nu_2(\\Gr{G}) \\geq \\ldots \\geq \\nu_n(\\Gr{G}), \\\\\n\\label{eq5:26.09.23}\n& \\delta_1(\\Gr{G}) \\leq \\delta_2(\\Gr{G}) \\leq \\ldots \\leq \\delta_n(\\Gr{G}).\n\\end{align}\n\n\n\\begin{example}\nConsider the complete bipartite graph $\\Gr{G} = \\CoBG{2}{3}$ with the adjacency matrix\n$$\\A(\\Gr{G})=\n\\begin{pmatrix}\n {\\bf{0}}_{2,2} & \\J{2,3} \\\\\n \\J{3,2} & {\\bf{0}}_{3,3}\n\\end{pmatrix}.$$\nThe spectra of $\\Gr{G}$ can be verified to be given as follows:\n\\begin{enumerate}\n\\item\nThe $\\A$-spectrum of $\\Gr{G}$ is\n\\begin{align}\n\\sigma_{\\A}(\\Gr{G})=\\Bigl\\{ -\\sqrt{6}, [0]^{3}, \\sqrt{6}\\Bigr\\},\n\\end{align}\nwith the notation that $[\\lambda]^m$ means that $\\lambda$ is an eigenvalue with multiplicity $m$.\n\n\\item The $\\LM$-spectrum of $\\Gr{G}$ is\n\\begin{align}\n\\sigma_{\\LM}(\\Gr{G})=\\Bigl\\{ 0, [2]^{2}, 3, 5\\Bigr\\} .\n\\end{align}\n\t\t\n\\item The $\\Q$-spectrum of $\\Gr{G}$ is\n\\begin{align}\n\\sigma_{\\Q}(\\Gr{G})=\\Bigl\\{ 0, [2]^{2}, 3, 5\\Bigr\\} .\n\\end{align}\n\n\\item The ${\\bf{\\mathcal{L}}}$-spectrum of $\\Gr{G}$ is\n\\begin{align}\n\\sigma_{{\\bf{\\mathcal{L}}}}(\\Gr{G})=\\Bigl\\{ 0, [1]^{3}, 2 \\Bigr\\} .\n\\end{align}\n\\end{enumerate}\n\\end{example}\n\n\\begin{remark}\nIf $\\Gr{H}$ is an induced subgraph of a graph $\\Gr{G}$, then $\\A(\\Gr{H})$ is a principal submatrix of $A(\\Gr{G})$.\nHowever, since the degrees of the remaining vertices are affected by the removal of vertices when forming the induced\nsubgraph $\\Gr{H}$ from the graph $\\Gr{G}$, this property does not hold for the other three associated matrices discussed\nin this paper (namely, the Laplacian, signless Laplacian, and normalized Laplacian matrices).\n\\end{remark}\n\n\\begin{definition}\nLet $\\Gr{G}$ be a graph, and let $\\CGr{G}$ be the complement graph of $\\Gr{G}$. Define the following matrices:\n\\begin{enumerate}\n\\item $\\overline{\\A}(\\Gr{G}) = \\A(\\overline{\\Gr{G}})$.\n\\item $\\overline{\\LM}(\\Gr{G}) = \\LM(\\overline{\\Gr{G}})$.\n\\item $\\overline{\\Q}(\\Gr{G}) = \\Q(\\overline{\\Gr{G}})$.\n\\item $\\overline{{\\bf{\\mathcal{L}}}}(\\Gr{G}) = {\\bf{\\mathcal{L}}}(\\overline{\\Gr{G}})$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\nLet $\\mathcal{X} \\subseteq \\Gmats$. The $\\mathcal{X}$-spectrum of a graph $\\Gr{G}$ is a list with $\\sigma_X(\\Gr{G})$\nfor every $X\\in \\mathcal{X}$.\n\\end{definition}\n\nObserve that if $\\mathcal{X} = \\{ X \\}$ is a singleton, then the $\\mathcal{X}$ spectrum is equal to the $X$-spectrum.\n\nWe now describe some important applications of the four matrices.\n\n\\subsubsection{Properties of the adjacency matrix}\n\\begin{theorem}[Number of walks of a given length between two fixed vertices]\n\\label{thm: number of walks of a given length}\nLet $\\Gr{G} = (\\Vertex, \\Edge)$ be a graph with a vertex set $\\Vertex = \\V{\\Gr{G}} = \\{ v_1, \\ldots, v_n\\}$, and\nlet $\\A = \\A(\\Gr{G})$ be the adjacency matrix of $\\Gr{G}$.\nThen, the number of walks of length $\\ell$, with the fixed endpoints $v_i$ and $v_j$, is equal to $(\\A^\\ell)_{i,j}$.\n\\end{theorem}\n\n\\begin{corollary}[Number of closed walks of a given length]\n\\label{corollary: Number of Closed Walks of a Given Length}\nLet $\\Gr{G} = (\\Vertex, \\Edge)$ be a simple undirected graph on $n$ vertices with an adjacency matrix\n$\\A = \\A(\\Gr{G})$, and let its spectrum (with respect to $\\A$) be given by $\\{\\lambda_j\\}_{j=1}^n$.\nThen, for all $\\ell \\in \\naturals$, the number of closed walks of length $\\ell$ in $\\Gr{G}$ is equal\nto $\\sum_{j=1}^n \\lambda_j^{\\ell}$.\n\\end{corollary}\n\n\\begin{corollary}[Number of edges and triangles in a graph]\n\\label{corollary: number of edges and triangles in a graph}\nLet $\\Gr{G}$ be a simple undirected graph with $n = \\card{\\V{\\Gr{G}}}$ vertices,\n$e = \\card{\\E{\\Gr{G}}}$ edges, and $t$ triangles. Let $\\A = \\A(\\Gr{G})$ be the\nadjacency matrix of $\\Gr{G}$, and let $\\{\\lambda_j\\}_{j=1}^n$ be its adjacency\nspectrum. Then,\n\\begin{align}\n& \\sum_{j=1}^n \\lambda_j = \\mathrm{tr}(\\A) = 0, \\label{eq: trace of A is zero} \\\\\n& \\sum_{j=1}^n \\lambda_j^2 = \\mathrm{tr}(\\A^2) = 2 e, \\label{eq: number of edges from A} \\\\\n& \\sum_{j=1}^n \\lambda_j^3 = \\mathrm{tr}(\\A^3) = 6 t. \\label{eq: number of triangles from A}\n\\end{align}\n\\end{corollary}\n\nFor a $d$-regular graph, the largest eigenvalue of its adjacency matrix is equal to~$d$.\nConsequently, by Eq.~\\eqref{eq: number of edges from A}, for $d$-regular graphs,\n$\\sum_j \\lambda_j^2 = 2e = nd = n \\lambda_1$. Interestingly, this turns to be\na necessary and sufficient condition for the regularity of a graph, which means\nthat the adjacency spectrum enables to identify whether a graph is regular.\n\\begin{theorem} \\cite[Corollary~3.2.2]{CvetkovicRS2010}\n\\label{theorem: graph regularity from A-spectrum}\nA graph $\\Gr{G}$ on $n$ vertices is regular if and only if\n\\begin{align}\n\\sum_{i=1}^n \\lambda_i^2 = n \\lambda_1,\n\\end{align}\nwhere $\\lambda_1$ is the largest eigenvalue of the adjacency matrix of $\\Gr{G}$.\n\\end{theorem}\n\n\\begin{theorem}[The eigenvalues of strongly regular graphs]\n\\label{theorem: eigenvalues of srg}\nThe following spectral properties are satisfied by the family of strongly regular graphs:\n\\begin{enumerate}[(1)]\n\\item \\label{Item 1: eigenvalues of srg}\nA strongly regular graph has at most three distinct eigenvalues.\n\\item \\label{Item 2: eigenvalues of srg}\nLet $\\Gr{G}$ be a connected strongly regular graph, and let its parameters be $\\SRG(n,d,\\lambda,\\mu)$.\nThen, the largest eigenvalue of its adjacency matrix is $\\Eigval{1}{\\Gr{G}} = d$ with multiplicity~1,\nand the other two distinct eigenvalues of its adjacency matrix are given by\n\\begin{align}\n\\label{eigs-SRG}\np_{1,2} = \\tfrac12 \\, \\Biggl( \\lambda - \\mu \\pm \\sqrt{ (\\lambda-\\mu)^2 + 4(d-\\mu) } \\, \\Biggr),\n\\end{align}\nwith the respective multiplicities\n\\begin{align}\n\\label{eig-multiplicities-SRG}\nm_{1,2} = \\tfrac12 \\, \\Biggl( n-1 \\mp \\frac{2d+(n-1)(\\lambda-\\mu)}{\\sqrt{(\\lambda-\\mu)^2+4(d-\\mu)}} \\, \\Biggr).\n\\end{align}\n\\item \\label{Item 3: eigenvalues of srg}\nA connected regular graph with exactly three distinct eigenvalues is strongly regular.\n\\item \\label{Item 4: eigenvalues of srg}\nStrongly regular graphs for which $2d+(n-1)(\\lambda-\\mu) \\neq 0$ have integral eigenvalues and the multiplicities\nof $p_{1,2}$ are distinct.\n\\item \\label{Item 5: eigenvalues of srg}\nA connected regular graph is strongly regular if and only if it has three distinct eigenvalues, where the largest\neigenvalue is of multiplicity~1.\n\\item \\label{Item 6: eigenvalues of srg}\nA disconnected strongly regular graph is a disjoint union of $m$ identical complete graphs $\\CoG{r}$, where $m \\geq 2$ and $r \\in \\naturals$.\nIt belongs to the family $\\srg{mr}{r-1}{r-2}{0}$, and its adjacency spectrum is $\\{ (r-1)^{[m]}, (-1)^{[m(r-1)]} \\}$, where superscripts\nindicate the multiplicities of the eigenvalues, thus having two distinct eigenvalues.\n\\end{enumerate}\n\\end{theorem}\n\nThe following result follows readily from Theorem~\\ref{theorem: eigenvalues of srg}.\n\\begin{corollary}\n\\label{corollary: cospectral SRGs}\nStrongly regular graphs with identical parameters $(n,d,\\lambda,\\mu)$ are cospectral.\n\\end{corollary}\n\n\\begin{remark}\n\\label{remark: NICS SRGs}\nStrongly regular graphs having identical parameters $(n, d, \\lambda, \\mu)$ are\ncospectral but may not be isomorphic. For instance, Chang graphs form a set\nof three nonisomorphic strongly regular graphs with identical parameters\n$\\srg{28}{12}{6}{4}$ \\cite[Section~10.11]{BrouwerM22}. Consequently, the three\nChang graphs are strongly regular NICS graphs.\n\\end{remark}\n\nAn important class of strongly regular graphs, for which $2d+(n-1)(\\lambda-\\mu)=0$, is given by the family of conference graphs.\n\\begin{definition}[Conference graphs]\n\\label{definition: conference graphs}\nA conference graph on $n$ vertices is a strongly regular graph with the parameters\n$\\srg{n}{\\tfrac12(n-1)}{\\tfrac14(n-5)}{\\tfrac14(n-1)}$, where $n$ must satisfy $n=4k+1$\nwith $k \\in \\naturals$.\n\\end{definition}\nIf $\\Gr{G}$ is a conference graph on $n$ vertices, then so is its complement $\\CGr{G}$; it is, however,\nnot necessarily self-complementary. By Theorem~\\ref{theorem: eigenvalues of srg}, the distinct eigenvalues of\nthe adjacency matrix of $\\Gr{G}$ are given by $\\tfrac12 (n-1)$, $\\tfrac12 (\\hspace*{-0.1cm}\n\\sqrt{n}-1)$, and $-\\tfrac12 (\\hspace*{-0.1cm} \\sqrt{n}+1)$ with multiplicities\n$1, \\tfrac12 (n-1)$, and $\\tfrac12 (n-1)$, respectively. In contrast to Item~\\ref{Item 4: eigenvalues of srg}\nof Theorem~\\ref{theorem: eigenvalues of srg}, the eigenvalues $\\pm \\tfrac12 (\\hspace*{-0.1cm} \\sqrt{n}+1)$ are\nnot necessarily integers. For instance, the cycle graph $\\CG{5}$, which is a conference graph, has an adjacency\nspectrum $\\bigl\\{2, \\bigl[\\tfrac12 (\\hspace*{-0.1cm} \\sqrt{5}-1) \\bigr]^{(2)},\n\\bigl[-\\tfrac12 (\\hspace*{-0.1cm} \\sqrt{5}+1) \\bigr]^{(2)} \\}$. Thus, apart from the largest eigenvalue, the\nother eigenvalues are irrational numbers.\n\n\n\\subsubsection{Properties of the Laplacian matrix}\n\\begin{theorem}\n\\label{theorem: On the Laplacian matrix of a graph}\nLet $\\Gr{G}$ be a finite, simple, and undirected graph, and let $\\LM$ be the Laplacian matrix of $\\Gr{G}$. Then,\n\\begin{enumerate}\n\\item \\label{Item 1: Laplacian matrix of a graph}\nThe Laplacian matrix $\\LM = {\\mathbf{N}} {\\mathbf{N}}^{\\mathrm{T}}$ is positive semidefinite,\nwhere ${\\mathbf{N}}$ is the oriented incidence matrix of $\\Gr{G}$ (see Definition~\\ref{definition: oriented incidence matrix} and \\cite[p.~185]{CvetkovicRS2010}).\n\\item \\label{Item 2: Laplacian matrix of a graph}\nThe smallest eigenvalue of $\\, \\LM$ is zero, with a multiplicity equal to the number of components in $\\Gr{G}$ (see \\cite[Theorem~7.1.2]{CvetkovicRS2010}).\n\\item \\label{Item 3: Laplacian matrix of a graph}\nThe size of the graph, $\\bigcard{\\E{\\Gr{G}}}$, equals one-half of the sum of the eigenvalues of $\\, \\LM$, counted with multiplicities (see\n\\cite[Eq.~(7.4)]{CvetkovicRS2010}).\n\\end{enumerate}\n\\end{theorem}\n\nThe following celebrated theorem provides an operational meaning of the $\\LM$-spectrum of graphs in counting their number of\nspanning subgraphs.\n\\begin{theorem}[Kirchhoff's Matrix-Tree Theorem \\cite{Kirchhoff1958}]\n\\label{theorem: number of spanning trees}\nThe number of spanning trees in a connected and simple graph $\\Gr{G}$ on $n$ vertices is determined by the $n-1$ nonzero\neigenvalues of the Laplacian matrix, and it is equal to $\\frac{1}{n} \\overset{n}{\\underset{\\ell=2}{\\prod}} \\, \\mu_\\ell(\\Gr{G})$.\n\\end{theorem}\n\n\\begin{corollary}[Cayley's Formula \\cite{Cayley1889}]\n\\label{corollary: number of spanning trees}\nThe number of spanning trees of $\\CoG{n}$ is $n^{n-2}$.\n\\end{corollary}\n\\begin{proof}\nThe $\\LM$-spectrum of $\\CoG{n}$ is given by $\\{0, [n]^{n-1}\\}$, and the result readily follows from Theorem~\\ref{theorem: number of spanning trees}.\n\\end{proof}\n\t\n\\subsubsection{Properties of the signless Laplacian matrix}\n\\begin{theorem}\n\\label{theorem: On the signless Laplacian matrix of a graph}\nLet $\\Gr{G}$ be a finite, simple, and undirected graph, and let $\\Q$ be\nthe signless Laplacian matrix of $\\Gr{G}$. Then,\n\\begin{enumerate}\n\\item \\label{Item 1: signless Laplacian matrix of a graph}\nThe matrix $\\Q$ is positive semidefinite. Moreover, it is a completely positive matrix, expressed as $\\Q = {\\mathbf{B}} {\\mathbf{B}}^{\\mathrm{T}}$, where\n${\\mathbf{B}}$ is the incidence matrix of $\\Gr{G}$ (see Definition~\\ref{definition: incidence matrix} and \\cite[Section~2.4]{CvetkovicRS2010}).\n\\item \\label{Item 2: signless Laplacian matrix of a graph}\nIf $\\Gr{G}$ is a connected graph, then it is bipartite if and only if the least eigenvalue of $\\Q$ is equal to zero. In this case,\n$0$ is a simple $\\Q$-eigenvalue (see \\cite[Theorem~7.8.1]{CvetkovicRS2010}).\n\\item \\label{Item 3: signless Laplacian matrix of a graph}\nThe multiplicity of 0 as an eigenvalue of $\\Q$ is equal to the number of bipartite components in $\\Gr{G}$ (see \\cite[Corollary~7.8.2]{CvetkovicRS2010}).\n\\item \\label{Item 4: signless Laplacian matrix of a graph}\nThe size of the graph $\\bigl| E(\\Gr{G}) \\bigr| $ is equal to one-half the sum of the eigenvalues of~$\\Q$, counted with multiplicities\n(see \\cite[Corollary~7.8.9]{CvetkovicRS2010}).\n\\end{enumerate}\n\\end{theorem}\nThe interested reader is referred to \\cite{OliveiraLAK2010} for bounds on the $\\Q$-spread (i.e., the difference between the largest and smallest\neigenvalues of the signless Laplacian matrix), expressed as a function of the number of vertices in the graph. In regard to\nItem~\\ref{Item 2: signless Laplacian matrix of a graph} of Theorem~\\ref{theorem: On the signless Laplacian matrix of a graph}, the interested\nreader is referred to \\cite{Cardoso2008} for a lower bound on the least eigenvalue of signless Laplacian matrix for connected non-bipartite graphs,\nand to \\cite{ChenH2018} for a lower bound on the least eigenvalue of signless Laplacian matrix for a general simple graph with a fixed number of\nvertices and edges.\n\n\\subsubsection{Properties of the normalized Laplacian matrix}\nThe normalized Laplacian matrix of a graph, defined in \\eqref{eq: normalized Laplacian matrix}, exhibits several\ninteresting spectral properties, which are introduced below.\n\\begin{theorem} \\cite{CvetkovicRS2010,CvetkovicRS2007}\n\\label{theorem: On the normalized Laplacian matrix of a graph}\nLet $\\Gr{G}$ be a finite, simple, and undirected graph, and let ${\\bf{\\mathcal{L}}}$ be\nthe normalized Laplacian matrix of $\\Gr{G}$. Then,\n\\begin{enumerate}\n\\item \\label{Item 1: normalized Laplacian matrix of a graph}\nThe eigenvalues of ${\\bf{\\mathcal{L}}}$ lie in the interval $[0,2]$ (see \\cite[Section~7.7]{CvetkovicRS2010}).\n\\item \\label{Item 2: normalized Laplacian matrix of a graph}\nThe number of components in $\\Gr{G}$ is equal to the multiplicity of~0 as an eigenvalue of ${\\bf{\\mathcal{L}}}$ (see \\cite[Theorem~7.7.3]{CvetkovicRS2010}).\n\\item \\label{Item 3: normalized Laplacian matrix of a graph}\nThe largest eigenvalue of ${\\bf{\\mathcal{L}}}$ is equal to~2 if and only if the graph has a bipartite component (see \\cite[Theorem~7.7.2(v)]{CvetkovicRS2010}).\nFurthermore, the number of the bipartite components of $\\Gr{G}$ is equal to the multiplicity of~2 as an eigenvalue of~${\\bf{\\mathcal{L}}}$.\n\\item \\label{Item 4: normalized Laplacian matrix of a graph}\nThe sum of its eigenvalues (including multiplicities) is less than or equal to the graph order $(n)$, with equality if and only\nif the graph has no isolated vertices (see \\cite[Theorem~7.7.2(i)]{CvetkovicRS2010}).\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsubsection{More on the spectral properties of the four associated matrices}\n\n\\noindent\n\nThe following theorem considers equivalent spectral properties of bipartite graphs.\n\\begin{theorem}\n\\label{theorem: equivalences for bipartite graphs}\nLet $\\Gr{G}$ be a graph. The following are equivalent:\n\\begin{enumerate}\n\\item \\label{Item 1: TFAE bipartite graphs}\n$\\Gr{G}$ is a bipartite graph.\n\\item \\label{Item 2: TFAE bipartite graphs}\n$\\Gr{G}$ does not have cycles of odd length.\n\\item \\label{Item 3: TFAE bipartite graphs}\nThe $\\A$-spectrum of $\\Gr{G}$ is symmetric around zero, and for every eigenvalue $\\lambda$ of $\\A(G)$,\nthe eigenvalue $-\\lambda$ is of the same multiplicity \\cite[Theorem~3.2.3]{CvetkovicRS2010}.\n\\item \\label{Item 4: TFAE bipartite graphs}\nThe $\\LM$-spectrum and $\\Q$-spectrum are identical (see \\cite[Proposition~7.8.4]{CvetkovicRS2010}).\n\\item \\label{Item 5: TFAE bipartite graphs}\nThe ${\\bf{\\mathcal{L}}}$-spectrum has the same multiplicity of $0$'s and $2$'s as eigenvalues (see\n\\cite[Corollary~7.7.4]{CvetkovicRS2010}).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\n\\label{remark: on connected bipartite graphs}\nItem~\\ref{Item 3: TFAE bipartite graphs} of Theorem~\\ref{theorem: equivalences for bipartite graphs} can be\nstrengthened if $\\Gr{G}$ is a connected graph. In that case, $\\Gr{G}$ is bipartite if and only if $\\lambda_1 = -\\lambda_n$\n(see \\cite[Theorem~3.2.4]{CvetkovicRS2010}).\n\\end{remark}\n\n\\begin{table}[hbt]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\textbf{Matrix} & \\textbf{\\# edges} & \\textbf{bipartite} & \\textbf{\\# components} & \\textbf{\\# bipartite components} & \\textbf{\\# of closed walks} \\\\\n\\hline\n$\\A$ & Yes & Yes & No & No & Yes \\\\\n\\hline\n$\\LM$ & Yes & No & Yes & No & No \\\\\n\\hline\n$\\Q$ & Yes & No & No & Yes & No \\\\\n\\hline\n${\\bf{\\mathcal{L}}}$ & No & Yes & Yes & Yes & No \\\\\n\\hline\n\\end{tabular}\n\\caption{Some properties of a finite, simple, and undirected graph that one can or cannot determine\nby the $X$-spectrum for $X\\in \\{\\A,\\LM,\\Q, {\\bf{\\mathcal{L}}} \\}$} \\label{table:properties_determined by the spectrum}\n\\end{table}\nTable~\\ref{table:properties_determined by the spectrum}, borrowed from \\cite{Butler2014}, lists properties of a graph that\ncan or cannot be determined by the $X$-spectrum for $X\\in \\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$. From the $\\A$-spectrum of a\ngraph $\\Gr{G}$, one can determine the number of edges and the number of triangles in $\\Gr{G}$\n(by Eqs.~\\eqref{eq: number of edges from A} and \\eqref{eq: number of triangles from A}, respectively), and whether the graph\nis bipartite or not (by Item~\\ref{Item 3: TFAE bipartite graphs} of Theorem~\\ref{theorem: equivalences for bipartite graphs}).\nHowever, the $\\A$ spectrum does not indicate the number of components (see Example~\\ref{example: ANICS graphs with 5 vertices}).\nFrom the $\\LM$-spectrum of a graph $\\Gr{G}$, one can determine the number of edges\n(by Item~\\ref{Item 3: Laplacian matrix of a graph} of Theorem~\\ref{theorem: On the Laplacian matrix of a graph}),\nthe number of spanning trees (by Theorem~\\ref{theorem: number of spanning trees}),\nthe number of components of $\\Gr{G}$ (by Item~\\ref{Item 2: Laplacian matrix of a graph} of\nTheorem~\\ref{theorem: On the Laplacian matrix of a graph}), but not the number of its triangles, and\nwhether the graph $\\Gr{G}$ is bipartite. From the $\\Q$-spectrum, one can determine whether the graph is bipartite, the number\nof bipartite components, and the number of edges (respectively, by Items~\\ref{Item 3: signless Laplacian matrix of a graph}\nand~\\ref{Item 4: signless Laplacian matrix of a graph} of Theorem~\\ref{theorem: On the signless Laplacian matrix of a graph}),\nbut not the number of components of the graph, and whether the graph is bipartite (see Remark~\\ref{remark: bipartiteness}).\nFrom the ${\\bf{\\mathcal{L}}}$-spectrum, one can determine the number of components and the\nnumber of bipartite components in $\\Gr{G}$ (by Theorem~\\ref{theorem: On the normalized Laplacian matrix of a graph}),\nand whether the graph is bipartite (by Items~\\ref{Item 1: TFAE bipartite graphs} and~\\ref{Item 5: TFAE bipartite graphs} of\nTheorem~\\ref{theorem: equivalences for bipartite graphs}). The number of closed walks in $\\Gr{G}$ is determined by\nthe $\\A$-spectrum (by Corollary~\\ref{corollary: Number of Closed Walks of a Given Length}), but not by the spectra with\nrespect to the other three matrices.\n\n\\begin{remark}\n\\label{remark: bipartiteness}\nBy Item~\\ref{Item 2: signless Laplacian matrix of a graph} of Theorem~\\ref{theorem: On the signless Laplacian matrix of a graph},\na connected graph is bipartite if and only if the least eigenvalue of its signless Laplacian matrix is equal to zero.\nIf the graph is disconnected and it has a bipartite component and a non-bipartite component, then the least eigenvalue of its\nsignless Laplacian matrix is equal to zero, although the graph is not bipartite.\nAccording to Table~\\ref{table:properties_determined by the spectrum}, the $\\Q$-spectrum alone does not determine whether a\ngraph is bipartite. This is due to the fact that the $\\Q$-spectrum does not provide information about the number of components in the\ngraph or whether the graph is connected.\nIt is worth noting that while neither the $\\LM$-spectrum nor the $\\Q$-spectrum independently determines whether a graph is bipartite,\nthe combination of these spectra does. Specifically, by Item~\\ref{Item 4: TFAE bipartite graphs} of\nTheorem~\\ref{theorem: equivalences for bipartite graphs}, the combined knowledge of both spectra enables to establish this property.\n\\end{remark}\n\n\\section{Graphs determined by their spectra}\n\\label{section: DS graphs}\n\nThe spectral determination of graphs has long been a central topic in spectral graph theory. A major open question in this area is:\n\"Which graphs are determined by their spectrum (DS)?\" This section begins our survey of both classical and recent results on spectral\ngraph determination. We explore the spectral characterization of various graph classes, methods for constructing or distinguishing\ncospectral nonisomorphic graphs, and conditions under which a graph’s spectrum uniquely determines its structure. Additionally, we\npresent newly obtained proofs of existing results, offering further insights into this field.\n\n\\begin{definition}\nLet $\\Gr{G},\\Gr{H}$ be two graphs. A mapping $\\phi \\colon \\V{\\Gr{G}} \\rightarrow \\V{\\Gr{H}}$ is a\n\\emph{graph isomorphism} if\n\\begin{align}\n\\{u,v\\} \\in \\E{\\Gr{G}} \\iff \\bigl\\{ \\phi(u),\\phi(v) \\bigr\\} \\in \\E{\\Gr{H}}.\n\\end{align}\nIf there is an isomorphism between $\\Gr{G}$ and $\\Gr{H}$, we say that these graphs are \\emph{isomorphic}.\n\\end{definition}\n\n\\begin{definition}\nA \\emph{permutation matrix} is a $\\{0,1\\}$--matrix in which each row and each column contains exactly one entry equal to $1$.\n\\end{definition}\n\n\\begin{remark}\nIn terms of the adjacency matrix of a graph, $\\Gr{G}$ and $\\Gr{H}$ are cospectral graphs if $\\A(\\Gr{G})$ and $\\A(\\Gr{H})$\nare similar matrices, and $\\Gr{G}$ and $\\Gr{H}$ are isomorphic if the similarity of their adjacency matrices is through\na permutation matrix ${\\bf{P}}$, i.e.\n\\begin{align}\nA(\\Gr{G}) = {\\bf{P}} \\, \\A(\\Gr{H}) \\, {\\bf{P}}^{-1}.\n\\end{align}\n\\end{remark}\n\n\n\\subsection{Graphs determined by their adjacency spectrum (DS graphs)}\n\\label{subsection: Graphs determined by their adjacency spectrum}\n\n\\begin{theorem} \\cite{vanDamH03}\n\\label{theorem: van Dam and Haemers, 2003 - thm1}\nAll of the graphs with less than five vertices are DS.\n\\end{theorem}\n\n\\begin{example}\n\\label{example: ANICS graphs with 5 vertices}\nThe star graph $\\SG{5}$ and a graph formed by the disjoint union of a length-4 cycle and an isolated vertex,\n$\\CG{4} \\DU \\CoG{1}$, have the same $\\A$-spectrum $\\{-2 , [0]^3 , 2\\}$. They are, however, not isomorphic\nsince $\\SG{5}$ is connected and $\\CG{4} \\DU \\CoG{1}$ is disconnected (see Figure~\\ref{fig:graphs with 5 vertices}).\n\\vspace*{-0.1cm}\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=8cm]{ANICS_graph_with_5_vertices.png}\n\\caption{The graphs $\\SG{4} = \\CoBG{1}{4}$ and $\\CG{4} \\DU \\CoG{1}$ (i.e., a union of a 4-length cycle\nand an isolated vertex) are cospectral and nonisomorphic graphs ($\\A$-NICS graphs) on five vertices.\nThese two graphs therefore cannot be determined by their adjacency matrix.}\n\\label{fig:graphs with 5 vertices}\n\\end{figure}\nIt can be verified computationally that all the connected nonisomorphic graphs on five vertices can be\ndistinguished by their $\\A$-spectrum (see \\cite[Appendix~A1]{CvetkovicRS2010}).\n\\end{example}\n\n\\begin{theorem} \\cite{vanDamH03}\n\\label{theorem: van Dam and Haemers, 2003 - thm2}\nAll the regular graphs with less than ten vertices are DS (and, as will be clarified later, also\n$\\mathcal{X}$-DS for every $\\mathcal{X} \\subseteq \\{\\A, \\LM, \\Q\\}$).\n\\end{theorem}\n\n\\begin{example}\n\\label{example: NICS regular graphs on 10 vertices}\n\\cite{vanDamH03} The following two regular graphs in Figure \\ref{fig:graphs with 10 vertices} are\n$\\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$-NICS.\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=12cm]{cospectral_and_nonisomorphic_4-regular_graphs.png}\n\\caption{$\\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$-NICS regular graphs with $10$ vertices.\nThese cospectral graphs are nonisomorphic because each of the two blue edges in $\\Gr{G}$ belongs\nto three triangles, whereas no such an edge exists in $\\Gr{H}$.}\\label{fig:graphs with 10 vertices}\n\\end{figure}\nThe regular graphs $\\Gr{G}$ and $\\Gr{H}$ in Figure~\\ref{fig:graphs with 10 vertices} can be verified to be cospectral\nwith the common characteristic polynomial $$P(x)= x^{10} - 20x^8 - 16x^7 + 110x^6 + 136x^5 - 180x^4 - 320x^3 + 9x^2 + 200x + 80.$$\nThese graphs are also nonisomorphic because each of the two blue edges in $\\Gr{G}$ belongs\nto three triangles, whereas no such an edge exists in $\\Gr{H}$. Furthermore, it is shown in Example~4.18 of \\cite{Sason2024}\nthat each pair of the regular NICS graphs on 10~vertices, denoted by $\\{\\Gr{G}, \\Gr{H}\\}$ and $\\{\\CGr{G}, \\CGr{H}\\}$,\nexhibits distinct values of the Lov\\'{a}sz $\\vartheta$-functions, whereas the graphs $\\Gr{G}$, $\\CGr{G}$, $\\Gr{H}$,\nand $\\CGr{H}$ share identical independence numbers~(3), clique numbers~(3), and chromatic numbers~(4).\nFurthermore, based on these two pairs of graphs, it is constructively shown in Theorem~4.19 of \\cite{Sason2024} that for\nevery even integer $n \\geq 14$, there exist connected, irregular, cospectral, and nonisomorphic graphs on $n$ vertices,\nbeing jointly cospectral with respect to their adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices,\nwhile also sharing identical independence, clique, and chromatic numbers, but being distinguished by their Lov\\'{a}sz\n$\\vartheta$-functions.\n\\end{example}\n\n\\begin{remark}\n\\label{remark: relations to Igal's paper 2023}\nIn continuation to Example~\\ref{example: NICS regular graphs on 10 vertices}, it is worth noting that closed-form\nexpressions for the Lov\\'{a}sz $\\vartheta$-functions of regular graphs, which are edge-transitive or strongly regular,\nwere derived in \\cite[Theorem~9]{Lovasz79_IT} and \\cite[Proposition~1]{Sason23}, respectively. In particular,\nit follows from \\cite[Proposition~1]{Sason23} that strongly regular graphs with identical four parameters\n$(n,d,\\lambda,\\mu)$ are cospectral and they have identical Lov\\'{a}sz $\\vartheta$-numbers, although they need not be\nnecessarily isomorphic. For such an explicit counterexample, the reader is referred to \\cite[Remark~3]{Sason23}.\n\\end{remark}\n\nWe next introduce friendship graphs to address their possible determination by their spectra with respect to\nseveral associated matrices.\n\\begin{definition}\n\\label{definition: friendship graph}\nLet $p\\in \\naturals$. \\emph{The friendship graph} $\\FG{p}$, also known as the \\emph{windmill graph}, is a graph\nwith $2p+1$ vertices, consisting of a single vertex (the central vertex) that is adjacent to all the other $2p$ vertices.\nFurthermore, every pair of these $2p$ vertices shares exactly one common neighbor, namely the central vertex (see\nFigure~\\ref{fig:friendship graph F4}). This graph has $3p$ edges and $p$ triangles.\n\\end{definition}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=3cm]{F4.png}\n\\caption{The friendship (windmill) graph $\\FG{4}$ has 9~vertices, 12 edges, and~4 triangles.}\\label{fig:friendship graph F4}\n\\end{figure}\nThe term friendship graph in Definition~\\ref{definition: friendship graph} originates from the\n\\emph{Friendship Theorem} \\cite{Erdos1963}. This theorem states that if $\\Gr{G}$ is a finite graph where\nany two vertices share exactly one common neighbor, then there exists a vertex that is adjacent to all other\nvertices. In this context, the adjacency of vertices in the graph can be interpreted socially as a representation\nof friendship between the individuals represented by the vertices (assuming friendship is a mutual relationship).\nFor a nice exposition of the proof of the Friendship Theorem, the interested reader is referred to Chapter~44 of\n\\cite{AignerZ18}.\n\n\\begin{theorem}\n\\label{theorem: special classes of DS graphs}\nThe following graphs are DS:\n\\begin{enumerate}[1.]\n\\item \\label{item 1: DS graphs}\nAll graphs with less than five vertices, and also all regular graphs with less than 10 vertices \\cite{vanDamH03} (recall\nTheorems~\\ref{theorem: van Dam and Haemers, 2003 - thm1} and~\\ref{theorem: van Dam and Haemers, 2003 - thm2}).\n\\item \\label{item 2: DS graphs}\nThe graphs $\\CoG{n}$, $\\CG{n}$, $\\PathG{n}$, $\\CoBG{m}{m}$ and $\\CGr{\\CoG{n}}$ \\cite{vanDamH03}.\n\\item \\label{item 3: DS graphs}\nThe complement of the path graph $\\CGr{\\PathG{n}}$ \\cite{DoobH02}.\n\\item \\label{item 4: DS graphs}\nThe disjoint union of $k$ path graph with no isolated vertices, the disjoint union of $k$ complete graphs with no isolated\nvertices, and the disjoint union of $k$ cycles (i.e., every 2-regular graph) \\cite{vanDamH03}.\n\\item \\label{item 5: DS graphs}\nThe complement graph of a DS regular graph \\cite{CvetkovicRS2010}.\n\\item \\label{item 6: DS graphs}\nEvery $(n-3)$-regular graph on $n$ vertices \\cite{CvetkovicRS2010}.\n\\item \\label{item 7: DS graphs}\nThe friendship graph $\\FG{p}$ for $p \\ne 16$ \\cite{CioabaHVW2015}.\n\\item \\label{item 8: DS graphs}\nSandglass graphs, which are obtained by appending a triangle to each of the pendant (i.e., degree-1) vertices of a path \\cite{LuLYY09}.\n\\item \\label{item 9: DS graphs}\nIf $\\Gr{H}$ is a subgraph of a graph $\\Gr{G}$, and $\\Gr{G} \\setminus \\Gr{H}$ denotes the graph obtained from $\\Gr{G}$\nby deleting the edges of $\\Gr{H}$, then also the following graphs are DS \\cite{CamaraH14}:\n\\begin{itemize}\n\\item $\\CoG{n} \\setminus (\\ell \\CoG{2})$ and $\\CoG{n} \\setminus \\CoG{m}$, where $m \\leq n-2$,\n\\item $\\CoG{n} \\setminus \\CoBG{\\ell}{m}$,\n\\item $\\CoG{n} \\setminus \\Gr{H}$, where $\\Gr{H}$ has at most four edges.\n\\end{itemize}\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsection{Graphs determined by their spectra with respect to various matrices (X-DS graphs)}\n\\label{subsection: Graphs determined by their X-DS spectrum}\n\n\\noindent\n\nIn this section, we consider graphs that are determined by the spectra of various associated matrices beyond the adjacency matrix spectrum.\n\n\\begin{definition}\nLet $\\Gr{G} , \\Gr{H}$ be two graphs and let $\\mathcal{X} \\subseteq \\Gmats$.\n\\begin{enumerate}\n\\item $\\Gr{G}$ and $\\Gr{H}$ are said to be \\emph{$\\mathcal{X}$-cospectral} if they have the same $X$-spectrum, i.e.\n$\\sigma_X(\\Gr{G}) = \\sigma_X(\\Gr{H})$.\n\\item\nNonisomorphic graphs $\\Gr{G}$ and $\\Gr{H}$ that are $\\mathcal{X}$-cospectral are said to be \\emph{$\\mathcal{X}$-NICS},\nwhere {\\em NICS} is an abbreviation of {\\em non-isomorphic and cospectral}.\n\\item A graph $\\Gr{G}$ is said to be \\emph{determined by its $\\mathcal{X}$-spectrum\n($\\mathcal{X}$-DS)} if every graph that is $\\mathcal{X}$-cospectral to $\\Gr{G}$ is also isomorphic to $\\Gr{G}$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{notation}\nFor a singleton $\\mathcal{X} = \\{ X \\}$, we abbreviate $\\{ X \\} $-cospectral, $\\{X\\}$-DS and $\\{X\\}$-NICS by $X$-cospectral,\n$X$-DS and $X$-NICS, respectively. For the adjacency matrix, we will abbreviate $\\A$-DS by DS.\n\\end{notation}\n\n\\begin{remark}\n\\label{remark: X,Y cospectrality}\n Let $\\mathcal{X} \\subseteq \\mathcal{Y} \\subseteq \\Gmats$. The following holds by definition:\n\\begin{itemize}\n \\item If two graph $\\Gr{G}, \\Gr{H}$ are $\\mathcal{Y}$-cospectral, then they are $\\mathcal{X}$-cospectral.\n \\item If a graph $\\Gr{G}$ is $\\mathcal{X}$-DS, then it is $\\mathcal{Y}$-DS.\n\\end{itemize}\n\\end{remark}\n\n\\begin{definition}\n\\label{definition: generalized spectrum}\nLet $\\Gr{G}$ be a graph. The \\emph{generalized spectrum} of $\\Gr{G}$ is the $\\{\\A, \\overline{\\A}\\}$-spectrum of $\\Gr{G}$.\n\\end{definition}\n\nThe following result on the cospectrality of regular graphs can be readily verified.\n\\begin{proposition}\n\\label{proposition: regular graphs cospectrality}\nLet $\\Gr{G}$ and $\\Gr{H}$ be regular graphs that are $\\mathcal{X}$-cospectral for {\\em some}\n$\\mathcal{X} \\subseteq \\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$. Then, $\\Gr{G}$ and $\\Gr{H}$ are\n$\\mathcal{Y}$-cospectral for {\\em every}\n$\\mathcal{Y} \\subseteq \\{\\A, \\overline{\\A}, \\LM, \\overline{\\LM}, \\Q, \\overline{\\Q}, {\\bf{\\mathcal{L}}}, \\overline{{\\bf{\\mathcal{L}}}} \\}$.\nIn particular, the cospectrality of regular graphs (and their complements) stays unaffected\nby the chosen matrix among $\\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$.\n\\end{proposition}\n\n\\begin{definition}\n\\label{definition: DGS}\nA graph $\\Gr{G}$ is said to be \\emph{determined by its generalized spectrum (DGS)} if it is uniquely determined by its\ngeneralized spectrum. In other words, a graph $\\Gr{G}$ is DGS if and only if every graph $\\Gr{H}$ with the same\n$\\{\\A, \\overline{\\A}\\}$-spectrum as $\\Gr{G}$ is necessarily isomorphic to $\\Gr{G}$.\n\\end{definition}\nIf a graph is not DS, it may still be DGS, as additional spectral information is available. Conversely, every DS graph\nis also DGS. For further insights into DGS graphs, including various characterizations, conjectures, and studies, we refer\nthe reader to \\cite{WangXu06,Wang13,Wang17}.\n\n\\vspace*{0.2cm}\nThe continuation of this section characterizes graphs that are $X$-DS, where $X \\in \\{\\LM, \\Q, \\mathcal{L}\\}$, with pointers\nto various studies. We first consider regular DS graphs.\n\n\\begin{theorem} \\cite[Proposition~3]{vanDamH03}\n\\label{theorem: regular DS graphs}\nFor regular graphs, the properties of being DS, $\\LM$-DS, and $\\Q$-DS are equivalent.\n\\end{theorem}\n\\begin{remark}\n\\label{remark: recurring approach}\nTo avoid any potential confusion, it is important to emphasize that in statements such as Theorem~\\ref{theorem: regular DS graphs},\nthe only available information is the spectrum of the graph. There is no indication or prior knowledge that the spectrum corresponds\nto a regular graph. In such cases, the regularity of the graph is not part of the revealed information and, therefore, cannot be used\nto determine the graph. This recurring approach --- stating that $\\Gr{G}$ is stated to be a graph satisfying certain properties (e.g., regularity,\nstrong regularity, etc.) and then examining whether the graph can be determined from its spectrum --- appears throughout this paper. It\nshould be understood that the only available information is the spectrum of the graph, and no additional properties of the graph beyond\nits spectrum are disclosed.\n\\end{remark}\n\n\\begin{remark}\n\\label{remark: DS regular graphs are not necessarily DS w.r.t. normalized Laplacian}\nThe crux of the proof of Theorem~\\ref{theorem: regular DS graphs} is that there are no two NICS\ngraphs, with respect to either $\\A$, $\\LM$, or $\\Q$, where one graph is regular and the other is\nirregular (see \\cite[Proposition~2.2]{vanDamH03}).\nThis, however, does not extend to NICS graphs with respect to the normalized Laplacian matrix $\\mathcal{L}$,\nand regular DS graphs are not necessarily $\\mathcal{L}$-DS. For instance, the cycle $\\CG{4}$ and the bipartite complete\ngraph $\\CoBG{1}{3}$ (i.e., $\\SG{3}$) share the same $\\mathcal{L}$-spectrum, which is given by $\\{0, 1^2, 2\\}$,\nbut these graphs are nonisomorphic (as $\\CG{4}$ is regular, in contrast to $\\CoBG{1}{3}$). It therefore follows\nthat the 2-regular graph $\\CG{4}$ is {\\em not} $\\mathcal{L}$-DS, although it is DS (see Item~\\ref{item 2: DS graphs}\nof Theorem~\\ref{theorem: special classes of DS graphs}).\nMore generally, it is conjectured in \\cite{Butler2016} that $\\CG{n}$ is $\\mathcal{L}$-DS if\nand only if $n>4$ and $4 \\hspace*{-0.1cm} \\not| \\, n$.\n\\end{remark}\n\n\\begin{theorem}\n\\label{theorem: L-DS graphs}\nThe following graphs are $\\LM$-DS:\n\\begin{enumerate}[1.]\n\\item $\\PathG{n},\\CG{n},\\CoG{n},\\CoBG{m}{m}$ and their complements \\cite{vanDamH03}.\n\\item The disjoint union of $k$ paths, $\\PathG{n_1} \\DU \\PathG{n_2} \\DU \\ldots \\DU \\PathG{n_k}$ each\nhaving at least one edge \\cite{vanDamH03}.\n\\item The complete bipartite graph $\\CoBG{m}{n}$ with $m,n\\geq2$ and $\\frac{5}{3}n<m$ \\cite{Boulet2009}.\n\\item \\label{stars: L-DS}\nThe star graphs $\\SG{n}$ with $n \\neq 3$ \\cite{OmidiT2007,LiuZG2008}.\n\\item Trees with a single vertex having a degree greater than~2 (referred to as starlike trees) \\cite{OmidiT2007,LiuZG2008}.\n\\item The friendship graph $\\FG{p}$ \\cite{LiuZG2008}.\n\\item The path-friendship graphs, where a friendship graph and a starlike tree are joined by merging their vertices of degree greater than~2 \\cite{OboudiAAB2021}.\n\\item The wheel graph $\\Gr{W}_{n+1} \\triangleq \\CoG{1} \\vee \\CG{n}$ for $n \\neq 7$ (otherwise, if $n=7$, then it is not $\\LM$-DS) \\cite{ZhangLY09}.\n\\item The join of a clique and an independent set on $n$ vertices, $\\CoG{n-m} \\vee \\, \\CGr{\\CoG{m}}$, where $m \\in \\OneTo{n-1}$ \\cite{DasL2016}.\n\\item Sandglass graphs (see also Item~\\ref{item 8: DS graphs} in Theorem~\\ref{theorem: special classes of DS graphs}) \\cite{LuLYY09}.\n\\item The join graph $\\Gr{G} \\vee \\CoG{m}$, for every $m \\in \\naturals$, where $\\Gr{G}$ is a disconnected graph \\cite{ZhouBu2012}.\n\\item The join graph $\\Gr{G} \\vee \\CoG{m}$, for every $m \\in \\naturals$, where $\\Gr{G}$ is an $\\LM$-DS connected graph on $n$ vertices and $m$ edges\nwith $m \\leq 2n-6$, $\\CGr{G}$ is a connected graph, and either one of the following conditions holds \\cite{ZhouBu2012}:\n\\begin{itemize}\n\\item $\\Gr{G} \\vee \\CoG{1}$ is $\\LM$-DS;\n\\item the maximum degree of $\\Gr{G}$ is smaller than $\\tfrac12 (n-2)$.\n\\end{itemize}\n\\item Specifically, the join graph $\\Gr{G} \\vee \\CoG{m}$, for every $m \\in \\naturals$, where $\\Gr{G}$ is an $\\LM$-DS tree on $n \\geq 5$ vertices\n(since, the equality $m=n-1$ holds for a tree on $n$ vertices and $m$ edges) \\cite{ZhouBu2012}.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\nIn general, a disjoint union of complete graphs is not determined by its Laplacian spectrum.\n\\end{remark}\n\n\n\\begin{theorem}\n\\label{theorem: Q-DS graphs}\nThe following graphs are $\\Q$-DS:\n\\begin{enumerate}[1.]\n\\item The disjoint union of $k$ paths, $\\PathG{n_1} \\DU \\PathG{n_2} \\DU \\ldots \\DU \\PathG{n_k}$ each\nhaving at least one edge \\cite{vanDamH03}.\n\\item The star graphs $\\SG{n}$ with $n \\geq 3$ \\cite{BuZ2012b,OmidiV2010}.\n\\item Trees with a single vertex having a degree greater than~2 \\cite{BuZ2012b,OmidiV2010}.\n\\item The friendship graph $\\FG{k}$ \\cite{WangBHB2010}.\n\\item The lollipop graphs, where a lollipop graph, denoted by $\\mathrm{H}_{n,p}$ where $n,p \\in \\naturals$ and $p<n$,\nis obtained by appending a cycle $\\CG{p}$ to a pendant vertex of a path $\\PathG{n-p}$ \\cite{HamidzadeK2010,ZhangLZY09}.\n\\item $\\Gr{G} \\vee \\CoG{1}$ where $\\Gr{G}$ is a either a $1$-regular graph, an $(n-2)$-regular graph of order $n$\nor a $2$-regular graph with at least $11$ vertices \\cite{BuZ2012}.\n\\item If $n \\geq 21$ and $0 \\leq q \\leq n-1$, then $\\CoG{1} \\vee (\\PathG{q} \\DU \\, (n-q-1) \\CoG{1})$ \\cite{YeLS2025}.\n\\item If $n \\geq 21$ and $3 \\leq q \\leq n-1$, then $\\CoG{1} \\vee (\\CG{q} \\DU \\, (n-q-1) \\CoG{1})$ is $\\Q$-DS if and\nonly if $q \\neq 3$ \\cite{YeLS2025}.\n\\item The join of a clique and an independent set on $n$ vertices, $\\CoG{n-m} \\vee \\, \\CGr{\\CoG{m}}$, where\n$m \\in \\OneTo{n-1}$ and $m \\neq 3$ \\cite{DasL2016}.\n\\end{enumerate}\n\\end{theorem}\n\nSince the regular graphs $\\CoG{n}$, $\\CGr{\\CoG{n}}$, $\\CoBG{m}{m}$ and $\\CG{n}$ are DS, they are also $\\mathcal{X}$-DS\nfor every $\\mathcal{X} \\subseteq \\{\\A, \\LM, \\Q \\}$ (see Theorem~\\ref{theorem: regular DS graphs}). This, however, does\nnot apply to regular ${\\bf{\\mathcal{L}}}$-DS graphs (see Remark~\\ref{remark: DS regular graphs are not necessarily DS w.r.t. normalized Laplacian}),\nwhich are next addressed.\n\n\\begin{theorem}\n\\label{theorem: X-DS friendship graphs}\nThe following graphs are ${\\bf{\\mathcal{L}}}$-DS:\n\\begin{itemize}\n\\item $\\CoG{n}$, for every $n \\in \\naturals$ \\cite{ButlerH2016}.\n\\item The friendship graph $\\FG{k}$, for $k \\geq 2$ \\cite[Corollary~1]{BermanCCLZ2018}.\n\\item More generally, $\\mathrm{F}_{p,q} = \\CoG{1} \\vee (p \\CoG{q})$ if $q \\geq 2$, or $q=1$ and $p \\geq 2$ \\cite[Theorem~1]{BermanCCLZ2018}.\n\\end{itemize}\n\\end{theorem}\n\n\\noindent\n\n\n\\section{Special families of graphs}\n\\label{section: special families of graphs}\nThis section introduces special families of structured graphs and it states conditions for their unique\ndetermination by their spectra.\n\n\\subsection{Stars and graphs of pyramids}\n\\label{subsection: Stars and graphs of pyramids}\n\n\\noindent\n\n\\begin{definition}\n\\label{definition: graphs of pyramids}\nFor every $k,n \\in \\naturals$ with $k<n$, define the graph $T_{n,k}=\\CoG{k} \\vee \\, \\overline{\\CoG{n-k}}$.\nFor $k=1$, the graph $T_{n,k}$ represents the \\emph{star graph} $\\SG{n}$. For $k=2$, it represents\na graph comprising $n-2$ triangles sharing a common edge, referred to as a \\emph{crown}. For $n,k$ satisfying\n$1<k<n$, the graphs $T_{n,k}$ are referred to as \\emph{graphs of pyramids} \\cite{KrupnikB2024}.\n\\end{definition}\n\n\\begin{theorem} \\cite{KrupnikB2024}\n\\label{thm: KrupnikB2024 - pyramids are DS}\nThe graphs of pyramids are DS for every $1<k<n$.\n\\end{theorem}\n\n\\begin{theorem} \\cite{KrupnikB2024}\n\\label{thm: KrupnikB2024 - DS star graphs}\nThe star graph $\\SG{n}$ is DS if and only if $n-1$ is prime.\n\\end{theorem}\n\nTo prove these theorems, a closed-form expression for the spectrum of $T_{n,k}$ is derived in \\cite{KrupnikB2024}, which\nalso presents a generalized result. Subsequently,\nusing Theorem~\\ref{thm: number of walks of a given length}, the number of edges and triangles in any graph cospectral with\n$T_{n,k}$ are calculated. Finally, Schur's theorem (Theorem~\\ref{theorem: Schur complement}) and Cauchy's interlacing theorem\n(Theorem~\\ref{thm:interlacing}) are applied in \\cite{KrupnikB2024} to prove Theorems~\\ref{thm: KrupnikB2024 - pyramids are DS}\nand~\\ref{thm: KrupnikB2024 - DS star graphs}.\n\n\\subsection{Complete bipartite graphs}\n\\label{subsection: Complete bipartite graphs}\nBy Theorem~\\ref{thm: KrupnikB2024 - DS star graphs}, the star graph $\\SG{n}=\\CoBG{1}{n-1}$ is DS if and only if $n-1$ is prime.\nBy Theorem~\\ref{theorem: special classes of DS graphs}, the regular complete bipartite graph $\\CoBG{m}{m}$ is DS for every $m \\in \\naturals$.\nHere, we generalize these results and provide a characterization for the DS property of $\\CoBG{p}{q}$ for every $p,q\\in \\naturals$.\n\n\\begin{theorem} \\cite{vanDamH03}\n\\label{thm:spectrum of CoBG}\nThe spectrum of the complete bipartite graph $\\CoBG{p}{q}$ is $\\bigl\\{-\\sqrt{pq}, [0]^{p+q-2}, \\sqrt{pq} \\bigr\\}$.\n\\end{theorem}\n\nThis theorem can be proved by Theorem~\\ref{theorem: Schur complement}. An alternative simple proof is next presented.\n\\begin{proof}\nThe adjacency matrix of $\\CoBG{p}{q}$ is given by\n\\begin{align}\n\\A(\\CoBG{p}{q}) =\n\\begin{pmatrix}\n\\mathbf{0}_{p,p} & \\J{p,q}\\\\\n\\J{q,p} & \\mathbf{0}_{q,q}\n\\end{pmatrix} \\in \\Reals^{(p+q) \\times (p+q)}\n\\end{align}\nThe rank of $\\A(\\CoBG{p}{q})$ is equal to 2, so the multiplicity of $0$ as an eigenvalue is $p+q-2$.\nBy Corollary~\\ref{corollary: number of edges and triangles in a graph}, the two remaining eigenvalues\nare given by $\\pm \\lambda$ for some $\\lambda \\in \\Reals$, since the eigenvalues sum to zero. Furthermore,\n\\begin{align}\n2\\lambda^2 = \\sum_{i=1}^{p+q} \\lambda_i^2 = 2 \\, \\card{\\E{\\CoBG{p}{q}}} = 2pq,\n\\end{align}\nso $\\lambda = \\sqrt{pq}$.\n\\end{proof}\n\nFor $p,q \\in \\mathbb{N}$, the arithmetic and geometric means of $p,q$ are, respectively, given by\n$\\AM{p,q}=\\tfrac12 (p+q)$ and $\\GM{p,q}= \\sqrt{ pq}$. The AM-GM inequality states that for\nevery $p,q \\in \\naturals$, we have $\\GM{p,q} \\le \\AM{p,q}$ with equality if and only if $p=q$.\n\n\\begin{definition}\n\\label{definition: AM minimizer}\nLet $p,q \\in \\naturals$. The two-elements multiset $\\{p,q\\} $ is said to be an \\emph{AM-minimizer} if\nit attains the minimum arithmetic mean for their given geometric mean, i.e.,\n\\begin{align}\n\\label{eq: AM minimizer}\n\\AM{p,q} &= \\min \\Bigl\\{\\AM{a,b}: \\; a,b \\in \\mathbb{N}, \\, \\GM{a,b}=\\GM{p,q} \\Bigr\\} \\\\\n\\label{eq2: AM minimizer}\n&= \\min \\Bigl\\{\\tfrac12 (a+b): \\; a,b \\in \\mathbb{N}, \\, ab=pq \\Bigr\\}.\n\\end{align}\n\\end{definition}\n\n\\begin{example}\n\\label{example: AM minimizer}\nThe following are AM-minimizers:\n\\begin{itemize}\n\\item $\\{k,k\\}$ for every $k\\in \\naturals $. By the AM-GM inequality, it is the only case where\n$\\GM{p,q} = \\AM{p,q}$.\n\\item $\\{p,q\\}$ where $p,q$ are prime numbers. In this case, the following family of multisets\n\\begin{align}\n\\Bigl\\{ \\{a,b\\} : \\, a,b \\in \\mathbb{N}, \\; \\GM{a,b}=\\GM{p,q} \\Bigr\\}\n\\end{align}\nonly contains the two multisets $\\{p,q\\},\\{pq,1\\}$, and $p+q \\leq pq < pq+1$ since $p,q \\geq 2$.\n\\item $\\{1,q\\}$ where $q$ is a prime number.\n\\end{itemize}\n\\end{example}\n\n\\begin{theorem}\n\\label{thm:when CoBG is DS?}\nThe following holds for every $p,q \\in \\naturals$:\n\\begin{enumerate}\n\\item \\label{thm:when CoBG is DS? - part1}\nLet $\\Gr{G}$ be a graph that is cospectral with $\\CoBG{p}{q}$. Then, up to isomorphism, $G = \\CoBG{a}{b} \\cup \\Gr{H}$\n(i.e., $\\Gr{G}$ is a disjoint union of the two graphs $\\CoBG{a}{b}$ and $\\Gr{H}$), where $\\Gr{H}$ is an empty graph and\n$a,b \\in \\naturals$ satisfy $\\GM{a,b} = \\GM{p,q}$.\n\\item \\label{thm:when CoBG is DS? - part2}\nThe complete bipartite graph $\\CoBG{p}{q}$ is DS if and only if $\\{p,q\\}$ is an AM-minimizer.\n\\end{enumerate}\n\\end{theorem}\n\\begin{remark}\n\\label{remark: complete bipartite graphs}\nItem~\\ref{thm:when CoBG is DS? - part2} of Theorem~\\ref{thm:when CoBG is DS?} is equivalent to Corollary~3.1 of\n\\cite{MaRen2010}, for which an alternative proof is presented here.\n\\end{remark}\n\n\\begin{proof}\n(Proof of Theorem~\\ref{thm:when CoBG is DS?}):\n\\begin{enumerate}\n\\item Let $\\Gr{G}$ be a graph cospectral with $\\CoBG{p}{q}$. The number of edges in $\\Gr{G}$ equals the number of edges\nin $\\CoBG{p}{q}$, which is $pq$. As $\\CoBG{p}{q}$ is bipartite, so is $\\Gr{G}$. Since $\\A(\\Gr{G})$ is of rank $2$,\nand $\\A(\\PathG{3})$ has rank $3$, it follows from the Cauchy's Interlacing Theorem (Theorem~\\ref{thm:interlacing})\nthat $\\PathG{3}$ is not an induced subgraph of $\\Gr{G}$. \\newline\nIt is claimed that $\\Gr{G}$ has a single nonempty connected component. Suppose to the contrary that $\\Gr{G}$ has (at least)\ntwo nonempty connected components $\\Gr{H}_1,\\Gr{H}_2$. For $i\\in \\{1,2\\}$, since $\\Gr{H}_i$ is a non-empty graph, $\\A(\\Gr{H}_i)$\nhas at least one eigenvalue $\\lambda \\ne 0$. Since $\\Gr{G}$ is a simple graph, the sum of the eigenvalues of $\\A(\\Gr{H}_i)$\nis $\\trace{\\A(\\Gr{H}_i)}=0$, so $\\Gr{H}_i$ has at least one positive eigenvalue. Thus, the induced subgraph $\\Gr{H}_1 \\cup \\Gr{H}_2$\nhas at least two positive eigenvalues, while $\\Gr{G}$ has only one positive eigenvalue, contradicting Cauchy's Interlacing Theorem. \\\\\nHence, $\\Gr{G}$ can be decomposed as $\\Gr{G} = \\CoBG{a}{b} \\cup \\, \\Gr{H}$ where $\\Gr{H}$ is an empty graph. Since $\\Gr{G}$\nand $\\CoBG{p}{q}$ have the same number of edges, $pq=ab$, so $\\GM{p,q}=\\GM{a,b}$.\n\\item First, we will show that if $\\{p,q\\}$ is not an AM-minimizer, then the graph $\\CoBG{p}{q}$ is not $\\A$-DS. This is done\nby finding a nonisomorphic graph to $\\CoBG{p}{q}$ that is $\\A$-cospectral with it. By assumption, since $\\{p,q\\}$ is not an\nAM-minimizer, there exist $a, b \\in \\naturals$ satisfying $\\GM{a,b} = \\GM{p,q}$ and $a + b < p+q$.\nDefine the graph $\\Gr{G}=\\CoBG{a}{b} \\vee \\, \\overline{\\CoG{r}}$ where $r=p+q-a-b$. Observe that $r \\in \\naturals$.\nThe $\\A$-spectrum of both of these graphs is given by\n\\begin{align}\n\\sigma_{\\A}(\\Gr{G}) = \\sigma_{\\A}(\\CoBG{p}{q}) = \\bigl\\{-\\sqrt{pq},[0]^{pq-2},\\sqrt{pq} \\bigr\\},\n\\end{align}\nso these two graphs are nonisomorphic and cospectral, which means that $\\Gr{G}$ is not $\\A$-DS. \\newline\nWe next prove that if $\\{p,q\\}$ is an AM-minimizer, then $\\CoBG{p}{q}$ is $\\A$-DS. Let $\\Gr{G}$ be a\ngraph that is cospectral with $\\CoBG{p}{q}$. From the first part of this theorem,\n$\\Gr{G}=\\CoBG{a}{b} \\cup \\, \\Gr{H}$ where $\\GM{a,b} = \\GM{p,q}$ and $\\Gr{H}$ is an empty graph. Consequently, it follows that\n$\\AM{a,b}=\\tfrac12(a+b) \\leq \\tfrac12(p+q) = \\AM{p,q}$.\nSince $\\{p,q\\}$ is assumed to be an AM-minimizer, it follows that $\\AM{a,b} \\ge \\AM{p,q}$, and thus equality holds.\nBoth equalities $\\GM{a,b} = \\GM{p,q}$ and $\\AM{a,b} = \\AM{p,q}$ can be satisfied simultaneously if and only if\n$\\{ a , b \\} = \\{ p , q \\}$, so $r=p+q-a-b=0$ and $\\Gr{G}=\\CoBG{p}{q}$.\n\\end{enumerate}\n\\end{proof}\n\n\n\\begin{corollary}\n\\label{cor: bipartite not DS}\nAlmost all of the complete bipartite graphs are not DS. More specifically, for every $n \\in \\naturals$, there exists\na single complete bipartite graph on $n$ vertices that is DS.\n\\end{corollary}\n\n\\begin{proof}\nLet $n \\in \\naturals$. By the \\emph{fundamental theorem of arithmetic}, there is a unique decomposition\n$n = \\prod_{i=1}^{k} p_i$ where $k\\in \\naturals$ and $\\{p_i\\}$ are prime numbers for every $1 \\le i \\le k$.\nConsider the family of multisets\n\\begin{align}\n\\set{D} = \\Bigl\\{ \\{a,b\\} : a,b \\in \\mathbb{N} , \\GM{a,b}=\\sqrt{n} \\Bigr\\}.\n\\end{align}\nThis family has $2^k$ members, since every prime factor $p_i$ of $n$ should be in the prime decomposition of $a$ or $b$.\nSince the minimization of $\\AM{a,b}$ under the equality constraint $\\GM{a,b}=\\sqrt{n}$ forms a convex optimization problem,\nonly one of the multisets in the family $\\set{D}$ is an AM-minimizer.\nThus, if $n = \\prod_{i=1}^{k} p_i$, then the number of complete bipartite graphs of $n$ vertices is $O(2^k)$, and\n(by Item~\\ref{thm:when CoBG is DS? - part2} of Theorem~\\ref{thm:when CoBG is DS?}) only one of them is DS.\n\\end{proof}\n\n\\subsection{Tur\\'{a}n graphs}\n\\label{subsection: Turan graphs}\nThe Tur\\'{a}n graphs are a significant and well-studied class of graphs in extremal graph theory, forming an important family\nof multipartite complete graphs. Tur\\'{a}n graphs are particularly known for their role in Tur\\'{a}n's theorem, which provides a\nsolution to the problem of finding the maximum number of edges in a graph that does not contain a complete subgraph of a given\norder \\cite{Turan1941}. Before delving into formal definitions, it is noted that the distinction of the Tur\\'{a}n graphs as\nmultipartite complete graphs is that they are as balanced as possible, ensuring their vertex sets are divided into parts of\nnearly equal size.\n\n\\begin{definition}\nLet $n_1, \\ldots, n_k$ be natural numbers. Define the \\emph{complete $k$-partite graph}\n\\begin{align}\n\\CoG{n_1, \\ldots, n_k}= \\bigvee_{i=1}^{k}\\overline{\\CoG{n_i}}.\n\\end{align}\nA graph is multipartite if it is $k$-partite for some $k \\geq 2$.\n\\end{definition}\n\n\\begin{definition}\n\\label{definition: Turan graph}\nLet $2 \\le k \\le n$. The \\emph{Tur\\'{a}n graph} $T(n,k)$\n(not to be confused with the graph of pyramids $T_{n,k}$) is\nformed by partitioning a set of $n$ vertices into $k$ subsets,\nwith sizes as equal as possible, and then every two vertices\nare adjacent in that graph if and only if they belong to different subsets.\nIt is therefore expressed as the complete $k$-partite graph\n$K_{n_1,\\dots,n_k}$, where $|n_i-n_j| \\leq 1$ for all $i,j \\in \\OneTo{k}$\nwith $i \\neq j$. Let $q$ and $s$ be the quotient and remainder, respectively,\nof dividing $n$ by $k$ (i.e., $n = qk+s$, $s \\in \\{0,1, \\ldots, k-1\\}$),\nand let $n_1 \\leq \\ldots \\leq n_k$. Then,\n\\begin{align}\n\\label{eq: n_i in Turan's graph}\nn_i=\n\\begin{cases}\nq, & 1\\leq i \\leq k-s,\\\\\nq+1, & k-s+1 \\leq i \\leq k.\n\\end{cases}\n\\end{align}\nBy construction, the graph $T(n,k)$ has a clique of order $k$ (any subset of vertices with\na single representative from each of the $k$ subsets is a clique of order $k$), but it cannot\nhave a clique of order $k+1$ (since vertices from the same subset are nonadjacent).\nNote also that, by \\eqref{eq: n_i in Turan's graph}, the Tur\\'{a}n graph $T(n,k)$ is an\n$(n-q)$-regular graph if and only if $n$ is divisible by $k$, and then $q = \\frac{n}{k}$.\n\\end{definition}\n\n\\begin{definition}\nLet $q,k \\in \\naturals$. Define the \\emph{regular complete multipartite graph},\n$\\mathrm{K}_{q}^{k}: = \\overset{k}{\\underset{i=1}{\\bigvee}} \\overline{\\CoG{q}}$, to be the $k$-partite\ngraph with $q$ vertices in each part. Observe that $\\mathrm{K}_{q}^{k} = T(kq,k)$.\n\\end{definition}\n\nLet $\\Gr{G}$ be a simple graph on $n$ vertices that does not contain a clique of order greater than\na fixed number $k \\in \\naturals$. Tur\\'{a}n investigated a fundamental problem in extremal graph theory of\ndetermining the maximum number of edges that $\\Gr{G}$ can have \\cite{Turan1941}.\n\\begin{theorem}[Tur\\'{a}n's Graph Theorem]\n\\label{theorem: Turan's theorem}\nLet $\\Gr{G}$ be a graph on $n$ vertices with a clique of order at most $k$ for some $k \\in \\naturals$.\nThen,\n\\begin{align}\n\\card{\\E{\\Gr{G}}} &\\leq \\card{\\E{T(n,k)}} \\\\\n&= \\biggl(1-\\frac1k\\biggr) \\, \\frac{n^2-s^2}{2} + \\binom{s}{2}, \\quad s \\triangleq n - k \\bigg\\lfloor \\frac{n}{k} \\bigg\\rfloor.\n\\end{align}\n\\end{theorem}\nFor a nice exposition of five different proofs of Tur\\'{a}n's Graph Theorem, the interested reader is referred\nto Chapter~41 of \\cite{AignerZ18}.\n\n\n\\begin{corollary}\n\\label{corollary:turan} Let $k \\in \\naturals$, and let $\\Gr{G}$ be a graph\non $n$ vertices where $\\omega(\\Gr{G})\\le k$ and $\\card{\\E{\\Gr{G}}}=\\card{\\E{T(n,k)}}$.\nLet $\\Gr{G}_{1}$ be a graph obtained by adding an arbitrary edge to\n$\\Gr{G}$. Then $\\omega(\\Gr{G}_1)>k$.\n\\end{corollary}\n\n\n\\subsubsection{The spectrum of the Tur\\'{a}n graph}\n\n\\begin{theorem} \\cite{EsserH1980}\n\\label{theorem: spectrum of multipartite graphs}\nLet $k\\in\\naturals$, and let $n_1 \\leq n_2 \\leq \\ldots \\leq n_k$ be natural numbers.\nLet $\\Gr{G} = \\CoG{n_1,n_2, \\dots, n_k}$ be a complete multipartite graph on $n = n_1 + \\ldots n_k$\nvertices. Then,\n\\begin{itemize}\n\\item $\\Gr{G}$ has one positive eigenvalue, i.e., $\\lambda_1(\\Gr{G}) > 0$ and $\\lambda_2(\\Gr{G})\\le 0$.\n\\item $\\Gr{G}$ has $0$ as an eigenvalue with multiplicity $n-k$.\n\\item $\\Gr{G}$ has $k-1$ negative eigenvalues, and\n\\begin{align}\nn_1 \\leq -\\lambda_{n-k+2}(\\Gr{G}) \\leq n_2 \\leq -\\lambda_{n-k+3}(\\Gr{G}) \\le n_3 \\leq \\ldots \\leq n_{k-1} \\leq -\\lambda_{n}(\\Gr{G}) \\le n_{k}.\n\\end{align}\n\\end{itemize}\n\\end{theorem}\n\n\\begin{corollary}\n\\label{corollary:Kqk-spectrum}\nThe spectrum of the regular complete $k$-partite graph $\\CoG{q, \\ldots, q} \\triangleq \\CoG{q}^k$ is given by\n\\begin{align}\n\\sigma_{\\A}(\\CoG{q}^{k})=\\Bigl\\{ [-q]^{k-1}, [0]^{(q-1)k}, q(k-1) \\Bigr\\}.\n\\end{align}\n\\end{corollary}\n\n\\begin{proof}\nThis readily follows from Theorem~\\ref{theorem: spectrum of multipartite graphs} by setting $n_1 = \\ldots = n_k = q$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma: Join-A-Spec} \\cite{Butler2008} Let $\\Gr{G}_{i}$\nbe $r_{i}$-regular graphs on $n_{i}$ vertices for $i\\in \\{1,2\\}$, with the adjacency spectrum\n$\\sigma_{\\A}(\\Gr{G}_1)=(r_{1}=\\mu_{1}\\ge\\mu_{2}\\ge...\\ge\\mu_{n})$\nand $\\sigma_{A}(\\Gr{G}_2) = (r_{2}=\\nu_{1}\\ge\\nu_{2}\\ge...\\ge\\nu_{n})$.\nThe $\\A$-spectrum of $\\Gr{G}_1\\vee \\Gr{G}_2$ is given by\n\\begin{align}\n\\sigma_{\\A}(\\Gr{G}_{1}\\vee \\Gr{G}_{2})=\\{ \\mu_{i} \\}_{i=2}^{n_{1}} \\cup \\{ \\nu_{i}\\}_{i=2}^{n_{2}} \\cup\n\\left\\{ \\frac{r_1+r_2 \\pm\\sqrt{(r_1-r_2)^{2}+4 n_1 n_2}}{2} \\right\\}.\n\\end{align}\n\\end{lemma}\n\n\\begin{theorem}\n\\label{theorem: A-spectrum of Turan graph}\nLet $q,s\\in \\naturals$ such that $n=kq+s$ and $0 \\le s \\leq k-1.$ The following\nholds with respect to the $\\A$-spectrum of $T(n,k)$:\n\\begin{enumerate}\n\\item \\label{item: irregular Turan graph}\nIf $1 \\leq s \\leq k-1$, then the $\\A$-spectrum of the irregular Tur\\'{a}n graph $T(n,k)$ is given by\n\\begin{align}\n\\sigma_{\\A}(T(n,k))=& \\biggl\\{ [-q-1]^{s-1}, [-q]^{k-s-1}, [0]^{n-k} \\biggr\\} \\nonumber \\\\\n\\label{eq: A-spectrum of irregular Turan graph}\n& \\cup \\Biggl\\{\\tfrac12 \\biggl[n-2q-1\\pm \\sqrt{\\Bigl(n-2(q+1)s+1\\Bigr)^2+4q(q+1)s(k-s)} \\biggr] \\Biggr\\}.\n\\end{align}\n\\item \\label{item: regular Turan graph}\nIf $s=0$, then $q = \\frac{n}{k}$, and the $\\A$-spectrum of the regular Tur\\'{a}n graph $T(n,k)$ is given by\n\\begin{align}\n\\label{eq: A-spectrum of regular Turan graph}\n\\sigma_{\\A}(T(n,k))=& \\Bigl\\{ [-q]^{k-1}, [0]^{n-k}, (k-1)q \\Bigr\\}.\n\\end{align}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nLet $1 \\leq s \\leq k-1$, and we next derive the $\\A$-spectrum of an irregular Tur\\'{a}n graph $T(n,k)$ in\nItem~\\ref{item: irregular Turan graph} of this theorem (i.e., its spectrum if $n$ is not divisible by $k$ since $s \\neq 0$).\nBy Corollary~\\ref{corollary:Kqk-spectrum}, the spectra of the regular\ngraphs $\\CoG{q}^{k-s}$ and $\\CoG{q+1}^{s}$ is\n\\begin{align}\n& \\sigma_{\\A}(\\CoG{q}^{k-s})=\\left\\{ [-q]^{k-s-1}, [0]^{(q-1)(k-s)}, q(k-s-1) \\right\\}, \\\\\n& \\sigma_{\\A}(\\CoG{q+1}^{s})=\\left\\{ [-q-1]^{s-1}, [0]^{qs}, (q+1)(s-1) \\right\\}.\n\\end{align}\nThe $(k-s)$-partite graph $\\CoG{q}^{k-s}$ is $r_1$-regular with $r_1=q(k-s-1)$, the\n$s$-partite graph $\\CoG{q+1}^{s}$ is $r_2$-regular with $r_2 = (q+1)(s-1)$, and\nby Definition~\\ref{definition: Turan graph}, we have $T(n,k) = \\CoG{q}^{k-s} \\vee \\CoG{q+1}^{s}$.\nHence, by Lemma~\\ref{lemma: Join-A-Spec}, the adjacency spectrum of $T(n,k)$ is given by\n\\begin{align}\n\\sigma_{\\A}(T(n,k)) &= \\sigma_{\\A}(\\CoG{q}^{k-s} \\vee \\CoG{q+1}^{s}) \\nonumber \\\\\n\\label{eq0: 23.12.2024}\n&=\\set{S}_1 \\cup \\set{S}_2 \\cup \\set{S}_3,\n\\end{align}\nwhere\n\\begin{align}\n\\label{eq1: 23.12.2024}\n\\set{S}_1 &= \\Bigl\\{ [-q]^{k-s-1}, [0]^{(q-1)(k-s)} \\Bigr\\}, \\\\\n\\label{eq2: 23.12.2024}\n\\set{S}_2 &= \\Bigl\\{ [-q-1]^{s-1}, [0]^{qs} \\Bigr\\}, \\\\\n\\set{S}_3 &= \\biggl\\{ \\frac{r_1+r_2 \\pm \\sqrt{(r_1-r_2)^2 + 4 n_1 n_2}}{2} \\biggr\\} \\nonumber \\\\\n\\label{eq3: 23.12.2024}\n&= \\Biggl\\{\\tfrac12 \\biggl[n-2q-1\\pm \\sqrt{\\Bigl(n-2(q+1)s+1\\Bigr)^2+4q(q+1)s(k-s)} \\biggr] \\Biggr\\},\n\\end{align}\nwhere the last equality holds since, by the equality $n=kq+s$ and the above expressions of $r_1$\nand $r_2$, it can be readily verified that $r_1+r_2 = n-2q-1$ and $r_1-r_2 = n-2(q+1)s+1$.\nFinally, combining \\eqref{eq0: 23.12.2024}--\\eqref{eq3: 23.12.2024} gives the $\\A$-spectrum\nin \\eqref{eq: A-spectrum of irregular Turan graph} of an irregular Tur\\'{a}n graph $T(n,k)$.\n\nWe next prove Item~\\ref{item: regular Turan graph} of this theorem, referring to a regular Tur\\'{a}n graph $T(n,k)$\n(i.e., $k|n$ or equivalently, $s=0$). In that case, we have $T(n,k)=\\CoG{q}^{k}$ where $q = \\frac{n}{k}$,\nso the $\\A$-spectrum in \\eqref{eq: A-spectrum of regular Turan graph} holds by Corollary~\\ref{corollary:Kqk-spectrum}.\n\\end{proof}\n\n\\begin{remark}\n\\label{remark: Turan - multiplicity of negative eigenvalues}\nIn light of Theorem~\\ref{theorem: A-spectrum of Turan graph}, if $k \\geq 2$, then the number of negative\neigenvalues (including multiplicities) of the adjacency matrix of the Tur\\'{a}n graph $T(n,k)$ is $k-1$\nif the graph is regular (i.e., if $k|n$), and it is $k-2$ otherwise (i.e., if the graph is irregular).\nIf $k=1$, which corresponds to an empty graph (having no edges), then all eigenvalues are zeros\n(having no negative eigenvalues). Furthermore, the adjacency matrix of $T(n,k)$ always has a single positive\neigenvalue, which is of multiplicity~1 irrespectively of the values of $n$ and $k$. We rely on these properties\nlater in this paper (see Section~\\ref{subsubsection: Turan graph is DS}).\n\\end{remark}\n\n\\begin{example}\n\\label{example: A-spectrum of Turan graph}\nBy Theorem~\\ref{theorem: A-spectrum of Turan graph}, let us calculate the $\\A$-spectrum of the\nTur\\'{a}n graph $T(17,7)$, and verify it numerically with the SageMath software \\cite{SageMath}. Having $n=17$\nand $k=7$ gives $q=2$ and $s=3$, which by Theorem~\\ref{theorem: A-spectrum of Turan graph} implies that\n\\begin{align}\n\\label{eq4: 23.12.2024}\n\\sigma_{\\A}\\bigl(T(17,7)\\bigr)= \\Bigl\\{ [-3]^2, [-2]^3, [0]^{10}, \\, 6(1+\\sqrt{2}), \\, 6(1-\\sqrt{2}) \\Bigr\\}.\n\\end{align}\nThat has been numerically verified by programming in the SageMath software.\n\\end{example}\n\n\\subsubsection{Tur\\'{a}n graphs are DS}\n\\label{subsubsection: Turan graph is DS}\nThe main result of this subsection establishes that all Tur\\'{a}n graphs are determined by their $\\A$-spectrum.\nThis result is equivalent to Theorem~3.3 in \\cite{MaRen2010}, while also presenting an alternative proof that\noffers additional insights.\n\\begin{theorem}\n\\label{theorem: Turan's graph is DS}\nThe Tur\\'{a}n graph $T(n,k)$ is $\\A$-DS.\n\\end{theorem}\n\nIn order to prove Theorem~\\ref{theorem: Turan's graph is DS}, we first introduce an auxiliary result from\n\\cite{Smith1970}, followed by several other lemmata.\n\\begin{theorem} \\cite[Theorem~1]{Smith1970}\n\\label{theorem:Smith_theorem_1}\nLet $\\Gr{G}$ be a graph. Then, the following statements are equivalent:\n\\begin{itemize}\n\\item $\\Gr{G}$ has exactly one positive eigenvalue.\n\\item $\\Gr{G}=\\Gr{H} \\cup \\CGr{\\CoG{m}}$ for some $m$, where $\\Gr{H}$ is a nonempty\ncomplete multipartite graph. In other words, the non-isolated vertices of $\\Gr{G}$\nform a complete multipartite graph.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{theorem: Turan's graph is DS}]\nLet $\\Gr{G}$ be a graph that is $\\A$-cospectral with $T(n,k)$. Denote $n=qk+s$\nfor $s,q\\in \\naturals$ such that $0\\le s<k$.\n\\begin{lemma}\n\\label{lemma:Tnk_DS_lemma_1}\nThe graph $\\Gr{G}$ doesn't have a clique of order $k+1$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose to the contrary that the graph $\\Gr{G}$ has a clique of order $k+1$, which means that\n$\\CoG{k+1}$ is an induced subgraph of $\\Gr{G}$. The complete graph $\\CoG{k+1}$ has\n$k$ negative eigenvalues ($-1$ with a multiplicity of $k$). On the other hand, $\\Gr{G}$ has\nat most $k-1$ negative eigenvalues, $n-k$ zero eigenvalues, and exactly one positive eigenvalue;\nindeed, this follows from Theorem~\\ref{theorem: A-spectrum of Turan graph} (see\nRemark~\\ref{remark: Turan - multiplicity of negative eigenvalues}), and since $\\Gr{G}$\nand $T(n,k)$ are $\\A$-cospectral graphs. Hence, by Cauchy's Interlacing Theorem,\nevery induced subgraph of $\\Gr{G}$ on $k+1$ vertices has at most $k-1$ negative\neigenvalues (i.e., those eigenvalues interlaced between the negative and zero eigenvalues of $\\Gr{G}$\nthat are placed at distance $k+1$ apart in a sorted list of the eigenvalues of $\\Gr{G}$ in decreasing order).\nThis contradicts our assumption on the existence of a clique of $k+1$ vertices because of the $k$ negative\neigenvalues of $\\CoG{k+1}$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:Tnk_DS_lemma_2}\nThe graph $\\Gr{G}$ is a complete multipartite graph.\n\\end{lemma}\n\n\\begin{proof}\nSince $\\Gr{G}$ has exactly one positive eigenvalue, which is of multiplicity one, we get\nfrom Theorem~\\ref{theorem:Smith_theorem_1} that $\\Gr{G} = \\Gr{H} \\cup \\CGr{\\CoG{\\ell}}$\nfor some $\\ell \\in \\naturals$, where $\\Gr{H}$ is a nonempty multipartite graph. We next\nshow that $\\ell=0$. Suppose to the contrary that $\\ell \\geq 1$, and let $v$ be an isolated\nvertex of $\\CGr{\\CoG{\\ell}}$. Since $\\Gr{H}$ is a nonempty graph, there exists a vertex\n$u\\in \\V{\\Gr{H}}$. Let $\\Gr{G}_1$ be the graph obtained from $\\Gr{G}$ by adding the\nsingle edge $\\{v,u\\}$. By Lemma~\\ref{lemma:Tnk_DS_lemma_1}, $\\Gr{G}$ does not have a clique\nof order $k+1$. Hence, $\\Gr{G}_1$ does not have a clique of order $k+1$ either, contradicting\nCorollary~\\ref{corollary:turan}. Hence, $\\Gr{G}=\\Gr{H}$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:Tnk_DS_lemma_3}\nThe graph $\\Gr{G}$ is a complete $k$-partite graph.\n\\end{lemma}\n\n\\begin{proof}\nBy Lemma \\ref{lemma:Tnk_DS_lemma_2}, $\\Gr{G}$ is a complete multipartite graph. Let\n$r$ be the number of partite subsets in the vertex set $\\V{\\Gr{G}}$. We show that $r=k$,\nwhich then gives that $\\Gr{G}$ is a complete $k$-partite graph. By\nLemma~\\ref{lemma:Tnk_DS_lemma_1}, $\\Gr{G}$ doesn't have a clique of order $k+1$.\nHence, $r\\le k$. Suppose to the contrary that $r<k$. Since $\\Gr{G}$ is a complete\n$r$-partite graph, the largest order of a clique in $\\Gr{G}$ is $r$.\nLet $\\Gr{G}_{1}$ be a graph obtained from $\\Gr{G}$ by adding an edge between\ntwo vertices within the same partite subset. The graph $\\Gr{G}_1$ becomes an $(r+1)$-partite\ngraph. Consequently, the maximum order of a clique in $\\Gr{G}_1$ is at most $r+1 \\leq k$.\nThe graph $\\Gr{G}_1$ has exactly one more edge than $\\Gr{G}$. Since $\\Gr{G}$ is\n$\\A$-cospectral to $T(n,k)$, it has the same number of edges as in $T(n,k)$.\nHence, $\\Gr{G}_1$ contains more edges than $T(n,k)$, while also lacking a clique of order $k+1$.\nThis contradicts Corollary~\\ref{corollary:turan}, so we conclude that $r=k$.\n\\end{proof}\n\nLet $n_1, n_2, \\dots , n_k \\in \\naturals$ be the number of vertices in each partite subset\nof the complete $k$-partite graph $\\Gr{G}$, i.e., $\\Gr{G} = \\CoG{n_1, n_2, \\dots , n_k}$.\nThen, the next two lemmata subsequently hold.\n\\begin{lemma}\n\\label{lemma:Tnk_DS_lemma_4}\nFor all $i \\in \\OneTo{k}$, $n_i \\le q+1$ .\n\\end{lemma}\n\n\\begin{proof}\nSuppose to the contrary that there exists a partite subset in the complete $k$-partite graph\n$\\Gr{G}$ with more than $q+1$ vertices. Let $\\set{P}_1$ be such a partite subset, and suppose\nwithout loss of generality that $n_1= \\card{\\set{P}_1} \\geq q+1$. By the pigeonhole principle, there\nexists a partite subset of $\\Gr{G}$ with at most $q$ vertices (since $\\sum_{i \\in \\OneTo{k}} n_i = n = kq+s$,\nwhere $0\\leq s \\leq k-1$). Let $\\set{P}_2$ be such a partite subset of $\\Gr{G}$, and suppose\nwithout loss of generality that $n_2 = \\card{\\set{P}_2} \\leq q$. Let $\\Gr{G}_{1}$\nbe the graph obtained from $\\Gr{G}$ by removing a vertex $v \\in \\set{P}_1$, adding\na new vertex $u$ to $\\set{P}_2$, and connecting $u$ to all the vertices outside its partite subset.\nThe new graph $\\Gr{G}_1$ is also $k$-partite, so it does not contain a clique of order greater than $k$.\nFurthermore, by construction, $\\Gr{G}_{1}$ has more edges than $\\Gr{G}$, so\n\\begin{align}\n\\label{eq1: 24.12.2024}\n\\bigcard{\\E{\\Gr{G}_1}} > \\bigcard{\\E{\\Gr{G}}} = \\bigcard{\\E{T(n,k)}}.\n\\end{align}\nHence, $\\Gr{G}_{1}$ is a graph with no clique of order greater than $k$,\nand it has more edges than $T(n,k)$. That contradicts Theorem~\\ref{theorem: Turan's theorem},\nso $\\{n_i\\}_{i=1}^k$ cannot include any element that is larger than $q+1$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:Tnk_DS_lemma_5}\nFor all $i \\in \\OneTo{k}$, $n_i \\geq q$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof of this lemma is analogous to the proof of Lemma~\\ref{lemma:Tnk_DS_lemma_4}.\nSuppose to the contrary that there exists a partite subset of $\\Gr{G}$ with less than $q$\nvertices. Let $\\set{P}_1$ be such a partite subset, so $p_1 \\triangleq \\bigcard{\\set{P}_1}<q$.\nBy the pigeonhole principle, there exists a partition with at least $q+1$ vertices.\nLet $\\set{P}_2$ be such a partite subset set, whose number of vertices is denoted by\n$p_{2} \\triangleq \\card{\\set{P}_2} \\geq q+1$.\nLet $\\Gr{G}_{1}$ be the graph obtained by removing a vertex $v \\in \\set{P}_2$,\nadding a new vertex $u$ to $\\set{P}_1$, and connecting the vertex $u$ to all\nthe vertices outside its partite subset. $\\Gr{G}_{1}$ is $k$-partite, so it does\nnot contain a clique of order greater than $k$, and $\\Gr{G}_{1}$ has more edges\nthan $\\Gr{G}$ so \\eqref{eq1: 24.12.2024} holds.\nHence, $\\Gr{G}_1$ is a graph with no clique of order greater than $k$,\nand it has more edges than $T(n,k)$. That contradicts Theorem~\\ref{theorem: Turan's theorem},\nso $\\{n_i\\}_{i=1}^k$ cannot include any element that is smaller than $q$.\n\\end{proof}\n\nBy Lemmata~\\ref{lemma:Tnk_DS_lemma_4} and~\\ref{lemma:Tnk_DS_lemma_5}, we conclude that\n$n_i \\in \\{q,q+1\\} $ for every $1 \\le i \\le k-1$. Let $\\alpha$ be the number of partite\nsubsets of $q$ vertices and $\\beta$ be the number of partite subsets of $q+1$ vertices.\nSince $\\Gr{G}$ has $n$ vertices, where $\\sum n_{i}=n$, it follows that $q\\alpha + (q+1)\\beta = n$.\nMoreover, $\\Gr{G}$ is $k$-partite, so it follows that $\\alpha+\\beta=k$. This gives the linear\nsystem of equations\n\\begin{align}\n\\begin{pmatrix}q & q+1\\\\\n1 & 1\n\\end{pmatrix}\\begin{pmatrix}\\alpha\\\\\n\\beta\n\\end{pmatrix}=\\begin{pmatrix}n\\\\\nk\n\\end{pmatrix},\n\\end{align}\nwhich has the single solution\n\\begin{align}\n\\alpha = k-s, \\quad \\beta=n-qk=s.\n\\end{align}\nHence, $\\Gr{G} = T(n,k)$, which completes the proof of Theorem~\\ref{theorem: Turan's graph is DS}.\n\\end{proof}\n\n\\begin{remark}\n\\label{remark: on the alternative proof by Ma and Ren}\nThe proof of Theorem~\\ref{theorem: Turan's graph is DS} is an alternative proof of Theorem~3.3 in \\cite{MaRen2010}.\nWhile both proofs rely on Theorem~\\ref{theorem:Smith_theorem_1}, which is Theorem~1 of \\cite{Smith1970}, our proof\nrelies on the adjacency spectral characterization in Theorem~\\ref{theorem: A-spectrum of Turan graph}, noteworthy\nin its own right, and further builds upon a sequence of results presented in Lemmata~\\ref{lemma:Tnk_DS_lemma_1}--\\ref{lemma:Tnk_DS_lemma_5}.\nOn the other hand, the proof of Theorem~3.3 in \\cite{MaRen2010} relies on Theorem~\\ref{theorem:Smith_theorem_1},\nbut then deviates substantially from our proof (see Lemmata~2.4 and~2.5 in \\cite{MaRen2010} and Theorem~3.1 in \\cite{MaRen2010},\nserving to prove Theorem~\\ref{theorem: Turan's graph is DS}).\n\\end{remark}\n\n\\subsection{Line graphs}\n\\label{subsection: Line graphs}\n\nAmong the various studied transformations on graphs, the line graphs of graphs are one of the most studied transformations\n\\cite{BeinekeB21}. We first introduce their definition, and then address the spectral graph determination properties.\n\\begin{definition}\n\\label{definition: line graph}\nThe {\\em line graph} of a graph $\\Gr{G}$, denoted by $\\ell(\\Gr{G})$, is a graph whose vertices\nare the edges in $\\Gr{G}$, and two vertices are adjacent in $\\ell(\\Gr{G})$ if the corresponding\nedges are incident in $\\Gr{G}$.\n\\end{definition}\nA notable spectral property of line graphs is that all the eigenvalues of their adjacency matrix\nare greater than or equal to~$-2$ (see, e.g., \\cite[Theorem~4.6]{BeinekeB21}). For the determination\nof all graphs whose spectrum is bounded from below by $-2$, the interested reader is referred to\n\\cite[Section~4.5]{BeinekeB21}.\n\nThe following theorem characterizes some families of line graphs that are DS.\n\\begin{theorem}\n\\label{theorem: DS line graphs}\nThe following line graphs are DS:\n\\begin{enumerate}[1]\n\\item \\label{Item 1: DS line graphs}\nThe line graph of the complete graph $\\CoG{k}$, where $k \\geq 4$ and $k \\neq 8$ (see\n\\cite[Theorem~4.1.7]{CvetkovicRS2010}),\n\\item \\label{Item 2: DS line graphs}\nThe line graph of the complete bipartite graph $\\CoBG{k}{k}$, where $k \\geq 2$ and\n$k \\neq 4$ (see \\cite[Theorem~4.1.8]{CvetkovicRS2010}),\n\\item \\label{Item 3: DS line graphs}\nThe line graph $\\ell(\\CGr{\\CG{6}})$ (see \\cite[Proposition~4.1.5]{CvetkovicRS2010}),\n\\item \\label{Item 4: DS line graphs}\nThe line graph of the complete bipartite graph $\\CoBG{m}{n}$, where $m+n \\geq 19$ and\n$\\{m,n\\} \\neq \\{2s^2+s, 2s^2-s\\}$ with $s \\in \\naturals$ (see\n\\cite[Proposition~4.1.18]{CvetkovicRS2010}).\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\n\\label{remark: triangular graphs}\nIn regard to Item~\\ref{Item 1: DS line graphs} of Theorem~\\ref{theorem: DS line graphs},\nthe line graphs of complete graphs are referred to as triangular graphs. These\nare strongly regular graphs with the parameters $\\srg{\\tfrac12 k(k-1)}{2(k-2)}{k-2}{4}$,\nwhere $k \\geq 4$. For $k=8$, the corresponding triangular graph is cospectral and nonisomorphic\nto the three Chang graphs (see Remark~\\ref{remark: NICS SRGs}), which are strongly regular graphs\n$\\srg{28}{12}{6}{4}$.\n\\end{remark}\n\nWe next prove the following result in regard to the Petersen graph, which appears in \\cite[Problem~4.3]{CvetkovicRS2010}\nand \\cite[Section~10.3]{BrouwerM22} (without a proof).\n\\begin{corollary}\n\\label{corollary: Petersen graph is DS}\nThe Petersen graph is DS.\n\\end{corollary}\n\\begin{proof}\nThe Petersen graph is known to be isomorphic to the complement of the line graph of the complete graph $\\CoG{5}$ (i.e., it\nis isomorphic to $\\CGr{\\ell(\\CoG{5})}$. By Item~\\ref{Item 1: DS line graphs} of Theorem~\\ref{theorem: DS line graphs}, the line\ngraph $\\ell(\\CoG{5})$ is DS. It is also a 6-regular graph (as the line graph of a $d$-regular graph is $(2d-2)$-regular, and\n$\\CoG{5}$ is a 4-regular graph). Consequently, by Item~\\ref{item 5: DS graphs} of Theorem~\\ref{theorem: special classes of DS graphs},\nthe complement of $\\ell(\\CoG{5})$ is also DS.\n\\end{proof}\n\nThe following definition and theorem provide further elaboration on Item~\\ref{Item 2: DS line graphs}\nof Theorem~\\ref{theorem: DS line graphs}.\n\\begin{definition} \\cite[Section~1.1.8]{BrouwerM22}\n\\label{definition: Lattice graphs}\nThe {\\em Hamming graph} $\\mathrm{H}(2,q)$, where $q \\geq 2$, has the vertex set $\\OneTo{q} \\times \\OneTo{q}$, and\nany two vertices are adjacent if and only if they differ in one coordinate (i.e., their Hamming distance\nis equal to~1). These are also referred to {\\em lattice graphs}, and denoted by $\\mathrm{L}_2(q)$. The\nLattice graph $\\mathrm{L}_2(q)$, where $q \\geq 2$, is also the line graph of the complete bipartite\ngraph $\\CoBG{q}{q}$, and it is a strongly regular graph with parameters $\\srg{q^2}{2(q-1)}{q-2}{2}$.\n\\end{definition}\n\n\\begin{theorem} \\cite{Shrikhande59}\n\\label{theorem: DS Lattice graphs}\nThe lattice graph $\\mathrm{L}_2(q)$ is a strongly regular DS graph for all $q \\neq 4$. For $q=4$, the graph $\\mathrm{L}_2(4)$\nis not DS since it is cospectral and nonisomorphic to the Shrikhande graph, which are nonisomorphic strongly regular graphs\nwith the common parameters $\\srg{16}{6}{4}{2}$.\n\\end{theorem}\n\nThe following result provides an interesting connection between $\\A$ and $\\Q$-cospectralities of graphs.\n\\begin{theorem} \\cite[Proposition~7.8.5]{CvetkovicRS2010}\nIf two graphs are $\\Q$-cospectral, then their line graphs are $\\A$-cospectral.\n\\end{theorem}\n\n\\subsection{Nice graphs}\n\\label{subsection: Nice graphs}\n\nThe family referred to as \"nice graphs\" has been recently introduced in \\cite{KovalK2024}.\n\\begin{definition}\n\\label{definition: nice graphs}\n\\noindent\n\\begin{itemize}\n\\item A graph $\\Gr{G}$ is \\emph{sunlike} if it is connected, and can be obtained from a cycle\n$\\mathrm{C}$ by adding some vertices and connecting each of them to some vertex in $\\mathrm{C}$.\n\\item Let $l,k\\in \\naturals$. A sunlike graph $\\Gr{G}$ is \\emph{$(l,k)$-nice} if it can be obtained by a cycle $\\CG{\\ell}$ and\n\\begin{itemize}\n\\item There is a single vertex $v_1 \\in \\V{\\CG{\\ell}}$ of degree $3$.\n\\item There are $k$ vertices $u_1, \\ldots, u_k \\in \\V{\\CG{\\ell}}$ of degree $4$. Let $\\set{U} = \\{u_1, \\ldots, u_k\\}$.\n\\item By starting a walk on $\\CG{\\ell}$ from $v_1$ at some orientation, then after 4 or 6 steps we get to a vertex $u_1 \\in \\set{U}$.\nThen, after another $4$ or $6$ steps from $u_1$ we get to $u_2 \\in \\set{U}$, and so on until we get to the vertex $u_k \\in \\set{U}$.\n\\end{itemize}\n\\end{itemize}\n\\end{definition}\n\n\\begin{theorem} \\cite{KovalK2024}\n\\label{theorem: Koval and Kwan, 2024}\nLet $l,k \\in \\naturals$ such that $l \\equiv 2 \\, (\\text{mod } 4)$. Let $\\Gr{G}$ be an $(l,k)$-nice graph.\nIf the order of $\\Gr{G}$ is a prime number greater than some $n_0 \\in \\naturals$, then the line graph\n$\\ell(\\Gr{G})$ is DS.\n\\end{theorem}\n\nA more general class of graphs is introduced in \\cite{KovalK2024}, where it is shown that for every sufficiently\nlarge $n \\in \\naturals$, the number of nonisomorphic $n$-vertex DS graphs is at least $e^{cn}$ for some positive\nconstant $c$ (see \\cite[Theorem~1.4]{KovalK2024}). This recent result represents a significant advancement in the\nstudy of Haemers' conjecture because the earlier lower bounds on the number of nonisomorphic $n$-vertex DS graphs\nwere all of the form of $e^{c \\sqrt{n}}$, for some positive constant $c$. As noted in \\cite{KovalK2024}, the first\nform of such a lower bound was derived by van Dam and Haemers \\cite[Proposition~6]{vanDamH09}, who proved that a\ngraph $\\Gr{G}$ is DS if every connected component of $\\Gr{G}$ is a complete subgraph, leading to a lower bound that\nis approximately of the form $e^{c \\sqrt{n}}$ with $c = \\sqrt{\\tfrac{2}{3}} \\, \\pi$.\nTherefore, the transition to a lower bound in \\cite{KovalK2024} that scales exponentially with $n$, rather than with\n$\\sqrt{n}$, is both remarkable and noteworthy.\n\n\n\\subsection{Friendship graphs and their generalization}\n\\label{subsection: The friendship graphs}\n\nThe next theorem considers whether friendship graphs (see Definition~\\ref{definition: friendship graph})\ncan be uniquely determined by the spectra of four of their associated matrices.\n\n\\begin{theorem}\n\\label{theorem: friendship graphs are X-DS}\nThe friendship graph $\\FG{p}$ satisfies the following properties:\nIt is DS if and only if $p \\ne 16$ (i.e., the friendship graph is DS unless\nit has 16~triangles) \\cite{CioabaHVW2015}, $\\LM$-DS \\cite{LinSM2010},\n$\\Q$-DS \\cite{WangBHB2010}, and ${\\bf{\\mathcal{L}}}$-DS \\cite{BermanCCLZ2018}.\n\\end{theorem}\n\nThe friendship graph $\\FG{p}$, where $p \\in \\naturals$, can be expressed in the\nform $\\FG{p}=\\CoG{1} \\vee (p\\CoG{2})$ (see Figure~\\ref{fig:friendship graph F4}).\nThe last observation follows from a property of a generalized friendship graph, which is defined as follows.\n\\begin{definition}\nLet $p,q\\in \\naturals$. The \\emph{generalized friendship graph} is given by\n$\\GFG{p}{q} = \\CoG{1} \\vee (p \\CoG{q})$. Note that $\\GFG{p}{2} = \\FG{p}$.\n\\end{definition}\n\nThe following theorem addresses the conditions under which generalized friendship graphs can be uniquely\ndetermined by the spectra of their normalized Laplacian matrix.\n\\begin{theorem}\n\\label{theorem: Berman's generalized friendship graph}\nThe generalized friendship graph $\\GFG{p}{q}$ is ${\\bf{\\mathcal{L}}}$-DS if and only if $q \\ge 2$, or $q=1$\nand $p=2$ \\cite{BermanCCLZ2018}.\n\\end{theorem}\n\n\\begin{corollary}\n\\label{corollary: friendship graph is DS w.r.t. normalized Laplacian}\nThe friendship graph $\\FG{p}$ is ${\\bf{\\mathcal{L}}}$-DS \\cite{BermanCCLZ2018}.\n\\end{corollary}\n\n\\subsection{Strongly regular graphs}\n\\label{subsection: strongly regular graphs}\n\nStrongly regular graphs with an identical vector of parameters $(n,d,\\lambda,\\mu)$ are cospectral, but may not be isomorphic (see,\nCorollary~\\ref{corollary: cospectral SRGs}, Remark~\\ref{remark: NICS SRGs}, and Theorem~\\ref{theorem: DS Lattice graphs}).\nFor that reason, strongly regular graphs are not necessarily determined by their spectrum. There are, however, infinite\nfamilies of strongly regular DS graphs:\n\\begin{theorem} \\cite[Proposition~14.5.1]{BrouwerH2011}\n\\label{theorem: DS SRG}\nIf $n \\neq 8$, $m \\neq 4$, $a \\geq 2$, and $\\ell \\geq 2$, then the disjoint union of identical complete graphs $a \\CoG{\\ell}$,\nthe line graph of a complete graph $\\ell(\\CoG{n})$, and the line graph of a complete bipartite graph with partite sets of equal\nsize $\\ell(\\CoBG{m}{m})$, as well as their complements, are strongly regular DS graphs.\n\\end{theorem}\n\nWe next show that, although connected strongly regular graphs are not generally DS, the property of strong regularity,\nas well as the four parameters that characterize strongly regular graphs can be determined by the spectrum of their adjacency\nmatrix.\n\n\\begin{theorem}\n\\label{theorem: on A-spectrum and SRGs}\nLet $\\Gr{G}$ be a connected strongly regular graph. Then, its strong regularity, the vector of parameters $(n,d,\\lambda,\\mu)$,\nLov\\'{a}sz $\\vartheta$-function $\\vartheta(\\Gr{G})$, number of edges and triangles, girth, and diameter can be all determined\nby its $\\A$-spectrum.\n\\end{theorem}\n\\begin{proof}\nThe order of a graph $n$ is determined by the $\\A$-spectrum, being the number of eigenvalues (including multiplicities).\nBy Theorem~\\ref{theorem: graph regularity from A-spectrum}, the regularity of a graph is determined by its $\\A$-spectrum.\nBy Item~\\ref{Item 5: eigenvalues of srg} of Theorem~\\ref{theorem: eigenvalues of srg}, a connected regular graph is\nstrongly regular if and only if it has three distinct eigenvalues. Hence, the strong regularity property of $\\Gr{G}$\nis determined by its $\\A$-spectrum. For such a connected regular, the largest eigenvalue is simple, $\\lambda_1=d$, and\nthe other two distinct eigenvalues of the adjacency matrix of $\\Gr{G}$ are given by $\\lambda_2$ and $\\lambda_n$ with\n$\\lambda_n<\\lambda_2$.\nWe next show that the number of common neighbors of any pair of adjacent vertices ($\\lambda$), and the number of common\nneighbors of any pair of nonadjacent vertices ($\\mu$) in $\\Gr{G}$ are, respectively, given by\n\\begin{align}\n\\label{eq1:23.09.23}\n& \\lambda = \\lambda_1 + (1+\\lambda_2)(1+\\lambda_n) - 1, \\\\\n\\label{eq2:23.09.23}\n& \\mu = \\lambda_1 + \\lambda_2 \\lambda_n,\n\\end{align}\nso, these parameters are explicitly expressed in terms of the adjacency spectrum of the strongly regular graph. Indeed, by\nTheorem~\\ref{theorem: eigenvalues of srg}, the second-largest and least eigenvalues of the adjacency matrix of $\\Gr{G}$ are given by\n\\begin{align}\n\\label{eq1:21.01.25}\n\\begin{dcases}\n\\lambda_2 = \\tfrac12 \\Bigl( \\lambda - \\mu + \\sqrt{(\\lambda-\\mu)^2 + 4(d-\\mu)} \\Bigr), \\\\[0.1cm]\n\\lambda_n = \\tfrac12 \\Bigl( \\lambda - \\mu - \\sqrt{(\\lambda-\\mu)^2 + 4(d-\\mu)} \\Bigr),\n\\end{dcases}\n\\end{align}\nfrom which it follows that (noting that $d = \\lambda_1$)\n\\begin{align}\n\\label{eq2:21.01.25}\n\\begin{dcases}\n\\lambda_2 + \\lambda_n = \\lambda - \\mu, \\\\\n\\lambda_2 \\lambda_n = \\mu - \\lambda_1.\n\\end{dcases}\n\\end{align}\nThis gives \\eqref{eq1:23.09.23} and \\eqref{eq2:23.09.23} from, respectively, the second equality in \\eqref{eq2:21.01.25}\nand by adding the two equalities in \\eqref{eq2:21.01.25}.\n\nThe Lov\\'{a}sz $\\vartheta$-function of a strongly regular graph is given by (see \\cite[Proposition~1]{Sason23})\n\\begin{align}\n\\label{eq: Lovasz SRG}\n\\vartheta(\\Gr{G}) = -\\frac{n \\lambda_n}{d - \\lambda_n},\n\\end{align}\nso $\\vartheta(\\Gr{G})$ is determined by its $\\A$-spectrum since the strong regularity property of $\\Gr{G}$ was first determined.\nThe number of edges of $\\Gr{G}$ of a $d$-regular graph is given by $\\tfrac12 nd$, and the number of triangles of the strongly regular\ngraph $\\Gr{G}$ is given by $\\tfrac16 nd \\lambda$, so they are both determined once the four parameters of the strongly regular graphs are\nrevealed. The diameter of a connected strongly regular graph is equal to~2 (note that complete graphs are excluded from the family of\nstrongly regular graphs). Finally, the girth of the strongly regular graph $\\Gr{G}$ is determined as follows \\cite{CameronL2010}:\n\\begin{enumerate}\n\\item If $\\lambda > 0$, then the girth of $\\Gr{G}$ is equal to~3;\n\\item If $\\lambda=0$ and $\\mu \\geq 2$, then the girth of $\\Gr{G}$ is equal to~4;\n\\item If $\\lambda=0$ and $\\mu=1$, then the girth of $\\Gr{G}$ is equal to~5.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{remark}\n\\label{remark: connected SRG}\nA strongly regular graph is connected if and only if $\\mu > 0$.\n\\end{remark}\n\nBy \\cite[Proposition~2]{vanDamH03}, no pair of $\\A$-cospectral graphs exists where one graph is regular, and\nthe other is not. The following result extends this observation to strong regularity.\n\\begin{corollary}\n\\label{corollary: cospectral graphs - one is SRG}\nThere are no two $\\A$-cospectral connected graphs where one is strongly regular and the other is not.\n\\end{corollary}\n\\begin{proof}\nFor a connected strongly regular graph, the strong regularity is determined by the $\\A$-spectrum.\n\\end{proof}\n\nAnother corollary that follows from Theorem~\\ref{theorem: on A-spectrum and SRGs} applies to strongly regular DS graphs.\n\\begin{corollary}\n\\label{corollary: SRG DS graphs}\nLet $\\Gr{G}$ be a connected strongly regular graph such that there is no other nonisomorphic strongly regular graph\nwith an identical vector of parameters $(n, d, \\lambda, \\mu)$. Then, $\\Gr{G}$ is a DS graph.\n\\end{corollary}\nCorollary~\\ref{corollary: SRG DS graphs} naturally raises the following question.\n\\begin{question}\n\\label{question: DS SRG}\nWhich connected strongly regular graphs are determined by their vector of parameters $(n, d, \\lambda, \\mu)$?\n\\end{question}\nA partial answer to Question~\\ref{question: DS SRG} is provided below.\n\nBy Corollary~\\ref{corollary: SRG DS graphs}, for connected strongly regular graphs, there exists an equivalence between\ntheir spectral determination (due to their regularity and in light of Theorem~\\ref{theorem: regular DS graphs}, based on\nthe spectrum of their adjacency, Laplacian, or signless Laplacian matrices) and the uniqueness of these graphs for the given\nparameter vector $(n, d, \\lambda, \\mu)$.\n\nThe study of the number of nonisomorphic strongly regular graphs corresponding to a given set of parameters has been extensively\nexplored. For example, by Theorem~\\ref{theorem: DS Lattice graphs}, there is a unique (up to isomorphism) strongly regular graph\nof the form $\\srg{q^2}{2(q-1)}{q-2}{2}$ for any given $q \\geq 2$ with $q \\neq 4$. Specifically, this implies the uniqueness of\n$\\srg{36}{10}{4}{2}$ (setting $q=6$). On the other hand, a computer search by McKay and Spence established that there are\n$32,548$ strongly regular graphs of the form $\\srg{36}{15}{6}{6}$, so none of them is DS (by Corollary~\\ref{corollary: SRG DS graphs}).\n\nFurther results on the uniqueness or non-uniqueness of strongly regular graphs with a given parameter vector $(n, d, \\lambda, \\mu)$\ncan be found in \\cite{Cameron2003} and the references therein. Infinite families of strongly regular DS graphs are presented in\nTheorem~\\ref{theorem: DS SRG}. Some known sporadic strongly regular DS graphs are listed in \\cite[Table~2]{vanDamH03}, with an\nupdate in \\cite[Table~1]{vanDamH09}). The uniqueness of further strongly regular graphs with given parameter vectors, making them\ntherefore DS graphs, was established, e.g., in \\cite{Brouwer83,BrouwerH92,StevanovicM2009,Coolsaet2006,Mesner56}.\n\nA combination of \\cite[Table~1]{vanDamH09} and Corollary~\\ref{corollary: Petersen graph is DS} implies that, apart from\ncomplete graphs on fewer than three vertices and all complete bipartite regular graphs (which are known to be DS, as stated in\nItem~\\ref{item 2: DS graphs} of Theorem~\\ref{theorem: special classes of DS graphs}), also all the seven currently known triangle-free\nstrongly regular graphs (see \\cite{StruikB2010}) are DS. These include:\n\\begin{itemize}\n\\item The Pentagon graph $\\CG{5}$ that is $\\srg{5}{2}{0}{1}$ (by Item~\\ref{item 2: DS graphs} of Theorem~\\ref{theorem: special classes of DS graphs},\nand see \\cite[Section~10.1]{BrouwerM22}),\n\\item The Petersen graph $\\srg{10}{3}{0}{1}$ (by Corollary~\\ref{corollary: Petersen graph is DS}, and see \\cite[Section~10.3]{BrouwerM22}),\n\\item Clebsch graph $\\srg{16}{5}{0}{2}$ (see \\cite[Section~10.7]{BrouwerM22}),\n\\item Hoffman-Singleton srg $\\srg{50}{7}{0}{1}$ (see \\cite[Section~10.19]{BrouwerM22}),\n\\item Gewirtz graph $\\srg{56}{10}{0}{2}$ (see \\cite[Section~10.20]{BrouwerM22}),\n\\item Mesner ($\\mathrm{M}_{22}$) graph $\\srg{77}{16}{0}{4}$ (see \\cite[Section~10.27]{BrouwerM22} and \\cite{Brouwer83}),\n\\item Higman-Sims graph $\\srg{100}{22}{0}{6}$ (see \\cite[Section~10.31]{BrouwerM22} and \\cite{Mesner56}).\n\\end{itemize}\n\nAn up-to-date list of strongly regular DS graphs --- strongly regular graphs that are uniquely determined by their parameter\nvectors --- as well as the number of strongly regular NICS graphs for given parameter vectors, is available on Brouwer's website\n\\cite{Brouwer}. An exclamation mark placed to the left of a parameter vector $(n,d,\\lambda,\\mu)$, without a preceding number, indicates\na strongly regular DS graph. In contrast, an exclamation mark preceded by a natural number greater than~1 specifies the number\nof strongly regular NICS graphs with the corresponding parameter vector. For example, as shown in \\cite{Brouwer}, strongly regular\ngraphs with the parameter vectors $(13,6,2,3)$, $(15,6,1,3)$, $(17,8,3,4)$, and $(21,10,3,6)$, among others, are DS graphs. On\nthe other hand, according to \\cite{Brouwer}, there are 15 strongly regular NICS graphs with the parameter vector $(25,12,5,6)$,\n10 strongly regular NICS graphs with the parameter vector $(26,10,3,4)$, and so forth.\n\nTo conclude, as strongly regular NICS graphs are not DS, $\\LM$-DS, or $\\Q$-DS, we were recently informed of ongoing research by\nCioaba \\textit{et al.} \\cite{CioabaGJM25}, which investigates the spectral properties of higher-order Laplacian matrices associated\nwith these graphs. This research demonstrates that the spectra of these new matrices can distinguish some of the strongly regular\nNICS graphs. However, in other cases, strongly regular NICS graphs remain indistinguishable even with the spectra of these\nhigher-order Laplacian matrices.\n\n\n\\section{Graph operations for the construction of cospectral graphs}\n\\label{section: graph operations}\n\nThis section presents such graph operations, focusing on unitary and binary transformations that enable the systematic\nconstruction of cospectral graphs. These operations are designed to preserve the spectral properties of the original\ngraph while potentially altering its structure, thereby producing non-isomorphic graphs with identical eigenvalues.\nBy employing these techniques, one can generate diverse examples of cospectral graphs, offering valuable tools for\ninvestigating the limitations of spectral characterization and exploring the boundaries between graphs that are or\nare not determined by their spectrum, which then relates the scope of the present section to Section~\\ref{section: special families of graphs}\nthat deals with graphs or graph families that are determined by their spectrum.\n\n\\subsection{Coalescence}\n\\label{subsection: Coalescence}\n\n\\noindent\n\nA construction of cospectral trees has been offered in \\cite{Schwenk1973}, implying that almost all trees are not DS.\n\n\\begin{definition}\n\\label{definition: Coalescence}\nLet $\\Gr{G}_1 = (V_1,E_1), \\Gr{G}_2=(V_2,E_2)$ be two graphs with $n_1$, $n_2$ vertices, respectively.\nLet $v_1 \\in V_1$ and $v_2 \\in V_2$ be an arbitrary choice of vertices in both graphs.\nThe \\emph{coalescence of $\\Gr{G}_1$ and $\\Gr{G}_2$ with respect to $v_1$ and $v_2$} is the graph with\n$n_1 + n_2 -1$ vertices, obtained by the union of $\\Gr{G}_1$ and $\\Gr{G}_2$ where $v_1$ and $v_2$ are\nidentified as the same vertex in the united graph.\n\\end{definition}\n\n\\begin{theorem}\n\\label{theorem: Coalescence}\nLet $\\Gr{G}_1 = (V_1,E_1), \\Gr{G}_2=(V_2,E_2)$ be two cospectral graphs, and let $v_1 \\in V_1$ and $v_2 \\in V_2$\nbe an arbitrary choice of vertices in both graphs.\nLet $\\Gr{H}_1$ and $\\Gr{H}_2$ be the subgraphs of $\\Gr{G}_1$ and $\\Gr{G}_2$ that are induced by $V_1 \\setminus \\{v_1\\}$\nand $V_2 \\setminus \\{v_2\\}$, respectively.\nLet $\\Gamma$ be a graph and $u\\in V(\\Gamma)$. If $\\Gr{H}_1$ and $\\Gr{H}_2$ are cospectral, then the coalescence of\n$\\Gr{G}_1$ and $\\Gamma$ with respect to $v_1$ and $u$ is cospectral to the coalescence of $\\Gr{G}_2$ and $\\Gamma$\nwith respect to $v_2$ and $u$.\n\\end{theorem}\n\nCombinatorial arguments that rely on the coalescence operation on graphs lead to a striking asymptotic result in\n\\cite{Schwenk1973}, stating that the fraction of $n$-vertex trees with cospectral and nonisomorphic mates, which\nare also trees, approaches one as $n$ tends to infinity. Consequently, the fraction of the $n$-vertex nonisomorphic\ntrees that are determined by their spectrum (DS) approaches zero as $n$ tends to infinity. In other words, this means\nthat almost all trees are not DS (with respect to their adjacency matrix) \\cite{Schwenk1973}.\n\n\n\\subsection{Seidel switching}\n\\label{subsection: Seidel switching}\n\n\\noindent\n\nSeidel switching is one of the well-known methods for the construction of cospectral graphs.\n\n\\begin{definition}\n\\label{definition: Seidel switching}\nLet $\\Gr{G}$ be a graph, and let $\\set{U} \\subseteq \\V{\\Gr{G}}$. Constructing a graph $\\Gr{G}_{\\set{U}}$ by preserving\nall the edges in $\\Gr{G}$ between vertices within $\\set{U}$, as well as all edges in $\\Gr{G}$ between vertices within\nthe complement set $\\cset{U} = \\V{\\Gr{G}} \\setminus \\set{U}$, while modifying adjacency and nonadjacency between any\ntwo vertices where one is in $\\set{U}$ and the other is in $\\cset{U}$, is referred to (up to isomorphism) as\n\\emph{Seidel switching} of $\\Gr{G}$ with respect to $\\set{U}$.\n\\end{definition}\nBy Definition~\\ref{definition: Seidel switching}, the Seidel switching of $\\Gr{G}$ with respect to $\\set{U}$ is equivalent\nto its Seidel switching with respect to $\\cset{U}$.\nLet $\\A(\\Gr{G})$ and $\\A(\\Gr{G}_{\\set{U}})$ be the adjacency matrices of a graph $\\Gr{G}$ and its Seidel switching\n$\\Gr{G}_{\\set{U}}$, and let $\\A_{\\set{U}}$ and $\\A_{\\cset{U}}$ be the matrices of $\\A(\\Gr{G})$ that, respectively,\nrefer to the adjacency matrices of the subgraphs of $\\Gr{G}$ induced by $\\set{U}$ and $\\cset{U}$. Then, for some\n$\\mathbf{B} \\in \\{0,1\\}^{\\card{\\cset{U}} \\times \\card{\\set{U}}}$, we get\n\\begin{align}\n\\A(\\Gr{G})=\n\\begin{pmatrix}\n\\A_{\\set{U}} & \\mathbf{B}^{\\mathrm{T}}\\\\\n\\mathbf{B} & \\A_{\\cset{U}}\n\\end{pmatrix},\n\\end{align}\nand by Definition~\\ref{definition: Seidel switching},\n\\begin{align}\n\\A(\\Gr{G}_{\\set{U}})=\n\\begin{pmatrix}\n\\A_{\\set{U}} & \\overline{\\mathbf{B}}^{\\mathrm{T}}\\\\\n\\overline{\\mathbf{B}} & \\A_{\\cset{U}}\n\\end{pmatrix},\n\\end{align}\nwhere $\\overline{\\mathbf{B}}$ is obtained from $\\mathbf{B}$ by interchanging zeros and ones. If $\\Gr{G}$\nis a regular graph, the following necessary and sufficient condition for $\\Gr{G}_{\\set{U}}$ to be a regular\ngraph of the same degree of its vertices.\n\\begin{theorem} \\cite[Proposition~1.1.7]{CvetkovicRS2010}\n\\label{theorem: d-regular graphs in Seidel switching}\nLet $\\Gr{G}$ be a $d$-regular graph on $n$ vertices. Then, $\\Gr{G}_{\\set{U}}$ is also $d$-regular if and\nonly if $\\set{U}$ induces a regular subgraph of degree $k$, where $\\card{\\set{U}} = n - 2(d-k)$.\n\\end{theorem}\nThe next result shows the relevance of Seidel switching for the construction of regular and cospectral graphs.\n\\begin{theorem} \\cite[Proposition~1.1.8]{CvetkovicRS2010}\n\\label{theorem: Seidel switching}\nLet $\\Gr{G}$ be a $d$-regular graph, $\\set{U} \\subseteq \\V{\\Gr{G}}$, and let $\\Gr{G}_{\\set{U}}$ be obtained\nfrom $\\Gr{G}$ by Seidel switching. If $\\Gr{G}_{\\set{U}}$ is also a $d$-regular graph, then $\\Gr{G}$ and\n$\\Gr{G}_{\\set{U}}$ are cospectral (and due to their regularity, they are $\\mathcal{X}$-cospectral for\nevery $\\mathcal{X} \\in \\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$).\n\\end{theorem}\n\n\\begin{remark}\n\\label{remark: Seidel switching}\nTheorem~\\ref{theorem: Seidel switching} provides a method for finding cospectral regular graphs. These graphs\nmay be, however, also isomorphic. If the graphs are nonisomorphic, then it gives a pair of NICS graphs.\n\\end{remark}\n\n\\begin{remark}\n\\label{remark: limitation of Seidel switching}\nA regular graph $\\Gr{G}$ on $n$ vertices cannot be switched into another regular graph if $n$ is odd (see\n\\cite[Corollary~4.1.10]{CvetkovicRS2010}), which means that the conditions in Theorem~\\ref{theorem: Seidel switching}\ncannot be satisfied for any regular graph of an odd order.\n\\end{remark}\n\n\\begin{remark}\n\\label{remark: equivalence relation of Seidel switching}\nSeidel switching determines an equivalence relation on graphs. This follows from the fact that switching\nwith respect to a subset $\\set{U} \\in \\V{\\Gr{G}}$, and then with respect to a subset $\\set{V} \\in \\V{\\Gr{G}}$,\nis the same as switching with respect to $(\\set{V} \\setminus \\set{U}) \\cup (\\set{U} \\setminus \\set{V})$ (see\n\\cite[p.~18]{CvetkovicRS2010}).\n\\end{remark}\n\n\\begin{example}\n\\label{example: NICS graphs by Seidel switching}\nThe Shrikhande graph can be obtained through Seidel switching applied to the line graph $\\ell(\\CoBG{4}{4})$ with respect to\nfour independent vertices of the latter (see \\cite[Example~1.2.4]{CvetkovicRS2010}). Both are 6-regular graphs (hence, they\nare cospectral graphs by Theorem~\\ref{theorem: Seidel switching}). Moreover, the former graph is a strongly regular graph\n$\\srg{16}{6}{2}{2}$, whereas the line graph $\\ell(\\CoBG{4}{4})$ is not. Consequently, these are nonisomorphic and cospectral\n(NICS) 6-regular graphs on 16 vertices.\n\\end{example}\n\n\\subsection{The Godsil and McKay method}\n\\label{subsection: Godsil and McKay method}\n\n\\noindent\n\nAnother construction of cospectral pairs of graphs was offered by Godsil and McKay in \\cite{GodsilM1982}.\n\n\\begin{theorem}\n\\label{theorem: Godsil and McKay method}\nLet $\\Gr{G}$ be a graph with an adjacency matrix of the form\n\\begin{align}\n\\A(\\Gr{G})= \\begin{pmatrix}\n{\\mathbf{B}} & {\\mathbf{N}} \\\\\n{\\mathbf{N}}^{\\mathrm{T}} & {\\mathbf{C}}\n\\end{pmatrix}\n\\end{align}\nwhere the sum of each column in ${\\mathbf{N}} \\in \\{0,1\\}^{b \\times c}$ is either $0, b$ or $\\frac{b}{2}$.\nLet $\\widehat{{\\mathbf{N}}}$ be the matrix obtained by replacing each column $\\underline{c}$ in ${\\mathbf{N}}$\nwhose sum of elements is $\\frac{b}{2}$ with its complement $\\mathbf{1}_n - \\underline{c}$. Then, the modified\ngraph $\\widehat{\\Gr{G}}$ whose adjacency matrix is given by\n\\begin{align}\n\\A(\\widehat{\\Gr{G}})\n= \\begin{pmatrix}\n{\\mathbf{B}} & \\widehat{\\mathbf{N}} \\\\\n{\\widehat{\\mathbf{N}}}^{\\mathrm{T}} & {\\mathbf{C}}\n\\end{pmatrix}\n\\end{align}\nis cospectral with $\\Gr{G}$.\n\\end{theorem}\nTwo examples of pairs of NICS graphs are presented in Section~1.8.3 of \\cite{BrouwerH2011}.\n\n\\subsection{Graphs resulting from the duplication and corona graphs}\n\\label{subsection: Graphs resulting from the duplication and corona graphs}\n\n\\begin{definition} \\cite{SampathkumarW1980}\n\\label{definition: duplication graph}\nLet $\\Gr{G}$ be a graph with a vertex set $\\V{\\Gr{G}}=\\{v_1, \\ldots, v_n\\}$, and\nconsider a copy of $\\Gr{G}$ with a vertex set $\\mathrm{\\Gr{G}} = \\{u_1, \\ldots, u_n\\}$,\nwhere $u_i$ is a duplicate of the vertex $v_i$. For each $i \\in \\OneTo{n}$, connect the\nvertex $u_i$ to all the neighbors of $v_i$ in $\\Gr{G}$, and then delete all edges in\n$\\Gr{G}$. Similarly, for each $i \\in \\OneTo{n}$, connect the vertex $v_i$ to all the\nneighbors of $u_i$ in the copied graph, and then delete all edges in the copied graph.\nThe resulting graph, which has $2n$ vertices is called the \\emph{duplication graph}\nof $\\Gr{G}$, and is denoted by $\\DuplicationGraph{\\Gr{G}}$ (see Figure~\\ref{fig:DG(C5)}).\n\\end{definition}\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.30\\textwidth]{DG_C5.png}\n\\caption{\\label{fig:DG(C5)} The duplication graph $\\DuplicationGraph{\\CG{5}}$ (see\nDefinition~\\ref{definition: duplication graph}).}\n\\end{figure}\n\n\\begin{definition} \\cite{FruchtH1970}\n\\label{definition: corona graph}\nLet $\\Gr{G}_1$ and $\\Gr{G}_2$ be graphs on disjoint vertex sets of $n_1$\nand $n_2$ vertices, and with $m_{1}$ and $m_{2}$ edges, respectively. The \\emph{corona}\nof $\\Gr{G}_1$ and $\\Gr{G}_2$, denoted by $\\Corona{\\Gr{G}_1}{\\Gr{G}_2}$, is a graph\non $n_1+n_1 n_2$ vertices obtained by taking one copy of $\\Gr{G}_1$\nand $n_1$ copies of $\\Gr{G}_2$, and then connecting, for each $i \\in \\OneTo{n_1}$,\nthe $i$-th vertex of $\\Gr{G}_1$ to each vertex in the $i$-th copy of $\\Gr{G}_2$\n(see Figure~\\ref{fig:C4 CORONA 2K1}).\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.30\\textwidth]{C4_CORONA_2K1.png}\n\\par\\end{centering}\n\\caption{\\label{fig:C4 CORONA 2K1} The corona graph $\\Corona{\\CG{4}}{(2\\CoG{1})}$\n(see Definition~\\ref{definition: corona graph}) consists of a single copy of $\\CG{4}$\n(represented by the black vertices) and four copies of $2\\CoG{1}$ (represented by the red vertices).}\n\\end{figure}\n\\end{definition}\n\n\\begin{definition} \\cite{HouS2010}\n\\label{definition: edge corona graph}\nThe \\emph{edge corona} of $\\Gr{G}_1$ and $\\Gr{G}_2$, denoted by\n$\\EdgeCorona{\\Gr{G}_1}{\\Gr{G}_2}$, is defined as the graph obtained by taking\none copy of $\\Gr{G}_1$ and $m_1 = \\bigcard{\\E{\\Gr{G}_1}}$ copies of $\\Gr{G}_2$,\nand then connecting, for each $j \\in \\OneTo{m_1}$, the two end-vertices of the\n$j$-th edge of $\\Gr{G}_1$ to every vertex in the $j$-th copy of $\\Gr{G}_2$ (see\nFigure~\\ref{fig:C4 edge corona P3}).\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.30\\textwidth]{C4_edge_corona_P3.png}\n\\par\\end{centering}\n\\caption{\\label{fig:C4 edge corona P3}\nThe edge-corona graph $\\EdgeCorona{\\CG{4}}{\\PathG{3}}$ (see\nDefinition~\\ref{definition: edge corona graph}.}\n\\end{figure}\n\\end{definition}\n\n\\begin{definition}\n\\label{definition: duplication corona graph}\nLet $\\Gr{G}_1$ and $\\Gr{G}_2$ be graphs with disjoint vertex sets of $n_1$\nand $n_2$ vertices, respectively. Let $Du(\\Gr{G}_1)$ be the duplication\ngraph of $\\Gr{G}_1$ with vertex set $\\V{\\Gr{G}_1} \\cup \\mathrm{U}(\\Gr{G}_1)$, where\n$\\V{\\Gr{G}_1}=\\{v_{1},\\ldots,v_{n_1}\\}$ and with the duplicated vertex set\n$\\mathrm{U}(\\Gr{G}_1)=\\{u_{1},\\ldots,u_{n_{1}}\\}$ (see Definition~\\ref{definition: duplication graph}).\nThe \\emph{duplication corona graph}, denoted by $\\DuplicationCorona{\\Gr{G}_1}{\\Gr{G}_2}$,\nis the graph obtained from $\\DuplicationGraph{\\Gr{G}_1}$ and $n_1$ copies of $\\Gr{G}_2$\nby connecting, for each $i \\in \\OneTo{n_1}$, the vertex $v_{i} \\in \\V{\\Gr{G}_1}$ of\nthe graph $\\DuplicationGraph{\\Gr{G}_1}$ to every vertex in the $i$-th copy of $\\Gr{G}_2$\n(see Figure~\\ref{fig:D.C}).\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.30\\textwidth]{D.C.png}\n\\par\\end{centering}\n\\caption{\\label{fig:D.C} The duplication corona graph $\\DuplicationCorona{\\CG{4}}{\\CoG{2}}$\n(see Definition~\\ref{definition: duplication corona graph}).}\n\\end{figure}\n\\end{definition}\n\n\n\\begin{definition}\n\\label{definition: duplication neighborhood corona}\nLet $\\Gr{G}_1$ and $\\Gr{G}_2$ be graphs with disjoint vertex sets of $n_1$\nand $n_2$ vertices, respectively. Let $\\DuplicationGraph{\\Gr{G}_1}$ be the duplication\ngraph of $\\Gr{G}_1$ with the vertex set $\\mathrm{V}(\\Gr{G}_1)\\cup \\mathrm{U}(\\Gr{G}_1)$, where\n$\\mathrm{V}(\\Gr{G}_1)=\\{v_1, \\ldots, v_{n_1}\\}$ and the duplicated vertex set\n$\\mathrm{U}(\\Gr{G}_1)=\\{u_1,\\ldots,u_{n_1}\\}$ (see Definition~\\ref{definition: duplication graph}).\nThe \\emph{duplication neighborhood corona}, denoted by $\\Gr{G}_1 \\boxbar \\Gr{G}_2$,\nis the graph obtained from $\\DuplicationGraph{\\Gr{G}_1}$ and $n_1$ copies of $\\Gr{G}_2$\nby connecting the neighbors of the vertex $v_i \\in \\V{\\Gr{G}_1}$ of $\\DuplicationGraph{\\Gr{G}_1}$\nto every vertex in the $i$-th copy of $\\Gr{G}_2$ for $i \\in \\OneTo{n_1}$ (see Figure~\\ref{fig:D.N.C}).\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.30\\textwidth]{Duplication_neighbours_corona.png}\n\\par\\end{centering}\n\\caption{\\label{fig:D.N.C} The duplication neighborhood corona $\\CoG{3} \\boxbar \\CoG{2}$\n(see Definition~\\ref{definition: duplication neighborhood corona}).}\n\\end{figure}\n\\end{definition}\n\n\n\\begin{definition}\n\\label{definition: duplication edge corona}\nLet $\\Gr{G}_1$ and $\\Gr{G}_2$ be graphs with disjoint vertex sets of $n_1$\nand $n_2$ vertices, respectively. Let $\\DuplicationGraph{\\Gr{G}_1}$ be the\nduplication graph of $\\Gr{G}_1$ with vertex set $\\V{\\Gr{G_1}} \\cup\n\\mathrm{U}(\\Gr{G}_1)$, where $\\V{\\Gr{G}_1}=\\{v_1, \\ldots, v_{n_1}\\}$ is the\nvertex set of $\\Gr{G}_1$ and $\\mathrm{U}(\\Gr{G}_1)=\\{u_1, \\ldots, u_{n_{1}}\\}$\nis the duplicated vertex set.\nThe \\emph{duplication edge corona}, denoted by $\\DuplicationEdgeCorona{\\Gr{G}_1}{\\Gr{G}_2}$,\nis the graph obtained from $\\DuplicationGraph{\\Gr{G}_1}$ and $\\bigcard{\\E{\\Gr{G}_1}}$\ncopies of $\\Gr{G}_2$ by connecting each of the two vertices $v_i, v_j \\in \\V{\\Gr{G}_1}$\nof $\\DuplicationGraph{\\Gr{G}_1}$ to every vertex in the $k$-th copy of $\\Gr{G}_2$\nwhenever $\\{v_i, v_j\\}=e_k \\in \\E{\\Gr{G}_1}$ (see Figure~\\ref{fig:D.E.C}).\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.30\\textwidth]{D.E.C.png}\n\\par\\end{centering}\n\\centering{} \\caption{\\label{fig:D.E.C}\nThe duplication edge corona $\\DuplicationEdgeCorona{\\CoG{3}}{\\CoG{2}}$ (see\nDefinition~\\ref{definition: duplication edge corona}).}\n\\end{figure}\n\\end{definition}\n\n\\begin{definition}\n\\label{definition: closed neighborhood corona}\nConsider two graphs $\\Gr{G}_1$ and $\\Gr{G}_2$ with $n_{1}$ and $n_{2}$\nvertices and, respectively. The \\emph{closed neighborhood corona} of\n$\\Gr{G}_1$ and $\\Gr{G}_2$, denoted by $\\ClosedNeighborhoodCorona{\\Gr{G}_1}{\\Gr{G}_2}$,\nis a new graph obtained by creating $n_{1}$ copies of $\\Gr{G}_2$. Each\nvertex of the $i^{th}$ copy of $\\Gr{G}_2$ is then connected to the\n$i^{th}$ vertex and neighborhood of the $i^{th}$ vertex of $\\Gr{G}_1$\n(see Figure~\\ref{fig:CNCP}).\n\\end{definition}\n\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.30\\textwidth]{C4andk3.png}\n\\par\\end{centering}\n\\caption{\\label{fig:CNCP} \\centering{The closed neighborhood corona\nof the 4-length cycle $\\CG{4}$ and the triangle $\\CoG{3}$, denoted by\n$\\ClosedNeighborhoodCorona{\\CG{4}}{\\CoG{3}}$ (see\nDefinition~\\ref{definition: closed neighborhood corona}).}}\n\\end{figure}\n\n\\begin{theorem} \\cite{AdigaRS2018}\n\\label{theorem: NICS graphs - 1}\nLet $\\Gr{G}_1,\\Gr{H}_1$ be $r_1$-regular, cospectral graphs, and let\n$\\Gr{G}_2$ and $H_{2}$ be $r_2$-regular, cospectral, and nonisomorphic (NICS)\ngraphs. Then, the following holds:\n\\begin{itemize}\n\\item The duplication corona graphs $\\DuplicationCorona{\\Gr{G}_1}{\\Gr{G}_2}$ and\n$\\DuplicationCorona{\\Gr{H}_1}{\\Gr{H}_2}$ are $\\{ \\A, \\LM, \\Q \\}$-NICS irregular graphs.\n\\item The duplication neighborhood corona $\\Gr{G}_1\\boxbar \\Gr{G}_2$ and\n$\\Gr{H}_1\\boxbar H_{2}$ are $\\{ \\A, \\LM, \\Q \\}$-NICS irregular graphs.\n\\item The duplication edge corona $\\DuplicationEdgeCorona{\\Gr{G}_1}{\\Gr{G}_2}$ and\n$\\DuplicationEdgeCorona{\\Gr{H}_1}{\\Gr{H}_2}$ are $\\{ \\A, \\LM, \\Q \\}$-NICS irregular graphs.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{question}\n\\label{question: NICS graphs - 1}\nAre the irregular graphs in Theorem~\\ref{theorem: NICS graphs - 1} also cospectral with\nrespect to the normalized Laplacian matrix?\n\\end{question}\n\n\\begin{theorem}\n\\label{theorem: NICS graphs - 2}\n\\cite{SonarS2024} Let $\\Gr{G}_1$ and $\\Gr{G}_2$ be cospectral regular graphs, and let\n$\\Gr{H}$ be an arbitrary graph. Then, the following holds:\n\\begin{itemize}\n\\item The closed neighborhood corona $\\ClosedNeighborhoodCorona{\\Gr{G}_1}{\\Gr{H}}$ and\n$\\ClosedNeighborhoodCorona{\\Gr{G}_2}{\\Gr{H}}$ are $\\{ \\A, \\LM, \\Q \\}$-NICS irregular graphs.\n\\item The closed neighborhood corona $\\ClosedNeighborhoodCorona{\\Gr{H}}{\\Gr{G}_{1}}$\nand $\\ClosedNeighborhoodCorona{\\Gr{H}}{\\Gr{G}_{2}}$ are $\\{ \\A, \\LM, \\Q \\}$-NICS irregular graphs.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{question}\n\\label{question: NICS graphs - 2}\nAre the irregular graphs in Theorem~\\ref{theorem: NICS graphs - 2} also cospectral with respect\nto the normalized Laplacian matrix?\n\\end{question}\n\n\\subsection{Graphs constructions based on the subdivision and bipartite incidence graphs}\n\\label{subsection: Graphs constructions based on the subdivision and bipartite incidence graphs}\n\\begin{definition} \\cite{CvetkovicRS2010}\n\\label{definition: subdivision graph}\nLet $\\Gr{G}$ be a graph. The \\emph{subdivision graph} of $\\Gr{G}$, denoted by $\\SubdivisionGraph{\\Gr{G}}$,\nis obtained from $\\Gr{G}$ by inserting a new vertex into every edge of $\\Gr{G}$.\nSubdivision is the process of adding a new vertex along an edge, effectively splitting the edge into two\nedges connected in series through the new vertex (see Figure~\\ref{fig:subdivition}).\n\\begin{figure}[hbt]\n\\centering{}\\includegraphics[width=0.30\\textwidth]{S_C4.png}\n\\caption{\\label{fig:subdivition} \\centering{The subdivision graph of a $4$-length cycle, denoted by\n$\\SubdivisionGraph{\\CG{4}}$, which is an $8$-length cycle $\\CG{8}$ (see Definition~\\ref{definition: subdivision graph}).}}\n\\end{figure}\n\\end{definition}\n\n\\begin{definition} \\cite{PisankiS13}\n\\label{definition: bipartite incidence graph}\nLet $\\Gr{G}$ be a graph. The \\emph{bipartite incidence graph} of $\\Gr{G}$, denoted by $\\BipartiteIncidenceGraph{\\Gr{G}}$,\nis a bipartite graph constructed as follows: For each edge $e \\in \\Gr{G}$, a new vertex $u_{e}$ is added to\nthe vertex set of $\\Gr{G}$. The vertex $u_e$ is then made adjacent to both endpoints of $e$ in $\\Gr{G}$\n(see Figure~\\ref{fig:R(C4)-1}).\n\\begin{figure}[H]\n\\begin{centering}\n\\includegraphics[width=0.30\\textwidth]{R_C4.png}\n\\par\\end{centering}\n\\centering{}\\caption{\\label{fig:R(C4)-1} \\centering{The bipartite incidence graph of a length-4 cycle $\\CG{4}$,\ndenoted by $\\BipartiteIncidenceGraph{\\CG{4}}$ (see Definition~\\ref{definition: bipartite incidence graph}).}}\n\\end{figure}\n\\end{definition}\n\nLet $\\Gr{G}_1$ and $\\Gr{G}_2$ be two vertex-disjoint graphs with $n_1$ and $n_2$ vertices, and $m_1$ and $m_2$ edges,\nrespectively. Four binary operations on these graphs are defined as follows.\n\\begin{definition}\n\\label{definition: subdivision-vertex-bipartite-vertex join}\nThe \\emph{subdivision-vertex-bipartite-vertex join} of graphs $\\Gr{G}_1$ and $\\Gr{G}_2$, denoted $\\SVBVJ{\\Gr{G}_1}{\\Gr{G}_2}$,\nis the graph obtained from $\\SubdivisionGraph{\\Gr{G}_1}$ and $\\BipartiteIncidenceGraph{\\Gr{G}_2}$\nby connecting each vertex of $\\V{\\Gr{G}_1}$ (that is the subset of the original vertices in the vertex set of $\\SubdivisionGraph{\\Gr{G}_1}$)\nwith every vertex of $\\V{\\Gr{G}_2}$ (that is the subset of the original vertices in the vertex set of $\\BipartiteIncidenceGraph{\\Gr{G}_2}$).\n\\end{definition}\nBy Definitions~\\ref{definition: subdivision graph}, \\ref{definition: bipartite incidence graph}, and~\\ref{definition: subdivision-vertex-bipartite-vertex join},\nit follows that the graph $\\SVBVJ{\\Gr{G}_1}{\\Gr{G}_2}$ has $n_1+n_2+m_1+m_2$ vertices and $n_1 n_2+2m_1+3m_2$ edges.\nFigure~\\ref{fig:subdivision-vertex-bipartite-vertex join} displays the graph $\\SVBVJ{\\CG{4}}{\\PathG{3}}$.\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.40\\textwidth]{C_c4_and_R_p3.png}\n\\par\\end{centering}\n\\caption{\\label{fig:subdivision-vertex-bipartite-vertex join} The graph $\\SVBVJ{\\CG{4}}{\\PathG{3}}$ (see Definition~\\ref{definition: subdivision-vertex-bipartite-vertex join}).\nThe black vertices represent the vertices of the length-4 cycle $\\CG{4}$ and the vertices of the path $\\PathG{3}$. The additional vertices in the\nsubdivision graph $\\SubdivisionGraph{\\CG{4}}$ are the four red vertices located at the bottom of this figure (as shown in Figure~\\ref{fig:subdivition}).\nSimilarly, the additional vertices in the bipartite incidence graph of the path $\\PathG{3}$ are the two red vertices located at the top of this figure.\nThe edges of the graph include the following: edges connecting each black vertex of $\\CG{4}$ to each black vertex of $\\PathG{3}$, edges between the black\nand red vertices at the bottom of the figure (that correspond to the subdivision of $\\CG{4}$), and the four (light blue) edges connecting the two top red\nvertices to the top black vertices of the figure (that correspond to the bipartite incidence graph of $\\PathG{3}$).}\n\\end{figure}\n\n\\begin{definition}\n\\label{definition: subdivision-edge-bipartite-edge join}\nThe \\emph{subdivision-edge-bipartite-edge join} of $\\Gr{G}_1$ and $\\Gr{G}_2$, denoted by $\\SEBEJ{\\Gr{G}_1}{\\Gr{G}_2}$,\nis the graph obtained from $\\SubdivisionGraph{\\Gr{G}_1}$ and $\\BipartiteIncidenceGraph{\\Gr{G}_2}$\nby connecting each vertex of $\\V{\\SubdivisionGraph{\\Gr{G}_1}} \\setminus \\V{\\Gr{G}_1}$ (that is the subset of the added vertices in the\nvertex set of $\\SubdivisionGraph{\\Gr{G}_1}$) with every vertex of $\\V{\\BipartiteIncidenceGraph{\\Gr{G}_2}} \\setminus \\V{\\Gr{G}_2}$\n(that is the subset of the added vertices in the vertex set of $\\BipartiteIncidenceGraph{\\Gr{G}_2}$).\n\\end{definition}\nBy Definitions~\\ref{definition: subdivision graph}, \\ref{definition: bipartite incidence graph}, and~\\ref{definition: subdivision-edge-bipartite-edge join},\nit follows that the graph $\\SEBEJ{\\Gr{G}_1}{\\Gr{G}_2}$ has $n_1+n_2+m_1+m_2$ vertices and $m_1 m_2 + 2m_1 + 3m_2$ edges.\nFigure~\\ref{fig:subdivision-edge-R-edge join} displays the graph $\\SEBEJ{\\CG{4}}{\\PathG{3}}$.\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.40\\textwidth]{subdivision-edge-R-edge-join.png}\n\\par\\end{centering}\n\\caption{\\label{fig:subdivision-edge-R-edge join} The graph $\\SEBEJ{\\CG{4}}{\\PathG{3}}$ (see Definition~\\ref{definition: subdivision-edge-bipartite-edge join}).\nIn comparison to Figure~\\ref{fig:subdivision-vertex-bipartite-vertex join}, edges connecting each black vertex of $\\CG{4}$ to each black vertex of $\\PathG{3}$\nare deleted and replaced in this figure by edges connecting each red vertex of $\\CG{4}$ to each red vertex of $\\PathG{3}$.}\n\\end{figure}\n\n\\begin{definition}\n\\label{definition: subdivision-edge-bipartite-vertex join}\nThe \\emph{subdivision-edge-bipartite-vertex join} of $\\Gr{G}_1$ and $\\Gr{G}_2$, denoted by $\\SEBVJ{\\Gr{G}_1}{\\Gr{G}_2}$,\nis the graph obtained from $\\SubdivisionGraph{\\Gr{G}_1}$ and $\\BipartiteIncidenceGraph{\\Gr{G}_2}$ by connecting each vertex\nof $\\V{\\SubdivisionGraph{\\Gr{G}_1}} \\setminus \\V{\\Gr{G}_1}$ (that is the subset of the added vertices in the vertex set of\n$\\SubdivisionGraph{\\Gr{G}_1}$) with every vertex of $\\V{\\Gr{G}_2}$ (that is the subset of the original vertices in the vertex\nset of $\\BipartiteIncidenceGraph{\\Gr{G}_2}$).\n\\end{definition}\nBy Definitions~\\ref{definition: subdivision graph}, \\ref{definition: bipartite incidence graph}, and~\\ref{definition: subdivision-edge-bipartite-vertex join},\nit follows that the graph $\\SEBVJ{\\Gr{G}_1}{\\Gr{G}_2}$ has $n_1+n_2+m_1+m_2$ vertices and $m_1 n_2 + 2m_1 + 3m_2$ edges.\nFigure~\\ref{fig:subdivision-edge-R-vertex join} displays the graph $\\SEBVJ{\\CG{4}}{\\PathG{3}}$.\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.40\\textwidth]{subdivision-edge-R-vertex-join.png}\n\\par\\end{centering}\n\\caption{\\label{fig:subdivision-edge-R-vertex join} The graph $\\SEBVJ{\\CG{4}}{\\PathG{3}}$ (see Definition~\\ref{definition: subdivision-edge-bipartite-vertex join}).\nIn comparison to Figure~\\ref{fig:subdivision-vertex-bipartite-vertex join}, edges connecting each black vertex of $\\CG{4}$ to each black vertex of $\\PathG{3}$\nare deleted and replaced in this figure by edges connecting each red vertex of $\\CG{4}$ to each black vertex of $\\PathG{3}$.}\n\\end{figure}\n\n\\begin{definition}\n\\label{definition: subdivision-vertex-bipartite-edge join}\nThe \\emph{subdivision-vertex-bipartite-edge join} of $\\Gr{G}_1$ and $\\Gr{G}_2$, denoted by $\\SVBEJ{\\Gr{G}_1}{\\Gr{G}_2}$,\nis the graph obtained from $\\SubdivisionGraph{\\Gr{G}_1}$ and $\\BipartiteIncidenceGraph{\\Gr{G}_2}$ by connecting each vertex\nof $\\V{\\Gr{G}_1}$ (that is the subset of the original vertices in the vertex set of $\\SubdivisionGraph{\\Gr{G}_1}$) with\nevery vertex of $\\V{\\BipartiteIncidenceGraph{\\Gr{G}_2}} \\setminus \\V{\\Gr{G}_2}$ (that is the subset of the added vertices in the vertex\nset of $\\BipartiteIncidenceGraph{\\Gr{G}_2}$).\n\\end{definition}\nBy Definitions~\\ref{definition: subdivision graph}, \\ref{definition: bipartite incidence graph}, and~\\ref{definition: subdivision-vertex-bipartite-edge join},\nit follows that the graph $\\SVBEJ{\\Gr{G}_1}{\\Gr{G}_2}$ has $n_1+n_2+m_1+m_2$ vertices and $n_1 m_2 + 2m_1 + 3m_2$ edges.\nFigure~\\ref{fig:subdivision-vertex-R-edge join} displays the graph $\\SVBEJ{\\CG{4}}{\\PathG{3}}$.\n\\begin{figure}[hbt]\n\\begin{centering}\n\\includegraphics[width=0.40\\textwidth]{subdivision-vertex-R-edge-join.png}\n\\par\\end{centering}\n\\caption{\\label{fig:subdivision-vertex-R-edge join} The graph $\\SVBEJ{\\CG{4}}{\\PathG{3}}$ (see Definition~\\ref{definition: subdivision-vertex-bipartite-edge join}).\nIn comparison to Figure~\\ref{fig:subdivision-vertex-bipartite-vertex join}, edges connecting each black vertex of $\\CG{4}$ to each black vertex of $\\PathG{3}$\nare deleted and replaced in this figure by edges connecting each black vertex of $\\CG{4}$ to each red vertex of $\\PathG{3}$.}\n\\end{figure}\n\nWe next present the main result of this subsection, which motivates the four binary graph operations introduced in\nDefinitions~\\ref{definition: subdivision-vertex-bipartite-vertex join}--\\ref{definition: subdivision-vertex-bipartite-edge join}.\n\\begin{theorem} \\cite{DasP2013}\n\\label{theorem: NICS - DasP2013} Let $\\Gr{G}_i$ , $\\Gr{H}_i$, where $i \\in \\{1,2\\}$,\nbe regular graphs, where $\\Gr{G}_1$ need not be different from $\\Gr{H}_1$.\nIf $\\Gr{G}_1$ and $\\Gr{H}_1$ are $\\A$-cospectral, and $\\Gr{G}_2$ and $\\Gr{H}_2$\nare $\\A$-cospectral, then\n\\begin{itemize}\n\\item $\\SVBVJ{\\Gr{G}_1}{\\Gr{G}_2}$ and $\\SVBVJ{\\Gr{H}_1}{\\Gr{H}_2}$\nare irregular $\\{ \\A, \\LM, {\\bf{\\mathcal{L}}} \\} $-NICS graphs.\n\\item $\\SEBEJ{\\Gr{G}_1}{\\Gr{G}_2}$ and $\\SEBEJ{\\Gr{H}_1}{\\Gr{H}_2}$\nare irregular $\\{ \\A, \\LM, {\\bf{\\mathcal{L}}} \\} $-NICS graphs.\n\\item $\\SEBVJ{\\Gr{G}_1}{\\Gr{G}_2}$ and $\\SEBVJ{\\Gr{H}_1}{\\Gr{H}_2}$\nare irregular $\\{ \\A, \\LM, {\\bf{\\mathcal{L}}} \\} $-NICS graphs.\n\\item $\\SVBEJ{\\Gr{G}_1}{\\Gr{G}_2}$ and $\\SVBEJ{\\Gr{H}_1}{\\Gr{H}_2}$\nare irregular $\\{ \\A, \\LM, {\\bf{\\mathcal{L}}} \\} $-NICS graphs.\n\\end{itemize}\n\\end{theorem}\n\nIn light of Theorem~\\ref{theorem: NICS - DasP2013}, the following questions naturally arises.\n\\begin{question}\n\\label{question: NICS - DasP2013}\nAre the graphs in Theorem~\\ref{theorem: NICS - DasP2013} also cospectral with respect to the signless Laplacian matrix (i.e., $\\Q$-cospectral)?\n\\end{question}\n\n\\subsection{Connected irregular NICS graphs}\n\\label{subsection: Connected irregular NICS graphs}\n\n\\noindent\n\nThe (joint) cospectrality of regular graphs with respect to their adjacency, Laplacian, signless Laplacian, and\nnormalized Laplacian matrices can be asserted by verifying their cospectrality with respect to only one of these matrices.\nIn other words, regular graphs are $X$-cospectral for some $X \\in \\{\\A, \\LM, \\Q, {\\bf{\\mathcal{L}}} \\}$\nif and only if they are cospectral with respect to all these matrices (see Proposition~\\ref{proposition: regular graphs cospectrality}).\nFor irregular graphs, this does not hold in general.\nFollowing \\cite{Butler2010}, it is natural to ask the following question:\n\\begin{question}\n\\label{question: Butler2010}\nAre there pairs of irregular graphs that are $\\{\\A,\\LM,\\Q,{\\bf{\\mathcal{L}}}\\}$-NICS,\ni.e., $X$-NICS with respect to every $X \\in \\{\\A, \\LM, \\Q, {\\bf{\\mathcal{L}}}\\}$?\n\\end{question}\nThis question remained open until two coauthors of this paper recently proposed a method for constructing pairs\nof irregular graphs that are $X$-cospectral with respect to every $X \\in \\{\\A, \\LM, \\Q, {\\bf{\\mathcal{L}}}\\}$,\nproviding explicit constructions \\cite{HamudB24}. Building on that work, another coauthor of this paper demonstrated\nin \\cite{Sason2024} that for every even integer $n \\geq 14$, there exist two connected, irregular\n$\\{\\A,\\LM,\\Q,{\\bf{\\mathcal{L}}}\\}$-NICS graphs on $n$ vertices with identical independence, clique, and chromatic numbers,\nyet distinct Lov{\\'a}sz $\\vartheta$-functions. We now present the preliminary definitions required to outline\nthe relevant results in \\cite{HamudB24} and \\cite{Sason2024}, and the construction of such cospectral irregular\n$X$-NICS graphs for all $X \\in \\{\\A, \\LM, \\Q, {\\bf{\\mathcal{L}}}\\}$.\n\n\\begin{definition}[Neighbors splitting join of graphs]\n\\label{definition: neighbors splitting join of graphs}\n\\cite{LuMZ2023} Let $\\Gr{G}$ and $\\Gr{H}$ be graphs with disjoint vertex sets, and let\n$\\V{\\Gr{G}} = \\{v_1, \\ldots, v_n\\}$. The {\\em neighbors splitting (NS) join}\nof $\\Gr{G}$ and $\\Gr{H}$ is obtained by adding vertices $v'_1, \\ldots, v'_n$\nto the vertex set of $\\Gr{G} \\vee \\Gr{H}$ and connecting $v'_i$ to $v_j$ if\nand only if $\\{v_i, v_j\\} \\in \\E{\\Gr{G}}$.\nThe NS join of $\\Gr{G}$ and $\\Gr{H}$ is denoted by $\\Gr{G} \\NS \\Gr{H}$.\n\\end{definition}\n\n\\begin{definition}[Nonneighbors splitting join of graphs \\cite{Hamud23,HamudB24}]\n\\label{definition: nonneighbors splitting join of graphs}\nLet $\\Gr{G}$ and $\\Gr{H}$ be graphs with disjoint vertex sets, and let\n$\\V{\\Gr{G}} = \\{v_1, \\ldots, v_n\\}$. The {\\em nonneighbors splitting (NNS) join}\nof $\\Gr{G}$ and $\\Gr{H}$ is obtained by adding vertices $v'_1, \\ldots, v'_n$\nto the vertex set of $\\Gr{G} \\vee \\Gr{H}$ and connecting $v'_i$ to $v_j$, with\n$i \\neq j$, if and only if $\\{v_i, v_j\\} \\not\\in \\E{\\Gr{G}}$.\nThe NNS join of $\\Gr{G}$ and $\\Gr{H}$ is denoted by $\\Gr{G} \\NNS \\Gr{H}$.\n\\end{definition}\n\n\\begin{remark}\nIn general, $\\Gr{G} \\NS \\Gr{H} \\not\\cong \\Gr{H} \\NS \\Gr{G}$ and\n$\\Gr{G} \\NNS \\Gr{H} \\not\\cong \\Gr{H} \\NNS \\Gr{G}$ (unless $\\Gr{G} \\cong \\Gr{H}$).\n\\end{remark}\n\n\n\\begin{example}[NS and NNS join of graphs \\cite{HamudB24}]\n\\label{example: NS and NNS Join of Graphs}\nFigure~\\ref{fig:BermanH23-Fig. 2.2} shows the NS and NNS\njoins of the path graphs $\\PathG{4}$ and $\\PathG{2}$, denoted by $\\PathG{4} \\NS \\PathG{2}$\nand $\\PathG{4} \\NNS \\PathG{2}$, respectively.\n\\vspace*{-0.1cm}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=11cm]{Hamud_and_Berman_Figure0.png}\n\\caption{The neighbors splitting (NS)\nand nonneighbors splitting (NNS) joins of the path graphs $\\PathG{4}$ and $\\PathG{2}$\nare depicted, respectively, on the left and right-hand sides of this figure.\nThe NS and NNS joins of graphs are, respectively, denoted by $\\PathG{4} \\NS \\PathG{2}$\nand $\\PathG{4} \\NNS \\PathG{2}$ \\cite{HamudB24}.}\\label{fig:BermanH23-Fig. 2.2}\n\\end{figure}\n\\end{example}\n\n\\begin{theorem}[Irregular $\\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$-NICS graphs]\n\\label{theorem: Berman and Hamud, 2023}\nLet $\\Gr{G}_1$ and $\\Gr{H}_1$ be regular and cospectral graphs, and let $\\Gr{G}_2$ and $\\Gr{H}_2$\nbe regular, nonisomorphic, and cospectral (NICS) graphs. Then, the following statements hold:\n\\begin{enumerate}[(1)]\n\\item \\label{Item 1: Berman and Hamud, 2023}\nThe NS-join graphs $\\Gr{G}_1 \\NS \\Gr{G}_2$ and $\\Gr{H}_1 \\NS \\Gr{H}_2$\nare irregular $\\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$-NICS graphs.\n\\item \\label{Item 2: Berman and Hamud, 2023}\nThe NNS join graphs $\\Gr{G}_1 \\NNS \\Gr{G}_2$ and $\\Gr{H}_1 \\NNS \\Gr{H}_2$\nare irregular $\\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$-NICS graphs.\n\\end{enumerate}\n\\end{theorem}\nThe proof of Theorem~\\ref{theorem: Berman and Hamud, 2023} is provided in \\cite{Hamud23,HamudB24}, and it relies\nheavily on the notion of the Schur complement (see Theorem~\\ref{theorem: Schur complement}).\nThe interested reader is referred to the recently published paper \\cite{HamudB24} for further details.\n\nRelying on Theorem~\\ref{theorem: Berman and Hamud, 2023}, the following result is stated and proved in \\cite{Sason2024}.\n\\begin{theorem}[On irregular NICS graphs]\n\\label{theorem: Sason - existence of NICS graphs with distinct theta functions}\nFor every even integer $n \\geq 14$, there exist two connected, irregular $\\{\\A, \\LM, \\Q, \\bf{\\mathcal{L}}\\}$-NICS graphs\non $n$ vertices with identical independence, clique, and chromatic numbers, yet their Lov\\'{a}sz $\\vartheta$-functions\nare distinct.\n\\end{theorem}\nThat result is in fact strengthened in Section~4.2 of \\cite{Sason2024}, and the interested reader is referred to that paper\nfor further details. The proof of Theorem~\\ref{theorem: Sason - existence of NICS graphs with distinct theta functions} is\nalso constructive, providing explicit such graphs.\n\n\n\\section{Open questions and outlook}\n\\label{section: summary and outlook}\n\nWe conclude this paper by highlighting some of the most significant open questions in the fascinating area of research related to\ncospectral nonisomorphic graphs and graphs determined by their spectrum, leaving them as topics for further future study.\n\n\\subsection{Haemers' conjecture}\n\\label{subsection: Haemers' conjecture}\n\nHaemers' conjecture \\cite{Haemers2016,Haemers2024}, a prominent topic in spectral graph theory, posits that almost all graphs\nare uniquely determined by their adjacency spectrum. This conjecture suggests that for large graphs, the probability of having two\nnon-isomorphic graphs, on a large number of vertices, sharing the same adjacency spectrum is negligible. The conjecture has inspired\nextensive research, including studies on specific graph families, cospectrality, and algebraic graph invariants, contributing to deeper\ninsights into the relationship between graph structure and eigenvalues. Haemers' conjecture is stated formally as follows.\n\n\\begin{definition}\n\\label{definition: Haemers' conjecture}\nFor $n \\in \\mathbb{N}$, let $I(n)$ to be the numbers of distinct graphs on $n$ vertices, up to isomorphism.\nFor $\\mathcal{X} \\subseteq \\Gmats$, let $\\alpha_{\\mathcal{X}}(n)$ be the number of $\\mathcal{X}$-DS graphs\non $n$ vertices, up to isomorphism, and (for the sake of simplicity of notation) let $\\alpha(n) \\triangleq \\alpha_{\\{ A \\}}(n)$\n\\end{definition}\n\n\\begin{conjecture}\n\\label{conjecture: Haemers' conjecture} \\cite{Haemers2016}\nFor sufficiently large $n$, almost all of the graphs are DS, i.e.,\n\\begin{align}\n\\label{eq: Heamers' conjecture}\n\\lim_{n \\rightarrow \\infty} \\frac{\\alpha(n)}{I(n)} = 1.\n\\end{align}\n\\end{conjecture}\n\nSeveral results lend support to this conjecture \\cite{Haemers2016, KovalK2024, WangW2024}, but a complete proof in its full generality remains elusive.\nBy \\cite{GodsilR2001}, the number of graphs with $n$ vertices up to isomorphism is\n\\begin{align}\nI(n) = \\bigl( 1 - \\epsilon(n) \\bigr) \\, \\frac{2^{n(n-1)/2}}{n!},\n\\end{align}\nwhere $\\underset{n \\rightarrow \\infty}{\\lim} \\epsilon(n) = 0$.\nBy Stirling's approximation for $n!$ and straightforward algebra, $I(n)$ can be verified to be given by\n\\begin{align}\nI(n) = 2^{\\frac{n(n-1)}{2} \\; \\bigl(1 - \\epsilon(n)\\bigr)}.\n\\end{align}\n\nIt is shown in \\cite{KovalK2024} that the number of DS graphs on $n$ vertices is at least $e^{cn}$ for some constant $c>0$ (see also the\ndiscussion in Section~\\ref{subsection: Nice graphs}).\nIn light of Remark~\\ref{remark: X,Y cospectrality}, Conjecture~\\ref{conjecture: Haemers' conjecture} leads to the following question.\n\\begin{question}\n\\label{question: generalized Haemers' conjecture}\nFor what minimal subset $\\mathcal{X} \\subset \\Gmats$ (if any) does the limit\n\\begin{align}\n\\label{eq: generalized Haemers' conjecture}\n\\lim_{n\\rightarrow \\infty} \\frac{\\alpha_{\\mathcal{X}}(n)}{I(n)} = 1\n\\end{align}\nhold?\n\\end{question}\n\nSome computer results somewhat support Haemers' Conjecture. Approximately $80\\%$ of the graphs with $12$ vertices are DS \\cite{BrouwerS2009}.\nA new class of graphs is defined in \\cite{WangW2024}, offering an algorithmic method for finding all the $\\{\\A,\\overline{\\A}\\}$-cospectral mates\nof a graph $\\Gr{G}$ in this class (i.e., graphs that are $\\{\\A,\\overline{\\A}\\}$-cospectral with $\\Gr{G}$).\nUsing this algorithm, they found that out of $10,000$ graphs with $50$ vertices, chosen uniformly at random from this class, at least $99.5\\%$\nof them were $\\{\\A, \\overline{\\A}\\}$-DS.\n\n\\subsection{DS properties of structured graphs}\n\\label{subsection: DS properties of structured graphs}\n\nThis paper explores various structures of graph families determined by their spectrum with respect to one or more of their associated matrices,\nas well as the related problem of constructing pairs of nonisomorphic graphs that are cospectral with respect to some or all of these matrices.\nSeveral questions are interspersed throughout the paper (see Questions~\\ref{question: DS SRG}, \\ref{question: NICS graphs - 1}, \\ref{question: NICS graphs - 2},\n\\ref{question: NICS - DasP2013}, and~\\ref{question: Butler2010}), which remain open for further study.\n\nIn addition to serving as a survey on spectral graph determination, this paper suggests a new alternative proof to Theorem~3.3 in \\cite{MaRen2010},\nasserting that all Tur\\'{a}n graphs are determined by their adjacency spectrum (see Theorem~\\ref{theorem: Turan's graph is DS}).\nThis proof is based on a derivation of the adjacency spectrum of the family of Tur\\'{a}n graphs (see Theorem~\\ref{theorem: A-spectrum of Turan graph}).\nSince these graphs are generally bi-regular multipartite graphs (i.e., their vertices have only two possible degrees, which in this case are consecutive\nintegers), it does not necessarily imply that Tur\\'{a}n graphs are determined by the spectrum of some other associated matrices, such as the Laplacian,\nsignless Laplacian, or normalized Laplacian matrices. Determining whether this holds is an open question.\n\nThe distance matrix of a connected graph is the symmetric matrix whose columns and rows are indexed by the graph vertices, and its entries are\nequal to the pairwise distances between the corresponding vertices. The distance spectrum is the multiset of eigenvalues of the distance matrix,\nand its characterization has been a subject of fruitful research (see \\cite{AouchicheH14,HuiqiuJJY21} for comprehensive surveys on the distance\nspectra of graphs, and \\cite{Banerjee23,HogbenR22} on the spectra of graphs with respect to variants of the distance matrix). Nonisomorphic graphs\nmay share an identical adjacency spectrum, as well as an identical distance spectrum, and therefore be graphs that are not determined by either\ntheir adjacency or distance spectrum.\nThis holds, e.g., for the Shrikhande graph and the line graph of the complete bipartite graph $\\CoBG{4}{4}$, which are nonisomorphic strongly\nregular graphs with the identical parameters $(16,6,2,2)$, sharing an identical $\\A$-spectrum given by $\\{6, [2]^6, [-2]^6\\}$ and an identical\ndistance spectrum given by $\\{24, [0]^9, [-4]^6\\}$. There exist, however, graphs that are determined by their distance spectrum but are not determined\nby their adjacency spectrum. In this context, \\cite{JinZ2014} proves that complete multipartite graphs are uniquely determined by their distance\nspectra (this result confirms a conjecture proposed in \\cite{LinHW2013} and extends the special case established there for complete bipartite graphs).\nA Tur\\'{a}n graph is, in particular, determined by its distance spectrum \\cite{JinZ2014}, and by its adjacency spectra (see\nTheorem~\\ref{theorem: Turan's graph is DS} here). On the other hand, while complete multipartite graphs are not generally determined by their\nadjacency spectra (e.g., for complete bipartite graphs, see Theorem~\\ref{thm:when CoBG is DS?}), they are necessarily determined by their distance\nspectra \\cite{JinZ2014}. Another family of graphs that are determined by their distance spectrum but are not determined by their adjacency spectrum,\nnamed $d$-cube graphs, is provided in \\cite{KoolenHI2016,KoolenHI2016b}. These graphs have their vertices represented by binary $n$-length tuples,\nwhere any two vertices are adjacent if and only if their corresponding binary $n$-tuples differ in one coordinate; it is also shown that these graphs\nhave exactly three distinct distance eigenvalues. The question of which multipartite graphs, or graphs in general, are determined by their distance spectra remains open.\n\nAnother newly obtained proof presented in this paper refers to a necessary and sufficient condition in \\cite{MaRen2010} for complete bipartite graphs\nto be $\\A$-DS (see Theorem~\\ref{thm:when CoBG is DS?} and Remark~\\ref{remark: complete bipartite graphs} here). These graphs are also bi-regular, and\nthe two possible vertex degrees can differ by more than one.\nBoth of these newly obtained proofs, discussed in Section~\\ref{section: special families of graphs}, provide insights into the broader question of which\n(structured) multipartite graphs are determined by their adjacency spectrum or, more generally, by the spectra of some of their associated matrices.\n\nEven if Haemers' conjecture is eventually proved in its full generality, it remains surprising when a new family of structured graphs is shown to be DS\n(or $X$-DS, more generally). This is because for certain structured graphs, such as strongly regular graphs and trees, their spectra often fail to uniquely\ndetermine them \\cite{vanDamH03,vanDamH09}. This stark contrast between the fact that almost all random graphs of high order are likely to be DS and the\nexistence of interesting structured graphs that are not DS has significant implications.\n\nIn addition to the questions posed earlier in this paper, we raise the following additional concrete question:\n\\begin{question}\n{\\em By Theorem~\\ref{theorem: Berman's generalized friendship graph}, the family of generalized friendship graphs $F_{p,q}$ is ${\\bf{\\mathcal{L}}}$-DS.\nAre these graphs also DS with respect to their other associated matrices?}\n\\end{question}\n\nWe speculate that the DS property of a graph correlates, to some extent, with the number of symmetries that the graph possesses, and we hypothesize\nthat the size of the automorphism group of a graph can partially indicate whether it is DS.\n\nA justification for this claim is that the automorphism group of a graph reflects its symmetries, which can influence the eigenvalues of its adjacency\nmatrix. Highly symmetric graphs (i.e., those with large automorphism groups) often exhibit eigenvalue multiplicities and patterns that are shared by other\nnonisomorphic graphs, making such graphs less likely to be DS. Conversely, graphs with trivial automorphism groups are typically less symmetric and may\nhave eigenvalues that uniquely determine their structure, increasing the likelihood that they are DS. As noted in \\cite{GodsilR2001}, almost all graphs\nhave trivial automorphism groups. This observation aligns with the conjecture that most graphs are DS, as the absence of symmetry reduces the likelihood\nof two nonisomorphic graphs sharing the same spectrum.\n\nIt is noted, however, that the DS property of graphs is not solely dictated by their automorphism groups. Specifically, a graph with a large\nautomorphism group can still be DS if its eigenvalues uniquely encode its structure (see Section~\\ref{subsection: strongly regular graphs}).\nIn contrast to these DS graphs, other graphs with trivial automorphism groups are not guaranteed to be DS; in such cases, the spectrum might\nnot capture enough structural information to uniquely determine the graph. Typically, graphs with a small number of distinct eigenvalues\nseem to be, in general, the hardest graphs to distinguish by their spectrum. As noted in \\cite{vanDamO11}, it seems that most graphs with\nfew eigenvalues (e.g., most of the strongly regular graphs) are not determined by their spectrum, which served as one of the motivations of\nthe work in \\cite{vanDamO11} in studying graphs whose normalized Laplacian has three eigenvalues.\n\nTo conclude, the size of the automorphism group of a graph can provide some indication of whether it is DS, but it is not a definitive criterion. 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Omidi, ``Graphs whose normalized Laplacian has three eigenvalues,''\n{\\em Linear Algebra and its Applications}, vol.~435, no.~10, pp.~2560--2569, November 2011.\n\\doilink{https://doi.org/10.1016/j.laa.2011.02.005}\n\n\\end{thebibliography}\n\n\\end{document}\n\n\n"}}},{"rowIdx":297,"cells":{"id":{"kind":"string","value":"2412.20841v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.20841v1"},"title":{"kind":"string","value":"Compact harmonic RCD$(K, N)$ spaces are harmonic manifolds"},"full_text":{"kind":"string","value":"\\documentclass[12pt]{article}\n\\usepackage{mathrsfs}\n\\usepackage{amsmath}\n\\usepackage{float}\n\\usepackage{cite}\n\\usepackage{esint}\n\\usepackage[all]{xy}\n\\usepackage{amssymb}\n\\usepackage{amsfonts}\n\\usepackage{amsthm}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\usepackage{bm}\n\\usepackage{indentfirst}\n\\usepackage{geometry}\n\\geometry{a4paper,scale=0.7}\n\\theoremstyle{plain}\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{ques}[thm]{Question}\n\\newtheorem{property}[thm]{Property}\n\\newtheorem{cor}[thm]{Corollary}\n\\theoremstyle{definition}\n\\newtheorem{defn}[thm]{Definition}\n\\theoremstyle{remark}\n\\newtheorem{remark}[thm]{Remark}\n\\newtheorem{exer}[thm]{Exercise}\n\\newtheorem{exmp}[thm]{Example}\n\\makeatletter\n\\makeatother\n\n\n\\numberwithin{equation}{section}\n\n\n\n\n\n\n\\begin{document}\n\\title{Compact harmonic RCD$(K,N)$ spaces are harmonic manifolds}\n\\author{\\it Zhangkai Huang \\thanks{Sun Yat-sen University: huangzhk27@mail.sysu.edu.cn}}\n\\date{\\small\\today}\n\\maketitle\n\n\n\n\\begin{abstract}\nThis paper focus on the properties and characterizations of harmonic RCD$(K,N)$ spaces, which are the counterparts of harmonic Riemannian manifolds in the non-smooth setting. We prove that a compact RCD$(K,N)$ space is isometric to a locally harmonic Riemannian manifold if it satisfies either of the following harmonicity conditions:\n\\begin{enumerate}\n\\item the heat kernel $\\rho(x,y,t)$ depends only on the variable $t$ and the distance between points $x$ and $y$.\n\\item the volume of the intersection of two geodesic balls depends only on their radii and the distance between their centers. \n\\end{enumerate}\n\n\n\\end{abstract}\n\n\\tableofcontents\n\n\\section{Introduction}\n\n\nIn $n$-dimensional Euclidean space $\\mathbb{R}^n$, there exist harmonic functions that depend only on the geodesic distance. For example when $n>2$, the function $f(x_1,\\ldots,x_n)=(x_1^2+\\cdots+x_n^2)^{1-n/2}$ is harmonic on $\\mathbb{R}^n\\setminus \\{0\\}$. Inspired by this property, Ruse sought to identify harmonic functions on Riemannian manifolds with similar characteristics in 1930. However, it turned out that such radial harmonic functions exist only in very special cases. He proposed, and indeed, the historical definition of such special classes of manifolds, which is stated as follows.\n\n\\begin{defn}\nA Riemannian manifold $(M^n,\\mathrm{g})$ is said to be \\textit{harmonic} if the volume density function $\\theta_p:=\\sqrt{|\\det (g_{ij})|}$ in the normal coordinate at each point $p$ is a radial function.\n\\end{defn}\n\nToday, numerous equivalent definitions about harmonic manifolds have been established (see for instance \\cite{B78,DR92,W50}). Below, we list a few of these definitions for clarity.\n\n\\begin{thm}\nA complete $n$-dimensional Riemannian manifold $(M^n,\\mathrm{g})$ is harmonic if and only if either of the following condition holds.\n\\begin{enumerate}\n\\item[$(1)$] For any point $p\\in M^n$ and the distance function $d_p:=\\mathsf{d}_{\\mathrm{g}}(p,\\cdot)$, $\\Delta d_p^2$ is radial on some small ball $B_r(p)$.\n\\item[$(2)$] For any $p\\in M^n$ there exists a nonconstant radial harmonic function in a punctured neighborhood of $p$.\n\\item[$(3)$] Every small geodesic sphere in $M^n$ has constant mean curvature.\n\\item[$(4)$] {Every harmonic function satisfies the mean value property}.\n\\end{enumerate}\n\\end{thm}\n\n\n\n\nWhen the space is simply connected, we have other equivalent characterizations of the space's harmonicity.\n\n\n\\begin{thm}[{\\cite[Thm. 3]{CH11}}, {\\cite[Thm. 1]{CH12}}]\\label{harm1}\nA connected, simply connected and complete Riemannian manifold is harmonic if and only if the volume of the intersection of two geodesic balls depends only on their radii and the distance between their centers.\n\\end{thm}\n\n\n\\begin{thm}[{\\cite[Thm. 1.1]{S90}}]\\label{harm2}\nA connected, simply connected and complete Riemannian manifold $(M^n,\\mathrm{g})$ is harmonic if and only if the heat kernel $\\rho(x,y,t)$ depends only on the variable $t$ and the distance between points $x$ and $y$, i.e., it is of the form $\\rho(x,y,t) = \\rho(\\mathsf{d}_{\\mathrm{g}}(x,y),t)$.\n\\end{thm}\n\nGiven a Riemannian manifold $(M^n,\\mathrm{g})$, let $H_i$ ($i=1,2$) be two functions on $M^n\\times M^n$ such that for any $x\\in M^n$ the functions $H_i^x(\\cdot):=H_i(x,\\cdot)$ ($i=1,2$) are $L^2$-integrable. Under these conditions, the convolution of $H_1$ and $H_2$, denoted as $H_1 \\ast H_2$, is defined by\n\\[\nH_1 \\ast H_2(x, y) =\\int_{M^n} H_1(x,z)H_2(y,z) \\,\\mathrm{d}\\mathrm{vol}_{\\mathrm{g}}(z).\n\\]\nA function $H:M^n\\times M^n\\rightarrow \\mathbb{R}$ is called radial kernel if $H(x, y)$ depends only on the geodesic distance between $x$ and $y$, that is, if $H =h\\circ \\mathsf{d}_{\\mathrm{g}}$, where $h:\\mathbb{R}^+\\rightarrow \\mathbb{R}$ is an arbitrary function.\n\nAs an application of Theorem \\ref{harm2}, through the approximation with characteristic functions, we can immediately deduce the following theorem.\n\n\\begin{thm}[{\\cite[Prop. 2.1]{S90}}]\\label{harm3}\nA connected, simply connected and complete Riemannian manifold is harmonic if and only if the convolution\nof the radial kernel functions $H_1=h_1\\circ \\mathsf{d}_{\\mathrm{g}}$ and $H_2=h_2\\circ \\mathsf{d}_{\\mathrm{g}}$ is a radial kernel function whenever $h_1,h_2$ are $L^2$-integrable functions on $\\mathbb{R}$ with\ncompact support.\n\\end{thm}\n\n\nThe primary objective of this paper, is to characterize non-smooth contexts, namely $\\mathrm{RCD}(K,N)$ metric measure spaces which are defined precisely in the following subsection and satisfy the conditions outlined in Theorem \\ref{harm1}-\\ref{harm3}.\n\\subsection{RCD spaces}\n\nIn this paper, by metric measure space we mean a triple $(X, \\mathsf{d},\\mathfrak{m})$ such that $(X, \\mathsf{d})$ is a complete\nseparable metric space and that $\\mathfrak{m}$ is a nonnegative Borel measure with full support on\n$X$ and being finite on any bounded subset of $X$.\n\nThe curvature-dimension condition CD$(K,N)$, which involves a lower Ricci curvature bound $K\\in\\mathbb{R}$ and an upper dimension bound $N\\in [1,\\infty)$ for metric measure spaces in a synthetic sense, was first introduced in \\cite{St06a,St06b} by Sturm and in \\cite{LV09} by Lott-Villani respectively. Subsequently, by adding a Riemannian structure into CD$(K,N)$ metric measure spaces, Ambrosio-Gigli-Savar\\'{e} \\cite{AGS14a}, Gigli \\cite{G13,G15}, Erbar-Kuwada-Sturm \\cite{EKS15} and\nAmbrosio-Mondino-Savar\\'{e} \\cite{AMS19} introduced the notion of RCD$(K, N)$ spaces. To be precise, a metric measure space is classified as an RCD$(K, N)$ space if it satisfies the CD$(K,N)$ condition and its $H^{1,2}$-Sobolev space is Hilbertian.\n\nAlthough RCD$(K,N)$ spaces are generally not smooth, they still possess some desirable analytical properties. For instance, Jiang-Li-Zhang \\cite{JLZ16} established the Gaussian estimates for the heat kernels on RCD$(K,N)$ spaces, which implies the existence of locally Lipschitz representation of the heat kernels. Furthermore, Bru\\`e-Semola \\cite{BS20} demonstrated that each RCD$(K,N)$ space $({X},\\mathsf{d},\\mathfrak{m})$ bears a unique integer $n\\in [1,N]$, which is called its essential dimension and is denoted by $n:=\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}({X})$, such that the the tangent space of $\\mathfrak{m}$-almost every point is $\\mathbb{R}^n$.\n\n\nIt is worth mentioning that De Philippis-Gigli \\cite{DG18} introduced a synthetic counterpart of volume non-collapsed Gromov-Hausdorff limit spaces of Riemannian manifolds with a constant dimension and a lower Ricci curvature bound, namely non-collapsed RCD$(K,N)$ spaces, meaning that the reference measure equals $N$-dimensional Hausdorff measure $\\mathscr{H}^N$. In this case it can be readily checked that in this case $N$ must be an integer and $N:=\\mathrm{dim}_{\\mathsf{d},\\mathscr{H}^N}({X})$. \n\nIt is natural to explore additional conditions that could improve the regularity of general RCD$(K,N)$ spaces. Recently, Brena-Gigli-Honda-Zhu \\cite{BGHZ23} demonstrated the equivalence between non-collapsed RCD$(K,N)$ spaces and weakly non-collapsed RCD$(K,N)$ spaces. Furthermore, the author \\cite{H23} proved that every RCD$(K,N)$ space is non-collapsed, given an isometrically heat kernel immersing condition. Additionally, in \\cite{H23}, the author demonstrated that a compact non-collapsed RCD$(K,n)$ space with eigenfunctions that realize an isometric immersion into Euclidean space is, in fact, smooth. \n\n\n\n\\subsection{Main results}\n\nIn this paper, we focus on the following specific subclasses of RCD$(K,N)$ spaces.\n\n\\begin{defn}[Strongly harmonic RCD$(K,N)$ space]\nAn RCD$(K,N)$ space $(X,\\mathsf{d},\\mathfrak{m})$ is said to be \\textit{strongly harmonic} if there exists a real valued function $H:[0,\\infty)\\times [0,\\infty)\\rightarrow \\mathbb{R}$ such that the heat kernel $\\rho$ of $(X,\\mathsf{d},\\mathfrak{m})$ satisfies\n\\begin{equation}\\label{eeqn1.1}\n\\rho(x,y,t)=H(\\mathsf{d}\\left(x,y\\right),t),\\ \\forall x,y\\in X,\\ \\forall t>0.\n\\end{equation}\n\\end{defn}\n\n\\begin{defn}[Radial symmetric RCD$(K,N)$ space]\nAn RCD$(K,N)$ space $(X,\\mathsf{d},\\mathfrak{m})$ is said to be \\textit{radial symmetric} if there exists a real valued function $F:[0,\\infty)\\times [0,\\infty)\\rightarrow \\mathbb{R}$ and non-constant eigenfunctions $\\{\\phi_i\\}_{i=1}^m$ such that\n\\begin{equation}\\label{eqn1.2}\n\\sum\\limits_{i=1}^m \\phi_i(x)\\phi_i(y)=F(\\mathsf{d}\\left(x,y\\right)),\\ \\forall x,y\\in X.\n\\end{equation}\n\\end{defn}\nWe prove the following three regularity results for the above subclasses of RCD$(K,N)$ spaces.\n\\begin{thm}\\label{mainthm2}\nLet $(X,\\mathsf{d},\\mathfrak{m})$ be a strongly harmonic $\\mathrm{RCD}(K,N)$ space. Then $\\mathfrak{m}=c\\mathscr{H}^n$ for some constant $c>0$, where $n=\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}(X)$ is the essential dimension of the space. In particular, $(X,\\mathsf{d},\\mathscr{H}^n)$ is a non-collapsed $\\mathrm{RCD}(K,n)$ space.\n\\end{thm}\n\n\n\\begin{thm}\\label{mainthm1}\nLet $(X,\\mathsf{d},\\mathscr{H}^n)$ be a non-collapsed radial symmetric $\\mathrm{RCD}(K,n)$ space. Then $(X,\\mathsf{d})$ is isometric to an $n$-dimensional smooth closed Riemannian manifold $(M^n,\\mathrm{g})$.\n\\end{thm}\n\\begin{cor}\\label{cor1.5}\nAssume $(X,\\mathsf{d},\\mathfrak{m})$ is a compact strongly harmonic $\\mathrm{RCD}(K,N)$ space with $\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}(X)=n$. Then $(X,\\mathsf{d})$ is isometric to an $n$-dimensional smooth closed Riemannian manifold $(M^n,\\mathrm{g})$.\n\\end{cor}\n\nAs an application of Corollary \\ref{cor1.5}, we are positioned to establish the following theorem, which bears a direct relationship with Theorem \\ref{harm1}.\n\n\\begin{thm}\\label{thm1.11}\nLet $(X,\\mathsf{d},\\mathfrak{m})$ be a compact $\\mathrm{RCD}(K,N)$ space. If in addition the measure of the intersection of two geodesic balls depends only on their radii and the distance between their centers, then the following holds. \\begin{enumerate}\n\\item[$(1)$] $\\mathfrak{m}=c\\mathscr{H}^n$ for some constant $c>0$ and $(X,\\mathsf{d},\\mathscr{H}^n)$ is a non-collapsed $\\mathrm{RCD}(K,n)$ space;\n\\item[$(2)$] $(X,\\mathsf{d})$ is isometric to an $n$-dimensional smooth closed Riemannian manifold $(M^n,\\mathrm{g})$.\n\\item[$(3)$] Moreover, if in the addtion that $X$ is simply-connected, then it is a harmonic manifold. \n\\end{enumerate}\n\n\\end{thm}\n \\subsection{Outline of the proofs}\nIn section \\ref{sec3}, we initially focus on the smoothness of compact, non-collapsed, radial symmetric RCD$(K,n)$ spaces $(X,\\mathsf{d},\\mathscr{H}^n)$. Our main goal is to establish that the map $(\\phi_1,\\ldots,\\phi_m):X\\rightarrow \\mathbb{R}^m$, constructed from eigenfunctions meeting the conditions in (\\ref{eqn1.2}), is an isometric immersion. The smoothness is then inferred from \\cite[Thm. 1.10]{H23}. Regarding the proof for the regularity of strongly harmonic RCD$(K,N)$ spaces $(X,\\mathsf{d},\\mathfrak{m})$, we employ a blow-up argument to demonstrate their non-collapsing nature and to enhance the upper bound $N$ to its essential dimension $n:=\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}({X})$. Subsequently, we establish the smoothness of compact, non-collapsed, strongly harmonic RCD$(K,n)$ spaces, leveraging the observation that they are radial symmetric.\n\nIn section \\ref{sec4}, we explore regularities of RCD$(K,N)$ spaces $(X,\\mathsf{d},\\mathfrak{m})$, where the volume of the intersection of two geodesic balls is only determined by their radii and the distance between their centers. Utilizing a blow-up argument demonstrate that these spaces are non-collapsed and that the upper bound $N$ can be refined to the essential dimension $n:=\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}({X})$. Furthermore, under the additional assumption of compactness for $X$, we examine the smoothness of the space. We observe that any geodesic can be extended to a length equal to its diameter. This observation allows us to construct bi-Lipschitz coordinate charts using distance functions, which shares some similarity of the bi-Lipschitz coordinates near regular points in Alexandrov spaces (see for instance \\cite{BGP92}). By mapping back to Euclidean space, we derive a key equation for any two distinct points $x,\\bar{x} \\in X$.\n\\[\n\\lim_{r\\rightarrow 0} \\,\\frac{1}{r}\\left(\\frac{\\mathscr{H}^n\\left(B_r(x)\\setminus B_{\\mathsf{d}(x,\\bar{x})}(\\bar{x})\\right)}{\\omega_n r^n}-\\frac{1}{2}\\right)=\\Delta \\mathsf{d}_{\\bar{x}}(x).\n\\]\n Notably, as shown in \\cite{HT03}, in a smooth setting, the second term in this equation corresponds to the mean curvature of the geodesic sphere of radius $\\mathsf{d}(x,\\bar{x})$ centered at $\\bar{x}$. Finally, under polar coordinates, the volume density function is found to be radial, which implies that the space is strongly harmonic and, consequently, smooth.\n\\ \n\\ \n\\ \n\n\\textbf{Acknowledgements} The author acknowledges the support of Fundamental Research Funds for the Central Universities, Sun Yat-sen University, Grant Number 24qnpy105. He is thankful to Prof. Shouhei Honda and Prof. Huichun Zhang for their insightful discussions. \n\\section{Preliminaries}\n\nIn this paper we consistently employ $C=C(k_1,\\ldots,k_m)$ to denote a positive constant that depends sorely on the parameters $k_1,\\ldots,k_m$. For convenience, we may occasionally abbreviate this to $C$, acknowledging that it may change from one line to another if necessary.\n\n\\subsection{Metric space}\nIn this subsection, we will review some fundamental notation pertaining to metric spaces. Let $(X, \\mathsf{d})$ be a metric space that is both complete and separable.\n\n\\begin{itemize}\n\\item We denote by $C(X)$ and $\\mathrm{Lip}(X,\\mathsf{d})$ the set of continuous function and Lipschitz functions on $(X,\\mathsf{d})$, respectively. The subsets $\\mathrm{Lip}_b(X,\\mathsf{d})$ and $\\mathrm{Lip}_c(X,\\mathsf{d})$ represent bounded Lipschitz functions and compactly supported Lipschitz functions, respectively.\n\\item For any $f\\in \\mathrm{Lip}(X,\\mathsf{d})$, the Lipschitz constant of $f$ is defined by\n\\[\n\\mathrm{Lip}\\mathop{f}:=\\sup_{x\\neq y} \\frac{|f(y)-f(x)|}{\\mathsf{d}\\left(y,x\\right)};\n\\]\nthe pointwise Lipschitz constant of $f$ at a point $x\\in X$ is defined by\n\\[\n\\mathrm{lip}\\mathop{f}(x):=\\limsup_{y\\rightarrow x} \\frac{|f(y)-f(x)|}{\\mathsf{d}\\left(y,x\\right)}\n\\]\nif $x$ is not isolated, and is understood to be 0 if $x$ is isolated.\n\\item We denote by $B_r^X(x)$ (or briefly $B_r(X)$) the open ball of radius $r>0$ centered at $x$, i.e. $\\{y\\in X:\\mathsf{d}\\left(y,x\\right)<r\\}$ and by $A_{r_1,r_2}^X(x)$ (or briefly $A_{r_1,r_2}(x)$) the annulus $\\{y\\in X:r_1<\\mathsf{d}\\left(x,y\\right)<r_2\\}$.\n\\item For any $n\\in [1,\\infty)$, we define $\\mathscr{H}^n_\\mathsf{d}$ (or simply $\\mathscr{H}^n$ when the context is clear and there is no risk of confusion) as the $n$-dimensional Hausdorff measure on $(X,\\mathsf{d})$. Specifically, we denote by $\\mathscr{L}^n$ the standard Lebesgue measure on $\\mathbb{R}^n$, which is consistent with the the $n$-dimensional Hausdorff measure. Additionally, we let $\\omega_n$ represent Lebesgue measure of the unit ball $B_1(0_n)$ in $\\mathbb{R}^n$.\n\\item If $(X,\\mathsf{d})$ is compact, then we define its diameter as:\n\\[\n\\mathrm{diam}(X,\\mathsf{d}):=\\sup_{x,y\\in X}\\mathsf{d}\\left(x,y\\right).\n\\]\n\\end{itemize}\n\\subsection{RCD space and heat kernel}\n\nThroughout this paper, when refering to a triple $(X,\\mathsf{d},\\mathfrak{m})$ as a metric measure space, we mean that $(X,\\mathsf{d})$ is a complete and separable metric space and $\\mathfrak{m}$ is a non-negative Borel measure which is finite on bounded sets. Now, let us consider $(X,\\mathsf{d},\\mathfrak{m})$ as a metric measure space. \n\nThe Cheeger energy $\\mathrm{Ch}:L^2(\\mathfrak{m})\\rightarrow [0,\\infty]$ is a convex and lower semi-continuous functional defined as\n\\[\n\\mathrm{Ch}(f):=\\inf_{\\{f_i\\}}\\left\\{\\int_X \\left(\\mathrm{lip}\\mathop{f_i}\\right)^2\\mathrm{d}\\mathfrak{m}\\right\\},\n\\]\nwhere the infimum is taken among all sequences $\\{f_i\\}\\subset \\mathrm{Lip}_b(X,\\mathsf{d})\\cap L^2(\\mathfrak{m})$ such that $\\|f_i-f\\|_{L^2(\\mathfrak{m})}\\rightarrow 0$. The Sobolev space $H^{1,2}(X,\\mathsf{d},\\mathfrak{m})$ is then defined as the set of $L^2$-integrable functions with finite Cheeger energy. Subsequently, unless it leads to ambiguity, we will use the simplified notations $L^p$ for $L^p(X,\\mathfrak{m})$ and $H^{1,2}$ for $H^{1,2}(X,\\mathsf{d},\\mathfrak{m})$.\n\nFor any function $f\\in H^{1,2}$, we can employ a minimizing sequence and apply Mazur's Lemma to uniquely identify a minimal relaxed slope $|\\nabla f|\\in L^2$ such that\n\\[\n\\mathrm{Ch}(f)=\\int_X |\\nabla f|^2\\, \\mathrm{d}\\mathfrak{m}.\n\\]\nThis minimal relaxed slope possesses the locality property, meaning that $|\\nabla f|=|\\nabla h|$ $\\mathfrak{m}$-a.e. on $\\{f=h\\}$. \n\nLet us further make a convention that all metric measure spaces under consideration in this paper is \\textit{infinitesimally Hilbertian}. This term signifies that the associated Sobolev spaces of these spaces are Hilbert spaces. It is noteworthy that, as demonstrated in \\cite{AGS14a,G15}, under this condition, for any two functions $f,h\\in H^{1,2}$, the following function is well-defined:\n\\[\n\\langle\\nabla f,\\nabla h\\rangle:=\\lim_{\\epsilon\\rightarrow 0}\\frac{|\\nabla f+\\epsilon h|^2-|\\nabla h|^2}{2\\epsilon}\\in L^1.\n\\]\nFurthermore, the domain of the Laplacian, denoted as $D(\\Delta)$ can be defined as the set of functions $f\\in H^{1,2}$ for which the following equation\n\\[\n\\int_X \\langle \\nabla f,\\nabla \\varphi\\rangle\\, \\mathrm{d}\\mathfrak{m}=-\\int_X h\\varphi\\, \\mathrm{d}\\mathfrak{m},\\ \\forall \\varphi\\in H^{1,2},\n\\]\nholds for some $h\\in L^2$. This $h$ is unique and is denoted by $\\Delta f$. As a localized version, for any open set $U\\subset X$ we denote by $D(\\Delta,U)$ the set of all $f\\in H^{1,2}(U,\\mathsf{d},\\mathfrak{m})$ such that the following holds for some $h\\in L^2(U)$, which is denoted by $\\Delta_U f$,\n\\[\n\\int_X \\langle \\nabla f,\\nabla \\varphi\\rangle \\,\\mathrm{d}\\mathfrak{m}=-\\int_X h\\varphi,\n \\ \\forall \\varphi\\in \\mathrm{Lip}_c(U,\\mathsf{d}).\n\\]\nHere by $f\\in H^{1,2}(U,\\mathsf{d},\\mathfrak{m})$ we mean that $\\varphi f\\in H^{1,2}$ for any $\\varphi\\in \\mathrm{Lip}_c(U,\\mathsf{d})$ and $f,|\\nabla f|\\in L^2(U)$.\n\n\nWe are now in a position to introduce the definition of RCD$(K,N)$ spaces. See \\cite{AGS15,AMS19,EKS15,G15} for details.\n\\begin{defn}\nFor any $K\\in \\mathbb{R}$, $N\\in [1,\\infty]$, an infinitesimally Hilbertian metric measure space $(X,\\mathsf{d},\\mathfrak{m})$ is said to be an RCD$(K,N)$ space if it satisfies the following conditions.\n\\begin{enumerate}\n\\item There exists $x\\in X$ and $C>1$ such that for any $r>0$ we have $\\mathfrak{m}(B_r(x))\\leqslant C\\exp(Cr^2)$.\n\\item Any $f\\in H^{1,2}$ satisfying $|\\nabla f|\\leqslant 1$ $\\mathfrak{m}$-a.e. has a 1-Lipschitz representative.\n\\item For any $f\\in D(\\Delta)$ with $\\Delta f\\in H^{1,2}$ and any $\\varphi \\in \\mathrm{Test}F\\left({X},\\mathsf{d},\\mathfrak{m}\\right)$ with\n $ \\varphi \\geqslant 0$, we have\n\\[\n\\frac{1}{2}\\int_X |\\nabla f|^2 \\Delta \\varphi\\mathop{\\mathrm{d}\\mathfrak{m}}\\geqslant \\int_X \\varphi \\left(\\frac{(\\Delta f)^2}{N}+\\langle \\nabla f,\\nabla \\Delta f\\rangle+K|\\nabla f|^2 \\right)\\mathrm{d}\\mathfrak{m},\n\\]\nwhere $\\mathrm{Test}F({X},\\mathsf{d},\\mathfrak{m})$ is the class of test functions defined by\n \\[\n\\mathrm{Test}F({X},\\mathsf{d},\\mathfrak{m}):=\\left\\{\\varphi\\in \\text{Lip}({X},\\mathsf{d})\\cap D(\\Delta)\\cap L^\\infty:\\Delta f\\in H^{1,2}\\cap L^\\infty\\right\\}.\n\\]\n\\end{enumerate}\n\n\nIf in addition $\\mathfrak{m}=\\mathscr{H}^N$, then $(X,\\mathsf{d},\\mathfrak{m})$ is said to be a non-collapsed RCD$(K,N)$ space.\n\\end{defn}\n\nThroughout this paper, when referring to an RCD$(K,N)$ space, we always mean that $N \\in (1,\\infty)$. For the remainder of this subsection, let us consider $(X, \\mathsf{d}, \\mathfrak{m})$ as a representative example of an RCD$(K,N)$ space.\n\nIt should be emphasized that RCD$(K,N)$ spaces satisfy the following Bishop-Gromov volume growth inequality (see \\cite[Thm. 5.31]{LV09}, \\cite[Thm. 2.3]{St06b}).\n\\begin{thm}\\label{BGineq}\nFor any $R>r>0$ $(R\\leqslant \\pi\\sqrt{(N-1)/K}$ if $K>0)$ it holds that\n\\[\n\\dfrac{\\mathfrak{m}\\left(B_R(x)\\right)}{\\mathfrak{m}\\left(B_r(x)\\right)}\\leqslant \\frac{V_{K,N}(R)}{ V_{K,N}(r)},\n\\]\nwhere $V_{K,N} (r)$ denotes the volume of a ball of radius\n$r$ in the $N$-dimensional model space with Ricci curvature $K$.\n\n\\end{thm}\n\n\n\n\n\n\n\nBuilding on the contributions by Sturm (see \\cite[Prop. 2.3]{St95}, \\cite[Cor. 3.3]{St96}) and Jiang-Li-Zhang (as detailed in Theorem \\ref{JLZ} below), it is established that RCD$(K,N)$ spaces possess locally Lipschitz continuous heat kernels. More precisely, there exists a non-negative function $\\rho$ on $X \\times X \\times (0,\\infty)$ such that the unique solution to the heat equation can be expressed as follows.\n\\[\n\\mathrm{h}_t f=\\int_X \\rho(\\cdot,y,t)f(y)\\,\\mathrm{d}\\mathfrak{m}(y),\\ \\forall f\\in L^2,\\ \\forall t>0,\n\\]\nwhere by heat equation we mean $\\mathrm{h}_t f$ satisfies that\n\\[\n\\frac{d }{d t} \\mathrm{h}_t f=\\Delta \\mathrm{h}_t f,\\ \\text{in }L^2;\\ \\ \\lim_{t\\downarrow 0}\\| \\mathrm{h}_t f-f\\|_{L^2}=0.\n\\]\n\n\n\\begin{thm}[Gaussian estimates for the heat kernel {\\cite[Thm. 1.1, Thm. 1.2]{JLZ16}}]\\label{JLZ}\nLet $\\rho$ be the heat kernel of $(X,\\mathsf{d},\\mathfrak{m})$. Then given any $\\epsilon>0$, there exist $C_i=C_i(K,N,\\epsilon)$ $(i=1,2,3,4)$ such that\n\\begin{equation}\\label{JLZineq}\nC_1^{-1}\\exp\\left(-\\frac{\\mathsf{d}^2\\left(x,y\\right)}{(4-\\epsilon )t}-C_2 t\\right)\\leqslant \\mathfrak{m}(B_{\\sqrt{t}}(x))\\rho(x,y,t)\\leqslant C_1\\exp\\left(-\\frac{\\mathsf{d}^2\\left(x,y\\right)}{(4+\\epsilon )t}+C_2 t\\right)\n\\end{equation}\nholds for any $x,y\\in X$ and\n\\[\n|\\nabla_x \\rho(x,y,t)|\\leqslant \\frac{C_3}{\\sqrt{t}\\mathop{\\mathfrak{m}(B_{\\sqrt{t}}(x))}}\\exp\\left(-\\frac{\\mathsf{d}\\left(x,y\\right)}{(4+\\epsilon)t}+C_4 t\\right)\n\\]\nholds for $\\mathfrak{m}$-a.e. $x,y\\in X$.\n\nIn particular, if $K=0$, then there exists $C_5=C_5(N,\\epsilon)$ such that $(\\ref{JLZineq})$ can be improved to\n\\[\nC_5^{-1}\\exp\\left(-\\frac{\\mathsf{d}^2\\left(x,y\\right)}{(4+\\epsilon )t}\\right)\\leqslant \\mathfrak{m}(B_{\\sqrt{t}}(x))\\rho(x,y,t)\\leqslant C_5\\exp\\left(-\\frac{\\mathsf{d}^2\\left(x,y\\right)}{(4-\\epsilon )t}\\right),\n\\]\nfor any $x,y\\in X$.\n\\end{thm}\n\n\\begin{remark}[Rescaled RCD spaces]\n For any\n$a, b\\in (0,\\infty)$, the rescaled metric measure space $(X, a\\mathsf{d}, b\\mathfrak{m})$ is an RCD$(a^{-2}K, N)$ space, and its heat\nkernel $\\tilde{\\rho}$ can be expressed as\n\\[\n\\begin{aligned}\n\\tilde{\\rho}:X\\times X\\times (0,\\infty)&\\longrightarrow (0,\\infty)\\\\\n (x,y,t)&\\longmapsto b^{-1}\\rho(x,y,a^{-2}t).\n\\end{aligned}\n\\]\n\\end{remark}\n\\subsection{Convergence of RCD spaces}\n\nWe omit the precise definition of pointed measure Gromov-Hausdorff (pmGH) convergence. The details about this definition and the following theorem can be found for instance in \\cite[Sec. 3]{GMS15}.\n\\begin{thm}\\label{GMS}\nAssume that $\\{(X_i,\\mathsf{d}_i,\\mathfrak{m}_i,x_i)\\}$ is a sequence of pointed $\\mathrm{RCD}(K,N)$ spaces such that\n\\[\n0<\\liminf_{i\\rightarrow \\infty}\\mathfrak{m}_i(B_1^{X_i}(x_i))\\leqslant \\limsup_{i\\rightarrow \\infty}\\mathfrak{m}_i(B_1^{X_i}(x_i))<\\infty.\n\\]\nThen this sequence has a subsequence $\\{(X_{i(j)},\\mathsf{d}_{i(j)},\\mathfrak{m}_{i(j)},x_{i(j)})\\}$ which $\\mathrm{pmGH}$ converges to a pointed $\\mathrm{RCD}(K,N)$ space $(X,\\mathsf{d},\\mathfrak{m},x)$.\n\\end{thm}\nConsequently, we can define the concept of \\textit{regular sets} as follows.\n\\begin{defn}[Regular set]\\label{regularset}\nLet $(X,\\mathsf{d},\\mathfrak{m})$ be an RCD$(K,N)$ space. The tangent space at $x\\in X$, denoted by $\\mathrm{Tan}(X,\\mathsf{d},\\mathfrak{m},x)$, is defined as\n\\[\n\\left\\{(Y,\\mathsf{d}_Y,\\mathfrak{m}_Y,y):\\exists r_i\\downarrow 0 \\text{, s.t. }\\left(X,\\frac{1}{r_i}\\mathsf{d},\\frac{\\mathfrak{m}}{\\mathfrak{m}(B_{r_i}(x))},x\\right)\\xrightarrow{\\mathrm{pmGH}}(Y,\\mathsf{d}_Y,\\mathfrak{m}_Y,y)\\right\\}.\n\\]\nThe $k$-dimensional regular set is then defined as\n\\[\n\\mathcal{R}_k:=\\left\\{x\\in X:\\mathrm{Tan}(X,\\mathsf{d},\\mathfrak{m},x)=\\left\\{\\left(\\mathbb{R}^k,\\mathsf{d}_{\\mathbb{R}^k},\\frac{1}{\\omega_k}\\mathscr{L}^k,0_k\\right)\\right\\}\\right\\}.\n\\]\n\\end{defn}\nFor the subsequent result concerning the existence of the \\textit{essential dimension} in RCD spaces, we refer to \\cite[Thm. 0.1]{BS20}.\n\\begin{thm}\\label{BS}\nLet $(X,\\mathsf{d},\\mathfrak{m})$ be an $\\mathrm{RCD}(K,N)$ space. There exists a unique integer $n\\in [1,N]$ such that $\\mathfrak{m}\\left(X\\setminus \\mathcal{R}_n\\right)=0$. This integer $n$ is referred to as the essential dimension of $(X,\\mathsf{d},\\mathfrak{m})$ and is denoted by $\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}(X)$. In particular, $\\mathfrak{m}$ can be represented as $\\theta \\mathscr{H}^n\\llcorner \\mathcal{R}_n$ for some Borel function $\\theta$ defined on $X$.\n\\end{thm}\n\\begin{remark}\\label{AHT}\nUnder the assumption of Theorem \\ref{BS}, we define the set\n\\[\n\\mathcal{R}_n^\\ast:=\\left\\{x\\in \\mathcal{R}_n: \\exists\\lim_{r\\downarrow 0}\\frac{\\mathfrak{m}(B_r(x))}{r^n}\\in (0,\\infty)\\right\\},\n\\]\nand assert that $\\mathfrak{m}\\left(\\mathcal{R}_n\\setminus \\mathcal{R}_n^\\ast\\right)=0$. Furthermore, for $\\mathfrak{m}\\text{-a.e. }x\\in \\mathcal{R}_n^\\ast$, the density function $\\theta$ is given by\n\\[\n\\theta(x)=\\lim_{r\\downarrow 0}\\frac{\\mathfrak{m}(B_r(x))}{\\omega_n r^n},\\ \\mathfrak{m}\\text{-a.e. }x\\in \\mathcal{R}_n^\\ast.\n\\] For further details, refer to \\cite[Thm. 4.1]{AHT18}.\n\\end{remark}\n\nSpecifically, in the case of non-collapsed RCD$(K,n)$ spaces, the following assertion regarding the density function and the essential dimension is valid.\n\\begin{thm}[{\\cite[Cor. 1.7]{DG18}}]\\label{DG18cor1.7} Let $(X,\\mathsf{d},\\mathscr{H}^n)$ be a non-collapsed\n$\\mathrm{RCD}(K, n)$ space. Then its essential dimension $\\mathrm{dim}_{\\mathsf{d},\\mathscr{H}^n}(X)$ equals $n$ (notably $n$ is an integer), and it holds that $\\theta(x)\\leqslant 1$ for any $x\\in X$. Moreover, equality $\\theta(x)=1$ occurs if and only if $x\\in \\mathcal{R}_n$.\n\\end{thm}\n\nIn the subsequent part of this subsection, we consider a sequence of pointed RCD$(K,N)$ spaces, denoted by $\\{(X_i, \\mathsf{d}_i, \\mathfrak{m}_i, x_i)\\}$, which converges in the pointed measured Gromov-Hausdorff (pmGH) sense to another pointed RCD$(K,N)$ space, namely $(X, \\mathsf{d}, \\mathfrak{m}, x)$.\n\nWe assume that readers are familiar with the definitions of $L^2$-weak and $L^2$-strong convergence and $L^2_{\\mathrm{loc}}$ (and their counterparts for Sobolev functions, namely $H^{1,2}$-weak and $H^{1,2}$-strong convergence, and $H^{1,2}_{\\mathrm{loc}}$-weak and $H^{1,2}_{\\mathrm{loc}}$-strong convergence) on various spaces. For reference, see \\cite{AST17,AH17,GMS15} and \\cite[Def. 1.1]{H15}. We conclude this subsection by presenting some useful results related to this topic.\n\\begin{thm}[Pointwise convergence of heat kernels {\\cite[Thm. 3.3]{AHT18}}]\\label{pchk}\nThe heat kernels $\\rho_i$ of $(X_i,\\mathsf{d}_i,\\mathfrak{m}_i)$ satisfy\n\\[\n\\lim_{i\\rightarrow \\infty} \\rho_i(x_i,y_i,t_i)=\\rho(x,y,t)\n\\]\nwhenever $X_i\\times X_i\\times (0,\\infty)\\ni (x_i,y_i,t_i)\\rightarrow (x,y,t)\\in X\\times X\\times (0,\\infty)$.\n\\end{thm}\n\n\\begin{thm}[Arzel\\`{a}-Ascoli theorem]\\label{AAthm}\nAssume $f_i\\in C(X_i)$ $(i\\in\\mathbb{N})$ such that the sequence $\\{f_i\\}$ meets the following two conditions.\n\\begin{enumerate}\n\\item[$1.$](Locally uniformly bounded) For any $R>0$ it holds that\n\\[\n\\sup_i \\sup_{y_i\\in B_R(x_i)}|f_i(y_i)|<\\infty.\n\\]\n\\item[$2.$](Locally equicontinuous)\nFor any $\\epsilon,R\\in (0,\\infty)$, there exists $\\delta\\in (0,1)$ such that for any $i\\in \\mathbb{N}$ it holds that\n\\[\n|f_i(y_i)-f_i(z_i)|<\\epsilon, \\ \\forall y_i,z_i\\in B_R(x_i) \\text{ such that } \\mathsf{d}_i(y_i,z_i)<\\delta.\n\\]\n\\end{enumerate}\nThen after passing to a subsequence, there exists $f\\in C(X)$ such that $\\{f_i\\}$ pointwisely converges to $f$ in the following sense:\n\\[\nf_i(y_i)\\rightarrow f(y) \\text{\\ whenever\\ } X_i\\ni y_i\\rightarrow y\\in X.\n\\]\n\\end{thm}\n\n\nUtilizing the Arzelà-Ascoli theorem discussed above, we can demonstrate the subsequent theorem concerning precompactness in the context of $L^2$-weak convergence.\n\\begin{thm}\nAssume $f_i\\in L^2(\\mathfrak{m}_i)$ $(i\\in\\mathbb{N})$ such that $\\sup_i \\|f_i\\|_{L^2(\\mathfrak{m}_i)}<\\infty$, then after passing to a subsequence $f_i$ $L^2$-weakly converges to some $f\\in L^2(\\mathfrak{m})$.\n\\end{thm}\n\nThe forthcoming theorem is instrumental in our subsequent arguments and is derived from \\cite{AH18}.\n \\begin{thm}[Stability of Laplacian on balls]\\label{AH18}\n Let $f_i\\in D(\\Delta, B_R(x_i))$ $(i\\in\\mathbb{N})$ such that $\\sup_i \\|\\Delta f_i\\|_{L^2\\left(B_R(x_i)\\right)}<\\infty$ and $\\{f_i\\}$ $L^2$-strongly converges to some $f\\in L^2(B_R(x),\\mathfrak{m})$ on $B_R(x)$. Then for any $r\\in (0,R)$ the following holds.\n \\begin{enumerate}\n \\item[1.] $f|_{B_r(x)}\\in D(\\Delta,B_r(x))$.\n \\item[2.] $\\{\\Delta_i f_i\\}$ $L^2$-weakly converges to $\\Delta f$ on $B_r(x)$.\n \\item[3.] $\\{f_i\\}$ $H^{1,2}$-strongly converges to $f$ on $B_r(x)$.\n \\end{enumerate}\n \\end{thm}\n\n\n\\subsection{Calculus on RCD$(K,N)$ spaces}\n\nLet $(X,\\mathsf{d},\\mathfrak{m})$ be an RCD$(K,N)$ space.\n\nFor the sake of brevity, we forgo detailing the definitions of the spaces of $L^p$-one forms, $L^p$-tensor fields of type (0,2) over a Borel set $A\\subset X$, denoted by $L^p(T^\\ast(A,\\mathsf{d},\\mathfrak{m}))$, $L^p((T^\\ast)^{\\otimes 2}(A,\\mathsf{d},\\mathfrak{m}))$, respectively. We also omit the definition of the pointwise Hilbert-Schimidt norm, denoted by $|\\cdot|_{\\mathsf{HS}}$, for $L^p$-tensor fields. Instead, we only report some crucial results on the calculus within RCD$(K,N)$ spaces. Details can be found in for instance \\cite{G18a,G18b}.\n\n\n\n\n\n\n\\begin{thm}[Exterior derivative]\nThe linear operator $d$, called the exterior derivative, defined by\n\\[\n\\begin{aligned}\nd:H^{1,2}&\\longrightarrow L^2(T^\\ast(X,\\mathsf{d},\\mathfrak{m}))\\\\\nf&\\longmapsto d\\,f\n\\end{aligned}\n\\]\nsatisfies that $|d\\, f|=|\\nabla f|$ $\\mathfrak{m}$-a.e. for any $f\\in H^{1,2}$. Moreover, the set $\\{d\\, f:f\\in H^{1,2}\\}$ is dense in $L^2(T^\\ast(X,\\mathsf{d},\\mathfrak{m}))$.\n\\end{thm}\n\n\\begin{thm}[The Hessian]\nFor any $f\\in \\mathrm{Test}F\\left({X},\\mathsf{d},\\mathfrak{m}\\right)$, there exists a unique $T\\in L^2\\left((T^\\ast)^{\\otimes 2}({X},\\mathsf{d},\\mathfrak{m})\\right)$, called the Hessian of $f$, denoted by $ \\mathop{\\mathrm{Hess}}f$, such that for any $f_i\\in \\mathrm{Test}F\\left({X},\\mathsf{d},\\mathfrak{m}\\right)$ $(i=1,2)$,\n\\begin{equation}\n{}{2T(\\nabla f_1,\\nabla f_2)= \\langle \\nabla f_1,\\nabla\\langle \\nabla f_2,\\nabla f\\rangle\\rangle +\\langle \\nabla f_2,\\nabla\\langle \\nabla f_1,\\nabla f\\rangle\\rangle-\\langle \\nabla f,\\nabla\\langle \\nabla f_1,\\nabla f_2\\rangle\\rangle }\n\\end{equation}\nholds for $\\mathfrak{m}$-a.e. $x\\in {X}$. Moreover, the following holds for any $f\\in \\mathrm{Test}F\\left({X},\\mathsf{d},\\mathfrak{m}\\right)$, $\\varphi\\in \\mathrm{Test}F({X},\\mathsf{d},\\mathfrak{m})$ with $\\varphi\\geqslant 0$. \n\n\n \n\\begin{equation}\\label{abc2.14}\n\\frac{1}{2}\\int_{X} \\Delta \\varphi \\cdot |\\nabla f|^2\\,\\mathrm{d}\\mathfrak{m}\\geqslant \\int_{X}\\varphi \\left(|\\mathop{\\mathrm{Hess}}f|_{\\mathsf{HS}}^2+ \\langle \\nabla \\Delta f,\\nabla f\\rangle+K|\\nabla f|^2\\right) \\mathrm{d}\\mathfrak{m}.\n\\end{equation}\n\n\n\\end{thm}\n\n\\begin{remark}\\label{rmk2.16}\nSince $\\mathrm{Test}F({X},\\mathsf{d},\\mathfrak{m})$ is dense in $D(\\Delta)$, for any $f\\in D(\\Delta)$, the Hessian $\\mathop{\\mathrm{Hess}}f$ is well-defined and belongs to $ L^2\\left((T^\\ast)^{\\otimes 2}({X},\\mathsf{d},\\mathfrak{m})\\right)$. Furthermore, if $f_i\\in D(\\Delta)\\cap \\mathrm{Lip}(X,\\mathsf{d})$ $(i=1,2)$, then we have $\\langle \\nabla f_1,\\nabla f_2 \\rangle\\in H^{1,2}$ and $\\mathrm{Hess}\\, f_1=\\mathrm{Hess}\\, f_2$ $\\mathfrak{m}$-a.e. on $\\{f_1=f_2\\}$. Additionally, the following holds for any $\n\\varphi\\in H^{1,2}$.\n\\begin{equation}\\label{11eqn2.16}\n\\langle \\nabla \\varphi, \\nabla \\langle \\nabla f_1,\\nabla f_2 \\rangle \\rangle= \\mathop{\\mathrm{Hess}}f_1\\left(\\nabla f_2,\\nabla\\varphi\\right)+ \\mathop{\\mathrm{Hess}}f_2\\left(\\nabla f_1,\\nabla\\varphi\\right) \\ \\ \\mathfrak{m}\\text{-a.e.}\n\\end{equation}\n\n We note that when $\\mathrm{diam}(X,\\mathsf{d})>2$, for any $f\\in D(\\Delta)$, the local $L^2$-norm of the Hessian of $f$ can be estimated as follows:\n\\begin{equation}\\label{eeeeeeqn2.5}\n\\begin{aligned}\n\\int_{B_1(x)}\\left|\\mathrm{Hess}\\, f\\right|_{\\mathsf{HS}}^2\\mathrm{d}\\mathfrak{m}\\leqslant&\\, C(K,N)\\left(\\int_{B_2(x)}\\left(\\Delta f\\right)^2\\mathrm{d}\\mathfrak{m}+\\inf_{m\\in\\mathbb{R}}\\int_{B_2(x)}\\left||\\nabla f|^2-m\\right|\n\\mathrm{d}\\mathfrak{m}\\right)\\\\\n&-2(K\\wedge0)\\int_{B_2(x)}|\\nabla f|^2\\,\\mathrm{d}\\mathfrak{m}.\n\\end{aligned}\n\\end{equation}\nIt is important to highlight that, due to the locality of the Hessian, even if a function $f$ belongs to $D(\\Delta,B_2(x))$, its Hessian on $B_1(x)$, which is still denoted by $\\mathrm{Hess}\\, f$, can be interpreted as $\\mathrm{Hess}\\,(\\varphi f)$, where $\\varphi$ is a non-negative cut-off function in $\\mathrm{Test}F(X,\\mathsf{d},\\mathfrak{m})$ such that $\\varphi=1$ on $B_1(x)$, that the support of $\\varphi$ is contained within $ B_2(x)$, and that $|\\Delta \\varphi|+|\\nabla \\varphi|\\leqslant C(K,N)$ $\\mathfrak{m}$-a.e. on $B_2(x)$. See \\cite[Thm. 6.7]{AMS14}, \\cite{G18b} and \\cite[Lem. 3.1]{MN14} for details. \n\n\nLet us consider a function $f\\in D(\\Delta,B_{2r}(x))$ for some $0<r< \\mathrm{diam}(X,\\mathsf{d})/2$, and examine $f/r$ on the rescaled RCD$(r^2K,N)$ space $\\left(X,\\mathsf{d}/r,\\mathfrak{m}/\\mathfrak{m}(B_r(x))\\right)$. Then it immediately follows from (\\ref{eeeeeeqn2.5}) that\n\\begin{equation}\\label{eeeeeeqn2.6}\n\\begin{aligned}\nr^2\\int_{B_r(x)}\\left|\\mathrm{Hess}\\, f\\right|_{\\mathsf{HS}}^2\\mathrm{d}\\mathfrak{m}\\leqslant& C(K,N)\\left(r^2\\int_{B_{2r}(x)}\\left(\\Delta f\\right)^2\\mathrm{d}\\mathfrak{m}+\\inf_{m\\in\\mathbb{R}}\\int_{B_{2r}(x)}\\left||\\nabla f|^2-m\\right|\n\\mathrm{d}\\mathfrak{m}\\right)\\\\\n&-2r^2(K\\wedge0)\\int_{B_{2r}(x)}|\\nabla f|^2\\,\\mathrm{d}\\mathfrak{m}.\n\\end{aligned}\n\\end{equation}\n\\end{remark}\n\n\nEquation (\\ref{eeeeeeqn2.5}) was first introduced in \\cite[Sec. 1.3]{BPS23}. We express our gratitude to Elia Bru\\'{e} for providing a comprehensive proof of this equation, which is reproduced below for the reader's convenience.\n\\begin{proof}[Proof of (\\ref{eeeeeeqn2.5})]\nWithout loss of generality, we may assume that $K$ is non-positive. Consider the auxiliary problem\n\\[\n\\left\\{\n\\begin{aligned}\n\\Delta \\tilde{f}=0\\ \\ \\text{on }B_2(x),\\ \\ \\ \\ \\\\\n \\tilde{f}-f\\in H_0^{1,2}(B_2(x)).\n\\end{aligned}\n\\right.\n\\]\nThe existence of such a solution is guaranteed by \\cite[Cor. 1.2]{BM95} and \\cite[Thm 10.12]{BB11}. Since RCD$(K,N)$ spaces satisfy (2,2)-Poincar\\'{e} inequality, they consequently fulfill the Sobolev inequality (see \\cite[Thm. 1]{R12} and \\cite[Thm. 5.1]{HK00}) and \\cite[Thm. 5.51]{BB11}). This allows us to directly estimate that \n\\[\n\\begin{aligned}\n\\int_{B_2(x)}|\\nabla (f-\\tilde{f})|^2\\,\\mathrm{d}\\mathfrak{m}&=\\int_{B_2(x)}( f- \\tilde{f})\\,\\Delta\n f\\,\\mathrm{d}\\mathfrak{m}\\\\\n&\\leqslant \\left(\\int_{B_2(x)}( f- \\tilde{f})^2\\,\\mathrm{d}\\mathfrak{m}\\right)^{\\frac{1}{2}}\\left(\\int_{B_2(x)}(\\Delta f)^2\\,\\mathrm{d}\\mathfrak{m}\\right)^{\\frac{1}{2}}\\\\\n&\\leqslant C(K,N)\\left(\\int_{B_2(x)}|\\nabla( f- \\tilde{f})|^2\\,\\mathrm{d}\\mathfrak{m}\\right)^{\\frac{1}{2}}\\left(\\int_{B_2(x)}(\\Delta f)^2\\,\\mathrm{d}\\mathfrak{m}\\right)^{\\frac{1}{2}},\n\\end{aligned}\n\\]\nwhich implies\n\\begin{equation}\\label{eeeeqn2.6}\n\\int_{B_2(x)}|\\nabla (f-\\tilde{f})|^2\\,\\mathrm{d}\\mathfrak{m}\\leqslant C(K,N)\\int_{B_2(x)}(\\Delta f)^2\\,\\mathrm{d}\\mathfrak{m}.\n\\end{equation}\nUsing (\\ref{abc2.14}) (\\ref{eeeeqn2.6}), taking the cut-off function $\\varphi$, and applying integral by parts we see\n\\begin{equation}\\label{eeeeeeeqn2.7}\n\\begin{aligned}\n&\\int_{B_1(x)}\\left|\\mathrm{Hess}\\, (f-\\tilde{f})\\right|_{\\mathsf{HS}}^2\\,\\mathrm{d}\\mathfrak{m}\\leqslant \\int_{B_2(x)}\\varphi\\left|\\mathrm{Hess}\\, (f-\\tilde{f})\\right|_{\\mathsf{HS}}^2\\,\\mathrm{d}\\mathfrak{m}\\\\\n\\leqslant &\\, C(K,N)\\int_{B_2(x)}\\left|\\nabla (f-\\tilde{f})\\right|^2\\,\\mathrm{d}\\mathfrak{m}-\\int_{B_2(x)}\\varphi\\, \\langle \\nabla \\Delta (f-\\tilde{f}),\\nabla (f-\\tilde{f}) \\rangle\\,\\mathrm{d}\\mathfrak{m}\\\\\n=&\\,C(K,N)\\int_{B_2(x)}\\left|\\nabla (f-\\tilde{f})\\right|^2\\,\\mathrm{d}\\mathfrak{m}+\\int_{B_2(x)}\\left(\\varphi\\, (\\Delta f)^2+\\langle\\nabla \\varphi,\\nabla (f-\\tilde{f})\\rangle\\,\\Delta f\\right)\\,\\mathrm{d}\\mathfrak{m}\\\\\n\\leqslant &\\,C(K,N)\\int_{B_2(x)}(\\Delta f)^2\\,\\mathrm{d}\\mathfrak{m}.\n\\end{aligned}\n\\end{equation}\nSimilarly, given that $\\Delta \\tilde{f}=0$ and applying (\\ref{eeeeqn2.6}), we deduce that for any $m\\in\\mathbb{R}$ the following equation holds.\n\n\n\\begin{equation}\\label{eeeeeeeeeeeqn2.7}\n\\begin{aligned}\n&\\int_{B_1(x)}\\left|\\mathrm{Hess}\\,\\tilde{f}\\right|_{\\mathsf{HS}}^2\\,\\mathrm{d}\\mathfrak{m}\\\\\n\\leqslant&\\ \\frac{1}{2} \\int_{B_2(x)}\\Delta \\varphi\\left( \\left|\\nabla \\tilde{f}\\right|^2-m\\right)\\,\\mathrm{d}\\mathfrak{m}-K\\int_{B_2(x)}\\varphi\\left|\\nabla \\tilde{f}\\right|^2\\,\\mathrm{d}\\mathfrak{m}\\\\\n\\leqslant& \\ C(K,N)\\int_{B_2(x)}\\left| \\left|\\nabla \\tilde{f}\\right|^2-m\\right|\\,\\mathrm{d}\\mathfrak{m}-K\\int_{B_2(x)}\\left|\\nabla \\tilde{f}\\right|^2\\,\\mathrm{d}\\mathfrak{m}\\\\\n=& \\ C(K,N)\\int_{B_2(x)}\\left| \\left|\\nabla \\tilde{f}\\right|^2-m\\right|\\,\\mathrm{d}\\mathfrak{m}-K\\int_{B_2(x)}\\left(|\\nabla f|^2-\\left|\\nabla (f-\\tilde{f})\\right|^2\\right)\\,\\mathrm{d}\\mathfrak{m}\\\\\n\\leqslant & \\ C(K,N)\\left(\\int_{B_2(x)}\\left| \\left|\\nabla \\tilde{f}\\right|^2-m\\right|\\,\\mathrm{d}\\mathfrak{m}+\\int_{B_2(x)}{(\\Delta f)}^2\\,\\mathrm{d}\\mathfrak{m}\\right)-K\\int_{B_2(x)}{|\\nabla f|}^2\\, \\mathrm{d}\\mathfrak{m}.\n\\end{aligned}\n\\end{equation}\nSince \n\\begin{equation}\\label{eeeeeqn2.9}\n\t\\int_{B_1(x)}\\left|\\mathrm{Hess}\\, f\\right|_{\\mathsf{HS}}^2\\mathrm{d}\\mathfrak{m}\\leqslant 2\\int_{B_1(x)}\\left(\\left|\\mathrm{Hess}\\, (f-\\tilde{f})\\right|_{\\mathsf{HS}}^2+\\left|\\mathrm{Hess}\\,\\tilde{f}\\right|_{\\mathsf{HS}}^2\\right)\\mathrm{d}\\mathfrak{m},\n\\end{equation}\nto conclude it suffices to show that \n\\begin{equation}\\label{eeeeqn2.10}\n\\inf_{m\\in\\mathbb{R}}\\int_{B_2(x)}\\left| \\left|\\nabla \\tilde{f}\\right|^2-m\\right|\\mathrm{d}\\mathfrak{m}\\leqslant \\inf_{m\\in\\mathbb{R}}\\int_{B_2(x)}\\left| |\\nabla f|^2-m\\right|\\mathrm{d}\\mathfrak{m}+C(K,N)\\int_{B_2(x)}(\\Delta f)^2\\,\\mathrm{d}\\mathfrak{m}.\n\\end{equation}\nLet $\\hat{m}$ be the real number such that \n\\[\n\\int_{B_2(x)}\\left| \\left|\\nabla f\\right|^2-\\hat{m}\\right|\\mathrm{d}\\mathfrak{m}=\\inf_{m\\in\\mathbb{R}}\\int_{B_2(x)}\\left| \\left|\\nabla f\\right|^2-m\\right|\\mathrm{d}\\mathfrak{m}.\n\\]\nThen the harmonicity of $\\tilde{f}$ gives that \n\\[\n\\begin{aligned}\n&\\inf_{m\\in\\mathbb{R}}\\int_{B_2(x)}\\left| \\left|\\nabla \\tilde{f}\\right|^2-m\\right|\\,\\mathrm{d}\\mathfrak{m}\\leqslant \\int_{B_2(x)}\\left| \\left|\\nabla \\tilde{f}\\right|^2-\\hat{m}\\right|\\,\\mathrm{d}\\mathfrak{m}\\\\\n\\leqslant &\\int_{B_2(x)}\\left| \\left|\\nabla f\\right|^2-\\hat{m}\\right|\\,\\mathrm{d}\\mathfrak{m}+\\int_{B_2(x)}\\left| \\left|\\nabla f\\right|^2-\\left|\\nabla\\tilde{f}\\right|^2\\right|\n\\,\\mathrm{d}\\mathfrak{m}\\\\\n=&\\int_{B_2(x)}\\left| \\left|\\nabla f\\right|^2-\\hat{m}\\right|\\,\\mathrm{d}\\mathfrak{m}+\\int_{B_2(x)}\\left|\\nabla (f-\\tilde{f})\\right|^2\\,\\mathrm{d}\\mathfrak{m},\n\\end{aligned}\n\\]\nwhich together with (\\ref{eeeeqn2.6}) yields (\\ref{eeeeqn2.10}). \n\nFinally, the conclusion is derived from (\\ref{eeeeeeeeeeeqn2.7})-(\\ref{eeeeqn2.10}). \n\\end{proof} \n\\begin{thm}[Canonical Riemannian metric \\cite{AHPT21,GP22}]\nThere exists a unique Riemannian metric $\\mathrm{g}\\in L^\\infty((T^\\ast)^{\\otimes 2}(X,\\mathsf{d},\\mathfrak{m}))$ such that $|\\mathrm{g}|_{\\mathsf{HS}}=\\sqrt{n}$ $\\mathfrak{m}$-a.e. and that for any $f_1, f_2\\in H^{1,2}\n$ it holds\nthat\n\\[\n\\mathrm{g}(\\nabla f_1,\\nabla f_2)=\\langle \\nabla f_1,\\nabla f_2\\rangle,\\ \\mathfrak{m}\\text{-a.e.}\n\\]\n\\end{thm}\n\nWe are now prepared to present a recent result in \\cite[Thm. 1.10]{H23}, which concludes this subsection. The theorem establishes that compact non-collapsed RCD$(K,n)$ spaces equipped with isometrically immersing eigenmaps are, in fact, smooth.\n\n\\begin{thm}\\label{H22}\nLet $(X,\\mathsf{d},\\mathscr{H}^n)$ be a compact non-collapsed $\\mathrm{RCD}(K,n)$ space and $g$ be the canonical Riemannian metric on it. If there exist a finite number of non-constant eigenfunctions $\\{\\phi_i\\}_{i=1}^m$ such that\n\\[\n\\mathrm{g}=\\sum\\limits_{i=1}^m d\\phi_i\\otimes d\\phi_i ,\n\\]\nthen $(X,\\mathsf{d})$ is isometric to an $n$-dimensional smooth closed Riemannian manifold $\\left(M^n,\\mathrm{g}\\right)$.\n\\end{thm}\n\nThis theorem played an important role in proving the smoothness of compact isometrically heat kernel immersing RCD$(K,N)$ spaces as demonstrated in \\cite{H23}. By ``isometrically heat kernel immersing\" we mean that there exists a function $t\\mapsto c(t)$ on $(0,\\infty)$ such that for each $t > 0$, the function $x \\mapsto (y \\mapsto \\sqrt{c(t)}\\, \\rho(x,y,t))$ provides an isometric immersion into $L^2$. \n\n\n\n\\section{Symmetric and harmonic RCD$(K,N)$ spaces}\\label{sec3}\nThe first part of this section is aimed at proving Theorem \\ref{mainthm1}. To achieve this, we require several key estimates concerning eigenvalues and eigenfunctions. For further details and background, refer to \\cite[Appendix]{AHPT21} and \\cite{ZZ19}.\n\\begin{prop}\\label{prop3.1}\nLet $(X, \\mathsf{d}, \\mathfrak{m})$ be a compact $\\mathrm{RCD}(K, N)$ space with $\\mathfrak{m}(X)=1$. Let $0=\\lambda_0<\\lambda_1\\leqslant \\lambda_2\\leqslant \\cdots \\rightarrow \\infty$ be all its eigenvalues counted with multiplicities and let $\\{\\varphi_i\\}_{i=0}^\\infty$ be the corresponding eigenfunctions, which form an $L^2$-orthonormal basis. Then, there exists a constant $C= C(K,N,\\mathrm{diam}(X, \\mathsf{d}))$, such that for all $i \\geqslant 1$ we have\n\\[\n\\|\\varphi_i\\|_{L^\\infty}\\leqslant C\\,\\lambda_i^{\\frac{N}{4}},\\ \\| |\\nabla \\varphi_i |\\|_{L^\\infty}\\leqslant C\\,\\lambda_i^{\\frac{N+2}{4}},\\ C^{-1} \\,i^{\\frac{2}{N}}\\leqslant \\lambda_i\\leqslant C\\,i^2.\n\\]\n\\end{prop}\n\nIn the rest of this section we use the notation of (\\ref{eqn1.2}). We let $\\mu_i$ be the corresponding eigenvalue of $\\phi_i$ ($i=1,\\ldots,m$) and use $C$ to denote the constant\n\\[\n C=C\\left(K,m,n,\\mathrm{diam}({X},\\mathsf{d}),\\mathscr{H}^n({X}),\\mu_1,\\ldots,\\mu_m,\\left\\|\\phi_1\\right\\|_{L^2},\\ldots,\\left\\|\\phi_m\\right\\|_{L^2}\\right).\n\\]\n\n\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{mainthm1}]\n\n\n\nLet us first show that\n\\begin{equation}\\label{eqn3.1}\n \\left|\\frac{F(0)-F(\\mathsf{d}\\left(x,y\\right))}{\\mathsf{d}^2\\left(x,y\\right)}\\right|\\leqslant C,\\ \\forall x,y\\in X.\n\\end{equation}\nLetting $x=y$ in (\\ref{eqn1.2}) we know\n\\begin{equation}\\label{eqn3.2}\n\\sum\\limits_{i=1}^m {\\phi_i}^2(x)=F(0),\\ \\forall x\\in X.\n\\end{equation}\nTherefore, it clearly follows from (\\ref{eqn1.2}) and (\\ref{eqn3.2}) that\n\\begin{equation}\\label{eqn3.3}\n\\sum\\limits_{i=1}^m (\\phi_i(x)-\\phi_i(y))^2=2\\,\\big(F(0)-F(\\mathsf{d}\\left(x,y\\right))\\big),\\ \\forall x,y\\in X.\n\\end{equation}\nAs a result, we have\n \\[\n\\frac{F(0)-F\\left(\\mathsf{d}\\left(x,y\\right)\\right)}{\\mathsf{d}^2\\left(x,y\\right)}=\\frac{1}{2}\\sum\\limits_{i=1}^m \\left(\\frac{\\phi_i(x)-\\phi_i(y)}{\\mathsf{d}\\left(x,y\\right)}\\right)^2,\\ \\forall x,y\\in X,\n \\]\n which together with the Proposition \\ref{prop3.1} yields (\\ref{eqn3.1}).\n\n From now on let us take an arbitrary but fixed\n\\[\nx_0\\in \\mathcal{R}_n(X)\\cap \\bigcap\\limits_{i,j=1}^m \\mathrm{Leb}(\\langle\\nabla \\phi_i,\\nabla\\phi_j\\rangle),\n\\]\nwhere $\\mathrm{Leb}(\\langle\\nabla \\phi_i,\\nabla\\phi_j\\rangle)$ is the Lebesgue point of the function $\\langle\\nabla \\phi_i,\\nabla\\phi_j\\rangle$. \n\n\n We first claim that\n \\begin{equation}\\label{eqn3.4}\n\\frac{1}{2} \\liminf_{r\\rightarrow 0} \\frac{F(0)-F(r)}{r^2}>0.\n \\end{equation}\n\n\n We prove by contradiction. Assume there exists a sequence $\\{r_l\\}\\subset (0,\\infty)$ such that $r_l\\rightarrow 0$, and that ${r_l}^{-2}(F(0)-F(r_l))\\to 0$ as $l\\to \\infty$. After passing to a subsequence, we have\n \\[\n (X_l,\\mathsf{d}_l,\\mathscr{H}^n,x_0):=\\left(X,\\frac{1}{r_l}\\mathsf{d},\\frac{1}{r_l^n}\\mathscr{H}^n_\\mathsf{d},x_0\\right)\\xrightarrow{\\mathrm{pmGH}} \\left(\\mathbb{R}^n,\\mathsf{d}_{\\mathbb{R}^n},\\mathscr{L}^n,0_n\\right).\n \\]\nFor convenience, we denote the gradient and the Laplacian on $(X_l,\\mathsf{d}_l,\\mathscr{H}^n)$ by $\\Delta_l,\\nabla_l$ respectively. Additionally, we use $B_r^l(x_0)$ to represent the ball $B_r^{X_l}(x_0)$. For each $i\\in \\{1,\\cdots,m\\}$ and $l\\in \\mathbb{N}$, let us set $\\varphi_{i,l}:=(\\phi_i-\\phi_i(x_0))/r_l$. Then we have\n\\[\n|\\nabla_l \\,\\varphi_{i,l}|=|\\nabla \\phi_i|,\\ \\Delta_l\\,\\varphi_{i,l} =r_l\\,\\Delta\\, \\phi_i=-r_l\\, \\mu_i \\,\\phi_i.\n\\]\nParticularly, the forthcoming estimates are straightforwardly extracted from Proposition \\ref{prop3.1}:\n\\begin{equation}\\label{eqn3.5}\n\\left\\|\\left|\\nabla_l\\, \\varphi_{i,l}\\right|\\right\\|_{L^\\infty(X_l)}\\leqslant C;\\ \\ \\left\\|\\Delta_l \\,\\varphi_{i,l}\\right\\|_{L^\\infty(X_l)}\\leqslant C\\,r_l\\rightarrow 0\\ \\ \\text{as }l\\rightarrow \\infty.\n\\end{equation}\n\n\n\n\n\nAccording to the Arzela-Ascoli Theorem (Theorem \\ref{AAthm}), for any $R>0$, and for each $i=1,\\ldots,m$, the sequence $\\{\\varphi_{i,l}\\}_l$ uniformly converges to a Lipschitz continuous function $\\varphi_i$ on $B_R(0_n)\\subset \\mathbb{R}^n$. Furthermore, (\\ref{eqn3.5}) and Theorem \\ref{AH18} indicate that each $\\varphi_i$ is a harmonic function satisfying $|\\nabla^{\\mathbb{R}^n} \\varphi_i|\\leqslant C$, and thus is a linear function on $\\mathbb{R}^n$. Therefore, for each $i,j$, since $x_0\\in \\mathrm{Leb}(\\langle\\nabla \\phi_i,\\nabla\\phi_j\\rangle)$, we have\n\\[\n\\begin{aligned}\nnt_{B_r(x_0)}\\langle\\nabla \\phi_i,\\nabla \\phi_j\\rangle\\mathop{\\mathrm{d}\\mathscr{H}^n}\\\\\n\\ &=\\frac{1}{\\omega_n}\\lim_{l\\rightarrow \\infty}\\int_{B^l_1(x_0)}\\langle\\nabla_l\\, \\varphi_{i,l},\\nabla_l\\, \\varphi_{j,l}\\rangle \\mathop{\\mathrm{d}\\mathscr{H}^n}=\\langle\\nabla^{\\mathbb{R}^n} \\varphi_i,\\nabla^{\\mathbb{R}^n} \\varphi_j\\rangle.\n\\end{aligned}\n\\]\nIn particular, for any $y\\in \\partial B_1(0_n)\\subset \\mathbb{R}^n$, by taking $y_l\\in \\partial B^l_1(x_0)=\\partial B_{r_l}(x_0)$ such that $y_l\\rightarrow y$, we observe from (\\ref{eqn3.3}) and the fact $\\mathsf{d}_l\\left(y_l,x_0\\right)\\rightarrow \\mathsf{d}_{\\mathbb{R}^n}(y,0_n)$ that\n\\begin{equation}\\label{eqn3.7}\n\\begin{aligned}\n\\lim_{l\\rightarrow \\infty} \\frac{F(0)-F\\left(\\mathsf{d}\\left(x_0,y_l\\right)\\right)}{\\mathsf{d}^2\\left(x_0,y_l\\right)}&=2\\lim_{l\\rightarrow \\infty}\\left(\\frac{r_l}{\\mathsf{d}\\left(x_0,y_l\\right)}\\right)^2\\sum\\limits_{i=1}^m \\left(\\frac{\\phi_i(x_0)-\\phi_i(y_l)}{r_l}\\right)^2\\\\\n\\ &=2\\sum\\limits_{i=1}^m {\\varphi_i}^2(y)=0.\n\\end{aligned}\n\\end{equation}\nAs a result, $|\\nabla^{\\mathbb{R}^n} \\varphi_1|\\equiv|\\nabla \\phi_1|(x_0)=0$. This is impossible since one can choose $x_0$ as the Lebesgue point of the set $\\{y\\in X: |\\nabla \\phi_1|(y)\\neq 0\\}$. Therefore we complete the proof of (\\ref{eqn3.4}). Additionally, from the above discussion we know $\\sum_i |\\nabla \\phi_i|^2>0$ for $\\mathscr{H}^n$-a.e. $x\\in X$.\n\n\nNext let us consider an arbitrary sequence $\\{r_l\\}$ with $r_l\\to 0$ such that the following limit exists.\n\\[\n\\frac{1}{2} \\,\\lim_{l\\to\\infty } \\frac{F(0)-F(r_l)}{{r_l}^2}=c>0.\n\\] \nThen according to the linearity of $\\varphi_i$ and (\\ref{eqn3.7}), on $\\mathbb{R}^n$ it holds that $\\sum_i {\\varphi_i}^2(y)=c\\,{|y|}^2$. It is easy to see that $\\sum_i |\\nabla \\phi_i|^2(x_0)=\\sum_i |\\nabla^{\\mathbb{R}^n} \\varphi_i|^2\\equiv nc$. Hence we deduce that\n\\begin{equation}\\label{eqn33.8}\n\\lim_{r \\downarrow 0} \\frac{F(0)-F(r)}{r^2}=2\\sum\\limits_{i=1}^m |\\nabla \\phi_i|^2(x_0):=2\\,nc>0,\n\\end{equation}\nbecause the sequence $\\{r_l\\}$ is taken to be arbitrary.\n\n\nFinally, we assert that\n\\begin{equation}\\label{eqn3.8}\n\\left|\\sum\\limits_{i=1}^m d\\,\\phi_i\\otimes d\\,\\phi_i-c\\,\\mathrm{g}\\right|_\\mathsf{HS}=0,\\ \\mathscr{H}^n\\text{-a.e.},\n\\end{equation}\nwhere $c$ is the constant obtained in (\\ref{eqn33.8}), and $\\mathrm{g}$ is the canonical Riemannian metric on $(X,\\mathsf{d},\\mathscr{H}^n)$.\n\nNow let $x_0$ be an arbitrary but fixed point in $\\mathrm{Leb}\\left(\\left|c\\,\\mathrm{g}-\\sum\\limits_{i=1}^m d\\,\\phi_i\\otimes d\\,\\phi_i\\right|_{\\mathsf{HS}}^2\\right)\\cap \\mathcal{R}_n$ and use the notations as above. Then since on $\\mathbb{R}^n$ it holds that $\\sum_i {\\varphi_i}^2(y)=c\\,{|y|}^2$, by applying the linearity of each $\\varphi_i$ and taking partial derivatives we know\n\\[\n\\sum_{i=1}^m\\frac{\\partial \\varphi_i }{\\partial y^\\alpha}\\,\\frac{\\partial \\varphi_i }{ \\partial y^\\beta}=c\\,\\delta_{\\alpha \\beta}, \n\\]\nwhich implies that $c\\,\\mathrm{g}_{\\mathbb{R}^n}=\\sum_i d\\,\\varphi_i\\otimes d\\,\\varphi_i$.\n\n\nFor each $i$, the $H^{1,2}$-strong convergence of the sequence $\\{\\varphi_{i,l}\\}_l$ on any $B_R(0_n)\\subset\\mathbb{R}^n$ (Theorem \\ref{AH18}) as well as (\\ref{eqn3.5}) implies that\n\\[\n\\begin{aligned}\n\\left|c\\,\\mathrm{g}-\\sum\\limits_{i=1}^m d\\,\\phi_i\\otimes d\\,\\phi_i\\right|_\\mathsf{HS}^2(x_0)\nnt_{B_r(x_0)}\\left|c\\,\\mathrm{g}-\\sum\\limits_{i=1}^m d\\,\\phi_i\\otimes d\\,\\phi_i\\right|_\\mathsf{HS}^2\\mathrm{d}\\mathscr{H}^n\\\\\n=\\ &\\frac{1}{\\omega_n}\\lim_{l\\rightarrow\\infty}\\int_{B_1^l(x_0)}\\left|c\\,\\mathrm{g}_{X_l}-\\sum\\limits_{i=1}^m d\\,\\varphi_{i,l}\\otimes d\\,\\varphi_{i,l}\\right|_\\mathsf{HS}^2\\mathrm{d}\\mathscr{H}^n\\\\\nnt_{B_1(0_n)}\\left|c\\,\\mathrm{g}_{\\mathbb{R}^n}-\\sum\\limits_{i=1}^m d\\,\\varphi_i\\otimes d\\,\\varphi_i\\right|_\\mathsf{HS}^2\\mathrm{d}\\mathscr{L}^n=0.\n\\end{aligned}\n\\]\nSince $x_0\\in\\mathrm{Leb}\\left(\\left|c\\, \\mathrm{g}-\\sum\\limits_{i=1}^m d\\,\\phi_i\\otimes d\\,\\phi_i\\right|_{\\mathsf{HS}}^2\\right)\\cap \\mathcal{R}_n$ is arbitrary, we have completed the proof of (\\ref{eqn3.8}). It now remains to apply Theorems \\ref{BS} and \\ref{H22} to reach our conclusion.\n\\end{proof}\n\n\\begin{remark}\nIn fact, the condition of radial symmetry in Theorem \\ref{mainthm1} can be \nreduced to the existence of a real valued function $F : [0,\\infty) \\times [0,\\infty)\\rightarrow\\mathbb{R}$ and non-constant eigenfunctions $\\{\\phi_i\\}_{i=1}^m$ such that for any $x\\in X$ there exists $\\epsilon_x>0$ such that\n\\[\n\\sum_{i=1}^m \\phi_i(x)\\phi_i(y) = F\\left(\\mathsf{d}\\left(x, y\\right)\\right), \\forall y\\in B_{\\epsilon_x}(x).\n\\]\n\\end{remark}\n\n\nNext, we focus on strongly harmonic RCD$(K,N)$ spaces. Specifically, we consider the RCD$(K,N)$ space $(X,\\mathsf{d},\\mathfrak{m})$ with $\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}(X)=n$ for the remainder of this section. Let us recall the definition of a strongly harmonic RCD$(K,N)$ space: there exists a real valued function $H:[0,\\infty)\\times [0,\\infty)\\rightarrow \\mathbb{R}$ such that the heat kernel $\\rho$ of $(X,\\mathsf{d},\\mathfrak{m})$ satisfies\n\\begin{equation}\\label{eeqn4.1}\n\\rho(x,y,t)=H\\left(\\mathsf{d}\\left(x,y\\right),t\\right),\\ \\forall x,y\\in X,\\ \\forall t>0.\n\\end{equation}\nWe begin with the proof of Theorem \\ref{mainthm2}, which parallels the proof of \\cite[Thm. 1.7]{H23}.\n\\begin{proof}[Proof of Theorem \\ref{mainthm2}]\n\nWe state that there exists a constant $\\tilde{c}>0$ such that\n\\begin{equation}\\label{eqn4.1}\n\\lim_{t\\downarrow 0} t^n H(rt,t^2)=\\tilde{c}\\mathop{(4\\pi )^{-\\frac{n}{2}}} \\exp\\left(-\\frac{r^2}{4}\\right), \\ \\forall r>0.\n\\end{equation}\n\n\nTake an arbitrary but fixed $x_0\\in \\mathcal{R}_n^\\ast(X)$. For any $\\{r_i\\}\\subset \\mathbb{R}$ with $r_i\\rightarrow 0$, after passing to a subsequence, consider the following pmGH convergence.\n\\[\n(X_i,\\mathsf{d}_i,\\mathfrak{m},x_0):=\\left(X,\\frac{1}{r_i}\\mathsf{d}_i,\\frac{\\mathfrak{m}}{\\mathfrak{m}(B_{r_i}(x_0))},x_0\\right)\\xrightarrow{\\mathrm{pmGH}} \\left(\\mathbb{R}^n,\\mathsf{d}_{\\mathbb{R}^n},\\frac{1}{\\omega_n}\\mathscr{L}^n,0_n\\right).\n\\]\nOn each $(X_i,\\mathsf{d}_i,\\mathfrak{m}_i)$, the heat kernel $\\rho_i$ satisfies that\n\\begin{equation}\\label{eeeqn4.2}\n\\rho_i(x_i,y_i,1)=\\mathfrak{m}(B_{r_i}(x_0))\\,\\rho(x_i,y_i,r_i^2), \\ \\forall x_i,y_i\\in X_i.\n\\end{equation}\nFor each $i$, we can take $x_i,y_i\\in B_{2r_i}(x_0)$ such that $\\mathsf{d}_i\\left(x_i,y_i\\right)=r_i$. Then after passing to a subsequence, we may assume that $X_i\\ni x_i\\rightarrow x\\in \\mathbb{R}^n$, $X_i\\ni y_i\\rightarrow y\\in \\mathbb{R}^n$.\n\nDue to Theorem \\ref{pchk}, we have\n\\begin{equation}\\label{eeeqn4.3}\n\\lim_{i\\rightarrow \\infty}\\rho_i(x_i,y_i,1)=(4\\pi )^{-\\frac{n}{2}}\\exp\\left(-\\frac{|x-y|^2}{4}\\right).\n\\end{equation}\nCombining (\\ref{eeqn4.1}), (\\ref{eeeqn4.2}) and (\\ref{eeeqn4.3}) then gives\n\\begin{equation}\\label{eqnaaa5.13}\n\\begin{aligned}\n\\ &\\vartheta_n(X,\\mathsf{d},\\mathfrak{m})(x_0)\\lim_{i\\rightarrow \\infty} r_i^{n} H(r_i|x-y|,r_i^2)\\\\\n=\\ &\\lim_{i\\rightarrow \\infty}\\mathfrak{m}(B_{r_i}(x_0))H\\left(r_i|x-y|,r_i^2\\right)=(4\\pi )^{-\\frac{n}{2}}\\exp\\left(-\\frac{|x-y|^2}{4}\\right).\n\\end{aligned}\n\\end{equation}\nSince the aforementioned equality does not depend on the choice of the sequence $r_i\\downarrow 0$, and the limit $\\lim\\limits_{t\\downarrow 0} t^n H(rt,t^2)$ does not depend on the choice of $x_0\\in \\mathcal{R}_n^\\ast(X)$, we thereby complete the proof of (\\ref{eqn4.1}). Indeed, we have also proved that\n\\[\n\\vartheta_n(X,\\mathsf{d},\\mathfrak{m})(x)=\\tilde{c}^{-1},\\ \\mathfrak{m}\\text{-a.e. }x\\in \\mathcal{R}_n^\\ast(X).\n\\]\n\nMoreover, if we initially take $x,y\\in \\mathbb{R}^n$ and then choose sequences $\\{x_i\\}$, $\\{y_i\\}$ such that $X_i\\ni x_i\\rightarrow x\\in \\mathbb{R}^n$, $X_i\\ni y_i\\rightarrow y\\in \\mathbb{R}^n$ in the above argument, we can find that\n\\[\n\\frac{\\mathsf{d}\\left(x_i,y_i\\right)}{r_i}=\\mathsf{d}_i\\left(x_i,y_i\\right)\\rightarrow |x-y|, \\ \\text{as}\\ i\\rightarrow \\infty.\n\\]\nAs a result, by following the calculations in (\\ref{eqnaaa5.13}), (\\ref{eqn4.1}) can be refined to\n\\begin{equation}\\label{717aaaaaa}\n\\lim_{t\\downarrow 0} t^n H(rt+o(t),t^2)=\\tilde{c}\\mathop{(4\\pi )^{-\\frac{n}{2}}} \\exp\\left(-\\frac{r^2}{4}\\right),\\ \\forall r>0.\n\\end{equation}\nAssume there exists a point $x_0$ such that $\\vartheta_n(X,\\mathsf{d},\\mathfrak{m})(x_0)\\neq\\tilde{c}$. For any $r_i\\downarrow 0$, after passing to a subsequence, there exists a pointed RCD$(0,n)$ space $\\left(X_\\infty,\\mathsf{d}_\\infty,\\mathfrak{m}_\\infty,x_\\infty\\right)$ such that \n\\[\n(X_i,\\mathsf{d}_i,\\mathfrak{m},x_0):=\\left(X,\\frac{1}{r_i}\\mathsf{d}_i,\\frac{\\mathfrak{m}}{\\mathfrak{m}(B_{r_i}(x_0))},x_0\\right)\\xrightarrow{\\mathrm{pmGH}} \\left(X_\\infty,\\mathsf{d}_\\infty,\\mathfrak{m}_\\infty,x_\\infty\\right).\n\\]\nLet $\n\\rho_\\infty$ be the heat kernel on $(X_\\infty,\\mathsf{d}_\\infty,\\mathfrak{m}_\\infty)$. For any $z_\\infty,w_\\infty \\in X_\\infty$, we can select two sequences $\\{z_i\\}$ and $\\{w_i\\}$ such that $X_i\\ni z_i\\rightarrow z_\\infty$, $X_i\\ni w_i\\rightarrow w_\\infty$. Similarly, we can show that the heat kernel on the limit space satisfies that\n\\begin{equation}\\label{eeqn4.5}\n\\begin{aligned}\n\\rho_\\infty(z_\\infty,w_\\infty,1)&=\\lim_{i\\rightarrow \\infty}\\rho_i(z_i,w_i,1)\\\\\n&=\\lim_{i\\rightarrow \\infty}\\mathfrak{m}(B_{r_i}(x_0))\\mathop{\\rho(z_i,w_i,r_i^2)}\\\\\n&=\\lim_{i\\rightarrow \\infty}\\mathfrak{m}(B_{r_i}(x_0))\\mathop{H(r_i\\,\\mathsf{d}_i\\left(z_i,w_i\\right),r_i^2).}\n\\end{aligned}\n\\end{equation}\nOwing to Theorem \\ref{JLZ} and (\\ref{717aaaaaa}), by letting $z_\\infty=x_\\infty$ and selecting $w_\\infty\\in \\partial B_1(x_\\infty)$, we observe from (\\ref{eeqn4.5}) that\n\\[\n(\\tilde{c}\\mathop{C})^{-1}\\leqslant\\lim_{i\\rightarrow \\infty}\\frac{\\mathfrak{m}(B_{r_i}(x_0))}{r_i^n}\\leqslant {\\tilde{c}}^{-1} \\mathop{C},\n\\]\nwhere $C=C(K,N)$ is a positive constant. Specifically, we know\n\\[\n(\\tilde{c}\\mathop{C})^{-1}\\leqslant\\liminf_{r\\rightarrow 0}\\frac{\\mathfrak{m}(B_{r}(x_0))}{r^n}\\leqslant \\limsup_{r\\rightarrow 0}\\frac{\\mathfrak{m}(B_{r}(x_0))}{r^n}\\leqslant {\\tilde{c}}^{-1} \\mathop{C}.\n\\]\nTherefore, applying \\cite[Thm. 2.4.3]{AT04} implies that $\\mathscr{H}^n \\ll \\mathfrak{m}$. This result, combined with Remark \\ref{AHT} then shows that $\\mathfrak{m}=\\tilde{c}^{-1}\\mathscr{H}^n$. Ultimately, it follows from \\cite[Thm. 1.5 and 2.22]{BGHZ23} that $(X,\\mathsf{d},\\mathscr{H}^n)$ is a non-collapsed RCD$(K,n)$ space.\n\n\\end{proof}\n\\begin{proof}[Proof of Corollary \\ref{cor1.5}]\nAccording to Theorem \\ref{mainthm2}, $\\mathfrak{m}=c\\mathscr{H}^n$ for some $c>0$, and $(X,\\mathsf{d},\\mathscr{H}^n)$ is an RCD$(K,n)$ space. Since the space can be rescaled, without loss of generality we may assume that $\\mathscr{H}^n(X)=1$. To conclude, it suffices to verify that $(X,\\mathsf{d},\\mathscr{H}^n)$ is a radial symmetric RCD$(K,n)$ space.\n\nRecall that for any $x,y\\in X$ any $t>0$, our assumption implies that\n\\[\n\\sum\\limits_{i=0}^\\infty e^{-\\mu_i t}\\phi_i(x)\\phi_i(y)=\\rho(x,y,t)=H\\left(\\mathsf{d}\\left(x,y\\right),t\\right).\n\\]\n\nLet $\\phi_1,\\ldots,\\phi_m$ be the $L^2$-orthonormal basis of eigenspace with corresponding eigenvalue $\\mu_1$. Given any two points $x,y\\in X$, for any $t>0$ we calculate that\n\\begin{equation}\\label{eeqn4.3}\n\\sum\\limits_{i=1}^m\\phi_i(x)\\phi_i(y)=e^{\\mu_1 t}(H(\\mathsf{d}(x,y),t)-1)-\\sum\\limits_{i=m+1}^\\infty e^{(\\mu_1-\\mu_i) t}\\phi_i(x)\\phi_i(y).\n\\end{equation}\nLet $N_0$ be the integer such that\n\\[\n\\mu_i\\geqslant 2\\, C_1(K,n)\\mathop{i^{\\frac{2}{n}}}\\geqslant 2\\mu_1,\\ \\forall i\\geqslant N_0.\n\\]\nThen the second term of the right hand side of (\\ref{eeqn4.3}) satisfies that\n\\begin{equation}\\label{eeqn4.4}\n\\begin{aligned}\n\\left|\\sum\\limits_{i=m+1}^\\infty e^{(\\mu_1-\\mu_i)t}\\phi_i(x)\\phi_i(y)\\right|&\\leqslant C_2(K,N) \\sum\\limits_{i=m+1}^\\infty e^{(\\mu_1-\\mu_i )t}\\mathop{i^{\\frac{n}{2}}}\\\\\n\\ &=\\sum\\limits_{i=m+1}^{N_0} e^{(\\mu_1-\\mu_i) t}\\mathop{i^{\\frac{n}{2}}}+\\sum\\limits_{i=N_0+1}^{\\infty} e^{(\\mu_1-\\mu_i) t}\\mathop{i^{\\frac{n}{2}}}\\\\\n\\ &\\leqslant \\sum\\limits_{i=m+1}^{N_0} e^{(\\mu_1-\\mu_i) t}\\mathop{i^{\\frac{n}{2}}}+\\sum\\limits_{i=N_0+1}^{\\infty} e^{C_1(K,N)i^{-\\frac{2}{n}}}\\mathop{i^{\\frac{n}{2}}}\\rightarrow 0\\ \\ \\text{as }t\\rightarrow \\infty.\n\\end{aligned}\n\\end{equation}\n\nFinally it follows from (\\ref{eeqn4.3}) and (\\ref{eeqn4.4}) that\n\\[\n\\sum\\limits_{i=1}^m\\phi_i(x)\\phi_i(y)=\\lim_{t\\rightarrow \\infty} e^{\\mu_1 t}(H\\left(\\mathsf{d}\\left(x,y\\right),t\\right)-1):=F\\left(\\mathsf{d}\\left(x,y\\right)\\right),\n\\]\nwhich shows that $(X,\\mathsf{d},\\mathscr{H}^n)$ is a radial symmetric RCD$(K,n)$ space. This completes the proof.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Smoothness Theorem \\ref{thm1.11}}\\label{sec4}\n The core of our analysis in this section is the proof of Theorem \\ref{thm1.11}. We first present several useful results.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Co-area formula and disintegration formula on RCD spaces}\nIn this section, we will frequently utilize the coarea formula. Let's review the version of the statement that we will require. For a more detailed discussion, refer to \\cite[Proposition 4.2]{M03}, and for the representation of the perimeter measure using Hausdorff measures, see \\cite{ABS19, BPS23}. Furthermore, we refer to for instance \\cite[Rmk. 3.5]{GH16} for an explanation of how the total variation measure of a Lipschitz function coincides with the product of its weak upper gradient and the reference measure.\n\n\\begin{thm}\\label{coareafor}Let $(X, \\mathsf{d}, \\mathscr{H}^n)$ be a non-collapsed $\\mathrm{RCD}(K, n)$ space. Let $v$ be a Lipschitz function on $X$. Then $\\{v > t\\}$ has finite perimeter for $\\mathscr{L}^1$-a.e. $t> 0$, and for every Borel function $f: X \\rightarrow [0, \\infty]$ it holds:\n\\[\n\\int_X f \\,|\\nabla v| \\, \\mathrm{d}\\mathscr{H}^n = \\int_{-\\infty}^\\infty \\left( \\int_{\\{v=t\\}} f \\, \\mathrm{d}\\mathscr{H}^{n-1} \\right) \\mathrm{d}t.\n\\]\n\\end{thm}\nWe may also need some knowledge concerning the disintegration formula of the distance function $\\mathsf{d}_x:=\\mathsf{d}\\left(x,\\cdot\\right)$ on a compact RCD$(K,N)$ space $(X,\\mathsf{d},\\mathfrak{m})$, where $x\\in X$ is a fixed point. We refer to for instance \\cite[Sec. 3]{CM20}.\n\nIt is worth pointing out the following geodesically non-branching property in the RCD setting. \n\n\\begin{thm}[{\\cite[Thm. 1.3]{D20}}]\\label{RCDnb}\n$\\mathrm{RCD}(K,N)$ spaces are non-branching. That is, there do not exist two unit speed geodesics $\\gamma_1\\neq \\gamma_2$ such that they are parameterized on the unit interval and that there exist $t\\in (0,1)$ such that\n\\[\n\\gamma_1(s)=\\gamma_2(s),\\ \\ \\forall s\\in [0,t].\n\\] \n\\end{thm}\n\nFor the remainder of this subsection, we assume that $(X,\\mathsf{d},\\mathfrak{m})$ is a compact RCD$(K,N)$ space with $\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}(X)\\geqslant 2$. Let us fix a point $x\\in X$ and define the set $\\Gamma$ as follows:\n\\[\n\\Gamma:=\\{(y,z)\\in X\\times X:\\mathsf{d}_x(y)-\\mathsf{d}_x(z)=\\mathsf{d}\\left(y,z\\right)\\}.\n\\]\nThe transpose of $\\Gamma$ is given by $\\Gamma^{-1}$, defined as $\\Gamma^{-1}:=\\{(y,z)\\in X\\times X: (z,y)\\in \\Gamma\\}$. We then define the transport relation $R$ as $R:=\\Gamma\\cup \\Gamma^{-1}$. Furthermore, we denote by $\\Gamma(y)=\\{z\\in X:(y,z)\\in \\Gamma\\}$ the section of $\\Gamma$ through $y$ in the first coordinate, and similarly for $R(y)$ (through either coordinate by symmetry). Subsequently, we introduce the forward branching point set, following the approach outlined in \\cite{C14}:\n\\[\n{\\mathcal{E}=\\{w\\in X:\\exists\\ (y,z)\\in \\Gamma(w)\\times \\Gamma(w) \\setminus R\\}. }\n\\]\nWe note here that by combining Theorem \\ref{RCDnb} with our assumption that $\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}(X)\\geqslant 2$, the backward branching point set in our context is precisely $\\{x\\}$. \n\nLet us consider the non-branched transport set $\\mathcal{T}:=X\\setminus (\\mathcal{E}\\cup\\{x\\})$, which is a $\\sigma$-compact set with $\\mathfrak{m}$-measure zero by \\cite[Lem. 4.3, Prop. 4.5]{C14}. \nWe also define the non-branched transport relation $\\mathcal{R}:=R\\cap (\\mathcal{T}\\times \\mathcal{T})$. According to \\cite[Thm. 4.6]{C14} (see also \\cite{BC13}), $\\mathcal{R}$ is an equivalence relation over $\\mathcal{T}$ and that for any $y\\in \\mathcal{T}$, $R(y)\\subset (X,\\mathsf{d})$ is a unit speed geodesic defined on a closed interval, starting from $x$.\n\nAs a result, the non-branched transport set $\\mathcal{T}$ is partitioned into a disjoint family of equivalence classes, denoted by $\\{X_\\alpha\\}_{\\alpha\\in Q}$, where $Q$ is an index set. It follows naturally to introduce the quotient map $\\mathfrak{Q}:\\mathcal{T}\\rightarrow Q$, which is induced by this partition: \n\\[\n\\alpha=\\mathfrak{Q}(y) \\iff y\\in X_\\alpha.\n\\]\nFurthermore, we can equip $Q$ with the quotient $\\sigma$-algebra $\\mathscr{Q}$, which is derived from the $\\sigma$-algebra $\\mathscr{X}$ over $X$ consisting of $\\mathfrak{m}$-measurable sets. This quotient $\\sigma$-algebra is defined by\n\\[\nA\\in \\mathscr{Q}\\iff \\mathfrak{Q}^{-1}(A)\\in \\mathscr{X},\n\\]\nand it represents the finest $\n\\sigma$-algebra on $Q$ for which $\\mathfrak{Q}$ is measurable. It is straightforward to verify that $\\mathfrak{Q}_\\sharp \\mathfrak{m}\\ll \\mathfrak{q}$. Owing to \\cite[Thm. 5.5]{C14}, for RCD$(K,N)$ spaces we have $\\mathfrak{m}(X\\setminus \\mathcal{T})=0$. After normalization, we can introduce the the quotient measure $\\mathfrak{q}$ as $\\mathfrak{q}:=\\mathfrak{Q}_\\sharp (\\mathfrak{m})$, meaning that $\\mathfrak{q}$ is obtained by pushing forward $\\mathfrak{m}(X)^{-1}\\mathfrak{m}$ via the quotient map $\\mathfrak{Q}$.\n\nIndeed, there is a rather explicit construction of the quotient map $\\mathfrak{Q}$ which embeds $Q$ a subset of $X$ and ensures that the map itself is $\\mathcal{A}$-measurable, where $\\mathcal{A}$ is the $\\sigma$-algebra generated by the analytic sets of $X$, as detailed in \\cite{CM17} and \\cite[Lem. 3.8]{CM201}.\n\nThus, by combining the classical disintegration theorem \\cite[Prop. 452F]{F03} with recent localization results (\\cite[Thm. 9.5]{BC13} and \\cite[Thm. 5.1]{CM17b}), we can summarize the above findings in the following theorem. \n\n\\begin{thm}\\label{thmdisinte}\nLet $(X,\\mathsf{d},\\mathfrak{m})$ be a compact $\\mathrm{RCD}(K,N)$ space with $\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}(X)\\geqslant 2$ and let $x\\in X$ be a fixed point. Then the measure $\\mathfrak{m}$ restricted to the non-branched transport set $\\mathcal{T}$ admits the following disintegration formula:\n\\begin{equation}\\label{disintefor}\n\\mathfrak{m}\\llcorner_\\mathcal{T}:=\\int_Q \\mathfrak{m}_\\alpha \\mathop{\\mathrm{d}\\mathfrak{q}(\n\\alpha)},\n\\end{equation}\nwhere $\\mathfrak{q}$ is a Borel probability measure over $Q\\subset X$ such that $\\mathfrak{Q}_\\sharp \\mathfrak{m}\\ll \\mathfrak{q}$. Moreover, $\\{\\mathfrak{m}_\\alpha\\}_{\\alpha\\in Q}$ satisfies the following properties.\n\\begin{enumerate}\n\\item For $\\mathfrak{q}$-a.e. $\\alpha\\in Q$, there exists a non-negative Borel function $h_\\alpha$ on $X_\\alpha$ such that $\\mathfrak{m}_\\alpha=h_\\alpha \\mathscr{H}^1\\llcorner_{X_\\alpha}$ and that $h_\\alpha$ is a $\\mathrm{CD}(K,N)$ density on $X_\\alpha$ (or equivalently by writing $\\overline{X}_\\alpha$ as the closure of $X_\\alpha$, $\\left(\\overline{X}_\\alpha,\\mathsf{d},h_\\alpha\\mathscr{H}^1\\right)$ verifies the $\\mathrm{CD}(K,N)$ condition). \\item For any $\\mathfrak{m}$-measurable set $B$, the map $\\alpha\\mapsto \\mathfrak{m}_\\alpha(B)$ is $\\mathfrak{q}$-measurable.\n\\item For any $\\mathfrak{m}$-measurable set $B$ and $\\mathfrak{q}$-measurable set $A$, the following disintegration formula holds:\n\\[\n\\mathfrak{m}(B\\cap \\mathfrak{Q}^{-1}(A))=\\int_A \\int_{X_\\alpha} \\chi_B\\mathop{\\mathrm{d}\\mathscr{H}^1}\\mathop{\\mathrm{d}\\mathfrak{q}(\\alpha).}\n\\]\n\\end{enumerate}\n\\end{thm}\n\\begin{remark}\\label{rmk4.6aaaaaaaaa}\nIn the second property of the family $\\{\\mathfrak{m}_\\alpha\\}_{\\alpha\\in Q}$, we omit the precise definition of the CD$(K,N)$ metric measure spaces, referring instead to \\cite{LV09,St06a,St06b} for details. We note that for all $\\alpha$ such that $h_\\alpha$ is a $\\mathrm{CD}(K,N)$ density on $X_\\alpha$, $h_\\alpha$ is locally semi-concave in the interior of $X_\\alpha$. That is, for every $z_0$ in the interior of $X_\\alpha$, there exists a constant $C(z_0,\\alpha)\\in \\mathbb{R}$ such that the function $z\\mapsto \\left(h_\\alpha-C(z_0,\\alpha)z^2\\right)$ is concave in a neighborhood (in $X_\\alpha$) of $z_0$. \n\n\nFinally, we recall that if $\\left([a,b],\\mathsf{d}_{\\mathbb{R}},h\\mathscr{H}^1\\right)$ is a CD$(K,N)$ metric measure space, then the derivative of $\\log h$ is bounded as follows:\n\\begin{equation}\\label{eqn4.6cccc}\n-(N-1)\\frac{V_{K,N}'(b-t)}{V_{K,N}(b-t)}\\leqslant (\\log h)'(t)\\leqslant (N-1)\\frac{V_{K,N}'(t-a)}{V_{K,N}(t-a)},\n\\end{equation}\nfor $t \\in (a,b)$ being a point of differentiability of $h$. See, for instance \\cite[Lem. A.9]{CM21}. \n\\end{remark}\n\n\n\\subsection{Proof of Theorem \\ref{thm1.11}}\n\nFor reader's convenience, we recall Theorem \\ref{thm1.11} as follows.\n\n\\begin{thm}\\label{thm5.1}\nLet $(X,\\mathsf{d},\\mathfrak{m})$ be a compact $\\mathrm{RCD}(K,N)$ space with $\\mathrm{dim}_{\\mathsf{d},\\mathfrak{m}}(X)=n$. If in addition the measure of the intersection of two geodesic balls depends only on their radii and the distance between their centers, then\n\\begin{enumerate}\n\\item[$(1)$] $\\mathfrak{m}=c\\mathscr{H}^n$ for some constant $c>0$ and $(X,\\mathsf{d},\\mathscr{H}^n)$ is a non-collapsed $\\mathrm{RCD}(K,n)$ space;\n\\item[$(2)$] $(X,\\mathsf{d})$ is isometric to an $n$-dimensional smooth closed Riemannian manifold $(M^n,\\mathrm{g})$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{remark}\\label{rmk4.8aaa}\nBy approximating with characteristic functions, one can demonstrate that the assumption of Theorem \\ref{thm5.1} is equivalent to the following condition: the convolution of two radial kernel functions $H_1=h_1\\circ \\mathsf{d}$ and $H_2=h_2\\circ \\mathsf{d}$ results in a radial kernel function, provided that $h_1,h_2$ are $L^2$-integrable functions on $\\mathbb{R}$ with compact support. This finding offers a generalization of Theorem \\ref{harm3} within the RCD framework.\n\n\n\\end{remark}\n\\begin{proof}[Proof of Theorem \\ref{thm5.1}(1)]\nFrom our assumption we deduce the existence of a function $\\theta:[0,\\mathrm{diam}(X,\\mathsf{d})]\\rightarrow [0,\\infty)$ such that\n\\[\n\\mathfrak{m}(B_r(x))=\\theta(r), \\forall x\\in X,\\ \\forall r\\in [0,\\mathrm{diam}(X,\\mathsf{d})].\n\\]\nThen Theorem \\ref{BS}, in conjunction with Remark \\ref{AHT} indicates the existence of a constant $c>0$ such that\n\\begin{equation}\\label{vpxbbctt}\n\\lim_{r\\rightarrow 0}\\frac{\\mathfrak{m}(B_r(x))}{\\omega_n r^n}=\\lim_{r\\rightarrow 0}\\frac{\\theta(r)}{\\omega_n r^n}=c.\n\\end{equation}\nFinally, following the argument as in the proof of Theorem \\ref{mainthm2}, we complete the proof of statement (1).\n\\end{proof}\n\\begin{remark}\nA metric measure space $(X,\\mathsf{d},\\mathfrak{m})$ is said to be ball-homogeneous if\n\\[\n\\mathfrak{m}(B_r(x))=\\theta(r), \\, \\forall x\\in X,\\ \\forall r\\in [0,\\infty).\n\\]\n\nTo the best of the author's knowledge, the following proposition represents the state-of-the-art in the regularity of ball-homogeneous RCD$(K,N)$ spaces. Nevertheless, it remains an open question whether a compact ball-homogeneous RCD$(K,N)$ space is isometric to a smooth closed Riemannian manifold, or even a Lipschitz manifold.\n\\end{remark}\n\\begin{prop}\nAssume $(X,\\mathsf{d},\\mathfrak{m})$ is a ball-homogeneous $\\mathrm{RCD}(K,N)$ space. Then statement $(1)$ of Theorem \\ref{thm5.1} also holds and $(X,\\mathsf{d},\\mathscr{H}^n)$ is a $C^\\alpha$-manifold, where $\\alpha$ is a constant depending only on $n$.\n\\end{prop}\n\\begin{proof}\nThe first part can be derived using the same proof as that of statement (1) in Theorem \\ref{thm5.1}.\n\nAs for the second part, let us denote by $\\left(X_r,\\mathsf{d}_r,\\mathscr{H}^n,x\\right)=\\left(X,\\mathsf{d}/r,\\mathscr{H}^n_{\\mathsf{d}}/r^n,x\\right)$ for convenience and claim that\n\\begin{equation}\\label{aaaaannnnnneqn5.6}\n\\lim_{r\\rightarrow 0}\\mathop{ \\sup_{x\\in X}}\\mathsf{d}_{\\mathrm{pmGH}}\\left(\\left(X_r,\\mathsf{d}_r,\\mathscr{H}^n,x\\right),\\left(\\mathbb{R}^n,\\mathsf{d}_{\\mathbb{R}^n},\\mathscr{L}^n,0_n\\right)\\right)= 0.\n\\end{equation}\nAssume the contrary, i.e. there exists $\\epsilon>0$, $r_i\\rightarrow 0$ and $\\{x_i\\}\\subset X$ such that\n\n\\begin{equation}\\label{11122222222222222}\n\\mathsf{d}_{\\mathrm{pmGH}}\\left(\\left(X_{r_i},\\mathsf{d}_{r_i},\\mathscr{H}^n,x_i\\right),\\left(\\mathbb{R}^n,\\mathsf{d}_{\\mathbb{R}^n},\\mathscr{L}^n,0_n\\right)\\right)>\\epsilon.\n\\end{equation}\nAfter passing to a subsequence, there exists an RCD$(0,n)$ space $(Y,\\mathsf{d}_Y,\\mathfrak{m}_Y,y)$ such that\n\\[\n\\left(X_{r_i},\\mathsf{d}_{r_i},\\mathscr{H}^n,x_i\\right)\\xrightarrow{\\mathrm{pmGH}} \\left(Y,\\mathsf{d}_Y,\\mathfrak{m}_Y,y\\right).\n\\]\n\nFrom (\\ref{vpxbbctt}) we know that for any $X_{r_i} \\ni z_i\\rightarrow z\\in Y$, it holds\n\\[\n\\mathfrak{m}_{Y}\\left(B^Y_r(z)\\right)=\\lim_{i\\rightarrow \\infty} \\frac{\\mathscr{H}_{\\mathsf{d}}^n(B^{X}_{r_i r}(z_i))}{r_i^n}=c\\ \\omega_n r^n, \\ \\forall r>0.\n\\]\n\nAs a consequence, following the same argument as in the proof of Theorem \\ref{thm5.1} (1), we deduce that $\\mathfrak{m}_Y=c\\mathscr{H}^n$ and that $\\left(Y,\\mathsf{d}_Y,\\mathscr{H}^n\\right)$ is a non-collapsed RCD$(0,n)$ space. Consequently, by \\cite[Thm. 1.6]{DG18}, it follows that $\\left(Y,\\mathsf{d}_Y\\right)$ is isometric to $\\left(\\mathbb{R}^n,\\mathsf{d}_{\\mathbb{R}^n}\\right)$. However, this leads to a contradiction with (\\ref{11122222222222222}).\n\nBy Theorem \\ref{DG18cor1.7} we know $X=\\mathcal{R}_n$. Finally, applying \\cite[Thm. A.1.2]{ChCo1} completes the proof.\n\\end{proof}\n\n \n In the remainder of this section, we focus on the second statement of Theorem \\ref{thm5.1}.\n \n\n\\begin{lem}\\label{step1} \nUnder the assumption of Theorem \\ref{thm5.1}, any geodesic with length less than the diameter of $X$ is locally extendible.\n\\end{lem}\n\\begin{proof}\nLet $D=\\mathrm{diam}(X,\\mathsf{d})$ and choose $x_0,y_0\\in X$ such that $\\mathsf{d}\\left(x_0,y_0\\right)=D$. Consider the unit speed geodesic $\\gamma_0:[0,D]\\rightarrow X$ from $x_0$ to $y_0$. Note that for any $r,s,t>0$ satisfying $r+s+t<D$, since $\\gamma_0(r+s+\\frac{t}{2})\\in B_{r+t}(\\gamma_0(s))\\setminus B_{r+s}(x_0)$, which is an open set, we have $ \\mathscr{H}^n\\left( B_{r+t}(\\gamma_0(s))\\setminus B_{r+s}(x_0)\\right)>0$.\n\n\n Given any arbitrary but fixed two points $x,y\\in X$ such that $e:=\\mathsf{d}\\left(x,y\\right)<D$, let $\\gamma:[0,e]\\rightarrow X$ be the unit speed geodesic from $x$ to $y$. For any $\\epsilon,e' \\in \\left(0,\\min \\{e,D-e\\} /4\\right)$, it follows the fact $\\mathscr{H}^n \\left( B_{e+\\epsilon}(\\gamma(e'))\\setminus B_{e+e'}(x)\\right)>0$ that $B_{e+\\epsilon}(\\gamma(e'))\\setminus B_{e+e'}(x)\\neq \\emptyset$. Consequently, for each $i\\in \\mathbb{N}$, we can select a point $z_i\\in B_{e+\\frac{1}{i}}(\\gamma(e'))\\setminus B_{e+e'}(x)$. By taking a subsequence, we may assume that $\\{z_i\\}$ converges to a point $z\\in X$. Thus we have\n\\[\n\\lim_{i\\to\\infty}\\left(e+\\frac{1}{i}+e'\\right)\\geqslant \\lim_{i\\to\\infty}\\mathsf{d}(z_i,\\gamma(e'))+e'\\geqslant \\lim_{i\\to\\infty}\\mathsf{d}(z_i,x)=\\mathsf{d}(z,x)\\geqslant e+e'.\n\\]\n\nIf we let $\\gamma':[0,e']\\rightarrow X$ be the unit speed geodesic from $y$ to $z$, then for any $s\\in [0,e]$, $s'\\in [0,e']$ it holds that\n\\[\n\\begin{aligned}\ne+e'&= \\mathsf{d}\\left(x,z\\right)\\leqslant \\mathsf{d}\\left(x,\\gamma(s)\\right)+\\mathsf{d}\\left(\\gamma(s),\\gamma'(s)\\right)+\\mathsf{d}\\left(\\gamma'(s),z\\right)\\\\\n \\ &=\\mathsf{d}\\left(\\gamma(s),\\gamma'(s)\\right)+s+e'-s'.\n\\end{aligned}\n\\]\nOn the other hand,\n\\[\n\\mathsf{d}\\left(\\gamma(s),\\gamma'(s)\\right)\\leqslant \\mathsf{d}\\left(\\gamma(s),y\\right)+\\mathsf{d}\\left(y,\\gamma'(s)\\right)=e-s+s'.\n\\]\nTherefore, $\\mathsf{d}\\left(\\gamma(s),\\gamma'(s)\\right)=e-s+s'$. Finally, owing to Theorem \\ref{RCDnb}, the curve $\\tilde{\\gamma}:[0,e+e']\\rightarrow X$ defined by\n\\[\n\\tilde{\\gamma}(s)=\n\\left\\{\\begin{array}{ll}\n\\gamma(s) &\\text{if $s\\in [0,e]$},\\\\\n\\gamma'(s-e')&\\text{if $s>e$.}\n\\end{array}\\right.\n\\]\nis a minimal geodesic from $x$ to $z$.\n\n\n\n\\end{proof}\n\\begin{lem}\\label{step2} Under the assumption of Theorem \\ref{thm5.1}, the injectivity radius of $X$ equals its diameter.\n\\end{lem}\n\\begin{proof}\nFirst, we demonstrate that for any $x,y\\in X$ with $0<\\mathsf{d}\\left(x,y\\right)=l<D$, there exists a unique unit speed geodesic from $x$ to $y$, denoted by $\\gamma_{x,y}$ for the remainder of this section.\n\nSuppose, for the sake of contradiction, that there are two geodesics $\\gamma_1,\\gamma_2$ from $x$ to $y$. By Lemma \\ref{step1} we can assume that the domain of $\\gamma_1$ can be extended to $[0,D]$ such that $\\gamma_1(0)=x$ and $\\gamma_1(l)=y$. For convenience, the extended curve is still denoted by $\\gamma_1$, and it remains a unit speed geodesic. \n\nGiven any $\\delta\\in (0,D-l)$, we calculate that\n\\[\nl+\\delta=\\mathsf{d}\\left(\\gamma_1(l+\\delta),\\gamma_2(0)\\right)\\leqslant \\mathsf{d}\\left(\\gamma_1(l+\\delta),\\gamma_2(l)\\right)+\\mathsf{d}\\left(\\gamma_2(0),\\gamma_2(l)\\right)=l+\\delta.\n\\]\nLet \n\\[\n\\eta(s)=\n\\left\\{\\begin{array}{ll}\n\\gamma_2(s) &\\text{if $s\\in [0,l]$},\\\\\n\\gamma_1(s-l)&\\text{if $l<s<D$.}\n\\end{array}\\right.\n\\]\nTherefore $\\eta$ is also a minimal geodesic from $x$ to $\\gamma_1(D)$ which agrees with $\\gamma_2$ on $[0,l]$. However, this leads to a contradiction with the non-branching property of $\\mathrm{RCD}(K,N)$ spaces (Theorem \\ref{RCDnb}).\n\\end{proof}\n\n\n\n\\begin{lem}\\label{step3} Under the assumption of Theorem \\ref{thm5.1}, given any fixed point $x_0\\in X$, we have the disintegration formula (\\ref{disintefor}) on $X\\setminus \\mathcal{T}_{x_0}$ with respect to $\\mathsf{d}_{x_0}$, where $\\mathcal{T}_{x_0}=\\{x_0\\}\\cup\\partial B_D(x_0)$ and $Q=\\partial B_{D/2}(x_0)$.\n\\end{lem}\n\\begin{proof}\n\nFor any $0<r_1<r_2<D$, let us set\n\\[\n\\begin{aligned}\nF_{r_2\\rightarrow r_1}:\\partial B_{r_2}(x_0)&\\longrightarrow \\partial B_{r_1}(x_0)\\\\\ny&\\longmapsto \\gamma_{x_0,y}(r_1),\n\\end{aligned}\n\\]\nwhich is well defined and is a bijection due to Lemmas \\ref{step1} and \\ref{step2}. Moreover, if $\\partial B_{r_2}(x_0)\\ni y_i\\rightarrow y\\in \\partial B_{r_2}(x_0)$, by Lemma \\ref{step2} we know $\\gamma_{x_0,y_i}$ must converge to $\\gamma_{x_0,y}$, which means that $F_{r_2\\rightarrow r_1}(y_i)\\rightarrow F_{r_2\\rightarrow r_1}(y)$. Hence $F_{r_2\\rightarrow r_1}$ is continuous. Similarly $\\left( F_{r_2\\rightarrow r_1}\\right)^{-1}$ (which is denoted by $F_{r_1\\rightarrow r_2}$ later on for convenience) is also continuous. Therefore $F_{r_2\\rightarrow r_1}$ is a homomorphism. Indeed, arguing by contradiction, we can show that $F_{r_2\\rightarrow r_1}$ and $F_{r_1\\rightarrow r_2}$ are uniform continuous.\n\nAs a result, the following map \n\\[\n\\begin{aligned}\n\\mathfrak{Q}:\\mathcal{T}_{x_0}&\\longrightarrow Q:=\\partial B_{D/2}(x_0)\\\\\ny&\\longmapsto F_{\\mathsf{d}\\left(y,x_0\\right)\\rightarrow D/2}(y),\n\\end{aligned}\n\\]\nis continuous and thus Borel. We also observe that in our context, each $\\alpha\\in \\partial B_{D/2}(x_0)$ uniquely determines a geodesic starting from $x_0$, which gives a partition \n\\[\nX\\setminus\\mathcal{T}_{x_0}=\\bigcup_{\\alpha\\in \\partial B_{D/2}(x_0)}X_\\alpha, \n\\]\nwhere\n\\[\n \\begin{aligned}X_\\alpha:(0,D)&\\longrightarrow X\\\\\n t&\\longmapsto F_{D/2\\rightarrow t}(\\alpha).\n\\end{aligned}\n\\]\n\nIn addition, Theorem \\ref{BGineq} ensures that $\\mathscr{H}^n\\left(\\partial B_D(x_0)\\right)=0$, which permits the application of Theorem \\ref{thmdisinte} to $X\\setminus \\left(\\mathcal{T}_{x_0}\\cup\\partial B_D(x_0)\\right)$. Consequently, using the identification\n\\[\nX\\setminus \\left(\\mathcal{T}_{x_0}\\cup\\partial B_D(x_0)\\right)\\ni y\\mapsto \\left(F_{\\mathsf{d}\\left(x_0,y\\right)\\rightarrow D/2}(y),\\mathsf{d}\\left(x_0,y\\right)\\right)\\in \\partial B_{D/2}(x_0)\\times (0,D),\n\\]\nwe can reformulate the disintegration formula as\n\\begin{equation}\\label{eqn4.10aa}\n\\int_X f\\mathop{\\mathrm{d}\\mathscr{H}^n}=\\int_{\\partial B_{D/2}(x_0)}\\int_{(0,D)}f\\left(\\alpha,t\\right)\\sigma_{x_0}\\left(\\alpha,t\\right)\\mathop{\\mathrm{d}t}\\mathop{\\mathrm{d}\\mathfrak{q}_{x_0}(\\alpha),}\\ \\forall f\\in L^1.\n\\end{equation}\n\\end{proof}\n\\begin{remark}\\label{rmk4.16}\nAs a direct consequence of (\\ref{eqn4.10aa}) and Theorem \\ref{coareafor}, we see that $\\partial B_R(x_0)$ is $\\mathscr{H}^{n-1}$-measurable for any $R\\in (0,D)$. Moreover, for any continuous function $f$ on $X$ it holds that \n\\[\n\\int_{\\partial B_{R}(x_0)}f\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}=\\int_{\\partial B_{D/2}(x_0)}f(\\alpha, R)\\sigma_{x_0}(\\alpha,R)\\mathop{\\mathrm{d}\\mathfrak{q}_{x_0}(\\alpha)}, \\ \\forall R\\in (0,D).\n\\]\nWe also observe that, according to Theorem \\ref{BGineq}, for any fixed point $y_0\\in X$, the function $r\\mapsto \\mathscr{H}^n(B_r(y_0))/V_{K,n}(r)$ is monotone decreasing. Therefore its derivative is non-positive, yielding that \n\\begin{equation}\\label{eqn4.13aaa}\n\\mathscr{H}^{n-1}(\\partial B_r(y_0))\\leqslant\\frac{V'_{K,n}(r)\\mathscr{H}^{n}( B_r(y_0))}{V_{K,n}(r)}\\leqslant C(K,n,D)r^{n-1}.\n\\end{equation}\n\\end{remark}\n\\begin{remark}\\label{rmk4.17}\nAccording to Remark \\ref{rmk4.6aaaaaaaaa} and \\cite[Prop. 4.7]{CM20}, it is known that for any $x\\in X$ and any $\\delta\\in (0,\\mathsf{d}(x_0,x))$, the following inequality holds:\n\\[\n\\|\\Delta \\,\\mathsf{d}_{x_0}\\|_{ L^\\infty(B_{\\delta/2}(x))}\\leqslant C(\\mathsf{d}(x_0,x),\\delta,K,N).\n\\]\nFurthermore, applying Theorem \\ref{BGineq} and (\\ref{eeeeeeqn2.6}) implies that \n\\[\nnt_{B_r(y)} |\\mathrm{Hess} \\mathop{\\mathsf{d}_{x_0}} |_{\\mathsf{HS}}^2\\mathop{\\mathrm{d}\\mathscr{H}^n}\\leqslant C(\\mathsf{d}(x_0,x),\\delta,K,N),\\ \\forall y\\in B_{\\delta/4}(x), \\ \\forall r\\in (0,\\delta/4).\n\\]\nAs a result, by letting $r\\rightarrow 0$ we deduce that \n\\[\n\\left\\||\\mathrm{Hess}\\mathop{\\mathsf{d}_{x_0}}|_\\mathsf{HS}\\right\\|_{L^\\infty(B_{\\delta/4}(x))}\\leqslant C(\\mathsf{d}(x_0,x),\\delta,K,n).\n\\]\n\n\\end{remark}\n\\begin{lem}\\label{step4}\nUnder the assumption of Theorem \\ref{thm5.1}, there exists a positive real number $\\epsilon_0$ depending on $K,n,D$ satisfying the following property: given any fixed point $x_0\\in X$, and any $\\epsilon\\in (0,\\epsilon_0)$, one can find $\\delta>0$ and $n$ points $\\{p_i\\}\\subset \\partial B_{D/3}(x_0)$ such that by letting $u_i:=\\mathsf{d}_{p_i}-D/3$, the map\n\\[\n\\begin{aligned}\n\\textbf{U}:B_\\delta(x_0)&\\longrightarrow \\mathbb{R}^n\\\\\ny&\\longmapsto(u_1(y),\\ldots,u_n(y)),\n\\end{aligned}\n\\] \nis bijective. Moreover, $\\textbf{U}$, $\\textbf{U}^{-1}|_{\\textbf{U}(B_\\delta(x_0))}$ are both $(1+C(K,n,D)\\epsilon)$-Lipschitz continuous.\n\\end{lem}\n\\begin{proof}\n\n\nFor any $p\\in \\partial B_{D/3}(x_0)$, $\\gamma_{p,x_0}$ can be extended to length $2D/3$. Moreover, Remarks \\ref{rmk4.16} and \\ref{rmk4.17} guarantees the existence of a constant $C=C(K,n,D)>1$ such that\n\\begin{equation}\\label{fffdajsflajlfad}\n\\|\\Delta \\,\\mathsf{d}_{p}\\|_{ L^\\infty(B_{D/6}(x_0))} +\\left\\||\\mathrm{Hess}\\mathop{\\mathsf{d}_{p}}|_\\mathsf{HS}\\right\\|_{L^\\infty(B_{D/6}(x_0))}\\leqslant C.\n\\end{equation}\nLet us first fix a point $p_1\\in \\partial B_{D/3}(x_0)$. Then with the aid of Theorems \\ref{AAthm} and \\ref{AH18}, $(\\mathsf{d}_{p_1}-D/3)/r$ uniformly converges to a linear function whose norm is 1 under the convergence (\\ref{aaaaannnnnneqn5.6}) as $r\\rightarrow 0$. In addition, by Gromov-Hausdorff approximation, there exists $r_1<\\epsilon$ and a point $p_2'\\in \\partial B_{r_1}(x_0)$ such that \n\\[\n\\left|\\mathsf{d}_{p_1}(p_2')-\\frac{D}{3}\\right|\\leqslant \\epsilon r_1.\n\\]\n\nSet $p_2=F_{r_1\\rightarrow D/3}(p_2')$. Then the $C$-Lipschitz continuity of $\\langle \\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2} \\rangle$ on $B_{r_1}(x_0)$ follows from the combination of (\\ref{11eqn2.16}) and (\\ref{fffdajsflajlfad}). Furthermore, the following calculation implies that $\\left|\\langle\\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2}\\rangle\\right|(x_0)\\leqslant C\\epsilon$. \n\\begin{equation}\\label{fdsajildajhf}\n\\begin{aligned}\n&\\ \\left|\\mathsf{d}_{p_1}(p_2')-\\frac{D}{3}\\right|\\\\\n=&\\ \\left|\\int_0^{r_1}\\langle\\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2}\\rangle(\\gamma_{x,p_2}(t))\\mathop{\\mathrm{d}t} \\right|\\\\\n\\geqslant&\\ r_1\\left|\\langle\\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2}\\rangle(x_0)\\right|-\\left|\\int_0^{r_1}\\left(\\langle\\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2}\\rangle(\\gamma_{x,p_2}(t)) -\\langle\\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2}\\rangle(x_0)\\right)\\mathop{\\mathrm{d}t}\\right|\\\\\n\\geqslant &\\ r_1\\left|\\langle\\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2}\\rangle(x_0)\\right|-\\frac{C}{2}(r_1)^2.\n\\end{aligned}\n\\end{equation}\n\n\nLet $\\gamma_1,\\gamma_2$ be limits of $\\gamma_{x,p_1}$, $\\gamma_{x,p_2}$ under the convergence (\\ref{aaaaannnnnneqn5.6}). Then thanks to the Lipschitz continuity of $\\langle\\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2}\\rangle$ and \\cite{HM17}, the angle $\\angle \\gamma_1 0_n \\gamma_2$ is uniquely determined by $\\langle\\nabla \\mathsf{d}_{p_1},\\nabla \\mathsf{d}_{p_2}\\rangle(x_0)$ and satisfies that $\\left|\\angle \\gamma_1 0_n \\gamma_2-\\pi/2\\right|\\leqslant C\\epsilon$. Similarly, one can identify points $p_3,\\ldots,p_n\\in \\partial B_{D/3}(x_0)$ such that \n\\[\n\\left|\\langle \\nabla\\mathsf{d}_{p_i},\\nabla \\mathsf{d}_{p_j} \\rangle\\right|(x_0)\\leqslant C\\epsilon,\\ \\forall i\\neq j.\n\\]\nTherefore, combining (\\ref{fffdajsflajlfad}) we have \n\\[\n\\max_{y\\in B_{4\\epsilon}(x_0)}\\left|\\langle \\nabla\\mathsf{d}_{p_i},\\nabla \\mathsf{d}_{p_j} \\rangle\\right|(y)\\leqslant 5\\,C\\epsilon,\\ \\forall i\\neq j,\n\\]\nwhich yields \n\\[\n\\max_{y\\in B_{4\\epsilon}(x_0)}\\left|g-\\sum_{i=1}^m d\\, \\mathsf{d}_{p_i}\\otimes d\\, \\mathsf{d}_{p_i}\\right|(y)\\leqslant 5\\,nC\\epsilon.\n\\]\nWe now let $u_i:=\\mathsf{d}_{p_i}-D/3$ and $\\textbf{U}:=(u_1,\\ldots,u_n)$ and claim that \n\\begin{equation}\\label{4.14}\n\\lim_{r\\rightarrow 0}\\sup_{y\\in B_{2\\epsilon}(x_0)}\\max_{w\\in \\partial B_{r}(y)} \\frac{|\\textbf{U}(w)-\\textbf{U}(y)|}{r}\\geqslant 1-C\\epsilon.\n\\end{equation}\n\nWe argue by contradiction. Assume there exists two sequences of points $\\{w_i\\}$ and $\\{y_i\\}$ such that $r_i=\\mathsf{d}(w_i,y_i)\\rightarrow 0$ and that $|\\textbf{U}(w_i)-\\textbf{U}(y_i)|<(1-C\\epsilon )r_i$. Then using (\\ref{aaaaannnnnneqn5.6}) and passing to a subsequence, we deduce that\n\\[\n(X_i,\\mathsf{d}_i,\\mathscr{H}^n,y_i):=\\left(X,\\frac{1}{r_i}\\mathsf{d},\\frac{1}{r_i^n}\\mathscr{H}^n,y_i\\right)\\xrightarrow{\\mathrm{pmGH}}\\left(\\mathbb{R}^n,\\mathsf{d}_{\\mathbb{R}^n},\\mathscr{L}^n,0_n\\right).\n\\]\nOwing to (\\ref{fffdajsflajlfad}) and Theorem \\ref{AH18}, for each $j=1,\\ldots,n$, $\\{u_j^i:=u_j/r_i\\}$ uniformly converges to a linear function $f_j:x\\mapsto\\alpha_j\\cdot x$ with $|\\alpha_j|=1$ on $\\mathbb{R}^n$. Therefore, we can estimate that\n\\[\n\\begin{aligned}\n\\omega_n \\sum_{j\\neq l} {(\\alpha_j\\cdot\\alpha_l)}^2=& \\int_{B_1(0_n)}\\left|\\mathrm{g}_{\\mathbb{R}^n}-\\sum_{j=1}^n d\\,f_j\\otimes d\\,f_j\\right|^2\\mathrm{d}\\mathscr{L}^n\\\\\n=&\\lim_{i\\rightarrow \\infty}\\int_{B_1^{X_i}(y_i)}\\left|g_i^{X_i}-\\sum_{i=j}^n d\\,u_j^i\\otimes d\\,u_j^i\\right|_{\\mathsf{HS}}^2 \\mathrm{d}\\mathscr{H}^n\\\\\n=&\\lim_{i\\rightarrow \\infty}\\frac{1}{r_i^n}\\int_{B_{r_i}(y_i)}\\left|g-\\sum_{j=1}^n d\\,u_j\\otimes d\\,u_j\\right|_{\\mathsf{HS}}^2 \\mathrm{d}\\mathscr{H}^n\\leqslant 25\\,\\omega_n C^2 \\epsilon^2.\n\\end{aligned}\n\\]\nLet us denote $b=(b_1,\\ldots,b_n)\\in \\partial B_1(0_n)$ as the limit point of $w_i$. Then \n\\[\n\\begin{aligned}\n(1-C\\epsilon)^2\\geqslant&\\ \\lim_{i\\rightarrow \\infty}\\left(\\frac{\\left|\\textbf{U}(w_i)-\\textbf{U}(y_i)\\right|}{r_i}\\right)^2\\\\\n=&\\ \\sum_{j=1}^n \\left(\\sum_{k=1}^n \\alpha_j\\cdot b\\right)^2= 1+\\sum_{k\\neq l}b_k b_l\\, {(\\alpha_k\\cdot\\alpha_l)}^2\\\\\n\\geqslant& \\,1-4C^2\\epsilon^2\\sum_{k\\neq l}\\frac{b_k^2+b_l^2}{2}\\geqslant 1-25nC^2\\epsilon^2.\n\\end{aligned}\n\\]\nTherefore to deduce the contradiction it suffices to set $\\epsilon_0=1/(25nC+C)$. Similarly, one can show that \n\\begin{equation}\\label{4.15}\n\\lim_{r\\rightarrow 0}\\sup_{y\\in B_{2\\epsilon}(x_0)}\\max_{w\\in \\partial B_{r}(y)} \\frac{|\\textbf{U}(w)-\\textbf{U}(y)|}{r}\\leqslant 1+C\\epsilon.\n\\end{equation}\nFinally by (\\ref{4.14}) and (\\ref{4.15}) it suffices to choose $\\delta>0$ such that \n\\[\n1-2C\\epsilon\\leqslant \\sup_{y\\in B_{2\\epsilon}(x_0)}\\max_{w\\in \\partial B_{r}(y)} \\frac{|\\textbf{U}(w)-\\textbf{U}(y)|}{r}\\leqslant 1+2C\\epsilon,\\ \\forall r<\\delta.\n\\]\n\\end{proof}\n\\begin{remark}\nIndeed, by employing a blow up argument, one can show that the choice of $\\delta$ is independent of $x_0$.\n\\end{remark}\nWe are now in a position to prove the following key proposition.\n\\begin{prop}\\label{step5} Under the assumption of Theorem \\ref{thm5.1}, given any fixed point $\\bar{x}\\in X$, it holds that \n\\begin{equation}\\label{1120}\n\\Delta \\,\\mathsf{d}_{\\bar{x}}(y)=\\Delta\\, \\mathsf{d}_{\\bar{x}}(z), \\ \\forall y,z \\in \\partial B_R(\\bar{x}),\\ \\forall R\\in (0,D).\n\\end{equation}\nIn other words, the value of $\\Delta\\, \\mathsf{d}_{\\bar{x}}$ only depends on the value of $\\mathsf{d}_{\\bar{x}}$.\n\\end{prop}\n\\begin{proof}\nSince we have the freedom to rescale the space, it suffices to consider equation (\\ref{1120}) near a point $x_0\\in X$ with $\\mathsf{d}(\\bar{x},x_0)=D/3$. Let us define $u_n=\\mathsf{d}_{\\bar{x}}-D/3$. Subsequently, we can choose a small $\\delta>0$ and construct a bi-Lipschitz map $\\textbf{U}=(u_1,\\ldots,u_n):B_{\\delta}(x_0)\\rightarrow \\mathbb{R}^n$ as described in Lemma \\ref{step4}. We claim that for $\\mathscr{H}^n$-a.e. $x\\in B_\\delta(x_0)$ the following holds.\n\\begin{equation}\\label{aaa4.16}\n\\lim_{t\\rightarrow 0} \\frac{1}{t}\\left(\\frac{\\mathscr{H}^n\\left(B_t(x)\\setminus B_{\\mathsf{d}(\\bar{x},x)}(x)\\right)}{\\omega_n t^n}-\\frac{1}{2}\\right)=\\Delta \\,\\mathsf{d}_{\\bar{x}}(x).\n\\end{equation}\n\nLet us take \n\\[\nx\\in \\bigcap_{j,k=1}^n \\mathrm{Leb}(\\langle \\mathrm{Hess} \\mathop{u_j},\\mathrm{Hess}\\mathop{ u_k}\\rangle)\\cap \\bigcap_{j,k,l=1}^n\\mathrm{Leb}(\\mathrm{Hess}\\mathop{u_j}(\\nabla u_k,\\nabla u_l)) \\cap B_\\delta(x_0),\n\\]\n and constants $a_{ij}$ ($i,j=1,\\ldots,n$) and $b^i_{jk}$ ($i,j,k=1,\\ldots,n$) such that the functions $v_i=\\sum_{j=1}^n a_{ij}(u_j-u_j(x))+\\sum_{j,k=1}^n b^i_{jk}(u_j-u_j(x)) (u_k-u_k(x))$ satisfy $v_i(x)=0$, $\\langle\\nabla v_i,\\nabla v_j\\rangle(x)=\\delta_{ij}$ and \n\\begin{equation}\\label{4.17aaaaaa}\nnt_{B_t(x)} |\\mathrm{Hess}\\mathop{v_i}|_{\\mathsf{HS}}^2\\,\\mathrm{d}\\mathscr{H}^n=0,\\ i=1,\\ldots,n. \n\\end{equation}\nTo determine these constants, it is sufficient to solve the following system of equations, the solution of which can be approached in a manner analogous to that described in \\cite[Sec. 4]{BMS23}.\n\\begin{equation}\\label{4.18c}\n\\sum_{k,l=1}^n a_{ik}a_{jl}\\langle \\nabla u_k,\\nabla u_l\\rangle(x)=\\delta_{ij},\n\\end{equation}\nand for each $i=1,\\ldots,n$,\n\\begin{equation}\\label{4.19c}\nnt_{B_t(x)}\\left(\\sum_{j,k,l=1}^n a_{ij}b^i_{kl} \\mathrm{Hess}\\mathop{u_j}(\\nabla u_k,\\nabla u_l)+\\sum_{j,k=1}^n a_{ij}a_{ik}\\langle \\mathrm{Hess} \\mathop{u_j},\\mathrm{Hess}\\mathop{ u_k}\\rangle\\right)\\mathop{\\mathrm{d}\\mathscr{H}^n}=0.\n\\end{equation}\nBy (\\ref{fffdajsflajlfad}), we may assume that \n\\begin{equation}\\label{vestimate}\n\\sum_{i=1}^n\\left(\\| |\\mathrm{Hess}\\mathop{v_i}|_{\\mathsf{HS}}\\|_{L^\\infty(B_{t_0}(x))}+\\| |\\nabla v_i|\\|_{L^\\infty(B_{t_0}(x))}+\\| \\Delta v_i\\|_{L^\\infty(B_{t_0}(x))}\\right)\\leqslant \\tilde{C}\n\\end{equation}\n for some constant $\\tilde{C}>0$ (which may vary from line to line) and some sufficiently small $t_0>0$. The subsequent discussion is predicated on the assumption that $t\\ll t_0$. \n\nFor convenience, let $\\textbf{V}=(v_1,\\ldots,v_n)$ and $r:=|\\textbf{V}|$. We recall that, from (\\ref{4.17aaaaaa}) (\\ref{4.18c}) and \\cite[Prop. 4.8]{BMS23}, it follows that\n\\begin{equation}\\label{eqn4.22}\nnt_{B_t(x)}\\left(\\sum_{i,j}|\\langle\\nabla v_i,\\nabla v_j\\rangle-\\delta_{ij}|+\\left||\\nabla r|-1\\right|+\\left|\\Delta r^2-2n\\right|\\right)\\mathop{\\mathrm{d}\\mathscr{H}^n}\\leqslant \\tilde{C}t\\mathop{\\xi(t)},\n\\end{equation} \nwhere $\\xi$ is a positive function that satisfies $\\lim_{t\\rightarrow 0}\\xi(t)=0$, and is given by the following expression:\n\\[\nnt_{B_s(x)} |\\mathrm{Hess}\\mathop{v_i}|_{\\mathsf{HS}}^2\\mathop{\\mathrm{d}\\mathscr{H}^n.}\n\\] \nTo conclude, we may need the following three lemmas.\n\\end{proof}\n\\begin{lem}\\label{hfouiaehofie}\nUnder the assumption of Theorem \\ref{thm5.1} and the setting in the proof of Proposition \\ref{step5} we have\n\\begin{equation}\\label{aabb4.24}\n\\begin{aligned}\n&\\lim_{t\\rightarrow 0}\\frac{1}{\\omega_n t^{n+1}}\\int_{B_{2t}(x)} \\left|\\left(\\det \\left(\\langle\\nabla v_i,\\nabla v_j \\rangle\\right)\\right)^{\\frac{1}{2}}-1\\right|\\mathop{\\mathrm{d}\\mathscr{H}^n}\\\\\n=&\\lim_{t\\rightarrow 0}\\frac{1}{\\omega_n t^{n+1}}\\int_{\\textbf{V}(B_{t}(x))\\cap \\{u_n\\geqslant u_n(x) \\}} \\left|\\left(\\det \\left(\\langle\\nabla v_i,\\nabla v_j \\rangle\\right)\\right)^{-\\frac{1}{2}}-1\\right|\\mathop{\\mathrm{d}\\mathscr{L}^n}=0.\n\\end{aligned}\n\\end{equation}\nTherefore, it holds that\n\\begin{equation}\\label{aabb4.25}\n\\mathscr{H}^n\\left(B_{t}(x)\\setminus B_{\\mathsf{d}(x,\\bar{x})}(x)\\right)=\\mathscr{L}^n\\left(\\textbf{V}\\mathop{(B_t(x))}\\cap\\ \\{u_n\\geqslant u_n(x)\\}\\right)+o(t^{n+1}).\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nFor the first limit in (\\ref{aabb4.24}), since $\\langle\\nabla v_i,\\nabla v_j\\rangle(x)=\\delta_{ij}$, it follows from (\\ref{vestimate}) that the matrix $\\left(\\langle\\nabla v_i,\\nabla v_j \\rangle\\right)(x)$ is of positive definite whenever $x$ is sufficiently close to $x_0$. Then the limit can be deduced directly from (\\ref{eqn4.22}). \n\nAs for the second limit in (\\ref{aabb4.24}), first following the argument used in the proof of Lemma \\ref{step4}, we can assume that the restrictions $\\textbf{V}|_{B_{t}(x)}$ and $\\textbf{V}^{-1}|_{\\textbf{V}(B_{t}(x))}$ are both $(1+\\tilde{C}t)$-Lipschitz maps. Then it suffices to apply a similar proof as in \\cite[Lem. 4.7]{H23} to verify that \n\\[\n\\textbf{V}_\\sharp (\\mathscr{H}^{n})=\\left(\\det \\left(\\langle\\nabla v_i,\\nabla v_j \\rangle\\right)\\right)^{-\\frac{1}{2}}\\mathscr{L}^n.\n\\]\nThis together with (\\ref{aabb4.24}) then implies (\\ref{aabb4.25}).\n\nnt_{B_{2t}(z)} |\\mathrm{Hess}\\mathop{v_i}|_{\\mathsf{HS}}^2\\mathop{\\mathrm{d}\\mathscr{H}^n}\\right)^{\\frac{1}{2}}\\mathop{\\mathrm{d}t}\\\\\n\n\\end{proof}\n\\begin{lem}\\label{step6} \nUnder the assumption of Theorem \\ref{thm5.1} and the setting in the proof of Proposition \\ref{step5} we have\n\\[\n\\begin{aligned}\n&\\int_{B_t(0_n)\\cap \\{u_n\\geqslant u_n(x)\\}} \\left(\\det \\left(\\langle\\nabla v_i,\\nabla v_j \\rangle\\right)\\right)^{-\\frac{1}{2}}\\mathop{\\mathrm{d}\\mathscr{L}^n}\\\\\n=&\\int_{\\textbf{V}(B_t(x))\\cap \\{u_n\\geqslant u_n(z)\\}} \\left(\\det \\left(\\langle\\nabla v_i,\\nabla v_j \\rangle\\right)\\right)^{-\\frac{1}{2}}\\mathop{\\mathrm{d}\\mathscr{L}^n}+ o(t^{n+1}).\n\\end{aligned}\n\\]\n\\end{lem}\n\\begin{proof}\n\n\n\n \n\n\n\n\n\n\nThe proof closely follows the approach outlined in \\cite[Lem. 3.5]{BMS23}. Let $\\tilde{r}=\\max\\{r,\\mathsf{d}_x\\}$ and $\\Omega_t:=B_t(x)\\cup \\textbf{V}^{-1}(B_t(0_n))$. Clearly we have $\\Omega_t=\\{\\tilde{r}\\leqslant t\\}$ and $B_{t-\\tilde{C}t^2}(x)\\subset\\Omega_t\\subset B_{t+\\tilde{C}t^2}(x)$. We claim that $\\mathscr{H}^n(\\Omega_t)-\\omega_n t^n\\geqslant -t^{n+1}\\xi(2t)$.\n\nUsing Theorem \\ref{coareafor} we see\n\\[\n\\int_{\\Omega_t}\\frac{r|\\nabla \\tilde{r}|}{\\tilde{r}^{n+1}}|\\nabla(\\tilde{r}-r)|^2\\mathop{\\mathrm{d}\\mathscr{H}^n}=\\int_0^t \\int_{\\partial \\Omega_s} \\frac{r}{s^{n+1}}|\\nabla(\\tilde{r}-r)|^2\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}=I_1+I_2,\n\\]\nwhere\n\\[\nI_1=\\int_0^t \\int_{\\partial \\Omega_s}\\frac{r}{s^{n+1}}(|\\nabla \\tilde{r}|^2+|\\nabla r|^2) \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}, \\ \\ I_2=-2\\int_0^t \\int_{\\partial \\Omega_s}\\frac{r}{s^{n+1}}\\langle\\nabla r,\\nabla \\tilde{r}\\rangle \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}.\n\\]\nDue to the locality of minimal relax slopes, we know $|\\nabla \\tilde{r}|=\\xi_{\\{r\\leqslant d_z\\}}+\\xi_{\\{r\\geqslant d_z\\}}|\\nabla r|$. Therefore applying (\\ref{eqn4.22}) we have\n\\[\n\\begin{aligned}\nI_1=2\\int_0^t \\int_{\\partial \\Omega_s}\\frac{r}{s^{n+1}} \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}+\\int_0^t \\int_{\\partial \\Omega_s}\\frac{r}{s^{n+1}}(|\\nabla \\tilde{r}|^2+|\\nabla r|^2-2) \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}=I_3+I_4,\n\\end{aligned}\n\\]\nwhere \n\\[\n\\begin{aligned}\n|I_4|&=\\left|\\int_0^t \\int_{\\partial \\Omega_s}\\frac{r}{s^{n+1}}(|\\nabla \\tilde{r}|^2+|\\nabla r|^2-2) \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}\\right|\\\\\n&\\leqslant 2\\int_0^t \\int_{\\partial \\Omega_s \\cap\\{\\tilde{r}\\geqslant \\mathsf{d}_x\\}}\\frac{r}{s^{n+1}}||\\nabla r|^2-1| \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}\\\\\n&\\leqslant \\int_0^t \\int_{\\partial \\Omega_s }\\frac{r}{s^{n+1}}||\\nabla r|^2-1| \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}= \\int_{ \\Omega_t }\\frac{r|\\nabla \\tilde{r}|}{\\tilde{r}^{n+1}}||\\nabla r|^2-1| \\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\\\\n&\\leqslant \\tilde{C}\\int_{ \\Omega_t }\\frac{1}{{\\tilde{r}}^{n}}||\\nabla r|^2-1| \\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\leqslant \\tilde{C}\\int_{B_{2t}(x)}\\frac{1}{{\\mathsf{d}_x}^{n}}||\\nabla r|^2-1| \\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\\\\n&\\leqslant \\frac{\\tilde{C}}{t^n} \\sum_{i=0}^\\infty 2^{in}\\int_{A_{2^{-i}\\, t,2^{-(i-1)}\\,t}(x)}||\\nabla r|^2-1| \\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\\\\n&\\leqslant \\frac{ \\tilde{C}}{t^n} \\sum_{i=0}^\\infty 2^{in}\\int_{B_{2^{-(i-1)}\\,t}(x)}||\\nabla r|^2-1| \\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\leqslant \\tilde{C}t\\mathop{\\xi(2t)}.\n\\end{aligned}\n\\]\nRegrading for $I_3$, we begin by substituting $r$ with $\\tilde{r}$ in the proof of \\cite[Prop. 4.8 i)]{BMS23} to observe that\n\\begin{equation}\\label{fauhfeouahfuoia}\nnt_{\\partial B_s(x)}\\tilde{r}\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\leqslant s+s^2\\mathop{\\xi(s)}\n\\end{equation}\nTherefore according to (\\ref{eqn4.13aaa}) and (\\ref{fauhfeouahfuoia}) we know \n\\[\n\\begin{aligned}\nI_3=2\\int_0^t \\frac{1}{s^{n+1}}\\int_{\\partial \\Omega_s}(r-s) \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}+2\\int_0^t \\frac{\\mathscr{H}^{n-1}(\\partial \\Omega_s)}{s^{n}}\\mathop{\\mathrm{d}s},\n\\end{aligned}\n\\]\nwith\n\\[\n\\begin{aligned}\n&\\left|\\int_0^t \\frac{1}{s^{n+1}}\\int_{\\partial \\Omega_s}(r-s) \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}\\right|\\\\\n=& \\int_0^t \\frac{1}{s^{n+1}}\\int_{\\partial \\Omega_s}(s-r) \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}\n={\\int_{\\Omega_t} \\frac{|\\nabla \\tilde{r}|}{\\tilde{r}^{n+1}}(\\tilde{r}-r)\\mathop{\\mathrm{d}\\mathscr{H}^{n}}}\\leqslant \\int_{B_{2t}(x)} \\frac{\\tilde{C}}{{\\mathsf{d}_x}^{n+1}}(\\tilde{r}-r)\\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\\\\n=&\\,\\tilde{C}\\int_0^{2t} \\frac{1}{s^{n+1}}\\int_{\\partial B_s(x)} (\\tilde{r}-r)\\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\mathop{\\mathrm{d}s}\\leqslant \\tilde{C}\\int_0^{2t} \\frac{1}{s^{n-1}}\\mathscr{H}^{n-1}(\\partial B_s(x))\\xi(s)\\mathop{\\mathrm{d}s}\\leqslant \\tilde{C}t\\mathop{\\xi(2t)}.\n\\end{aligned}\n\\]\n\nRecalling that for $\\mathscr{L}^1$-a.e. $s\\in (0,t)$, the exterior unit normal of $\\partial \\Omega_s$ coincides $\\mathscr{H}^{n-1}$-a.e. with $\\nabla \\tilde{r}$ (as shown in \\cite[Prop. 6.1]{BPS23b}), and applying the Gauss-Green formula from \\cite{BPS23} along with (\\ref{eqn4.22}), we obtain\n\\[\n\\begin{aligned}\nI_2&=-\\int_0^t \\frac{1}{s^{n+1}}\\int_{\\Omega_s} \\Delta r^2 \\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\mathop{\\mathrm{d}s}\\\\\n&=-2n\\int_0^t \\frac{\\mathscr{H}^n(\\Omega_s)}{s^{n+1}}\\mathop{\\mathrm{d}s}-\\int_0^t \\frac{1}{s^{n+1}} \\int_{\\Omega_s} (\\Delta r^2-2n) \\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\mathop{\\mathrm{d}s},\n\\end{aligned}\n\\]\nwith\n\\[\n\\left|\\int_0^t \\frac{1}{s^{n+1}} \\int_{\\Omega_s} (\\Delta r^2-2n) \\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\mathop{\\mathrm{d}s}\\right|\\leqslant \\tilde{C}t\\xi(2t).\n\\]\nNow by recalling the conclusion of \\cite[Lem. 3.5]{BMS23} we get\n\\[\n\\begin{aligned}\n&\\int_{\\Omega_t}\\frac{r|\\nabla \\tilde{r}|}{\\tilde{r}^{n+1}}|\\nabla(\\tilde{r}-r)|^2\\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\leqslant \\tilde{C}\\int_{B_{2t}(x)}\\frac{r}{\\tilde{r}^{n+1}}|\\nabla(\\tilde{r}-r)|^2\\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\\\\n\\leqslant&\\ \\tilde{C}\\int_0^{2t} \\frac{1}{s^{n+1}}\\int_{\\partial B_s(x)}r|\\nabla(\\tilde{r}-r)|^2\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}\\\\\n\\leqslant&\\ \\tilde{C} \\int_0^{2t} \\frac{1}{s^{n+1}}\\int_{\\partial B_s(x)}r|\\nabla(\\mathsf{d}_x-r)|^2\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\mathop{\\mathrm{d}s}\\leqslant \\tilde{C}t\\xi(2t).\n\\end{aligned}\n\\]\nFinally from the Newton-Leibniz rule for $\\mathscr{L}^1$-integrable functions we know \n\\[\n\\begin{aligned}\n&\\frac{1}{t^n}\\int_{\\Omega_t}|\\nabla\\tilde{r}|\\mathop{\\mathrm{d}\\mathscr{H}^n}-\\lim_{s\\to 0}\\frac{1}{s^n}\\int_{\\Omega_s}|\\nabla\\tilde{r}|\\mathop{\\mathrm{d}\\mathscr{H}^n}\\\\\n=&-n\\int_0^t \\frac{1}{s^{n+1}}\\int_{\\Omega_s}|\\nabla \\tilde{r}|\\mathop{\\mathrm{d}\\mathscr{H}^n}\\mathop{\\mathrm{d}s}+\\int_0^t \\frac{\\mathscr{H}^{n-1}(\\partial \\Omega_s)}{s^{n}}\\mathop{\\mathrm{d}s}\\\\\n=&-n\\int_0^t \\frac{1}{s^{n+1}}\\int_{\\Omega_s}\\left||\\nabla \\tilde{r}|-1\\right|\\mathop{\\mathrm{d}\\mathscr{H}^n}\\mathop{\\mathrm{d}s}-\\int_0^t \\frac{n\\,\\mathscr{H}^n(\\Omega_s)}{s^{n+1}}\\mathop{\\mathrm{d}s}+\\int_0^t \\frac{\\mathscr{H}^{n-1}(\\partial \\Omega_s)}{s^{n}}\\mathop{\\mathrm{d}s}\\\\\n=&-n\\int_0^t \\frac{1}{s^{n+1}}\\int_{B_{2s}(x)}\\left||\\nabla \\tilde{r}|-1\\right|\\mathop{\\mathrm{d}\\mathscr{H}^n}\\mathop{\\mathrm{d}s}-\\int_0^t \\frac{n\\,\\mathscr{H}^n(\\Omega_s)}{s^{n+1}}\\mathop{\\mathrm{d}s}+\\int_0^t \\frac{\\mathscr{H}^{n-1}(\\partial \\Omega_s)}{s^{n}}\\mathop{\\mathrm{d}s}\\\\\n \\leqslant&\\ \\tilde{C}t\\xi(2t).\n\\end{aligned}\n\\]\n\nSince a combination of (\\ref{vestimate}) and the $(1+\\tilde{C}t)$-bi-Lipschitz property of $\\mathbf{V}$ implies that $\\lim_{t\\to 0}t^{-n}\\int_{\\Omega_t}|\\nabla \\tilde{r}|\\mathop{\\mathrm{d}\\mathscr{H}^n}=\\omega_n$, and $t^{-n}\\int_{\\Omega_t}||\\nabla\\tilde{r}|-1|\\mathop{\\mathrm{d}\\mathscr{H}^n}\\leqslant \\tilde{C}t\\xi(2t)$, we obtain $\\mathscr{H}^n(\\Omega_t)-\\omega_n t^n\\leqslant \\tilde{C}t^{n+1}\\xi(2t)$. Following the argument presented in the proof of Lemma \\ref{hfouiaehofie}, we deduce that $|\\mathscr{H}^n(\\textbf{V}^{-1}B_t(0_n))-\\omega_n t^n|\\leqslant t^{n+1}\\xi(t)$. Additionally, since it is established in \\cite[Prop. 4.1]{BMS23} that $|\\mathscr{H}^n(B_t(x))-\\omega_n t^n|\\leqslant t^{n+1}\\xi(t)$, we conclude the proof by observing that\n\\[\n\\begin{aligned}\n\\int_{B_t(0_n)\\triangle \\textbf{V}(B_t(x))} \\left(\\det \\left(\\langle\\nabla v_i,\\nabla v_j \\rangle\\right)\\right)^{-\\frac{1}{2}}\\mathop{\\mathrm{d}\\mathscr{L}^n}= \\mathscr{H}^n\\left(\\textbf{V}^{-1}(B_t(0_n))\\triangle B_t(x)\\right)=o(t^{n+1}).\n\\end{aligned}\n\\]\n\n\n\n \n\\end{proof}\n\\begin{lem}\nUnder the assumption of Theorem \\ref{thm5.1} and the setting in the proof of Proposition \\ref{step5} we have for $\\mathscr{H}^n$-a.e. $x\\in B_\\delta(x_0)$ it holds that\n\\begin{equation}\\label{a4.26}\n2\\mathscr{L}^n\\left(B_t(0_n)\\cap \\{u_n\\geqslant u_n(x)\\}\\right)-\\omega_n t^n=2\\,\\omega_n \\Delta\\, \\mathsf{d}_{\\bar{x}}(x)(t^{n+1}+o(t^{n+1})).\n\\end{equation}\n\\end{lem}\n\\begin{proof}\nTo facilitate the subsequent discussion, let us first establish the notation. We denote by $c=u_n(x)$, $f=u_n=\\mathsf{d}_{\\bar{x}}-D/3$, $\\{u_n\\geqslant c\\}=\\{f\\geqslant c\\}$ for simplicity, and by $A=(a_{ij})$, $A^{-1}=(\\tilde{a}_{ij})$, $g^{ij}=\\langle\\nabla u_i ,\\nabla u_j\\rangle(x)$, $(g_{ij})=(g^{ij})^{-1}_{ij}$.\n\\end{proof}\nDue to \\cite{HT03}, the left-hand side of (\\ref{a4.26}) corresponds precisely to the mean curvature of the level set $\\{f=c\\}$ at the origin in Euclidean space. This mean curvature can be expressed as follows (with all partial derivatives evaluated at the origin).\n\\begin{equation}\\label{e4.29}\n\\begin{aligned}\n&\\left(\\left|\\nabla^{\\mathbb{R}^n} f\\right|\\right)^{-1}\\Delta^{\\mathbb{R}^n} f-\\left(\\left|\\nabla^{\\mathbb{R}^n} f\\right|\\right)^{-3}\\mathrm{Hess}^{\\mathbb{R}^n}\\mathop{f}\\left(\\nabla^{\\mathbb{R}^n} f,\\nabla^{\\mathbb{R}^n} f\\right)\\\\\n=&\\left(\\sum_i {f_i}^2\\right)^{-\\frac{1}{2}}\\sum_i f_{ii}-\\left(\\sum_i {f_i}^2\\right)^{-\\frac{3}{2}}\\sum_{ij}f_{ij}f_i f_j.\n\\end{aligned}\n\\end{equation}\nAccording to (\\ref{4.18c}) and (\\ref{4.19c}), we see\n\\begin{equation}\\label{4.30}\nf_i=\\tilde{a}_{ni},\\ f_{ij}=-2\\tilde{a}_{nk}\\tilde{a}_{\\alpha j}b^k_{\\alpha\\beta}\\tilde{a}_{\\beta i}.\n\\end{equation}\nMoreover, since $a_{ij}g^{jk}a_{kl}=\\delta_{il}$, we know that $\\sum_i {f_i}^2=g^{nn}=1$. Let us recall that in \\cite{H18}, Han proved that\n\\[\n\\Delta v_i=\\mathrm{Tr}(\\mathrm{Hess}\\, v_i)=\\langle \\mathrm{Hess} \\,v_i ,\\mathrm{g}\\rangle,\\ \\mathscr{H}^n\\mathrm{-a.e.},\n\\]\nwhich yields that for each $i$\n\\[\nnt_{B_t(x)}\\Delta v_i\\mathop{\\mathrm{d}\\mathscr{H}^n}=\\sum_{j} a_{ij}\\Delta u_j(x)+2\\sum_{j,k} b^{i}_{jk}g^{jk}.\n\\]\nAs a result, one may check that\n\\[\n\\sum_{i} f_{ii}=\\Delta u_n (x).\n\\]\nFor the second term in (\\ref{e4.29}), for each $i,\\tilde{\\alpha},\\tilde{\\beta}$, notice that \n\\[\n\\mathrm{Hess} \\mathop{v_i}(\\nabla u_{\\tilde{\\alpha}},\\nabla u_{\\tilde{\\beta}})=\\sum_{j}a_{ij}\\mathrm{Hess}\\mathop{u_j}(\\nabla u_{\\tilde{\\alpha}},\\nabla u_{\\tilde{\\beta}})+2\\sum_{k,l}b^i_{kl}g^{\\tilde{\\alpha}k}g^{\\tilde{\\beta}l}.\n\\]\nSince\n\\[\nnt_{B_t(x)}\\mathrm{Hess} \\mathop{v_i}(\\nabla u_{\\tilde{\\alpha}},\\nabla u_{\\tilde{\\beta}})\\mathop{\\mathrm{d}\\mathscr{H}^n}=0,\n\\] \ncombining (\\ref{4.18c}), (\\ref{4.19c}) and (\\ref{4.30}) we calculate that\n\\[\n\\begin{aligned}\n\\sum_{i,j} f_{ij}f_i f_j&=-2\\sum_{i,j,k,\\alpha,\\beta}\\tilde{a}_{ni}\\tilde{a}_{nj}\\tilde{a}_{nk}\\tilde{a}_{\\alpha j}b^k_{\\alpha\\beta}\\tilde{a}_{\\beta i}\\\\\n&=\\sum_{i,j,k,l,\\alpha,\\beta,\\tilde{\\alpha},\\tilde{\\beta}}\\tilde{a}_{ni}\\tilde{a}_{nj}\\tilde{a}_{nk}\\tilde{a}_{\\alpha j}\\tilde{a}_{\\beta i}g_{\\tilde{\\alpha}\\alpha}g_{\\tilde{\\beta}\\beta}a_{kl}\\mathrm{Hess}\\mathop{u_l}(\\nabla u_{\\tilde{\\alpha}},\\nabla u_{\\tilde{\\beta}})(x)\\\\\n&=\\sum_{i,j,k,l,\\alpha,\\beta,\\tilde{\\alpha},\\tilde{\\beta}}\\tilde{a}_{ni}\\tilde{a}_{nj}\\tilde{a}_{\\alpha j}\\tilde{a}_{\\beta i}g_{\\tilde{\\alpha}\\alpha}g_{\\tilde{\\beta}\\beta}\\mathrm{Hess}\\mathop{u_n}(\\nabla u_{\\tilde{\\alpha}},\\nabla u_{\\tilde{\\beta}})(x)\\\\\n&=\\mathop{\\mathrm{Hess}\\mathop{u_n}}(\\nabla u_n,\\nabla u_n)(x)=0.\n\\end{aligned} \n\\]\nTherefore we conclude.\n\n\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm5.1}(2)]\nAccording to Lemmas \\ref{step3} and \\ref{step4}, Remark \\ref{rmk4.16}, Proposition \\ref{step5} and \\cite[Prop. 4.7]{CM20}, there exists a positive Lipschitz function $\\omega$, which is independent of the choice of the base point $x\\in X$, defined on $(0,D)$ such that for any $r\\in (0,D)$ it holds that\n\\[\n\\mathrm{d}\\mathscr{H}^{n-1}\\llcorner_{\\partial B_r(x)}=\\omega(r)\\,\\mathrm{d}\\mathfrak{q}_x.\n\\]\n\n\n\nLet us set\n\\[\n\\begin{aligned}\nR_x:\\mathrm{Lip}(X,\\mathsf{d})&\\longrightarrow C([0,D])\\\\\nnt_{\\partial B_r(x)}f\\,\\mathrm{d}\\mathscr{H}^{n-1}\\right),\n\\end{aligned}\n\\]\nand\n\\[\n\\begin{aligned}\nA_x:C([0,D])&\\longrightarrow \\mathrm{Lip}(X,\\mathsf{d})\\\\\nf&\\longmapsto \\left(y\\mapsto f(\\mathsf{d}(x,y))\\right).\n\\end{aligned}\n\\]\nWe claim that $\\tilde{\\rho}:(x,y,t)\\mapsto R_x A_x \\rho(x,\\cdot,t)(y)$ is also a heat kernel. \n\nFirst we observe that for any test function $f$, since $\\int_{\\partial B_{D/2}(x)}\\,\\mathrm{d}\\mathfrak{q}_x(\\alpha)=1$, we have\n\\[\nnt_{B_{\\mathsf{d}(x,y)}(x)} f \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}} = \\int_{\\partial B_{D/2}(x)} f(\\alpha,\\mathsf{d}(x,y))\\,\\mathrm{d}\\mathfrak{q}_x(\\alpha),\n\\]\nwhich implies that $R_x A_x f\\in \\mathrm{Lip}(X,\\mathsf{d})$. Moreover, \nfor any test function $\\varphi$, applying the co-area formula with base point $x$ and the Gauss-Green formula in \\cite{BPS23} shows\n\\[\n\\begin{aligned}\n&\\int_X \\Delta \\varphi \\, R_x A_x f\\mathop{\\mathrm{d}\\mathscr{H}^{n}}=\\int_X \\langle\\nabla \\varphi,\\nabla \\mathsf{d}_x\\rangle (A_x f)'(\\mathsf{d}_x)\\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\\\\n=&\\int_0^D \\omega(r)\\left[\\int_{B_{D/2}(x)}\\frac{\\partial}{\\partial r} f(\\alpha,r)\\mathop{\\mathrm{d}\\mathfrak{q}_x(\\alpha)} \\int_{B_{D/2}(x)}\\frac{\\partial}{\\partial r}\\varphi(\\alpha,r)\\mathop{\\mathrm{d}\\mathfrak{q}_x(\\alpha)}\\right]\\mathop{\\mathrm{d}r}\\\\\n=&\\int_0^D \\left[\\int_{\\partial B_r(x)}\\langle\\nabla f,\\nabla \\mathsf{d}_x\\rangle\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}} \\frac{\\partial}{\\partial r}\\left(\\int_{B_{D/2}(x)}\\varphi(\\alpha,r)\\mathop{\\mathrm{d}\\mathfrak{q}_x(\\alpha)}\\right)\\right]\\mathop{\\mathrm{d}r}\\\\\n=&\\int_0^D \\left[\\int_{B_r(x)} \\Delta f\\mathop{\\mathrm{d}\\mathscr{H}^{n}} \\frac{\\partial}{\\partial r}\\left(\\int_{B_{D/2}(x)}\\varphi(\\alpha,r)\\mathop{\\mathrm{d}\\mathfrak{q}_x(\\alpha)}\\right)\\right]\\mathop{\\mathrm{d}r}\\\\\n=&-\\int_0^D \\left[\\frac{\\partial}{\\partial r}\\left(\\int_{B_r(x)} \\Delta f\\mathop{\\mathrm{d}\\mathscr{H}^{n}}\\right)\\int_{B_{D/2}(x)}\\varphi(\\alpha,r)\\mathop{\\mathrm{d}\\mathfrak{q}_x(\\alpha)}\\right]\\mathop{\\mathrm{d}r}\\\\\n=&-\\int_0^D \\left[\\int_{\\partial B_r(x)} \\Delta f\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}}\\int_{B_{D/2}(x)}\\varphi(\\alpha,r)\\mathop{\\mathrm{d}\\mathfrak{q}_x(\\alpha)}\\right]\\mathop{\\mathrm{d}r}=\\int_X \\varphi\\, R_x A_x \\Delta f\\mathop{\\mathrm{d}\\mathscr{H}^{n}}.\n\\end{aligned}\n\\] \n\nRecall that the set of test functions is dense in $H^{1,2}$. Therefore, we deduce that $R_x A_x f\\in D(\\Delta)$ and $\\Delta R_x A_x f=R_x A_x \\Delta f$. This finding verifies that\n\n\\begin{equation}\\label{4.34}\n\\frac{\\partial}{\\partial t} \\tilde{\\rho}(x,\\cdot,t)=\\frac{\\partial}{\\partial t}R_x A_x\\rho(x,\\cdot,t)=R_x A_x\\frac{\\partial}{\\partial t}\\rho(x,\\cdot,t)=R_x A_x \\Delta \\rho(x,\\cdot,t)=\\Delta \\tilde{\\rho}(x,\\cdot,t).\n\\end{equation}\nEspecially for any $f\\in \\mathrm{Lip}(X,\\mathsf{d})$, let \n\\[\n \\tilde{\\mathrm{h}}_t f:x\\mapsto \\int_X \\tilde{\\rho}(y,x,t)f(y)\\mathop{\\mathrm{d}\\mathscr{H}^n(y)},\n\\]\nwe will show that \n\\begin{equation}\\label{4.32}\n\\left\\| \\tilde{\\mathrm{h}}_t f-f\\right\\|_{L^2}\\rightarrow 0\\ \\ \\text{ as $t\\rightarrow 0$}.\n\\end{equation}\nThe first step is to show that \n\\begin{equation}\\label{4.31}\n \\left\\|\\int_X \\varphi \\, \\tilde{\\mathrm{h}}_t f \\mathop{\\mathrm{d}\\mathscr{H}^n}-\\int_X \\varphi f \\mathop{\\mathrm{d}\\mathscr{H}^n}\\right\\|_{L^2}\\rightarrow 0\\ \\ \\text{as }\\ t\\rightarrow 0,\\ \\forall \\varphi\\in L^2.\n\\end{equation}\n For any $y\\in X$, it is clear that $\\mathrm{Lip}(R_y A_y f)\\leqslant \\mathrm{Lip}\\,f$. Then the Bakry-\\'Emery estimate (see for instance \\cite[Thm. 6.1.4]{G18a}) implies $\\left\\|\\left|\\nabla \\mathrm{h}_t (R_y A_y f)\\right|\\right\\|_{L^\\infty}\\leqslant \\exp(-Kt)\\left\\| \\mathrm{h}_t(|\\nabla R_y A_y f|^2)\\right\\|_{L^\\infty}\\leqslant \\exp(-Kt)\\,\\mathrm{Lip}\\,f$. As a result, the $L^2$-convergence $\\mathrm{h}_t (R_y A_y f)\\rightarrow R_y A_y f$ can be improved to uniform convergence. In particular, we have $(\\mathrm{h}_t (R_y A_y f))(y)\\rightarrow (R_y A_y f)(y)=\\lim_{z\\rightarrow y} (R_y A_y f)(z)=f(y)$.\n\nAccording to Fubini's theorem,\n\\[\n\\begin{aligned}\n\\int_X \\varphi \\, \\tilde{\\mathrm{h}}_t f \\mathop{\\mathrm{d}\\mathscr{H}^n}&=\\int_X\\int_X \\varphi(y) R_y A_y\\rho(y,\\cdot,t)(x)f(x) \\mathop{\\mathrm{d}\\mathscr{H}^n(x)}\\mathop{\\mathrm{d}\\mathscr{H}^n(y)}\\\\\nnt_{\\partial B_r(y)} \\rho(y,z,t)\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}(z)}\\int_{\\partial B_r(y)}f(x) \\mathop{\\mathrm{d}\\mathscr{H}^{n-1}(x)}\\mathop{\\mathrm{d}\\mathscr{H}^n(y)}\\\\\n&=\\int_X \\varphi(y) \\int_0^D\\int_{\\partial B_r(y)} \\rho(y,z,t)(R_y A_y f)(z)\\mathop{\\mathrm{d}\\mathscr{H}^{n-1}(z)}\\mathop{\\mathrm{d}\\mathscr{H}^n(y)}\\\\\n&=\\int_X \\varphi(y) (\\mathrm{h}_t (R_y A_y f))(y)\\mathop{\\mathrm{d}\\mathscr{H}^n(y)}.\n\\end{aligned}\n\\]\nThus (\\ref{4.31}) is derived from the observation that $\\int_X (\\mathrm{h}_t (R_y A_y f))^2(y)\\mathop{\\mathrm{d}\\mathscr{H}^n(y)}\\leqslant \\mathscr{H}^n(X)(\\sup_X f)^2<\\infty$, combined with the application of dominated convergence theorem. In particular, one can prove (\\ref{4.32}) through the following calculation.\n\\[\n\\begin{aligned}\n\\int_X (\\tilde{\\mathrm{h}}_t f)^2 \\mathop{\\mathrm{d}\\mathscr{H}^n}&=\\int_X \\tilde{\\mathrm{h}}_t f(y) (\\mathrm{h}_t (R_y A_y f))(y)\\mathop{\\mathrm{d}\\mathscr{H}^n(y)}\\\\\n&=\\int_X (\\mathrm{h}_t (R_y A_y f))^2(y)\\mathop{\\mathrm{d}\\mathscr{H}^n(y)}\\rightarrow \\int_X f^2\\mathop{\\mathrm{d}\\mathscr{H}^n},\\ \\ \\text{as}\\ t\\rightarrow 0.\n\\end{aligned}\n\\]\nDue to (\\ref{4.34}), (\\ref{4.32}) and the uniqueness of the solution to the heat equation, for any $s,t>0$, since $\\mathrm{h}_s f$ is a test function, it follows that $\\tilde{\\mathrm{h}}_t\\mathrm{h}_s f=\\mathrm{h}_{t+s}f$. Moreover, by letting $s\\rightarrow 0$ and using dominated convergence theorem again, we see $\\tilde{\\mathrm{h}}_t f=\\mathrm{h}_{t}f$. Ultimately, considering the density of the set of test functions in $H^{1,2}$ (and hence in $L^2$) we can conclude\n\\[\n\\rho(x,y,t)=\\tilde{\\rho}(y,x,t),\\ \\forall x,y\\in X,\\ \\forall t>0.\n\\] \nThis verifies the strong harmonicity of $(X,\\mathsf{d},\\mathscr{H}^n)$, which in fact, implies the smoothness due to Corollary \\ref{cor1.5}. \n\n\n\\end{proof}\n\n\n\\bibliographystyle{alpha}\n\\bibliography{reff}\n\n\n\n\\bigskip\n\\end{document}\n\n\n"}}},{"rowIdx":298,"cells":{"id":{"kind":"string","value":"2501.04026v2"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2501.04026v2"},"title":{"kind":"string","value":"Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I"},"full_text":{"kind":"string","value":"\\documentclass{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage[utf8]{inputenc}\n\\usepackage[utf8]{inputenc}\n\\usepackage{dirtytalk}\n\\usepackage{tikz}\n\\usepackage{geometry}\n \\geometry{\n a4paper,\n total={170mm,257mm},\n left=20mm,\n top=20mm,\n }\n\\usepackage{graphicx}\n\\usepackage{enumitem}\n\\usepackage{parskip}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{xcolor}\n\\usepackage{hyperref}\n\\newlist{myitemize}{itemize}{1}\n\\setlist[myitemize,1]{leftmargin = 0.5in}\n\\hypersetup{colorlinks = true, linkcolor = red, linktocpage}\n\\newcommand{\\eqname}[1]{\\tag*{#1}}\n\n\n\\theoremstyle{plain}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem*{thm*}{Theorem}\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{hyp}[thm]{Hypothesis}\n\\theoremstyle{definition}\n\\newtheorem{defn}[thm]{Definition}\n\\newtheorem{conj}[thm]{Conjecture}\n\\newtheorem{exmp}[thm]{Example}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem*{note}{Note}\n\\setlength{\\parindent}{1cm}\n\n\n\n\\title{\\textbf{\\small{COUNTING THE NUMBER OF INTEGRAL FIXED POINTS OF A DISCRETE DYNAMICAL SYSTEM WITH APPLICATIONS FROM ARITHMETIC STATISTICS, I}}}\n\\author{\\footnotesize{BRIAN KINTU}}\n\\date{\\small{\\textit{Happily Dedicated: Dept. of MCS \\& Presidency of Meric S. Gertler at the University of Toronto}}}\n\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\n\\small{In this article, we inspect a surprising relationship between the set of fixed points of a polynomial map $\\varphi_{d, c}$ defined by $\\varphi_{d, c}(z) = z^d + c$ for all $c, z \\in \\mathbb{Z}$ and the coefficient $c$, where $d > 2$ is an integer. Inspired greatly by the elegance of the counting problems along with the very striking results of Bhargava-Shankar-Tsimerman and their collaborators in arithmetic statistics and by an interesting point-counting result of Narkiewicz on rational periodic points of $\\varphi_{d, c}$ for any odd degree $d>2$ in arithmetic dynamics, we prove that for any given prime integer $p\\geq 3$, the average number of distinct integral fixed points of any $\\varphi_{p, c}$ modulo $p$ is 3 or 0, as $p$ tends to infinity. Motivated further by the same down-to-earth work of (BST) and by a conjecture of Hutz on rational periodic points of $\\varphi_{p-1, c}$ for any given prime integer $p\\geq 5$ in arithmetic dynamics, we then also prove unconditionally that the average number of distinct integral fixed points of any $\\varphi_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$, as $p$ tends to infinity. Moreover, we also show that a density of $0\\%$ of integer polynomials $\\varphi_{p, c}(x)$ have three fixed points modulo $p$, as $c$ tends to infinity; and also show this same density on integer polynomials $\\varphi_{p-1, c}(x)$ with one or two fixed points modulo $p$. Consequently, a density of $100\\%$ of integer polynomials $f(x): = x^p -x + c$ gives rise to odd prime degree-$p$ number fields $K_{f}:=\\mathbb{Q}[x]\\slash (f(x))$; and similarly a density of $100\\%$ of integer polynomials $g(x): = x^{p-1}-x +c$ induces even degree-$p-1$ number fields $L_{g}:=\\mathbb{Q}[x]\\slash (g(x))$. Since each of the fields $K_{f}$ and $L_{g}$ comes naturally with the ring $\\mathcal{O}_{K_{f}}$ and $\\mathcal{O}_{L_{g}}$ of integers, resp., applying a density result of Bhargava-Shankar-Wang on our integer irreducibles $f(x)$ and $g(x)$ yields that a density equal to $\\zeta(2)^{-1}$ of polynomials $f(x)$ is such that $\\mathbb{Z}[x]\\slash (f(x))$ is the ring $\\mathcal{O}_{K_{f}}$; and similarly a density equal to $\\zeta(2)^{-1}$ of polynomials $g(x)$ is such that $\\mathbb{Z}[x]\\slash (g(x))$ is the ring $\\mathcal{O}_{L_{g}}$ of integers.}\n\\end{abstract}\n\n\\begin{center}\n\\tableofcontents\n\\end{center}\n\\begin{center}\n \\section{Introduction}\n\\end{center}\n\\noindent\nConsider any morphism $\\varphi: {\\mathbb{P}^N(K)} \\rightarrow {\\mathbb{P}^N(K)} $ of degree $d \\geq 2$ defined on a projective space ${\\mathbb{P}^N(K)}$ of dimension $N$, where $K$ is a number field. Then for any $n\\in\\mathbb{Z}$ and $\\alpha\\in\\mathbb{P}^N(K)$, we call $\\varphi^n = \\underbrace{\\varphi \\circ \\varphi \\circ \\cdots \\circ \\varphi}_\\text{$n$ times}$ the $n^{th}$ \\textit{iterate of $\\varphi$} and call $\\varphi^n(\\alpha)$ the \\textit{$n^{th}$ iteration of $\\varphi$ on $\\alpha$}. By convention, $\\varphi^{0}$ acts as the identity map, i.e., $\\varphi^{0}(\\alpha) = \\alpha$ for every point $\\alpha\\in {\\mathbb{P}^N(K)}$. The everyday philosopher may want to know (quoting here Devaney \\cite{Dev}): \\say{\\textit{Where do points $\\alpha, \\varphi(\\alpha), \\varphi^2(\\alpha), \\ \\cdots\\ ,\\varphi^n(\\alpha)$ go as $n$ becomes large, and what do they do when they get there?}} So now, for any given integer $n\\geq 0$ and any given point $\\alpha\\in {\\mathbb{P}^N(K)}$, we then call the set consisting of all the iterates $\\varphi^n(\\alpha)$ the \\textit{(forward) orbit of $\\alpha$}; and which in the theory of dynamical systems we usually denote by:\n\\begin{equation}\n\\mathcal{O}^{+}(\\alpha):= \\{\\varphi^n(\\alpha) : n \\in \\mathbb{Z}_{\\geq 0} \\}.\n\\end{equation}\n\nOne of the principal goals in the area of \\say{arithmetic dynamics}, a newly emerging area of mathematics concerned with studying number-theoretic properties of discrete dynamical systems, is to classify all the points $\\alpha\\in\\mathbb{P}^N(K)$ according to the behavior of their forward orbits $\\mathcal{O}^{+}(\\alpha)$. In this direction, we call any point $\\alpha\\in {\\mathbb{P}^N(K)}$ a \\textit{periodic point of $\\varphi$}, whenever $\\varphi^n (\\alpha) = \\alpha$ for some $n\\in \\mathbb{Z}_{\\geq 0}$; and call any integer $n\\geq 0$ such that $\\varphi^n (\\alpha) = \\alpha$ a \\textit{period of $\\alpha$}, and the smallest such positive integer $n\\geq 1$ is then called the \\textit{exact period of $\\alpha$}. We denote the set of all periodic points of $\\varphi$ by Per$(\\varphi, {\\mathbb{P}^N(K)})$; and for any given point $\\alpha\\in$Per$(\\varphi, {\\mathbb{P}^N(K)})$ we then call the set of all iterates of $\\varphi$ on $\\alpha$, \\textit{a periodic orbit of $\\alpha$}. In Sect. \\ref{sec4}, \\ref{sec5}, \\ref{sec6}, \\ref{sec7} and \\ref{sec8}, we study counting questions that are greatly inspired by all the beautiful work of Bhargava-Shankar-Tsimerman and their collaborators in the area of \\say{arithmetic statistics}, an area of mathematics concerned with studying the distributions of arithmetic objects (quantities); and among such questions includes the natural question: \\say{\\textit{How many distinct fixed orbits can any $\\varphi_{p,c}$ and $\\varphi_{p-1,c}$ acting independently on the space $\\mathbb{Z} / p\\mathbb{Z}$ via iteration have on average, as $p\\to \\infty$?}} In doing so, we first prove that conditioning on a theorem of Narkiewicz \\ref{theorem 3.2.1} yields the following main theorem on maps $\\varphi_{p,c}$ for any given prime $p\\geq 3$; and which we state later more precisely as Theorem \\ref{2.2}:\n\n\\begin{thm}\\label{Binder-Brian1}\nLet $p\\geq 3$ be any fixed prime integer, and assume Theorem \\ref{theorem 3.2.1}. Let $\\varphi_{p, c}$ be a map defined by $\\varphi_{p, c}(z) = z^p+c$ for all $c, z\\in\\mathbb{Z}$. The number of distinct integral fixed points of any $\\varphi_{p,c}$ modulo $p$ is $3$ or $0$.\n\\end{thm}\n\n\nInspired further by the activity and by all down-to-earth work of Bhargava-Shankar-Tsimerman in arithmetic statistics, and by an intriguing unresolved conjecture of Hutz \\ref{conjecture 3.2.1} on rational periodic points of maps $\\varphi_{d, c}$ of any even degree $d> 2$ (though not more importantly in our case attempting to prove his Conjecture \\ref{conjecture 3.2.1}) and by recent work of Panraksa \\cite{par2} in arithmetic dynamics, we revisit the setting in Sect. \\ref{sec2} and consider in Sect. \\ref{sec3} any $\\varphi_{p-1,c}$ of any even degree $p-1$ iterated on $\\mathbb{Z}\\slash p\\mathbb{Z}$ for any prime integer $p\\geq 5$. In doing so, we prove unconditionally the following main theorem on any $\\varphi_{p-1,c}$; which we state more precisely as Theorem \\ref{6.0.2}:\n\n\\begin{thm}\\label{Binder-Brian2}\nLet $p\\geq 5$ be any fixed prime integer, and let $\\varphi_{p-1, c}$ be a map defined by $\\varphi_{p-1, c}(z) = z^{p-1}+c$ for all $c, z\\in\\mathbb{Z}$. Then the number of distinct integral fixed points of any map $\\varphi_{p-1,c}$ modulo $p$ is $1$ or $2$ or $0$.\n\\end{thm}\n\n\\noindent Notice that the obtained count in Thm \\ref{Binder-Brian2} on the number of distinct integral fixed points of any $\\varphi_{p-1,c}$ modulo $p$ is independent of $p$ (and hence independent of the degree of $\\varphi_{p-1,c})$ in each of the three possibilities. Moreover, we may also observe that the expected total count (namely, $1+2+0 =3$) in Thm \\ref{Binder-Brian2} on the number of distinct integral fixed points in the whole family of maps $\\varphi_{p-1,c}$ modulo $p$ is also independent of $p$ and deg$(\\varphi_{p-1,c})$; an observation which somewhat surprisingly coincides not only with a similar observation on the count in each of the two possibilities in Thm \\ref{Binder-Brian1} but also coincides with a similar observation on the expected total count (namely, $3+0 =3$) on the number of distinct integral fixed points in the whole family of maps $\\varphi_{p,c}$ modulo $p$. \n\nSince we know the inclusion $\\mathbb{Z}\\hookrightarrow\\mathbb{Z}_{p}$ of rings, and so the space $\\mathbb{Z}_{p}$ of all $p$-adic integers is evidently a much larger space than $\\mathbb{Z}$. So then, inspired by the work of Adam-Fares \\cite{Ada} in arithmetic dynamics and again by a\\say{counting-application} philosophy in arithmetic statistics, we again inspect in a paper \\cite{BK3} the aforementioned relationship where it is $\\mathbb{Z}_{p}$ that's considered. Interestingly, we again obtain the same counting and asymptotics in the setting when $\\varphi_{p-1,c}$ is iterated on $\\mathbb{Z}_{p}\\slash p\\mathbb{Z}_{p}$, and get very different counting and asymptotics in the case when $\\varphi_{p,c}$ iterated on $\\mathbb{Z}_{p}\\slash p\\mathbb{Z}_{p}$. Motivated by a $K$-rational periodic point-counting result of Narkiewicz \\cite{Narkie} on maps $\\varphi_{p,c}$ defined over any real algebraic number field $K$ of degree $n\\geq 2$, in a forthcoming work \\cite{BK2}, we revisit the counting setting in Section \\ref{sec2} and then consider any $\\varphi_{p^{\\ell},c}$ defined over any real algebraic number $K\\slash \\mathbb{Q}$ of any degree $n\\geq 2$ for any prime $p\\geq 3$ and any integer $\\ell \\geq 1$. In doing so, we show that conditioning on Narkiewicz' theorem and again using the same elementary counting technique, we can obtain a fixed integral point-counting result that is not only independent of the degree $n$ of any real $K$ and any degree $p^{\\ell}$ of any map $\\varphi_{p^{\\ell},c}$, but is also very analogous to Theorem \\ref{Binder-Brian1} for any real number field $K$, any odd prime $p$ and any integer $\\ell$. More in that work \\cite{BK2}, we again revisit the counting setting in Section \\ref{sec3} and then consider any polynomial map $\\varphi_{(p-1)^{\\ell},c}$ defined over any algebraic number $K\\slash \\mathbb{Q}$ (where $K$ needn't be a real algebraic number field) of any degree $n\\geq 2$, for any prime integer $p\\geq 5$ and any integer $\\ell \\geq 1$. In doing so, we find that we can again obtain a fixed integral point-counting result that's not only independent of both the degree $n$ of any number field $K$ and any prime $p$, but is also very similar to Theorem \\ref{Binder-Brian2} for any given algebraic number field $K$, any prime $p$ and any integer $\\ell$. \n\n \nIt's worth mentioning that there are several authors in the literature who have also done an explicit study on orbits; and among such authors includes, Silverman \\cite{Sil} on the number of integral points in forward orbits of rational functions of degree $\\geq 2$ defined over the field $\\mathbb{C}$, and Wade \\cite{Wade} on the average number of integral points in forward orbits of rational function of degree $\\geq 2$ defined over $K$. In his 1993 beautiful paper \\cite{Sil}, Silverman showed [\\cite{Sil}, Theorem A] that the forward orbit of a rational function of degree $\\geq 2$ whose second iterate is not a polynomial over $\\mathbb{C}$, contains only finitely many integer points. Motivated by a point-counting result of Silverman \\cite{Sil} and conditioning on a standard height uniform conjecture in arithmetic geometry, Wade \\cite{Wade} established a zero-average result on the number of integral points in the forward orbit of a rational function of degree $\\geq 2$ defined over $K$. Now observe that in each of the works \\cite{Sil} and \\cite{Wade}, the focus is on understanding very well integral interiority of the forward orbits. In our case, the focus is on understanding a somewhat less thorny problem, namely, as we see from the above (and as we also did in \\cite{BK0} where we considered any arbitrary family of quadratic maps $\\varphi_{2,c}$ iterating on $\\mathbb{Z}\\slash p\\mathbb{Z}$ for any prime $p\\geq 2$): the problem of counting fixed orbits (on average), and consequently hope to understand the statistical behavior along with the question of the meaning, of the achieved count. Such a problem may supposedly be of some interesting insight in the area of classical dynamical systems, since one of the main objectives in that area is to understand \\textit{all} orbits via topological and analytic techniques, and in doing so, one may not only find that orbits can easily get very complicated but also interesting yet also important statistical questions concerning measuring the complexity of a dynamical system, in particular, the question of determining the topological entropy of a given system (which the reader need not worry about at all, since we won't be inspecting such a question in this article), may become intractable. \n\n\\begin{rem}\nLoosely speaking, recall that \\say{topological entropy} is a nonnegative statistic that gives a way of measuring the exponential growth rate of distinguishable orbits of a dynamical system as time progresses. You may see independently in more great details about topological entropy in the work of Adler \\cite{Adl} and Bowen \\cite{Ruf}.\n\\end{rem}Now before we proceed any further in our discussion, let's first take a very quick look at the following classical example of a polynomial morphism with a rational periodic point and a rational periodic orbit: \n\n\\begin{exmp} \\label{ex 1.1}\nConsider $\\varphi_{2, -21/16}$ defined by $\\varphi_{2, -21/16}(z) = z^2 -21/16$ for all $c,z \\in\\mathbb{Q}$. Then if we compute all iterations of $\\varphi_{2, -21/16}$ on $z = 1/4$, we get that $\\mathcal{O}^{+}(1/4)$ consists of points $\\varphi^0(1/4) = 1/4$, $\\varphi^1(1/4) = -5/4$, $\\varphi^2(1/4) = 1/4$ and so on; and moreover the points in $\\mathcal{O}^{+}(1/4)$ form a sequence of the following form:\n\\begin{center}\n$\\{1/4 \\longrightarrow -5/4 \\} \\longrightarrow \\{1/4 \\longrightarrow -5/4\\}\\longrightarrow \\{1/4 \\longrightarrow -5/4\\} \\longrightarrow \\cdots$ \\end{center}So in this example, $z = 1/4$ is a periodic point of a map $\\varphi_{2, -21/16}$ with exact period 2 and from the above definition, $\\mathcal{O}^{+}(1/4)$ is a rational periodic orbit (i.e., $\\mathcal{O}^{+}(1/4)$ is a periodic orbit consisting of rational points).\n\\end{exmp} \n\nIn addition to the notion of a periodic point and a periodic orbit, we also have in dynamical systems a more complicated but somewhat related notion of a preperiodic point and a preperiodic orbit. We call a point $\\alpha\\in {\\mathbb{P}^N(K)}$ a \\textit{preperiodic point of $\\varphi$}, whenever $\\varphi^{m+n}(\\alpha) = \\varphi^{m}(\\alpha)$ for some integers $m\\geq 0$ and $n\\geq 1$. In this case, the smallest integers $m\\geq 0$ and $n\\geq 1$ such that the equation $\\varphi^{m+n}(\\alpha) = \\varphi^{m}(\\alpha)$ happens, are called the \\textit{preperiod} and \\textit{eventual period of $\\alpha$}, respectively. Again, we denote the set of preperiodic points of $\\varphi$ by PrePer$(\\varphi, {\\mathbb{P}^N(K)})$. For any given preperiodic point $\\alpha$ of $\\varphi$, we then call the set of all iterates of $\\varphi$ on $\\alpha$, \\textit{the preperiodic orbit of $\\alpha$}.\nNow observe for $m=0$, we have $\\varphi^{n}(\\alpha) = \\alpha $ and so $\\alpha$ is a periodic point of period $n$. Hence, every periodic point of $\\varphi$ is also a preperiodic point of $\\varphi$, i.e., Per$(\\varphi, {\\mathbb{P}^N(K)}) \\subseteq$ PrePer$(\\varphi, {\\mathbb{P}^N(K)})$; however, it need not be that PrePer$(\\varphi, {\\mathbb{P}^N(K)})\\subseteq$ Per$(\\varphi, {\\mathbb{P}^N(K)})$ as illustrated by the following classical example: \n\\begin{exmp}\nConsider $\\varphi_{2, -29/16}$ defined by $\\varphi_{2, -29/16}(z) = z^2 -29/16$ for all $c,z\\in \\mathbb{Q}$. Then if we compute all iterations of $\\varphi_{2, -29/16}$ on $z = 3/4$, we get that $\\mathcal{O}^{+}(3/4)$ consists of points $\\varphi^0(3/4) = 3/4$, $\\varphi^1(3/4) = -5/4$, $\\varphi^2(3/4) = -1/4$ and so on; and moreover the points in $\\mathcal{O}^{+}(3/4)$ form a sequence of the following form: \n\\begin{center}\n$3/4 \\longrightarrow -5/4 \\longrightarrow \\{-1/4 \\longrightarrow -7/4 \\longrightarrow 5/4\\} \\longrightarrow \\{-1/4 \\longrightarrow -7/4 \\longrightarrow 5/4\\}\\longrightarrow \\cdots$ \n\\end{center}\nSo in this example, we conclude that $z = 3/4$ is a rational preperiodic point of $\\varphi_{2, -29/16}$ with preperiod $m=2$ and eventual period $n=3$; and $\\mathcal{O}^{+}(3/4)$ is a preperiodic orbit. Also, since preperiod $m= 2 \\neq 0$, then $z = 3/4$ is not a periodic point. As another example of a preperiodic point, recall in Example \\ref{ex 1.1} that $z = 1/4$ is a periodic point of $\\varphi_{2, -21/16}$, and since every periodic point is a preperiodic point, then $z = 1/4$ is a $\\mathbb{Q}$-preperiodic point of $\\varphi_{2, -21/16}$. Other interesting examples of $\\mathbb{Q}$-preperiodic points may be found in Poonen's work \\cite{Poonen}.\n\\end{exmp}\n\nIn the year 1950, Northcott \\cite{North} used the theory of height functions to show that not only is the set PrePer$(\\varphi, {\\mathbb{P}^N(K)})$ always finite, but also for a given morphism $\\varphi$ the set PrePer$(\\varphi, {\\mathbb{P}^N(K)})$ can be computed effectively. Forty-five years later, in the year 1995, Morton and Silverman conjectured that PrePer$(\\varphi, \\mathbb{P}^N(K))$ can be bounded in terms of degree $d$ of $\\varphi$, degree $D$ of $K$, and dimension $N$ of the space ${\\mathbb{P}^N(K)}$. This celebrated conjecture is called the \\textit{Uniform Boundedness Conjecture}; which we then restate here as the following conjecture:\n\n\\begin{conj} \\label{silver-morton}[\\cite{Morton}]\nFix integers $D \\geq 1$, $N \\geq 1$, and $d \\geq 2$. There exists a constant $C'= C'(D, N, d)$ such that for all number fields $K/{\\mathbb{Q}}$ of degree at most $D$, and all morphisms $\\varphi: {\\mathbb{P}^N}(K) \\rightarrow {\\mathbb{P}^N}(K)$ of degree $d$ defined over $K$, the total number of preperiodic points of a morphism $\\varphi$ is at most $C'$, i.e., \\#PrePer$(\\varphi, \\mathbb{P}^N(K)) \\leq C'$.\n\\end{conj}\n\\noindent Note that a special case of Conjecture \\ref{silver-morton} is when the degree $D$ of a number field $K$ is $D = 1$, dimension $N$ of a space $\\mathbb{P}^N(K)$ is $N = 1$, and degree $d$ of a morphism $\\varphi$ is $d = 2$. In this case, if $\\varphi$ is a polynomial morphism, then it is a quadratic map defined over the field $\\mathbb{Q}$. Moreover, in this very special case, in the year 1995, Flynn and Poonen and Schaefer conjectured that a quadratic map has no points $z\\in\\mathbb{Q}$ with exact period more than 3. This conjecture of Flynn-Poonen-Schaefer \\cite{Flynn} (which has been resolved for cases $n = 4$, $5$ in \\cite{mor, Flynn} respectively and conditionally for $n=6$ in \\cite{Stoll} is, however, still open for all cases $n\\geq 7$ and moreover, which also Hutz-Ingram \\cite{Ingram} gave strong computational evidence supporting it) is restated here formally as the following conjecture. Note that in this same special case, rational points of exact period $n\\in \\{1, 2, 3\\}$ were first found in the year 1994 by Russo-Walde \\cite{Russo} and also found in the year 1995 by Poonen \\cite{Poonen} using a different set of techniques. We restate here the anticipated conjecture of Flynn-Poonen-Schaefer as the following conjecture: \n \n\\begin{conj} \\label{conj:2.4.1}[\\cite{Flynn}, Conjecture 2]\nIf $n \\geq 4$, then there is no quadratic polynomial $\\varphi_{2,c }(z) = z^2 + c\\in \\mathbb{Q}[z]$ with a rational point of exact period $n$.\n\\end{conj}\nNow by assuming Conjecture \\ref{conj:2.4.1} and also establishing interesting results on preperiodic points, in the year 1998, Poonen \\cite{Poonen} then concluded that the total number of rational preperiodic points of any quadratic polynomial $\\varphi_{2, c}(z)$ is at most nine. We restate here formally Poonen's result as the following corollary:\n\\begin{cor}\\label{cor2}[\\cite{Poonen}, Corollary 1]\nIf Conjecture \\ref{conj:2.4.1} holds, then $\\#$PrePer$(\\varphi_{2,c}, \\mathbb{Q}) \\leq 9$, for all quadratic maps $\\varphi_{2, c}$ defined by $\\varphi_{2, c}(z) = z^2 + c$ for all points $c, z\\in\\mathbb{Q}$.\n\\end{cor}\n\nSince Per$(\\varphi, {\\mathbb{P}^N(K)}) \\subseteq$ PrePer$(\\varphi, {\\mathbb{P}^N(K)})$ and so if the size of PrePer$(\\varphi, \\mathbb{P}^N(K))$ is bounded above, then the size of Per$(\\varphi, \\mathbb{P}^N(K))$ is also bounded above and moreover bounded above by the same upper bound. So then, we may extract out the following periodic version of Conjecture \\ref{silver-morton}, and the reason we do so, is because in Sections \\ref{sec2} and \\ref{sec3} we study a dynamical setting in which $K$ is replaced with $\\mathbb{Z}$, $N=1$ and degree $d=p$ for any given prime integer $p> 2$ and $d = p-1$ for any any given prime $p>3$, respectively; in the attempt of understanding (and not claiming to prove) the possibility and validity of a periodic version of Conjecture \\ref{silver-morton}.\n\n\\subsection*{History on the Connection Between the Size of Per$(\\varphi_{d, c}, K)$ and the Coefficient $c$}\n\nIn the year 1994, Walde and Russo not only proved [\\cite{Russo}, Corollary 4] that for a quadratic map $\\varphi_{2,c}$ defined over $\\mathbb{Q}$ with a periodic point, the denominator of a rational point $c$, denoted as den$(c)$, is a square but they also proved that den$(c)$ is even, whenever $\\varphi_{2,c}$ admits a rational cycle of length $\\ell \\geq 3$. Moreover, Walde-Russo also proved [\\cite{Russo}, Cor. 6, Thm 8 and Cor. 7] that the size \\#Per$(\\varphi_{2, c}, \\mathbb{Q})\\leq 2$, whenever den$(c)$ is an odd integer. \n\nThree years later, in the year 1997, Call-Goldstine \\cite{Call} proved that the size of the set PrePer$(\\varphi_{2,c},\\mathbb{Q})$ of rational preperiodic points of a quadratic map $\\varphi_{2, c}$ can be bounded above in terms of the number of distinct odd primes dividing the denominator of a point $c\\in\\mathbb{Q}$. We restate here formally this result of Call and Goldstine as the following theorem. Note that in this theorem, $GCD(a, e)$ refers to the greatest common divisor of $a$, $e \\in \\mathbb{Z}$:\n\n\\begin{thm}\\label{2.3.1}[\\cite{Call}, Theorem 6.9]\nLet $e>0$ be an integer and let $s$ be the number of distinct odd prime factors of e. Define $\\varepsilon = 0$, $1$, $2$, if $4\\nmid e$, if $4\\mid e$ and $8 \\nmid e$, if $8 \\mid e$, respectively. Let $c = a/e^2$, where $a\\in \\mathbb{Z}$ and $GCD(a, e) = 1$. If $c \\neq -2$, then the total number of $\\mathbb{Q}$-preperiodic points of $\\varphi_{2, c}$ is at most $2^{s + 2 + \\varepsilon} + 1$. Moreover, a quadratic map $\\varphi_{2, -2}$ has exactly six rational preperiodic points.\n\\end{thm}\n\\noindent\nNow recall that the set Per$(\\varphi_{2,c}, \\mathbb{Q})$ is always a subset of PrePer$(\\varphi_{2,c}, \\mathbb{Q})$, and since we now know from Theorem \\ref{2.3.1} that in the case of quadratic maps $\\varphi_{2,c}$ defined over $\\mathbb{Q}$, Call and Goldstine have shown that the maximum number of elements in PrePer$(\\varphi_{2,c}, \\mathbb{Q})$ is at most $2^{s + 2 + \\varepsilon} + 1$ for any rational point $c\\neq -2$. Hence, for any $c\\neq -2$, the size of Per$(\\varphi_{2,c}, \\mathbb{Q})$ is equal to $2^{s + 2 + \\varepsilon} + 1$ or is strictly less than $2^{s + 2 + \\varepsilon} + 1$; and so for any $c\\neq -2$, the size of Per$(\\varphi_{2,c}, \\mathbb{Q})$ is bounded above by a constant $2^{s + 2 + \\varepsilon} + 1$ depending only on the number of distinct odd prime factors of den$(c)$ and on the quantity $\\varepsilon$. On the other hand, for $c = -2$ (in which case den$(c) = 1$ and so has no distinct prime factors), Theorem \\ref {2.3.1} tells us that \\#PrePer$(\\varphi_{2,-2}, \\mathbb{Q}) = 6$ and so \\#Per$(\\varphi_{2,-2}, \\mathbb{Q}) \\leq 6$. \n\nEight years later, after the work of Call-Goldstine, in the year 2005, Benedetto \\cite{detto} conducted a detailed analysis of the filled Julia set and then improved substantially the bounds on the size of PrePre$(\\varphi, K)$ which had been established by earlier work of Call-Goldstine and earlier works of several other people in the literature. In particular, by working only with polynomial maps $\\varphi$ and only in dimension $N=1$, however, allowing arbitrary degree $d\\geq 2$ and an arbitrary global field $K$ (i.e., $K$ is a number field or a function field defined over a finite field), Benedetto established the following remarkable result on the relationship between the size of the set PrePre$(\\varphi, K)$ and the number of bad primes of $\\varphi$ in $K$ (note that when the morphism $\\varphi = \\varphi_{d,c}$ for any point $c\\in K$, then the \\textit{bad primes} of $\\varphi_{d,c}$ in $K$ are those primes that in fact divide the denominator of a $K$-point $c$):\n\n\\begin{thm}\\label{main} [\\cite{detto}, Main Theorem]\nLet $K$ be a global field, $\\varphi\\in K[z]$ be a polynomial of degree $d\\geq 2$ and $s$ be the number of bad primes of $\\varphi$ in $K$. The number of preperiodic points of $\\varphi$ in $\\mathbb{P}^N(K)$ is at most $O(\\text{s log s})$. \n\\end{thm}\n\n\\noindent\nSince from our earlier discussion we know that the set of periodic points of any given morphism $\\varphi$ is always a subset of the set of preperiodic points of $\\varphi$, so then, if $\\varphi$ is a polynomial morphism of degree $d\\geq 2$ defined over a global field $K$, then Thm \\ref{main} also shows that the total number of periodic points of the underlying morphism $\\varphi$ is in fact also $\\ll \\text{s log s}$, i.e., at most a positive constant times $\\text{s log s}$. Notice that setting $\\varphi(z) = \\varphi_{d, c}(z) = z^d + c\\in K[z]$ and $K = \\mathbb{Q}$, then by Thm \\ref{main} the size of Per$(\\varphi_{d, c}, \\mathbb{Q})$ is $\\ll \\text{s log s}$. More to this, since Thm \\ref{main} applies to any $\\varphi(z) = \\varphi_{d, c}(z)$ of arbitrary degree $d\\geq 2$ defined over $\\mathbb{Q}$, it then follows that Thm \\ref{main} can be applied to any $\\varphi_{d, c}(z)$ of arbitrary prime or even degree $d\\geq 2$; and as such one must then obtain the upper bound in Thm \\ref{main} on the number of $\\mathbb{Q}$-periodic points and hence on the number of $\\mathbb{Z}$-periodic points. Hence, we again we see from Thm \\ref{main} that the size of the set Per$(\\varphi_{d, c}, \\mathbb{Q})$ and thus of Per$(\\varphi_{d, c}, \\mathbb{Z})$, is bounded above using arithmetic information on the prime divisors of the coefficients of the polynomial $\\varphi_{d, c}(z)$. \n\nSeven years after the work of Benedetto, in the year 2012, Narkiewicz's work \\cite{Narkie} not only showed that any $\\varphi_{d,c}$ defined over $\\mathbb{Q}$ with odd degree $d\\geq 3$ has no rational periodic points of exact period $n > 1$, but his also showed that the total number of $\\mathbb{Q}$-preperiodic points is at most 4. We restate this result here as the following: \n\n\\begin{thm} \\label{theorem 3.2.1}\\cite{Narkie}\nFor any integer $n > 1$ and any odd integer $d\\geq 3$, there is no $c\\in \\mathbb{Q}$ such that $\\varphi_{d,c}$ defined by $\\varphi_{d, c}(z)$ for all $c,z \\in \\mathbb{Q}$ has rational periodic points of exact period $n$. Moreover, $\\#PrePer(\\varphi_{d, c}, \\mathbb{Q}) \\leq 4$. \n\\end{thm} \n\n\\begin{rem}\nThe first part of Thm \\ref{theorem 3.2.1} is proved by observing that for each odd degree $d\\geq 3$, the polynomial $\\varphi_{d, c}(z)$ is non-decreasing on $\\mathbb{Q}$ and so by elementary mathematical analysis one then expects the forward orbit $\\mathcal{O}_{\\varphi_{d,c}} (z)$ of each rational point $z$ to form a non-decreasing sequence of iterations. Hence, it is immediately evident that the polynomial $\\varphi_{d, c}(z)$ can only have rational points of exact period $n = 1$ (with no preperiod). The upper bound 4 is obtained by counting the number of rational roots of $\\varphi_{d, c}(z)-z$. Notice Thm \\ref{theorem 3.2.1} shows that the number of rational points $z$ that satisfy the equation $z^d - z + c = 0$ is bounded above by 4; which then also means that the number of such rational points and hence such $\\mathbb{Z}$-points is equal to 4 or strictly less than 4. \n\\end{rem}\n\nSeven years after the work of Benedetto, in the year 2012, Narkiewicz's work \\cite{Narkie} not only showed that any $\\varphi_{d,c}$ over $\\mathbb{Q}$ with odd degree $d\\geq 3$, has no $\\mathbb{Q}$-periodic points of exact period $n \\geq 2$, but also showed that the total number of preperiodic points is $\\leq 4$. Three years after the work \\cite{Narkie}, in the year 2015, Hutz \\cite{Hutz} developed an algorithm for determining all $\\mathbb{Q}$-preperiodic points of morphisms over ${\\mathbb{P}^N(K)}$, and then made the following:\n\n\\begin{conj} \\label{conjecture 3.2.1}[\\cite{Hutz}, Conjecture 1a]\nFor any integer $n > 2$, there is no even degree $d > 2$ and no point $c \\in \\mathbb{Q}$ such that the polynomial map $\\varphi_{d, c}$ has rational points of exact period $n$.\nMoreover, \\#PrePer$(\\varphi_{d, c}, \\mathbb{Q}) \\leq 4$. \n\\end{conj}\n\n\\noindent On the note whether any theoretical progress has yet been made on Conjecture \\ref{conjecture 3.2.1}, more recently, Panraksa \\cite{par2} proved among many other results that the quartic polynomial $\\varphi_{4,c}(z)\\in\\mathbb{Q}[z]$ has rational points of exact period $n = 2$. Moreover, he also proved that $\\varphi_{d,c}(z)\\in\\mathbb{Q}[z]$ has no rational points of exact period $n = 2$ for any $c \\in \\mathbb{Q}$ with $c \\neq -1$ and $d = 6$, $2k$ with $3 \\mid 2k-1$. The interested reader may find these mentioned results of Panraksa in his unconditional Thms 2.1, 2.4 and also see his Thm 1.7 conditioned on the abc-conjecture in \\cite{par2}.\n\n\nTwenty-eight years later, after the work of Walde-Russo, in the year 2022, Eliahou and Fares proved [\\cite{Shalom2}, Theorem 2.12] that the denominator of a rational point $-c$, denoted as den$(-c)$ is divisible by 16, whenever a quadratic map $\\varphi_{2,-c}$ defined by $\\varphi_{2, -c}(z) = z^2 - c$ for all points $c, z\\in \\mathbb{Q}$ admits a rational cycle of length $\\ell \\geq 3$. Moreover, Eliahou and Fares also proved [\\cite{Shalom2}, Proposition 2.8] that the size \\#Per$(\\varphi_{2, -c}, \\mathbb{Q})\\leq 2$, whenever den$(-c)$ is an odd integer. Motivated by also the work of Call-Goldstine, Eliahou-Fares \\cite{Shalom2} also proved that the size of the set Per$(\\varphi_{2, -c}, \\mathbb{Q})$ can be bounded above simply by using information on den$(-c)$, namely, information in terms of the number of distinct primes dividing den$(-c)$. Moreover, the authors \\cite{Shalom1} also showed that the upper bound is 4, whenever $c\\in \\mathbb{Q^*} = \\mathbb{Q}\\setminus\\{0\\}$. We restate here their results as the following:\n\n\\begin{cor}\\label{sha}[\\cite{Shalom2, Shalom1}, Corollary 3.11 and Corollary 4.4, resp.]\nLet $c\\in \\mathbb{Q}$ such that den$(c) = d^2$ with $d\\in 4 \\mathbb{N}$. Let $s$ be the number of distinct primes dividing $d$. Then, the total number of $\\mathbb{Q}$-periodic points of $\\varphi_{2, -c}$ is at most $2^s + 2$. Moreover, for $c\\in \\mathbb{Q^*}$ such that the den$(c)$ is a power of a prime number. Then, $\\#$Per$(\\varphi_{2, c}, \\mathbb{Q}) \\leq 4$.\n\\end{cor}\n\n\n\\begin{rem}\nSince every point in a given $\\mathbb{Q}$-periodic cycle of $\\varphi_{2,c}$ is again a $\\mathbb{Q}$-periodic point of $\\varphi_{2,c}$, then it seems somewhat reasonable to infer from the foregoing history that what Walde-Russo initially proved and later improved by Eliahou-Fares is, precisely the claim that den$(c)$ is divisible by 2, whenever $\\#$Per$(\\varphi_{2, c}, \\mathbb{Q})\\geq 3$ or $\\#$Per$(\\varphi_{2, -c}, \\mathbb{Q})\\geq 3$. On the other hand, what both Walde-Russo and Eliahou-Fares also proved is the claim that the number $\\#$Per$(\\varphi_{2, c}, \\mathbb{Q})\\leq 2$ or $\\#$Per$(\\varphi_{2, -c}, \\mathbb{Q})\\leq 2$, respectively, whenever den$(c)$ is an odd integer. So now, it's highly plausible to assert that what the authors in \\cite{Russo, Call, Shalom1, Shalom2} have studied is this very surprising relationship between the total number of elements in Per$(\\varphi_{2, c}, \\mathbb{Q})$ or Per$(\\varphi_{2, -c},\\mathbb{Q})$ and denominator of a generic coefficient $c$ or $-c$ of $\\varphi_{2, c}(z)$ or $\\varphi_{2, -c}(z)$, respectively. For $d\\geq 2$, we've seen that what the authors in \\cite{detto, Narkie, Hutz, par2} studied, is this same relationship between size of Per$(\\varphi_{d, c}, \\mathbb{Q})$ and the coefficients of $\\varphi_{d, c}(z)$.\n\\end{rem}\n\n\\noindent The purpose of this article is to again inspect further the above connection, independently in the case of polynomial maps $\\varphi_{p, c}$ of odd prime degree $p$ defined over $\\mathbb{Z}$ for any given prime integer $p\\geq 3$ and in the case of polynomial maps $\\varphi_{p-1, c}$ of even degree $p-1$ defined over $\\mathbb{Z}$ for any given prime integer $p\\geq 5$; and doing so from a spirit that's truly inspired and guided by some of the many striking developments in arithmetic statistics.\n\n\\section{On the Number of Integral Fixed Points of any Family of Polynomial Maps $\\varphi_{p,c}$}\\label{sec2}\n\nIn this section, we use very elementary number-theoretic facts to count the number of distinct integral fixed points of any integral polynomial map $\\varphi_{p,c}$, first by modulo 3 and then by modulo $p$ for any given prime integer $p\\geq 3$. For any given integer $c\\in \\mathbb{Z}$ and any given prime integer $p\\geq 3$, we then define a counting function \n\\begin{equation}\\label{N_{c}}\nN_{c}(p) := \\#\\{ z\\in \\mathbb{Z} / p\\mathbb{Z} : \\varphi_{p,c}(z) - z \\equiv 0 \\ \\text{(mod $p$)}\\}.\n\\end{equation}\\noindent We now first prove the following theorem on the number of distinct integral fixed points of any $\\varphi_{3, c}$ modulo $3$:\n\\begin{thm} \\label{2.1}\nGiven any family of cubic maps $\\varphi_{3, c}$ defined by $\\varphi_{3, c}(z) = z^3 + c$ for all $c, z\\in\\mathbb{Z}$, and let $N_{c}(3)$ be defined as in \\textnormal{(\\ref{N_{c}})}. Then for any coefficient $c = 3t$, we have $N_{c}(3) = 3$; otherwise, $N_{c}(3) = 0$ for every $c \\neq 3t$.\n\\end{thm}\n\\begin{proof}\nLet $f(z)= \\varphi_{3, c}(z)-z = z^3 - z + c$, and for every coefficient $c = 3t$, we then have $f(z)= z^3 - z + 3t$. So now, reducing both sides of $f(z)$ modulo 3, we have $f(z)\\equiv z^3 - z$ (mod 3). But now for any $z\\in \\mathbb{Z}$, we then by Fermat's Little Theorem obtain $z^3-z\\equiv 0$ (mod 3), which also means $z(z^2-1)\\equiv 0$ (mod 3). This also means $z\\equiv 0$ (mod 3) or $z^2 - 1\\equiv 0$ (mod 3), since $\\mathbb{Z} / 3\\mathbb{Z}$ is an integral domain. Moreover, by a standard fact in elementary number theory, $z^2 -1 \\equiv 0$ (mod 3) implies $z\\equiv \\pm 1$ (mod $3$). This then means $f(z)=0$ has exactly three unique solutions $z\\equiv -1, 0, 1$ (mod 3); and so every integer solution of $f(z)=0$ is of the form $z_{1} = 3r-1$, $z_{2} = 3s$ and $z_{3} = 3u+1$ (mod $3$) for some $r, s, u\\in \\mathbb{Z}$. Hence, $\\#\\{ z\\in \\mathbb{Z} / 3\\mathbb{Z} : \\varphi_{3,c}(z) - z \\equiv 0 \\text{ (mod 3)}\\} = 3$, and thus $N_{c}(3) = 3$. To see the second part, we first note that since the coefficient $c\\neq 3t$ for any integer $t$, then this means $c\\not \\equiv 0$ (mod $3$). But then, the cubic polynomial $z^3 - z + c\\not \\equiv 0$ (mod $3$) for every integral point $z\\in \\mathbb{Z} / 3\\mathbb{Z}$ and so $f(z)\\not \\equiv 0$ (mod 3) for every $z\\in \\mathbb{Z} / 3\\mathbb{Z}$. Hence, the cubic polynomial $\\varphi_{3,c}(x)-x$ has no roots in $\\mathbb{Z} / 3\\mathbb{Z}$ for every $c$ indivisible by $3$, and so the number $N_{c}(3) = 0$. This then completes the whole proof, as desired.\n\\end{proof} \nWe now generalize Theorem \\ref{2.1} to any map $\\varphi_{p, c}$ for any given prime integer $p\\geq 3$. That is, assuming Theorem \\ref{theorem 3.2.1}, we show that the number of distinct integral fixed points of any $\\varphi_{p, c}$ modulo $p$ is either 3 or 0:\n\n\\begin{thm} \\label{2.2}\nLet $p\\geq 3$ be any fixed prime integer and assume Theorem \\ref{theorem 3.2.1}. Consider any family of polynomial maps $\\varphi_{p, c}$ defined by $\\varphi_{p, c}(z) = z^p + c$ for all points $c, z\\in\\mathbb{Z}$, and let the number $N_{c}(p)$ be defined as in \\textnormal{(\\ref{N_{c}})}. Then for every coefficient $c = pt$, we have $N_{c}(p) = 3$; otherwise, we have $N_{c}(p) = 0$ for every $c \\neq pt$.\n\\end{thm}\n\\begin{proof}\nWe give a similar argument as in the foregoing proof. Again, let $f(z)= z^p - z + c$, and for every $c = pt$, we have $f(z)= z^p - z + pt$. Now if we reduce both sides of $f(z)$ modulo $p$, we have $f(z)\\equiv z^p - z$ (mod $p$). But now for any $z\\in \\mathbb{Z}$, we then know by Fermat's Little Theorem that $z^p-z\\equiv 0$ (mod $p$), which also means that $z(z^{p-1}-1)\\equiv 0$ (mod $p$). This also means that $z\\equiv 0$ (mod $p$) or $z^{p-1} - 1\\equiv 0$ (mod $p$), since $\\mathbb{Z} / p\\mathbb{Z}$ is an integral domain. Moreover, a standard fact in elementary number theory shows that $z^{p-1} -1 \\equiv 0$ (mod $p$) holds for all $z\\in \\{1, 2, \\cdots, p-1 \\}$; and so $f(z)\\equiv 0$ (mod $p$) for all $z\\in \\{1, 2, \\cdots, p-1 \\}$. Now recall from Theorem \\ref{theorem 3.2.1} that $\\#\\{ z\\in \\mathbb{Z} : \\varphi_{p,c}(z) - z = 0\\}\\leq 4$, and moreover since good reduction (mod $p$) preserves periodicity of points [\\cite{Silverman}, Corollary 2.20], hence together with the solution $z\\equiv 0$ (mod $p$), then at most three additional values of $z\\in \\{1, 2, \\cdots, p-1 \\}$ are desired for the congruence $f(z)\\equiv 0$ (mod $p$). But now because of the first part of Theorem \\ref{2.1}, then for all primes $p\\geq 3$ the polynomial $f(z)$ (mod $p$) has three roots in $\\mathbb{Z} / p\\mathbb{Z}$. Hence, the number $\\#\\{ z\\in \\mathbb{Z} / p\\mathbb{Z} : \\varphi_{p,c}(z) - z \\equiv 0 \\text{ (mod $p$)}\\} = 3$, and so the number $N_{c}(p) = 3$, as needed. To see the second part, we first note that since the coefficient $c\\neq pt$ for any integer $t$, then this means $c\\not \\equiv 0$ (mod $p$). But then, the polynomial $z^p - z + c\\not \\equiv 0$ (mod $p$) for every point $z\\in \\mathbb{Z} / p\\mathbb{Z}$ and so $f(z)\\not \\equiv 0$ (mod $p$) for every integral point $z\\in \\mathbb{Z} / p\\mathbb{Z}$. Hence, we then see that the polynomial $\\varphi_{p,c}(x)-x$ has no roots in $\\mathbb{Z} / p\\mathbb{Z}$ for every $c$ indivisble by $p$ and thus the number $N_{c}(p) = 0$, as also needed. This then completes the whole proof, as required.\n\\end{proof}\n\n\n\\begin{rem}\\label{rem2.3}\nWith now Theorem \\ref{2.2}, we may then to each distinct integral fixed point of $\\varphi_{p,c}$ associate an integral fixed orbit. So then, a dynamical translation of Theorem \\ref{2.2} is the claim that the number of distinct integral fixed orbits that $\\varphi_{p,c}$ has when iterated on the space $\\mathbb{Z} / p\\mathbb{Z}$, is 3 or 0. Notice that in both coefficient cases $c\\equiv 0$ (mod $p)$ and $c\\not \\equiv 0$ (mod $p)$ that we considered in Theorem \\ref{2.2}, it may also follow that the expected total number of distinct integral fixed points (fixed orbits) in the whole family of maps $\\varphi_{p,c}$ modulo $p$ is $3 + 0 =3$.\n\\end{rem}\n\\section{On Number of Integral Fixed Points of any Family of Polynomial Maps $\\varphi_{p-1,c}$}\\label{sec3}\n\nAs in Section \\ref{sec2}, we also in this section wish unlike in Section \\ref{sec2} to count unconditionally the number of distinct integral fixed points of any $\\varphi_{p-1,c}$, first by modulo $5$ and then count by modulo $p$ for any given fixed prime integer $p\\geq 5$. To this end, for any given $c\\in \\mathbb{Z}$ and any given prime $p\\geq 5$, we define a counting function \n\\begin{equation}\\label{M_{c}}\nM_{c}(p) := \\#\\{ z\\in \\mathbb{Z} / p\\mathbb{Z} : \\varphi_{p-1,c}(z) - z \\equiv 0 \\ \\text{(mod $p$)}\\}.\n\\end{equation}\\noindent We then first prove the following theorem on the number of distinct integral fixed points of any $\\varphi_{4, c}$ modulo $5$:\n\\begin{thm} \\label{6.0.1}\nLet $\\varphi_{4, c}$ be defined by $\\varphi_{4, c}(z) = z^4 + c$ for all $c, z\\in\\mathbb{Z}$, and let $M_{c}(5)$ be defined as in \\textnormal{(\\ref{M_{c}})}. Then $M_{c}(5) = 1$ or $2$ for every $c\\equiv 1 \\ (mod \\ 5)$ or $c = 5t$, respectively; otherwise, $M_{c}(5) = 0$ for every $c\\equiv -1\\ (mod \\ 5)$. \n\\end{thm}\n\\begin{proof}\nLet $g(z)= \\varphi_{4,c}(z)-z = z^4 - z + c$ and for every coefficient $c = 5t$, reducing both sides of $g(z)$ modulo 5, we then have $g(z)\\equiv z^4 - z$ (mod 5); and so the reduced polynomial $g(z)$ modulo $5$ is now a polynomial defined over a finite field $\\mathbb{Z}\\slash5\\mathbb{Z}$ of 5 distinct elements. So now, it is a well-known fact about polynomials over finite fields that the quartic polynomial $h(x):=x^4 -1$ has 4 distinct nonzero roots in $\\mathbb{Z}\\slash5\\mathbb{Z}$; and so we have $z^4 = 1$ for every nonzero $z\\in \\mathbb{Z}\\slash5\\mathbb{Z}$. But now, the reduced polynomial $g(z)\\equiv 1 - z$ (mod 5) for every nonzero $z\\in \\mathbb{Z}\\slash5\\mathbb{Z}$; from which it then follows the polynomial $g(z)$ has a nonzero root in $\\mathbb{Z}\\slash5\\mathbb{Z}$, namely, $z\\equiv 1$ (mod 5). Moreover, since $z$ is also a linear factor of $g(z)\\equiv z(z^3 - 1)$ (mod 5), then it also follows that $z\\equiv 0$ (mod 5) is a root of $g(z)$ modulo $5$. Hence, the number $\\#\\{ z\\in \\mathbb{Z} / 5\\mathbb{Z}: \\varphi_{4,c}(z) - z \\equiv 0 \\text{ (mod 5)}\\} = 2$ for every coefficient $c\\equiv 0$ (mod $5$) and so the number $M_{c}(5) = 2$. To see the possibility that the number $M_{c}(5) = 1$ for every coefficient $c\\equiv 1$ (mod 5), we first note that with $c\\equiv 1$ (mod 5) and also recalling that $z^4 = 1$ for every nonzero $z\\in \\mathbb{Z}\\slash5\\mathbb{Z}$, then after reducing $g(z)= \\varphi_{4,c}(z)-z = z^4 - z + c$ modulo $5$, we then have $g(z)\\equiv 2 - z$ (mod 5) and so $g(z)$ has a root in $\\mathbb{Z}\\slash5\\mathbb{Z}$, namely, $z\\equiv 2$ (mod 5) and thus $M_{c}(5) = 1$. To see the second part, we first note that since the coefficient $c \\equiv -1$ (mod 5), we then obtain $z^4 - z + c\\equiv -z$ (mod 5) since $z^4 = 1$ for every nonzero $z\\in \\mathbb{Z}\\slash5\\mathbb{Z}$; and so the reduced polynomial $g(z) \\equiv -z$ (mod 5). But now, notice that the point $z\\equiv 0$ (mod 5) is a root of $g(z)$ modulo $5$ for every coefficient $c\\equiv -1$ (mod $5$) and for every coefficient $c\\equiv 0$ (mod $5$) as seen from the first part; which then clearly is impossible, since $-1\\not \\equiv 0$ (mod $5$). Hence, the quartic polynomial $\\varphi_{4,c}(x)-x$ has no roots in $\\mathbb{Z} / 5\\mathbb{Z}$ for every coefficient $c\\equiv -1$ (mod $5$) and so we obtain that the number $M_{c}(5) = 0$, as desired.\n\\end{proof} \nWe now generalize Theorem \\ref{6.0.1} to any $\\varphi_{p-1, c}$ for any given prime integer $p\\geq 5$. Specifically, we show that the number of distinct integral fixed points of any polynomial map $\\varphi_{p-1, c}$ modulo $p$ is equal to $1$ or $2$ or $0$:\n\n\\begin{thm} \\label{6.0.2}\nLet $p\\geq 5$ be any fixed prime integer, and $\\varphi_{p-1, c}$ be any polynomial map defined by $\\varphi_{p-1, c}(z) = z^{p-1}+c$ for all $c, z\\in\\mathbb{Z}$. Let the number $M_{c}(p)$ be defined as in \\textnormal{(\\ref{M_{c}})}. Then we have $M_{c}(p) = 1$ or $2$ for every coefficient $c\\equiv 1 \\ (mod \\ p)$ or $c = pt$, respectively; otherwise, we have $M_{c}(p) = 0$ for every point $c\\equiv -1\\ (mod \\ p)$.\n\\end{thm}\n\\begin{proof}\nWe give a similar argument as in the foregoing proof. As before, let $g(z)= z^{p-1} - z + c$ and for every coefficient $c = pt$, reducing both sides of $g(z)$ modulo $p$, we then have $g(z)\\equiv z^{p-1} - z$ (mod $p$); and so the reduced polynomial $g(z)$ modulo $p$ is now a polynomial defined over a finite field $\\mathbb{Z}\\slash p\\mathbb{Z}$ of $p$ distinct elements. So now, as before we have, the polynomial $h(x):=x^{p-1} -1$ has $p-1$ distinct nonzero roots in $\\mathbb{Z}\\slash p\\mathbb{Z}$; and so we have $z^{p-1} = 1$ for every nonzero $z\\in \\mathbb{Z}\\slash p\\mathbb{Z}$. But now, the reduced polynomial $g(z)\\equiv 1 - z$ (mod $p$) for every nonzero point $z\\in \\mathbb{Z}\\slash p\\mathbb{Z}$, and hence the polynomial $g(z)$ has a nonzero root in $\\mathbb{Z}\\slash p\\mathbb{Z}$, namely, $z\\equiv 1$ (mod $p$). Moreover, since $z$ is also a linear factor of the reduced polynomial $g(z)\\equiv z(z^{p-2} - 1)$ (mod $p$), then it also follows that $z\\equiv 0$ (mod $p$) is a root of the polynomial $g(z)$ modulo $p$. Thus, the number $\\#\\{ z\\in \\mathbb{Z} / p\\mathbb{Z} : \\varphi_{p-1,c}(z) - z \\equiv 0 \\text{ (mod $p$)}\\} = 2$ for every $c\\equiv 0$ (mod $p$) and so have $M_{c}(p) = 2$. To see the number $M_{c}(p) = 1$ for every coefficient $c\\equiv 1$ (mod $p$), we note that with $c\\equiv 1$ (mod $p$) and also that $z^{p-1} = 1$ for every nonzero $z\\in \\mathbb{Z}\\slash p\\mathbb{Z}$, then reducing $g(z)= \\varphi_{p-1,c}(z)-z = z^{p-1} - z + c$ modulo $p$, we then have $g(z)\\equiv 2 - z$ (mod $p$) since $z^{p-1} = 1$ for every nonzero point $z\\in \\mathbb{Z}\\slash p\\mathbb{Z}$; and hence $g(z)$ has a root in $\\mathbb{Z}\\slash p\\mathbb{Z}$, namely, $z\\equiv 2$ (mod $p$) and so $M_{c}(p) = 1$. To see the second part, we note that with $c \\equiv -1$ (mod $p$), we then obtain $z^{p-1} - z + c\\equiv -z$ (mod $p$) since $z^{p-1} = 1$ for every nonzero $z\\in \\mathbb{Z}\\slash p\\mathbb{Z}$; and so the polynomial $g(z) \\equiv -z$ (mod $p$). But now, notice $z\\equiv 0$ (mod $p$) is a root of $g(z)$ modulo $p$ for every $c\\equiv -1$ (mod $p$) and for every $c\\equiv 0$ (mod $p$) as seen from the first part; which then clearly is impossible, since $-1\\not \\equiv 0$ (mod $p$). Hence, we see $\\varphi_{p-1,c}(x)-x$ has no roots in $\\mathbb{Z} / p\\mathbb{Z}$ for every coefficient $c\\equiv -1$ (mod $p$) and so $M_{c}(p) = 0$, as desired.\n\\end{proof}\n\n\n\\begin{rem}\nWith now Theorem \\ref{6.0.2}, we may then to each distinct integral fixed point of $\\varphi_{p-1,c}$ associate an integral fixed orbit. So then, a dynamical translation of Theorem \\ref{6.0.2} is the claim that the number of distinct integral fixed orbits that $\\varphi_{p-1,c}$ has when iterated on the space $\\mathbb{Z} / p\\mathbb{Z}$, is equal to $1$ or $2$ or $0$. Again, observe that in all of the possible coefficient cases $c\\equiv 1, 0, -1$ (mod $p)$ that we considered in Theorem \\ref{6.0.2}, it may then also follow that the expected total number of distinct integral fixed points (fixed orbits) in the whole family of maps $\\varphi_{p-1,c}$ modulo $p$ is $1 + 2 + 0 =3$; which somewhat surprising coincides with the expected total number three of distinct integral fixed points (fixed orbits) in the whole family of maps $\\varphi_{p,c}$ (mod $p$) as remarked in \\ref{rem2.3}. \n\\end{rem}\n\n\\section{On the Average Number of Fixed Points of any Polynomial Map $\\varphi_{p,c}$ \\& $\\varphi_{p-1,c}$}\\label{sec4}\n\nIn this section, we wish to inspect independently the behavior of the counting functions $N_{c}(p)$ and $M_{c}(p)$ as $p$ tends to infinity. First, we wish to ask and answer precisely the question: \\say{\\textit{What is the average value of $N_{c}(p)$ as $p \\to \\infty$?}} The following corollary shows that the average value of the function $N_{c}(p)$ is $3$ or 0, as $p\\to \\infty$:\n\\begin{cor}\\label{4.1}\nLet $p\\geq 3$ be any prime integer. Then the average value of the function $N_{c}(p)$ exists and is equal to $3$ or $0$, as $p \\to \\infty$. More precisely, we have\n\\begin{myitemize}\n \\item[\\textnormal{(a)}] Avg $N_{c = pt}(p) := \\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 3, \\ p\\mid c}N_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 3, \\ p\\mid c}1}}} = 3.$ \n \n \\item[\\textnormal{(b)}] \\text{Avg} $N_{c\\neq pt}(p):= \\lim\\limits_{p \\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 3, \\ p\\nmid c}N_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 3, \\ p\\nmid c}1}}} = 0$. \n\\end{myitemize}\n\n\\end{cor}\n\\begin{proof}\nSince from Theorem \\ref{2.2} we know that the number $N_{c}(p) = 3$ for any prime $p\\mid c$, we then obtain that $\\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 3, \\ p\\mid c}N_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 3, \\ p\\mid c}1}}} = 3\\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 3, \\ p\\mid c}1}{\\Large{\\sum\\limits_{p\\geq 3, \\ p\\mid c}1}}} = 3$. Hence, the average value of $N_{c}(p)$, namely, Avg $N_{c = pt}(p)$ is equal to 3, which shows $\\textnormal{(a)}$. To see $\\textnormal{(b)}$, we recall from Theorem \\ref{2.2} that $N_{c}(p) = 0$ for any prime $p\\nmid c$; and so we then obtain $\\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 3, \\ p\\nmid c}N_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 3, \\ p\\nmid c}1}}} = 0$; and hence Avg $N_{c\\neq pt}(p) = 0$. This then completes the whole proof, as desired.\n\\end{proof}\n\\begin{rem} \\label{re5}\nFrom arithmetic statistics to arithmetic dynamics, Corollary \\ref{4.1} shows that any map $\\varphi_{p,c}$ iterated on the space $\\mathbb{Z} / p\\mathbb{Z}$, has on average three or no distinct integral fixed orbits, as the degree $p$ tends to infinity.\n\\end{rem}\n\nNow we also wish to ask and also answer precisely the question: \\say{\\textit{What is the average value of $M_{c}(p)$ as $p \\to \\infty$?}} The following corollary shows that the average value of the function $M_{c}(p)$ is $1$ or $2$ or 0, as $p\\to \\infty$:\n\\begin{cor}\\label{cor5}\nLet $p\\geq 5$ be any prime integer. Then the average value of the function $M_{c}(p)$ exists and is equal to $1$ or $2$ or $0$, as $p\\to\\infty$. Specifically, we have \n\\begin{myitemize}\n \\item[\\textnormal{(a)}] Avg $M_{c-1 = pt}(p) := \\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 5, \\ p\\mid (c-1)}M_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 5, \\ p\\mid (c-1)}1}}} = 1.$ \n\n \\item[\\textnormal{(b)}] Avg $M_{c= pt}(p) := \\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 5, \\ p\\mid c}M_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 5, \\ p\\mid c}1}}} = 2.$\n \n \\item[\\textnormal{(c)}] \\text{Avg} $M_{c+1= pt}(p):= \\lim\\limits_{p \\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 5, \\ p\\mid (c+1)}M_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 5, \\ p\\mid (c+1)}1}}} = 0$. \n\\end{myitemize}\n\n\\end{cor}\n\\begin{proof}\nSince from Theorem \\ref{6.0.2} we know that the number $M_{c}(p) = 1$ for any prime $p$ such that $p\\mid (c-1)$, we then obtain $\\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 5, \\ p\\mid (c-1)}M_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 5, \\ p\\mid (c-1)}1}}} = \\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 5, \\ p\\mid (c-1)}1}{\\Large{\\sum\\limits_{p\\geq 5, \\ p\\mid (c-1)}1}}} = 1$; and so the average value of $M_{c}(p)$, namely, Avg $M_{c-1 = pt}(p) = 1$. Similarly, since $M_{c}(p) = 2$ or $0$ for any prime $p$ such that $p\\mid c$ or $p$ such that $p\\mid (c+1)$ resp., we then also obtain $\\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 5, \\ p\\mid c}M_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 5, \\ p\\mid c}1}}} = 2$ or $\\lim\\limits_{p\\to\\infty} \\Large{\\frac{\\sum\\limits_{p\\geq 5, \\ p\\mid (c+1)}M_{c}(p)}{\\Large{\\sum\\limits_{p\\geq 5, \\ p\\mid (c+1)}1}}} = 0$, resp.; and so the average value Avg $M_{c = pt}(p) = 2$ or Avg $M_{c+1 = pt}(p) = 0$, resp., as desired. This then completes the whole proof, as required.\n\\end{proof} \n\\begin{rem} \\label{re5}\nAs before, from arithmetic statistics to arithmetic dynamics, Corollary \\ref{cor5} then demonstrates that any $\\varphi_{p-1,c}$ iterated on the space $\\mathbb{Z} / p\\mathbb{Z}$, has on average one or two or no fixed orbits, as $p$ tends to infinity.\n\\end{rem}\n\\section{On the Density of Monic Integer Polynomials $\\varphi_{p,c}(x)$ with the number $N_{c}(p) = 3$}\\label{sec5}\nIn this section, we wish to ask and answer: \\say{\\textit{For $p\\geq 3$ a prime integer, what is the density of monic integer polynomials $\\varphi_{p, c}(x) = x^p + c$ with exactly three integral fixed points modulo $p$?}} The following corollary shows that there are very few integer polynomials $\\varphi_{p,c}(x) = x^p + c$ with exactly three integral fixed points modulo $p$:\n\\begin{cor}\\label{Thm 4.1}\nLet $p\\geq 3$ be a prime integer. Then the density of monic polynomials $\\varphi_{p,c}(x) = x^p + c\\in \\mathbb{Z}[x]$ with the number $N_{c}(p) = 3$ exists and is equal to $0 \\%$, as $c\\to \\infty$. More precisely, we have \n\\begin{center}\n $\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ \\text{and} \\ N_{c}(p) \\ = \\ 3\\}}{\\Large{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}} = \\ 0.$\n\\end{center}\n\\end{cor}\n\\begin{proof}\nSince the defining condition $N_{c}(p) = 3$ is as we proved in Thm \\ref{2.2}, determined whenever the coefficient $c$ is divisible by any given prime $p\\geq 3$, hence, we may count $\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] : 3\\leq p\\leq c \\ \\text{and} \\ N_{c}(p) \\ = \\ 3\\}$ by simply counting the number $\\# \\{\\varphi_{p,c}(x)\\in \\mathbb{Z}[x] : 3\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\ \\text{for \\ any \\ fixed} \\ c \\}$. Thus, we then have that \n\\begin{center}\n$\\Large{\\frac{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ \\text{and} \\ N_{c}(p) \\ = \\ 3\\}}{\\Large{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}} = \\Large{\\frac{\\# \\{\\varphi_{p,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\ \\text{for any fixed} \\ c \\}}{\\Large{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}}$. \n\\end{center}\\indent Moreover, for any fixed $c$, we also have that the size \n\\begin{center}\n$\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] : 3\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\} = \\# \\{p : 3\\leq p\\leq c \\text{ and } p\\mid c \\} = \\sum_{3\\leq p\\leq c, \\ p\\mid c}1 = \\omega (c)$,\n\\end{center}where the counting function $\\omega(c)$ is the number of distinct prime factors of $c$. And if we also rewrite the size $\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] : 3\\leq p\\leq c \\} = \\sum_{3\\leq p\\leq c} 1 = \\pi(c)$, where $\\pi(.)$ is the prime-counting function, we then obtain \n\\begin{center}\n$\\Large{\\frac{\\# \\{\\varphi_{p,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\ \\text{for any fixed} \\ c \\}}{\\Large{\\# \\{\\varphi_{p,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}} = \\frac{\\omega(c)}{\\pi(c)}$.\n\\end{center}Now recall from analytic number theory that for any positive integer $c$, we have $2^{\\omega(c)}\\leq \\sigma (c) \\leq 2^{\\Omega(c)}$, where $\\sigma(c)$ is the divisor function and $\\Omega(c)$ counts the total number of prime factors of $c$, with respect to their multiplicity. So now taking logarithms, we then see that the inequality $2^{\\omega(c)}\\leq \\sigma (c) \\leq 2^{\\Omega(c)}$ yields $\\omega(c)\\leq \\frac{\\text{log} \\ \\sigma(c)}{\\text{log} \\ 2}$; and hence yielding that $\\frac{\\omega(c)}{\\pi(c)} \\leq \\frac{\\text{log} \\ \\sigma(c)}{\\text{log} \\ 2 \\cdot \\pi(c)}$. Moreover, for all $\\epsilon >0$, we know $\\sigma(c) = o(c^{\\epsilon})$ and so log $\\sigma(c) =$ log $o(c^{\\epsilon})$, which then yields that $\\frac{\\omega(c)}{\\pi(c)} \\leq \\frac{\\text{log} \\ o(c^{\\epsilon})}{\\text{log} \\ 2 \\cdot \\pi(c)}$. But now for any fixed $\\epsilon$, if we take limit on both sides of the foregoing inequality as $c\\to \\infty$, we then obtain that $\\lim\\limits_{c\\to\\infty} \\frac{\\text{log} \\ o(c^{\\epsilon})}{\\text{log} \\ 2 \\cdot \\pi(c)} = 0$ and so we have $\\lim\\limits_{c\\to\\infty} \\frac{\\omega(c)}{\\pi(c)} \\leq 0$. Hence, we have \n\\begin{center}\n$\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ \\text{and} \\ N_{c}(p) \\ = \\ 3\\}}{\\Large{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}} \\leq 0$.\n\\end{center}Moreover, we also observe that the size $\\# \\{\\varphi_{p,c}(x)\\in \\mathbb{Z}[x] : 3\\leq p\\leq c \\ \\text{and} \\ N_{c}(p) \\ = \\ 3\\}\\geq 1$, and so the quantity \n\\begin{center}\n$\\Large{\\frac{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ \\text{and} \\ N_{c}(p) \\ = \\ 3\\}}{\\Large{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}}\\geq \\frac{1}{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}} = 0$.\n\\end{center}Hence, we see $\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ \\text{and} \\ N_{c}(p) \\ = \\ 3\\}}{\\Large{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}} \\geq 0$ and which when combined with the above limit, we the obtain $\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ \\text{and} \\ N_{c}(p) \\ = \\ 3\\}}{\\Large{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}} = \\ 0$. This then completes whole proof, as desired.\n\\end{proof}\\noindent Note that one may also certainly interpret Corollary \\ref{Thm 4.1} by saying that the probability of choosing randomly a monic integer polynomial $\\varphi_{p,c}(x)$ in the space $\\mathbb{Z}[x]$ with exactly three integral fixed points modulo $p$, is zero.\n\\section{On Densities of Monic Integer Polynomials $\\varphi_{p-1,c}(x)$ with $M_{c}(p) = 1$ \\& $M_{c}(p) = 2$}\\label{sec6}\n\nIn this section, motivated by: \\say{\\textit{What is the density of polynomials $\\varphi_{p-1,c}(x)\\in \\mathbb{Z}[x]$ with two integral fixed points modulo $p$?}}, we prove in the corollary below that the density of such polynomials $\\varphi_{p-1,c}(x)$ exists and is zero:\n\\begin{cor}\\label{6.1}\nLet $p\\geq 5$ be a prime integer. The density of monic polynomials $\\varphi_{p-1,c}(x)=x^{p-1}+c\\in \\mathbb{Z}[x]$ with the number $M_{c}(p) = 2$ exists and is equal to $0\\%$, as $c\\to \\infty$. More precisely, we have \n\\begin{center}\n $\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x]\\ : \\ 5\\leq p\\leq c \\ and \\ M_{c}(p) \\ = \\ 2\\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x]\\ : \\ 5\\leq p\\leq c \\}}}} = \\ 0.$\n\\end{center}\n\\end{cor}\n\\begin{proof}\nBy applying the exact same reasoning as in the Proof of Corollary \\ref{Thm 4.1}, we then see the desired limit.\n\\end{proof} \\noindent Note that one may again interpret Corollary \\ref{6.1} by saying that the probability of choosing randomly a monic polynomial $\\varphi_{p-1,c}(x) = x^{p-1}+c$ in the space $\\mathbb{Z}[x]$ with exactly two integral fixed points modulo $p$, is zero.\n\nThe following immediate corollary shows that the probability of choosing randomly a monic integer polynomial $\\varphi_{p-1,c}(x)=x^{p-1}+c$ in the space $\\mathbb{Z}[x]$ with exactly one integral fixed point modulo $p$, is also zero:\n\n\\begin{cor}\\label{6.2}\nLet $p\\geq 5$ be a prime integer. The density of monic polynomials $\\varphi_{p-1,c}(x)=x^{p-1}+c\\in \\mathbb{Z}[x]$ with the number $M_{c}(p) = 1$ exists and is equal to $0\\%$, as $c\\to \\infty$. More precisely, we have \n\\begin{center}\n $\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x]\\ : \\ 5\\leq p\\leq c \\ and \\ M_{c}(p) \\ = \\ 1\\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x]\\ : \\ 5\\leq p\\leq c \\}}}} = \\ 0.$\n\\end{center}\n\\end{cor}\n\\begin{proof}\nAs before, $M_{c}(p) = 1$ is as we proved in Theorem \\ref{6.0.2}, determined whenever the coefficient $c$ is such that $c-1$ is divisible by any fixed prime $p\\geq 5$; and so we may count $\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] : 5\\leq p\\leq c \\ \\text{and} \\ M_{c}(p) \\ = \\ 1\\}$ by simply counting the number $\\# \\{\\varphi_{p-1,c}(x)\\in \\mathbb{Z}[x] : 5\\leq p\\leq c \\ \\text{and} \\ p\\mid (c-1) \\ \\text{for \\ any \\ fixed} \\ c \\}$. But now, since $c-1<c$, then if the number $\\# \\{p : 5\\leq p\\leq c \\ \\text{and} \\ p\\mid (c-1) \\}< \\# \\{p : 5\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\}$, we then have that\n\\begin{center}\n$\\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\ \\text{and} \\ p\\mid (c-1) \\ \\text{for any fixed} \\ c\\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\}}}} < \\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\ \\text{for any fixed} \\ c \\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\}}}}.$ \n\\end{center}Letting $c$ tend to infinity on both sides of the above inequality and then applying Corollary \\ref{6.1}, we then have \n\\begin{center}\n$\\lim\\limits_{c\\to\\infty}\\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\ \\text{and} \\ p\\mid (c-1)\\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\}}}} \\leq \\lim\\limits_{c\\to\\infty}\\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\}}}}$ = 0; \n\\end{center} from which it then follows that \n\\begin{center}\n $\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x]\\ : \\ 5\\leq p\\leq c \\ \\text{and} \\ M_{c}(p) \\ = \\ 1\\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x]\\ : \\ 5\\leq p\\leq c \\}}}} = \\ 0$, and hence showing the limit as desired in this case.\n\\end{center}Otherwise, if the number $\\# \\{p : 5\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\}< \\# \\{p : 5\\leq p\\leq c \\ \\text{and} \\ p\\mid (c-1) \\}$, we then have that\n\\begin{center}\n$\\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\ \\text{and} \\ p\\mid c \\ \\text{for any fixed} \\ c\\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\}}}} < \\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\ \\text{and} \\ p\\mid (c-1) \\ \\text{for any fixed} \\ c \\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\}}}}.$\n\\end{center}So now, taking limit as $c\\to \\infty$ on both sides of the above inequality and applying Corollary \\ref{6.1} and then applying a similar argument as in the Proof of Corollary \\ref{Thm 4.1} to obtain an upper bound zero, we then obtain\n\\begin{center}\n$\\lim\\limits_{c\\to\\infty}\\Large{\\frac{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\ \\text{and} \\ p\\mid (c-1)\\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\}}}} = 0$, as also needed. This then completes the whole proof.\n\\end{center}\n\\end{proof}\n\n\n\n\n\\section{On Density of Integer Monics $\\varphi_{p,c}(x)$ with $N_{c}(p) = 0$ and $\\varphi_{p-1,c}(x)$ with $M_{c}(p) = 0$}\\label{sec7}\n\nRecall in Corollary \\ref{Thm 4.1} that a density of $0\\%$ of monic integer (and hence monic rational) polynomials $\\varphi_{p,c}(x)$ have exactly three integral fixed points modulo $p$; and so the density of monic integer polynomials $\\varphi_{p,c}(x)$ that are reducible modulo $p$ is $0\\%$. Now, we also ask: \\say{\\textit{What is the density of monic integer polynomials $\\varphi_{p,c}(x)$ that don't have integral fixed points modulo $p$?}} The corollary below shows that the probability of choosing randomly an integer polynomial $\\varphi_{p,c}(x) = x^p+c$ such that $\\mathbb{Q}[x]\\slash (\\varphi_{p, c}(x)-x)$ is a degree-$p$ number field, is 1: \n\\begin{cor}\\label{8.1}\nLet $p\\geq 3$ be a prime integer. Then the density of monic polynomials $\\varphi_{p,c}(x) = x^p + c\\in \\mathbb{Z}[x]$ with the number $N_{c}(p) = 0$ exists and is equal to $100 \\%$, as $c\\to \\infty$. More precisely, we have \n\\begin{center}\n $\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p,c}(x)\\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\ and \\ N_{c}(p) \\ = \\ 0 \\}}{\\Large{\\# \\{\\varphi_{p,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 3\\leq p\\leq c \\}}}} = \\ 1.$\n\\end{center}\n\\end{cor}\n\\begin{proof}\nSince the number $N_{c}(p) = 3$ or $0$ for any given prime integer $p\\geq 3$ and since we also proved the density in Corollary \\ref{Thm 4.1}, we then obtain the desired density (i.e., we obtain that the limit exists and is equal to 1). \n\\end{proof}\n\\noindent Note that the foregoing corollary also shows that there are infinitely many polynomials $\\varphi_{p,c}(x)$ over $\\mathbb{Q}$ such that for $f(x) := \\varphi_{p,c}(x)-x = x^p-x+c$, the induced quotient ring $K_{f} = \\mathbb{Q}[x]\\slash (f(x))$ is an odd prime degree-$p$ number field. Comparing the densities in Corollaries \\ref{Thm 4.1} and \\ref{8.1}, one may then observe that in the whole family of monic integer polynomials $\\varphi_{p,c}(x) = x^p +c$, almost all such monics have no integral fixed points modulo $p$ (i.e., have no rational roots); and hence almost all such monic polynomials $f(x)$ are irreducible over $\\mathbb{Q}$. But this may then imply that the average value of $N_{c}(p)$ in the whole family of monic polynomials $\\varphi_{p,c}(x)$, is zero.\n\nWe also recall in Cor. \\ref{6.1} and \\ref{6.2} that a density of $0\\%$ of monic integer (and thus rational) polynomials $\\varphi_{p-1,c}(x)$ has $M_{c}(p) = 2$ or $1$, resp.; and so the density of polynomials $\\varphi_{p-1, c}(x)\\in \\mathbb{Z}[x]$ that are reducible modulo $p$ is $0\\%$. We now ask: \\say{\\textit{What is the density of monic polynomials $\\varphi_{p-1,c}(x)\\in \\mathbb{Z}[x]$ with no integral fixed points (mod $p$)?}} The following corollary shows that the probability of choosing randomly a monic integer polynomial $\\varphi_{p-1,c}(x) = x^{p-1}+c\\in \\mathbb{Z}[x]$ so that $\\mathbb{Q}[x]\\slash (\\varphi_{p-1, c}(x)-x)$ is a number field of even degree $p-1$, is 1:\n\\begin{cor} \\label{5.1}\nLet $p\\geq 5$ be a prime integer. The density of monic polynomials $\\varphi_{p-1, c}(x) = x^{p-1}+c\\in \\mathbb{Z}[x]$ with the number $M_{c}(p) = 0$ exists and is equal to $100 \\%$, as $c\\to \\infty$. More precisely, we have \n\\begin{center}\n $\\lim\\limits_{c\\to\\infty} \\Large{\\frac{\\# \\{\\varphi_{p-1, c}(x)\\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\ and \\ M_{c}(p) \\ = \\ 0 \\}}{\\Large{\\# \\{\\varphi_{p-1,c}(x) \\in \\mathbb{Z}[x] \\ : \\ 5\\leq p\\leq c \\}}}} = \\ 1.$\n\\end{center}\n\\end{cor}\n\\begin{proof}\nRecall that the number $M_{c}(p) = 1, 2$ or $0$ for any given prime $p\\geq 5$ and since we also proved the densities in Cor. \\ref{6.1} and \\ref{6.2}, we now obtain the desired density (i.e., we get that the limit exists and is equal to 1).\n\\end{proof}\n\\noindent As before, Corollary \\ref{5.1} also shows that there are infinitely many monic polynomials $\\varphi_{p-1,c}(x)$ over $\\mathbb{Q}$ such that for $g(x) := \\varphi_{p-1,c}(x)-x = x^{p-1}-x+c$, the induced quotient ring $L_{g} = \\mathbb{Q}[x]\\slash (g(x))$ is an algebraic number field of even degree $p-1$. Again, if we compare the densities in Cor. \\ref{6.1}, \\ref{6.2} and \\ref{5.1}, we may then see that in the whole family of monic polynomials $\\varphi_{p-1,c}(x) = x^{p-1} +c\\in \\mathbb{Z}[x]$, almost all such monics have no integral fixed points modulo $p$ (i.e., have no $\\mathbb{Q}$-roots); and so almost all monic polynomials $g(x)$ are irreducible over $\\mathbb{Q}$. Consequently, this may imply that the average value of $M_{c}(p)$ in the whole family of monics $\\varphi_{p-1,c}(x)$, is zero.\n\nAs always a central theme in algebraic number theory that whenever one is studying an algebraic number field $K$ of some interest, one must simultaneously try to describe very precisely what the associated ring $\\mathcal{O}_{K}$ of integers is; and this is because $\\mathcal{O}_{K}$ is classically known to describe naturally the arithmetic of the underlying number field. However, accessing $\\mathcal{O}_{K}$ in practice from a computational point of view is known to be an extremely involved problem. In our case here, the following corollary shows that the probability of choosing randomly a monic integer polynomial $f(x)$ such that the quotient $\\mathbb{Z}[x]\\slash (f(x))$ is the ring of integers of $K_{f}$, is $\\approx 60.7927\\%$:\n\\begin{cor} \\label{8.2}\nAssume Corollary \\ref{8.1}. When integer polynomials $f(x)$ are ordered by height $H(f) = |c|^{1\\slash p}$ as defined in \\cite{sch1}, the density of polynomials $f(x)$ such that $\\mathbb{Z}[x]\\slash (f(x))$ is the ring of integers of $K_{f}$ is $\\zeta(2)^{-1}$. \n\\end{cor}\n\n\\begin{proof}\nSince from Corollary \\ref{8.1} we know that there are infinitely many monic polynomials $f(x)$ over $\\mathbb{Q}$ (and hence over $\\mathbb{Z}$) such that $K_{f} = \\mathbb{Q}[x]\\slash (f(x))$ is an algebraic number field of degree deg$(f)$; and moreover associated to $K_{f}$, is the ring $\\mathcal{O}_{K_{f}}$ of integers. So now, applying a remarkable result of Bhargava-Shankar-Wang [\\cite{sch1}, Theorem 1.2] to the underlying family of monic integer polynomials $f(x)$ ordered by height $H(f) = |c|^{1\\slash p}$ such that $\\mathcal{O}_{K_{f}} = \\mathbb{Z}[x]\\slash (f(x))$, we then obtain that the density of such polynomials $f(x)$ is $\\zeta(2)^{-1} \\approx 60.7927\\%$. \n\\end{proof}\n\nAs with $K_{f}$, every number field $L_{g}$ induced by a polynomial $g$, is naturally equipped with the ring of integers $\\mathcal{O}_{L_{g}}$, and which as we again mention may be difficult to compute in practice. In the following corollary we again take great advantage of a density result of (BSW) [\\cite{sch1}, Thm 1.2] to then show that the probability of choosing randomly a polynomial $g(x)\\in \\mathbb{Z}[x]$ such that $\\mathbb{Z}[x]\\slash (g(x))$ is the ring of integers of $L_{g}$, is $\\approx 60.7927\\%$:\n\n\\begin{cor}\nAssume Corollary \\ref{5.1}. When integer polynomials $g(x)$ are ordered by height $H(g) = |c|^{1\\slash (p-1)}$ as defined in \\cite{sch1}, the density of polynomials $g(x)$ such that $\\mathbb{Z}[x]\\slash (g(x))$ is the ring of integers of $L_{g}$ is $\\zeta(2)^{-1}$. \n\\end{cor}\n\n\\begin{proof}\nBy applying the same reasoning as in the Proof of Corollary \\ref{8.2}, we then obtain the desired density.\n\\end{proof}\n\n\\section{On the Number of Number fields $K_{f}$ \\& $L_{g}$ with Bounded Absolute Discriminant}\\label{sec8}\n\nRecall we saw from Corollary \\ref{8.1} that there is an infinite family of irreducible monic integer polynomials $f(x) = x^p-x + c$ such that field $K_{f}$ associated to $f$ is an algebraic number field of odd prime degree $p$. Moreover, we also saw from Corollary \\ref{5.1} that one can always find an infinite family of irreducible monic integer polynomials $g(x) = x^{p-1}-x + c$ such that the field extension $L_{g}$ over $\\mathbb{Q}$ arising from $g$ is an algebraic number field of even degree $p-1\\geq 4$. In this section, we wish to study the problem of counting number fields; a problem that's originally from and is of very serious interest in arithmetic statistics. Inspired by work \\cite{sch}, we wish to count here primitive number fields with bounded absolute discriminant and as a result we have the following:\n\n\\begin{cor}\\label{cor6}\nAssume Corollary \\ref{8.1}, and let $K_{f} = \\mathbb{Q}[x]\\slash (f(x))$ be a primitive number field with discriminant $\\Delta(K_{f})$. Then up to isomorphism classes of number fields, we have that $\\# \\{ K_{f} \\ : \\ |\\Delta(K_{f})| < X \\} \\ll X^{p\\slash(2p-2)}$. \n\\end{cor}\n\n\\begin{proof}\nFrom Corollary \\ref{8.1}, we know that there are infinitely many monics $f(x) = x^p - x + c$ over $\\mathbb{Q}$ (and so over $\\mathbb{Z}$) such that $K_{f}$ is a number field of odd prime degree $p\\geq 3$. But now, when the underlying number fields $K_{f}$ are primitive, we may then use the reasoning of Bhargava-Shankar-Wang in \\cite{sch} to show that up to isomorphism classes of number fields the total number of such primitive number fields with $|\\Delta(K_{f})| < X$, is bounded above by the number of monic integer polynomials $f(x)$ of degree $p\\geq 3$ with height $H(f) \\ll X^{1\\slash(2p-2)}$ and vanishing subleading coefficient. Now since $H(f) = |c|^{1\\slash p}$ and so $|c| \\ll X^{p\\slash(2p-2)}$, we then have that the size $\\# \\{f(x)\\in \\mathbb{Z}[x] \\ : H(f) \\ll X^{1\\slash(2p-2)} \\} = \\# \\{f(x)\\in \\mathbb{Z}[x] \\ : |c| \\ll X^{p\\slash(2p-2)} \\} \\ll X^{p\\slash(2p-2)}$. Hence, we obtain that the number $\\# \\{ K_{f} : |\\Delta(K_{f})| < X \\} \\ll X^{p\\slash(2p-2)}$ up to isomorphism classes of number fields, as needed.\n\\end{proof}\n\nBy applying a similar counting argument of Bhargava-Shankar-Wang in \\cite{sch} as in Cor. \\ref{cor6}, we then also obtain the following corollary on the number of primitive number fields $L_{g}$ with bounded absolute discriminant:\n\n\\begin{cor}\nAssume Corollary \\ref{5.1}, and let $L_{g} = \\mathbb{Q}[x]\\slash (g(x))$ be a primitive number field with discriminant $\\Delta(L_{g})$. Then up to isomorphism classes of number fields, we have that $\\# \\{ L_{g} \\ : \\ |\\Delta(L_{g})| < X \\} \\ll X^{(p-1)\\slash(2p-4)}$. \n\\end{cor}\n\nWe recall in algebraic number theory that an algebraic number field $K$ is called \\say{\\textit{monogenic}} if there exists an algebraic number $\\alpha \\in K$ such that the ring $\\mathcal{O}_{K}$ of integers is the subring $\\mathbb{Z}[\\alpha]$ generated by $\\alpha$ over $\\mathbb{Z}$, i.e., $\\mathcal{O}_{K}= \\mathbb{Z}[\\alpha]$. In Corollary \\ref{cor6}, we counted primitive number fields $K_{f}$ (i.e., $K_{f} = \\mathbb{Q}(\\alpha)$ for some $\\alpha \\in K$) with $|\\Delta(K_{f})| < X$. Now we wish to count in the case when our number fields $K_{f}$ are monogenic with $|\\Delta(K_{f})| < X$ and such that the associated Galois group Gal$(K_{f}\\slash \\mathbb{Q})$ is the symmetric group $S_{p}$. A hard counting task that we very easily tackle here by taking great advantage of a result of Bhargava-Shankar-Wang [\\cite{sch1}, Corollary 1.3]:\n\n\\begin{cor}\\label{8.3}\nAssume Corollary \\ref{8.1}. Then the number of isomorphism classes of algebraic number fields $K_{f}$ of degree $p\\geq 3$ and with $|\\Delta(K_{f})| < X$ that are monogenic and have associated Galois group $S_{p}$ is $\\gg X^{\\frac{1}{2} + \\frac{1}{p}}$.\n\\end{cor}\n\n\\begin{proof}\nSince Cor. \\ref{8.1} also means the existence of infinitely many monic polynomials $f(x)$ over $\\mathbb{Q}$ (and so over $\\mathbb{Z}$) such that $K_{f}$ is a prime degree-$p$ number field. Now applying Bhargava-Shankar-Wang's result [\\cite{sch1}, Cor. 1.3] to the underlying number fields $K_{f}$ with $|\\Delta(K_{f})| < X$ that are monogenic and have associated Galois group $S_{p}$, we then obtain that the number of isomorphism classes of such number fields $K_{f}$ is $\\gg X^{\\frac{1}{2} + \\frac{1}{p}}$, as desired.\n\\end{proof}\n\nWe again wish to take great advantage of that same result of Bhargava-Shankar-Wang [\\cite{sch1}, Corollary 1.3] to also immediately count in the following corollary the number of fields $L_{g}$ that are monogenic with absolute discriminant $|\\Delta(L_{g})| < X$ and such that the associated Galois group Gal$(L_{g}\\slash \\mathbb{Q})$ is the symmetric group $S_{p-1}$: \n\n\\begin{cor}\nAssume Corollary \\ref{5.1}. Then the number of isomorphism classes of algebraic number fields $L_{g}$ of degree $p-1\\geq 4$ and $|\\Delta(L_{g})| < X$ that are monogenic and have associated Galois group $S_{p-1}$ is $\\gg X^{\\frac{1}{2} + \\frac{1}{p-1}}$.\n\\end{cor}\n\n\\begin{proof}\nBy applying the same reasoning as in the Proof of Corollary \\ref{8.3}, we then obtain the desired count.\n\\end{proof}\n\n\\addcontentsline{toc}{section}{Acknowledgements}\n\\section*{\\textbf{Acknowledgements}}\nI'm deeply indebted to my long-time great advisors, Dr. Ilia Binder and Dr. Arul Shankar, for all their boundless generosity, friendship and for all the inspiring weekly conversations and along with Dr. Jacob Tsimerman for always very uncomprimisingly supporting my professional and philosophical-mathematical research endeavours. I'm very grateful to Dr. Shankar for bringing my attention to a Number theory course (MAT1200HS, 2023/24) and for strongly suggesting it to me. This suggestion would've never easily be realized if the course instructor at that time had not been unsurpassably generous in terms of allowing anyone to attend his class and in terms of his friendliness; and so I'm once again deeply indebted to Dr. Shankar for everything and to that instructor, namely, Dr. Tsimerman for his very enlightening Algebraic Number Theory class and conversations. Not to forget, I am very grateful to the very vibrant Dept. of Mathematical and Computational Sciences (MCS) at the University of Toronto, Mississuaga; and in particular, I'm very grateful and deeply indebted to Dr. Yael Karshon (for the truly invaluable advice about the state of professional mathematics), Dr. Ilia Binder, Dr. Arul Shankar, Dr. Marina Tvalavadze, Dr. Alex Rennet, Dr. Michael Gr\\\"{o}echenig, Dr. Julie Desjardins, Dr. Duncan Dauvergne, Dr. Ke Zhang, Dr. Jaimal Thind, for everything. I'm truly very grateful to the Office of the President at the UofT for the amazing hospitality during my in-person visit to converse with President Meric S. Gertler on \\say{Leadership and Duty}. I'm very grateful to Dr. Meric for the very enlightening great conversations on the insurmountable importance of collaboration, inclusion and diversity in educational settings, and also more importantly for not only envisioning with Rose M. Patten for more representation of minority groups in U of T Science and Mathematics departments but also for taking very serious practical steps toward realizing such a vision with integrity, as I've thoroughly witnessed during the fantastic and progressive chairship of Dr. Binder with his team in Dept. of MCS. Lastly, I’m very grateful and deeply indebted to the former U of T Mississauga Registrar and Director of Enrolment Management, namely, Lorretta Neebar, for everything. Last but not the least, I'm truly very grateful and indebted to Dr. Michael Bumby for the great life-philosophical conversations, and more importantly for being a truly an amazing life coach and a great friend.\nAs a graduate research student, this work and my studies are hugely and wholeheartedly funded by Dr. Binder and Dr. Shankar. As part of the first harvest of an upcoming long harvest, I very happily dedicate this article to the Department of Mathematical and Computational Sciences (MCS) and the Presidency of Dr. Meric S. Gerlter at the University of Toronto! Any opinions expressed in this article belong solely to me, the author, Brian Kintu; and should never be taken at all as a reflection of the views of anyone that’s been happily acknowledged by me.\n\n\\bibliography{References}\n\\bibliographystyle{plain}\n\n\\noindent Dept. of Math. and Comp. Sciences (MCS), University of Toronto, Mississauga, Canada \\newline\n\\textit{E-mail address:} \\textbf{brian.kintu@mail.utoronto.ca}\\newline \n\\date{\\small{\\textit{January 1 \\& 30, 2025.}}}\n\n\\end{document}\n\n\n"}}},{"rowIdx":299,"cells":{"id":{"kind":"string","value":"2412.21021v1"},"paper_link":{"kind":"string","value":"http://arxiv.org/abs/2412.21021v1"},"title":{"kind":"string","value":"On the Algebraic Connectivity of Token Graphs and Graphs under Perturbations"},"full_text":{"kind":"string","value":"\\documentclass[11pt,a4paper]{article}\n\n\\usepackage{soul}\n\\usepackage[normalem]{ulem}\n\n\\usepackage{epsf,epsfig,amsfonts,amsgen,amsmath,amstext,amsbsy,amsopn,lineno}\n\\usepackage{amsmath,amsthm,amsbsy}\n\\usepackage{color}\n\\usepackage{cite}\n\\usepackage{subfig}\n\\usepackage{float}\n\\usepackage{graphicx,tikz}\n\\usepackage{mathrsfs}\n\\usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=black]{hyperref}\n\\usepackage{hyperref} 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\\allowdisplaybreaks \n\\def\\proof{\\par\\noindent{\\bf Proof.~}}\n\\def\\Tproof{\\par\\noindent{\\bf Proof}}\n\\def\\qed{\\hfill$\\Box$\\vspace{12pt}}\n\n\n\\newcommand{\\red}[1]{\\textcolor{red}{#1}}\n\\newcommand{\\blue}[1]{\\textcolor{blue}{#1}}\n\\newcommand{\\purple}[1]{\\textcolor{purple}{#1}}\n\\newcommand{\\green}[1]{\\textcolor[rgb]{0.00,0.50,0.50}{#1}} \\newcommand{\\orange}[1]{\\textcolor[rgb]{1.00,0.50,0.00}{#1}} \n\\makeatletter\n\\newenvironment{breakablealgorithm}\n {\n \\begin{center}\n \\refstepcounter{algorithm} \\hrule height.8pt depth0pt \\kern2pt \\renewcommand{\\caption}[2][\\relax]{ {\\raggedright\\textbf{\\ALG@name~\\thealgorithm} ##2\\par} \\ifx\\relax##1\\relax \\addcontentsline{loa}{algorithm}{\\protect\\numberline{\\thealgorithm}##2} \\else \\addcontentsline{loa}{algorithm}{\\protect\\numberline{\\thealgorithm}##1}\n \\kern2pt\\hrule\\kern2pt\n }\n }{\n \\kern2pt\\hrule\\relax \\end{center}\n}\n\\makeatother\n\n\n\\begin{document}\n\\title{On the Algebraic Connectivity of Token Graphs\\\\\nand Graphs under Perturbations\\thanks{This research has been supported by National Natural Science Foundation of China (No.~12471334, No.~12131013), and Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSZ009). C. Dalf\\'o and M. A. Fiol are funded by AGAUR from the Catalan Government under project 2021SGR00434 and MICINN from the Spanish Government under project PID2020-115442RB-I00.\nM. A. Fiol's research is also supported by a grant from the Universitat Polit\\`ecnica de Catalunya, reference AGRUPS-2024.}\\\\\n{\\small Dedicated to Professor Fuji Zhang on the occasion of his 88th birthday.}}\n\\author{X. Song$^{a,b}$, C. Dalf\\'o$^b$, M. A. Fiol$^c$, S. Zhang$^{a}$\\\\ \n$^a${\\small School of Mathematics and Statistics, Northwestern Polytechnical University}\\\\\n {\\small Xi'an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University} \\\\\n {\\small Xi'an, Shaanxi, P.R. China, {\\tt songxd@mail.nwpu.edu.cn, sgzhang@nwpu.edu.cn}}\\\\\n$^b${\\small Departament de Matem\\`atica, Universitat de Lleida} \\\\\n\t\t{\\small Igualada (Barcelona), Catalonia, {\\tt{cristina.dalfo@udl.cat}}}\\\\\n$^c${\\small Departament de Matem\\`atiques, Universitat Polit\\`ecnica de Catalunya} \\\\\n \t{\\small Barcelona Graduate School, Institut de Matem\\`atiques de la UPC-BarcelonaTech (IMTech)}\\\\\n \t{\\small Barcelona, Catalonia, {\\tt{miguel.angel.fiol@upc.edu}}}\n}\n\\date{}\n\\maketitle\n\n\\begin{abstract}\nGiven a graph $G=(V,E)$ on $n$ vertices and an integer $k$ between 1 and $n-1$, the $k$-token graph $F_k(G)$ has vertices representing the $k$-subsets of $V$, and two vertices are adjacent if their symmetric difference is the two end-vertices of an edge in $E$. Using the theory of Markov chains of random walks and the interchange process, it was proved that the algebraic connectivities (second smallest Laplacian eigenvalues) of $G$ and $F_k(G)$ coincide, but a combinatorial/algebraic proof has been shown elusive. In this paper, we use the latter approach and prove that such equality holds for different new classes of graphs under perturbations, such as extended cycles, extended complete bipartite graphs, kite graphs, and graphs with a cut clique.\nKite graphs are formed by a graph (head) with several paths (tail) rooted at the same vertex and with exciting properties. For instance, we show that the different eigenvalues of a kite graph are also eigenvalues of its perturbed graph obtained by adding edges.\nMoreover, as a particular case of one of our theorems, we generalize a recent result of Barik and Verma \\cite{bv24} about graphs with a cut vertex of degree $n-1$. \nAlong the way, we give conditions under which the perturbed graph $G+uv$, with $uv\\in E$, has the same algebraic connectivity as $G$.\n\\end{abstract}\n\n\\noindent{\\em Keywords:} Token graph, Laplacian spectrum, Algebraic connectivity, Binomial matrix, Kite graph, Cut clique.\n\t\n\\noindent{\\em MSC2010:} 05C15, 05C10, 05C50.\n\n\\section{Introduction}\n\nLet $G=(V,E)$ be a graph on $n=|V|$ vertices, with Laplacian matrix $\\L=\\L(G)$. This is a symmetric positive semidefinite and singular matrix with eigenvalues $\\lambda_1(=0)\\le\\lambda_2\\le\\cdots\\le\\lambda_n$. Since its introduction by Fiedler \\cite{Fiedler_1973}, the algebraic connectivity of $G$, $\\alpha(G)=\\lambda_2$, together with its eigenvector $\\x$ (or Fiedler vector), has deserved much attention in the literature. See, for instance, the comprehensive survey by Abreu \\cite{a07}. Moreover, the behavior of $\\alpha(G)$ and $\\x$ under graph perturbations has been considered.\nRecently, some papers have dealt with the algebraic connectivity of token graphs.\nThe $k$-token graph $F_k(G)$ has vertices representing the different configurations of $k$ indistinguishable tokens in different vertices of $G$. Two configurations are adjacent if one can be obtained from the other by moving a token along an edge from\nits current position to an unoccupied vertex. It was shown that the Laplacian spectrum of a graph $G$ is contained in the spectrum of its $k$-token graph; see \nDalf\\'o, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Mart\\'inez \\cite{Dalfo_2021}. Moreover, the authors proved that $\\alpha(F_k(G)) = \\alpha(G)$ for some families of graphs, such as the complete graphs and the paths. From their results, they conjectured that this was always the case.\nLater, they realized that this corresponded to the so-called `Aldous spectral graph conjecture' already proved by Caputo, Liggett, and Richthammer \\cite{clr10}. The proof was based on Markov chains of random walks and the interchange process. However, until now, a combinatorial/algebraic proof is elusive. Some advances towards such a proof have been recently made; see, for instance, Barik and Verma \\cite{bv24}, Dalf\\'o and Fiol \\cite{Dalfo_2024}, and Reyes, Dalf\\'o, Fiol, and Messegu\\'e \\cite{Reyes_2025}. \nIn this article, our contribution is a further step in this direction by proving the equality $\\alpha(F_k(G))=\\alpha(G)$ for different new infinite families of graphs.\nBesides, along the way, we give the conditions under which the perturbed graph $G+uv$ \n(where $uv$ is a new edge) has the same algebraic connectivity as $G$.\n\nThis paper is structured as follows. The following section gives the preliminary results and notations used in this work. The new results are presented in the further sections. Thus, in Section \\ref{sec:+edges}, we derive results about the algebraic connectivities of a graph and the same graph after\nadding one or more new edges. \nIn Section \\ref{sec:kite}, we give some results about the so-called kite graphs, that is, with a `head', which is a general graph, and a `tail,' which is a starlike tree (roughly speaking, a tree with paths). In the same section, we provide conditions under which the algebraic connectivity does not change when kite graphs are `perturbed' by adding some edges. In Section \\ref{sec:cut-clique}, we compute the algebraic connectivity of graphs with a cut clique $K_r$ with maximum degree. This result generalizes a theorem of Barik and Verma \\cite{bv24} about the algebraic connectivity and cut vertices. \n\n\\section{Preliminaries}\n\\label{sec:prelim}\nLet $G=(V,E)$ be a graph on $n$ vertices. Let $V'\\subset V$. Denote by $G[V']$ the {\\em induced subgraph} of $G$ whose vertex set is $V'$ and whose edge set consists of all edges of $G$ with both ends in $V'$. If $G'=(V',E')$ is an induced subgraph of $G$, the {\\em degree} of $G'$, denoted by $d_{G}(G')$, is the number of edges between $V'$ and $V\\setminus V'$. In particular, the degree of a vertex $v$ is denoted by $d_G(v)$. \nWe denote by $G_1\\cup G_2$ the {\\em union} of two simple graphs $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$, with vertex set $V_1\\cup V_2$ and edge set $E_1\\cup E_2$. If $G_1$ and $G_2$ are disjoint, we refer to their union as a {\\em disjoint union}.\nGiven an integer $k$ such that $1\\le k\\le n-1$, the {\\em $k$-token graph} of $G$, denoted by $F_k(G)$, has ${n\\choose k}$ vertices representing the configurations of $k$ indistinguishable tokens\nplaced at distinct vertices of $G$. Moreover, two configurations are adjacent whenever one\nconfiguration can be reached from the other by moving one token along an edge of $G$ from its current position to an unoccupied vertex, see \nFabila-Monroy, Flores-Peñaloza, Huemer, Hurtado, Urrutia, and Wood \\cite{ffhhuw12}.\nAs an example, Figure \\ref{fig:Y+F2(Y)} shows the graph $Y$ on 5 vertices, and its 2-token graph $F_2(Y)$ with ${5\\choose 2}=10$ vertices.\n\n\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=1.1,auto,swap]\n \\vertex (1) at (0.4,2.25) [fill,label=above:{$1$}]{};\n \\vertex (2) at (1.6,2.25) [fill,label=above:{$2$}]{};\n \\vertex (3) at (1,1) [fill,label=left:{$3$}]{};\n \\vertex (4) at (1,-0.2) [fill,label=left:{$4$}]{};\n \\vertex (5) at (1,-1.4) [fill,label=left:{$5$}]{};\n \\draw[line width=0.6pt](1)--(3);\n \\draw[line width=0.6pt](2)--(3);\n \\draw[line width=0.6pt](3)--(4);\n \\draw[line width=0.6pt](4)--(5);\n \\vertex (12) at (6.5,2.55) [fill,label=above:{$12$}]{};\n \\vertex (34) at (6.5,1.3) [fill,label=above:{$34$}]{};\n \\vertex (35) at (6.5,0.1) [fill]{};\n \\node () at (6.6,-0.05) [label=above left:{$35$}]{};\n \\vertex (45) at (6.5,-1.5) [fill,label=left:{$45$}]{};\n \\vertex (13) at (5.2,2.05) [fill,label=left:{$13$}]{};\n \\vertex (14) at (5.2,0.8) [fill,label=left:{$14$}]{};\n \\vertex (15) at (5.2,-0.4) [fill,label=left:{$15$}]{};\n \\vertex (23) at (7.8,2.05) [fill,label=right:{$23$}]{};\n \\vertex (24) at (7.8,0.8) [fill,label=right:{$24$}]{};\n \\vertex (25) at (7.8,-0.4) [fill,label=right:{$25$}]{};\n \\draw[line width=0.6pt](12)--(13);\n \\draw[line width=0.6pt](12)--(23);\n \\draw[line width=0.6pt](23)--(24);\n \\draw[line width=0.6pt](13)--(14);\n \\draw[line width=0.6pt](14)--(34);\n \\draw[line width=0.6pt](34)--(24);\n \\draw[line width=0.6pt](24)--(25);\n \\draw[line width=0.6pt](34)--(35);\n \\draw[line width=0.6pt](14)--(15);\n \\draw[line width=0.6pt](35)--(15);\n \\draw[line width=0.6pt](35)--(25);\n \\draw[line width=0.6pt](35)--(45);\n \\end{tikzpicture}\n \\caption{The graph $Y$ and its $2$-token graph $F_2(Y)$.}\\label{fig:Y+F2(Y)}\n \\end{center}\n\\end{figure}\n\nFor a matrix $\\M$ with order $n$, let $\\lambda_1(\\M)\\le \\lambda_2(\\M)\\le \\cdots \\le \\lambda_n(\\M)$ be its eigenvalues. \nDenote by $\\Phi(\\M)$ the characteristic polynomial of $\\M$.\nLet $\\L=\\L(G)$ be the Laplacian matrix of $G$. This matrix is positive semidefinite and singular, with eigenvalues $(0=)\\lambda_1 \\le \\lambda_2\\le \\cdots\\le \\lambda_n$. Its second smallest eigenvalue $\\lambda_2$ is\nknown as the {\\em algebraic connectivity} of $G$ (see Fiedler \\cite{Fiedler_1973}), and we denote it by $\\alpha(G)$.\nAn eigenvector corresponding to the algebraic connectivity is called a {\\em Fiedler vector}. \n\nLet $\\j$ be the all-1 (column) vector. Given a graph $G$ of order $n$, we define a vector $\\x$ to be an \\emph{embedding} of $G$ if $\\x \\in W_n$, where $W_n:=\\{\\y:\\y^{\\top}\\j = 0\\}$. \nFor a vertex $v\\in V$, the entry of $\\x$ corresponding to $v$ is denoted by $\\x_v$.\nThe value $\\frac{\\x^{\\top}\\L(G)\\x}{\\x^{\\top}\\x}$ is known as the \\emph{Rayleigh quotient}, and \n\\begin{equation}\n \\alpha(G)\\le \\frac{\\x^{\\top}\\L(G)\\x}{\\x^{\\top}\\x}=\\frac{\\sum_{uv\\in E(G)}(\\x_u-\\x_v)^2}{\\x^{\\top}\\x}\\label{eq-rayleigh} \n\\end{equation}\nfor any vector $\\x\\in W_n$ and $\\x\\ne \\vec0$, where equality holds if and only if $\\x$ is a Fiedler vector of $G$.\n\nGiven some integers $n$ and $k$ (with $k\\in [n]=\\{1,2,\\ldots\\}$), the $(n,k)$-\\emph{binomial matrix} $\\B$ is a ${n \\choose k}\\times n$ matrix whose rows are the characteristic vectors of the $k$-subsets of $[n]$ in a given order. Thus, if the $i$-th $k$-subset is $A$, then\n$$\n(\\B)_{ij}=\n\\left\\lbrace\n\\begin{array}{ll}\n\t1 & \\mbox{if } j\\in A,\\\\\n\t0 & \\mbox{otherwise.}\n\\end{array}\n\\right.\n$$\n\n \n\nDenote by $\\O$ the all-$0$ matrix with the appropriate dimension.\nLet $\\I_n$ and $\\J_n=\\j\\j^{\\top}$ be the identity matrix and the all-$1$ matrix with order $n$, respectively.\n\nNow, let us describe a pair of results on the interlacing of graph eigenvalues.\n\n\\begin{lemma}[Horn and Johnson \\cite{Horn_2013}]\n\\label{le-principal_submatrix}\nLet $\\M'$ be a Hermitian square matrix of order $n-1$, $\\z\\in \\mathbb{C}^{n-1}$, and $a\\in \\mathbb{R}$, and let \n$$\n\\M=\\left({\\begin{array}{cccc}\n \\M'&\\z\\\\\n \\z^*&a\n \\end{array}}\n \\right),\n$$\nwhere $\\z^*$ is the conjugate transpose of $\\z$. \nThen, the eigenvalues of $\\M'$ interlace the eigenvalues of $\\M$. That is, we have the inequalities\n\\[\n\\lambda_1(\\M)\\le \\lambda_1(\\M')\\le \\lambda_2(\\M)\\le \\cdots\\le \\lambda_{n-1}(\\M)\\le \\lambda_{n-1}(\\M')\\le \\lambda_n(\\M),\n\\]\nin which $\\lambda_i(\\M')=\\lambda_{i+1}(\\M)$ if and only if there is a nonzero vector $\\w \\in \\mathbb{C}^{n-1}$ such that $\\M'\\w=\\lambda_i(\\M')\\w$, with $\\z^*\\w=0$, and $\\M'\\w=\\lambda_{i+1}(\\M)\\w$. \n\\end{lemma}\n\\begin{lemma}[Horn and Johnson \\cite{Horn_2013}]\n\\label{le-matrix_add}\n Let $\\M_1$ and $\\M_2$ be two Hermitian matrices of order $n$. Then,\n \\[\n \\lambda_i(\\M_1)+\\lambda_1(\\M_2)\\le \\lambda_i(\\M_1+\\M_2)\\le \\lambda_i(\\M_1)+\\lambda_n(\\M_2)\n \\]\n for $i=1,\\ldots,n$, with equality in the upper bound if and only if there is nonzero vector $\\x$ such that $\\M_1\\x=\\lambda_i(\\M_1)\\x$, $\\M_2\\x=\\lambda_n(\\M_2)\\x$, and $(\\M_1+\\M_2)\\x=\\lambda_i(\\M_1+\\M_2)\\x$; equality in the lower bound occurs if and only if there is a nonzero vector $\\x$ such that $\\M_1\\x=\\lambda_i(\\M_1)\\x$, $\\M_2\\x=\\lambda_1(\\M_2)\\x$, and $(\\M_1+\\M_2)\\x=\\lambda_i(\\M_1+\\M_2)\\x$.\n\\end{lemma}\n\nLet us now show some results concerning a graph after adding a new edge $uv$ to it.\n\n\\begin{lemma}[Cvetkovi\\'c, Doob, and Sachs \\cite{Cvetkovic_1980}]\n\\label{le-interlacing}\nLet $G$ be a graph and $G'=G+uv$. Then, the Laplacian eigenvalues of $G$ and $G'$ interlace, that is, \n\\[\n0=\\lambda_1(G)= \\lambda_1(G')\\le \\lambda_2(G)\\le \\lambda_2(G')\\le \\cdots\\le \\lambda_n(G) \\le \\lambda_{n}(G').\n\\]\n\\end{lemma}\n\n\\begin{lemma}[Xue, Lin, and Shu \\cite{Xue_2019}]\n\\label{le-Xue}\nLet $G$ be a graph and $G'=G+uv$. If $\\alpha(G')=\\alpha(G)$, then there exists a Fiedler vector $\\x$ of $G$ such that $\\x_u=\\x_v$. \n\\end{lemma}\n\\begin{lemma}[Merris \\cite{Merris_1998}]\n\\label{le-Merris}\nLet $G$ be a graph and $G'=G + uv$ or $G'=G - uv$. Let $\\lambda$ be a Laplacian eigenvalue of $G$ corresponding to an eigenvector $\\x$. If $\\x_u=\\x_v$, then $\\lambda$ is also a Laplacian eigenvalue of $G'$ with eigenvector $\\x$, where $G'$ is the graph obtained from $G$ by deleting or adding an edge $uv$ depending on whether or not it is an edge of $G$. \n\\end{lemma}\n\nFrom Lemma \\ref{le-Merris}, we \nget that $\\lambda=\\alpha(G)$ is also a Laplacian eigenvalue of $G+uv$ corresponding to $\\x$ if $\\x_u=\\x_v$. Then, it must be $\\alpha(G+uv)\\le \\alpha(G)$ and, since $\\alpha(G+uv)\\ge \\alpha(G)$, we conclude that\n$\\alpha(G)=\\alpha(G+uv)$.\nCombining Lemmas \\ref{le-Xue} and \\ref{le-Merris}, we can obtain the following lemma, which gives a \\textbf{necessary and sufficient condition} for $\\alpha(G)=\\alpha(G+uv)$.\n\n\\begin{lemma}\n\\label{le-adding_edge_iff}\nLet $G$ be a graph on $n$ vertices.\nThen, $\\alpha(G)=\\alpha(G+uv)$ if and only if there exists a Fiedler vector $\\x$ of $G$ such that $\\x_u=\\x_v$. \n\\end{lemma}\nOf course, by applying repeatedly Lemma \\ref{le-adding_edge_iff}, if $G'$\nis obtained from $G$ by adding $r$ edges $u_iv_i$, we have that $\\alpha(G)=\\alpha(G')$ if and only if there exists a Fiedler vector $\\x$\nof $G$ such that $\\x_{u_i}=\\x_{v_i}$ for $i=1,\\ldots,r$.\n\nSome preliminary results related to token graphs are presented below.\n\\begin{lemma}[Dalf\\'o, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and\nZaragoza Mart\\'{\\i}ınez \\cite{Dalfo_2021}]\n\\label{le-Dalfo2021}\nConsider a graph $G(\\cong F_1(G))$ and its $k$-token graph $F_k(G)$, with corresponding Laplacian matrices $\\L(G)$ and $\\L(F_k(G))$, and the $(n,k)$-binomial matrix $\\B$. The following statements hold:\n\\begin{itemize}\n\\item[$(i)$] If $\\x$ is a $\\lambda$-eigenvector of $\\L(G)$, then $\\B\\x$ is a $\\lambda$-eigenvector of $\\L(F_k(G))$.\n\\item[$(ii)$] If $\\w$ is a $\\lambda$-eigenvector of $\\L(F_k(G))$ and $\\B^{\\top}\\w\\ne 0$, then $\\B^{\\top}\\w$ is a $\\lambda$-eigenvector of $\\L(G)$.\n\\end{itemize}\n\\end{lemma}\n\nAs a consequence of Lemma \\ref{le-Dalfo2021}, we have the following result.\n\n\\begin{theorem}[Dalf\\'o, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and\nZaragoza Mart\\'{\\i}ınez \\cite{Dalfo_2021}]\n\\label{th-Dalfo2021}\n Let $G$ be a graph on $n$ vertices and let $k$ be an integer such that $1\\le k\\le n-1$. Then, the spectrum of $G$ is contained in the spectrum of its $k$-token graph $F_k(G)$. \n\\end{theorem}\n\n\\begin{theorem}[Dalf\\'o and Fiol \\cite{Dalfo_2024}, Dalf\\'o, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and\nZaragoza Mart\\'{\\i}ınez \\cite{Dalfo_2021}, Reyes, Dalf\\'o, Fiol, and Messegu\\'e \\cite{Reyes_2025}]\n\\label{th-Dalfo2024}\nFor each of the following classes of graphs, the algebraic connectivity of a token graph $F_k(G)$ satisfies the following statements.\n\\begin{itemize}\n\\item[$(i)$]\nLet $T_n$ be a tree on $n$ vertices. Then,\n$\\alpha(F_k(T_n))=\\alpha(T_n)$ for every $n$ and $k=1,\\ldots,n-1$.\n\\item[$(ii)$]\nLet $G$ be a graph such that $\\alpha(F_k(G))=\\alpha(G)$. Let $T_G$ be a graph in which each vertex of $G$ is the root vertex of some (possibly empty) tree. Then, \n$\\alpha(F_k(T_G))=\\alpha(T_G)$.\n\\item[$(iii)$]\nLet $G=K_{n_1,n_2}$ be a complete bipartite graph on $n=n_1+n_2$ vertices, with $n_1\\le n_2$. Then, $\\alpha(F_k(G))=\\alpha(G)=n_1$ for every $n_1,n_2$ and $k=1,\\ldots,n-1$.\n\\item[$(iv)$]\nLet $G=K_n$ be a complete graph on $n$ vertices. Then, $\\alpha(F_k(G))=\\alpha(G)=n$ for every $k=1,\\ldots,n-1$.\n\\item[$(v)$]\nLet $C_n$ be a cycle on $n$ vertices. Then, $\\alpha(F_k(C_n))=\\alpha(C_n)$ for $k=1,2$.\n\\end{itemize}\n\\end{theorem}\n\\begin{lemma}[Barik and Verma \\cite{bv24}]\nLet $G$ be a graph on $n\\ge 4$ vertices, and $H$ be the graph formed by adding a\npendant vertex (say $v$, with the corresponding edge) to graph $G$. Then, for any integer k such that $2\\le k\\le n/2$,\n$$\n\\alpha(F_k(H)) \\le \\min\\{\\alpha(F_{k-1}(G)) + 1, \\alpha(F_k(G)) + 1\\}.\n$$\n\\end{lemma}\n\\begin{theorem}[Barik and Verma \\cite{bv24}]\n\\label{theBarik}\nLet $G$ be a graph on $n$ vertices and $v$ be a cut vertex in $G$ such that $d_G(v)=n-1$. Then, for any integer $k$ such that $2\\le k\\le \\frac{n}{2}$,\n$$\n\\alpha(F_k(G))=\\alpha(G)=1.\n$$\n\\end{theorem}\n\n\\section{Adding edges}\n\\label{sec:+edges}\nIn this section and the following ones, we present the new results of the paper. We begin by deriving results about the algebraic connectivities of a graph and the same graph after adding a new edge $uv$.\n\n\n\\subsection{A basic result}\n\n\\begin{theorem}\n\\label{th-add_edge}\nLet $G=(V,E)$ be a graph with order $n$ such that, for some $k$ with $2\\le k\\le n/2$, we have $\\alpha(F_k(G))=\\alpha(G)$. Consider adding an edge\n$uv$, for $u,v\\in V$, getting the new graph $G+uv$. If $\\alpha(G+uv)=\\alpha(G)$, then $\\alpha(F_k(G+uv))=\\alpha(G+uv)$.\n\\end{theorem}\n\\begin{proof}\n Note that $F_k(G)$ is a spanning subgraph of $F_k(G+uv)$ with \n \\[\n E(F_k(G+uv))\\backslash E(F_k(G))=\\{A_{r}A_{s} : A_{r}=\\{u,u_1,\\ldots,u_{k-1}\\}, A_{s}=\\{v,u_1,\\ldots,u_{k-1}\\}\\},\n \\]\n where $u_1,\\ldots,u_{k-1}\\in V(G)\\backslash \\{u,v\\}$. \n Then, by Lemma \\ref{le-interlacing},\n \\begin{equation}\n \\alpha(F_k(G+uv)){\\ge}\\alpha(F_k(G)).\\label{eqthe2'}\n \\end{equation}\n Since $\\alpha(G+uv)=\\alpha(G)$, by Lemma \\ref{le-adding_edge_iff}, there exists a Fiedler vector $\\x$ of $G$ such that $\\x_{u}=\\x_{v}$. \n Let $\\y=\\B\\x$, where $\\B$ is the $(n,k)$-binomial matrix. \n It follows from the hypothesis $\\alpha(F_k(G))=\\alpha(G)$ and Lemma \\ref{le-Dalfo2021}($i$) that $\\y$ is a Fiedler vector of $F_k(G)$. Moreover, observe that\n $$\n \\y_{A_r}-\\y_{A_s}=\\x_u+\\sum_{i=1}^{k-1}\\x_{u_i}-\\left(\\x_v+\\sum_{i=1}^{k-1}\\x_{u_i}\\right)=\\x_u-\\x_v=0.\n $$\n Hence, we get \n \\begin{align}\n \\alpha(F_k(G))=\\frac{\\y^{\\top}\\L(F_k(G))\\y}{\\y^{\\top}\\y}&=\\frac{\\sum_{A_rA_s\\in E(F_k(G))}(\\y_{A_r}-\\y_{A_s})^2}{\\y^{\\top}\\y}\\notag\\\\\n &=\\frac{\\sum_{A_rA_s\\in E(F_k(G+uv))}(\\y_{A_r}-\\y_{A_s})^2}{\\y^{\\top}\\y}\\notag\\\\\n &\\ge \\alpha(F_k(G+uv)).\\label{eqthe3'}\n \\end{align}\n Thus, from \\eqref{eqthe2'}, \\eqref{eqthe3'}, and the hypothesis, we conclude that $\\alpha(F_k(G+uv))=\\alpha(F_k(G))=\\alpha(G)=\\alpha(G+uv)$.\n \n Alternatively, we have\n \\begin{equation*}\n \\alpha(F_k(G+uv))\\ge\\alpha(F_k(G))=\\alpha(G)=\\alpha(G+uv).\n \\label{eq1'}\n \\end{equation*}\n Then, the result follows since, by Theorem \\ref{th-Dalfo2021}, it must be $\\alpha(F_k(G+uv))\\le \\alpha(G+uv)$.\\qed\n\\end{proof}\n\n\n\\begin{example}\nThe graph $Y=(V,E)$ of Figure \\ref{fig:Y+F2(Y)} has algebraic connectivity $\\alpha(Y)=0.5188$ with corresponding eigenvector $\\x=(-0.5969,-0.5969,-2.8772,0.4812,1)^{\\top}$. Then, since $\\x_1=\\x_2$, its `extended' graph $Y+12$ has the same algebraic connectivity\n$\\alpha(Y+12)=\\alpha(Y)$, with the same eigenvector $\\x$.\nThis graph is shown in Figure \\ref{fig:F2(Y+12)} together with its 2-token graph $F_2(Y+12)$. Notice that $F_2(Y)$ is a spanning subgraph of $F_2(Y+12)$, where the `new' edges induced by $12$ are $A_rA_s\\in \\{13\\sim 23, 14\\sim 24, 15\\sim 25\\}$. Since $\\alpha(F_2(Y))=\\alpha(Y)$, Theorem \\ref{th-add_edge} implies that $\\alpha(F_2(Y+12))=\\alpha(Y+12)=\\alpha(Y)=0.5188$. \n\\end{example}\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=1.1,auto,swap]\n \\vertex (1) at (0.4,2.25) [fill,label=above:{$1$}]{};\n \\vertex (2) at (1.6,2.25) [fill,label=above:{$2$}]{};\n \\vertex (3) at (1,1) [fill,label=left:{$3$}]{};\n \\vertex (4) at (1,-0.2) [fill,label=left:{$4$}]{};\n \\vertex (5) at (1,-1.4) [fill,label=left:{$5$}]{};\n \\draw[line width=0.6pt](1)--(3);\n \\draw[blue,line width=0.6pt](1)--(2);\n \\draw[line width=0.6pt](2)--(3);\n \\draw[line width=0.6pt](3)--(4);\n \\draw[line width=0.6pt](4)--(5);\n \\vertex (12) at (6.5,2.55) [fill,label=above:{$12$}]{};\n \\vertex (34) at (6.5,1.3) [fill,label=above:{$34$}]{};\n \\vertex (35) at (6.5,0.1) [fill]{};\n \\node () at (6.6,-0.05) [label=above left:{$35$}]{};\n \\vertex (45) at (6.5,-1.5) [fill,label=left:{$45$}]{};\n \\vertex (13) at (5.2,2.05) [fill,label=left:{$13$}]{};\n \\vertex (14) at (5.2,0.8) [fill,label=left:{$14$}]{};\n \\vertex (15) at (5.2,-0.4) [fill,label=left:{$15$}]{};\n \\vertex (23) at (7.8,2.05) [fill,label=right:{$23$}]{};\n \\vertex (24) at (7.8,0.8) [fill,label=right:{$24$}]{};\n \\vertex (25) at (7.8,-0.4) [fill,label=right:{$25$}]{};\n \\draw[line width=0.6pt](12)--(13);\n \\draw[line width=0.6pt](12)--(23);\n \\draw[line width=0.6pt](23)--(24);\n \\draw[line width=0.6pt](13)--(14);\n \\draw[line width=0.6pt](14)--(34);\n \\draw[line width=0.6pt](34)--(24);\n \\draw[line width=0.6pt](24)--(25);\n \\draw[line width=0.6pt](34)--(35);\n \\draw[line width=0.6pt](14)--(15);\n \\draw[line width=0.6pt](35)--(15);\n \\draw[line width=0.6pt](35)--(25);\n \\draw[line width=0.6pt](35)--(45);\n \\draw[blue,line width=0.6pt](13)--(23);\n \\draw[blue,line width=0.6pt](14)--(24);\n \\draw[blue,line width=0.6pt](15)--(25);\n \\end{tikzpicture}\n \\caption{The graph $Y+12$ and its 2-token graph $F_2(Y+12)$.}\\label{fig:F2(Y+12)}\n \\end{center}\n\\end{figure}\n\n\\subsection{Extended graphs with pendant vertices}\n\nFrom the result in Theorem \\ref{th-add_edge},\nit is natural to consider graphs satisfying $\\alpha(G)=\\alpha(G+uv)$ for some edge $uv$.\nA family of such graphs is given in the following Lemma \\ref{le-Shao2008}, whose statement\ncan be made more explicit by first computing the value of a particular eigenvalue. With this aim, \ngiven a vertex subset $V'\\subseteq V$, let $\\L_{[V']}(G)$ denote the principal submatrix of $\\L(G)$ whose rows and columns correspond to $V'$. \nWhen $G$ is a path $P_{r+1}$ with vertices $u_0,u_1,\\ldots,u_r$ and $V_r=\\{u_1,\\ldots,u_r\\}$, $\\L_{[V_r]}(G)$ is the $r\\times r$ tridiagonal matrix\n$$\n \\L_{[V_r]}(P_{r+1}) =\n {\\scriptsize\\left(\n \\begin{array}{cccccc}\n 2 & -1 & 0 & 0 & 0 & 0\\\\\n -1 & 2 & -1 & 0 & 0 & 0\\\\\n 0 & -1 & \\ddots & \\ddots & 0 & 0\\\\\n 0 & 0 & \\ddots & \\ddots & -1 & 0\\\\\n 0 & 0 & 0 & -1 & 2 & -1 \\\\\n 0 & 0 & 0 & 0 & -1 & 1 \n \\end{array}\n \\right)},\n $$\nwith eigenvalues\n\\begin{equation}\n\\label{eq:theta-k}\n\\theta_k=2+2\\cos\\left(\\frac{2k\\pi}{2r+1}\\right),\\quad\\mbox{for $k=1,\\ldots,r$}.\n\\end{equation}\n(See Yueh \\cite[Th. 1]{y05}). \nThe minimum eigenvalue is obtained when $k=r$ (since $\\theta_k$ is decreasing with $k$), so we have that\n $\\lambda_1(\\L_{[V_r]}(P_{r+1}))=\\theta_r$, and the following result of \n Shao, Guo, and Shan \\cite{Shao_2008} reads as follows.\n \n\\begin{lemma}[Shao, Guo, and Shan \\cite{Shao_2008}]\n\\label{le-Shao2008}\nLet $v$ be a vertex in a connected graph $H$ and suppose that $s(\\ge 2)$ new paths (with equal length $r$) $vv_{i1}v_{i2}\\cdots v_{ir}$ (for $i=1,\\ldots,s$ and $r\\ge 1$) are attached to $H$ at $v$ to form a new graph $G$. \nLet $G^+$ be the graph obtained from $G$ by arbitrarily adding edges among the vertices $v_{1j},v_{2j},\\ldots,v_{sj}$ for any given $j=1,\\ldots,r$. \nIf $\\alpha(G)\\neq 2+2\\cos\\left(\\frac{2r\\pi}{2r+1}\\right)$,\nthen $\\alpha(G^+)=\\alpha(G)$.\n\\end{lemma}\n\n\\begin{example}\nA simple example of Lemma \\ref{le-Shao2008} is obtained when $G=Y$, the graph of Figure \\ref{fig:Y+F2(Y)}, where $s=2$ and $r=1$. Then, as $\\alpha(Y)=0.5188\\neq \\theta_1=1$, the graph $G^+=Y+12$ of Figure \\ref{fig:F2(Y+12)} satisfies $\\alpha(G^+)=\\alpha(G)$.\nMoreover, Figure \\ref{fig:example} is an example when the statement of Lemma \\ref{le-Shao2008} does not hold because the necessary conditions fail. In the latter graph, we have $\\alpha(G^+)=0.2679$, which is different from $\\alpha(G^+-v_{13}v_{23})=0.1981$. This is because $\\alpha(G)=\\alpha(G^+-\\{v_{13}v_{23}, v_{11}v_{31}\\})=\\lambda_1(\\L_{[V_r]}(P_{r+1}))=\\theta_r=0.1981$ (see Table \\ref{tab1} for $r=3$).\n\\begin{table}\n\\begin{center}\n{\\small\n\\begin{tabular}{|c||cccccccccc|}\n\\hline\n$r$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\\\\n\\hline\n$\\theta_r$ & 1 & 0.3820 & 0.1981 & 0.1206 & 0.0810 & 0.0581 & 0.0437 & 0.0341 & 0.0273 & 0.0223\\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\caption{The values of $\\theta_r=\\lambda_1(\\L_{[V_r]}(P_{r+1}))$ for different values of $r$.}\n\\label{tab1}\n\\end{table}\n\\end{example}\n\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=1.1,auto,swap]\n \\vertex (0) at (1,1) [fill,label=above:{$v$}]{};\n \\vertex (11) at (2.2,1.8) [fill,label=above:{$v_{11}$}]{};\n \\vertex (12) at (3.4,2.3) [fill,label=above:{$v_{12}$}]{};\n \\vertex (13) at (4.6,2.7) [fill,label=above:{$v_{13}$}]{};\n \\vertex (21) at (2.7,1.3) [fill,label=above:{$v_{21}$}]{};\n \\vertex (22) at (3.8,1.45) [fill,label=above:{$v_{22}$}]{};\n \\vertex (23) at (5,1.5) [fill,label=right:{$v_{23}$}]{};\n \\vertex (31) at (2.3,0.7) [fill,label=below:{$v_{31}$}]{};\n \\vertex (32) at (3.5,0.4) [fill,label=below:{$v_{32}$}]{};\n \\vertex (33) at (4.6,0.1) [fill,label=below:{$v_{33}$}]{};\n \\vertex (41) at (-0.2,1.2) [fill,label=above:{$v_{41}$}]{};\n \\vertex (51) at (0,0.2) [fill,label=below:{$v_{51}$}]{};\n \\draw[line width=0.6pt](0)--(11);\n \\draw[line width=0.6pt](0)--(21);\n \\draw[line width=0.6pt](0)--(31);\n \\draw[line width=0.6pt](0)--(41);\n \\draw[line width=0.6pt](0)--(51);\n \\draw[line width=0.6pt](11)--(12);\n \\draw[line width=0.6pt](12)--(13);\n \\draw[line width=0.6pt](13)--(23);\n \\draw[line width=0.6pt](11)--(31);\n \\draw[line width=0.6pt](21)--(22);\n \\draw[line width=0.6pt](22)--(23);\n \\draw[line width=0.6pt](31)--(32);\n \\draw[line width=0.6pt](32)--(33);\n \\draw[line width=0.6pt](41)--(51);\n \\end{tikzpicture}\n \\caption{An example of $G^+$ where $H=K_3$ and $s=r=3$ in Lemma \\ref{le-Shao2008}.}\\label{fig:example}\n \\end{center}\n\\end{figure}\nIt has been shown in Kirland \\cite{Kirkland_2000} that, for a connected graph $G$ with $n$ vertices and a cut vertex $v$, we have $\\alpha(G)\\le 1$ and the equality holds if and only if $d_G(v)=n-1$. We get the following theorem by combining this with \nprevious results.\n\\begin{theorem}\n\\label{th-addedge-pendant}\n Let $G$ be a graph on $n$ vertices. Suppose that $v_1,v_2,\\ldots,v_s$ (for $s\\ge2$) are $s$ pendant vertices of $G$ adjacent to a common vertex $v$ with $d_G(v)\\ne n-1$. \n Let $G^+$ be a graph obtain from $G$ by adding any $t$ $\\left(\\mbox{for } 0\\le t\\le \\frac{s(s-1)}{2}\\right)$ edges among $v_1,v_2,\\ldots,v_s$. \n If $\\alpha(F_k(G))=\\alpha(G)$, then $\\alpha(F_k(G^+))=\\alpha(G^+)$. \n\\end{theorem}\n\\begin{proof}\n Note that $v$ is a cut vertex of $G$. Since $d_G(v)\\ne n-1$, we have $\\alpha(G)<1=\\theta_1$. It follows from Lemma \\ref{le-Shao2008} and Theorem \\ref{th-add_edge} that the results hold.\\qed\n\\end{proof}\n\n\n\nThe following corollary holds directly from Theorems \\ref{th-Dalfo2024}$(i)$ and \\ref{th-addedge-pendant}.\n\\begin{corollary}\n Let $T$ be a tree on $n$ vertices. Suppose that $v_1,v_2,\\ldots,v_s$ (for $s\\ge2$) are $s$ pendant vertices of $T$ adjacent to a common vertex $v$ with $d_T(v)\\ne n-1$. \n Let $T^+$ be a graph obtain from $T$ by adding any $t$ $\\left(\\mbox{for } 0\\le t\\le \\frac{s(s-1)}{2}\\right)$ edges among $v_1,v_2,\\ldots,v_s$. \n Then, $\\alpha(F_k(T^+))=\\alpha(T^+)$.\n\\end{corollary}\n\n\\subsection{Extended cycles}\nLet $G=(V,E)=C_n$ be a cycle with $n$ vertices, $v_0,v_1,\\ldots v_{n-1}$, and let $C^+_n=(V,E^+)$ be a graph obtained from $C_n$ by adding some edges in $E_0=\\{v_iv_j:i+j=\\nu\\}$, where $\\nu=n$ if $n$ is odd and $\\nu\\in \\{n,n-1\\}$ otherwise, so that $E^+=E\\cup E_0$.\nGuo, Zhang, and Yu \\cite{Guo_2018} showed \n that there exists a Fiedler vector $\\x$ of $C_n$ such that $\\x_{v_i}=\\x_{v_{n-i}}$ (for $i=1,\\ldots,(n-1)/2$), and an integer $s$ (with $1\\le s\\le (n-3)/2$) such that $\\x_{v_0}>\\x_{v_1}>\\cdots>\\x_{v_s}>0\\ge \\x_{v_{s+1}}>\\cdots>\\x_{v_{\\frac{n-1}{2}}}$ for odd $n$. (For even $n$, there is a similar result). Consequently, they proved that \nadding the edges in $E_0$ does not change the algebraic connectivity: $\\alpha(C_n)=\\alpha(C^+_n)$.\nCombining this fact with Theorems \\ref{th-add_edge} and \\ref{th-Dalfo2024}$(v)$, we obtain the following result. \n\\begin{theorem}\n Let $C_n$ be a cycle with $n$ vertices, and $C^+_n$ a graph obtained from $C_n$ by adding some edges in $E_0=\\{v_iv_j:i+j=\\nu\\}$. \n Then, $\\alpha(F_{k}(C^+_n))=\\alpha(C^+_n)$ for $k=1,2$.\n\\end{theorem}\n\n\n\\subsection{Extended complete bipartite graphs}\nNow, we consider adding new edges to complete bipartite graphs.\n\\begin{theorem}\n\\label{th-bip-addedge1}\n Let $K_{n_1,n_2}$ be a complete bipartite graph on $n=n_1+n_2$ vertices, with bipartition $(X,Y)$ such that $|X|=n_1\\le n_2=|Y|$. Let $K^+_{n_1,n_2}$ be a graph obtained from $K_{n_1,n_2}$ by adding any $t$ $\\left(\\mbox{for } 0\\le t\\le \\frac{n_1(n_1-1)}{2}\\right)$ edges among $X$. Then, $\\alpha(F_k(K^+_{n_1,n_2}))=\\alpha(K^+_{n_1,n_2})=n_1$ for every $n_1,n_2$ and $k=1,\\ldots,n-1$.\n\\end{theorem}\n\\begin{proof}\n Denote by $K^*_{n_1,n_2}$ the graph obtained from $K_{n_1,n_2}$ by adding all edges among $X$, that is, $G[X]\\cong K_{n_1}$.\n It can be shown that the spectrum of $K^*_{n_1,n_2}$ is $\\{0,n_1^{n_2-1},n^{n_1}\\}$.\n Thus, $\\alpha(K^*_{n_1,n_2})=n_1$. \n It follows from Lemma \\ref{le-interlacing} that \n \\begin{equation*}\n \\alpha(K^*_{n_1,n_2})\\ge \\alpha(K^+_{n_1,n_2})\\ge\\alpha(K_{n_1,n_2}).\n \\end{equation*}\n Note that $\\alpha(K^*_{n_1,n_2})=\\alpha(K_{n_1,n_2})=n_1$. We obtain $\\alpha(K^+_{n_1,n_2})=n_1=\\alpha(K_{n_1,n_2})$.\n Then, the result follows from Theorems \\ref{th-Dalfo2024}$(iii)$ and \\ref{th-add_edge}.\\qed\n\\end{proof}\nHow about adding edges among the vertices in $Y$ in the complete bipartite graph? We consider adding all edges among $Y$ based on the following lemma. \nFor a vertex $v\\in V(G)$, let $S_v:=\\{A\\in V(F_k(G)):v\\in A\\}$ and $S_v':=\\{B\\in V(F_k(G)):v\\notin B\\}$. \nLet $H_v$ and $H_v'$ be the subgraphs of $F_k(G)$ induced by $S_v$ and $S_v'$, respectively. Note that $H_v\\cong F_{k-1}(G-v)$ and $H_v'\\cong F_k(G-v)$. \n\\begin{lemma}[Dalf\\'o and Fiol \\cite{Dalfo_2024}]\n \\label{le-embedding}\n Given a vertex $v\\in V(G)$ and an eigenvector $\\w$ of $F_k(G)$ such that $\\B^{\\top}\\w=\\vec0$, let $\\y_v:=\\w|_{S_v}$ and $\\y_v':=\\w|_{S_v'}$.\n Then, $\\y_v$ and $\\y_v'$ are embeddings of $H_v$ and $H_v'$, respectively.\n\\end{lemma}\n\\begin{theorem}\n\\label{th-bip-addedge2}\n Let $G=K_{n_1,n_2}$ be a complete bipartite graph on $n=n_1+n_2$ vertices, with bipartition $(X,Y)$ such that $2\\le n_1\\le n_2$, where $|X|=n_1$ and $|Y|=n_2$. Let $K^*_{n_1,n_2}$ be a graph obtained from $K_{n_1,n_2}$ by adding all edges among $Y$. Then, $\\alpha(F_k(K^*_{n_1,n_2}))=\\alpha(K^*_{n_1,n_2})=n_2$ for every $n_1,n_2$ and $k=1,\\ldots,n-1$.\n\\end{theorem}\n\n\\begin{proof}\n Note that the spectrum of $K^*_{n_1,n_2}$ is $\\{0,n_2^{n_1-1},n^{n_2}\\}$.\n Thus, $\\alpha(K^*_{n_1,n_2})=n_2$.\n Recall that $\\alpha(F_k(K^*_{n_1,n_2}))\\le \\alpha(K^*_{n_1,n_2})=n_2$. \n Let $\\w$ be a Fiedler vector of $F_k(K^*_{n_1,n_2})$. Without loss of generality, suppose that $\\w^{\\top}\\w=1$. \n It follows from Lemma \\ref{le-Dalfo2021}($ii$) that, if $\\B^{\\top}\\w\\ne\\vec0$, then $\\alpha(F_k(K^*_{n_1,n_2}))$ is also an eigenvalue of $K^*_{n_1,n_2}$, so that $\\alpha(F_k(K^*_{n_1,n_2}))=\\w^{\\top}\\L(F_k(K^*_{n_1,n_2}))\\w\\ge \\alpha(K^*_{n_1,n_2})=n_2$.\n Then, it suffices to show $\\alpha(F_k(K^*_{n_1,n_2}))\\ge n_2$ when $\\B^{\\top}\\w=\\vec0$.\n We prove it by induction on $n_1(\\ge2)$ and $k$. For $k=1$, the claim holds, since $F_1(K^*_{n_1,n_2})\\cong K^*_{n_1,n_2}$. \n For $n_1=2$, note that the claim is true if $k=1$. Then, we assume that $\\alpha(F_{k-1}(K^*_{n_1,n_2}))\\ge n_2.$\n Let $v$ be a vertex in $X$. \n Thus, $G-v\\cong K_{n_2+1}$ is a complete graph.\n Let $\\y_v:=\\w|_{S_v}$ and $\\y_v':=\\w|_{S_v'}$ such that, by Lemma \\ref{le-embedding}, are embeddings of $H_v\\cong F_{k-1}(G-v)$ and $H_v'\\cong F_k(G-v)$. Then,\nfrom (\\ref{eq-rayleigh}), the hypothesis, and Lemma \\ref{th-Dalfo2024}$(iv)$, we have \n\\[\n\\frac{\\sum_{(A,B)\\in E(H_v)}\\left((\\y_v)_A-(\\y_v)_B\\right)^2}{\\sum_{A\\in V(H_v)}((\\y_v)_A)^2}\\ge \\alpha(H_v)=\\alpha(F_{k-1}(G-v))\\ge n_2\n\\] \nand \n\\[\n\\frac{\\sum_{(A,B)\\in E(H_v')}((\\y_v')_A-(\\y_v')_B)^2}{\\sum_{A\\in V(H_v')}((\\y_v')_A)^2}\\ge \\alpha(H_v')=\\alpha(F_k(G-v))=n_2+1.\n\\]\nThus, \n \\begin{eqnarray*}\n \\alpha(F_{k}(K^*_{n_1,n_2}))&=&\\w^{\\top}\\L(F_k(K^*_{n_1,n_2}))\\w\\\\\n &=&\\sum_{(A,B)\\in E(F_k(K^*_{n_1,n_2}))}(\\w_A-\\w_B)^2\\\\\n &\\ge &\\sum_{(A,B)\\in E(H_v)}\\left((\\y_v)_A-(\\y_v)_B\\right)^2+\\sum_{(A,B)\\in E(H_v')}\\left((\\y_v')_A-(\\y_v')_B\\right)^2\\\\\n &\\ge &\\alpha(F_{k-1}(G-v))\\sum_{A\\in V(H_v)}\\left((\\y_v)_A\\right)^2+\\alpha(F_k(G-v))\\sum_{A\\in V(H_v')}\\left((\\y_v')_A\\right)^2\\\\\n &\\ge &n_2\\left[\\sum_{A\\in V(H_v)}(\\w_A)^2+\\sum_{A\\in V(H_v')}(\\w_A)^2\\right]\\\\\n &=&n_2. \n \\end{eqnarray*}\n For $n_1>2$ and $k>1$, we obtain the claim using a similar approach as above. \n Together with $\\alpha(F_k(K^*_{n_1,n_2}))\\le n_2$, the result holds.\n \\qed\n\\end{proof}\n\n\n\n\n\\section{Kite graphs}\n\\label{sec:kite}\nThe graphs in Lemma \\ref{le-Shao2008} suggest the following definition. \n\n\\begin{definition}\nA {\\em kite} $K(H,T)$ is a graph consisting of a {\\em head} $H$ that is a general graph, \nand a {\\em tail} $T$ consisting of $s\\ge 2$ paths of equal length $r$ with the common end-vertex $v\\in V(H)$.\n\\end{definition}\n\nExamples of kite graphs are shown in Figures \\ref{fig:Kite2}, \\ref{fig:kite}, and \\ref{fig:ex-th-kite}. Thus, general kite graphs appear in Lemma \\ref{le-Shao2008}, whereas the graphs in Theorem \\ref{th-addedge-pendant} correspond to kite graphs with tail length $r=1$.\nBefore proving the main theorems about kite graphs, we first need some results.\n\\begin{lemma}[Bapat and Pati \\cite{Bapat_1998}]\n\\label{le-Bapat1998}\n Let $G$ be a connected graph. Let $W$ be a set of vertices of $G$ such that $G- W$ is disconnected. \n Let $G_1$ and $G_2$ be two components of $G-W$. \n Suppose that $\\lambda_1(\\L_{[V(G_1)]}(G))\\le \\lambda_1(\\L_{[V(G_2)]}(G))$. Then, one of the two following conditions holds:\n \\begin{itemize}\n\\item[$(i)$] \n$\\alpha(G)< \\lambda_1(\\L_{[V(G_2)]}(G))$;\n\\item[$(ii)$] \n$\\alpha(G)=\\lambda_1(\\L_{[V(G_1)]}(G))= \\lambda_1(\\L_{[V(G_2)]}(G))$.\n \\end{itemize} \n \\end{lemma}\nThe following result gives a sufficient condition for having the equality\nin Lemma \\ref{le-Bapat1998}$(ii)$.\n\\begin{lemma}\\label{le-suffcondition}\n Let $G$ be a connected graph with a cut vertex $v$. \n Let $G_1,G_2,\\ldots,G_r$ be $r$ components of $G-v$ with $\\lambda_1(\\L_{[V(G_1)]}(G))\\le \\lambda_1(\\L_{[V(G_2)]}(G))\\le \\cdots \\le\\lambda_1(\\L_{[V(G_r)]}(G))$. \n If $\\lambda_1(\\L_{[V(G_1)]}(G))=\\lambda_1(\\L_{[V(G_2)]}(G))$, then $\\alpha(G)=\\lambda_1(\\L_{[V(G_1)]}(G))=\\lambda_1(\\L_{[V(G_2)]}(G))$. \n\\end{lemma}\n\\begin{proof}\nIt follows from Lemma \\ref{le-principal_submatrix} that \n\\begin{equation}\n\\label{eq:ineq-lambdas}\n\\lambda_1(\\L(G))\\le \\lambda_1(\\L_{[V(G-v)]}(G))\\le \\lambda_2(\\L(G))\\le \\lambda_2(\\L_{[V(G-v)]}(G)).\n\\end{equation}\nSince $\\lambda_1(\\L_{[V(G_1)]}(G))=\\lambda_1(\\L_{[V(G_2)]}(G))$, we have $\\lambda_1(\\L_{[V(G-v)]}(G))= \\lambda_2(\\L_{[V(G-v)]}(G))$. \nThis is because $\\L_{[V(G-v)]}(G)$ is a block diagonal matrix with blocks $\\L_{[V(G_i)]}(G)$ for $i=1,\\ldots,r$ and, hence, the spectrum of the former is the union of the spectra of the latter.\nThen, \\eqref{eq:ineq-lambdas} gives $\\alpha(G)=\\lambda_2(\\L(G))=\\lambda_1(\\L_{[V(G_1)]}(G))=\\lambda_1(\\L_{[V(G_2)]}(G))$. \\qed\n\\end{proof}\nWe next give a sufficient and necessary condition for $\\alpha (G)=2+2\\cos(\\frac{2r\\pi}{2r+1})$ if $G$ is a kite graph.\n\\begin{lemma}\n\\label{le-kite-iff}\n Let $G=K(H,T)$ be a kite graph with a tail of length $r$ formed with $s(\\ge 2)$ paths. Then, $\\alpha (G)=2+2\\cos(\\frac{2r\\pi}{2r+1})$ if and only if $\\lambda_1(\\L_{[V(H-v)]}(H))\\ge 2+2\\cos(\\frac{2r\\pi}{2r+1}).$\n\\end{lemma}\n\\begin{proof}\n Note that \\[\\L_{[V(G-v)]}(G)=\\diag\\left(\\L_{[V(H-v)]}(H),\\L_{[V_{r}]}(P_{r+1}),\\ldots,\\L_{[V_{r}]}(P_{r+1})\\right),\\] \n and $\\lambda_1(\\L_{[V_{r}]}(P_{r+1}))=2+2\\cos(\\frac{2r\\pi}{2r+1})$. \n If $\\lambda_1(\\L_{[V(H-v)]}(H))\\ge \\lambda_1(\\L_{[V_{r}]}(P_{r+1}))$, then it follows from Lemma \\ref{le-suffcondition} that $\\alpha(G)=\\lambda_1(\\L_{[V_{r}]}(P_{r+1}))=2+2\\cos(\\frac{2r\\pi}{2r+1})$.\n \nConversely, suppose, by way of contradiction, that $\\lambda_1(\\L_{[V(H-v)]}(H))<\\lambda_1(\\L_{[V_{r}]}(P_{r+1}))$. Then, it follows from Lemma \n\\ref{le-Bapat1998} that $\\alpha(G)<\\lambda_1(\\L_{[V_{r}]}(P_{r+1}))=2+2\\cos(\\frac{2r\\pi}{2r+1})$, a contradiction.\nHence, we obtain $\\lambda_1(\\L_{[V(H-v)]}(H))\\ge 2+2\\cos(\\frac{2r\\pi}{2r+1})$.\\qed\n\\end{proof}\n\n\n\\begin{proposition}\n\\label{prop:LvsS}\n Let $G$ be a kite graph $K(H,T)$ with head $H$ and tail $T$ with $s(\\ge 2)$ paths of length $r$, with vertices $v_{i1},v_{i2},\\ldots,v_{ir}$ for $i=1,\\ldots,s$. Let $G^+$ be a graph obtained from $G$ by adding some edges among the vertices of each set $U_j=\\{v_{2j},v_{3j},\\ldots,v_{sj}\\}$ for each given $j\\in \\{1,\\ldots,r\\}$. Let $\\L$ be the Laplacian matrix of $G$. Let $\\SS=(s_{ij})$ be the matrix, indexed also by the vertices of $G$, with entries\n $$\n s_{uv}=\\left\\{\n \\begin{array}{lll}\n 1 & \\mbox{if } u=v\\in V(H)\\cup\\{v_{11},v_{12}, \\ldots,v_{1r}\\},\\\\\n\\frac{1}{s-1} & \\mbox{if } u,v\\in U_j\\quad \\mbox{for every } j=1,\\ldots,r,\\\\\n 0 & \\mbox{elsewhere.}\n \\end{array}\n \\right.\n $$\n Then, the following statements hold. \n \\begin{itemize}\n\\item[$(i)$]\nThe matrices $\\L$ and $\\SS$ commute: $\\L\\SS=\\SS\\L$.\n\\item[$(ii)$]\nFor every eigenvalue $\\lambda$ of $G$, there exists a corresponding eigenvector $\\x$ of $G$ such that $\\x_{v_{2j}}=\\x_{v_{3j}}=\\cdots=\\x_{v_{sj}}$ for $1\\le j\\le r$.\n\\item[$(iii)$]\nEvery different eigenvalue of $G$ appears also as an eigenvalue of \n$G^{+}$ (not necessarily including multiplicities). In particular, $\\alpha(G^{+})=\\alpha(G)$.\n\\end{itemize}\n \\end{proposition}\n \\begin{proof}\n $(i)$ For simplicity, assume that $H=\\{v\\}$ (if not, the proof is basically the same). Then, the block matrices $\\L$ and $\\SS$ are\n $$\n \\L=\n {\\scriptsize{\\left(\n \\begin{array}{cccccc}\n d_v & -\\j^{\\top} & \\vec0 & \\vec0 & \\cdots & \\vec0\\\\\n -\\j & 2\\I & -\\I & \\O & \\cdots & \\O\\\\\n \\vec0 & -\\I & \\ddots & \\ddots & \\cdots & \\vdots\\\\\n \\vec0 & \\O & \\ddots & \\ddots & -\\I & \\O\\\\\n \\vdots & \\vdots & \\vdots & -\\I & 2\\I & -\\I \\\\\n \\vec0 & \\O & \\cdots & \\O & -\\I & \\I \n \\end{array}\n \\right)}},\\quad\\mbox{and}\\quad\n \\SS=\n \\diag(1,\\SS_{11},\\ldots,\\SS_{rr})\n $$\n where $\\SS_{11}=\\cdots=\\SS_{rr}=\\diag\\left(1,\\frac{1}{s-1}\\J\\right)$. Then, the computation of $\\L\\SS$ and $\\SS\\L$ shows that the commutativity comes from $\\j^{\\top}\\SS_{11}=\\j^{\\top}$ and $\\SS_{11}\\j=\\j$.\\\\\n $(ii)$ Let $\\y$ be a $\\lambda$-eigenvector of $G$ so that $\\L\\y=\\lambda\\y$. First, notice that the vector $\\x=\\SS\\y$ satisfy the condition since\n \\begin{align}\n \\x_{v} &=\\y_{v},\\quad\\text{ if } u\\in V(H)\\cup\\{v_{11},v_{12},\\ldots,v_{1r}\\};\n \\label{eq-kite-vec1}\\\\\n \\x_{v_{2j}} &=\\x_{v_{3j}}=\\cdots=\\x_{v_{sj}}=\\frac{\\y_{v_{2j}}+\\cdots+\\y_{v_{sj}}}{s-1},\\quad \\text{ if } j=1,\\ldots, r.\n \\label{eq-kite-vec2}\n \\end{align}\nMoreover, since $\\y\\neq \\vec0$ and all the paths of the tail are isomorphic, there is no loss of generality if we assume that $\\y_u\\neq 0$ for some $u$ in \\eqref{eq-kite-vec1}. Then,\n$\\x\\neq \\vec0$, and\n $$\n \\L\\x=\\L\\SS\\y=\\SS\\L\\y=\\lambda\\SS\\y=\\lambda\\x,\n $$\n so that $\\x$ is also a $\\lambda$-eigenvector of $G$.\\\\\n $(iii)$ As happens for the algebraic connectivity in Lemma \\ref{le-adding_edge_iff}, if a $\\lambda$-eigenvector $\\x$ of $G$ has equal components $\\x_u=\\x_v$, as in \\eqref{eq-kite-vec2}, the graph $G^{+}$, obtained by adding the edge $uv$,\n has the same eigenvector $\\x$ and eigenvalue $\\lambda$. \\qed\n \\end{proof}\n\n\\begin{example}\nThe kite graph $G$ of Figure \\ref{fig:Kite2} (left) \n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=1.1,auto,swap]\n \\vertex (1) at (0,1) [fill,label=above:{$1$}]{};\n \\vertex (2) at (-1,2) [fill,label=above:{$2$}]{};\n \\vertex (3) at (-2,1) [fill,label=left:{$3$}]{};\n \\vertex (4) at (-1,0) [fill,label=below:{$4$}]{}; \n \\vertex (5) at (1,2) [fill,label=above:{$5$}]{};\n \\vertex (8) at (2,2) [fill,label=above:{$8$}]{};\n \\vertex (11) at (3,2) [fill,label=above:{$11$}]{};\n \\vertex (6) at (1,1) [fill,label=above:{$6$}]{};\n \\vertex (9) at (2,1) [fill,label=above:{$9$}]{};\n \\vertex (12) at (3,1) [fill,label=above:{$12$}]{};\n \\vertex (7) at (1,0) [fill,label=below:{$7$}]{};\n \\vertex (10) at (2,0) [fill,label=below:{$10$}]{};\n \\vertex (13) at (3,0) [fill,label=below:{$13$}]{};\n \\draw[line width=0.6pt](1)--(2);\n \\draw[line width=0.6pt](2)--(3);\n \\draw[line width=0.6pt](3)--(4);\n \\draw[line width=0.6pt](1)--(4);\n \\draw[line width=0.6pt](1)--(5);\n \\draw[line width=0.6pt](1)--(6);\n \\draw[line width=0.6pt](1)--(7);\n \\draw[line width=0.6pt](5)--(8);\n \\draw[line width=0.6pt](8)--(11);\n \\draw[line width=0.6pt](6)--(9);\n \\draw[line width=0.6pt](9)--(12);\n \\draw[line width=0.6pt](7)--(10);\n \\draw[line width=0.6pt](10)--(13);\n \\vertex (1') at (7,1) [fill,label=above:{$1$}]{};\n \\vertex (2') at (6,2) [fill,label=above:{$2$}]{};\n \\vertex (3') at (5,1) [fill,label=left:{$3$}]{};\n \\vertex (4') at (6,0) [fill,label=below:{$4$}]{}; \n \\vertex (5') at (8,2) [fill,label=above:{$5$}]{};\n \\vertex (8') at (9,2) [fill,label=above:{$8$}]{};\n \\vertex (11') at (10,2) [fill,label=above:{$11$}]{};\n \\vertex (6') at (8,1) [fill,label=above:{$6$}]{};\n \\vertex (9') at (9,1) [fill,label=above:{$9$}]{};\n \\vertex (12') at (10,1) [fill,label=above:{$12$}]{};\n \\vertex (7') at (8,0) [fill,label=below:{$7$}]{};\n \\vertex (10') at (9,0) [fill,label=below:{$10$}]{};\n \\vertex (13') at (10,0) [fill,label=below:{$13$}]{};\n \\draw[line width=0.6pt](1')--(2');\n \\draw[line width=0.6pt](2')--(3');\n \\draw[line width=0.6pt](3')--(4');\n \\draw[line width=0.6pt](1')--(4');\n \\draw[line width=0.6pt](1')--(5');\n \\draw[line width=0.6pt](1')--(6');\n \\draw[line width=0.6pt](1')--(7');\n \\draw[line width=0.6pt](5')--(8');\n \\draw[line width=0.6pt](8')--(11');\n \\draw[line width=0.6pt](6')--(9');\n \\draw[line width=0.6pt](9')--(12');\n \\draw[line width=0.6pt](7')--(10');\n \\draw[line width=0.6pt](10')--(13');\n \\draw[line width=0.6pt](12')--(13');\n \\draw[line width=0.6pt](6')--(7');\n \\draw[line width=0.6pt](9')--(10');\n \\end{tikzpicture}\n \\caption{A kite graph $G=(H,T)$ with head $H=C_4$ and tail $T$ with $s=3$ paths of length $r=3$, and its extended graph $G^+$.}\\label{fig:Kite2}\n \\end{center}\n\\end{figure}\nhas Laplacian matrix \n$$\n\\L = \n\\left({\\scriptsize\\begin{array}{ccccccccccccc}\n5 & -1 & 0 & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n -1 & 0 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n -1 & 0 & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \n\\\\\n -1 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \n\\\\\n -1 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 & 0 & 0 & -1 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 \n\\end{array}}\\right)\n$$\nwith algebraic connectivity $\\alpha(G)=0.19086$. Then,\nwe can add the edges $2\\sim 3$, $5\\sim 6$, or $8\\sim 9$ to obtain $G^+$, shown in the same figure (right), keeping the same value of $\\alpha(G)$, as shown in Proposition \\ref{prop:LvsS}. Thus, a Field vector is $\\y =(\n0,0,0,0,-0.44504,0,0.44504,$ $- 0.80193,0,0.80193,-1,0,1)^{\\top}$. If we multiply it by the matrix $\\SS$ \n$$\n\\SS = \n\\\\\n\\left({\\scriptsize\\begin{array}{ccccccccccccc}\n1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{1}{2} & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{1}{2} & 0 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{1}{2} \n\\\\\n 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & \\frac{1}{2} \n\\end{array}}\\right),\n$$\nthen $\\L\\SS=\\SS\\L$ and \nwe get\n\\begin{align*}\n&\\x=\\SS\\y=\\\\\n&(0,0,0,0,-0.44504,0.22252,0.22252,-0.80193,0.40096,0.40096,-1,0.5,0.5)^{\\top}.\n\\end{align*}\n In fact, every eigenvalue in\n $$\n \\spec G=\\{6.271,\n3.342,\n3.246^2,\n2.810,\n2,\n1.554^2,\n1.211,\n0.364,\n0.198^2,0\\}\n$$\nappears (in boldface) in \n\\begin{align*}\n &\\spec G^+=\\\\\n&\\{{\\bf 6.271},\n5.246,\n3.554,\n{\\bf 3.342},\n{\\bf 3.246},\n{\\bf 2.810},\n2.198,\n{\\bf 2},\n{\\bf 1.554},\n{\\bf 1.211},\n{\\bf 0.364},\n{\\bf 0.198},\n{\\bf 0}\\}.\n\\end{align*}\n\\end{example}\n\nNow, we are ready to obtain the result on the token graph of kite graphs.\n\\begin{theorem}\n\\label{th-kite}\n Let $G=K(H,T)$ be a kite graph with a tail of length $r$ formed with $s(\\ge 2)$ paths $v,v_{i1},v_{i2},\\ldots,v_{ir}$ for $i=1,\\ldots,s$. Let $K(H,T^+)$ be the graph obtained from $G$ by adding some edges between the vertices $v_{1j},v_{2j},\\ldots,v_{sj}$, for each $j$ of $T$, and $K^+(H,T)$ be the graph obtained from $G$ by adding some edges to $H$, and between the vertices $v_{2j},\\ldots,v_{sj}$ with $1\\le j\\le r$. Suppose that $\\alpha(H)=\\alpha(F_k(H))$, then the following statements hold.\n\\begin{itemize}\n\\item[$(i)$] \nIf $\\alpha (G)\\neq2+2\\cos(\\frac{2r\\pi}{2r+1})$, then $\\alpha(F_k(K(H,T^+)))=\\alpha(K(H,T^+))=\\alpha(G)$. \n\\item[$(ii)$] \nIf $\\alpha (G)=2+2\\cos(\\frac{2r\\pi}{2r+1})$, then $\\alpha(F_k(K^+(H,T)))=\\alpha(K^+(H,T))=\\alpha(G)=2+2\\cos(\\frac{2r\\pi}{2r+1})$.\n\\end{itemize}\n\\end{theorem}\n\\begin{proof}\n($i$) Note that the result by Shao, Guo, and Shan \\cite{Shao_2008} in Lemma \\ref{le-Shao2008} reads as $\\alpha(K(H,T^+))=\\alpha(G)$. \nCombining it with Theorems \\ref{th-add_edge} and \\ref{th-Dalfo2024}$(ii)$, and the hypothesis, we obtain $\\alpha(F_k(K(H,T^+)))=\\alpha(K(H,T^+))=\\alpha(G)$.\n\n($ii$) Let $G'\\cong K(H^+,T)$ be the graph obtained from $G$ by adding some edges to $H$, where $|V(H)|=h$. We first prove that $\\alpha(K(H^+,T))=\\alpha(K(H,T))$. \nFrom Lemma \\ref{le-kite-iff}, we get that $\\lambda_1(\\L_{[V(H-v)]}(H))\\ge \\lambda_1(\\L_{[V_{r}]}(P_{r+1}))=2+2\\cos(\\frac{2r\\pi}{2r+1})$.\nBesides, observe that $G'$ is the graph obtained by adding some edges in $H$. \nDenote these additional edges by $E'$ if they are between the vertices of $H-v$, and by $E''$ if they have the end-vertex $v$. \nLet $H'$ be the graph with vertex set $V(H')=V(H)$ and edge set $E'$.\nThen, \n$\\L_{[V(H-v)]}(G')=\\L_{[V(H-v)]}(H)+\\L(H')+\\I''$,\nwhere $\\I''$ is a diagonal matrix of order $h-1$, with $|E''|$ ones and $h-1-|E''|(\\ge 1)$ zeros.\nIt follows from Lemma \\ref{le-matrix_add} with $i=1$ and the hypothesis that \n\\begin{align}\n\\lambda_1(\\L_{[V(H-v)]}(G')) &\\ge \\lambda_1(\\L_{[V(H-v)]}(H))+\\lambda_1(\\L(H'))+\\lambda_1(\\I'')\\nonumber\\\\\n & \\ge\\lambda_1(\\L_{[V(H-v)]}(H))\\ge2+2\\cos\\left(\\frac{2r\\pi}{2r+1}\\right).\n\\label{eq:2nd-ineq}\n\\end{align}\n Then, from \\eqref{eq:2nd-ineq} and Lemma \\ref{le-suffcondition}, we have that \n $$\n \\alpha(G)=2+2\\cos\\left(\\frac{2r\\pi}{2r+1}\\right)=\\alpha(G').\n $$\n\n Next, let $G^+\\cong K^+(H,T)$ be the graph obtained from $G'$ by adding some edges between the vertices $v_{2j},\\ldots,v_{sj}$ with $1\\le j\\le r$. It follows from Proposition \\ref{prop:LvsS}$(iii)$ that $\\alpha(G^+)=\\alpha(G')=\\alpha(G)$. \n Together with Theorem \\ref{th-Dalfo2024}$(ii)$ (since $\\alpha(H)=\\alpha(F_k(H))$ implies $\\alpha(G)=\\alpha(F_k(G))$) and Theorem \\ref{th-add_edge}, we obtain $\\alpha(F_k(K^+(H,T)))=\\alpha(K^+(H,T))=\\alpha(G)=2+2\\cos\\left(\\frac{2r\\pi}{2r+1}\\right)$.\n \\qed\n\\end{proof}\n\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=0.9,auto,swap]\n \\vertex (v) at (0,1) [fill]{};\n \\node () at (-0.1,1) \n [label=above:{$v$}]{};\n \\vertex (11) at (-1,1) [fill,label=above:{$v_{11}$}]{};\n \\vertex (12) at (-2,0.9) [fill,label=above:{$v_{12}$}]{};\n \\vertex (13) at (-3,0.9) [fill,label=above:{$v_{13}$}]{};\n \\vertex (14) at (-4,0.35) [fill,label=above:{$v_{14}$}]{};\n \\vertex (21) at (-1,0.5) [fill]{};\n \\node () at (-1,0.35) [label=above:{$v_{21}$}]{};\n \\vertex (22) at (-2,0.3) [fill,label=above:{$v_{22}$}]{};\n \\vertex (23) at (-2.8,0.05) [fill,label=above:{$v_{23}$}]{};\n \\vertex (24) at (-3.3,-0.5) [fill]{};\n \\node () at (-3.4,-0.5) [label=above:{$v_{24}$}]{};\n \\vertex (31) at (-0.45,0) [fill,label=below:{$v_{31}$}]{};\n \\vertex (32) at (-1.5,-0.2) [fill,label=below:{$v_{32}$}]{};\n \\vertex (33) at (-2.3,-0.4) [fill]{};\n \\node () at (-2.2,-0.3) [label=below:{$v_{33}$}]{};\n \\vertex (34) at (-2.7,-0.9) [fill,label=below:{$v_{34}$}]{};\n \\vertex (1) at (0.7,2.3) [fill,label=above:{$u_{1}$}]{};\n \\vertex (2) at (2,2.9) [fill,label=above:{$u_{2}$}]{};\n \\vertex (3) at (3.4,2.4) [fill,label=above:{$u_{3}$}]{};\n \\vertex (4) at (2.8,1) [fill,label=below:{$u_{4}$}]{};\n \\vertex (5) at (1.5,0.4) [fill,label=below:{$u_{5}$}]{};\n \\draw[line width=0.6pt](v)--(11);\n \\draw[line width=0.6pt](11)--(12);\n \\draw[line width=0.6pt](12)--(13);\n \\draw[line width=0.6pt](13)--(14);\n \\draw[line width=0.6pt](v)--(21);\n \\draw[line width=0.6pt](21)--(22);\n \\draw[line width=0.6pt](22)--(23);\n \\draw[line width=0.6pt](23)--(24);\n \\draw[line width=0.6pt](v)--(31);\n \\draw[line width=0.6pt](31)--(32);\n \\draw[line width=0.6pt](32)--(33);\n \\draw[line width=0.6pt](33)--(34);\n \\draw[line width=0.6pt](v)--(1);\n \\draw[line width=0.6pt](1)--(2);\n \\draw[line width=0.6pt](2)--(3);\n \\draw[line width=0.6pt](3)--(4);\n \\draw[line width=0.6pt](4)--(5);\n \\draw[line width=0.6pt](v)--(5);\n \\draw[line width=0.6pt](1)--(5);\n \\draw[line width=0.6pt](1)--(3);\n \\draw[line width=0.6pt](2)--(4);\n \\draw[line width=0.6pt](2)--(5);\n \\draw[line width=0.6pt](v)--(4);\n \\vertex (v') at (8,1) [fill]{};\n \\node () at (7.9,1)\n [label=above:{$v$}]{};\n \\vertex (11') at (7,1) [fill,label=above:{$v_{11}$}]{};\n \\vertex (12') at (6,0.9) [fill,label=above:{$v_{12}$}]{};\n \\vertex (13') at (5,0.9) [fill,label=above:{$v_{13}$}]{};\n \\vertex (14') at (4,0.35) [fill,label=above:{$v_{14}$}]{};\n \\vertex (21') at (7,0.5) [fill]{};\n \\node () at (7,0.35) [label=above:{$v_{21}$}]{};\n \\vertex (22') at (6,0.3) [fill,label=above:{$v_{22}$}]{};\n \\vertex (23') at (5.2,0.05) [fill,label=above:{$v_{23}$}]{};\n \\vertex (24') at (4.7,-0.5) [fill]{};\n \\node () at (4.6,-0.5) [label=above:{$v_{24}$}]{};\n \\vertex (31') at (7.45,0) [fill,label=below:{$v_{31}$}]{};\n \\vertex (32') at (6.5,-0.2) [fill,label=below:{$v_{32}$}]{};\n \\vertex (33') at (5.7,-0.4) [fill]{};\n \\node () at (5.8,-0.3)\n [label=below:{$v_{33}$}]{};\n \\vertex (34') at (5.3,-0.9) [fill,label=below:{$v_{34}$}]{};\n \\vertex (1') at (8.7,2.3) [fill,label=above:{$u_1$}]{};\n \\vertex (2') at (10,2.9) [fill,label=above:{$u_2$}]{};\n \\vertex (3') at (11.4,2.4) [fill,label=above:{$u_3$}]{};\n \\vertex (4') at (10.8,1) [fill,label=below:{$u_{4}$}]{};\n \\vertex (5') at (9.5,0.4) [fill,label=below:{$u_{5}$}]{};\n \\draw[line width=0.6pt](v')--(11');\n \\draw[line width=0.6pt](11')--(12');\n \\draw[line width=0.6pt](12')--(13');\n \\draw[line width=0.6pt](13')--(14');\n \\draw[line width=0.6pt](v')--(21');\n \\draw[line width=0.6pt](21')--(22');\n \\draw[line width=0.6pt](22')--(23');\n \\draw[line width=0.6pt](23')--(24');\n \\draw[line width=0.6pt](v')--(31');\n \\draw[line width=0.6pt](31')--(32');\n \\draw[line width=0.6pt](32')--(33');\n \\draw[line width=0.6pt](33')--(34');\n \\draw[line width=0.6pt](v')--(1');\n \\draw[line width=0.6pt](1')--(2');\n \\draw[line width=0.6pt](2')--(3');\n \\draw[line width=0.6pt](3')--(4');\n \\draw[line width=0.6pt](4')--(5');\n \\draw[line width=0.6pt](v')--(5');\n \\draw[line width=0.6pt](1')--(5');\n \\draw[line width=0.6pt](1')--(3');\n \\draw[line width=0.6pt](2')--(4');\n \\draw[line width=0.6pt](2')--(5');\n \\draw[line width=0.6pt](v')--(4');\n \\draw[blue,line width=0.6pt](21')--(31');\n \\draw[blue,line width=0.6pt](23')--(33');\n \\draw[blue,line width=0.6pt](24')--(34');\n \\draw[blue,line width=0.6pt](3')--(v');\n \\draw[blue,line width=0.6pt](2')--(5');\n \\draw[blue,line width=0.6pt](4')--(1');\n \\end{tikzpicture}\n \\caption{A kite graph $G=K(H,T)$ and its extended graph $G^+$.}\\label{fig:kite}\n \\end{center}\n\\end{figure}\n\n\n\\begin{example}\n Figure \\ref{fig:ex-th-kite} is an example of \\eqref{eq:2nd-ineq} with $E'=\\{36,46\\}$ and $E''=\\{13,14\\}$. Note that $\\L_{[V(H-v)]}(G^+)=\\L_{[V(H-v)]}(H)+\\L(H')+\\I''$ with \n \\begin{eqnarray*}\n \\L_{[V(H-v)]}(H)\n &=&{\\scriptsize{\\left({\\begin{array}{ccccc}\n 3&-1&0&-1&0\\\\\n -1&2&-1&0&0\\\\\n 0&-1&2&-1&0\\\\\n -1&0&-1&3&-1\\\\\n 0&0&0&-1&2\n \\end{array}}\n \\right)}},\\\\\n \\L(H')&=&{\\scriptsize{\\left({\\begin{array}{ccccc}\n 0&0&0&0&0\\\\\n 0&1&0&0&-1\\\\\n 0&0&1&0&-1\\\\\n 0&0&0&0&0\\\\\n 0&-1&-1&0&2\n \\end{array}}\n \\right)}},\n ~\\text{and } \\I''={\\scriptsize{\\left({\\begin{array}{ccccc}\n 0&0&0&0&0\\\\\n 0&1&0&0&0\\\\\n 0&0&1&0&0\\\\\n 0&0&0&0&0\\\\\n 0&0&0&0&0\n \\end{array}}\n \\right).}}\n \\end{eqnarray*}\n Moreover, $\\lambda_1(\\L_{[V(H-v)]}(G^+))=0.747>0.284=\\lambda_1(\\L_{[V(H-v)]}(H))$.\n\\end{example}\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=1.1,auto,swap]\n \\vertex (1) at (0,1) [fill]{};\n \\node () at (0.1,1) [label=above:{$1$}]{};\n \\node () at (0.1,1) [label=below:{$v$}]{};\n \\vertex (01) at (1,1.2) [fill]{};\n \\vertex (02) at (2,1.4) [fill]{};\n \\vertex (03) at (3,1.6) [fill]{};\n \\vertex (04) at (1,0.8) [fill]{};\n \\vertex (05) at (2,0.6) [fill]{};\n \\vertex (06) at (3,0.4) [fill]{};\n \\vertex (2) at (-0.7,1.8) [fill,label=above:{$2$}]{};\n \\vertex (3) at (-1.7,1.8) [fill,label=above:{$3$}]{};\n \\vertex (4) at (-2.4,1) [fill,label=left:{$4$}]{};\n \\vertex (5) at (-1.7,0.2) [fill,label=below:{$5$}]{};\n \\vertex (6) at (-0.7,0.2) [fill,label=below:{$6$}]{}; \n \\node () at (-2.4,1.5) [label=above:{$H$}]{};\n \\node () at (1.2,1.5) [label=above:{$T$}]{};\n \\draw[line width=0.6pt](1)--(01);\n \\draw[line width=0.6pt](01)--(02);\n \\draw[line width=0.6pt](02)--(03);\n \\draw[line width=0.6pt](1)--(04);\n \\draw[line width=0.6pt](04)--(05);\n \\draw[line width=0.6pt](05)--(06);\n \\draw[line width=0.6pt](1)--(2);\n \\draw[line width=0.6pt](2)--(3);\n \\draw[line width=0.6pt](3)--(4);\n \\draw[line width=0.6pt](4)--(5);\n \\draw[line width=0.6pt](5)--(6);\n \\draw[line width=0.6pt](1)--(6);\n \\draw[line width=0.6pt](2)--(5);\n \\draw[blue,line width=0.6pt](1)--(3);\n \\draw[blue,line width=0.6pt](1)--(4);\n \\draw[blue,line width=0.6pt](3)--(6);\n \\draw[blue,line width=0.6pt](4)--(6);\n \\end{tikzpicture}\n \\caption{A example for Theorem \\ref{th-kite}.}\\label{fig:ex-th-kite}\n \\end{center}\n\\end{figure}\n\nThis theorem and the following ones in this section can be generalized for superkite graphs. A superkite graph $K(H,{\\cal T})$ has head $H$ and supertail ${\\cal T}=Q^s$ formed by $s(\\ge 2)$ trees isomorphic to $Q$ with the same root $v\\in V(H)$. Then, basically, the same proof of Theorem \\ref{th-kite} leads to the following result.\n\\begin{theorem}\n \\label{th-superkite}\n Let $G$ be a superkite graph $K(H,{\\cal T})$ with tail ${\\cal T}=Q^s$, such that $H$ is not a complete graph with $h=|V(H)|$, with $\\alpha(H)=\\alpha(F_k(H))$, and \n $$\n \\lambda_1(\\L_{[V(H-v)]}(H))\\ge \\lambda_1(\\L_{[V(Q-v)]}(Q)).\n $$\n Let $G^+$ be the graph obtained from $G$ by adding some edges to $H$. Then, $\\alpha(G^+)=\\alpha(F_k(G^+))$.\n\\end{theorem}\nThe following two corollaries deal with the cases in which the head of a kite graph is a cycle and a complete bipartite graph, respectively.\n\\begin{corollary}\n\\label{th-kite-cycle}\n Let $G$ be a kite graph $K(H,T)$ with tail length $r$, such that $H\\cong C_h$ is a cycle with order $h$ $(\\mbox{for } h\\le 2r+1)$ and $\\alpha(H)=\\alpha(F_k(H))$. \n Let $G^+$ be the graph obtained from $G$ by adding some edges to $H$, and between the vertices $v_{2j},\\ldots,v_{sj}$ with $1\\le j\\le r$. Then, \n $\\alpha(G^+)=\\alpha(F_k(G^+))$. \n\\end{corollary}\n\\begin{proof}\nIt is known that the characteristic polynomials of $\\L_{[V']}(C_{h})$, where $V'=V(C_h-v)$, and $\\L(P_{h})$ are related as $x\\cdot\\Phi(\\L_{[V']}(C_{h}))=\\Phi(\\L(P_{h}))$ (see Guo \\cite{Guo_2008}).\nHence, $\\lambda_i(\\L_{[V']}(C_{h}))=\\lambda_{i+1}(\\L(P_{h}))$ for $i=1,\\ldots,h-1$. Then, since the Laplacian eigenvalues of the path $P_{\\ell}$ of length $\\ell$ are $\\tau_k=2-2\\cos(\\frac{k\\pi}{\\ell})$ for $k=0,1,\\ldots,\\ell$, we have, with $H=C_h$,\n\\begin{align*}\n\\lambda_1(\\L_{[V(H-v)]}(H))=\\lambda_2(\\L(P_h))&\\ge\\lambda_2(\\L(P_{2r+1}))=2-2\\cos\\left(\\frac{\\pi}{2r+1}\\right)\\\\\n &=2+2\\cos\\left(\\frac{2r\\pi}{2r+1}\\right)=\n\\lambda_1(\\L_{[V_r]}(P_{r+1})).\n \\end{align*}\n From Lemma \\ref{le-kite-iff} and Theorem \\ref{th-kite}, the result holds. \\qed\n\\end{proof}\n\n\\begin{corollary}\n\\label{th-kite-bipartite}\n Let $G$ be a kite graph $K(H,T)$ with root vertex $v$, tail length $r$, and head $H\\cong K_{h_1,h_2}$ being a complete bipartite graph with vertex set $V=V_1\\cup V_2$, on $h=h_1+h_2$ vertices, where $h_i=|V_i|$.\n Let $G^+$ be the graph obtained from $G$ by adding some edges to $H$, and between the vertices $v_{2j},\\ldots,v_{sj}$ with $1\\le j\\le r$. \n If $v\\in V_i$ for some $i\\in\\{1,2\\}$, and \n $$\n \\frac{1}{2}\\left(h-\\sqrt{h^2-4h_i}\\right)\\ge 2+2\\cos\\left(\\frac{2r\\pi}{2r+1}\\right),\n $$\n then \n $\\alpha(G^+)=\\alpha(F_k(G^+))$.\n\\end{corollary}\n\\begin{proof}\n It follows from Theorem \\ref{th-Dalfo2024}($iii$) that $\\alpha(F_k(K_{h_1,h_2}))=\\alpha(K_{h_1,h_2})$. \n Moreover, it can be shown that, if $v\\in V_i$, then $\\lambda_1(\\L_{V(H-v)}(H))=\\frac{1}{2}\\left(h-\\sqrt{h^2-4h_i}\\right)$ for $i=1,2$.\n Then, we have $\\lambda_1(\\L_{V(H-v)}(H))\\ge 2+2\\cos(\\frac{2r\\pi}{2r+1})=\\lambda_1(\\L_{[V_{r}]}(P_{r+1}))$.\n Therefore, the result holds by Lemma \\ref{le-kite-iff} and Theorem \\ref{th-kite}.\n \\qed\n\\end{proof}\n\n\\begin{figure}\n \\begin{center}\n \\begin{tikzpicture}[scale=1.1,auto,swap]\n \\vertex (v) at (1,1) [fill]{};\n \\vertex (11) at (0,1) [fill]{};\n \\vertex (12) at (-1,0.9) [fill]{};\n \\vertex (13) at (-2,0.9) [fill]{};\n \\vertex (21) at (0,0.5) [fill]{};\n \\vertex (22) at (-1,0.3) [fill]{};\n \\vertex (23) at (-1.8,0.05) [fill]{};\n \\vertex (31) at (0.45,0) [fill]{};\n \\vertex (32) at (-0.5,-0.2) [fill]{};\n \\vertex (33) at (-1.3,-0.4) [fill]{};\n \\vertex (1) at (1.8,1.9) [fill]{};\n \\vertex (2) at (3,1.5) [fill]{};\n \\vertex (3) at (3,0.5) [fill]{};\n \\vertex (4) at (1.8,0.1) [fill]{};\n \\draw[line width=0.6pt](v)--(11);\n \\draw[line width=0.6pt](11)--(12);\n \\draw[line width=0.6pt](12)--(13);\n \\draw[line width=0.6pt](v)--(21);\n \\draw[line width=0.6pt](21)--(22);\n \\draw[line width=0.6pt](22)--(23);\n \\draw[line width=0.6pt](v)--(31);\n \\draw[line width=0.6pt](31)--(32);\n \\draw[line width=0.6pt](32)--(33);\n \\draw[line width=0.6pt](v)--(1);\n \\draw[line width=0.6pt](1)--(2);\n \\draw[line width=0.6pt](2)--(3);\n \\draw[line width=0.6pt](3)--(4);\n \\draw[line width=0.6pt](4)--(v);\n \\vertex (2v) at (7,1) [fill]{};\n \\vertex (211) at (6,1) [fill]{};\n \\vertex (212) at (5,0.9) [fill]{};\n \\vertex (213) at (4,0.9) [fill]{};\n \\vertex (221) at (6,0.5) [fill]{};\n \\vertex (222) at (5,0.3) [fill]{};\n \\vertex (223) at (4.2,0.05) [fill]{};\n \\vertex (231) at (6.45,0) [fill]{};\n \\vertex (232) at (5.5,-0.2) [fill]{};\n \\vertex (233) at (4.7,-0.4) [fill]{};\n \\vertex (21) at (7.35,1.9) [fill]{};\n \\vertex (22) at (8.4,1.9) [fill]{};\n \\vertex (23) at (8.2,1) [fill]{};\n \\vertex (24) at (7.9,0.2) [fill]{};\n \\draw[line width=0.6pt](2v)--(211);\n \\draw[line width=0.6pt](211)--(212);\n \\draw[line width=0.6pt](212)--(213);\n \\draw[line width=0.6pt](2v)--(221);\n \\draw[line width=0.6pt](221)--(222);\n \\draw[line width=0.6pt](222)--(223);\n \\draw[line width=0.6pt](2v)--(231);\n \\draw[line width=0.6pt](231)--(232);\n \\draw[line width=0.6pt](232)--(233);\n \\draw[line width=0.6pt](2v)--(23);\n \\draw[line width=0.6pt](2v)--(22);\n \\draw[line width=0.6pt](21)--(22);\n \\draw[line width=0.6pt](21)--(23);\n \\draw[line width=0.6pt](21)--(24);\n \\draw[line width=0.6pt](24)--(2v);\n \\end{tikzpicture}\n \\caption{The example in Corollary \\ref{th-kite-cycle} (left) and Corollary \\ref{th-kite-bipartite} (right) with $r=3$ and $s=3$.}\\label{fig:kite_example}\n \\end{center}\n\\end{figure}\n\n\\section{Graphs with a cut clique}\n\\label{sec:cut-clique}\nThe notion of cut clique generalizes the idea of cut vertex.\nThus, a clique $K_r$ of a graph $G$ is a {\\em cut clique} if the removal of (the vertices and edges of) $K_r$ disconnects $G$ in two or more components.\nThen, the following result is a natural generalization of Kirland's result \\cite{Kirkland_2000} for cut vertices.\n\\begin{proposition}\n\\label{prop-cut-clique}\nFor a connected graph $G$ with $n$ vertices and a cut clique $K_r$, we have $\\alpha(G)\\le r$ and the equality holds if and only if \n$d_G(K_r)=r(n-r)$. \n\\end{proposition}\n\\begin{proof}\n It follows from Fiedler \\cite[\\S 3.12]{Fiedler_1973} that \n $$\n \\alpha(G)\\le \\min\\{\\alpha(G-K_r)+|V(K_r)|,\\alpha(K_r)+|V(G-K_r)|\\}.\n $$\n Then, since $G-K_r$ is disconnected, we have $\\alpha(G)\\le r$.\n\nWe next show that, if $d_G(K_r)=r(n-r)$, then $\\alpha(G)=r$ .\n Let $G_1,\\ldots,G_s$ be all components of $G-K_r$. \n Notice that $\\L(G)=\\L_1+\\L_2$, where $\\L_1$ is the Laplacian matrix of the graph with vertex set $V(G)$ and edge set $E(G_1\\cup\\cdots\\cup G_s)$, and $\\L_2$ is the Laplacian matrix of $G$ removing all the edges of $G_1\\cup\\cdots\\cup G_s$. That is,\n $$\\L(G)={\\scriptsize{\\left({\\begin{array}{ccccccc}\n \\L(G_1)&&&0&\\ldots&0\\\\\n &\\ddots&&&\\ddots&\\\\\n &&\\L(G_s)&0&\\ldots&0\\\\\n 0&\\ldots&0&0&\\ldots&0\\\\\n &\\ddots&& &\\ddots&\\\\\n 0&\\ldots&0& 0& \\ldots&0\n \\end{array}}\n \\right)}}+{\\scriptsize{\\left({\\begin{array}{ccccccc}\n r\\I& & &-1&\\ldots&-1\\\\\n &\\ddots & &&\\ddots&\\\\\n &&r\\I&-1&\\ldots&-1\\\\\n -1&\\ldots&-1&n-1&\\ldots&-1\\\\\n &\\ddots&& &\\ddots&\\\\\n -1&\\ldots&-1& -1& \\ldots&n-1\n \\end{array}}\n \\right)}}. $$\n Since the spectrum of $\\L_2$ is $\\{0^1,r^{n-r-1},n^r\\}$, we have that $\\lambda_2(\\L_2)=r$. \n It follows from Lemma \\ref{le-matrix_add} with $i=2$ that \n $\\alpha(G)=\\lambda_2(\\L(G))\\ge\\lambda_1(\\L_1)+\\lambda_2(\\L_2)=r.$\n Together with $\\alpha(G)\\le r$, we conclude that $\\alpha(G)=r$. \n\n Otherwise, if $d_{G}(K_r)\\ne r(n-r)$, then there exists $u\\in V(K_r)$ and $v\\in V(G_1\\cup\\cdots\\cup G_s)$ such that $uv\\notin E(G)$.\n Denote by $n_1$ and $n_2$ the order of $G_1\\cup\\cdots\\cup G_{s-1}$ and $G_s$, respectively.\n Let $G^*$ be a graph obtained from $G$ by adding edges such that $G^*[V(G_1\\cup\\cdots\\cup G_{s-1})]$ and $G^*[V(G_{s})]$ are complete graphs, and $d_{G^*}(K_r)=r(n-r)$.\n Note that $$\\L(G^*)=\\left({\\begin{array}{ccc}\n (n_1+r)\\I_{n_1}-\\J_{n_1}&\\O&-\\J_{n_1\\times r}\\\\\n \\O&(n_2+r)\\I_{n_2}-\\J_{n_2}&-\\J_{n_2\\times r}\\\\\n -\\J_{r\\times n_1}&-\\J_{r\\times n_2}&n\\I_{r}-\\J_{r}\n \\end{array}}\n \\right).$$\n It can be shown that $\\Phi(\\L(G^*))=x(x-r)(x-r-n_1)^{n_1-1}\n (x-r-n_2)^{n_2-1}(x-n)^r$. Thus, \n the spectrum of $\\L(G^*)$ is $\\{0^1,r^1,(r+n_1)^{n_1-1},(r+n_2)^{n_2-1},n^r\\}$.\n Moreover, it is easily checked that \n \\[\n \\x=\\Big(\\underbrace{1,\\ldots,1}_{n_1},\\underbrace{-\\frac{n_1}{n_2},\\ldots,-\\frac{n_1}{n_2}}_{n_2},\\underbrace{0,\\ldots,0}_r\\Big)^{\\top}\n \\]\n is a Fiedler vector of $G^*$. Furthermore, since $r$ has multiplicity 1, any eigenvector $\\y$ of $G^*$ with eigenvalue $\\alpha(G^*)$ must be a multiple of $\\x$.\n It follows from Lemma \\ref{le-interlacing} that $\\alpha(G^*)\\ge\\alpha(G^*-uv)$. \n If the equality holds and $\\w$ is a Fiedler vector of $G^*-uv$, we have $\\w_u=\\w_v$ from Lemma \\ref{le-adding_edge_iff}.\n Moreover, $\\w$ also is a Fiedler vector of $G^*$ with $\\w_u=\\w_v$, a contradiction. \n Hence, the inequality must be strict. \n Finally, since $G^*-uv$ can be obtained by adding edges in $G$, we conclude that $\\alpha(G)\\le\\alpha(G^*-uv)<\\alpha(G^*)=r.$\n\\qed\n\\end{proof}\n\\begin{proposition}\n\\label{prop-subgraph}\n Let $G_1$ and $G_2$ be two edge-disjoint graphs with the same $n$ vertices. \n Let $G=G_1\\cup G_2$. Then, $F_k(G)=F_k(G_1)\\cup F_k(G_2)$. \n\\end{proposition}\n\\begin{proof}\n It suffices to consider that there is exactly one edge $uv$ in $G_2$, that is, $G_1=G-uv$. \n The result holds since $F_k(G-uv)$ is a spanning subgraph of $F_k(G)$ with \n \\begin{align*}\n E(F_k(G))\\backslash E(F_k(G-uv)) &=\\{A_{r}A_{s} : A_{r}=\\{u,u_1,\\ldots,u_{k-1}\\}, A_{s}=\\{v,u_1,\\ldots,u_{k-1}\\}\\}\\\\\n & =E(F_k(G_2)). \\mbox{\\hskip 7.8cm \\qed} \n \\end{align*}\n \\end{proof}\n\n\\begin{theorem}\n\\label{th-clique}\nLet $G$ be a graph on $n$ vertices, with a cut clique $K_r$ such that $G-K_r$ results in some disjoint graphs $G_1,\\ldots,G_s$, for $s\\ge 2$, and \n degree $d_G(K_r)=r(n-r)$ (so that there are {\\bf all} the edges between the vertices of $K_r$ and the vertices of each $G_i$). Then, $\\alpha(G)=\\alpha(F_k(G))=r$.\n\\end{theorem}\n\n\\begin{proof}\n By Theorem \\ref{th-Dalfo2021} and Proposition \\ref{prop-cut-clique}, we get\n \\begin{equation}\n \\alpha(F_k(G))\\le \\alpha(G)=r.\\label{eq-th-clique_1}\n \\end{equation}\n Let $H_1$ be a graph with vertex set $V(G)$ and edge set $E(H_1)=\\{uv:uv\\in G_1\\cup\\cdots\\cup G_s\\}$, and let $H_2$ be a graph obtained from $G$ by removing all the edges of $E(H_1)$. Note that $H_1$ and $H_2$ are the graphs with Laplacian matrices $\\L_1$ and $\\L_2$, respectively, given in the proof of Proposition \\ref{prop-cut-clique}, and $H_2$ is the graph obtained from a complete bipartite graph by adding all edges in the partition with order $r$.\n By Theorems \\ref{th-bip-addedge1}, and \\ref{th-bip-addedge2}, we obtain $\\alpha(F_k(H_2))=r$. \n From Proposition \\ref{prop-subgraph}, we have $\\L(F_k(G))=\\L(F_k(H_1))+\\L(F_k(H_2))$. \n Then, it follows from Lemma \\ref{le-matrix_add} with $i=2$ that\n \\begin{equation*}\n \\alpha(F_k(G))=\\lambda_2(F_k(G))\\ge \\lambda_1(F_k(H_1))+\\lambda_2(F_k(H_2))=\\alpha(F_k(H_2))=r.\n \\end{equation*}\n Combining it with (\\ref{eq-th-clique_1}), this completes the proof.\\qed\n\\end{proof}\n\nIn particular, with $r=1$, the vertex of $K_1$ is a cut vertex of degree $n-1$, and we reobtain a result of Barik and Verma \\cite{bv24} (see Theorem \\ref{theBarik} in the Preliminaries). \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{thebibliography}{99}\n\\bibitem{a07}\nN. M. M. de Abreu, Old and new results on algebraic connectivity of graphs, {\\em Linear Algebra Appl.} {\\bf 423} (2007) 53--73.\n\n\\bibitem{Bapat_1998}\nR. B. Bapat and S. Pati, \nAlgebraic connectivity and the characteristic set of a graph, \n{\\em Linear Multilinear Algebra} {\\bf 45} (1998) 247--273. \n\n\\bibitem{bv24}\nS. Barik and P. Verma, \nSpectral properties of token graphs, \n{\\em Linear Algebra Appl.} {\\bf 687} (2024) 181--206.\n\n\\bibitem{clr10}\nP. Caputo, T. M. Liggett, T. Richthammer, Proof of Aldous’ spectral gap conjecture, {\\em J. Amer. Math. Soc.} {\\bf 23(3)} (2010) 831--851.\n\n\n\\bibitem{Cvetkovic_1980}\nD. M. Cvetkovi\\'c, M. Doob, and H. Sachs, \n{\\em Spectra of Graphs},\n{\\em Academic Press, New York}, 1980.\n\n\\bibitem{Dalfo_2021}\nC. Dalf\\'o, F. Duque, R. Fabila-Monroy, M. A. Fiol, C. Huemer, A. L. Trujillo-Negrete, and F. J. Zaragoza Mart\\'inez, \nOn the Laplacian spectra of token graphs,\n{\\em Linear Algebra Appl.} {\\bf 625} (2021) 322--348. \n\n\\bibitem{Dalfo_2024}\nC. Dalf\\'o and M. A. Fiol, \nOn the algebraic connectivity of some token graphs, \n{\\em J. Algebraic Combin.} {\\bf 60} (2024) 45--56.\n\n\\bibitem{ffhhuw12}\nR. Fabila-Monroy, D. Flores-Peñaloza, C. Huemer, F. Hurtado, J. Urrutia, and D. R. Wood, Token graphs, {\\em Graphs Combin.} {\\bf 28(3)} (2012) 365--380.\n\n\\bibitem{Fiedler_1973}\nM. Fiedler,\nAlgebraic connectivity of graphs,\n{\\em Czech. Math. J.} {\\bf 23(2)} (1973) 298--305.\n\n\n\\bibitem{Guo_2008}\nJ. Guo, \nA conjecture on the algebraic connectivity of connected graphs with fixed girth,\n{\\em Discrete Math.} {\\bf 308} (2008) 5702--5711. \n\n\n\\bibitem{Guo_2018}\nS. Guo, R. Zhang, and G. Yu, \nHamiltonian graphs of given order and minimum algebraic connectivity, \n{\\em Linear Multilinear Algebra} {\\bf 66(3)} (2018) 459--468.\n\n\\bibitem{Horn_2013}\nR. A. Horn and C. R. Johnson, \n{\\em Matrix Analysis},\nCambridge University Press, Cambridge, 2013.\n\n\\bibitem{Kirkland_2000}\nS. Kirkland, \nA bound on the algebraic connectivity of a graph in terms of the number of cutpoints, \n{\\em Linear Multilinear Algebra} {\\bf 47} (2000) 93--103. \n\n\\bibitem{Merris_1998}\nR. Merris, \nLaplacian graph eigenvectors, \n{\\em Linear Algebra Appl.} {\\bf 278} (1998) 221--236.\n\n\\bibitem{Reyes_2025}\nM. A. Reyes, C. Dalf\\'o, M. A. Fiol, and A. Messegu\\'e,\nA general method to find the spectrum and eigenspaces of the $k$-token graph of a cycle, and $2$-token through continuous fractions, \n{\\em Discrete Appl. Math.} {\\bf 360} (2025) 353--365. \n\n\\bibitem{Shao_2008}\nJ. Shao, J. Guo, and H. Shan, \nThe ordering of trees and connected graphs by algebraic connectivity,\n{\\em Linear Algebra Appl.} {\\bf 428} (2008) 1421--1438. \n \n\\bibitem{Xue_2019}\nJ. Xue, H. Lin, and J. Shu, \nThe algebraic connectivity of graphs with given circumference, \n{\\em Theoret. Comput. Sci.} {\\bf 772} (2019) 123--131.\n\n\\bibitem{y05}\nW.-C. Yueh, \nEigenvalues of several tridiagonal matrices, \n\\textit{Appl. Math. E-Notes} \\textbf{5} (2005) 66--74.\n\n\\end{thebibliography}\n\\end{document}\n\n\n\n\n"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":2,"numItemsPerPage":100,"numTotalItems":591,"offset":200,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc3NjM2NjU1NSwic3ViIjoiL2RhdGFzZXRzL2V0aHotc3B5bGFiL21hdGhfbGF0ZXgiLCJleHAiOjE3NzYzNzAxNTUsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.hbPmBCIrwC04OsLTIafy_VXdKk23jA_9pHH1HZs_MNOHB2RedsF_egD11ozooVkWP3prkBFVoXn0wONcPJbyBg","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":591,"numBytesParquet":27690937}]},"discussionsStats":{"closed":0,"open":0,"total":0},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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2412.17882v1 | http://arxiv.org/abs/2412.17882v1 | The Frobenius problem for Binomial Coefficients | \documentclass[12pt]{article}
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2412.17311v1 | http://arxiv.org/abs/2412.17311v1 | Dualizing involutions on the $n$-fold metaplectic cover of $\GL(2)$ | \documentclass[11 point, a4paper, reqno]{amsart}
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2412.17421v1 | http://arxiv.org/abs/2412.17421v1 | Hereditary properties of finite ultrametric spaces | \documentclass[11pt,english]{article}
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2412.17416v1 | http://arxiv.org/abs/2412.17416v1 | Minimum spanning paths and Hausdorff distance in finite ultrametric spaces | \documentclass[12pt]{amsart}
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2412.17437v1 | http://arxiv.org/abs/2412.17437v1 | Existence of solution to modified Gursky-Streets equation | \documentclass[psamsfonts]{amsart}
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2412.17500v1 | http://arxiv.org/abs/2412.17500v1 | Oriented Ramsey numbers of some oriented graphs with one cycle | \documentclass[11pt]{article}
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2412.17480v1 | http://arxiv.org/abs/2412.17480v1 | Width bounds and Steinhaus property for unit groups of continuous rings | \pdfoutput=1
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2412.17554v2 | http://arxiv.org/abs/2412.17554v2 | Growth-Optimal E-Variables and an extension to the multivariate Csiszár-Sanov-Chernoff Theorem | \documentclass[11pt]{article}
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2412.17545v1 | http://arxiv.org/abs/2412.17545v1 | Lattice 3-polytopes of lattice width 2 and corresponding toric hypersurfaces | \documentclass[12pt,a4paper]{amsart}
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2412.17642v1 | http://arxiv.org/abs/2412.17642v1 | Characterization of Double-Arborescences and their Minimum-Word-Representants | \documentclass{llncs}
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2412.17668v1 | http://arxiv.org/abs/2412.17668v1 | The Graph Coloring Game on $4\times n$-Grids | \documentclass{article}
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2412.17735v1 | http://arxiv.org/abs/2412.17735v1 | Colouring t-perfect graphs | \documentclass[a4paper, 12pt]{article}
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2412.17775v1 | http://arxiv.org/abs/2412.17775v1 | The Calderón problem for the logarithmic Schrödinger equation | \documentclass[a4paper, 10pt, twoside, notitlepage]{amsart}
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2412.17916v1 | http://arxiv.org/abs/2412.17916v1 | Data-Driven Priors in the Maximum Entropy on the Mean Method for Linear Inverse Problems | \documentclass[onefignum,onetabnum]{siamonline220329}
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2412.17930v2 | http://arxiv.org/abs/2412.17930v2 | Runs in Paperfolding Sequences | \documentclass[12pt,reqno]{article}
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2412.18075v1 | http://arxiv.org/abs/2412.18075v1 | How many crossing changes or Delta-moves does it take to get to a homotopy trivial link? | \documentclass[12pt,letterpaper]{amsart}
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2412.18361v1 | http://arxiv.org/abs/2412.18361v1 | On a generalized Monge-Ampère equation on closed almost Kähler surfaces |
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2412.18436v2 | http://arxiv.org/abs/2412.18436v2 | Fundamental solutions for parabolic equations and systems: universal existence, uniqueness, representation | \documentclass[11pt, letterpaper]{amsart}
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2412.18427v1 | http://arxiv.org/abs/2412.18427v1 | Global Bifurcation Curve for Fourth-Order MEMS/NEMS Models with Clamped Boundary Conditions | \documentclass[reqno]{amsart}
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2412.18425v1 | http://arxiv.org/abs/2412.18425v1 | Computing the k-binomial complexity of generalized Thue--Morse words | \documentclass[a4paper]{article}
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2412.18480v1 | http://arxiv.org/abs/2412.18480v1 | Diameter bounds for distance-regular graphs via long-scale Ollivier Ricci curvature | \documentclass[a4paper,11pt]{article}
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2412.18555v1 | http://arxiv.org/abs/2412.18555v1 | Analysis of non-overlapping models with a weighted infinite delay | \documentclass{ws-m3as}
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2412.18692v1 | http://arxiv.org/abs/2412.18692v1 | Most subrings of $\mathbb{Z}^n$ have large corank | \documentclass[12pt]{amsart}
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2412.18761v1 | http://arxiv.org/abs/2412.18761v1 | Tail Dependence of Multivariate Archimedean Copulas |
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2412.18809v1 | http://arxiv.org/abs/2412.18809v1 | Existence and uniqueness of Generalized Polarization Tensors vanishing structures | \documentclass[10pt,reqno]{article}
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2412.18841v1 | http://arxiv.org/abs/2412.18841v1 | Splitting the difference: Computations of the Reynolds operator in classical invariant theory | \documentclass[11pt]{amsart}
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2412.18949v1 | http://arxiv.org/abs/2412.18949v1 | Metric Space Recognition by Gromov-Hausdorff Distances to Simplexes | \documentclass[leqno]{article}
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2412.18968v1 | http://arxiv.org/abs/2412.18968v1 | Boundary Blow-up Solutions of Second Order Quasilinear Equation on Infinite Cylinders | \documentclass[A4paper, 11pt]{amsart}
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2412.19008v1 | http://arxiv.org/abs/2412.19008v1 | On the minimal parabolic induction | \documentclass[a4paper]{article}
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2412.19054v2 | http://arxiv.org/abs/2412.19054v2 | Regularized neural network for general variational inequalities involving monotone couples of operators in Hilbert spaces | \begin{filecontents*}{example.eps}
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2412.19044v1 | http://arxiv.org/abs/2412.19044v1 | A New Adaptive Control Scheme for Unstable Heat Equation with Unknown Control Coefficient | \documentclass[11pt]{article}
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2412.19100v1 | http://arxiv.org/abs/2412.19100v1 | Constrained stochastic linear quadratic control under regime switching with controlled jump size | \documentclass[12pt]{amsart}
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2412.19095v1 | http://arxiv.org/abs/2412.19095v1 | On Laplacian and Distance Laplacian Spectra of Generalized Fan Graph & a New Graph Class | \documentclass[12pt]{article}
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2412.19126v2 | http://arxiv.org/abs/2412.19126v2 | Polycyclic Codes over the Product Ring $\mathbb{F}_q^l$ and their Annihilator Dual | \documentclass[11pt,a4paper]{article}
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2412.19223v1 | http://arxiv.org/abs/2412.19223v1 | Functional identities involving additive maps on division rings | \documentclass{amsart}
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2412.19258v1 | http://arxiv.org/abs/2412.19258v1 | Complexity and Structural Results for the Hull and Convexity Numbers in Cycle Convexity for Graph Products | \documentclass[12pt]{article}
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2412.19277v1 | http://arxiv.org/abs/2412.19277v1 | A study on the dual of $C(X)$ with the topology of (strong) uniform convergence on a bornology | \documentclass[reqno, 12pt]{amsart}
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2412.19334v1 | http://arxiv.org/abs/2412.19334v1 | The cuspidal cubic and line arrangements with only triple points | \documentclass[11pt,english]{amsart}
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2501.06197v1 | http://arxiv.org/abs/2501.06197v1 | A Physics-informed Sheaf Model | \documentclass{article}
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2412.19410v1 | http://arxiv.org/abs/2412.19410v1 | Game theoretical asymptotic mean value properties for non-homogeneous $p$-Laplace problems | \documentclass[a4paper, 10pt]{amsart}
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2412.19397v1 | http://arxiv.org/abs/2412.19397v1 | Well-Posedness of Second-Order Uniformly Elliptic PDEs with Neumann Conditions | \documentclass[12pt]{article}
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2412.19486v1 | http://arxiv.org/abs/2412.19486v1 | A group and the completion of its coset semigroup | \def\CTeXPreproc{Created by ctex v0.2.13, don't edit!}
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2412.19536v1 | http://arxiv.org/abs/2412.19536v1 | Potential Vector Fields in $\mathbb R^3$ and $α$-Meridional Mappings of the Second Kind $(α\in \mathbb R)$ |
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2412.19558v1 | http://arxiv.org/abs/2412.19558v1 | Pretabular Tense Logics over S4t |
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2412.19618v1 | http://arxiv.org/abs/2412.19618v1 | On Counting Constructions and Isomorphism Classes of $I$-Graphs | \documentclass[12pt]{article}
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2412.19605v1 | http://arxiv.org/abs/2412.19605v1 | Infinitary combinatorics in condensed math and strong homology | \documentclass{amsart}
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2412.19681v1 | http://arxiv.org/abs/2412.19681v1 | Casimir Radial Parts via Matsuki Decomposition | \pdfoutput=1
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2412.19714v2 | http://arxiv.org/abs/2412.19714v2 | Fractional nonlinear Schrödinger and Hartree equations in modulation spaces | \documentclass[12pt,a4paper,oneside,reqno]{amsart}
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2412.19752v1 | http://arxiv.org/abs/2412.19752v1 | A random walk among random graphs | \documentclass[12pt]{report}
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2412.19895v1 | http://arxiv.org/abs/2412.19895v1 | The c-Entropy optimality of Donoghue classes | \documentclass{amsproc}
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2501.00047v3 | http://arxiv.org/abs/2501.00047v3 | $σ$-Sets and $σ$-Antisets | \documentclass{article}
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\usepackage{amsthm}
\newtheore... |
2412.19956v1 | http://arxiv.org/abs/2412.19956v1 | $L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve | \documentclass[11pt,a4paper,reqno]{amsart}
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\AtBeginDocument{\pagenumbering{arabic}}
\usepackag... |
2412.19960v1 | http://arxiv.org/abs/2412.19960v1 | Numerical Linear Algebra: Least Squares, QR and SVD | \documentclass[12pt]{article}
\input{headlines}
\title{\large\sffamily\bfseries Lecture Notes \\
\LARGE{Numerical Linear Algebra}\\ \large{Least Squares, QR and SVD}}
\author{\Large\sffamily Davoud Mirzaei\\ Uppsala University}
\date{\sffamily September 1, 2023}
\begin{document}
\setlength{\abovedisplayskip}{3pt}
... |
2412.19959v1 | http://arxiv.org/abs/2412.19959v1 | Numerical Solution of Initial Value Problems | \documentclass[12pt]{article}
\input{headlines}
\title{\large\sffamily\bfseries Lecture Notes \\
\LARGE{Numerical Solution of ODEs}}
\author{\Large\sffamily Davoud Mirzaei\\ Uppsala University}
\date{\sffamily February 16, 2023}
\begin{document}
\setlength{\abovedisplayskip}{3pt}
\setlength{\belowdisplayskip}{3pt... |
2412.20001v2 | http://arxiv.org/abs/2412.20001v2 | Colouring signed Kneser, Schrijver, and Borsuk graphs | \documentclass[11pt, a4paper]{article}
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\usepackage{graphics}
\usepackage{pict2e}
\usepackage{cite}
\usepackage{subfigure}
\us... |
2412.20097v1 | http://arxiv.org/abs/2412.20097v1 | Parameter spaces for cross-diffusive-driven instability in a reaction-diffusion system on an annular domain |
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\markboth{G. Yigit et al.}{Instability in a reaction-diffusion system}
\title{Parameter space... |
2412.20137v1 | http://arxiv.org/abs/2412.20137v1 | On convergence of Thurston's iteration for transcendental entire functions with infinite post-singular set | \documentclass[10pt,reqno,a4paper]{amsart}
\setcounter{tocdepth}{2}
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\renewcommand{\tocsubsection}[2]{\hspace{2em}\oldtocsubsection{#1}{#2}}... |
2412.20174v1 | http://arxiv.org/abs/2412.20174v1 | Explicit bounds on common projective torsion points of elliptic curves | \documentclass[11pt]{amsart}
\usepackage{amssymb,amsthm,amsmath,amstext}
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2412.20282v1 | http://arxiv.org/abs/2412.20282v1 | Invariance of intrinsic hypercontractivity under perturbation of Schrödinger operators |
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\newtheorem{algorithm}[theorem]{Algor... |
2412.20353v1 | http://arxiv.org/abs/2412.20353v1 | Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds | \documentclass[twoside,a4paper,11pt]{amsart}
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2412.20339v2 | http://arxiv.org/abs/2412.20339v2 | On the formal ribbon extension of a quasitriangular Hopf algebra |
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2412.20443v4 | http://arxiv.org/abs/2412.20443v4 | Monogenic trinomials and class numbers of related quadratic fields | \documentclass[12]{amsart}
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\newcommand{\ord}{\operatorname{ord}}
\newtheor... |
2412.20483v1 | http://arxiv.org/abs/2412.20483v1 | Curvature, area and Gauss-Bonnet formula of the Moyal sphere | \documentclass[a4paper,12pt]{article}
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\usepackage{graphicx,hyperref}
\usepackage{amsthm,float,subfigure}
\newtheorem{prop}{Property}
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2501.01383v2 | http://arxiv.org/abs/2501.01383v2 | Electrical networks and data analysis in phylogenetics | \documentclass[11pt,reqno]{amsart}
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\usepackage{yhmath}
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\usepackage[... |
2412.20562v1 | http://arxiv.org/abs/2412.20562v1 | Some Necessary and Sufficient Conditions for Diophantine Graphs | \documentclass{article}
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\usepackage{color}
\usepackage{lineno}
\usepackage{multirow}
\usepackage{epstopdf}
\usepackage{rota... |
2412.20589v1 | http://arxiv.org/abs/2412.20589v1 | Countable models of weakly quasi-o-minimal theories I | \documentclass{amsart}
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final,
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\the... |
2412.20652v1 | http://arxiv.org/abs/2412.20652v1 | Hyperbolic knots with arbitrarily large torsion order in knot Floer homology |
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\newtheorem{claim}[theorem]{Claim}
\newtheorem{conj... |
2412.20642v2 | http://arxiv.org/abs/2412.20642v2 | Direct and inverse spectral problems for the Schrodinger operator with double generalized Regge boundary conditions |
\documentclass[reqno,12pt,centertags]{article}
\usepackage{amsmath,amsthm,amscd,amssymb,latexsym,upref,stmaryrd}
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\usepackage[numbers,sort&compress]{natbib}
\usepackage{amsmath}
\newcommand{\highl... |
2412.20703v2 | http://arxiv.org/abs/2412.20703v2 | The Restricted Inverse Optimal Value Problem under Weighted Bottle-neck Hamming distance on trees |
\documentclass[a4paper,fleqn]{cas-dc}
\def\tsc#1{\csdef{#1}{\textsc{\lowercase{#1}}\xspace}}
\tsc{WGM}
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\usepackage{amsfonts, amssymb, amsmath, cite,bm}
\usepackage{algorithm}
\usepackage{algorithmic}... |
2412.20700v1 | http://arxiv.org/abs/2412.20700v1 | Optimal rolling of fair dice using fair coins | \documentclass[12pt]{article}
\usepackage[backend=biber,style=numeric,]{biblatex}
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\newtheorem{lemma}{Lemma}
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\usepackage{etoolbox}
\preto\tabular{\setcounter{alglinecounter}{0}}
\ne... |
2412.20782v1 | http://arxiv.org/abs/2412.20782v1 | A randomisation method for mean-field control problems with common noise | \documentclass[
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\usepackage{amsmath}
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\usepackage{listings}
\usepackage{lmodern} \usepack... |
2412.20778v2 | http://arxiv.org/abs/2412.20778v2 | An accurate approach to determining the spatiotemporal vehicle load on bridges based on measured boundary slopes | \documentclass[preprint,12pt]{elsarticle}
\usepackage{amssymb}
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\newdefinition{rmk}{Remark}
\newproof{proof}{P... |
2412.20775v6 | http://arxiv.org/abs/2412.20775v6 | On Spectral Graph Determination | \documentclass[11pt,twoside,reqno]{amsart}
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\usepackage{amsmath, amssymb, ... |
2412.20841v1 | http://arxiv.org/abs/2412.20841v1 | Compact harmonic RCD$(K, N)$ spaces are harmonic manifolds | \documentclass[12pt]{article}
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\usepackage{hyperref}
\usepackage{bm}
\usepackage{indentfirst}
\us... |
2501.04026v2 | http://arxiv.org/abs/2501.04026v2 | Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I | \documentclass{article}
\usepackage[utf8]{inputenc}
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\usepackage{graphicx}
\usepackage{enumitem}
\usepackage{parskip}
\usepackage{am... |
2412.21021v1 | http://arxiv.org/abs/2412.21021v1 | On the Algebraic Connectivity of Token Graphs and Graphs under Perturbations | \documentclass[11pt,a4paper]{article}
\usepackage{soul}
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\usepackage{amsmath,amsthm,amsbsy}
\usepackage{color}
\usepackage{cite}
\usepackage{subfig}
\usepackage{float}
\usepackage{graphicx,tikz}
\usepackage{mathrs... |
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