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https://mathoverflow.net/questions/210676 | 8 | Consider a smooth, connected and complete Riemannian manifold $M$. It is well known that harmonic functions defined on some open subset of $M$ are $C^\infty$.
I'm interested in knowing whether there are quantitative $C^1$ estimates for such harmonic functions which depend only on some bound from below on the curvatur... | https://mathoverflow.net/users/58975 | $C^1$ regularity of harmonic functions on Riemannian manifolds | For the people interested in the solution to this problem, Gigli encouraged Nuñez Zimbrón and De Philippis to look for a solution to this problem, which now can be found here <https://arxiv.org/pdf/1909.05220.pdf>
| 3 | https://mathoverflow.net/users/510475 | 452743 | 181,952 |
https://mathoverflow.net/questions/452708 | 3 | Let $n$ be any integer greater than $2^{10^6}$. Given any $s\le (\log\_2 n)/1000$ integers $1=q\_1\le q\_2\le \cdots q\_{s-1}\le q\_s=n$. Prove that
$$\min\_\ell\left(\sum\_{i=1}^\ell q\_i\right)\left(\sum\_{i=\ell+1}^s\frac{1}{q\_i}\right)\le \frac{1}{64}$$
| https://mathoverflow.net/users/510447 | Min problem on integers | Let us denote
$$\sigma\_\ell:=\sum\_{i=1}^\ell q\_i\qquad\text{and}\qquad\tau\_\ell:=\sum\_{i=\ell+1}^s\frac{1}{q\_i}.$$
Then
$$\prod\_{\ell=1}^{s-1}\left(\frac{q\_\ell}{q\_{\ell+1}}\cdot\frac{\sigma\_{\ell+1}}{\sigma\_\ell}\cdot\frac{\tau\_{\ell-1}}{\tau\_\ell}\right)=\frac{q\_1}{q\_s}\cdot\frac{\sigma\_{s}}{\sigma\_1... | 9 | https://mathoverflow.net/users/11919 | 452748 | 181,955 |
https://mathoverflow.net/questions/452696 | 1 | We consider the minimal surface equation $$
(1+|\nabla u|^2) \, \Delta u=\sum\_{i,j=1}^n\partial\_iu \, \partial\_ju \, \partial\_{ij}u\quad\hbox{in $B\_1\subset\mathbb R^n.$}
$$
If $u\in C^2(B\_1)$ is a positive solution of above equation, does the Harnack inequality hold in $B\_{1/2}$? That is, is there a constant $C... | https://mathoverflow.net/users/105893 | Harnack inequality for the minimal surface equation | The minimal surface equation is uniformly elliptic, at least in the sense that its linearization at any solution is uniformly elliptic. It will be convenient to rewrite the equation as
$$\nabla \cdot \left(\frac{\nabla u}{\sqrt{1 + |\nabla u|^2}}\right) = 0$$
(this is the standard way that the equation is written on eu... | 1 | https://mathoverflow.net/users/109533 | 452752 | 181,957 |
https://mathoverflow.net/questions/452753 | 1 | Let $X=\operatorname{Spec}(A)$ be an affine [Dedekind domain](https://en.m.wikipedia.org/wiki/Dedekind_domain#Alternative_definitions) with field of fractions $K$. Let $\widetilde{A}$ be the integral closure of $A$ in separable closure $ K^{\text{sep}}$. A closed point $x$ of $X$ is a nonzero prime ideal $\mathfrak{p}$... | https://mathoverflow.net/users/108274 | Field of fractions of etale stalk of Dedekind domain (Example from Milne's LEC) | As Milne (sort of) explains, the following two data are exactly the same:
1. A connected étale neighbourhood $U \to X$ with a lift $\bar x \to U$ of $\bar x \to X$, up to shrinking Zariski-locally around the image of $\bar x$ in $U$ (I suppose you could call these 'Zariski germs of connected étale neighbourhoods');
2... | 4 | https://mathoverflow.net/users/82179 | 452758 | 181,958 |
https://mathoverflow.net/questions/452760 | -3 | In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers:
>
> In the language of set theory, what we are doing here is starting with the space N × N of ordered
> pairs (a, b) of natural numbers. Then we place an equivalence relation ∼ on these pairs ... | https://mathoverflow.net/users/510487 | Analysis I, simpler proof of Tao's construction of the integers | In fact one doesn't need the replacement axiom at all in order to implement this set-theoretic construction of the integers. The entire construction can be undertaken in Zermelo set theory, which lacks the replacement axiom.
Specifically, for a fixed copy of the natural numbers $\mathbb{N}$, one can form the set of p... | 13 | https://mathoverflow.net/users/1946 | 452763 | 181,960 |
https://mathoverflow.net/questions/452730 | 1 | Let $V > 0$ and let $\Phi(\cdot)$ be the standard normal CDF.
Consider the infimum of
$$f(x\_1, x\_2,x\_3, p\_1, p\_2, p\_3) := p\_1 \Phi(x\_1) + p\_2 \Phi(x\_2) + p\_3 \Phi(x\_3)$$
with respect to $x\_1, x\_2, x\_3, p\_1, p\_2$ and $p\_3$ satisfying
$\begin{align}
&p\_1 \geq 0,\\\\
&p\_2 \geq 0,\\\\
&p\_3 \geq 0,\... | https://mathoverflow.net/users/145647 | Showing that the infimum is a minimum | $\newcommand{\R}{\mathbb R}\newcommand{\tx}{\tilde x}$Indeed, the infimum (say $f^\*$) in question is attained for each $V\in(0,\infty)$.
To prove this, for each $j\in\{1,2,3\}$ let $(x\_j^n)\_{n=1}^\infty$ and $(p\_j^n)\_{n=1}^\infty$ be sequences such that for each $n$ all your conditions are satisfied with $x\_1^n... | 2 | https://mathoverflow.net/users/36721 | 452766 | 181,963 |
https://mathoverflow.net/questions/452663 | 4 | Suppose that $(X,\rho)$ is a compact [doubling metric space](https://en.wikipedia.org/wiki/Doubling_space). Does there necessarily exist an $\epsilon>0$ and a maximal $\epsilon$-net $\{x\_i\}\_{i=1}^n\subseteq X$ such that the map
$$
\begin{aligned}
\Phi:(X,\rho) & \rightarrow (\mathbb{R}^n,|\cdot|\_2) \\
x&\mapsto \bi... | https://mathoverflow.net/users/36886 | Bi-Lipschitz embeddings of compact doubling spaces | No. There are many compact, doubling metric spaces with no bi-Lipschitz embedding in a Hilbert space. Some examples:
* The closed unit ball in the (continuous) Heisenberg group with its Carnot-Carath'eodory metric. (See, e.g., Theorem 6.1 of "Differentiability of Lipschitz maps from metric spaces to Banach spaces" an... | 2 | https://mathoverflow.net/users/510495 | 452767 | 181,964 |
https://mathoverflow.net/questions/452761 | 2 | Let $M$ be a type $\rm{II}\_{1}$ factor with trace $\tau$, acting by the GNS representation on $B(L^{2}(M,\tau))$. Let $R\subset M \subset B(L^{2}(M,\tau))$ be a hyperfinite $\rm{II}\_{1}$ subfactor of $M$.
>
> **Question:** Is there a norm one Banach space projection $\Phi: B(L^{2}(M,\tau))\rightarrow R$ such that... | https://mathoverflow.net/users/6269 | Hyperexpectations from injective subfactors of a type $II_1$ factor | Such a projection $\Phi$ actually always exists. In the terminology of Definition 2.2 of [N. Ozawa and S. Popa, On a class of II$\_1$ factors with at most one Cartan subalgebra. Ann. of Math. 172 (2010), 713-749] the existence of $\Phi$ is saying that $R$ is amenable relative to $\mathbb{C} 1$ inside $M$ (see Theorem 2... | 3 | https://mathoverflow.net/users/159170 | 452775 | 181,965 |
https://mathoverflow.net/questions/451346 | 2 | Let $(X\_{jn})\_{1\leq j \leq n}$, $n\in \mathbb N$, be a triangular array of random vectors in $\mathbb R^d$ (the $X\_{jn}$ are understood to be independent in $j$ for fixed $n$.). We say that the triangular array is *null* if $X\_{jn} \overset{p}{\to} 0$, as $n \to \infty$, uniformly in $j$, i.e.,:
\begin{equation}
\... | https://mathoverflow.net/users/479236 | Convergence of the row sums in a triangular null array with zero mean | $\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}$Let us rephrase the question a bit. For each natural $n$, let $X\_{1,n},\dots,X\_{j\_n,n}$ be independent zero-mean random vectors in $\R^d$ such that (i) for each real $\ep>0$
\begin{equation\*}
\max\_{j\in J\_n} P(\|X\_{j,n}\|>\ep)\to0 \tag{0}\label{0}
\end{eq... | 1 | https://mathoverflow.net/users/36721 | 452788 | 181,969 |
https://mathoverflow.net/questions/452692 | 1 | (Asking again in a new question because the [previous version](https://mathoverflow.net/q/452519/36721) had insufficient conditions, as pointed out in the [answer there](https://mathoverflow.net/a/452659/36721).)
Define the densities:
$$p(\phi;\theta,r) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi+\theta... | https://mathoverflow.net/users/510206 | Monotone likelihood ratio of a family of densities with convexity property | $\newcommand{\ep}{\varepsilon}$The "convex" part of this conjecture is not true in general.
Indeed, suppose it is true. Then (letting $x:=\phi$, $t:=\theta\_1$, and $\theta\_2\downarrow\theta\_1=t$) we see that for any strictly increasing convex smooth function $g$ with $g'''\ge0$ and all $x$ and $t$ in $(0,\pi/2)$ w... | 1 | https://mathoverflow.net/users/36721 | 452792 | 181,970 |
https://mathoverflow.net/questions/452794 | 0 | (Asking a final time in a new question because the [previous version](https://mathoverflow.net/q/452519/36721) had insufficient conditions, as pointed out in the [answer there](https://mathoverflow.net/a/452659/36721).)
Define the densities:
$$p(\phi;\theta,r) = \Big(f\big(r\cos(\phi-\theta)\big) - f\big(r\cos(\phi... | https://mathoverflow.net/users/510206 | Sign Regularity of a Density Kernel with Convexity Properties | $\newcommand{\ep}{\varepsilon}$This conjecture is not true in general.
Indeed, suppose it is true. Then (letting $x:=\phi$, $t:=\theta\_1$, and $\theta\_2\downarrow\theta\_1=t$) we see that for any strictly increasing convex smooth function $g$ with $g'''>0$ and $g''''>0$ and all $x$ and $t$ in $(0,\pi/2)$ we would h... | 1 | https://mathoverflow.net/users/36721 | 452800 | 181,974 |
https://mathoverflow.net/questions/452803 | 1 | I'm looking for references for two facts that are stated without proof in the paper:
>
> *Talagrand, M.*, Are all sets of
> positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
> Geometric aspects of functional analysis. Israel seminar (GAFA)
> 1992-94. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 7... | https://mathoverflow.net/users/380543 | Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand | $\newcommand{\R}{\mathbb R}\renewcommand{\th}{\theta}\newcommand{\ep}{\varepsilon} $This indeed follows immediately from [Theorem 15](https://doi.org/10.1007/BF02392556) of Talagrand, Regularity of Gaussian processes, Acta Math. 159, No. 1-2, 99-149 (1987).
---
Details: The mentioned theorem by Talagrand states t... | 2 | https://mathoverflow.net/users/36721 | 452804 | 181,977 |
https://mathoverflow.net/questions/452789 | 7 | Let $A\_n$ be the coinvariant algebra of the symmetric group $S\_n$.
This algebra has vector space dimension $n!$.
$A\_n$ is the quotient algebra of the polynomial ring $K[x\_1,...,x\_n]$ by the elementary symmetric polynomials $e\_i$. Since $e\_1=x\_1+....+x\_n$, one can "eliminate" the variable $x\_n$ to obtain an al... | https://mathoverflow.net/users/61949 | Basis parametrized by the symmetric group elements for the coinvariant algebra | So the Artin basis already does this. There is a natural bijection of the symmetric group $S\_n$ with sequences $(a\_1,\ldots,a\_{n-1})$ of integers $a\_i$ with $0\leq a\_i\leq n-i$. Namely, let $w:[n]\to [n]$ be a permutation. Then define
$$a\_i(w)=\#\{j>i\mid w(i)>w(j)\}$$
This is known as the Lehmer code. It's a lit... | 8 | https://mathoverflow.net/users/62135 | 452809 | 181,978 |
https://mathoverflow.net/questions/452811 | 11 | I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does the statement that there exist embedding functors (monomorphisms in $\mathbf{Cat}$) $\mathcal{F}: \mathcal{C} \rightarrow... | https://mathoverflow.net/users/510524 | Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property? | One can take some of the standard violations of CSB with other kinds of mathematical structures and transfer them to categories.
For example, with linear orders, we have the two linear orders $$\langle\mathbb{Q},\leq\rangle\qquad \langle\mathbb{Q}^{\geq 0},\leq\rangle,$$
which each order-embed into each other, but th... | 21 | https://mathoverflow.net/users/1946 | 452814 | 181,982 |
https://mathoverflow.net/questions/452580 | 2 | I am reading "The Uncertainty Principle" by Fefferman (Bull. AMS, 1983) and have some issues following the arguments. In Lemma $C$ we have the following setting:
>
> Let $Q^0\subseteq \mathbb{R}^n$ be some dyadic cube, $V: \mathbb{R}^n \rightarrow (-\infty, 0]$ some function, $p>1$ and
> $$ V^+\_Q(x):= \sup\_{\_{Q'... | https://mathoverflow.net/users/91098 | Maximal function on small cubes | By Holder’s inequality
$$ \frac{1}{\lvert Q\rvert}\int\_{Q}V^{+}dm \leq \frac{1}{\lvert Q\rvert} \lvert Q\rvert^{1/q}\cdot \left(\int\_{Q} \left(V^{+}\right)^{p}dm \right)^{1/p} = \left(\lvert Q\rvert^{-1}\int\_{Q} \left(V^{+}\right)^{p}dm \right)^{1/p}.$$
The RHS term is $\lVert V^{+}\rVert\_{p}$ (up to the $Q$-norm... | 2 | https://mathoverflow.net/users/8857 | 452819 | 181,984 |
https://mathoverflow.net/questions/452805 | 0 | Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in [this](https://math.stackexchange.com/questions/4301414/if-f-n-is-cauchy-and-converges-a-e-to-f-there-is-a-cauchy-subsequence) thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue mea... | https://mathoverflow.net/users/99469 | Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable? | Let
* $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, and $(E, | \cdot |)$ a Banach space.
* $S (X)$ the space of $\mu$-simple functions from $X$ to $E$.
* $\mathcal C :=\mathcal A \otimes \mathcal B$ the product $\sigma$-algebra of $\mathcal A$ and $\mathcal B$.
* $\lamb... | 0 | https://mathoverflow.net/users/99469 | 452821 | 181,986 |
https://mathoverflow.net/questions/452816 | 1 | Let $X$ be an extremally disconnected compact Hausdorff space *with no open points*, and $f:X\to\mathbb{C}$ be a non-constant continuous function. Let $D\_f$ be the linear span of the functions of the form $$u\to \gamma(u) f(\sigma(u))$$ where $\sigma:X\to X$ is a homeomorphism, and $\gamma:X\to\mathbb{C}$ is continuou... | https://mathoverflow.net/users/164350 | Subspaces generated by the orbits of the group of isometries on $C(K)$ | Take two extremally disconnected spaces $Y$ and $Z$ without open points and with different cardinality (say $Z$ is the one with larger cardinality). Define $X$ as the disjoint union of $Y$ and $Z$ and let $f$ be the indicator function of $Y$.
For every sequence $(g\_n)$ in $D\_f$ the union of the supports $\{g\_n \no... | 5 | https://mathoverflow.net/users/102946 | 452822 | 181,987 |
https://mathoverflow.net/questions/452843 | 0 | The real hyperbolic fixed points of $\mathrm{SL}\_2(\mathbb{Z})$ are the points $x\in\mathbb R\smallsetminus\mathbb{Q}$ with
$$
\frac{ax+b}{cx+d}=x
$$
for some $\left(\begin{array}{2} a&b\\ c&d\end{array}\right)\in\mathrm{SL}\_2(\mathbb{Z})$. These are special quadratic numbers and I wonder, what their properties are.
... | https://mathoverflow.net/users/473423 | Hyperbolic fixed points of SL(2,Z) | Am I missing anything?
Any fixed point $x$ is a root of quadratic equation $px^2+qx+r=0$ with $p,q,r\in\mathbb Z$, where $D=q^2-4pr$ is not a square; we assume that $(p,q,r)=1$. This $x$ must also satisfy $cx^2+(d-a)x-b=0$, which yields $c=\nu p$, $b=-\nu r$, $d=\nu q+a$ for an integer $\nu$. So we need
$$
1=\left|\... | 8 | https://mathoverflow.net/users/17581 | 452845 | 181,991 |
https://mathoverflow.net/questions/451815 | 1 | Let $X\_t, Y\_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields.
Is there some analogue of the following fact in finite dimensions (with no time dependence, identifying Lie group elements with exponentials of algebra elements could also b... | https://mathoverflow.net/users/125989 | Commuting time dependent vector fields and pullback invariance | Let $\phi^t$, $\psi^t$ be the respective flows of two vector fields $A$, $B$ at time $t$ on a manifold $M$ (assume to fix ideas that these flows do exist for every time).
If $A, B$ do ***not*** depend on time, then of course one has an equivalence between the 4 properties
i) $[A,B]=0$;
ii) $A$ is invariant by every... | 2 | https://mathoverflow.net/users/105095 | 452864 | 181,998 |
https://mathoverflow.net/questions/452865 | 3 | Is this true that finitely generated flat module over an integral domain is projective.
If Yes, please provide a proof.
| https://mathoverflow.net/users/510594 | On Flat and Projective Modules over integral domain | Yes, this is true. Lemma 5 on page 249 of Cartier, "Questions de rationalité des diviseurs en géométrie algébrique" says that over an integral domain, whether or not it is Noetherian, if the localisation of a finitely generated module at every maximal ideal is free, then the module is projective. Over a local ring, eve... | 5 | https://mathoverflow.net/users/460592 | 452867 | 181,999 |
https://mathoverflow.net/questions/452866 | 2 | $\newcommand{\std}{\mathrm{std}}\newcommand{\SL}{\mathrm{SL}}\newcommand{\mmod}{/\!\!/}$Fix the base field to be the complex numbers $\mathbf{C}$. Let $\std = \mathbf{A}^2$ denote the standard representation of $\SL\_2$, so that there is a natural action of $\SL\_2^{\times 3}$ on $\std^{\otimes 3}$. Let $Y$ denote the ... | https://mathoverflow.net/users/102390 | Pairs of quadratic forms and $\mathbf{A}^8/\mathrm{SL}_2^{\times 3}$ | $\newcommand{\std}{\mathrm{std}}\newcommand{\SL}{\mathrm{SL}}\newcommand{\mmod}{/\!\!/}$
A lowbrow answer to your question is given by Bhargava's theory of $2 \times 2 \times 2$ cubes explained in his [Higher Composition Laws Paper](https://annals.math.princeton.edu/wp-content/uploads/annals-v159-n1-p03.pdf).
In part... | 6 | https://mathoverflow.net/users/422 | 452870 | 182,000 |
https://mathoverflow.net/questions/452443 | 2 | Let $F$ be an order unit Banach space with order unit $e$ and topological dual space $F^\*$ ordered by the dual cone. Let $E\subset F^\*$ be a closed subspace that separates points of $F$ and such that the dual cone intersected with $E$ is spanning for $E$. Consider the initial topology $\sigma(E,F)$ induced on $E$ by ... | https://mathoverflow.net/users/493899 | Decomposition of weak* convergent nets into positive weak* convergent nets | Based on the comment by @BillJohnson:
If $0\le \omega\in E$ then $\|\omega\| = \omega(e)$ implying that a net of positive elements in $E$ converges to $0$ in weak$^\*$ topology if and only if it converges to $0$ in norm. Thus, a positive answer to the question would imply that a net (not necessarily positive) converges... | 2 | https://mathoverflow.net/users/493899 | 452874 | 182,001 |
https://mathoverflow.net/questions/452783 | 1 | Oguiso writes[1]
>
> **Theorem 1.1** *Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A\_k$ be the generic fiber of $f$. Then, $\rho(A\_K)= 1$. Here $\rho(A\_K)$ is the Picard number of $A\_K$ over $K$.*
>
>
> It can happen that $\rho(X\_t) \ge... | https://mathoverflow.net/users/111897 | Are horizontal divisors on abelian fibered hyperkähler manifolds proportional in $NS(X)$ up to vertical divisors? | This is simply false, already for K3 surfaces with an elliptic fibration. The quotient of $NS(X)$ by the subgroup generated by the vertical components + the zero section is the Mordell-Weil group of the generic fiber, it can very well be nontrivial. See for instance *Elliptic surfaces* by Schütt and Shioda, Adv.Stud. P... | 2 | https://mathoverflow.net/users/40297 | 452878 | 182,004 |
https://mathoverflow.net/questions/452737 | 0 | Let $\frak{g}$ be a semisimple Lie algebra and $U(\frak{g})$ its universal enveloping algebra. The adjoint action of $\frak{g}$ on itself extends to an action of $\frak{g}$ on $U(\frak{g})$. How does this action interact with the PBW basis of $U(\frak{g})$? More precisely: are the irreducible submodules of $U(\frak{g})... | https://mathoverflow.net/users/507923 | Adjoint action on the universal enveloping algebra and the PBW theorem | Since the sum of all the trivial subrepresentations of $U(\mathfrak{g})$ with the adjoint action is its centre, if you could do this then the centre must have a monomial basis but it does not.
For example even if $\mathfrak{g}=\mathfrak{sl}\_2$ with basis $e,h,f$ in that order then the centre consists of polynomials ... | 6 | https://mathoverflow.net/users/345 | 452889 | 182,009 |
https://mathoverflow.net/questions/452887 | 10 | Is there a Bell number $B\_n$ of the form $2^k$ for some $k>1$? If there is, are there infinitely many?
| https://mathoverflow.net/users/101817 | Can a Bell number be a power of 2? | No. It's easy to see that $B\_n=4$ is impossible, and $B\_n$ is never divisible by 8. This follows from the fact that $B\_n$ is periodic modulo 8 with period 24. See, e.g., W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, [Arithmetic properties of Bell numbers to a composite modulus I](http://matwbn.icm.edu.pl/ksi... | 25 | https://mathoverflow.net/users/10744 | 452891 | 182,010 |
https://mathoverflow.net/questions/452855 | 17 | On Mathematics Stack Exchange, I asked the following question: [Why are infinite-dimensional vector spaces usually equipped with additional structure?](https://math.stackexchange.com/questions/4751895/why-are-infinite-dimensional-vector-spaces-usually-equipped-with-additional-stru) Although it received one good answer,... | https://mathoverflow.net/users/144779 | Why do infinite-dimensional vector spaces usually have additional structure? | Here is a supplement to the nice [answer](https://math.stackexchange.com/a/4751952/87579) that you got at MSE.
Much of the theory of infinite dimensional vector spaces is motivated by solving concrete problems in **analysis**. To solve differential equations, it is often profitable to use vector spaces of functions, ... | 15 | https://mathoverflow.net/users/494541 | 452892 | 182,011 |
https://mathoverflow.net/questions/452895 | 9 | What is known about the (asymptotic?) behaviour of the number of equivalence classes of functions $n\to n$, where two functions are considered equivalent if they differ by a permutation of $n$, i.e., $f\sim g$ if there is a permutation $\sigma$ of $n$ so that $f=\sigma g\sigma^{-1}$?
I am guessing this a known or ele... | https://mathoverflow.net/users/2225 | How many functions are there from a set to itself, up to isomorphism? | The OEIS page linked by Sam Hopkins also refers to page 308 of Finch, Mathematical Constants (which refers back to the Meir and Moon paper). Finch writes that the generating function
$$
P(x)=\sum\_1^{\infty}P\_nx^n
$$
satisfies
$$
1+P(x)=\prod\_1^{\infty}(1-T(x^k)))^{-1}
$$
where
$$
T(x)=\sum\_1^{\infty}T\_nx^n
$$
is g... | 11 | https://mathoverflow.net/users/3684 | 452901 | 182,015 |
https://mathoverflow.net/questions/452893 | 6 | Let $f(x)$ be a real-valued Riemann integrable function supported in $[0,1]$ with range in $[0,1]$. Let $\alpha$ be irrational. Consider the weighted Riemann sum
$$S\_N:=\frac{1}{N}\sum\_{k=1}^Nf\left(\frac{k}{N}\right)e^{2\pi i \alpha k}.$$
What can be said about $\lim\_{N\to\infty} S\_N$? It seems like a kind of ... | https://mathoverflow.net/users/499631 | Twisted Riemann sums | $\newcommand\abs[1]{\lvert#1\rvert}$As hinted at by [Achim Krause](https://mathoverflow.net/questions/452893/twisted-riemann-sums#comment1171477_452893), the limit is $0$ for all such functions $f$, however the approximation procedure is somewhat subtle.
First let us show it is $0$ for all step functions $f$. By line... | 11 | https://mathoverflow.net/users/173490 | 452902 | 182,016 |
https://mathoverflow.net/questions/452454 | 5 | In Shelah's [paper](https://shelah.logic.at/papers/679/) [679](https://arxiv.org/abs/math/0003163), he proves primitive recursive bounds for the polynomial Hales-Jewett theorem and thus for the polynomial van der Waerden theorem.
How about for the multidimensional polynomial HJ and multidimensional polynomial vdW the... | https://mathoverflow.net/users/119533 | Primitive recursive bounds for multidimensional polynomial vdW / HJ | It is a standard fact that the (linear) Hales--Jewett theorem tensorizes to yield its multidimensional version. The same observation applies to the polynomial version.
I will describe the idea for the two-dimensional case: assume that $n=2k$ is an even integer, $A$ is a finite alphabet and $c:A^{[n]^2}\to [r]$ is an ... | 4 | https://mathoverflow.net/users/510630 | 452922 | 182,020 |
https://mathoverflow.net/questions/452907 | 0 | Prove that $p\mid\genfrac[]0{}{p^w}k$ where $p$ is an odd prime, $w \in \mathbb{N}$, $1<k<p^w$ and $k \neq p^v$ for some positive integer $v<w$. This has to be already done I just can't find where. Where in the literature has this been stated?
| https://mathoverflow.net/users/265714 | Prime divisibility of Stirling numbers of first kind | (The $w=1$ case is well-known, so we suppose $w\ge2$ in what follows; we also let $p$ denote any prime, which need not be odd.) See the remarks following the proof of Theorem 2.2 in Fredric T. Howard, [“Congruences for the Stirling numbers and associated Stirling numbers”](https://eudml.org/doc/206267), *Acta Arithmeti... | 1 | https://mathoverflow.net/users/118745 | 452925 | 182,022 |
https://mathoverflow.net/questions/452911 | 2 | In [this post](https://mathoverflow.net/q/397649), an answer claims that the normal bundle of the Veronese decomposes into a filtration, such that the associated graded is $$\bigoplus\_{i=2}^d S^i T,$$ where $T$ is the tangent bundle. But I thought it was actually $$\bigoplus\_{i=2}^d S^i T \otimes \mathcal{O}(i-1)$$..... | https://mathoverflow.net/users/510609 | Normal bundle of veronese as iteration extension of symmetric powers | Your third exact sequence is incorrect --- the correct form is
$$
0 \to S^{d-1}V \otimes \mathcal{O}(d-1)
\to S^{d}V \otimes \mathcal{O}(d)
\to S^dT
\to 0.
$$
| 2 | https://mathoverflow.net/users/4428 | 452926 | 182,023 |
https://mathoverflow.net/questions/452920 | 10 | In Proposition 7.2.1.14 of *Higher Topos Theory*, Lurie asserts the following:
>
> Let $\mathcal{X}$ be an $\infty$-topos and let $\tau\_{\leq0}:\mathcal{X}\to\tau\_{\leq0}\mathcal{X}$ denote a left adjoint to the inclusion. A morphism $\phi:U\to X$ is an effective epimorphism if and only if $\tau\_{\leq 0}$ is an ... | https://mathoverflow.net/users/144250 | Effective epimorphisms and 0-truncations (HTT, 7.2.1.14) | Here's an easy way to resolve the circularity. Proposition 7.2.1.13 is only used in the proof of 7.2.1.14 to establish the following statement:
(1) If $f\colon V\to X$ is a monomorphism and is surjective on $\tau\_{\leq 0}$, then it is an isomorphism.
This in turn follows from:
(2) If $f\colon V\to X$ is a monomo... | 9 | https://mathoverflow.net/users/20233 | 452929 | 182,025 |
https://mathoverflow.net/questions/452932 | 0 | Below we use [Bochner measurability](https://www.wikiwand.com/en/Bochner_measurable_function) and [Bochner integral](https://www.wikiwand.com/en/Bochner_integral). Let
* $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces,
* $(E, | \cdot |)$ a Banach space,
* $S (X)$ the space... | https://mathoverflow.net/users/99469 | How to construct this sequence that converges a.e. in product measure and that has a very particular form? | $\newcommand\C{\mathcal C}\newcommand\A{\mathcal A}\newcommand\B{\mathcal B}$Using the $\sigma$-finiteness condition, truncation of $f$, and your *attempt*, we see that without loss of generality $f=1\_C$ for some $C\in\C$.
But in this case the result follows by (say) Theorem 1.4 of [this paper](https://link.springer... | 2 | https://mathoverflow.net/users/36721 | 452934 | 182,026 |
https://mathoverflow.net/questions/452930 | 4 | A 1950 result of Tur'an establishes an equivalence between any prime number theorem of the form $\operatorname{li}(x)-\pi(x)= O(xe^{-C(\log x)^\alpha}) \ (x \to \infty)$ and a certain class of zero-free regions of $\zeta(s)$. See: P. Tur'an, On the remainder-term of the prime-number formula, II, Acta Math. Acad. Sci. H... | https://mathoverflow.net/users/17218 | Zero-free regions of $\zeta(s)$ equivalent to prime number theorems with error bound | Yes. All of the results quoted in the answer are stated in Pintz's paper "[On the remainder term of the prime number formula. II. On a theorem of Ingham.](https://zbmath.org/0447.10038)" (Acta Arith. 37, 209-220 (1980)). Below, your $(\alpha,\beta)$ correspond to my $(1/(a+1),b/(a+1))$.
1. One direction is older than... | 10 | https://mathoverflow.net/users/31469 | 452936 | 182,027 |
https://mathoverflow.net/questions/452937 | 3 | I mean specifically real-valued Schwartz distributions on the real line. That is linear functionals on $C^{\infty}\_c(\mathbb{R})$ continuous in the canonical LF topology. My question is, what are all of these that have weak derivative 0?
Right now the only examples I know are distributions corresponding to const... | https://mathoverflow.net/users/38783 | What real distributions solve $f'=0$? | You can take Fourier transforms $\widehat{f'}(t)=it\widehat{f}(t)=0$ to conclude that $\widehat{f}$ is supported by $\{0\}$, so $\widehat{f}=\sum\_{j=0}^n c\_j \delta^{(j)}$, so $f$ is a polynomial and then $f=c$.
(If $f$ is locally integrable with distributional derivative in $L^1\_{\textrm{loc}}$, then $f$ is absol... | 4 | https://mathoverflow.net/users/48839 | 452938 | 182,028 |
https://mathoverflow.net/questions/452279 | 0 | Let $\hat{g}$ and $\bar{g}$ be two smooth Riemannian metrics defined, say, on $\mathbb{R}^n.$ Consider a smooth function $\xi$ that acts as an interpolation function between the two metrics above on an annulus $B\_{2R}(0)\backslash B\_R(0).$
If we define $g=\xi\hat{g} + (1-\xi)\bar{g},$ which is clearly a smooth Riem... | https://mathoverflow.net/users/100597 | Curvature tensor of interpolation of two metrics | After some Googling, the trick seems to be to consider the difference between the two metrics $ h = \tilde{g}-\bar{g}.$ Thus, $g= \bar{g} + \xi h.$ Using the notation above, we can write the Christoffel symbols of $g$ schematically as:
$\Gamma(g) = \Gamma(\bar{g}) + \frac{1}{2}g \* \bar{\nabla}(\xi h),$
where $ h =... | 0 | https://mathoverflow.net/users/100597 | 452940 | 182,030 |
https://mathoverflow.net/questions/452942 | 4 | Let $E$ be an elliptic curve over a number field $K$ and $p$ a prime. Suppose that $E$ has a $K$-rational $p$-torsion, which gives the short exact sequence $0\to\mathbb{Z}/p\to E[p]\to\mu\_p\to0$ of Galois modules. Let us assume that this sequence splits, i.e. $E[p]\simeq\mathbb{Z}/p\times\mu\_p$ as Galois modules. Set... | https://mathoverflow.net/users/44005 | Primes of bad reductions for quotients of elliptic curves | As has already been remarked, but more generally, if $K$ is a number field and $A/K$ and $B/K$ are abelian varieties that are isogenous over $K$, then the criterion of Neron-Ogg-Shafarevich has as a quick corollary that $A$ and $B$ have the same set of primes of bad reduction. The paper with the proof is by Serre and T... | 7 | https://mathoverflow.net/users/11926 | 452944 | 182,032 |
https://mathoverflow.net/questions/452906 | 0 | I have a question about affine Coxeter groups when reading Humphreys's book:
Let $(W,S)$ be an irreducible affine Coxeter group, $M=(m\_{ij})$ be its Coxeter matrix, and $\{\alpha\_s\}\_{s\in S}\in V$ be the system of simple roots in the standard geometric realization, so the $\{\alpha\_s\}$ are linearly independent.... | https://mathoverflow.net/users/70248 | A question on irreducible affine Coxeter groups | When you ask for a *direct* proof, I'm not sure what I'm allowed to use. But if I am allowed to know how to construct the affine root system from an appropriate finite root system, the fact that $W'$ is non-trivial becomes easy:
If you use the standard construction of an affine root system for your Coxeter group, the... | 5 | https://mathoverflow.net/users/5519 | 452954 | 182,037 |
https://mathoverflow.net/questions/336330 | 11 | $\def\inv{\mathrm{inv}}\def\Acyc{\mathrm{Acyc}}$Let $G$ be a graph whose vertices are numbered $\{ 1,2, \ldots, n \}$. Given an orientation $\omega$ of $G$, define the inversions of $\omega$, written $\inv(\omega)$, to be the set of edges $(i,j)$ with $i<j$, which are oriented $i \leftarrow j$. Define one orientation $... | https://mathoverflow.net/users/297 | When is the poset of acyclic orientations of a graph a lattice? | Vincent Pilaud's recent paper ["Acyclic reorientation graphs and their lattice quotients"](https://arxiv.org/abs/2111.12387) is a thorough answer to this question and every question like it. In particular, here is the answer to the particular question which is asked:
Given a directed acyclic graph $G$, the **transiti... | 6 | https://mathoverflow.net/users/297 | 452969 | 182,042 |
https://mathoverflow.net/questions/452971 | 4 | *Note: We view the sphere $S^1$ as $[0,1]$ with the endpoints identified, and equip it with its usual addition structure, and Lebesgue measure.*
**Question:** Does there exist an absolute constant $C > 0$ such that for all $L^1$ functions $f: S^1 \to \mathbb R$,
$$\sup\_{t \in S^1} \int\_{S^1} |f(x + t) - f(x)| \, ... | https://mathoverflow.net/users/173490 | The maximal difference between a function and translates of itself | Using the condition $\int\_{S^1}dx=1$ and Jensen's inequality, we have
$$\sup\_{t\in S^1} \int\_{S^1}dx\,|f(x + t)-f(x)|
\ge\int\_{S^1}dt\, \int\_{S^1}dx\,|f(x + t)-f(x)| \\
=\int\_{S^1}dx\, \int\_{S^1}dt\,|f(x)-f(x + t)|
\ge\int\_{S^1}dx\, \Big|f(x)-\int\_{S^1}dt\,f(x + t)\Big| \\
=\int\_{S^1}dx\, \Big|f(x)-\int\_{S^... | 5 | https://mathoverflow.net/users/36721 | 452977 | 182,046 |
https://mathoverflow.net/questions/452989 | 0 | Suppose that $G=(V,E)$ is a simple, undirected graph. We say that $D\subseteq V$ is *dominating* if for all $v\in V\setminus D$ there is $d\in D$ such that $\{v,d\}\in E$. We say $D$ is *minimal dominating* if for all $d\in D$ we have that $D\setminus\{d\}$ is no longer dominating. There are graphs that [do not have mi... | https://mathoverflow.net/users/8628 | Minimal dominating sets in flat graphs | For flat graphs, dominating sets satisfy Zorn's condition: the intersection of a chain of dominating sets $D:=\cap D\_\alpha$ is dominating. Indeed, for any vertex $v$ the finite set $\{v\cup N(v)\}$ has non-empty intersection with every $D\_\alpha$, thus this finite non-empty intersection stabilizes and its limit is c... | 2 | https://mathoverflow.net/users/4312 | 452991 | 182,051 |
https://mathoverflow.net/questions/452970 | 5 | I am interested in the hypersurface $X\subset\mathbb{A}^4\_{\mathbb{F}\_{5^n}}$ defined by
$$
X = \{x^3 + 3xy^2 + z^3 + 3zw^2 + 1 = 0\}
$$
over a finite field $\mathbb{F}\_{5^n}$ with $5^n$ elements. Via some computer experiment I have noticed that when $n$ is odd the number of points of $X$ is equal to the number of p... | https://mathoverflow.net/users/510696 | Points on affine hypersurface over finite field | Note that the defining equation for $X$ can be rewritten as
$$(x+y)^3+(x-y)^3+(z+w)^3+(z-w)^3=-2.$$
As the linear transformation $(x,y,z,w)\mapsto(x+y,x-y,z+w,z-w)$ is invertible over any field of characteristic not equal to $2$, the number of $\mathbb{F}\_{5^n}$-points on $X$ is equal to the number of $\mathbb{F}\_{... | 10 | https://mathoverflow.net/users/51424 | 452992 | 182,052 |
https://mathoverflow.net/questions/452990 | 1 | I have recently learned about relative Poincaré duality in the book *Weil conjectures, perverse sheaves and $\ell$-adic Fourier transform* by Kiehl and Weissauer (2001). The reference is section II.7. When trying to apply the statement to some toy examples however, I end up with an absurd conclusion. Thus, there must b... | https://mathoverflow.net/users/125617 | Confusion about relative Poincaré duality in the context of $\ell$-adic cohomology | In your last paragraph you don't have $K\simeq K^\vee$. Let $i$ be the inclusion of the closed point, so $K=i\_!\mathbf Q\_\ell$. Then by local Verdier duality
$$K^\vee = RHom(i\_!\mathbf Q\_\ell,\mathbf Q\_\ell)=i\_\ast RHom(\mathbf Q\_\ell,i^!\mathbf Q\_\ell)$$which is a skyscraper sheaf concentrated in degree 2.
| 4 | https://mathoverflow.net/users/1310 | 452994 | 182,053 |
https://mathoverflow.net/questions/234507 | 11 | Let $A$ be a C\*-algebra. We identify $A$ with its canonical image in the bidual $A^{\*\*}$. Consider the following conditions:
(1) $A$ is a von Neumann algebra.
(2) There is a multiplicative conditional expectation from $A^{\*\*}$ onto $A$, that is, a map $\pi\colon A^{\*\*}\to A$ that is a \*-homomorphism and suc... | https://mathoverflow.net/users/24916 | von Neumann algebras as C*-algebras with multiplicative conditional expectation $A^{**}\to A$ | This is a seven-year-old question, so I'm not sure if an answer is still meaningful at this point, but no, $A$ needs not be a vNa. There is even a commutative counterexample. In fact, fix any monotone complete commutative $C^\*$-algebra $A$ which is not a vNa. Then $A$ is injective in the category of unital commutative... | 5 | https://mathoverflow.net/users/504602 | 452997 | 182,055 |
https://mathoverflow.net/questions/452993 | 2 | Do there exist positive integers $A, B, C$ such that all seven numbers $$A, B, C, A+B, B+C, A+C, A+B+C$$ are perfect squares?
| https://mathoverflow.net/users/4312 | 3-dimensional Boolean cube of Squares | If such $(A, B, C)$ exist, then $(\sqrt{A}, \sqrt{B}, \sqrt{C})$ are sides of a perfect cuboid (see [here](https://en.m.wikipedia.org/wiki/Euler_brick)). Such a cuboid has not yet been found.
| 4 | https://mathoverflow.net/users/507773 | 452999 | 182,057 |
https://mathoverflow.net/questions/452557 | 5 | I would like to find a reference that describes the semantics of constructive higher order logic *with function types* in toposes. In particular, it seems that if we are to take function types as primitive, then we need the **definite description operator** $\iota x. P(x)$, which produces a term that satisfies $\exists... | https://mathoverflow.net/users/136535 | Topos semantics of constructive higher order logic | Categorical logic texts such as Lambek & Scott's "Introduction to higher-order categorical logic" and Johstone's "Elephant" usually focus on the categorical side of things and are often a bit cavalier about the details of the internal language – enough so to require significant adjustments before one can actually think... | 6 | https://mathoverflow.net/users/1176 | 453003 | 182,059 |
https://mathoverflow.net/questions/453007 | 5 | Consider the Fermat cubic
$$
X = \{x\_0^3+\dots +x\_n^3 = 0\}\subset\mathbb{P}^n\_{\mathbb{F}\_{q}}
$$
over a finite field $\mathbb{F}\_{q}$ with $q$ elements.
If $q \equiv 2 \mod 3$ then the projection $\pi:X\rightarrow \mathbb{P}^{n-1}\_{\mathbb{F}\_{q}}$ defined by $\pi(x\_0,\dots,x\_n) = (x\_0,\dots,x\_{n-1})$ yi... | https://mathoverflow.net/users/510696 | Fermat cubic hypersurfaces over finite fields | A. Weil, in "Numbers of solutions of equations in finite fields" (Bull. Am. Math. Soc. 55, 497-508 (1949)), proved that the number of $\mathbb{F}\_q$-points when $q\equiv 1 \bmod 3$ is
$$\frac{q^n-1}{q-1} +\frac{1}{q-1} \sum\_{\chi\_0,\ldots,\chi\_n} J\_0(\chi\_0,\ldots,\chi\_n)$$
where the sum is over $(n+1)$-tuples o... | 12 | https://mathoverflow.net/users/31469 | 453010 | 182,060 |
https://mathoverflow.net/questions/452903 | 3 | *Based upon discussion at [Math.SE](https://math.stackexchange.com/questions/4751580/what-separation-is-required-to-ensure-extremally-disconnected-spaces-are-sequent)*
Consider the property [extremally disconnected](https://topology.pi-base.org/properties/P000049), for which the closure of any open set remains open.
... | https://mathoverflow.net/users/73785 | Must US extremally disconnected spaces be sequentially discrete? | Here's an example, I think.
Let $X=(\omega\_1+1)\times(\omega\_0+1)$ and let $Y$ denote the subset $\omega\_1\times(\omega\_0+1)$.
Define a topology on $X$ by declaring every set of the form $Y\setminus C$ with $C$ countable open. So $Y$ will be an open subspace. We ensure that $Y$ has the co-countable topology by spec... | 4 | https://mathoverflow.net/users/5903 | 453013 | 182,061 |
https://mathoverflow.net/questions/453016 | 3 | The following lemma is due to Campana, [The class $\mathcal C$ is not stable by small deformations](https://link.springer.com/article/10.1007/BF01459236)
>
> Let $\mathcal X\rightarrow \Delta$ be a smooth family, if $K\_{X\_0}$ is nef and big, then so is every $K\_{X\_t}$ for $t$ sufficiently small.
>
>
>
Here... | https://mathoverflow.net/users/141609 | semiample of canonical bundle in a smooth family (Campana's proof) | Let $f:X\rightarrow \Delta $ be your family. $\ (\*)$ implies that $f\_\*K\_{X/\Delta }^{N}$ is a vector bundle on $\Delta $, with fiber $H^0(X\_t, K\_{X\_t}^N)$ at $t\in\Delta$. The canonical homomorphism $\ f^\*f\_\*K\_{X/\Delta }^{N}\rightarrow K\_{X/\Delta }^{N}$ is surjective on $X\_0$ by hypothesis, hence also on... | 5 | https://mathoverflow.net/users/40297 | 453017 | 182,063 |
https://mathoverflow.net/questions/451528 | 0 | We have proposed a new approach to solve the maximum edge bi-clique problem, however, we couldn't succeed to find real-world datasets (graph or bipartite graph datasets) to test our approach. Does anyone have an idea where I could find some datasets for this problem?
| https://mathoverflow.net/users/509350 | Real-world datasets for testing the maximum edge bi-clique problem | Since this problem has many applications in data mining, you probably should take a look at datasets used as benchmarks in papers on the topic. As an example, [this recent paper](https://ieeexplore.ieee.org/abstract/document/10121476) mostly uses graphs from the [KONECT Project](http://konect.cc/).
| 0 | https://mathoverflow.net/users/8193 | 453023 | 182,064 |
https://mathoverflow.net/questions/453004 | 10 | Do there exist positive integers $x,y,z$ such that
$$
(x+y)(xy-1)=z^2+1
$$
In my previous question [Can you solve the listed smallest open Diophantine equations?](https://mathoverflow.net/questions/400714), I discuss the smallest equations for which we do not know if they have any integer solutions. Some equations (t... | https://mathoverflow.net/users/89064 | Positive integers such that $(x+y)(xy-1)=z^2+1$ | As you have already noticed, we may assume that $x \equiv 3 \pmod{4}$, $y \equiv 2 \pmod{4}$. Let $p \equiv 3 \pmod{4}$ be a prime divisor of $y + 1 \equiv 3 \pmod{4}$ such that $\nu\_p(y+1) \equiv 1 \pmod{2}$. Consider the equation
$$(x + y)(xy - 1) = z^2 + 1$$
modulo $p$.
$$(x + y)(xy - 1) \equiv (x - 1)(-x - 1) \equ... | 12 | https://mathoverflow.net/users/507773 | 453024 | 182,065 |
https://mathoverflow.net/questions/453006 | 4 | Is there a general algorithm that can compute in finite time all rational points on any curve of genus $g\geq 2$ whose Jacobian has rank $0$?
If not, for what special cases such algorithm is known? For genus $g=2$, we have Chabauty0 command in Magma, is it always guaranteed to work? For genus $g=3$, natural families ... | https://mathoverflow.net/users/89064 | Algorithm for computing rational points if the rank of Jacobian is 0 | There is an algorithm due to Bjorn Poonen (Computing torsion points on curves, Experiment. Math. 10 (2001), no.3, 449–465) that, given a (not necessarily rational) base-point $P\_0$ on the curve, finds all (geometric) points $P$ on the curve such that $[P-P\_0]$ is torsion in the Jacobian. This solves an even harder pr... | 5 | https://mathoverflow.net/users/21146 | 453031 | 182,067 |
https://mathoverflow.net/questions/453043 | 5 | How many solutions are there to $n\_1^2-n\_2^2-n\_3^2+n\_4^2=k$ where $n\_i\in [1,N]$ and $n\_i,k\in \mathbb Z$?
For $k=0$ by paucity we know it should be $\ll N^2 \log N$ but what about different $k$?
| https://mathoverflow.net/users/479223 | How many solutions are there to $n_1^2-n_2^2-n_3^2+n_4^2=k$? | Let us allow $n\_i\in[-N,N]$ for simplicity; this will not change the order of magnitude of the number of solutions. So let us consider
$$R(k,N):=\#\left\{(n\_1,n\_2,n\_3,n\_4)\in\{-N,\dotsc,N\}^4:n\_1^2-n\_2^2-n\_3^2+n\_4^2=k\right\}.$$
Let us also assume that $k$ is odd, and $N\gg\sqrt{k}$ with a sufficiently large i... | 6 | https://mathoverflow.net/users/11919 | 453048 | 182,073 |
https://mathoverflow.net/questions/452924 | 5 | Let $p$ be a prime number and let $k$ be a field with $char(k)\neq p$ such that all finite extensions have degree coprime to $p$. (For example, we can take $k=\mathbb{R}$ and $p\neq 2$ or let $k$ the union of $\mathbb{F}\_{l^{(p^n)}}$ for all $n\geq 1$.)
Let $C$ be a regular affine curve over $k$ with (regular) proje... | https://mathoverflow.net/users/127489 | $p$-divisibility of Picard groups | $\newcommand{\wt}{\widetilde}$
$\newcommand{\mr}{\mathrm}$
The question has a positive answer, in fact, regularity of $C$ is not needed. The proof as written below works under the assumption that $C$ is (geometrically) irreducible and reduced, but both these conditions could be relaxed to some extent. (Note that $\mr... | 3 | https://mathoverflow.net/users/519 | 453055 | 182,077 |
https://mathoverflow.net/questions/453036 | 9 | We work in ZFC.
Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$.
A field $E$ is called real-closed if it has unique ordering $x\geq 0$ iff $\exists\_{y\in E}\ x = y^2$ and all polynomials of odd deg... | https://mathoverflow.net/users/150060 | Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH? | In [On ultra powers of Boolean algebras](http://topology.nipissingu.ca/tp/reprints/v09/tp09205.pdf) (Topology Proceedings 9 (1984) 269-291) Alan Dow proved (Corollary 2.3) that $\neg\mathsf{CH}$ implies there are two fields of the form $C(\mathbb{N})/M$ that are not isomorphic (as ordered sets, and hence as ordered fie... | 13 | https://mathoverflow.net/users/5903 | 453063 | 182,079 |
https://mathoverflow.net/questions/453059 | 3 | The integral I want to calculate is defined as
$$
P(s)=\int\_{-\infty}^{\infty}{\rm d}x\int\_{-\infty}^{\infty}{\rm d}y\ \delta\left(\frac{(x+y)^2+4x^2y^2}{(x+y)^2+(x+y)^4}-s\right)e^{-\left(x^2+y^2\right)/2a^2}
$$
This integral seems no explicit solution (?) since the quartic terms in the $\delta$ function. So I just ... | https://mathoverflow.net/users/482984 | Asymptotic behavior of the integral that contains $\delta$ function | $$P(s)=2\int\_{-\infty}^{\infty}{\rm d}p\int\_{-\infty}^{\infty}{\rm d}q\ \delta\left(\frac{p^2+(p^2-q^2)^2}{p^2+4p^4}-s\right)e^{-\left(p^2+q^2\right)/a^2}$$
$$\qquad=2\int\_{0}^{\infty}\frac{{\rm d}x}{\sqrt{x}}(x+4x^2)\int\_{0}^{\infty}\frac{{\rm d}y}{\sqrt{y}}\ \delta\left(x+(x-y)^2-s(x+4x^2)\right)e^{-\left(x+y\rig... | 5 | https://mathoverflow.net/users/11260 | 453064 | 182,080 |
https://mathoverflow.net/questions/453071 | 3 | Let $X$ be a smooth, complex projective variety of dimension $n$. Let $F$ be a vector bundle over $X$ with rank$F=r \leq n$. Let $s$ be a regular global section of $F$ and $Z$ be the scheme of zeroes of $s$ [Here by regular I mean codim$Z$=r]. Then we have the corresponding exact Koszul complex given by :
$0 \to \wed... | https://mathoverflow.net/users/187857 | Koszul complex for bundles of rank higher than the dimension | Yes, this complex is exact. To prove exactness at point $x \in X$, note that the question is local, so one may assume that $F \cong \mathcal{O}^{\oplus r}$ is a trivial bundle and $s = (s\_1,\dots,s\_r)$ is a sequence of functions. Note further that the Koszul complex can be written as a tensor product
$$
\bigotimes\_{... | 7 | https://mathoverflow.net/users/4428 | 453072 | 182,082 |
https://mathoverflow.net/questions/452598 | 5 | I'm interested in connected matroids $M$ on the ground set $[n]$ for which there is no connected matroid on $[n]$ of the same rank but with a strictly smaller set of bases (by inclusion). Equivalently, the matroid polytope of $M$ has dimension $n-1$ and does not contain any other matroid polytope of dimension $n-1$. Ob... | https://mathoverflow.net/users/19864 | "Minimal" connected matroids | In general there does not seem to be a nice classification. You're asking for connected matroids of a given rank which are minimal in the "weak map order" (a weak map between matroids of the same rank is an inclusion of bases). In [Weak maps of combinatorial geometries, Theorem 6.10](https://www.ams.org/journals/tran/1... | 3 | https://mathoverflow.net/users/94696 | 453074 | 182,083 |
https://mathoverflow.net/questions/314946 | 4 | Let $k$ be a $p$-adic field, $G$ a connected reductive group over $k$ with minimal parabolic $P\_0$ containing a maximal split torus $A\_0$. Let $W = N\_G(A\_0)(k)/Z\_G(A\_0)(k)$ be the Weyl group, and $S \subset W$ the simple reflections from $P\_0$.
For $\theta, \Omega \subset S$, we have the standard parabolic su... | https://mathoverflow.net/users/38145 | The quotients of double cosets $P_\theta \backslash P_\theta w P_\Omega$ are algebraic varieties over $k$ | I originally gave an [answer](https://mathoverflow.net/a/315163) that relied on a claim in [Casselman - Introduction to the theory of admissible representations of $p$-adic reductive groups](https://personal.math.ubc.ca/%7Ecass/research/pdf/p-adic-book.pdf) that, as you pointed out by linking to [Errata for Casselman's... | 2 | https://mathoverflow.net/users/2383 | 453078 | 182,084 |
https://mathoverflow.net/questions/453077 | 0 | Let $\mathbf{G}$ be a connected semisimple algebraic group defined over a field $k$. Let $T$ be a maximal torus of $\mathbf{G}$ defined over $k$, and let $S \subset T$ be a maximal $k$-split torus. Let $\Phi = \Phi(T,\mathbf{G})$ be the absolute root system of $\mathbf{G}$, and let $\Phi\_{\text{rel}} = \Phi(S,\mathbf{... | https://mathoverflow.net/users/510773 | Calculating relative root systems | Since $W(G, S)(k) = (N\_G(S)/C\_G(S))(k) = N\_G(S)(k)/C\_G(S)(k)$ is finite, there is what Borel, in [Linear algebraic groups](https://doi.org/10.1007/978-1-4612-0941-6), §14.7, calls an admissible scalar product on $S^\* \otimes\_{\mathbb Z} \mathbb R$, obtained simply by averaging any random scalar (i.e., inner) prod... | 3 | https://mathoverflow.net/users/2383 | 453080 | 182,085 |
https://mathoverflow.net/questions/453091 | 5 | I've read in a couple of different places (a paper and a blog) the following fact:
if $F$ is a surface, properly embedded in a three-dimensional handlebody of genus at least two, then $F$ is either compressible or boundary compressible.
Neither of these sources included a reference or a proof and I am a bit of a thre... | https://mathoverflow.net/users/156387 | Properly embedded surfaces in handlebodies are compressible or boundary compressible? | Suppose that the surface $F$ is properly embedded in the handlebody $V$. We induct on the genus of $V$.
In the base case $V$ has genus zero and so is a three-ball. In this case the desired result follows from a theorem of Alexander. (Briefly, suppose that $F$ is not a sphere or a disk. Sweepout the ball $V$ by planes... | 8 | https://mathoverflow.net/users/1650 | 453100 | 182,089 |
https://mathoverflow.net/questions/453103 | 10 | I could name on the spot a bunch of abelian categories satisfying AB5 but I cannot think of any that satisfies AB5\*. That is, it should have all limits and the cofiltered limits are exact. Is there any example that I should have in mind?
| https://mathoverflow.net/users/496941 | Abelian categories satisfying AB5* | The snarky response would be "the opposite category of any of the categories you could name on the spot". The less-snarky response is to observe that some of these are quite natural. For example, $\mathrm{Vect}^{\mathrm{op}}\_{\mathbb{F}\_p}$ agrees with the category of profinite $\mathbb{F}\_p$ vector spaces, since $\... | 14 | https://mathoverflow.net/users/39747 | 453104 | 182,090 |
https://mathoverflow.net/questions/453081 | 8 | Let $k=\mathbb{C}$ be the field of complex numbers. I consider the (DG) algebra $A:=k[x]/(x^2)$ such that $\deg(x)=-1$. My question is how to compute the periodic cyclic homology, Hochschild homology and Hochschild cohomology of this (DG) algebra? There are some references on computing those (co)homology of the ring of... | https://mathoverflow.net/users/41650 | How to compute the periodic cyclic homology of this algebra | You can use a derived version of the HKR theorem, i.e.
$HH(A) = \text{Sym}\_A^\bullet(\mathbb{L}\_A[1])$ where $\mathbb{L}\_A$ is the cotangent complex (over $k$). I'm not sure about a reference though.
Since $A$ is quasi-smooth, its cotangent complex will be in degrees $[-1, 0]$, and one can check using the fact tha... | 8 | https://mathoverflow.net/users/6059 | 453106 | 182,091 |
https://mathoverflow.net/questions/453099 | 1 | I'm considering a problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int\_{\mathbb R^2} \exp u(x) \, d x< +\infty$.
LEMMA 1.1 (Ding). If $u$ is a solution of $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int\_{\mathbb R^2} \exp u(x) \, d x<$ $+\infty$,... | https://mathoverflow.net/users/469129 | Time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2}\exp u(x) \, dx< +\infty$ | I think your question is just basic real analysis, not at the research level, you should ask this question on stack exchange instead of overflow.
Since $\int\_{\mathbb{R}^2} e^u \, dx <\infty$, for every $\epsilon>0$, there exists $\delta>0$ such that if $|A|<\delta$, then $\int\_A e^u \, dx <\epsilon$. By Chebyshev'... | 2 | https://mathoverflow.net/users/166368 | 453107 | 182,092 |
https://mathoverflow.net/questions/453133 | 3 | ### Problem
I am trying to invert an equation of the form:
$R(l\_0)=(\int\_{0}^{l\_0} \rho(x) \, dx)(\int\_{l\_0}^{l} \rho(x) \, dx)$
where $0\leq l\_0 \leq l$
I.e. I want to find $\rho(x)$ given $R(l\_0)$ via some kind of integral transform, series solution, or discretized approximation. What I do not want to do... | https://mathoverflow.net/users/510808 | How to find the inverse of a product of two integral equations | Let $F(y) = \int\_0^y \rho(x) dx$. Then your equation reduces to:
$$R(y) = F(y)(F(l) - F(y))$$
Note that for any value of $F(y)$, we get that $R(y) \leq \frac{F(l)^2}{4}$; this is easily proven by completing the square. As $F$ is an antiderivative, it is continuous, so by the intermediate value theorem (using $F(0)... | 4 | https://mathoverflow.net/users/44191 | 453139 | 182,096 |
https://mathoverflow.net/questions/453122 | 2 | I would like to prove the following but I couldn't manage to do it. Let $a>b>0$ be two real numbers. Let $f$ be the function defined as:
$$\forall \sigma>0, ~\forall x\in\mathbb{R},~f\_\sigma(x):=\sigma e^{-x^2/\sigma^2}+\sqrt{\pi}x\text{erf}\left(\frac{x}{\sigma}\right).$$
**I would like to show that $f\_a-f\_b$ i... | https://mathoverflow.net/users/510079 | Log-concavity of the difference of the second anti-derivative of Gaussians | This conjecture is true.
Indeed, for $h:=\ln(f\_a-f\_b)$ and real $x\ne0$ we have
\begin{equation}
h'(x)=R(x):=\frac{F(x)}{G(x)},
\end{equation}
where
\begin{equation}
F(x):=\sqrt{\pi }
\left(\text{erf}\left(\frac{x}{a}\right)-\text{erf}\left(\frac{x}{b}\right)\right),
\end{equation}
\begin{equation}
G(x):= a e^... | 2 | https://mathoverflow.net/users/36721 | 453145 | 182,100 |
https://mathoverflow.net/questions/453123 | 39 | One of the cornerstones of the mathematical formulation of General Relativity (GR) is the result (due to Choquet-Bruhat and others) that the initial value problem for the Einstein field equations is solvable and it admits a essentially unique maximal (globally hyperbolic) development (GHD).
The proofs I have seen of ... | https://mathoverflow.net/users/493244 | How much of mathematical General Relativity depends on the Axiom of Choice? | The dependence on AC through the use of Zorn's lemma in the proof of the Choquet-Bruhat–Geroch theorem on the existence of a maximal globally hyperbolic development for solutions of the Einstein equations has been eliminated around the same time by Jan Sbierski, in [On the Existence of a Maximal Cauchy Development for ... | 32 | https://mathoverflow.net/users/2622 | 453152 | 182,102 |
https://mathoverflow.net/questions/275409 | 8 | I asked this question on [math.stackexchange](https://math.stackexchange.com/q/2348425/275190), but without much success.
Assume that $R$ is a ring (commutative, with unit) and that $M$, $N$ are two $R$-modules. Let $(e\_i)\_{i\in I}$ and $(f\_j)\_{j\in J}$ be families of elements in $M$ and $N$, respectively. In $M\... | https://mathoverflow.net/users/70808 | linear independent families in a tensor product | [This answer](https://mathoverflow.net/a/232343/22989) to a related question gives a way of constructing counterexamples.
For a similar but more concrete example, let $k$ be a field and $R=k[x,y]/(x^2,xy,y^2)$, so $R$ is a $3$-dimensional algebra over $k$, spanned by $1$, $x$ and $y$.
Let $M$ be the $5$-dimensional... | 6 | https://mathoverflow.net/users/22989 | 453155 | 182,103 |
https://mathoverflow.net/questions/453167 | 2 | Let $\mathcal{C}$ be a small category and let $\mathcal{M} = \operatorname{sPre}(C)$ be the model category of simplicial presheaves on $\mathcal{C}$ with the injective model structure.
Let $S$ be a set of morphisms in $\mathcal{M}$ that is stable under pullback and let $L\_S\mathcal{M}$ be the left Bousfield localizati... | https://mathoverflow.net/users/133676 | Is a left Bousfield localization of simplicial presheaves a locally cartesian closed model category? | Section 2 of [this paper of Rezk](https://arxiv.org/abs/0901.3602) addresses exactly the question of when the localization by S yields a Cartesian model category. For that the relevant property is that that if you take the product of a map in S and a representable object, the result must be in the saturation of S.
Th... | 4 | https://mathoverflow.net/users/184 | 453176 | 182,108 |
https://mathoverflow.net/questions/453175 | 1 | Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|\_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of Lebesgue measurable functions $f:\mathbb R^d \to \mathbb R$ such that
$$
\|f\|\_{\tilde L^p} := \sup\_{x \in \mathbb R^d} \... | https://mathoverflow.net/users/99469 | Is there a version of dominated convergence theorem for local $L^p$ spaces? | A counterexample is given by $f\_n=1\_{(n,\infty)}$, $g=1$, and $f=0$.
Then all the conditions on $f\_n,g,f$ hold, but $\|f\_n-f\|\_{\tilde L^p} \not\to0$.
| 5 | https://mathoverflow.net/users/36721 | 453180 | 182,110 |
https://mathoverflow.net/questions/453185 | 6 | Given $f \in L^1 (\mathbb R^d)$, and $\varepsilon > 0$, define the *minimal function* $m\_\varepsilon f$ by
$$m\_\varepsilon f(x) := \inf\_B \frac1{|B|} \int\_B |f| ,$$
where the infimum is taken over all balls $B$ containing $x$ of radius less than or equal to $\varepsilon$, and the integral is with respect to Leb... | https://mathoverflow.net/users/173490 | Is the Hardy Littlewood “minimal function” comparable to the original function in $L^1$ norm? | This is not true. Take $\epsilon=1$ and, on the real line, $f\_\delta=\delta^{-1} \chi\_{(0, \delta)}$, so that $\|f\_\delta\|\_1=1$. Then $0 \leq m\_1 f\_\delta \leq \chi\_{(0, \delta)}$, by choosing small balls outside $(0, \delta)$ and balls of radius 1 otherwise. Thus $\|m\_1 f\_\delta\|\_1 \leq \delta$.
| 4 | https://mathoverflow.net/users/150653 | 453189 | 182,112 |
https://mathoverflow.net/questions/452544 | 3 | **Problem:**
Let $f\colon \mathopen[0,1\mathclose] \to \mathbb{R}$ be defined as
$$
f(x) = \frac{e^{\rho x}-1}{e^{\rho x}-1+e^{\rho (1-\gamma x)}-e^{\rho (1-\gamma) x}}
$$
where $\rho$ and $\gamma$ are strictly positive real parameters. I am interested in showing that, for any fixed $\rho$, there exists a value $\gam... | https://mathoverflow.net/users/510325 | How to establish regions of convexity/concavity of a ratio of exponential polynomials? | This conjecture is true.
Indeed, letting $r:=\rho$, $a:=\gamma$, $t:=rx$, and $u:=e^r-1$, we see
\begin{equation\*}
f(x)=R(t):=\frac{e^t-1}{(1+u)e^{-a t}-e^{t-a t}+e^t-1}. \tag{10}\label{10}
\end{equation\*}
So, the problem can be restated as follows:
>
> Show that for each real $u>0$ there is some real $a\_u... | 1 | https://mathoverflow.net/users/36721 | 453192 | 182,114 |
https://mathoverflow.net/questions/453157 | 3 | I am a bit rusty in my differential geometry and I would like to confirm that my reasoning below holds, and I have some related questions (and all references to related concepts are of interest to me).
Let $G$ be a compact, connected Lie group endowed with the Riemannian metric induced by the Killing form, I would li... | https://mathoverflow.net/users/44134 | Eigenforms of the Laplacian on Lie groups | Here are a few brief comments, but, as you suspect, an enormous amount is known about the Laplacian on functions and forms on compact Lie groups.
• Presumably, you know that the Killing form is non-degenerate only in the semi-simple case, i.e., when the center of $G$ is discrete, so I assume that you are only interes... | 2 | https://mathoverflow.net/users/13972 | 453195 | 182,116 |
https://mathoverflow.net/questions/452916 | 5 | I am interested in the following graph invariant: for a given graph $G=(V,E)$, $c(G)$ is defined to be the smallest number of vertices such that I can recreate the connectivity of $G$ by disconnecting and relabelling vertices for reuse once they have been connected to all vertices in their neighborhood in $G$.
That is,... | https://mathoverflow.net/users/510619 | Connectivity graph invariant | I think what you describe is exactly the "node-search number" as described by Kirousis and Papadimitriou (<https://www.sciencedirect.com/science/article/pii/0012365X85900469>). As shown in that paper, it is indeed equivalent to the interval thickness, or (which is the same) the pathwidth plus one.
| 2 | https://mathoverflow.net/users/45545 | 453203 | 182,119 |
https://mathoverflow.net/questions/453210 | 8 | Let $f(T) = \sum a\_n T^n \in \mathbf{F}\_p [[ T ]]$ be a power series. We'll say that the coefficients of $f(T)$ are equidistributed modulo $p$ if for every residue class $a$ modulo $p$, we have
$$ \lim\_{X \to \infty} \dfrac{1}{X} \cdot \# \{n<X : a\_n \equiv a \mod p \} = \dfrac{1}{p}.$$
Suppose that $f(T), g(T) ... | https://mathoverflow.net/users/394740 | Is the product of two equidistributed power series equidistributed? | No. Let $f(T)$ be chosen uniformly at random from all power series with constant term nonzero and let $g(T) = 1/ f(T) $. Then $g(T)$ is also chosen uniformly at random from all power series with constant term nonzero. (Here uniformly means each possible sequence of first $n$ coefficients has equal probability, and the ... | 14 | https://mathoverflow.net/users/18060 | 453211 | 182,121 |
https://mathoverflow.net/questions/453117 | 3 | Is there a reasonable/canonical way to mollify a Borel probability measure without changing its marginals. Let $\pi \in \mathcal{P}(\mathbb{R}^2)$ with marginals $\mu,\nu$. I want to smooth out $\pi$ up to scale $\varepsilon$, which is equivalent to cutting it off at frequency $|k|\approx \varepsilon^{-1}$ in Fourier s... | https://mathoverflow.net/users/106281 | Mollifying a measure without changing its marginals | $\newcommand{\R}{\mathbb R}\newcommand{\ep}{\varepsilon}\newcommand{\tX}{\tilde X}\newcommand{\tY}{\tilde Y}\newcommand{\tpi}{\tilde\pi} $Here is how this can be done in an explicit way, at least when the densities (say $p$ and $q$) of $\mu$ and $\nu$ are everywhere $>0$.
Let $(X,Y)$ be a random point in $\R^2$ with ... | 3 | https://mathoverflow.net/users/36721 | 453237 | 182,129 |
https://mathoverflow.net/questions/453242 | 2 | I've been trying to understand various questions to do with sigma algebras on uncountable product spaces.
Let $T$ be an uncountable set and for each $t \in T$, let $\Omega\_t$ be a topological space. Let $\Omega := \prod\_{t \in T}\Omega\_t$ be the product space equipped with the product topology in the usual way. Th... | https://mathoverflow.net/users/122587 | Relationship between Baire sigma algebra and Borel sigma algebra of an uncountable product | Let all factor spaces be nontrivial compact Hausdorff spaces. Then every continuous function is [determined by countably many coordinates](https://mathoverflow.net/questions/364062/can-a-continuous-real-valued-function-on-a-large-product-space-depend-on-uncount), and so is, consequently, every Baire measurable set.
I... | 3 | https://mathoverflow.net/users/35357 | 453246 | 182,132 |
https://mathoverflow.net/questions/452382 | 13 | We endow ${\cal P}(\omega)$ with an equivalence relation by saying that $A\simeq\_{\text{fin}} B$ iff the symmetric difference $A\Delta B$ is finite. The resulting set of equivalence classes is denoted ${\cal P}(\omega)/(\text{fin})$, and by $$[A]\_{\simeq\_{\text{fin}}} \leq [B]\_{\simeq\_{\text{fin}}} \text{ if and o... | https://mathoverflow.net/users/8628 | Can any poset of cardinality $\leq 2^{\aleph_0}$ be embedded in ${\cal P}(\omega)/(\text{fin})$? | Here is an attempt at a 'definitive summary'.
To begin with positive results: $\mathsf{CH}$ implies a “yes” answer to
this question. The fastest way to see this is to first embed a given partial
order $(P,\le)$ (of cardinality at most $\mathfrak{c}$) into its power set
via $p\mapsto\{x:x\le p\}$ and then take the Boo... | 7 | https://mathoverflow.net/users/5903 | 453258 | 182,137 |
https://mathoverflow.net/questions/453231 | 8 | Let $E/\mathbb{Q} = E\_{a,b}$
$$\displaystyle y^2 = x^3 + ax + b, a,b \in \mathbb{Z}$$
be an elliptic curve defined over the field of rational numbers, and let $n \geq 3$ be an integer. Let $K\_n$ be the $n$-torsion field of $E$, namely the field obtained by adjoining all of the coordinates of $n$-torsion points of $E$... | https://mathoverflow.net/users/10898 | $n$-torsion fields of an elliptic curve defined over $\mathbb{Q}$ | No. The modular curve $X\_E(n)$, which paramaterizes pairs $(E',\iota)$, where $E'$ is another elliptic curve and $\iota \colon E'[n] \to E[n]$ is an isomorphism (of group schemes, or equivalently, of Galois modules) is a twist of $X(n)$, and thus has genus 0 for $n = 2,3,4,5$ and genus 1 for $n = 6$. In particular, fo... | 10 | https://mathoverflow.net/users/2 | 453261 | 182,138 |
https://mathoverflow.net/questions/453255 | 9 | Let $K$ be an imaginary quadratic number field (so there are no real embeddings) with ring of integers $\mathcal{O}\_K$ . Let $w$ be the number of units in $K$ and $h$ be the class number of $K$. Let $\mathfrak{f}$ be a non-zero integral ideal in $K$, and let $\mathcal{I}\_{\mathfrak{f}}$ be the ray class group mod $\m... | https://mathoverflow.net/users/307675 | Questions about ray class groups | **1.** Surely, by "class group modulo $\mathfrak{f}$", the authors mean "ray class group modulo $\mathfrak{f}$". I guess the authors omit "ray", because $K$ has no real embedding, so there are no "rays" here.
**2.** Let $K^\times\_\mathfrak{f}$ be the multiplicative group of elements $\alpha\in K^\times$ congruent to... | 8 | https://mathoverflow.net/users/11919 | 453269 | 182,141 |
https://mathoverflow.net/questions/453274 | 2 | Let $M$ be the projective cover (e.g, [Gleason1958](https://doi.org/10.1215/ijm/1255454110)) of the Cantor set $\{-1,1\}^{\mathbb{N}}$. Let $\textrm{homeo}(M)$ denote the group of all homeomorphisms of $M$.
Some of the $\gamma\in\textrm{homeo}(M)$ are extensions of the homeomorphisms of the Cantor set. Some others are ... | https://mathoverflow.net/users/164350 | Homeomorphisms of the projective cover of the Cantor set | I claim that every autohomeomorphism of $M$ is conjugate to an extension of an autohomeomorphism of the Cantor space.
Let $B$ be the Stone space of the Cantor space. Let $\overline{B}$ denote its Boolean completion. We shall show that every automorphism of $\overline{B}$ is conjugate to an automorphism of $B$. Recall... | 2 | https://mathoverflow.net/users/22277 | 453286 | 182,145 |
https://mathoverflow.net/questions/453067 | 5 | Suppose $C\leq A,B$ are finitely generated groups of finite exponent. Can $A$ and $B$ be [amalgamated](https://en.wikipedia.org/wiki/Amalgamation_property) over $C$ in a group of finite exponent? What about if $A,B$ are periodic, can we find a periodic amalgam?
Note that if we assume that $A$ and $B$ are finite, then... | https://mathoverflow.net/users/54415 | Amalgamation of finitely generated finite exponent groups | It's not hard to construct counter-examples for large exponents.
Consider the following two automorphisms $\xi, \eta$ of the free group $F=F(x,y)$ of rank $2$, defined by
$$\xi(x)=y,~\xi(y)=x \text{ and } \eta(x)=x,~\eta(y)=yx.$$
The composition $\xi \circ \eta \in Aut(F)$ is the so-called Fibonacci automorphism $\va... | 3 | https://mathoverflow.net/users/7644 | 453288 | 182,146 |
https://mathoverflow.net/questions/453268 | 5 | Suppose one has a filter (a collection of subsets closed under increasing the size of the set and under finite intersection) $F$ on a ring $R$. Say that $F$ is (ring) ideal-like if for every set $U \subset R$ in $F$, there exists a $U'\in F$ such that $U' + U' \subset U$, and there exists a $U'' \in F$ such that $U'' R... | https://mathoverflow.net/users/484180 | Ideal-like filter on a ring not generated by ring ideals | This does not hold for all commutative rings, but it does hold for Noetherian rings and for valuation rings (assuming the convention that filters don't contain $\emptyset,$ or else $\mathcal{P}(R)$ is a trivial counterexample for any ring).
Suppose $R$ is a commutative ring, $F$ is an ideal-like filter on $R,$ and $U... | 6 | https://mathoverflow.net/users/109573 | 453291 | 182,148 |
https://mathoverflow.net/questions/453282 | 9 | Given a cardinal $\kappa\geq \aleph\_0$, is there a poset $(P,\leq)$ with $|P| = \kappa$ such that every poset of cardinality $\kappa$ can be order-embedded into $(P,\leq)$?
| https://mathoverflow.net/users/8628 | Universal poset for cardinals $\kappa \geq \aleph_0$ | The easy case occurs when $\kappa^{<\kappa}=\kappa$. Under GCH, this includes every regular cardinal. As Emil mentions in the comments, the result for this case follows from general model-theoretic considerations of saturated models.
But for definiteness, let me describe a concrete construction. We construct a univer... | 6 | https://mathoverflow.net/users/1946 | 453305 | 182,153 |
https://mathoverflow.net/questions/453034 | 4 | Do there exist 3+1 dimensional spacetimes (i.e. Lorentzian manifolds with signaure (1,3)), for which the Chern–Pontryagin scalar
\begin{equation}
K\_2= \epsilon^{\mu\nu\rho\sigma}R^{\alpha}{}\_{\beta\mu\nu}R^{\beta}{}\_{\alpha\rho\sigma}\,,
\end{equation}
is a non-zero constant? $R^{\alpha}{}\_{\beta\mu\nu}$ is Riemann... | https://mathoverflow.net/users/165560 | Metric with a constant Chern–Pontryagin scalar | I found a solution:
\begin{equation}
g\_{\mu\nu} = \left(
\begin{array}{cccc}
\frac{A}{(1-t z)^2 (x y-1)^2} & 0 & 0 & 0 \\
0 & -\frac{A}{(1-t z)^2 (x y-1)^2} & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1 \\
\end{array}
\right)\,.
\end{equation}
The Chern-Pontryagin scalar is $K\_2 = -16/|A|$ for $xy>1,tz>1$.
**Upda... | 3 | https://mathoverflow.net/users/165560 | 453306 | 182,154 |
https://mathoverflow.net/questions/453281 | 2 | According to the [Wikipedia](https://en.wikipedia.org/wiki/Vertex_enumeration_problem) page on the issue, the vertex enumeration problem is NP-hard.
However, double description and reverse linear search are algorithms listed to solve the problem. Moreover, Fukuda and Avis have published a library for it that works we... | https://mathoverflow.net/users/159040 | Is the problem of vertex enumeration from an H-representation of a polytope NP-hard? | As noted in the comments this is not a decision problem. But more than that typically we discuss P and NP in terms of polynomial time with respect to input size. In this problem, the input is the collection of inequalities/half-spaces. However, the output is the collection of vertices which can be exponential in the nu... | 2 | https://mathoverflow.net/users/51668 | 453324 | 182,162 |
https://mathoverflow.net/questions/453198 | 13 | There seems to be some combinatorial fact that every subset $A$ of $G=(\mathbb{Z}/p)^{\times n}$ of cardinality $\frac{p^n-1}{p-1}+1$ containing $\vec{0}$ satisfies $(p-1)A=G$. ($p$ is a prime number.)
Is that true and known in the literature?
I would appreciate a reference or a proof.
Thanks!
| https://mathoverflow.net/users/510890 | Subsets of $(\mathbb{Z}/p)^{\times n}$ | There is an alternative elementary way to prove this and see where the number $\frac{p^n-1}{p-1}+1$ comes from.
**Lemma:** If $|A|\geq \frac{p^n-1}{p-1}+1$ then every point in $\mathbb F\_p^n$ lies in a line that passes from two distinct points in $A$.
Proof: If this was false then there would be a point $P\notin A... | 14 | https://mathoverflow.net/users/2384 | 453328 | 182,165 |
https://mathoverflow.net/questions/453323 | 5 | I think my question applies to most games, but for the sake of concreteness, I shall consider one specific game in this question. We consider the game posed by Ilias Kastanas in his paper *On the Ramsey property for sets of reals*: We first fix some $X \subseteq [\omega]^\omega$.
1. Player I start by choosing an infi... | https://mathoverflow.net/users/146831 | Uniform strategy on Kastanas' game | This is a great question — definitely enjoyed.
Assuming the axiom of choice, then the answer is yes.
**Theorem.** Assume there is a well ordering of the real numbers. If player I has a winning strategy, then there is a uniform winning strategy.
**Proof.** Suppose that $\sigma$ is a winning strategy for player I. ... | 6 | https://mathoverflow.net/users/1946 | 453330 | 182,166 |
https://mathoverflow.net/questions/453312 | 19 | Last year, Andrej Bauer [gave a talk](https://www.youtube.com/watch?v=4CBFUojXoq4&list=FLuLR2oAJOXHh0nBVDDuj3Lg&index=24&t=71s) showing that there is a topos in which the set of Dedekind reals is (sub)countable, and thus, you cannot prove that $\mathbb{R}$ is uncountable without LEM. He claims in [this abstract](https:... | https://mathoverflow.net/users/502850 | Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel? | Please allow me to list some basic observations that might clear up things. I work constructively (without excluded middle) and without the axiom of choice, and assuming powersets are available.
(One can contemplate working in a predicative setting such as CZF, where the Dedekind reals form a class, but I fail to see... | 24 | https://mathoverflow.net/users/1176 | 453340 | 182,168 |
https://mathoverflow.net/questions/453097 | 4 | Let $I$ be the (nerve of the) walking isomorphism (a simplicial set). Consider the inclusion $\Delta^1 \to I$ and the inclusion $\partial \Delta^1 \to \Delta^1$. Take their [pushout-product](https://ncatlab.org/nlab/show/pushout-product) to obtain a map
$$\Delta^1 \times \Delta^1 \cup\_{\partial \Delta^1 \times \Delt... | https://mathoverflow.net/users/2362 | Is $\Delta^1 \times \Delta^1 \cup_{\partial \Delta^1 \times \Delta^1} \partial \Delta^1 \times I \to \Delta^1 \times I$ inner anodyne? | As requested, here is a proof that the map in question is inner anodyne that does not proceed by explicitly filling inner horns.
By definition (more or less), a map is inner anodyne iff it has the LLP wrt all inner fibrations; but if moreover the codomain of that map is a quasi-category, as in our example, it suffice... | 1 | https://mathoverflow.net/users/57405 | 453341 | 182,169 |
https://mathoverflow.net/questions/453349 | 2 | I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought that $sup$ norm would be nice measurable norm to complete the RKHS of mine (just as classical Wiener measure on $C[0,1]... | https://mathoverflow.net/users/508638 | Reference for Wiener type measure on $C(T)$ when $T$ is open | I think you can adapt the proof in [these notes of mine](https://arxiv.org/abs/1607.03591v2), Theorem 4.44.
The first step in the proof of my notes is to pick, for each $n$, a finite-rank projection $P\_n$ such that for all finite-rank projections $P \perp P\_n$, we have
$$\tilde{\mu}\left(\left\{ h \in H : \|Ph\|\_W... | 2 | https://mathoverflow.net/users/4832 | 453353 | 182,173 |
https://mathoverflow.net/questions/453361 | 3 | Suppose that graphs $A$ and $B$ with $V(A)=V(B)$ have Hadwiger numbers $a$ and $b$. That is, $K\_a$ and $K\_b$ are the largest clique minors of $A$ and $B$, respectively.
Are there upper bounds on the Hadwiger number of the union graph $A\cup B$ (with edge set $E(A\cup B)=E(A)\cup E(B)$) using $a$ and $b$ like $a+b$ or... | https://mathoverflow.net/users/511540 | Bounding the size of clique minor of the union of two graphs | Take a complete graph $G\_n$, replace each edge $uv$ to a path $uxv$. Imagine that $ux\in A$, $vx\in B$, and that's for every edge. It looks that both $A,B$ are forests, so have small Hadwiger number.
| 7 | https://mathoverflow.net/users/4312 | 453362 | 182,174 |
https://mathoverflow.net/questions/453343 | 20 | Cross post from [Maths stack exchange](https://math.stackexchange.com/questions/4495174/a-difficult-integral-for-the-chern-number?noredirect=1#comment10099463_4495174)
---
The integral
$$
I(m)=\frac{1}{4\pi}\int\_{-\pi}^{\pi}\mathrm{d}x\int\_{-\pi}^\pi\mathrm{d}y\phantom{,} \frac{m\cos(x)\cos(y)-\cos x-\cos y}{\l... | https://mathoverflow.net/users/490549 | A difficult integral for the Chern number | If Stokes' theorem counts as a standard technique, then here's an answer:
Introduce a "vector potential"
\begin{equation}
A\_i = \frac{1-\hat n\_z}{\hat n\_x^2 + \hat n\_y^2}(\hat n\_x \partial\_i \hat n\_y - \hat n\_y \partial\_i \hat n\_x)\,,\qquad i=x,y\,,
\end{equation}
which is defined everywhere, except where $... | 24 | https://mathoverflow.net/users/165560 | 453369 | 182,177 |
https://mathoverflow.net/questions/453236 | 3 | Let $G$ and $H$ two connected linear Lie groups. Is $G\ltimes H$ also linear?
| https://mathoverflow.net/users/509535 | Semidirect product of two linear groups | No. There is a semidirect product $\mathbf{R}\ltimes (\mathbf{R}\times\mathbf{R}/\mathbf{Z})$ that is not linear. Namely, the quotient of the Heisenberg group by a lattice of its center has this form.
Replacing $\mathbf{R}$ with $\mathbf{C}$ also yields an example in the realm of complex Lie groups.
These are class... | 6 | https://mathoverflow.net/users/14094 | 453378 | 182,179 |
https://mathoverflow.net/questions/453370 | 10 | We say a measurable function $f: \mathbb R^n \to \mathbb R$ is *essentially continuous* if the inverse image of any open set $O$ differs from an open set by a set of null measure, in the sense that there exists an open subset $U$ of $\mathbb R^n$ such that $\mu(f^{-1} (O) \, \Delta \, U) = 0$ where $\Delta$ denotes the... | https://mathoverflow.net/users/173490 | A topological characterisation of a.e. continuity | I adapt the proof of a more general result: Theorem 4.12 in Chapter XI of K. Kuratowski and A. Mostowski’s book “*Set Theory: with an introduction to descriptive set theory*” (see page 408).
**Claim.** *$f$ is essentially continuous iff there exists a null set $N$ such that $f|({\bf R}^n-N)$ is continuous.*
**Proof... | 2 | https://mathoverflow.net/users/118745 | 453383 | 182,182 |
https://mathoverflow.net/questions/452795 | 3 | I have a question about following argument I found
in [these notes](https://www.google.de/url?sa=t&source=web&rct=j&opi=89978449&url=https://people.math.rochester.edu/faculty/doug/otherpapers/WebbMF.pdf&ved=2ahUKEwiiu9Lu7d6AAxVg_rsIHU-KCSEQFnoECBAQAQ&usg=AOvVaw1OZ96p6a5V9Y-wW04DRo45) on Mackey functors:
>
> **(2.1)... | https://mathoverflow.net/users/108274 | Mackey coset decomposition formula | The induction functor preserves all *(weakly) connected* limits (note that this covers pullbacks, but not products - and of course, because it doesn't preserve the terminal object, it couldn't do both !).
There is a more general way to argue, but in this case you can say as follows (there is also probably a more conc... | 3 | https://mathoverflow.net/users/102343 | 453392 | 182,187 |
https://mathoverflow.net/questions/417448 | 23 | Cardinalities in non-standard models of $\mathsf{ZF}$ are not generally reflected externally, but certain internal facts about cardinalities are always externally reflected (e.g., any internally infinite set is externally infinite; if $|A| \leq |B|$ internally, then the same holds externally; the internal powerset of a... | https://mathoverflow.net/users/83901 | Can a non-standard model $M$ of $\mathsf{ZF}$ contain an internally infinite set that is externally smaller than $\aleph_0^M$? | The answer is yes. Let $M$ be a countable transitive model of $\mathsf{ZF}$ containing an amorphous set $A$ (i.e., every subset of $A$ is either finite or co-finite).
Let $T$ be the theory consisting of the elementary diagram of $M$ together with a fresh constant $c$ and the sentences $c \in \omega$ and $n < c$ for e... | 8 | https://mathoverflow.net/users/83901 | 453394 | 182,188 |
https://mathoverflow.net/questions/453267 | 4 | I have a second-order ODE with an unknown parameter $p$,
$$\frac{y''}{(1+y'^{2})^{\frac{3}{2}}} -p - A(x-B)^2 =0,$$
where $x$ is the independent variable, $y$ is the unknown function, $p$ is unknown parameter to be determined, and $A$ and $B$ are known constants.
I am given $y$ values at three $x$ points, $y\_0(x=x\_... | https://mathoverflow.net/users/511129 | How to solve an ODE with an unknown parameter but given y values at three points | According to the suggestions from @AVK,
I built the problem in MATLAB(but I have no MATLAB to test)
I have read that example and try to adapt it to my case .
$$
\frac{y''}{(1+y'^{2})^{\frac{3}{2}}} -p - A(x-B)^2 =0
$$
given
$$
\begin{equation}
\begin{cases}
y(x\_a)=YA \\
y(x\_c)=YC \\
y(x\_b)=YB
\end{cases}\,
\... | 2 | https://mathoverflow.net/users/511129 | 453405 | 182,192 |
https://mathoverflow.net/questions/453093 | 2 | Let $R$ be a commutative Noetherian ring, and $\phi: R \to R$ be a ring homomorphism. For an $R$-module $M$, let $^{\phi}M$ be the $R$-module defined via restriction of scalars via $\phi$, i.e., as abelian groups, $M$ and $^{\phi}M$ are the same, but the $R$-action on $^{\phi}M$ is given by $r\*x:=\phi(r)x, \forall r \... | https://mathoverflow.net/users/135389 | Finding the mistake in an argument concerning $F$-finite $F$-split local Cohen--Macaulay ring of dimension $1$ | @uno, you are correct in the comments above, $F^e\_\*(-)$ is not $R$-linear. In particular, the mistake is the line right here:
>
> The exact sequence $0\to R \xrightarrow{x} R\to R/xR\to 0$ then gives rise to exact sequence $0\to F^e\_\* R \xrightarrow{x} F^e\_\* R \to F^e\_\*(R/xR)\to 0$ implying $F^e\_\*R/xF^e\_... | 1 | https://mathoverflow.net/users/3521 | 453408 | 182,193 |
https://mathoverflow.net/questions/453398 | 1 | *(This question is a repost of a deleted question I asked, because the previous version had several elements missing)*
**Setting**
For fixed $N \in \mathbb{N}$, **I wish to compute the entries of a matrix $P\_N$** that is $2^N \times 3^N$-dimensional, real-valued and satisfies:
1. $P\_N^T \textbf{1} = \textbf{1}$... | https://mathoverflow.net/users/511514 | Methods to solve for a matrix whose entries satisfy certain properties | There are zero or infinitely many solutions depending on where the non-zero entries have to be. So there is no general-purpose answer.
I don't think your equations are properly stated, as $\boldsymbol 1$ seems to have at least two different meanings. I'll interpret your intention as that every row sums to $3^N/2^N$ a... | 1 | https://mathoverflow.net/users/9025 | 453409 | 182,194 |
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