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https://mathoverflow.net/questions/449301 | 8 | $\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from [Proposition 7](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-31/issue-2/A-class-of-Wasserstein-metrics-for-probability-distributions/10.1307/mmj/1029003026.full) and this [recent answer](https://mathoverflow.net/a/... | https://mathoverflow.net/users/36721 | On a matrix inequality | I'll try to answer both questions at once. First, let's find the reason why the LHS is positive at all. I claim that the spectrum $\mu\_1\le\mu\_2\le\dots\le\mu\_n$ of $2\sqrt{A^{1/2}BA^{1/2}}$ is dominated by the spectrum $\lambda\_1\le\lambda\_2\le\dots\le\lambda\_n$ of $A+B$ elementwise. Indeed, for each $k$, there ... | 5 | https://mathoverflow.net/users/1131 | 451309 | 181,504 |
https://mathoverflow.net/questions/451310 | 3 | Let $K$ be a number field. For each ideal $I$ of the ring of integers $\mathcal{O}\_K$ let $N\_K(I)$ denote the norm of $I$. For a prime $\mathfrak{p}\subset \mathcal{O}\_K$ above the rational prime $p\in \mathbb{Z}$, set $\deg(\mathfrak p):=[\mathcal{O}\_K/\mathfrak{p}:\mathbb{Z}/p]$. Consider the following limit:
$$
... | https://mathoverflow.net/users/92433 | Density of prime ideals of a given degree | This is fairly classical, and the answer is zero for any $n>1$, and $1$ for $n=1$. The reason basically comes down to the fact that asymptotic-wise, almost all prime powers are prime.
Here is the proper proof. Firstly, by Chebotarev, a positive density of primes in $\Bbb Z$ split completely in $O\_K$. This alone impl... | 11 | https://mathoverflow.net/users/30186 | 451313 | 181,506 |
https://mathoverflow.net/questions/451317 | 9 | I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\dashv\Gamma$ for schemes through Isbell duality. As I understand it, Isbell duality relates appropriate "algebraic" and "... | https://mathoverflow.net/users/509047 | Isbell Duality and Dualizing Scheme Objects | *Stone Spaces* by Peter Johnstone (CUP 1982) is about just this subject.
It is an *exceptionally* well written book. It would be better that you read it than that I make any attempt to summarise it for you.
By "exceptionally" I mean in contrast to the usual way in which research level pure maths books are written, ... | 6 | https://mathoverflow.net/users/2733 | 451332 | 181,509 |
https://mathoverflow.net/questions/451334 | 2 | I already posted this question on mathstackexchange [there](https://math.stackexchange.com/questions/4739910/composition-of-gysin-and-restriction-maps-on-ell-adic-cohomology), but I figured that it may have more replies here.
---
I follow the notations of Milne's lectures notes on etale cohomology, most specifica... | https://mathoverflow.net/users/125617 | Composition of Gysin and restriction maps on $\ell$-adic cohomology | Yes: $\pi ^\*\pi \_\*$ is the cup-product with $c\_c(N)$, the $c$-th Chern class of the normal bundle of $Y$ in $X$. This is Theorem VII.4.1 in SGA 5.
| 4 | https://mathoverflow.net/users/40297 | 451338 | 181,511 |
https://mathoverflow.net/questions/451337 | 4 | Consider two domains
$$
\begin{aligned}
D\_1&=\{x=(x\_1,x\_2,...,x\_n)\in\mathbb{R}^n:x\_n\leq 0\},\\
D\_2&=\{x=(x\_1,x\_2,...,x\_n)\in\mathbb{R}^n:x\_n\leq \psi(x\_1,x\_2,...,x\_{n-1})\},
\end{aligned}
$$
where $ \psi:\mathbb{R}^{n-1}\to\mathbb{R} $ is a smooth bounded function. I want to construct a conformal map $ \... | https://mathoverflow.net/users/241460 | Conformal maps between two given domains | Any conformal map in dimensions $\ge 3$ is necessary a superposition of inversions and isometries (see e.g. the link suggested by Daniele Tampieri in his comment), so it takes the boundary of $D\_1$ to a hyperplane or (a part of) a sphere. Therefore, the graph of the $\psi$ is a part of a sphere provided a conformal ma... | 7 | https://mathoverflow.net/users/14515 | 451375 | 181,516 |
https://mathoverflow.net/questions/451377 | 1 | By Hausdorff-Bernstein-Widder theorem, any completely monotonic function on the half line $\mathbb{R}\_{\geq 0}:=[0,\infty)$ is given by the Laplace transform of a positive measure on $\mathbb{R}\_{\geq 0}$.
My question is following:
Is there a positive definite function on $\mathbb{R}\_{\geq 0}$ which is not given... | https://mathoverflow.net/users/509119 | Positive definite but not completely monotonic function on the upper half line | Counterexample: If $f(x)=e^x$ for real $x\ge0$, then $f$ is positive definite. However, $f$ is not the Laplace transform of a positive measure on $[0,\infty)$ -- because otherwise $f$ would be nonincreasing.
| 0 | https://mathoverflow.net/users/36721 | 451381 | 181,518 |
https://mathoverflow.net/questions/451311 | 0 | Let $\mathbf H$ be an infinite dimensional Hilbert space.
I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H \times \mathbf H\to \mathbf H \times \mathbf H$ such that the spectrum of $A$ is discrete but the spectrum of $MA$ is... | https://mathoverflow.net/users/152870 | Spectrum of a product of a symmetric positive definite matrix and a positive definite operator | Here's a way to construct such an example:
For each integer $n \ge 1$ consider the matrices $A\_n, M\_n \in \mathbb{C}^{2 \times 2}$ given by
$$
M\_n =
e\_1 e\_1^T
=
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
\qquad \text{and} \qquad
A\_n =
f\_n f\_n^T
,
$$
where $e\_1 \in \mathbb{C}^2$ if the first c... | 1 | https://mathoverflow.net/users/102946 | 451400 | 181,522 |
https://mathoverflow.net/questions/451401 | 13 | Let $M$ be a simply connected, (finite dimensional) smooth manifold. Is it possible that $M$ is homotopy equivalent to $M\times M,$ without $M$ being contractible? This would imply $\pi\_n(M)\times\pi\_n(M)\cong \pi\_n(M)$ for all $n\in\mathbb{N}.$ I know there are groups which satisfy $G\times G\cong G,$ but this is a... | https://mathoverflow.net/users/40323 | Can a simply connected manifold satisfy $\simeq \times $? | Thanks to [Dave Benson](https://mathoverflow.net/questions/451401/can-a-simply-connected-manifold-satisfy-simeq-%c3%97#comment1167394_451420) for pointing out an algebra error in the first draft of this answer and a missing detail.
Suppose $X$ is homotopy equivalent to a finite dimensional, simply-connected, and nonc... | 9 | https://mathoverflow.net/users/134512 | 451420 | 181,525 |
https://mathoverflow.net/questions/451425 | 3 | This is probably going to lead nowhere, but maybe it be possible to use the matrix logarithm to invert matrices?
For positive definite matrices, we have that the logarithm exists and
$$
\log(A^{-1})= - \log(A)
$$
So very brutally applying the first taylor approximation everywhere (assuming this works for matrices)
$$
... | https://mathoverflow.net/users/122659 | Inverting a matrix using the Matrix logarithm | Numerically, inverting a matrix by computing matrix exponentials and logarithms doesn't really work well, because (1) typically methods to compute matrix exponentials and logarithms are much more expensive than methods to compute the inverse, and (2) there are branch points in the logarithm which may create stability p... | 7 | https://mathoverflow.net/users/1898 | 451427 | 181,528 |
https://mathoverflow.net/questions/451394 | 8 | Let $\pi\colon X \rightarrow Y$ be a Serre fibration. Define $\Sigma\_f\pi \colon \Sigma\_f X \rightarrow Y$ be the fiberwise unreduced suspension of $\pi$. Thus $\Sigma\_f X = X \times [0,1] / {\sim}$, where $\sim$ identifies $(x,0)$ with $(x',0)$ whenever $\pi(x) = \pi(x')$ and also $(x,1)$ with $(x',1)$ whenever $\p... | https://mathoverflow.net/users/509127 | Is the fiberwise suspension of a Serre fibration a Serre fibration? | Lemma 6 in Section 3 of
*Vandembroucq, Lucile*, [**Fibrewise suspension and Lusternik-Schnirelmann category**](https://doi.org/10.1016/S0040-9383(02)00007-1), Topology 41, No. 6, 1239-1258 (2002). [ZBL1009.55002](https://zbmath.org/?q=an:1009.55002).
(also available at <https://core.ac.uk/download/pdf/82041491.pdf>... | 7 | https://mathoverflow.net/users/8103 | 451428 | 181,529 |
https://mathoverflow.net/questions/451030 | 2 | I've been following the works of [Totaro](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-48/issue-1/The-topology-of-smooth-divisors-and-the-arithmetic-of-abelian/10.1307/mmj/1030132736.full), [Pereira](https://www.ams.org/journals/jag/2006-15-01/S1056-3911-05-00417-0/), and [Bogomolov/Pirutka/S... | https://mathoverflow.net/users/111293 | Varieties with disjoint prime divisors | Take a smooth curve $C$ of genus $\geq 2$ and consider a smooth divisor $D \subset C \times C$ which is $2$-divisible in $\operatorname{Pic}(C \times C)$ and which is transverse to both factors (by Bertini-type arguments one see that there is plenty of them).
Then there exists a double cover $f \colon X \to C \times ... | 4 | https://mathoverflow.net/users/7460 | 451434 | 181,531 |
https://mathoverflow.net/questions/362997 | 8 | First let me recall the combinatorial theory of the characters of $\mathfrak{gl}\_m$, a.k.a., Schur polynomials. For a partition $\lambda$, a *semistandard Young tableaux* of shape $\lambda$ is a filling of the boxes of (the Young diagram of) $\lambda$ with positive integers such that entries strictly increase down col... | https://mathoverflow.net/users/25028 | Bender-Knuth involutions for symplectic (King) tableaux | Let the hyperoctahedral group $\mathbb{S}\_n\wr\mathbb{S}\_2$ act naturally on the set $\{1, 2, ..., n\} \cup \{1', 2', ..., n'\}$. Then, we can say $\mathbb{S}\_n\wr\mathbb{S}\_2$ is generated by the transposition $(1\quad 1')$ and the permutations $(i\quad i\!+\!1)(i'\quad i\!+\!1')$ for $i = 1, ..., n-1$. We want to... | 3 | https://mathoverflow.net/users/206706 | 451444 | 181,534 |
https://mathoverflow.net/questions/451445 | 2 | Suppose using the lebesgue outer measure $\lambda^{\*}$, we restrict $A$ to sets [measurable in the Caratheodory sense](https://en.m.wikipedia.org/wiki/Carath%C3%A9odory%27s_criterion), defining the Lebesgue measure $\lambda$.
**Question:**
Does there exist an explicit and *bijective* $f:\mathbb{R}\to\mathbb{R}$, w... | https://mathoverflow.net/users/87856 | Finding an explicit & bijective function that satisfies the following properties? | $\newcommand\R{\mathbb R}\newcommand\la{\lambda}$No. Indeed, let $G:=\{(x,f(x))\colon x\in\R\}$. Then, by the Tonelli theorem,
$$\la(G)=\int\_\R dx\,\int\_{[f(x),f(x)]}dy=\int\_\R dx\,0=0.$$
So, for all real $x\_1,x\_2,y\_1,y\_2$ such that $-\infty<x\_1<x\_2<\infty$ and $-\infty<y\_1<y\_2<\infty$ we have
$$\la(([x\_1,x... | 5 | https://mathoverflow.net/users/36721 | 451448 | 181,535 |
https://mathoverflow.net/questions/451452 | 5 | Suppose $\lambda^{\*}$ is the Lebesgue outer measure.
**Question:**
Does there exist an explicit $f:\mathbb{R}\to\mathbb{R}$, where:
1. The range of $f$ is $\mathbb{R}$
2. For all real $x\_1,x\_2,y\_1,y\_2$, where $-\infty<x\_1<x\_2<\infty$ and $-\infty<y\_1<y\_2<\infty$:
$$\lambda^{\*}(\left([x\_1,x\_2]\times[y\... | https://mathoverflow.net/users/87856 | Is it known that there is any function $f:\mathbb{R}\to\mathbb{R}$ at all, whose graph has positive outer measure on every rectangle in the plane? | The answer is yes.
We construct $f$ by transfinite recursion using a well ordering of the reals. (So this may not be explicit enough for you.)
In fact, we can make the function $f$ bijective, with the graph of $f$ having full outer measure in every rectangle.
**Theorem.** There is a bijective function $f:\mathbb{... | 8 | https://mathoverflow.net/users/1946 | 451454 | 181,537 |
https://mathoverflow.net/questions/451449 | 2 | Let $f : \mathbb R^d \to \mathbb R$ be Lipschitz and $[f] := \sup\_{x,y \in \mathbb R^d; x\neq y} \frac{|f(x) - f(y)|}{|x-y|}$ its Lipschitz constant. By Rademacher theorem, $f$ is differentiable a.e., so $\nabla f$ is defined a.e.
>
> Is it true that $\|\nabla f\|\_{L^\infty} = [f]$?
>
>
>
Thank you so much f... | https://mathoverflow.net/users/99469 | Is the Lipschitz constant of $f$ equal to $\|\nabla f\|_{L^\infty}$? | Yes, Theorem 1.41 of my book *Lipschitz Algebras* (second edition). But you must interpret ``$\|\nabla f\|\_{L^\infty}$'' to mean $\|\,|\nabla f|\,\|\_{L^\infty}$ (sup of the norm of the gradient taken in $\mathbb{R}^n$).
| 4 | https://mathoverflow.net/users/23141 | 451457 | 181,539 |
https://mathoverflow.net/questions/451460 | 10 | Let $L$ be the Lazard ring, i.e., the underlying ring of the universal one-dimensional formal group law. Let $M$ be the ring $\mathbb{Z}[c\_4, c\_6, 1/6]$ of Weierstrass curves over $\mathbb{Z}[1/6]$. There is a natural surjective ring map $L[1/6] \rightarrow M$ classifying the formal group law of a Weierstrass curve.
... | https://mathoverflow.net/users/509184 | With 6 inverted, is the ring of Weierstrass curves a quotient of the Lazard ring by a regular sequence? | I'll refer to my notes on formal groups at <https://strickland1.org/courses/formalgroups/fg.pdf>. There are results about the formal group law of an elliptic curve in Section 19. That is written in terms of the general homogeneous Weierstrass form
$$ y^2 z + a\_1 x y z + a\_3 y z^2 - x^3 - a\_2 x^2 z - a\_4 x z^2 - a\_... | 12 | https://mathoverflow.net/users/10366 | 451468 | 181,543 |
https://mathoverflow.net/questions/451473 | 6 | Suppose we have two curves $C/\mathbb{Q}$ and $C'/\mathbb{Q}$ which are twists of each other i.e. they are isomorphic over a field extension $K/\mathbb{Q}$.
Suppose that $C$ has good reduction at a prime $l$. Is it true that $C'$ has good reduction at $l$ if and only if $l$ is unramified in $K$? It seems to me that t... | https://mathoverflow.net/users/478525 | Good and bad reduction for twists of algebraic curves | Let $B$ be a Dedekind scheme with function field $K$. (Think of $B=\mathrm{Spec } \ \mathbb{Z}\_{p}$ for simplicity, so that $K=\mathbb{Q}\_p$.) Let $C$ be a smooth proper geometrically connected curve over $K$ of genus at least one with good reduction over $B$. Let $\mathcal{C}\to B$ be its (unique) smooth proper mode... | 6 | https://mathoverflow.net/users/4333 | 451497 | 181,550 |
https://mathoverflow.net/questions/451504 | 5 | Let $\mathbb F\_p$ denote the finite prime field of $p$ elements. What reference can be recommended for an analysis of the structure of the group of units of the power series ring $\mathbb F\_p[[x]]$? There must be much known more generally when the finite field is replaced by any other field but I am most interested i... | https://mathoverflow.net/users/124943 | Reference request for the group of units of a power series ring in one variable | In general the multiplicative group $1+x\,A[\![x]\!]\leq U(A[\![x]\!])$ is isomorphic to the additive group of the ring $W(A)$ of big Witt vectors of $A$. If $A$ is $p$-local for some prime $p$, then $W(A)$ is additively isomorphic to a product of copies of the ring $W\_{p^\infty}(A)$ of $p$-typical Witt vectors, with ... | 8 | https://mathoverflow.net/users/10366 | 451509 | 181,554 |
https://mathoverflow.net/questions/451487 | 1 | I am interested in the value function of a quadratic program of the form
$$
v(y)=\min\_x \frac{1}{2} x^\top Q(y) x,
$$
subject to a linear equality constraint
$$
E(y)x=d(y),
$$
and a linear inequality constraint
$$
A x \preceq b,
$$
where $\preceq$ denotes component-wise inequalities.
Notice that $Q$, $E$ and $d$ all... | https://mathoverflow.net/users/91545 | Does the value function of a quadratic program stay convex when adding constraints? | Not necessarily as written in the generality you want. Suppose that we are in $\mathbb R^1$, there is no linear constraint, and $Q(y)$ is some positive function of $y$. Then $v(y)\equiv 0$, which is convex. Now add the condition $x<-1$. Then $v(y)=Q(y)$ and that can be absolutely anything.
On the other hand, you, pro... | 4 | https://mathoverflow.net/users/1131 | 451513 | 181,555 |
https://mathoverflow.net/questions/451500 | 4 | OEIS [A109388](https://oeis.org/A109388) $\{a\_n\}\_{n\ge1}$ is an integer sequence with $a\_n=\binom{n}{\lfloor \frac{n}{3} \rfloor}\times 2^{n-\lfloor\frac{n}{3}\rfloor}$, I noticed that OEIS says
>
> $a\_n$ is the **size** of the largest antichain in the partial ordering
> $(0,1,a)^n$ where $0$ and $1$ are less ... | https://mathoverflow.net/users/507284 | Largest antichain in partial ordering in OEIS | The poset $(0,1,a)^n$ is isomorphic to the face lattice of an $n$-dimensional cube, with the empty face removed. This lattice can be proved to be Sperner by the same argument Lubell used to prove the Spernicity of the boolean algebra. A reference is the slides [here](https://math.mit.edu/~rstan/transparencies/garsiafes... | 7 | https://mathoverflow.net/users/2807 | 451530 | 181,557 |
https://mathoverflow.net/questions/451544 | 6 | Let $A$, $B$ be $n\times n$ unitary complex matrices, such that for all indices $i,j$ we have $|a\_{ij}|=|b\_{ij}|$. Does there then exist diagonal unitary matrices $D,D’$ such that $DAD’=B$?
This can also be phrased as, take a unitary matrix, and sprinkle norm one signs on all of its entries. If the resulting matrix... | https://mathoverflow.net/users/128502 | Sprinkling signs in unitary matrices | There are [5 inequivalent Hadamard matrices](https://en.wikipedia.org/wiki/Hadamard_matrix#Equivalence_and_uniqueness) of order 16; if I understand correctly that's a counterexample.
| 11 | https://mathoverflow.net/users/1898 | 451545 | 181,559 |
https://mathoverflow.net/questions/451538 | 1 | Let $G$ be a (split) reductive group over a $p$-adic field $F$, and $\mathbf{1}$ the trivial representation of $G(F)$. Under the (conjectural) Langlands correspondence, this should correspond to an $L$-parameter
$$
\varphi:WD\_F\longrightarrow G^\vee(\mathbb{C}),
$$
where $WD\_F$ is the Weil-Deligne group and $G^\vee(\... | https://mathoverflow.net/users/32746 | How to assign the $L$- and $A$-parameters for the trivial representation | The L-parameter of the trivial representation is the norm map $W\_F \to \mathbf{C}^\times$ composed with the cocharacter $\mathbf{C}^\times \to G^{\vee}(\mathbf{C})$ given by the half-sum of positive coroots. So e.g. for $GL\_n$ it sends $w \in W\_F$ to $\mathrm{diag}(|w|^{(n-1)/2},|w|^{(n-3)/2},\dots,|w|^{(1-n)/2})$.
... | 3 | https://mathoverflow.net/users/496798 | 451547 | 181,561 |
https://mathoverflow.net/questions/451404 | 3 | I came across the book "Cohen-Macaulay Representations" by Graham J. Leuschke and Roger Wiegand, and now I'm wondering if this is an active area of research.
If yes, then
>
> what are some of the active problems that people are interested in solving, and what connections does it have to other fields?
>
>
>
| https://mathoverflow.net/users/338456 | Cohen-Macaulay Representations | Maximal Cohen-Macaulay modules, which seems to be at the center of this book, play a very important role in the theory of *non-commutative resolutions singularities*. Using MCM modules, Van-den-Bergh gave a very interesting reformulation of Bridgeland's proof of Bondal-Orlov conjecture in dimension 3. See [Three dimens... | 3 | https://mathoverflow.net/users/37214 | 451549 | 181,563 |
https://mathoverflow.net/questions/451520 | 1 | It is shown in Moriya (Multiplicative formality of operads and
Sinha’s spectral sequence for long knots, 2.1) that there exists a left proper model category structure on non-symmetric operads over $k$-chain complexes $\textrm{Ch}(k)$, where $k$ is a field. He gives a direct proof of the theorem, and I am not sure wheth... | https://mathoverflow.net/users/140013 | Left Proper model structure on the category of non-symmetric operads in chain complexes | Let $k$ be a commutative ring and let $\mathrm{Ch}(k)$ be the category of *non-negatively* graded chain complexes of $k$-modules. We endow it with the projective model structure. Weak equivalences are quasi-isomorphisms and fibrations are maps which are surjective in *positive* degrees. As a consequence of Theorem 1.11... | 4 | https://mathoverflow.net/users/12166 | 451551 | 181,564 |
https://mathoverflow.net/questions/451562 | 1 | Given a uniform planar convex region C, let us consider 2 centers of mass - the center of mass of the region as a whole and the center of mass of its boundary alone (assuming its boundary to have constant linear density). It is obvious that for all regular polygonal regions, both centers coincide. Even for other shapes... | https://mathoverflow.net/users/142600 | When do the centers of mass of a uniform convex planar region as a whole and of its boundary alone coincide? | In Euclidean triangle geometry the Spieker point $S$ is the centroid of the triangle sides, and it is known to be the
anticomplement of the incircle center $I$ (<https://faculty.evansville.edu/ck6/encyclopedia/ETC.html>), which means it is in line with
$I$ and the triangle centroid $C$, such that
$$\overline{S C}~=~\fr... | 2 | https://mathoverflow.net/users/20804 | 451563 | 181,566 |
https://mathoverflow.net/questions/451550 | 1 | Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (<https://stacks.math.columbia.edu/tag/07DR>). Then, $R$ is the homomorphic image of a localization of some polynomial ring over $k$.
My question is: Does there exist a surjective $k$-algebra homomorphi... | https://mathoverflow.net/users/174552 | On "minimal presentation" of local rings essentially of finite type over a field | If $R$ is the local ring of the point $P$ on a smooth $n$-dimensional $k$-variety $V$ then any such homomorphism would be an isomorphism. So it cannot exist if $V$ is irrational. For example, take $V$ to be an elliptic curve.
| 2 | https://mathoverflow.net/users/8726 | 451572 | 181,568 |
https://mathoverflow.net/questions/450246 | 5 | Given integers $k,n\ge 1$, I shall write $\Bbb{Z}\_k^n := (\Bbb{Z}/k\Bbb{Z})^n$.
Fix $k\ge 3$. Let $r\_k(\Bbb{Z}\_k^n)$ denote the cardinality of the largest $A\subset \Bbb{Z}\_k^n$, such that $A$ does not contain any $k$-term progression $P\subset A$ with $|P| = k$.
It is clear that $r\_k(\Bbb{Z}\_k^1)=k-1$. Induc... | https://mathoverflow.net/users/130484 | Beating trivial bound for $k$-AP-free sets in characteristic $k$ | I think that this is known. A good source to check recent things would be
<https://arxiv.org/abs/2211.02588>.
| 3 | https://mathoverflow.net/users/955 | 451578 | 181,569 |
https://mathoverflow.net/questions/451480 | 3 | I'm currently reading "Bordism of Elementary Abelian Groups via Inessential Brown-Peterson Homology" by Hanke ([arXiv:1503.04563](https://arxiv.org/abs/1503.04563)) and have come across some notation that I'm not familiar with. Let $\phi: \mathbb{Z}/3 \rightarrow (\mathbb{Z}/3)^2$ be a group homomorphism and $S^5$ the ... | https://mathoverflow.net/users/503849 | Understanding $(\mathbb{Z}/3)^2 \times_{\mathbb{Z}/3} M$ | This has been more or less said in the comments, but it seems like someone should write an answer.
If $H$ is a subgroup of $G$ and $H$ has a left action on a space $X$, then a space $G\times \_HX$ is defined, a quotient space of $G\times X$, by declaring $(gh,x)\sim (g,hx)$. Note that we are using the right action of... | 3 | https://mathoverflow.net/users/6666 | 451579 | 181,570 |
https://mathoverflow.net/questions/451577 | 3 | Corollary 3.3 in Chapter IV of "Ample subvarieties of algebraic varieties" by R. Hartshorne asserts the following:
Let $X$ be a smooth projective variety and $Y\subset X$ a smooth subvariety of dimension at least three. Assume that $Y$ is a strict complete intersection in $X$ then the natural map
$$
Pic(X)\rightarrow... | https://mathoverflow.net/users/14514 | A question on "Ample subvarieties of algebraic varieties" | I suspect that you are supposed to view the projective variety $X$ as being given with a chosen projective embedding $X\subset \mathbb P^n$, and therefore a distinguished ample divisor $\mathcal{O}\_X(1) = \mathcal{O}\_{\mathbb P^n}(1)|\_X$. Then the complete intersection $Y$ should be cut out by sections of $\mathcal{... | 9 | https://mathoverflow.net/users/104695 | 451582 | 181,571 |
https://mathoverflow.net/questions/451591 | 0 | Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$].
For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, we consider the *$c$-transform*
$$\tag{1}
T\_c \, : \, L^1(\mu) \,\ni\, \varphi \,\mapsto\, \varphi^c, \quad\text{where}\quad \var... | https://mathoverflow.net/users/472548 | Continuity of generalised Legendre transform | $\newcommand\vpi\varphi$The answer is no. E.g., suppose that $X=Y=\mathbb R$, $c=0$, and $\vpi\_a(x)=\min(a,\max(0,x-a))$ for $a\ge0$ and all real $x$. Then, as $a\to\infty$, we have $\vpi\_a\to0$ uniformly on compact sets, whereas $T\_c(\vpi\_a))=-a\not\to0=T\_c(0)$ in $L^1(\nu)$ for any probability measure $\nu$ over... | 2 | https://mathoverflow.net/users/36721 | 451596 | 181,574 |
https://mathoverflow.net/questions/451601 | 0 | Let $X$ and $Y$ be $\sigma$-compact spaces, and $\mu$ [resp. $\nu$] be a regular Borel probability measure on $X$ [resp. $Y$].
For a bounded continuous $c:X\times Y\rightarrow\mathbb{R}$, consider the *$c$-transform*
$$\tag{1}
T\_c \, : \, L^1(\mu) \,\ni\, \varphi \,\mapsto\, \varphi^c, \quad\text{where}\quad \varphi... | https://mathoverflow.net/users/472548 | Generalised Lebesgue transform continuous wrt. strict topology? | $\newcommand\vpi\varphi$The answer, which is a modification of the [previous answer](https://mathoverflow.net/a/451596/36721), is still no.
E.g., suppose that $X=Y=\mathbb R$, $c=0$, and $\vpi\_a(x)=\min(1,\max(0,x-a))$ for $a\ge0$ and all real $x$.. Then, as $a\to\infty$, we have $\vpi\_a\to0$ in the (generalised) s... | 1 | https://mathoverflow.net/users/36721 | 451604 | 181,577 |
https://mathoverflow.net/questions/451569 | -3 | [The question has been edited]
Can we have an effectively generated consistent theory $T$, that extends $\sf PA$, such that:
$T + \\ \forall \mathcal S \, [\exists x: \operatorname {Proof}\_T(x, \ulcorner \mathcal S \urcorner) \oplus \exists y: \operatorname {Proof}\_T(y, \operatorname {neg}(\ulcorner \mathcal S \u... | https://mathoverflow.net/users/95347 | Can there be an effectively generated consistent theory that extends PA and be consistently extended by its own completeness and consistency? | The answer to the edited question is No, there is no such theory $T$, because the incompleteness theorem is provable in PA and hence also in $T$. The theory $T$ will prove that there is no effective complete consistent theory of arithmetic, and in particular, $T$ will prove that $T$ itself (as defined by its effective ... | 3 | https://mathoverflow.net/users/1946 | 451606 | 181,578 |
https://mathoverflow.net/questions/451566 | 4 | Let $S\_1, \ldots, S\_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota\_i$ be the inversion in $S\_i$. Viewing the conformal group of $\mathbb{S}^{n-2} = \mathbb{R}^{n-2} \cup \{\infty\}$ as an index-2 subgroup of $\mathrm{O}(n-1,1)$ yields a repre... | https://mathoverflow.net/users/163543 | What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres? | I computed the matrices for $n=3,...,10$ in Mathematica and found the factorizations of the characteristic polynomials:
$$-(1 + x) (1 - 18 x + x^2),$$
$$ 1 - 68 x - 122 x^2 - 68 x^3 +
x^4,$$ $$-(1 + x) (1 - 228 x - 314 x^2 - 228 x^3 + x^4),$$
$$1 - 710 x - 2033 x^2 - 2708 x^3 - 2033 x^4 - 710 x^5 +
x^6,$$ $$ -(1 ... | 5 | https://mathoverflow.net/users/1345 | 451611 | 181,579 |
https://mathoverflow.net/questions/451345 | 2 | For definition of productive set, see [here](https://encyclopediaofmath.org/wiki/Productive_set) and [here](https://en.wikipedia.org/wiki/Creative_and_productive_sets), that is defined with computability, or computable function. Restricting computable function as function of polynomial computational complexity, is ther... | https://mathoverflow.net/users/14024 | The counterpart of productive set with polynomial computational complexity | Jie Wang published some papers on the topic, see Polynomial time productivity, approximations, and levelability. SIAM Journal on Computing, 21:1100-1111, 1992 and On p-creative sets and p-completely creative sets. Theoretical Computer Science, 85:1-31, 1991.
| 3 | https://mathoverflow.net/users/509457 | 451621 | 181,585 |
https://mathoverflow.net/questions/451610 | 3 | I'm reading the documentation of this [package: Manopt](https://manoptjl.org/v0.1/manifolds/hyperbolic/), and they claim that in the hyperboloid model for $\mathbb{H}^d$ the parallel transport between tangent spaces $T\_x$ and $T\_y$ is given for any $u\in T\_x$ by
$$
P\_{x\mapsto y}:\,
u
\mapsto
u
-
\frac{
\l... | https://mathoverflow.net/users/36886 | Reference: parallel transport in the hyperboloid model | I doubt you will find this in a modern "peer-reviewed" work: deriving the formula is suitable as a homework exercise for a course in semi-Riemannian geometry.
Here's a sketch of proof:
1. The hyperboloid model realizes $\mathbb{H}^n$ as a hypersurface in Minkowski space $\mathbb{R}^{1,n}$. This hypersurface is tota... | 7 | https://mathoverflow.net/users/3948 | 451622 | 181,586 |
https://mathoverflow.net/questions/451617 | 1 | I guess the chances are slim but still curious about the integral in the title.
Let $f : [0, \infty) \to \mathbb{R}$ be a locally "square-integrable" function on $[0,\infty)$.
Then, for any $\epsilon \in (0,1)$, is it possible to estimate the following integral?:
\begin{equation}
\int\_{\epsilon}^1 \Bigl\lvert \int... | https://mathoverflow.net/users/56524 | Estimating the integral $\int_{\epsilon}^1 \Bigl\lvert \int_0^x \frac{f(y)}{\lvert x-y\rvert^{1/2}} dy\Bigr\rvert^2 dx$ for $L^2$ function $f(y)$? | First, we can observe that your integral depends solely on the behavior of $f$ on the interval $[0, 1]$. Its values outside that region do not affect the expression. So we may multiply by the cutoff function $\chi\_{[0, 1]}$.
Thus we may consider this problem for elements of $L^2(\mathbb{R})$ with compact support.
... | 3 | https://mathoverflow.net/users/508939 | 451626 | 181,587 |
https://mathoverflow.net/questions/451595 | 1 | I ended up needing the following statement, which seems true but to which I couldn't find a proof nor a counterexample. Let $q \in [4/3, 2]$ and $\lambda \in \mathbb{R}^n$ with $\|\lambda\|\_q = 1$. Is it always possible to find an index decomposition $I \subseteq [n]$ such that both the following hold ($\lesssim$ mean... | https://mathoverflow.net/users/186347 | Decomposition of a vector in parts with bounded $\ell^1$ and $\ell^2$ norms | $\newcommand\la\lambda\newcommand\La\Lambda\newcommand{\R}{\mathbb R}\newcommand{\cc}{\mathsf c}$As suggested by [Aleksei Kulikov](https://mathoverflow.net/questions/451595/decomposition-of-a-vector-in-parts-with-bounded-ell1-and-ell2-norms#comment1167553_451595), your desired statement does not hold in general.
Inde... | 2 | https://mathoverflow.net/users/36721 | 451632 | 181,588 |
https://mathoverflow.net/questions/451291 | 4 | Is there a Hamiltonian $H:\mathbb{R}^{2n} \to \mathbb{R}$ with the following property:
For two regular values $a<b$ for which $[a,b]$ consists of regular values, the dynamics of $X\_H$ on $H^{-1}(a)$ is not topological equivalent to the dynamic of $X\_H$ on $H^{-1}(b)$?
Here $X\_H$ is the hamiltonian vector field a... | https://mathoverflow.net/users/36688 | Dynamical analogue of Morse theory | This is again with the caveat that I have not done the calculations. Let $a>1$ be an irrational number. Let $H:\mathbb{R}^4\rightarrow \mathbb R$ be a function such that
$
H(x,y,z,w)=(x^2+y^2+z^2+w^2)
$
when $x^2+y^2+z^2+w^2=1$ and
$
H(x,y,z,w)=x^2+y^2+z^2+aw^2
$
when $x^2+y^2+z^2+aw^2=2$. You can convince your... | 4 | https://mathoverflow.net/users/12156 | 451638 | 181,592 |
https://mathoverflow.net/questions/451576 | 4 | Let the structure $\mathfrak{A} = (A, R\_1, ..., R\_n)$ be strongly acceptable iff $\mathfrak{A}$ is an acceptable structure (in the sense of Moschovakis' *Elementary Induction on Abstract Structures*), $\mathfrak{A}$ has a countable domain, there is isomorphic copy of the natural numbers $\mathcal{N}^{\mathfrak{A}}$ "... | https://mathoverflow.net/users/509398 | Let $\pi$ be a $ℍ_$-recursive projection of $ℍ_$ into . What does $ℍ_{(, Domain(\pi))}$ contain? | The following result will immediately give the answer to your second question:
>
> **Proposition.** Let $\mathfrak{M}$ be strongly acceptable and $\pi\colon \mathrm{HYP}\_\mathfrak{M}\to\mathfrak{M}$ be a $\mathrm{HYP}\_\mathfrak{M}$-recursive projection of domain $D\_\pi$. Then $\mathrm{HYP}\_{\mathfrak{M},D\_\pi}... | 3 | https://mathoverflow.net/users/48041 | 451640 | 181,593 |
https://mathoverflow.net/questions/451619 | 1 | Suppose you have a sequence of continuous stochastic processes $X\_N$ with $X\_N(0)=0$, and that $X\_N$ converge weakly on the space of continuous functions, to a stochastic process $X$. Suppose $X\_N$ are so that for all $N$ you sample a random variable $Z$ and then you run the process. Furthermore, for all time $t>0$... | https://mathoverflow.net/users/479223 | Is deterministic evolution preserved under weak converge of stochastic processes? | Following the discussion in the comments, the statement is not true, even if we stipulate that we only want $X$ to have the desired properties with respect to its own natural filtration.
To see this, let $X\_t$ be the process that is identically $0$ up to time $1$ and from then on is equal to either $t - 1$ or $-t + ... | 3 | https://mathoverflow.net/users/173490 | 451655 | 181,595 |
https://mathoverflow.net/questions/451649 | 11 | Let $k$ be a field with algebraic closure $\bar{k}$.
Recall that a *gerbe* over $k$ is an algebraic stack $\mathcal{G}$ over $k$ such that the groupoid $\mathcal{G}(\bar{k})$ is connected. We say that $\mathcal{G}$ is *neutral* if $\mathcal{G}(k)$ is non-empty.
Now assume that $k = \mathbb{F}\_q$ is a finite field.... | https://mathoverflow.net/users/5101 | Gerbes over finite fields | Suppose that $x$ is an object of $\mathcal{G}$ over a finite extension $k'/k$. Denote $\sigma : k' \to k'$ the $q$-power frobenius. Let $x^\sigma$ be the pullback of of $x$ by $\sigma$. As $\mathcal{G}$ is a gerbe over $k$, after replacing $k'$ by a further finite extension, we may assume there is an isomorphism $\alph... | 15 | https://mathoverflow.net/users/152991 | 451659 | 181,596 |
https://mathoverflow.net/questions/451631 | 1 | I would like to know if the following differential operator on $(0,\infty)$ is well-known or derived from such one:
\begin{align}
L := \frac{1}{2}\frac{d^{2}}{dx^{2}} - a x^{b} \frac{d}{dx} \quad (a,b > 0).
\end{align}
I also would like to know if the eigenfunctions are represented by some special functions.
Thank you ... | https://mathoverflow.net/users/509468 | Is a differential operator $\frac{1}{2}\frac{d^{2}}{dx^{2}} - a x^{b} \frac{d}{dx}$ well-known? | *Comments*
With Maple, solving $Lf = \lambda f$,
$\bullet\;$ In case $b=1$ we get Kummer functions $M$ and $U$.
$$
f \left( x \right) =C\_1\,{{\rm M}\left({\frac {a+\lambda}{2\,a}
},\,{\frac{3}{2}},\,a{x}^{2}\right)}x+C\_2\,x{{\rm U}\left({
\frac {a+\lambda}{2\,a}},\,{\frac{3}{2}},\,a{x}^{2}\right)}
$$
$\bull... | 2 | https://mathoverflow.net/users/454 | 451661 | 181,597 |
https://mathoverflow.net/questions/451507 | 5 | Let $\mathcal{C}$ be a symmetric monoidal category. One can imagine a theorem
*Tannakian reconstruction:* If $\mathcal{B}$ is a braided monoidal category and $F:\mathcal{B}\to \mathcal{C}$ is a functor of ~~braided~~ monoidal categories (with [???] conditions), then $\mathcal{B}\simeq A\text{-Mod}$ for a quasitriangu... | https://mathoverflow.net/users/119012 | Tannakian reconstruction for braided categories | Beware that the forgetful functor on the category of representations of a quasi-triangular Hopf algebra is typically **not** a braided functor. Rather, the general pattern is that you can reconstruct a bialgebra from a monoidal functor, and then additional structures/properties of the category you started with induces ... | 2 | https://mathoverflow.net/users/13552 | 451662 | 181,598 |
https://mathoverflow.net/questions/451665 | 7 | I have been lead to believe, due to various conversations and presentations, that there is a standard notion of an enriched 2-category (indeed, even an enriched n-category). However, after searching I can find no reference for such a construction (although I did find [a paper](https://arxiv.org/pdf/2205.12235.pdf) wher... | https://mathoverflow.net/users/137577 | Enriched 2-categories | There is a standard notion of a [bicategory](https://ncatlab.org/nlab/show/bicategory) [enriched](https://ncatlab.org/nlab/show/enriched+bicategory) in a [monoidal bicategory](https://ncatlab.org/nlab/show/monoidal+bicategory), which is a categorification of the notion of category enriched in a monoidal category. The s... | 8 | https://mathoverflow.net/users/152679 | 451668 | 181,601 |
https://mathoverflow.net/questions/451657 | 15 | I am looking for an analytic function $F: \mathbb{R} \rightarrow (0,\infty)$ with $\int\_{\mathbb{R}} F(x) \, dx = 1$ and the property, that $\sum\limits\_{k=0}^{\infty} |c\_k| \varepsilon^k (2k)! < \infty$ for some $\varepsilon > 0$, if $\sum\limits\_{k=0}^{\infty} c\_k x^k$ is the series representation of $F(x)$ arou... | https://mathoverflow.net/users/409412 | A kernel 'more analytic' than $\exp(-x^2)$ | $\newcommand{\eps}{\varepsilon}$
I will show that if $\sum |c\_n| C^n n! < \infty$ for all $C > 0$ then there's no such function $F$ (if we only know it for some fixed $C$ then there are such functions). Note that this condition is equivalent to having a minimal exponential type. The proof is based on the following v... | 12 | https://mathoverflow.net/users/104330 | 451688 | 181,612 |
https://mathoverflow.net/questions/451667 | 4 | Let $A$ be an abelian variety over a field $k$ with group operation $m\colon A\times A\to A$, and let $A'$ be the dual abelian variety. I know that $A'(k)$ is isomorphic to the subgroup $\operatorname{Pic}^0(A)$ of $\operatorname{Pic}(A)$ composed of the line bundles $\mathscr{L}$ satisfying $m^\*\mathscr{L}\simeq \mat... | https://mathoverflow.net/users/131975 | If $A$ is an abelian variety over $k$ and $S$ is a $k$-scheme, what's $A'(S)$ geometrically? | Let $f:A \to \operatorname{Spec}k$ be your abelian variety and let $P:=\operatorname{Pic}\_{A/k}$ denote the relative Picard functor of $A/k$, i.e. the functor $(Sch/k)^{op} \to Sets$ taking $S$ to $\operatorname{Pic}(A\_S)/\operatorname{Pic}(S)$. This functor is an fppf sheaf and coincides with the rigidified Picard f... | 3 | https://mathoverflow.net/users/158721 | 451690 | 181,613 |
https://mathoverflow.net/questions/451656 | 5 | I was reading *The symplectic Floer homology of a Dehn twist* by P. Seidel, which you can find [here](https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1996/0003/0006/MRL-1996-0003-0006-a010.pdf).
In Lemma 3(ii) the following topological property of Dehn twists is stated without proof:
>
> Let $\Sig... | https://mathoverflow.net/users/509477 | Reference for a property of Dehn twists | The [Primer](https://press.princeton.edu/books/hardcover/9780691147949/a-primer-on-mapping-class-groups-pms-49) has "model proofs" for various steps of the proof.
[Bleiler's notes on Casson's lectures](https://www.cambridge.org/core/books/automorphisms-of-surfaces-after-nielsen-and-thurston/2AD58B246E36B971CCB92BD5B923... | 4 | https://mathoverflow.net/users/1650 | 451699 | 181,617 |
https://mathoverflow.net/questions/451703 | 10 | Let $x\_1,x\_2,\ldots, x\_k \in [0,1]$ be irrational numbers.
I'm interested in what, if anything, can be said about the values $\{nx\_i \bmod 1: n\in \mathbb{N}\}$.
Specifically, I'm interested if there are constraints that we can place on the $x\_i$'s such that for any set of $y\_i$'s and any $\varepsilon \in (0,1/... | https://mathoverflow.net/users/155604 | Continuous variant of the Chinese remainder theorem | Your guess is correct. For $\alpha = (\alpha\_1, \alpha\_2, \ldots, \alpha\_d) \in \mathbb{R}^d$, the values of $m \alpha \bmod 1$ are equidistributed (and in particular dense) in $(\mathbb{R}/\mathbb{Z})^d$ if and only if, for all nonzero $k \in \mathbb{Z}^d$, we have $k \cdot \alpha \not\in \mathbb{Z}$. See, for exam... | 13 | https://mathoverflow.net/users/297 | 451704 | 181,618 |
https://mathoverflow.net/questions/451710 | 3 | Let $G$ be a finite group and $K$ a field with field extension $L$ ($K$ perfect and $L$ finite field extension first for simplicity),
Let $S$ be a simple $KG$ module.
Viewed as a $LG$-module $S$ decomposes as a direct sum of simple $LG$-modules $L\_l$.
>
> Question 1: Let $i \geq 1$ be fixed. Do we have $Ext\_{KG}^... | https://mathoverflow.net/users/61949 | Extensions for simple modules over group algebras | Let $G$ be the alternating group $A\_4$, let $K=\mathbb{F}\_2$, and $L=\mathbb{F}\_4$. Then there are two non-trivial one dimensional simple $LG$-modules, say $S\_1$ and $S\_2$. These lie over a single two dimensional $KG$-module $S$, so extending the field on $S$ gives $S\_1\oplus S\_2$. Let $i=1$. Neither $S\_1$ nor ... | 3 | https://mathoverflow.net/users/460592 | 451713 | 181,619 |
https://mathoverflow.net/questions/451716 | 2 | Let $G$ be a connected Lie group. Let $\Gamma$ a lattice in $G$ not necessarily uniform (cocompact). Is it true that $\Gamma$ is finitely generated?
| https://mathoverflow.net/users/509535 | About finitely generated lattices in Lie groups | J. Lie Theory 30 (2020), no. 1, 33–40, arXiv:1903.04828 Gelander and Slutsky, "On the minimal size of a generating set for lattices in Lie groups", Corollary 1.6 says yes, and says it is "well known but non-trivial".
| 2 | https://mathoverflow.net/users/460592 | 451717 | 181,620 |
https://mathoverflow.net/questions/451719 | 2 | Inspired by [An algebraic Hamiltonian vector field with a finite number of periodic orbits (2)](https://mathoverflow.net/questions/210996/an-algebraic-hamiltonian-vector-field-with-a-finite-number-of-periodic-orbits-2) we ask if there is a 1 dimensional analytic foliation of $\mathbb{R}^4$ which has at least 1 compact ... | https://mathoverflow.net/users/36688 | A 1 dimensional foliation of $\mathbb{R}^4$ with few compact leaves | Yes, there is. Conceptually, imagine a flow on $\mathbb{R}^3$ with a single closed orbit on the unit circle in the plane $z=0$, while every other trajectory has an increasing z-coordinate. It's then easy to embed that flow in $\mathbb{R}^4$ to get the foliation you want.
More precisely, consider the vector field $$f(... | 2 | https://mathoverflow.net/users/1227 | 451720 | 181,621 |
https://mathoverflow.net/questions/451714 | 3 | Let $X=\mathbb{R}$, and $\mathcal{A}:=\mathbb{R}[x]$ be the subalgebra (of $C(X)$) of univariate polynomials.
Given $\varphi\in C\_b(X)$ and $K\subset X$ compact, we know from Stone-Weierstrass that
$$\tag{1} \forall\,\varepsilon>0 \, : \ \exists\, p\_\varepsilon\in\mathcal{A} \quad\text{such that}\quad \sup\nolimi... | https://mathoverflow.net/users/160714 | Global control of locally approximating polynomial in Stone-Weierstrass? | Yes, this works. Let's say $K\subseteq [-1,1]$. Start out by finding a polynomial $q$ such that $\varphi-\epsilon/2 \le q \le \varphi-\epsilon/4$ on $[-3,3]$. We can then take $p(x)=q(x)-(\epsilon/4)(x/2)^{2N}$. This will approximate $\varphi$ on $K$ with the desired accuracy for any choice of $N$, and also $p\le q\le\... | 4 | https://mathoverflow.net/users/48839 | 451722 | 181,622 |
https://mathoverflow.net/questions/451707 | 4 | Let $\mathcal{C}$ be a (sufficiently complete and cocomplete) closed monoidal category with internal hom $[-,-]$. Let $F : \mathcal{A} \to \mathcal{C}$ be a functor obtained as the left Kan extension of $F' : \mathcal{B} \to \mathcal{C}$ along a fully faithful functor $i : \mathcal{B} \to \mathcal{A}$. If $G : \mathcal... | https://mathoverflow.net/users/322094 | Relationship between Kan extensions and internal hom | The analogous adjunction for the enriched category of functors can be deduced from the adjunction for the ordinary category of functors using the Yoneda lemma.
Indeed, to establish a natural isomorphism
$$[F,G]≅[F',G∘i]$$
it suffices (by the Yoneda lemma) to establish an isomorphism
$$\def\cC{{\cal C}}\cC(V,[F,G])≅\cC(... | 3 | https://mathoverflow.net/users/402 | 451723 | 181,623 |
https://mathoverflow.net/questions/451693 | 3 | Given $S = \{1, 2, \ldots, n\}$, consider partitions of $S$ of the form $(R, R')$ where $R \subset S$ and $R'$ is $S \setminus R$, the complement of $R$ in $S$. The goal is to list 2-part partitions of $S$ such that
1. the sequence $(Q, Q'), (R, R')$ is allowed in the list if either $Q' \subset R$ or $R \subset Q'$;
... | https://mathoverflow.net/users/14807 | Finding an inclusion-based path through 2-part set partitions | Here is a construction using [Gray codes](https://en.wikipedia.org/wiki/Gray_code)
Run the standard $n-1$st Gray code backwards, getting all subsets $S\_1$, ..., $S\_{2^{n-1}}=\varnothing$ of $\{1,...,n-1\}$.
Replace each odd-numbered subset with its complement in $\{1,...,n\}$.
1. and 3. are straightforward.
2. ... | 3 | https://mathoverflow.net/users/41291 | 451725 | 181,625 |
https://mathoverflow.net/questions/451729 | 9 | For the presheaf topos $\mathrm{PSh}(C)$, the subobject classifier is the presheaf $\Omega$ such that
* For $c \in C$, $\Omega(c)$ is the set of all subobjects of the functor $\mathrm{Hom}(-, c)$
* For $f: c \to c'$, $\Omega(f)(F) = \{g: a \to c | gf \in F\}$, where a subfunctor $F < \mathrm{Hom}(-, c')$, and the def... | https://mathoverflow.net/users/148161 | What is known about the homotopy type of the classifier of subobjects of simplicial sets? | It’s not hard to check that the subobject classifier $\Omega$ is trivially fibrant, and so its homotopy type is trivial. Orthogonality of $\Omega \to 1$ against a map $i : A \to B$ corresponds to the property that any subobject $A' \to A$ extends to a subobject $B' \to B$ such that $i^\*B' = A'$; when $i$ is mono, this... | 19 | https://mathoverflow.net/users/2273 | 451730 | 181,626 |
https://mathoverflow.net/questions/451735 | 6 | This question can be phrased in different settings. I will discuss a spectral formulation and the equivalent random walk version. The question came up naturally in [recent work](https://arxiv.org/abs/2302.06021) with Devriendt and Ottolini. We have no particular reason to think the inequality is true except that it wou... | https://mathoverflow.net/users/138664 | Diameter bound for graphs: spectral and random walk versions | The conjectured inequality is false in General.
---
Proof: Let $\ell=\lfloor n^{1/2}/2 \rfloor$ and let
$G\_1,G\_2,\dots,G\_\ell$ be disjoint cliques of size $\ell$. Let $K$ be a clique on $n-\ell^2 >n/2$ nodes. Connect every node in $G\_i$ to every node in $G\_{i+1}$ for $i<\ell$, and connect every node in $G\_\... | 6 | https://mathoverflow.net/users/7691 | 451747 | 181,630 |
https://mathoverflow.net/questions/451753 | 3 | Let $X$ be a centered semimartingale that has continuous sample paths almost surely. Is it then true that $X$ is a continuous semimartingale? Meaning that $X$ has a decomposition $X=M+V$ where $M$ is a continuous local martingale and $V$ is a continuous finite variation process?
| https://mathoverflow.net/users/479223 | Is a semimartingale that is continuous a continuous semimartingale? | I found an answer - the answer is yes. Rogers and Williams page 358.
| 3 | https://mathoverflow.net/users/479223 | 451755 | 181,633 |
https://mathoverflow.net/questions/451750 | 2 | Suppose we have an elliptic curve $E$ over $K$, an $l$-adic field. Say that $|j(E)|>1$ where $|.|$ is the $l$-adic valuation. By the theory of the Tate curve $E$ is isomorphic over $L$ to a Tate curve $E\_q$, where $L/K$ is a field extension of $K$ of degree at most 2.
In the case $[L:K]=2$, then $E(L)\cong E\_q(L) \... | https://mathoverflow.net/users/478525 | Twist of the Tate Curve | Chris provided a reference, but for those who don't have a copy of the book:
$$
E(K) \cong \bigl\{ u\in L^\*/q^{\mathbb Z} : \operatorname{\textsf{Norm}\_{L/K}}(u) \in q^{\mathbb Z}/q^{2\mathbb Z} \bigr\}.
$$
| 5 | https://mathoverflow.net/users/11926 | 451756 | 181,634 |
https://mathoverflow.net/questions/448774 | 10 | Let the [trace norm](https://en.wikipedia.org/wiki/Matrix_norm#Schatten_norms) of $X$ be
$$\Vert X\Vert\_1 := \operatorname{tr} \left(\,(X^\dagger X)^{1/2}\right)$$
and let the operator inequality $A \leq B$ denote that the operator $B-A$ is positive semidefinite.
---
If the **quantum states** (finite-dimensi... | https://mathoverflow.net/users/51613 | Does approximate equality of quantum states imply operator inequality in a large subspace? | Let $\sigma$ be represented by a PD matrix $A$ and $\rho$ by $A+B$. Note that $|B|\ge B$ (in the sense of PSD matrices) and has the same $1$-norm. Also $\Pi |B|\Pi\ge \Pi B \Pi$, so to dominate $\Pi B\Pi $ is always easier than to dominate $\Pi|B|\Pi$. Thus, we can replace $B$ by $|B|$ and assume that $B$ is PD as well... | 6 | https://mathoverflow.net/users/1131 | 451757 | 181,635 |
https://mathoverflow.net/questions/451764 | 1 | I'm having a tough time on this problem, suppose we have $v\_i$ for $i=1\cdots n$ unit vectors sampled from a $d$-dimensional hypersfere. How can we evaluate this average over these vectors?
$$
\mathbb{E}\_v\frac{1}{n^4}\left(\sum\_{i,j=1}^{n,n}(v\_i\cdot v\_j)^2\right)^2=?
$$
I was able to determine that
$$
\mathbb{E}... | https://mathoverflow.net/users/58126 | Average of cosine similarity between $n$-vectors sampled from $d$-dim hypersphere | **Hint**: Note that the sum under the expectation can be written as $$\sum\_{i,j,k,l}\frac{(z\_i.z\_j)^2(z\_k.z\_l)^2}{\|z\_i\|^2\|z\_j\|^2\|z\_k\|^2\|z\_l\|^2},$$ where $z\_i$ are standard normal random vectors in $R^d$. You have to now consider the cases where $2,3$ or $4$ of these indices are same, and the correspon... | 3 | https://mathoverflow.net/users/64194 | 451768 | 181,637 |
https://mathoverflow.net/questions/451770 | 1 | Let $\mathbb{Q}\_{p}$ be a p-adic field such that $ p \neq 2 $. We knew that for every $ n=2m $ there exists exactly one unramified extension $ K $ of $ \mathbb{Q}\_{p} $ of degree $ n $, obtained by adjoining $ (p^n -1)$-th roots of unity to $ \mathbb{Q}\_{p}$
. Moreover $ Gal(K / \mathbb{Q}\_p) $ is isomorphic to $Ga... | https://mathoverflow.net/users/215016 | Unramified extension over $ \mathbb{Q}_{p} $ | You did not specify $f$: it should be equal to $n$. A particular generator of the resulting cyclic Galois group is the Frobenius automorphism, which acts on the $(p^n−1)$-th roots of unity by $x\mapsto x^p$. The quadratic subextension is generated by the $(p^2-1)$-th roots of unity. As $p^2-1$ is divisible by $8$, this... | 3 | https://mathoverflow.net/users/11919 | 451771 | 181,638 |
https://mathoverflow.net/questions/451774 | 3 | I have a soft question regarding the Jacobian of vector fields and isolated equilibria, and what they imply about local behavior of nearby integral curves near.
Let $V:U \subset\_{open} \mathbb{R}^n \to \mathbb{R}^n$ be a smooth vector field. Let $x^\* \in U$ be an isolated equilibrium of $V$. That is, $V(x^\*)=0$ an... | https://mathoverflow.net/users/141449 | What does the Jacobian of a vector field at an equilibrium tell you about local behavior of integral curves when the Jacobian is not a stable? | The Jacobian alone doesn't have the information you need. For example, consider the two vector fields
$$f(x, y) = (-x^2, -y)$$
and
$$g(x, y) = (-x^3, -y)$$.
They have an isolated equilibrium at the origin, with the same Jacobian: $\begin{bmatrix}
0 & 0 \\
0 & -1
\end{bmatrix}$.
But the equilibrium in $f$ isn't st... | 7 | https://mathoverflow.net/users/1227 | 451776 | 181,639 |
https://mathoverflow.net/questions/451778 | 6 | Is the following statement true?
Let $N$ be sufficiently large, and choose $t$ uniformly randomly in $\{1,2,\ldots,N\}$. Then
$$\Pr[\gcd(t, N)>N^{3/4}] < N^{-1/16}.$$
This is the "dual" question to
[Lower bounding the probability that $\gcd(t,N)≤B$, for a random $t$ and fixed (large) $N$](https://mathoverflow.net/q... | https://mathoverflow.net/users/155604 | Probability of large gcd | The statement is true in the stronger form that
$$\Pr[\gcd(t, N)>N^{3/4}] < N^{-1/2}.$$
Indeed, the probability that $\gcd(t,N)$ equals a given $k\mid N$ is at most $1/k$. For $k>N^{3/4}$, this is less than $N^{-3/4}$, hence the probability in question is less than $d^\*(N)N^{-3/4}$, where $d^\*(N)$ is the number of di... | 7 | https://mathoverflow.net/users/11919 | 451779 | 181,641 |
https://mathoverflow.net/questions/451629 | 0 | Is there a formal name for the point which is the reflection of the incenter about the circumcenter of a triangle?
| https://mathoverflow.net/users/265714 | Name this geometric point? | The reflection of the incenter about the circumcenter is the *Bevan point* of the triangle. Quoting from the [mathworld](https://mathworld.wolfram.com/BevanPoint.html) page:
>
> The Bevan point $V$ of a triangle $\triangle ABC$ is the circumcenter of the [excentral triangle](https://mathworld.wolfram.com/ExcentralT... | 2 | https://mathoverflow.net/users/83134 | 451782 | 181,644 |
https://mathoverflow.net/questions/451488 | 6 | I want to calculate the integral defined as
$$
P(s)=\iint \mathrm dx \, \mathrm dy\ \ \delta\left(\frac{(x+y)^2+4x^2y^2}{(x+y)^2+(x+y)^4}-s \right).
$$
The integration is taken within the rectangle $-a\le x,y\le a$. All I know is that $P(0)=0$. Is it possible to explicitly carry out this integral?
| https://mathoverflow.net/users/482984 | Integral of the $\delta$ function | Let's see how far we can get with the original problem. First, note that, since $x^2 y^2= \frac{1}{16} [(x+y)^4 + (x-y)^4 -2 (x+y)^2 (x-y)^2]$, the integrand is symmetric under reflections with respect to the $y=x$ axis and the $y=-x$ axis. Therefore, we can restrict the integration to the region $x+y \geq 0$, $x-y \le... | 3 | https://mathoverflow.net/users/134299 | 451795 | 181,646 |
https://mathoverflow.net/questions/451524 | 5 | Let $V\subset \mathbb{A}^m$ and $W\subset \mathbb{A}^n$ be affine varieties defined over an arbitrary field. Let $f:V\to \mathbb{A}^n$ be a morphism given by polynomials of degree $\leq D$.
Is it true that
$$\deg(f^{-1}(W)) \leq \deg(V) \deg(W) D^{\dim(\overline{f(V)})}?$$
Here, by "degree", we mean "sum of the deg... | https://mathoverflow.net/users/398 | Degree of the preimage of a variety | [I'm editing my original answer slightly so as to work in affine space, for the sake of clarity, since, in the proof of Lemma 3 in the comments, I am really working in affine space. If one needs degree bounds of the kind I'm giving in projective space (not that I do), one can just deduce them from the bounds in affine ... | 3 | https://mathoverflow.net/users/398 | 451800 | 181,648 |
https://mathoverflow.net/questions/451817 | 2 | Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form
$$ \|\phi\otimes\phi\|\_{\dot{H}^1(D\times D)} \lesssim \|\phi\|\_{\dot{H}^{1/2}(D)}^2. $$
Many thanks.
| https://mathoverflow.net/users/168601 | Controlling the tensor product of functions in $H^1$ with lower derivatives | Assuming the inequality you hoped for is $\| \phi \otimes \phi \|\_{\dot{H}^1} \lesssim \|\phi\|\_{\dot{H}^{1/2}}^2$, I claim that this is still impossible.
Let $D = [0,2\pi]$ again. Set
$$ \phi\_N(x) = \sum\_{k = 1}^N k^{-1/2} \sin(kx) $$
Then
$$ \|\phi\_N\|\_{\dot{H}^{1/2}}^2 \approx \sum\_{k = 1}^N 1 = N $$
But
$$... | 2 | https://mathoverflow.net/users/3948 | 451821 | 181,650 |
https://mathoverflow.net/questions/451823 | 9 | Let: $\rho$ be a non-trivial zero of the Riemann zeta function, $\Lambda$ be the von-Mangoldt function and $\psi(x) =\sum\_{n \leq x} \Lambda(n)$. What is the best known upper bound for
$$f(x, T) := \psi(x) - x + \sum\_{|\rho| \leq T} \frac{x^{\rho}}{\rho}$$
for $\frac{T}{\log T} \leq x \leq T$, with or without the... | https://mathoverflow.net/users/507786 | On the error term of the Riemann explicit formula | Without any restriction on $x$ and $T$ (aside from $x,T \ge 2$) one has
$$f(x,T) \ll \frac{x}{T}\log^2 x + \log x,$$
which goes back to Landau. A modern reference is Theorem 12.5 in Montgomery and Vaughan's book (p. 400). By using the Brun-Titchmarsh inequality when bounding the error incurred from Perron one can do be... | 19 | https://mathoverflow.net/users/31469 | 451825 | 181,651 |
https://mathoverflow.net/questions/451804 | 0 | Is there a probability distribution $\mu$ (with reasonably nice density $f$ on $\mathbb{R}$) such that the Fourier transform (aka. characteristic function) $\psi\_\mu(t) = \int\_{\mathbb{R}} e^{itx} \, d\mu(x) = \int\_{\mathbb{R}} e^{itx} f(x) \, dx$ converges to zero for $t \rightarrow \pm\infty$ faster than $\exp(-ct... | https://mathoverflow.net/users/409412 | A probability distribution, with Fourier transform smaller than $C \exp(-ct^2)$ | Yes. E.g., if $f(x)=\dfrac{1-\cos x}{\pi x^2}$ for real $x\ne0$, then $\psi\_\mu(t)=\max(0,1-|t|)$ for real $t$ (which latter $=0$ if $|t|\ge1$).
| 1 | https://mathoverflow.net/users/36721 | 451826 | 181,652 |
https://mathoverflow.net/questions/451833 | 2 | Let $G = (V,E)$ be a simple, undirected graph. We say that $v\neq w\in V$ *lie on a common cycle* if there is an integer $n\geq 3$ and an injective graph homomorphism $f: C\_n\to V$ such that $v,w\in \text{im}(f)$.
We call a function $p:\omega\to V(G)$ an $\omega$-*walk* if for all $n\in \omega$ we have $\{p(n),p(n+1... | https://mathoverflow.net/users/8628 | Does $(\omega, E)$ with the cycle condition have an $\omega$-path? | The answer to both questions in negative.
Let $G$ be the graph on $\omega$ with $nE0$ and $nE1$ for all $n\geq 2$ (and no other edges). Clearly all $m\neq n$ with $n,m\geq 2$ are on a cycle of length $4$, given by $nE0EmE1En$. However any $\omega$-walk must go through either $0$ or $1$ infinitely many times.
| 4 | https://mathoverflow.net/users/49381 | 451835 | 181,654 |
https://mathoverflow.net/questions/451465 | 2 | It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial\_t \rho = \partial\_x \left(\rho\_\infty \partial\_x\left(\frac{\rho}{\rho\_\infty}\right)\right) \label{1}\tag{1}$$ where $\rho\_\infty$ (say for instance of the form $\rho\_\infty(x) \propto \mathrm{e}^{... | https://mathoverflow.net/users/163454 | Reference request: analysis of a nonlinear Fokker-Planck type equation | Let me rewrite the equation (3) as
$$
\partial\_t \rho = \partial\_x \left(\rho\, \partial\_x\log\left(\frac{\rho}{\mathcal{F}[\rho]}\right)\right) \label{3'}\tag{3'}.
$$
Then, this equation is a gradient flow with respect to the metric tensor after Otto inducing the Wasserstein distance iff there exists a driving free... | 2 | https://mathoverflow.net/users/13400 | 451846 | 181,658 |
https://mathoverflow.net/questions/451751 | 8 | Humphreys conjecture describes the support variety of tilting modules using the correspondence between two-sided cells and nilpotent orbits. But the support variety can also be given by Lusztig–Vogan bijection, as in the work [Conjectures on tilting modules and antispherical $p$-cells](https://arxiv.org/abs/1812.09960)... | https://mathoverflow.net/users/509570 | What is the remaining difficulty in the proof of the Humphreys conjecture (on the support variety of tilting modules)? | The paper [Silting complexes of coherent sheaves and the Humphreys conjecture](https://arxiv.org/abs/2106.04268) by Achar and Hardesty proves this conjecture in full generality (for $p \ge h$).
| 7 | https://mathoverflow.net/users/919 | 451847 | 181,659 |
https://mathoverflow.net/questions/451844 | 0 | Given a number $n$ and an Interval $I = [ \; \lfloor n^{1/4} \rfloor, \lfloor n^{(1/3) \rfloor \;} ]$, can we say anything about the distribution of $\{ n \mod b \;\;| \; b \in I \}$?
1. In particular, if I wanted the residue to be close to any region in $[0, b-1]$, say close to the "top" of the residue classes aroun... | https://mathoverflow.net/users/63939 | Residues distribution modulo an interval | Note that $n \equiv b-1 \pmod b$ implies $b | n+1$, $n \equiv b-2 \pmod b$ implies $b | n+2$, and so on. Therefore you can factor $n+1$, $n+2$, and so on, until one of them has a factor in $I$. You can do something similar for other target regions.
While I don't have a proof, empirically it seems like you only need t... | 1 | https://mathoverflow.net/users/498835 | 451853 | 181,661 |
https://mathoverflow.net/questions/451860 | 2 | Consider the continuous and injective mapping
\begin{eqnarray\*}
\varphi:[0,1] &\rightarrow& \mathbb{R}^2, \\
t &\mapsto& (x(t),y(t)),
\end{eqnarray\*}
such that $x(0)<x(1)$, and
\begin{equation\*}
\big( (x(t)-x(s)\big)\big( y(t)-y(s)\big) \ge 0,\quad \forall t,s\in [0,1].
\end{equation\*}
My intuition is that $x(... | https://mathoverflow.net/users/488929 | A continuous injection from $[0,1]$ to $\mathbb{R}^2$ | If $x(t\_0)\notin [x(0),x(1)]$ for some $t\_0\in (0,1)$, then for a small positive $c$ we have $z(t\_0)\notin [z(0),z(1)]$ where $z=x+cy$. Thus $z$ is not monotone, therefore not injective. But two points with the same value of $z$ contradict to your second assumption (it is crucial here that $(x, y)$ is injective).
| 6 | https://mathoverflow.net/users/4312 | 451862 | 181,663 |
https://mathoverflow.net/questions/451834 | 3 | Denote the (unsigned) [*Stirling numbers of the $1^{st}$-kind*](https://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind) by ${n \brack k}$ and define
$$\mathbf{F}\_a(q)=\sum\_{m\geq1}\frac{q^{am}}{(1-q^m)^{2a}} \qquad \text{and} \qquad
\mathbf{G}\_b(q)=\sum\_{m\geq1}\frac{m^bq^m}{1-q^m}.$$
I have encountered... | https://mathoverflow.net/users/66131 | $q$-series and Stirling of the 1st kind |
>
> **QUESTION.** Is this true? Or, can you provide a reference to it.
> $$\mathbf{F}\_a(q)=\frac1{(2a-1)!}\mathbf{G}(\mathbf{G}^2-1^2)(\mathbf{G}^2-2^2)(\mathbf{G}^2-3^2)\cdots(\mathbf{G}^2-(a-1)^2);$$
> where we adopt an umbral notation $\mathbf{G}^j$ to stand for $\mathbf{G}\_j$ and multiply accordingly.
>
>
>
... | 2 | https://mathoverflow.net/users/46140 | 451865 | 181,664 |
https://mathoverflow.net/questions/451864 | 8 | I posted the following question on [MSE](https://math.stackexchange.com/questions/4734216/direct-proof-of-unique-invariant-distribution-for-ergodic-positive-recurrent-ma), feeling that it perhaps isn't *research level* mathematics, but didn't get any bites. So, I am crossposting here.
The following *ergodic theorem* ... | https://mathoverflow.net/users/509687 | Direct proof of unique invariant distribution for ergodic, positive-recurrent Markov chain | Yes, uniqueness can be proved without appealing to probabilistic arguments.
Generally speaking, one can study properties of Markov chains by arguments from functional analysis and operator theory, since Markov chains can be described by positive operators on appropriately chosen ordered Banach spaces.
Here's an arg... | 14 | https://mathoverflow.net/users/102946 | 451868 | 181,665 |
https://mathoverflow.net/questions/451870 | 6 | Given that $F$ is a field, let $F\_n$ be the completion of $F$ with respect to roots of degree $n$ polynomials. For example this would make $\mathbb{Q}\_2$ the field of (ruler and compass) constructible numbers. Clearly $F\_n \subseteq F\_m$ whenever $n \leq m$ as any degree $n$ polynomial can be multiplied by $x^{m-n}... | https://mathoverflow.net/users/120665 | If degree $N$ polynomials always have a root, does there still exist an irreducible polynomial of degree $N+1$? | The answer to the first question is yes: For $n\ge5$ let $L$ be a Galois extension of $\mathbb Q$ with Galois group the alternating group $A\_n$. Suppose that $L$ is contained in the composition $E$ of (w.l.o.g. finitely many) splitting fields of polynomials of degree $\le n-1$.
This yields a group $G$ which is a sub... | 4 | https://mathoverflow.net/users/18739 | 451879 | 181,669 |
https://mathoverflow.net/questions/451861 | 3 | Let $H^3$ be the Heisenberg manifold. It is known that the first betti number of $H^3\times S^1$ is odd and therefore it does not support any Kähler metric. Now let $I=(0,1)$ or $I=[0,1]$, does it still hold that $H^3\times I$ admits no Kähler metric?
| https://mathoverflow.net/users/167284 | Does $H^3\times I$ admit a Kähler metric? | Note that $S^3$ embeds in $S^4$ and $S^3$ is the total space of the Euler class one circle bundle over $S^2$. It follows that the Euler class one circle bundle over $\Sigma\_g$ embeds in $S^4$ for all $g \geq 0$, see Proposition $7.2$ of [*Smoothly Embedding Seifert Fibered Spaces in $S^4$*](https://arxiv.org/abs/1810.... | 9 | https://mathoverflow.net/users/21564 | 451881 | 181,670 |
https://mathoverflow.net/questions/451871 | 1 | Let $X$ be a set of $N>0$ elements (with the counting measure) and consider a family of measurable functions $f\_i:X\to [0,1]$, for $i\in \mathbb N$.
Any function $f\_i$ can be seen as a point in the hypercube $[0,1]^N$, therefore by using a mesh of sub-hypercubes (all of the same diameter) it is possible to find **f... | https://mathoverflow.net/users/65980 | Approximating a family of measurable functions | $\newcommand\de\delta\newcommand\N{\mathbb N}$Such an approximation is impossible in such generality.
Indeed, let $f\_i:=(1+r\_i)/2$ for $i\in\mathbb N$, where $(r\_i)\_{i\in\mathbb N}$ is the [Rademacher system](https://en.wikipedia.org/wiki/Rademacher_system) (of real-valued functions on $X:=[0,1]$). Then the $f\_i... | 2 | https://mathoverflow.net/users/36721 | 451888 | 181,671 |
https://mathoverflow.net/questions/293557 | 6 | Let $\mathbf{\Omega}=(\Omega,\mathcal{F},(\mathcal{F}\_t)\_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space satisfying the Usual Conditions.
Let $P \colon [0,\infty) \times \mathbb{R} \times \mathcal{B}(\mathbb{R}) \to [0,1]$ be a family of Markov transition probabilities on $\mathbb{R}$ (that is: $P(t... | https://mathoverflow.net/users/15570 | Does the strong Markov property imply the "really strong Markov" property? | (I'll only do the case in which $S:\Omega\to[0,\infty)$. A truncation argument reduces the general case to this special case.)
Consider a bounded $\mathcal F\otimes
\mathcal B[0,\infty)$ measurable function $F$. Then for $\mathcal F\_\tau$ measurable $S:\Omega\to[0,\infty)$,
$$
\Bbb E[F(\cdot,S) \mid\mathcal F\_\tau]... | 3 | https://mathoverflow.net/users/42851 | 451891 | 181,673 |
https://mathoverflow.net/questions/451869 | 6 | Let $n$ be the positive integer. Let $A$ and $B$ be sets of divisors of $n$ less and more than $\sqrt{n}$ respectively.
Consider bipartite graph $(A, B)$, where two vertices are connected when one divides another. Denote $M(n)$ number of perfect matchings in this graph.
Is $M(n) > 0$ for all $n$(maybe excluding squ... | https://mathoverflow.net/users/509688 | Pair matching between divisors less and more than $\sqrt{N}$ | Here is a proof that $M(n)>0$.
Denote $[\alpha]=\{0,1,\dots,\alpha\}$. All divisors of $n$ correspond, in a natural way, to the points in a parallelepiped $P=[\alpha\_1]\times\dots\times [\alpha\_k]$. For $a=(a\_i), b=(b\_i)\in P$ we write $a\leq b$ if $a\_i\leq b\_i$ for all $i$. Let $S$ and $L$ denote the sets of p... | 6 | https://mathoverflow.net/users/17581 | 451894 | 181,675 |
https://mathoverflow.net/questions/451867 | 4 | Let $\mathbb{G}$ be a discrete quantum group (in the sense of Vaes-Kustermans) with function algebra $(\ell^\infty(\mathbb{G}), \Delta)$. A (right) action of $\mathbb{G}$ on a von Neumann algebra $M$ is an injective, unital, normal $\*$-homomorphism $\alpha: M \to M \otimes \ell^\infty(\mathbb{G})$ such that
$$(\iota \... | https://mathoverflow.net/users/470427 | Examples of discrete quantum group actions on commutative von Neumann algebras | The dual coideals for Podleś sphere algebras are commutative coideal subalgebras of $\ell^\infty \widehat{\mathit{SU}}\_q(2)$. They can be thought as ‘quantized’ $L(T)$ for the 1-dimensional toruses $T < \mathit{SU}(2)$ sitting as coisotropic subgroups.
| 4 | https://mathoverflow.net/users/9942 | 451912 | 181,681 |
https://mathoverflow.net/questions/451913 | 5 | Presumably it was Hilbert who discovered [Hilbert polynomials](https://en.wikipedia.org/wiki/Hilbert_series_and_Hilbert_polynomial) - where did they first appear?
The basic theorem is that for a finitely generated graded module $M = \bigoplus\_k M\_k$ over the ring of polynomials in finitely many variables over a fie... | https://mathoverflow.net/users/1508 | Discovery of Hilbert polynomial | In the 1st chapter of Eisenbud's book that you had mentioned, he discusses four fundamental theorems of Hilbert that appeared in 1890 and 1893 (see p. 26): the basis theorem, the Nullstellensatz, the polynomial nature of Hilbert series, and the syzygy theorem. The history section of the Wikipedia page on the syzygy the... | 11 | https://mathoverflow.net/users/3272 | 451916 | 181,682 |
https://mathoverflow.net/questions/451501 | 3 | Let me start with the following
**Illustration:** Let $G$ be a compact group, and let $\pi:G\to H$ be its (surjective) continuous homomorphism onto a (compact) group $H$. So we can think that $H$ is the quotient group of $G$ modulo the kernel $K=\operatorname{Ker}\pi$ of the mapping $\pi$
$$
H=G/K
$$
(and $\pi$ is ju... | https://mathoverflow.net/users/18943 | Is there an operation in topology analogous to the operation of averaging over a compact subgroup in harmonic analysis? | As stated (i.e., without assuming metrizability of $X$), the answer is negative: Let $Y=\alpha\mathbb N$ be the Alexandrov (one point) compactification, $X=\beta\mathbb N$ the Stone-Cech compactification and $\pi:X \to Y$ the unique continuous extension to $\beta\mathbb N$ of the inclusion $\mathbb N\hookrightarrow\alp... | 6 | https://mathoverflow.net/users/21051 | 451931 | 181,687 |
https://mathoverflow.net/questions/451859 | 6 | In this question, the matrices are square and real and the polynomials have real coefficients, but feel free to mention other fields if that is interesting.
Let $\chi(M)$ denote the characteristic polynomial of a matrix $M$.
For two pairs of matrices, let us write $(A\_1,B\_1)\stackrel\chi\sim (A\_2,B\_2)$ if
$$ \c... | https://mathoverflow.net/users/9025 | Eigenvalues of polynomials of two matrices | The problem of simultaneous conjugation has been studied for more than 50 years. First off, it has been proved that the ring of $GL(n)$-invariants on $M(n)\times M(n)$ or, more generally, on $M(n)^m$ is generated by the coefficients of the characteristic polynomials $\chi(p(A\_1,\ldots,A\_m))$ where $p$ runs through al... | 4 | https://mathoverflow.net/users/89948 | 451952 | 181,693 |
https://mathoverflow.net/questions/451792 | 3 | Can the prime number theorem be obtained from the explicit formula,
$\psi(x)=x-\sum\_{\zeta(\rho)=0}\frac{x^\rho}{\rho}+O(1)$?
Here, $\psi(x)=\sum\_{k=1}^\infty\sum\_{p^k<x}\log p$
| https://mathoverflow.net/users/8435 | Prime number theorem via the explicit formula | Expanding on my comments above, most textbooks will show how the Prime Number Theorem follows from $\psi(x)\sim x$. This does not require the full strength of the Explicit Formula for $\psi(x)$, and most textbooks will prove the PNT before the Explicit Formula. One needs more than just the real part of each $\rho$ is $... | 8 | https://mathoverflow.net/users/6756 | 451958 | 181,694 |
https://mathoverflow.net/questions/451897 | 4 | The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form
$$\||x|^\gamma u\|\_{L^p} \leq C\||x|^\alpha \nabla u\|\_{L^q}^a \||x|^\beta u\|\_{L^r}^{1-a},$$
for any $u \in C\_c^\infty(\mathbb{R}^d)$ (there are constraints on all the paramete... | https://mathoverflow.net/users/146531 | Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight | I'll try to keep your notation except for three things: I prefer to have all parameters non-negative not to get confused myself, I'd rather have $z=(x,y)\in\mathbb R^2$ coordinates on the plane to avoid typing too many subscripts, and I'll replace $L^r$ by $L^q$ because $r$ is too handy for other things like the distan... | 6 | https://mathoverflow.net/users/1131 | 451959 | 181,695 |
https://mathoverflow.net/questions/451954 | 1 | Let $A$ be a filtered algebra and let $G$ be its associated graded algebra. As discussed in this [question](https://mathoverflow.net/questions/429814/properties-of-a-filtered-algebra-that-can-be-concluded-from-properties-of-its-as), if $G$ is Frobenius, then $A$ is also Frobenius.
If $G$ is a symmetric Frobenius alge... | https://mathoverflow.net/users/507923 | Associated graded algebras and symmetric Frobenius algebras | If we assume that the algebras you are interested in are finite dimensional, then the statement that you refer to about Frobenius algebras can be found in Bongale, "Filtered Frobenius Algebras" and "Filtered Frobenius Algebras II".
Now, a Frobenius algebra has a Nakayama automorphism, which effects a permutation $\nu... | 1 | https://mathoverflow.net/users/460592 | 451968 | 181,696 |
https://mathoverflow.net/questions/451895 | 2 | Under appropriate regularity conditions it is well-known that Maximum Likelihood Estimation (MLE) produces asymptotically efficient estimators in the sense that their asymptotic covariance is given by the inverse of Fisher information, i.e.
$$
\sqrt n(\theta-\tilde\theta)\overset{d}{\to}\mathcal N(0,I^{-1}(\theta))
$$
... | https://mathoverflow.net/users/125801 | Why MLEs are asymptotically efficient whereas method of moment estimators are not? | The comment by [kjetil b halvorsen](https://mathoverflow.net/questions/451895/why-mles-are-asymptotically-efficient-whereas-method-of-moment-estimators-are-no?noredirect=1#comment1168436_451895) seems to have a good point. In view of the ensuing discussion, it may be of use to detail, clarify, and complement some of th... | 3 | https://mathoverflow.net/users/36721 | 451969 | 181,697 |
https://mathoverflow.net/questions/451967 | 3 | I consider an SDE of the form $dX\_t=b(X\_t) \, dt + \sigma(X\_t) \, dW\_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$.
How can I prove that there exists a *unique* family of transition kernels $\mathscr{K}=\{k(dx\_t,t\mid x\_s,s) \mid s\leq t\}$ with the property that a stochastic process $(X\_t)\_{... | https://mathoverflow.net/users/351083 | Each diffusion SDE is associated to a *unique* family of transition kernels | there are unique corresponding forward/backward equations (Fokker Plank) to an SDE, and unique solutions for them that correspond to transition kernels. See the nice notes here [Lecture 10: Forward and Backward equations for SDEs](https://cims.nyu.edu/%7Eholmes/teaching/asa19/handout_Lecture10_2019.pdf) where they give... | 5 | https://mathoverflow.net/users/99863 | 451970 | 181,698 |
https://mathoverflow.net/questions/451786 | 4 | I am reading Demailly's notes on pseudodifferential operators on manifolds. And I cannot understand a statement he had made when he tried to prove that the image of an elliptic differential operator is closed. [The notes is available here](https://www-fourier.ujf-grenoble.fr/%7Edemailly/analytic_geometry_2019/pseudodif... | https://mathoverflow.net/users/480953 | A problem on Demailly's proof of finiteness theorem of elliptic differential operator | $\def\eps{\varepsilon}\def\rbbR{\mathbb R^+}\def\bbNo{\mathbb N\_0}\def\r#1{{\rm#1}}\def\seq#1{\langle\,#1\,\rangle}$It seems to me that Demailly's unhappy vague notation is the main cause of confusion. Actually, the argument in this part of the proof goes e.g. as follows. Using separability of $F=L^2$ and density of $... | 2 | https://mathoverflow.net/users/12643 | 451980 | 181,702 |
https://mathoverflow.net/questions/451981 | 2 | Let $ X $ be a variety with an automorphism $ \phi : X \rightarrow X $. Suppose there is a short exact sequence of vector bundles $ 0 \rightarrow F \rightarrow E \rightarrow G \rightarrow 0 $ on $ X $ such that there are isomorphisms $ f: F \rightarrow \phi^\* F $ and $ g : G \rightarrow \phi^\* G $. Is there any reaso... | https://mathoverflow.net/users/152391 | Pullback of a vector bundle extension class | For a counterexample, one can take $X$ an elliptic curve over a field of characteristic not $2$, $\phi$ the involution sending each point to its inverse under the group law, $F$ and $G$ both $\mathcal O\_X$, $f$ and $g$ the obvious isomorphisms, $E$ any nontrivial extension.
If $E$ corresponds to a class $\alpha \in ... | 8 | https://mathoverflow.net/users/18060 | 451983 | 181,704 |
https://mathoverflow.net/questions/450644 | 3 | In the book *Gaussian Measures in Finite and Infinite Dimensions* by Stroock, there is a theorem with a comment
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> The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, there is no truly probabilistic or real analytic proof of it.
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> **Theorem 2.2.1** If $X$ and $Y$ are indepen... | https://mathoverflow.net/users/477203 | Is there a real/functional analytic proof of Cramér–Lévy theorem? | I will just turn my comment into answer since this seems to still be an open problem as mentioned in "Regularized distributions and entropic stability of Cramer’s characterization of the normal law."
There they try to approach this question using entropy and prove Cramers for an example with noise but also give a cou... | 1 | https://mathoverflow.net/users/99863 | 451989 | 181,706 |
https://mathoverflow.net/questions/451986 | 6 | Suppose I have two different ring structures on the same domain $\langle R,+,\cdot,0,1\rangle$, $\langle R,\oplus,\otimes,\bar 0,\bar 1\rangle$ and I throw the structures together into a single common structure
$$\langle R,+,\cdot,0,1,\oplus,\otimes,\bar0,\bar1\rangle$$
where I have all four operations together. Let us... | https://mathoverflow.net/users/1946 | Normal form for terms in language with two ring structures | There is no normal form besides that any subterm using $+,\cdot,0,1$—treating its subterms whose topmost functions are $\oplus,\otimes,\bar0,\bar1$ as black boxes—can be written as a polynomial (i.e., a possibly empty commutative sum of possibly empty commutative products), and vice versa. As the question puts it, “we ... | 9 | https://mathoverflow.net/users/12705 | 452001 | 181,710 |
https://mathoverflow.net/questions/451955 | 5 | Nevanlinna in his book *Analytic functions* seems to state the following (at the very end of Ch. X): For every compact Riemann surface $X$ of genus $g\geq 2$ there is a non-constant holomorphic map $f : X \to Y$ to some hyperelliptic Riemann surface $Y$. How is this proved?
I explain in more detail. Nevanlinna proves... | https://mathoverflow.net/users/25510 | Existence of a holomorphic map between Riemann surfaces | I think this is what Picard says, translated in modern algebraic geometry language:
You have a curve $X$ with a map $\pi :X\rightarrow \mathbb{P}^1$.
Assume for simplicity that for each $z\in \mathbb{P}^1$, $\pi ^{-1}(z)$ contains at most one ramification point, with ramification index 2. Choose 6 branch points of $\... | 4 | https://mathoverflow.net/users/40297 | 452002 | 181,711 |
https://mathoverflow.net/questions/452014 | -6 | Let $\gamma$ denote the imaginary part of a non-trivial zero of the Riemann zeta function. Do there exist some function $f$ such that $\gamma\_{n+1} - \gamma\_n > f(n)>0$ for all large $n$? To be more explicit, do there exist some $c > 1$ such that $\gamma\_{n+1}-\gamma\_n > n^{-c}$ for all $n>n\_0$ ?
| https://mathoverflow.net/users/507786 | On gaps between consecutive zeros of the Riemann zeta function | Your first question can be reformulated as follows: [Are the nontrivial zeros of the Riemann zeta simple?](https://mathoverflow.net/questions/59770/are-the-nontrivial-zeros-of-the-riemann-zeta-simple) We don't know the answer to that question, even under the Riemann hypothesis.
| 1 | https://mathoverflow.net/users/11919 | 452018 | 181,713 |
https://mathoverflow.net/questions/451476 | 15 | Suppose that $M$ and $N$ are closed connected oriented surfaces. It is well-known that **if $f \colon M \to N$ has degree $d > 0$, then $\chi(M) \le d \cdot \chi(N)$.**
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> What is an elementary proof of this?
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I found references to some H.Kneser's articles (1928, 1929, 1930). However, I am not sure that ... | https://mathoverflow.net/users/76500 | Maximal degree of a map between orientable surfaces | The best, elementary, self-contained, and clarifying proof of Kneser's result, including the desired inequality, is due to Richard Skora. See
Skora, Richard, The degree of a map between surfaces, Math. Ann. 276(1987), no.3, 415–423.
He careful handles the cases when the surfaces are not necessarily orientable, usin... | 11 | https://mathoverflow.net/users/1822 | 452019 | 181,714 |
https://mathoverflow.net/questions/451971 | 1 | Consider the root system $R$ for a Coxeter system $(W,S)$ of type $A\_n$ with a choice of simple roots. Denote by $I(w)$ for $w\in W$ the set of positive roots $\beta\in R^+$ such that $w(\beta)$ is a negative root.
A subset $A\subseteq I(w)$ is said to be closed if for each $\beta,\gamma\in A$, if $\beta+\gamma\in I... | https://mathoverflow.net/users/62135 | Linear independence of reciprocals of products of closed sets of roots in type $A$ inversion sets | I found a counterexample. Every subset of $I(s\_2s\_3s\_1s\_2)$ is closed, but there is a dependence relation for subsets of size $3$, where if we clear denominators we get
$$(\alpha\_1+\alpha\_2)-\alpha\_2+(\alpha\_2+\alpha\_3)-(\alpha\_1+\alpha\_2+\alpha\_3)=0$$
| 1 | https://mathoverflow.net/users/62135 | 452039 | 181,720 |
https://mathoverflow.net/questions/451975 | 6 | Consider $$S\_N:=\sum\_{n=1}^N \exp\left(2\pi i\left(\frac12n^2x+\alpha n\right)\right)$$
where $\alpha$ is irrational. For certain $x$ (say integer) we can get that this is bounded for all $N$. I am curious, for what $x\in \mathbb R$ is this sum "small" - that is $S\_N=o(\sqrt N)$?
Is the Lebesgue measure of such ... | https://mathoverflow.net/users/479223 | Is the Lebesgue measure of the $x$ so that this exponential sum is $o(\sqrt{N})$ positive? | Ok, apologies if this is overkill, but [this paper](https://arxiv.org/abs/2011.09306) shows that for almost every (irrational) $\alpha \in \mathbb{R}$, only a measure $0$ set of $x \in \mathbb{R}$ satisfy $S\_{N,\alpha}(x) = o(\sqrt{N})$ as $N \to \infty$.
Indeed, that paper,
Metric theory of Weyl sums
Changhao ... | 8 | https://mathoverflow.net/users/129185 | 452052 | 181,723 |
https://mathoverflow.net/questions/452020 | 3 | Can a (finite dimenaional) $\mathbb{K}$-algebra $A$ be equipped with more than one Frobenius structure $\lambda:A \to \mathbb{K}$? Of course we identify two structures $\lambda$ and $\lambda'$ if they differ by a scalar multiple.
If it can what is a good example? If we restrict to filtered Frobenius algebras can this... | https://mathoverflow.net/users/507923 | An algebra with more than one Frobenius algebra structure | If $A$ is a Frobenius $K$-algebra and $\lambda\colon A\to K$ is a Frobenius form, then the Frobenius forms are the mappings of the form $a\mapsto \lambda(ua)$ with $u$ a unit of $A$.
One way to see that is being Frobenius means $A\_A\cong \hom\_K({}\_AA,K)$ (where $M\_A$ means $M$ is a right $A$-module and ${}\_AM$ m... | 9 | https://mathoverflow.net/users/15934 | 452061 | 181,728 |
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