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1 vote
1 answer
106 views

Iterated optimal transport

Suppose we are interested in two consecutive transport plans (in the Kantorovich formulation). That is, we are given finite sets $X$, $Y$ and $Z$, endowed with probability measures $\mu_X$, $\mu_Y$ ...
1 vote
0 answers
89 views

The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup

I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results. Let $A$ be a finitely generated abelian group,...
8 votes
2 answers
596 views

If a semigroup embeds into a group, then is it a subdirect product of groups?

The title has it all: Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups? If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
16 votes
2 answers
1k views

Identifying short introductory book on non-commutative geometry I read c.2008

I’m trying to identify a book I remember reading and enjoying in grad school around 2008–9; I’ve forgotten its author and title, and haven’t been able to find a book matching my memories in half an ...
51 votes
7 answers
11k views

How is representation theory used in modular/automorphic forms?

There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and ...
3 votes
0 answers
152 views

What is known about the word problem on free algebraic models?

Consider the fragment of first-order logic with equality and universal quantification as the only logical symbols; we call this logic the logic of universal algebra. I am interested in languages $\...
2 votes
1 answer
358 views

q-polynomials in terms of a basis

Consider the polynomials $$f_n(q)=\prod_{j=1}^n(1+q^j) \qquad \text{and} \qquad g_m(q)=1+q+q^2+\cdots+q^m.$$ I'll list a few examples to motivate my question. Direct calculations show that $$f_1=g_1, \...
7 votes
1 answer
847 views

Caccioppoli-Leray Inequality for De Giorgi's theorem proof

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
3 votes
1 answer
327 views

Holomorphic homotopy conjecture

Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
7 votes
0 answers
265 views

Herbrand's consistency proof

Jacques Herbrand's thesis "Investigations in proof theory: The properties of true propositions" (or in the original French "Recherches sur la théorie de la démonstration", with the ...
1 vote
0 answers
95 views

Reference for structure of subhomogeneous C*-algebras?

Recall that a C*-algebra is subhomogeneous if there is some $n$ such that the dimension of $H$ is at most $n$ for every irreducible representation $\pi \colon A \to \mathcal{B}(H)$ on some Hilbert ...
2 votes
1 answer
315 views

Recommendation for books on boundary-value problems that include perturbed boundaries and many solved problems

I am looking for a book or resource that contains applied math analytical methods and a lot of solved problems in Boundary-Value Problems for second-order PDEs, and if it could be related to wave-...
4 votes
3 answers
320 views

Examples of IF-groups

I have seen that several authors say that an infinite group $G$ is an IF-group (or has the IF-property) if every subgroup of infinite index in $G$ is free (for instance, see https://arxiv.org/pdf/1607....
3 votes
0 answers
389 views

Status of motives in higher category theory: motives and algebraic cycles through a higher categorical perspective

A while ago this interesting question was asked Derived Algebraic Geometry and Chow Rings/Chow Motives. Primary question: Have there been any recent developments/advances on the above question? If not,...
1 vote
1 answer
331 views

Finding a special solution in a solution set over F2

Given a solution set of a linear system of the following form $$ \{ \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
13 votes
3 answers
670 views

How algebraic can the dual of a topological category be?

(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at ...
0 votes
0 answers
78 views

Definition of Moore-Penrose inverse for unbounded self-adjoint operators?

I know there is a concept of Moore-Penrose or pseudoinverse of a matrix. I would like to know if one can define it for densely defined unbounded self-adjoint operators on Hilbert spaces as well. ...
1 vote
1 answer
85 views

$H^2$-elliptic regularity (up to the boundary) for operators with lower order terms for Lipschitz/convex domains

Let $\Omega$ be a bounded domain which is Lipschitz or convex. Given an elliptic operator of the form $$\langle Au, v \rangle = a_{ij}u_{x_i}v_{x_j} + b_i u_{x_i}v + cuv$$ are there any elliptic ...
5 votes
0 answers
156 views

Reference Request: Completeness of the space of all Whittaker models(a lemma in JPSS1981)

$\DeclareMathOperator\GL{GL}$There is a lemma in the proofs of local converse theorem stated as Suppose $F$ is a non-archimedean local field, $\psi$ is a non-trivial addtive character on $F$. $N_n$ ...
11 votes
2 answers
517 views

Knots having the same Alexander module which are not S-equivalent

As is discussed in this answer, S-equivalence class can be regarded as the Alexander module plus Blachfield pairing, an isomorphism of some duals of Alexander modules. There are examples of knots ...
1 vote
1 answer
222 views

Sum over three squares

Let $x$ be a sufficiently large number. Is there an explicit or asymptotic formula for the following sum $$\sum_{\substack{n\leq x\\ n=a^2+b^2+c^2}} 1.$$ Any reference would be helpful.
2 votes
0 answers
83 views

Beyond the Bring Radical: What is known about "generating radicals" for roots of polynomials of a given degree?

Famously, there is no general solution by radicals to find roots of polynomials (real, say) with degree $d\geq 5$. Somewhat less famously, there is a general solution[?] in degree $5$ using the so-...
3 votes
2 answers
2k views

Proving that every strongly connected tournament T on at least 4 vertices contains distinct vertices u, v such that T-u and T-v are strongly connected

I have a two part question: Is there a simple proof that every strongly connected tournament $T$ on $n\geq 4$ vertices contains distinct $u,v\in V(T)$ such that $T-u$ and $T-v$ are strongly connected?...
3 votes
1 answer
162 views

Ind-completion commutes with category product

$\def\A{\mathcal{A}} \def\C{\mathcal{C}} \def\D{\mathcal{D}} \def\ind{\operatorname{Ind}} \def\op{\mathrm{op}} \def\Hom{\operatorname{Hom}}$I am trying to understand the following result from ...
3 votes
1 answer
109 views

Literature request: Covariance operators for Gaussian measures

I am looking to answer the question: If $\mathcal{B}$ is a separable Banach space and $R: \mathcal{B}^*\to\mathcal{B}$ is a symmetric and positive operator, then $\phi: \mathcal{B}^*\to\mathbb{R}, \...
0 votes
0 answers
101 views

Identities for Prime Coefficients of Certain Cusp Forms

While working with Fourier expansions of cusp forms of congruence subgroups of the modular group, I observed the following patterns in their prime coefficients. Let $a(n)$ be the Fourier coefficients ...
23 votes
2 answers
3k views

Formula expressing symmetric polynomials of eigenvalues as sum of determinants

The trace of a matrix is the sum of the eigenvalues and the determinant is the product of the eigenvalues. The fundamental theorem of symmetric polynomials says that we can write any symmetric ...
2 votes
2 answers
362 views

Size of $\zeta'(s)$ at its zeros

How large can the derivative of the Riemann zeta function be at its zeros? More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
3 votes
0 answers
61 views

Is this bipartite equivalent of 1-walk-regular graphs known?

A graph $G$ is 1-walk-regular if for each vertex $v$ the number of closed walks of length $\ell$ starting (and ending) at $v$ depends only on $\ell$ but not on $v$. for each edge $vw$ the number of ...
3 votes
1 answer
488 views

Strict inequality in decoupling inequality

I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032. Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ ...
1 vote
0 answers
53 views

Moduli space of stable bundles with fixed Chern class is bounded

This is an excerpt on a set of lecture notes I'm going over: Classically, a lot of interest in stable vector bundles is due to the fact that stability allows the study of moduli of vector bundles via ...
9 votes
2 answers
493 views

Reference Request for global Hölder continuity of solutions to elliptic PDEs

This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
1 vote
0 answers
52 views

How large can the normalizer of $\mathrm{Ad}(G)$ in $\mathrm{GL}(\mathfrak{g})$ be?

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ad{Ad}$Let $G$ be a real Lie group with Lie algebra $\mathfrak g$ (say reductive/semisimple if it makes the question easier). I am interested in ...
2 votes
2 answers
242 views

A necessary and sufficient condition for three diagonals of a hexagon to be concurrent

When talking about the condition for the three diagonals of a hexagon to be concurrent, we will think of Brianchon's theorem. Using software, I discovered a necessary and sufficient trigonometric ...
7 votes
2 answers
851 views

Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs

Where can I find a (readable and self-contained) proof of the following result? Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...
-1 votes
1 answer
61 views

Asking for some references on correlations of joint optimization problems

Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
0 votes
0 answers
67 views

Projection of a gaussian random vector onto a convex body

Let $K \subset \mathbb{R}^n$ denote a convex body. Let $\Pi_K$ denote the projection onto $K$, $$ \Pi_K(y) = \mathrm{arg\,min}_{x \in K} \|y - x\|, $$ where $\|\cdot\|$ denotes the usual Euclidean ...
17 votes
3 answers
2k views

Optimal 8-vertex isoperimetric polyhedron?

I know from Marcel Berger's Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry (p.531) that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...
8 votes
1 answer
428 views

Wishart matrices: are eigenvalues and eigenvectors independent?

Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries. In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ ...
1 vote
0 answers
52 views

High order parabolic PDEs on manifolds: Reference request

I recently became interested in parabolic PDEs of order 4 on surfaces. However, I have a very little background in parabolic PDEs. I discovered Lunardi's book (Analytic semigroups and optimal ...
1 vote
0 answers
86 views

Gamma convergence via density argument: Looking for references

I am looking for a reference or result dealing with Gamma via density argument. Let me elaborate more my wish. I am actually trying to establish the Gamma convergence (precisely only the liminf) of a ...
17 votes
4 answers
2k views

Eulerian number identity

The Eulerian number $A(n,m)$ is defined as the number of permutations $\sigma \in S_n$ having precisely $m$ descents, i.e. indices $i$ such that $\sigma(i)>\sigma(i+1)$. The wikipedia entry on ...
11 votes
2 answers
1k views

A generalization of the law of tangents

The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. Let $a$, $b$, and $c$ be the lengths of the three ...
2 votes
1 answer
79 views

What is weak convergence of random permutons?

In various papers on permutons you can find statements similar to this (see Maazoun's thesis) For any $n$ let $\sigma_n$ be a random permutation of size $n$. TFAE: $(\mu_{\sigma_n})_n$ converges in ...
7 votes
1 answer
360 views

Strictification for closed monoidal categories

The strictification theorem for monoidal categories states that every monoidal categorically is monoidally equivalent to a strict monoidal category. Is there a strictification theorem for closed ...
16 votes
2 answers
1k views

CH in non-set theoretic foundations

I asked this question on stack exchange and got little attention, barring a nice example I intend to look into. The original post can be found here: https://math.stackexchange.com/q/4941233/1053681 I ...
1 vote
1 answer
298 views

maximal sets of vertices that avoids a clique

I am looking for some known algorithm that finds, for a given graph, all the maximal sets of vertices that avoid a clique of some given size $k$. I'd prefer one written in MATLAB, but other languages ...
3 votes
1 answer
159 views

Reference request: generalized Jacobian variety for higher dimensional variety

Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate ...
3 votes
1 answer
6k views

About eigen-functions of the Gaussian kernel

If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
3 votes
1 answer
135 views

Concentration of sample median for iid Gaussians

Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...

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