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5
votes
1answer
145 views

Is every matrix involution over a UFD diagonalisable?

Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$). Is every involution in $\mathrm{GL}_n(A)$ diagonalisable? This is of ...
4
votes
1answer
255 views

Hartshorne's proof of Halphen's theorem

Apologies if this is not quite at the level of MathOverflow, but it has already been asked at MSE and gone unresolved for several years despite a bounty. Hartshorne states the theorem as follows: ...
3
votes
0answers
55 views

Negation-quantifier-negation blocks in nonclassical logic: reference request

I'm looking for references to discussion of a certain question in the literature on non-classical first-order logics. I suspect it must have been investigated thoroughly, but I can't seem to find ...
0
votes
0answers
55 views

What is a general way to express the volume of some subset of $\mathbb{R}^n$ [closed]

I saw a theorem which states that Let $B \subset \mathbb{R}^n$ be any subset of $\mathbb{R}^n$, then it is true that for any $c > 0$, $$\text{volume}(cB) = c^n\times \text{volume}(B)$$ How is ...
3
votes
0answers
43 views

Virtually abelian fundamental groups equivalent to nonnegative curvature

This is a follow up question inspired by Fundamental groups of compact manifolds with non-negative Ricci curvature. In dimensions 3 and 2 (and 1) a manifold has a virtually abelian fundamental group ...
1
vote
2answers
94 views

On an angle distribution of a random linear subspace of a given dimension

$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
1
vote
0answers
121 views

Hodge's conjecture as a quasi-isomorphism between two complexes of sheaves

A version of Hodge's conjecture due to Beilinson, expects that the Betti cycles class map $H_{\mathcal{M}}^i(X,\mathbb{Q}(j))\rightarrow hom_{MHS}(\mathbb{Q}(0),H^{i}(X,\mathbb{Q}(j) ))$ is surjective ...
6
votes
0answers
203 views

Best explicit bound on $\zeta'(1+it)/\zeta(1+it)$

Assume the Riemann hypothesis. We know that $$\left|\frac{\zeta'(1+it)}{\zeta(1+it)}\right| \leq 2 \log \log t + O(1)$$ (see, e.g., Thm. 13.13 in Montgomery-Vaughan). What is the best explicit bound ...
4
votes
0answers
63 views

Generalizations of the idea of automorphic

The notion of an automorphic form/representation (and sometimes, of Langlands program in tandem) has been extended in many directions - from arithmetic to geometric to topological - but two versions I ...
4
votes
0answers
76 views

Can the injective envelope ever be injective for $*$-homomorphisms?

The answers to the question "Is the injective envelope functorial" resoundingly remind us that the injective envelope of a C$^*$-algebra really belongs in the category of completely positive ...
0
votes
0answers
18 views

Comparing two 3D-tensors (+normalization) [closed]

I want to compare two different tensors array1 and array2. So, I want to use Mean Squared Error. Therefore, I use the below formulation. Is it correct? $$ \frac{1}{n*k*z}\cdot \frac{\left \| x-\hat{x} ...
3
votes
1answer
137 views

Split monomorphisms of modules - does the finite case imply the infinite case?

Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for ...
1
vote
1answer
54 views

Lower-bound for $\mathbb E[e^{-b(v^\top X - c)^2}]$, when $X$ is log-concave in high-dimensions

Let $d$ be a large positive integer. Fix a unit-vector $v \in \mathbb R^d$, and scalars $b,c \in \mathbb R$ with $b > 0$. Let $X$ be a log-concave random vector in $\mathbb R^d$ normalized so that ...
5
votes
0answers
149 views

A problem concerning a divergent function on $[0, 1]$

This problem was posted on another forum and was given at the 1992 Miklós Schweitzer Competition. This competition is known for its very difficult problems and this one seems no exception. I also can'...
0
votes
0answers
22 views

Regrouping the leaf nodes of the WordNet DAG

Motivation I am trying to find a criterion to regroup the classes of the ImageNet challenge dataset, one of the most important datasets used in Machine Learning. The ImageNet dataset has 1000 classes ...
0
votes
0answers
112 views

A 2 dimensional integral in polar coordinate

Recently I got stuck on a 2 dimensional integral in polar coordinate, the expression is the following: $I(x)=\lim_{\xi\rightarrow0^+}\int_0^\infty dr\int_{-\pi/2}^{\pi/2}dt\frac{2\xi ^{2-2 x}r^{2x+1} \...
3
votes
0answers
43 views

Simplicial spaces and reflexive coequalisers

Let $X_\bullet$ be a simplicial space. Consider the reflexive coequaliser of $X_1\rightrightarrows X_0$, which we call $X$. Then we clearly have a map $\varphi\colon |X_\bullet|\to X$, where $|X_\...
5
votes
1answer
86 views

Isometric imbedding of a 2-disk into Euclidean 3-space

Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
5
votes
1answer
172 views

The localization of the span category

Suppose one has a model category $C$ with its class of weak equivalences $W$. It is possible to form a separate homotopical category $(C_{\operatorname{span}},W_{\operatorname{span}})$ which has ...
2
votes
1answer
61 views

Is a cap an Alexandrov space?

Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
12
votes
1answer
394 views

Representing $x^3-2$ as a sum of two squares

Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers. Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
0
votes
0answers
21 views

Is this gradient-descent-like algorithm which creates sparity

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a class $C^{1,1}$-function, $\lambda\geq 0$ be a ''learning rate'', $\lambda\geq 0$ be some "sparity generating parameter", $T\in \mathbb{N}$ be ...
0
votes
0answers
46 views

How to prove the Ky Fan inequality and its opposite

How do I prove that if $A$ and $B$ are Hermitian matrices with eigenvalues $a_1>a_2>\dots >a_n$ and $b_1>b_2>\dots >b_n$ and the eigenvalues of the sum are $c_1>c_2>\dots >...
2
votes
1answer
131 views

Is the consecutive sum set large in general?

$\DeclareMathOperator\CSS{CSS}$It is well known that for a set $A$ of integers, if $\gcd(A) = d$, then the set of (integer) linear combinations of $A$ is $d\mathbb{Z}$. I'm looking for a probability ...
2
votes
0answers
44 views

Can $\delta(G)$ get arbitrarily large in relation to $\eta(G)$?

For any finite, simple, undirected graph $G$, let $\eta(G)$ be the maximum $n$ such that the complete graph $K_n$ is a minor of $G$, and let $\delta(G)$ be the minimum degree of $G$. In certain graphs ...
1
vote
0answers
62 views

A problem about using the moving plane method to prove radial symmetry of the $C^{2}$ global solution of a elliptic PDE in $R^{2}$

Recently I'm learning the use of moving plane method to prove radial symmetry of $C^{2}$ global solution of a PDE in $R^{2}$, and I'm reading a paper where this method is applied: precisely I'm ...
0
votes
0answers
60 views

$\mathbb{R}^n$-flow, cross-section and Whitney theorem

For a $\mathbb{R}$-flow (X, $\Phi_{\mathbb{R}}$), the (local) cross-section is well defined (recall that a subset $S\subset X$ is a cross section of time $\xi>0$ if $S\cap \Phi_{[-\xi, \xi]}(x)=\{x\...
0
votes
0answers
23 views

A convergence question in $L^2$ construction of Brownian motion

I feel confused with a particular step in the $L^2$ consturction of Brownian motion. Let $\{\xi_n \sim N(0,1)\}_{n\geq 1}$ be a sequence of i.i.d Gaussian random variables on some probability space $(\...
4
votes
0answers
53 views

Conjugacy classes in normalized unit group of a group ring

Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
10
votes
1answer
349 views

How quasirandom are the nonabelian finite simple groups?

A group is $d$-quasirandom if every nontrivial complex representation has dimension at least $d$. Gowers introduced quasirandomness in this paper and proved that every nonabelian finite simple group ...
2
votes
0answers
58 views

Probability calculation of rooted trees

Given a rooted tree $T_r$ (up to isomorphism), define the probability $P(T_r)$ as the probability of ending up with $T_r$ if one starts with a single (root) vertex and incrementally connects new ...
2
votes
0answers
111 views

Nearby cycle is tamely ramified?

Let $S$ be a henselization of a closed point $s$ in a smooth algebraic curve $C$ over some finite field $\mathbb{F}_q$. Then we can consider nearby cycles over $S$. Let $s$ be the closed point of $S$ ...
4
votes
0answers
117 views

Legendre-Fenchel transform

Suppose $F:\mathbb R^n\to \mathbb R$ is a convex continuous function. Moreover, for any $x\in \mathbb R^n$, $$ \limsup_{\lambda\to\infty} \frac {|F(\lambda x)|}{\lambda}<\infty. $$ I would like to ...
7
votes
2answers
200 views

Is there an efficient generalized algorithm to find at least one binary word with the maximum rotational imbalance and the full $\{0, 1\}$-balance?

Assuming that $x$ is a sequence of $l$ bits (i.e. a binary word of length $l$) and $0 \le m < l$, let $R(x, m)$ denote the result of the left bitwise rotation (i.e. the left circular shift) of $x$ ...
3
votes
0answers
83 views

Smooth proper varieties over the integers that are not toric

Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric? By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by ...
3
votes
1answer
167 views

Relation between cohomological dimensions of manifolds

$\DeclareMathOperator\Ch{Ch}$Let $M$ be a connected manifold of finite type. We denote $\Ch_{\mathbb{Q}}(M),$ $\Ch_{\mathbb{Z}}(M)$ and $\Ch_{\mathbb{\pm}\mathbb{Z}}(M)$ by cohomological dimensions of ...
0
votes
0answers
30 views

Continuous piecewise linear mapping that preserve the orientation defined by the homology generator

Suppose I have a topological $d$-manifold $M$ embedded in $\mathbb R^D$. I also have a mapping that is piecewise linear over the subsets of the manifold. Specifically, each piece of the mapping may ...
8
votes
0answers
120 views

Key ideas behind p-adic Baker's theorem

I'm trying to understand Kunrui Yu's series of papers [1 2 3] on lower bounds of linear forms of p-adic logarithms (i.e., p-adic Baker's theorem). I know the proof of the usual Baker's theorem through ...
7
votes
0answers
147 views

In need of help with parsing non-Archimedean function theory

My current work revolves around studying functions from the $p$-adic integers to the $q$-adic rationals, where $p$ and $q$ are distinct primes ("$(p,q)$-adic functions", as I call them). I'...
-4
votes
1answer
226 views

Why do we need to represent integers as the sum of three cubes? [closed]

It is conjectured that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to $$ a^3+b^3+c^3=k. $$ Some cases for integer $k$ becomes too hard like $42$ which it ...
3
votes
1answer
111 views

Isoperimetric inequality for $\epsilon$-expansion of a set only along a certain subspace

Let $\gamma_n$ be the standard gaussian distribution on $\mathbb R^n$. Let $V$ be a $k$-dimensional subspace of $\mathbb R^n$. Finally let $A$ be any (nonempty) Borel subset of $A$ with $\gamma_n(A) = ...
1
vote
1answer
116 views

How to prove that the L-infinity norm is smaller than the Besov norm?

Suppose we have a distribution $u\in B_{\infty,\infty}^\alpha$, the Besov space with regularity coefficient $\alpha>0$. How to prove the folowing inequality? $$ \|u\|_{L^\infty}\leqslant c\|u\|_{B_{...
8
votes
2answers
234 views

example of "really" non-existent transferred model structure

I am looking for an example where a transferred model structure fails to exist, even if one is willing to work with semi-model category. But let me be more precise: Let's say I have a combinatorial ...
-6
votes
0answers
184 views

Messing around with $e+\pi$

This question originates from the conjecture that $e+\pi$ is transcendental, and that $e$ is conjectured not to be a period. Jianming Wan in his paper Degrees of periods states that the transcendence ...
0
votes
0answers
21 views

Constant in Brascamp-Lieb inequality being $1$ when reduced to Loomis-Whitney inequality?

The question is basically that, since I heard that the Loomis-Whitney inequality is a special case of the Brascamp-Lieb inequality, I would like to check the constant factor in B-L inequality is ...
2
votes
1answer
77 views

Subset which maximizes $\frac{\int_E\min(p(x), q(x))}{\int_E\max(p(x), q(x))}$?

Let $p(x), q(x)$ be two p.d.f.s of distributions on $\mathbb{R}$. I am interested in finding the subset $E$ that maximizes the quantity $$\frac{\int_{E}\min(p(x),q(x))\mathrm{d}x}{\int_{E}\max(p(x),q(...
3
votes
0answers
49 views

Clarifications involving automorphisms of projective planes and lines?

I have been learning some classical projective geometry recently and I am hoping to gain some clarity regarding various different automorphism groups. There are three different levels of generality ...
15
votes
1answer
515 views

Conjectures inspired by AI

Today in Nature a paper described how AI guided mathematicians to make highly non-trivial conjectures, which they managed to prove, one in Knot Theory involving a new invariant, the other in ...
6
votes
2answers
465 views

Is the injective envelope functorial?

Let $A$ and $B$ be unital $C^*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^*$-...
6
votes
0answers
84 views

Bounds on exponential and character sums of ratio of linear recurrences

Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\chi$ be a non-trivial additive character of $\mathbb{F}_q$, and let $\psi$ be a non-trivial multiplicative character of $\mathbb{F}_q$. Also,...

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