# All Questions

130,894
questions

**5**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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,...