# All Questions

155,593
questions

3
votes

1
answer

133
views

### At what ordinal $\chi$ does $\mathrm{L}_\chi$ contain a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?

Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, ...

3
votes

0
answers

102
views

### Can one formalize the prevalence of the Big Five systems of reverse math?

Simpson's systems of second order arithmetic turn out to be five in
number; to simplify notation let's denote them A, B, C, D, E. What
seems to be an empirical observation is that most theorems in
...

3
votes

0
answers

61
views

### Local Class field theory and Artin map for the Weil group

I am searching a reference for local class field theory that use the Weil group instead of the absolute Galois group. In particular that the Artin map is an isomorphism between the multiplicative ...

0
votes

0
answers

17
views

### Why is the $\alpha$-divergence unique in positive measure space $\mathcal{M}$?

In this article https://bsi-ni.brain.riken.jp/database/file/298/303.pdf (S. Amari 2009), it is said that a $f$-divergence (eq. 17) which can be written by a decomposable Bregman divergence (eq. 53) ...

2
votes

0
answers

29
views

### Fractional Dehn Twist coefficient of monodromy of rational open book

Given an open book $(S,h)$, the fractional Dehn twist coefficient $c(h)$ in some sense measures the difference between $h$ and its Thurston representative $g$. More specifically, one can consider the ...

2
votes

1
answer

91
views

### On an integer factoring algorithm based on smooth class number of quadratic fields

We got an algorithm and toy implementation of integer factoring algorithm
based on smooth class number of quadratic fields.
It is close to the elliptic curve factorization method (ECM) and
succeeds if ...

-1
votes

0
answers

78
views

### Possible pattern in prime numbers? [closed]

Introduction: As we know by now that there seems to be no simplistic patterns for all prime numbers. However it doesn't seem to be fair that how could a set of numbers be so arrogant in exposing ...

-1
votes

0
answers

121
views

### Non-computable numbers in constructive mathematics

Edited in order to take into account feedback from comments:
If we have an uncountable formal language L with well-defined semantics, can every set in the set-theoretic universe V be explicitly ...

0
votes

0
answers

133
views

### Possible implications of the bound $\sum_{n\leq x}\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)=O\left(x\right)$

Let $\lambda(n)$ be the Liouville function and consider the Goldbach-type problem $\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$. Assume the Riemann hypothesis, the ...

0
votes

0
answers

41
views

### Tamari lattice and bicategory coherence

Reference links:
Tamari lattice (Wikipedia): https://en.wikipedia.org/wiki/Tamari_lattice
Associahedra: https://en.wikipedia.org/wiki/Tamari_lattice#/media/File:Tamari_lattice.svg
The Tamari lattice ...

0
votes

0
answers

12
views

### Chebyshev approximation via iterated weighted least squares fits

I have the task of finding a Chebyshev approximation for a time-series; I want to check different types of functions, e.g. polynomials, rational functions, harmonics, etc.
I know that the Remez ...

-1
votes

0
answers

37
views

### Ask for help in understanding a derivation relating to the Gaussian Distribution

Does anyone know how equation (1.14) in the following content is derived? Note that $K$ is the variance of the Gaussian distribution $p(z) = \frac{1}{\sqrt{2\pi K}}e^{-\frac{z^2}{2K}}$.
These ...

1
vote

0
answers

47
views

### Relation between the field and $\mathbb{Z}$-algebra generated by eigenvalues of modular form

Cross-posted from MSE: https://math.stackexchange.com/questions/4944262/relation-between-the-field-and-mathbbz-algebra-generated-by-eigenvalues-of
Let $f$ be a cusp form of weight $k\in\mathbb{Z}$ for ...

1
vote

0
answers

78
views

### Evaluating the difference of weighted binomial coefficients

I encountered the following type of sum:
$$
\begin{align}
\left[
\sum_{k=1}^{t}\binom{k+i-2}{i-1}\binom{t-k+l_1-i}{l_1-i}\sum_{s=k}^{t}\binom{t-s+l_2-j+1}{l_2-j+1}\binom{s+j-3}{j-2}
\right] \tag{1} \\
...

3
votes

1
answer

114
views

### Can we see quantifier elimination by comparing semirings?

This question came up while reading the paper Hales, What is motivic measure?. Broadly speaking, I'm interested in which ideas from motivic measure make sense in arbitrary first-order theories (or ...

-1
votes

0
answers

86
views

### Why does the integrand $\frac{f(x)-f(y)}{|x-y|^\alpha}$ often appear when studying regularity?

Certain spaces (such as Besov, Holder, or Sobolev spaces) often measure regularity by function increments or difference quotients, and their norms contain integrands similar to
$$\frac{f(x)-f(y)}{|x-y|...

1
vote

1
answer

95
views

### Block-diagonal embedding of $U(n)$ into $U(mn)$

What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...

4
votes

1
answer

180
views

### Integral points on homogeneous spaces over a DVR

Let $R$ be a DVR (possibly mixed characteristic) with fraction field $K$. Let $V \to \operatorname{Spec} R$ be a smooth affine scheme with a transitive action of $GL_{n,R}$ so that each geometric ...

4
votes

1
answer

140
views

### When can a generalized connected sum be aspherical

Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$...

0
votes

0
answers

21
views

### Group or semi-ring on polytope family

I want to determine how to provide structure to the polytope family; I guess the main article where this is discussed is the one by Peter Mcmullen entitled “The polytope Algebra”, here they talk about ...

0
votes

1
answer

108
views

### Can PA define functions related to higher theories?

Working in $\sf ZFC$ we can define a function $F$ from naturals to naturals such that $F(0) = \, ^\circ\ulcorner r({\sf Z}) \urcorner$, where $r({\sf Z})$ is the Rosser sentence of Zermelo set theory $...

3
votes

0
answers

57
views

### A "resampling identity" for the Bessel(3) process

I've come across the following resampling identity and was wondering if this is known since it seems rather natural. Take $X$ a two-sided Brownian motion conditioned to always stay below $1$. (So if ...

-6
votes

1
answer

117
views

### Actual infinitesimals for solving Vitali paradox

Has anyone tried to use actual infinitesimals to solve paradoxes about non-measurability? In Vitali paradox, for example, they divide a set with measure 1 into $\infty$ subsets of zero measure and ...

4
votes

0
answers

228
views

### On a simple alternative correction to Roos' theorem on $\varprojlim^1$

Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...

1
vote

0
answers

74
views

### Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces

Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...

4
votes

1
answer

129
views

### Lie algebra cohomology and Lie groups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}\DeclareMathOperator\Sp{Sp}$Let $G$ be a complex Lie group and $\mathfrak{g}$ its Lie algebra (which is over $\mathbb{C}$). My first question is:
(...

4
votes

1
answer

353
views

### 4-color theorem for hypergraphs

Question. Does every hypergraph that does not admit a complete minor with $5$ elements have a coloring with $4$ colors?
Below are the definitions to make this precise.
If $H = (V, E)$ is a hypergraph ...

0
votes

0
answers

80
views

### How can I prove this stronger version of Fedder's Criterion?

I was reading Fedder's original paper which proved what is now known as "Fedder's criterion". I noticed that the abstract stated something which is a priori stronger than what is proved in ...

2
votes

0
answers

91
views

### What's the number of facets of a $d$-dimensional cyclic polytope?

A face of a convex polytope $P$ is defined as
$P$ itself, or
a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...

2
votes

0
answers

86
views

### A nonzero cuspidal automorphic representation has a nonzero Fourier-coefficients?

$\DeclareMathOperator\GL{GL} \DeclareMathOperator\Sp{Sp} \DeclareMathOperator\Ind{Ind}\DeclareMathOperator\B{B}$Let $F$ be a number field and $G_n$ the symplectic group over a $2n$-dimensional ...

3
votes

0
answers

46
views

### Rational model for composition of linear isometries

There is a composition map on spaces of linear isometries (over $\mathbb{C}$ say)
$$
\mathcal{L}(\mathbb{C}^k, \mathbb{C}^\ell) \times \mathcal{L}(\mathbb{C}^\ell, \mathbb{C}^m) \longrightarrow \...

-1
votes

0
answers

39
views

### Every graph with a unique vertex of maximum degree is Class 1

Let $G$ be graph such that there is a unique vertex $v \in V(G)$ of maximum degree $\Delta(G)$. Prove that $\chi'(G) = \Delta(G)$ where $\chi'(G)$ is the chromatic index of $G$, i.e., the minimum ...

3
votes

1
answer

86
views

### Finding the non-trivial block of a finite dimensional algebra via GAP

Let $A$ be a finite dimensional $K$-algebra that has a block decomposition $A=A_1 \times A_2 \times \dots\times A_n$. (we can assume that $A$ is a quiver algebra if that helps, meaning all simple $A$-...

1
vote

0
answers

79
views

### Dirac operator on $\operatorname{Spin}(7)$, $G_2$ and $\operatorname{SU}(3)$ manifolds

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let's take a $\Spin(7)$ manifold $M$ (the $\Spin(7)$ structure can have torsion), then the standard Dirac operator from negavtive spinors to ...

1
vote

0
answers

44
views

### A generalization of Barrow's inequality

More than seven years ago. I posted this problem in stackexchange:
Let $ABC$ be a triangle, $P$ be arbitrary point inside of $ABC$. Let $A_1B_1C_1$ be the tangential traingle of $ABC$. Let $A'$, $B'$,...

3
votes

1
answer

182
views

### Bounds on relative entropy for MLE in Bernoulli coin tosses

In the context of estimating the parameter $p$ from a dataset of $n$ i.i.d Bernoulli coin tosses, we often use the relative entropy $D(p \parallel \hat{p})$ to measure the performance of an estimator $...

0
votes

0
answers

32
views

### How to calculate the expected distance between two points [closed]

I am trying to calculate the expected distance a mobile user. so if $x_{start}$ and $y_{start}$ are the initial positions and $x_{end}$ and $y_{end}$ are the final positions of the mobile user (which ...

0
votes

1
answer

220
views

### Carleson's theorem: proof of a lemma

I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...

3
votes

0
answers

92
views

### When is a ring complete with respect to its nilradical?

Let $R$ be a commutative ring and let $I$ be its nilradical. When is $R$ complete with respect to $I$?
For example, if $I$ is finitely generated, there exists $N$ such that $I^N = 0$ and thus $R$ is ...

23
votes

8
answers

2k
views

### Object of proven finiteness, yet with no algorithm discovered?

I explain my title by two examples in number theory:
The rational points on elliptic curve over number fields forms a finitely generated abelian group, so its rank is an integer, but so far we do not ...

0
votes

0
answers

6
views

### Convergence of a recursively defined sequence (discrete selector mutator equation)

Let $\beta \in (0,1)$ and let $(u_n(k))_{n,k
\geq 0}$ be recursively defined by $u_0(k) = \mathbf 1_{k=0}$ and, for $n, k \geq 0$ :
$$u_{n+1}(k) = \beta u_n(k-1) \mathbf 1_{k \geq 1} + (1-\beta) \...

3
votes

2
answers

191
views

### What is the expected size of the smallest hitting set?

Suppose we pick $n$ subsets of size $j$ of an $N$-element set $S$ uniformly at random. A hitting set is a subset of $S$ that intersects all our subsets. I am interested in the smallest size of an ...

0
votes

1
answer

51
views

### Logan's theorem in compressed sensing

In some research papers in the nuclear magnetic resonance field Ref:, Logan's theorem is used to provide a justification for randomized sampling of free induction decay curves which are converted to ...

5
votes

0
answers

94
views

### Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?

In the setting of Banach spaces, it is well known that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...

1
vote

0
answers

28
views

### Moments on the Stiefel manifold

Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...

0
votes

0
answers

33
views

+50

### Numerical implemenation of denoising data using maximum entropy

I am trying to reproduce a denoising approach from a paper titled "Near-optimal smoothing using a maximum entropy criterion" (Link). This paper is from 1992 and also checked the PhD thesis ...

1
vote

0
answers

61
views

### Density structure on Noetherian space

I'm confused about Example 2.23. in Voevodsky's Homotopy theory of simplicial sheaves in completely decomposable topologies
I'm considering this example: over an algebraically closed field $k$, let $...

2
votes

0
answers

90
views

+100

### GIT semi-stability on graded Artinian local $\Bbbk$-algebras

Let $\Bbbk$ be a algebraically closed field of characteristic zero. A graded Artinian local $\Bbbk$-algebra is $(A,\mathfrak{m},\bigoplus A_i)$ such that $(A,\mathfrak{m})$ is an Artianian local $\...

2
votes

1
answer

102
views

### What is the fastest algorithm for classical period finding?

Let $N$ be a positive integer, and choose an integer $a$ such that $\gcd(a,N)=1$. Then $a^r \equiv 1 \,\text{mod}\, N$ for some $r$. What is the current fastest classical algorithm for finding the ...

2
votes

0
answers

36
views

+50

### Uniqueness for a nonlinear kinetic PDE-system with heat transfer coupling in one dimension

I am currently trying to understand the following article "Thermalization of a rarefied gas with total energy conservation: existence, hypocoercivity, macroscopic limit" (2021) by Favre, ...