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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, ...
Frode Alfson Bjørdal's user avatar
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 ...
Mikhail Katz's user avatar
  • 15.8k
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 ...
Mario's user avatar
  • 253
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) ...
aaaa's user avatar
  • 1
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 ...
Dongtai He's user avatar
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 ...
joro's user avatar
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-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 ...
Arish altmish's user avatar
-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 ...
Pan Mrož's user avatar
  • 179
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 ...
Yep's user avatar
  • 1
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 ...
Buschi Sergio's user avatar
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 ...
Manfred Weis's user avatar
  • 12.8k
-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 ...
Chao Yang's user avatar
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 ...
1.414212's user avatar
  • 337
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} \\ ...
Haimu Wang's user avatar
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 ...
Noah Schweber's user avatar
-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|...
CBBAM's user avatar
  • 565
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 ...
Alonso Perez-Lona's user avatar
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 ...
Dori Bejleri's user avatar
  • 2,960
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 $M\#_S N$...
TopologyStudent's user avatar
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 ...
Wrloord's user avatar
  • 229
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 $...
Zuhair Al-Johar's user avatar
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 ...
Martin Hairer's user avatar
-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 ...
Марат Рамазанов's user avatar
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 ...
FShrike's user avatar
  • 871
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\...
Bogdan's user avatar
  • 1,434
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: (...
Qwert Otto's user avatar
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 ...
Dominic van der Zypen's user avatar
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 ...
Anon's user avatar
  • 425
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 ...
A. H.'s user avatar
  • 35
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 ...
Andrew's user avatar
  • 969
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 \...
Niall Taggart's user avatar
-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 ...
nina99's user avatar
  • 1
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 $A$-...
Mare's user avatar
  • 26.3k
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 ...
Partha's user avatar
  • 893
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 $A'$, $B'$,...
Đào Thanh Oai's user avatar
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 $...
entropy07's user avatar
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 ...
user7341333's user avatar
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 ...
Alexander's user avatar
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 ...
Joshua Mundinger's user avatar
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 ...
J.Li's user avatar
  • 1,013
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) \...
Olivier's user avatar
  • 468
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 ...
HenrikRüping's user avatar
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 ...
ACR's user avatar
  • 791
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 ...
Qingping Zeng's user avatar
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, ...
Drew Brady's user avatar
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 ...
ACR's user avatar
  • 791
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 $...
Xiong Jiangnan's user avatar
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 $\...
Display Name's user avatar
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 ...
Jackson Walters's user avatar
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, ...
kumquat's user avatar
  • 55

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