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Is there a prime between q and 2q, for sufficiently large q?

Chebechev's theorem says there is a prime between n and 2n, for natural n. Well, is there a prime between q and 2q, where q is rational? Of course, for sufficienlty large n.
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  • 97
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0 answers
9 views

An infinite moving average is stationary iff its innovations are stationary

Let $(a_j)_{j \in \mathbb{N}_0}$ be a real-valued sequence such that $\sum_{j = 0}^\infty a_j^2 < \infty$. Further, define an infinite moving average time series $X = \{ X(t), t \in \mathbb{Z}\}$ ...
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0 votes
0 answers
14 views

Differentiability of interpolation operator

Consider the map $T:\ell^2\rightarrow H^2(\mathbb{R})$ defined by $$ T(x) = \operatorname{argmin}\, \sum_{n=1}^{\infty}\, 2^{-n}\|f(x_n)-x_n\| + \|f\|_{H^2}. $$ Is this map differentiable and if so, ...
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  • 5,124
2 votes
1 answer
17 views

Relationship between KL, chi-squared, and Hellinger

There are many well-known relationships between the KL divergence, chi-squared ($\chi^2$) divergence, and the Hellinger metric. In the paper "Assouad, Fano, and Le Cam" by Bin Yu, the author ...
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  • 21
0 votes
0 answers
16 views

Alternative definition of physical states

Suppose that we have a vertex operator algebra $V$ with a conformal element $\omega$ and the associated conformal field $$ Y(\omega,z) = \sum_{k\in \mathbb{Z}} L_kz^{-k-2}\,, $$ where $L_k$ satisfy ...
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  • 864
3 votes
0 answers
12 views

Partitioning convex polygons into triangles of equal area and perimeter

This post is based on https://math.stackexchange.com/questions/2822589/dissect-square-into-triangles-of-same-perimeter and On a possible variant of Monsky's theorem Question 1: Is this statement ...
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0 votes
0 answers
28 views

Isometric embeddings of $c_0$ into metric spaces

Are there any nice and useful criteria or theorems which assert when a given metric space $M$ contains an isometric (not necessarily linear) copy of the Banach space $c_0$ or its unit ball $B_{c_0}$? (...
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0 votes
0 answers
38 views

Szemeredi's regularity lemma for countably infinite graphs?

Consider the following version of Szemeredi's regularity lemma found in the Fox and Lovasz paper, "A tight lower bound for Szemeredi's regularity lemma", arXiv: 1403.1768v1 [math.CO] 7 Mar ...
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1 vote
0 answers
47 views

Proof without sieves: Equivalent grothendieck topologies have the same sheaves

I'm currently learning about sheaf theory with Notes on Grothendieck topologies, fibered categories and descent theory. And in page 35, there is the following definition of a refinement and a ...
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2 votes
0 answers
18 views

Homotopy colimits in subcategories of combinatorial model categories

We know that, in a combinatorial simplicial model category $\mathbf{M}$, we can find a regular cardinal $\lambda$ large enough that all $\lambda$-filtered homotopy colimits can be computed as strict ...
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2 votes
0 answers
39 views

Filtrations of the irreducible representations of the symmetric groups

For a partition $\lambda$ of $n$, write $S^{\lambda}$ for the irreducible $S_n$-representation that corresponds to $\lambda$ (a.k.a. the Specht module). For two integers $d<n$ write $Par_d(n) = \{\...
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  • 4,779
2 votes
0 answers
37 views

Functional inverse problem based on a variational principle

I am trying to solve an inverse problem based on variational principle. I will first present a forward problem that is already solved, and then present the inverse problem that I am trying currently ...
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  • 91
1 vote
0 answers
22 views

What are some other methods for partitioning an n-dimensional space based on a set of points in that space?

So this is a very general question, but I'm curious if there are any other methods for partitioning an n-dimensional space based on the location of a set of points, either randomly chosen or specified,...
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  • 11
2 votes
0 answers
36 views

Fickle's argument for Mazur manifolds

In the page 482 of his article, Fickle wrote the following argument: Let $Y$ be a homology $3$-sphere. Next Add a $2$-handle to $Y \times [0,1]$ and produce a $4$-manifold with boundary $S^1 \times S^...
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0 votes
0 answers
48 views

Looking for proof of Serre's mass formula

Let $F$ be a finite field extension of the $p$-adic numbers $\mathbb{Q}_p$, whose residue field has $q$ elements. Let $\mathfrak{p}$ be the prime ideal of $F$. Given a finite field extension $K/F$, ...
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-4 votes
0 answers
46 views

How to Compute the Permutation Matrix Partial Derivative: $\frac{\partial\Pi^k}{\partial k}$ [closed]

How can we take the partial derivative of this forward cyclic permutation matrix $\Pi^k$, where $k$ is the real variable? $\frac{\partial\Pi^k}{\partial k}=?$ where $$ \Pi=\left[ \begin{matrix} 0 &...
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0 votes
0 answers
29 views

Does an isogeny between tori induce an isomorphism of the Lie algebras of their lft Néron models?

Let $f:T_1 \to T_2$ be an isogeny of tori over a number field $K$. Does $f$ induce an isomorphism of the Lie algebras of the lft Néron models of $T_1$ and $T_2$ ? This would hold if $f$ extends to an ...
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0 votes
0 answers
31 views

Lower bound on the number of balanced graphs

Let $\alpha>1$ be a constant and define $B_n$ as the number of (labeled) balanced graphs with $n$ vertices and $\left\lceil \alpha n\right\rceil $ edges. The paper Strongly Balanced Graphs and ...
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  • 101
3 votes
0 answers
81 views

When is a generalised Baumslag-Solitar group linear?

$\DeclareMathOperator\BS{BS}$The linearity of the Baumslag-Solitar groups $\BS(m, n)=\langle a, t\mid t^{-1}a^mt=a^n\rangle$ is completely understood, and it may be phrased as: $\BS(m, n)$ is linear ...
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  • 2,124
6 votes
2 answers
150 views

Is every smooth projective variety contained in a chain of smooth projective varieties of increasing dimension?

Let $X ⊆ \mathbb{P}^n$ be a smooth projective variety (over $\mathbb{C}$). I think we can find a chain of irreducible varieties $X = X_0 ⊆ X_1 ⊆ X_2 ⊆ \cdots ⊆ X_k = \mathbb{P}^n$ whose dimension ...
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2 votes
0 answers
64 views

Explicit bound on $\zeta(s)$ inside a zero-free region?

Does anybody know of a place in the literature where one can find an explicit result of the form $|\zeta(\sigma+it)|\leq C \log t$ for $t$ within a zero-free region (assuming $t$ is larger than an ...
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  • 16.8k
3 votes
1 answer
95 views

modularity lifting theorems for non-compact unitary groups

I am reading David Geraghty's paper, 'Modularity lifting theorems for ordinary Galois representations'(https://link.springer.com/article/10.1007/s00208-018-1742-4) and I have a related question, which,...
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  • 41
4 votes
0 answers
69 views

Reference Request: RSK under restriction

I am curious about the following result concerning the Robinson-Schensted insertion procedure. I can formulate a proof via the Schützenberger evacuation operator, but I have struggled to find such an ...
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2 votes
0 answers
34 views

Perfect matching decomposition algorithm for bipartite regular graphs

It is a well-known result that a bipartite graph can be decomposed into edge-disjoint perfect matchings if and only if it is regular. Now here comes the question. Given a bipartite regular graph, is ...
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  • 21
0 votes
0 answers
21 views

Likelihood ratio of non-trivial cycles in an inhomogeneous random square lattice graph embedded on a toroidal surface

Consider a square lattice (random) graph $G$ embedded on a toroidal surface. Each edge $(i, j)$ of the graph has an associated likelihood probability $p_{ij}$. The probabilities $p_{ij}$ come from a ...
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0 votes
0 answers
95 views

Possible research directions in random matrix theory

I'm a second year PhD student in applied mathematics with research interests in random matrix theory and numerical linear algebra. I'm now working on writing a research proposal. What I've done so far ...
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4 votes
0 answers
130 views

Generalization of IMO5 from 1987

The following question appeared as question 5 on the IMO in 1987: Prove that for all $n \geq 3$ one can find $n$ distinct points on the Euclidean plane with the property that the distance between any ...
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-4 votes
0 answers
65 views

give me formal definition of $\omega_1$, which is first uncountable ordinal [closed]

is it $\omega_1 :\equiv \mu\alpha\left(\aleph_0 \lt |\alpha| \right)$? or $\bigcup\left\{\alpha | \ |\alpha| = \aleph_0 \right\}$?
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0 votes
0 answers
23 views

Analytic formula for Inf-convolution of norms

Let $n$ be a positive integer, $p \in [2,\infty]$, and $r \ge 0$. Question. What is an analyitic formula for the infimal-convolution $$ \gamma_r(x):=\inf_{y \in \mathbb R^n}\|y-x\|_2+\lambda \|y\|_p $$...
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  • 5,840
0 votes
0 answers
22 views

Maximize entropy under Kulback-Leibler divergence

I posed this question in math.stackexchange.com, but have not received any answer. I would like to try my luck here. In this question, it is to solve \begin{align} \max_p &-\int dy\,p(y)\ln p(y) \\...
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  • 2,029
2 votes
0 answers
72 views

Two equivalent definitions of semisimplicity of group representations, proof by Zorn's lemma, a “counterexample” from the Fourier transform theory

Consider a representation $A$ of a group $G$ in a complex vector space ${\mathbb{V}}$: $$ A:~~G~\longrightarrow~\operatorname{GL}({\mathbb{V}})~~, $$ and let ${\mathbb{V}}$ be decomposable into a ...
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2 votes
0 answers
102 views

Discrimination between set of binary distributions

Suppose we know two sets of distributions $A=\{p_1,p_2,\cdots,p_k\}$ and $B=\{q_1,q_2,\cdots,q_k\}$. We are given $C=\{r_1,r_2,\cdots,r_k\}$ such that $r_i=p_i$ for all $i$ or $r_i=q_i$ for all $i$. ...
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  • 1,323
0 votes
0 answers
18 views

Vector version of concentration of Lipshitz functions on sphere (Levy's Lemma)

Levy's Lemma asserts Lipshitz functions of vectors chosen uniformly from the unit hypersphere concentrate: Lemma. Suppose $f:\mathbb{S}^{d-1} \to \mathbb{R}$ is $L$-Lipshitz on the unit hypersphere. ...
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  • 101
0 votes
0 answers
30 views

Correlation between vectors after matrix-multiplication

Consider $x$ and $y$ two $N\times 1$ complex vectors, and $T$ a $N\times N$ complex random matrix. Each element of $T$ is chosen in a complex normal distribution. Let us also define the (Pearson) ...
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1 vote
0 answers
33 views

Algorithm to compute S-units in imaginary quadratic number field

What efficient algorithms are there to compute the $S$-units of a given imaginary quadratic field $K$, where $S$ is a finite set of non-archimedean primes? Computing $S$-units are implemented in ...
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  • 539
4 votes
2 answers
183 views

Is there a simple expression for $\sum_{k =1}^{(p-3)/2} \frac{1\cdot 3\cdots (2k-1)}{2\cdot 4 \cdots 2k\cdot(2k+1)} \bmod p$?

Let $p \equiv 1 \pmod 4$ be a prime and $E_n$ denote the $n$-th Euler number. While investigating $E_{p-1} \pmod{p^2}$ I have encountered this summation (modulo $p$) \begin{align*} \sum_{k =1}^{\frac{...
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0 votes
0 answers
53 views

If all moment of X are greater than all moment of Y, can we said something about their probability?

Consider a continuous probability distribution $f$ and two random variables $X, Y$ both are greater than equal to $0$ and they are not identical random variables. Let's say one can show that $E[X]^k \...
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2 votes
0 answers
57 views

Maximal subsets of affine or projective space with no three collinear points

Let ${\mathbb A}^n_q$ and ${\mathbb P}^n_q$ be affine and projective spaces of dimension $n$ over a field of order $q$. Say that a subset of either ${\mathbb A}^n_q$ or ${\mathbb P}^n_q$ is generic if ...
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0 votes
0 answers
30 views

Set of ideals, whose Jacobson radical & nilradicals coincide - II

This question is related to a previous question . Let $R$ be a unital commutative ring and $\mathcal{S}$ be the set of all ideals of $R$ whose Jacobson radicals and nilradicals coincide. Consider the ...
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  • 1,271
4 votes
0 answers
62 views

Find $\mathbb{Z}$ -basis of module over Dedekind domain provided its pseudobasis

Let $K$ be number field of degree $d$. Suppose we are given module $ \mathcal{M}$ in form: \begin{equation}\label{key} \mathcal{M} = v_1 \cdot \mathfrak{a}_1 \oplus v_2 \cdot \mathfrak{a}_2 \...
-1 votes
0 answers
32 views

Euler vs. Hamiltonian path or circuit for mapping a bus route

Let's say that we have to pick up and drop off children at different stops along a bus route. Would a Euler path and circuit be more practical, or a Hamiltonian path or circuit for a mapping algorithm?...
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2 votes
1 answer
65 views

Characterizing algebraic tangle by their double branched covers

Montesinos proved that the double branched cover $\Sigma(T)$ of an algebraic tangle $T$ in a $3$-ball is a graph manifold. I wonder if the converse true: Is $T$ algebraic if $\Sigma(T)$ is a graph ...
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  • 2,212
1 vote
0 answers
23 views

What can we say about $\mathbb{E}[\mathrm{tr}A^{1/2}]$ for $A=\frac{1}{C}\sum_{i=1}^\infty c_i \alpha_i\alpha_i^\top \in\mathbb{R}^{m\times m}$?

Suppose we are given a summable sequence $(c_i)_{i\in\mathbb{N}}$ with $\sum_{i=1}^\infty c_i = C<\infty$ and independent $m$-dimensional, standard Gaussian vectors $\alpha_i\sim\mathcal{N}(0,I_m)$,...
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0 votes
0 answers
68 views

Spacings of Satake parameters under Ramanujan conjecture

I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match: the distribution of spacings between Satake parameters of an L-function $F$ ...
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3 votes
1 answer
107 views

Equidistribution of distances of integer points to a circle

I have noticed in the following graph that the euclidean distance between points $k \in\mathbb{Z}^2\cap C_7^1$ ($C_7^1:={}$Circle with radius 7 and shell with thickness 1) and the nearest point on the ...
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  • 159
1 vote
0 answers
52 views

Simplicial approximation theorem for toric varieties

Given abstract simplicial complexes $K$ and $L$, one constructs topological spaces $|K|$ and $|L|$. Simplicial approximation theorem says for any continuous map $f: |K|\to |L|$ that there exists ...
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2 votes
3 answers
182 views

How to prove this (corollary of) hyperplane separation theorem?

$X$ is a nonempty convex subset of $\mathbb{R}^n$ whose element is $x=\left(x_1,...,x_n\right)$. The theorem is as follows. If for each $x\in X$, there is an $i \in \left\{1,...,n\right\}$ such that $...
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  • 129
4 votes
0 answers
102 views

Spanier-Whitehead dual of space of natural transformations

Let $F, G: \mathcal{J} \to \mathsf{Sp}$ be continuous functors between $\sf{Sp}$-enriched categories, where $\sf{Sp}$ denotes any of the point-set models for spectra (i.e., orthogonal spectra). ...
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1 vote
0 answers
63 views

Locally symmetric spaces dependence on number field

A special case of locally symmetric spaces is the moduli space of abelian varieties of a given dimension $g$ (over a given base field $k$), lets call it $\mathcal{A}_g$ and is a $k$-scheme. For any ...
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0 votes
0 answers
55 views

Can we have such an external function on some $L(A)$?

Working within $\sf ZF-Reg.$, is there anything that forbids having a set $A$ and a transitive model $M$ of $\sf ZF+ V=L(A)$, that admits an external bijection $j$ between two limit stages $L_\alpha(A)...
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