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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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### 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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### 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, ...
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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### 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 ...
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 ...
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 $B_{c_0}$? (...
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 ...
1 vote
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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### 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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### 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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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 $i$. ...
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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### 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) ...
1 vote
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 ...
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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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). ...
1 vote
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 ...
### 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)...