# All Questions

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### Did Lebesgue like non-measurable set or not?

I was surprised by the following paragraph in Bressoud's A radical approach to Lebesgue's theory of integration, quoted by Caicedo's in his comment to this question: Vitali's nonmeasurable set, ...
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### Integral involving exp(i*cos(x)) and exp(i*sin(x))

Does somebody know if there is a formula for the integral $\int \limits_0^{2\pi}e^{i a cos x+i b sin x }e^{imx }dx$ if m is integer. The formulas 3.937(1,2) in [Gradshteyn, I. S., & Ryzhik, I. M. ...
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1 vote
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### Serre functors and global dimensions

Let $k$ be a field. Let $\mathcal{C}$ be an abelian category (over $k$). $\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that $$\operatorname{Ext}^i(M,N) =0$$ ...
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### Ind-completion commutes with category product

$\def\A{\mathcal{A}} \def\C{\mathcal{C}} \def\D{\mathcal{D}} \def\ind{\operatorname{Ind}} \def\op{\mathrm{op}} \def\Hom{\operatorname{Hom}}$I am trying to understand the following result from ...
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### Do you think e equal to the sum of all 1 cubic meter existence in the high dimension universe? [closed]

Do you think e equal to the sum of all 1 cubic meter existence in the high dimension universe ? (What is the ultimate meaning of e ? )
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### Rigorous statistical mechanics: difficulty of realistic models

Soft question: I am a mathematician self-learning statistical mechanics. The (mathematical) literature is concentrated on lattice models like the Ising model and the lattice-gas model. I understand ...
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### Optimal Stratification of Time-Evolving Relational Structures with Constrained Update Mechanisms

Let $(S, T, \preceq)$ be a totally ordered set of timestamps, where $\preceq$ is the natural ordering on timestamps. Define a relational structure $R$ as a tuple $(I, A, V, \tau)$ where: $I$ is a ...
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### Galois cohomology for rational torsion of elliptic curves

Let $E/K$ be an elliptic curve over a number field. Let $M=K(E[p])$. I want to know $H^1(M/K,E[p])$: for $p=2$, it is $0$, but what about the case $p>2$? Is it always zero? In fact, I want to know ...
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### Is a continuum in the plane regular for the Dirichlet problem at all points?

As the title asks. Let me elaborate; suppose $\mathcal K$ is a continuum (compact, connected) set of $\mathbb C$ (with at least two points!). Let's say that $g(z;a)$ is the green's function of the ...
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### Projections of closed geodesics under the modular function

In the answers to this question it was shown that for closed geodesics on $\mathbb{H}^2/\Gamma(2)$, the projection under the modular function $\lambda$ is an immersed topological component of a real ...
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### Why the hyperbolic Laplacian?

In the theory of automorphic forms there is the weight $k\in\mathbb{Z}$ Laplacian \begin{align*} \Delta_k:=-y^2 \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+iky\frac{\...
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### Excesive couples in simplicial homology [migrated]

I am reading "Elements of topological algebra" by Munkres and here is an exercise: Prove, that if K1 and K2 are ...
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### Direct characterization of finite-dimensional $1$-injective Banach spaces

It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...
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1 vote
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### Bounding the Bloch-Kato Selmer group of a twisted symmetric power of a Tate module

Let $E$ be an elliptic curve over $\mathbb{Q}$, with good reduction at $p$, and let $V = H^1_{et}(\overline{E}, \mathbb{Q}_p)$ be (the dual of) its (rationalized) Tate module. Let $S^nV$ denote its $n$...
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### What does the decision boundary of a hypothesis in a logistic regression actually signifies and how to visualise it?

Suppose I am solving a linear regression of the form $h_\theta(x) = \theta^T x$ which is shown on the left of the image. Now if I solve it as a logistic regression problem by using a sigmoid function ...
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### Field extensions and completions at possibly infinite places

In Serre's Corps Locaux, Chapter 2 §3, is presented a classical proof. We are in an "ABKL" setup, where $K/L$ is finite, $A$ is Dedekind, $B$ is the integral closure of $A$ and $B$ is $A$-...
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### Exponential sums over a linear subspace

I'm looking into certain type of exponential sums, which are summed over a linear subspace, and I couldn't find a good reference for that. The (simplified) setting is the following. Let $p$ be a prime,...
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### Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$

I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by: $$Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$$ This operator ...
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### Is Morava K-theory of a classifying space of a compact Lie group a Noetherian ring?

Let $p$ be a prime and $n > 1$ a height. My conventions for Morava K-theory are that $K_p(n)^*(pt)=\mathbb{F}_p[v_n,v_n^{-1}]$, $|v_n|$ (the degree of $v_n$) is $2(p^n-1)$. Question: If $G$ be a ...
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### Is $\overline{\mathcal{M}}_{g,n}$ a Koszul space?

In https://arxiv.org/abs/1902.06318 Dotsenko proved that $\overline{\mathcal{M}}_{0,n+1}$ is a Koszul space, i.e. it is both formal and coformal. Equivalently, a space is Koszul if it is formal and ...
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### representation of finite $p$-group [closed]

Suppose $G$ is a finite non-abelian $p$-group, $F_{p}$ is finite field of $p$ elements. If there is a injective homomorphism of $G$ to $GL(n,F_{p})$,then $$n\geq \frac{|G|}{p} ?$$
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