# Tagged Questions

This tag is for questions seeking external references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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### Infinitely generated torsion free modules over PID

Let $R$ be a PID and $\mathbf{V}$ a torsion-free $R$-module, not necessarily finitely generated. If I understand it correctly, every rank 1 submodule of $\mathbf{V}$ is isomorphic to a submodule of ...
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### What's about $\sum_{n=1}^\infty e^{-p_n u}$, where $p_n$ is the nth-prime number?

I am assuming that the following function, for which I am asking as reference request, should be known in the literature, since Glaisher studied the Prime Zeta Function, and my computation is the ...
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### What area of statistics deals with such kind of problems?

Consider $2$ samples from the starting normal distribution with parameters $\mu=0, \sigma = 1$ with size $N$. Find the variance of the random variable $\xi$ equal to average sum of $1$st sample - ...
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### Cohomology groups of a non-degenerate algebraic variety.

Let $X\subset\mathbb{P}^{n}$ be an algebraic variety. Let us suppose that $X$ is non-degenerate (it is not contained in any hyperplane of $\mathbb{P}^{n}$). I have read that (at least for curves) the ...
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### Convergence of the Euler product

Suppose that the Riemann Hypothesis is true. It is well known that then the Dirichlet series $$\sum_{n=1}^\infty\frac{\mu(n)}{n^s}$$ converges in the half-plane ${\rm {Re}}\, s>\frac{1}{2}$. Does ...
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### Capacity vs measure of a set - intuitive understanding

There is a concept of measure of "largeness" of a set, called capacity. The intuition is, instead of physical largeness (measured by Hausdorff or Lebesgue measure), capacity measures how good a given ...
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### How to relearn undergrad and tackle grad mathematics? Want to become a better mathematician!

I am a student who has just completed their degree in pure math. Unfortunately, my undergrad was a very... Unpleasant time for me due to personal reasons. Although math is accepted as a very ...
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### A proof of $|J_{\nu}(x)|\leq x/(2\nu-1)$

I am looking for a proof of the following inequality for Bessel functions : $$|J_{\nu}(x)|\leq \frac{x}{2\nu-1} \quad \left(\text{for}~\nu>1,~0\leq x \leq \frac{\pi}{2}\right).$$ Many thanks !
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### Does equivalence of algebraic categories imply bi-interpratibility of their theories?

By an algebraic theory $\mathcal{T}$ I mean any category with finite products such that the objects are given by all finite powers of some object $X$. Let $Alg\mathcal{T}$ be the concrete category of ...
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### How to derive the five-segment axiom of Tarski's geometry from Hilbert's axioms?

We are trying to prove that in an arbitrary Hilbert plane (assuming Hilbert's axioms of Group I:Incidence, II:Order and III:Congruence https://en.wikipedia.org/wiki/Hilbert%27s_axioms), Tarski's ...
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### Elliptic equation with discontinuity

I'm considering the boundary value problem $$u''(x)+u(x)=f(x) \; \mbox{on}\; \Omega=(-1,0)\cup (0,1)$$ with boundary conditions $u_x(-1)=u_x(1)=0$ and ...
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### Book recommendations for introductory Bayesian statistics?

Anyone here have some recommendations for a good book introducing the reader to Bayesian statistics? Let me mention my background. My undergraduate majors were in Actuarial Science and Statistics, ...
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### Studying Polynomials using abstract mathematics

What abstract mathematics topics, like Galois theory, ring theory, field theory, etc, and what specific topics can one study to understand polynomials. The history of investigations into Polynomial ...
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### Comparing Patrick Billingsley's Aniversary Edition to previous editions, and to Robert B. Ash's book.

I'm reading some of the reviews at amazon to the Anniversary edition of Billingsley's 'Probability and Measure', and several users state that the book is riddled with new typos, and plain errors, ...
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### Limits and Puiseux series expansions

This is a follow-up to my infinite sum question. I'm now faced with calculating: $$\lim_{n\to\infty} \left( 5n+3 \right) \left( 1 - \sum_{k=0}^n \frac{(\frac{3}{6})_k}{(\frac{13}{6})_k}\right)$$ ...
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### Set equipped with some operation which is such that it is not known is it a group and is of some importance in mathematics

Well, I know of some examples of groups which are trivial enough and of some which maybe are not so trivial. It could be the case that we could construct some operation on some set which is such that ...
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### Reference request for Grothendieck's work on “Integration with values in a topological group”

Recently I was reading the available part of the second part of W. Scharlau's book on Alexandre Grothendieck (see here). There I found, An anecdote survives about Grothendieck's arrival in Nancy: ...
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### Boolean function analysis on random graphs?

Random graphs have some properties that are determined in some random way such as edge probabilities in the interval $[0,1]$. Ryan O'Donnell's book "Analysis of Boolean Functions" (2014) has analysis ...
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### Constructing the asymptotes of a hyperbola by compass and straightedge.

Is it possible to construct the asymptotes of a hyperbola by compass and straightedge? And if so, how to construct them? I have no idea how to approach the first question. It seems it should be ...
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### Gap probability for i.i.d. random variables

Given a set $\{X_1,\ldots,X_N\}$ of real i.i.d. random variables, drawn from a common parent pdf $p_X(x)$, what is the probability that, given one random variable taking value in $(t-dt,t)$, there are ...
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### Can't Remember a Book about Binomial Sums and Hypergeometry

Some time ago I had come across a website which had the online version of a book about techniques dealing with the solution of sums involving binomial coefficients, and something with the word ...
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### If $f$ and $g$ are bounded, then every solution of the autonomous system of differential equations is defined for $t \in \mathbb R$.

Consider the system of autonomous differential equations (autonomous system of differential equations?) $$x' = f(x,y)$$ $$y' = g(x,y)$$ where $x=x(t)$ and $y=y(t)$ Let $f$ and $g$ have first ...
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### Luna-Vust theory for embeddings of homogenous spaces

I'm interested in the theory of Luna and Vust of embeddings of homogenous spaces like presented in D. Luna, Th. Vust: Plongements d'espaces homogènes, Comment. Math. Helvetici 58 (1983) 186-245. ...
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### Rings that cannot be representations rings

Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo ...
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### Up-to-date Matrix Cookbook

My copy of the Matrix cookbook is dated November 15, 2012, and is the newest copy I've been able to find. Identities may not change overtime, but the approach to an error-free presentation can be ...
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### Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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### Reference request: planar Cayley graphs

In 1896, Maschke classified all finite groups that admit a planar Cayley graph. The paper is here: http://www.jstor.org/stable/pdf/2369680.pdf I've been trying to digest this paper, but I'm finding ...
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### Every $\mathcal{C}^1$ manifold can be made smooth?

I heard of a theorem saying that each $\mathcal{C}^k$-manifold with $k\geq 1$ can be made into a smooth manifold, i.e. $\mathcal{C}^{\infty}$ (by restriction of the atlas). However, I cannot find ...
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### Do you know which book this is?

A professor wrote that there is a book that takes a historical approach to mathematical logic: There is a book I’d think you’d enjoy, that takes a historical approach. Can’t remember what it’s ...
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### Reference request: monotone and strongly monotone with respect to derivatives

Recall, let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be monotone if $$\langle F(u)-F(v), u-v\rangle\geq 0, \quad \forall u,v\in H;$$ strongly monotone if there exists ...
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### Covering vertices of a graph by cycles (not necessarily disjoint)

As title says, I want to find $k$ cycles of a graph $G$ such that each vertex appears at least once in a cycle. Let's count a single vertex as a cycle. To be more specific, I'd like to have some ...
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### Module over an infinite dimensional algebra

I have two question related to infinite dimensional algebra I have been seen a lot of example about Module over a finite dimensional $k-$algebra, but I could not find a literature about Module ...
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### Elements of $C(K)^{**}$, do they have a name?

Let $K$ be a compact (Hausdorff) space, and let $C(K)$ be the Banach algebra of contunous functions on $K$ (with the usual $\sup$-norm). The enveloping von Neumann algebra of $C(K)$ is its second dual ...
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### Is $|z_1|^2 - |z_2|^2 = 1$ a conic section?

In complex analysis of one variable, I'm aware that $$||z-a|-|z-b||= 2c$$ is the equation of a hyperbola with foci $a,b$ such that $a, b, c \in \mathbb{C}$. Now if we move up in dimension, i.e. we ...
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### Explain following Congruences in elementry way

While studding David M Burton I am felling difficulties with Linear Congruence is there any another way expertise this area (online resources). And how can I show that $21x \equiv 49\ (mod\ 10)$ can ...