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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40 views

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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1answer
27 views

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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1answer
37 views

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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52 views

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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57 views

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 ...
5
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0answers
76 views

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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1answer
15 views

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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1answer
40 views

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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0answers
13 views

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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0answers
11 views

Elliptic equation with discontinuity

I'm considering the boundary value problem \begin{equation} u''(x)+u(x)=f(x) \; \mbox{on}\; \Omega=(-1,0)\cup (0,1) \end{equation} with boundary conditions $u_x(-1)=u_x(1)=0$ and ...
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1answer
15 views

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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0answers
14 views

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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0answers
31 views

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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1answer
19 views

Limits and Puiseux series expansions

This is a follow-up to my infinite sum question. I'm now faced with calculating: $$\lim_{n\to\infty} \left( 5n+3 \right) \left( 1 - \sum_{k=0}^n \frac{(\frac{3}{6})_k}{(\frac{13}{6})_k}\right)$$ ...
2
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1answer
46 views

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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0answers
46 views

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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46 views

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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1answer
24 views

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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0answers
25 views

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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1answer
30 views

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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1answer
47 views

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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1answer
82 views

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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1answer
80 views

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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30 views

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 ...
2
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1answer
24 views

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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0answers
33 views

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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1answer
47 views

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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0answers
28 views

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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0answers
16 views

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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0answers
9 views

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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1answer
33 views

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 ...
2
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0answers
40 views

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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0answers
29 views

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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0answers
9 views

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 ...
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1answer
19 views

$C(A)$ is $\|\cdot\|_2$-dense in $\ell_2(A)$

Let $A \neq \varnothing$ and $\cal {F}$$(A) = \{F \subset A \mid F$ is finite$\}$. Define $\ell_2 (A) =L^2(A, 2^A, \mu_C)$, with $\mu_C$ the counting measure. Let $C(A) = \{f: A \to \Bbb C, \exists ...
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1answer
42 views

Taylor series for $(x^n + x^m + 1)^s$

Using the trinomial theorem one gets that $$ [x^t](x^n + x^m + 1)^s = \sum_{\substack{i + j + k = s\\mi + nj = t}}{s \choose i,j,k}. $$ I was wondering if one could point out to me a reference (I'm ...
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0answers
19 views

Recommend 2nd logic and discrete math books?

I passed all the required undergraduate math for the computer science program at my university. It didn't include a course in complex analysis and the advanced required courses were about discrete ...
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1answer
159 views

Name of the class of graphs obtained by deleting $\mathcal{Q}_d$ from $\mathcal{Q}_n$

Let $\mathcal{Q}_n$ denote the $n$-cube graph. I would like to know if there is a name for the class of graphs obtained by deleting a ${\bf single}$ arbitrary copy of $\mathcal{Q}_d$ from ...
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0answers
27 views

Is there a formula for the number of proper k-colorings of a graph $G$ up to isomorphism?

In Richard Stanley's Algebraic Combinatorics book, he gives a formula for the inequivalent colorings of a set $X$ up to action by some permutation group $G$, namely: $$ \frac{1}{|G|}\sum_{\pi \in G} ...
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0answers
24 views

Arithmetic in a dihedral extension

Let $\;$L = $\Bbb Q$[$\sqrt[4]{2}$, i ]$\;$ which is a dihedral extension of the rationals. There are three quadratic and five quartic intermediate fields between L and $\Bbb Q$. The following ...
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0answers
25 views

Difficulty during self-studying unique set proofs

I have been following Velleman's How to prove it and working through it on my own. I am working full time now so I can only study after work without any other help. It's been going fairly ok until I ...
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3answers
44 views

Reference request: integration of *one*-forms along curves on a differentiable manifold.

Could somebody please direct me to a book/lecture notes with an introduction to integration of one-forms along curves in a differentiable/Riemannian manifold -- preferably leaning more towards ...
2
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1answer
26 views

The significance of CW-complexes in homotopy theory

I try to understand the significance of CW-complexes in homotopy theory, in particular with respect to the classical models structure on $\mathbf{Top}$. Why do we chose Serre cofibrations for the ...
0
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1answer
30 views

Extending fast growing functions to the reals “naturally”

There are a lot of incredibly fast growing functions defined on the natural numbers. Typical examples start with tetration, further hyper operators, Ackermann, and then there is monsters like the ...
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1answer
35 views

Is there a statistical measure of bitwise entropy?

(Somewhat inspired by this website, particularly Section III. Also, I might be using a different definition of entropy than usual; what I am using is closest to the physics definition (the one I ...
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5answers
894 views

Real analysis for a non-mathematician.

I'm currently in an engineering program, so most of my mathematical education has been applied in nature (multivariable calculus, ODEs, PDEs, probability). The only real "theory"-based courses I've ...
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1answer
64 views

Proving ZFC is consistent

I've heard from a friend that we can actually prove the consistency of ZFC if we assume at least one inaccessible cardinal exists. How is this carried out, precisely? Googling doesn't help and my ...
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1answer
20 views

Regularity of solution to an elliptic PDE

Let $R$ be the shifted open unit cube \begin{equation} \Big(-\frac{1}{2}, \frac{1}{2} \Big)^3 \subset \mathbb{R}^3, \end{equation} and let $k \in \mathbb{C}$ be a constant with $\textrm{Re}(k) ...
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1answer
48 views

Reference for category of Lie algebras?

Are there any references which deal with categorical aspects of Lie algebras? I'm looking for constructions like kernels, products, coproducts (limits and colimits in general) etc. My goal is to ...