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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Reference Request: Sequence of Positive Integers Excluding a Quadratic Integer Sequence

I remember that I have seen a similar problem to this one here: The set of natural numbers that don't belong to a set. If my memory serves me right, that problem is about a sequence excluding ...
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0answers
29 views

Requirement for Algebraic topology.

I am graduate student. I wil learn algebraic topology next semester which is my second semester. Book learned is written by Hatcher. In vacation , I want to study requirements for algebraic topology....
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1answer
14 views

Reference Request - Statistics Book with exercises

I'm looking for an as complete as possible statistics book with exercises, including the following topics: Probability Review Random Variables and Samples Descriptive Statistics Estimation (...
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0answers
22 views

Different versions of Cauchy's integral formula and related matters

Having been banging my head against a brick wall, as it were, for some considerable time, failing to prove something, I am putting myself through a rapid revision course in various areas, one of which ...
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0answers
83 views

Finding $1/x^2 + 1/x^3 + 1/x^5 + \dots $

The following function came up in my work: $$ f(x)=\sum_{p\text{ prime}}\frac{1}{x^p}=\frac{1}{x^2}+\frac{1}{x^3}+\frac{1}{x^5}+\frac{1}{x^7}+\frac{1}{x^{11}}+\cdots. $$ Naturally, this converges for ...
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0answers
16 views

Methods for finding decompositions of Hilbert-Schmidt integral operators

For a Hilbert-Schmidt integral operator $$(Kf)(x) = \int_Y k(x,y)f(y) dy$$ a decomposition (called Hilbert-Schmidt decomposition) of the following form exists: $$k(x,y) = \sum_n \sigma_n u_n(x)v_n(...
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0answers
19 views

What are “boundaries” (as defined here) really called and where can I learn more?

My guess is that boundaries (perhaps under a different name) in graph theory are probably defined like this: Definition 0. Let $G$ denote a graph, $A$ denote a subset of $G$. Then a candidate ...
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1answer
28 views

Good Books on relations and functions [on hold]

What are the books you would recommend to starters on the topics of Relations and functions. In your opinion why is this book better than the others.
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2answers
148 views
+50

Future learning for a math graduate in applied mathematics references

As a mathematics graduate with focus on programming we did a whole lot of coding of some mathematical statements (as well as proving them), but yet rarely giving real life examples and applications ...
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1answer
60 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

(Note: This question has been cross-posted from MO.) Let $\sigma$ be the classical sum-of-divisors function. A number is said to be perfect if $\sigma(N)=2N$. If $q^k n^2$ is an odd perfect number ...
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2answers
29 views

Non model-theoretic, constructive proof that it is valid to introduce new unique constants in a first order theory with equality

I'm currently reading through Mendelson's `Introduction to Mathematical Logic', and one of the proofs has left me dissatisfied. In general, I am fine with seeing metamathematical results proven ...
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1answer
142 views
+100

Is anyone talking about “ball bundles” of metric spaces?

In differential geometry: Each smooth manifold $M$ is equipped with a tangent bundle $TM,$ which is a manifold equipped with a projection back to $M$ Given a smooth map $f : M \rightarrow N$ between ...
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0answers
27 views

Additive combinatorics modulo $N$: Reference request

For integers $N, t \geq 1$, would you know of any special sets $A$ of integers in literature for which either an explicit formula (hopefully nice enough) or good estimate is known for the number $$ \#\...
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0answers
16 views
+100

References on Rogers-Shephard inequality

If $K\subset \mathbb{R}^n$ is a convex body, let $K'$ be the convex hull of $K$ and $-K$. One of Rogers-Shephard inequalities asserts: $$\operatorname{vol}(K') \le 2^n \operatorname{vol}(K).$$ ...
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0answers
17 views

Wavelet reference for PDE

Can anyone recommend a very readable introduction to wavelets for use in theoretical PDE/harmonic analysis? I'm frustrated with the account in Lemarie-Rieusset's Navier-Stokes book since he provides ...
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0answers
31 views

references for concrete computations in Lie groups for abstract toplogical concepts

A Lie group is a smooth manifold whose tangent space at its origin is its Lie algebra. Taking an example for lie group such as SL(2), and due to above facts we should then be able to translate the ...
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0answers
47 views

Looking for a right book for Algebraic Topology - is Dieck's textbook a good choice?

I self-study Algebraic Topology. I use Hatcher's textbook Algebraic Topology and soon I'm going to end reading Chapter 3. I know that there is one more chapter about homotopy theory but I'd like to ...
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1answer
32 views

A book on Vector Calculus with emphasis on geometrical intuition

I am a physicist trying to learn vector calculus in a way that is a mixture of the way mathematicians learn it with the way that physicist learn it in order to be able to learn Differential Geometry ...
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3answers
292 views

Cover a line segment randomly with smaller line segments [on hold]

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon). But the problem when the circle is changed to a line segment doesn't seem to have been ...
4
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1answer
41 views

Special chains in $\mathbb{R}^n$

We had a topology exam yesterday, where the following was a question: Prove or disprove that there is a chain (ordered by set inclusion) of discrete subsets of $\mathbb{R}^n$ containing uncountably ...
1
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1answer
14 views

Books with problems on extremal principle

Now, I've seen and read quite a few problem solving books but Arthur Engel's 'Problem Solving Strategies' is the only one I've seen where the extremal principle is treated. (Unlike the pigeonhole ...
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1answer
91 views

Hailstone collatz max sequence length upper bound of $260.5+x^{.43}$?

Let the Collatz function be defined as if $x$ even $c(x)=x/2$, if $x$ odd then $c(x)=3x+1$ over the naturals. Each operation is defined as a step. For example $3$ goes $(3,10,5,16,8,4,2,1)$ and takes ...
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0answers
14 views

Problem in finding introductory material (matrix spectra)

I am looking for introductory material on: 1) matrix eigenvalue spectra and useful matrix algebra theorems that can be applied in the field. 2) Statistics of random matrices (i.e. ensembles, ...
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1answer
41 views

Is the Euler prime of an odd perfect number a repunit, or otherwise?

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$). (That is, $2N=\sigma(N)$ where $\sigma$ is the ...
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1answer
92 views

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i'm not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, ...
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0answers
12 views

Solving infinite system of linear equations

I want recommendations for books,PDFs and so on that treat the problem of solving infinite system of linear equations. I'm not familiar with functional analysis, so I wish explanations that only ...
3
votes
1answer
96 views
+50

Does PA prove that Con(PA) implies Con(ZF-I) and Con(NFU)?

I read from many sources that PA and ZF-I (a suitable axiomatization of ZF minus Infinity plus its negation) are bi-interpretable, but is PA enough to prove that they are equiconsistent? Specifically ...
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0answers
46 views

Online Video Lectures for Graduate Level Mathematics?

I am wondering if there is a nice compilation of good video lectures in graduate level mathematics? I mean a website, or a forum, or maybe something like course-era (which is mostly undergraduate ...
2
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1answer
40 views

Soft Question: Good strategies for writing a technical book or textbook

Most people would agree that doing technical work, whether it be pure or applied, and learning the background knowledge necessary to do this work comprises most of the literature and curriculum in ...
5
votes
1answer
219 views

Name for Number of Ancestors/Descendants of Vertex in Directed Acyclic Graph

Let $G = (V, E)$ be a directed acyclic graph. For each vertex $v \in V$, define the ancestors of $v$ to be the set of vertices $u \in V$ such that there exists a directed path from $u$ to $v$. ...
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1answer
34 views

User directed prime testing for smallish integers (100-1000 decimal digits)

I have become interested recently in (A1) what one can do, if anything, about ~100 digit numbers with no easy factors and no access to anything but basic calculators/software that can cope with ...
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0answers
11 views

Reference for *optimal* stopping theorem for supermartingales

Can anyone introduce a good reference about optimal (not optional!) stopping times for submartingales / supermartingales? I am looking for some theorem like the one mentioned in this question. I ...
0
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1answer
22 views

Textbooks for algebraic invariant theory

I'm currently learning algebraic invariant theory from Hilbert's lectures (Theory of algebraic invariants) and while I find them very clear and enjoyable, they don't have any exercises and I'm also ...
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0answers
19 views

Green's function and strong Markov property for stopped Brownian motion

Let $X(t)$ be a Brownian motion in $\mathbb{R}^n$, stopped at some fixed time $T$. Is there a notion of Green's function for such a Brownian motion? I am guessing that there is, and $G(x, y) : = \...
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3answers
340 views

Historic proof of the area of a circle

The area of a circle radius $R$ is $\pi R^2$ which is quite easy to prove with integral calculus. Consider a ring of radius $\mathrm{d}r$ at a distance $r$ from the centre. This ring has area $2\pi r ...
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0answers
10 views

reference request on Wong Sequence

I am trying understand properties of matrix pencils and pair of matrices etc and I need to learn Wong Sequence , so could anyone tell me name of books, authentic papers, article to start with? Thanks
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0answers
16 views

Known results on the relationship between automorphisms and spectrum of a graph?

I recently saw this post from Ed Pegg on Math Stack Exchange about integral graphs with trivial automorphism groups. I am interested in trying to construct smaller such graphs - at the very least, I ...
0
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1answer
16 views

A $C^1$-function, s.t. approximation by the Trapezoidal rule is more accurate than by Simpson's rule?

Find values $a, b \in \mathbb{R}$ and a function $f \in C^{1}[a,b]$, such that the approximation of $\int_{a}^{b} f(x)dx$ by the Trapezoidal rule $T(f)$ is better than the approximation by the Simpson ...
0
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1answer
22 views

Reference: Gaussianity of linear functional of Gaussian process

My question is similar to this one, but I'm looking for a reference rather than derivation. I've been told, inserting my own commentary in square brackets, If you take $X$ in $C([a,b])$ [i.e., $X$...
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0answers
37 views

Reference Request: Lie Theory For Quantum Field Theory

I have encountered the section on non-Abelian gauge theories in Peskin and Schroeder's QFT textbook, and although I am comfortable with the derivation of the Yang-Mills Lagrangian they present, the ...
2
votes
2answers
177 views

Have historians responded to Raju's critique?

C. K. Raju has made some outrageous criticisms of the traditional take on Euclid in particular and Western history in general. Yes he has a book published on the subject with an apparently respectable ...
2
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1answer
52 views

Good book on Spherical Trigonometry

Possible approach/content: Modern Practical (Navigation/Geodesy) unifies with Euclidean/Hyperbolic Trigonometry
3
votes
1answer
39 views

Is there no proof of Dirichlet's results on quadratic residues without analysis?

Wikipedia states that all known proofs of Dirichlet's results $$ L(1) = -\frac{\pi}{\sqrt q}\sum_{n=1}^{q-1} \frac{n}{q} \left(\frac{n}{q}\right) \gt 0 $$ and $$ L(1) = \frac{\pi}{\left(2-\left(\...
2
votes
1answer
22 views

Rellich-Kondrachov

I read an article about the Rellich-Kondrachov embedding theorem in Sobolev spaces: https://en.wikipedia.org/wiki/Rellich%E2%80%93Kondrachov_theorem. Nevertheless, when I checked the refererence in ...
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1answer
36 views

Writing one formal power series as a function of another

Suppose we have a formal power series $x(t)=t+\sum_{k=2}^\infty x_k(t^k/k!)$. In principle, this can be inverted to obtain $g(x)=x+\sum_{k=2}^\infty g_k(x^k/k!)$ such that $x(g(x))=x$. The specific ...
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1answer
26 views

Curve in a disk

I take a curve $\vec\gamma:[a,b]\longrightarrow \Delta$, where $\Delta \subseteq \mathbb{R}^2$ is the disk of radius $r>0$. If the curve has length $L>0$ does exist an upper bound (in terms of $...
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0answers
57 views

What is the best book on mathematical logic [on hold]

What is the best book on mathematical logic, the most complete, the most formal, and the most up to date? PS: price doesn't matter.
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2answers
25 views

curl-free vector field on 3-torus

Let $U$ be an open and simply-connected subset of $ \mathbb{R}^3$. Then for every curl-free vector field $v \: \colon U \to \mathbb{R}^3$ there is a potential $\phi \in C^{\infty}(U; \mathbb{R})$ such ...
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0answers
54 views

Introductory Book on Faltings' Proof of the Mordell Conjecture

I'm currently reading Diamond and Shurman's book a First Course in Modular Forms and I've found it to be a wonderful introduction to the modularity theorem. Is there a similar introductory book for ...
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1answer
18 views

How to find the invariant forms of a finite group

Let $G\subset GL(n,\mathbb{Z})$. I am looking for an algorithm that finds all symmetric matrices $F$ left invariant by G, ie $$g^TFg=F\quad \forall g\in G.$$ I have found lists of these invariants for ...