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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0
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0answers
79 views

Closed-from for the series: $\sum_{k=0}^{\infty} \frac{1}{(k!)!}$

As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)!}$$ Here I don't mean the double factorial (treated here) ...
1
vote
1answer
62 views

Is there a closed-form for $\sum_{k=0}^{\infty} \frac{1}{(k!)^2}$?

As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)^2}$$ Trying on WolframAlpha, I get the value $2....
0
votes
0answers
49 views

Is potential theory worth learning these days/what are good intro texts?

Is potential theory worth learning these days? I've been told by my peers that it's completely dead and not worth learning. It seems like it has applications just about anywhere from physics to ...
5
votes
1answer
228 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 $v$. ...
3
votes
2answers
50 views

How did Euler prove the partial fraction expansion of the cotangent function?

As far as we know, Euler was the first to prove $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$ I've seen several modern proofs of it and they all ...
11
votes
3answers
415 views

A topology on the set of lines?

Of course any set $X$ can have a topology, but are there more natural topologies, metrics or similar on the set of straight lines in $\mathbb R^2$?
2
votes
1answer
28 views

Two definitions of double categories?

A double category can be defined as a category object in $\mathbf{Cat}$ the category of small categories. We can also define a double category as four categories satisfying some compatibility ...
3
votes
1answer
49 views

irreducible components of subscheme

Let $f : X \to Y$ be a closed immersion of (noetherian) schemes. Is there any "general" result on $f$ out there ensuring that $X$ has the same number of irreducible components as $Y$ ?
1
vote
1answer
39 views

Method for solving collection of simple PDEs

How would you go about evaluating the following collection of simple PDEs: $$\frac{\partial A_3}{\partial y} - \frac{\partial A_2}{\partial z} = yz$$ $$\frac{\partial A_1}{\partial z} - \frac{\...
8
votes
2answers
147 views

Is there any book similar to “Halmos Naive Set Theory” in Category Theory?

When I wanted to learn set theory in high school, I found Halmos Naive Set Theory book very readable and understandable. But now, at university, I have been searching for a similar book in category ...
33
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1answer
4k views

GRE past papers

As it is required for most students who wish to do a Ph.D in maths in the US to sit the GRE subject specific mathematics exam, I hope this question will be of interest to the mathematical community ...
0
votes
1answer
52 views

Pure geometry - Textbook(s) or course(s) [closed]

I've recently discovered the beauty of pure mathematics and I played around with various geometry problems I could find here and there (past papers in IMO or STEP exams and stuff I could find right ...
0
votes
1answer
20 views

Diameter of a Topological Manifold

I know that for a Riemannian Manifold is defined the concept of diameter. I wuold know if it's defined a similar concept for a most general Topological Manifold. Thanks in advance.
8
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5answers
3k views

Proofs of Sylow theorems

It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know.
2
votes
0answers
51 views

computational insight behind why connections fix the shape of surface

Based on a video lecture, I had some queries. If we just have a manifold [M-set,O-topology,A-atlas] say $S^2$, this manifold represents a football or a potato equally. But once we choose a connection $...
2
votes
1answer
39 views

Hopf fibration and exact sequence in homotopy

I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils ...
0
votes
1answer
42 views

Introduction to subfactor theory

I have almost no knowledge about subfactor theory but I would like to understand what it is. As a self-learner, I do not know where to start. Could you suggest introductory text/paper/book to study ...
9
votes
2answers
102 views

Favourite problem books at university level

As background let me start by stating what I perceive to be the point of problem books, or to put the matter in perhaps more acceptable way, how I define problem books. A large majority of textbooks ...
4
votes
1answer
73 views

Existence of solutions of $\Delta u = f$ such that $|u|_{L^\infty} < \infty$ if $|f|_{L^\infty} < \infty$.

I am struggling with figuring out the details of proposition 7.1. in the paper Curvature and Uniformization - R. Mazzeo and M. Taylor. Setting is as follows. Let $\Omega$ be a noncompact Riemann ...
6
votes
2answers
211 views

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 ...
-4
votes
0answers
68 views

Is the “Sophomore's dream” number rational or irrational?

I wanted to edit the wikipedia page of the "Sophomore's dream" page and wondered is the number rational, algebraic or transcendental (or belongs to another catagory) Link to wikipedia page https://en....
1
vote
2answers
863 views

Book on Infinite series

I am looking to broaden my knowledge on infinite series I would like to know the proofs or the method with which mathematicians came to conclusion on why to use various tests and how did they propose ...
1
vote
1answer
82 views

Least symmetric group having a certain Abelian group as subgroup

Given an Abelian group $G\simeq\bigoplus_{k}\mathbb Z_{p^{n_k}_{k}}$, where $p_1\leq p_2\leq ...$ are primes, how to calculate the least symmetric group $S_n$ having a subgroup isomorphic to $G$?
3
votes
2answers
107 views

Is there a theory of “rings” with partially defined multiplication?

Consider the abelian group $R [[\mathbb{Z}^d]]$ of all formal Laurent series over a commutative ring $R$ (a typical element has the form $\sum_{v \in \mathbb{Z}^d} \lambda_v \cdot X_1^{v_1} \dots X_d^{...
2
votes
0answers
53 views

Different versions of Cauchy's integral formula and the $\overline{\partial}$ question

Since this question has not attracted any answers yet, I have substantially rewritten it in the hopes that I will make it clearer. The first time students are exposed to Cauchy's Theorem, they are ...
18
votes
3answers
1k views

Looking for student's guide to diagram chasing

I'm teaching myself some category theory, and I find that I'm very slow with diagram chasing. It takes me some times a very long time to decide whether adding an arrow to a diagram preserves the ...
0
votes
1answer
31 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 ...
1
vote
3answers
55 views

Why does $p$ (is true) strictly agree with $p$ while $p$ (is false) strictly disagrees?

Let's make the truth table: $$\begin{array}{|c|c|c|} \hline p&(p) \text{ is true}&(p) \text{ is false}\\ \hline T&T&F\\ F&F&T\\\hline \end{array}$$ "$p$ is true" strictly ...
28
votes
2answers
5k views

Soviet Russian mathematics books

The introductory part of this book briefly describes the popularity of mathematics in Soviet Russia. It touches on Russian mathematical circles and generally how society in Russia took to mathematics ...
1
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0answers
53 views

Brownian motion hitting probability and Martin capacity

Consider a Brownian motion $B_t$ in $\mathbb{R}^n, n\geq 3$ and the ball $B(0, r)$ of radius $r$ around the origin. Let $\overline{C}$ be a compact set inside $B(0, r)$ such that $C$ is open in $B(0, ...
0
votes
1answer
53 views

How to tackle a mathematical problem? [closed]

I am just interested in knowing what should be the general mindset while solving/tackling a mathematical problem. I know that there is no one way of approaching the problem, but still. I was searching ...
1
vote
1answer
26 views

minimum distance of a linear codes

My question is about computing the minimum distance (weight) of a linear code. Assume that we have the generating matrix of the code. Then we can easily compute the weights of each row and of course ...
1
vote
1answer
59 views

Is $\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}$ already known?

Is it already shown or at least conjectured that $$\zeta(2n+1)\notin (2\pi)^{2n+1}\mathbb{Q}?$$ You have any names and years who proved or conjectured it?
0
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0answers
42 views

What facts are known about the hypothetical smallest divergent integer in the collatz conjecture?

If there is a divergent integer in the collatz conjecture then there must be a smallest divergent number by the WOP. We can observe some properties of this number such as it must be odd because if it ...
10
votes
0answers
125 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 ...
1
vote
0answers
31 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 ...
0
votes
1answer
37 views

Good Books on relations and functions [closed]

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.
3
votes
1answer
65 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 ...
0
votes
2answers
41 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 ...
13
votes
1answer
197 views

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 ...
0
votes
0answers
38 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 $$ \#\...
0
votes
0answers
22 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 ...
1
vote
0answers
35 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 ...
1
vote
0answers
58 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 ...
2
votes
1answer
46 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 ...
0
votes
3answers
297 views

Cover a line segment randomly with smaller line segments [closed]

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
votes
1answer
44 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
vote
1answer
18 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 ...
1
vote
1answer
99 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 ...
0
votes
0answers
18 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, ...