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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23 views

Reference for Ramsey Numbers

Just wondering about diagonal Ramsey numbers $R(n)$. Can anyone provide reference on either of the following? Have there been any notable attempts to make sense of $R(n)$ by using non-combinatorial ...
16
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5answers
2k views

Beginner's text for Algebraic Number Theory

What's good book for learning Algebraic Number Theory with minimum prerequisites? Assume that the reader has done an basic abstract algebra course.
2
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0answers
50 views

$V$-bundles and vector bundles

I am looking for more information on $V$-bundles. They are hard to search for as either vector bundles come up or something like GL($V$)-bundles come up. I am looking for some nice expository ...
3
votes
0answers
60 views

Reference Request: Group Theory via the Group Action Perspective

I am looking for a higher undergraduate or graduate level textbook that introduces group actions after groups just as many textbooks introduce modules after rings. I think the semigroup/semigroup ...
1
vote
0answers
47 views

What families of transcendental equations do we have solved?

I'm particularly interested in transcendental equations but searching in internet gives me only results about the classical linear-exponential equation (which is solved with Lambert's W) and its ...
37
votes
3answers
1k views

Small primes attract large primes

$$ \begin{align} 1100 & = 2\times2\times5\times5\times11 \\ 1101 & =3\times 367 \\ 1102 & =2\times19\times29 \\ 1103 & =1103 \\ 1104 & = 2\times2\times2\times2\times ...
0
votes
3answers
51 views

Function on $\mathbb Z^2$ whose value equals the average of values at adjacent points $\Rightarrow$ function is constant

This is a reference request. I am not asking for a proof. If I remember correctly, there is a theorem that states that if a bounded [criterion added after editing] function $f:\mathbb Z^2\to\mathbb ...
5
votes
0answers
31 views

Are the ring of integers of the constructible numbers a Euclidean domain?

I suspect that since Euclid uses the Euclidean Algorithm to perform division on constructible numbers in Elements, the ring of integers of the constructible numbers are a Euclidean Domain, but I have ...
3
votes
1answer
38 views

What is the Wedderburn decomposition of $\mathbb{R}[D_{2n}]$?

I have been looking everywhere and can't seem to find a general formula for the Wedderburn decomposition of the real group ring of the dihedral group ring of order $2n$, $\mathbb{R}[D_{2n}]$. Does ...
-1
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2answers
26 views

Differential Equations applications in Computer Science

I'm writing a project on differential equations and their applications on several scientific fields (such as electrical circuits, polulation dynamics, oscillations etc) but i'm mainly interested in DE ...
2
votes
1answer
73 views

Request derivative and integral problems bank [closed]

I need derivative and integral problems in many numbers (I prefer >100 questions, multiple sources are okay). Scope: Start from high school material then raise to pre-college. Derivative, ...
4
votes
1answer
70 views

Casson handles neighborhoods are representable by $D^2$-bundles over $S^2$.

On 250 page of Scorpan's book Wild world of 4-manifolds. there is a construction of an exotic $\mathbb{R}^4$. It starts from taking manifold $M = \mathbb{C}P^2 \# 9 \overline{\mathbb{C}P}^2$ and ...
2
votes
1answer
52 views

Gradient descent method with random perturbation

Suppose there is a function $f:\mathbb R^n \to \mathbb R$. One way to find a stationary value is to solve the ODE $\dot x = - \nabla f(x)$, and look at $\lim_{t\to\infty} x(t)$. However I want to ...
3
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0answers
42 views

Generating Sets for Subgroups of $(\Bbb Z^n,+)$.

The question Finite Generated Abelian Torsion Free Group is a Free Abelian Group led me to conjecture and prove an interesting thing about generating sets for $\Bbb Z^n$ and certain subgroups. If ...
13
votes
2answers
2k views

Category Theory vs. Universal Algebra - Any References?

After seeing the answer to the question Category theory, a branch of abstract algebra, I would like to ask Are there literature discussing the difference/indifference/comparison between category ...
3
votes
2answers
159 views

Modules that have finitely many submodules

Drawing the lattice of submodules of a given module helps me to gain some intuition about the structure of module. Sometimes, however, it is not possible to draw in neat manner; For example vector ...
5
votes
3answers
404 views

From a deterministic discrete process to a Markov chain: conditions?

When will a probabilistic process obtained by an "abstraction" from a deterministic discrete process satisfy the Markov property? Example #1) Suppose we have some recurrence, e.g., $a_t=a^2_{t-1}$, ...
2
votes
1answer
21 views

Survey on large deviation bounds of queuing delay in CSMA scheduling

I am trying to do some literature survey on the theoretical guarantees in uplink scheduling algorithms. I found there exist a series of papers from UIUC and UC Berkeley by L.Jiang, J. Walrand, R. ...
2
votes
1answer
144 views

Is there any new improvement in the proof or disproof of the twin prime conjucture?

I think this is not the first question about twin primes here, but my own is the latest one! I am a postgraduate student in Mathematics interested in the field of number theory. While searching on ...
1
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2answers
39 views

Notation Clarification: $M\odot N$ for von Neumann algebras $M$ and $N$

Given a Banach space $E$, $y\in E$, $\phi\in E^{*}$, I am led to the understanding that $y\odot \phi$ denotes the operator in $B(E)$ defined by $$x\mapsto \phi(x)y,\text{ for all } x\in E$$ Now I ...
0
votes
0answers
6 views

General theory of Galerkin approximations for evolution equations

I'm studying parabolic evolution equations from Lawrence Evans's book and I encounter the Galerkin method for finding weak solutions. I wonder if there is a general theory (for abstract equations on ...
1
vote
2answers
226 views

Cryptography textbook

Might come as a rather strange request but does anyone know a textbook on cryptography that is small and short, say around 300 pages max. I am tired of having a sore shoulder from carrying 5 heavy ...
1
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2answers
31 views

Reference book for Brownian Motion

I want to know about books for reading Brownian motion. I am aware of measure theoretic probability theory.
1
vote
1answer
52 views

Special Properties of Real Matrices With Real Distinct Eigenvalues

Are there any special properties of real matrices (not necessarily symmetric) with "real" distinct eigenvalues, other than the well-known properties like being diagonalizable, which has nothing to do ...
3
votes
0answers
20 views

Intuition for homotopy (co)limits in triangulated categories

The following definition is taken from Daniel Murfet's Triangulated Categories Part I notes. Let $\mathcal T$ be a triangulated category with countable coproducts. Suppose we are given a ...
1
vote
0answers
21 views

Analysis for Lie groups

So my goal would be to learn some Lie algebras. I was told that I should study firstly Lie groups, I will have better picture and more motivation in mind. For now, I don't want to study it in depth. ...
8
votes
5answers
115 views

$32$ Goldbach Variations - Papers presenting a single gem in number theory or combinatorics from different point of view

A short time ago I found the nice paper Thirty-two Goldbach Variations written by J.M. Borwein and D.M. Bradley. It presents $32$ different proofs of the Euler sum identity \begin{align*} ...
2
votes
1answer
32 views

A compact, connected, abelian Lie group is a torus?

How to prove that a compact, connected, abelian Lie group is a torus? It seems very intuitive. Any reference?
1
vote
0answers
25 views

Pre-College Algebra Book

I am looking for a high school/ pre-college level Algebra book that is self contained for self-study. Nothing special, I don't want a book about number theory, but a book in preparation of high school ...
0
votes
0answers
37 views

Soviet Optimization books

I am aware of an answer on Soviet math books here: Soviet Russian Mathematical Books and the book by Boris Polyak on non linear optimization. I am also aware of a few books by Kantorovich which I do ...
0
votes
0answers
14 views

“One-sided” Morita equivalence and Hochschild homology

Suppose $A$ and $B$ are $k$-algebras. Then we have the Hochschild homologies $HH(A) = HH(A,A)$ and $HH(B) = HH(B,B)$. Now suppose that $P$ is an $A$-$B$ bimodule and $Q$ is a $B$-$A$ bimodule so ...
6
votes
2answers
287 views

Elementary Abelian $p$-Subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number. If $G$ is a finite group, an elementary abelian $p$-subgroup of $G$ of rank $r$ is any group $E\subset G$ such that $E\cong (\mathbb{Z}/p)^{\oplus r}.$ Let $\mathcal{E}_r$ ...
2
votes
1answer
22 views

Dual Cone Construction $\{z \; | \;z \perp v \text{ for some } v \in \Lambda \}$

In a linear algebra computation, in order to estimate the second eigenvalue we consider a collection of vectors. Let $\Lambda$ be a cone in $\mathbb{R}^d$ then $$ \Lambda' = \Big\{z \;\Big| \;z ...
6
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3answers
78 views

Arbitrarily large values for $|Li(x) - \pi(x)|$

I was wondering whether there are arbitrarily large values for the $|Li(x) - \pi(x)|$. I do know that $Li(x) - \pi(x)$ changes sign infinitely often, but this does not imply that the difference stays ...
0
votes
0answers
21 views

Representing an order relation with a real-valued function

Let $\succeq$ be a relation on a set $X$. The function $u: X\to \mathbb{R}$ represents the relation $\succeq$ if $x\succeq y \iff u(x)\geq u(y)$. I am looking for a good reference on questions such ...
1
vote
1answer
24 views

What is the name of the transform which finds the number of ways to make partitions of the given sizes?

I'm looking for the name of a transform which takes a sequence giving the number of 'prime' elements of a given size to the number of ways to make a number out of a sum of 'prime' elements, up to ...
2
votes
0answers
113 views

Kock Frobenius algebras and 2D TQFTs

My mathematical career consists of selfstudy of the first 7 chapters of Lee topological manifolds. I want to read the book Kock Frobienius algebras and 2d TQFTs. Can you suggest which books and what ...
18
votes
5answers
669 views

Big list of serious but fun “unusual” books

I would like to have some suggestions about serious (that is, with good mathematical content) but fun books that cover topics (or propose problems) in "recreational mathematics"; in any other field ...
1
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0answers
59 views

Please guide me books and online materials for this course

I have recently taken Course on Numerical Analysis. It is correspondence course. So i to do self study. I will be glad if someone mentions online videos and elementary books which contains following ...
10
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1answer
373 views

Can the “inducing” vector norm be deduced or “recovered” from an induced norm?

Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ...
0
votes
0answers
13 views

Reference Request for Cartier Modules, Dieudonne Modules

The question is pretty self-explanatory from the title. I'd be very grateful if someone could recommend a readable, fairly comprehensive introduction to these topics. Many thanks
3
votes
0answers
28 views

Reading references about the process of proving a theorem, or doing mathematics in general

Doing mathematics is a very unique process, that puzzles non-mathematicians. All PhD students or professional mathematicians regularly hear things like "What? There are still theorems to prove? In ...
1
vote
2answers
55 views

Looking for a Great book on Proving, on Mathematical Logic in general

I'm looking for a good/great book on Mathematics. More specifically one that focuses on how I should go about to prove various things, e.g. given equations what are the methods I can use in order to ...
3
votes
1answer
66 views

Hypergraph Colorability

I'm interested in hypergraphs for which there are known (nontrivial) lower bounds on the chromatic number. If someone could point me to existing literature (survey papers etc) on this topic that would ...
1
vote
0answers
20 views

Finding locally and globally closing loops in a graph with toroidal topology

I have a two dimensional square lattice (with periodic boundary) with loops on it, i.e., collections of connected links which form closed loops. The lattice has the topology of a 2-torus and therefore ...
11
votes
4answers
4k views

How to do well on Math Olympiads

I'm a high school student who really likes maths and I'm quite good at school. I want to start training maths by myself but I think I need some guidelines. I want to do well on IMO but I don't know ...
16
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10answers
2k views

Problem books in higher mathematics

Are there any problem book series on different topics with proper solutions? I have already found some in analysis (Problems in Mathematical Analysis 1-3) but not any in less popular topics.
4
votes
0answers
54 views

Good problem books at a relatively advanced level?

I have been searching for problem books on advanced topics. By advanced I am referring to the undergraduate level and above. I am looking for something analogous to the olympiad type problem books ...
1
vote
1answer
254 views

Book covering introduction to mathematical proofs

I am looking for some introductory books covering mathematical proofs, axioms, propositions, proof techniques etc in general.
5
votes
1answer
116 views

$\sin x$ as a sum involving fractional parts

Does there exist a formula giving a sense to the formal equation $$ \sin x=-\pi\sum_{n=1}^{+\infty}\frac{\mu(n)}{n}\left\{\frac{nx}{2\pi}\right\}, $$ where $\mu$ is the Möbius function, $\{\cdot\}$ ...