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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31
votes
5answers
1k views

Why does the Hilbert curve fill the whole square?

I have never seen a formal definition of the Hilbert curve, much less a careful analysis of why it fills the whole square. The Wikipedia and Mathworld articles are typically handwavy. I suppose the ...
12
votes
5answers
2k views

What is a good text in intermediate set theory?

I've been working my way through Enderton's Elements of Set Theory for a while, and I feel I have a decent grasp on some of the basics of elementary set theory. My question is, where should I look to ...
26
votes
1answer
9k views

Who discovered this number-guessing paradox?

In this math.se post I described in some detail a certain paradox, which I will summarize: $A$ writes two distinct numbers on slips of paper. $B$ selects one of the slips at random (equiprobably), ...
20
votes
2answers
3k views

Good introductory books on homological algebra

Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative ...
10
votes
2answers
828 views

Explaining the method of characteristics

I am learning about solving p.d.e.s by the method of characteristics at the moment. I was given an "algorithm" to solve these problems but I want to know also what is going on, how it works and what ...
17
votes
2answers
5k views

ArcTan(2) a rational multiple of $\pi$?

Consider a $2 \times 1$ rectangle split by a diagonal. Then the two angles at a corner are ArcTan(2) and ArcTan(1/2), which are about $63.4^\circ$ and $26.6^\circ$. Of course the sum of these angles ...
17
votes
4answers
2k views

Alternative proof that the parity of permutation is well defined?

I learned the following theorem about the properties of permutation from Gallian's Contemporary Abstract Algebra. When I tried to reconstruct the proof myself, I found that it suffices to prove the ...
13
votes
1answer
6k views

How to self study Linear Algebra

I have no idea if this question is appropriate for this forum, but I hope you guys can overlook the fact that I asked it on a wrong forum (if I did) and still help me answer it (of course, if this is ...
9
votes
6answers
955 views

Recommend a statistics fundamentals book

To give you some background, I have a grasp on the basics of statistics and probability theory and even remember touching Bayes theorem at the university data mining course. But being a few years away ...
12
votes
5answers
2k views

The definition of metric space,topological space

I have read some books in analysis. All of them define metric space, topological space or vector space directly, without any reason. Therefore, I want to know the background of the definition - the ...
5
votes
2answers
176 views

One-dimensional Noetherian UFD is a PID

I am looking for a reference which has a self-contained (elementary, that is, at the "undergraduate algebra level") proof of the the fact that any one-dimensional Noetherian UFD is a PID. Does anyone ...
3
votes
5answers
564 views

Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
27
votes
3answers
2k views

Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers

A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with ...
8
votes
1answer
1k views

Effective Upper Bound for the Number of Prime Divisors

Let $\omega(n) = \sum_{p \mid n} 1$. Robin proves for $n > 2$, \begin{align} \omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}}. \end{align} Is there a similar ...
8
votes
7answers
1k views

Casual book on abstract algebra

A friend of mine, who is a high school math teacher and majored in math in college, recently asked me for a good book to read on Abstract Algebra (presumably, group theory). She is looking for ...
4
votes
1answer
401 views

Where can I learn more about commutative hyperoperations?

I just learned about commutative hyperoperations, and they look interesting. However, the wikipedia page doesn't link to more information. Is there an article or book where I can learn more? I'm ...
3
votes
2answers
1k views

Suggest books in calculus to improve problem solving skills

Suggest some books to cover the topics of calculus taught in first year of under graduation so that I can improve my problem solving skills in it ( I've knowledge of calculus and pre calculus of high ...
9
votes
3answers
1k views

Three finite groups with the same numbers of elements of each order

There exist pairs of finite groups $G$ and $H$ such that $G$ and $H$ are not isomorphic, yet they have the same number of elements of each order. For example, if $p$ is an odd prime, then the group ...
8
votes
2answers
287 views

What is known about the quotient group $\mathbb{R} / \mathbb{Q}$?

Let $G = \mathbb{R} / \mathbb{Q}$. Is this an interesting group to study? Is it isomorphic to any more natural mathematical objects?
6
votes
5answers
2k views

Distribution theory book

I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the ...
7
votes
4answers
330 views

What can I do with proper classes?

There are standard tricks, constructions and techniques in ZFC when working with proper classes; for instance one can form the cartesian product of a pair of classes without difficulty, or more ...
6
votes
1answer
1k views

Order of cyclic groups and the Euler phi function

According to Wikipedia, a cyclic number (in group theory) is one which is coprime to its Euler phi function and is the necessary and sufficient condition for any group of that order to be cyclic. Why ...
1
vote
1answer
128 views

$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$ then $f$ is linear. I know this is a well-known ...
8
votes
5answers
1k views

Can anybody recommend me a topology textbook? [duplicate]

Possible Duplicate: choosing a topology text Introductory book on Topology I'm a graduate student in Math. But I never learnt Topology during my undergraduate study. Next semester, I am ...
63
votes
3answers
18k views

phd qualifying exams

Where can I find phd qualifying exams questions.Is there any website that keeps a collection of such problems? I need it for doing some revision of the basic topics.I know of a book but that do not ...
149
votes
14answers
35k views

Is computer science a branch of mathematics?

I have been wondering, is computer science a branch of mathematics? No one has ever adequately described it to me. It all seems very math-like to me. My second question is, are there any books about ...
44
votes
8answers
69k views

What is the best book to learn probability?

Question is quite straight... I'm not very good in this subject but need to understand at a good level.
39
votes
1answer
1k views

Counterexample Math Books

I have been able to find several counterexample books in some math areas. For example: $\bullet$ Counterexamples in Analysis, Bernard R. Gelbaum, John M. H. Olmsted $\bullet$ Counterexamples in ...
26
votes
9answers
7k views

Introductory texts on manifolds

I was studying some hyperbolic geometry previously and realised that I needed to understand things in a more general setting in terms of a "manifold" which I don't yet know of. I was wondering if ...
27
votes
1answer
3k views

How to compute rational or integer points on elliptic curves

This is an attempt to get someone to write a canonical answer, as discussed in this meta thread. We often have people come to us asking for solutions to a diophantine equation which, after some clever ...
70
votes
3answers
3k views

Is there a definitive guide to speaking mathematics?

Is there a definitive guide to speaking mathematics to avoid ambiguity? I'm writing a program to generate text for a variety of mathematical expressions and would like to code it so that it adheres ...
28
votes
4answers
8k views

Rudin or Apostol

I have an option to choose between the two books Mathematical Analysis by Tom Apostol and Principles of Mathematical Analysis by Walter Rudin as I was gifted Rudin by a friend and ended up buying the ...
28
votes
7answers
13k views

Game theory - self study

I want to self study game theory. Which math-related qualifications should I have? And can you recommend any books? Where do I have to begin?
24
votes
8answers
3k views

Books on classical geometry

I'm curious to whether you guys have any tips on book concerning classical euclidean geometry. I'd like somewhat of an advanced treatment, around the same level as Coxeter's "Geometry revisited". I'd ...
17
votes
6answers
7k views

suggest textbook on calculus

I read single variable calculus this semester, and the course is using Thomas Calculus as the textbook. But this book is just too huge, a single chapter contains 100 exercise questions! Now I'm ...
31
votes
12answers
825 views

What are the theorems in mathematics which can be proved using completely different ideas?

I would like to know about theorems which can give different proofs using completely different techniques. Motivation: When I read from the book Proof from the Book, I saw there were many ...
30
votes
3answers
4k views

Reference request: introduction to commutative algebra

My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura ...
11
votes
1answer
1k views

When can we find holomorphic bijections between annuli?

I'm self-studying some complex analysis, and apparently holomorphic bijections between two annuli exist precisely when the ratios of the radii are the same. More exactly, if ...
28
votes
3answers
1k views

What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the ...
15
votes
5answers
11k views

Good book for high school algebra

I'm gonna take a Calculus course next year, my professor suggest me to review high school algebra. I want to know, which book is good for refresh knowledge on high school algebra?
10
votes
3answers
2k views

Best book on axiomatic set theory.

Which is the best book on axiomatic set theory? I am interested in a book that is suitable for graduate studies and it is very mathematically rigorous.
8
votes
6answers
4k views

Best measure theoretic probability theory book?

I'm looking for a clear way to learn measure theoretic probability theory. Any suggestions?
11
votes
5answers
5k views

Introductory text for calculus of variations

I am currently working on problems that require familiarity with calculus of variations. I am fairly new to this field. Please suggest a good introductory book for the same that could help me pick up ...
9
votes
6answers
7k views

Linear Programming Books

Do you know of a good book on linear programming? To be more specific, i am taking linear optimization class and my textbook sucks. Teacher is not too involved in this class so can't get too much help ...
7
votes
2answers
794 views

Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms? I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm ...
13
votes
9answers
4k views

More Theoretical and Less Computational Linear Algebra Textbook

I found what seems to be a good linear algebra book. However, I want a more theoretical as opposed to computational linear algebra book. The book is Linear Algebra with Applications 7th edition by ...
22
votes
1answer
5k views

The Determinant of a Sum of Matrices

Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$, \begin{align} \det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma}, \end{align} ...
14
votes
1answer
2k views

Fekete's lemma for real functions

The following result, which I know under the name Fekete's lemma is quite often useful. It was, for example, used in this answer: Existence of a limit associated to an almost subadditive sequence. ...
12
votes
6answers
8k views

Prerequisites/Books for A First Course in Linear Algebra

What mathematical knowledge do I need to begin studying linear algebra? In particular, how much calculus do I need to know? Also, do you have a favorite linear algebra book you can recommend?
6
votes
5answers
955 views

Differential Geometry of curves and surfaces: bibliography?

Dear all, next year, I will probably teach a one-semester course of Differential Geomtry of curves and surfaces. Its content must be something along the lines of the first four chapters of Do Carmo's ...