This tag is for questions where the poster seeks references (books, articles, etc.) about a particular subject. Please do not use this as the only tag for a question.

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4
votes
8answers
533 views

Order of cyclic groups

Wikipedia says: It is known that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if n is 1 or 2 or 4 or $p^k$ or $2p^k$ for an odd prime number p and k ≥ 1. The statement seems provable ...
2
votes
2answers
662 views

Computation of the probability density function for $(X,Y) = \sqrt{2 R} ( \cos(\theta), \sin(\theta))$

Let $R$ be a almost surely non-negative continuous random variable with absolutely continuous measure, and $\Theta$ be an independent random variable, uniformly distributed on the interval $[0, 2 ...
1
vote
2answers
257 views

mathematical analysis books with many examples [duplicate]

I am looking for analysis books that can explain thing clearly and contain many good examples to help me understand the math. I don't possess a very in-depth knowledge of mathematics. Also, it's ...
43
votes
3answers
6k 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 ...
38
votes
25answers
14k views

Best Maths books for non-mathematicians

I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often ...
128
votes
14answers
22k 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 ...
56
votes
5answers
2k views

What are the issues in modern set theory?

This is spurred by the comments to my answer here. I'm unfamiliar with set theory beyond Cohen's proof of the independence of the continuum hypothesis from ZFC. In particular, I haven't witnessed ...
26
votes
28answers
2k views

Classical texts that should not be missing from any shelf [closed]

It seems to me as if many modern texts are rather streamlined. They are designed not to expect too much from the reader but they often miss the depth of respective classical literature. The purpose ...
28
votes
7answers
5k views

Good books on Math History

I'm trying to find good books on the history of mathematics, dating as far back as possible. There was a similar question here Good books on Philosophy of Mathematics, but mostly pertaining to ...
25
votes
9answers
22k views

What is a good book for learning math from the ground up?

I am wondering which books are recommended for learning math from the ground up-- from rather basic math to advanced math (middle school -> graduate school). I am about to finish my masters of ...
10
votes
5answers
5k views

Resources/Books for Discrete Mathematics

I am going to a Computer Science Course in University next year. I heard that Discrete Mathematics is whats required for Comp Sci so, I am looking for resources/books that I can read to get started ...
59
votes
3answers
2k 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 ...
14
votes
6answers
2k views

List of problem books in undergraduate and graduate mathematics

I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, ...
13
votes
4answers
2k views

What algebraic topology book to read after Hatcher's?

I've currently finished chapter 2 of his book and done all the exercises of in chapter 0, 1 and 2. Was wondering when I finished reading this book what book do I read next in algebraic topology?
26
votes
3answers
928 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 ...
25
votes
3answers
1k views

Asymptotic (divergent) series

MOTIVATION. After having read in detail an article by Alf van der Poorten I read a very short paper by Roger Apéry. I am interested in finding a proof of a series expansion in the latter, which is in ...
22
votes
5answers
776 views

Axiom of choice - to use or not to use

I was wondering if there are examples of results in mathematics that were first proven using axiom of choice and later someone found a proof of the result without using the axiom of choice.
13
votes
9answers
1k views

Reference for general-topology

Though there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or ...
7
votes
4answers
7k views

Best Algebraic Topology book/Alternative to Allen Hatcher free book?

Allen Hatcher seems impossible and this is set as the course text? So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online. ...
23
votes
4answers
985 views

Book/tutorial recommendations: acquiring math-oriented reading proficiency in German

I'm interested in others' suggestions/recommendations for resources to help me acquire reading proficiency (of current math literature, as well as classic math texts) in German. I realize that ...
15
votes
5answers
828 views

What is the modern axiomatization of (Euclidean) plane geometry?

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
27
votes
5answers
703 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 ...
8
votes
7answers
3k views

Any good Graduate Level linear algebra textbook for practice/problem solving?

I am looking for good graduate linear algebra books that contain practice problems with solutions (which is better) or hints to solve the problems. By the way, two graduate courses I am gonna take are ...
8
votes
1answer
675 views

Toward “integrals of rational functions along an algebraic curve”

In a talk by V.I. Arnold, this is said: When I was a first-year student at the Faculty of Mechanics and Mathematics of the Moscow State University, the lectures on calculus were read by the ...
3
votes
5answers
2k views

Could you recommend some classic textbooks on ordinary/partial differential equation?

I love R.Courant and F.John's Introduction to Calculus and Analysis because of its wide coverage, precise description and friendly written style. Are there any classic textbooks like it on ODE/PDE? ...
19
votes
4answers
2k views

Primes of the form $n^2+1$ - hard?

I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book ...
15
votes
1answer
828 views

Learning path to the proof of the Weil Conjectures and étale topology

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...
14
votes
2answers
709 views

A general formula for the $n$-th derivative of a parametrically defined function

Letting $$\begin{align*}x&=f(t)\\y&=g(t)\end{align*}$$ the following expressions are well known: $$\begin{align*}\frac{\mathrm dy}{\mathrm dx}&=\frac{g^\prime (t)}{f^\prime ...
13
votes
1answer
3k 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 ...
6
votes
5answers
5k views

what is the best book for Pre-Calculus?

i have missed pre-calculus knowledge in my school but i was good at maths, and now i am a computer science student, i am feeling bad being bad in maths, so i am looking for the best Pre-Calculus book, ...
6
votes
5answers
2k views

Need Help: Any good textbook in undergrad multi-variable analysis/calculus?

This semester, I will be taking a senior undergrad course in advanced calculus "real analysis of several variables", and we will be covering topics like: -Differentiability. -Open mapping theorem. ...
5
votes
7answers
5k views

Good introductory book on Calculus on Manifolds

I have already taken up to Multivariable Calculus, Linear Algebra and Diff Eq. I want to learn Calculus on Manifolds by myself, could you recommend a good introductory book on this subject? Should I ...
16
votes
1answer
7k 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), ...
12
votes
1answer
552 views

Properties of Haar measure

Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and ...
9
votes
2answers
475 views

Is there any way to read articles without subscription?

This is not mathematician question but I think it's related. How I can get access to some of "Software: Practice and Experience" articles without subscription? Any advice is welcome. Sorry if I'm ...
8
votes
1answer
233 views

reference for operator algebra

I am taking a course on operator algebra this semester. My instructor has suggested a reference "Kadinson and Ringrose." Are there any other good/standard references for this subject that I can look ...
6
votes
3answers
1k views

What is the prerequisite knowledge for learning Galois theory?

What is the prerequisite knowledge for learning Galois theory? I don't know what a ring is.
6
votes
4answers
697 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 ...
7
votes
3answers
315 views

Introductory text for lattice theory

What would be your recommendation for the text which could be useful for someone starting with lattice theory? Suggestion for both books and online materials/lecture notes are welcome. This was ...
7
votes
1answer
325 views

Holomorphic function of a matrix

A statement is made below. The questions are: (a) Is the statement true? (b) If it is, does it appear in the literature? Here is the statement. For any matrix $A$ in $M_n(\mathbb C)$, write ...
6
votes
1answer
261 views

Scott's trick without the Axiom of Regularity

Scott's trick is a method for constructing a set as a subset of a proper class. Specifically, let $A$ be a class (proper or otherwise), and let $$S(A)=\{x\in A:\forall y\in ...
4
votes
2answers
200 views

A conjecture relating Multiple Zeta Values and the Polya Enumeration Theorem

Let me state my motivation. I believe that the Polya Enumeration Theorem and Multiple Zeta Values (the classic being the Basel problem and the values of the Riemann zeta function at the even ...
13
votes
2answers
380 views

Status of the classification of non-finitely generated abelian groups.

From the Wikipedia on abelian groups: By contrast, classification of general infinitely-generated abelian groups is far from complete. How far are we from a classification exactly? It seems ...
12
votes
3answers
1k views

Squares in arithmetic progression

It is easy to find 3 squares (of integers) in arithmetic progression. For example, $1^2,5^2,7^2$. I've been told Fermat proved that there are no progressions of length 4 in the squares. Do you know ...
11
votes
2answers
819 views

Every partial order can be extended to a linear ordering

How do I show that every partial order can be extended to a linear ordering? I think that I manage to prove that claim for finite set, how can I prove it for infinite set? Thank you.
10
votes
2answers
426 views

Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as $$ D(n) = \sum_{k=1}^{n}d(k) , $$ where $$ d(n) = \sum_{k|n}^{n}1. $$ One can observe the following pattern in the values of ...
9
votes
2answers
609 views

Are there higher-dimensional analogues of sectional curvature?

I recently learned that on Riemannian manifolds, one can define the sectional curvature (http://en.wikipedia.org/wiki/Sectional_curvature) of a (2-dimensional) plane section. I was wondering if a ...
6
votes
4answers
327 views

Divergent series and $p$-adics

If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct. Surely ...
4
votes
1answer
346 views

Inverse Galois problem for small groups

I am looking for a list of all small groups (maybe order $\leq 20$) realized as the Galois groups of a polynomial over $\mathbb{Q}$, with proof. Any idea where I could find these? Partial answers or ...
3
votes
2answers
302 views

List of functions not integrable in elementary terms

When teaching integration to beginning calculus students I always tell them that some integrals are "impossible" (with a bit of expansion on what that actually means). However I must admit that the ...