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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21
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3answers
2k views

Books to study for Math GRE, self-study, have some time.

I just graduated from a regional university in the US with a minor in mathematics. There is a masters program overseas, for economics, that I want to attend but they require applicants to take the ...
27
votes
3answers
3k 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 ...
15
votes
10answers
2k 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 ...
15
votes
7answers
4k views

Good books on “advanced” probabilities

what are some good books on probabilities and measure theory? I already know basic probabalities, but I'm interested in sigma-algrebas, filtrations, stopping times etc, with possibly examples of ...
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 ...
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.
10
votes
7answers
4k 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
7answers
6k 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 ...
12
votes
5answers
4k views

Geometry Book Recommendation?

Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...
11
votes
1answer
888 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 ...
8
votes
6answers
5k 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
689 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 ...
7
votes
6answers
3k views

Best measure theoretic probability theory book?

I'm looking for a clear way to learn measure theoretic probability theory. Any suggestions?
28
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 ...
7
votes
6answers
598 views

Book on combinatorial identities

Do you know any good book that deals extensively with identities obtained using combinatorial and/or probabilistic arguments (e.g., by solving the same combinatorial or probability problem in two ...
19
votes
1answer
8k 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), ...
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 ...
11
votes
1answer
1k 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. ...
11
votes
6answers
6k 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?
14
votes
5answers
8k views

Which calculus text should I use for self-study?

I am 36 years old, and have forgotten a lot of math from high school, of which I only took up to Algebra 2. However I am teaching myself mathematics and am now, as an adult, completely fascinated ...
14
votes
8answers
2k views

Reference request: is mathematics discovered or created?

I have to write a short monograph as an assignment for a course on the philosophy of science. Being a math student, of course I want to opt for something math-related. After some initial ideas which ...
8
votes
1answer
663 views

$C_0(X)$ is not the dual of a complete normed space

Let $X$ be any locally compact Hausdorff space and assume that it is not compact. I've heard that the Banach space $(C_0(X),\|\!\cdot\!\|_\infty)$ is not isometrically isomorphic to the (norm) dual of ...
3
votes
5answers
452 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 ...
2
votes
1answer
259 views

Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
11
votes
1answer
593 views

Radius of convergence of power series

Given a meromorphic function on $\mathbb{C}$, is the radius of convergence in a regular point exactly the distance to the closest pole? As Robert Israel points out in his answer, that this is of ...
9
votes
5answers
1k 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 ...
9
votes
2answers
401 views

How can I calculate $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$?

Calculating with Mathematica, one can have $$\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt=\frac{\pi}{4}.$$ How can I get this formula by hand? Is there any simpler idea than using $u = \sin ...
6
votes
3answers
8k views

Proof a graph is bipartite if and only if it contains no odd cycles

How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
5
votes
1answer
781 views

Getting the grip of geometry and Algebra; books and resources for a beginner

I was very interested in math since the day I was introduced to numbers. Sadly, the curricula here are driving me the opposite way. Is there any free online courses that would start from, say, ...
4
votes
1answer
328 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
860 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 ...
26
votes
3answers
1k 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
4answers
133 views

Bernoulli Number analog using Cosine

I know that Bernoulli Numbers can be found with the generating function $$\frac{x}{e^x-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}x^n$$ I was wondering if any work has been done using a similar equation ...
8
votes
2answers
266 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?
7
votes
4answers
373 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 ...
7
votes
2answers
2k views

What is operator calculus?

I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus. I have searched ...
2
votes
1answer
207 views

Transition between field representation

In a number of papers related to efficient implementation of the AES Sbox, people are computing stuff (the multiplicative inverse for instance) in GF(($2^4$)$^2$) instead of GF($2^8$). In some cases ...
11
votes
4answers
3k views

Looking to understand the rationale for money denomination

Money is typically denominated in a way that allows for a greedy algorithm when computing a given amount $s$ as a sum of denominations $d_i$ of coins or bills: $$ s = \sum_{i=1}^k n_i ...
6
votes
1answer
322 views

Known bounds and values for Ramsey Numbers

Is there a good online reference that lists known bounds on Ramsey numbers (and is relatively up to date)? The wikipedia page only has numbers for $R_2(n,m)$. I am specifically interested in known ...
5
votes
1answer
641 views

The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
4
votes
5answers
1k 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 ...
3
votes
2answers
116 views

Ultrafilter Lemma implies Compactness/Completeness of FOL

Apologies if this has been asked somewhere before, but I didn't see what I was looking for after several pages of Google results. I was reading Jech's The Axiom of Choice and was introduced to the ...
6
votes
2answers
569 views

References about Sierpinski's Theorem regarding Darboux functions

I am writing something about the following two theorems: Every function $f: \Bbb{R} \to \Bbb{R}$ can be written $f=f_1+f_2$ where $f_1,f_2:\Bbb{R} \to \Bbb{R}$ both have the Darboux property. ...
4
votes
8answers
603 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
830 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
474 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 ...
0
votes
1answer
137 views

Primary decomposition of $I = (x^2, y^2, xy)$

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find ...
73
votes
15answers
15k views

Mathematical equivalent of Feynman's Lectures on Physics?

I'm slowly reading through Feynman's Lectures on Physics and I find myself wondering, is there an analogous book (or books) for math?
48
votes
28answers
20k 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 ...
141
votes
14answers
29k 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 ...