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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33
votes
7answers
12k views

Good 1st PDE book for self study

What is a good PDE book suitable for self study? I'm looking for a book that doesn't require much prerequisite knowledge beyond undergraduate-level analysis. My goal is to understand basic ...
21
votes
9answers
1k views

Very good linear algebra book.

I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), ...
17
votes
4answers
1k views

probability textbooks

Has anyone compiled a moderately comprehensive list on the web or elsewhere of textbooks on probability For students who have not been introduced to the subject before That introduce both discrete ...
17
votes
2answers
380 views

How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?

I'm considering the transfer-function $$ t(x) = \log(1 + \exp(x)) $$ and find the beginning of the power series (simply using Pari/GP) as $$ t(x) = \log(2) + 1/2 x + 1/8 x^2 – 1/192 x^4 + 1/2880 x^6 - ...
6
votes
7answers
10k views

Good First Course in real analysis book for self study

Does anybody know of a good book in real analysis for self study for a beginner? What about Analysis 1 by Terence Tao?
3
votes
1answer
464 views

Characterization of integers which has a $2$-adic square root

Does anyone know an "elementary" proof of the following theorem? Let $k \neq 0$ be a rational integer. Then $k$ admits a square root in $\mathbb{Z}_2$ if $k = 4^a (8b+1)$ for some $a \in \mathbb{N}$, ...
21
votes
7answers
11k views

Book recommendation on plane Euclidean geometry

I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
62
votes
5answers
2k views

Defining a manifold without reference to the reals

The standard definition I've seen for a manifold is basically that it's something that's locally the same as $\mathbb{R}^n$, without the metric structure normally associated with $\mathbb{R}^n$. ...
16
votes
14answers
7k views
17
votes
6answers
8k views

What are good books/other readings for elementary set theory?

I am looking to expand my knowledge on set theory (which is pretty poor right now -- basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to ...
17
votes
6answers
2k views

Reference for Algebraic Geometry

I tried to learn Algbraic Geometry through some texts, but by Commutative Algebra, I left the subject; many books give definitions and theorems in Commutative algebra, but do not explain why it is ...
8
votes
5answers
798 views

What are or where can I find style guidelines for writing math?

I am a scientist writing my first manuscript with a substantial amount of mathematical methodological documentation. I am using LaTeX, but this is not my question. I would like to find a list of ...
5
votes
4answers
449 views

Can the principle of explosion be removed from constructive logic?

Classical logic has the theorem ($p\wedge\lnot p)\rightarrow q$, which I will call EFQ ("ex falso quodlibet"). Constructive logic often has the principle built in, in the form of an axiom ...
17
votes
3answers
2k 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 ...
6
votes
3answers
748 views

Looking for a Calculus Textbook

I want to start signal processing and I need a book that satisfies my mathematical requirements: I am in the third grade of high school and I don't know any useful thing about limit, differential, ... ...
3
votes
4answers
1k views

Multivariable Calculus Book Reference

I am looking for a multivariable calculus book that is really physics oriented. Anyone know of any? EDIT: My wife is looking to brush up on multivariable at the same time she needs to brush up on ...
12
votes
1answer
988 views

Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly

While reading David Williams's "Probability with Martingales", the following statement caught my fancy: Every subset of $\mathbb{R}$ which we meet in everyday use is an element of Borel ...
159
votes
37answers
14k views

Fun but serious mathematics books to gift advanced undergraduates.

I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but ...
64
votes
5answers
3k 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 ...
31
votes
10answers
13k views

Good books on mathematical logic?

I just started to learn mathematical logic. I'm a graduate student. I need a book with relatively more examples. Any recommendation?
26
votes
3answers
2k views

Consequences of Degree Theory

I'm preparing a presentation on an overview of algebraic and differential topology, and my introduction includes some motivational material on Degree Theory. I have two fundamental and invaluable ...
37
votes
15answers
9k views

Good Book On Combinatorics

What is your recommendation for an in-depth introductory combinatoric book? A book that doesn't just tell you about the multiplication principle, but rather shows the whole logic behind the questions ...
21
votes
6answers
4k views

Complete undergraduate bundle-pack [closed]

First of all I'm sorry if this is not the right place to post this. I like math a lot. But I'm not sure if i want to do a math major in college. My question is: Can you give me a list of books that ...
33
votes
7answers
7k 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 ...
30
votes
16answers
13k views

What are good books to learn graph theory?

What are some of the best books on graph theory, particularly directed towards an upper division undergraduate student who has taken most the standard undergraduate courses? I'm learning graph theory ...
16
votes
9answers
5k views

A Book for abstract Algebra

I am self learning abstract algebra. I am using the book Algebra by Serge Lang. The book has different definitions for some algebraic structures. (For example, according to that book rings are defined ...
35
votes
2answers
2k views

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$?

Is there an explicit isomorphism between $L^\infty[0,1]$ and $\ell^\infty$? In some sense, this is a follow-up to my answer to this question where the non-isomorphism between the spaces $L^r$ ...
26
votes
13answers
10k views

Complex Analysis Book

I want a really good book on Complex Analysis, for a good understanding of theory. There are many complex variable books that are only a list of identities and integrals and I hate it. For example, I ...
31
votes
1answer
464 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
7
votes
2answers
599 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 ...
21
votes
4answers
3k 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 ...
10
votes
3answers
3k views

Good problem book on Abstract Algebra

I am currently self-studying abstract algebra from Artin. In that background, I am looking for a problem book in a spirit somewhat similar to Problems in Mathematical Analysis by AMS so that I have a ...
15
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 ...
6
votes
2answers
156 views

There is no norm in $C^\infty ([a,b])$, which makes it a Banach space.

Does anyone knows a reference, which proves the following: Let $a,b\in \mathbb{R}$ with $a<b$. There is no norm in the space $C^\infty([a,b])$, which makes it a Banach space.
15
votes
2answers
3k 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 ...
5
votes
1answer
799 views

What is a good book to learn number theory?

What would be a good book to learn basic number theory? If possible a book which also has a collection of practice problems? Thanks.
2
votes
2answers
756 views

How many $N$ digits binary numbers can be formed where $0$ is not repeated

How many $N$ digits binary numbers can be formed where $0$ is not repeated. Note - first digit can be $0$. I am more interested on the thought process to solve such problems, and not just the answer. ...
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 ...
7
votes
3answers
263 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
812 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
97 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
940 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 ...
53
votes
3answers
11k 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 ...
66
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 ...
24
votes
8answers
2k 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 ...
25
votes
1answer
2k 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 ...
21
votes
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
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 ...