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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9
votes
6answers
756 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 ...
6
votes
1answer
610 views

Treatise on non-elementary integrable functions

All of us mathematicians after some time (and trial-and-error, of course) we are able to guess with reasonable accuracy whether or not a given function is elementary integrable (test yourself: ...
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 ...
13
votes
2answers
2k 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.
3
votes
1answer
686 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 $a \in \mathbb{N}$, ...
34
votes
9answers
11k views

References for the multivariable calculus

Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing ...
50
votes
17answers
14k 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 ...
35
votes
16answers
14k 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 ...
11
votes
8answers
1k views

Book about technical and academic writing

I'm in the process of writing my Master's Thesis on automata theory. The writing must be in English which isn't my mother tongue. So the question is, given that this is my first time long (hundred ...
13
votes
1answer
14k views

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ...
16
votes
1answer
1k 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 ...
2
votes
2answers
1k 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. ...
5
votes
2answers
1k views

Introductory book on Topology [duplicate]

Possible Duplicate: choosing a topology text best book for topology? What book would you recommend for an undergraduate wanting to learn the basics of topology? I've come across several ...
92
votes
4answers
14k views

Books that every student “needs” to go through

I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden). I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on ...
75
votes
6answers
4k 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 ...
33
votes
12answers
17k 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?
32
votes
4answers
6k views

Learning Roadmap for Algebraic Topology

I am now considering about studying algebraic topology. There are a lot of books about it, and I want to choose the most comprehensive book among them. I have a solid background in Abstract Algebra, ...
47
votes
8answers
18k 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 ...
25
votes
14answers
11k views
18
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 ...
12
votes
6answers
6k 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 ...
4
votes
1answer
668 views

Single Variable Calculus Reference Recommendations

This question is a generalization of the common question asking for calculus references. It is here to abstract away the repetition, and give a canonical resource for calculus references. I'm ...
11
votes
3answers
4k 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 ...
25
votes
4answers
4k 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 ...
18
votes
2answers
423 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 - ...
17
votes
4answers
2k 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 ...
9
votes
7answers
12k 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
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 ...
36
votes
7answers
10k 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 ...
23
votes
7answers
17k 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 ...
65
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 $\mathbb{R}^n$. ...
19
votes
10answers
7k 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 ...
17
votes
6answers
3k 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 ...
12
votes
7answers
5k 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 ...
9
votes
4answers
1k 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 ...
32
votes
1answer
554 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 ...
16
votes
3answers
801 views

Reference for combinatorial game theory.

What is a good reference material for elementary combinatorial game theory? By combinatorial game theory I mean chiefly the study of zero-sum, deterministic two-player games (perhaps even more ...
15
votes
4answers
552 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
234 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.
5
votes
4answers
546 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 ...
5
votes
1answer
1k 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.
7
votes
3answers
963 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, ... ...
166
votes
37answers
16k 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 ...
42
votes
15answers
64k 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 ...
28
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 ...
25
votes
6answers
6k 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 ...
24
votes
2answers
3k 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 ...
21
votes
8answers
6k 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 ...
36
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 $L^r$ ...
8
votes
7answers
8k 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 ...