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
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1answer
377 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 A,\operatorname{rank}(x)\le\...
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 non-...
8
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 ...
7
votes
3answers
1k 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, ... ...
6
votes
1answer
2k views

What's a good book on advanced linear algebra?

I'm taking an advanced linear algebra course and I'm a little confused about books. The teacher said we could use any book we wanted to, but he recomended just Hoffman and Kunze and also Kostrikin, ...
4
votes
1answer
424 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 ...
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
4answers
353 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 ...
0
votes
1answer
92 views

Proof of Parseval's identity

I am seeking a proof for the following statement of the Parseval identity: If $f$ is a holomorphic function defined on the ball $B(0,r)$, with power series $f(z) = \sum_n a_n z^n$, then $2 \pi \...
179
votes
37answers
17k 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 ...
26
votes
6answers
7k 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 ...
36
votes
4answers
7k 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, ...
27
votes
9answers
8k 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 ...
72
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 ...
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 ...
14
votes
4answers
9k views

Difficulty level of Courant's book

I am currently studying Introduction to Calculus and Analysis by Richard Courant and Fritz John.I would like to compare Courant's book with Apostol's and Spivak's in terms of difficulty of the ...
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 ...
32
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 (...
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 ...
12
votes
7answers
6k 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 ...
22
votes
4answers
2k views

Good book on evaluating difficult definite integrals (without elementary antiderivatives)?

I am very interested in evaluating difficult definite integrals without elementary antiderivatives by manipulating the integral somehow (e.g. contour integration, interchanging order of integration/...
17
votes
2answers
1k views

$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$

Let $A$ be an integral domain of finite Krull dimension. Let $\mathfrak{p}$ be a prime ideal. Is it true that $$\operatorname{height} \mathfrak{p} + \dim A / \mathfrak{p} = \dim A$$ where $\dim$ ...
9
votes
1answer
4k views

High-level linear algebra book

Please, recommend high-level and modern books on linear algebra (not for first reading). Like Kostrikin, Manin "Linear algebra and geometry" or respective chapters of Lang "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.
15
votes
6answers
2k views

Algebraic Geometry Text Recommendation

I need to learn about Algebraic Geometry (perhaps from in the context of finite fields) and am looking for a recommendation for a text. Now, I've already done a search and checked out what was ...
11
votes
6answers
8k 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 ...
12
votes
5answers
3k 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 ...
12
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 ...
10
votes
7answers
13k 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?
7
votes
2answers
880 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 ...
9
votes
2answers
4k 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 ...
24
votes
1answer
6k 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. ...
10
votes
2answers
341 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.
8
votes
1answer
923 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 ...
8
votes
1answer
521 views

References on density of subsets of $\mathbb{N}$

I'm interested in books (maybe chapter(s) of a book), articles, results or whatever on the concept of density of subsets $A \subset \mathbb{N}$, often defined as: $$\lim_{n \rightarrow \infty} \frac{|...
3
votes
5answers
627 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 ...
16
votes
5answers
11k 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 ...
5
votes
2answers
214 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 ...
14
votes
2answers
997 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 ...
10
votes
4answers
168 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 $$\...
9
votes
3answers
2k views

All real functions are continuous

I've heard that within the field of intuitionistic mathematics, all real functions are continuous (i.e. there are no discontinuous functions). Is there a good book where I can find a proof of this ...
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 ...
5
votes
2answers
320 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 ...
11
votes
2answers
266 views

Is it possible that every set can be specified?

Is it possible for there to be a model of ZFC with the property that, for every set $S$ in the model, there is a unary predicate in the language of ZFC such that $S$ is the is the only set satisfying ...
9
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 ...
7
votes
3answers
612 views

Reference request for algebraic Peter-Weyl theorem?

It seems that, for $GL_n$, and possibly for something like complex reductive groups $G$ in general, there's an algebraic version of the Peter-Weyl theorem, which might say that the coordinate ring of $...
5
votes
1answer
1k 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, 'grade-...
9
votes
1answer
490 views

Some maps of the land of mathematics?

This question is motivated by a little anecdote. I was at home teaching some secondary school math to a relative. At some relax time, he glanced at a book I had over the table - it was some text about ...