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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24
votes
9answers
17k 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 ...
58
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 ...
11
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, ...
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 ...
25
votes
3answers
918 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 ...
21
votes
5answers
756 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.
6
votes
4answers
6k 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
871 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 ...
12
votes
9answers
927 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 ...
6
votes
3answers
17k views

Eigenvectors of real symmetric matrices are orthogonal

Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = ...
15
votes
1answer
709 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" ...
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 ...
3
votes
5answers
1k 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 ...
14
votes
2answers
661 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 ...
12
votes
1answer
2k 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
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
4k 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
1answer
494 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
454 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 ...
6
votes
4answers
631 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 ...
5
votes
6answers
1k views

Best measure theoretic probability theory book?

I'm looking for a clear way to learn measure theoretic probability theory. Any suggestions?
8
votes
1answer
201 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 ...
8
votes
1answer
385 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 ...
6
votes
1answer
245 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 ...
6
votes
1answer
294 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 ...
5
votes
5answers
4k 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, ...
13
votes
2answers
323 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 ...
11
votes
2answers
723 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
403 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
5answers
837 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 ...
6
votes
4answers
317 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 ...
5
votes
3answers
289 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 ...
4
votes
1answer
555 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, ...
2
votes
1answer
175 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 ...
8
votes
2answers
207 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?
6
votes
1answer
827 views

Good resources (book or otherwise) to learn/study basic Combinatorics

I'm currently studying basic Combinatorics for a college course and my professor is awful (and that is being generous). Therefore I'm looking for good resources to learn basic Combinatorics so that I ...
3
votes
1answer
268 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 ...
2
votes
3answers
261 views

Text recommendation for introduction to linear algebra

I have learned linear algebra before, but with my right brain - basically nothing left, so, anyone can recommend me good textbook so I can learn it again? The text should subject to the conditions ...
2
votes
2answers
580 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 ...
2
votes
2answers
471 views

Constructing dependent product (right adjoint to pullback) in a locally cartesian closed category

I've been trying to find a proof that the pullback functors in a locally cartesian closed category have right adjoints (used to model the notion of indexed product inside a category (rather than ...
2
votes
1answer
845 views

Are there any good algebraic geometry books to recommend? [duplicate]

Possible Duplicate: (undergraduate) Algebraic Geometry Textbook Recomendations I am interested in algebraic number theory and I am recently acquainted with the theory of valuations, which ...
2
votes
5answers
1k views

Where can I find a review of discrete math

I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago.
9
votes
2answers
320 views

The history of set-theoretic definitions of $\mathbb N$

What representations of the natural numbers have been used, historically, and who invented them? Are there any notable advantages or disadvantages? I read about Frege's definition not long ago, ...
5
votes
1answer
372 views

Why are these two definitions of a perfectly normal space equivalent?

I've been skimming through some topology textbooks recently. Some sources, (such as Munkres' Topology and Willard's General Topology) define a space $(X,\mathcal{T})$ to be perfectly normal iff $X$ is ...
5
votes
1answer
519 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
2answers
143 views

Are there any foundations in which the universe itself gets dynamically extended?

In the foundations I'm familiar with, you don't ever "create" anything. Take ZFC for example. If we need the real numbers, we search in our static universe for a set $\mathbb{R}$ of the correct ...
3
votes
1answer
180 views

Ideals in ring of continuous functions $\mathcal{C}[0,1]$ … NBHM- Algebra

I would like to compile all questions I have encountered with Ideals in the ring $\mathcal{C}[0,1]$ of all continuous real valued functions and ask if there are any gaps. Question is to see if : ...
1
vote
2answers
1k views

Ellipse fitting methods.

I have set of points and want to fit ellipse to this set. I have found only function which fits ellipse in least squares sense. In this set of points there are some noise points which should not be ...
48
votes
4answers
5k 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 ...