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.

learn more… | top users | synonyms (3)

8
votes
1answer
173 views

Representations of SL(2) as an algebraic group

I understand the finite-dimensional representations of $\text{SL}(2,\mathbb C)$ as a Lie group and their correspondence with Lie algebra representations of $\mathfrak{sl}(2,\mathbb C)$. Does anyone ...
1
vote
0answers
42 views

Software for knotted $\mathbb{S}^2$'s in $\mathbb{S}^4$

According to the work of J. Scott Carter you can draw pictures of knotted surfaces in four-space in several different ways. I know the man is a real artist in this, but did anybody come across some ...
4
votes
1answer
518 views

Eigenvalues of anti-circulant matrices using 1-circulant matrices

Is there any theorem to find the eigenvalues of any anti-circulant matrix using the equivalent (with the same first row) circulant matrix. I found out that, for any anti-circulant matrix, the ...
1
vote
2answers
160 views

Central limit theorems, Almost sure invariance principles and Brownian motion

In a paper I was reading on dynamics, I came across a proof of a central limit theorem in a certain situation using brownian motion and an almost sure invariance principle. I am not very experienced ...
1
vote
1answer
1k views

Algorithm for finding limits of compositions of simple functions?

There are two questions: Define the set $S$. Compute the limit of functions $f/g$ for functions $f,g\in S$, where $S$ is defined in the following way. All constant function are in $S$, $f(n) = ...
5
votes
4answers
745 views

Where can I find good, free resources on differential equations?

I'd like to know if there are any good online books, lecture notes, videos, tutorials, or similar that are free to the public (on differential equations). Suggestions are welcome!
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 ...
1
vote
0answers
205 views

What is a Hodge-Tate decomposition of an algebraic group?

I want some reference to learn basic things about Hodge-Tate decomosition, and what back ground I need.
64
votes
20answers
48k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
80
votes
16answers
20k 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?
1
vote
0answers
136 views

Books on multi-linear algebra for CS students

I was wondering if anybody had any good recommendations on books on multi-linear algebra for graduate and ~applied computer science students. More specifically, I am looking for reference texts that ...
19
votes
3answers
543 views

Getting the name of combinatorial problems

I'll often find myself with some combinatorial problem that's obviously been studied before. For example, "Find the smallest set(s) of positive integers such that every integer from 1 to n is the sum ...
2
votes
1answer
412 views

Illustration of vector calculus vs. differential forms

I am looking for a nice illustration of how vector calculus relates to differential forms. A demonstration that employs physics is appreciable (e.g. electromagnetism). In particular, while dualizing ...
11
votes
1answer
376 views

Is it possible to deduce a model for hyperbolic geometry from a synthetic set of axioms a la Euclid/Hilbert/Tarski?

Motivation I learned from Emil Artin's book Geometric Algebra that the standard incidence axioms of affine geometry (two points determine a unique line, parallel postulate, no three collinear points ...
4
votes
3answers
1k views

What are the symmetries of the tetrahedron?

Suppose I like combinatorics, and want to count how many ways to paint the faces of a tetrahedron using a pallet of $x$ colors. I don't want to over count cases where I could just rotate one painted ...
2
votes
0answers
467 views

Properties of the universal cover of CW-complexes

Let $Y$ be a CW-complex and $X$ its universal cover. Could you give me a proof (or a referece) for the following fact: $X$ is contractible $\Leftrightarrow$ $H_i(X)=0$ $\forall i\geq2$ ...
3
votes
1answer
67 views

Collection of congruencies

I have been going around various questions based on number theory in this forum, and what I have found is that congruencies serve as an important tool in many of the questions and actually simplify ...
8
votes
1answer
541 views

What is the primary source of Hilbert's famous “man in the street” statement?

I read somewhere a long time ago that Hilbert once said words (no doubt in German) to the effect that any mathematician worth his salt ought to be able to explain his results to any man in the street. ...
4
votes
1answer
788 views

What is a Weil-Deligne representation?

Can anyone explain (or give reference) Weil-Deligne representations, and how they are linked with Galois representations of p-adic field (on l-adic vector spaces or else) ?
0
votes
1answer
77 views

Low distortion embeddings (reference request)

I read about the Johnson Lindenstrauss Lemma. I googled and found that low distortion embeddings is a live subject, but it seems that many interesting results are already known. Is there a book on ...
24
votes
0answers
492 views

Dedekind Sum Congruences

For $a,b,c \in \mathbb{N}$, let $a^{\prime} = \gcd(b,c)$, $b^{\prime} = \gcd(a,c)$, $c^{\prime} = \gcd(a,b)$ and $d = a^{\prime} b^{\prime} c^{\prime}$. Define $\mathfrak{S}(a,b,c) = a^{\prime} ...
10
votes
4answers
2k views

Understanding the Musical Isomorphisms in Vector Spaces

I am trying to solidify my understanding of the muscial isomorphisms in the context of vector spaces. I believe I understand the definitions but would appreciate corrections if my understanding is not ...
8
votes
3answers
2k views

What is the connection between linear algebra and geometry?

I am currently studying linear algebra. Yet, I found discussions about linear algebra usually explain things in a geometric fashion. I am quite confused on how to link up these two topics. Can ...
5
votes
0answers
221 views

Reference: Geometric group theory

H. Bass has studied existence of lattices on trees. Can someone suggest a (readable) reference for lattices on graphs?
10
votes
2answers
2k views

Prerequisites for Atiyah Macdonald

I am currently doing a one semester course on groups and rings where we have learned about (so far): Definitions of groups, subgroups, cyclic and normal subgroups, the symmetric group, homomorphisms, ...
3
votes
1answer
75 views

characterization of residual finiteness for finitely generated groups

Where can I find a proof of the fact that a finitely generated group is residually finite if and only if it acts faithfully on a locally finite rooted tree?
4
votes
2answers
325 views

A sum involving the Möbius function

I am trying to find some work done on the following: $$\sum_{d \vert n}\frac{2^{\omega(d)}}{d}\mu(d)$$ where $\omega(d)$ is the number of distinct prime factors of $d$ and $\mu$ is the Möbius ...
7
votes
3answers
249 views

“Great Theorems” references

I read a great book a few years ago that gave itself this description: For disciplines as diverse as literature, music and art, there is a tradition of examining masterpieces - the "great novels", ...
17
votes
1answer
929 views

What was the last mathematical paper published in Latin?

From an answer to a previous question I learned that Peano published in Latin as long as 1889. What was the last mathematical paper/book of recognized importance published in Latin?
6
votes
2answers
498 views

Liouville Theorem

Liouville theorem for superharmonic functions states that Any bounded function $f:\mathbb R^n\to\mathbb R$ admitting an inequality $\Delta f\leq 0$ on $\mathbb R^n$ is a constant function. Here ...
2
votes
0answers
375 views

Learning analysis through topology

One of my supervisors once mentioned that when he was learning analysis he learnt it backwards. He learnt topology first and then saw analysis after, instead of the usual approach of doing everything ...
8
votes
4answers
1k views

Where can one find (freely, online) mathematical articles with a fighting chance to be understood by high school students and undergraduates?

I am an undergraduate non-math major. I just finished my university's engineering calculus series, looking forward to linear algebra in the coming semester. To be frank, I always despised math because ...
5
votes
1answer
231 views

Why has the extreme value function never been defined? –Or has it?

Just as when x and y are arbitrary real numbers, we often wish to consider their distance apart, and use the absolute value function to do so (namely, by means of the expression |x – y|), so also when ...
4
votes
2answers
3k 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 ...
12
votes
1answer
314 views

Trigonometric identity

It is well known (?) that if $\alpha+\beta+\gamma=\pi$ then $4\sin\alpha\sin\beta\sin\gamma = \sin(2\alpha)+\sin(2\beta)+\sin(2\gamma)$ (I think I've seen it in some late-19th-century books, and I ...
4
votes
1answer
105 views

For a ring of char $p$ where $p>0$ is a prime, what does $R^{1/p}$ mean?

If $R$ is a ring of characteristic $p\gt 0$, what does $R^{1/p}$ mean? I am not sure how to search for it, since I don't know a name for it. From the notation, it seems to be a ring consisting of the ...
10
votes
3answers
5k views

Is there an English translation of Diophantus's Arithmetica available?

It should be in the public domain (obviously), so I'd thought I could find the English text on the web somewhere. Apparently not?
19
votes
2answers
686 views

Books on topology and geometry of Grassmannians

Grassmannians are a pretty useful subject in numerous fields of mathematics (and physics). In fact, it was the first non-trivial higher-dimensional example that was given in an introductory projective ...
3
votes
0answers
203 views

Formally undecidable problems on finitely presented quandles

In the literature, one sometimes sees the claim that finitely presented quandles (in particular, knot quandles) are "hard to deal with". Hence, a great deal of effort has gone into studying finite ...
3
votes
3answers
316 views

Introduction to Information Theory

I'm studying bioengineering but in conversations and reading I've found that a great background in information theory as it applies to probability, statistics, random process, causation and inference ...
10
votes
1answer
442 views

Proof of Hölder inequality by differentiation

I need a reference where we can read a proof of the inequality $\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$ where $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$ for $L^p$-spaces of a measure ...
5
votes
1answer
272 views

Comparison between Rademacher average and random average

Let $X$ be a Banach space. We let $j=e^{2i\pi/3}$. Let $(\epsilon_i)$ be a sequence of independent Rademacher variables on a fixed probability space $\Omega$. Let $(\varphi_i)$ be a sequence of ...
3
votes
3answers
1k views

Numerical Analysis References

Could anyone suggest any good (perhaps online ref papers) reference material on numerical analysis focusing on determining accuracy/estimated errors, rates/orders of convergence especially when ...
16
votes
1answer
1k views

A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded

The title is a claim my classmate made during our summer vacation :D He showed me a TeX file describing a proof of his claim, and it contains a fairly short but elegant proof. He says that the ...
8
votes
1answer
5k views

Understanding the definition of dense sets

I am recently confused about the definition of dense sets. What I learned is as the following In topology and related areas of mathematics, a subset $A$ of a topological space $X$ is called dense ...
2
votes
2answers
249 views

How to compute the formal group law of K-Theory

Could anyone point me to a reference where the formal group law of (topological or motivic) K-theory is computed in as much detail as possible?
13
votes
5answers
2k views

Software to draw links or knots

I am looking for software that can aid me in drawing knots and links. There are of course (examples) knotplotters all over the web, but they can only draw specific knots. What I am looking for is the ...
4
votes
1answer
466 views

Which branch of mathematics is this and what are the introductory references?

I am self-studying a physics textbook on waves. While discussing solutions to linear homogeneous ODEs, the author talked about the exponential as "irreducible" solutions and on a footnote, said that ...
8
votes
1answer
279 views

An infinite series involving the Zeta Function

I am wondering if anyone knows how to evaluate either of the following sums in terms of known constants: $$\sum_{k=2}^{\infty}-\frac{\zeta^{'}(k)}{\zeta(k)},$$ and ...
3
votes
1answer
573 views

Proof of Kirchhoff's theorem for directed nonsimple graphs?

There's a marvelous theorem in graph theory that reduces the count of spanning trees for a graph to a computation of determinant of a naturally-defined matrix (the Laplacian matrix): ...