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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44
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0answers
1k views

Theorem that von Neumann proved in five minutes.

In "How To Solve It", George Pólya writes: "There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it ...
32
votes
0answers
2k views

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
16
votes
0answers
346 views

Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology?

Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a ...
16
votes
0answers
385 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} ...
15
votes
0answers
531 views

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty ...
14
votes
0answers
139 views

References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
12
votes
0answers
361 views

Normalizers of automorphism groups

In abstract groups $\Gamma$ the normalizer $N_\Gamma(S)$ of a subset $S\subseteq\Gamma$ is the subgroup of all $x \in \Gamma$ that commute with $S$, i.e. $xS = Sx$, i.e. $x\ y\ x^{-1} \in S $ for all ...
11
votes
0answers
259 views

Anecdote about mathematicians leaping to tops of problems and then building a staircase down?

I've run across this cute little story before, and now for the life of me I can't find it anywhere. It goes something like: Two people are looking out onto a mathematical landscape, and there are ...
11
votes
0answers
235 views

On the Constant Rank Theorem and the Frobenius Theorem for differential equations.

Recently I was reading chapter $4$ (p. $60$) of The Implicit Function Theorem: History, Theorem, and Applications (By Steven George Krantz, Harold R. Parks) on proof's of the equivalence of the ...
11
votes
0answers
166 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
11
votes
0answers
1k views

Which is better strategy to learn and read books, traditionally one by one OR re-read carefully on perfect books

(Just focus on how to learn and master the stuff pretty well, not involve the aspect of courses or exam) Because recently I always feel that the time and energy are pretty limited, I want to try ...
10
votes
0answers
269 views

Maximal ideal space of $C^*$-algebra of Riemann integrable functions

Let $R([0,1])$ be the unital commutative $C^*$-algebra of complex valued Riemann integrable functions on $[0,1]$ with pointwise operations and the supremum norm. In the 1980 paper The Gelfand space ...
9
votes
0answers
99 views

Size of connected regions on a randomly-colored infinite chessboard

Consider an infinite chessboard where each square is colored white with probability $p$ and black with probability $1-p$. Suppose without loss of generality that the square at $(0,0)$ is white. We ...
9
votes
0answers
163 views

Reference on standard types

This question is about what I presume is a basic construction in type theory. The finite types are defined as follows: 0 is a finite type; if $\sigma, \tau$ are finite types, then so is ...
8
votes
0answers
152 views

Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...
8
votes
0answers
128 views

Cantor-Schröder-Bernstein without elements

The Cantor-Schröder-Bernstein Theorem for the category of sets has several "analogues" for other categories as well, for example measurable spaces and operator algebras (but not for arbitrary ...
8
votes
0answers
265 views

How much time is reasonable to complete baby Rudin?

I've been teaching myself math for more than a year. My current aim is towards algebraic topology and differential geometry. Apart from a messy (by which i mean some rigorous and some not) ...
8
votes
0answers
137 views

Where to go after _To Mock A Mockingbird_?

So long ago I read Raymond Smullyan's delightful To Mock A Mockingbird, a gentle introduction to combinatory logic (representing combinators as 'birds' singing back and forth to each other). I fell ...
8
votes
0answers
335 views

Is it possible to learn mathematics right from the source instead of reading textbooks. By studying the masters and not their pupils

i was wondering if mathematics learning process require the use of textbooks. When i was a high school student, i read as a preparation for university, Legendre book on Elements of geometry and ...
8
votes
0answers
107 views

Reference for l-adic Lie algebras

I don't know much at all about Lie algebras or representation theory, and I'm trying to read Ribet's `Review of Abelian l-adic Representations and Elliptic Curves'. Is there a standard reference for ...
8
votes
0answers
95 views

Mean hitting time: reference request

After answering the question Expectation of a stopping time uniquely determined by a function I was looking for the literature on the mean hitting/exit time for a discrete-time Markov process. In Meyn ...
8
votes
0answers
269 views

Stokes theorem on Lipschitz-manifolds?

I was wondering if Stokes' theorem could be formulated in a setting which could be easily applied in situations where the traditional form cannot, such as on manifolds with corners like a rectangle or ...
8
votes
0answers
221 views

What are D-branes (in a topological field theory)?

In the past couple years, I've read many words pertaining to D-branes without feeling I have really comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
7
votes
0answers
120 views

Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
7
votes
0answers
85 views

Reference request on the occasion of Maria Agnesi's birthday

Today's Google doodle on google.co.uk, google.de, google.it, google.fr, google.ch, google.at and probably several others but not on google.com celebrates Maria Gaetana Agnesi, whose birthday is today. ...
7
votes
0answers
98 views

Reference request: a differential equation arising in geometry

$$ \frac{d\beta}{d\alpha} = \frac {\sin\beta}{\sin\alpha} $$ In what contexts (if any) is this equation known to occur?
7
votes
0answers
96 views

To whom is the proof that $A_n$ is simple for $n\geqslant 5$ due, in Rotman's book?

The proof in Rotman's book, Introduction to the Theory of Groups, that $A_n$ is simple consists of the observation that $A_n$ is generated by the $3$-cycles, and hence that if a normal subgroup $H\lhd ...
7
votes
0answers
151 views

Adelic/Idelic method for number fields

I'm searching for a place where is developed all the machinery of adeles and ideles of a number field; the functional equation for the zeta functions is gained by idelic integration, and it's seen the ...
7
votes
0answers
117 views

Elementary references on Robinson Arithmetic, Prim. Recursive fns etc.

I'm in the middle of revising my freely available and much-downloaded introductory notes Gödel Without (Too Many) Tears. (They are a sort of cut down version of part of my Gödel book, and I'm ...
7
votes
0answers
140 views

Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and ...
7
votes
0answers
256 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
7
votes
0answers
198 views

Generalizing the natural numbers - has this been studied before?

The ordinals numbers generalize the natural numbers and satisfy a generalized induction principle. However, the algebraic properties of the ordinal numbers aren't so good. For example, ordinal sums ...
7
votes
0answers
218 views

Kakeya Needle problem video

I'm intruiged by the Kakeya Needle problem, described here on Wikipedia. Wikipedia has a nice animation of a needle turning through a hypo-cycloid: What I'm searching for is a visualisation of the ...
7
votes
0answers
187 views

Citation for subset complement result

Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
7
votes
0answers
247 views

What is the class of languages accepted by DFAs whose transition monoids are transitive permutation groups?

In the Wiki page A permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states. ..... A formal language is p-regular ...
7
votes
0answers
120 views

Two cones over a projective variety

Let $A$ be a commutative graded algebra over a field $k$ and $X=\operatorname{Proj}(A)$ is a smooth scheme, then $E=\oplus_{i \geq 0} \mathcal O_X(i)$ is a quasi-coherent sheaf of algebras on $X$. I ...
7
votes
0answers
188 views

Ideal of the pullback of a closed subscheme

Let $f : X \to Y$ be a morphism of schemes and $J \subseteq \mathcal{O}_Y$ a quasi-coherent ideal. Let $I$ denote the image of $f^* J \to f^* \mathcal{O}_Y = \mathcal{O}_X$. Then $I \subseteq ...
7
votes
0answers
142 views

Measure-driven differential equations

Background: I need some help to understand the concept behind measure-driven differential equations. The solution of an ordinary differential equation is continuous. In order to describe discontinuous ...
7
votes
0answers
165 views

Reference suggestion: eigenvalues of tridiagonal matrices

I would like to ask for a reference on the problem of computing the eigenvalues/eigenvectors of tridiagonal matrices (not necessarily with constant diagonals). I have seen authors use continued ...
7
votes
0answers
305 views

Klein's Erlangen program taken seriously

Felix Klein suggested in his Erlangen program a way to classify geometries based on group theory. According to Wikipedia, we have the following definition: A Klein geometry is a pair (G, H) where ...
7
votes
0answers
116 views

Higher Order Coarea Formula

I was wondering, if there is a generalization of the coarea formula to higher order derivatives, which would allow one, for example, to relate the norm of the Hessian of a real-valued function $u$ to ...
7
votes
0answers
466 views

“The Galois group of $\pi$ is $\mathbb{Z}$”

Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question: The Galois group of $\pi$ is $\mathbb{Z}$. In what sense/framework is ...
6
votes
0answers
45 views

Representation theorem for Heyting algebras?

A fundamental theorem by Stone asserts that any Boolean algebra is isomorphic to a subalgebra of the archetypical Boolean algebras, that is the power sets of a set $X$ (equipped with intersection, ...
6
votes
0answers
110 views

Category Theory textbook (learning through guided discovery Dummit and Foote)

sorry for asking the same question in a slightly different angle, I want a book in Category Theory similar to Dummit and Foote's book in Abstract Algebra. I want it to have tons of examples and ...
6
votes
0answers
40 views

Recurrence relations with multiple roots of auxiliary equation

Suppose we have a homogeneous linear recurrence relation of the form $$u_{n+r}+a_1u_{n+r-1}+\dots +a_ru_n=0$$ The auxiliary equation is then $$f(x)=x^r+a_1x^{r-1}+\dots +a_r=0$$ It is well known that ...
6
votes
0answers
71 views

A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
6
votes
0answers
149 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
6
votes
0answers
185 views

Improper integral with trig functions

A colleague of mine came across this equation on an old exam: $$ \int_0^{\pi}{\cos n\theta\over\cos\theta-\cos\phi}\,d\theta={\pi\sin n\phi\over\sin\phi} $$ It said on the exam that students were ...
6
votes
0answers
528 views

Optimization / personalization within clusters

I have the following optimization problem: I have a (random and very noisy) objective function f(A, P), where A is a vector of "observable" parameters of the input and P is the parameters that I can ...
6
votes
0answers
151 views

Point set topology from an algebraic perspective?

I got this idea of viewing a topology as an operation on a ring of sets. Let $\mathcal R = (\mathcal P(X), \cap, \triangle)$ be a ring of sets. ($\triangle$ is the symmetric difference operation and ...