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)

40
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 ...
25
votes
0answers
885 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 ...
22
votes
0answers
430 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} ...
16
votes
0answers
491 views

Which universities teach true infinitesimal calculus?

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
16
votes
0answers
186 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 ...
14
votes
0answers
111 views

Representability of diagonal of $\mathscr{M}_g$

Let $\mathscr{M}_g$ be the moduli stack of genus $g$ curves ($g \geq 2$). That is, $\mathscr{M}_g$ is the category whose objects are proper smooth morphisms $f: C \to S$ whose geometric fibers are ...
13
votes
0answers
446 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 ...
13
votes
0answers
188 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 ...
12
votes
0answers
128 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 ...
11
votes
0answers
148 views

References on filter quantifiers

This post is primarily a reference request. In combinatorics and other areas, we use filter quantifiers to simplify the statements of various definitions, theorems and proofs. The general idea is ...
11
votes
0answers
124 views

How to name these “ideals”?

Background. Let $\mathcal{C}$ be a symmetric monoidal category with unit $\mathbf{1}$. A subobject of $\mathbf{1}$ is just a monomorphism $I \to \mathbf{1}$. We may also call this an ideal of ...
11
votes
0answers
181 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 ...
11
votes
0answers
286 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 ...
10
votes
0answers
276 views

Analysis of the function $\prod\limits_{n=-\infty}^{ \infty }(1-e^{-a{(x+n)^2} })$

While playing on pruduct series, I noticed interesting function that it has similar behavior to $C.\sin^2(\pi x)$ function . I need help about some results below. $$f(x)=\prod\limits_{n=-\infty}^{ ...
10
votes
0answers
186 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 ...
10
votes
0answers
443 views

Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.

Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$. The following is from the book "Sobolev spaces" ...
10
votes
0answers
299 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
65 views

What can we learn about a magma by studying these monoids?

Given a magma $(X,*)$, we get three monoids in the following way. First, define a pair of functions $L,R : X \rightarrow (X \rightarrow X).$ $$(Lx)(y) = x*y,\quad (Rx)(y) = y*x$$ Then each of the ...
9
votes
0answers
196 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 ...
8
votes
0answers
144 views

A question on popularization of math: inspiring the beauty of mathematics while making New Year's wishes

Many fellow students of mine today shared by various means the following picture: . I was told that this picture is supposed to communicate the beauty of math in a funny way, but I really can't ...
8
votes
0answers
65 views

The least subset of $\mathbb{R}_{>0}$ that includes $1$, and is closed under addition, multiplication, reciprocation, and exponentiation.

Let $S$ denote the least subset of $\mathbb{R}_{>0}$ that includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
8
votes
0answers
139 views

Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, ...
8
votes
0answers
185 views

Representation theory of the general linear group over a finite prime field

The irreducible modules of $\operatorname{GL}_n(\mathbb C)$ over $\mathbb C$ are completely classified and well-understood via Schur-Weyl duality, the algebraic Peter-Weyl theorem and the entire ...
8
votes
0answers
92 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 ...
8
votes
0answers
96 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. ...
8
votes
0answers
212 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 ...
8
votes
0answers
134 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 ...
8
votes
0answers
230 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 ...
8
votes
0answers
287 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 ...
8
votes
0answers
289 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 ...
8
votes
0answers
161 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
282 views

A few standard results (on metrizability and relative separation strength) without choice?

I've been going back over some results from Munkres's Topology, and I'm curious about some things.... I know that Choice principles have some connection to the separation axioms (in ZF, at ...
8
votes
0answers
225 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 ...
8
votes
0answers
114 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
99 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
301 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
231 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
117 views

Is Stokes theorem on $n$-cells equivalent to Stokes theorem with singularities

Could you tell me if Stokes theorem for $n$- cells is equivalent to Stokes theorem for manifolds with singularities (where the set of singularities has measure zero/ is negligible)? I mean, would it ...
7
votes
0answers
101 views

An Additional Rule for Calculus

Background The rules for differentiating elementary functions (arithmetic, exponential, trigonometric, etc.) together with the chain rule for differentiating compositions of functions are often ...
7
votes
0answers
133 views

A More Advanced Version Of Aluffi

Paulo Aluffi's Book, Algebra, Chapter 0 aims to teach basic algebra from a categorical viewpoint. The first chapters of the book, however, introduce groups and rings using very basic categorical ...
7
votes
0answers
74 views

Bousfield–Kan spectral sequence for homotopy colimits

Let $\mathcal{J}$ be a small category and let $X : \mathcal{J} \to \mathbf{sSet}$ be a diagram. We define its homotopy colimit $\newcommand{\hocolim}{\mathop{\mathrm{ho}{\varinjlim}}}\hocolim X$ ...
7
votes
0answers
176 views

Irreducibility of cyclotomic polynomials via schemes

A few months ago, someone told me there existed a scheme theoretic proof of the irreducibility of cyclotomic polynomials. I've tried coming up with a proof, and when that didn't really yield anything ...
7
votes
0answers
283 views

How can I have a copy of this old paper?

How can I have a copy of this old paper and a translation of it? Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459.
7
votes
0answers
147 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
89 views

Raising a partial function to the power of an ordinal

Consider a set $X$, and let $f : X \rightarrow X$ denote a partial function. Then for natural $n$, we can define $f^n$ as iterated composition, e.g. $f^2 = f \circ f$. Now suppose that $X$ is also ...
7
votes
0answers
97 views

Reference request: certain special LFTs

$\newcommand{\cs}{\operatorname{cs}}$ My actual question is at the bottom. Let $$ a\diamond b = \frac{a+b}{1+ab} $$ and $$ a\dagger=\dfrac{1-a}{1+a}. $$ Then $$ a\dagger\dagger = a $$ and ...
7
votes
0answers
105 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 ...
7
votes
0answers
116 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
187 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
613 views

Solution manual to Larson's “Problem Solving through Problems”

I am working through Larson's "Problem Solving through Problems" (http://math.la.asu.edu/~ifulman/mat194/problem-solving.pdf) but many of the problems have neither solutions nor sources included. Does ...