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.
13
votes
0answers
647 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 ...
12
votes
0answers
154 views
Is there an axiomatic approach to ordinal arithmetic?
I've always wondered, is there an axiomatic approach to the arithmetic of ordinal numbers?
If so, I imagine it would be on par with set theory in terms of its proof-theoretic strength.
12
votes
0answers
295 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} ...
11
votes
0answers
159 views
Any proof to $\pi^{e}$'s irrationality?
I've searched for this for a while but get nothing...
There are plenty of proofs to irrationality of $e$,$\pi$,$e^{\pi}$. However, I can't find a proof for $\pi^e$. More, when searching for this I ...
11
votes
0answers
80 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 ...
10
votes
0answers
171 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 ...
10
votes
0answers
227 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 ...
10
votes
0answers
307 views
Hermite's solution of the general quintic in terms of theta functions
Can someone point me at or produce a translation or modern exposition of Hermite's solution of the general quintic in terms of theta functions? (the "before" and "after" steps are on the mathworld ...
9
votes
0answers
86 views
Basic categories cheat sheet
Has anyone come across a cheat sheet containing basic properties of the most well-known categories (i.e. does it have (co)products, (co)equalizers, (co)limits, etc?)?
8
votes
0answers
112 views
Hao Wang's $\mathfrak S$ system: a “transfinite type” theory?
Long ago, while I was reading a book ($*$) about the various way to build set theories (Zermelo-Freankel, Von Neumann–Bernays–Gödel, and type theories), I read about a variant of type theory with ...
8
votes
0answers
58 views
Expression of basis vectors of permutation modules in different bases.
Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
8
votes
0answers
164 views
Infinite dimensional constant rank theorem
Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
8
votes
0answers
218 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 ...
7
votes
0answers
138 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
154 views
Addition formula for $f_n(x+y)$ in closed form.
$n$ is a positive integer.
$$f_n(x)^n+\left(\frac{df_n(x)}{dx}\right)^n=1$$
$f_n(0)=0$,
$f_n'(0)=1$ then
I am looking for the addition formula for $f_n(x+y)$ in closed form.
if $n=1$ then
...
7
votes
0answers
75 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 ...
7
votes
0answers
119 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 ...
7
votes
0answers
69 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 ...
7
votes
0answers
156 views
Can all subseries of an infinite series be pairwise independent over $\mathbb{Q}$?
I'm wondering about a simple question that has multiple possible variants depending on a few parameters. The prototypical one would be:
Does there exist an infinite series such that any two ...
7
votes
0answers
155 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 ...
6
votes
0answers
154 views
Priority of the content of a note by Lebesgue from 1905
I refer to a note by Lebesgue Remarques sur la définition de l'intégrale, Bull.Sci.Math. 29 (1905) 272-275 not very known (see pdf for an exposition in English).
It is a pedagogical note containing a ...
6
votes
0answers
70 views
The center of a simply connected semisimple Lie group
I am learning about Lie groups, and I have the following basic question:
Every Lie group $G$ has a (unique) universal covering group $ \bar G $ that is simply connected, and such that the covering ...
6
votes
0answers
117 views
Alternative construction of the tensor product (or: pass this secret)
The paper Tensor products and bimorphisms by B. Banachewski and E. Nelson studies tensor products (defined by classifying bimorphisms) in concrete categories. It is quite interesting that their main ...
6
votes
0answers
87 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 ...
6
votes
0answers
191 views
Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?
Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem
Does the following generalization of that fact also hold?
Theorem: ...
6
votes
0answers
82 views
van der Waerden's original proof
I am looking for a book/site which has the english translation of the proof of van der Waerden's theorem as presented by van der Waerden himself. In other words is the translation of the paper:
...
6
votes
0answers
75 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 ...
6
votes
0answers
122 views
Articles on ideas in the history of mathematics notation?
I'm teaching a course this term on the history of scripts (writing systems) and rather than talking interminably about Semitic and Chinese and their spawn, I'd like to give students a more varied ...
6
votes
0answers
246 views
Resource: Group Theory
There is a website providing recent thesis in finite geometry. Is there any website with collection of recent thesis on (finite) group theory.
I want to see the Ph.D. thesis of Raymond T. Shepherd, ...
6
votes
0answers
92 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 ...
6
votes
0answers
260 views
Existence of non-atomic probability measure
The Question
Let $X$ be a set. Let $\mathcal{F}\subseteq P(X)$ be a $\sigma$-algebra. (Or, if it makes a difference, let $X$ be a topological space and $\mathcal{F}$ the Borel sets.) When can we ...
6
votes
0answers
369 views
How to use Hardy and Wright's text and what corresponding exercises/problem books can I do?
I have just started out with Hardy and Wright's An Introduction to the Theory of Numbers today. I find the lack of exercises in the book as a departure from the style of the textbooks we are so ...
6
votes
0answers
578 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 ...
5
votes
0answers
46 views
+100
Amenable group rings embeddable in skew fields
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent:
(1) the group ring $K[G]$ is a domain;
(2) $K[G]$ is ...
5
votes
0answers
179 views
Theory of structures with infinite Partial orders. $\langle H, \{\sqsubset_i \}_{i\in I} \rangle$
My recent interests have led me to have to deal with particular structures I have never seen before. Sets equipped with an infinite numbers of partial orders $\{\sqsubset_i:i\in I\}$.
I'm a bit ...
5
votes
0answers
74 views
Model theory in terms of type spaces/Lindenbaum algebras
Are there any good references that go into some detail of known 'translations' between properties of the type space of a model and the model theoretic properties of the model? All I seem to find are ...
5
votes
0answers
202 views
Problem of Scottish Book
Does anyone know if the problem 50 to Banach written in The Scottish Book is resolved? The problem is:
Prove that the integral of denjoy is a Baire functional in the space M ( that is to say, in ...
5
votes
0answers
34 views
“Inverse problem” for Brauer groups
This question is really just a curiosity, but I'm really interested in the answer.
Given a field $K$, we can form the set$^*$ $Br(K)$ consisting of equivalence classes of finite-dimensional central ...
5
votes
0answers
97 views
Algorithm to calculate multiple integral.
One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
5
votes
0answers
174 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 ...
5
votes
0answers
90 views
Coarse moduli space with no autmorphisms is also a fine moduli space
I'm working in the category of schemes over an algebraically closed field $k$, $Sch_k$. Suppose I have a contravariant functor $F:Sch_k\rightarrow (Set)$ which has a coarse moduli space $M$ (which is ...
5
votes
0answers
92 views
Emil Artin's proof for Wedderburn's Little Theorem
I am looking through different proofs for Wedderburn's Little Theorem, which states that every finite division ring is necessarily a field.
I would like to read Emil Artin's proof for this theorem:
...
5
votes
0answers
84 views
Fun problems with binary operations.
Does anyone know of a book / internet resource containing lots of problems relating to properties of binary operations?
An example of the sort of problem I'm looking for is:
Let * be a binary ...
5
votes
0answers
85 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 ...
5
votes
0answers
186 views
Estimator for sum of independent and identically distributed (iid) variables
Consider the Chernoff bound described in Theorem 1 of this paper:
Theorem 1. Let $X_1,\ldots,X_n$ be discrete, independent random variables such that $E[X_i] = 0$ and $|X_i|<1$ for all $i$. Let ...
5
votes
0answers
116 views
Has the $\Gamma$-like function $f_p(n) = 1^{\ln(1)^p} \cdot 2^{\ln(2)^p} \cdot \ldots \cdot n^{\ln(n)^p} $ been discussed anywhere?
In an older fiddling with the gamma-function (expanding on the idea of sums of consecutive like-powers of logarithms, similarly as the bernoulli-polynomials for the sums of like powers of consecutive ...
5
votes
0answers
96 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 ...
5
votes
0answers
111 views
How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?
This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
5
votes
0answers
146 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 ...
5
votes
0answers
127 views
Navier-Stokes Equation and turbulence, current status of research?
What is the currect research status of solving Navier-Stokes Equation, any up-to-date review/good paper on this topic?
Or direct numerical simulation is still the best way to understand the ...
