Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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8 votes
5 answers
476 views

Reference for : a Fréchet nuclear space is Montel

I'm looking for a reference to cite regarding the property presented in the title: "Closed and bounded sets of a nuclear Fréchet space are compact" Thank you in advance for the help!
15 votes
2 answers
2k views

Categorification of probability theory: what does a "probability sheaf" tell us (if anything) about probability theory?

Disclaimer: I only have a superficial knowledge of what category theory and related subjects are concerned with. So, my understanding is that category theory and related fields of higher mathematics ...
15 votes
0 answers
415 views

Grothendieck dessins d'enfants - current surveys or text you can recommend?

I was recommended this forum to be the leading site for algebraic geometry, so I would like to ask you a question about Grothendieck dessins d´enfants. My background is in maps on surfaces (graph ...
5 votes
0 answers
127 views

On a reference for computing global spectrum of $A_n$-curve singularities, by H.Dao and E.Faber

This question is about chasing down a reference in a paper relating to non-commutative crepant resolutions and Cohen-Macaulay representation theory. Allow me to first give a minor introduction. Let $(...
37 votes
2 answers
6k views

What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
6 votes
1 answer
502 views

Directed graph minor theorems

In proving the graph minor theorem, Robertson and Seymour proved a stronger statement, namely that the directed graph minor theorem is true, using the definition A directed graph is a minor of ...
4 votes
1 answer
238 views

Latest progress on Tarski numbers

Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group? The second question is the same as in the title: What is the latest ...
0 votes
0 answers
173 views

Generalization of elementary symmetric polynomials

The elementary symmetric polynomials (ESPs) are defined as - \begin{align*} E_{1}^{1} &= X_1, \\ E_{1}^{2} &= X_1 + X_2, \\ E_{2}^{2} &= X_1 X_2, \\ E_{2}^{3} &= X_1 X_2 + X_1 X_3 + ...
9 votes
0 answers
351 views

When is an increasing union a colimit?

Let's consider a diagram $\Phi: \lambda \to \mathcal{T}_*$ $$ X_0 \to X_1 \to \cdots \to X_\xi \to X_{\xi+1} \to \cdots $$ of pointed spaces, indexed by some ordinal $\lambda$, in which each $X_\xi$ ...
2 votes
0 answers
53 views

Free resolutions of universal enveloping algebras for simple, finite dimensional Lie algebras

I'm currently studying Anick's resolution on the context of universal enveloping algebras for certain Lie algebras, namely some of the smallest cases: $A_1,A_2,A_3,B_2,G_2$, and so on. What are some ...
5 votes
1 answer
250 views

A graph similar to the Bruhat graph, what is it called?

The weak Bruhat graph (or 1-skeleton of the permutohedron) $B_n$ can be constructed as follows: the vertices of $B_n$ are the permutations of the tuple $(1,...,n)$, two are joined by an edge, if they ...
9 votes
1 answer
655 views

Learning from unsuccessful attempts at the Poincaré conjecture

This is reposted from MSE, but perhaps it is more appropriate to post it here. Let me know if I'm wrong. Among the many unsuccessful attempts at solving the Poincaré conjecture, I'm wondering if there ...
1 vote
0 answers
125 views

Angular excitations and Schrodinger operators with radial potential in N-dimensions

Can someone please explain the following in mathematical language? "First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves....
2 votes
0 answers
139 views

Reference for the $3$-series of an elliptic formal group law

The $3$-series of the formal group law of the Weierstrass curve $y^2 = x^3 + a_2 x^2 + a_4 x$ begins $$ [3](z) = 3 z - 8 a_2 z^3 + (24 a_2^2 - 96 a_4) z^5 - (72 a_2^3 - 288 a_2 a_4) z^7 + (216 a_2^4 - ...
10 votes
2 answers
376 views

Reference request: Recent progress on the conjugacy problem for torsion-free one-relator groups?

I am aware that the Spelling Theorem of B. B. Newman implies that one-relator groups with torsion are hyperbolic, and thus have a solvable conjugacy problem. My understanding is that for one-relator ...
7 votes
1 answer
323 views

Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. ...
4 votes
1 answer
191 views

Computation of the Lusztig a-function

See for example https://www.sciencedirect.com/science/article/pii/0021869387901542 for the definition of the Lusztig a-function. Question 1: Is there a table for the values of Lusztig's a-function ...
5 votes
1 answer
3k views

Rigorous multivariate differentiation of integral with moving boundaries (Leibniz integral rule)

The Leibniz integral rule, in its multivariate form, deals with differentiation of the following sort: $$ \frac{\partial}{\partial t} \int_{D(t)} F({\bf x}, t) \, d{\bf x} \, , \qquad D(t)\in \mathbb{...
5 votes
2 answers
218 views

Reference for Cochran-Orr-Teichner's filtrations on knot concordance

I would be interested in recommendations for easily accessible/somehow simply readable texts to Cochran-Orr-Teichner's filtrations on knot concordance: Tim D. Cochran, Kent E. Orr, and Peter Teichner....
5 votes
2 answers
370 views

Conjecture about minimal number of edge crossings in complete bipartite graphs

I am interested in the status of the conjecture about the minimum number of edge crossings $cr(K_{m,n})$ in a drawing of the complete bipartite graph $K_{m,n}$. The Wikipedia article https://en....
2 votes
0 answers
203 views

Poincaré Recurrence Theorem for flows

Someone knows a book that have the proof of the Poincaré Recurrence Theorem for flows when the set have finite volume and when don't?
2 votes
0 answers
68 views

Linearly dependent points and the uniform position theorem

One proof of the uniform position theorem (as stated in p. 109 or p. 113 in Section III.1 of "Geometry of Algebraic Curves") uses a monodromy argument. While this gives us something even ...
9 votes
2 answers
286 views

Schur Weyl duality for the supergroup $\text{GL}(m|n)$

Let $G$ be the supergroup $\text{GL}(m|n)$. It has a tautological representation $V= \mathbb{C}^{m|n}$. For every natural number $d$ we have a natural map $$\Phi_d:\mathbb{C} S_d\to \text{End}_G(V^{\...
50 votes
15 answers
11k views

Explicit computations using the Haar measure

This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very ...
1 vote
0 answers
99 views

Dislocations and Random Matrix Theory

Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help. By ...
2 votes
3 answers
246 views

Diophantine equation of a factorial type

I'm interested in nontrivial solutions of Diophantine equations of the type $$a^2b^3 = \frac{c!}{(c-k)!} $$ For various values of k fixed, and of course $a,b,c \in \mathbb{Z^+}$ Does anyone have any ...
4 votes
1 answer
332 views

Reference to a Classical Regularity Theorem

(Edited) I need a reference to the following result: If $u \in H^2(B_1^+) \cap {\rm Lip}(B_1^+)$ satisfies \begin{cases} {\rm div}(F(x,u,\nabla u)) = F_0(x,u,\nabla u) \quad & {\rm in} \ B_1^+ ...
4 votes
1 answer
346 views

Blow up the diagonal of a symmetric product space

Consider the complex blow-up of the diagonal $\triangle\subset Sym^2(\mathbb{C}^n)$. It is known that the blown-up space is smooth when $n=1,2$. I was wondering if that is still smooth for $n\geq3$? ...
6 votes
2 answers
269 views

RSK and crystal operators

Is there a good reference on how RSK (and the 3 other variants) interact with crystal operators on the semi-standard tableaux $(P,Q)$ in the image? That is, we have biwords, $W$ which are in ...
1 vote
0 answers
111 views

Rowmotion for general lattices

Let $L$ be a finite lattice and $x \in L$ with covers $r_1,...,r_l$ in $L$. One can define $row(x):= \min \{ y | y \leq r_1 \lor \cdots \lor r_l $ and $ y \nleq r_1 \lor \cdots \lor \overline{r_t} \...
3 votes
1 answer
294 views

Reference for Function-Valued Random Variables?

Question: Are there any good references for facts about function-valued random variables? In particular for facts like the following: Let $X$ be a topological space, $Y$ be a random variable with ...
24 votes
7 answers
7k views

What are some good resources for mathematical translation?

I am currently in the process of translating a lecture on the étale topology by John Hubbard from French into English (and from transparencies into Beamer). For the most part, the translation is ...
26 votes
2 answers
4k views

Why did Robertson and Seymour call their breakthrough result a "red herring"?

One of the major results in graph theory is the graph structure theorem from Robertson and Seymour https://en.wikipedia.org/wiki/Graph_structure_theorem. It gives a deep and fundamental connection ...
1 vote
1 answer
87 views

Convergence properties of related series

Let $u_m = \ln ^2 m$. Does there exist a non-increasing sequence of positive numbers $\{g_n\}_{n \in \mathbb{N}}$, $g_n \to 0$, such that $$\sum\limits_{n \in \mathbb{N} } g_n = \infty, \ \ \ \ \...
3 votes
1 answer
458 views

Example of an intersection complex not concentrated in a single degree

I'm having trouble finding references for in-depth examples of perverse sheaves, so answers in the form of such a reference would be most helpful. I want to construct an example of an intersection ...
2 votes
1 answer
359 views

Reference request on Gentzen's proof of the consistency of PA

I've always been interested in having a good understanding of Gentzen's proof of the consistency of arithmetic. Being more precise, he showed that $PRA + WF(\epsilon_0) \vdash Con(PA)$. I am ...
3 votes
0 answers
102 views

Is there a source in which Demazure's function $p$ defined in SGA3, exp. XXI, is calculated?

Suppose that $\mathcal{R}=(M,R,M^*,R^*)$ is a root datum. In section 1.2 of SGA3, exp. XXI, Demazure defines the $\mathbb Z$-linear map $p:M\to M^*$ by $$p(x)=\sum_{u\in R^*}(u,x)u$$ and proves many ...
9 votes
1 answer
667 views

Curious anti-commutative ring

Has anyone seen the ring $\Lambda[x_0, x_1, x_2, \ldots]/(x_i x_j - (i+1) x_0 x_{i+j})$ in some natural context? Here $\Lambda[x_0, x_1, x_2, \ldots]$ is the (graded-)commutative algebra (either over ...
1 vote
1 answer
248 views

When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?

Let $A$ be an $n \times n$ real symmetric matrix. Let $$ M = \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} $$ where $X$ is a real invertible $n \times n$ matrix. I am interested in finding ...
4 votes
1 answer
100 views

Literature on the polynomials and equations, in structures with zero-divisors

I need literature about zeroes of polynomials and equation resolution in associative algebraic structures with zero-divisors, but I am having difficulties to find it. For example, there is literature ...
1 vote
1 answer
112 views

Pairing up vertices in a graph

Given a connected (undirected) graph with an even number of vertices, consider how many ways are there to pair up vertices so that each pair is connected by an edge. Is there a known classification of ...
3 votes
0 answers
88 views

Error rate implying regularity

My question is a bit general/vague. It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (...
16 votes
4 answers
2k views

Inverse problem of Chern Classes

For my graduate (master) thesis I am studying the theory of Chern Classes. As a possible personal development the only sensible idea I have so far, and which I frankly think is impossible, is to work ...
2 votes
0 answers
75 views

Maximal order of $x^n-d$ and its dependence on $d$

It's well known that the structure of the maximal order of $\mathbb{Q}[\sqrt{d}]$ depends on $d$ modulo $4$: (assuming $d$ is squarefree), the maximal order is $\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\...
-2 votes
3 answers
763 views

Oldest abstract algebra book with exercises?

Per the title, what are some of the oldest abstract algebra books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of ...
2 votes
1 answer
96 views

There is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon

Conjecture 1: With $n\ge 5$, given general n-polygon, there is no general method to construct n-regular polygon such that the given n-polygon inscribed the n-regular polygon (with one and only one ...
2 votes
1 answer
142 views

English translation of "Une inégalité pour martingales à indices multiples et ses applications"

Does anyone know of a English translation of "Une inégalité pour martingales à indices multiples et ses applications" by Renzo Cairoli. Or could translate the statement of the martingale ...
10 votes
2 answers
808 views

Fundamental group of a compact branched cover

My problem originates from the following classical result, proved, as far as I know, by Grauert and Remmert: Theorem. Let $Y$ be a compact complex manifold, $B \subset Y$ be a connected submanifold ...
4 votes
2 answers
558 views

From Zurab's integral representation for the Apéry's constant to almost impossible integrals

I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
3 votes
0 answers
93 views

Origins of the ``baby Freiman'' theorem

It is a basic folklore fact from the area of additive combinatorics that a subset $A$ of an abelian group satisfies $|2A|<\frac32\,|A|$ if and only if $A$ is contained in a coset of a (finite) ...

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