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

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1
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0answers
19 views

a continuous analogue of a graph theory question

I am reading a paper and it mentions a continuous analogue of a related graph theory question that people concern. The question is that suppose $E\subset Q=[0,1]^2$ has lebesgue measure $|E|>0$, is ...
3
votes
2answers
823 views

Unreasonable application of mathematics to the other areas [on hold]

What are some papers or talks on the philosophy of mathematics which contains some statements about the unnecessary and unreasonable application of mathematics in other areas of science? I found ...
4
votes
1answer
222 views

Is there any good survey on the hook length formula and related topics?

I am recently doing some research related to the hook length formula. The hook formula counts the number of Young tableaux of certain type. I find there are plenty of research already been done and ...
0
votes
0answers
42 views

Simultaneous extension of modules

Let $R$ be a commutative ring. Suppose $R$-modules $X,A,B,C$ and $Y$ are given such that the outer two rows and the outer two columns in the following diagram are exact. $\hskip1in$ Does it ...
0
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0answers
23 views

Higher dimensional analogs of logarithmic density

For a set $A\subseteq \mathbb{N}$ its lower/upper asymptotic/logarithmic densities are given by \begin{align*} \underline{d}(A)=\liminf_{N\to\infty} \frac{|A\cap [1,N]|}{N},\\ \bar{d}(A)=\limsup_{N\to\...
6
votes
0answers
125 views

Nonabelian $H^2$ and Galois descent

I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved. Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety ...
0
votes
0answers
24 views

Triangulation in which every Triangle has at least one Boundary edge

I am looking for information about Triangulation of a bounded (polygonal) planar region in which every triangle has at least one Boundary edge, that is every triangle has at least one edge which is ...
3
votes
1answer
133 views

Reference: Finsler Derivative?

On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...
1
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1answer
68 views

Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$

It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise ...
0
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0answers
36 views

Interplay between the Cayley graph of a finite group and its realizability over Q

As the structure of any finitely generated, and thus any finite, group, can be described by its Cayley graph, have partial results on the realizability of such a group over $Q$ been obtained ...
5
votes
1answer
365 views

Reference request - Jacquet module and asymptotic of matrix coefficients

Hello, I would like to know some nice references about the relation between asymptotics of matrix coefficients of representations of reductive groups over local fields, and the pairing between the ...
3
votes
0answers
348 views

Fermat-Wiles “first case” in extensions of cyclotomic fields

I fell on the following fact : Let $p$ be an odd prime, let $K$ denote the $p$-th cyclotomic field, let $L$ be an extension of $K$ with finite degree not divisible by $p,$ and assume that the prime ...
4
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0answers
70 views

What is know about maps between loop spaces of Spheres? - Reference request

What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values ...
1
vote
1answer
198 views

When is the sum of complemented subspaces complemented?

Let $X$ be a Banach space. Question. Suppose $X_1,...,X_n$ are complemented subspaces of $X$. When is the sum $X_1+...+X_n$ complemented? Further, suppose we know some projections $P_1,...,P_n$ onto $...
1
vote
1answer
401 views

Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

I would like to know a reference of the following statement (or counter example). Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is ...
3
votes
0answers
53 views

Symmetries of modular categories coming from quantum groups

This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided ...
2
votes
1answer
91 views

Finite groups of planar homeomorphsims

Let $G$ be a finite subgroup of the group $H$ of orientation-preserving homeomorphisms of the plane that fix the origin. Is $G$ conjugate in $H$ to a group of rotations? I've been told this result ...
-1
votes
0answers
98 views

(Dynamical) mean field theory for mathematicians? [on hold]

I am looking for a readable introduction/tutorial on dynamical mean-field theory, written for someone who doesn't know anything about particle physics. My physics background is non-existent beyond ...
5
votes
0answers
130 views

Classification of representation-finite algebras up to stable equivalence of Morita type

Assume $K$ is an algebraically closed field. I wanted to ask if there is a classification of the representation-finite $K$-algebras up to stable equivalence of Morita type (at least for some small ...
1
vote
1answer
151 views

Could I affirm that $f$ is not identically 0?

Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric: $ d(x,y)=\sum_{i\geq 1}\frac{|...
2
votes
1answer
87 views

Number of vectors of fixed norm

Let $P$ and $Q$ be two even, unimodular, positive definite quadratic forms of rank $n$. Let $r_{k}(P)$ be the number of vectors of norm $k$, in symbols: $$ r_k(P)=\textrm{cardinality of }\{v\in \...
7
votes
2answers
451 views

The p-adic valuation of a linear recurrence

Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely, $$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$ for some $a_1, \ldots, a_k \in \...
3
votes
0answers
61 views

Turán's inequalities for Hermite functions

Given $\lambda \in \mathbb{R}$ let $H_{\lambda}(x)$ be the solution of the Hermite differential equation: $$ \frac{d^{2}}{dx^{2}} H_{\lambda}(x)-x\frac{d}{dx}H_{\lambda}(x)+\lambda H_{\lambda}(x)=0, ...
1
vote
1answer
187 views

Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
3
votes
3answers
319 views

Jacobi's theorem on sums of two squares (reference request)

One of Jacobi's theorems states that the number of representations of a positive integer $n$ as a sum of two squares of integers equals $$4(d_1(n)-d_3(n)),$$ where the function $d_i$ counts the number ...
15
votes
3answers
951 views

DG categories in algebraic geometry - guide to the literature?

Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric ...
1
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0answers
43 views

Combination of certain linear-programming topics new?

Consider the combination of the following topics, aimed at a future book on Linear Programming: Generalization of certain parts of the polyhedron theory and of the Simplex Algorithm to arbitrary ...
9
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0answers
109 views

Existence of flat connections via characteristic classes, for nice groups

I have two questions about what I write below (which honestly seems pretty elementary). Is it true (more or less)? Is there a clean reference that I can cite. Let $G$ be a compact Lie group, $M$ a ...
1
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0answers
65 views

Law of large numbers for random functions?

Is there a version of the law of large numbers for random functions of the type: $h(X_j,\hat{\theta}_n)$, where $X_1,\dots,X_n$ are i.i.d. random variables, with distribution $F$, and $\hat{\theta}_n =...
8
votes
1answer
751 views

Automorphism group of a finite group

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = \mathbb{Z}/m\...
1
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0answers
22 views

Reference Request: $M_t/M_t/1/K$ queue length distributions

I am investigating functionals defined over sequences of discrete probability distributions related to dynamical/stochastic system performance. As an initial step, I am searching for references that ...
4
votes
2answers
195 views

Invariant theory for parabolics

Let $G$ be a connected reductive group over $\mathbb{C}$ of (reductive) rank $\ell$. Let $P$ be a parabolic of $G$ and let $P=LN$ denote the Levi decomposition. Let $\mathfrak{g}, \mathfrak{p}, \...
0
votes
1answer
66 views

Reference request: Existence of plurisubharmonic potential for positive (1,1)-current on Stein (affine) manifold

I am looking for a reference for the following fact, which I believe should be true. Let $X$ be a Stein manifold (or smooth affine variety over $\mathbb{C}$). If $\omega$ is a positive closed $(1,...
3
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0answers
94 views

Does anyone have Delzell's Thesis on Bad Points of Forms?

Since a number of papers (e.g. this one) treating denominators in Hilbert's 17th problem point to E.G. Strauss's unpublished letter to G. Kriesel or to Chapter 5 of Delzell's Thesis, which contains an ...
4
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0answers
187 views

Zeros of polynomials modulo a non-prime

Suppose I have a set $S$ and I want to find a polynomial $p$ such that $p(s) = 0 \mod n$ if $s \in S$, and that it is non-zero modulo $n$ otherwise. In the literature such an $S$ is sometimes called ...
5
votes
0answers
174 views

Homotopical interpretation of flatness?

I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like: A map of commutative algebras is flat if pushing it out ...
1
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1answer
101 views

Gaussian bounds on Dirichlet heat kernel

Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0 $ such that $$\frac{c}{t^{n/2}} e^{-\frac{1}{4t}d(x, y)^2} \leq ...
6
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0answers
172 views

Jets of sections of vector bundles expressed by symmetrized iterated covariant derivatives - who did it first?

The (non-unique) bundle isomorphism between the bundle $J^r E$ of $r$-th order jets of sections of a vector bundle $\pi:E\rightarrow M$ and the direct sum $$\bigoplus^r_{k=0}\vee^kT^*M\otimes E\...
4
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2answers
252 views

Critical points in $ZF$ without Choice

Recall the definition of critical point for set theory: A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to ...
0
votes
0answers
70 views

Laplace transform (or characteristic functional) of atomic random measure

A random (nonnegative Radon) measure $M$ (on $\mathbb R^n$, say) has its law characterized by the Laplace transform $\mathbb E\exp(-\int \varphi(x)\ M(dx))$, $\varphi\in C_c^+(\mathbb R^n)$ (...
3
votes
1answer
148 views

Reference or counter-example for Closed Graph Theorem for multivalued maps in general topological spaces

Could someone be so kind to point me in the direction of a citeable proof of the following version of the Closed Graph Theorem? (i.e. assuming this is true, could someone give me a literature ...
2
votes
1answer
93 views

$H^1$-continuity of Laplace's equation with respect to boundary data

Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...
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0answers
77 views

Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let $$ \mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} . $$ We ...
6
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0answers
246 views

Coefficients in expansion of a classical symmetric polynomial

If we expand \begin{equation} P_3(x_1,\ldots, x_n):=\Pi_{1\leq i<j<k\leq n} (x_i+x_j+x_k), \end{equation} then \begin{equation} P_3 = \sum_\alpha \sum_{\mathcal{O}(\alpha)} c_\alpha x_1^{\...
1
vote
1answer
91 views

Stability of the critical points set

Let $F:\mathbb{S}^{2}\times\lbrack0,1]\rightarrow\mathbb{R}$ be a smooth ($C^{\infty}$) function and $f_{t}(x)=F(x,t)$. Suppose that $f_{0}=f_{1\text{ }}$is the projection over $z$-axis, so point $P=(...
6
votes
1answer
343 views

How to describe morphisms to a weighted projective space (bundle)?

The case of an usual projective space (bundle) is well known (Grothendiek,EGA II, Publ.Math. IHES, 8, 1961; or Hartshorne, Alg.Geom.).The more general case of toric varieties has been considered by D. ...
2
votes
0answers
426 views

Space of derivations of holomorphic (analytic) functions

Let $M$ be a (real) smooth manifold, and $p \in M$. The space of (linear) derivations $D:C^{\infty}(p) \to \mathbb{R}$ (i.e., maps satisfying $D(f+g) = D(f)+ D(g)$ and $D(fg)=D(f)g(p) + f(p)D(g)$) ...
5
votes
2answers
512 views

Finite groups with a character having very few nonzero values?

A number theorist I know (who studies Galois representations) raised a question recently: Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
0
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0answers
40 views

Brownian motion hitting probability [closed]

$B_t$ is a Brownian process, starting from the origin in $\mathbb{R}^n$. Let $\theta_X(T)$ denote the probability that the particle hits the set $X \subset\mathbb{R}^n$ within time $T$. Keeping $T$ ...
1
vote
1answer
155 views

Finding many disjoint sub-trees with many leaves

Let $T$ be a rooted binary tree with $L$ leaves, and let $\ell$ be a natural number smaller than $L$. The question is what is the maximal number of disjoint rooted sub-trees with at least $\ell$ ...