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

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5
votes
1answer
330 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 ...
1
vote
0answers
16 views

Lattice points in regular simplex

Suppose we are given a regular (closed) simplex $S$ in a vector space $V$ of dimension $n$, whose vertices have integer values. Then for a lattice $L$, is there a sufficient criterion, for $S$ to ...
1
vote
2answers
61 views

List of tensor product spaces with uniform crossnorms

Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products $H:=\bigotimes_{j=1}^n H^{(j)}$ and ...
1
vote
1answer
383 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 ...
2
votes
0answers
87 views

Semistability of a sheaf on nodal curve

Suppose $X$ is a projective, connected, nodal curve (can be reducible) over an algebraically closed field $k$ of arbitrary characteristic. Let $F$ be a pure sheaf on $X$ and denote by $\pi^{*}(F)$ its ...
0
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0answers
51 views

Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15): Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the ...
16
votes
3answers
2k views

Analysis from a categorical perspective

I have not studied category theory in extreme depth, so perhaps this question is a little naive, but I have always wondered if analysis could be taught naturally using categories. I ask this because ...
3
votes
0answers
35 views

Real interpolation of weighted Sobolev spaces with different weights

Let $\Omega \subseteq \mathbb{R}^n$ be open and let $w_0$ and $w_1$ be measurable and almost everywhere positive and finite functions defined on $\Omega$. Let $L^2_{w_0}(\Omega)$ be the weighted ...
3
votes
0answers
111 views
+50

Optimal instructions for the modular construction of rectlinear Lego structures

Let $X$ be a compact (or periodic) union of integer translates of unit cubes such that the interior of $X$ is connected. (If it makes any difference, suppose that the dimension $n$ of $X$ is 3.) I am ...
15
votes
4answers
3k views

Classification of finite groups of isometries

Consider the problem of classifying the finite groups of isometries of $\mathbb{R}^n$. For $n=2$ it is cyclic and dihedral groups. For $n=3$ they are well known, probably from Kepler and are related ...
4
votes
2answers
502 views

Bounding the minimal maximum norm of a solution of a linear system.

I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...
14
votes
2answers
590 views

Geometric interpretation of exceptional Symmetric spaces

Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...
5
votes
3answers
181 views

Text for studying group representations in the context of (abstract) harmonic analysis

I would like to study elements of representation theory as I often encounter it when reading texts on harmonic analysis. I was therefore curious if someone could recommend a book for this. When ...
6
votes
2answers
364 views

Kazhdan's property (T) vs. residual finiteness

I have asked this question already on mathstackexchange but got no answer (see http://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
2
votes
1answer
72 views

Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$. I assume that my ODE ...
2
votes
0answers
23 views

Points of continuity of a lower semicontinuous function have non empty interior

Let $X\subset \mathbb R^d$ having non empty interior and let $f:X\to\mathbb{R}$ be lower semicontinuous. I know that the set of discontinuities of such a function is contained in a meager set, and ...
4
votes
1answer
289 views

Understanding a particular approximation for Stirling's number of the second kind

I noticed and employed (without a problem) an approximation for Stirling's number of the second kind found on Wikipedia (http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind), in ...
6
votes
0answers
117 views

$\Bbb A^1$-Localisation of Schemes, and $\Bbb A^1$-Rigid Schemes

Question 1: Are there some publications or preprints that provide $\Bbb A^1$-fibrant replacements of certain classes of (smooth) schemes? Of course, smooth schemes that are $\Bbb A^1$-fibrant are ...
10
votes
3answers
847 views

Complete (possibly official) list of “What is…” articles from the Notices of the AMS

Does it exist online and where can one find it? (For example, these two sources are not official; is the longer one complete?)
12
votes
0answers
455 views

Generalization of Witten's computation of the volume of moduli space

Let $\Sigma$ be a Riemann surface, and let $X:=\operatorname{Hom}(\pi_1(\Sigma),\operatorname{SU}(2))/\operatorname{SU}(2)$ be the $\operatorname{SU}(2)$ character variety of $\pi_1(\Sigma)$. There ...
1
vote
1answer
178 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 ...
4
votes
1answer
220 views

What are Motivic homotopy types?

There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it. I would like to know the reference in which Grothendieck did it, ...
1
vote
0answers
70 views

Applications of the Weak and Weak$^*$ topologies to PDEs?

Chapter $3$ of Functional Analysis, Sobolev Spaces and Partial Differential Equations by Haim Brezis constructs and explains the Weak and Weak$^*$ topologies over a Banach Space $E$. The most ...
3
votes
1answer
135 views

Range of random walk

I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact ...
7
votes
1answer
168 views

When is a mapping the proximity operator of some convex function?

Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ? That is, given $p : ...
4
votes
1answer
121 views

Minimal Nagata-like compactification

Working over a field $k$, Nagata's compactification theorem implies that any separated scheme $X$ of finite type over $k$ admits a compactification (a dense open immersion $i \colon ...
12
votes
5answers
495 views

Applications of space filling curves

I am seeking articles where a space filling curve has been used as a theoretical application, such as in the study of general orthogonal polynomials.
1
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0answers
65 views

Diagonal of Green's Function

I am looking to numerically calulate the diagonal of Green's function. I am interested in Green's functions of elliptic PDEs and in those that arise from stochastic processes (discrete and ...
2
votes
1answer
163 views

What is the extension of the truth-table degrees to Baire Space called?

Recall that for sets $A, B \in 2^\omega$ that we say $A \leq_{tt} B$ if there is a total Turing functional $F \colon 2^\omega \to 2^\omega$ such that $F(B)=A$. This is called truth-table ...
2
votes
0answers
151 views

Euler-Maclaurin Formula Review

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely Generalizations to broader ...
2
votes
0answers
56 views

Strengthening of the local smoothing estimates for the free Laplacian

The classical local-smoothing estimates for the free Laplacian asserts that: $$\Vert e^{-it\Delta}f\Vert_{L^2((-\infty,+\infty)\,;\,H^{1/2}(B))}\leq C_B\cdot\Vert f\Vert_{L^2}$$ where ...
2
votes
0answers
47 views

On two functions with isodirectional gradients

Let $U\subset \mathbb{R}^n$ be open and $f,g:U \to \mathbb{R}$ be two $C^1$ functions whose gradients are always in the same direction, i.e. $\forall i,j \in \left\{1,...,n\right\}$ \begin{equation} ...
0
votes
0answers
34 views

Where can I find this article of Doléans-Dade?

I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade. I could not find a pdf version online, and my university library does not have a printed version. Thank ...
5
votes
2answers
242 views

Stabilization of the pencil of skew symmetric matrices by the orthogonal group

During my researches I've come across the following question. Let $A$ and $B$ be a couple of square $k\times k$ skew symmetric matrices on $\mathbb R$. Let us consider the (real) pencil generated by ...
7
votes
0answers
817 views

The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
2
votes
0answers
77 views

The integral of $\exp(-|x-a|)$ over an even dimensional sphere

I'm after a reference for an integral. For $m$ a positive integer and $R>0$ let $S^{2m}_R\subset \mathbb{R}^{2m+1}$ denote the radius $R$ sphere of dimension $2m$. Suppose that $a$ lies inside ...
6
votes
4answers
467 views

Minimum negative eigenvalue of zero-one matrices

The following question must have been answered decades ago. For $n$ fixed, what is the most negative eigenvalue among all trace zero zero-one matrices (that is, all entries are either zero or one, ...
3
votes
1answer
213 views

Small resolutions are automatically crepant?

Page 17 of the following survey: http://arxiv.org/abs/1103.5380 makes the claim that small resolutions, meaning resolutions such that the exceptional set is in codimension at least two, are ...
2
votes
0answers
257 views

An explicit formula for $\zeta(2m+1)$ with good convergence

The question: Is the following formula known? $$\zeta(2m+1)=\frac{(-1)^m 2^{4m+2}\pi^{2m}}{2^{2m}-1} \sum\limits_{k=1}^m \frac{(2^{2k}-1)b_{2k}}{2^{2k}(2k)!}\cdot \sum\limits_{v=k}^m ...
0
votes
0answers
60 views

Order of vanishing of Laplace's equation with potential

Consider the equation $-\Delta u + V u = 0$ with Dirichlet boundary conditions on the bounded domain $\Omega \subseteq \mathbb{R}^n$, where $V$ is a smooth potential. Let $V \leq 0$, and bounded on ...
2
votes
0answers
100 views

Can infinitely many orbifolds be “added up” to form a fractal space?

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it. Intuitively, an orbifold, from what I understand, is a ...
2
votes
6answers
2k views

Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...
6
votes
0answers
112 views

Weak* continuity of positive parts, again

Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of ...
7
votes
0answers
120 views

Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...
7
votes
1answer
249 views

Abstract result on partitions of unity?

A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...
0
votes
0answers
48 views

Matrix concentration inequality

Let $X \in \mathbb{R}^{n \times d}$ be a fixed matrix and $W \in \mathbb{R}^{n \times d}$ be a random matrix with elements $w_{ij} = x_{ij} + \epsilon_{ij}$, where $\epsilon_{ij}$ are iid subgaussian ...
9
votes
2answers
235 views

Finite objects for which isomorphism is NP-hard or harder?

Are there finite objects for which deciding isomorphism is NP-hard or harder? Graphs and groups are not solutions. Searching the web didn't return answer for me. Partial result based on Chow's ...
11
votes
1answer
593 views

On a result attributed to W. Ljunggren and T. Nagell

I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation $$\frac{x^{n}-1}{x-1} = y^{2}$$ doesn't admit solutions in ...
6
votes
1answer
308 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. ...
1
vote
2answers
295 views

Expository articles on Algebraic Number Theory

I am about to start learning Algebraic Number Theory and thus was looking for some expository articles on this subject. So far I have found two such articles: Dickson, L. E.. (1917). Fermat's Last ...