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

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3
votes
1answer
132 views

Irreducibility of family of polynomials

Consider the following family of polynomials over $\mathbb{Q}$: $$f_n = x^n - x^{n-1} - \dots - 1$$ Notice that these polynomials satisfy the recurrence $$ f_{n+1} = x f_n - 1 $$ I would like to ...
7
votes
0answers
103 views

Alternative definitions of Sobolev spaces on non-compact Riemannian manifolds

SHORT VERSION: Does the Meyers-Serrin theorem hold on complete, non-compact Riemannian manifolds, i.e. $W^{k,p}(M) = H^{k,p}(M)$? My guess is that this holds for the special case $k=1$ (and all $p\geq ...
0
votes
0answers
23 views

Article Using Kullback Leibler Divergence to Measure Divergence of Observation from Distribution

I am currently attempting to compare an observed distribution to a theoretical distribution, and my current approach is to normalize the two and find the Kullback Leibler Divergence. I am beginning to ...
3
votes
1answer
134 views

A Combinatorial Identity Involving Characters of $S_n$ (Reference Request?)

It is a well-known exercise that $C_n = \chi_{(n,n)}(1)=\chi_{(n,n)}^{1^n}$ where $C_n$ is the $n$th Catalan number and $\chi_{(n,n)}^{1^n}$ is the character of the irrep $(n,n)$ on conjugacy class $1^...
5
votes
1answer
195 views

Numbers divisible only by primes of the form 4k+1

Let $A(N)$ denote the number of positive integers $n\le N$ composed of prime numbers $p\equiv 1\pmod 4$ only. Is there an asymptotic formula for $A(N)$ (as $N$ tends to infinity)?
4
votes
1answer
338 views

Higgs fields whose determinant have only simple zeros

Is the following property true for every stable holomorphic bundle of rank 2 with trivial determinant on a compact Riemann surface: The space of trace-free Higgs fields, whose determinant have only ...
56
votes
16answers
8k views

Examples of algorithms requiring deep mathematics to prove correctness

I am looking for examples of algorithms for which the proof of correctness requires deep mathematics ( far beyond what is covered in a normal computer science course). I hope this is not too broad.
0
votes
0answers
151 views

Regularity for a div-curl system

Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...
5
votes
0answers
178 views

Equivalence of algebraic and topological monodromy representations?

Does anyone know of a reference for the following fact? Let $M_g$ denote the moduli stack of genus g curves, let $A_g$ denote the moduli stack of abelian varieties, and let $U_g \rightarrow A_g$ ...
2
votes
0answers
30 views

First eigenvalue for strictly convex domains

Let $M^n$ be a compact Riemannian manifold with boundary, suppose 1). $Ric(M)\ge (n-1)$ and 2). the principle curvatures of the boundary is bounded from below by $h\ge 0$. Is there any results on the ...
0
votes
0answers
37 views

Fisher metric for shift-invariant probabilities

I'm just discovering what seems to be the tremendous heuristic value of the (century-old, more or less) canonical Riemannian metric (Fisher metric) on the $n$-dimensional simplex $\Sigma_n:=\{(p_i)_{i=...
5
votes
1answer
235 views

How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?

I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...
4
votes
2answers
245 views

Determining a function is harmonic from mean value property for just three(?) radii

A couple days ago I posted this on MSE (here) but in retrospect it might be more appropriate for this site. This theorem is well-known (maybe it can be called Morera's theorem): A continuous ...
1
vote
0answers
160 views

on universal homeomorphisms between schemes

We are taught since when we are young that schemes are cool because they take into account "nilpotents". This means also that we can distinguish between schemes which have the same underlying ...
4
votes
1answer
227 views

How to construct an abelian variety with CM by a given CM field?

Let $F$ be a totally real number field, and let $K$ be a quadratic extension of $F$ which cannot be embedded into $\mathbb{R}$. Then $K$ is a so called CM field. For instance, take $F = \mathbb{Q}(\...
2
votes
0answers
47 views

level sets portrait near a critical point

Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a smooth ($C^{\infty}$) function and $O$ be an isolated critical point of $f$. I am looking at the local level sets diagram near $O$ from topological ...
7
votes
1answer
171 views

Posets obtained from a semigroup by the definition $x \leq y \iff x \cdot y = x$

A po-groupoid is a groupoid $\langle A,\cdot\rangle $ such that the relation defined by $$ x \leq y \text{ if and only if } x \cdot y = x $$ is a partial order on $A$, the order related to $\langle ...
0
votes
1answer
83 views

Partitioning an orthogonal matrix into full rank square submatrices

Let $U$ be an $n \times n$ orthogonal matrix. Given an arbitrary partition ${\mathcal P}_c=\{y_1,y_2,\ldots,y_k\}$ of the columns of $U$, does there always exist a corresponding partition ${\mathcal ...
1
vote
0answers
37 views

Probability for a SRW to be at some place in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
10
votes
1answer
278 views

Uncle of Witt algebra

A Witt algebra W is an infinite-dimensional Lie-algebra defined by the generator relations: W: $[L_{j},L_{k}]:=(j-k)\cdot L_{j+k}$ And my first thought was: What about the analogous algebra defined by ...
1
vote
1answer
267 views

Further research on $\mathrm L_{\infty}$

In the mathoverflow question , "Godel's Constructible Universe in Infinitary Logics (A Possible Solution to $HOD$ Problem), Prof Hamkins answered user46667's question 2 What is $\mathrm L_{\infty}$...
4
votes
1answer
89 views

Syzygy between covariants of pairs of ternary quadratic forms

In the book Nonlinear Computational Geometry, Page 208 (or page 15 of the online version on the author's website: http://www.loria.fr/~petitjea/papers/imaconics.pdf), Remark 5.1, Petitjean states that ...
3
votes
1answer
66 views

Explicit generators of the Lie algebra $spin(9)$

It is well known that the Lie group $Spin(9)$ acts on the vector space $\mathbb{R}^{16}$ (see e.g. Harvey's book "Spinors and calibrations".) It is convenient to identify this vector space with the ...
0
votes
0answers
77 views

Feynman-Kac for heat equation on a compact manifold with boundary

It is known that for any open $\Omega \subset \mathbb{R}^n$, given $f \in L^2(\Omega)$, $x \in \Omega$, one has $$e^{t\Delta}f(x) = \mathbb{E}_x(f(\omega(t))\psi_\Omega(\omega, t)), $$ where $\Delta $ ...
8
votes
1answer
202 views

Restriction of irreducible unitary representation to normal subgroup of finite index

Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...
0
votes
0answers
70 views

Presheaves of Dendroidal Sets?

Are there any references available for presheaves of dendroidal sets? Seems like a natural extension of simplicial presheaves.
4
votes
1answer
383 views

Three dimensional representations of Alternating group

The alternating group $A_5$ has $2$ irreducible representation of degree $3$. The characters for these representations have irrational values. I guess the ring of invariants of these representations ...
0
votes
1answer
125 views

Bounds on the smallest real positive root of a polynomial

I'm trying to find upper and lower bounds of the smallest positive root of a polynomial, stated in terms of its coefficients. As I appreciate it might be a very general problem, My specific interest ...
1
vote
0answers
30 views

distance from the mean of a normal distribution to the span of a random sample

Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from ...
0
votes
1answer
250 views

How to locate articles from Expositiones Mathematicae from 1988 and 1992

Unfortunately, my university library seems unable to find the following (I've tried the interlibrary loan tool but this particular journal is somehow outside its scope): W. C. Waterhouse, ...
6
votes
1answer
219 views

Infinite families in stable homotopy groups

I will be very grateful for any advise or reference on the following. 1- How much is known about infinite families in ${_2\pi_*^s}$, the $2$-component of the stable homotopy ring? 2- How much is ...
4
votes
1answer
127 views

Dye's Theorem for real von Neumann algebras

Dye's classical theorem from 1955 states that if $\theta$ is a projection orthoisomorphism (i.e., a bijection between the projective lattices that preserves orthogonality - it then automatically ...
5
votes
2answers
224 views

Definition of Hecke operators on orthogonal modular forms

In his paper Automorphic forms with singularities on Grassmannians, Borcherds poses Problem 16.5: "Describe how the correspondence in this paper behaves under the action of Hecke operators." Since ...
9
votes
1answer
334 views

Computational complexity of computing simplicial homology

Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell ...
2
votes
0answers
83 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 ...
2
votes
1answer
107 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\...
0
votes
0answers
95 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 ...
6
votes
1answer
294 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 ...
4
votes
1answer
156 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 ...
8
votes
0answers
166 views

What is known 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 ...
3
votes
0answers
67 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 ...
1
vote
1answer
157 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
2answers
946 views

Unreasonable application of mathematics to the other areas [closed]

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 ...
3
votes
1answer
96 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 \...
3
votes
0answers
65 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, ...
2
votes
0answers
47 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 ...
10
votes
0answers
124 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
vote
0answers
67 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 =...
1
vote
0answers
25 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 ...
3
votes
3answers
335 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 ...