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

**3**

votes

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**16**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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 ...