Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [reference-request]

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

1
vote
0answers
32 views

Representation as $n=p^2+q^2-r^2$

What is known about the number of representations of a positive integer $n$ as $$ \rho(n) = \# \{ (p,q,r): n=p^2+q^2-r^2\}, $$ where all the variables are primes? What about the average number of ...
4
votes
0answers
23 views

Normalizers of subsystem subgroups of Lie groups

Let $G$ be a semisimple complex Lie group, and let $H$ be a subgroup corresponding to a subset of the extended Dynkin diagram of $G$ (à la Borel - de Siebenthal). I would like to know if there is a ...
2
votes
0answers
35 views

Algebra dimension computation in GAP

How does GAP compute the dimension of a matrix algebra over the rational numbers? I am curious about the run time. For example, the manual https://www.gap-system.org/Manuals/doc/ref/chap62.html does ...
2
votes
0answers
24 views

Convergence of Eigenvalues and Eigenvectors for Uniformly Form-Bounded Operators

Suppose that $A$ is an operator on a dense domain $D(A)\subset L^2$ with compact resolvent, and with quadratic form $q(f,g):=\langle f,Ag\rangle$. Let $(r_n)_{n\in\mathbb N}$ be a sequence of ...
1
vote
0answers
62 views

Known lower bounds on the minimum average distance problem?

Given a set $S\subseteq \{0,1\}^d$ of the Boolean hypercube of dimension $d$, define the average distance of $S$ as $$ \bar{d}(S) = \frac{1}{\lvert S\rvert^2} \sum_{x,y\in S} d_H(x,y)\tag{1} $$ where $...
5
votes
0answers
55 views

Laplace Beltrami eigenvalues on surface of polytopes

The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...
-1
votes
1answer
88 views

Do you know the reference for this law?

I am graduate student. As you know, the convolution operation satisfy the below equation due to commutative law. a(n)*b(n)*c(n) = a(n)*c(n)*b(n) In addition, the muliplication operation also ...
4
votes
2answers
68 views

Complexity of Random Delaunay Triangulation in 3D

My question: Is the number of cells in a three-dimensional Poisson-Delaunay triangulation with $n$ vertices $\mathcal O(n)$ with probability one? which is equivalent to the question Is the ...
3
votes
1answer
63 views

Incompressible Navier-Stokes equation with heat conduction

How does the incompressible Navier-Stokes system read with heat conduction? Where can I find an existence result for its weak solutions?
13
votes
2answers
445 views

Is a matrix similar to its transpose over $\mathbb{Z}_p$?

Is every $n \times n$ matrix with entries in $\mathbb{Z}_p$ (or even $\mathbb{Z}$) conjugate to its transpose via a matrix in $GL_n(\mathbb{Z}_p)$? On the one hand, I know the analogous fact is false ...
0
votes
0answers
36 views

Computing derivative of certain path integrals

Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $\mathbf{w}$, and (2) input data ${\bf x} \in {\mathbb R}^d$. Fix $i \in [d]$, consider the following ...
2
votes
0answers
57 views

Nerve theorem for locally infinite covers by subcomplexes

Let $Y$ be a simplicial complex and let $\{Y_i\}_{i\in I}$ be a set of subcomplexes of $Y$ such that $\bigcup_{i\in I}Y_i=Y$. Let $\mathcal N$ be the nerve of this covering, and assume that for each ...
5
votes
1answer
118 views

Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials

$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}_n$ be the symmetric group, $\SYT_n$ be the set of standard young tableaux of size $n$. For $u\in \...
16
votes
1answer
437 views

Where's the best place for an algebraic geometer to learn some algebraic number theory?

There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- ...
6
votes
1answer
182 views

Theory of surfaces in $\mathbb{R}^3$ as level sets

Is there a book that treats the classical theory of surfaces in $\mathbb{R}^3$ from the point of view of level sets of a function? I seem to remember someone telling me that such a book exists, but I ...
6
votes
0answers
222 views

measure of generic reals in forcing extensions

It is well-known that if $V[G]$ is a generic extension by adding a Cohen real, then the set $\{r \in V[G]: r$ is Cohen generic over $V\}$ has measure zero. On the other hand, if $V[G]$ is a generic ...
11
votes
1answer
262 views

Map of Grassmannians associated with a Veronese embedding

I'm quite sure this should be classically known, however I am not an expert on the topic and I was unable to find a precise reference in the huge literature concerning Veronese embeddings and ...
0
votes
0answers
43 views

Energy estimates involving test functions for weak solutions of PDE problems

I was reading an article on Arxiv.org about Navier-Stokes system ([Breit]) and I stumble on this sentence on the second page: "A weak (in the PDE sense) solution satisfying the energy inequality ...
2
votes
0answers
84 views

Invariants of the group $SO(2)$

Let $V_d$ be the complex vector space of binary forms of degree $d$ endowed with the natural action of the special orthogonal group $SO(2).$ Consider the corresponding action of the group $SO(2)$ on ...
2
votes
1answer
34 views

Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is “uniform” across $P \overline{N}$

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup ...
5
votes
1answer
168 views

Cubic surfaces and configurations of 6 points

A smooth cubic surface $X\subset \mathbb{P}^3$ is isomorphic to $\mathbb{P}^2$ blown up at six points, so there should be a rational map ${\rm Hilb}^6\mathbb{P}^2\dashrightarrow H^0(\mathbb{P}^3,\...
1
vote
0answers
67 views
+50

Bifurcations due to a nonlinearity parameter

Suppose we want to analyze the behavior of the system $$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x},t;\varepsilon),\quad \mathbf{x}\in\mathbb{R}^n,\quad t\in\mathbb{R}^+,\quad\varepsilon\in\mathbb{R}^+, $$ ...
7
votes
1answer
135 views

Finite subgroups of $PSU(3)$

I'm looking for a reference to a classification or description of finite subgroups of $SU(3)$ that contain the center, or equivalently $PSU(3)$. Can anyone point me in the right direction?
7
votes
1answer
266 views

Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?

Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...
11
votes
3answers
250 views

Finite groups with few conjugacy classes of maximal subgroups

Let $c$ be a positive integer, $G$ a finite group with at most $c$ conjugacy classes of maximal subgroup. What can we say about $G$? Same question, but this time $G$ is a finite group with at most $c$...
1
vote
0answers
127 views

Reference to a particular result of Scholl and Faltings

Let $f=\sum_{n\geq 1} a_n q^n$ be a normalized eigenform which is supersingular and crystalline at a prime $p$ and let $V_f$ be the associated crystalline representation, then it follows from the work ...
0
votes
0answers
9 views

How to Create Point-Optimal Objective Functions

Here is a problem that has originated from some IP research i'm working on. You are given a polyhedron $P$ in standard matrix inequality form of $Ax \le b$, $x \in \mathbb{R}^n$ as well as a point $...
1
vote
0answers
36 views

When does a collection of sets forming a geometric lattice give the flats of a matroid?

Say we have a matroid on a finite set $X$. The collection of its flats forms a geometric lattice under $\subseteq$, where the join is given by intersection. This question is about the converse to ...
1
vote
0answers
36 views

Diffusion generators with gradient vector fields

Let $\mathcal{A}$ be a second order operator on an $n$-dimensional smooth manifold $M$, expressed in Hörmander form as $$\mathcal{A}=X_0+\frac{1}{2}\sum_i^kX_i^2,$$ where $X_0,X_1,...,X_k$ are ...
3
votes
0answers
53 views

Name of a binary matroid coming from the cycle space of a graph

In some of my recent work, I have 'discovered' a binary matroid which I will describe below. Given a graph $G$, let $H_1(G, \mathbb{Z}/2\mathbb{Z})$ denote the cycle space. This is a vector space ...
0
votes
1answer
61 views

Reference request: Moving source to initial condition and vice versa in PDE problem

I am trying to find references in the literature that connect solutions of two problems given bellow. They deal with deterministic conservation laws. Inhomogeneous Cauchy problem: $$(1) \hspace{1cm} ...
2
votes
1answer
83 views

Reference for Minkowski functional when 0 is not in the interior

The Minkowski functional on a normed linear space $E$ is usually defined for convex (or sometimes even non convex) subsets $C$ of $E$ such that $0 \in \operatorname{int}(C)$. Is there any standard ...
4
votes
3answers
253 views

Elliptic regularity on compact manifold without boundary

Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this: For any $u\in H^1(M)$, ...
1
vote
1answer
76 views

Basis of cone lattice

I only want to know whether a construction that I use appears in literature and maybe has a name already. Let $V$ be a $\mathbb Q$ vector space of dimension $d\in\mathbb N$. A subset $C\subset V$ is ...
0
votes
0answers
14 views

Non-asymptotic tail-bounds for Hotelling $T^2$ statistic

Let $X_1,\ldots,X_n$ be an i.i.d sample from a distribution on $\mathbb R^p$ with mean $\mu = 0 \in \mathbb R^p$ and $p$-by-$p$ covariance matrix $\Sigma$ of rank $r \le p$. Consider the centered ...
1
vote
0answers
77 views

stochastical stable

Given dynamic $f: S^1 \to S^1$ with Lebegue measure $dm$ on $S^1$. Assume it has unique SRB probability measure $\frac{d\mu_f}{dm} dm $. Given left shift space $([-\epsilon, \epsilon]^{\otimes \...
5
votes
0answers
70 views

“middle” partial denominator in continued fraction expansion of square roots

Suppose $d$ is a positive integer that is not a perfect square such that the negative Pell equation, $x^{2}-dy^{2}=-1$ has no solution. Then we know the minimal period of the continued fraction ...
2
votes
1answer
65 views

Hoeffding's inequality for Hilbert space valued random elements

Suppose that $\mathbb H$ is a separable Hilbert space and $X_1,\ldots,X_n$ are independent zero mean $\mathbb H$-valued random elements such that $\|X_i\|\le s$ for each $1\le i\le n$, where $\|\cdot\|...
7
votes
1answer
259 views

Products of Catalan numbers

Let $c(n)=\frac{1}{n+1}\binom{2n}{n}$ be the Catalan number. It seems that a product $\prod_{n\in I} c(n)$, where $I\subset\mathbb N_{>1}$, is never a Catalan number. Is this a (known) fact?
2
votes
0answers
75 views

Function equation over general number fields

Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions $$L(s, \chi)?$$ I only find references for the case ...
10
votes
2answers
292 views

Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?

Good morning, I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
17
votes
1answer
309 views

Curves over number fields with everywhere good reduction

My question is the following:$\newcommand{\Q}{\Bbb Q} \newcommand{\Z}{\Bbb Z}$ What is known about number fields $K$ fulfilling the condition $C_{g,K}$ "there is a smooth projective curve of ...
3
votes
0answers
118 views

How can I get my hands on McKay's “Finite p-Groups” lecture notes?

The notes I'm talking about are these. I emailed Peter Cameron, but he has since moved to a different university, and has no copies himself. I also emailed the school manager at Queen Mary, but they ...
3
votes
1answer
81 views

Is coprimality in $NC$?

Following reference https://pdfs.semanticscholar.org/e86e/8d7a267a29b9ad4ca112828109adfec55e8b.pdf claims integer coprimality is in $NC$ and it also has one citation. Is this claim valid?
1
vote
0answers
35 views

Extension of a result about measurable, additive functionals

Let $W$ be a set, and let $v$ be a finitely additive probability measure on $2^W$. Equip $2^W$ with the Borel sigma-algebra $\mathcal{B}$ generated by the sub-basic sets of the form $\{a: w \in a\}$ ...
1
vote
0answers
40 views

Generating totally ordered free commutative monoids

Let’s say I have a set $A$. I build the free commutative monoid $M$ generated by $A$. When can a well-order on $A$ be extended to $M$, in a way that is compatible with its monoid structure? I am ...
12
votes
1answer
223 views

Tilting Objects in BGG Categories $\mathcal{O}$

Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by ...
7
votes
1answer
126 views

Deuring's result on elliptic curves. Any proof reference

I have heard of this result from Deuring 1941 paper: Given $\mathbb F_p$ ($p$ prime number) and any number $n$ in the Hasse interval $[p+1-2\sqrt p, p+1+2\sqrt p]$ there is an elliptic curve over $\...
2
votes
1answer
87 views

Diophantine equation for generating computably enumerable set

By Matiyasevich's theorem, each member of computably enumerable set can be obtain from a diophantine equation system. For prime numbers, this system of diophantine equation is found. My question is: ...
0
votes
0answers
82 views

A reference for studying special ring

A topological space $X$ is called profinite if it is compact, Hausdorff, and has a basis of open–closed sets. Also a commutative ring $R$ with 1 is called a topological ring it there is a topology on ...