Questions tagged [reference-request]

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

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6
votes
1answer
216 views

Orbifolds are Thom-Mather stratified spaces

Where can I find a proof of (or if it is even true) that an (effective) orbifold is a Thom-Mather stratified space? edit: after some search, I found the proof should be contained in either GIBSON, C....
6
votes
1answer
188 views

Is the projectivization of a topological vector space Tychonoff?

Let $E$ be a locally convex topological vector space over $\mathbb{R}$. The projectivization $PE$ is the quotient of $E\backslash\{0_{E}\}$ with respect to the equivalence relation $e\sim f$ if $e=\...
5
votes
1answer
144 views

Permutations of a group that are eventually left translations

$\DeclareMathOperator\FSym{FSym}\DeclareMathOperator\Sym{Sym}$Notation: for $X$ a set, $\Sym(X)$ the group of permutations of $X$, and let $\FSym(X)$ be the subgroup of finitely supported permutations ...
5
votes
0answers
73 views

C^*-algebra theory with all the Koszul signs

I was wondering if someone knows of a reference in which $\mathbb{Z}_2$-graded $C^*$-algebra theory is developed using the sign convention $(ab)^* = (-1)^{|a||b|}b^* a^*$. I would be most enthusiastic ...
3
votes
1answer
66 views

Array power-means / generalisation of Hölder inequality

Summary: I have (I think) a generalisation of Hölder's inequality for positive reals that I can neither prove nor find references to. Pointers would be appreciated or, indeed, a counterexample. ...
4
votes
1answer
123 views

Parabolic equation with Cauchy boundary condition

Consider the domain $[0,1] \times [0,T]$ and the uniformly parabolic operator $L -\partial_t$ with smooth coefficient. I would like to obtain the existence of the problem \begin{equation} \left\{\...
3
votes
0answers
32 views

Reproducing kernel Hilbert space of Matérn kernels

I am trying to read a recent paper titled "Interpolation and learning with scale dependent kernels" by Pagliana, Ruidi, De Vito, and Rosasco. (The paper can be found on ArXiv) On the top of ...
6
votes
1answer
145 views

$2$-dimensional complete local normal domain with rational singularity that has exactly one exceptional curve

Let $(R, \mathfrak m)$ be a complete local normal domain of dimension $2$ with residue field $R/\mathfrak m$ algebraically closed and characteristic $0$. Assume Spec$(R)$ has rational singularity, let ...
0
votes
0answers
100 views

The strong twin conjecture can be transformed into the unsolvability of a particular Diophantine equation

Let us consider the strong twin conjecture: For all positive integer $n$ there exist a prime $p$ such that $$n+4<p<2^n2^4$$ and $p$ is a prime and $p+2$ is a prime Since the inequalities and the ...
3
votes
2answers
275 views

When are two resolutions of a coherent sheaf homotopic

Let $\mathcal{F}$ be a coherent sheaf on a projective manifold $X$. It is well known that one can construct a resolution of $\mathcal{F}$ by holomorphic vector bundles (locally free sheaves). Are two ...
11
votes
3answers
405 views

An open triangle problem in plane geometry

Some years ago, I asked some 'famous' people in an advanced Plane Geometry forum about the following: Let $ABC$ be arbitrary triangle, how can one construct a point $P$ in the plane such that $P$ is ...
3
votes
0answers
80 views

Polytope algebra and toric vareties

Let $\Pi$ denote the McMullen polytope algebra (over the field of rationals $\mathbb{Q}$) generated by convex polytopes with rational vertices in $\mathbb{R}^n$. For a simple polytope $P$ let us ...
4
votes
0answers
154 views

Two triangles have the same centroid theorem

Let $\triangle ABC$ and $\triangle A'B'C'$ be two triangles. The line through $A$ and perpendicular to $AA'$ meets the line through $B'$ and perpendicular to $BB'$ at $A_b$; The line through $A$ and ...
3
votes
0answers
65 views

Radial-energy decomposition of a velocity field in 2D: is anyone able to show that the lemma below is true (…or false)?

In the book “Vorticity and Incompressible Flow” by Majda and Bertozzi, there is the following lemma. Lemma 3.2. Any smooth incompressible vector field $v$ in $\mathbb{R}^2$ with vorticity $$\omega=\...
1
vote
0answers
193 views

Globalization of a local field

I am reading the paper ''Endoscopic classification of representations of quasi-split unitary groups'' by Chung Pang Mok, and cannot come up with the proof of theorem 7.2.1. Here is the statement. ...
2
votes
1answer
88 views

Asymptotic growth of ternary partitions of integers $3n$

Consider the binary partitions of $2n$ in powers of $2$, denoted by $b(2n)$, with the generating function $$\sum_{n\geq0}b(2n)\,x^n=\frac1{1-x}\prod_{k\geq0}\frac1{1-x^{2^n}}.$$ A result of De Bruijn ...
10
votes
2answers
532 views

Reference for combinatorics with view towards representation theory/algebraic geometry

I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly ...
1
vote
0answers
62 views

Number of $k$-cycles in a random Mallows permutation

It is well-known that for uniform random permutations, the number of $k$ cycles for fixed $k$ is distributed as a Poisson random variable with mean $1/k$. I am looking for similar results on the ...
1
vote
0answers
72 views

Separating two sets by a $\boldsymbol{\Delta}_3^0$ set

Let $X$ be a Polish space and $A,B\subseteq X$ be such that $A\cap B = \emptyset$, we know that if there is no $\boldsymbol{\Delta}_2^0$ set separating $A$ from $B$ then there exists a Cantor set $C\...
4
votes
0answers
66 views

Possible number of zeros of a stable perturbation of a germ $(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$

Let $f:(\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0)$ be an analytic germ. Assume that it has isolated zero at 0, that is, $f^{-1}(0)=\{0\}$, what is more, assume that the dimension of the local algebra* $Q(...
0
votes
1answer
120 views

Matrix representations of Lie groups of type $B_n$

For the Lie algebra $\mathfrak{so}(2n+1, \mathbb{C})$, there is a matrix representation given by the following matrices: \begin{align} \left( \begin{matrix} 0 & x & y \\ -y^T & A & B \\...
5
votes
1answer
275 views

Proof that every three-dimensional Einstein manifold has constant curvature

In pseudo-Riemannian geometry it is well known that every three-dimensional Einstein manifold has constant curvature. A proof of this is sketched here. Question. Does anyone know where in the ...
12
votes
2answers
742 views

Two interpretations of a sequence: an opportunity for combinatorics

The sequence that is addressed here is resourced from the most useful site OEIS, listed as A014153, with a generating function $$\frac1{(1-x)^2}\prod_{k=1}^{\infty}\frac1{1-x^k}.$$ In particular, look ...
7
votes
1answer
203 views

On a certain double integral appearing in the Fourier series coefficients of $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series

The following integral appears naturally within the computation of the Fourier series coefficients of a real analytic $\mathrm{SL}_2(\mathbb{C})$-Eisenstein series: \begin{align*} \int_{-\infty}^{\...
0
votes
0answers
103 views

What is the inverse Fourier transform of $\operatorname{sinc} \Big{(} \prod_{k=1}^{m} \big{(} k(z-m+1) -z) \big{)} \Big{)} $?

For a certain interpolation problem, I'm looking into a sequence of functions of the form $$f_{m}(z) = \operatorname{sinc} \Bigg{(} \prod_{k=1}^{m} \big{(} k(z-m+1) -z) \big{)} \Bigg{)} . $$ Here, $m&...
2
votes
1answer
214 views

Congruence residues of integer partitions

Consider the number of integer partitions $p(n)$ of $n$ whose (product) generating function reads $$\sum_{n\geq0}p(n)\,x^n=\prod_{k\geq1}\frac1{1-x^k}.$$ There are many congruences for $p(n)$ ...
5
votes
0answers
110 views

Higher homotopy groups of an orbifold

Given an orbifold $\mathcal{O}$, I have seen many ways to define the orbifold fundamental group: Thinking of $\mathcal{O}$ as a groupoid $\mathcal{G}$, $\pi_1^{orb}(\mathcal{O})$ can be defined as ...
3
votes
1answer
515 views

A (mild?) question on the number of monomials

Let $[n]_q=\frac{1-q^n}{1-q}$ with $[0]_q=0$. Recall the $q$-factorials $[n]_q!=[1]_q[2]_q\cdots[n]_q$ (with $[0]_q!=1$) and the $q$-binomials $$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\,[n-k]_q!}.$$ Now, ...
5
votes
0answers
100 views

Laplacian spectrum and measured Gromov-Hausdorff convergence of Riemannian manifolds with boundary

In the paper "Collapsing of Riemannian manifolds and eigenvalues of Laplace operator" by Kenji Fukaya, it is proven that the spectrum of the Laplacian is continuous with respect to measured ...
1
vote
0answers
90 views

Reference for global theory of Schrödinger operators

Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
1
vote
0answers
63 views

Four incenters lie on a circle-Does this theorem have a name?

When I read the new paper 100 CHARACTERIZATIONS OF TANGENTIAL QUADRILATERALS-section 7, I remember that I posed some problem associated with tangential quadrilateral from 2014 in here and 2015 in here ...
0
votes
0answers
38 views

Measure and other properties of nodal domains of Laplacian

Let $(\phi_k,\lambda_k)$ be the couple of eigenfunctions and eigenvalues of the the Laplacian operator on $\Omega \subset \mathbb R^n$. The nodal set of $\phi_k$ is the set $$\mathcal N_k = \{x \in \...
2
votes
0answers
43 views

A.e. global existence of solution to 'encased' n-body problem already somewhere in the literature?

In the study of the Newtonian n-body problem, it seems that Von Zeipel's theorem and Saari's theorem concerning the improbability of collision singularities ought to lend themselves to a nice ...
2
votes
0answers
129 views

Counterexample to mostow rigidity theorem

I am looking for an example of $M$ and $N$ two orientable hyperbolic complete without boundary 3-manifolds ( with infinite hyperbolic volume) such that $\pi_{1}M\cong \pi_{1}N$ but $M$ is not ...
1
vote
0answers
69 views

Decomposing a planar graph

Thomassen proved that the vertex set of every planar graph can be decomposed into two sets inducing a 1-degenerate graph and a 2-degenerate graph, respectively (C. Thomassen, Decomposing a planar ...
1
vote
0answers
71 views

Set of integer non-negative matrices with positive diagonals

This is essentially a reference request/name inquiry. Is there a name for the set $M_k$ formed by $k$ by by $k$ matrices with non-negative integer entries and positive values on the diagonal? Related, ...
1
vote
0answers
52 views

Partially hyperbolic systems and specification

Let $f: M \rightarrow M$ be a $C^{1+\alpha}$ diffeomorphism on a Riemannian compact manifold. Suppose that $f$ admits a dominated splitting $T M=E \oplus F$ with $E\ll F$, where $E$ is uniformly ...
4
votes
0answers
259 views

Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$

Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified. Is there any characterization of $\Gamma$ such that $\Gamma$...
2
votes
0answers
50 views

Eisenstein theorem for algebraic power series in positive characteristic

Eisenstein proved that if a power series $\sum_{n\ge0}a_nz^n$ over $\mathbb C$ is algebraic over $\overline{\mathbb Q}(z)$, then it exists positive integers $a$ and $b$ such that for all $n\in\mathbb ...
2
votes
1answer
57 views

Elliptic regularity when the Lagrangian is possibly infinite

I want to solve variational problems of the form $$\inf_u \int_{-1}^1 \phi(u'(x)) \text{ with } u(-1)=u(1) = 0,$$ where $\phi(p)$ is convex and is allowed to take on the value $+\infty$ for some ...
2
votes
2answers
221 views

Direct calculation of the Fisher distance via Riemannian geodesics

I'm looking for a reference for a direct calculation of the Fisher distance (to avoid overloading the term "metric") $d_F(x,y) := 2 \cos^{-1} \sum_i \sqrt{x_i y_i}$ as the geodesic distance ...
0
votes
2answers
180 views

Question about $\theta$ and the Riemann Hypthesis - reference request

It is well known that the Riemann Hypothesis implies the following: $|\theta(x) - x| = O(x^{1/2 + \epsilon})$ for all $\epsilon > 0$. where $\theta$ is the first Chebyshev function; that is, $\...
2
votes
1answer
83 views

Change of variable and boundary data for Laplace equation

Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem $$ \begin{cases} -\Delta u = 0 & x \in \Omega \\ u = 1 & x \in \partial \Omega \end{cases} $$ Does it make ...
5
votes
1answer
235 views

Analogue of the second Hardy-Littlewood conjecture for numbers of divisors?

Let $f(n)$ denote the proposition "There exists some $k>1$ such that $$ \sum_{m=k}^{k+n-1}\tau(m) < \sum_{m=1}^n\tau(m) $$ where $\tau(m)$ is the number of the divisors of $m$." (This ...
3
votes
0answers
128 views

de Rham currents/distributions on manifolds with boundaries

My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
2
votes
1answer
154 views

Viscosity solutions of $(-\Delta)^s u = 0$ in $\Omega $ with non-homogeneous data $u = 1$ in $\mathbb R^n \setminus \Omega$

Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem $$ (1) \quad \begin{cases} (-\Delta)^s u +\lambda u= 0 & x \in \Omega \\ u = 1 & x \in \mathbb R^n \...
2
votes
0answers
45 views

Second order estimates for Dirichlet problem for complex Monge-Ampere equation

Let $\Omega\subset \mathbb{C}^n$ be a bounded pseudo-convex domain. Let $0<f\in C^{\infty}(\bar\Omega)$, $\phi\in C^\infty(\partial \Omega)$. Consider the Dirichlet problem for the complex Monge -...
4
votes
0answers
74 views

"Dual" of a CP map

Let $M,N$ be von Neumann algebras, and let $\phi:M\rightarrow N$ be a normal completely positive map. I am interested in conditions when there is a "dual" normal completely positive map $\...
7
votes
4answers
180 views

Best source for classification of right-angled hyperbolic hexagons

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose ...
2
votes
0answers
182 views

Has anyone written down an approach to the Lenard-Magri integrability scheme via algebraic geometry?

I’ve been thinking about the algebro-geometric meaning of the Lenard-Magri scheme of getting an integrable system from a pair of compatible Poisson structures. I think one might be able to prove a ...