# Tagged Questions

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

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votes

**1**answer

61 views

### Smoluchowski-Poisson dynamics with atomic measures

"Smoluchowski-Poisson dynamics" is just a tentative provisional name I give to the following transport equation:$$\partial_t m+\nabla_x\cdot(um)=0$$where $u(x,t)\in\mathbb R^n$ ($x\in\mathbb R^n$, ...

**1**

vote

**0**answers

63 views

### On the unipotent conjugacy classes in $SU(3,q^2)$

Consider the special unitary group of degree 3 over a finite field $\mathbb{F}_{q^2}$, $q=p^n$ a prime power, and $U$ its Sylow $p$-subgroup (we way fix it to be the subgroup of upper triangular ...

**9**

votes

**1**answer

282 views

### A strengthening of Frankl's union-closed sets conjecture?

A Frankl family is a nonempty finite family $\mathcal F$ of nonempty finite sets such that $A,B\in\mathcal F\implies A\cup B\in\mathcal F.$ Define $d_\mathcal F(x)=|\{A\in\mathcal F:x\in A\}|$ and ...

**2**

votes

**0**answers

103 views

### Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...

**5**

votes

**1**answer

89 views

### Reference request: normalization of intertwining operators for GL(2, C)

Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral
$$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ ...

**1**

vote

**1**answer

47 views

### Seeking a specific proof of endpoint boundedness of Riesz potential

The Riesz potential is defined by
$$I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
Once $f\in L^{d/\alpha}(\mathbb{R}^n)$, then $I_\alpha f(x)\in BMO$. ...

**3**

votes

**1**answer

352 views

### Who is the original author of this simple paradoxical decomposition?

Paradoxical decompositions of sets usually require the axiom of choice; Hausdorff or Banach-Tarski are well-known examples. A paradoxical decomposition of a point set without the axiom of choice has ...

**12**

votes

**1**answer

764 views

### Reverse-engineer forcing: am I reinventing the wheel?

In the course of a project I’m working on, I’ve started playing around with a sort of “reverse-engineering” forcing. It seems interesting, but
I have a sinking feeling I’m reinventing the wheel; does ...

**3**

votes

**0**answers

71 views

### Shifts in the decomposition of Bott-Samelson bimodules

Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that ...

**4**

votes

**1**answer

96 views

### Uniqueness of hyperbolic rescaling

Let $X$ be a compact oriented surface of genus at least two, equipped with a Riemannian metric $g$. By the uniformization theorem for Riemann surfaces, there is a conformal universal covering map ...

**7**

votes

**1**answer

237 views

### Ring of invariants for the regular representation

The symmetric group $S_n$ acts on $\mathbb C^n$ by permuting the coordinates. In this case the ring of invariants is generated by elementary symmetric polynomials in n-variables. Now consider the ...

**4**

votes

**1**answer

160 views

### Improvement of a bound on divisor distributions from “Divisors” (Hall and Tenenbaum)?

In the classic text referred to in the title of this question, the bound
$$
H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x})
$$
is given, where ...

**1**

vote

**0**answers

30 views

### Comparison between the entrance measure and the harmonic measure

Consider the standard two-dimensional Brownian motion, and define $\tau(A)$ to be the hitting time of $A\subset \mathbb{R}^2$. Let $hm_A$ be the harmonic measure (from infinity) on $A$. Let $B(r)$ be ...

**2**

votes

**0**answers

77 views

### Heat kernel on manifold with boundary

Let $M$ be a Riemannian manifold with boundary, let $\mathcal{V}$ be a metric vector bundle over $M$ and let $L$ be a formally self-adjoint Laplace type operator, acting on sections of $\mathcal{V}$. ...

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vote

**0**answers

66 views

### Gaussian heat kernel bounds on Riemannian manifolds

I wish to know if we have Gaussian lower and upper bounds for the heat kernel,i.e. $$
t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_1t}} \lesssim p_t(x,y) \lesssim t^{-n/2}e^{-\frac{\rho(x,y)^2}{C_2t}},
$$
on a ...

**26**

votes

**1**answer

871 views

### Should axiomatic set theory be translated into graph theory?

Recently I saw the abstract of a paper by Nash-Williams: ``Should axiomatic set theory be translated into graph theory?''. The abstract, taken from Mathscinet says the following:
The author ...

**2**

votes

**2**answers

228 views

### The space $L^p(\partial\Omega)$ in cited references

The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...

**4**

votes

**2**answers

138 views

### first order languages over graphs (and other discrete models)

A lot of research has been devoted to the study of first order language of graphs (FO). Formulas in this language are constructed using variables $x, y,\dots$ ranging over the vertices of a graph, the ...

**4**

votes

**0**answers

103 views

### Geometric Characterization of Martingales

Recently I've read a paraphrasing from Ito saying that he sometimes thinks of martingales as Geodesics in a very large dimensional manifold.
My question is, is there any research studying this idea?
...

**4**

votes

**1**answer

301 views

### Looking for a rare paper

I am looking for the rare paper " Shinziro Mori, Allgemeine Z.P.I.-Ringe, J. Sci. Hirosima Univ. Ser. A. 10 (1940), 117–136. ". Can anyone help me to find it or any help?

**4**

votes

**1**answer

209 views

### Relations between some works by Deligne-Mostow and Thurston

happy new year 2016!
A coworker and I are interested in the relations between the works of Deligne and Mostow ([DM] and [M]) on the monodromy of Appell-Lauricella hypergeometric functions (Publ. ...

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vote

**0**answers

89 views

### Monotonicity of the Hellinger integral/distance

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral and distance are
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$ and ...

**1**

vote

**0**answers

22 views

### Elliptic Equation with Wentzell boundary condition

I'm looking for a reference showing how to obtain a priori estimate for solutions to a linear second-order elliptic equation with Wentzell boundary condition in a bounded domain in $H^1$ space.
The ...

**6**

votes

**1**answer

137 views

### Travelling waves for nonlinear Schrödinger equation

Consider the following nonlinear Schrödinger equation:
$$
-\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi,
$$
where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...

**13**

votes

**2**answers

218 views

### Generating functions for objects with irrational sizes

A problem I'm investigating concerns a combinatorial class in which the 'atoms' have irrational sizes. It seems likely that this is something that has been considered before, but I haven't been able ...

**8**

votes

**1**answer

116 views

### The scope of correspondence principle in quantum chaos

My understanding of the so-called correspondence principle in quantum chaos, is that it is a connection between the behaviour of a classical Hamiltonian system (chaotic/completely integrable) and the ...

**1**

vote

**1**answer

56 views

### “Convergence speed” results for the Langevin process

The Langevin process is defined by the following stochastic differential equation:
$$ \dot X = - \nabla \phi + \sqrt 2 dW_t $$
Its equilibrium distribution is the following:
$$ p_\infty (x) \propto ...

**9**

votes

**2**answers

710 views

### Algebraic independance of exponentials

First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution ...

**11**

votes

**0**answers

464 views

### Arguments against Freiling's argument against Continuum Hypothesis

Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...

**12**

votes

**1**answer

334 views

### On convergence of convex bodies

Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any ...

**9**

votes

**2**answers

311 views

### Adaptive version of the Azuma–Hoeffding inequality

The Azuma inequality states that if we have a martingale $X_1,\ldots,X_N$ that satisfies a bounded difference condition:
$$|X_k - X_{k-1}| \leq c_k$$
Then:
$$\Pr\left[X_N - X_0 \geq \sqrt{2\sum_kc_k^2 ...

**2**

votes

**0**answers

73 views

### Kruskal-Katona for multisets?

Following Fedor Petrov's remarks, here is a "set-theoretic version" of the
question I asked a while ago.
For integer $n\ge 1$, denote by $\mathcal M_n$ the family of all (finite)
multisets with ...

**4**

votes

**2**answers

142 views

### Frobenius Groups of Automorphisms

Recently, I am looking different papers on the topic
$$\mbox{Frobenius groups of automorphisms of a group.}$$
But I am familiar with Frobenius groups only; not with their (faithful) actions on groups. ...

**1**

vote

**0**answers

96 views

### Full version of Soucaliuc's research announcement “Réflexion entre deux diffusions conjuguées”

Florin Soucaliuc published the following research announcement in 2002 containing some results from his thesis on reflected diffusion processes:
[1] F. Soucaliuc, Réflexion entre deux diffusions ...

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votes

**0**answers

155 views

### Shortest path to inspect a polyhedron

This is a variant of two as-yet unsolved MO questions cited below.
Let $P$ be a closed polyhedron in $\mathbb{R}^3$.
The task is to find a shortest path $\sigma$ on the surface of $P$ from which
all ...

**7**

votes

**0**answers

162 views

### A definition of the homotopy colimit of a coherent diagram

Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in ...

**7**

votes

**1**answer

330 views

### On (a generalization of) the Gauss Circle Problem

Most (if not all) references I read about the Gauss Circle Problem that proves a bound below $O(R^{2/3})$ reduces the GCP to the Dirichlet Divisor Problem by the well known expression of $r_2(n)$, the ...

**9**

votes

**0**answers

292 views

### Is there an “Erlangen Program” for Graph Theory?

There are certain graph theoretic problems (especially optimization problems), whose solution-subgraph (i.e. the set of vertices and edges)), is invariant under certain modifications (especially ...

**3**

votes

**2**answers

89 views

### About Kurosch-Ore theorem

Where can I find the proof of Kurosch-Ore theorem in lattice theory?
The statement of this theorem is: Let $L$ be a modular lattice with $0$ and $1$ that satisfies both chain conditions. Then for any ...

**8**

votes

**2**answers

256 views

### Proofs of main probability results from other fields

Making connections between different areas is very exciting and probability has already made connections with other fields (BM used in proving complex analysis and PDE results).
To keep it short, I ...

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votes

**1**answer

179 views

### Is Lax-Milgram true without the separability assumption?

I read the Lax-Milgram Theorem in the Navier-Stokes Equations by Temam:
Let $X$ be a separable Hilbert space (norm $\|\cdot\|_X$) and let
$$
a:X\times X\to\Bbb{R}
$$
be a bilinear continuous ...

**2**

votes

**1**answer

138 views

### Stiefel-Whitney classes of tensor product $\xi^m \otimes \eta^n$, computation

Let $\xi^m$ and $\eta^n$ be vector bundles over a paracompact base space. Where can I find a reference to the Stiefel-Whitney classes of the tensor product $\xi^m \otimes \eta^n$ being computed as ...

**4**

votes

**1**answer

209 views

### If $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form, is $n$ a square?

(I have asked a similar question in MSE four days ago, but did not receive any answers. I have therefore cross-posted it to this site, hoping to get some responses.)
An odd perfect number $N$ is ...

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vote

**1**answer

162 views

### Generalizing Dedekind's Factorization Theorem

A classical theorem due to Dedekind states the following:
Let $O_{K}$ be the ring of integers of a number field $K$, and assume
$K$ is generated by adjoining the algebraic integer $\alpha$ to
...

**3**

votes

**0**answers

174 views

### Quartics containing twisted cubics

The set of quartic surfaces in $\mathbb{P}^3$ containing at least one twisted cubic forms the divisor on the space of all quartic spaces (parameterized by $\mathbb{P}^{34}$).
I'am wondering how one ...

**2**

votes

**2**answers

342 views

### A logarithmic cotangent inequality

I must be a terrible googling searcher but I cannot find a reference to the following inequality:
$$ \forall_{\phi\in(0;\frac \pi 4)}\ \ln(\cot(\phi)))\, <\, \cot(2\!\cdot\!\phi) $$
I have just ...

**2**

votes

**2**answers

228 views

### Behaviour of cohomology groups under extension of scalars

Let $\hat{R}\to R$ be a homomorphism of commutative unital rings and let
$\hat{M}$ be an $\hat{R}G$-module for a group $G$. Does the $R$-module isomorphism $$H^n(G,\hat{M}\otimes R)\cong ...

**8**

votes

**2**answers

444 views

### What is an ordered structure, in general?

This is basically a reference request, but the post is going to be relatively long (and a little bit verbose): I apologize in advance for that.
Premise. There are several examples of "ordered ...

**1**

vote

**1**answer

89 views

### Exponential sum estimates similar to the one for $\sum_p (\log p) e(p \alpha)$, but for different sequences

Obtaining a non-trivial estimate for $\sum_p (\log p) e(p \alpha)$ over the minor arcs is one of the estimates required for obtaining the ternary Goldbach for $n$ sufficiently large via the circle ...

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**0**answers

42 views

### “Harmonic oscillator” with $p$-Laplacian

I wonder if there is any literature on the eigenvalue problem for the "$p$-harmonic oscillator" $$-(|u'|^{p-2}u')'(x)+(x^2-\lambda) |u(x)|^{p-2} u(x)=0$$ in $L^p(\mathbb R)$, $p\in(1,\infty)$. Are ...