All Questions

Filter by
Sorted by
Tagged with
0 votes
1 answer
72 views

Reverse Pinsker's inequality for smooth density classes

Suppose we are given a class of probability density functions $\mathcal{F}$ so that for every $f \in \mathcal{F}$ we have $\alpha \leq f \leq \beta$ for some positive $\alpha, \beta \in \mathbb{R}_+$ ...
spacetimewarp's user avatar
9 votes
1 answer
435 views

Does proper forcing preserve properness under PFA?

I'm interested in forcing classes $\Gamma$ which preserve membership in themselves, i.e. for all posets $\mathbb{P}, \mathbb{Q}\in \Gamma$, we have $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}\in\Gamma$. ...
Ben Goodman's user avatar
0 votes
0 answers
24 views

Set of enclosed convex polyhedra in a graph

Given a straight-line graph embedded in $\mathbb{R}^3$ with known vertex coordinates and edges and no edge intersections, is it possible to find all the enclosed convex polyhedra inside? If so, is ...
n1ps's user avatar
  • 1
1 vote
0 answers
23 views

Genericity of local representation with a non-generic local A-parameter

Let $\pi$ be an irreducible smooth representation of a classical $p$-adic group. Suppose that $\pi$ has a local L-parameter associated to some non-generic local A-parameter $\psi$. Then I am wondering ...
Andrew's user avatar
  • 875
3 votes
1 answer
89 views

Extending curves on a surface to a basis for its first homology satisfying intersection criteria

The title suggests a broader scope of inquiry, but my question mostly pertains to the following example: Let $(Y, \mathcal{Z}, \phi)$ be a bordered 3-manifold with Heegaard diagram $\mathcal{H}$ of ...
contingent's user avatar
3 votes
1 answer
114 views

Does the union of fractional Sobolev spaces fills $L^p$?

Let $\Omega\subset \Bbb R^d$ be any open set. Recall that for $s\in (0,1)$, the fractional Sobolev space $W^{s,p}(\Omega)$ is the collection of function in $L^p(\Omega$ such that \begin{align*} \iint_{...
Guy Fsone's user avatar
  • 1,033
2 votes
0 answers
43 views

Relationship between the homology of two types of tensor products of $\mathbb{Z}/ 2 \mathbb{Z}$-graded objects?

Let's consider a $2$-periodic complex $F$ of free $R$-modules, which is just a $\mathbb{Z} / 2 \mathbb{Z}$-graded complex $$F_1 \xrightarrow{d_1} F_0 \xrightarrow{d_0} F_1$$ (really the arrow $d_0$ ...
Rellek's user avatar
  • 401
0 votes
0 answers
46 views

Some calculation about Chern connection

The Chern connection is the unique connection satisfying $\nabla^{0,1}=\bar{\partial}$ and $$ \partial_k\langle u, v\rangle=\left\langle\nabla_k u, v\right\rangle+\left\langle u, \nabla_{\bar{k}} v\...
Elio Li's user avatar
  • 719
0 votes
0 answers
25 views

Elliptic regularity for Dirichlet problem

Let $\overline{M}=M \cup \partial M$ be a compact manifold with boundary, where $\partial M$ is the boundary of $\overline{M}$ and $M$ is the interior of $\overline{M}$. Let $P$ be an injective ...
user505117's user avatar
3 votes
0 answers
65 views

Expansion of Schubert polynomials into standard elementary monomials

I have an explicit formula for expressing any Schubert polynomial in terms of standard elementary monomials that may or may not be cancelation-free. I haven't determined this yet, but it seems likely ...
Matt Samuel's user avatar
  • 2,008
1 vote
0 answers
48 views

Pontryagin's maximum principle for discrete systems: reference request for general case

I am reading the articles: Optimal control for systems described by difference systems, Hubert Halkin, Advances in Control Systems, Vol 1, Academic Press, New York-London, 1964, Pages 173-196, ...
ExpressionCoder's user avatar
1 vote
1 answer
44 views

Linearized operator of higher order $p$ Laplacian

The $p$th Laplacian is defined as $-\Delta_pv= \text{div}(|Dv|^{p−2}|Dv|)$. My question is whether there are any analogous notions of $p$th $m$-Laplacian for $m$ even and odd. For the $p$th bi-...
Sarthak's user avatar
  • 73
0 votes
1 answer
41 views

$L^\infty$ estimate for elliptic PDE with mixed boundary conditions

Take $\Omega$ to be a bounded smooth domain with boundary $\partial\Omega = \Gamma_1 \cup \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are disjoint. Consider the problem $$\Delta u = f \quad\text{in $\...
BBB's user avatar
  • 31
1 vote
1 answer
111 views

Chromatic number of the insert-and-shift graph on $S_n$

Let $S_n$ be the collection of bijections $\varphi:\{1,\ldots,n\}\to \{1,\ldots,n\}$. In an earlier question, the insert-and-shift graph structure was introduced on $S_n$ and the resulting graph is ...
Dominic van der Zypen's user avatar
-3 votes
0 answers
48 views

Bijective proof that kC(n,k)=nC(n-1,k-1) [closed]

I have an exercise that I tried and I really can't do it I'm completely stuck, so I have to prove the equality kC(n,k)=nC(n-1,k-1) with A BIJECTIVE PROOF so by finding two sets as well as a bijection ...
Zappa's user avatar
  • 1
0 votes
0 answers
66 views

Groups $P$ of order $p^5$ with $\Omega_1(P)=P$

I have been working with (particular) groups $P$ of order $p^5$. In fact, the ones that interest me the most are those that satisfy $$\langle x\in P\mid x^p=1\rangle=:\Omega_1(P)=P.$$ After a search ...
Antoine's user avatar
  • 143
3 votes
0 answers
72 views

The criterion for dimensional conjecture for universal Galois deformation rings

I’m writing to ask a question about Mazur’s dimensional conjecture in Lemma 7.5 of the paper [Galatius S, Venkatesh A. Derived Galois deformation rings. Advances in Mathematics. 2018 Mar 17;327:470-...
Nobody's user avatar
  • 817
17 votes
0 answers
2k views

Global character of ABC/Szpiro inequalities

In [M24] it is asserted that "considering $abc$ triples of the form $(1,p^n,1+p^n)$ for a prime number $p$ and an arbitrarily large positive integer $n$, one can verify that ABC/Szpiro ...
Jon23's user avatar
  • 297
0 votes
0 answers
138 views

Why $k((x,t))$ can not be a local field?

If $k$ is a finite field, then $k((x))$ is a local field, and we can define a discrete valuation on $k((x))$ with respect to which it is complete. It is sometimes called a 1-dimensional local field. I ...
MAS's user avatar
  • 872
6 votes
1 answer
150 views

Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces

I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
Steve's user avatar
  • 61
1 vote
1 answer
58 views

Finding closed form roots for pseudo-trinomial

I have the below function: $$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) ...
Arthur's user avatar
  • 11
1 vote
0 answers
98 views

Mulitplicity one property for $\mathcal{D}'$ and $L^2$ over a homogeneous space

Let $G$ and $G_0$ be Lie groups, and suppose that a homogeneous space $X=G/G_0$ have a $G$-invariant measure. It is known (E.G.F. Thomas showed) that there is an admissible parametrization $\{\mathcal{...
user509119's user avatar
4 votes
2 answers
164 views

Simple proof that exactness implies strong mixing

Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
Uagi's user avatar
  • 63
2 votes
1 answer
205 views

Order on Euclidean space in which a finite poset embeds

Fix positive integers $k$ and $n$. For which finite posets $(X,\lesssim)$ with $\#X=k$ does there exist an order embedding $\phi\colon(X,\lesssim)\to (\mathbb{R}^n,\le)$, where $\le$ is the standard ...
ABIM's user avatar
  • 5,019
6 votes
3 answers
453 views

Evaluating the infinite product $\prod_{k\geq 2}(1-\frac{1}{k^3})$

Does anyone know how to evaluate the infinite product $$ \prod_{k = 2}^{\infty} \left( 1 - \frac{1}{k^3} \right)? $$ I know that a generalized quadratic version has a nice closed form $$ \frac{\sin(\...
kodlu's user avatar
  • 10.1k
5 votes
0 answers
196 views

Theories of manifolds w/ extra structure and singularities

Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
Will Sawin's user avatar
  • 135k
1 vote
1 answer
140 views

Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves

The following passage is from a thesis I'm reading: Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
Johannes's user avatar
0 votes
0 answers
23 views

Is there a variant of the crossing lemma for multigraphs with arbitrary embedding?

Suppose $G$ is a graph with $m=|E(G)|$ edges and $n=|V(G)|$ vertices. Suppose $sim(G)$, the simplification of $G$ contains $ m' >> 3n $ edges. Call the set of edges corresponding to an edge $uv$...
Hao S's user avatar
  • 181
5 votes
1 answer
342 views

Second-order ordinal definability

As is familiar, a set $S$ is ordinal definable ($S \in \mathsf{OD}$) just in case there exists a formula $\varphi(x, \vec{z})$ of the first-order language of set theory with parameters $\vec{z} = z_1, ...
Beau Madison Mount's user avatar
1 vote
1 answer
97 views

Particular example of a quadratic extension of a nonunital ring

I want to construct a concrete non-unital ring $R$ with the following properties: $R$ is a noncommutative non-unital ring with a right unite $r$ i.e $t.r=t$ for any $t\in R$. $S\subset R$ is a ...
GSM's user avatar
  • 343
3 votes
1 answer
174 views
+100

Systems of (hyperbolic) 2nd order PDEs with lower order constraints

Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\...
Daniel Castro's user avatar
0 votes
0 answers
30 views

Gradient-based optimization of $n$ functions

I appreciate the willingness of everyone to assist me in advance. I am faced with a set of $n$ distinct convex optimization problems, each defined as follows: \begin{equation} \max\limits_{x \in \...
raian's user avatar
  • 1
2 votes
1 answer
157 views

Reference request: Best book to cite on a property of the family of Cauchy distributions

Kai Lai Chung once began a section of a textbook on probability by writing "Everybody knows" that $$ e^x = \sum_{n\,=\,0}^\infty \frac{x^n}{n!}. $$ (with those quotation marks). Other ...
Michael Hardy's user avatar
2 votes
0 answers
48 views

How many ways to win a game between two teams with arbitrary player skills

Suppose we have $n\geq 4$ players $p_1,\cdots,p_n$ of a game between two teams: team $A$ and team $B$ (disjoint sets, each with two or more players, so that $|A|+|B|=n$). Assume that each player $p_i$ ...
bernardorim's user avatar
7 votes
2 answers
277 views

Integral means vs infinite convex combinations

Let $(X,\mathcal A, \mu)$ be a probability space, $\mathbb E$ a Banach space, and $f:X\to\mathbb E$ a Bochner integrable function. Does there exist a sequence $(x_k)_{k\ge 1} $ in $X$, and a ...
Pietro Majer's user avatar
  • 56.5k
1 vote
1 answer
50 views

Equivalent condition for the Pick matrix being positive semidefinite

On the wikipedia page of the Nevanlinna-Pick theorem the following claim appears: Let $\lambda_1,\lambda_2,f(\lambda_1),f(\lambda_2)\in\mathbb{D}$. The matrix $P_{ij}:=\frac{1-f(\lambda_i)\overline{f(\...
JustSomeGuy's user avatar
0 votes
1 answer
99 views

Regular value theorem for Banach manifolds without surjectivity

It is well-known (e.g. Lang, Fundamentals of differential geometry, Prop. 2.3 in Chapter II) that the following extension of the regular value theorem holds for Banach manifolds: Let $\phi : M\...
Martin's user avatar
  • 141
3 votes
1 answer
200 views

Are there atoms in the lattice of intermediate logics?

A few days ago I stumbled upon this question on MS. The question is: Does the lattice of intermediate logics have an atom, i.e. an element that is strictly stronger than IPC but not strictly stronger ...
Navid's user avatar
  • 31
5 votes
0 answers
135 views

A puzzle with magic Egyptian tilings

Background I've recently been devising a puzzle that incorporates elements from Egyptian fractions, magic squares, and tilings. The objective of the puzzle is to tessellate a square with sides of ...
Max Muller's user avatar
  • 4,485
-3 votes
0 answers
41 views

Probability of sequence of estimated values belonging to a set

Suppose I have estimated residuals of an ARCH(1) process: $\hat{\varepsilon}_1, \dots, \hat{\varepsilon}_n$ from the sample of length $n$. On the other hand, I have "true" residuals: $\...
Grigori's user avatar
  • 17
0 votes
0 answers
33 views

Permutation for the fast computing of the $k$-th partition of $n$ in reverse lexicographic order

Let $a(n)$ be A000041 (i.e., $a(n)$ is the number of partitions of $n$ (the partition numbers)). Let $b(n)$ be A000070. Here $$ b(n) = \sum\limits_{i=0}^{n}a(i) $$ Let $c(n)$ be $k-1$ where $k$ is the ...
Notamathematician's user avatar
1 vote
0 answers
186 views

On $(k,\ell)$-sumfree sets

Call a set $\mathcal S \subset \mathbb N$ to be $(k,\ell)$-sumfree if there are no non-trivial solutions to the equation $$x_1+\dots +x_k = y_1+\dots +y_\ell$$ in the set (for distinct $x_i$'s and $...
Sayan Dutta's user avatar
1 vote
1 answer
85 views

Characterization of Fellerian kernels

This question concerns Feller Markov kernels, similar to Vanessa's question. Terminology By 'Markov kernel' $N:E\to F$, we adopt exactly the same definition as Vanessa, with the exception that $E,F$ ...
Ano2Math5's user avatar
  • 103
1 vote
0 answers
124 views
+50

Specific type of PDE

While deriving the SHJB equation for a specific case I stumbled upon this (using Einstein's convention for repeated indices): $$\rho J = (x_i \tilde u_i-K_i) + (\alpha_i + \beta_{ij}x_j - R_{ijk} \...
Gennaro Marco Devincenzis's user avatar
-1 votes
0 answers
51 views

Continuous version of ergodic with integral

Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
Sean's user avatar
  • 311
0 votes
0 answers
74 views

I'm looking for the NLab page on particle species

This is just a reference request. I came across an NLab page on particle species described as certain vector bundles. But I can't seem to find it again when I searched recently. If someone can point ...
Mozibur Ullah's user avatar
8 votes
1 answer
469 views

Is there a better name for the "Mayer-Vietoris Octahedral axiom" and has it been studied?

$\newcommand{\K}{\mathcal{K}}$Say $\K$ is a triangulated category with suspension $\Sigma:\K\simeq \K$. In Iversen's book "Cohomology of Sheaves", he doesn't exactly examine triangulated ...
FShrike's user avatar
  • 569
-3 votes
0 answers
117 views

Extending the proof of Maschke's Theorem from finite groups to algebras

In the theory of representations of a finite group there is Maschke's Theorem that any finite-dimensional representation of a finite group $G$ can be decomposed into a direct sum of irreducible ...
Dale's user avatar
  • 431
5 votes
1 answer
214 views

Parabolic subgroups of reductive group as stabilizers of flags

$\DeclareMathOperator\GL{GL}$Let $G$ be a linear algebraic group (probably reductive will be needed). Consider a faithful representation $G \to \GL(V)$. Given a parabolic subgroup $P < G$, we can ...
a_g's user avatar
  • 497
-4 votes
0 answers
81 views

What is the Measure of the Permutations of the Real Numbers? [closed]

Since the permutations of the real numbers would form a set of cardinality $\aleph_2$, do we just say it has infinite measure? It would seem to be the case for a Lebesgue measure. Is there a standard ...
Carl Gueck's user avatar

15 30 50 per page