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Hardy-Littlewood in Sobolev Spaces

For an application in kinetic theory I have to apply the Hardy-Littlewood-Sobolev(H-L-S) for $q=\infty$. The dimension is $3$ and H-L-S inequality says that for $1<p<q<\infty$ and $0<\nu&...
Sanchit's user avatar
  • 81
1 vote
0 answers
118 views

On the exponent of a certain matrix $A$ in characteristic $p > 0$

Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$. Suppose that further the $(m,n)$-component $a_{m,n}$ ...
Pierre's user avatar
  • 563
2 votes
0 answers
100 views

Component Groups of Reductive Groups

Suppose $G$ is a reductive group that is not necessarily connected and $Z \subset G$ is a central subgroup. Suppose $G^0$ is the identity component of $G$. Is it true that $G/G^0Z= \pi_0(G/Z)$? I can ...
Alexander's user avatar
  • 861
1 vote
0 answers
50 views

Estimate on Covariant Derivatives of Coordinate Derivatives

I am currently reading Topping's lecture notes on Ricci flow. At one point in the narrative (page 65) he says that using the fact that $\bigg(\frac{\partial}{\partial t}\nabla - \nabla \frac{\...
Hollis Williams's user avatar
12 votes
1 answer
432 views

Quasimorphisms and Bounded Cohomology: Quantitative Version?

Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. ...
BharatRam's user avatar
  • 939
5 votes
2 answers
567 views

Compact complex affine Kähler manifold is a torus

Before giving a motivation let me ask the precise question firstly. By a complex affine manifold I mean a complex manifold $M$ with the property that there exists an holomorphic atlas for which ...
J.E.M.S's user avatar
  • 437
0 votes
1 answer
85 views

Linear intersection number and chromatic number for infinite graphs

Given a hypergraph $H=(V,E)$ we let its intersection graph $I(H)$ be defined by $V(I(H)) = E$ and $E(I(H)) = \{\{e,e'\}: (e\neq e'\in E) \land (e\cap e'\neq \emptyset)\}$. A linear hypergraph is a ...
Dominic van der Zypen's user avatar
5 votes
1 answer
202 views

Spectral radius for multiple linear operators

Suppose that $X$ is a finite dimensional Hilbert space. Let $A_{1},\dots,A_{r}:X\rightarrow X$ be linear operators. Then define the multi-spectral radius of $(A_{1},\dots,A_{r})$ to be $$\limsup_{n\...
Joseph Van Name's user avatar
6 votes
0 answers
130 views

Integral geometry for general closed smooth manifolds

Let $M$ be a closed smooth manifold of dimension $2n$ for some positive integer $n$. Let $\mathit{Diff}(M)$ be the group of diffeomorphisms of $M$. Let $L$ be a closed embedded $n$-dimensional ...
rori's user avatar
  • 257
1 vote
0 answers
150 views

Do we believe that the distribution of spacings of successive critical zeros of zeta is log-symmetric?

Let $\gamma^{+}(T)$ be the imaginary part of the critical zero of $\zeta$ closest to $1/2+iT$ with $\gamma^{+}(T)\ge T$ and define similarly $\gamma^{-}(T)$ with a reversed inequality. Let $g(T)$ be ...
Sylvain JULIEN's user avatar
2 votes
0 answers
139 views

Global solution of second order ODE defined on riemannian manifold

Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the ...
Foivos's user avatar
  • 345
0 votes
0 answers
56 views

Are total graph of power of cycles homeomorphic to powers of cycles?

Is the total graph associated to powers of cycles homeomorphic to powers of cycles themselves? I think yes, because the total graph associated to cycles is homeomorphic to cycles(i think?)So, does ...
vidyarthi's user avatar
  • 2,007
4 votes
0 answers
463 views

Comparing real topological K-theory and algebraic K-theory

Let $R$ be a commutative unital ring and let $i$ be a non-negative integer such that $K^i_{alg}(R)$ is finitely generated abelian group. Is it possible that there does not exist weak homotopy type of ...
rori's user avatar
  • 257
1 vote
1 answer
247 views

A totally geodesic triangulation

Let $M$ be a compact orientable $n$ dimensional Riemannian manifold. Is there a triangulation of $M$ such that every $k$ dimensional face of each simplex is a totally geodesic submanifold, $\forall k ...
Ali Taghavi's user avatar
5 votes
1 answer
331 views

Why restrict to $\Sigma_1^0$ formulas in $RCA_0$ induction?

I recently asked this question over on math.se, warmly welcomed by crickets. I hope it's appropriate here. I'm reading Stillwell's Reverse Mathematics, and the induction axiom was just introduced. ...
luqui's user avatar
  • 585
2 votes
0 answers
97 views

Joining metrics of positive Ricci curvature

Let $M$ be a smooth manifold such that there is a closed submanifold $S\subset M$ with a Riemannian metric $g_S$ given by the restriction of a Riemannian metric on $M$ satisfying $\mathrm{Ricci}_{g_S}(...
L.F. Cavenaghi's user avatar
2 votes
0 answers
127 views

Status of locality of perfectoidness for uniform rings

Let $k$ be a perfectoid field of zero characteristic. Recall that a Tate $k$-algebra is called uniform if the set of power-bounded elements is bounded. Let $(A, A^+)$ be a uniform complete affinoid $k$...
rori's user avatar
  • 257
2 votes
0 answers
100 views

Solutions of a partial differential equations

I'm looking for solutions of a PDE of the form : $ P(t,x):[0,\infty]*[0,b]\rightarrow[0,1]$ $$ \partial_t P(t,x)= \partial_x [(1+2x) P(t,x)]$$ $$ P(0,x)=\delta_x (0)$$ $$P(t,b)=g(t)$$ Where $b$ is a ...
user avatar
3 votes
1 answer
351 views

Solution singular PDE

I've been studying the following singular PDE $$ \mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$ in $\Omega \subset \mathbb{R}^{2}$. Do you know any reference, ...
user127742's user avatar
5 votes
0 answers
214 views

Generalizing the formula between Wu class and the Steenrod square

I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy $$ Sq^{d-j}(x_j)=u_{d-j} x_j, \text{ for any } x_j \in H^j(M^d;\mathbb Z_2) . \tag{eq.1}$$ ...
wonderich's user avatar
  • 10.3k
1 vote
0 answers
47 views

Density of different types of critical points in an algebra of elementary embeddings

Suppose that $j,k:V_{\lambda}\rightarrow V_{\lambda}$ are elementary embeddings. Let $\mathrm{crit}_{n}(j,k)$ denote the $n$-th element in $\{\mathrm{crit}(\ell)\mid\ell\in\langle j,k\rangle\}$. ...
Joseph Van Name's user avatar
2 votes
0 answers
153 views

Condition on a Lie groupoid to be represented by manifold/group or an action groupoid

Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions. When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some ...
Praphulla Koushik's user avatar
1 vote
0 answers
93 views

Has this result about the number of permutations of a given cycle type (or centralizers) been proved?

I was playing with the cardinality of conjugacy classes of the symmetric groups, which we know is the number of permutations of a given cycle type and there is a natural one-to-one correspondence ...
Anthonny's user avatar
  • 151
5 votes
1 answer
377 views

Oscillation and Hölder continuity

Where can I find a proof of the following fact? If $$w(u,x_{0},r)=\sup _{B_{r}(x_{0})}u-\inf _{B_{r}(x_{0})}u$$ for some function $u(x)$ satisfies $$ w\left(u,x_{0},{\tfrac {r}{2}}\right)\leq \lambda ...
user avatar
1 vote
0 answers
42 views

Density of critical points subalgebras of the algebras of elementary embeddings

Let $j:V_{\lambda}\rightarrow V_{\lambda}$ be an elementary embedding. Then $\{\mathrm{crit}(k)\mid k\in\langle j\rangle\}$ has order type $\omega$, so let $\mathrm{crit}_{n}(j)$ denote the $n$-th ...
Joseph Van Name's user avatar
2 votes
0 answers
85 views

$n$-point map $S$ from $k$-manifold $X$ to $\mathbb{R}^k$

It is known that for every continuous map $M:S^1\to \mathbb{R}$ there are infinitely many double points. Also by the Borsuk-Ulam theorem, this is true for each continuous map $N:S^n\to \mathbb{R}^n$, $...
MasM's user avatar
  • 289
6 votes
0 answers
140 views

On computability of sheafification

The question will feature some imprecise words but I believe that an expert could parse it to a precise question more or less uniquely. Assume I have a reasonable topological space (say the ...
rori's user avatar
  • 257
2 votes
2 answers
143 views

Expectation of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Given that following two random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
Felipe Augusto de Figueiredo's user avatar
12 votes
2 answers
2k views

Can we do better than Azuma-Hoeffding when the variance is small?

The Azuma-Hoeffding Inequality says that if $X_1,X_2, \ldots$ is a martingale and the differences are bounded by constants, $\|X_i - X_{i-1}\| \le 1$ say, then we should not expect the difference $\|...
Daron's user avatar
  • 1,761
-1 votes
1 answer
133 views

Integral representation of the Digamma function vs. Integral representation of the Polygamma function [closed]

Yesterday I asked for the derivation of the Integral representation of the Digamma-Function: https://math.stackexchange.com/questions/3113119/integral-representation-of-digamma-function Thanks again @...
ansebene's user avatar
2 votes
1 answer
97 views

p.d.f. of $\left| \frac{\textbf{x}^{H} \textbf{y} }{\| \textbf{x} \|^2} \right|^2$, where $\textbf{x}$ and $\textbf{y}$ are complex Gaussians?

Given that the random variables $\textbf{x} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{x}^{2}\textbf{I}_{M})$ and $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ are ...
Felipe Augusto de Figueiredo's user avatar
4 votes
0 answers
66 views

Periodic modules in Frobenius algebras

Let $A$ be a finite dimensional Frobenius algebra and assume there exists an indecomposable periodic module $M$, that is $\Omega^n(M) \cong M$ for some $n$. Question: Does this imply that there is ...
Mare's user avatar
  • 25.8k
3 votes
2 answers
242 views

Reference request: $\alpha$-Hölder spaces as double duals

If $(X,d)$ is a complete metric space, we define the $\alpha$-Hölder class $\Lambda_\alpha(X)$ as the subset of $C_b(X)$ satisfying that $$ \sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}. $$ ...
Adrián González Pérez's user avatar
1 vote
1 answer
119 views

Sobolev extension operators

Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary and that $M\subset \tilde{M}$ with $(\tilde{M},g)$ also a compact Riemannian manifold with smooth boundary. Let us consider a one-...
Ali's user avatar
  • 4,077
1 vote
0 answers
137 views

Proof that two vectors can not have the same power spectrum when one is a permutation (excluding rotations) of the other?

The power spectrum being the absolute value of the DFT of the vector. Has it been proven that two vectors can not have the same power spectrum if one is a permutation of the other? Where, in this ...
kreitz's user avatar
  • 53
7 votes
0 answers
229 views

Completeness is a conformal invariant

In this article about completeness of semi-riemannian manifolds the author writes (in section 2.2) that it is unkown if the following statement holds: A compact indefinite manifold which is conformal ...
JS.'s user avatar
  • 893
3 votes
1 answer
332 views

Peak sets and Choquet boundary of a function algebra

I have two problems to ask. Let $A$ be a function algebra of $C(K)$. $t\in K$ is said to be a peak point of $A$ if $\exists~f\in A$ s.t. $|f(t)|=\|f\|$ and $|f(s)|<|f(t)|$ for any $s\neq t$. ...
Tanmoy Paul's user avatar
2 votes
1 answer
140 views

Must $q$ be analytic?

I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that $$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$ which also takes $\mathbb{R}^+ \to \...
Richard Diagram's user avatar
2 votes
0 answers
193 views

Shattering with sinusoids

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \{-1,1\}$ such that $\sum\...
Rajesh D's user avatar
  • 704
1 vote
1 answer
450 views

Is there a way to turn a non convex set to a convex one? [closed]

Perhaps the question is rather vague, but if we are given a non-convex set S, can we construct some invertible mapping f so that f(x) becomes a convex set? In my problem , I am given a set of matrix ...
蔣聞哲's user avatar
2 votes
1 answer
57 views

parametrize triangles meeting certain conditions

Consider triangles with angles alpha, beta, gamma such that gamma = 2 alpha, and sides (a,b,c) are integers. I want to parametrize such triangles by a single integer or rational variable.
Michael Beeson's user avatar
1 vote
0 answers
85 views

non zero differential in a spectral sequence

This is the situation: Let $A = R_* \otimes C_*$ be an $R$-module where $C_*$ is a finitely generated graded ($*\geq 0$) vector space over a field $F$ which is also bounded above, and $R$ is a ...
Vitolo's user avatar
  • 81
1 vote
1 answer
912 views

Monotonicity, Convexity, and Smoothness of the KL-Divergence between Two Brownian Motions with Different Initializers

We consider the two distributions $$ p_t = p_0 * N(0, tI),\quad q_t = q_0 * N(0, t I), $$ where $*$ denotes the convolution between two densities, while $p_0$ and $q_0$ have the same mean and ...
Minkov's user avatar
  • 1,117
1 vote
0 answers
196 views

Find wrapping angle of helix on a torus

I need some help in calculating the wrapping angle of a spiral helix wrapped on a torus with constant angle against all the meridians of the torus. The wrapping angle (or the angle measured around and/...
Niculae George Razvan's user avatar
6 votes
2 answers
318 views

Number of integer partitions modulo 3

Motivated by Parity of number of partitions of $n!/6$ and $n!/2$, I asked my computer (and my FriCAS package for guessing) for an algebraic differential equation for the number of integer partitions ...
Martin Rubey's user avatar
  • 5,533
2 votes
0 answers
54 views

example in $L^p_{s}-$Sobolev spaces

We define $L^p-$ Sobolev spaces as follows: $$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$ where $\langle \...
Math Learner 's user avatar
0 votes
1 answer
51 views

Minimizing the set of "faulty" edges in a map between the vertex sets of $2$ graphs

The starting point of this question is the fact that for some simple, undirected graphs $G, H$ there is no graph homomorphism $f:G\to H$. This is the case for instance if $\chi(G)>\chi(H)$. ...
Dominic van der Zypen's user avatar
6 votes
1 answer
428 views

A question about Poincare duality

Let $k$ be a field. Let $C$ be a small category and assume that for every $i\geq 0$ we have a functor $H^i:C\rightarrow FinDimVect_k$. Assume that there is a function $dim:Obj(C)\rightarrow \mathbb{Z}...
paul's user avatar
  • 365
0 votes
0 answers
92 views

How many lattices require exactly 3 elements to generate them?

This question by Moshe Newman: How many different lattices are there on n points, that require exactly 3 elements to generate them? This sequence seems to start 0,0,1,0,4,3 (for n = 1 to 6) and seems ...
David S. Newman's user avatar
6 votes
0 answers
206 views

Can one use forcing as a step to prove the Keisler-Shelah isomorphism theorem?

Can one use forcing (perhaps at the expense of using larger ultrafilters) to prove the Keisler-Shelah isomorphism theorem? My idea is to use an ultrafilter $U$ on a set $I$ such that if $M=V^{I}/U$, ...
Joseph Van Name's user avatar

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