All Questions
152,891
questions
3
votes
0
answers
167
views
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&...
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 $a_{m,n}$ ...
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 ...
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{\...
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. ...
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 ...
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 ...
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\...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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.
...
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}(...
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$...
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)$$
Where $b$ is a ...
3
votes
1
answer
351
views
Solution singular PDE
I've been studying the following singular PDE
$$
\mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$
in $\Omega \subset \mathbb{R}^{2}$.
Do you know any reference, ...
5
votes
0
answers
214
views
Generalizing the formula between Wu class and the Steenrod square
I know that on the tangent bundle of $M^d$, the corresponding Wu class and the Steenrod square satisfy
$$
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}$$
...
1
vote
0
answers
47
views
Density of different types of critical points in an algebra of elementary embeddings
Suppose that $j,k:V_{\lambda}\rightarrow V_{\lambda}$ are elementary embeddings. Let $\mathrm{crit}_{n}(j,k)$ denote the $n$-th element in $\{\mathrm{crit}(\ell)\mid\ell\in\langle j,k\rangle\}$. ...
2
votes
0
answers
153
views
Condition on a Lie groupoid to be represented by manifold/group or an action groupoid
Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions.
When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some ...
1
vote
0
answers
93
views
Has this result about the number of permutations of a given cycle type (or centralizers) been proved?
I was playing with the cardinality of conjugacy classes of the symmetric groups, which we know is the number of permutations of a given cycle type and there is a natural one-to-one correspondence ...
5
votes
1
answer
377
views
Oscillation and Hölder continuity
Where can I find a proof of the following fact?
If
$$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 ...
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 ...
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 $N:S^n\to \mathbb{R}^n$, $...
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 ...
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 $\textbf{y} \sim \mathcal{CN}(\textbf{0}_{M},\sigma_{y}^{2}\textbf{I}_{M})$ ...
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 $\|...
-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 @...
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 ...
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 ...
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
$$
\sup_{x \neq y} \frac{|f(x) - f(y)|}{|x - y|^\alpha}.
$$
...
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-...
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 ...
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 ...
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 $s\neq t$. ...
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 \...
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\...
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 ...
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.
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 ...
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 ...
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/...
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 ...
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 \...
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 $\chi(G)>\chi(H)$.
...
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}...
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 ...
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 $M=V^{I}/U$, ...