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Serre duality for non-compact Riemann surfaces

Suppose $X$ is a Riemann surface. If $X$ is compact, then Serre duality tells us that we have an isomorphism in sheaf cohomology $$ H^1(X,E) \cong H^0(X,\Omega\otimes E^\ast)^\ast $$ Can we say ...
Aidan's user avatar
  • 488
1 vote
0 answers
40 views

Continuity of the constant in maximal Sobolev regularity

Let $\Omega \subset \mathbb R^n$ be a smooth, bounded domain. For each pair $(p, q) \in (1, \infty)^2$, maximal regularity asserts that there is some $\widetilde K(p, q) > 0$ such for all $f \in L^...
Keba's user avatar
  • 303
4 votes
0 answers
93 views

Does the permutohedron satisfy any minimal distortion property for graph metric vs Euclidean distance?

We can look on the permutohedron as a kind of "embedding" of the Cayley graph of $S_n$ to the Euclidean space. (That Cayley graph is constructed by the standard generators, i.e. ...
Alexander Chervov's user avatar
0 votes
1 answer
54 views

Hahn-Mazurkiewicz with finite one-dimensional Hausdorff measure

Suppose that there is a continuous surjection from $[0,1]$ to a metric space $(X,d)$. If $(X,d)$ has finite one-dimensional Hausdorff measure, must there exist a Lipschitz surjection from $[0,1]$ to $(...
Kevin's user avatar
  • 133
4 votes
1 answer
116 views

Examples of Noetherian integral group ring

I need to study the integral group ring of the fundamental group of a manifold. My knowledge of group and ring theory is very limited. I am looking for some examples of groups $G$ for which $\Bbb ZG$ ...
Random's user avatar
  • 927
7 votes
0 answers
86 views

Is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?

In short, the question is in the title: is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis? A bit of context: Given a topological space $X$, a family $\...
Cla's user avatar
  • 755
1 vote
1 answer
108 views

Example of triangulated category with vanishing $K_0$

Let $R$ be a ring, let $\operatorname{Perf}(R)$ the category of perfect modules over $R$. Suppose we have $E$ an perfect $R$-module (concentrated in degree $0$) such that its class $[E]\in K_0(R)$ is ...
cellular's user avatar
  • 1,005
3 votes
0 answers
83 views

Preservation of Kan extensions

I am currently studying the theory of kan extensions more seriously, but I'm surprised of the apparent absence if theorems of preservation/reflection. What I have in mind is something along the lines ...
whatisandwhatshouldneverbe's user avatar
0 votes
0 answers
40 views

Prove the orthogonality of vector spherical harmonics

We define $S_a^{lm} = \Big( - \frac{1}{\sin \theta} Y^{lm}_{,\varphi}, \sin \theta\ Y^{lm}_{,\theta} \Big)$ $Y_a^{lm} = \Big( Y^{lm}_{,\theta}, Y^{lm}_{,\varphi} \Big)$ to be the axial vector ...
AleNekro97's user avatar
0 votes
0 answers
68 views

Quasi polynomial algorithm for NP complete problem [closed]

I know that quasi polynomial algorithm is neither polynomial nor exponential. But I want to know if we find such algorithm for NP complete problem, will it be of any use? Or is there such algorithm ...
user's user avatar
  • 19
0 votes
1 answer
87 views

How to understand the unique continuation result

Let $E$ be the closure of $C_c^{\infty}\left(\mathbb{R}^N\right)$ ($N \geqq 3)$ under the norm $$ \|u\|_E=\left(\int_{\mathbb{R}^N}|\nabla u|^2\right)^{1 / 2}. $$ Suppose $K(x) \in C^1\left(\mathbf{R}^...
Davidi Cone's user avatar
5 votes
1 answer
114 views

Finding the point within a convex n-gon that maximizes the least angle subtended there by an edge of the n-gon

For any point P in the interior of a convex polygon, the sum of the angles subtended by the edges of the polygon is obviously 2π. Given a convex polygon, how does one algorithmically find the point (...
Nandakumar R's user avatar
  • 5,473
2 votes
0 answers
63 views

What Cayley graphs arise as nodes+edges from "nice" polytopes and when are these polytopes convex?

The Permutohedron is a remarkable convex polytope in $R^n$, such that its nodes are indexed by permutations and edges correspond to the Cayley graph of $S_n$ with respect to the standard generators, i....
Alexander Chervov's user avatar
8 votes
0 answers
119 views

On Lemma 5.5.16 of Cisinski's "Higher Categories and Homotopical Algebra"

I have a question regarding Section 5 of Cisinski's "Higher Categories and Homotopical Algebra". Let us write $\mathbf{sSet}$ and $\mathbf{bisSet}$ for the categories of simplicial sets and ...
Keisuke Hoshino's user avatar
3 votes
1 answer
101 views

Concentration of measure on spheres with respect to a unitary of trace approximately zero

Cross-posted from MSE, where it hasn’t received any answer yet: This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
David Gao's user avatar
  • 1,262
4 votes
0 answers
106 views

Do all nonnegative integers appear in A051521?

For every positive integer $n$, $\tau(n)$ is the number of divisors of $n$. If we list the ratio of each positive integer $n$ to $\tau(n)$,they form a rational sequence 1,1,3/2,4/3,5/2,3/2,… Because $\...
Tong Lingling's user avatar
0 votes
0 answers
22 views

Is the average of two viscosity sub-solutions to linear elliptic equations is also a sub-solution?

Let $b\in C_b(R;R)$. Consider the following LINEAR equation on $R^2$: \begin{equation} u-\partial_{xx}^2 u + (b(x+y)-b(x)) \partial_y u=f\in C^\infty_c(R^2). \tag{1} \end{equation} Assume that $...
Guohuan Zhao's user avatar
5 votes
0 answers
93 views

Points of the sheaf topos over Blass' category

There is a site $\textsf{Blass}$ used for (constructive) non-standard analysis, whose objects are sets equipped with a filter, and morphisms are continuous functions defined up to a small set. (It is ...
Trebor's user avatar
  • 1,021
9 votes
1 answer
260 views

Maybe a folklore natural map between reflexive pullbacks

In the introduction of [HK04], it is proposed that for a morphism between varieties $f:X'\to X$, and a coherent sheaf $\mathcal{F}$ on $X$, there is a natural map $\alpha:f^*(\mathcal{F}^{\vee\vee})\...
nariri's user avatar
  • 318
5 votes
1 answer
143 views

Proof that the inclusion $\Delta \to \mathbf{Pos}$ preserves colimits

I am trying to to prove that the inclusion $\Delta \to \mathbf{Pos}$ preserves colimits, where $\mathbf{Pos}$ is the category of partially ordered sets and monotone map, and $\Delta$ is the full ...
Calin Tataru's user avatar
3 votes
1 answer
105 views

Converting an algebraic equation into a ODE

I'm working on a method to solve algebraic equations by converting them into ordinary differential equations (ODEs) and then integrating these ODEs over time. Given an algebraic equation $f(x(t), t) = ...
Joe's user avatar
  • 31
3 votes
0 answers
150 views

A relative Abel-Jacobi map on cycle classes

I have a question about relativizing a classical cohomological construction that I think should be easy for someone well versed in such manipulations. Background: Suppose $X$ is a smooth projective ...
Asvin's user avatar
  • 7,648
1 vote
0 answers
113 views

Is a symmetric monoidal category ("tensor-category" in P. Deligne & J.S. Milne's vocabulary) neccessarily locally small?

Let $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (I will denote this by just $(\mathcal{C},\otimes)$) be a tensor-category (in P. Deligne & J.S. Milne's vocabulary, see https://www.jmilne.org/math/...
Ben123's user avatar
  • 163
0 votes
1 answer
88 views

Analyzing a Dirichlet series with log-oscillating terms via Fourier methods

I am investigating the series $S(z)$ defined as follows: $$ S(z) = \sum_{n=1}^{\infty} n^{-a}\cos(b\ln(n)), $$ where $z = a + bi \in \mathbb{C}$, with $0 < a < 1$, and $b \in \mathbb{R}$. I want ...
swami's user avatar
  • 369
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
438 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
25 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
27 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
93 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
117 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
44 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
26 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
49 views

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

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
47 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
44 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
115 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
50 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
68 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
76 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
26 votes
1 answer
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
  • 387
0 votes
0 answers
143 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
152 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
101 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
169 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
209 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
473 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
198 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

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