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11 votes
3 answers
816 views

Conditions under which $\lim_{s\to1^+}\sum_{n=1}^{\infty}\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n}$

I was working with some Dirichlet series and I realized that I have never seen any general conditions under which \begin{equation} \sum_{n=1}^{\infty}\frac{a_n}{n}=\lim_{s\to1^+}\sum_{n=1}^{\infty}\...
0 votes
1 answer
86 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 ...
26 votes
1 answer
5k views

1-Wasserstein distance between two multivariate normal

The $p$-Wasserstein between two measures $\nu_1$ and $\nu_2$ on $X$ is given by $$d_p(\nu_{1},\nu_{2})=\left(\underset{\pi\in\Gamma(\nu_{1},\nu_{2})}{\inf}\int_{\mathbf{\mathcal{X}}^{2}}d(x,y)^p\pi(dx,...
0 votes
0 answers
64 views

Quasi polynomial algorithm for NP complete problem

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 ...
4 votes
0 answers
104 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 $\...
3 votes
0 answers
81 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 ...
0 votes
0 answers
37 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 ...
4 votes
1 answer
175 views

Equivariant projective embeddings with optimal dimension

Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find ...
2 votes
0 answers
61 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....
1 vote
0 answers
72 views

In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?

Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements: $\lambda$ being a random large prime such as $w^\lambda > 2\times m$ $1 < n < m−1$. m is ...
1 vote
4 answers
1k views

Covering of a graph via independent sets

I suspect that a topic such as this may have been considered before: if so, I hope that someone can point me to a reference on the subject. I have a graph G with an upper bound d on its maximum ...
1 vote
0 answers
127 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} \...
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) ...
7 votes
0 answers
112 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 ...
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 $\alpha, \beta \in \mathbb{R}_+$ ...
0 votes
0 answers
20 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 $...
5 votes
1 answer
211 views

In a topological group, is $G/A\to G/B$ a covering map if $A$ is open in $B$?

Let $G$ be a (Hausdorff) topological group, let $A,B$ be closed subgroups of $G$ such that $A$ is an open subgroup in $B$. Then we have an open continuous map $f:G/A\to G/B$, with typical fiber $B/A$. ...
7 votes
1 answer
361 views

Integral refinements of rigid cohomology

Disclaimer: I know absolutely nothing about p-adic cohomology, so it is possible that even the premises of this question are incorrect. But it turns out that I need to apply the theory of rigid ...
12 votes
2 answers
633 views

Polynomial inequalities of the form $\int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \,dx$

Let $P_n$ denote all (real or complex) polynomials $f(x)=\sum_{k=0}^n a_k x^k$. I'm interested in inequalities of the form $$ \int_0^1 |f(x)|^2 x \,dx \geq C \int_0^1 |f(x)|^2 \, dx, \quad \text{for ...
4 votes
0 answers
86 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 ...
9 votes
1 answer
2k views

Chain rule for distributional derivative

Let $V \subset H \subset V^*$ be a Gelfand triple (eg. $H^1 \subset L^2 \subset H^{-1}$). Let $u \in L^2(0,T;V)$ have a distributional derivative $u' \in L^2(0,T;V^*)$. So $\int_0^T u(t)\varphi'(t) = ...
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{...
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) = ...
7 votes
0 answers
154 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})\...
5 votes
1 answer
137 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 ...
2 votes
1 answer
220 views

Why do we get a connected 2-regular graph?

In reading "PUBLIC-KEY CRYPTOSYSTEM BASED ON ISOGENIES" by Rostovtsev and Stolbunov, they claim on page 8 that the set $U=\{E_i(\mathbb{F}_p)\}$ of elliptic curves with a specific prime $l$ ...
25 votes
4 answers
6k views

Singular Homology/Cohomology as a derived functor?

Hello, Learning some Alg.geometry and Sheaf theory, I got used to the notion that cohomology arises naturally as a derived functor of some sort. This has led me thinking, singular cohomology, from ...
1 vote
1 answer
257 views
+50

Show convergence of sets

Consider these sets $$ A\equiv \bigcap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\} $$ $$ C_n(L_n)\equiv \{x \in X: d(p_n, [\ell(x), u(x)])\leq L_n\} $$ ...
11 votes
1 answer
844 views

Cohomological bounds for scalar curvature of an extremal Kähler metric

There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...
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 ...
4 votes
1 answer
381 views

Cluster algebras of type A and X

I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces. Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
6 votes
3 answers
454 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(\...
3 votes
0 answers
143 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 ...
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, ...
1 vote
1 answer
114 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 ...
1 vote
0 answers
109 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/...
2 votes
3 answers
521 views

Solving the unknotting problem by pulling both ends of the string

It is an open question as to whether there is a polynomial time algorithm for recognizing the unknot. Consider the following procedure for doing so on an actual physical string: Suppose there is a ...
9 votes
1 answer
2k views

What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...
0 votes
0 answers
115 views
+50

What is the strength of adding this de-schematizing inference rule to Ackermann's set theory?

Language: first order logic with equality, membership, and a constant symbol $W$. Axioms: Extensionality: $\forall z \, (z \in x \leftrightarrow z \in y) \to x=y$ Comprehension: $\exists x \forall y \,...
9 votes
1 answer
436 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 $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}\in\Gamma$. ...
128 votes
10 answers
19k views

Are there any very hard unknots?

Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
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 ...
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 ...
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 ...
2 votes
1 answer
528 views

Another functional inequality

Is there some general solution to the functional inequality: $$ f(xy) \leq y f(x) + x f(y)$$ Where $x,y\in[0,1]$? I can find many particular solutions but I just wonder if there is a more general ...
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 $d_0$ ...
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\...
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 $\...
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 ...
4 votes
1 answer
5k views

Should a job application research statement include a "research plan"?

I am wondering whether I should include a "research plan" as part of my research statement for the academic job market. My concern is that my research plan may not exactly match the projects being ...

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