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Bounding the degree of the Weierstrass polynomial of a product of a holomorphic function and a polynomial

In brief. For a fixed holomorphic function $v$, I want to bound the degree $q$ of the Weierstrass polynomial $Q$ in the Weierstrass decomposition $v^TP = uQ$, in terms of the degree $p$ of the ...
Sébastien Loisel's user avatar
1 vote
0 answers
51 views

Hardness of an optimization problem when some variables are fixed

Given a general optimization problem, I would like to know what we can say about the hardness of the problem when a subset of its variables are fixed. With the two (related) examples, it is clear that ...
Ro. Cohof's user avatar
0 votes
0 answers
162 views

Gluing faces of n-cube

Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$. Let $f_0$ and $f_1$ be faces ...
mahu's user avatar
  • 53
2 votes
0 answers
46 views

Size of set of positive integers no sum of two distinct elements giving square

Question: find the size of a maximal subset $A$ of $[n]=\{1,\cdots,n\}$ satisfying that for any distinct elements $x,y\in A$, $x+y$ is not a perfect square. Consider a graph with $n$ vertices: $x$ and ...
Haoran Chen's user avatar
8 votes
0 answers
443 views

Sheaf of compact Hausdorff spaces but not a condensed anima

Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{...
Qi Zhu's user avatar
  • 435
1 vote
1 answer
132 views

A non-example of a graded Frobenius algebra

Take the class of finite dimensional graded algebras $A = \sum_i A_i$ satisfying $|A_n| = 1$ where $A_n \neq 0$ and $A_m = 0$, for all $m > n$. What is an example in this class that is not ...
Lorenzo Del Vecchiopontopolos's user avatar
2 votes
0 answers
180 views

Approximating $L^p$ functions by eigenfunctions of Laplacian

I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932. In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
ze min jiang's user avatar
2 votes
1 answer
247 views

On Dirac/ Clifford matrices

Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$. The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition $$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$ where $\{A,B\}=...
asv's user avatar
  • 21.8k
5 votes
0 answers
373 views

A (possible) Lie algebra extension of the Lie algebra of a foliation

Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is ...
Ali Taghavi's user avatar
2 votes
0 answers
54 views

Finite (schema) axiomatizability of representable cylindric algebras

If we know that the class of all representable cylindric algebras of dimension $\alpha$ (for any ordinal number $\alpha>2$) is NOT finitely (schema) axiomatizable*, then does it (perhaps trivially) ...
Âloh's user avatar
  • 63
-1 votes
1 answer
194 views

Does a symmetric monoidal functor between cartesian monoidal categories automatically preserve products? [closed]

Suppose $\cal A, B$ are cartesian monoidal categories, that is, categories equipped with a choice of finite products (including the nullary one, $1$). Suppose, moreover, that $F: \cal A \to B$ is a ...
seldon's user avatar
  • 1,083
1 vote
3 answers
168 views

Can a disk that is limit of disks in a pseudoconvex domain be partially contained in the boundary?

A holomorphic disk is the image of an injective holomorphic map $f:\mathbb D \to \mathbb C^n$ from the unit disk $\mathbb D \subset \mathbb C$ to $\mathbb C^n$. Let $\Omega$ be a pseudoconvex domain ...
hife's user avatar
  • 133
2 votes
0 answers
144 views

Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$

Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$ I would like to prove (or disprove) ...
Guy Fsone's user avatar
  • 1,101
1 vote
0 answers
131 views

Large-deviation inequalities for a class of simple random multivariate polynomials

Let $N$ be a large positive integer and let $[N] := \{1,2,\ldots,N\}$. For any $k$, let $K_{N,k}$ denote the collection of $k$-element subsets of $[N]$. Let $x=(x_1,\ldots,x_N)$ be a uniformly random ...
dohmatob's user avatar
  • 6,853
2 votes
0 answers
199 views

Not a twin prime pair test using $\gcd$ only

Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$. Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
Notamathematician's user avatar
-1 votes
1 answer
148 views

Strong law of large numbers for a sequence of random variables in different probability spaces

Is it known whether the following version of the strong law of large numbers holds? For each $k\in\mathbb{N}$, let $\Omega_k$ be a finite set and $\mu_k$ be a probability measure on $\Omega_k$. Let $(...
Aleksi's user avatar
  • 1
2 votes
0 answers
60 views

Klyachko-type inequalities for shifted Schur structure constants

Klyachko (+Knutson-Tao) provide a set of inequalities that are necessary and sufficient for the Littlewood-Richardson coefficient $c^{\lambda}_{\mu\nu}$ to be non-zero. Is there a similar result for ...
Per Alexandersson's user avatar
34 votes
6 answers
4k views

Why study finite topological spaces?

In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage: … this means that some concepts that I use freely and naturally in my personal thinking are foreign to ...
Wahome's user avatar
  • 737
0 votes
0 answers
161 views

Markov process with time varying transition kernels

I cross post this question from StackExchange as it may be more appropriate. I am interested in studying the evolution of a variable $\alpha_t\in [0,1]$ governed by the following stochastic dynamical ...
Francesco Bilotta's user avatar
1 vote
0 answers
116 views

Is this class of $p$-groups large?

Call a $p$-group $G$ good if for each subgroups $H, H_1, H_2\subseteq G$ for which $H_1\subseteq H$, $H_2\subseteq H$, $|H_1| = |H_2| = |H|/p$, $H_1\not= H_2$, $H'\not=\{e\}$ holds we have that there ...
solver6's user avatar
  • 291
1 vote
0 answers
125 views

Is the Frobenius semisimple on the de-Rham cohomology?

Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. ...
Richard's user avatar
  • 785
1 vote
0 answers
76 views

How to construct a $C^1$ variation of the unique, minimal geodesic at its first conjugate point

Motivation for the question: This is a question that I encountered when reading Proposition 1.8 of the paper "Optimal transport and curvature" by Alessio Figalli and Cedric Villani, whose ...
Chee's user avatar
  • 984
1 vote
1 answer
100 views

Orbit closure of two elliptic Möbius transformations

Let $g_1$ and $g_2$ be two elliptic Möbius transformations of infinite order in $\mathrm{Aut}(\bar{\mathbb{C}})$. If $\mathrm{Fix} (g_1) \cap \mathrm{Fix} (g_2) = \emptyset$, then can we deduce that $...
Chicken feed's user avatar
2 votes
1 answer
225 views

Boundary points in $\overline{\operatorname{conv}\{z_i\}_{i\in I}}$

Let $X$ be an infinitely-dimensional Banach space and $\{z_i\}_{i\in I}$ be a set of linearly independent points in $X_{\leq 1}$, the closed unit ball of $X$. $I$ the index set is not necessarily ...
Sanae Kochiya's user avatar
4 votes
1 answer
179 views

A "simple" space with closed retracts but non-unique sequential limits

This question asking how KC ("Kompacts are Closed") and RC ("Retracts are Closed") are distinct has some good discussion, including a now-published example by Banakh and Stelmakh ...
Steven Clontz's user avatar
6 votes
1 answer
310 views

Surjectivity of a class of integrals in dimensions two

Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
MathLearner's user avatar
1 vote
1 answer
269 views

Question on ideal triangulation and geodesic lamination

Q1. Does a closed hyperbolic surface admit an ideal triangulation? Here, an ideal triangulation of a surface means a partition of a surface by geodesics such that each component of the complement ...
one potato two potato's user avatar
1 vote
0 answers
114 views

What is $H^*(\mathbb{CP}^{2^N-1}/\Sigma_n;\mathbb{Z})$ when $N=\binom{n}{2}$?

$H^*((S^3)^N/\Sigma_n;\mathbb{Q})$ is computed here. It makes a little more sense to compute $H^*((S^2)^N/\Sigma_n;\mathbb{Q})$ given that global phase is irrelevant. The proof is exactly the same. ...
Jackson Walters's user avatar
1 vote
0 answers
108 views

Manyfold iterated exponential sum with growing conductor

Let $\varphi(x_1,\dots, x_k)$ be some smooth function with partial derivatives of magnitude $\asymp 1$ for $x_i\asymp 1$. For concreteness, as it doesn't appear to add much extra structure beyond the ...
Mayank Pandey's user avatar
3 votes
0 answers
122 views

How to find the right path of integration to get the asymptotic partition formula

I am trying to understand how the asymptotic partition formula $p(n) \sim \frac{e^{\pi\sqrt{\frac{2n}{3}}}}{4n\sqrt3} $ was derived for a project and have been reading and following many papers. I am ...
Aadi Deepchand's user avatar
3 votes
1 answer
256 views

Sum of squares of chromatic roots of a bipartite graph

Given a graph $G = (V, E)$, we can calculate its chromatic polynomial $P(G, k)$, and it has $n$ (complex) roots, also known as chromatic roots. It is a well-known fact that the sum of chromatic roots ...
Jing Guo's user avatar
1 vote
1 answer
143 views

$L^1$ error between indicator function and smoothed out version

For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is, $$f_r(x) = \frac{1}{\sqrt{\pi}}\...
Staki42's user avatar
  • 101
4 votes
1 answer
210 views

Is Stokes equation a saddle point problem or a minimum problem?

Consider $\Omega \subset \mathbb{R}^2 $ (or $\mathbb{R}^3$). The well known stationary Stokes equations in the incompressible case are \begin{equation} \begin{cases} - \Delta u + \nabla p = f \text{ ...
tommy1996q's user avatar
9 votes
0 answers
295 views

How acyclic models led to idea of model categories

The Wikipedia article about Acyclic models notices that the way that they were used in the proof of the Eilenberg–Zilber theorem laid the foundation stone to the idea of the model category. Could ...
user267839's user avatar
  • 5,966
1 vote
0 answers
105 views

Poisson summation for solutions of the Burgers equation in the form 1/x

Long story short: I'm looking for a good way of showing that the Fourier transform of $1/x$ is a sign function. Motivation and why this has been a problem: I'm dealing with an equation similar to the ...
Rafael's user avatar
  • 93
6 votes
2 answers
803 views

Conjecture about partitions of the powerset without the empty set

I would like to have some ideas about possibilities of proving or disproving the following conjecture: For any partition $\mathcal{F}=\{\mathcal{A_1},\ldots,\mathcal{A_m} \}$ of the powerset without ...
Fabius Wiesner's user avatar
2 votes
1 answer
381 views

Impredicativity, definition, recursion and conservatism

Suppose we in an impredicative framework isolate the fixed point $$Gx\leftrightarrow A(G,x)$$ from a $Gx$ obtained by $\Pi^1_1$-comprehension as equivalent to $\forall K((A(K,x)\to Kx)\to Kx)$, where $...
Frode Alfson Bjørdal's user avatar
2 votes
1 answer
225 views

Isomorphic symmetric squares of non-isomorphic curves

Is it possible for the symmetric squares of a pair of non-isomorphic curves $C_1, C_2$ defined over a field $K$ to be isomorphic? EDIT: as the user @abx has mentioned, there exist such examples in the ...
kindasorta's user avatar
  • 2,907
2 votes
0 answers
50 views

Counting permutations of $X^2$ that induce 4 quasigroup operations up to isotopy

Let $X$ be a finite set. Recall that a binary operation $\ast$ on $X$ is said to be a quasigroup operation if there are binary operations $/,\backslash$ where $(x/y)\ast y=(x\ast y)/y=x$ and $x\ast(x\...
Joseph Van Name's user avatar
2 votes
0 answers
158 views

Topos of sheaves on a scheme considered as a functor

The spectrum of a ring $R$ can be defined as $\operatorname{Spec} R := \operatorname{Hom}(R, -)\colon \mathrm{fpRing} \to \mathrm{Set}$ ($\mathrm{fpRing}$ are commutative finitely presentable rings). ...
Arshak Aivazian's user avatar
2 votes
1 answer
369 views

Tests for determining membership of exponential family

Provided an arbitrary random variable $X$ with probability density/mass function $f$, are there any tests to determine if $f$ forms an exponential family? Certainly, if $f$ can be written in the form $...
Aaron Hendrickson's user avatar
1 vote
0 answers
84 views

Number of polyhedral covers of a triangulation of $S^2$

For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)? Under polygonal cover, ...
Kregnach's user avatar
  • 183
14 votes
0 answers
241 views

Unitary group of a von Neumann algebra: is it a retract of $U(H)$?

Let $M\subset B(H)$ be a properly infinite von Neumann algebra (the case I care about is $M=$ hyperfinite $\mathrm{III}_1$). Consider the unitary groups $U(M)$ and $U(H)$ in their strong operator ...
André Henriques's user avatar
1 vote
0 answers
120 views

Large Tate-Shafarevich group of an elliptic curve with the form $E_{p,n}:y^2=x^3+p^nx$

Let $p$ be a prime number and $n$ be positive integer. Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve. LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$. This is the biggest size of $Sha(...
Duality's user avatar
  • 1,541
3 votes
0 answers
70 views

Is every finite spectrum $X$ $K(h)$-locally equivalent to a finite spectrum $Y$ with $\dim (K(h)_\ast Y) = \dim ((H\mathbb F_p)_\ast Y)$?

Let $X$ be a finite spectrum and $K = K(h)$ be the $h$th Morava $K$-theory at the prime $p$. Then $\dim_{K_\ast} K_\ast X$ is increasing in $h$, and eventually constant at $\dim H_\ast(X,\mathbb F_p)$....
Tim Campion's user avatar
7 votes
1 answer
178 views

Homogeneous metric connections on 3-dimensional Lie groups

Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle. Considering the moduli space of connections $\mathscr{B}$...
Matteo Bruno's user avatar
6 votes
1 answer
213 views

Is there a subgroup of a non-abelian $p$-group $G$ with a large nilpotency class?

Let $G$ be a non-abelian $p$-group ($p\ne2$). Does there exist a group $H\subset G$ such that both 1, 2 are satisfied? $|H| = |G|/p$. $c(H)\geq c(G) - 1$.
solver6's user avatar
  • 291
7 votes
1 answer
275 views

Does the inner automorphism group of the fundamental group of a closed aspherical manifold always have an element of infinite order?

Let $\pi_1$ be the fundamental group of a closed aspherical manifold of dimension $n$. In particular, $\pi_1$ is finitely presented, torsion-free and its cohomology is finitely generated and satisfies ...
Jordi Daura's user avatar
4 votes
0 answers
246 views

Dynamical obstruction for a vector field to have a Harmonic divergence

Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
Ali Taghavi's user avatar
12 votes
0 answers
168 views

Can the optimal packing density in $\mathbb{Z}^d$ be irrational?

For a finite $S \subset \mathbb{Z}^d$, let $d_p(S)$ be its optimal packing density. That is, the maximal lower asymptotic density of $A+S$, where $A \subset \mathbb{Z}^d$ is such that $(a_1+S)\cap (...
Arsenii Sagdeev's user avatar

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