Newest Questions
159,074 questions
3
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0
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87
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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 ...
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 ...
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 ...
2
votes
0
answers
46
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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 ...
8
votes
0
answers
443
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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 $$ \mathbf{...
1
vote
1
answer
132
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A non-example of a graded Frobenius algebra
Take the class of finite dimensional graded algebras $A = \sum_i A_i$ satisfying $|A_n| = 1$ where $A_n \neq 0$ and $A_m = 0$, for all $m > n$. What is an example in this class that is not ...
2
votes
0
answers
180
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Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
2
votes
1
answer
247
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On Dirac/ Clifford matrices
Let $(\eta^{\mu\nu})=\operatorname{diag}(+1,-1,-1,-1)$.
The Dirac matrices $\gamma^\mu$, $\mu=0,1,2,3$ satisfy by definition
$$\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}\tag{1}\label{1}$$
where $\{A,B\}=...
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 ...
2
votes
0
answers
54
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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) ...
-1
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1
answer
194
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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 ...
1
vote
3
answers
168
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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 ...
2
votes
0
answers
144
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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) ...
1
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0
answers
131
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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 ...
2
votes
0
answers
199
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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$...
-1
votes
1
answer
148
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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 $(...
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 ...
34
votes
6
answers
4k
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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 ...
0
votes
0
answers
161
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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 ...
1
vote
0
answers
116
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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 ...
1
vote
0
answers
125
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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)$. ...
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 ...
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 $...
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 ...
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 ...
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 ...
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 ...
1
vote
0
answers
114
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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.
...
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 ...
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 ...
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 ...
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}}\...
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{ ...
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 ...
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 ...
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 ...
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 $...
2
votes
1
answer
225
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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 ...
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\...
2
votes
0
answers
158
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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). ...
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
$...
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, ...
14
votes
0
answers
241
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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 ...
1
vote
0
answers
120
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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(...
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 $\dim H_\ast(X,\mathbb F_p)$....
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 $\mathscr{B}$...
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?
$|H| = |G|/p$.
$c(H)\geq c(G) - 1$.
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 ...
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 ...
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 (...