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$L^p$ domination of mixed partial derivatives by the unmixed ones?

Is it true that for each real $p\ge1$ there is some real $C_p$ such that for all smooth real-valued functions $u$ compactly supported on $S:=(0,1)^2$ one has $$\|D_1D_2u\|_p\le C_p(\|D_1^2u\|_p+\|D_2^...
Iosif Pinelis's user avatar
2 votes
0 answers
76 views

Are the roots of an infinitely divisible probability infinitely divisible themselves?

Let $\mu$ be an infinitely divisible probability on a topological group $G$. If $\nu ^{* n} = \mu$ for some $n$, is $\nu$ an infinitely divisible probability too? A sufficient criterion would be to ...
Alex M.'s user avatar
  • 5,207
2 votes
2 answers
704 views

Bochner's theorem for measures of positive type

Is there a version of Bochner's theorem characterizing measures of positive type on a locally compact group? By a measure of positive type on the group $\Gamma$, I mean a measure $\mu$ satisfying $\...
Evan DeCorte's user avatar
2 votes
0 answers
335 views

Enumerating certain types of permutation polynomials

Given a prime power $q$, I would like to enumerate (preferably up to isomorphism*) all the permutation polynomials $f(x)$ on $K = GF(q^3)$ satisfying the following conditions: $f(ax) = af(x)$ for all ...
Anurag's user avatar
  • 1,157
2 votes
2 answers
284 views

Morphisms between fundamental groups of Lie groups

Let $X$ be a compact connected manifold. Since $\mathbb T^1$ is an Eilenberg-MacLane space $K(\mathbb Z,1)$, it follows that for every morphism $\varphi\colon\pi_1(X)\to\pi_1(\mathbb T^1)$ there is a ...
William of Baskerville's user avatar
2 votes
1 answer
1k views

Symplectic block-diagonalization of a complex symmetric matrix

This is a follow-up question to the one asked here: Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...
Fabian's user avatar
  • 280
2 votes
1 answer
140 views

Is there a lower bound on the size of a supertransitive model of ZFC?

In a posting to mathstackexchange I've alluded to the concept of supertransitive model. Now $M$ is a supertransitive model of a set $Q$ of first order sentences, denoted by $M \models^{sptr} Q$, is ...
Zuhair Al-Johar's user avatar
2 votes
1 answer
274 views

On semi-discrete Wasserstein distance

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$, where $\nu$ has a bounded support. Consider the $2-$Wasserstein distance below: $$...
user111097's user avatar
2 votes
1 answer
129 views

noncompact Riemannian homogeneous is trivial vector bundle over compact homogeneous

Is it true that a manifold $ E $ admits a metric with respect to which the isometry group is transitive ($ E $ is Riemannian homogeneous) if and only if $ E $ is the total space of a $ K $ equivariant ...
Ian Gershon Teixeira's user avatar
2 votes
3 answers
16k views

Cycle of length 4 in an undirected graph

Can anyone give me a hint for an algorithm to find a simple cycle of length 4 (4 edges and 4 vertices that is) in an undirected graph, given as an adjacency list? It needs to use $O(v^3)$ operations (...
user15816's user avatar
2 votes
1 answer
603 views

Distance matrices

We say that a matrix $M\in\mathbb{R}^{n\times n}$ is a distance matrix on a metric space $(X,d)$, if there exist $x_1,\cdots,x_n \in X$ such that $M=[d(x_i,x_j)]_{n\times n}$. Question. For which ...
Mahdi's user avatar
  • 1,991
2 votes
1 answer
244 views

A question about pushforward measures and continuous Borel isomorphisms

It is fairly well known that if $\mu$ and $\nu$ are nonatomic measures on the standard Borel spaces $(X,B)$ and $(Y,C)$ such that $\mu(X)=\nu(Y)$. If $X$ and $Y$ are uncountable, then there exists a ...
O-Schmo's user avatar
  • 33
2 votes
2 answers
303 views

A graphic representation of classical unitals on 28 points

I would like to understand the geometry of the classical unitals. They are block designs containing $q^3+1$ points and whose blocks have cardinality $q+1$, where $q$ is a prime power. For $q=2$ (if I ...
Taras Banakh's user avatar
  • 40.8k
2 votes
1 answer
969 views

Is a polynomial decay sufficient for a smooth function to be in $\mathcal{F}(L^1)$?

Background: I have a function $g(\omega)\in C^{\infty}(\mathbb{R})$, which vanishes like $O(|\omega|^{-\beta})$ at infinity for some $\beta>0$. This answer states that functions that decays "too ...
Chen Wang's user avatar
  • 243
2 votes
1 answer
268 views

Minimum length path touching $n$ circles

Given $n$ non-overlapping circles of radius $1$ (i.e., the distance between the centers of any two circles is greater than $2$), how to find the minimum length path (the path can be of any form) that ...
lchen's user avatar
  • 459
2 votes
1 answer
182 views

Lattices without nontrivial dense elements

This question arose from another one of mine, Homotopy type of some lattices with top and bottom removed. An element $d$ of a bounded lattice $L$ is called $\mathit{dense}$ if $$ \forall x\in L\ (d\...
მამუკა ჯიბლაძე's user avatar
2 votes
1 answer
251 views

Approximating rational values in $]0,1[$ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions. Are there positive integers $m<n \in \mathbb{N}$, such that for ...
Dominic van der Zypen's user avatar
2 votes
0 answers
481 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory) Clearly my ...
Fabrice Pautot's user avatar
2 votes
1 answer
430 views

dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative). The resolutions ...
lks8271's user avatar
  • 165
2 votes
1 answer
415 views

Is full Replacement provable in Z + Ordinal Replacement?

$\text{Ordinal Replacement:}$ if $\phi(x,y)$ is a formula in two free variables $x,y$, then: $\forall x \ [ordinal(x) \to \exists! y \ (ordinal (y) \wedge \phi(x,y)) ] \to \forall A \ (\forall x \...
Zuhair Al-Johar's user avatar
2 votes
1 answer
196 views

Finite $k$-set-respecting splitting of $\mathbb{N}$

Motivation. My sons participated in a large football tournament recently; everyone wanted to be in a team with everyone else at least once. Tricky! Formulation of the question. For any positive ...
Dominic van der Zypen's user avatar
2 votes
0 answers
275 views

Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues

In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
dtn's user avatar
  • 145
2 votes
1 answer
109 views

Strongly rigid connected $k$-regular graphs

It is easy to see that vertex-transitive graphs must be regular. This question looks for regular graphs that are "the opposite" of vertex-transitive. Question. Is there an integer $N\in\mathbb{N}$ ...
Dominic van der Zypen's user avatar
2 votes
1 answer
186 views

Relation between Riemannian and Cayley-graph distance in a finite Coxeter group

Background: Let $W$ be a finite reflection group of rank $n$, acting on $\mathbb{R}^n$. The reflecting hyperplanes of $W$ meet the unit sphere $S^{n-1}\subset\mathbb{R}^n$, inducing a simplicial ...
benblumsmith's user avatar
  • 2,831
2 votes
1 answer
165 views

Global centralizers in Jordan-Chevalley decomposition in bad characteristic

Let $G$ be an affine algebraic group defined over an algebraically closed field $k$ of arbitrary characteristic, and write $\mathfrak{g}$ for its Lie algebra. Given $X\in\mathfrak{g}$, it has (...
Jason's user avatar
  • 53
2 votes
1 answer
103 views

To find the convex planar region minimizing diameter when area and perimeter are given

The basic question is to find that planar convex region for which diameter is a minimum when area and perimeter are specified. A partial answer is given here: http://nandacumar.blogspot.com/2012/11/...
Nandakumar R's user avatar
  • 5,463
2 votes
2 answers
2k views

Does the Euler product formula diverge for any zero of the Riemann zeta function?

Simple question (but not for me): Does the Euler product formula diverge for any zero of the Riemann zeta function? The reason why I ask this is that I heard we should not use the Euler product ...
Seongsoo Choi's user avatar
2 votes
0 answers
104 views

Multiplicativity of the analytic index (or of kernel bundle)

What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators. In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
Ho Man-Ho's user avatar
  • 1,087
2 votes
0 answers
257 views

Codimension restrictions on intersections

This is a question I stumbled across earlier this week. I see a similar one has been asked here. Suppose I have a smooth quasi projective variety $X$ over a field $K$, and I call $\text{Chow}^r(X\...
user avatar
2 votes
0 answers
227 views

when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective?

Let $G$ be a finite group with a subgroup $K$. Given $[\beta]\in H^2(K,\mathbb{C}^*)$ is the an obstruction which checks whether or not $[\beta]$ is the restriction of some $[\alpha]\in H^2(G, \mathbb{...
Ofir's user avatar
  • 243
2 votes
2 answers
775 views

Strong differentiability and the inverse function theorem in Banach spaces

I am trying to prove the strong differentiability version of the Inverse Function Theorem for Banach spaces, but I am not sure if it is true. I am interested in this because it is a kind of punctual ...
Pedro G. Mattos's user avatar
2 votes
1 answer
292 views

Projective variety of general type such that $S^m \Omega_X^1$ is globally generated

Let $X$ be a smooth complex projective variety of general type; in my applications, I work with a surface, but let me ask this question in full generality. Assume that for some $m \geq 1$ the vector ...
Francesco Polizzi's user avatar
2 votes
1 answer
910 views

Infinite Grassmannian does not have the homotopy type of a finite-dimensional complex

Is there a proof that $BO(k)$ is not of the homotopy type of a finite dimensional complex? The Grassmannian $BO(k) := \{ k\text{-dim subspaces of } \mathbb{R}^\infty \}$ classifies the $k$-...
Jan Steinebrunner's user avatar
2 votes
0 answers
166 views

Singularity of the solution of a PDE whose coefficients have zeros

The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post): $$\mathcal{A}p=0, \quad p\in C^2(\...
S.Surace's user avatar
  • 1,675
2 votes
1 answer
269 views

sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

This question was also asked on MSE. Does there exist an asymptotic estimate for the following sum over primes $$ \sum_{p\leq x} \frac{\tau(p-1)}{p}\;, $$ where $\tau(n)=\sum_{d|n}1$ is the divisor ...
PITTALUGA's user avatar
  • 215
2 votes
2 answers
337 views

Do splines preserve monotonicity?

Start with a monotone nonincreasing function and sample it at finite set of points $x_0, ..., x_n$, $x_i<x_{i+1}$ so that $f(x_i)<f(x_{i+1})$. If you approximate $f$ with a linear spline then ...
Michael's user avatar
  • 2,175
2 votes
3 answers
478 views

A good introduction to the study of the Thue Equation

Hi, I am interested in studying the Thue equation, where we are concerned with a binary form $F(x,y) = a_0 x^r + a_1 x^{r-1}y + \cdots + a_r y^r$ and solutions of the form $$F(x,y) = h$$ for some ...
Stanley Yao Xiao's user avatar
2 votes
1 answer
481 views

Time interval of existence of an SDE solution with locally Lipschitz drift

Consider the stochastic ODE $$ dX = F(X) \, dt + dB $$ where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
Hausdorff's user avatar
2 votes
0 answers
157 views

Quantum invariant: The canonical $2$-tensor

In Chapter XVI Kassel introduces a proper definition of a quantum universal enveloping algebra of a Lie algebra $\mathfrak{g}$. (See definition XVI.5.1). Notice that a quantum enveloping algebra has a ...
Mathematician 42's user avatar
2 votes
2 answers
682 views

Is there a definition of $\log(x)$ for quaternion/octonion $x$?

I'm trying to implement $\log({\bf q})$ in python, where ${\bf q} = (q_0,\ldots,q_7) \in \mathbb{O}$ is an octonion. There is a well known definition of $\log({\bf q})$ for quaternions ${\bf q} = (s,v)...
Dieter Kadelka's user avatar
2 votes
2 answers
406 views

What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?

What is the solution, $f(n)$, of the following functional equation: $$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$ where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? ...
mark's user avatar
  • 153
2 votes
1 answer
531 views

Now that I got a mutant-discriminating invariant...

...what can I do with the darn thing? Background: I read that still no Vassiliev Invariant with mutant-discriminating power is known (correct me if this is outdated). Now, my research lead to a whole ...
Hauke Reddmann's user avatar
2 votes
1 answer
220 views

Possible symmetry groups of power terms

Previously asked and bountied at MSE: Let $\mathfrak{E}=(\mathbb{N};\mathit{exp})$ be the algebra in the sense of universal algebra consisting of the natural numbers with just exponentiation. To each ...
Noah Schweber's user avatar
2 votes
1 answer
234 views

Non-isomorphic line bundles detected by sub-curves?

This seems like it should be easy, but unfortunately I don't see how to do it. Let $X$ be a variety; I'm happy to assume that $X$ is quasiprojective. If $L_1$ and $L_2$ are two non-isomorphic line ...
user84144's user avatar
  • 2,769
2 votes
0 answers
257 views

A relation of the prime counting function $\pi$ to counting the ordered ways of a number $n$ as a sum of two primes and two questions?

The definitions are from these two questions: https://math.stackexchange.com/questions/3164216/a-series-related-to-prime-numbers https://math.stackexchange.com/questions/4349186/trying-to-understand-...
mathoverflowUser's user avatar
2 votes
2 answers
166 views

Theoretical/Practical Implications of DFT Eigenvectors

Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of ...
ABB's user avatar
  • 3,898
2 votes
0 answers
279 views

infinite dimensional germs of schemes and tangent spaces

(The question of the type "how to define?") Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as ...
Dmitry Kerner's user avatar
2 votes
1 answer
155 views

Does the path category of a quiver determine the quiver up to isomorphism?

Let $G$ and $G'$ be quivers. If their path categories $Path[G]$ and $Path[G']$ are isomorphic, does is follow that $G$ is isomorphic to $G'$?
user1005113's user avatar
2 votes
1 answer
227 views

curve through a point avoiding an hypersurface, II

Inspired by this question: Suppose given an algebraic curve $C \subset \mathbb{A}^2$, and a point $x \in C$. Can you find another (closed) curve $D \subset \mathbb{A}^2$ such that $C \cap D = x$?
Vivek Shende's user avatar
  • 8,663
2 votes
1 answer
1k views

Condition for doubly non-negative matrices to be completely positive

Consider a doubly non-negative matrix $A$ of order $n$. $A$ is completely positive if and only if $A$ can be factorized into $BB^{T}$ where all entries in $B$ are non-negative. $B$ is $n\times k$. The ...
Pawan Aurora's user avatar

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