All Questions
17,157
questions
2
votes
2
answers
179
views
$L^p$ 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^...
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 ...
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 $\...
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 ...
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 ...
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$ ...
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 ...
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:
$$...
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 ...
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 (...
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 ...
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 ...
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 ...
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 ...
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 ...
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\...
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 ...
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 ...
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 ...
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 \...
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 ...
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}$ ...
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}$ ...
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 ...
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 (...
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/...
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 ...
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 ...
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\...
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{...
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 ...
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 ...
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$-...
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(\...
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 ...
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 ...
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 ...
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 "...
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 ...
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)...
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? ...
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 ...
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 ...
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 ...
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-...
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 ...
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 ...
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 $G'$?
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 $C \cap D = x$?
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 ...