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3 votes
1 answer
92 views

Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?

We endow the space $\mathcal P_2^a (\mathbb R^d)$ of absolutely continuous probability measures with finite second moment with the Wasserstein distance $W_2$. Let $\mathcal H (\mu)$ be the relative ...
0 votes
1 answer
35 views

Norm of a $2$-tuple of operators

Let $E$ be a complex Hilbert space and $K_1,K_2$ are bounded linear operators on $E$. Let $\omega(K_1)$ and $\omega(K_2)$ be the numerical radius of $K_1$ and $K_2$ respectively. That is \begin{align*}...
3 votes
1 answer
355 views

Ask for a generating function or an explicit expression of a triangle of positive integers

Preliminaries I encountered the following triangle of positive integers: $c_{n,k}$ $n=1$ $n=2$ $n=3$ $n=4$ $n=5$ $n=6$ $n=7$ $n=8$ $k=0$ $1$ $3$ $15$ $105$ $315$ $3465$ $45045$ $45045$ $k=1$ $5$ $...
2 votes
0 answers
47 views

Nielsen–Thurston classification and configuration spaces

Viewing the $n$-strand braid group as the mapping class group of an $n$-punctured disk, braids can be classified as periodic, reducible, or pseudo-Anosov. The same group is also the fundamental group ...
4 votes
0 answers
283 views

References and upper bounds for the SONNAT tiling game?

Introduction In a video released about a month ago, Pembesita describes1 a tiling game called SONNAT: Same Orientation Neighbour Not Allowed, Tiling. In the single-player game2, the player may employ ...
3 votes
1 answer
118 views

Comparing Kummer maps to étale homotopy at finite level

$\DeclareMathOperator\Mor{Mor}\DeclareMathOperator\Hom{Hom}\newcommand{\et}{\mathrm{et}}\newcommand{\top}{\mathrm{top}}$In Voevodsky's paper "Étale topologies of schemes over fields of finite ...
2 votes
1 answer
96 views

Teaching suggestions for Kleene fixed point theorem

I will take over two lectures from a colleague in which we discuss fixed point theory in the context of complete partial orders, and culminates in showing the Kleene fixed point theorem (see f.e. ...
1 vote
1 answer
184 views

Prove the limit of the integral

Suppose: $f:[0,\frac{\pi}{2}]\to \mathbb{R}$, $f(x)\in C^2[0,\frac{\pi}{2}]$, $\gamma=\lim_{n\to \infty}(1+1/2+\dotsb+1/n-\ln{n})$. Prove:$$ \lim_{s\to +\infty}\left(\int_{0}^{\frac{\pi}{2}} \frac{\...
1 vote
0 answers
72 views

Trivial representation of a maximal torus

Let G be a connected compact Lie group and $T\subset G$ a maximal torus. For an irreducible representation $V_\lambda$ of $G$, the multiplicity of the trivial representation of $T$ in $V_\lambda$ is ...
1 vote
1 answer
317 views

Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$. Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$. For ...
1 vote
0 answers
89 views

Extremally disconnected sets as building blocks for compact Hausdorff spaces

Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
1 vote
0 answers
1k views

Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero

Using the following definition of Dirichlet L-function $$ L(1,\chi)=\begin{cases} \dfrac{2\pi h}{w\sqrt{m}} & \textit{if}\ \chi(-1)=-1 \\\\ \dfrac{2 h \log{|\epsilon|}}{w\sqrt{m}} & \textit{...
-1 votes
0 answers
100 views

Prove a generalization of hairy-ball theorem

I found an interesting question below: Prove that the 6-sphere admits no continuous field of tangent 3-planes.
12 votes
1 answer
2k views

Hobbled rook tour – Hamiltonian cycle on square grid

Consider a square grid of even side length ($2n \times 2n$). It is easy to see that there must exist a Hamiltonian cycle on the corresponding grid graph. Such a cycle is called balanced if the number ...
2 votes
1 answer
185 views

Combination of simple tensors - II

This is a follow-up question to Combination of simple tensors. I am interested in devising an alternative norm (I mean, other than the usual $\pi$ or $\epsilon$ norms) in the tensor product of two ...
0 votes
1 answer
40 views

An expansion for 2d Euler equation

Let $R>0$ be a large constant, such that for any $x \in \Omega$, $\Omega \subset B_R(x)$. Consider the following problem in $\mathbb{R}^2$: $$ -\varepsilon^2 \Delta u=1_{\{u>a\}} \text { in }\, ...
2 votes
0 answers
384 views

Semigroup transformation (symmetry?) and hamiltionan dynamic. Noether Theorem generalization?

In reasoning about symmetries of dynamical systems usually there is an Legrangian $ L(p,q) $ and symmetry transformation $s' = f(s)$ where $s = p$ or $q$. If $f(s)$ represent continuous symmetry of ...
15 votes
2 answers
2k views

Clifford PBW theorem for quadratic form

$\DeclareMathOperator\Cl{Cl}$Update Feb 3 '12: now with a question 2 which is much more elementary (and should be well-known!). Let $k$ be a commutative ring with $1$. Let $L$ be a $k$-module, and $g:...
2 votes
1 answer
291 views

Is it possible to find an upper bound and a lower bound for $(f(\xi)-f(\tilde{\xi}))^T(\frac{\partial f}{\partial \xi}(\bar{\xi}))(\xi-\tilde{\xi})$?

For a system engineering problem I have to solve the problem below, but since I am not a mathematician, I am not sure if I have enough knowledge to solve it. Problem definition: Let $f(\xi) \in \...
4 votes
2 answers
149 views

Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function

Consider the function $$\phi_{a,b}(t)=\frac{|t|}{e^{a|t|}-e^{-b|t|}}, \ \ t\in\mathbb{R},$$ where $0<a<b$. Can $\phi_{a,b}$ be the Fourier transform of a positive function for some $a<b$?
1 vote
0 answers
62 views

Uniform approximation over compacts using weighted function spaces

I'm interested in approximations over the so-called weighted function spaces. Let $(X,\tau_X)$ be some completely regular Hausdorff topological space. Additionally, consider some map $\psi: X \to (0,\...
-3 votes
0 answers
44 views

Prove the formula of the Area of a convex region D [closed]

I'm looking into a problem in Introduction to the Mathematics of Medical Imaging: If the boundary of a convex region $D$ is given parametrically by: $(x(\theta),y(\theta)) = h_D(\theta) \omega + h'_D(\...
0 votes
0 answers
78 views

A criterion for when a symmetrized decomposable tensor is nonzero

In the problem I am interested in, one has a collection of vector spaces, all $2$-dimensional, denoted by $V_{i, j}$, where $1 \leq i \neq j \leq n$. In other words, the set of indices is $$S = \{(i, ...
4 votes
3 answers
780 views

Strictifying strong monoidal functors

Let $C_1$ and $C_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C_1 \rightarrow C_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal ...
3 votes
1 answer
219 views

Does $f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$ imply $f=0$?

Let $\beta \in (0, 1)$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that $$ f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s, \quad \...
11 votes
1 answer
2k views

Entropy of first return map and suspension flows

There are some well know formulas of Abramov about derived systems. Firstly let $(X,\mu,f)$ be a probability preserving system and $A\subset X$ is measurable such that $\bigcup_{n\ge0}f^nA=X$. Let $\...
2 votes
1 answer
379 views

Chinese remainder theorem for target interval

Given $n$ pairwise coprime natural numbers $m_{1}, \dots, m_{n}$ with remainders $y_{i}$, for all $i \leq n$. Furthermore, we have a target interval $I := \left[ a, b \right]$, with $1 \leq a < b \...
10 votes
2 answers
778 views

Is there a two-variable prime-representing polynomial (in the sense of Jones-Sato-Wada-Wiens)?

In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the ...
10 votes
2 answers
719 views

Is cohomology with local coefficients a representable functor?

It is well known that the functor of cohomology is representable. More precisely, given $n\ge1$ and abelian group $G$, we have $H^n(X;G)\simeq[X,K(G,n)]$. (Here we probably need some ``nice'' ...
1 vote
0 answers
80 views

Extension of MMP from the central fiber to some neighborhood

I was reading Professor Kollár's paper deformation of varieties of general type (here: https://arxiv.org/abs/2101.10986 ) There is a theorem about the extension of MMP step when the central fiber has ...
3 votes
1 answer
207 views

Van der Waerden's classical paper on Hilbert polynomials and Bezout's theorem

Is it possible to obtain an electronic copy of Van der Waerden's "On Hilbert series of composition of ideals and generalisation of the Theorem of Bezout", Proceedings of the Koninklijke ...
0 votes
0 answers
73 views

Can a Tangent Space always be expressed with “more structure” than just a vector space (e.g. a choice of basis for Stiefel manifold)

I'm currently trying to read about the Stiefel manifold, or set of all $p$ orthonormal $n$-dimensional vectors embedded in $\mathbb{R}^{n\times p}$. $$\mathcal{V}_p(\mathbb{R}^n) = \{U \in \mathbb{R}^{...
0 votes
0 answers
40 views

Orbits/affine spaces in GAP

Another GAP-related question. I need to compute the orbits of a lot (probably, hundreds of thousands) groups acting on $\mathbb{F}_2$-vectors spaces of dimension 23 or 22. The groups range from (...
3 votes
1 answer
138 views

Request for non-Einstein positive constant scalar curvature Kähler surfaces

I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature. There are of course the Fano (del Pezzo) Kähler-...
4 votes
1 answer
208 views

Randomly removing length 1 intervals in an interval (a fragmentation process)

Short version: Start with a closed interval of length $t>0$ and repeatedly remove a random and uniformly distributed subinterval of length $1$ so long as this is possible. For $t$ large, what is ...
0 votes
0 answers
32 views

Inertia indices in GAP

Not sure that this is the right place, but I could not find a GAP specific forum. Does anyone know if there is a built-in function in GAP to find the inertia indices of a symmetric matrix, say, over ...
5 votes
1 answer
570 views

Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution

Examples of infinite dimensional involutions Edit 2/25/23, as suggested by YCOR below: (Start) The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
7 votes
0 answers
80 views

Is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?

In short, the question is in the title: is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis? A bit of context: Given a topological space $X$, a family $\...
-1 votes
0 answers
51 views

Continuous version of ergodic with integral

Let $f\in L^1(\mathbb{T}^n)$ such that $f \geq 0$ and $f$ might have a singularity, but along some curve or line $\xi(t): \mathbb{R}\to \mathbb{T}^n$ the value $f(\xi(t))$ is defined and continuous, ...
2 votes
1 answer
108 views

Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$

Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that $$ f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(...
-1 votes
0 answers
34 views

simpler question from sylvesters eqn [closed]

i think i can simplify my problem from sylvesters equation (AX - XB = C) to this: (D_1) X - X(D_2) = 0 where D_1 is diagonal nxn D_2 is diagonal mxm X is nxm D and D_2 are diagonal matrices and they ...
3 votes
1 answer
99 views

Concentration of measure on spheres with respect to a unitary of trace approximately zero

Cross-posted from MSE, where it hasn’t received any answer yet: This question arose out of my attempt to understand how a unitary of trace approximately zero acts on the unit sphere of a $n$-...
15 votes
2 answers
671 views

On sums of independent random variables in Banach spaces

Let $(\xi_n)_{n\ge 1}$, $(\eta_n)_{n\ge 1}$ be independent mean-zero random variables with values in a Banach space $X$ such that $$\sum_n\mathbb P(\xi_n\in A)\le\sum_n\mathbb P(\eta_n\in A)$$for any ...
1 vote
0 answers
114 views

Bounding dimensions of Galois cohomology

Let $G$ be the absolute Galois group of the rationals, and $V$ a finite dimensional $p$-adic representation. Is there a way to provide a general bound on $H^i(G, V)$ in terms of $\dim_{\mathbb{Q}_p} V$...
-4 votes
0 answers
49 views

Is 3d writhe for tight rational tangles quantized?

The 3d writhe of (simple) ideal (i.e. tight) knots is quantized, as Stasiak and colleagues have shown years ago. What's the current state of progress on the question whether the writhe of rational ...
7 votes
2 answers
1k views

What is a good definition of a mathematical structure?

At the moment I am writing a textbook in Foundations of Mathematics for students and trying to give a precise definition of a mathematical structure, which is the principal notion of structuralist ...
0 votes
0 answers
46 views

Constructing squares using linear operations when a sizeable residue is given

Given $x\in\{0,1,\dots,2^k-1\}$ and given $x^2\bmod p$ where $p$ is a prime at in $[2^k,2^{k+1}]$ is it possible to construct $x^2$ using only at most $O(2^{k})$ linear in $x$ operations (that is ...
1 vote
1 answer
51 views

Equivalent condition for the Pick matrix being positive semidefinite

On the wikipedia page of the Nevanlinna-Pick theorem the following claim appears: Let $\lambda_1,\lambda_2,f(\lambda_1),f(\lambda_2)\in\mathbb{D}$. The matrix $P_{ij}:=\frac{1-f(\lambda_i)\overline{f(\...
2 votes
1 answer
126 views

Example of a conditionally convergent series $\sum_{n=1}^\infty b_n$ such that $n^2(b_n-b_{n+1})$ is bounded

Let $(b_n)_{n \in \mathbb{N}}$ be a real sequence such that $(nb_n)$ is bounded. I know that if the series $\sum_{n=1}^\infty b_n$ is conditionally convergent, then $(n^2b_n)_n$ is not bounded. But, ...
3 votes
1 answer
115 views

Does the union of fractional Sobolev spaces fills $L^p$?

Let $\Omega\subset \Bbb R^d$ be any open set. Recall that for $s\in (0,1)$, the fractional Sobolev space $W^{s,p}(\Omega)$ is the collection of function in $L^p(\Omega$ such that \begin{align*} \iint_{...

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