Related papers: A covariance equation

We consider the following conjecture (from Huang, et al): Let $\Delta^+$ denote the upper half disc in $\mathbb{C}$ and let $\gamma = ( - 1, 1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic…

A parametric curve $\gamma$ of class $C^n$ on the $n$-sphere is said to be nondegenerate (or locally convex) when $\det\left(\gamma(t),\gamma'(t),\cdots,\gamma^{(n)}(t)\right)>0$ for all values of the parameter $t$. We orthogonalize this…

Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $\rho \colon G(R)^+ \to GL_n (K)$ over an…

A classical discovery known as Fenchel's conjecture and proved in the 1950s, shows that every co-compact Fuchsian group $F$ has a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered orientable…

We reformulate Fourier-space crystallography in the language of cohomology of groups. Once the problem is understood as a classification of linear functions on the lattice, restricted by a particular group relation, and identified by gauge…

We study traces of elements of fractional Sobolev spaces $H_p^\alpha(\mathbb{R}^n)$ on closed subsets $\Gamma$ of $\mathbb{R}^n$, given as the supports of suitable measures $\mu$. We prove that if these measures satisfy localized upper…

Let $\Gamma$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^\Gamma)$ to denote the free part of the Bernoulli shift action of $\Gamma$ on $k^\Gamma$. Seward and Tucker-Drob showed that there exists a free…

Suppose $G$ is an amenable locally compact group. If $\{F_\gamma\} = \{F_\gamma\}_{\gamma\in\Gamma}$ is a F\o{}lner net for $G$, associate it with the net $\{\chi_{F_\gamma} / |F_\gamma|\} \subset L_1(G) \subset L_\infty^*(G)$. Thus, every…

Let A be an introverted algebra with mean value. We prove that its spectrum \Delta (A) is a compact topological semigroup, and that the kernel K(\Delta (A)) of \Delta (A) is a compact topological group over which the mean value on A can be…

M. Goresky, G. Harder, and R. MacPherson defined weighted cohomologies of arithmetic groups \Gamma in a real group G, with coefficients in certain local systems, associated to arbitrary upper and lower weight profiles. The author shows,…

We propose an abstract framework for the homogenization of random functionals which may contain non-convex terms, based on a two-scale $\Gamma$-convergence approach and a definition of Young measures on micropatterns which encodes the…

Let $\rho$ be a quasiconformal mapping on the plane with complex dilatation $\mu$. We show that if $\mu$ satisfies a certain Carleson measure condition, then one can transfer $H^{\infty}$ on the upper half plane onto the corresponding space…

\begin{abstract} We state the following weighted Hardy inequality \begin{equation*} c_{o, \mu}\int_{{\R}^N}\frac{\varphi^2 }{|x|^2}\, d\mu\le \int_{{\R}^N} |\nabla\varphi|^2 \, d\mu + K \int_{\R^N}\varphi^2 \, d\mu \quad \forall\, \varphi…

$\mu$ being a nonnegative measure satisfying some log-Sobolev inequality, we give conditions on F for the measure $\nu=e^{-2F} \mu$ to also satisfy some log-Sobolev inequality. Explicit examples are studied.

Let $G$ be a locally compact abelian group, and let $\widehat{G}$ denote its dual group, equipped with a Haar measure. A variant of the uncertainty principle states that for any $S \subset G$ and $\Sigma \subset \widehat{G}$, there exists a…

We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $\mu(\om)$ of (not necessarily compact) basic semi-algebraic sets$\om\subset\R^n$. We obtain two monotone (non increasing and non…

The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups…

Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $\nu$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of…

Arbitrary representations of a commutative unital ($\ast$-) $\mathbb{F}$-algebra $A$ as a subalgeba of $\mathbb{F}^X$ are considered, where $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$ and $X\neq\emptyset$. The Gelfand spectrum of $A$ is…

Let $E_G$ be a $\Gamma$--equivariant algebraic principal $G$--bundle over a normal complex affine variety $X$ equipped with an action of $\Gamma$, where $G$ and $\Gamma$ are complex linear algebraic groups. Suppose $X$ is contractible as a…