Related papers: Polarization, sign sequences and isotropic vector …

We investigate the sharp endpoint extension inequality for the moment curve in finite fields. We determine the optimal constant and characterize the maximizers in two complementary regimes: (i) low dimensions $d\leq 20$; (ii) large field…

The 355 optically polarized QSOs have redshifts from 0.061 to 3.94 and are spread out over the sky except for a 60 degree band centered on the Galactic Equator. The data we analyze was measured, collected and published by others. Here, we…

It is well known that the vertices of any Euclidean simplicial regular polytope determine an optimal packing of equal spherical balls. We prove a stability version of optimal order of this result.

Visibility $V$ and distinguishability $D$ quantify wave-ray duality: $V^2 + D^2 \le 1$. We join them to polarization $P$ via the Polarization Coherence Theorem, a tight equality: $P^2 = V^2 + D^2$.

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$…

We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication…

We investigate a class of sharp Fourier extension inequalities on the planar curves $s=|y|^p$, $p>1$. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that…

Among subsets of Euclidean space with prescribed measure, for which sets is the $L^q$ norm of the Fourier transform of the indicator function maximized? Various partial results concerning this question are established, including the…

We prove Wehrl-type $L^2(G)-L^{p}(G)$ inequalities for matrix coefficients of vector-valued holomorphic discrete series of $G$, for even integers $p=2n$. The optimal constant is expressed in terms of Harish-Chandra formal degrees for the…

We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We propose and rigorously…

We give an explicit sequence of polarizations such that for every measurable function, the sequence of iterated polarizations converge to the symmetric rearrangement of the initial function.

We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical…

In this paper we solve the polarization problem for real Hilbert spaces, a long-standing conjecture that had remained open for nearly three decades. We also confirm that the only extremal configurations are orthonormal sets. These are…

We obtain all extreme and exposed points of the closed unit ball of the space of bilinear forms $T:\ell_{\infty}^{2}\times\ell_{\infty}^{2}\rightarrow \mathbb{R}.$ We also show that any (norm one) bilinear form $T:\ell_{\infty…

We provide symmetrization results in the form of mass concentration comparisons for fractional singular elliptic equations in bounded domains, coupled with homogeneous external Dirichlet conditions. Two types of comparison results are…

Let $L$ be a finite-dimensional real normed space, and let $B$ be the unit ball in $L$. The sign sequence constant of $L$ is the least $t>0$ such that, for each sequence $v_1, \ldots, v_n \in B$, there are signs $\varepsilon_1, \ldots,…

Let $c<3p/16$ be a prime or $c=1$. Let $E$ be a $\mathbb{Z}[\sqrt{-cp}]$-oriented supersingular elliptic curve defined over $\mathbb{F}_{p^2}$. There exists a $c$-isogeny from $E$ to $E^p$ with kernel $G \subset E[c]$. Given an Eichler…

Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\lambda(1+u)^{p} & {in}\ \ \B, %0<u\leq 1 & {in}\ \ \B, u=\frac{\partial u}{\partial n} =0 & {on}\ \ \partial \B {array}. $$ has…

In this paper we deal with a best approximation of a vector with respect to a closed semi-algebraic set $C$ in the space $\mathbb{R}^n$ endowed with a semi-algebraic norm $\nu$. Under additional assumptions on $\nu$ we prove…