Mathematika最新文献

In this note, we show existence and regularity of periodic tilings of the Euclidean space into equal cells containing a ball of fixed radius, which minimize either the classical or the fractional perimeter. We also discuss some qualitative properties of minimizers in dimensions 3 and 4.

In the previous paper (Skriganov, J. Complexity 56 (2020), 101428), Stolarsky's invariance principle, known in the literature for point distributions on Euclidean spheres, has been extended to the real, complex, and quaternionic projective spaces and the octonionic projective plane. Geometric features of these spaces as well as their models in terms of Jordan algebras have been used very essentially in the proof. In the present paper, a new pure analytic proof of the extended Stolarsky's invariance principle is given, relying on the theory of spherical functions on compact Riemannian symmetric manifolds of rank one.

Recently there have been several works estimating the number of $$ matrices with elements from some finite sets $$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial upper bound $$, where $$ is the cardinality of $$. Here we show that even for arbitrary sets $$, some recent results from additive combinatorics enable us to obtain a stronger bound with a power saving.

We show that any finite family of pairwise intersecting balls in $$ can be pierced by $$ points improving the previously known estimate of $$. As a corollary, this implies that any 2-illuminable spiky ball in $$ can be illuminated by $$ directions. For the illumination number of convex spiky balls, that is, cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of $$, which can be compared with the previous $$ established only for the centrally symmetric cap bodies. We also prove the lower bounds of $$ for the three problems above.

We obtain sharp embeddings from the Sobolev space $$ into the space $$ and determine the extremal functions. This improves on a previous estimate of the sharp constants of these embeddings due to Kalyabin.

We develop a calculus of Berezin–Toeplitz operators quantizing exotic classes of smooth functions on compact Kähler manifolds and acting on holomorphic sections of powers of positive line bundles. These functions (classical observables) are exotic in the sense that their derivatives are allowed to grow in ways controlled by local geometry and the power of the line bundle. The properties of this quantization are obtained via careful analysis of the kernels of the operators using Melin and Sjöstrand's method of complex stationary phase. We obtain a functional calculus result, a trace formula, and a parametrix construction for this larger class of functions. These results are crucially used in proving a probabilistic Weyl-law for randomly perturbed (standard) Berezin–Toeplitz operators in Oltman (arXiv:2207.09599).

If four people with Gaussian-distributed heights stand at Gaussian positions on the plane, the probability that there are exactly two people whose height is above the average of the four is exactly the same as the probability that they stand in convex position; both probabilities are $$. We show that this is a special case of a more general phenomenon: The problem of determining the position of the mean among the order statistics of Gaussian random points on the real line (Youden's demon problem) is the same as a natural generalization of Sylvester's four point problem to Gaussian points in $$. Our main tool is the observation that the Gale dual of independent samples in $$ itself can be taken to be a set of independent points (translated to have barycenter at the origin) when the distribution of the points is Gaussian.

In this paper, we present a unique four-dimensional body of constant width based on the classical notion of focal conics.

A set of points with finite density is constructed in $$, with $$, by adding points to a Poisson process such that any line segment of length $$ in $$ will contain one of the points of the set within distance $$ of it. The constant implied by the big-$$ notation depends on the dimension only.

Let $$ and let $$ and $$ be two convex bodies in $$ such that their orthogonal projections $$ and $$ onto any $$-dimensional subspace $$ are directly congruent, that is, there exists a rotation $$ and a vector $$ such that $$. Assume also that the 2-dimensional projections of $$ and $$ are pairwise different and they do not have $$-symmetries. Then $$ and $$ are congruent. We also prove an analogous more general result about twice differentiable functions on the unit sphere in $$.