Mathematika最新文献

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 $$.

In recent decades, the use of ideas from Minkowski's Geometry of Numbers has gained recognition as a helpful tool in bounding the number of solutions to modular congruences with variables from short intervals. In 1941, Mahler introduced an analogue to the Geometry of Numbers in function fields over finite fields. Here, we build on Mahler's ideas and develop results useful for bounding the sizes of intersections of lattices and convex bodies in $$, which are more precise than what is known over $$. These results are then applied to various problems regarding bounding the number of solutions to congruences in $$, such as the number of points on polynomial curves in low-dimensional subspaces of finite fields. Our results improve on a number of previous bounds due to Bagshaw, Cilleruelo, Shparlinski and Zumalacárregui. We also present previous techniques developed by various authors for estimating certain energy/point counts in a unified manner.

We prove the existence of infinitely many $$ such that the difference of harmonic numbers $$ approximates 1 well

A monoid $$ is right coherent if every finitely generated subact of every finitely presented right $$-act itself has a finite presentation; it is weakly right coherent if every finitely generated right ideal of $$ has a finite presentation. We show that full and partial transformation monoids, symmetric inverse monoids and partition monoids over an infinite set are all weakly right coherent, but that none of them is right coherent. Left coherency and weak left coherency are defined dually, and the corresponding results hold for these properties. In order to prove the non-coherency results, we give a presentation of an inverse semigroup which does not embed into any left or right coherent monoid.

This paper presents two general criteria to determine spaceability results in the complements of unions of subspaces. The first criterion applies to countable unions of subspaces under specific conditions and is closely related to the results of Kitson and Timoney [J. Math. Anal. Appl. 378 (2011), 680–686]. This criterion extends and recovers some classical results in this theory. The second criterion establishes sufficient conditions for the complement of a union of Lebesgue spaces to be $$-spaceable, or not, even when they are not locally convex. We use this result to characterize measurable subsets having positive measure. Armed with these results, we have improved existing results in environments such as Lebesgue measurable function sets, spaces of continuous functions, sequence spaces, nowhere Hölder function sets, Sobolev spaces, non-absolutely summing operator spaces and even sets of functions of bounded variation.