Mathematika最新文献

Let $��G�=�(�V�,�E�)�$ be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices $��x_1�,�\cdots �,�x_k�$, take $�x_{�k�+�1�}$ to be any vertex maximizing the sum of distances to the vertices already chosen and iterate, keep adding the “most remote” vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices $��m�\ll �\left|�V�\right|�$. We prove that this suggests that the graph G is, in a suitable sense, “m-dimensional” by exhibiting an explicit 1-Lipschitz embedding $��\varphi �:�V�\to �{\ell }^1��\left(�R^m�\right)��$ with good properties.

We investigate games played between Maker and Breaker on an infinite complete graph whose vertices are coloured with colours from a given set, each colour appearing infinitely often. The players alternately claim edges, Maker's aim being to claim all edges of a sufficiently colourful infinite complete subgraph and Breaker's aim being to prevent this. We show that if there are only finitely many colours, then Maker can obtain a complete subgraph in which all colours appear infinitely often, but that Breaker can prevent this if there are infinitely many colours. Even when there are infinitely many colours, we show that Maker can obtain a complete subgraph in which infinitely many of the colours each appear infinitely often.

There has recently been some interest in optimizing the error term in the asymptotic for the fourth moment of Dirichlet L-functions and a closely related mixed moment of L-functions involving automorphic L-functions twisted by Dirichlet characters. We obtain an improvement for the error term of the latter.

Let X be a set of n points in $�R^d$ in general position. What is the maximum number of vertices that $��\mathrm{conv}�\left(�T�\right(�X�\left)�\right)�$ can have among all the possible permissible projective transformations T? In this paper, we investigate this and other related questions. After presenting several upper bounds, obtained by using oriented matroid machinery, we study a closely related problem (via Gale transforms) concerning the maximal number of minimal Radon partitions of a set of points. The latter led us to a result supporting a positive answer to a question of Pach and Szegedy asking whether balanced 2-colorings of points in the plane maximize the number of induced multicolored Radon partitions. We also discuss a related problem concerning the size of topes in arrangements of hyperplanes as well as a tolerance-type problem of finite sets.

Assuming the Riemann hypothesis and a hypothesis on small gaps between zeta zeros (see equation (ES 2K) below for a precise definition), we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith [J. Math. Phys. 60 (2019), no. 8, 083509], which states that for any positive integer K and real number $��a�>�0�$,

Shephard (Canad. J. Math. 26 (1974), 302–321) proved a decomposition theorem for zonotopes yielding a simple formula for their volume. In this note, we prove a generalization of this theorem yielding similar formulae for their intrinsic volumes. We use this result to investigate geometric extremum problems for zonotopes generated by a given number of segments. In particular, we solve isoperimetric problems for d-dimensional zonotopes generated by d or $��d�+�1�$ segments, and give asymptotic estimates for the solutions of similar problems for zonotopes generated by sufficiently many segments. In addition, we present applications of our results to the ℓ₁ polarization problem on the unit sphere and to a vector-valued Maclaurin inequality conjectured by Brazitikos and McIntyre in 2021.

We introduce a new operation between nonnegative integrable functions on $�R^n$, that we call geometric combination; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Prékopa–Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the 0-sum, we get an alternative proof of the log-Brunn–Minkowski inequality for unconditional convex bodies and for convex bodies with n symmetries.

We prove a conjecture of D. Oberlin on the dimension of unions of lines in $�R^n$. If $��d�⩾�1�$ is an integer, $��0�⩽�\beta �⩽�1�$, and L is a set of lines in $�R^n$ with Hausdorff dimension at least $��2�(�d�-�1�)�+�\beta �$, then the union of the lines in L has Hausdorff dimension at least $��d�+�\beta �$. Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear → linear argument of Bourgain and Guth.

Recently, K.-Y. Wu introduced affine subspace concentration conditions for the cone volumes of polytopes and proved that the cone volumes of centered, reflexive, smooth lattice polytopes satisfy these conditions. We extend the result to arbitrary centered polytopes.

The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem of Adams, Bush and Frick [1] that if $��f�:�S^1�\to �R^{�2�k�+�1�}�$ is a continuous function on the unit circle S¹ in $�C$ such that $��f�(�-�z�)�=�-�f�\left(�z�\right)�$ for all $��z�\in �S^1�$, then there is a finite subset X of S¹ of diameter at most $��\pi �-�\pi �/�(�2�k�+�1�)�$ (in the standard metric in which the circle has circumference of length 2π) such the convex hull of $��f�\left(�X�\right)�$ contains $��0�\in �R^{�2�}$