Archiv der Mathematik最新文献

Rédei and Megyesi proved that the number of directions determined by a p-element subset of ({mathbb F}_p^2) is either 1 or at least (frac{p+3}{2}). The same result was independently obtained by Dress, Klin, and Muzychuk. We give a new and short proof of this result using a lemma proved by Kiss and the author. The new proof relies on a result on polynomials over finite fields.

In this note, we provide a characterization for the set of extreme points of the Lipschitz unit ball in a specific vectorial setting. While the analysis of the case of real-valued functions is covered extensively in the literature, no information about the vectorial case has been provided up to date. Here, we aim at partially filling this gap by considering functions mapping from a finite metric space to a strictly convex Banach space that satisfy the Lipschitz condition. As a consequence, we present a representer theorem for such functions. In this setting, the number of extreme points needed to express any point inside the ball is independent of the dimension, improving the classical result from Carathéodory.

In this note, we study the regularity of the Berezin transform on the generalized Hartogs triangles. By introducing a rotation invariant weight function, we show the unboundedness of the Berezin transform of weighted Hilbert spaces defined on the generalized Hartogs triangles.

Let p be an odd prime number. We show that the modular isomorphism problem has a positive answer for finite p-groups whose center has index (p^3), which is a strong contrast to the analogous situation for (p = 2).

Denote by ({mathbb {G}}(k,n)) the Grassmannian of linear subspaces of dimension k in ({mathbb {P}}^n). We show that if (varphi :{mathbb {G}}(l,n) rightarrow {mathbb {G}}(k,n)) is a nonconstant morphism and (l not =0,n-1), then (l=k) or (l=n-k-1) and (varphi ) is an isomorphism.

The solvable conjugacy class graph of a finite group G, denoted by (Gamma _{sc}(G)), is a simple undirected graph whose vertices are the non-trivial conjugacy classes of G and two distinct conjugacy classes C, D are adjacent if there exist (x in C) and (y in D) such that (langle x, yrangle ) is solvable. In this paper, we discuss certain properties of the genus and crosscap of (Gamma _{sc}(G)) for the groups (D_{2n}), (Q_{4n}), (S_n), (A_n), and ({{,mathrm{mathop {textrm{PSL}}},}}(2,2^d)). In particular, we determine all positive integers n such that their solvable conjugacy class graphs are planar, toroidal, double-toroidal, or triple-toroidal. We shall also obtain a lower bound for the genus of (Gamma _{sc}(G)) in terms of the order of the center and number of conjugacy classes for certain groups. As a consequence, we shall derive a relation between the genus of (Gamma _{sc}(G)) and the commuting probability of certain finite non-solvable group.

Given an uncountable, compact metric space X, we show that there exists no reproducing kernel Hilbert space that contains the space of all continuous functions on X.

Recall that D is a half-factorial domain (HFD) when D is atomic and every equation (pi _1cdots pi _k = rho _1 cdots rho _ell ), with all (pi _i) and (rho _j) irreducible in D, implies (k=ell ). We explain how techniques introduced to attack Artin’s primitive root conjecture can be applied to understand half-factoriality of orders in real quadratic number fields. In particular, we prove that (a) there are infinitely many real quadratic orders that are half-factorial domains, and (b) under the generalized Riemann hypothesis, ({mathbb {Q}}(sqrt{2})) contains infinitely many HFD orders.

Let (mathbb {F}) be an algebraically closed field of characteristic zero. We prove that functorial equivalence over (mathbb {F}) and perfect isometry between blocks of finite groups do not imply each other.