Given a group (G, ·) and positive integers r, s ≤ |G|, we denote with μ_G (r,s) the least possible size of the sumsets AB = {a · b : a ∈ A and b ∈ B}, where A, B run over all subsets of G, such that |A| = r and |B| = s. Let H = Q × (Z/2Z)^k × C be a Hamiltonian group, where k is a non-negative integer, Q is the quaternion group of 8 elements and C is a cyclic group of odd order. We present an explicit formula for μ_H(r,s).

The Lipschitz classes Lip(α, p, w), 0 < α ≤ 1 are defined for the weighted Lebesgue spaces L^p_w with Muckenhoupt weights, and the degree of approximation by matrix transforms of f ∈ Lip (α, p, w) is estimated by n^−α.

We give a simple proof of the boundedness of the fractional maximal operator providing in this way an alternative approach to the one given by C. Capone, D. Cruz Uribe and A. Fiorenza in The fractional maximal operator and fractional integrals on variable L^p spaces, Rev. Mat. Iberoam. 23 (2007), no. 3, 743-770.

For a compact Riemannian manifold (Mⁿ,g) with boundary and dimension n, with n ≥ 2, we study the existence of metrics in the conformal class of g with scalar curvature R_g and mean curvature h_g on the boundary. In this paper we find sufficient and necessary conditions for the existence of a smaller metric g' < g with curvatures R_g' = R_g and h_g' = h_g. Furthermore, we establish the uniqueness of such a metric g' in the conformal class of the metric g when R_g ≥ 0.

In this paper, we determine two families R and G of subsets of a hyperring R and sufficient conditions such that two geometric spaces (R, R) and (R, G) are strongly transitive. Moreover, we prove that the relations Γ and α are strongly regular equivalence relations on a hyperfield or a hyperring such that (R, +) has an identity element.

Affine connections compatible with a symplectic structure are defined and conditions for two compatible connections on a Kähler manifold to form a dualistic structure are given. The special symplectic case is detailed.

Recently, various Uzawa methods were proposed based on different matrix splitting for solving nonsingular saddle point problems, and the necessary and sufficient condition of the convergence for those Uzawa methods were derived. Motivated by their results, in this paper we give the semi-convergence analysis of the inexact Uzawa method which is applied to solve singular saddle point problems under certain conditions.

In this note we consider Haar type systems as unconditional bases for Lorentz spaces defined on spaces of homogeneous type. We also give characterizations of these spaces in terms of the Haar coefficients. The basic tools are the Rubio de Francia extrapolation technique and the characterization of weighted Lebesgue spaces with Haar bases.

In this paper we present a model for the study of the total balancedness of packing and covering games, concerning some aspects of graph theory. We give an alternative proof of van Velzen’s characterization of totally balanced covering games. We introduce new types of graph perfection, which allows us to give another approach to the open problem of characterizing totally balanced packing games.

The aim of this paper is to relate the star order in operators in a Hilbert space with certain norm inequalities. We are showing inequalities of the type ‖BXA‖₂ ≤ ‖XBA‖₂ (or ‖BXA₂‖ ≥ ‖XBA‖₂), which are already known under the assumption that A = ψ(B), with ψ a positive increasing (or decreasing, respectively) function defined on the spectrum of B. In this work, we will study this type of inequalities with the hypothesis that A ≤^∗ B, where A ≤^∗ B if A^∗A = B^∗A and AA^∗ = BA^∗.

In this paper, we obtain some Hájek-Rényi inequalities for sequences of Hilbert valued random variables which are associated, negatively associated and φ-mixing. As applications, we give some almost sure convergence theorems for these dependent sequences in Hilbert space. These results extend and improve some well-known results.

Let G be an abelian group with n elements. Let S be a sequence of elements of G, where the repetition of elements is allowed. Let |S| be the cardinality, or the length of S. A sequence S ⊆ G with |S| ≥ 2 is barycentric or has a barycentric-sum if it contains one element a_j such that Σ_{ai ∈ S}a_i = |S|a_j. This paper is a survey on the following three barycentric constants: the k-barycentric Olson constant BO(k, G), which is the minimum positive integer t ≥ k ≥ 3 such that any subset of t elements of G contains a barycentric subset with k elements, provided such an integer exists; the k-barycentric Davenport constant BD(k, G), which is the minimum positive integer t such that any subsequence of t elements of G contains a barycentric subsequence with k terms; the barycentric Davenport constant BD(G), which is the minimum positive integer t ≥ 3 such that any subset of t elements of G contains a barycentric subset. New values and bounds on the above barycentric constants when G = Z_n is the group of integers modulo n are also given.

Let M be a finite von Neumann algebra with a faithful normal trace τ. Denote by L^p(M)_sh the skew-Hermitian part of the non-commutative L^p space associated with (M,τ). Let 1 < p < ∞, z ∈ L^p(M)_sh and S be a real closed subspace of L^p(M)_sh. The metric projection Q : L^p(M)_sh → S is defined for every z ∈ L^p(M)_sh as the unique operator Q(z) ∈ S such that ||z − Q(z)||_p = min_y∈S ||z − y||_p.

We show the relation between metric projection and metric geometry of homogeneous spaces of the unitary group U_M of M, endowed with a Finsler quotient metric induced by the p-norms of τ, ||x||_p = τ(|x|^p)^1/p, p an even integer. The problem of finding minimal curves in such homogeneous spaces leads to the notion of uniformly bounded metric projection. Then we show examples of metric projections of this type.

A compact Klein surface with boundary of algebraic genus g ≥ 2 has at most 12(g−1) automorphisms. When a surface attains this bound, it has maximal symmetry, and the group of automorphisms is then called an M^∗-group. If a finite group G of odd order acts on a bordered Klein surface X of algebraic genus g ≥ 2, then |G| ≤ 3(g − 1). If G acts with the largest possible order 3(g − 1), then G is called an O^∗-group. In this paper, using the results about some normal subgroups of the extended modular group Γ, we obtain some results about O^∗-groups. Also, we give the relationships between O^∗-groups and M^∗-groups.

A formula for the generating function of prime repunits is given in terms of a Lambert series using S. Golomb’s formula.

We consider the class of complex Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals, of finite dimensional complex semisimple Lie algebras. A list of Kac-Moody types for indecomposable nilpotent complex Lie algebras of dimension ≤ 7 is given.

The existence of right inverses of the divergence as an operator from H¹₀(Ω)ⁿ to L²₀(Ω) is a problem that has been widely studied because of its importance in the analysis of the classic equations of fluid dynamics. When Ω is a bounded domain which is star-shaped with respect to a ball B, a right inverse given by an integral operator was introduced by Bogovskii, who also proved its continuity using the Calderón-Zygmund theory of singular integrals.

In this paper we give an alternative elementary proof of the continuity using the Fourier transform. As a consequence, we obtain estimates for the constant in the continuity in terms of the ratio between the diameter of Ω and that of B. Moreover, using the relation between the existence of right inverses of the divergence with the Korn and improved Poincaré inequalities, we obtain estimates for the constants in these two inequalities. We also show that one can proceed in the opposite way, that is, the existence of a continuous right inverse of the divergence, as well as estimates for the constant in that continuity, can be obtained from the improved Poincaré inequality. We give an interesting example of this situation in the case of convex domains.

In this paper we prove that given a generalized dihedral group D_H and a generating subset S, if S ∩ H = ∅ then the Cayley digraph Cay(D_H, S) is Hamiltonian. The proof we provide is via a recursive algorithm that produces a Hamilton circuit in the digraph.

We give a conceptual proof of the fact that if M is a complete submanifold of a space form, then the maximal integral manifolds of the nullity distribution of its second fundamental form through points of minimal index of nullity are complete.

The finite model property of the variety of S-algebras was proved by X. Caicedo using Kripke model techniques of the associated calculus. A more algebraic proof, but still strongly based on Kripke model ideas, was given by Muravitskii. In this article we give a purely algebraic proof for the finite model property which is strongly based on the fact that for every element x in a S-algebra the interval [x, S(x)] is a Boolean lattice.

Let p, q be distinct prime numbers, and k an algebraically closed field of characteristic 0. Under certain restrictions on p, q, we discuss the structure of semisimple Hopf algebras of dimension p²q². As an application, we obtain the structure theorems for semisimple Hopf algebras of dimension 9q² over k. As a byproduct, we also prove that odd-dimensional semisimple Hopf algebras of dimension less than 600 are of Frobenius type.