Let (X,d) be a compact metric space and let ℳ(X) denote the space of all finite signed Borel measures on X. Define I:ℳ(X)→ℝ by and set M(X)=sup I(μ), where μ ranges over the collection of signed measures in ℳ(X) of total mass 1. The metric space (X,d) is quasihypermetric if for all n∈ℕ, all α1,…,αn∈ℝ satisfying ∑ i=1nαi=0 and all x1,…,xn∈X, the inequality ∑ i,j=1nαiαjd(xi,xj)≤0 holds. Without the quasihypermetric property M(X) is infinite, while with the property a natural semi-inner product structure becomes available on ℳ0(X), the subspace of ℳ(X) of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of (X,d), the semi-inner product space structure of ℳ0(X) and the Banach space C(X) of continuous real-valued functions on X; conditions equivalent to the quasihypermetric property; the topological properties of ℳ0(X) with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-* topology and the measure-norm topology on ℳ0(X); and the functional-analytic properties of ℳ0(X) as a semi-inner product space, including the question of its completeness. A later paper [P. Nickolas and R. Wolf, Distance geometry in quasihypermetric spaces. II, Math. Nachr., accepted] will apply the work of this paper to a detailed analysis of the constant M(X).

In this paper, some new coincidence point theorems in continuous function spaces are presented. We show the hybrid mapping version and multivalued version of both Lou’s fixed point theorem (Proc. Amer. Math. Soc.127 (1999)) and de Pascale and de Pascale’s fixed point theorem (Proc. Amer. Math. Soc.130 (2002)). Our new results encompass a number of previously known generalizations of the theorems. Two examples are presented.

In this paper, we characterize quadratic number fields possessing unique factorization in terms of the power cancellation property of torsion-free rank-two abelian groups, in terms of Σ-unique decomposition, in terms of a pair of point set topological properties of Eilenberg–Mac Lane spaces, and in terms of the sequence of rational primes. We give a complete set of topological invariants of abelian groups, we characterize those abelian groups that have the power cancellation property in the category of abelian groups, and we characterize those abelian groups that have Σ-unique decomposition. Our methods can be used to characterize any direct sum decomposition property of an abelian group.

We identify the majority of Siegel modular eigenforms in degree four and weights up to 16 as being Duke–Imamoḡlu–Ikeda or Miyawaki–Ikeda lifts. We give two examples of eigenforms that are probably also lifts but of an undiscovered type.

In this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

Let G be a group. A subset N of G is a set of pairwise noncommuting elements if xy⁄=yx for any two distinct elements x and y in N. If ∣N∣≥∣M∣ for any other set of pairwise noncommuting elements M in G, then N is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements in a three-dimensional general linear group. Moreover, we show how to modify a given maximal subset of pairwise noncommuting elements into another maximal subset of pairwise noncommuting elements that contains a given ‘generating element’ from each maximal torus.

We prove sharp estimates for the dilation operator f(x)⟼f(λx), when acting on Wiener amalgam spaces W(Lp,Lq). Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces Mp,q, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.

Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math.13(6) (2009)] studied an implicit viscosity method for approximating solutions of variational inequalities by solving hierarchical fixed point problems. The approximate solutions are a net (xs,t) of two parameters s,t∈(0,1), and under certain conditions, the iterated lim t→0lim s→0xs,t exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of (xs,t) as (s,t)→(0,0) jointly in the norm topology. In this paper we further study the behaviour of the net (xs,t); in particular, we give a negative answer to this problem.

In this paper, we consider the existence of nontrivial solutions for the nonlinear fractional differential equation boundary-value problem (BVP) where 1<α≤2, η∈(0,1),β∈ℝ=(−∞,+∞), βηα−1≠1, Dα is the Riemann–Liouville differential operator of order α, and f:[0,1]×ℝ→ℝ is continuous, q(t):[0,1]→[0,+∞) is Lebesgue integrable. We give some sufficient conditions for the existence of nontrivial solutions to the above boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. Particularly, we do not use the nonnegative assumption and monotonicity on f which was essential for the technique used in almost all existed literature.

We characterize surjective nonexpansive mappings between unit spheres of ℒ∞(Γ)-type spaces. We show that such mappings turn out to be isometries and can be extended to linear isometries on the whole space ℒ∞(Γ).

Qualified difference sets (QDS) composed of unions of cyclotomic classes are discussed. An exhaustive computer search for such QDS and modified QDS that also possess the zero residue has been conducted for all powers n=4,6,8 and 10. Two new families were discovered in the case n=8 and some new isolated systems were discovered for n=6 and n=10.

Let n∈ℕ and let A and B be rings. An additive map h:A→B is called an n-Jordan homomorphism if h(an)=(h(a))n for all a∈A. Every Jordan homomorphism is an n-Jordan homomorphism, for all n≥2, but the converse is false in general. In this paper we investigate the n-Jordan homomorphisms on Banach algebras. Some results related to continuity are given as well.

We investigate the connection between measures of noncompactness of a bounded subset of a given Banach space and the corresponding measures of noncompactness of an ultrapower of this subset. The Kuratowski, Hausdorff and separation measures of noncompactness are considered. We prove that in the first two cases the measures of a subset are equal to the respective measures of ultrapowers of this subset. In the case of separation measure of noncompactness, the equality is not necessarily fulfilled.