For a prime power q, let 𝔽q be the finite field of q elements. We show that 𝔽*q⊆d𝒜2 for almost every subset 𝒜⊂𝔽q of cardinality ∣𝒜∣≫q1/d. Furthermore, if q=p is a prime, and 𝒜⊆𝔽p of cardinality ∣𝒜∣≫p1/2(log p)1/d, then d𝒜2 contains both large and small residues. We also obtain some results of this type for the Erdős distance problem over finite fields.

For a locally compact group G, let LUC(G) denote the space of all left uniformly continuous functions on G. Here, we investigate projectivity, injectivity and flatness of LUC(G) and its dual space LUC(G)* as Banach left modules over the group algebra as well as the measure algebra of G.

Let C be a bounded, closed, convex subset of a uniformly convex Banach space X. We investigate the existence of common fixed points for pointwise Lipschitzian semigroups of nonlinear mappings Tt:C→C, where each Tt is pointwise Lipschitzian. The latter means that there exists a family of functions αt:C→[0,∞) such that for x,y∈C. We also demonstrate how the asymptotic aspect of the pointwise Lipschitzian semigroups can be expressed in terms of the respective Fréchet derivatives.

A variational principle is established to provide a new formulation for convex Hamiltonian systems. Using this formulation, we obtain some existence results for second-order Hamiltonian systems with a variety of boundary conditions, including nonlinear ones.

We characterize certain properties in a matrix ordered space in order to embed it in a C*-algebra. Let such spaces be called C*-ordered operator spaces. We show that for every self-adjoint operator space there exists a matrix order (on it) to make it a C*-ordered operator space. However, the operator space dual of a (nontrivial) C*-ordered operator space cannot be embedded in any C*-algebra.

For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz–Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilical point, which was developed by Choe [‘Sufficient conditions for constant mean curvature surfaces to be round’, Math. Ann.323(1) (2002), 143–156]. Using the concept of the rotation index at the interior and boundary umbilical points and applying the Poincaré–Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than four vertices in a domain of bounded by (spacelike or timelike) totally umbilical surfaces is part of a (spacelike) plane or a hyperbolic plane. Moreover, we prove that the only immersed spacelike disk type capillary surface inside a de Sitter surface in is part of (spacelike) plane or a hyperbolic plane.

Our main aim is to investigate the properties of harmonic ν-Bloch mappings. Firstly, we establish coefficient estimates and a Landau theorem for harmonic ν-Bloch mappings, which are generalizations of the corresponding results in Bonk et al. [‘Distortion theorems for Bloch functions’, Pacific. J. Math.179 (1997), 241–262] and Chen et al. [‘Bloch constants for planar harmonic mappings’, Proc. Amer. Math. Soc.128 (2000), 3231–3240]. Secondly, we obtain an improved Landau theorem for bounded harmonic mappings. Finally, we obtain a Marden constant for harmonic mappings.

For a Banach algebra 𝒜 and a character ϕ on 𝒜, we introduce and study the notion of essential ϕ-amenability of 𝒜. We give some examples to show that the class of essentially ϕ-amenable Banach algebras is larger than that of ϕ-amenable Banach algebras introduced by Kaniuth et al. [‘On ϕ-amenability of Banach algebras’, Math. Proc. Cambridge Philos. Soc.144 (2008), 85–96]. Finally, we characterize the essential ϕ-amenability of various Banach algebras related to locally compact groups.

The paper deals with the classification of Leibniz central extensions of a filiform Lie algebra. We choose a basis with respect to which the multiplication table has a simple form. In low-dimensional cases isomorphism classes of the central extensions are given. In the case of parametric families of orbits, invariant functions (orbit functions) are provided.

In this note, we prove that the Gauss–Picard modular group PU(2,1;Θ1) has Property (FA). Our result gives a positive answer to a question by Stover [‘Property (FA) and lattices in SU(2,1)’, Internat. J. Algebra Comput.17 (2007), 1335–1347] for the group PU(2,1;Θ1).

Complete minimal immersions satisfying the Omori–Yau maximum principle are investigated. It is shown that the limit set of a proper immersion into a convex set must be the whole boundary of the convex set. In case of a nonproper and nonplanar immersion we prove that the convex hull of the immersion is a half-space or ℝ3.

Let λ be a regular ordinal with λ≥ω1. Then we prove that (λ+1)×λ is not base-countably metacompact. This implies that base-κ-paracompactness is not an inverse invariant of perfect mappings, which answers a question asked by Yamazaki.

For a Banach algebra 𝒜 and a character ϕ on 𝒜, we have recently introduced and studied the notion of ϕ-pseudo-amenability of 𝒜. Here, we give some characterizations of this notion in terms of derivations from 𝒜 into various Banach 𝒜-bimodules.

A set A⊆ℤ is called an asymptotic basis of ℤ if all but finitely many integers can be represented as a sum of two elements of A. Let A be an asymptotic basis of integers with prescribed representation function, then how dense A can be? In this paper, we prove that there exist a real number c>0 and an asymptotic basis A with prescribed representation function such that for infinitely many positive integers x.

A group G is said to be a B(n,k) group if for any n-element subset A of G, ∣A2∣≤k. In this paper, characterizations of B(5,16) groups and B(5,17) groups are given.

For a finite group G, we denote by μ(G) the minimum degree of a faithful permutation representation of G. We prove that if G is a finite p-group with an abelian maximal subgroup, then μ(G/G′)≤μ(G).

We discuss the distribution properties of hybrid sequences whose components stem from Niederreiter–Halton sequences on the one hand, and Kronecker sequences on the other. In this paper, we give necessary and sufficient conditions on the uniform distribution of such sequences, and derive a result regarding their discrepancy. We conclude with a short summary and a discussion of topics for future research.

A submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

Here we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover, we show that every rank r>2 weakly uniform vector bundle with splitting type a1,1=⋯=ar,s=0 is trivial and every rank r>2 uniform vector bundle with splitting type a1>⋯>ar splits.