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.

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.

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.

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.

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).

This paper deals with a sufficient condition for the infinite product of rational numbers to be an irrational number. The condition requires only some conditions for convergence and does not use other properties like divisibility. The proof is based on an idea of Erdős.

We investigate the tempered representations derived from the principal series of SLℓ(F) and their geometric structure. In particular, we give the parameterization for special representations and prove the tempered part of the Aubert–Baum–Plymen conjecture for the toral cases of SLℓ(F).

Generalizing the concept of right bounded rings, a module MR is called bounded if annR(M/N)≤eRR for all N≤eMR. The module MR is called fully bounded if (M/P) is bounded as a module over R/annR(M/P) for any ℒ2-prime submodule P◃MR. Boundedness and right boundedness are Morita invariant properties. Rings with all modules (fully) bounded are characterized, and it is proved that a ring R is right Artinian if and only if RR has Krull dimension, all R-modules are fully bounded and ideals of R are finitely generated as right ideals. For certain fully bounded ℒ2-Noetherian modules MR, it is shown that the Krull dimension of MR is at most equal to the classical Krull dimension of R when both dimensions exist.

Roth’s theorem on the consistency of the generalized Sylvester equation AX−YB=C is a special case of a rank minimization theorem.

In this note, we will prove that a finite-dimensional Lie algebra L over a field of characteristic zero, admitting an abelian algebra of derivations D≤Der(L), with the property for some n>1, is necessarily solvable. As a result, we show that if L has a derivation d:L→L such that Ln⊆d(L), for some n>1, then L is solvable.

Let G be a group. A subset X of G is a set of pairwise noncommuting elements if xy≠yx for any two distinct elements x and y in X. If |X|≥|Y | for any other set of pairwise noncommuting elements Y in G, then X 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 for some p-groups of maximal class. Specifically, we determine this cardinality for all 2 -groups and 3 -groups of maximal class.

Let P be a topological property. A space X is said to be star P if whenever 𝒰 is an open cover of X, there exists a subspace A⊆X with property P such that X=St(A,𝒰), where St(A,𝒰)=⋃ {U∈𝒰:U∩A≠0̸}. In this paper we construct an example of a pseudocompact Tychonoff space that is not star Lindelöf, which gives a negative answer to Alas et al. [‘Countability and star covering properties’, Topology Appl.158 (2011), 620–626, Question 3].

In this note we show that there exist a semiprime ring R, a strictly ordered artinian, narrow, unique product monoid (S,≤) and a monoid homomorphism ω:S⟶End(R) such that the skew generalized power series ring R[[S,ω]] is semicommutative but R[[S,ω]] is not reversible. This answers a question posed in Marks et al. [‘A unified approach to various generalizations of Armendariz rings’, Bull. Aust. Math. Soc.81 (2010), 361–397].

Let A be a domain of the boundary of a (weakly) pseudoconvex domain Ω of ℂn and M a smooth, closed, maximally complex submanifold of A. We find a subdomain E of ℂn, depending only on Ω and A, and a complex variety W⊂E such that bW=M in E. Moreover, a generalization to analytic sets of depth at least 4 is given.

Let k≥2 be an integer. A natural number n is called k-perfect if σ(n)=kn. For any integer r≥1, we prove that the number of odd k-perfect numbers with at most r distinct prime factors is bounded by (k−1)4r3.

In this paper, we show that the character of any monotonically Lindelöf generalized ordered (GO) space is not greater than ω1, which gives a negative answer to a question posed by Levy and Matveev [‘Some questions on monotone Lindelöfness’, Questions Answers Gen. Topology26 (2008), 13–27, Question 51].

Only three of the 15 973 distinct six-element semigroups are presently known to be nonfinitely based. This paper introduces a fourth example.

The max-plus algebra is well known and has useful applications in the investigation of discrete event systems and affine equations. Structural matrix rings have been considered by many authors too. This article introduces more general structural matrix semirings, which include all matrix semirings over the max-plus algebra. We investigate properties of ideals in this construction motivated by applications to the design of centroid-based classification systems, or classifiers, as well as multiple classifiers combining several initial classifiers. The first main theorem of this paper shows that structural matrix semirings possess convenient visible generating sets for ideals. Our second main theorem uses two special sets to determine the weights of all ideals and describe all matrix ideals with the largest possible weight, which are optimal for the design of classification systems.

Let X be a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially bounded ι-measurable functions f on X vanishing at infinity. We introduce and study a locally convex topology β1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with . Next, by showing that β1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove that L1 (G) , the group algebra of a locally compact Hausdorff topological group G, equipped with the convolution multiplication is a complete semitopological algebra under the β1 (G) topology.