In this paper, a general construction of a skew-product dynamical system, for which the skew-product dynamical system studied by Hahn is a special case, is given. Then the ergodic and topological properties (of a special type) of our newly defined systems (called Milnes-type systems) are investigated. It is shown that the Milnes-type systems are actually natural extensions of dynamical systems corresponding to some special distal functions. Finally, the topological centre of Ellis groups of any skew-product dynamical system is calculated.

The deep Preiss theorem states that a Lipschitz function on a nonempty open subset of an Asplund space is densely Fréchet differentiable. However, the simpler Fabian–Preiss lemma implies that it is Fréchet intermediately differentiable on a dense subset and that for a large class of Lipschitz functions this dense subset is residual. Results are presented for Asplund generated spaces.

A number of well-known results of Ghahramani and Loy on the essential amenability of Banach algebras are generalized. It is proved that a symmetric abstract Segal algebra with respect to an amenable Banach algebra is essentially amenable. Applications to locally compact groups are given.

In this paper, we establish bounds on the norm of multiplication operators on the Bloch space of the unit disk via weighted composition operators. In doing so, we characterize the isometric multiplication operators to be precisely those induced by constant functions of modulus 1. We then describe the spectrum of the multiplication operators in terms of the range of the symbol. Lastly, we identify the isometries and spectra of a particular class of weighted composition operators on the Bloch space.

We prove that in any compact symmetric space, G/K, there is a dense set of a1,a2∈G such that if μj=mK*δaj*mk is the K-bi-invariant measure supported on KajK, then μ1*μ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1Ka2K, has nonempty interior in G.

Suppose that X is an infinite set and I(X) is the symmetric inverse semigroup defined on X. If α∈I(X), we let dom α and ran α denote the domain and range of α, respectively, and we say that g(α)=|X/dom α| and d(α)=|X/ran α| is the ‘gap’ and the ‘defect’ of α, respectively. In this paper, we study algebraic properties of the semigroup . For example, we describe Green’s relations and ideals in A(X), and determine all maximal subsemigroups of A(X) when X is uncountable.

In this note we study the zeros of solutions of differential equations of the form u′′+pu=0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given.

Let E be a totally real set on a Stein open set Ω on a complete noncompact Kähler manifold (M,g) with nonnegative holomorphic bisectional curvature such that (Ω,g) has bounded geometry at E. Then every function f in a Cp class with compact support on Ω and -flat on E up to order p−1,p≥2 (respectively, in a Gevrey class of order s>1, with compact support on Ω and -flat on E up to infinite order) can be approximated on compacts subsets of E by holomorphic functions fk on Ω with degree of approximation equal k−p/2 (respectively, exp (−c(s)k1/2(s−1)) ).

Morozov’s discrepancy principle is one of the simplest and most widely used parameter choice strategies in the context of regularization of ill-posed operator equations. Although many authors have considered this principle under general source conditions for linear ill-posed problems, such study for nonlinear problems is restricted to only a few papers. The aim of this paper is to apply Morozov’s discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems under general source conditions.

A group G satisfies the second Engel condition [X,Y,Y ]=1 if and only if x commutes with xy, for all x,y∈G. This paper considers the generalization of this condition to groups G such that, for fixed positive integers r and s, xr commutes with (xs)y for all x,y∈G. Various general bounds are proved for the structure of groups in the corresponding variety, defined by the law [Xr,(Xs)Y]=1.