In this paper, two gradient properties of explicit convex functions are given.

Pommerenke initiated the study of linearly invariant families of locally schlicht holomorphic functions defined on the unit disk The concept of linear invariance has proved fruitful in geometric function theory. One aspect of Pommerenke's work is the extension of certain results from classical univalent function theory to linearly invariant functions. We propose a definition of a related concept that we call hyperbolic linear invariance for locally schlicht holomorphic functions that map the unit disk into itself. We obtain results for hyperbolic linearly invariant functions which generalize parts of the theory of bounded univalent functions. There are many similarities between linearly invariant functions and hyperbolic linearly invariant functions, but some new phenomena also arise in the study of hyperbolic linearly invariant functions.

D. Przeworska-Rolewicz developed an algebra-based theory around linear, not necessarily continuous, operators D: X → X which admit a right inverse, the elementary example being D = d/dt or, more generally, where ai are constants. We give conditions for the right invertibility of D in the case where ai are functions, or more generally, where D is the Lie or convariant derivative associated with a vector field on a (Banach) manifold M.

We give a characterization of complex extreme measurable selections for a suitable set-valued map. We use this result to obtain necessary and sufficient conditions for a function to be a complex extreme point of the closed unit ball of Lp (ω, Σ, ν X), where (ω, σ, ν) is any positive, complete measure space, X is a separable complex Banach space, and 0 < p < ∞.

If R is a 2-group of symplectic type with exponent 4, then R is isomorphic to the extraspecial group , or to the central product 4 o 21+2n of a cyclic group of order 4 and an extraspecial group, with central subgroups of order 2 amalgamated. This paper gives an explicit description of a projective representation of the group A of automorphisms of R centralizing Z(R), obtained from a faithful representation of R of degree 2n. The 2-cocycle associated with this projective representation takes values which are powers of −1 if R is isomorphic to and powers of otherwise. This explicit description of a projective representation is useful for computing character values or computing with central extensions of A. Such central extensions arise naturally in Aschbacher's classification of the subgroups of classical groups.

Let υ∈ be a sequence of DiPema-Majda approximate solutions to the 2-d incompressible Euler equations. We prove that if the vorticity sequence is weakly compact in the Hardy space H1 (R2) then a subsequence of υ∈ converges strongly in the energy norm to a solution of the Euler equations.

An existence variety of regular semigroups is a class of regular semigroups which is closed under the operations of forming all homomorphic images, all regular subsemigroups and all direct products. In this paper we generalize results on varieties of inverse semigroups to existence varieties of orthodox semigroups.

We give a revised and updated exposition of the theory of full dualities initiated by Clark, Davey, Krauss and Werner, introducing the (stronger) notion of a strong duality. All known full dualities turn out to be strong. A series of theorems which provide necessary and sufficient conditions for a strong duality to exist is proved. All full dualities in the literature can be obtained from these results and many new strong dualities can be derived. In particular, we show that within congruence distributive varieties every duality can be upgraded to a strong duality. Amongst the new strong dualities are the dualities of Davey, Priestley and Werner for the varieties of pseudocomplemented distributive lattices which are either strong as they stand or can easily be made strong by the addition of partial operations to the dual structures.

In this final contribution to the investigation of commutator laws in groups, we answer some of the questions left open in the previous two papers. The principal result is the independence of the Jacobi-Witt-Hall type laws from the so-called standard set of laws. The main results of the earlier papers are summarised. An interlude corrects some of the numerous printing errors in the second of our papers.

In this paper, we obtain some trace inequalities for arbitrary finite positive definite operators. Finally an open question is presented.

The main result of this paper is an upper bound for the number of maximal subgroups in finite solvable groups. Our result improves an earlier one of Cook, Wiegold and Williamson [1]. At the end, we use our bound to deduce an estimation for the total number of subgroups in finite solvable groups.

Throughout Section 2 of [1], X should be taken as a completely regular Hausdorff KR-space. The lines 1 through 26 in the proof of Theorem 2.1 of the paper should be replaced by the following lines.