The quantum double construction is applied to the group algebra of a finite group. Such algebras are shown to be semi-simple and a complete theory of characters is developed. The irreducible matrix representations are classified and applied to the explicit construction of R-matrices: this affords solutions to the Yang-Baxter equation associated with certain induced representations of a finite group. These results are applied in the second paper of the series to construct unitary representations of the Braid group and corresponding link polynomials.

If a class Z of morphisms in a monoidal category A is closed under tensoring with the objects of A then the category obtained by inverting the morphisms in Z is monoidal. We note the immediate properties of this induced structure. The main application describes monoidal completions in terms of the ordinary category completions introduced by Applegate and Tierney. This application in turn suggests a “change-of-universe” procedure for category theory based on a given monoidal closed category. Several features of this procedure are discussed.

In this paper we study nonuniform dichotomy concepts of linear evolutionary processes which are defined in a general Banach space and whose norms can increase no faster than an exponential. Connections between the dichotomy concepts and (B, D) admissibility properties are established. These connections have been partially accomplished in an earlier paper by the authors for the case when the process was a semigroup of class C0 and (B, D) = [(Lp, Lq).

The properties of the degree of a matrix of rational functions are obtained in a simplified way, which enables them to be generalised to matrices whose elements are not necessarily rational functions. On the basis of these results a theory of realisations is developed, which similarly generalises the theory of state space realisations of a matrix of rational functions.

In this paper, we give some characterisations of homomorphisms on von Neumann algebras by linear preservers. We prove that a bounded linear surjective map from a von Neumann algebra onto another is zero-product preserving if and only if it is a homomorphism multiplied by an invertible element in the centre of the image algebra. By introducing the notion of tr-rank of the elements in finite von Neumann algebras, we show that a unital linear map from a linear subspace ℳ of a finite von Neumann algebra ℛ into ℛ can be extended to an algebraic homomorphism from the subalgebra generated by ℳ into ℛ; and a unital self-adjoint linear map from a finite von Neumann algebra onto itself is completely tr-rank preserving if and only if it is a spatial *-automorphism.

Let S and X be any two sets; then a mapping Γ which assigns to each point t in S a set Γ(t) of points in X is called a multifunction from S into X. A selector for Γ is a function f from S into X such that f(t) ∈ Γ(t) for each t. We introduce here a class of multifunctions which is both well-supplied with measurable selectors and yet is comprehensive enough to include those kinds of multifunction which have been most commonly studied before. Hence in order to show that a multifunction with non-empty values, which may arise naturally in an implicit function problem, has a measurable selector, it is sufficient to show that it is of Souslin type.

Various n-th derivative characterisations involving different kinds of oscillations of F(p,q,s) functions are established, and the mean growth of derivatives of F(p,q,s) functions is considered. Moreover, inclusion relations between certain analytic function spaces are discussed.

Let X be a real uniformly convex Banach space satisfying the Opial's condition, C a bounded closed convex subset of X, and T: C → C an asymptotically non-expansive mapping. Then we show that for each x in C, the sequence {Tnx} almost converges weakly to a fixed point y of T, that is,

This implies that {Tnx} converges weakly to y if and only if T is weakly asymptotically regular at x, that is, weak- . We also present a weak convergence theorem for asymptotically nonexpansive semigroups.

We prove under quite general assumptions the existence of a positive solution to the equation Δu + f(x, u) + g(x)x.∇u = 0 in exterior domains of Rn (n ≥ 3).

This paper studies the unique range set of meromorphic functions and shows that the set S = {w | w13 + w11 + 1 = 0} is unique range set of meromorphic functions with 13 elements.

Let P*(α, β) denote the class of functions

analytic and univalent in |z| < 1 for which

where α є [0, 1), β є (0, 1].

Sharp results concerning coefficients, distortion theorem and radius of convexity for the class P*(α, β) are determined. A comparable theorem for the classes C*(α, β) and P*(α, β) is also obtained. Furthermore, it is shown that the class P*(α, ß) is closed under ‘arithmetic mean’ and ‘convex linear combinations’.

The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

The one-sided Laplace transform is defined on a space of generalised functions called transformable Boehmians. The space of one-sided Laplace transformable distributions is shown to be a proper subspace of transformable Boehmians. Some basic properties of the Laplace transform are investigated. An inversion formula and an Abelian theorem of the final type are obtained.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\langle X,d \rangle $ be a metric space. We characterise the family of subsets of $X$ on which each locally Lipschitz function defined on $X$ is bounded, as well as the family of subsets on which each member of two different subfamilies consisting of uniformly locally Lipschitz functions is bounded. It suffices in each case to consider real-valued functions.

We consider in this paper the implicit complementarity problem imposed by quasi-variational inequalities and stochastic optimal control. The principal result is an existence theorem for the implicit complementarity problem in Hilbert spaces.

For non-negative integers n we determine the roots of the trinmial , with a ≠ 0, over a finite field of characteristic p.

In the paper by Aderogba and Blake [1], the first order approximation to the shape of the interface between two immiscible liquids, at which surface tension acted, was obtained in the case of a point force directed parallel to the interface. In the case of the normal force no solution was obtained due to the logarithmic singularity associated with problems of this type. However, the problem is not physically well-posed in the case of the normal force. Surface tension is not suitable because it cannot alone balance the induced stress on the interface due to a point force. We need an additional force to balance the action of the point force on the interface. The obvious solution is to include a density difference Δρ* between the two fluids

where and are the densities of the lower and upper fluid respectively.

In this paper we study the variational-like inequalities, which generalise some results of Parida, Sahoo and Kumar, and we also investigate the quasivariational-like inequalities. We establish some existence theorems of a solution for the above problem.

The two generator restricted Burnside group of exponent five is shown to have order 534 and class 12 by two independent methods. A consistent commutator power presentation for the group is given.

Let G be a finite group and let H≤G. We refer to |H||CG(H)| as the Chermak–Delgado measure ofH with respect to G. Originally described by Chermak and Delgado, the collection of all subgroups of G with maximal Chermak–Delgado measure, denoted 𝒞𝒟(G), is a sublattice of the lattice of all subgroups of G. In this paper we note that if H∈𝒞𝒟(G) then H is subnormal in G and prove that if K is a second finite group then 𝒞𝒟(G×K)=𝒞𝒟(G)×𝒞𝒟(K) . We additionally describe the 𝒞𝒟(G≀Cp) where G has a nontrivial centre and p is an odd prime and determine conditions for a wreath product to be a member of its own Chermak–Delgado lattice. We also examine the behaviour of centrally large subgroups, a subset of the Chermak–Delgado lattice.