We give a characterisation for the extension of uniformly smooth norms from subspaces Y of superreflexive spaces X to uniformly smooth norms on all of X. This characterisation is applied to obtain results in various contexts.

B.H. Neumann recently proved some implications for semigroup laws in groups. This may help in the solution of a problem posed by G.M. Bergman in 1981.

In this article we investigate the total torsion of closed lines of curvature on a surface in 3 and obtain the following results.

(1) The total torsion of a closed line of curvature on a surface is kπ where k is an integer. Conversely, if the total torsion of a closed curve is kπ for an integer k, then the curve can appear as a line of curvature on a surface. In particular, if the total torsion of a closed curve is 2kπ, then it can appear as a line of curvature on a closed, oriented surface of genus 1.

(2) The total torsion of a closed line of curvature on an ovaloid is zero.

We study homomorphisms form the ring of Siegel modular forms of a given degree to the ring of elliptic modular forms for a congruence subgroup. These homomorphisms essentially arise from the restriction of Siegel modular forms to modular curves. These homomorphisms give rise to linear relations among the Fourier coefficients of a Siegel modular form. We use this technique to prove that dim .

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.

Dedicated to Professor Kôzô Yabuta on the occasion of his 60th birthday

J. Kinnunen proved that of P > 1, d ≤ 1 and f is a function in the Sobolev space W1,P(Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function ℳf belong to LP(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.

We investigate which numbers are expressible as differences of two conjugate algebraic integers. Our first main result shows that a cubic, whose minimal polynomial over the field of rational numbers has the form x3 + px + q, can be written in such a way if p is divisible by 9. We also prove that every root of an integer is a difference of two conjugate algebraic integers, and, more generally, so is every algebraic integer whose minimal polynomial is of the form f (xe) with an integer e ≥ 2.

In this paper we adopt the notion of orthogonality introduced by the author in a previous article. We establish a characterisation for orthogonality in the spaces , where S is a set of positive integers and C is the field of complex numbers. Generalisations of the usual characterisation of orthogonality in the Hilbert spaces , via inner products, are obtained.

For an even Dirichlet character ψ, we obtain a formula for L (1, ψ) in terms of a sum of Dirichlet L-Series evaluated at s = 2 and s = 3 and a rapidly convergent numerical series involving the central binomial coefficients. We then derive a class number formula for real quadratic number fields by taking L (s, ψ) to be the quadratic L-series associated with these fields.

In linear viscoelasticity, the fundamental model is the Boltzmann caual integral equation which defines how the stress σ(t) at time t depends on the earlier history of the shear rate via the relaxation modulus (kernel) G (t). Physical reality is achieved by requiring that the form of the relaxation modulus G (t) gives the Boltzmann equation fading memory, so that changes in the distant past have less effect now than the same changes in the more recent past. A popular choice, though others have previously been proposed and investigated, is the assumption that G (t) be a completely monotone function. This assumption has much deeper ramifications than have been identified, discussed or exploited in the rheological literature. The purpose of this paper is to review the key mathematical properties of completely monotone functions, and to illustrate how these properties impact on the theory and application of linear viscoelasticity and polymer dynamics. A more general representation of a completely monotone function, known in the mathematical literature, but not the rheological, is formulated and discussed. This representation is used to derive new rheological relationships. In particular, explicit inversion formulas are derived for the relationships that are obtained when the relaxation spectrum model and a mixing rule are linked through a common relaxation modulus.

We prove that there exists a soluble, saturated and s-closed formation ℱ of groups such that the class (Gℱ | G is a group, Gℱ is the ℱ-residual of G) is not closed under subdirect products. This result a negative answer to an open problems recently proposed by L.A. Shemetkov in 1998.

Using the Zermelo Principle, we establish a common fixed point theorem for two progressive mappings on a partially ordered set. This result yields the Browder–Göhde–Kirk fixed point theorem for nonexpansive mappings.

We discuss some results on CS modules, SI rings, and SC rings. Then we consider the question of when, over a right SC ring R, every right R-module is CS. In Theorem 3.1 we show that this is the case if and only if R is a right countably Σ-CS ring. In light of this, we give an example showing that a result proved by Chen (2000) is incorrect. Furthermore, Theorem 4.1 shows that the assumptions of Chen (2000) can be weakened considerably.

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

The main purpose of this paper is to generalise a supersolvability theorem of O. U. Kramer to a saturated formation containing the class of supersolvable groups. As applications, we generalise some results recently obtained by some scholars.

We prove the convergence of an iterative method to a fixed point of a nonexpansive mapping in a uniformly smooth Banach space. We are able to relax one of the control conditions of P.L. Lions (1977).

Under suitable assumptions we establish the formulas for calculating generalised gradients and generalised directional derivatives in the Clarke sense of the supremum and the infimum of an infinite family of Lipschitz functions. From these results we derive the results ensuring such a supremum or infimum are an invex function when all functions of the invex. Applying these results to a class of mathematical programs, we obtain necessary and sufficient conditions for optimality.

Loops are nonassociative algebras which can be investigated by using their multiplication groups and inner mapping groups If the inner mapping group of a loop is finite and Abelian, then the multiplication group is a solvable group. It is clear that not all finite Abelian groups can occur as inner mapping groups of loops. In this paper we show that certain finite Abelian groups with a special structure are not isomorphic to inner mapping groups of finite loops. We use our results and show how to construct solvable groups which are not isomorphic to multiplication groups of loops.

We describe the analytic continuation of the heat kernel on the Heisenberg group ℍn(ℝ). As a consequence, we show that the convolution kernel corresponding to the Schrödinger operater eisL is a smooth function on ℍn(ℝ) \ Ss, where Ss = {(0, 0, ±sk) ∈ ℍn(ℝ) : k = n, n + 2, n + 4,…}. At every point of Ss the convolution kernel of eisL has a singularity of Calderón–Zygmund type.

A set-valued mapping Φ : X → 2Y acting between topological spaces X and Y is said to be “lower demicontinuous” if the interior of the closure of the set Φ−1(V): = {x ∈ X : Φ(x) ∩ V ≠ ∅} is dense in the closure of Φ−1(V) for each open set V in Y. Čoban, Kenderov and Revalski (1994) showed that for every densely defined lower demicontinuous mapping Φ acting from a Baire space X into subsets of a monotonely Čech-complete space Y, there exist a dense and Gδ subset X1 ⊆ X and an usco mapping G: X1 → 2Y such that G (x) ⊆ Φ*(x), for every x ∈ X1, where the mapping Φ*: X → 2Y is the extension of Φ defined by, W is a neighbourhood of x}.

In this paper we present a proof of the above result with the notion of monotone Čcech-completeness replaced by the weaker notion of partition completeness. In addition, we observe that if the range space also lies is Stegall's class then we may assume that the mapping G is single-valued on X1.