In this paper we first obtain a basic formula for the conditional analytic Feynman integral of the first variation of a functional on Wiener space. We then apply this basic result to obtain several integration by parts formulas for conditional analytic Feynman integrals and conditional Fourier-Feynman transforms.

Banach and Fréchet algebras with a Laurent series generator are investigated leading, via the discrete Beurling algebras, to functional analytic characterisations of the holomorphic function algebras on the annulus as well as the C∞-algebra on the unit circle.

In this paper, we construct a new maximal independent system of units in cyclotomic function fields and their subfields. We also calculate its index in the full units group and show that it is smaller than the index of Feng-Yin's system.

We complete the determination of the groups of positive deficiency which occur as lattices in connected Lie groups. The torsion free groups among them are 3-mainfold groups. We show that any other torsion free 3-manifold group which is such a lattice is the group of an aspherical closed geometric 3-manifold.

We prove that for positive integers n and r satisfying 1 < r < n, with the single exception of n = 4 and r = 2, there exists a constant weight Gray code of r-sets of Xn = {1, 2, …, n} that admits an orthogonal labelling by distinct partitions, with each subsequent partition obtained from the previous one by an application of a permutation of the underlying set. Specifically, an r-set A and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. We prove that for all n and r as stated, and taken modulo , there exists a list of the distinct r-sets of Xn with |Ai ∩ Ai+1| = r − 1 and a list of distinct partitions such that πi is orthogonal to both Ai and Ai+1, and πi+1 = πiλi for a suitable permutation λi of Xn.

The notions of a Baire-1 and a weak Baire-1 multifunction are defined and a striking analogy between Baire-1 multifunctions and classical Baire-1 functions is established. A selection theorem is presented which asserts that if X is a metrisable space, Y a Polish space and F: X → 2Y/{∅} a closed-valued, weak Baire-1 multifunction, then F admits a Baire-1 selection. Using the machinery developed we prove that if X is a Banach space with separable dual, then every weak* usco, defined on a completely metrisable space Z, which values are weakly* compact subsets of the dual, is norm lower semicontinuous on a dense Gδ set.

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.

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 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 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.

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.

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.

Willis has established a structure theory of locally compact totally disconnected groups. An important feature of this theory is the notion of a tidy subgroup. In this paper we provide several results regarding these subgroups.

An infinite family of finite semigroups is studied. It is shown that most of them do not generate a quasivariety which admits a natural duality.

Let rk (n) denote the number of representations of n as the sum of k squares. We give elementary proofs of relations between rk (n) and rk (m) where n = 4λm and 4 ∤ m, when k = 5, 7, 9 and 11. The relations, which were first stated without proof by Stieltjes, are of the form rk (n) = Crk (m) where C depends on λ and on the residue of m modulo 8. They have recently been include by S. Cooper in a more complete description of the relations between rk (n) and rk (n′) where n′ is the squarefree part of n, when k = 5, 7, 9 and 11.

A Banach space X has the average distance property if there exists a unique real number r such that for each positive integer n and all x1,…,xn in the unit sphere of X there is some x in the unit sphere of X such that

We show that lp does not have the average distance property if p > 2. This completes the study of the average distance property for lp spaces.

A new class of bilinear permutation polynomials was recently identified. In this note we determine the class of permutation polynomials which represents the functional inverse of the bilinear class.