On compact oriented differentiable manifolds, we define a well behaved Riemann type integral which coincides with the Lebesgue integral on nonnegative functions, and such that the exterior derivative of a differentiable (not necessarily continuously) exterior form is always integrable and the Stokes formula holds.

The concept of a permutation polynomial function over a commutative ring with 1 can be generalized to multiplace functions in two different ways, yielding the notion of a k-ary permutation polynomial function (k > 1, k ∈ N) and the notion of a strict k-ary permutation polynomial function respectively. It is shown that in the case of a residue class ring Zm of the integers these two notions coincide if and only if m is squarefree.

By simple geometrical considerations new proofs for some classical results are given and also new theorems about approximation by continued fractions are derived. This geometrical approach presents an instructive visualisation of the nature of proximation theorems.

The object of the present paper is first to derive an interesting unification (and generalization) of a fairly large number of finite summation formulas including, for example, those that appeared in this Journal recently. We then briefly remark on its various (known or new) special cases which are associated with certain classes of hypergeometric polynomials in one and two variables. We also give several further generalizations (involving multiple series with essentially arbitrary terms) which are shown to be applicable in the derivation of analogous summation formulas for hypergeometric series (and polynomials) in three and more variables. Finally, with a view to presenting relevance of these types of results in various seemingly diverse areas of applied sciences and engineering, some indication of applicability is provided.

In this paper it is proved that the principal series of representations of Γ = Z2*…*Z2 may be analytically continued to give uniformly bounded representations on the same Hilbert space, and that these representations are irreducible. Further, the reducibility of the restrictions to Γ ⊂ SL(2, Qp) of the irreducible unitary representations of SL(2, Qp) is examined.

We consider the distribution μ of numbers whose binary digits are generated from infinitely many tosses of a biased coin. It is shown that, if E has positive μ measure, then some n-fold sum of E with itself must contain an interval. This contrasts with the known result that all convolution powers of μ are singular.

It is proved that a finite soluble group of order n has at most (n − 1)/(q − 1) maximal subgroups, where q is the smallest prime divisor of n.

In this paper we study some properties of vector measures with values in various topological vector spaces. As a matter of fact, we give a necessary condition implying the Pettis integrability of a function f: S → E, where S is a set and E a locally convex space. Furthermore, we prove an iff condition under which (Q, E) has the Pettis property, for an algebra Q and a sequentially complete topological vector space E. An approximating theorem concerning vector measures taking values in a Fréchet space is also given.

We have defined and studied some pseudogroups of local diffeomorphisms which generalise the complex analytic pseudogroups. A 4-dimensional (or 8-dimensional) manifold modelled on these ‘Further pseudogroups’ turns out to be a quaternionic (respectively octonionic) manifold.

We characterise compact Further manifolds as being products of compact Riemann surfaces with appropriate dimensional spheres. It then transpires that a connected compact quaternionic (H) (respectively O) manifold X, minus a finite number of circles (its ‘real set’), is the orientation double covering of the product Y × P2, (respectively Y×P6), where Y is a connected surface equipped with a canonical conformal structure and Pn is n-dimensonal real projective space.

A corollary is that the only simply-connected compact manifolds which can allow H (respectively O) structure are S4 and S2 × S2 (respectively S8 and S2×S6).

Previous authors, for example Marchiafava and Salamon, have studied very closely-related classes of manifolds by differential geometric methods. Our techniques in this paper are function theoretic and topological.

We study the equation dY(t)/dt = f(Y(t), Eh(Y(t))) for random initial conditions, where E denotes the expected value. It turns out that in contrast to the deterministic case local Lipschitz continuity of f and h are not sufficient to ensure uniqueness of the solutions. Finally we also state some sufficient conditions for uniqueness.

Let L be an integer lattice, and S a set of lattice points in L. We say that S is optimal if it minimises the number of rectangular sublattices of L (including degenerate ones) which contain an even number of points in S. We show that the resolution of the Hadamard conjecture is equivalent to the determination of |S| for an optimal set S in a (4s-1) × (4s-1) integer lattice L. We then specialise to the case of 1 × n integer lattices, characterising and enumerating their optimal sets.

The Voronoi region and covering radius of the lattice E*6 are determined, and the normalised second moment is calculated, confirming the estimate given by Conway and Sloane.

We give sufficient conditions for an analytic function from Rn to R to be analytically equivalent to a rational regular function.