Let P be an IP-set of integers namely for a certain sequence The main questions studied here are : (1) Under what conditions on (an ) is Pα dense modulo 1 for every irrational α? (2) Under what conditions on (an ) is Pα (considered as a sequence ordered in a way to be subsequently defined) uniformly distributed modulo 1 for every irrational α?

Let B be a Boolean algebra and G a group of automorphisms of B. Define an equivalence relation ∼ on B by letting x ∼ y if there are x 1, x 2,…,xn , y 1, y 2, …y n in B such that x is the disjoint union of the x i , y is the disjoint union of the y i , and for each i there is a member of G taking x i to y i . The equivalence classes under ∼ are called equidecomposability types. Addition of equidecomposability types is given by (x) + (y) = (x V y) provided x ∧ y = 0. An example is given of a complete Boolean algebra B and a group G of automorphisms of B with X, Y ∊ B such that (X) + (X) = (Y) + (Y) but (X) ≠ (Y), answering a question of Wagon (see [5 p. 231 problem 14]). Moreover B may be taken to be the algebra of Borel subsets of Cantor space modulo sets of the first category. It is also remarked that in this case equidecomposability types do not form a weak cardinal algebra.

The Ramanujan-Bailey identity that establishes a symmetric relation between two hypergeometric series 3 F 2 is extended to a symmetric relation between two hypergeometric series 4 F 3. We also give a symmetric model that makes this symmetry evident.

Let H(D) be the linear space of analytic functions on a domain D of ℂ endowed with the topology of locally uniform convergence and let H‘(D) be the topological dual space of H(D). For domains D which are symmetric with respect to the real axis we use the notation Furthermore, denote by S the set of all univalent mappings f defined on the unit disk Δ which are normalized by f (0) = 0 and f‘(0) =1.

Let K be a number field and let OK denote the integers of K. The locally free class groups, Cl(OK [G]), furnish a fundamental collection of invariants of a finite group, G. In this paper I will construct some new, non-trivial homomorphisms, called restricted determinants, which map the NGH-invariant idèlic units of Ok([Hab] to Cl(OK [G]). These homomorphisms are constructed by means of the Horn-description of Cl(OK [G]), which describes the locally free class group in terms of the representation theory of G, and the technique of Explicit Brauer Induction, which was introduced in [5].

Two families of associated Wilson polynomials are introduced. Both families are birth and death process polynomials, satisfying the same recurrence relation but having different initial conditions. Contiguous relations for generalized hypergeometric functions of the type 7 F 6 are derived and used to find explicit representations for the polynomials and to compute the corresponding continued fractions. The absolutely continuous components of the orthogonality measures of both families are computed. Generating functions are also given.

The subject of this paper is the use of the theory of Schwartz distributions and approximate identities in studying the functional equation

The aj ’s and b are complex-valued functions defined on a neighbourhood, U, of 0 in R m , hj . U → R n with hj (0) = 0 and fj , g: R n → C for 1 ≦ j ≦ N. In most of what follows the aj 's and hj 's are assumed smooth and may be thought of as given. The fj ‘s, b and g may be thought of as the unknowns. Typically we are concerned with locally integrable functions f 1, … , fN such that, for each s in U, (1) holds for a.e. (almost every) x ∈ R n , in the sense of Lebesgue measure.

An analytic isomorphism of C*-algebras is a C*-isomorphism which maps one distinguished subalgebra, the analytic subalgebra, onto another. A strict partial order of a topological group acting on a topological space determines the analytic subalgebra of the transformation group C*-algebra as a certain non-self-adjoint subalgebra of the C*-algebra. When the group action is free and locally parallel, this analytic subalgebra is locally a subfield of compact operators contained in a reflexive algebra whose subspace lattice is determined by the group order. If in addition the group has the dominated convergence property, an analytic isomorphism of such transformation group C*-algebras induces a homeomorphism of the transformation spaces which maps orbits to orbits. In particular, the C*-algebras for two regular foliations of the plane are analytically isomorphic only if the foliations are topologically conjugate. In the case of parallel actions, a quotient of the group of analytic automorphisms is isomorphic to the second Čech cohomology of a transversal for the action.

The importance of finite limits in completeness conditions has been long recognized. One has only to consider elementary toposes, pretoposes, exact categories, etc., to realize their ubiquity. However, often pullbacks suffice and in a sense are more natural. For example it is pullbacks that are the essential ingredient in composition of spans, partial morphisms and relations. In fact the original definition of elementary topos was based on the notion of partial morphism classifier which involved only pullbacks (see [6]). Many constructions in topos theory, involving left exact functors, such as coalgebras on a cotriple and the gluing construction, also work for pullback preserving functors. And pullback preserving functors occur naturally in the subject, e.g. constant functors and the Σα. These observations led Rosebrugh and Wood to introduce partial geometric morphisms; functors with a pullback preserving left adjoint [9]. Other reasons led Kennison independently to introduce the same concept under the name semi-geometric functors [5].

Recently, M. Hofmeister [4] counted all nonisomorphic double coverings of a graph by using its Ζ 2 cohomology groups, and J. Kwak and J. Lee [5] did the same work for some finite-fold coverings. In this paper, we give an algebraic characterization of isomorphic graph bundles, from which we get a formula to count all nonisomorphic graph-bundles. Some applications to wheels are also discussed.