In this paper it is shown that the construction of measures on standard spaces via Loeb measures and the standard part map does not depend on the full structure of the internal algebra being used. A characterization of universal Loeb measurability is given for completely regular Hausdorff spaces, and the behavior of this property under various topological operations is investigated.

We prove a homogenization formula for a sub-Laplacian are left invariant Hörmander vector fields) on a connected Lie group Gof polynomial growth. Then using a rescaling argument inspired from M. Avellanedaand F. H. Lin [2], we prove Harnack inequalities for the positive solutions of the equation (∂/∂t+ L)u= 0. Using these inequalities and further exploiting the algebraic structure of Gwe prove that the Riesz transforms , are bounded on Lq,1 < q <+∞ and from L1to weak-L1.

This paper gives another construction of (0, p)-forms on a complex analytic space and of the operator. This construction is independent of the one in [1] and apart from the general result of Section 1 of [1], it can be read independently. As in [1], the hypotheses on S are the following: S has normal singularities, its singular locus X is smooth, the exceptional divisor in a desingularization of S is irreducible.

For the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer polynomials, Hahn polynomials or continuous symmetric Hahn polynomials. We give a group theoretic explanation of this phenomenon and relate the properties of the polynomials p to the properties of the corresponding representation. The method is extended to the case of intertwining functions.

Infinite sequences of exchangeable binary random variables have strong positive dependence properties; in particular, we show they are strong FKG. If the infinite exchangeable sequence is allowed to have multiple values this is no longer true. Positive dependence conditions such as association still have natural application in this context. We establish necessary and sufficient conditions for an infinite exchangeable sequence to be associated. This result shows that exchangeable Polyà urn processes are associated. We also establish necessary and sufficient conditions for finite exchangeable sequences to be weakly associated. The match set distribution of a random permutation has recently been shown to be associated by an extensive analysis of cases. Our result easily yields the weak association of such distributions.

Let (E, T) be a Hausdorff topological vector space whose topological dual separates points of E, X be a non-empty weakly compact convex subset of E and W be the relative weak topology on X. If F: (X, W) → 2(E,T) is continuous (respectively, upper semi-continuous if £ is locally convex), approximation and fixed point theorems are obtained which generalize the corresponding results of Fan, Park, Reich and Sehgal-Singh-Smithson (respectively, Ha, Reich, Park, Browder and Fan) in several aspects.

Let A be a Banach algebra and let B be a linear subspace of A. Recall that A has the Dunford Pettis property if whenever ƒn→ 0 weakly in A* and φn → 0 weakly in A* then φn(ƒn) → 0. Bourgain showed that H∞ has the Dunford Pettis property using the theory of ultraproducts. The Dunford Pettis property is related to the notion of Bourgain algebra, denoted Bb, introduced by [6] Cima and Timoney. The algebra Bb is the set of ƒ in A such that if ƒn → 0 weakly in B then dist(ƒƒn, B) —> 0. Bourgain showed [2] that a closed subspace X of C(L)y where L is a compact Hausdorff space, has the Dunford Pettis property if Xb — C(L). Cima and Timoney proved that Bb is a closed subalgebra of A and that if B is an algebra then B⊂Bb. In this paper we study the Bourgain algebra associated with various algebras of functions on the unit circle T.

Analysis on sparse parts of the Banach algebra of bounded analytic functions is given. It is proved that Sarason's theorem for QC-level sets cannot be generalized to general Douglas algebras.

In this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in [10]–[31] and is related to work in [3]–[4].

An ordered structure is o-minimal if every definable subset is the union of finitely many points and open intervals. A theory is o-minimal if all its models are ominimal. All theories considered will be o-minimal. A theory is said to be n-ary if every formula is equivalent to a Boolean combination of formulas in n free variables. (A 2-ary theory is called binary.) We prove that if a theory is not binary then it is not rc-ary for any n. We also characterize the binary theories which have a Dedekind complete model and those whose underlying set order is dense. In [5], it is shown that if T is a binary theory, is a Dedekind complete model of T, and I is an interval in , then for all cardinals K there is a Dedekind complete elementary extension of , so that . In contrast, we show that if T is not binary and is a Dedekind complete model of T, then there is an interval I in so that if is a Dedekind complete elementary extension of .

Extreme points of positive functionals and spectral states on real commutative Banach algebras are investigated and characterized as multiplicative functionals extending the well-known results from complex to real Banach algebras. As an application a new and short proof of the existence of the Shilov boundary of a real commutative Banach algebra with nonempty maximal ideal space is given.

In this paper a q-analogue of Gegenbauer's addition formula for Bessel functions is obtained by using the orthogonality relation for the q-Ultraspherical polynomials of Rogers'. Also some product formulas and an integral representation for the Hahn-Exton q-Bessel functions are obtained.

We first prove a nonlinear ergodic theorem for nonexpansive semigroups without convexity in a Hilbert space. Further we prove a fixed point theorem for non-expansive semigroups without convexity which generalizes simultaneously fixed point theorems for left amenable semigroups and left reversible semigroups.

In [6] and [7], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [9], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to k-setcontraction maps, k < 1. Other versions for completely continuous maps were given by Fournier-Peitgen [2] and G. Fournier [1].

The purpose of this paper is to generalise some of these results to upper semi continuous multivalued maps which are K-set contractions, k < 1, and differentiable at the origin and infinity.