Let denote the q-shifted factorial, and set Recently, Gasper [7] showed that some bibasic summation formulas derived by Carlitz [5], Al-Salam and Verma [1], and Wm. Gosper could be extended to the indefinite bibasic summation formula where p and q are independent bases and a, b, c are arbitrary parameters.

This paper is concerned with a combinatorial construction which mysteriously “mimics” or “models” the decomposition of certain reducible representations of orthogonal groups. Although no knowledge of representation theory is needed to understand the body of this paper, a little familiarity is necessary to understand the representation theoretic motivation given in the introduction. Details of the proofs will most easily be understood by people who have had some exposure to Schensted's algorithm or jeu de tacquin.

The most usual diameters in the world are those of a sphere and they all contain its centre. More generally, a chord of a convex body in Rd is called a diameter if there are two parallel supporting hyperplanes at the two endpoints of the chord.

It is easily seen that there are points on at least two diameters. From a result of Kosiński [6] proved in a more general setting it follows that every convex body has a point lying on at least three diameters. Does a typical convex body behave like a sphere and contain a point on infinitely or even uncountably many diameters?

But what is a typical convex body? The space 𝒦 of all convex bodies (d-dimensional compact convex sets) in Rd, equipped with the Hausdorff metric, is a Baire space.

In 1980 Jim Lawrence suggested a construction Λ which assigns to a given rank r oriented matroid M on n points a rank n + r oriented matroid Λ(M) on 2n points such that the face lattice of Λ(M) is polytopal if and only if M is realizable. The Λ-construction generalized a technique used by Perles to construct a nonrational polytope [10]. It was used by Lawrence to prove that the class of polytopal lattices is strictly contained in the class of face lattices of oriented matroids (unpublished) and by Billera and Munson to show that the latter class is not closed under polarity. See [4] for a discussion of this construction and both of these applications.

The concept of a mitotic group was introduced in [3] by Baumslag, Dyer and Heller who showed that mitotic groups were acyclic. In [8] one of the authors introduced the concept of a pseudo-mitotic group, a concept weaker than that of a mitotic group, and showed that pseudo-mitotic groups were acyclic and that the group Gnof homeomorphisms of Rn with compact support is pseudo-mitotic. In our present paper we develop techniques to prove pseudomitoticity of certain other homeomorphism groups. In [5] Kan and Thurston observed that the group of set theoretic bijections of Q with bounded support is acyclic. A natural question is to decide whether the group of homeomorphisms of Q (resp. the irrationals I ) with bounded support is acyclic or not. In the present paper we develop techniques to answer this question in the affirmative.

The homotopy groups of any group-like space are equipped with a Samelson product satisfying, up to sign, the identities of a graded Lie bracket. We shall compute the Samelson product in two kinds of spaces of selfhomotopy equivalences arising when adding a homotopy or a homology group to a space.

First, let A→ X be a cofibration with a Moore space M(G,n) as cofibre. For the monoid autA (X) of maps under A homotopic (rel. A) to the identity, the Samelson product is a pairing

πn+i(G;X)⨂πn+j(G;X) → πn+i+j(G;X)

of homotopy groups with coefficients [1] in G. Theorem 2.1 computes this pairing in terms of a homomorphism associated to a α ∈ πi(autAX)).

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

Dans ce travail, nous construisons des solutions pour une certaine classe d'équations semi-linéaires complexes dans le plan. Plus précisément on considère prés d'un point XQde R2 l'équation

où Pest un opérateur différentiel d'ordre m(m≧ 1) à coefficients C∞ complexes, et o ù ƒest à valeurs complexes, analytique en et seulement C∞ en x.En supposant alors que Pest de type principal près de xoet vérifie la condition de Nirenberg-Trêves sous elliptique (que nous noterons ((P), voir [5]), nous construisons une solution locale de (*), de classe C∞ (Théorème 2.1).

Ce résultat échappe évidemment aux théorèmes classiques d'existence de Hamilton-Jacobi et de Cauchy-Kowalevsky.

Let be a sequence of independent Ornstein-Uhlenbeck processes with coefficients lk and ƛk,i.e., Xk(.)is a Gaussian process with EXk(t) =0 and

The process Y(.)was first studied by Dawson (1972) as the stationary solution of the infinite array of stochastic differential equations

where are independent Wiener processes (cf. also [6],[19],and [1]).

In this part, we shall be concerned with the structure of projections in a simple σ-unital C*-algebra with the FS property, and in the associated multiplier and corona algebras. We shall also consider the closed ideal structure of the corona algebra. Most of results appear to be new even for separable simple AF algebras, and are technically independent of the previous parts I and II ([37] and [38]). The whole work develops after finding a new property of a σ-unital (nonunital) simple C*-algebra with FS, which was not known even for a separable simple AF algebra. We relate this new property to the structure of the multiplier and corona algebras from vairous points of view.