Pandey and Chaudhary [13] recently developed the theory of Hilbert transform of Schwartz distribution space (DLp)',p > 1 in one dimension using Parseval's types of relations for one dimensional Hilbert transform [17] and noted that their theory coincides with the corresponding theory for the Hilbert transform developed by Schwartz [16] by using the technique of convolution in one dimension.

The corresponding theory for the Hilbert transform in n-dimension is considerably harder and will be successfully accomplished in this paper.

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to prove that the class of polytopal lattices is strictly contained in the class of face lattices of oriented matroids (unpublished) and by Billera and Munson to show that the latter class is not closed under polarity.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.

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 X be a space with two metrics d1 and d2. Let S :(X, d1) → (X, d2) be continuous. We say S has the generalized pseudoorbit shadowing property with respect to the metrics d1 and d2 if for every every δ-pseudo-orbit in d1 can be ∊-shadowed by a true orbit in d2, i.e., if {x0, x1,…} satisfies for all i ≧ 0, then for all i ≧ 0. The main result of this note shows that certain Markov operators P : L1→ L1 have the generalized shadowing property on weakly compact subsets of the space of probability density functions, where d1 is the metric of norm convergence and d2 is the metric of weak convergence. An important class of such operators are the Frobenius-Perron operators induced by certain expanding and nonexpanding maps on the interval. When there is exponential convergence of the iterates to the density, we can express δ in terms of ∊. We also show that, unlike the situation in the space X itself, the generalized shadowing property is valid for all parameters in families of maps and that there is stability of the shadowing property.

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.

Varadarajan [13] named a map f: A → X a cyclic map when there exists a map F: X × A → X such that for the folding map ∇X: X ∨ X → X. He defined the generalized Gottlieb set G(A, X) of the homotopy classes of the cyclic maps F: A → X and studied the fundamental properties of G(A, X) If A is a co-Hopf space, then the Varadarajan set G(A, X) has a group structure [13]. The group G(A,X) is a generalization of G(X) and Gn(X) of Gottlieb [2,3]. Some authors studied the properties of the Varadarajan set, its dual and related topics [4, 5, 6, 7,12,15,16,17].

Let E be a Banach Lattice. We will consider E to be weakly sequentially complete and to have a weak unit u. Thus we may represent E as a lattice of real valued functions defined on a measure space (χ, , μ). There is a set R ⊂ χ such that R supports a maximal invariant function Φ for a postive contraction T on E [5]. Let N = χ — R be the complement of R. Akcoglu and Sucheston showed that where E+ is the positive cone of E. If in addition a monotone condition (UMB) is satisfied, then the same authors showed [4] that converges in norm.

There is an extensive body of literature on the bicyclic biquadratic fields. These fields provide the simplest examples of abelian noncyclic extensions of Q. In sharp contrast, there is a dearth of literature on the bicyclic bicubic extensions of the rational numbers. These fields together with the abelian noncyclic octic extensions provide the next simplest abelian noncyclic extensions.

In this article, we shall study abelian bicyclic bicubic extensions of Q of degree 9. Hasse [4, v-ix] has stated as important objectives: the computation of an integral basis, the determination of class number and the calculation of fundamental units for abelian fields. In this article, we will solve the first problem completely, and show that the solution to the unit problem leads to a solution of the class number problem. Moreover, we shall give a method for determining the unit group up to a subgroup which has index 1 or 3 and so determine the class number up to a factor of 3.

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].

Let Fm be the free Lie algebra of rank m over a field K of characteristic 0 freely generated by the set ﹛x1,… ,xm﹜, m ≧ 2. Cohn [7] proved that the automorphism group Aut Fm of the K-algebra Fm is generated by the following automorphisms: (i) automorphisms which are induced by the action of the general linear group GLm (= GLm(K)) on the subspace of Fm spanned by ﹛x1, … ,xm﹜; (ii) automorphisms of the form x1 → x1 +f(x2,… ,xm),Xk → xk, k ≠ 1, where the polynomial f(x2,…,xm) does not depend on x1.

Recently MacCluer and Shapiro [6] have characterized the compact composition operators in Hardy and weighted Bergman spaces of the disc, and MacCluer [5] has made an extensive study of these opertors in the unit ball of Cn. Angular derivatives and Carleson measures have played an essential role in these studies. In this article we study composition operators in poly discs and characterize those operators which are bounded or compact in Hardy and weighted Bergman spaces. In addition to Carleson measure theorems resembling those of [5], [6], we give necessary and sufficient conditions satisfied by the maps inducing bounded or compact composition operators. We conclude by considering the role of angular derivatives on the compactness question explicitly.

In [6] we studied the polynomials which are generalizations of the classical (generalized) Laguerre polynomials These polynomials were shown to be orthogonal on the interval [0, ∞) with respect to the inner product where a > — 1,M ≧0 and N ≧0.

Let be a compact Riemann surface, be the complement of a nonvoid finite subset of and A() be the ring of finite sums of meromorphic functions in with finite divisor. In this paper it is proved that every nonzero f ∈ A() can be decomposed as a product αβ, where α is either a unit or a product of powers of irreducible elements of A(), uniquely determined by f up to multiplication by units, and β is a product of functions of the type eφ – 1, with φ holomorphic and nonconstant in . Furthermore, a similar result is obtained for a certain class of subrings of A().

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 7F6 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.

By now the literature concerning the representation of disjointly additive functionals and operators is quite extensive. A few entries on the subject are [6, 7, 8, 11, 20, 21]. In [7, 8, 17] further references can be found, in [7] the “prehistory” of the subject is also discussed.

To quote a typical result, we may take a 1967 theorem of Drewnowski and Orlicz ([6] Th. 3.2, [17] 12.4) which asserts that, under proper assumptions, an abstract modular (= disjointly countably additive functional) p on a “substantial“ subspace D of L° can be realized by the formula .

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)).

In this paper we will study the Fubini theorem for stochastic integrals with respect to semimartingales in Hilbert space.

Let (Ω, , P) he a probability space, (X, , μ) a measure space, H and G two Hilbert spaces, L(H, G) the space of bounded linear operators from H into G, Z an H-valued semimartingale relative to a given filtration, and φ: X × R+ × Ω → L(H, G) a function such that for each t ∈ R+ the iterated integrals are well-defined (the integrals with respect to μ are Bochner integrals). It is often necessary to have sufficient conditions for the process Y1 to be a version of the process Y2 (e.g. [1], proof of Theorem 2.11).

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 Rm, hj. U → Rn with hj(0) = 0 and fj, g: Rn → 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 f1, … , fN such that, for each s in U, (1) holds for a.e. (almost every) x ∈ Rn, in the sense of Lebesgue measure.

Let G be a group and n(≧ 2) an integer. We say that G belongs to the class of groups Pn if every product of n elements can be reordered, i.e. for all n-tuples , there exists a non-trivial element σ in the symmetric group Σn such that Let P denote the union of the classes Pn, n ≧ 2. Clearly every finite group belongs to P and each class Pn is closed with respect to forming subgroups and factor groups.

(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﹜.