We consider expansions with respect to the multi-dimensional Hermite functions and to the multi-dimensional Hermite polynomials. They are respectively eigenfunctions of the Harmonic oscillator $\cal{L}= -\Delta +|x|^2$ and of the Ornstein-Uhlenbeck operator ${\bf L} = -\Delta +2x \cdot \nabla.$ The corresponding heat semigroups and Riesz transforms are considered and results on both aspects (polynomials and functions) are obtained.

In the study of the structure of regular semigroups, it is customary to impose several conditions restricting the behaviour of ideals, idempotents or elements. In a few instances, one may represent them as subdirect products of some much more restricted types of regular semigroups, e.g., completely (0-) simple semigroups, bands, semilattices, etc. In particular, studying the structure of completely regular semigroups, one quickly distinguishes certain special cases of interest when these semigroups are represented as semilattices of completely simple semigroups. In fact, this semilattice of semigroups may be built in a particular way, idempotents may form a subsemigroup, ℋ may be a congruence, and so on.

The concept of weak relative-injectivity of modules was introduced originally in [10], where it is shown that a semiperfect ring R is such that every cyclic right module is embeddable essentially in a projective right R-module if and only if R is right artinian and every indecomposable projective right R-module is uniform and weakly R-injective. We show that in the above characterization the requirement that indecomposable projective right R-modules be uniform is superfluous (Theorem 1.11). In this paper we further the study of weak relative-injectivity by considering the class of rings for which every right module is weakly injective relative to every finitely generated right module. We refer to such rings as right weakly-semisimple rings. The class of right weakly-semisimple rings includes properly all semisimple rings and is a subclass of the class of right QI-rings. A ring R is said to be a right QI-ring if every quasi-injective right R-module is injective. QI-rings have been studied in [2], [3], [4], [6], [7], [8], [11], among others.

The concept of (V*) set was introduced, as a dual companion of that of (V)-set, by Pelczynski in his important paper [14]. In the same paper, the so called properties (V) and (V*) are defined by the coincidence of the (V) or (V*) sets with the weakly relatively compact sets. Many important Banach space properties are (or can be) defined in the same way; that is, by the coincidence of two classes of bounded sets. In this paper, we are concerned with the study of the class of (V*) sets in a Banach space, and its relationship with other related classes. To this general study is devoted Section I. A (as far as we know) new Banach space property (we called it property weak (V*)) is defined, by imposing the coincidence of (V*) sets and weakly conditionally compact sets. In this way, property (V*) is decomposed into the conjunction of the weak (V*) property and the weak sequential completeness. In Section II, we specialize to the study of (V*) sets in Banach lattices. The main result in the section is that every order continuous Banach lattice has property weak (V*), which extends previous results of E. and P. Saab ([16]). Finally, Section III is devoted to the study of (V*) sets in spaces of Bochner integrable functions. We characterize a broad class of (V*) sets in L1(μ, E), obtaining similar results to those of Andrews [1], Bourgain [6] and Diestel [7] for other classes of subsets. Applications to the study of properties (V*) and weak (V*) are obtained. Extension of these results to vector valued Orlicz function spaces are also given.

In this paper, we show that a commutative Noetherian ring which satisfies the radical formula must be of dimension at most one. From this we give a characterization of commutative Noetherian rings that satisfy the radical formula.

A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we write

The unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if

¶ y- z ¶≥ 2a

The statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.

Let Γ denote the modular group, consisting of the Möbius transformations

As usual we denote the above transformation by the matrix remembering that V and – V represent the same transformation. If N is a positive integer we let Γ0(N) denote the transformations for which c ≡ 0 mod N. Then Γ0(N) is a subgroup of index

the product being taken over all prime divisors of N.

Let πc(x) be the number of the integers n≤x such that [nc] is prime. We shall prove that $$π_{c}\tfrm{(}x\tfrm{)&2_Gt;}{x\over \log x}$$ for 1<c<45/38. This improves the former range 1<c<13/11.

Let A and B be regular semisimple commutative Banach algebras; that is to say, regular Banach function algebras. A linear map T denned from A into B is said to be separating or disjointness preserving if f.g = 0 implies Tf.Tg = 0, for all f, g ∈ A In this paper we prove that if A satisfies Ditkin's condition then a separating bijection is automatically continuous and its inverse is separating. If also B satisfies Ditkin's condition, then it induces a homeomorphism between the structure spaces of A and B.

We study the rate of growth of some integral means of the derivatives of a Blaschke product and we generalize several classical results. Moreover, we obtain the rate of growth of integral means of the derivative of functions in the model subspace KB generated by the Blaschke product B.

The problem of Josephus is the following. We are given two positive integers n, q. There are n places arranged around a circle, and numbered clockwise 1, 2, …, n. Each of n people takes one of the places, then (please excuse this, but we didn't invent the problem!) every qth one is executed, until just one remains. More precisely, the occupant of place q is ‘removed’ first, and in general, if some place j has just been vacated, then the qth one of the places clockwise around from j that are still occupied will be vacated next. One question is this: if you would like to be the last survivor, then into what place should you go initially? We denote the answer to this question by Jq(n). For example, if n = 5 and q = 2, the order of execution is 2, 4, 1, 5, 3, and J2(5) = 3. Other questions have been raised about the problem, and it has an extensive literature (see [1]–[10]). In this paper we deal with the Jq(n)'s.

Let S be a regular semigroup. Given x ∈ S, we shall say that a ∈ S is an associate of x if xax = x. The set of associates of x ∈ S will be denoted by A(x). Now suppose that S has an identity element 1. Let H1 denote the group of units of S. Then we say that u ∈ S is a unit associate of x whenever u ∈ A(x)∩Hl. In what follows we shall write U(x) = A(x)∩=H1, and we shall say that S is unit regular [1, 3] if (∀ x∈S)U(x)≠ ∅. Examples of unit regular semigroups include the full transformation semigroup on a finite set [1] and the semigroup of endomorphisms of a finite–dimensional vector space [3]. In this paper we shall be concerned with semigroups that are unit orthodox (i.e. unit regular and orthodox), and we shall describe completely the structure of those semigroups that are uniquely unit orthodox (i.e. orthodox and uniquely unit regular in the sense that, for every x∈S, the set U(x) is a singleton). It is worthy of mention that neither of the examples cited above is of this type.

Introductory Remarks. Recently a number of studies (Chen & Saffman [2], Jones & Toland [7,11], Hogan [5]) have been made of periodic capillary-gravity waves which form the free surface of an ideal fluid contained in a channel of infinite depth. However, little work appears to have been done on the corresponding problem when the depth is finite. The most significant contributions appear to be those of Reeder & Shinbrot [9], Barakat & Houston [1] and Nayfeh [8] all of whom confined themselves to Wilton ripples (see §1.3). Yet there are sound reasons why such a study should be made. For quite apart from the unsolved problem regarding the type of capillary-gravity waves which may occur at finite depths, the consideration of the finite depth problem may be regarded as a first step in the study of solitary capillary-gravity waves. In this paper, a new integral equation for the infinite depth problem, due to J. F. Toland and the author, is adapted to be of use in tackling the finite depth problem. Using this we obtain results for the exact equations of motion which answer rigorously the questions of existence and multiplicity of small amplitude solutions of the periodic capillary-gravity wave problem of finite depth.

The convergence of least squares approximations for dual orthogonal series in Hilbert space is established, thus providing a theorem applicable to practically all dual orthogonal series (such as dual trigonometric series, dual Bessel series, etc.) that have appeared in the literature. Our results establish for such dual series the existence of a sequence of functions satisfying in the L2norm the dual series relation, with an error tending to zero and, in particular, rigorously justify the calculations in [2] which showed least squares to be a practical approximation procedure for dual trigonometric equations. In fact, the desire to provide a rigorous convergence theorem for [2] motivated this study.

The purpose of this note is to characterize those Banach lattices (E, ∥·∥) which have the property:

an operator T: E → c0 is a Dunford-Pettis operator if and only if T is regular (*)

(i.e., T is the difference of two positive operators). Our characterization generalizes a theorem recently proved by Holub [6] and Gretsky and Ostroy [4], who have remarked that the space L1[0, 1] has the property (*). The main result presented here is the following theorem.

Face 2-colourable triangulations of complete tripartite graphs $K_{n,n,n}$ correspond to biembeddings of Latin squares. Up to isomorphism, we give all such embeddings for $n=3,4,5$ and 6, and we summarize the corresponding results for $n=7$. Closely related to these are Hamiltonian decompositions of complete bipartite directed graphs $K^*_{n,n}$, and we also give computational results for these in the cases $n=3,4,5$ and 6.

In the present paper we prove that every 2-local derivation on a semi-finite von Neumann algebra is a derivation.

Let R be a ring (with identity). We shall call R a local ring if R is aright noetherian ring such that the Jacobson radical M is a maximal ideal (and so is the only maximal ideal), and R/M is a simple artinian ring. A local ring R with maximal ideal M is called regular if there exists a chain

of ideals Mi of such that Mi–1/Mi is generated by a central regular element of R/Mi (1 ≦ i ≦ n). For such a ring R, Walker [6, Theorem 2. 7] proved that R is prime and n is the right global dimension of R, the Krull dimension of R, the homological dimension of theR-module R/M and the supremum of the lengths of chains of prime ideals of R. Such regular local rings will be called n-dimensional. The aim of this note is to give examples of regular local rings. These arise as localizations of universal enveloping algebras of nilpotent Lie algebras over fields and localizations of group algebras of certain finitely generated finite-by-nilpotent groups.

Let ℋ be a complex Hilbert space and B(ℋ) be the algebra of all bounded linear opeators on ℋ. An operator T ∈ B(ℋ) is said to be p-hyponormal if (T*T)p–(TT*)p. If p = 1, T is hyponormal and if p = ½ is semi-hyponormal. It is well known that a p-hyponormal operator is p-hyponormal for q≤p. Hyponormal operators have been studied by many authors. The semi-hyponormal operator was first introduced by D. Xia in [7]. The p-hyponormal operators have been studied by A. Aluthge in [1]. Let T be a p-hyponormal operator and T=U|T| be a polar decomposition of T. If U is unitary, Aluthge in [1] proved the following properties.

All representations and characters studied in this paper are taken over the complex numbers, and all groups considered are finite. For basic definitions concerning projective representations see [1].

If G is a group and or is a cocycle of G we denote by Proj(G, α) = {ξ1, …, ξt} the set of irreducible projective characters of G with cocycle α, where of course t is the number of α-regular conjugacy classes of G; ξ1, (1) is called the degree of ξ1. Also as normal, M(G) will denote the Schur multiplier of G, [α] the cohomology classof α, and [1] the cohomology class of the trivial cocycle of G.