In a distributive lattice L with 0 the set of all ideals of the form (x]* can be made into a lattice A0(L) called the lattice of annulets of L. A 0(L) is a sublattice of the Boolean algebra of all annihilator ideals in L. While the lattice of annulets is no more than the dual of the so-called lattice of filets (carriers) as studied in the theory of l-groups and abstractly for distributive lattices in [1, section4] it is a useful notion in its own right. For example, from the basic theorem of [3] it follows that A 0(L) is a sublattice of the lattice of all ideals of L if and only if each prime ideal in L contains a unique minimal prime ideal.

Characterisations of finite groups in which normality is a transitive relation are presented in the paper. We also characterise the finite groups in which every subgroup is either permutable or coincides with its permutiser as the groups in which every subgroup is permutable.

For a locally compact group $G$ with left Haar measure and a Young function ${\rm\Phi}$, we define and study the weighted Orlicz algebra $L_{w}^{{\rm\Phi}}(G)$ with respect to convolution. We show that $L_{w}^{{\rm\Phi}}(G)$ admits no bounded approximate identity under certain conditions. We prove that a closed linear subspace $I$ of the algebra $L_{w}^{{\rm\Phi}}(G)$ is an ideal in $L_{w}^{{\rm\Phi}}(G)$ if and only if $I$ is left translation invariant. For an abelian $G$, we describe the spectrum (maximal ideal space) of the weighted Orlicz algebra and show that weighted Orlicz algebras are semisimple.

In 1985 John Reade determined the spectrum of C1 regarded as an operator on the space c0 of all null sequences normed by ║x║ = supn≧0|xn|. It is the purpose of this paper to determine the spectrum of C1 regarded as an operator on the space bv0 of all sequences x such that xk → 0 as k → ∞ and .

In thispaper we find many families in the product space ℍ2×ℝ of complete embedded, simply connected, minimal and surfaces with constant mean curvature H such that |H|≤1/2. We study complete surfaces invariant either by parabolic or by hyperbolic screw motions. We study the notion of isometric associate immersions. We exhibit an explicit formula for a Scherk-type minimal surface. We give a one-parameter family of entire vertical graphs of mean curvature 1/2. We prove a generalized Bour lemma that can be applied to ℍ2×ℝ,𝕊2×ℝ and to Heisenberg’s space to produce a family of screw motion surfaces isometric to a given one.

Using Fourier transforms, we give a new proof of certain identites for the fundamental solutions of the iterated Dirac operators and Ḏl = (Σ/Σx0 + Ḏ)l. Based on the close relationship between the fundamental solutions and the conformal weights we then give a simple proof of B. Bojarski's results on the conformal covariance of Ḏl. We also prove a new conformal covatiance result of D.

A basic theory for probabilistic convergence spaces based on filter convergence is introduced. As in Florescu's previous theory of probabilistic convergence structures based on nets, one is able to assign a probability that a given filter converges to a given point. Various concepts and theorems pertaining to convergence spaces are extended to the realm of probabilistic convergence spaces, and illustrated by means of examples based on convergence in probability and convergence almost everywhere. Diagonal axioms due to Kowalsky and Fischer are also studied, first for convergence spaces and then in the setting of probabilistic convergence spaces.

If C is a group which satisfies the maximal condition for normal subgroups, then C may be cancelled from a group A in direct products if and only if the infinite cyclic group can be cancelled from A. Finitely generated torsion free nilpotent groups of class 2 satisfy a Remak Krull Schmidt condition.

One elementary proof of the spectral theorem for bounded self-adjoint operators depends on an elementary construction for the square root of a bounded positive self-adjoint operator. The purpose of this paper is to give an elementary construction for the unbounded case and to deduce the spectral theorem for unbounded self-adjoint operators. In so far as all our results are more or less immediate consequences of the spectral theorem there is little is entirely new. On the other hand the elementary approach seems to the author to provide a deeper insight into the structure of the problem and also leads directly to the spectral theorem without appealing first to the bounded case. Besides this, our methods for proving uniqueness of the square root and of the spectral family seem to be new even in the bounded case. In particular there is no need to invoke representation theorems for linear functionals on spaces of continuous functions.

Some results on fixed point of asymptotically regular multivalued mapping are obtained in metric spaces. The structure of common fixed points and coincidence points of a pair of compatible multivalued mappings is also discussed. Our work generalizes known results of Aubin and Siegel, Dube, Dube and Singh, Hardy and Rogers, Hu, Iseki, Jungck, Kaneko, Nadler, Ray and Shiau, Tan and Wong.

The methods developed up to now in the theory of plates do not lend themselves easily to the solution of problems in the case of plates of arbitrary shape. Such methods lend themselves only to the solution of problems for plates of certain special simple shapes. Moreover, because of the complicated mathematical tools involved in these methods, the methods are not suitable for engineers. Hence there arises the need for the formulation of an approximate theory which may appear quite suitable for attacking problems on plates of arbitrary shape. In the present paper the author develops a new method which aims at the solution of the problem quite independently of the shape of the plate.

Let Xi, i = 1, 2, 3,··· be a sequence of independent and identically distributed random variables and write Sn = X1+X2+…+Xn. If the mean of Xi is finite and positive, we have Pr(Sn ≦ x) → 0 as n → ∞ for all x1 – ∞ < x < ∞ using the weak law of large numbers. It is our purpose in this paper to study the rate of convergence of Pr(Sn ≦ x) to zero. Necessary and sufficient conditions are established for the convergence of the two series where k is a non-negative integer, and where r > 0. These conditions are applied to some first passage problems for sums of random variables. The former is also used in correcting a queueing Theorem of Finch [4].

We present some questions that we feel are important and interesting for the theory of finite p-groups, and survey known results regarding these questions.

Mike, the other year you asked me how I saw the future of p-group theory. All I can offer in answer are more questions.

This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

Suppose we are given a “Thue equation” f(x, y) = 1, where f is a binary form with coefficients in a function field K of characteristic zero. A typical result is that if f is of degree at least 5 and has no multiple factors, then every solution x = (x, y) of the equation with components in K has H(x)≤90H(f) + 250g. Here g is the genus of K and H(x), H(f) are suitably defined heights. No assumption is made that x be “integral” in some sense. As an application, bounds are derived for “integral” solutions of hyperelliptic equations over K.

This paper proposes matrix variate generalization of Kummer-Beta family of distributions which has been studied recently by Ng and Kotz. This distribution is an extension of Beta distribution. Its characteristic function has been derived and it is shown that the distribution is orthogonally invariant. Some results on distribution of random quadratic forms have also been derived.

The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. The associated Toeplitz algebras have a universal property, and their representations are faithful if the generating isometries satisfy a joint properness condition. When applied to right-angled Artin groups this yields a uniqueness theorem for the C*-algebra generated by a collection of isometries such that any two of them either *-commute or else have orthogonal ranges. The analogous result fails to hold for the nonabelian Artin groups of finite type considered by Brieskorn and Saito, and Deligne.

A number of authors have studied busy period problems for particular cases of the general single-server queueing system. For example, using the now standard notation of Kendall [7], GI/M/1 was studied by Conolly [3] and Takacs [18]. Earlier work on M/M/1 includes that of Ledermann and Reuter [8] and Bailey [1]. Kendall [6], Takacs [17], and Prabhu [11] have considered M/G/1.

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of R into R(N)R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is quasi-injective. The following results are obtained: M is quasi-injective if and only if M is pseudo-injective and M2 is CS. Furthermore, if M is a direct sum of uniform modules, then M is quasi-injective if and only if M is pseudo-injective. As a consequence of this it is shown that over a right Noetherian ring R, quasi-injective modules are precisely pseudo-injective CS modules.

The essential cover Mk of a class M is defined as the class of all essential extensions of rings belonging to M. M is called essentially closed if Mk = M. Every class M has a unique essential closure, i.e. a smallest essentially closed class containing M.

Let M be a hereditary class of (semi)prime rings. Then M is proved to be a (weakly) special class if and only if M is essentially closed. A main result is that Mk is the smallest (weakly) special class containing M. Further it is shown that the upper radical UM determined by M, is hereditary if and only if UM has the intersection property with respect to Mk.