We construct a suitable representation of a C*-algebra that carries single elements to rank one operators. We also prove an abstract spectral theorem for compact elements in the algebra. This leads naturally to an abstract definition of Cp-classes of compact elements in the algebra.

We consider the weighted $L_p$ solvability for divergence and nondivergence form parabolic equations with partially bounded mean oscillation (BMO) coefficients and certain positive potentials. As an application, global regularity in Morrey spaces for divergence form parabolic operators with partially BMO coefficients on a bounded domain is established.

Birkhoff and Pierce [2] introduced the concept of an ƒ-ring and showed that an l-ring is an f-ring if and only if it is a subdirect product of totallyordered rings. An l-ideal of an f-ring R is an algebraic ideal which is at the same time a lattice ideal of R. Structure spaces (i.e. sets of prime ideals endowed with the so-called hull-kernel or Stone topology) for ordinary rings have been studied by many authors. In this paper we consider certain analogues for ƒ-rings, and give characterisations of ƒ-rings for which these structure spaces are discrete.

The problem considered is that of approximating irrationals α by rationals p/q where p and q avoid certain congruence classes mod 2k for certain integers k. Results are obtained which give close bounds on a number c such that |α - p/q| < c/q2 has infinitely many solutions where p and q can be expressed as the sum of three squares.

The existence of positive solutions of some semilinear elliptic equations of the form −Δu = λf(u) is studied. The major results are a nonexistence theorem which gives a λ* = λ*(f,Ω) > 0 below which no positive solutions exist and a lower bound theorem for umax for Ω a ball. As a corollary of the nonexistence theorem that describes the dependence of the number of solutions on λ, two other nonexistence theorems, and an existence theorem are also proved.

The best lattice quantizers seem to be duals of extreme lattices. The quantizing constant associated with the dual lattice of Barnes's senary form φ6 is found, together with a new type of quantizing technique. The quantizing constant is better than expected in the sense that it is better than D*6 even though D6 provides a denser packing. This is the smallest dimension for which this occurs.

We consider spectral radius algebras associated with C0 contractions. When the operator A is algebraic, we describe all invariant subspaces that are common for operators in its spectral radius algebra ℬA. When the operator A is not algebraic, ℬA is weakly dense and we characterize a set of rank-one operators in ℬA that is weakly dense in ℒ(ℋ).

Two local nilpotent properties of an associative or alternative ring A containing an idempotent are shown. First, if A = A11 + A10 + A01 + A00 is the Peirce decomposition of A relative to e then if a is associative or semiprime alternative and 3-torsion free then any locally nilpotent ideal B of Aii generates a locally nilpotent ideal 〈B〉 of A. As a consequence L(Aii) = Aii ∩ L(A) for the Levitzki radical L. Also bounds are given for the index of nilpotency of any finitely generated subring of 〈B〉. Second, if A(x) denotes a homotope of A then L(A) ⊆ L(A(x)) and, in particular, if A(x) is an isotope of A then L(A) = L(A(x)).

It is shown that in a Grothendieck space with the Dunford-Pettis property, the class of well-bounded operators of type (B) coincides with the class of scalar-type spectral operators with real spectrum. It turns out that in such Banach spaces, analogues of the classical theorems of Hille-Sz. Nagy and Stone concerned with the integral representation of C0-semigroups of normal operators and strongly continuous unitary groups in Hilbert spaces, respectively, are of a very special nature.

If M2n is a cohomology CPn and P is an odd prime, let Gp be the cyclic group of order p. A Type I I0Gp action on M2n is an action with fixed point set a codimension-2 submanifold and an isolated point. A Type I I0 Gp action is standard if it is regular and the degree of the fixed codimension-2 submanifold is one. If n is odd and M2n admits a standard Gp action of Type I I0, then every Type I I0Gp action M2n is standard and so, if n is odd, CPn admits a Gp action of Type I I0 if and only if the action is standard.

In this paper we treat transcendental meromorphic solutions of some algebraic differential equations. We consider the number of distinct transcendental meromorphic solutions. Algebraic relations between meromorphic solutions and comparisons of the growth of transcendental meromorphic solutions are also discussed.

We relate a previous result of ours on families of Diophantine equations having only trivial solutions with a result on the approximation of an algebraic number by products of rational numbers and units. We compare this approximation with a Liouville type estimate, and with an estimate arising from a lower bound for a linear combination of logarithms.

Rates of convergence to normality of O(N-½) are obtained for a standardized sum of m random variables selected at random from a finite set of N random variables in two cases. In the first case, the sum is randomly normed and the variables are not restricted to being independent. The second case is an alternative proof of a result due to von Bahr, which deals with independent variables. Both results derive from a rate obtained by Höglund in the case of sampling from a finite population.

In a well-known paper, Hall and Higman proved the reduction theorem on a coprime order operator group acting on a finite group. This theorem plays an important role in local analysis of finite group theory. In this paper, we generalize the Hall-Higman reduction theorem by dropping the restrictive hypothesis (|G|, |H|) = 1 and determine the detailed structure of G completely.

In [2] Hauptfleisch proved that if A, B, H, K are Abelian groups, φ:A → H and ψ:B → K are epimorphisms, then every central group extension G of H by K is homomorphic image of a central loop extension L of A by B. The aim of the present note is to prove (using almost the same argument as in [2])

Regular maps of type {p, q}r and the associated groups Gp,q,r are considered for small values for p, q and r. In particular, it is shown that the groups G4,6,6 and G5,5,6 are Abelian-by-infinite, and there are infinitely many regular maps of each of the types {4, 6}6, {5, 5}6, {5, 6}6 and {6, 6}6.

Let ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

The purpose of this paper is to generalize some results in a recent paper by Tomiuk and the author ([11]).

In a recent paper by P. D. Finch and myself [1], the solution for the limiting distribution of a moving average queueing system was obtained. In this paper the system is generalised to the case of batch arrivals in batches of size ρ > 1.

Let S be a simplex in En which is homothetic to a given simplex S*, which contains no point of the integral lattice in its interior, and which has maximal volume V(S). We conjecture that V(S) > nn/n!, and establish the conjecture for n < 3.