Let X be an infinite, locally finite, almost transitive graph with polynomial growth. We show that such a graph X is the inverse limit of an infinite sequence of finite graphs satisfying growth conditions which are closely related to growth properties of the infinite graph X.

1991 Mathematics Subject Classification. Primary 05C25, Secondary 20F8.

We give a discrete analogue of the harmonic morphism between two Riemannian manifolds. Roughly speaking, this is a mapping between two graphs preserving local harmonic functions. We characterize harmonic morphisms in terms of horizontal conformality. Many examples including coverings, non-complete extended p-sums and collapsings are given. Introducing the horizontal and vertical Laplacians, the Green kernel estimates are obtained for the harmonic morphism. As applications, a general and sharp estimate of the Green kernel for an infinite tree is obtained.

We investigate the function R(T,σ), which denotes the error term in the asymptotic formula for \int_0^T|\log\zeta(σ + it)|^2dt. It is shown thatR(T,σ) is uniformly bounded for σ \ge 1 and almost periodic in the sense of Bohr for fixed σ \ge 1; hence R(T,σ) = Ω(1) when T \to \infty. In case {1 \over 2}<σ<1 is fixed we can obtain the bound R(T,σ) \ll_ϵ T\,^{(9-2σ)/8+ϵ}.

We obtain two refinements of the so called local duality of ultrapowers, that is, the ultrapower version of the well-known principle of local reflexivity.

1991 Mathematics Subject Classification. 46B07, 46B08.

In the present paper, for a large family of topological semigroups namely, compactly cancellative and right cancellative foundation semigroups S, we study the topological centers of the Banach algebrasLUC(S) * and M_a(S)**. We also give a generalization of a known result of Lau and Lorsert by showing that for such topological semigroups the topological center of LUC(S)* (M_a(S)**, respectively) is the same as M(S) (M_a(S), respectively).

We show that if F is a free Lie algebra of rank at least 2 and if G is a non-trivial finite group of automorphisms of F then the fixed point subalgebra F^G is not finitely generated. Some similar results are proved for relatively free Lie algebras.

In this paper we study fundamental properties of spectra of log-hyponormal operators on a Hilbert space. In particular we show that log-hyponormal operators are normaloid and quasi-similar log-hyponormal operators have the same spectra and essential spectra.

Considering odd-dimensional complex projective space as a complex contact manifold, one may ask which of the Calabi (Veronese) imbeddings can be positioned by a holomorphic congruence as integral submanifolds of the complex contact structure. It is first shown that when the first normal space is the whole normal space, this is impossible. It is also shown to be impossibile for a Calabi surface (complex dimension 2) in complex projective space of dimension 9 where one has both a first and second normal space. However when the complex dimension of the submanifold is odd and the whole normal space consists of the first and second normal spaces, then there is a holomorphic congruence positioning the Calabi imbedding as an integral submanifold of the complex contact structure.

In this paper we deduce the existence of analytic structure in a neighbourhood of a maximal ideal M in the spectrum of a commutative Banach algebra, A, from homological assumptions. We assume properties of certain of the cohomology groups H^n(A,A/M), rather than the stronger conditions on the homological dimension of the maximal ideal the first author has considered in previous papers. The conclusion is correspondingly weaker: in the previous work one deduces the existence of a Gel'fand neighbourhood with analytic structure, here we deduce only the existence of a metric neighbourhood with analytic structure. The main method is to consider products of certain co-cycles to deduce facts about the symmetric second cohomology, which is known to be related to the deformation theory of algebras.

1991 Mathematics Subject Classification. 46J20, 46M20.

We investigate the boundedness of solutions of a second order nonlinear differential system.

We prove that exponentially harmonic morphisms are precisely the Riemannian submersions with minimal fibres.

1991 Mathematics Subject Classification 58E20.

In this note we investigate the diophantine equation<$$>x^y\pm y^x=\prod n_i!<$$>

where x and y are odd and greater than 1. We prove that this equation has no integer solutions.

In this paper we study quasilinear second order boundary value problems with multivalued right hand side and Dirichlet boundary conditions. We prove three existence theorems. The first two deal with the “convex” and “nonconvex” problems respectively, while the third establishes the existence of extremal solutions. For the first two the proof is based on the theory of nonlinear operators of monotone type, while the proof of the third uses a fixed point argument.

Let R be an associative ring with 1, and let I be a nilpotent two-sided ideal of R. Assume further that there exists z \in Z(R) such that z, z^2-1 \in R^*. Let m \in ℕ with m \geq 3. In this paper we describe all liftings of the elementary group{\tf="times-b"E}_m(R/I\,) to the general linear group{\tf="times-b"GL}_m(R) , i.e. all splittings of the natural projection{\tf="times-b"E}_m(R) + {\tf="times-b"M}_m(I\,) \rightarrow {\tf="times-b"E}_m(R/I\,) .

Let L^2_a (D, d\sigma d\theta /2\pi ) be a complete weighted Bergman space on the open unit disc D, where d\sigma is a positive finite Borel measure on [0, 1). We show the following : when \phi is a continuous function on the closed unit disc \bar {D}, T_\phi is compact if and only if \phi = 0 on \partial D.

1991 Mathematics Subject Classification 47B35, 47B07.

We say that the operator T on a Hilbert space H into itself is strongly stable if \left\Vert T^nx\right\Vert →0 as n→∞, for allx∈H . If T is a contraction, then T is said to be cs-stable if T has C_0 completely non-unitary part. This note considers the strong stability of operators A⊗B and the p-hyponormality of operators A⊗B. It is shown that the contraction A⊗B is cs-stable if and only if so are the contractionscA and c^{−1}B for some scalar c andA⊗B is p-hyponormal if and only if A andB are. We also characterize p-hyponormal A⊗B for which the commutator |A⊗B|_{2p}−|A^*⊗B^*|^{2p} is compact.

The problem considered is the determination of “lower bounds” of matrix operators on the spaces\ell_p(w) or d(w,p). Under fairly general conditions, the solution is the same for both spaces and is given by the infimum of a certain sequence. Specific cases are considered, with the weighting sequence defined by w_n = 1/n^\alpha . The exact solution is found for the Hilbert operator. For the averaging operator, two different upper bounds are given, and for certain values of p and \alpha it is shown that the smaller of these two bounds is the exact solution.

In this paper, groups are investigated in which all subgroups, all normal subgroups, or all characteristic subgroups have a proper supplement. This supplement can be either an arbitrary subgroup, a normal or a characteristic subgroup, resulting in nine classes of groups. Properties of these classes are studied such as containment and closure properties, and characterizations for several of these classes are given.

1991 Mathematics Subject Classification Primary 20E34, Secondary 20E15.

It is shewn that, if N(P) be the number of solutions of theindeterminate equation

ax^3+by^3+cz^3+dw^3 =0 \qquad (a,b,c,d \neq0)

for which \vert x \vert,\vert y\vert, \vert z\vert, \vert w\vert \leq P, then

N(P) = KP^2 + o(P^2),

where, to within a term O(P),KP^2 is the contribution to N(P) corresponding to therational lines in the projective surface defined by the equation. This proves a conjecture made byHeath-Brown, who has studied N(P) under the assumption of the Riemann Hypothesisfor certain Hasse-Weil L-functions. The remainder term o(P^2)in the formula represents O(P^{ {4 \over 3}+ϵ}),O(P^ {{5 \over 3}+ϵ}),or O(P^{2}/^3 \sqrt{ \log P}) according as the surface contains three, one, or norational lines.

Let A be a noetherian local ring, x a non-unit element of A, B=A/(x). Let E be the Koszul complex associated to an arbitrary set of generators of the ideal (x) of A. Assume that H1(E) is a free B-module. Then A is Gorenstein if and only if B is also.

1991 Mathematics Subject Classification 13H10, 13D03.