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.

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).

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.

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.

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.

A subgroup H of a group G is said to be c-supplemented inG if there exists a subgroup K of G such thatHK=G and H \cap K is contained in Core_G (H). We follow Hall's ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.

A well known result of B. Osofsky asserts that if R is a left (or right) perfect, left and right selfinjective ring thenR is quasi-Frobenius. It was subsequently conjectured by Carl Faith that every left (or right) perfect, left selfinjective ring is quasi-Frobenius. While several authors have proved the conjecture in the affirmative under some restricted chain conditions, the conjecture remains open even if R is a semiprimary, local, left selfinjective ring withJ(R)^3=0. In this paper we construct a local ring R withJ(R)^3=0 and characterize when R is artinian or selfinjective in terms of conditions on a bilinear mapping from a D-D-bimodule toD , where D is isomorphic to R/J(R). Our work shows that finding a counterexample to the Faith conjecture depends on the existence of aD -D-bimodule over a division ring D satisfying certain topological conditions.

We describe all possible coradically graded pointed Hopf algebras of dimension p^5 (wherep is an odd prime number) over an algebraically closed field of characteristic 0.

The structure of an inverse monoid can be determined by the complete set of Schützenberger graphs of a presentation. Necessary and sufficient conditions for a collection of inverse X-graphs to be the complete set of Schützenberger graphs of some inverse monoid presentation are established and decidability results are obtained. Conditions for a single inverse X-graph to be a Schu¨tzenberger graph for some presentation are also obtained, and both problems are restricted to the case of Clifford monoids and E-unitary inverse monoids. Decidability and undecidability results are obtained for the case of finite graphs. It is also proved that the problem of embedding a finite inverse X-graph in the Cayley graph of a group is undecidable.

An interesting class of submanifolds of Hermitian manifolds is the class of slant submanifolds which are submanifolds with constant Wirtinger angle. In [1–4,7,8] slant submanifolds of complex projective and complex hyperbolic spaces have been investigated. In particular, it was shown that there exist many proper slant surfaces in CP^2 and inCH^2 and many proper slant minimal surfaces in C2. In contrast, in the first part of this paper we prove that there do not exist proper slant minimal surfaces inCP^2 and in CH^2. In the second part, we present a general construction procedure for obtaining the explicit expressions of such slant submanifolds. By applying this general construction procedure, we determine the explicit expressions of special slant surfaces of CP^2 and of CH^2. Consequently, we are able to completely determine the slant surface which satisfies a basic equality. Finally, we apply the construction procedure to prove that special θ-slant isometric immersions of a hyperbolic plane into a complex hyperbolic plane are not unique in general.

Let A be a noetherian connected graded ring with a balanced dualizing complex R. If A has cohomological dimension and Krull dimension 2, then

(1) R is Auslander;

(2) \rm{Cdim} M=\rm{Kdim} M for all noetherian graded A-modules M.

In particular, ifA is AS-Gorenstein of injective and Krull dimension 2, then

(3)A is Auslander-Gorenstein;

(4) A is 2-pure with a self-injective artinian quotient ring;

(5) A has a residue complex.

(1,3,4) generalize a result of Levasseur [7, 5.13] and (5) generalizes a result of Ajitabh [1, 3.12].

Let(π_{λ}, ℋ_{λ}) be a unitary highest weight representation of the connected Lie group G and ℊ its Lie algebra. Thenℊ contains an invariant closed convex cone W_{\rm{max}} such that, for each X∈W_{\rm{max}}^0, the selfadjoint operatori·dπ_{λ}(X) is bounded from above. We show that for each suchX , the space ℋ_{λ}^{∞} of smooth vectors for the action of G on ℋ_{λ} coincides with the set𝒟^{∞}(dπ_{λ}(X)) of smooth vectors for the generally unbounded operator dπ_{λ}(X).

Let X be a complex infinite dimensional Banach space. We use σ_a(T) andσ_{ea}(T) , respectively, to denote the approximate point spectrum and the essential approximate point spectrum of a bounded operator T onX . Also, \pi _a(T) denotes the set <$>{\rm{iso} σ_a(T)\backslash σ_{ea}(T)}<$>. An operator T onX obeys the a-Browder's theorem provided that<$>σ_{ea}(T) =σ_a(T\,)\backslash π_a(T)<$> . We investigate connections between the Browder's theorems, the spectral mapping theorem and spectral continuity.