On some riesz bases

In this paper we consider three problems concerning systems of vector exponentials. In the first part we prove a conjecture of V. Komornik raised in [14] on the independence of the movement of a rectangular membrane in different points. It was independently proved by M. Horváth [9] and S. A. Avdonin (personal communication). The analogous problem for the circular membrane was partly solved in [3] — the complete solution is given in [10]. In the second part we fill in a gap in the theory of Blaschke-Potapov products developed in the paper [19] of Potapov. Namely we prove that the Blaschke-Potapov product is determined by its kernel sets up to a multiplicative constant matrix. In the third part of the present paper we give a multidimensional generalization of the notion of sine type function developed by Levin [16], [17] and by our generalization we prove the multidimensional variant of the Levin-Golovin basis theorem [16], [6].     

Irregular lil behaviour of lacunary trigonometric series

Research supported by Hungarian National Foundation for Scientific Research, grant no. 1808.     

Dominating cliques in P5-free graphs

All induced connected subgraphs of a graphG contain a dominating set of pair-wise adjacent vertices if and only if there is no induced subgraph isomorphic to a path or a cycle of five vertices inG. Moreover, the problem of finding any given type of connected dominating sets in all members of a classG of graphs can be reduced to the graphsG∈G that have a cut-vertex or do not contain any cutsetS dominated by somes∈S. This research was supported in part by the “AKA” Research Fund of the Hungarian Academy of Sciences.     

Cartan connections for CR manifolds

We first follow the construction of Chern–Moser and prove that Chern–Moser connections are indeed isomorphic to Tanaka connections in dimension three. Then, we show that a compact simply connected spherical CR manifold is diffeomorphic to S ³.     

A Note on the Green Invariants in Finite Group Modular Representative Theory

As in Finite Group Modular Representation Theory, let be a commutative complete noetherian ring with an algebraically closed residue field k. Let G be a finite group and let N be a normal subgroup of G. First suppose that V is an indecomposable -module, so that Inf ^G _G/N (V) is an indecomposable G-module. We relate the Green invariants of V as an -module to those of Inf ^G _G/N (V) as an G-module. Secondly, let V and W be indecomposable G-modules. Assume that W is an endo-permutation lattice and that is also an indecomposable G-module. We relate the Green invariants of the G-modules V and . (This situation arises under important Morita equivalences.) Received: December 11, 2006. Revised: August 22, 2007.     

Fully k-Primary Rings

Let k be a positive integer. The structure of rings all of whose ideals are n-primary for some positive integer n ≤ k is studied, and several examples of such rings are constructed. Rings all of whose nonzero ideals are n-primary for some positive integer n ≤ k is also considered.     

New type Paley-Wiener theorems for the modified multidimensional Mellin transform

New type Paley-Wiener theorems for the modified multidimensional Mellin and inverse Mellin transforms are established. The supports of functions are described in terms of their modified Mellin (or inverse Mellin) transform without passing to the complexification. Acknowledgments and Notes. The work is supported by the Kuwait University research grant SM 112.     

The centralizer algebras of lie color algebras

In his 1977 paper, V.G. Kac classified the finite-dimensional simple complex Lie superalgebras. After Kac’s paper, M. Scheunert initiated the study of a generalization of Lie superalgebras - the Lie color algebras. We construct some new families of simple Lie color algebras. Following the work of A. Berele and A. Regev and A.N. Sergeev, who studied the general linear and sq(n)-series superalgebra cases, and the work of G. Benkart, C. Lee Shader, and A. Ram, who studied the orthosymplectic cases, we examine the centralizer algebras of some classical Lie superalgebras and their Lie color algebra counterparts acting on tensor space and derive Schur-Weyl duality results for these algebras and their centralizers.     

Semi-invariants of actions of multiplier hopf algebras

Let A be a multiplier Hopf algebra which acts on an algebra R. In this paper we study semi-invariants of this action. This idea has proved interesting in the case thatA is a Hopf algebra.     

Quasi- and weakly-quasi-first-countable spaces

In this paper, we study two countability properties that are weaker than the first-countability property and following the pattern of Michael [3], we obtain the spaces with such properties as quotient images of metric spaces by particular kinds of maps.     

On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation

One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of Kähler manifolds with nonnegative holomorphic bisectional curvature, $\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of K?hler manifolds with nonnegative holomorphic bisectional curvature, for some a > 67(n + 4)². We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincaré inequality.     

On Prime Submodules of a Finitely Generated Free Module Over a Commutative Ring

In this article, we give a full characterization of prime submodules of a finitely generated free module F over a commutative ring R with identity. Also we study the existence of primary decomposition of a submodule of F and characterize the minimal primary decomposition of this submodule. We also describe the structure of prime submodules of a module over a Dedekind domain.     

Remarks on the length of conjugacy classes of finite groups

Let G be a finite group. The question of how certain arithmetical conditions on the lengths of the conjugacy classes of G influence the group structure has been studied by several authors. In this paper we study restrictions on the structure of a finite group in which the lengths of conjugacy classes are not divisible by p ² for some prime p. We generalise and provide simplified proofs for some earlier results.     

Sequences of polynomials whose zeros lie on fixed lemniscates

For fixed integersk2 we study sequences of polynomialsP _n (z) with the following properties: (i) degP _n ; (ii) the zeros of all theP _n (z) lie on a certain lemniscate withk ₁k foci, one of which is the origin; (iii) theP _n (z) can be cut in such a way that the zeros of the lower part all lie on the unit circle and those of the upper part lie on a lemniscate having the foci in (ii) excluding the origin. Several special cases and examples are considered.     

On the normalizer property for integral group rings

Let R(G) denote the intersection of all nonnormal subgroups of a group G. In this note, we prove that for every finite group G, if R(G) is not trivial, then the normalizer property holds forG.