Salem numbers defined by Coxeter transformation

A real algebraic integer alfa > 1 is called a Salem number if all its remaining conjugates have modulus at most 1 with at least one having modulus exactly 1. It is known ([12], [10], [5]) that the spectral radii of Coxeter transformation defined by stars, which are neither of Dynkin nor of extended Dynkin type, are Salem numbers. We prove that the spectral radii of the Coxeter transformation of generalized stars are also Salem numbers. A generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3

On zeros of reciprocal polynomials

The purpose of this paper is to show that all zeros of the reciprocal polynomia

A new construction of Salem polynomials

An earlier result of the author on the zeros of reciprocal polynomials is applied to give a new construction of Salem number

On a theorem of V. Dlab

A new proof of Dlab's theorem asserting that the left regular representation of an algebra is filtered by the standard modules if and only if the right regular representation of it is filtered by the proper standard modules, is given

Additive functions on trees

The motivation of considering positive additive functions on trees was the characterization of extended Dynkin graphs (see I. Reiten [R]) and the application of additive functions in the representation theory (see H. Lenzing and I. Reiten [LR] and T. H¨ubner [H]). We consider graphs equipped with functions of integer values, i.e.valued graphs (see also [DR]). Methods are given for the construction of additive functions on valued trees (in particular on Euclidean graphs) and for the characterization of their structure. We introduce the concept of almost additive functions, which are additive on each vertex of a graph except for one (called exceptional vertex). On (valued) trees (with fixed exceptional vertex) the almost additive functions are unique up to rational multiples. For valued trees a necessary and sufficient condition is given for the existence of positive almost additive functions

On construction of some classes if quasi-hereditary algebras

Inspired by the work of Mirollo and Vilonen [MV] describing the categories of perverse sheaves as module categories over certain finite dimensional algebras, Dlab and Ringel introduced [DR2] an explicit recursive construction of these algebras in terms of the algebras A(gamma): In particular, they characterized the quasi-hereditary algebras of Cline-Parshall-Scott [PS] and constructed them in this way. The present paper provides a characterization of lean algebras and some other special classes of algebras in terms of this recursive process

Construction of self-dual binary [2^(2k),2^(2k-1),2^k] -codes

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