On Poincare Polynomials of Hyperbolic Lie Algebras

We have general frameworks to obtain Poincare polynomials for Finite and also Affine types of Kac-Moody Lie algebras. Very little is known however beyond Affine ones, though we have a constructive theorem which can be applied both for finite and infinite cases. One can conclusively said that theorem gives the Poincare polynomial P(G) of a Kac-Moody Lie algebra G in the product form P(G)=P(g) R where g is a precisely chosen sub-algebra of G and R is a rational function. Not in the way which theorem says but, at least for 48 hyperbolic Lie algebras considered in this work, we have shown that there is another way of choosing a sub-algebra in such a way that R appears to be the inverse of a finite polynomial. It is clear that a rational function or its inverse can not be expressed in the form of a finite polynomial. Our method is based on numerical calculations and results are given for each and every one of 48 Hyperbolic Lie algebras. In an illustrative example however, we will give how above-mentioned theorem gives us rational functions in which case we find a finite polynomial for which theorem fails to obtain.Comment: 14 pages, 7 Figures, Plain Te

Simulated Annealing for Location Area Planning in Cellular networks

LA planning in cellular network is useful for minimizing location management cost in GSM network. In fact, size of LA can be optimized to create a balance between the LA update rate and expected paging rate within LA. To get optimal result for LA planning in cellular network simulated annealing algorithm is used. Simulated annealing give optimal results in acceptable run-time.Comment: 7 Pages, JGraph-Hoc Journa

Linear Codes associated to Determinantal Varieties

We consider a class of linear codes associated to projective algebraic varieties defined by the vanishing of minors of a fixed size of a generic matrix. It is seen that the resulting code has only a small number of distinct weights. The case of varieties defined by the vanishing of 2 x 2 minors is considered in some detail. Here we obtain the complete weight distribution. Moreover, several generalized Hamming weights are determined explicitly and it is shown that the first few of them coincide with the distinct nonzero weights. One of the tools used is to determine the maximum possible number of matrices of rank 1 in a linear space of matrices of a given dimension over a finite field. In particular, we determine the structure and the maximum possible dimension of linear spaces of matrices in which every nonzero matrix has rank 1.Comment: 12 pages; to appear in Discrete Mat