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Byline: Damir Vukicevic (1), Milan Randic (2), Alexandru T. Balaban (3) Keywords: polycyclic benzenoid hydrocarbons; algebraic vs. geometric Kekule structures Abstract: A rigorous proof is provided for the theorem on the necessary and sufficient conditions for the algebraic Kekule structure of a peri-condensed or corona-condensed benzenoid hydrocarbon to correspond to more than one geometric Kekule structure. Author Affiliation: (1) Department of Mathematics, University of Split, Nikole Tesle 12, 21000, Split, Croatia (2) 3225 Kingman Road, Ames, IA, 50014, USA (3) Department of Marine Sciences, Texas A &amp M University at Galveston, 5007 Avenue U, Galveston, TX, 77551, USA Article History: Registration Date: 27/10/2004

A tabu search heuristic for the design of self-healing rings (SHRs), to protect the transmission of telecommunication demands, in a zonal network is proposed. This search has demonstrated its efficiency by improving significantly upon the best solutions that could be found with a greedy constructive heuristic over a large set of real and randomly generated instances.

We introduce linear cyclic codes over the ring [Mathematical Expression Omitted], where [u.sup.2] = 0 and study them by analogy with the [Z.sub.4] case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of [Z.sub.4]-codes. Unlike the [Z.sub.4] case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map. Index Terms - Codes over rings, cyclic codes, Gray map, self-dual codes.

This paper discusses a design of a hybrid ring -- mesh network in a survivable communication network. Given a set of traffic demands, the problem is to assign each traffic demand to rings and mesh such that the cost of add -- drop multiplexer (ADM) and digital cross-connect system (DCS) equipment required is minimized. This assignment problem can be considered together with the fibre routing of nodes on rings and mesh. As a solution procedure, a tabu search is developed with recency-based short-term and frequency-based long-term memory structures. In computational experiments, the proposed tabu search is compared with the solutions obtained by the branch and bound procedure of CPLEX. We see that the tabu search provides a nearly optimal solution within sufficiently short time periods for all test problems with a gap of approximately 1 -- 4 % from the lower bound. Keywords: communication network design; ADM ring; DCS mesh; tabu search

In this paper, we study self-dual codes over the ring [Z.sub.2k] of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce Type II codes over [Z.sub.2k] which are closely related to even unimodular lattices, as a remarkable class of self-dual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extremal Type II code over [Z.sub.6] of length 24 is constructed, which gives a new construction of the Leech lattice. The complete and symmetrized weight enumerators in genus g of codes over [Z.sub.2k] are introduced, and the MacWilliams identities for these weight enumerators are given. We investigate the groups which fix these weight enumerators of Type II codes over [Z.sub.2k] and we give the Molien series of the invariant rings of the groups for small cases. We show that modular forms are constructed from complete and symmetrized weight enumerators of Type II codes. Shadow codes over [Z.sub.2k] are also introduced. Index Terms - Codes over [Z.sub.2k], even unimodular lattices, invariant rings, Type II codes.