The Goldman-Rota identity and the Grassmann scheme

We inductively construct an explicit (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. The main step is a constructive, linear algebraic interpretation of the Goldman-Rota recurrence for the number of subspaces of a finite vector space. This interpretation shows that the up operator on subspaces has an explicitly given recursive structure. Using this we inductively construct an explicit orthogonal symmetric Jordan basis with respect to the up operator and write down the singular values, i.e., the ratio of the lengths of the successive vectors in the Jordan chains. The collection of all vectors in this basis of a fixed rank forms a (common) orthogonal eigenbasis for the elements of the Bose-Mesner algebra of the Grassmann scheme. We also pose a bijective proof problem on the spanning trees of the Grassmann graphs.Comment: 19 Page

Secure Communication using Compound Signal from Generalized Synchronizable Chaotic Systems

By considering generalized synchronizable chaotic systems, the drive-auxiliary system variables are combined suitably using encryption key functions to obtain a compound chaotic signal. An appropriate feedback loop is constructed in the response-auxiliary system to achieve synchronization among the variables of the drive-auxiliary and response-auxiliary systems. We apply this approach to transmit analog and digital information signals in which the quality of the recovered signal is higher and the encoding is more secure.Comment: 7 pages (7 figures) RevTeX, Please e-mail Lakshmanan for figures, submitted to Phys. Lett. A (E-mail: [email protected]

Rich Variety of Bifurcations and Chaos in a Variant of Murali-Lakshmanan-Chua Circuit

A very simple nonlinear parallel nonautonomous LCR circuit with Chua's diode as its only nonlinear element, exhibiting a rich variety of dynamical features, is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua(MLC). By constructing a two-parameter phase diagram in the $(F-\omega)$ plane, corresponding to the forcing amplitude (F) and frequency $(\omega)$, we identify, besides the familiar period-doubling scenario to chaos, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, Farey sequences, and so on. The chaotic dynamics is verified by both experimental as well as computer simulation studies including PSPICE.Comment: 4 pages, RevTeX 4, 5 EPS figure

Approximate Linear Time ML Decoding on Tail-Biting Trellises in Two Rounds

A linear time approximate maximum likelihood decoding algorithm on tail-biting trellises is prsented, that requires exactly two rounds on the trellis. This is an adaptation of an algorithm proposed earlier with the advantage that it reduces the time complexity from O(mlogm) to O(m) where m is the number of nodes in the tail-biting trellis. A necessary condition for the output of the algorithm to differ from the output of the ideal ML decoder is reduced and simulation results on an AWGN channel using tail-biting rrellises for two rate 1/2 convoluational codes with memory 4 and 6 respectively are reporte