Formal orthogonal polynomials revisited. Applications

From some results concerning the formal orthogonal polynomials, already proved in [5], we develop new properties of generalized adjacent polynomials which correspond to a change in the weight function. A new structure of the singular blocks is given. These results have a direct application to Lanczos methods, theG and -algorithms.     

On sparse spanners of weighted graphs

Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v V, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.The work of G. Das and D. Joseph was supported by NSF PYI Grant DCR-8402375. The work of D. Dobkin was supported by NSF Grant CCR-8700917. The work of J. Soares was supported by CNPq proc 203039/87.4 (Brazil) and NSF Grant CCR-9014562. This research was accomplished while G. Das was a student at the University of Wisconsin-Madison. A preliminary version was presented at the Second Scandinavian Workshop on Algorithm Theory, Bergen, Norway, 1990, under the title Generating Sparse Spanners for Weighted Graphs, and proceedings appear in the series Lecture Notes in Computer Science, Springer-Verlag. The preliminary version also appears as Princeton University Technical Report CS-TR-261-90, and as University of Wisconsin-Madison Computer Sciences Technical Report 882.     

A one-parameter family of hamiltonian structures for the KP hierarchy and a continuous deformation of the nonlinear WKP algebra

The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of W_KP admitting a central extension for generic values of the parameter, reducing naturally to W _n for special values of the parameter, and contracting to the centrally extended W₁₊, W and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic tow _KP. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of which contracts to a new nonlinear algebra of the W-type.Address after October 1993: Queen Mary and Westfield College, UK