The general theory of bivariate orthogonal polynomials goes back to the work of Jackson [12]. Special examples have arisen in studies related to symmetric groups [4, 16, 22]), as extensions of one variable polynomials [7, 15] and as eigenfunctions of partial differential equations [19, 14, 13, 21]. An updated account of the theory can be found in the books [5, 25]. A major difficulty encountere...

Decoupled rowwise ordering is an ordering scheme for 2-dimensional grids, which is tailored for preconditioning 5-point diierence equations arising from discretizations of partial diierential equations. This paper describes the ordering scheme and implementations of a conjugate gradient solver and SSOR preconditioners which use the decoupled rowwise and the red black ordering schemes on the CM-...

In this paper we investigate the problem of measuring social mobility when the social status of individuals is given by their rank. In order to sensibly represent the rank mobility of subgroups within a given society, we address the problem in terms of partial permutation matrices which include standard (“global”) matrices as a special case. We first provide a characterization of a partial orde...

We prove that the group of left self-distributivity, a cousin of Thompson’s group F and of Artin’s braid group B∞ that describes the geometry of the identity x(yz) = (xy)(xz), admits a bi-invariant linear ordering. To this end, we define a partial action of this group on finite binary trees that preserves a convenient linear ordering.

This paper studies likelihood ratio ordering between two parallel systems with exponential components. We introduce a new partial order and extend the classical Dykstra’s theorem to higher dimension. Some other results on likelihood ratio ordering of parallel systems are also derived.

We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fräıssé’s conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering, quasiordered by the...

Ordering composite hypotheses in a Bayesian network based on its associated a posteriori probabilities can be exponentially hard. This paper discusses a qualitative reasoning approach which reduces the computational complexity of deriving a partial ordering of composite hypotheses. Such a reasoning makes use of the meta-knowledge about the causal relationships among individual hypotheses to ded...