iDEA Collection:
http://hdl.handle.net/1860/1173
2014-04-24T04:10:36ZConservative dilations of dissipative multidimensional systems: the commutative and non-commutative settings
http://hdl.handle.net/1860/2731
Title: Conservative dilations of dissipative multidimensional systems: the commutative and non-commutative settings
Authors: Ball, Joseph A.; Kaliuzhnyi-Verbovetskyi, Dmitry S.
Abstract: We establish the existence of conservative dilations for various
types of dissipative non-commutative N-dimensional (N-D) systems. As a
corollary, a criterion of existence of conservative dilations for corresponding
dissipative commutative N-D systems is obtained. We point out the cases
where this criterion is always fulfilled, and the cases where it is not always
fulfilled.2008-01-01T00:00:00ZTwo variable orthogonal polynomials on the bicircle and structured matrices
http://hdl.handle.net/1860/2700
Title: Two variable orthogonal polynomials on the bicircle and structured matrices
Authors: Geronimo, Jeffrey S.; Woerdeman, Hugo
Abstract: We consider bivariate polynomials orthogonal on the bicircle with respect to a
positive linear functional. The lexicographical and reverse lexicographical orderings are used to
order the monomials. Recurrence formulas are derived between the polynomials of different degrees.
These formulas link the orthogonal polynomials constructed using the lexicographical ordering with
those constructed using the reverse lexicographical ordering. Relations between the coefficients in the
recurrence formulas are derived and used to give necessary and sufficient conditions for the existence
of a positive linear functional. These results are then used to construct a class of two variable
measures supported on the bicircle that are given by one over the magnitude squared of a stable
polynomial. Applications to Fej´er–Riesz factorization are also given.2007-01-01T00:00:00ZSchur function analogs for a filtration of the symmetric function space
http://hdl.handle.net/1860/1947
Title: Schur function analogs for a filtration of the symmetric function space
Authors: Morse, Jennifer; Lapointe, Luc
Abstract: We consider a filtration of the symmetric function space given by At(k)
, the linear
span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k.
We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur
functions in the subspaces At(k)
. We prove several properties for the k-Schur functions including
that they form a basis for these subspaces that reduces to the Schur basis when k is large. We
also show that the connection coefficients for the k-Schur function basis with the Macdonald
polynomials belonging to At(k)
are polynomials in q and t with integral coefficients. In fact, we
conjecture that these integral coefficients are actually positive, and give several other conjectures
generalizing Schur function theory.2001-11-17T00:00:00ZOrder ideals in weak subposets of Young’s lattice and associated unimodality conjectures
http://hdl.handle.net/1860/1946
Title: Order ideals in weak subposets of Young’s lattice and associated unimodality conjectures
Authors: Morse, Jennifer; Lapointe, Luc
Abstract: The k-Young lattice Y k is a weak subposet of the Young lattice containing partitions whose
first part is bounded by an integer k > 0. The Y k poset was introduced in connection with generalized
Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine
symmetric group ˜ Sk+1 by a maximal parabolic subgroup. We prove a number of properties for Y k
including that the covering relation is preserved when elements are translated by rectangular partitions
with hook-length k. We highlight the order ideal generated by an m× n rectangular shape. This order
ideal, Lk(m, n), reduces to L(m, n) for large k, and we prove it is isomorphic to the induced subposet
of L(m, n) whose vertex set is restricted to elements with no more than k−m+1 parts smaller than m.
We provide explicit formulas for the number of elements and the rank-generating function of Lk(m, n).
We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications
connect to recent work on sieved q-binomial coefficients.2004-05-07T00:00:00Z