On the eigenvalues of matrices with given upper triangular part

We give necessary and sufficient conditions for a set of numbers to be the eigenvalues of a completion of a matrix prescribed in its upper triangular part.Partially supported by the NSF Grant DMS-8701615-02Partially supported by the NSF Grant DMS-8802836 and United States-Israel Binational Science Foundation Grant 88-00304/I.     

Automorphisms of the KdV-subvariety

Partially supported by NSF Grant DMS-8705813 and Army Grant DAAL-03-87-K-0110.     

A Morera type theorem forL 2 functions in the Heisenberg group

We obtain a Morera type characterization for the square integrable CR functions on the Heisenberg group. Partially supported by NSF Grant DMS-90524569. $Partially supported by NSF Grant DMS-9000968.     

Modules of piecewise polynomials and their freeness

Partially supported by NSF Grant DMS-9004123 and ARO through MSI Cornell (DAAG 29-85-C-0018)     

A note on schur functions

We extend the Vandermonde determinant and then extend the classical presentation of Schur functions as quotients of such determinants. Partially supported by NSA Grant MDA904-91-H-0050 and by NSF Grant DMS-9101488.     

The spread of a partial order

The deficiency of a partial order is the number of incomparable pairs. Another measure, the spread of a partial order is defined and its relation to the deficiency is established. In certain cases which arise naturally in the analysis of various sorting and selection algorithms where the partial order is only implicitly defined, the spread of the partial order is easier to estimate directly and provides a convenient way to estimate the deficiency.Partially supported by NSF Grant DCR-85-05053.Partially supported by NSF Grant DMS-85-05004.     

The equations of conjugacy classes of nilpotent matrices

Partially supported by NSF Grant     

Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field

Partially supported by NSF Grant DMS-9123742     

A note on subgroups generated by reflections in Coxeter groups

Partially supported by NSF Grant DMS 8502310.     

Rigid versus non-rigid cyclic actions

Partially supported by NSF Grant MCS 8514551     

On the number of generators of a finite group

Partially supported by NSF Grant DMS 8700961.     

Exponential sums on (G m ) n

Partially supported by NSF Grant No. DMS-8803085     

A recurrence for linear extensions

The number e(P) of linear extensions of a finite poset P is expressed in terms of e(Q) for certain smaller posets Q. The proof is based on M. Schützengerger's concept of promotions of linear extensions.Partially supported by NSF Grant #DMS-8700995.Partially supported by NSF Grant #DMS-8401376.     

On the Shintani zeta function for the space of binary quadratic forms

Partially supported by NSF Grant DMS-8803085, DMS-8610730     

Bounds on Scott rank for various nonelementary classes

Partially supported by NSF Grant DMS-8602557     

Some invariants of Kakutani equivalence

We introduce some invariants of Kakutani equivalence and using them we prove that any two distinct cartesian powers of the horocycle flow are inequivalent. Partially supported by NSF Grant MCS74-19388.     

Normal forms for periodic orbits of real vector fields

We consider normal forms of real vector fields near periodic orbits and provide sufficient conditions for their smooth linearization. In addition, the main results also assert the existence of vertical local foliations, whose leaves are all transversal to the periodic orbit.     

On the regularity of optimal controls

We outline some recent results on the regularity of optimal controls. We formulate the general regularity problem for open-loop and closed-loop controls, and explain how results for the open-loop case have implications for the closed-loop case as well. We then describe a number of results on the regularity of open-loop controls.Partially supported by a fellowship from the Alfred P. Sloan Foundation.Partially supported by NSF Grant No. DMS83-01678-01.     

The cartesian square of the horocycle flow is not loosely Bernoulli

We give an example of an algebraic non-loosely Bernoulli flow. Namely, we prove that the cartesian square of the classical horocycle flow is not loosely Bernoulli. Partially supported by the Sloan Foundation and NSF Grant MCS74-19388.     

Positivity and strict contractivity of functions of operators

Carathéodory class functionsf(z) are described having the property that the self-adjoint part off(A) is positive definite for every contractionA whose spectral radius is less than 1. Analogous results are obtained for bounded analytic functions in the unit disc, and for the Nevanlinna class. Applications to Markov chains are indicated.Partially supported by the US Air Force Grant AFOSR-94-0293.Partially supported by the NSF Grant DMS-9500924.