Variational Analysis of the Abscissa Mapping for Polynomials via the Gauss-Lucas Theorem

Consider the linear space ? ⁿ of polynomials of degree n or less over the complex field. The abscissa mapping on ? ⁿ is the mapping that takes a polynomial to the maximum real part of its roots. This mapping plays a key role in the study of stability properties for linear systems. Burke and Overton have shown that the abscissa mapping is everywhere subdifferentially regular in the sense of Clarke on the manifold ? ⁿ of polynomials of degree n. In addition, they provide a formula for the subdifferential. The result is surprising since the abscissa mapping is not Lipschitzian on ? ⁿ . A key supporting lemma uses a proof technique due to Levantovskii for determining the tangent cone to the set of stable polynomials. This proof is arduous and opaque. It is a major obstacle to extending the variational theory to other functions of the roots of polynomials. In this note, we provide an alternative proof based on the Gauss-Lucas Theorem. This new proof is both insightful and elementary.     

Convergence of Bieberbach polynomials in domains with interior cusps

We extend the results on the uniform convergence of Bieberbach polynomials for domains with certain interior zero angles (outward pointing cusps) and show that they play a special role in the problem. Namely, we construct a Keldysh-type example on the divergence of Bieberbach polynomials at an outward pointing cusp and discuss thecritical order of tangency at this interior zero angle, separating the convergent behavior of Bieberbach polynomials from the divergent one for sufficiently thin cusps. Research of both authors was supported in part by the National Science Foundation grant DMS-9707359. Research of the second author was also supported in part by the National Science Foundation grant DMS-9970659.     

Local Linear Convergence for Alternating and Averaged Nonconvex Projections

The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness, for example), we prove that von Neumann’s method of “alternating projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method of “averaged projections” converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly. Research of A.S. Lewis supported in part by National Science Foundation Grant DMS-0504032. Research of D.R. Luke supported in part by National Science Foundation Grant DMS-0712796.     

Spectral conditioning and pseudospectral growth

Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii’s analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra. J. V. Burke’s research was supported in part by National Science Foundation Grant DMS-0505712. A. S. Lewis’s research was supported in part by National Science Foundation Grant DMS-0504032. M. L. Overton’s research was supported in part by National Science Foundation Grant DMS-0412049.     

A robust sequential quadratic programming method

The sequential quadratic programming method developed by Wilson, Han and Powell may fail if the quadratic programming subproblems become infeasible, or if the associated sequence of search directions is unbounded. This paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems. Furthermore, questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed.Work supported in part by the National Science Foundation under Grant No. DMS-8602399 and by the Air Force Office of Scientific Research under Grant No. ISSA-860080.Work supported in part by the National Science Foundation under Grant No. DMS-8602419.     

Index formulas for geometric Dirac operators in Riemannian foliations

With regards to certain Riemannian foliations we consider Kasparov pairings of leafwise and transverse Dirac operators. Relative to a pairing with a transversal class we commence by establishing an index formula for foliations with leaves of nonpositive sectional curvature. The underlying ideas are then developed in a more general setting leading to pairings of images under the Baum-Connes map in geometricK-theory with transversal classes. Several ideas implicit in the work of Connes and Hilsum-Skandalis are formulated in the context of Riemannian foliations. From these we establish the notion of a dual pairing inK-homology and a theorem of the Grothendieck-Riemann-Roch type.R. G. D. was supported by The National Science Foundation under Grant No. DMS-9304283.J. F. G. and F. W. K. were supported in part by The National Science Foundation under Grant No. DMS-9208182.F. W. K. was also supported in part by an Arnold O. Beckman Research Award from the Research Board of the University of Illinois.     

Verifiability and strong verifiability

Research supported by National Science Foundation Grant No. DMS-8908670.     

An <Emphasis Type="Italic">ABC</Emphasis> inequality for Mahler’s measure

We prove an upper bound for the Mahler measure of the Wronskian of a collection of N linearly independent polynomials with complex coefficients. If the coefficients of the polynomials are algebraic numbers we obtain an inequality for the absolute Weil heights of the roots of the polynomials. This later inequality is analogous to the abc inequality for polynomials, and also has applications to Diophantine problems. Research supported in part by the National Science Foundation (DMS-06-03282) and the Erwin Schr?dinger Institute. Author’s address: Department of Mathematics, University of Texas, Austin, Texas 78712, USA     

Weak sharp minima revisited, part II: application to linear regularity and error bounds

The notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. It has been studied extensively by several authors. This paper is the second of a series on this subject where the basic results on weak sharp minima in Part I are applied to a number of important problems in convex programming. In Part II we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming. We obtain both new results and reproduce several existing results from a fresh perspective. We dedicate this paper to our friend and mentor Terry Rockafellar on the occasion of his 70th birthday. He has been our guide in mathematics as well as in the backcountry and waterways of the Olympic and Cascade mountains. Research supported in part by the National Science Foundation Grant DMS-0203175.     

Subgradient of distance functions with applications to Lipschitzian stability

The paper is devoted to studying generalized differential properties of distance functions that play a remarkable role in variational analysis, optimization, and their applications. The main object under consideration is the distance function of two variables in Banach spaces that signifies the distance from a point to a moving set. We derive various relationships between Fréchet-type subgradients and limiting (basic and singular) subgradients of this distance function and corresponding generalized normals to sets and coderivatives of set-valued mappings. These relationships are essentially different depending on whether or not the reference point belongs to the graph of the involved set-valued mapping. Our major results are new even for subdifferentiation of the standard distance function signifying the distance between a point and a fixed set in finite-dimensional spaces. The subdifferential results obtained are applied to deriving efficient dual-space conditions for the local Lipschitz continuity of distance functions generated by set-valued mappings, in particular, by those arising in parametric constrained optimization. Dedicated to Terry Rockafellar in honor of his 70th birthday. This research was partially supported by the National Science Foundation under grant DMS-0304989 and by the Australian Research Council under grant DP-0451158.     

Finite-Term Relations for Planar Orthogonal Polynomials

We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar domain, with sufficiently regular boundary, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that Dirichlet’s problem with boundary polynomial data has a polynomial solution. This, and an additional compactness assumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. To Peter Duren on the occasion of his seventieth birthday The first author was partially supported by the National Science Foundation Grant DMS- 0350911. Received: October 15, 2006. Revised: January 22, 2007.     

Limit theorems for sums of dependent random variables occurring in statistical mechanics

Summary By the use of conditioning, we extend previously obtained results on the asymptotic behavior of partial sums for certain triangular arrays of dependent random variables, known as Curie-Weiss models. These models arise naturally in statistical mechanics. The relation of these results to multiple phases, metastable states, and other physical phenomena is explained.Alfred P. Sloan Research Fellow. Research supported in part by National Science Foundation Grant MPS 76-06644A01Alfred P. Sloan Research Fellow. Research supported in part by National Science Foundation Grant MCS 77-20683 and by U.S.-Israel Binational Science FoundationResearch supported in part by National Science Foundation Grant PHY77-02172     

Chains in the Bruhat order

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees. A.P. was supported in part by National Science Foundation grant DMS-0201494 and by Alfred P. Sloan Foundation research fellowship. R.S. was supported in part by National Science Foundation grant DMS-9988459.     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

On the distribution of imaginary parts of zeros of the Riemann zeta function,II

We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Montgomery’s pair correlation function and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. The first author is supported by National Science Foundation Grant DMS-0555367. The second author is partially supported by the National Science Foundation and the American Institute of Mathematics (AIM). The third author is supported by National Science Foundation Grant DMS-0456615.     

Co-volume methods for degenerate parabolic problems

Summary A complementary volume (co-volume) technique is used to develop a physically appealing algorithm for the solution of degenerate parabolic problems, such as the Stefan problem. It is shown that, these algorithms give rise to a discrete semigroup theory that parallels the continuous problem. In particular, the discrete Stefan problem gives rise to nonlinear semigroups in both the discreteL ¹ andH ^–1 spaces.The first author was supported by a grant from the Hughes foundation, and the second author was supported by the National Science Foundation Grant No. DMS-9002768 while this work was undertaken. This work was supported by the Army Research Office and the National Science Foundation through the Center for Nonlinear Analysis.     

Mixed hyperbolic-elliptic systems in self-similar flows

From the observation that self-similar solutions of conservation laws in two space dimensions change type, it follows that for systems of more than two equations, such as the equations of gas dynamics, the reduced systems will be of mixed hyperbolic-elliptic type, in some regions of space. In this paper, we derive mixed systems for the isentropic and adiabatic equations of compressible gas dynamics. We show that the mixed systems which arise exhibit complicated nonlinear dependence. In a prototype system, the nonlinear wave system, this behavior is much simplified, and we outline the solution to some typical Riemann problems.Dedicated to Constantine Dafermos on his 60^th birthdayResearch supported by the National Science Foundation, grant DMS-9970310.Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-9973475 (POWRE).Research supported by the Department of Energy, grant DE-FG-03-94-ER25222 and by the National Science Foundation, grant DMS-0103823.     

On the dimension of bivariate spline spaces of smoothnessr and degreed=3r+1

Summary We consider the well-known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally. To date, dimension formulae for such spaces have been established only whend3r+2, (except for the special case wherer=1 andd=4). In this paper we establish dimension formulae for allr1 andd=3r+1 for almost all triangulations.Dedicated to R. S. Varga on the occasion of his sixtieth birthdaySupported in part by National Science Foundation Grant DMS-8701121Supported in part by National Science Foundation Grant DMS-8602337     

Consolidation rates for two interacting systems in the plane

Summary This paper is a sequel of a paper of Cox and Griffeath “diffusive clustering in the two dimensional voter model”. We continue our study of the voter model and coalescing random walks on the two dimensional integer lattice. Some exact asymptotics concerning the rate of clustering in the former process and the coalescence rate of the latter are derived. We use these results to prove a limit law, announced in that earlier paper, concerning the size of the largest square centered at the origin which is of solid color at a large time t. Partially supported by the National Science Foundation under Grant DMS-831080 Partially supported by the National Science Foundation under Grant DMS-841317 Partially supported by the National Science Foundation under Grant DMS-830549     

New differential properties of sup-type functions

In this paper, we study the directional derivative, subderivative, and subdifferential of sup-type functions without any compactness assumption on the index set. As applications, we provide an estimate of the Lipschitz modulus for sup-type functions.