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.     

On the number of solutions to systems of Pell equations

We prove that, for positive integers a, b, c and d with c≠d, a>1, b>1, the number of simultaneous solutions in positive integers to ax²−cz²=1, by²−dz²=1 is at most two. This result is the best possible one. We prove a similar result for the system of equations x²−ay²=1, z²−bx²=1.     

An Algebraic Perspective on Multivariate Tight Wavelet Frames

Recent advances in real algebraic geometry and in the theory of polynomial optimization are applied to answer some open questions in the theory of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations merge recent advances in real algebraic geometry and wavelet frame theory and lead to an affirmative answer to the long-standing open question of the existence of tight wavelet frames in dimension d=2. They also provide, for every d, efficient numerical methods for checking the existence of tight wavelet frames and for their construction. A class of counterexamples in dimension d=3 show that, in general, the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Stronger sufficient conditions for determining the existence of tight wavelet frames in dimension d≥3 are derived. The results are illustrated on several examples.     

Non-negative hereditary polynomials in a free *−algebra

We prove a non-negative-stellensatz and a null-stellensatz for a class of polynomials called hereditary polynomials in a free *-algebra.Partially supported by NSF, DARPA and Ford Motor Co.Partially supported by NSF grant DMS-0140112Partially supported by NSF grant DMS-0100367     

Viability for Differential Inclusions on Graphs

Let X be a Banach space, I a nonempty, bounded interval and let $K:I\leadsto X$ be a given multi-valued function. We rephrase the concept of tangent set introduced by Cârj? et al. (Trans Amer Math Soc, 2008; Viability, Invariance and Applications. North-Holland Mathematics Studies, vol. 207. Elsevier, 2007) by saying that a bounded set $E\subseteq X$ is right tangent to at if $\liminf_{h\downarrow0}{1}/{h}\,\text{\rm dist}(\xi+hE;K(\tau+h))=0$ . Next, by using a tangency condition expressed in the terms of this concept, we establish several necessary and sufficient conditions for viability referring to differential inclusions on graphs.     

High-electric-field limit for the Vlasov–Maxwell–Fokker–Planck system

In this paper we derive the high-electric-field limit of the three-dimensional Vlasov–Maxwell–Fokker–Planck system. We use the relative entropy method which requires the smoothness of the solution of the limit problem. We obtain convergences of the electro-magnetic field, charge and current densities.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

Iterative Methods for Pseudomonotone Variational Inequalities and Fixed-Point Problems

In this paper, we introduce an iterative scheme for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive some necessary and sufficient conditions for strong convergence of the sequences generated by the proposed scheme.     

Irreducibility criteria for compositions of polynomials with integer coefficients

We provide irreducibility criteria for some classes of compositions of polynomials with integer coefficients of the form \(F\circ G\), with F being a quadratic irreducible polynomial and G a polynomial of arbitrary degree.