Addendum to “The Extremal Truncated Moment Problem”

Due to a technical problem, we accidentally omitted a paragraph from the proof of one of the main results in [1]. In this Addendum we provide the portion of the proof that did not appear in [1]. This is an addendum to .     

Asymptotics of Regular Solutions to the Camassa–Holm Problem

Computational Mathematics and Mathematical Physics - A periodic boundary value problem is considered for a modified Camassa–Holm equation, which differs from the well-known classical equation...     

Laurent–Jacobi Matrices and the Strong Hamburger Moment Problem

Let be the linear space of the Laurent polynomials and suppose that <, < is a positive-definite Hermitian inner product in with the additional property that . Starting from the five-term recurrence relation for orthogonal Laurent polynomials with respect to <, <, we derive Laurent–Jacobi matrices and for the multiplication operator and its inverse in . These matrices are real and symmetric, and generates a symmetric operator in the Hilbert space ² with natural basis { e _n } _{n = 0} . We show that this operator has deficiency indices (0, 0) or (1, 1) and that every self-adjoint extension A in ² has simple spectrum with generating vector e ₀. Let E be the spectral measure of A. Then the measure _{e 0} given by _{e 0}() =<E() e ₀, e ₀< for all Borel sets in , satisfies forf,g. In this way, we obtain a solution _{e 0} of the Strong Hamburger Moment Problem (SHMP) for which is dense in L ²( _{e 0}). Some results concerning the relation between the deficiency indices andthe set of all solutions of the SHMP are established. Finally, we give an analogue of a theorem by M. H. Stone which tells us which self-adjoint operators are generatedby a Laurent–Jacobi matrix with deficiency indices (0, 0).     

The Spectral Carathéodory-Fejér Problem

The problem of the title is to construct an analytic k × k matrix-valued function in the unit disc with a number of prescribed derivatives at 0 and with spectral radius bounded by 1. We show that the problem can be reduced to an interpolation problem for the symmetrized polydisc , and thereby show that, in the case of derivatives of orders 0 and 1 being prescribed, the problem is equivalent to the infinitesimal Kobayashi extremal problem for , which is solved completely in the case k = 2.     

On An Extremal Hypergraph Problem Of Brown, Erdős And Sós

Let f_r(n,v,e) denote the maximum number of edges in an r-uniform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemerédi and of Erdős, Frankl and R?dl, we partially resolve a problem raised by Brown, Erdős and Sós in 1973, by showing that for any fixed 2≤k<r, we have

Regular Spaces and Hyper-spaces

We are concerned with the construction of regular spaces and hyper-spaces,and the characterization of continuous linear operators in such spaces.     

Ideal-comparability over Regular Rings

We introduce the concept of ideal-comparability condition for regular rings. Let I be an ideal of a regular ring R. If R satisfies the I-comparability condition, then R is one-sided unit-regular if and only if so is R/I. Also, we show that a regular ring R satisfies the general comparability if and only if the following hold： （1） R/I satisfies the general comparability; （2） R satisfies the general I-comparability condition; （3） The natural map B（R） → B（R/I） is surjective.     

On the Carathéodory–Fejér Interpolation Problem for Generalized Schur Functions

The solutions of the Carathéodory–Fejér interpolation problem for generalized Schur functions can be parametrized via a linear fractional transformation over the class of classical Schur functions. The linear fractional transformation of some of these functions may have a pole (simple or multiple) in one or more of the interpolation points or not satisfy one or more interpolation conditions, hence not all Schur functions can serve as a parameter. The set of excluded parameters is characterized in terms of the related Pick matrix.Research was supported by the Summer Research Grant from the College of William and MarySubmitted: June 26, 2002 Revised: January 31, 2003     

Strongly Regular Rings and SF-rings

in this paper, new characteristic properties of strongly regular rings are' given.Relations between certain generalizations of duo rings are also considered. The followingconditions are shown to be equivalent: (1) R is a strongly regular ring; (2) R is a left SFring such that every product of two independent closed left ideals of R is zero; (3) R is aright SF-ring such that every product of two independent closed left ideals of R is zero; (4)R is a left SF-ring whose every special left annihilator is a quasi-ideal; (5) R is a right SFring whose every special left annihilator is a quasi-ideal; (6) R is a left SF-ring whose everymaximal left ideal is a quasi-ideal; (7) R is a right SF-ring whose every maximal left ideal isa quasi-ideal; (8) R is a left SF-ring such that the set N(R) of all nilpotent elements of R isa quasi-ideal; (9) R is a right SF-ring such that N(R) is a quasi-ideal.     

The Cauchy Problem for Nonlinear Parabolic Equations with Lévy Laplacian

A solution to the Cauchy problem for a rather general class of nonlinear parabolic equations involving the infinite-dimensional Laplacian Δ_L of the form , where f is a real function defined on R³ is presented. Mathematics Subject Classifications (2000) 35R15, 46G05.     

The Perturbed Problem on the Boussinesq System of Rayleigh-Bénard Convection

In this paper,the infinite Prandtl number limit of Rayleigh-B′enard convection is studied.For well prepared initial data,the convergence of solutions in L∞(0,t;H2(G)) is rigorously justified by analysis of asymptotic expansions.     

Existence Conditions of Negative Eigenvalues in the Regular Sturm–Liouville Boundary Value Problem and Explicit Expressions for Their Number

Computational Mathematics and Mathematical Physics - For the regular Sturm–Liouville boundary value problem with general nonseparated self-adjoint boundary conditions, conditions for the...     

Multiple Boundary Concentrating Solutions to Dirichlet Problem of Hénon Equation

Abstract  Let Ω be the unit ball centered at the origin in . We study the following problem

Excursions Away from a Regular Point for One-Dimensional Symmetric Lévy Processes without Gaussian Part

The characteristic measure of excursions away from a regular point is studied for a class of symmetric Lévy processes without Gaussian part. It is proved that the harmonic transform of the killed process enjoys Feller property. The result is applied to prove extremeness of the excursion measure and to prove several sample path behaviors of the excursion and the h-path processes.     

An Optimal Control Problem Associated with SDEs Driven by Lévy-Type Processes

Abstract

In this article, we consider an optimal control problem associated with jump type stochastic differential equations driven by Lévy-type processes. The problem arises from portfolio optimization for the pair of the wealth process and the cumulative consumption process in (incomplete) financial market models. We establish the existence and the uniqueness of (constrained) viscosity solutions to the associated the integro-differential Hamilton–Jacobi–Bellman equation.     

Controllability of the Vallée–Poussin Problem for Impulsive Differential Systems

Necessary and sufficient conditions for the controllability of multipoint boundary-value problems for linear semihomogeneous and nonhomogeneous impulsive differential systems are established. The set of all controls solving such problems is described. Moreover, we provide sufficient conditions for the time-optimal control of the Vallée–Poussin problem and obtain the Pontryagin maximum principle in sufficient form.     

A Solution to a Problem of Jacobson,Kézdy and Lehel

We solve a problem proposed by Jacobson, Kézdy, and Lehel [4] concerning the existence of forbidden induced subgraph characterizations of line graphs of linear k-uniform hypergraphs with sufficiently large minimal edge-degree. Actually, we prove that for each k3 there is a finite set Z(k) of graphs such that each graph G with minimum edge-degree at least 2k²–3k+1 is the line graph of a linear k-uniform hypergraph if and only if G is a Z(k)-free graph.Acknowledgments. We thank the anonymous referees, whose suggestions helped to improve the presentation of the paper.Winter 2002/2003 DIMACS Award is gratefully acknowledged2000 Mathematics Subject Classification: 05C65 (05C75, 05C85)     

Stability in the Rotating Bénard Problem and Its Optimal Lyapunov Functions

Stability analysis of the rotating Bénard problem gives a spectral instability threshold of the purely conducting solution that can be expressed as a critical Rayleigh number R ² depending on the Taylor number T ². The definition of a functional which can be used to prove Lyapunov stability up to the threshold of spectral instability (optimal Lyapunov function) is an important step forward both, for a proof of nonlinear stability and for the investigation of the basin of attraction of the equilibrium. In previous works a Lyapunov function was found, but its optimality could be proven only for small T ². In this work we describe the reason why this happens, and provide a weaker definition of Lyapunov function which allows to prove that, for the linearized system, the spectral instability threshold is also the Lyapunov stability threshold for every value of T ².