A note on Poisson brackets for orthogonal polynomials on the unit circle

The connection of orthogonal polynomials on the unit circle to the defocusing Ablowitz–Ladik integrable system involves the definition of a Poisson structure on the space of Verblunsky coefficients. In this paper, we compute the complete set of Poisson brackets for the monic orthogonal and the orthonormal polynomials on the unit circle, as well as for the second kind polynomials and the Wall polynomials. This answers a question posed by Cantero and Simon (J Approx Theory 158(1):3–48, 2009), for the case of measures with finite support. We also show that the results hold for the case of measures with periodic Verblunsky coefficients.     

Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in L ² under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.     

On the minimum principle for dipolar materials with stretch

In the present paper we generalize the results obtained by Iesan and Quintanilla for microstretch elastic bodies in order to cover dipolar elastic materials with stretch. For the boundary value problem considered in this context, we prove a generalized existence result and, also, an extension of the principle of minimum potential energy.     

Classification of Harmonic Functions in the Exterior of the Unit Ball

We solve the Laplace equation in an exterior infinite spherical domain with nonlinear (quadratic) boundary conditions on the spherical boundary. We linearize the problem and, under the additional assumption that the distinguishing function is spherically symmetric, write the solution by using the formal power series method with recursion of the series coefficients. Applying the Poincaré--Perron theorem, we describe the space of convergent formal power series and calculate its dimension. Estimating the roots of the fourth-degree characteristic polynomial corresponding to the given problem, we also calculate the dimension of the space of functions whose gradient at each point of the sphere is orthogonal to the linear combination of an axially symmetric dipole and a quadrupole. In conclusion, we state several unsolved problems arising in geophysical applications.     

Periodic Solutions of the Abelian Higgs Model and Rigid Rotation of Vortices

The metric and potential energy on the reduced moduli space of selfdual vortices in the Abelian Higgs model on are computed in a certain limit, first identified by Bradlow. In this limit it is proved that the Higgs field is asymptotic to a standard holomorphic section. These results are then used to prove a theorem asserting the existence of time-periodic solutions of the Abelian Higgs model on which represent two vortices in rigid rotation about one another. The theorem answers affirmatively the question, raised by Jaffe and Taubes, of whether a balance between the inter-vortex attraction and the centrifugal repulsion provides for the existence of such solutions (as it does in the classical two body problem for point particles.) The starting point of the analysis is the adiabatic limit system, i.e. the Hamiltonian system defined by restricting the Abelian Higgs model to the moduli space of self-dual vortices. The Hamiltonian consists of a potential energy term and kinetic energy term which is given by the metric on the moduli space induced from . It is shown under two assumptions on the metric and potential energy that the adiabatic limit system admits periodic solutions of the required type. Periodic solutions to the full system are then obtained by an application of the implicit function theorem. Explicit examples where the assumptions on the adiabatic limit system hold are provided by the computations of the metric and potential in the Bradlow limit. Submitted: March 1998.     

On the convergence rates of subdivision algorithms for box spline surfaces

Dahmen and Micchelli [8] have shown that in general the coefficients of the refined control nets of a box spline surface converge to the surface at (at least) the rate of the refinement. The purpose of this article is to show that under mild additional assumptions the convergence rate is even quadratic. Although this rate is in general best possible, we point out under what circumstances even higher rates are obtained (locally).     

On the Characterization of Preinvex Functions

In Refs. [J. Math. Anal. Appl. 258:287–308, [2001]; J. Math. Anal. Appl. 256:229–241, [2001]], Yang and Li presented a characterization of preinvex functions and semistrictly preinvex functions under a certain set of conditions. In this note, we show that the same results or even more general ones can be obtained under weaker assumptions; we also give a characterization of strictly preinvex functions under mild conditions. This research was supported by the National Natural Science Foundation of China under Grants 70671064 and 60673177, and the Education Department Foundation of Zhejiang Province Grant 20070306. The authors thank Professor F. Giannessi for valuable comments on the original version of this paper.     

SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN R~n

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs. The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.     

A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems

A finite difference method for a time-dependent convection-diffusion problem in one space dimension is constructed using a Shishkin mesh. In two recent papers (Clavero et al. (2005) [2] and Mukherjee and Natesan (2009) [3]), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in [2] and [3] and under a slightly less restrictive condition on the mesh.     

Time-Dependent Loss of Derivatives for Hyperbolic Operators with Non Regular Coefficients

In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension N ≥ 1. We will suppose the coefficients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradifferential calculus with parameters will be the main tool to get energy estimates in Sobolev spaces and these estimates will present a time-dependent loss of derivatives.     

SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN $R^n$

This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.     

Perturbation of Orthogonal Polynomials on an Arc of the Unit Circle

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szeg recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The polynomials then live essentially on the are {e^iθ : α ≤ θ ≤ 2 π − α) where cos(α/2) [formula] with α (0, π). We analyze the orthogonal polynomials by comparing them with the orthogonal polynomials with constant reflection coefficients, which were studied earlier by Ya. L. Geronimus and N. I. Akhiezer. In particular, we show that under certain assumptions on the rate of convergence of the reflection coefficients the orthogonality measure will be absolutely continuous on the are. In addition, we also prove the unit circle analogue of M. G. Krein′s characterization of compactly supported nonnegative Borel measures on the real line whose support contains one single limit point in terms of the corresponding system of orthogonal polynomials.     

Multidimensional high-field limit of the electrostatic Vlasov-Poisson-Fokker-Planck system

In this paper, the high-field limit of the Vlasov-Poisson-Fokker-Planck system for charged particles is rigorously derived. The first result is obtained in any space dimension by using modulated energy techniques. It requires the smoothness of the solutions of the limit problem. In dimension 2, it is possible to handle more general data by using methods developed for a diagonal defect measures theory. The convergence of the concentration of particles is obtained in the space of bounded measures. In both cases, the limit of the sequence of densities of distribution functions is shown to solve a nonlinear system of partial differential equations which is related to Ohm's law.     

广义Korteweg-de Vries-Burgers方程组的谱方法及其误差估计

引言 近十多年来,偏微分方程的谱方法发展迅速,主要是由于在谱方法计算中应用了快速Pourier变换(FFT),减少了计算量,使之具有实用价值.谱方法的另一优点是具有“无穷阶”的快敛速,即,若原微分方程的解是无穷可微的,则合适的(半离散)谱方法逼近的收敛性比N~(-1)的任何幂次都快(这里N是所取基函数的个数).郭本瑜提供了证明KdV-Burgers方程谱方法格式产格误差估计的技巧,并在[6]中推广到二维涡度方程,证     

On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback

The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24-52] where the system is claimed to be nondissipative.     

About partial boundary dissipation to Timoshenko system with delay

We consider the Timoshenko model with partial dissipative boundary condition with delay, and we prove that the solution decays exponentially to zero, provided the wave speed are equal; this improve earlier result due to Bassam et al and Muñoz Rivera and Naso. Moreover, consider the exponential stability to the corresponding semilinear problems.     

A note for tightening 0–1 models

We describe two approaches for 0–1 program model tightening that are based on the coefficient increasing and reduction methods proposed in Dietrich, Escudero and Chance (1993). We present some characterizations for the new formulations to be tighter than the original model. It can be shown that tighter models can be obtained even when applying any of both approaches to a redundant constraint; see Escudero and Muñoz (1998). We also present some situations where these approaches cannot be applied.     

The ice flow behavior in the neighborhood of the grounding line. Non-Newtonian case

In this paper we address one of the problems that has attracted much interest in the glaciological scientific community which is the grounding line dynamics. The grounding line is the line where transition between ice attached to the solid ground and ice floating over the sea takes place. We analyze a mathematical model describing the ice flow near the grounding line where the ice is considered a non-Newtonian fluid. This generalizes the results obtained in [M.A. Fontelos, A.I. Muñoz, A free boundary problem in glaciology: The motion of grounding lines, Interfaces Free Bound. (9) (2007) 67–93] for the Newtonian case and allows us to consider a more realistic rheological model. We prove the existence and uniqueness (in a class to be defined) of weak solutions with moving grounding lines and zero contact angle and also determine the shape and asymptotic properties of the free boundary. Finally some finite element numerical simulations will illustrate the local and global behavior of the problem solutions.     

Minimal algebras and 2-step nilpotent Lie algebras in dimension 7

We use the methods of Bazzoni and Muñoz (Trans Am Math Soc 364:1007–1028, 2012) to give a classification of 7-dimensional minimal algebras, generated in degree 1, over any field ${\mathbf{k}}$ of characteristic ${{\rm char}(\mathbf{k})\neq 2}$ , whose characteristic filtration has length 2. Equivalently, we classify 2-step nilpotent Lie algebras in dimension 7. This classification also recovers the real homotopy type of 7-dimensional 2-step nilmanifolds.     

Uniform decay in a Timoshenko-type system with past history

Fernández Sare and Rivera [H.D. Fernández Sare, J.E. Muñoz Rivera, Stability of Timoshenko systems with past history, J. Math. Anal. Appl. 339 (1) (2008) 482–502] considered the following Timoshenko-type system

ρ₁φ_tt−K(φ_x+ψ)_x=0,