DARBOUX TRANSFORMATION OF THE DISCRETE ABLOWITZ-LADIK EIGENVALUE PROBLEM

§1. Introduction Darboux transformation is successfully used by some authors to solve quite a few soliton equations in recent years. In this paper, we are going to use this approach to the study of some kind of discrete soliton equations. Consider the discrete Ablowitz-Ladik eigenvalue problem with four potentials:     

BEHAVIOR OF SOLUTIONS TO A DEGENERATE DIFFUSION PROBLEM

1.IntroductionInthispaper,weareconcernedwiththebehaviorofsolutionsofthefollowingproblemwheretheinitialdata"o(x)satisfiesnoEW"oo(a,b)and"o(x)20a.e.in(a,6).(1.4)Definition1.AnonnegativefunctionuELoo([0,co);W'loo(n))withfi=(a,b)iscalledaweaksolutionoftheproblem(1.1)--(1.3)ifthefollowingconditionsarefulfilled:(i)acEL'(flx[0,F])foranyT>0,(n)usatisfiestheX..d.yvaluecondition(l.2)intheusualsense.(iii)FOranytestfunctiongbEC2)1(fix[0, co))withgb(a,t)~op(b,t)~0,thereholdstheintegralequalityRecellt…     

A NEGATIVE ANSWER TO A PROBLEM OF FREMLIN AND MENDOZA

This article studies some convergence results for the McShane integral of func- tions mapping the interval[0,1]into a Banach space X from the point of view of an open problem propose by D.H.Fremlin and J.Mendoza in[2].also the anthors give a negative answer to this open problem.     

AN APPLICATION OF THE EXPONENTIAL CUBIC SPLINES TO NUMERICAL SOLUTION OF A SELF-ADJOINT PERTURBATION PROBLEM

We present a class of the second order optimal splines difference schemes derived from ex-ponential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem.We prove an optimal error estimate and give illustrative numerical example.     

SOLUTION OF A RESEARCH PROBLEM

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ON A PROBLEM OF WHITNEY

In 1944 H.Whitney raised a problem:Let M be an open smooth n-manifold.Doesthere exist an imbedding of M into R~(2n) with no limit point set?Introducing a sort of Morsenumber for open manifolds and using Whitney trick,the author gives a direct proof ofthe affirmative answer to it.     

STABILITY OF A KIND OF COMPOUND BOUNDARY VALUE PROBLEM WITH RESPECT TO THE PERTURBATION OF BOUNDARY CURVE

In this paper, we discuss the stability of general compound boundary value prob-lems combining Riemann boundary value problem for an open arc and Hilbert bound-ary value problem for a unit circle with respect to the perturbation of boundary curve.     

A NEW INTEGRABLE SYMPLECTIC MAP ASSOCIATED WITH A DISCRETE MATRIX SPECTRAL PROBLEM

A hierarchy of lattice soliton equations is derived from a discrete matrix spectral problem. It is shown that the resulting lattice soliton equations are all discrete Liouville integrable systems. A new integrable symplectic map and a family of finite-dimensional integrable systems are given by the binary nonli-nearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resulting lattice soliton equations.     

WELL-POSEDNESS OF A PARABOLIC INVERSE PROBLEM

1.IntroductionItiswellknownthatinverseproblemsinpartialdifferentialequations,mostofwhichhavenotyetbeensolveduptonow,remainasachallengeinappliedmathematics.Therefore,manymathematiciansstudiedvariousinverseproblemsforparabolicequa-tions.FOrasimplesurveywereferto[1,2,8,9]foridentifyingcoefficients,[7]foridentifyingboundaryvalues,[4,10]foridentifyingsourcetermsofparabolicequations.Wehavenotincludedalotofpapersconcerningthecomputationalmethodsusedforsolvinginverseparabolicproblems.Inthispaperthein…     

GLOBAL STABILITY OF LARGE SOLUTIONS TO THE 3D MAGNETIC BéNARD PROBLEM

In this paper,we consider the 3 D magnetic Bénard problem.More precisely,we prove that the large solutions are stable under certain conditions.And we obtain the equivalent condition with respect to this stability condition.Finally,we also establish the stability of 2 D magnetic Bénard problem under 3 D perturbations.     

CORRECTION TO “ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES

The assertion of Th.1 in[1]should be replaced bylimsup n→∞ a_nn~(k/(2k m)=∞.(A)Since the proof of Th.1 in[1]is somewhat in error,we give here a sketch ofproof of(A).Choose f∈C_ka with f(x)≥a>0 for ‖x‖≤ε>0,and define h_δ(x)=f(x) e_(kδ)(x),where e_(kδ)(x),as well as d and C_(kα)~(n)(d) to appear in the following,are thesame as in[1].Choose ■>0 so that h_δ∈C_(kα) for δ∈(0,■).For each δ in(0,■),thereexists an integer n such that h_δ∈C_(kα)~(n)(d).Hence an integer N can be found such that     

SOLUTION OF A RESEARCH PROBLEM ON TREES OF SUBSETS

ＳＯＬＵＴＩＯＮＯＦＡＲＥＳＥＡＲＣＨＰＲＯＢＬＥＭＯＮＴＲＥＥＳＯＦＳＵＢＳＥＴＳ ＷＵＳＨＩＱＵＡＮＡｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｓｏｌｖｅａｒｅｓｅａｒｃｈｐｒｏｂｌｅｍｏｎｔｒｅｅｓｏｆｓｕｂｓｅｔｓｐｏｓｅｄｂｙＦ．Ｒ．Ｍｃ?..     

THEORY OF SPECTRAL DECOMPOSITIONS WITH RESPECT TO THE IDENTITY FOR CLOSED OPERATORS

In this paper,the auther discusses some properties of closed operators acting on aBanach space with the spectral decomposition property with respect to the identity(abbrev.SDI).First,some equivalent conditions are given for a closed operator T to have the SDI.Next,for every hyperinvariant subspace Y of T with the SDI,it is proved that thecoinduced operator=T/Y has the SDI.Finally,properties of maximal nets of hyperinva-riant subspaces are discussed.     

A CHARACTERISTIC OF RIEMANNIAN SPACES ADMITTING QUASI-CONCIRCULAR TRANSFORMATION

In this paper, we have considered some properties of quasi-umbilical hypersurfaces of a Riemannian space and obtained a characteristic of Riemannian spaces admitting quasi-concircular transformation.     

REGULARITY OF SOLUTIONS TO THE DIRICHLET PROBLEM FOR DEGENERATE ELLIPTIC EQUATION

In this paper, the author studies the regularity of Solutions to the Dirichlet problem for equation Lu = f, where L is a second order degenerate elliptic operator in divergence form in Ω, a bounded open subset of Rn (n ≥3).     

THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF INITIAL VALUE PROBLEM TO SYSTEM OF GAS DYNAMICS WITH VISCOSITY

This paper considers the system where with where provided orThe traveling wave solution of problem (A), (B) is proved to be asymptoically stable when (C)_1, (C)_2 are satisfied and rarefaction wave solution is proved to be asymptotical when (C)_3, (C)_4 are satisfied, provided that the initial disturbance is suitably small and of zero constant component. The proof is given by the elemental L~2 energy method.     

REGULATION OF A FISHERY: FROM A LOCAL OPTIMAL CONTROL PROBLEM TO AN “INVARIANT DOMAIN‘ APPROACH

ABSTRACT. This study aims at regulating a fishery by reducing its catch and effort variations around a reference equilibrium point. A simple bioeconomic continuous time model is considered and the control variable acts on the rate of change of effort. Two approaches have been implemented: the first one is a local optimal approach that sets the problem in the classical linear-quadratic framework; it is extended to an “invariant domain‘ approach, whose robustness constitutes a major advantage.     

THE INITIAL VALUE PROBLEM OF A SYSTEM OF CONSERVATION LAWS

§ 1. Introduction It is very important to study the theory of the multidimensional hyperbolic conservation laws. A typical model is the following initial value problem     

A NOTE ON THE SOLUTION TO NEUMANN BOUNDARY VALUE PROBLEM OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION

The author of this paper, by means of the semi-rank theory, establish a new comparative theorem and give the existence of maximal and minimal solutions to Neumann boundary value problems of second order nonlinear differential equation in ordered Banach spaces when the upper and lower solutions in the reversed order of the problem are given.     

EXISTENCE OF GLOBAL SMOOTH SOLUTION TO INITIAL BOUNDARY VALUE PROBLEM FOR A VISCOELASTIC MODEL WITH RELAXATION

This paper is concerned with the initial boundary value problem for a vis-coelastic model with relaxation. Under the only assumption that the C^0-norm of theinitial data is small, without smallness hypothesis for the C^1-norm, the existence of theglobal smooth solution to the corresponding initial boundary value problem is proved.The analysis is based on some a priori estimates obtained by the “maximum principle” offirst-order quasilinear hyperbolic system.