THE FRACTIONAL QUADRATIC-FORM IDENTITY AND BI-HAMILTONIAN STRUCTURES OF AN INTEGRABLE COUPLING OF THE FRACTIONAL COUPLED BURGERS HIERARCHY

Based on a general isospectral problem of fractional order and the fractional quadratic-form identity by Yue and Xia, the new integrable coupling of fractional coupled Burgers hierarchy and its fractional bi-Hamiltonian structures are obtained.     

STARLIKENESS AND SUBORDINATION PROPERTIES OF A LINEAR OPERATOR

In this paper, we introduce a new class of p-valent analytic functions defined by using a linear operator Lαk. For functions in this class Hαk(p, λ; h) we estimate the coefficients.Furthermore, some subordination properties related to the operator Lαkare also derived.     

THE CAUCHY PROBLEM OF THE HARTREE EQUATION

In this paper, we systematically study the wellposedness, illposedness of the Hartree equation, and obtain the sharp local wellposedness, the global existence in Hs, s≥1 and the small scattering result in Hs for 2 〈γ〈 n and s ≥ 2/γ - 1. In addition, we study the nonexistence of nontrivial asymptotically free solutions of the Hartree equation.     

THE SENSITIVITY OF THE EXPONENTIAL OF AN ESSENTIALLY NONNEGATIVE MATRIX

This paper performs perturbation analysis for the exponential of an essentially nonnegative matrix which is perturbed in the way that each entry has a small relative perturbation. For a general essentially nonnegative matrix, we obtain an upper bound for the relative error in 2-norm, which is sharper than the existing perturbation results. For a triangular essentially nonnegative matrix, we obtain an upper bound for the relative error in entrywise sense. This bound indicates that, if the spectral radius of an essentially nonnegative matrix is not large, then small entrywise relative perturbations cause small relative error in each entry of its exponential. Finally, we apply our perturbation results to the sensitivity analysis of RC networks and complementary distribution functions of phase-type distributions.     

NUMERICAL LOCALIZATION OF ELECTROMAGNETIC IMPERFECTIONS FROM A PERTURBATION FORMULA IN THREE DIMENSIONS

This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.     

INSTABILITY TO DIFFERENTIAL EQUATIONS OF FIFTH AND SIXTH ORDER WITH VARIABLE DEVIATING ARGUMENTS

In this paper, we discuss the instability of the zero solutions to certain nonlinear differential equations of fifth and sixth order with multiple retarded arguments. By the Lyapunov＇s direct method, sufficient conditions for instability of the zero solutions are obtained.     

UNSTABLE SOLUTIONS TO A CLASS OF VECTOR DIFFERENTIAL EQUATIONS OF SIXTH ORDER

In this paper, we consider a class of nonlinear vector differential equations of sixth order. By constructing appropriate Lyapunov functions, the non-existence of periodic solutions is established. Moreover, we provide an example to show the feasibility of our results. Our results extend and improve two related results in the previous literature from scalar cases to vectorial cases.     

A REMARK ON THE BLOW-UP CRITERION OF STRONG SOLUTIONS AND REGULARITY FOR WEAK SOLUTIONS OF NAVIER-STOKES EQUATIONS

We give blow-up criteria of strong solutions of Navier-Stokes equations with its initial data in Besov spaces and consider the regularity of Leray-Hopf solutions of the equation.     

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A CLASS OF NONLINEAR FRACTIONAL ORDER DIFFERENTIAL EQUATIONS WITH DELAY

A class of nonlinear fractional order differential equations with delay is investigated in this paper. Using Leray-Schauder fixed point theorem and the contraction mapping theorem, we obtain some sufficient conditions for the existence and uniqueness of solutions to the fractional order differential equations.     

EXISTENCE AND MULTIPLICITY OF SOLUTIONS TO BOUNDARY-VALUE PROBLEMS OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, using two fixed-point theorems, we consider the existence and mul- tiplicity results of solutions to a nonlinear two point boundary value problem. In argument, the properties of the Green function play an important role.     

EXTINCTION AND PERMANENCE OF TWO COMPETING SPECIES WITH IMPULSES

In this paper we consider a nonautonomous two dimensional Lotka-Volterra system with impulsive effect. We prove that one of components is driven to extinction while the other stabilizes at a certain solution of an impulsive logistic equation under some conditions.     

CONSENSUS FORMATION OF TWO-LEVEL OPINION DYNAMICS

Opinion dynamics have received significant attention in recent years. This paper proposes a bounded confidence opinion model for a group of agents with two different confidence levels. Each agent in the population is endowed with a confidence interval around her opinion with radius αd or(1- α)d, where α∈(0, 1/2] represents the differentiation of confidence levels. We analytically derived the critical confidence bound dc = 1/(4α) for the two-level opinion dynamics on Z. A single opinion cluster is formed with probability 1 above this critical value regardless of the ratio p of agents with high/low confidence. Extensive numerical simulations are performed to illustrate our theoretical results. Noticed is a clear impact of p on the collective behavior: more agents with high confidence lead to harder agreement. It is also experimentally revealed that the sharpness of the threshold dc increases with α but does not depend on p.     

PYRAMIDAL TRAVELING FRONTS OF BISTABLE REACTION-DIFFUSION EQUATIONS WITH DELAY

This paper is concerned with nonplanar traveling fronts for delayed reaction- diffusion equation with bistable nonlinearity in RTM （m〉 3）. By the comparison principle and super- and subsolutions technique, we establish the existence of pyra- midal traveling fronts.     

BIFURCATIONS OF DEGENERATE SINGULAR POINT OF A PREDATOR-PREY SYSTEM WITH HOLLING-IV FUNCTIONAL RESPONSE

A predator-prey system with Holling-IV functional response is investigated. It is shown that the system has a positive equilibrium, which is a cusp of co-dimension 2 under certain conditions. When the parameters vary in a small neighborhood of the values of parameters, the model undergoes the Bogdanov-Takens bifurcation. Dif- ferent kinds of bifurcation phenomena are exhibited, which include the saddle-node bifurcation, the Hopf bifurcation and the homo-clinic bifurcation. Some computer simulations are presented to illustrate the conclusions.     

DECAY RATES TOWARD STATIONARY WAVES OF SOLUTIONS FOR DAMPED WAVE EQUATIONS

Abstract This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the half space R＋{utt-txx＋ut＋f（u）x=0,t〉0,x∈R＋,u（0,x）=u0（x）→u＋,asx→＋∞,ut（0,x）=u1（x）,u（t,0）=ub.For the non-degenerate case f]（u＋） 〈 0, it is shown in [1] that the above initialboundary value problem admits a unique global solution u（t,x） which converges to the stationary wave φ（x） uniformly in x ∈ R＋ as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. Moreover, by using the space-time weighted energy method initiated by Kawashima and Matsumura [2], the convergence rates （including the algebraic convergence rate and the exponential convergence rate） of u（t, x） toward φ（x） are also obtained in [1]. We note, however, that the analysis in [1] relies heavily on the assumption that f＇（ub） 〈 0. The main purpose of this paper is devoted to discussing the case of f＇（ub）= 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.     

EXISTENCE OF MULTIPLE SOLUTIONS TO A FRACTIONAL DIFFERENCE BOUNDARY VALUE PROBLEM WITH PARAMETER

By establishing the corresponding variational framework, and using critical point theory, we give the existence of multiple solutions to a fractional difference boundary value problem with parameter. Under some suitable assumptions we obtain some results which ensure the existence of well precise interval of parameter for which the problem admits multiple solutions.     

UNIQUENESS AND VALUE DISTRIBUTION ON DIFFERENCE POLYNOMIALS OF MEROMORPHIC FUNCTIONS

In this paper, we study the value distribution question of certain difference polynomials of meromorphic functions, and investigate the uniqueness questions of meromorphic functions whose certain difference polynomials share a small functions. The results in this paper improve and extend the corresponding results in Zhang [16] and Chen [1].     

REAL ROOT ISOLATION OF SPLINE FUNCTIONS

In this paper, we propose an algorithm for isolating real roots of a given univariate spline function, which is based on the use of Descartes＇ rule of signs and de Casteljau algorithm. Numerical examples illustrate the flexibility and effectiveness of the algorithm.     

BIFURCATION OF BOUNDED SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS DEPENDING ON A PARAMETER

Using the notion of an isolated invariant set and an isolating block, an existence criterion of bifurcation points of nonstationary bounded solutions to ordinary differential systems depending on a parameter is given.     

EXISTENCE OF THREE SOLUTIONS TO NONLINEAR DISCRETE DIRICHLET PROBLEM INVOLVING p-LAPLACIAN

In this paper, we study a nonlinear Dirichlet boundary value problem for difference equations involving the p-Laplacian with two positive parameters. One result on the existence of at least three solutions for this problem is obtained using critical point theory.