Darboux Transformation and Exact Solutions of the Myrzakulov-I Equation

The Myrzakulov-I equation is a 2＋1-dimensional generalization of the Heisenberg ferromagnetic equation and has a non-isospectral Lax pair. The Darboux transformation with non-constant spectral parameter is constructed and an extra constraint on the spectral parameter for the existence of the Darboux transformation is derived. Explicit expressions of the solutions of the Myrzakulov-I equation are presented.

Soliton resolution and asymptotic stability of N-solutions for the defocusing Kundu–Eckhaus equation with nonzero boundary conditions

In this paper,we address interesting soliton resolution,asymptotic stability of N-soliton solutions and the Painleve asymptotics for the Kundu-Eckhaus(KE)equation with nonzero boundary conditions iq_(t)+q_(xx)-2(l|q|^(2)-1)q+4β^(2)(lql^(4)-1)q+4iβ(lql^(2))_(x)q=0,q(x,0)=q_(0)(x)-±1,x→±∞.The key to proving these results is to establish the formulation of a Riemann-Hilbert(RH)problem associated with the above Cauchy problem and find its connection with the RH problem of the defocusing NLS equation.With the■-steepest descent method and the results of the defocusing NLS equation,we find complete leading order approximation formulas for the defocusing KE equation on the whole(x,t)half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region,Zakharov-Shabat asymptotics in a solitonless region and the Painlevéasymptotics in two transition regions.

Curvature Estimates for Minimal Submanifolds of Higher Codimension

The authors derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau’s results and Ecker-Huisken’s results are generalized to higher codimension. In this way, Hildebrandt-Jost-Widman’s result for the Bernstein type theorem is improved.

MODIFIED NEWTON＇S ALGORITHM FOR COMPUTING THE GROUP INVERSES OF SINGULAR TOEPLITZ MATRICES

Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton＇s iteration can be done efficiently. We show that the convergence of the modified Newton iteration is still very fast. Numerical results are presented to demonstrate the fast convergence of the proposed method.

ON SPACELIKE AUSTERE SUBMANIFOLDS IN PSEUDO-EUCLIDEAN SPACE

In this article, we construct some spacelike austere submanifolds in pseduoEuclidean spaces. We also get some indefinite special Lagrangian submanifolds by constructing twisted normal bundle of spacelike austere submanifolds in pseduo-Euclidean spaces.

On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

In this paper, we introduce the notion of a （2＋1）-dimenslonal differential equation describing three-dimensional hyperbolic spaces （3-h.s.）. The （2＋1）-dimensional coupled nonlinear Schrodinger equation and its sister equation, the （2＋1）-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s, The （2 ＋ 1 ）-dimensional generalized HF model：St=（1/2i[S,Sy]＋2iσS）x,σx=-1/4i tr（SSxSy）, in which S ∈ GLc（2）/GLc（1）×GLc（1）,provides another example of （2＋1）-dimensional differential equations describing 3-h.s. As a direct con-sequence, the geometric construction of an infinire number of conservation lairs of such equations is illustrated. Furthermore we display a new infinite number of conservation lairs of the （2＋1）-dimensional nonlinear Schrodinger equation and the （2＋1）-dimensional derivative nonlinear Schrodinger equation by a geometric way.

On Singular Sets of Local Solutions to p-Laplace Equations

The author proves that the on the singular set of a local solution to existence of an optimal control problem. right-hand term of a p-Laplace equation is zero the equation. Such a result is used to study the

Partial energies monotonicity and holomorphicity of Hermitian pluriharmonic maps

In this paper, we introduce the notion of Hermitian pluriharmonic maps from Hermitian manifold into Kiihler manifold. Assuming the domain manifolds possess some special exhaustion functions and the vecotor field V = JMδJM satisfies some decay conditions, we use stress-energy tensors to establish some monotonicity formulas of partial energies of Hermitian pluriharmonic maps. These monotonicity inequalities enable us to derive some holomorphicity for these Hermitian pluriharmonic maps.

Global Exact Boundary Controllability for Cubic Semi-linear Wave Equations and Klein-Gordon Equations

The authors prove the global exact boundary controllability for the cubic semilinear wave equation in three space dimensions,subject to Dirichlet,Neumann,or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem.The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method,which reduces the global exact boundary controllability problem to a local one.The proof is carried out in line with [2,15].Then a constructive method that has been developed in [13] is used to study the local problem.Especially when the region is star-complemented,it is obtained that the control function only need to be applied on a relatively open subset of the boundary.For the cubic Klein-Gordon equation,similar results of the global exact boundary controllability are proved by such an idea.

The Cartesian analytical solutions for the Ndimensional compressible Navier-Stokes equations with density-dependent viscosity

In this paper,we prove the existence of general Cartesian vector solutions u=b(t)+A(t)x for the Ndimensional compressible Navier–Stokes equations with density-dependent viscosity,based on the matrix and curve integration theory.Two exact solutions are obtained by solving the reduced systems.

Betti Numbers of Locally Standard 2-Torus Manifolds

Let Mn be a smooth closed n-manifold with a locally standard （Z2）n-action. This paper deals with the relationship among the rood 2 Betti numbers of Mn, the mod 2 Betti numbers and the h-vector of the orbit space of the action.

The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth

The paper is devoted to the Cauchy problem of backward stochastic superparabolic equations with quadratic growth.We prove two Ito formulas in the whole space.Furthermore,we prove the existence of weak solutions for the case of onedimensional state space,and the uniqueness of weak solutions without constraint on the state space.

CONDITION NUMBER FOR WEIGHTED LINEAR LEAST SQUARES PROBLEM

In this paper, we investigate the condition numbers for the generalized matrix inversion and the rank deficient linear least squares problem： minx ｜｜Ax- b｜｜2, where A is an m-by-n （m ≥ n） rank deficient matrix. We first derive an explicit expression for the condition number in the weighted Frobenius norm ｜｜ [AT,βb] ｜｜F of the data A and b, where T is a positive diagonal matrix and β is a positive scalar. We then discuss the sensitivity of the standard 2-norm condition numbers for the generalized matrix inversion and rank deficient least squares and establish relations between the condition numbers and their condition numbers called level-2 condition numbers.

A Second Order Accurate in Time, Energy Stable Finite Element Scheme for the Flory-Huggins-Cahn-Hilliard Equation

In this paper, we propose and analyze a second order accurate in time, masslumped mixed finite element scheme for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. The standard backward differentiation formula (BDF)stencil is applied in the temporal discretization. In the chemical potential approximation,both the logarithmic singular terms and the surface diffusion term are treatedimplicitly, while the expansive term is explicitly updated via a second-order Adams-Bashforth extrapolation formula, following the idea of the convex-concave decompositionof the energy functional. In addition, an artificial Douglas-Dupont regularizationterm is added to ensure the energy dissipativity. In the spatial discretization, the masslumped finite element method is adopted. We provide a theoretical justification of theunique solvability of the mass lumped finite element scheme, using a piecewise linearelement. In particular, the positivity is always preserved for the logarithmic argumentsin the sense that the phase variable is always located between -1 and 1. In fact, thesingular nature of the implicit terms and the mass lumped approach play an essentialrole in the positivity preservation in the discrete setting. Subsequently, an unconditionalenergy stability is proven for the proposed numerical scheme. In addition, theconvergence analysis and error estimate of the numerical scheme are also presented.Two numerical experiments are carried out to verify the theoretical properties.

On Galois Extension of Hopf Algebras

Let H be a cosemisimple Hopf algebra over a field k, and π ： A→ H be a surjective cocentral bialgebra homomorphism of bialgebras. The authors prove that if A is Galois over its coinvariants B=LH Ker π and B is a sub-Hopf algebra of A, then A is itself a Hopf algebra. This generalizes a result of Cegarra [3] on group-graded algebras.

Mean Curvature Flow with Convex Gauss Image

In this paper,the mean curvature flow of complete submanifolds in Euclidean space with convex Gauss image and bounded curvature is studied.The confinable property of the Gauss image under the mean curvature flow is proved,which in turn helps one to obtain the curvature estimates.Then the author proves a long time existence result.The asymptotic behavior of these solutions when t→∞is also studied.