The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.     

Rigidity of higher elliptic genera

We prove some general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes. We also obtain the vanishing of some higher elliptic genera.Both authors are supported in part by NFS     

The mixed problem for a nonlinear degenerate elliptic equation of second order

The present paper deals with the mixed boundary value problem for a nonlinear elliptic equation with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the uniqueness and existence of solutions of the above problem for the nonlinear elliptic equation by the extremum principle and the method of parameter extension. The complex method is used to discuss the corresponding problem for degenerate elliptic complex equation of first order and then that of second order.     

Boundary value problems for two types of degenerate elliptic systems in R~4

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R~4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system's solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R~4 are derived.     

Convergence of the arithmetic-geometric mean procedure for the complex variables and the calculation of the complete elliptic integrals with complex modulus

The convergence of the arithmetic-geometric mean procedure is checked for complex variables. The procedure is shown to be useful for the evaluation of the complete elliptic integrals of the first and second kinds with complex modulus. It is suggested that the procedure will be useful also for the numerical calculation of the elliptic integrals and the Jacobian elliptic functions with complex modulus in general.     

Refined elliptic tropical enumerative invariants

We suggest a new refined (i.e., depending on a parameter) tropical enumerative invariant of toric surfaces. This is the first known enumerative invariant that counts tropical curves of positive genus with marked vertices. Our invariant extends the refined rational broccoli invariant invented by L. Göttsche and the first author, though there is a serious difference between the invariants: our elliptic invariant counts weights assigned partly to individual tropical curves and partly to collections of tropical curves, and our invariant is not always multiplicative over the vertices of the counted tropical curves as was the case for other known tropical enumerative invariants of toric surfaces. As a consequence we define elliptic broccoli curves and elliptic broccoli invariants as well as elliptic tropical descendant invariants for any toric surface.     

EXTENDABILITY OF SOLUTIONS FOR THE LINEAR SYSTEM OF ELLIPTIC TYPE EQUATIONS

The solutions of linear system of elliptic type equations with first order isdiscussed by using the method of several complex analysis and, a series of newextended results of the solutions for the system of elliptic type are obtained.     

On the stable difference scheme for source identification nonlocal elliptic problem

Approximation of source identification problem for elliptic equation with integral-type nonlocal condition is discussed. The first order of accuracy difference scheme for elliptic nonlocal identification problem is studied. By using spectral resolution of a self-adjoint operator, we establish stability inequalities for solution of constructed scheme. Subsequently, the difference scheme for approximate solution of multidimensional boundary value problem with integral-type nonlocal and first kind boundary conditions is investigated on stability. Numerical test examples are presented.     

Primitive divisors of elliptic divisibility sequences

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.     

Sharp bounds for the first eigenvalue of symmetric Markov processes and their applications

By adopting a nice auxiliary transform of Markov operators, we derive new bounds for the first eigenvalue of the generator corresponding to symmetric Markov processes. Our results not only extend the related topic in the literature, but also are efficiently used to study the first eigenvalue of birth-death processes with killing and that of elliptic operators with killing on half line. In particular, we obtain two approximation procedures for the first eigenvalue of birth-death processes with killing, and present qualitatively sharp upper and lower bounds for the first eigenvalue of elliptic operators with killing on half line.     

Meromorphic Traveling Wave Solutions of the Complex Cubic-Quintic Ginzburg-Landau Equation

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie??s lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.     

Boundary value problems for two types of degenerate elliptic systems in <Emphasis Type="Italic">R</Emphasis><Superscript>4</Superscript>

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R ⁴ is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system’s solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R ⁴ are derived.     

Calculation of the elliptic integrals of the first and second kinds with complex modulus

Summary. Numerical procedures for calculating the elliptic integrals of the first and the second kind with complex modulus and their analytic continuations are presented. The corresponding results for the elliptic integral of the third kind are given in Appendices. Received September 9, 1996 / Revised version received June 2, 1998     

A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

Based on the homogeneous balance method,the Jacobi elliptic expansion method and the auxiliary equation method,the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations.New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple.The method is also valid for other (1+1)-dimensional and higher dimensional systems.     

Noncoercive Boundary Value Problems for Elliptic Partial Differential and Differential-Operator Equations

In this paper we find conditions guarantee that irregular boundary value problems for elliptic differential-operator equations of the second order in an interval are fredholm. We apply this result to find some algebraic conditions guarantee that irregular boundary value problems for elliptic partial differential equations of the second order in cylindrical domains are fredholm. Apparently this is the first paper where the regularity of an elliptic boundary value problem is not satisfied on a manifold of the dimension equal to dimension of the boundary. Nevertheless the problem is fredholm and the resolvent is compact. It is interesting to note that the considered boundary value problems for elliptic equations in a cylinder being with separating variables are noncoercive.     

ON BOUNDARY VALUE PROBLEMS FOROVERDETERMINED ELLIPTIC SYSTEMS OF TWO COMPLEX VARIABLES

In this paper,based on previous results,the Riemann-Hilbert boundary valueproblems with general forms for overdetermined elliptic equations of first order are consi-dered.The characteristic of modified function space is given.It is proved that there existsa unique solution for modified problem of the problem which we discuss.By the way,itis pointed out that there are great differences between overdetermined elliptic systems andfirst order elliptei systems in the plane.     

The Green-matrix for elliptic systems which satisfy the Legendre-Hadamard condition

We study a generalization of the Green-matrix for strongly elliptic systems of second order to elliptic systems which satisfy the weaker Legendre-Hadamard condition. The main result is that even in this more general case a Green-matrix G exists, moreover we have the standard local estimates for G and its first partial derivatives.     

The commutativity of integrals of motion for quantum spin chains and elliptic functions identities

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show their liner independence for the number of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.      

Perfect powers in elliptic divisibility sequences

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations.     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.