Lower Bounds for Catalan's Equation

This paper begins with a short historical survey on Catalan's equation, namely x^p-y^q=1, where p andq are prime numbers and x, y are non-zero rational integers. It is conjectured that the only solution is the trivial solution 3²-2³=1. We prove that there is no non-trivial solution with p orq smaller than 30000. The tools to reach such a result are presented. A crucial role is played by a recent estimate of linear forms in two logarithms obtained by Laurent, Mignotte and Nestrenko. The criteria used are also quite recent. We give information on the enormous amount of computation needed for the verification.     

矩阵方程X-A*X-2A=Q的正定解及其扰动分析

研究非线性矩阵方程X-A*X-2A=Q(Q>0)的Hermite正定解及其扰动问题.给出了该方程存在唯-Hermite正定解的充分条件及解的迭代计算公式.在此条件下,给出了该唯一解的扰动界及正定解条件数的一种表达式,并用数值例子对所得结果进行了说明.     

A Hilbert space approach to maximum entropy reconstruction

We examine here the problem of reconstructing an X-ray attenuation function from measurements of its integrals. The approach that is taken is to maximize the difference of the entropy and the residual error in meeting the measurements. The solution of this optimization problem is constrained by requiring that the solution lie in a certain weakly compact subset of L², to be determined by physical information. We show that the constrained optimization problem is well-posed: there exists a unique solution (even when the measured data are inconsistent) and the solution depends continuously on the measurements. In the course of proving this, we show that the entropy functional is continuous on L². We further demonstrate that the solution of the optimization problem for a special case, must be piecewise constant.     

Regular splittings and the discrete Neumann problem

Summary Iterative methods are discussed for approximating a solution to a singular but consistent square linear systemAx=b. The methods are based upon splittingA=M–N withM nonsingular. Monotonicity and the concept of regular splittings, introduced by Varga, are used to determine some necessary and some sufficient conditions in order that the iterationx ⁱ⁺¹=M^–1Nxⁱ+M^–1b converge to a solution to the linear system. Finally, applications are given to solving the discrete Neumann problem by iteration which are based upon the inherent monotonicity in the formulation.This research was supported by the U. S. Army Research Office-Durham under contract no. DAHCO4 74 C 0019.     

A regularization procedure for the auto‐correlation equation

The paper deals with the auto‐correlation equation and its regularization by means of a Lavrent'ev regularization procedure in L². The solution of this quadratic integral equation of the first kind and of the regularized equation of the second kind are obtained by reduction to a boundary value problem for the Fourier transform of the solution. We prove convergence of the approximate solution to the exact solution and derive a stability estimate for the error. Copyright © John Wiley & Sons, Ltd.     

The pointwise estimates of a conservative difference scheme for Burgers' equation

In this article, we are concerned with the numerical analysis of a nonlinear implicit difference scheme for Burgers' equation. A priori estimation of the analytical solution is provided in the sense of L^∞ -norm when the initial value is bounded in H¹-norm. Conservation, boundedness, and unique solvability are proved at length. Inspired by the method of the priori estimation for the analytical solution, we prove the convergence and stability of the difference scheme in L^∞ -norm. Finally, numerical examples are carried out to verify our theoretical results.     

Large time behavior for the incompressible magnetohydrodynamic equations in half‐spaces

The weighted L^r‐asymptotic behavior of the strong solution and its first‐order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half‐space. Further, the L^∞‐decay rates of the second‐order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.     

The negative extended KdV equation with self-consistent sources: soliton, positon and negaton

The negative extended KdV equation with self-consistent sources (eKdV⁻ESCSs) is firstly presented and the associated linear auxiliary equations are derived. The generalized binary Darboux transformation (DT) is applied to construct some new solutions of the eKdV⁻ESCSs such as singular N-soliton solution, N-soliton solution with finite amplitude, N-positon solution and N-negaton solution. The properties of these solutions are analyzed. Moreover, the interactions of two solitons, positon and negaton, positon and soliton, and two positons are discussed.     

带有不连续系数的线性输运方程差分格式的收敛性

提出了一个基于三角形网格的显式差分格式逼近带有不连续系数的线性输运方程. 通过对数值解的有界性、TVD(total variation decreasing)和空间、时间方向的平移估计, 利用Kolmogorov紧性原理证明了数值解在L¹_loc模下收敛于初值问题的唯一弱解.从而得到了初值问题解的存在唯一性和关于初值的稳定性. 数值算例表明本文提出的格式计算方便而且比 Lax-Friedrichs格式更有效.       

Bright soliton solution to a generalized Burgers–KdV equation with time-dependent coefficients

We consider a generalized Burgers–KdV type equation with time-dependent coefficients incorporating a generalized evolution term, the effects of third-order dispersion, dissipation, nonlinearity, nonlinear diffusion and reaction. The exact bright soliton solution for the considered model is obtained by using a solitary wave ansatz in the form of sech^s function. The physical parameters in the soliton solution are obtained as functions of the time varying coefficients and the dependent exponents. The dependent exponents and the temporal variations of the model coefficients satisfy certain parametric conditions as shown by the obtained soliton solution. This solution may be useful to explain some physical phenomena in genuinely nonlinear dynamical systems that are described by Burgers–KdV type models.     

Generalized dynamic programming methods in integer programming

When regarded as a shortest route problem, an integer program can be seen to have a particularly simple structure. This allows the development of an algorithm for finding thek ^th best solution to an integer programming problem with max{O(kmn), O(k logk)} operations. Apart from its value in the parametric study of an optimal solution, the approach leads to a general integer programming algorithm consisting of (1) problem relaxation, (2) solution of the relaxed problem parametrically by dynamic programming, and (3) generation ofk ^th best solutions until a feasible solution is found. Elementary methods based on duality for reducingk for a given problem relaxation are then outlined, and some examples and computational aspects are discussed.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates and asymptotic behavior of extremal function to Hardy-Sobolev type inequality on the H-type group*

This paper is devoted to discuss the regularity of the weak solution to a class of non-linear equations corresponding to Hardy-Sobolev type inequality on the H-type group. Combining the Serrin's idea and the Moser's iteration, L^p estimates of the weak solution are obtained, which generalize the results of Garofalo and Vassilev in [6, 14]. As an application, asymptotic behavior of the weak solution has been discussed. Finally, doubling property and unique continuation of the weak solution are given. *This material is based upon work funded by Zhejiang Provincial Natural Science Foundation of China under Grant No. Y606144.     

Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

The subject of the paper is the derivation of error estimates for the combined finite volume-finite element method used for the numerical solution of nonstationary nonlinear convection-diffusion problems. Here we analyze the combination of barycentric finite volumes associated with sides of triangulation with the piecewise linear nonconforming Crouzeix-Raviart finite elements. Under some assumptions on the regularity of the exact solution, the L ²(L ²) and L ²(H ¹) error estimates are established. At the end of the paper, some computational results are presented demonstrating the application of the method to the solution of viscous gas flow.     

Conform and mixed finite element schemes for the Dirichlet problem for the Bilaplacian in plane domains with corners

In plane domains with corners for the Bilaplacian a uniquely solvable conform variational principle is studied on weighted Sobolev spaces which is equivalent to the standard Dirichlet problem in the weak form. Clamped plates under point forces near corners are handled by this approach. With weighted Hsieh-Clough-Tocher elements on regular triangulations as conform C¹-finite elements a new error analysis is performed without higher regularity assumptions on the exact solution than given by the data and the boundary. The rate of convergence of the error depends on the eigenvalue with smallest imaginary part of a clamped infinite wedge since this eigenvalue describes the singularity of the exact solution in a sector with same angle. Using different spaces of trial and test functions in the standard Galerkin procedure it is shown that the error in the weighted energy norm does not pollute. For convex corners asymptotic error estimates, are proved yielding convergence for a mixed method in hydrodynamics where the solution of a system of 2^nd order and its Laplacian are approximated simultaneously by C⁰-finite elements being piecewise polynomials.     

A stationary iterative pseudoinverse algorithm

Iterative methods applied to the normal equationsA ^T Ax=A ^T b are sometimes used for solving large sparse linear least squares problems. However, when the matrix is rank-deficient many methods, although convergent, fail to produce the unique solution of minimal Euclidean norm. Examples of such methods are the Jacobi and SOR methods as well as the preconditioned conjugate gradient algorithm. We analyze here an iterative scheme that overcomes this difficulty for the case of stationary iterative methods. The scheme combines two stationary iterative methods. The first method produces any least squares solution whereas the second produces the minimum norm solution to a consistent system. This work was supported by the Swedish Research Council for Engineering Sciences, TFR.     

Homogenization of the parabolic cauchy problem in the Sobolev class H<Superscript>1</Superscript>(R<Superscript>d</Superscript>)

Homogenization in the small period limit for the solution ue of the Cauchy problem for a parabolic equation in R^d is studied. The coefficients are assumed to be periodic in R^d with respect to the lattice ɛG. As ɛ → 0, the solution u ɛ converges in L2(R^d) to the solution u0 of the effective problem with constant coefficients. The solution u ɛis approximated in the norm of the Sobolev space H ¹(R^d) with error O( ɛ); this approximation is uniform with respect to the L2-norm of the initial data and contains a corrector term of order ɛ. The dependence of the constant in the error estimate on time t is given. Also, an approximation in H ¹(R^d) for the solution of the Cauchy problem for a nonhomogeneous parabolic equation is obtained.     

关于Hall问题

Hall在T型差集的研究中,产生了如下问题:方程 q~m=p~n 2,p,q是素数,m>1,n>1,(1)除开p=5,n=2,q=3,m=3以外,是否还有其他的解?这是一个未决问题。最近,孙琦和周小明在研究不定方程 a~x b~y=c~z,a,b,c是不同的素数,max(a,b,c)=19时,顺便给出了方程(1)的一个结果,即     

Enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations

Summary A solution of a nonlinear equation in Hilbert spaces is said to be a simple singular solution if the Fréchet derivative at the solution has one-dimensional kernel and cokernel. In this paper we present the enlargement procedure for resolution of singularities at simple singular solutions of nonlinear equations. Once singularities are resolved, we can compute accurately the singular solution by Newton's method. Conditions for which the procedure terminates in finite steps are given. In particular, if the equation defined in ⁿ is analytic and the simple singular solution is geometrically isolated, the procedure stops in finite steps, and we obtain the enlarged problem with an isolated solution. Numerical examples are given.This research is partially supported by Grant-in-Aid for Encouragment of Young Scientist No. 60740119, the Ministry of EducationDedicated to Professor Seiiti Huzino on his 60th birthday     

Weighted interpolation in Paley-Wiener spaces and finite-time controllability

This paper considers the solution of weighted interpolation problems in model subspaces of the Hardy space H² that are canonically isometric to Paley-Wiener spaces of analytic functions. A new necessary and sufficient condition is given on the set of interpolation points which guarantees that a solution in H² can be transferred to a solution in a model space. The techniques used rely on the reproducing kernel thesis for Hankel operators, which is given here with an explicit constant. One of the applications of this work is to the finite-time controllability of diagonal systems specified by a C₀ semigroup.