完全非线性一致椭圆方程的外问题

利用Perron方法得到了完全非线性一致椭圆方程外问题具有渐近性质的粘性解的存在性.     

拟线性双曲方程组一类广义Riemann问题的整体间断解

对一阶拟线性双曲方程组一类广义Riemann问题，该文在一定条件下证明了，包含一个接触间断和一个中心波以及包含一个接触间断和一个激波的间断解的整体存在性和唯一性.     

The domination numbers of the 5 × n and 6 × n grid graphs

The k × n grid graph is the product P_k × P_n of a path of length k ? 1 and a path of length n ? 1. We prove here formulas found by E. O. Hare for the domination numbers of P₅ × P_n and P₆ × P_n. © 1993 John Wiley & Sons, Inc.     

An Algorithm for Minimax Solution of Overdetermined Systems of Non-linear Equations

The problem of minimizing the maximum residual of a system ofnon-linear equations is studied in the case where the numberof equations is larger than the number of unknowns. It is supposedthat the functions defining the problem have continuous firstderivatives, and the algorithm is based on successive linearapproximations to these functions. The resulting linear systemsare solved in the minimax sense, subject to bounds on the solutions,the bounds being adjusted automatically, depending on the goodnessof the linear approximations. It is proved that the method alwayshas sure convergence properties. Some numerical examples aregiven.     

Boundary-Value Problems for Second-Order Equations and for Systems of First-Order Equations

The oblique derivative problem for harmonic functions under violation of the Shapiro–Lopatinsky condition is considered as well as some multi-dimensional analogues of the Cauchy–Riemann system. These problems are reduced to nonelliptic pseudo-differential equations. A method generalizing the regularization of singular integral equations is also presented.     

The crossing number of C5 × Cn

We prove that the crossing number of C₅ × C_n is 3n, which is consistent with the general conjecture that the crossing number of C_m × C_n is (m − 2)n, for 3 ≤ m ≤ n. © 1996 John Wiley & Sons, Inc.     

完全非线性抛物方程障碍问题的强解

在自然结构条件下证明了具有初值和非线性斜边值的二阶完全非线性抛物方程障碍问题W~(2.1)_∝强解的存在唯一性．     

A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations

Summary A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations is presented in this paper, based on the non-zero value of the topological degree. Further, while the method does not compute the topological degree, it takes care of keeping its non-zero value during the bisections and thus results in a fast bisection algorithm.     

On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics

We give a simple proof of the existence of solutions of the dispersion management and diffraction management equations for a zero average dispersion, respectively, diffraction. These solutions are found as maximizers of non-linear and non-local variational problems which are invariant under a large non-compact group. Our proof of the existence of a maximizer is rather direct and avoids the use of Lions’ concentration compactness argument or Ekeland’s variational principle. The existence of diffraction managed solitons in the discrete case is shown under the weakest possible assumptions on the diffraction profile, and the existence of dispersion managed solitons in the continuous case is shown under very mild conditions on the dispersion profile, which cover all physically relevant cases.     

Mixed Boundary Value Problems for some Linear and Non-linear Elliptic Difference Equations

We obtain a priori bounds on difference quotients for some mixedboundary value problems associated with uniformly elliptic differenceequations over rectangular regions. These results are appliedto mildly quasi-linear and mildly non-linear problems to establishexistence and uniqueness criteria. Our estimates are usefulin the numerical computation of these solutions.     

Boundary Value Problems for Systems of Functional Differential Equations

Algorithms for finding an approximate solution of boundary value problems for systems of functional ordinary differential equations are studied. Sufficient conditions for consistency and convergence of these methods are given. In the last section, a construction of methods of arbitrary order is presented.     

Operator Interpretation of Resonances Arising in Spectral Problems for 2 × 2 Operator Matrices

We consider operator matrices ${\rm{H = }}\left( {{_{B_{10} }^{A{_0}}} {_{A_{1} }^{B{_{01}}}} } \right)$ with self - adjoint entries A_i, i = 0,1, and bounded B₁₀ = B₁₀ acting in the orthogonal sum μ=μ₀ ⊕ μ₁ of Hilbert spaces μ₀ and μ₁. We are especially interested in the case where the spectrum of, say, A₁ is partly or totally embedded into the continuous spectrum of A₀ and the transfer function V₁(z) = B₁₀ where (χ - A₀)^?1 B₀₁, admits analytic continuation (as an operator - valued function) through the cuts along branches of the continuous spectrum of the entry A₀ into the unphysical sheet(s) of the spectral parameter plane. The values of χ in the unphysical sheets where M₁^?1(z) and consequently the resolvent (H - z)^?1 have poles are usually called resonances. A main goal of the present work is to find non - selfadjoint operators whose spectra include the resonances as well as to study the completeness and basis properties of the resonance eigenvectors of M₁(z) in μ₁. To this end we first construct an operator - valued function V₁(Y) on the space of operators in μ₁ possessing the property: V₁(Y)?₁ = V₁(z)?₁ for any eigenvector ?₁ of Y corresponding to an eigenvalue z and then study the equation H₁ = A₁ + V₁(H₁). We prove the solvability of this equation even in the case where the spectra of A₀ and A₁ overlap. Using the fact that the root vectors of the solutions H₁ are at the same time such vectors for M₁(z), we prove completeness and even basis properties for the root vectors (including those for the resonances).     

Initial and Nonlinear Oblique Boundary Value Problems for Fully Nonlinear Parabolic Equations

We consider the initial and nonlinear oblique derivative bouodary value problem for fully nonlinear uniformly parabolic partial differential equations of second order. The parabolic operators satisfy natural structure conditions which have been introduced by Krylov. The nonlinear boundary operalors satisfy certain natural structure conditions also. The existence and uniqueness of classical solution are proved when the initial boundary values and the coefficients of the equation are suitable smooth.     

Degenerate Nonlinear Boundary Value Problems for Fully Nonlinear Elliptic Equations of Second Order

In this paper, we will consider the degenerate nonlinear boundary value problems for nonlinear elliptic equations. We extend Lieberman & Trudinger's results from nondegenerate oblique derivative boundary values to degenerate boundary values.     

一般形式的一阶椭圆组的非线性Riemann问题

讨论一般形式的一阶线性和拟线性椭圆型方程的非线性Riemann问题,通过把这些问题转化为奇异积分方程,利用压缩原理和广义压缩原理来证明在某些假设条件下所讨论问题的解的存在性．     

Models and Algorithms for Carsharing Systems and Related Problems

In a Carsharing System, a fleet of cars is distributed at specified stations in an urban area, users can take and return cars at any time and station. For operating such a system in a satisfactory way, the stations have to keep a good ratio between the total number of places and the number of cars in each station, in order to refuse as few customer requests as possible. In this work, we propose to model the resulting problem of balancing the load of the stations as a General Pickup and Delivery Problem. As problems of this type are known to be hard, we discuss possible heuristic approaches both for the static (offline) and the dynamic (online) version of the problem, and give approximation results for special cases.     

Nonlinear Degenerate Oblique Boundary Value Problems for Second Order Fully Nonlinear Elliptic Equations

In this paper we study the existence theorem for solution of the nonlinear degenerate oblique boundary value problems for second order fully nonlinear elliptic equations F(x, u, Du, D²u) = 0 \quad x ∈ Ω, G(x, u, D, u) = 0, \qquad x ∈ ∂Ω where F (x, z, p, r) satisfies the natural structure conditions, G (x, z, q) satisfies G_q ≥ 0, G_x ≤ - G_0 < 0 and some structure conditions, vector τ is nowhere tangential to ∂Ω. This result extends the works of Lieberman G. M., Trudinger N. S. [2], Zhu Rujln [1] and Wang Feng [6].