Necessary and sufficient second order optimality conditions for multiobjective problems with C1 data

In this paper, we obtain necessary and sufficient second order optimality conditions for multiobjective problems using second order directional derivatives. We propose the notion of second order KT-pseudoinvex problems and we prove that this class of problems has the following property: a problem is second order KT-pseudoinvex if and only if all its points that satisfy the second order necessary optimality condition are weakly efficient. Also we obtain second order sufficient conditions for efficiency.     

On the symmetry classification and conservation laws for quasilinear differential equations of second order

We obtain a sufficient condition for the absence of tangent transformations admitted by quasilinear differential equations of second order and a sufficient condition for the linear autonomy of the operators of the Lie group of transformations admitted by weakly nonlinear differential equations of second order. We prove a theorem concerning the structure of conservation laws of first order for weakly nonlinear differential equations of second order. We carry out the classification by first-order conservation laws for linear differential equations of second order with two independent variables.     

抛物型方程基于POD方法的时间二阶中心差的时间二阶精度简化有限元格式

本文将特征正交分解(Proper Orthogonal Decomposition,简记为POD)方法应用于抛物型方程通常时间二阶中心差的时间二阶精度有限元格式(简称为通常格式),简化其为一个自由度极少但具有时间二阶精度的有限元格式,并给出简化的时间二阶中心差的时间二阶精度有限元格式(简称为简化格式)解的误差分析.数值...     

非线性规划问题的二阶对称对偶性

本文讨论了二阶凸和二阶凹条件下的二阶对称对偶问题,并利用有效性和真有效性概念证明了弱对偶、强对偶、逆对偶及自对偶定理。     

On a second order parallel variable transformation approach

In this paper we present a second order PVT (parallel variable transformation) algorithm converging to second order stationary points for minimizing smooth functions, based on the first order PVT algorithm due to Fukushima (1998). The corresponding stopping criterion, descent condition and descent step for the second order PVT algorithm are given.     

Superconvergence of three dimensional Morley elements on cuboid meshes for biharmonic equations

In the present paper, superconvergence of second order, after an appropriate postprocessing, is achieved for three dimensional first order cuboid Morley elements of biharmonic equations. The analysis is dependent on superconvergence of second order for the consistency error and a corrected canonical interpolation operator, which help to establish supercloseness of second order for the corrected canonical interpolation. Then the final superconvergence is derived by a standard postprocessing. For first order nonconforming finite element methods of three dimensional fourth order elliptic problems, it is the first time that full superconvergence of second order is obtained without an extra boundary condition imposed on exact solutions. It is also the first time that superconvergence is established for nonconforming finite element methods of three dimensional fourth order elliptic problems. Numerical results are presented to demonstrate the validity of the theoretical results.     

A second order <Emphasis Type="Italic">η</Emphasis>-approximation method for constrained optimization problems involving second order invex functions

A new approach for obtaining the second order sufficient conditions for non-linear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order η-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order η-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second orderη-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem.     

一类二阶变系数线性微分方程的积分因子解法

通过寻求积分因子f1(x),f2(x),求解一类二阶线性微分方程,包括二阶常系数线性微分方程和二阶Euler方程.     

From finitary to infinitary second‐order logic

A back and forth condition on interpretations for those second‐order languages without functional variables whose non‐logical vocabulary is finite and excludes functional constants is presented. It is shown that this condition is necessary and sufficient for the interpretations to be equivalent in the language. When applied to second‐order languages with an infinite non‐logical vocabulary, excluding functional constants, the back and forth condition is sufficient but not necessary. It is shown that there is a class of infinitary second‐order languages whose non‐logical vocabulary is infinite for which the back and forth condition is both necessary and sufficient. It is also shown that some applications of the back and forth construction for second‐order languages can be extended to the infinitary second‐order languages. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Treatment of Second Order Differential-Algebraic Systems

We consider the numerical treatment of systems of second order differential-algebraic equations (DAEs). The classical approach of transforming a second order system to first order by introducing new variables can lead to difficulties such as an increase in the index or the loss of structure. We show how we can compute an equivalent strangeness-free second order system using the derivative array approach and we present Runge-Kutta methods for the direct numerical solution of second order DAEs. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second order duality in fuzzy mathematical programming with fuzzy coefficients

In this paper, the concept of second order convexity of fuzzy mappings with fuzzy order relation under ranking function is used to formulate a second order duality for fuzzy mathematical programming problems with fuzzy coefficients, where all the fuzzy mappings are twice differentiable. Also, Wolfe’s dual of these problems under ranking function is investigated, and finally, an advantage of second order duality with respect to first order duality is stated.     

利用二阶方向导数证明极值的充分条件

介绍二元函数二阶方向导数的概念与计算方法.利用线性代数中的二次型知识,对二元函数在驻点处是否取得极值的充分性定理给出有几何意义的证明.     

一类非线性偏积分微分方程二阶差分全离散格式

给出了数值求解一类非线性偏积分微分方程的二阶全离散差分格式.采用了二阶向后差分格式,积分项的离散利用了Lubich的二阶卷积求积公式,给出了稳定性的证明、误差估计及收敛性的结果.     

椭圆方程在Orlicz类中的正则估计(英文)

The purpose of this paper is to give the element proof of the regularity estimates in Orlicz classes for the second order derivatives of the solutions to the general second order elliptic equations.The global regularities in Orlicz for the second order derivatives of the solutions of the Dirichlet problems are also given.     

Notes on Duality in Second Order and p -Order Cone Optimization

Recently, the so-called second order cone optimization problem has received much attention, because the problem has many applications and the problem can in theory be solved efficiently by interior-point methods. In this note we treat duality for second order cone optimization problems and in particular whether a nonzero duality gap can be obtained when casting a convex quadratically constrained optimization problem as a second order cone optimization problem. Furthermore, we also discuss the p -order cone optimization problem which is a natural generalization of the second order case. Specifically, we suggest a new self-concordant barrier for the p -order cone optimization problem.     

一类二阶微分方程的异宿解

利用变分法获得了一类二阶微分方程异宿解存在的一个充分条件.将连续情况下的二阶微分方程异宿解研究推广到离散情况下来研究.     

Oscillation of even order nonlinear delay dynamic equations on time scales

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor’s Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.     

Sur l'existence de modèles du second ordre non dénombrables avec domaine du premier ordre dénombrable

We generalize to second order logic a result of Keisler concerning second order arithmetic. We prove that for any countable second order model, verifying certain axioms, there exists an elementary extension having the same domain of intrepretation for individuals and whose domain of interpretation for relations is uncountable. The axioms we ask for are the comprehension scheme, a choice scheme and a pairing scheme that allow us not to have explicitly a pairing function in the language.     

Nonlinear free surface condition due to second order diffraction by a pair of cylinders

An analysis of the non-homogeneous term involved in the free surface condition for second order wave diffraction on a pair of cylinders is presented. In the computations of the nonlinear loads on offshore structures the most challenging task is the computation of the free surface integral. The main contribution to this integrand is due to the nonhomogeneous term present in the free surface condition for second order scattered potential. In this paper, the free surface condition for the second order scattered potential is derived. Under the assumption of large spacing between the two cylinders, waves scattered by one cylinder may be replaced in the vicinity of the other cylinder by equivalent plane waves together with non-planner correction terms. Then solving a complex matrix equation, the first order scattered potential is derived and since the free surface term for second order scattered potential can be expressed in terms of the first order potentials, the free surface term can be obtained using the knowledge of first order potentials only.     

Error estimates for shooting methods in two-point boundary value problems for second-order equations

This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem.