Dual extremum principles for geometrically exact finite strain beams

Both necessary and sufficient conditions for the existence of two complementary-dual extremum principles for geometrically exact finite strain (one-dimensional) beam models are investigated by means of two different approaches. One is based on the results published by Gao and Strang, and the other relies on the approach proposed by Noble and Sewell. While the former is limited to beam models restricted to moderate large deformations, the latter is valid for arbitrarily large deformations (and strains). The numerical implementation of the complementary-dual extremum principles can lead to simple true global upper bounds of the error of the approximate solutions.     

带摩擦的弹性接触问题广义变分不等原理的简化证明

在弹性摩擦接触问题中 ,从变分原理出发来研究接触问题 ,可以将摩擦力纳入问题的能量泛函 .为了得到摩擦约束弹性接触问题的能量泛函 ,日前大多是用拉格朗日乘子法 ,但拉格朗日方法用在变分不等问题中 ,要利用非线性泛函分析和凸分析来证明 ,证明复杂 .本文利用向量分析的工具及巧妙的变换 ,对带摩擦约束的弹性接触问题的广义变分不等原理进行了严格的证明 ,由于只用到向量分析 ,简化了证明 .     

The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations

A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan's equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.

     

企业科技行为系统的评价体系研究

现代企业科技行为属于高阶非线性系统行为，对其运行的评价应采用系统评价的思想和方法进行。本文确定了现代企业科技行为系统评价的基本准则，构建了评价的指标体系；提出了各指标值的归一化处理方法以及各指标权重的确定方法     

Maximum principle for functional equations in the space of discontinuous functions of three variables

The paper is devoted to the maximum principles for functional equations in the space of measurable essentially bounded functions. The necessary and sufficient conditions for validity of corresponding maximum principles are obtained in a form of theorems about functional inequalities similar to the classical theorems about differential inequalities of the Vallee Poussin type. Assertions about the strong maximum principle are proposed. All results are also true for difference equations, which can be considered as a particular case of functional equations. The problems of validity of the maximum principles are reduced to nonoscillation properties and disconjugacy of functional equations. Note that zeros and nonoscillation of a solution in a space of discontinuous functions are defined in this paper. It is demonstrated that nonoscillation properties of functional equations are connected with the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Simple sufficient conditions of nonoscillation, disconjugacy and validity of the maximum principles are proposed. The known nonoscillation results for equation in space of functions of one variable follow as a particular cases of these assertions. It should be noted that corresponding coefficient tests obtained on this basis cannot be improved. Various applications to nonoscillation, disconjugacy and the maximum principles for partial differential equations are proposed.     

Comparison principles for viscosity solutions of elliptic equations via fuzzy sum rule

A comparison principle for viscosity sub- and super-solutions of second order elliptic partial differential equations is derived using the “fuzzy sum rule” of non-smooth calculus. This method allows us to weaken the assumptions made on the function F when the equation F(x,u,Du,D²u)=0 is under consideration.     

Genericity of Well-Posedness, Perturbations and Smooth Variational Principles

We show that three important topics in nonlinear analysis and optimization are intimately related: the theory of perturbations, the notion of well-posedness and variational principles in the sense of Ekeland, Borwein–Preiss and Deville–Godefroy–Zizler. The concept of genericity and the new notion of flexible perturbation play a key role in these connections. This notion enables one to consider topologies on spaces of functions which have been introduced recently. A link between the Asplund and Ekeland–Lebourg methods and the Palais–Smale condition, another important topic in nonlinear analysis, is pointed out.     

The heat kernel and Hardy’s theorem on symmetric spaces of noncompact type

For symmetric spaces of noncompact type we prove an analogue of Hardy’s theorem which characterizes the heat kernel in terms of its order of magnitude and that of its Fourier transform.     

Exact solutions to the equations of perfect gases through Lie group analysis and substitution principles

In this paper we consider the equations that govern the motion of perfect gases. We explicitly characterize some classes of steady solutions in two and three space dimensions, by introducing invertible point transformations suggested by Lie group analysis; moreover, by using various transformations known as substitution principles, new steady and unsteady solutions are constructed.     

Variational principles for nonlinear piezoelectric materials

Summary In the present paper, we consider the behavior of nonlinear piezoelectric materials by generalization for this case of the Hashin-Shtrikman variational principles. The new general formulation used here differs from others, because, it gives the possibility to evaluate the upper and lower Hashin-Shtrikman bounds for specific physical nonlinearities of piezoelectric materials. Geometrical nonlinearities are not considered.This work was sponsored by PAPIIT, DGAPA-UNAM IN103301. PNCB IBMFQ 09-2004 and, completed while one of the authors (RRR) was visiting the Laboratoire de Modélisation en Mécanique from Université Pierre et Marie Curie, Paris, while another one (G.A.M) benefited from the Max-Plank Award for International Cooperation (2002–2005).