功率型变分原理和功能型拟变分原理及其应用

自从钱伟长建立了功率型变分原理以来,功率型变分原理和功能型变分原理在理论方面和应用方面有什么区别和联系,成为学术界关注的课题.应用变积方法,根据Jourdain原理和d’Alembert原理,建立了不可压缩黏性流体力学的功率型变分原理和功能型拟变分原理,推导了不可压缩黏性流体力学的功率型变分原理的驻值条件和功能型拟变分原理的拟驻值条件.研究了不可压缩黏性流体力学的功率型变分原理在有限元素法中的应用.研究表明,功率型变分原理与Jourdain原理相吻合,功能型变分原理与d’Alembert原理相吻合.功率型变分原理直接在状态空间中研究问题,不仅在建立变分原理的过程中可以省略在时域空间中的一些变换,而且给动力学问题有限元素法的数值建模带来方便.     

非线性弹性理论变分原理的统一理论

本文旨在介绍和讨论非线性弹性理论的几个主要变分原理——古典的势能原理,余能原理以及目前争论甚多的另两个余能原理(Levinson原理和Fraeijs de Veubeke原理),同时给出了和这些原理相对应的广义变分原理.本文单一地从虚功原理出发,系统推导并严格论证了这些原理,并且指出了各原理间的内在联系.出发点是一个,采取不同的变量和Legendre变换就导致不同的原理.这样,各变分原理在统一的框架里构成一个有机的整体.文中未涉及的其它原理也同样可以纳入这框架.给出的关系图使读者更能看清各原理间的纵横关系.     

粘性流体力学的变分原理和广义变分原理

本文建立了不可压缩和可压缩粘性流体力学问题的变分原理,即最大功率消耗原理和它们的广义变分原理.     

饱和多孔介质耦合系统的变分原理

本文采用变积方法,建立了等温准静态下饱和多孔介质的六类变量的广义变分原理.在此基础上,通过引入约束条件得到各级变分原理,其中包括五类变量,四类变量,三类变量和二类变量的变分原理.除得到文献中已有的变分原理外,本文给出了许多新的变分原理,为建立饱和多孔介质的有限元模型提供了基础.     

非线性弹性体的弹性力学变分原理

作者自1978年以后,曾发表了一系列有关弹性力学的变分原理和广义变分原理的文章如[1](1978),[6](1980),[2]、[3](1983),[4]、[5](1984),都是指线性应力应变关系的线性弹性体的.在1985年出版的广义变分原理中,初步推广至非线性弹性体,但并未进行较全面的探讨.本文特别讨论非线性应力应变关系的弹性体的变分原理和广义变分原理,这里有不少问题是值得注意的,有时,它对线性弹性体的变分原理,有指导意义.当应变很小,其高次项可以略去时,本文所得结论,都能近似地化简为通常线性理论的结果.     

关于板的动力学的最小转换能量原理和最小值原理

本文首先通过Laplace变换导出了关于具有三个广义位移并考虑转动惯量效应时各向异性的线弹性板在动力学中的转换虚功原理和三个最小转换能量原理及其在原空间时间域中用原函数表示的对应形式,然后通过引进相容权函数的集合推导出关于空间时间域的三个最小值原理.在上述两组最小值原理中各有两个均为板在静力学中最小势能原理和最小余能原理的推广形式;而另一个最小值原理,在静力学中便没有对应形式.     

Tangential extremal principles for finite and infinite systems of sets, I: basic theory

In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization.     

On the variational principles of mechanics

Variational principles, generalizing the classical d’Alembert–Lagrange, Hölder, and Hamilton–Ostrogradskii principles, are established. After the addition of anisotropic dissipative forces and taking the limit, when the coefficient of viscous friction tends to infinity, these variational principles transform into the classical principles, which describe the motion of systems with constraints. New variational relations are established for searching for the periodic trajectories of the reversible equations of dynamics.     

Pigeonhole and Choice Principles

We shall investigate certain set‐theoretic pigeonhole principles which arise as generalizations of the usual (finitary) pigeonhole principle; and we shall show that many of them are equivalent to full AC. We discuss also several restricted cases and variations of those principles and relate them to restricted choice principles. In this sense the pigeonhole principle is a rich source of weak choice principles. It is shown that certain sequences of restricted pigeonhole principles form implicational hierarchies with respect to ZF. We state also several open problems in order to indicate the extent to which the whole subject of pigeonhole principles requires further exploration.     

Invariance principles for hybrid systems with memory

Hybrid systems with memory are dynamical systems exhibiting both delayed and hybrid dynamics. Such systems can be described by hybrid functional inclusions. Classical invariance principles play an instrumental role in proving stability and convergence of dynamical systems. Invariance principles for general hybrid systems with delays, however, remain an open topic. In this paper, we prove invariance principles for hybrid systems with memory, using both Lyapunov–Razumikhin function and Lyapunov–Krasovskii functional methods. These invariance principles are then applied to derive two stability results as corollaries.     

线弹性动力学的某些一般定理及广义与广义分区变分原理*

从四维空间思想出发,在四种时端条件下,系统地推导得出了弹性动力学有关的一般定理,如:可能功作用量原理,虚位移原理,虚应力一动量原理,互易定理及由此导出的位移互等定理与始末时刻条件关系定理等;得出了线弹性动力学的位能作用量变分原理,余能作用量变分原理,动力问题的胡-鹫原理,H-R原理及本构关系变分原理.Hamilton原理,Toupin原理及有关文献如[5]、[17]～[24]的工作均可作为文中一般结果的特例.对应于有限元分析.在空间分区,时间分区及时空均分区情况.给出了动力学问题的分区位能作用量原理.分区余能作用量原理,分区混合能作用量原理及相应的分区广义变变分原理.导出了分区原理的一般形式.若去掉时间维及有关量,文中有关结果可转化为静力问题中有关的相应结果.     

Maximum principles for a time–space fractional diffusion equation

In this paper, we focus on maximum principles of a time–space fractional diffusion equation. Maximum principles for classical solution and weak solution are all obtained by using properties of the time fractional derivative operator and the fractional Laplace operator. We deduce maximum principles for a full fractional diffusion equation, other than time-fractional and spatial-integer order diffusion equations.     

On patched variational principles with application to elliptic and mixed elliptic-hyperbolic problems

Summary Variational principles are important tools for the approximate solution of boundary-value problems. There are many types of variational principles, and each has its advantages and disadvantages. In this paper we show how to use a combination of variational principles, each for a given subregion of the underlying region of space, so as to best utilize the chief benefits of the individual principles. Such a patched principle is particularly useful in solving transonic flow problems, where we use different principles in the elliptic and hyperbolic regions. We present the results of some numerical experiments for the Tricomi problem. These seem to indicate that our patched principle, when used in conjunction with the finite element method, leads to accuracy which is second-order in the mesh spacing, as compared to the standard numerical methods of solving this problem, which are only first-order.     

Large deviations for Glauber dynamics of continuous gas

This paper is devoted to the large deviation principles of the Glauber-type dynamics of finite or infinite volume continuous particle systems.We prove that the level-2 empirical process satisfies the large deviation principles in the weak convergence topology,while it does not satisfy the large deviation principles in the T-topology.     

Axiomatization of local-global principles for pp-formulas in spaces of orderings

We use a model theoretic approach to investigate properties of local-global principles for positive primitive formulas in spaces of orderings, such as the existence of bounds and the axiomatizability of local-global principles. As a consequence we obtain various classes of special groups satisfying local-global principles for all positive primitive formulas, and we show that local-global principles are preserved by some natural constructions in special groups.Mathematics Subject Classification (2000): 11E81, 03C65Acknowledgement This research was undertaken while both authors were partially suported by the European RTN Network (HPRN-CT-2001-00271) on real algebraic and analytic geometry.     

有限位移理论线弹性力学二类和三类混合变量的变分原理及其应用

提出了有限位移理论线弹性力学二类混合变量和三类混合变量的变分原理.考虑已知边界条件的变化并应用有限位移理论的功的互等定理,在导出上述两类变分原理的过程中起到了关键作用和桥梁作用.首先,考虑已知位移边界条件的变化和应用功的互等定理,导出了二类混合变量的最小势能原理.用类似的方法,导出了二类混合变量的驻值余能原理.应用应变能密度和应力余能密度的关系式于上述两个变分原理,得到三类混合变量的变分原理.然后,给出了二类和三类混合变量的虚功原理和虚余功原理.同时,应用拉氏乘子法导出了广义变分原理.以一个算例说明了在某些情况下拉氏乘子法会失效,介绍了构成广义变分原理泛函的半逆法.最后,应用二类混合变量最小势能原理计算了一大挠度悬臂梁的弯曲.     

Variational principles of second, first and intermediate kinds for non-holonomic mechanics

The differential variational principles of second kind for non-holonomic mechanics are given, from which a number of integral variational principles of second kind are set up. From the latter, the general relation of δq′-δq and the general form of integral variational principles of the first kind and intermediate kinds are derived. Thus not only all previous relations of δq′-δq and integral variational principles are unified but also the existance of the variational principles of intermediate kinds are pointed out. Project supported by the National Natural Science Foundation of China (Grant No. 19272064).     

Dynamic programming and principles of optimality

A sequential decision model is developed in the context of which three principles of optimality are defined. Each of the principles is shown to be valid for a wide class of stochastic sequential decision problems. The relationship between the principles and the functional equations of dynamic programming is investigated and it is shown that the validity of each of them guarantees the optimality of the dynamic programming solutions. As no monotonicity assumption is made regarding the reward functions, the results presented in this paper resolve certain questions raised in the literature as to the relation among the principles of optimality and the optimality of the dynamic programming solutions.     

Generalized Maximum Principles and Stochastic Completeness for Pseudo-Hermitian Manifolds*

In this paper, the authors establish a generalized maximum principle for pseudoHermitian manifolds. As corollaries, Omori-Yau type maximum principles for pseudoHermitian manifolds are deduced. Moreover, they prove that the stochastic completeness for the heat semigroup generated by the sub-Laplacian is equivalent to the validity of a weak form of the generalized maximum principles. Finally, they give some applications of these generalized maximum principles.