N-S方程一般近似惯性流形构造和逼近

1．引言为了用有限维常微分方程来研究Navier七lobes（N七）方程的长时间动力学行为,Foias,Sell和Temaml］引入了耗散系统惯性流形的概念．但惯性流形存在的一个相当苛刻的条件一谱间隔条件一是包括N习方程在内的很多耗散系统无法满足的．因此,Foias,Manlea和TemamZ］随后又提出了近似惯性流形的概念．近似惯性流形也是一个光滑的Lipschitz流形,所有原方程的解在时间充分大时,将被吸引进入该流形的一个三邻域中．因其存在性不需要谱间隔条件来保证,从而可证明包括N－S方程在内的一大类耗散系统存在近似惯性流形．利用近似惯性…     

On the lexicographic minimax approach to location problems

When locating public facilities, the distribution of travel distances among the service recipients is an important issue. It is usually tackled with the minimax (center) solution concept. The minimax solution concept, despite the most commonly used in the public sector location models, is criticized as it does not comply with the major principles of the efficiency and equity modeling. In this paper we develop a concept of the lexicographic minimax solution (lexicographic center) being a refinement of the standard minimax approach to location problems. We show that the lexicographic minimax approach complies with both the Pareto-optimality (efficiency) principle (crucial in multiple criteria optimization) and the principle of transfers (essential for equity measures) whereas the standard minimax approach may violate both these principles. Computational algorithms are developed for the lexicographic minimax solution of discrete location problems.     

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.     

Pragmatic inquiry

Inquiry into how people might act to solve a problem, which involves powerful communities of stakeholders, cannot be undertaken using only the principles of scientific inquiry. Science is useful for inquiry into physical activity but less so for inquiry into designing appropriate social systems such as a strategic plan or the appropriate solution to a social conflict. This paper argues that complex problem solving needs to draw on a set of pragmatic principles, which it explains. Pragmatism is an epistemology intended to deal with alternative interpretations of the same physical reality. It is increasingly coming back into the attention of soft operational researchers. After briefly explaining which thread of pragmatism is being drawn upon, this paper identifies five interrelated principles for undertaking a pragmatic inquiry. To explain these interrelations, the principles are designed into a linked system of pragmatic inquiry which is applied in a small case study.     

Subgame Consistent Cooperative Solutions in Stochastic Differential Games

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that: (i) the extension of the solution policy to a later starting time and to any possible state brought about by the prior optimal behavior of the players would remain optimal; (ii) all players do not have incentive to deviate from the initial plan. In this paper, we develop a mechanism for the derivation of the payoff distribution procedures of subgame consistent solutions in stochastic differential games with transferable payoffs. The payoff distribution procedure of the subgame consistent solution can be identified analytically under different optimality principles. Demonstration of the use of the technique for specific optimality principles is shown with an explicitly solvable game. For the first time, analytically tractable solutions of cooperative stochastic differential games with subgame consistency are derived.     

A participatory multi-criteria approach for power generation and transmission planning

It is well-known that reinsurance can be an effective risk management solution for financial institutions such as the insurance companies. The optimal reinsurance solution depends on a number of factors including the criterion of optimization and the premium principle adopted by the reinsurer. In this paper, we analyze the Value-at-Risk based optimal risk management solution using reinsurance under a class of premium principles that is monotonic and piecewise. The monotonic piecewise premium principles include not only those which preserve stop-loss ordering, but also the piecewise premium principles which are monotonic and constructed by concatenating a series of premium principles. By adopting the monotonic piecewise premium principle, our proposed optimal reinsurance model has a number of advantages. In particular, our model has the flexibility of allowing the reinsurer to use different risk loading factors for a given premium principle or use entirely different premium principles depending on the layers of risk. Our proposed model can also analyze the optimal reinsurance strategy in the context of multiple reinsurers that may use different premium principles (as attributed to the difference in risk attitude and/or imperfect information). Furthermore, by artfully imposing certain constraints on the ceded loss functions, the resulting model can be used to capture the reinsurer’s willingness and/or capacity to accept risk or to control counterparty risk from the perspective of the insurer. Under some technical assumptions, we derive explicitly the optimal form of the reinsurance strategies in all the above cases. In particular, we show that a truncated stop-loss reinsurance treaty or a limited stop-loss reinsurance treaty can be optimal depending on the constraint imposed on the retained and/or ceded loss functions. Some numerical examples are provided to further compare and contrast our proposed models to the existing models.     

一个刚性守恒律方程组的全隐式差分方法

１．引言本文研究如下非线性刚性守恒律方程组的全隐式差分逼近． 方程（1．１）中的源项ｇ（ｕ,ｖ）定义为 ｇ（ｕ,ｖ）＝ｖ－（１－μ）ｆ（ｕ）,（１．２）其中ｆ是ｕ的一个给定函数,δ是一个小正参数,称为松弛时间,μ是参数．方程组（１．１）频繁出现于粘弹性力学中． 在零松弛时间限（δ→０）下,从（１．１）可得到如下方程组该方程组通常称为“平衡”模型,而方程组（１．１）称为“非平衡”模型． 文中将假设μ满足 ０＜ μ＜ １,（１．４）以便保证拟稳定性条件［１９,２０］和次特征条件［１１,２,３］： λ１≤λ＊…     

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.     

Discrete and continuous maximum principles for parabolic and elliptic operators

The major qualitative properties of linear parabolic and elliptic operators/PDEs are the different maximum principles (MPs). Another important property is the stabilization property (SP), which connects these two types of operators/PDEs. This means that under some assumptions the solution of the parabolic PDE tends to an equilibrium state when t→∞, which is the solution of the corresponding elliptic PDE. To solve PDEs we need to use some numerical methods, and it is a natural requirement that these qualitative properties are preserved on the discrete level. In this work we investigate this question when a two-level discrete mesh operator is used as the discrete model of the parabolic operator (which is a one-step numerical procedure for solving the parabolic PDE) and a matrix as a discrete elliptic operator (which is a linear algebraic system of equations for solving the elliptic PDE). We clarify the relation between the discrete parabolic maximum principle (DPMP), the discrete elliptic maximum principle (DEMP) and the discrete stabilization property (DSP). The main result is that the DPMP implies the DSP and the DEMP.     

A Maximum Principle for Stochastic Control with Partial Information

Abstract

We study the problem of optimal control of a jump diffusion, that is, a process which is the solution of a stochastic differential equation driven by Lévy processes. It is required that the control process is adapted to a given subfiltration of the filtration generated by the underlying Lévy processes. We prove two maximum principles (one sufficient and one necessary) for this type of partial information control. The results are applied to a partial information mean-variance portfolio selection problem in finance.     

Stability of solutions to parameterized nonlinear complementarity problems

n such that x≥0,  F(x,u)-v≥0 , and F(x,u)-v ^T·x=0 where these are vector inequalities. We characterize the local upper Lipschitz continuity of the (possibly set-valued) solution mapping which assigns solutions x to each parameter pair (v,u). We also characterize when this solution mapping is locally a single-valued Lipschitzian mapping (so solutions exist, are unique, and depend Lipschitz continuously on the parameters). These characterizations are automatically sufficient conditions for the more general (and usual) case where v=0. Finally, we study the differentiability properties of the solution mapping in both the single-valued and set-valued cases, in particular obtaining a new characterization of B-differentiability in the single-valued case, along with a formula for the B-derivative. Though these results cover a broad range of stability properties, they are all derived from similar fundamental principles of variational analysis. Received March 30, 1998 / Revised version received July 21, 1998 Published online January 20, 1999     

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.     

Second-order elliptic integro-differential equations: viscosity solutions' theory revisited

The aim of this work is to revisit viscosity solutions' theory for second-order elliptic integro-differential equations and to provide a general framework which takes into account solutions with arbitrary growth at infinity. Our main contribution is a new Jensen–Ishii's lemma for integro-differential equations, which is stated for solutions with no restriction on their growth at infinity. The proof of this result, which is of course a key ingredient to prove comparison principles, relies on a new definition of viscosity solution for integro-differential equation (equivalent to the two classical ones) which combines the approach with test-functions and sub-superjets.     

Variational Principles of Contact Elastostatics

In this paper, the variational principles of contact elastostatics,which were proposed and proved by Fichera (1964) and Duvaut& Lions (1972) are developed in an engineering fashion withoutuse of functional analysis. The theory contains a number ofnew elements, but the elegant existence proofs of the French-Italianschool are missing. A start is made by extending the principleof virtual work to normal and frictional contact in such a mannerthat it needs no longer be known beforehand whether—inthe case of normal contact—actual contact is or is notestablished, or—in the case of frictional contact—slipdoes or does not occur. Then the principle of minimal potentialenergy is set up for a non-linear elastic body in contact witha rigid base. Uniqueness and minimality of the solution areproved under certain conditions, the Reissner principle is established,and the principle of minimal complementary energy is derived.Finally the principles are cast in what is termed surface mechanicalform, and two examples are given: the variational principlefor normal half-space contact problems, and a new principlefor time-dependent frictional half-space contact. Upon discretization,these principles provide a quadratic object function to be minimizedunder linear or quadratic inequality constraints. The positionsof the contact area and of the regions of slip and adhesionappear as by-products of the calculation.     

A Metaheuristic to Solve a Location-Routing Problem with Non-Linear Costs

The paper deals with a location-routing problem with non-linear cost functions. To the best of our knowledge, a mixed integer linear programming formulation for the addressed problem is proposed here for the first time. Since the problem is NP-hard exact algorithms are able to solve only particular cases, thus to solve more general versions heuristics are needed. The algorithm proposed in this paper is a combination of a p-median approach to find an initial feasible solution and a metaheuristic to improve the solution. It is a hybrid metaheuristic merging Variable Neighborhood Search (VNS) and Tabu Search (TS) principles and exploiting the synergies between the two. Computational results and conclusions close the paper.     

Variants of simulated annealing for the examination timetabling problem

This paper is concerned with the use of simulated annealing in the solution of the multi-objective examination timetabling problem. The solution method proposed optimizes groups of objectives in different phases. Some decisions from earlier phases may be altered later as long as the solution quality with respect to earlier phases does not deteriorate. However, such limitations may disconnect the solution space, thereby causing optimal or near-optimal solutions to be missed. Three variants of our basic simulated annealing implementation which are designed to overcome this problem are proposed and compared using real university data as well as artificial data sets. The underlying principles and conclusions stemming from the use of this method are generally applicable to many other multi-objective type problems.     

Kernels and Choquet capacities

Summary M. Brelot showed that the capacity corresponding to a function-kernel is a Choquet capacity, provided that the kernel satisfies the principle of equilibrium, the weak domination principle and the adjoint kernel satisfies the weak principle of equilibrium. This result is not applicable for a series of important kernels in potential theory (e.g. the fundamental solution of the heat equation, or the Kolmogorov equation), since the above principles no longer hold in this situation. New principles for function kernels guaranteeing that the capacity is a Choquet capacity are introduced and applied in the framework of balayage spaces. In particular, polar and adjoint polar sets are shown to coincide in this context.     

Stability and accuracy functions in a coalition game with bans,linear payoffs and antagonistic strategies

A coalition game with a finite number of players in which initial coefficients of linear payoff functions are subject to perturbations is considered. For any efficient solution which may appear in the game, appropriate measures of the quality are introduced. These measures correspond to the so-called stability and accuracy functions defined earlier for efficient solutions of a generic multiobjective combinatorial optimization problem with Pareto and lexicographic optimality principles. Various properties of such functions are studied. Maximum norms of perturbations for which an efficient in sense of equilibrium solution preserves the property of being efficient are calculated.     

On solution uniqueness of elliptic boundary value problems

In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213–224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations.     

Behavior of Stochastic Shear Beams Under Random Loading via Stochastic Variational Principles

In this study we derive, apparently for the first time in the literature, some exact solutions for the shear beams with stochastic flexibility, when these beams are acted upon by the random loading. The importance of these solutions lies in the fact that they can serve as benchmark solutions, to which the approximate solutions of various nature can be compared. Then we formulate stochastic variational principles, the first principle governs the behavior of the mean displacement, whereas the second principle is satisfied by the displacements covariance function. These variational principles allow us to formulate approximate techniques for the cases in which the exact solution is presently unavailable. In particular, we develop a stochastic version of the Rayleigh–Ritz method. Several examples are evaluated to shed more light on the probabilistic behavior of randomly excited structures possessing random flexibility.