Perfection and Stability of Stationary Points with Applications in Noncooperative Games

It is well known that an upper semi-continuous compact- and convex-valued mapping ö from a nonempty compact and convex set X to the Euclidean space of which X is a subset has at least one stationary point, being a point in X at which the image ö (x)has a nonempty intersection with the normal cone at x.In many circumstances there may be more than one stationary point.In this paper we refine the concept of stationary point by perturbing simultaneously both the set X and the solution concept.In case a stationary point is the limit of a sequence of perturbed solutions on a sequence of sets converging continuously to X we say that the stationary point is stable with respect to this sequence of sets and the mapping which defines the perturbed solution.It is shown that stable stationary points exist for a large class of perturbations.A specific refinement, called robustness, is obtained if a stationary point is the limit of stationary points on a sequence of sets converging to X.It is shown that a robust stationary point always exists for any sequence of sets which starts from an interior point and converges to X in a continuous way.We also discuss several applications in noncooperative game theory.We first show that two well known refinements of the Nash equilibrium, namely, perfect Nash equilibrium and proper Nash equilibrium, are special cases of our robustness concept.Further, a third special case of robustness refines the concept of properness and a robust Nash equilibrium is shown to exist for every game.In symmetric bimatrix games, our results imply the existence of a symmetric proper equilibrium.Applying our results to the field of evolutionary game theory yields a refinement of the stationary points of the replicator dynamics.We show that the refined solution always exists, contrary to many well known refinement concepts in the field that may fail to exist under the same conditions.noncooperative games;stationary point;stability;equilibrium analysis

eXtended Variational Quasicontinuum Methodology for Lattice Networks with Damage and Crack Propagation

Lattice networks with dissipative interactions are often employed to analyze materials with discrete micro- or meso-structures, or for a description of heterogeneous materials which can be modelled discretely. They are, however, computationally prohibitive for engineering-scale applications. The (variational) QuasiContinuum (QC) method is a concurrent multiscale approach that reduces their computational cost by fully resolving the (dissipative) lattice network in small regions of interest while coarsening elsewhere. When applied to damageable lattices, moving crack tips can be captured by adaptive mesh refinement schemes, whereas fully-resolved trails in crack wakes can be removed by mesh coarsening. In order to address crack propagation efficiently and accurately, we develop in this contribution the necessary generalizations of the variational QC methodology. First, a suitable definition of crack paths in discrete systems is introduced, which allows for their geometrical representation in terms of the signed distance function. Second, special function enrichments based on the partition of unity concept are adopted, in order to capture kinematics in the wakes of crack tips. Third, a summation rule that reflects the adopted enrichment functions with sufficient degree of accuracy is developed. Finally, as our standpoint is variational, we discuss implications of the mesh refinement and coarsening from an energy-consistency point of view. All theoretical considerations are demonstrated using two numerical examples for which the resulting reaction forces, energy evolutions, and crack paths are compared to those of the direct numerical simulations.Comment: 36 pages, 23 figures, 1 table, 2 algorithms; small changes after review, paper title change

Consistency in Multi-Viewpoint Architectural Design of Enterprise Information Systems

Different stakeholders in the design of an enterprise information system have their own view on that design. To help produce a coherent design this paper presents a framework that aids in specifying relations between such views. To help produce a consistent design the framework also aids in specifying consistency rules that apply to the view relations and in checking the consistency according to those rules. The framework focuses on the higher levels of abstraction in a design, we refer to design at those levels of abstraction as architectural design. The highest level of abstraction that we consider is that of business process design and the lowest level is that of software component design. The contribution of our framework is that it provides a collection of basic concepts that is common to viewpoints in the area of enterprise information systems. These basic concepts aid in relating viewpoints by providing: (i) a common terminology that helps stakeholders to understand each others concepts; and (ii) a basis for defining re-usable consistency rules. In particular we define re-usable rules to check consistency between behavioural views that overlap or are a refinement of each other. We also present an architecture for a tool suite that supports our framework. We show that our framework can be applied, by performing a case study in which we specify the relations and consistency rules between the RM-ODP enterprise, computational and information viewpoints

Semantic verification of Behavior Conformance

This paper introduces a formal yet practical method to verify whether the behavior design of a distributed application conforms to the behavior design of the enterprise in which the application is embedded. The method allows both enterprise architects and application architects to talk about designs in their own terms, and introduces a common set of terms as the linking pin between enterprise and application designs. The formal semantics of these common terms allows us to verify the conformance between an enterprise and its applications formally and automatically

A goal-oriented requirements modelling language for enterprise architecture

Methods for enterprise architecture, such as TOGAF, acknowledge the importance of requirements engineering in the development of enterprise architectures. Modelling support is needed to specify, document, communicate and reason about goals and requirements. Current modelling techniques for enterprise architecture focus on the products, services, processes and applications of an enterprise. In addition, techniques may be provided to describe structured requirements lists and use cases. Little support is available however for modelling the underlying motivation of enterprise architectures in terms of stakeholder concerns and the high-level goals that address these concerns. This paper describes a language that supports the modelling of this motivation. The definition of the language is based on existing work on high-level goal and requirements modelling and is aligned with an existing standard for enterprise modelling: the ArchiMate language. Furthermore, the paper illustrates how enterprise architecture can benefit from analysis techniques in the requirements domain

An approach to relate business and application services using ISDL

This paper presents a service-oriented design approach that allows one to relate services modelled at different levels of granularity during a design process, such as business and application services. To relate these service models we claim that a 'concept gap' and an 'abstraction gap' need to be bridged. The concept gap represents the difference between the conceptual models used to construct service models by different stakeholders involved in the design process. The abstraction gap represents the difference in abstraction level at which service models are defined. Two techniques are presented that bridge these gaps. Both techniques are based on the Interaction System Design Language (ISDL). The paper illustrates the use of both techniques through an example

Wavelet-based Adaptive Techniques Applied to Turbulent Hypersonic Scramjet Intake Flows

The simulation of hypersonic flows is computationally demanding due to large gradients of the flow variables caused by strong shock waves and thick boundary or shear layers. The resolution of those gradients imposes the use of extremely small cells in the respective regions. Taking turbulence into account intensives the variation in scales even more. Furthermore, hypersonic flows have been shown to be extremely grid sensitive. For the simulation of three-dimensional configurations of engineering applications, this results in a huge amount of cells and prohibitive computational time. Therefore, modern adaptive techniques can provide a gain with respect to computational costs and accuracy, allowing the generation of locally highly resolved flow regions where they are needed and retaining an otherwise smooth distribution. An h-adaptive technique based on wavelets is employed for the solution of hypersonic flows. The compressible Reynolds averaged Navier-Stokes equations are solved using a differential Reynolds stress turbulence model, well suited to predict shock-wave-boundary-layer interactions in high enthalpy flows. Two test cases are considered: a compression corner and a scramjet intake. The compression corner is a classical test case in hypersonic flow investigations because it poses a shock-wave-turbulent-boundary-layer interaction problem. The adaptive procedure is applied to a two-dimensional confguration as validation. The scramjet intake is firstly computed in two dimensions. Subsequently a three-dimensional geometry is considered. Both test cases are validated with experimental data and compared to non-adaptive computations. The results show that the use of an adaptive technique for hypersonic turbulent flows at high enthalpy conditions can strongly improve the performance in terms of memory and CPU time while at the same time maintaining the required accuracy of the results.Comment: 26 pages, 29 Figures, submitted to AIAA Journa