The concept of equilibrium plays a central role in various applied sciences, such as physics (especially, mechanics), economics, engineering, transportation, sociology, chemistry, biology and other fields. If one can formulate the equilibrium problem in the form of a mathematical model, solutions of the corresponding problem can be used for forecasting the future behavior of very complex systems and, also, for correcting the the current state of the system under control. This book presents a unifying look on different equilibrium concepts in economics, including several models from related sciences.- Presents a unifying look on different equilibrium concepts and also the present state of investigations in this field- Describes static and dynamic input-output models, Walras, Cassel-Wald, spatial price, auction market, oligopolistic equilibrium models, transportation and migration equilibrium models- Covers the basics of theory and solution methods both for the complementarity and variational inequality problems- The methods are illustrated by applications and exercises to economic equilibrium models

The concept of equilibrium plays a central role in various applied sciences, such as physics (especially, mechanics), economics, engineering, transportation, sociology, chemistry, biology and other fields. If one can formulate the equilibrium problem in the form of a mathematical model, solutions of the corresponding problem can be used for forecasting the future behavior of very complex systems and, also, for correcting the the current state of the system under control. This book presents a unifying look on different equilibrium concepts in economics, including several models from related sciences.- Presents a unifying look on different equilibrium concepts and also the present state of investigations in this field- Describes static and dynamic input-output models, Walras, Cassel-Wald, spatial price, auction market, oligopolistic equilibrium models, transportation and migration equilibrium models- Covers the basics of theory and solution methods both for the complementarity and variational inequality problems- The methods are illustrated by applications and exercises to economic equilibrium models

While the classic model checking problem is to decide whether a finite system satisfies a specification, the goal of parameterized model checking is to decide, given finite systems ����(n) parameterized by n ∈ ℕ, whether, for all n ∈ ℕ, the system ����(n) satisfies a specification. In this book we consider the important case of ����(n) being a concurrent system, where the number of replicated processes depends on the parameter n but each process is independent of n. Examples are cache coherence protocols, networks of finite-state agents, and systems that solve mutual exclusion or scheduling problems. Further examples are abstractions of systems, where the processes of the original systems actually depend on the parameter. The literature in this area has studied a wealth of computational models based on a variety of synchronization and communication primitives, including token passing, broadcast, and guarded transitions. Often, different terminology is used in the literature, and results are based on implicit assumptions. In this book, we introduce a computational model that unites the central synchronization and communication primitives of many models, and unveils hidden assumptions from the literature. We survey existing decidability and undecidability results, and give a systematic view of the basic problems in this exciting research area.

While the classic model checking problem is to decide whether a finite system satisfies a specification, the goal of parameterized model checking is to decide, given finite systems ����(n) parameterized by n ∈ ℕ, whether, for all n ∈ ℕ, the system ����(n) satisfies a specification. In this book we consider the important case of ����(n) being a concurrent system, where the number of replicated processes depends on the parameter n but each process is independent of n. Examples are cache coherence protocols, networks of finite-state agents, and systems that solve mutual exclusion or scheduling problems. Further examples are abstractions of systems, where the processes of the original systems actually depend on the parameter. The literature in this area has studied a wealth of computational models based on a variety of synchronization and communication primitives, including token passing, broadcast, and guarded transitions. Often, different terminology is used in the literature, and results are based on implicit assumptions. In this book, we introduce a computational model that unites the central synchronization and communication primitives of many models, and unveils hidden assumptions from the literature. We survey existing decidability and undecidability results, and give a systematic view of the basic problems in this exciting research area.

Variational inequalities proved to be a very useful and powerful tool for in­ vestigation and solution of many equilibrium type problems in Economics, Engineering, Operations Research and Mathematical Physics. In fact, varia­ tional inequalities for example provide a unifying framework for the study of such diverse problems as boundary value problems, price equilibrium prob­ lems and traffic network equilibrium problems. Besides, they are closely re­ lated with many general problems of Nonlinear Analysis, such as fixed point, optimization and complementarity problems. As a result, the theory and so­ lution methods for variational inequalities have been studied extensively, and considerable advances have been made in these areas. This book is devoted to a new general approach to constructing solution methods for variational inequalities, which was called the combined relax­ ation (CR) approach. This approach is based on combining, modifying and generalizing ideas contained in various relaxation methods. In fact, each com­ bined relaxation method has a two-level structure, i.e., a descent direction and a stepsize at each iteration are computed by finite relaxation procedures.

The concept of equilibrium plays a central role in various applied sciences, such as physics (especially, mechanics), economics, engineering, transportation, sociology, chemistry, biology and other fields. If one can formulate the equilibrium problem in the form of a mathematical model, solutions of the corresponding problem can be used for forecasting the future behavior of very complex systems and, also, for correcting the the current state of the system under control. This book presents a unifying look on different equilibrium concepts in economics, including several models from related sciences.- Presents a unifying look on different equilibrium concepts and also the present state of investigations in this field- Describes static and dynamic input-output models, Walras, Cassel-Wald, spatial price, auction market, oligopolistic equilibrium models, transportation and migration equilibrium models- Covers the basics of theory and solution methods both for the complementarity and variational inequality problems- The methods are illustrated by applications and exercises to economic equilibrium models

The concept of equilibrium plays a central role in various applied sciences, such as physics (especially, mechanics), economics, engineering, transportation, sociology, chemistry, biology and other fields. If one can formulate the equilibrium problem in the form of a mathematical model, solutions of the corresponding problem can be used for forecasting the future behavior of very complex systems and, also, for correcting the the current state of the system under control. This book presents a unifying look on different equilibrium concepts in economics, including several models from related sciences.- Presents a unifying look on different equilibrium concepts and also the present state of investigations in this field- Describes static and dynamic input-output models, Walras, Cassel-Wald, spatial price, auction market, oligopolistic equilibrium models, transportation and migration equilibrium models- Covers the basics of theory and solution methods both for the complementarity and variational inequality problems- The methods are illustrated by applications and exercises to economic equilibrium models

While the classic model checking problem is to decide whether a finite system satisfies a specification, the goal of parameterized model checking is to decide, given finite systems ����(n) parameterized by n ∈ ℕ, whether, for all n ∈ ℕ, the system ����(n) satisfies a specification. In this book we consider the important case of ����(n) being a concurrent system, where the number of replicated processes depends on the parameter n but each process is independent of n. Examples are cache coherence protocols, networks of finite-state agents, and systems that solve mutual exclusion or scheduling problems. Further examples are abstractions of systems, where the processes of the original systems actually depend on the parameter. The literature in this area has studied a wealth of computational models based on a variety of synchronization and communication primitives, including token passing, broadcast, and guarded transitions. Often, different terminology is used in the literature, and results are based on implicit assumptions. In this book, we introduce a computational model that unites the central synchronization and communication primitives of many models, and unveils hidden assumptions from the literature. We survey existing decidability and undecidability results, and give a systematic view of the basic problems in this exciting research area.

While the classic model checking problem is to decide whether a finite system satisfies a specification, the goal of parameterized model checking is to decide, given finite systems ����(n) parameterized by n ∈ ℕ, whether, for all n ∈ ℕ, the system ����(n) satisfies a specification. In this book we consider the important case of ����(n) being a concurrent system, where the number of replicated processes depends on the parameter n but each process is independent of n. Examples are cache coherence protocols, networks of finite-state agents, and systems that solve mutual exclusion or scheduling problems. Further examples are abstractions of systems, where the processes of the original systems actually depend on the parameter. The literature in this area has studied a wealth of computational models based on a variety of synchronization and communication primitives, including token passing, broadcast, and guarded transitions. Often, different terminology is used in the literature, and results are based on implicit assumptions. In this book, we introduce a computational model that unites the central synchronization and communication primitives of many models, and unveils hidden assumptions from the literature. We survey existing decidability and undecidability results, and give a systematic view of the basic problems in this exciting research area.

Variational inequalities proved to be a very useful and powerful tool for in­ vestigation and solution of many equilibrium type problems in Economics, Engineering, Operations Research and Mathematical Physics. In fact, varia­ tional inequalities for example provide a unifying framework for the study of such diverse problems as boundary value problems, price equilibrium prob­ lems and traffic network equilibrium problems. Besides, they are closely re­ lated with many general problems of Nonlinear Analysis, such as fixed point, optimization and complementarity problems. As a result, the theory and so­ lution methods for variational inequalities have been studied extensively, and considerable advances have been made in these areas. This book is devoted to a new general approach to constructing solution methods for variational inequalities, which was called the combined relax­ ation (CR) approach. This approach is based on combining, modifying and generalizing ideas contained in various relaxation methods. In fact, each com­ bined relaxation method has a two-level structure, i.e., a descent direction and a stepsize at each iteration are computed by finite relaxation procedures.