This book focuses on a coherent representation of the main approaches to analyze the dynamics of cellular automata. Cellular automata are an inevitable tool in mathematical modeling. In contrast to classical modeling approaches as partial differential equations, cellular automata are straightforward to simulate but hard to analyze. In this book we present a review of approaches and theories that allow the reader to understand the behavior of cellular automata beyond simulations. The first part consists of an introduction of cellular automata on Cayley graphs, and their characterization via the fundamental Cutis-Hedlund-Lyndon theorems in the context of different topological concepts (Cantor, Besicovitch and Weyl topology). The second part focuses on classification results: What classification follows from topological concepts (Hurley classification), Lyapunov stability (Gilman classification), and the theory of formal languages and grammars (Kůrka classification). These classifications suggest to cluster cellular automata, similar to the classification of partial differential equations in hyperbolic, parabolic and elliptic equations. This part of the book culminates in the question, whether properties of cellular automata are decidable. Surjectivity, and injectivity are examined, and the seminal Garden of Eden theorems are discussed. The third part focuses on the analysis of cellular automata that inherit distinct properties, often based on mathematical modeling of biological, physical or chemical systems. Linearity is a concept that allows to define self-similar limit sets. Models for particle motion show how to bridge the gap between cellular automata and partial differential equations (HPP model and ultradiscrete limit). Pattern formation is related to linear cellular automata, to the Bar-Yam model for Turing pattern, and Greenberg-Hastings automata for excitable media. Also models for sandpiles, the dynamics of infectious diseases and evolution of predator-prey systems are discussed. Mathematicians find an overview about theory and tools for the analysis of cellular automata. The book contains an appendix introducing basic mathematical techniques and notations, such that also physicists, chemists and biologists interested in cellular automata beyond pure simulations will benefit.
This trusted commentary by Jac J. Muller provides careful expositions of Philippians and Philemon that are characterized by a desire to balance the requirements of exacting scholarship and the need to relate Scripture to personal faith. In treating insightfully and comprehensively these Pauline epistles, Muller first provides essential historical and critical background to each text, including its date and place of origin, its occasion and purpose, and its style, structure, and general themes. The commentaries themselves then proceed through the texts section-by-section, making clear Paul's message both to his original audiences and to readers today. Long included in the widely acclaimed New International Commentary on the New Testament series and now available as an independent work, this volume continues to hold extraordinary value for scholars, pastors, students, and general readers alike. "This is a painstaking and helpful commentary." - The British Weekly Jac J. Muller was professor of New Testament at the University of Stellenbosch, South Africa. His other books explored themes in New Testament studies and church ministry.
This is the only book that teaches all aspects of modern mathematical modeling and that is specifically designed to introduce undergraduate students to problem solving in the context of biology. Included is an integrated package of theoretical modeling and analysis tools, computational modeling techniques, and parameter estimation and model validation methods, with a focus on integrating analytical and computational tools in the modeling of biological processes. Divided into three parts, it covers basic analytical modeling techniques; introduces computational tools used in the modeling of biological problems; and includes various problems from epidemiology, ecology, and physiology. All chapters include realistic biological examples, including many exercises related to biological questions. In addition, 25 open-ended research projects are provided, suitable for students. An accompanying Web site contains solutions and a tutorial for the implementation of the computational modeling techniques. Calculations can be done in modern computing languages such as Maple, Mathematica, and MATLAB?.
Diese wissenschaftliche Studie setzt sich mit der Frage nach der Bedeutung von Vätern für die Entwicklung von Sozialkompetenz auseinander. Sie basiert auf der mit der Note 1,0 bewerteten Diplomarbeit des Autoren, welche nachträglich um einzelne Aspekte ergänzt wurde. In einem ausführlichen Theorieteil werden Definition der Vaterrolle und die entwicklungspsychologische Bedeutung von Vätern im Allgemeinen, Definition und Bedeutung von Sozialkompetenz sowie bisherige Erkenntnisse zum Zusammenhang von väterlichem Engagement und der Sozialkompetenz des Kindes detailliert dargelegt. Im Sinne eines weiteren Mosaiksteines zum Verständnis des väterlichen Einflusses wird im empirischen Teil die Bedeutung von Vätern für die Entwicklung selbstbezogener Sozialkompetenz untersucht. Die daraus abgeleiteten bedeutungsvollen Verhaltensweisen und Beziehungsaspekte werden anschließend auf bestehende empirische Ergebnisse sowie ihre Implikationen für die Praxis hin diskutiert.
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