Infinite Words is an important theory in both Mathematics and Computer Sciences. Many new developments have been made in the field, encouraged by its application to problems in computer science. Infinite Words is the first manual devoted to this topic. Infinite Words explores all aspects of the theory, including Automata, Semigroups, Topology, Games, Logic, Bi-infinite Words, Infinite Trees and Finite Words. The book also looks at the early pioneering work of Büchi, McNaughton and Schützenberger. Serves as both an introduction to the field and as a reference book. Contains numerous exercises desgined to aid students and readers. Self-contained chapters provide helpful guidance for lectures.

The book contains a detailed treatment of thermodynamic formalism on general compact metrizable spaces. Topological pressure, topological entropy, variational principle, and equilibrium states are presented in detail. Abstract ergodic theory is also given a significant attention. Ergodic theorems, ergodicity, and Kolmogorov-Sinai metric entropy are fully explored. Furthermore, the book gives the reader an opportunity to find rigorous presentation of thermodynamic formalism for distance expanding maps and, in particular, subshifts of finite type over a finite alphabet. It also provides a fairly complete treatment of subshifts of finite type over a countable alphabet. Transfer operators, Gibbs states and equilibrium states are, in this context, introduced and dealt with. Their relations are explored. All of this is applied to fractal geometry centered around various versions of Bowen’s formula in the context of expanding conformal repellors, limit sets of conformal iterated function systems and conformal graph directed Markov systems. A unique introduction to iteration of rational functions is given with emphasize on various phenomena caused by rationally indifferent periodic points. Also, a fairly full account of the classicaltheory of Shub’s expanding endomorphisms is given; it does not have a book presentation in English language mathematical literature.

A central aim and ever-lasting dream of computer science is to put the development of hardware and software systems on a mathematical basis which is both firm and practical. Such a scientific foundation is needed especially for the construction of reactive programs, like communication protocols or control systems. For the construction and analysis of reactive systems an elegant and powerful theory has been developed based on automata theory, logical systems for the specification of nonterminating behavior, and infinite two-person games. The 19 chapters presented in this multi-author monograph give a consolidated overview of the research results achieved in the theory of automata, logics, and infinite games during the past 10 years. Special emphasis is placed on coherent style, complete coverage of all relevant topics, motivation, examples, justification of constructions, and exercises.

A certain category of infinite strings of letters on a finite alphabet is presented here, chosen among the 'simplest' possible one may build, both because they are very deterministic and because they are built by simple rules (a letter is replaced by a word, a sequence is produced by iteration). These substitutive sequences have a surprisingly rich structure. The authors describe the concepts of quantity of natural interactions, with combinatorics on words, ergodic theory, linear algebra, spectral theory, geometry of tilings, theoretical computer science, diophantine approximation, trancendence, graph theory. This volume fulfils the need for a reference on the basic definitions and theorems, as well as for a state-of-the-art survey of the more difficult and unsolved problems.

This book is concerned with the role played by modules of infinite length when dealing with problems in the representation theory of groups and algebras, but also in topology and geometry; it shows the intriguing interplay between finite and infinite length modules.

The Beauty of the Infinite is a splendid extended essay in "theological aesthetics." David Bentley Hart here meditates on the power of a Christian understanding of beauty and sublimity to rise above the violence -- both philosophical and literal -- characteristic of the postmodern world. The book begins by tracing the shifting use and nature of metaphysics in the thought of Kant, Hegel, Nietzsche, Heidegger, Lyotard, Derrida, Deleuze, Nancy, Levinas, and others. Hart pays special attention to Nietzsche's famous narrative of the "will to power" -- a narrative largely adopted by the world today -- and he offers an engaging revision (though not rejection) of the genealogy of nihilism, thereby highlighting the significant "interruption" that Christian thought introduced into the history of metaphysics. This discussion sets the stage for a retrieval of the classic Christian account of beauty and sublimity, and of the relation of both to the question of being. Written in the form of a dogmatica minora, this main section of the book offers a pointed reading of the Christian story in four moments, or parts: Trinity, creation, salvation, and eschaton. Through a combination of narrative and argument throughout, Hart ends up demonstrating the power of Christian metaphysics not only to withstand the critiques of modern and postmodern thought but also to move well beyond them. Strikingly original and deeply rewarding, The Beauty of the Infinite is both a constructively critical account of the history of metaphysics and a compelling contribution to it.

In this last work, Kenan Osborne addresses the intersection between new scientific insights into the origin of the human species and the growing awareness of a multicultural and multi-religious world with our contemporary understanding of God. After a review of current presentations of Trinitarian theology, he analyzes in detail the biblical record for the names of God and develops a cogent description of the thinking about God in the first six centuries. Complementing his 2015 volume The Infinity of God and A Finite World, A Franciscan Approach, this present work challenges theologians and believers in two distinct ways: Do the terms "Father" and "Son" have any essential meaning for divinity? From a human standpoint, God is essentially neither a "Father" nor a "Son." Nor do these two words have some exclusive meaning when they refer to divinity. What then do we mean when we talk about God? Second, in many theological textbooks, the term "infinite" is considered as an "attribute" of God. Infinity is in no way an attribute, even a divine attribute. It is rather an essential description of God, as the Franciscan philosophical theologian John Duns Scotus argues. And if this is so, can we really understand God, or are all our views and descriptions of an infinite God partial insights into a transcendent infinite God who embraces all human creatures from the beginning of homo sapiens sapiens to the present?

This book constitutes the proceedings of the 41st International Conference on Application and Theory of Petri Nets and Concurrency, PETRI NETS 2020, which was supposed to be held in Paris, France, in June 2020. The conference was held virtually due to the COVID-19 pandemic. The 17 regular and 6 tool papers presented together in this volume were carefully reviewed and selected from 56 submissions. The focus of the conference is on following topics: application of concurrency to system design; languages and synthesis; semantics; process mining and applications; extensions and model checking; tools.