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Author : David I. Spivak
ISBN : 9780262320535
Genre : Mathematics
File Size : 72.79 MB
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An introduction to category theory as a rigorous, flexible, and coherent modeling language that can be used across the sciences. Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. Information is inherently dynamic; the same ideas can be organized and reorganized in countless ways, and the ability to translate between such organizational structures is becoming increasingly important in the sciences. Category theory offers a unifying framework for information modeling that can facilitate the translation of knowledge between disciplines. Written in an engaging and straightforward style, and assuming little background in mathematics, the book is rigorous but accessible to non-mathematicians. Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs—categories in disguise. After explaining the “big three” concepts of category theory—categories, functors, and natural transformations—the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with solutions. Category Theory for the Sciences is intended to create a bridge between the vast array of mathematical concepts used by mathematicians and the models and frameworks of such scientific disciplines as computation, neuroscience, and physics.

Author : Benjamin C. Pierce
ISBN : 0262660717
Genre : Computers
File Size : 23.47 MB
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Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial * Applications * Further Reading

Author : Francis Borceux
ISBN : 0521441781
Genre : Mathematics
File Size : 28.68 MB
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First of a 3-volume work giving a detailed account of what should be known by all working in, or using category theory. Volume 1 covers basic concepts.

Author : Jaime Gómez-Ramirez
ISBN : 9789400777385
Genre : Medical
File Size : 22.66 MB
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The purpose of the book is to advance in the understanding of brain function by defining a general framework for representation based on category theory. The idea is to bring this mathematical formalism into the domain of neural representation of physical spaces, setting the basis for a theory of mental representation, able to relate empirical findings, uniting them into a sound theoretical corpus. The innovative approach presented in the book provides a horizon of interdisciplinary collaboration that aims to set up a common agenda that synthesizes mathematical formalization and empirical procedures in a systemic way. Category theory has been successfully applied to qualitative analysis, mainly in theoretical computer science to deal with programming language semantics. Nevertheless, the potential of category theoretic tools for quantitative analysis of networks has not been tackled so far. Statistical methods to investigate graph structure typically rely on network parameters. Category theory can be seen as an abstraction of graph theory. Thus, new categorical properties can be added into network analysis and graph theoretic constructs can be accordingly extended in more fundamental basis. By generalizing networks using category theory we can address questions and elaborate answers in a more fundamental way without waiving graph theoretic tools. The vital issue is to establish a new framework for quantitative analysis of networks using the theory of categories, in which computational neuroscientists and network theorists may tackle in more efficient ways the dynamics of brain cognitive networks. The intended audience of the book is researchers who wish to explore the validity of mathematical principles in the understanding of cognitive systems. All the actors in cognitive science: philosophers, engineers, neurobiologists, cognitive psychologists, computer scientists etc. are akin to discover along its pages new unforeseen connections through the development of concepts and formal theories described in the book. Practitioners of both pure and applied mathematics e.g., network theorists, will be delighted with the mapping of abstract mathematical concepts in the terra incognita of cognition.

Author : Michael Barr
ISBN : UOM:39015034447873
Genre : Mathematics
File Size : 35.20 MB
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A wide coverage of topics in category theory and computer science is developed in this text, including introductory treatments of cartesian closed categories, sketches and elementary categorical model theory, and triples. Over 300 exercises are included.

Author : Grzegorz Rozenberg
ISBN : 9810214626
Genre : Computers
File Size : 79.92 MB
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The book is a very up-to-date collection of articles in theoretical computer science, written by leading authorities in the field. The topics range from algorithms and complexity to algebraic specifications, and from formal languages and language-theoretic modeling to computational geometry. The material is based on columns and articles that have appeared in the EATCS Bulletin during the past two to three years. Although very recent research is discussed, the largely informal style of writing makes the book accessible to readers with little or no previous knowledge of the topics.

Author : Duane W. Hybertson
ISBN : 1420072528
Genre : Technology & Engineering
File Size : 81.7 MB
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Systems engineering (SE) is experiencing a significant expansion that encompasses increasingly complex systems. However, a common body of knowledge on how to apply complex systems engineering (CSE) has yet to be developed. A combination of people and other autonomous agents, crossing organization boundaries and continually changing, these hybrid systems are less predictable while being more self-organizing and adaptive than traditional systems. The growing pains of this evolution and the ever-widening reach of SE technology require an effective foundation for integrating traditional and complex engineering methods, addressing machine and human interaction, as well as scaling up and down, from nano scale to the macro system-of-systems level. Model-oriented Systems Engineering Science: A Unifying Framework for Traditional and Complex Systems addresses solutions to that expansion and integration problem. This text takes advantage of better-understood systems science (SS) to support the transition, identifying and using commonalities between complex systems and other sciences, such as biology, sociology, cognitive science, organizational theory, and computational science. The author defines Model-oriented Systems Engineering Science (MOSES), an organized system that selects appropriate information from these disciplines and unifies it into a coherent framework. The result is a seamless approach to the class of systems across the extended scope of the new SE—a foundation upon which to develop an enhanced and unified SE. Modeling orientation (MO) provides a common perspective on the entire SES/SE enterprise, including all supporting sciences, engineering for the full range of traditional, complex, and hybrid systems, and their management. This book extends existing modeling approaches into an MO that views all science artifacts and engineering artifacts as models of systems. It organizes them into a virtual structured repository called the "SE model space"—effectively a container for the accumulating body of SE and SES knowledge in the form of models and patterns. By organizing and integrating all these elements into a common framework, the author makes the material not only easily accessible but also immediately applicable, and provides a well-grounded basis for future growth and evolution of the SE discipline.

Author : P. L. Curien
ISBN : 038754495X
Genre : Computers
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"The papers in this volume were presented at the fourth biennial Summer Conference on Category Theory and Computer Science, held in Paris, September3-6, 1991. Category theory continues to be an important tool in foundationalstudies in computer science. It has been widely applied by logicians to get concise interpretations of many logical concepts. Links between logic and computer science have been developed now for over twenty years, notably via the Curry-Howard isomorphism which identifies programs with proofs and types with propositions. The triangle category theory - logic - programming presents a rich world of interconnections. Topics covered in this volume include the following. Type theory: stratification of types and propositions can be discussed in a categorical setting. Domain theory: synthetic domain theory develops domain theory internally in the constructive universe of the effective topos. Linear logic: the reconstruction of logic based on propositions as resources leads to alternatives to traditional syntaxes. The proceedings of the previous three category theory conferences appear as Lecture Notes in Computer Science Volumes 240, 283 and 389"--PUBLISHER'S WEBSITE.

Author : Carol S. Aneshensel
ISBN : 9781452287164
Genre : Social Science
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Carol S. Aneshensel's Second Edition of Theory-Based Data Analysis for the Social Sciences presents the elaboration model for the multivariate analysis of observational quantitative data. Two complementary strategies are used: an exclusionary strategy and an inclusive strategy. The primary emphasis is on the translation of theory into a logical analytic strategy and the interpretation of results. The elaboration model is applied with case studies drawn from newly published research serving as prototypes for aligning theory and the data analytic plan used to test it. The second application of the elaboration model is in the form of original data analysis presented in two Analysis Journals that are integrated throughout the text and implement the full elaboration model. Using real data, not contrived examples, the text provides a step-by-step guide through the process of integrating theory with data analysis in order to arrive at meaningful answers to research questions.

Author : Ernest G. Manes
ISBN : UCAL:B4406139
Genre : Automates - Congrès
File Size : 67.41 MB
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Presents the results of a symposium which brought together scientists interested in applying modern algebraic techniques to problems in control & computer science.

Author : Stuart A. Umpleby
ISBN : 0891167358
Genre : Technology & Engineering
File Size : 81.20 MB
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This work contains the proceedings of a meeting held by 18 American and Soviet scholars on the state of cybernetics and systems theory in their two countries. American interest focused on the observation of systems, whilst Soviet interest focused on mathematical modelling.

Monograph( based very largely upon results original to the Czechoslovakian authors) presents an abstract account of the theory of automata for sophisticated readers presumed to be already conversant in the language of category theory. The seven chapters are punctuated at frequent intervals by exampl

"Et moi, ... , si j'avait su comment en revenir, je One service mathematics bas rendered the human race. It bas put common sense back n'y serais point all~.' where it belongs, on the topmost shelf next to lu1esVeme the dusty canister labelled 'discarded nonsense'~ Eric T. Bell 1be series is divergent; therefore we may be able to do something with it O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari ties abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sci ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser vice topology has rendered mathematical physics ... '; 'One service logic has rendered computer science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d 'etre of this series.

Author : Paul Weingartner
ISBN : 3540407448
Genre : Computers
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The contributions collected here address the question of whether the usage of logic in the sciences requires a deviation from classical mathematical logic. The monograph provides a succinct picture of recent research in alternative logics as they have been developed for applications in the sciences.

Author : Anatolii M. Samoilenko
ISBN : 0792307038
Genre : Mathematics
File Size : 32.26 MB
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Deals with structure theorems and models for finite and countable families of self-adjoint operators, which satisfy commutative, non-commutative, Lie and more general relations. This book is useful for mathematicians and physicists whose work involves spectral theory, Lie algebras and probability theory.

Author : Semen G. Gindikin
ISBN : 0792320379
Genre : Mathematics
File Size : 39.25 MB
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One service mathematics has rendered the 'Et moi, .. ., si j'avait su comment cn rcvenir, human race. It has put common sense back. je n'y serais point aile.' where it bdongs, on the topmost shelf neAt Jules Verne to the dusty canister labelled 'discarded non· sense'. The series is divergent; therefore we may be Eric T. Bdl able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. :; 'One service category theory has rendered mathematics .. .'. All a,rguably true. And all statements obtainable this way form part of the raison d'etre of this series.