Mathematical Modelling By Help Of Category Theory
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Mathematical Modelling By Help Of Category Theory
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Author : Dmitrii Legatiuk
language : en
Publisher: Springer Nature
Release Date : 2025-03-19
Mathematical Modelling By Help Of Category Theory written by Dmitrii Legatiuk and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-03-19 with Mathematics categories.
This monograph offers a novel structural perspective on the modelling of engineering problems, utilizing abstract mathematics in the form of category theory. Specifically, the book aims to enhance the understanding of mathematical modelling by developing a category theory-based framework. Category theory is employed to establish clear relationships between mathematical models and their complexities. The theory is then extended to encompass coupled mathematical models, incorporating more advanced categorical structures. To bridge theory and practice, the book presents engineering applications of the abstract categorical framework, providing various modelling examples from real-world engineering scenarios. Additionally, it introduces initial concepts for automatic model generation and error detection in modelling. The theory developed here demonstrates the practical utility of category theory, making this book a valuable resource for researchers in applied mathematics and engineering, particularly those focused on theoretical foundations of modelling.
Mathematical Modelling By Help Of Category Theory
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Author : Dmitrii Legatiuk
language : en
Publisher:
Release Date : 2025
Mathematical Modelling By Help Of Category Theory written by Dmitrii Legatiuk and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025 with Engineering mathematics categories.
This monograph offers a novel structural perspective on the modelling of engineering problems, utilizing abstract mathematics in the form of category theory. Specifically, the book aims to enhance the understanding of mathematical modelling by developing a category theory-based framework. Category theory is employed to establish clear relationships between mathematical models and their complexities. The theory is then extended to encompass coupled mathematical models, incorporating more advanced categorical structures. To bridge theory and practice, the book presents engineering applications of the abstract categorical framework, providing various modelling examples from real-world engineering scenarios. Additionally, it introduces initial concepts for automatic model generation and error detection in modelling. The theory developed here demonstrates the practical utility of category theory, making this book a valuable resource for researchers in applied mathematics and engineering, particularly those focused on theoretical foundations of modelling.
Mathematical Modelling By Help Of Category Theory
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Author : Dmitrii Legatiuk
language : en
Publisher:
Release Date : 2024*
Mathematical Modelling By Help Of Category Theory written by Dmitrii Legatiuk and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024* with categories.
Solution of any engineering problem starts with a modelling process, which provides a model representing a system under consideration. The critical question for practical use of models is to assess if the model and results of using it can be trusted. To address this question, several approaches have been proposed in the past. However, some of these approaches deal with the practical aspects of models, or in other words, they address correctness of model implementation, while other approaches look at the models from the system engineering point of view, what makes them difficult to transfer to other fields of applications. Moreover, if we look at the modelling process more generally, then we could realise that (physics-based or mathematical) models are created by formalising specific sets of modelling assumptions in terms of mathematical expressions. In this context, if a mistake has been made and one specific modelling assumption has been violated, then the model can be wrong already before even starting using it in concrete applications and computations. Thus, it is necessary to develop tools and methods for detecting such modelling errors related to the conceptualisation of the model as early as possible. This dissertation aims at supporting the understanding of mathematical modelling by proposing a more abstract point of view on models and coupling of models, namely by developing a category theory-based modelling framework. In this context, categories of mathematical models are introduced, and their structure is discussed. In particular, the use of category theory allowed us to introduce clear relations between mathematical models, which are provided by the model property named complexity. Additionally, the notion of model convertibility is introduced, which establish a connection between different model formulations used in practical applications, e.g. integral and differential. After that, all basic notions are uplifted to the case of coupled mathematical models. As expected, coupled models require a more technical treatment, because the structure of categories of coupled mathematical models is more complicated. In particular, it is shown that not every comparison of coupled models makes sense. To support practical applications of the abstract modelling framework, this dissertation presents also engineering perspective on the abstract categorical construction, as well as provides various modelling examples coming from engineering practice. Additionally,based on the category theory-based modelling framework, first ideas towards automatic model generation and conceptual modelling error detection are proposed in this dissertation. Thus, a connection to type theory and functional programming is established in this way. Finally, an extension of the category theory-based modelling framework beyond mathematical models is proposed at the end of the dissertation. This extension sumvi marises knowledge about models gathered by help of category theory-based approach and addresses models from the abstract algebraic point of view. Hence, consideration of very general models is possible in this case. Therefore, this dissertation indicates that category theory and tools of abstract mathematics are indeed useful in practical applications.
Categories For The Working Philosopher
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Author : Elaine M. Landry
language : en
Publisher: Oxford University Press
Release Date : 2017
Categories For The Working Philosopher written by Elaine M. Landry and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Mathematics categories.
This is the first volume on category theory for a broad philosophical readership. It is designed to show the interest and significance of category theory for a range of philosophical interests: mathematics, proof theory, computation, cognition, scientific modelling, physics, ontology, the structure of the world. Each chapter is written by either a category-theorist or a philosopher working in one of the represented areas, in an accessible waythat builds on the concepts that are already familiar to philosophers working in these areas.
Basic Category Theory
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Author : Tom Leinster
language : en
Publisher: Cambridge University Press
Release Date : 2014-07-24
Basic Category Theory written by Tom Leinster and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-24 with Mathematics categories.
A short introduction ideal for students learning category theory for the first time.
Category Theory For The Sciences
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Author : David I. Spivak
language : en
Publisher: MIT Press
Release Date : 2014-10-10
Category Theory For The Sciences written by David I. Spivak and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-10 with Mathematics categories.
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.
An Introduction To Mathematical Modeling
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Author : Edward A. Bender
language : en
Publisher: Courier Corporation
Release Date : 2012-05-23
An Introduction To Mathematical Modeling written by Edward A. Bender and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-23 with Mathematics categories.
Employing a practical, "learn by doing" approach, this first-rate text fosters the development of the skills beyond the pure mathematics needed to set up and manipulate mathematical models. The author draws on a diversity of fields — including science, engineering, and operations research — to provide over 100 reality-based examples. Students learn from the examples by applying mathematical methods to formulate, analyze, and criticize models. Extensive documentation, consisting of over 150 references, supplements the models, encouraging further research on models of particular interest. The lively and accessible text requires only minimal scientific background. Designed for senior college or beginning graduate-level students, it assumes only elementary calculus and basic probability theory for the first part, and ordinary differential equations and continuous probability for the second section. All problems require students to study and create models, encouraging their active participation rather than a mechanical approach. Beyond the classroom, this volume will prove interesting and rewarding to anyone concerned with the development of mathematical models or the application of modeling to problem solving in a wide array of applications.
An Invitation To Applied Category Theory
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Author : Brendan Fong
language : en
Publisher: Cambridge University Press
Release Date : 2019-07-18
An Invitation To Applied Category Theory written by Brendan Fong and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-07-18 with Computers categories.
Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.
Elements Of Category Theory
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Author : Emily Riehl
language : en
Publisher: Cambridge University Press
Release Date : 2022-02-10
Elements Of Category Theory written by Emily Riehl and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-02-10 with Mathematics categories.
This book develops the theory of infinite-dimensional categories by studying the universe, or ∞-cosmos, in which they live.
Sets For Mathematics
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Author : F. William Lawvere
language : en
Publisher: Cambridge University Press
Release Date : 2003-01-27
Sets For Mathematics written by F. William Lawvere and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-01-27 with Mathematics categories.
In this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra.