This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.

This book offers a gentle introduction to type-2 fuzzy sets and, in particular, interval type-2 fuzzy sets and their application in biological modeling. Interval type-2 fuzzy modeling is a comparatively recent direction of research in fuzzy modeling. As the modeling of biological problems is inherently uncertain, the use of fuzzy sets in this field is a natural choice. The coverage begins with a succinct review of type-1 fuzzy basic theory, before providing a comprehensive and didactic explanation of type-2 fuzzy set components. In turn, Fuzzy Rule-Based Systems, or FRBS, are shown for both types, interval type-2 and type-1 fuzzy sets. Applications include the pharmacological models, prediction of prostate cancer stages, a model for HIV population transfer (asymptomatic to symptomatic), an epidemiological disease caused by HIV, some models in population growth, included the Malthus Model, and an epidemic model refers to COVID-19. The book is ideally suited to graduate students in mathematics and related fields, professionals, researchers, or the public interested in interval type-2 fuzzy modeling. Largely self-contained, it can also be used as a supplementary text in specialized graduate courses.

This book gathers together selected contributions presented at the 3rd Moroccan Andalusian Meeting on Algebras and their Applications, held in Chefchaouen, Morocco, April 12-14, 2018, and which reflects the mathematical collaboration between south European and north African countries, mainly France, Spain, Morocco, Tunisia and Senegal. The book is divided in three parts and features contributions from the following fields: algebraic and analytic methods in associative and non-associative structures; homological and categorical methods in algebra; and history of mathematics. Covering topics such as rings and algebras, representation theory, number theory, operator algebras, category theory, group theory and information theory, it opens up new avenues of study for graduate students and young researchers. The findings presented also appeal to anyone interested in the fields of algebra and mathematical analysis.

This book presents a philosophy of science, based on panenmentalism: an original modal metaphysics, which is realist about individual pure (non-actual) possibilities and rejects the notion of possible worlds. The book systematically constructs a new and novel way of understanding and explaining scientific progress, discoveries, and creativity. It demonstrates that a metaphysics of individual pure possibilities is indispensable for explaining and understanding mathematics and natural sciences. It examines the nature of individual pure possibilities, actualities, mind-dependent and mind-independent possibilities, as well as mathematical entities. It discusses in detail the singularity of each human being as a psychical possibility. It analyses striking scientific discoveries, and illustrates by means of examples of the usefulness and vitality of individual pure possibilities in the sciences.

This book provides an informal and geodesic introduction to factorization homology, focusing on providing intuition through simple examples. Along the way, the reader is also introduced to modern ideas in homotopy theory and category theory, particularly as it relates to the use of infinity-categories. As with the original lectures, the text is meant to be a leisurely read suitable for advanced graduate students and interested researchers in topology and adjacent fields.

This monograph introduces involutive categories and involutive operads, featuring applications to the GNS construction and algebraic quantum field theory. The author adopts an accessible approach for readers seeking an overview of involutive category theory, from the basics to cutting-edge applications. Additionally, the author's own recent advances in the area are featured, never having appeared previously in the literature. The opening chapters offer an introduction to basic category theory, ideal for readers new to the area. Chapters three through five feature previously unpublished results on coherence and strictification of involutive categories and involutive monoidal categories, showcasing the author's state-of-the-art research. Chapters on coherence of involutive symmetric monoidal categories, and categorical GNS construction follow. The last chapter covers involutive operads and lays important coherence foundations for applications to algebraic quantum field theory. With detailed explanations and exercises throughout, Involutive Category Theory is suitable for graduate seminars and independent study. Mathematicians and mathematical physicists who use involutive objects will also find this a valuable reference.

This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.

The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and non-substantial ontology in which all entities can be treated as transformations, and in which objects are merely the sources and aims of these transformations. Thus, in a rather surprising way, when employed as a formal ontology, category theory can unite seemingly disparate disciplines in contemporary science and the humanities, such as physics, mathematics and philosophy, but also computer and complex systems science.

This book collects and coherently presents the research that has been undertaken since the author's previous book Module Theory (1998). In addition to some of the key results since 1995, it also discusses the development of much of the supporting material. In the twenty years following the publication of the Camps-Dicks theorem, the work of Facchini, Herbera, Shamsuddin, Puninski, Prihoda and others has established the study of serial modules and modules with semilocal endomorphism rings as one of the promising directions for module-theoretic research. Providing readers with insights into the directions in which the research in this field is moving, as well as a better understanding of how it interacts with other research areas, the book appeals to undergraduates and graduate students as well as researchers interested in algebra.

In a fragment entitled Elementa Nova Matheseos Universalis (1683?) Leibniz writes "the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined"; in another fragment he takes the mathesis to be "the science of all things that are conceivable." Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between "arbitrary objects" ("objets quelconques"). It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the "reasons" ("Grunde") of others, and the latter are "consequences" ("Folgen") of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification.

This book lays out the theory of Mordell-Weil lattices, a very powerful and influential tool at the crossroads of algebraic geometry and number theory, which offers many fruitful connections to other areas of mathematics. The book presents all the ingredients entering into the theory of Mordell-Weil lattices in detail, notably, relevant portions of lattice theory, elliptic curves, and algebraic surfaces. After defining Mordell-Weil lattices, the authors provide several applications in depth. They start with the classification of rational elliptic surfaces. Then a useful connection with Galois representations is discussed. By developing the notion of excellent families, the authors are able to design many Galois representations with given Galois groups such as the Weyl groups of E6, E7 and E8. They also explain a connection to the classical topic of the 27 lines on a cubic surface.Two chapters deal with elliptic K3 surfaces, a pulsating area of recent research activity which highlights many central properties of Mordell-Weil lattices. Finally, the book turns to the rank problem-one of the key motivations for the introduction of Mordell-Weil lattices. The authors present the state of the art of the rank problem for elliptic curves both over Q and over C(t) and work out applications to the sphere packing problem. Throughout, the book includes many instructive examples illustrating the theory.

This edited book brings together research work in the field of constructive semantics with scholarship on the phenomenological foundations of logic and mathematics. It addresses one of the central issues in the epistemology and philosophy of mathematics, namely the relationship between phenomenological meaning constitution and constructive semantics. Contributing authors explore deep structural connections and fundamental differences between phenomenology and constructivism. Papers are drawn from contributions to a prestigious workshop held at the University of Friedrichshafen. Readers will discover insight into structural connections between the phenomenological concept of meaning constitution and constructivist concepts of meaning. Discussion ranges from more specific conceptualizations in the philosophy of logic and mathematics to more general considerations in epistemology, inferential semantics and phenomenology. Questions such as a possible phenomenological understanding of the relationship between structural rules and particle rules in dialogical logic are explored. Significant aspects of both phenomenology and dialectics, and dialectics and constructivism emerge. Graduates and researchers of philosophy, especially logic, as well as scholars of mathematics will all find something of interest in the expert insights presented in this volume.

Many believe mathematics is only about calculations, formulas, numbers, and strange letters. But mathematics is much more than just crunching numbers or manipulating symbols. Mathematics is about discovering patterns, uncovering hidden structures, finding counterexamples, and thinking logically. Mathematics is a way of thinking. It is an activity that is both highly creative and challenging. This book offers an introduction to mathematical reasoning for beginning university or college students, providing a solid foundation for further study in mathematics, computer science, and related disciplines. Written in a manner that directly conveys the sense of excitement and discovery at the heart of doing science, its 25 short and visually appealing chapters cover the basics of set theory, logic, proof methods, combinatorics, graph theory, and much more. In the book you will, among other things, find answers to: What is a proof? What is a counterexample? What does it mean to say that something follows logically from a set of premises? What does it mean to abstract over something? How can knowledge and information be represented and used in calculations? What is the connection between Morse code and Fibonacci numbers? Why could it take billions of years to solve Hanoi's Tower? Logical Methods is especially appropriate for students encountering such concepts for the very first time. Designed to ease the transition to a university or college level study of mathematics or computer science, it also provides an accessible and fascinating gateway to logical thinking for students of all disciplines.

Large linear systems of equations arise in most scientific problems where mathematical models are used. The most efficient methods for solving these equations are iterative methods. The first part of this book contains basic and classical material from the study of linear algebra and numerical linear algebra. The second half of the book is unique among books on this topic, because it is devoted to the construction of preconditioners and iterative acceleration methods of the conjugate gradient type. This book is for graduate students and researchers in numerical analysis and applied mathematics.

This edited book focuses on concepts and their applications using the theory of conceptual spaces, one of today's most central tracks of cognitive science discourse. It features 15 papers based on topics presented at the Conceptual Spaces @ Work 2016 conference. The contributors interweave both theory and applications in their papers. Among the first mentioned are studies on metatheories, logical and systemic implications of the theory, as well as relations between concepts and language. Examples of the latter include explanatory models of paradigm shifts and evolution in science as well as dilemmas and issues of health, ethics, and education. The theory of conceptual spaces overcomes many translational issues between academic theoretization and practical applications. The paradigm is mainly associated with structural explanations, such as categorization and meronomy. However, the community has also been relating it to relations, functions, and systems. The book presents work that provides a geometric model for the representation of human conceptual knowledge that bridges the symbolic and the sub-conceptual levels of representation. The model has already proven to have a broad range of applicability beyond cognitive science and even across a number of disciplines related to concepts and representation.

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

This book constitutes the thoroughly revised selected papers from the 16th International Conference on Formal Aspects of Component Software, FACS 2019, held in Amsterdam, The Netherlands, in October 2019. The 9 full papers presented together with 9 full papers and 3 short papers as well as 2 other papers were carefully reviewed and selected from 27 submissions. FACS 2019 is concerned with how formal methods can be used to make component-based and service-oriented software development succeed. Formal methods have provided a foundation for component-based software by successfully addressing challenging issues such as mathematical models for components, composition and adaptation, or rigorous approaches to verification, deployment, testing, and certification.

This book constitutes the refereed proceedings of the International Symposium on Logical Foundations of Computer Science, LFCS 2020, held in Deerfield Beach, FL, USA, in January 2020. The 17 revised full papers were carefully reviewed and selected from 30 submissions. The scope of the Symposium is broad and includes constructive mathematics and type theory; homotopy type theory; logic, automata, and automatic structures; computability and randomness; logical foundations of programming; logical aspects of computational complexity; parameterized complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logics; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple-agent system logics; logics of proof and justification; non-monotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; mathematical fuzzy logic; system design logics; other logics in computer science.

This book offers a review of the vibrant areas of geometric representation theory and gauge theory, which are characterized by a merging of traditional techniques in representation theory with the use of powerful tools from algebraic geometry, and with strong inputs from physics. The notes are based on lectures delivered at the CIME school "Geometric Representation Theory and Gauge Theory" held in Cetraro, Italy, in June 2018. They comprise three contributions, due to Alexander Braverman and Michael Finkelberg, Andrei Negut, and Alexei Oblomkov, respectively. Braverman and Finkelberg's notes review the mathematical theory of the Coulomb branch of 3D N=4 quantum gauge theories. The purpose of Negut's notes is to study moduli spaces of sheaves on a surface, as well as Hecke correspondences between them. Oblomkov's notes concern matrix factorizations and knot homology. This book will appeal to both mathematicians and theoretical physicists and will be a source of inspiration for PhD students and researchers.

In this volume, Lambek and Scott reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher-order logic, and cartesian closed categories, are essentially the same. Part II demonstrates that another formulation of higher-order logic, (intuitionistic) type theories, is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and the algebraic language of category theory are given.The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludeds with a set of exercises.

This monograph initiates a theory of new categorical structures that generalize the simplicial Segal property to higher dimensions. The authors introduce the notion of a d-Segal space, which is a simplicial space satisfying locality conditions related to triangulations of d-dimensional cyclic polytopes. Focus here is on the 2-dimensional case. Many important constructions are shown to exhibit the 2-Segal property, including Waldhausen's S-construction, Hecke-Waldhausen constructions, and configuration spaces of flags. The relevance of 2-Segal spaces in the study of Hall and Hecke algebras is discussed. Higher Segal Spaces marks the beginning of a program to systematically study d-Segal spaces in all dimensions d. The elementary formulation of 2-Segal spaces in the opening chapters is accessible to readers with a basic background in homotopy theory. A chapter on Bousfield localizations provides a transition to the general theory, formulated in terms of combinatorial model categories, that features in the main part of the book. Numerous examples throughout assist readers entering this exciting field to move toward active research; established researchers in the area will appreciate this work as a reference.

This short textbook provides a succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. It will be suitable for all mathematics undergraduates coming to the subject for the first time. The book is based on lectures given at the University of Cambridge and covers the basic concepts of logic: first order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. There are also chapters on recursive functions, the axiom of choice, ordinal and cardinal arithmetic and the incompleteness theorems. Dr Johnstone has included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. Consequently the book, while making an attractive first textbook for those who plan to specialise in logic, will be particularly valuable for mathematics and computer scientists whose primary interests lie elsewhere.

This LNCS book is part of the FOLLI book series and constitutes the proceedings of the 7th International Workshop on Logic, Rationality, and Interaction, LORI 2019, held in Chongqing, China, in October 2019. The 31 papers presented in this book were carefully reviewed and selected from 56 submissions. They focus on the following topics: agency; argumentation and agreement; belief revision and belief merging; belief representation; cooperation; decision making and planning; natural language; philosophy and philosophical logic; and strategic reasoning.

This book provides an accessible introduction to the state of the art of representation theory of finite groups. Starting from a basic level that is summarized at the start, the book proceeds to cover topics of current research interest, including open problems and conjectures. The central themes of the book are block theory and module theory of group representations, which are comprehensively surveyed with a full bibliography. The individual chapters cover a range of topics within the subject, from blocks with cyclic defect groups to representations of symmetric groups. Assuming only modest background knowledge at the level of a first graduate course in algebra, this guidebook, intended for students taking first steps in the field, will also provide a reference for more experienced researchers. Although no proofs are included, end-of-chapter exercises make it suitable for student seminars.