In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Goettingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations. Chapter 8 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

Wavelets are mathematical functions that divide data into different frequency components, and then study each component with a resolution matched to its scale. First generation wavelets have proved useful in many applications in engineering and computer science. However they cannot be used with non-linear, data-adaptive decompositions and non-equispaced data.

"Second Generation Wavelets and their Applications" introduces "second generation wavelets" and the lifting transform that can be used to apply the traditional benefits of wavelets into a wide range of new areas in signal processing, data processing and computer graphics. This book details the mathematical fundamentals of the lifting transform and illustrates the latest applications of the transform in signal and image processing, numerical analysis, scattering data smoothing and rendering of computer images.

The papers on rough set theory and its applications placed in this volume present a wide spectrum of problems representative to the present. stage of this theory. Researchers from many countries reveal their rec.ent results on various aspects of rough sets. The papers are not confined only to mathematical theory but also include algorithmic aspects, applications and information about software designed for data analysis based on this theory. The volume contains also list of selected publications on rough sets which can be very useful to every one engaged in research or applications in this domain and sometimes perhaps unaware of results of other authors. The book shows that rough set theory is a vivid and vigorous domain with serious results to its credit and bright perspective for future developments. It lays on the crossroads of fuzzy sets, theory of evidence, neural networks, Petri nets and many other branches of AI, logic and mathematics. These diverse connec tions seem to be a very fertile feature of rough set theory and have essentially contributed to its wide and rapid expansion. It is worth mentioning that its philosophical roots stretch down from Leibniz, Frege and Russell up to Popper. Therefore many concepts dwelled on in rough set theory are not entirely new, nevertheless the theory can be viewed as an independent discipline on its own rights. Rough set theory has found many interesting real life applications in medicine, banking, industry and others."

Contents and treatment are fresh and very different from the standard treatments Presents a fully constructive version of what it means to do algebra The exposition is not only clear, it is friendly, philosophical, and considerate even to the most naive or inexperienced reader

This book explores the limits of our knowledge. The author shows how uncertainty and indefiniteness not only define the borders confining our understanding, but how they feed into the process of discovery and help to push back these borders. Starting with physics the author collects examples from economics, neurophysiology, history, ecology and philosophy. The first part shows how information helps to reduce indefiniteness. Understanding rests on our ability to find the right context, in which we localize a problem as a point in a network of connections. New elements must be combined with the old parts of the existing complex knowledge system, in order to profit maximally from the information. An attempt is made to quantify the value of information by its ability to reduce indefiniteness. The second part explains how to handle indefiniteness with methods from fuzzy logic, decision theory, hermeneutics and semiotics. It is not sufficient that the new element appears in an experiment, one also has to find a theoretical reason for its existence. Indefiniteness becomes an engine of science, which gives rise to new ideas.

This book highlights a number of recent research advances in the field of symplectic and contact geometry and topology, and related areas in low-dimensional topology. This field has experienced significant and exciting growth in the past few decades, and this volume provides an accessible introduction into many active research problems in this area. The papers were written with a broad audience in mind so as to reach a wide range of mathematicians at various levels. Aside from teaching readers about developing research areas, this book will inspire researchers to ask further questions to continue to advance the field. The volume contains both original results and survey articles, presenting the results of collaborative research on a wide range of topics. These projects began at the Research Collaboration Conference for Women in Symplectic and Contact Geometry and Topology (WiSCon) in July 2019 at ICERM, Brown University. Each group of authors included female and nonbinary mathematicians at different career levels in mathematics and with varying areas of expertise. This paved the way for new connections between mathematicians at all career levels, spanning multiple continents, and resulted in the new collaborations and directions that are featured in this work.

Formal methods for the specification and verification of hardware and software systems are becoming more and more important as systems increase in size and complexity. The aim of the book is to illustrate progress in formal methods, based on Petri net formalisms. It contains a collection of examples arising from different fields, such as flexible manufacturing, telecommunication and workflow management systems.The book covers the main phases in the life cycle of design and implementation of a system, i.e., specification, model checking techniques for verification, analysis of properties, code generation, and execution of models. These techniques and their tool support are discussed in detail including practical issues. Amongst others, fundamental concepts such as composition, abstraction, and reusability of models, model verification, and verification of properties are systematically introduced.

Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights. This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features: Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments Includes a large number of exercises of varying levels of difficulty, supplementing each chapter The text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas. Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.

This volume contains the accounts of papers delivered at the Nato Advanced Study Institute on Finite and Infinite Combinatorics in Sets and Logic held at the Banff Centre, Alberta, Canada from April 21 to May 4, 1991. As the title suggests the meeting brought together workers interested in the interplay between finite and infinite combinatorics, set theory, graph theory and logic. It used to be that infinite set theory, finite combinatorics and logic could be viewed as quite separate and independent subjects. But more and more those disciplines grow together and become interdependent of each other with ever more problems and results appearing which concern all of those disciplines. I appreciate the financial support which was provided by the N. A. T. O. Advanced Study Institute programme, the Natural Sciences and Engineering Research Council of Canada and the Department of Mathematics and Statistics of the University of Calgary. 11l'te meeting on Finite and Infinite Combinatorics in Sets and Logic followed two other meetings on discrete mathematics held in Banff, the Symposium on Ordered Sets in 1981 and the Symposium on Graphs and Order in 1984. The growing inter-relation between the different areas in discrete mathematics is maybe best illustrated by the fact that many of the participants who were present at the previous meetings also attended this meeting on Finite and Infinite Combinatorics in Sets and Logic.

Semigroups, Automata, Universal Algebra, Varieties

This monograph is a comprehensive account of formal matrices, examining homological properties of modules over formal matrix rings and summarising the interplay between Morita contexts and K theory. While various special types of formal matrix rings have been studied for a long time from several points of view and appear in various textbooks, for instance to examine equivalences of module categories and to illustrate rings with one-sided non-symmetric properties, this particular class of rings has, so far, not been treated systematically. Exploring formal matrix rings of order 2 and introducing the notion of the determinant of a formal matrix over a commutative ring, this monograph further covers the Grothendieck and Whitehead groups of rings. Graduate students and researchers interested in ring theory, module theory and operator algebras will find this book particularly valuable. Containing numerous examples, Formal Matrices is a largely self-contained and accessible introduction to the topic, assuming a solid understanding of basic algebra.

Includes detailed applications of cybersecurity and forensics for real life problems Addresses the challenges and solutions related to implementation of cybersecurity in multiple domains of smart computational technologies Includes the latest trends and area of research in cybersecurity and forensics Offers both quantitative and qualitative assesmnet of the topics Includes case studies that will be helpful for the researchers

"Knot theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject." - Ed Witten, Recipient of the Fields Medal "I spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It's a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field." - Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis Knot theory has proven to be a fascinating area of mathematical research, dating back about 150 years. Encyclopedia of Knot Theory provides short, interconnected articles on a variety of active areas in knot theory, and includes beautiful pictures, deep mathematical connections, and critical applications. Many of the articles in this book are accessible to undergraduates who are working on research or taking an advanced undergraduate course in knot theory. More advanced articles will be useful to graduate students working on a related thesis topic, to researchers in another area of topology who are interested in current results in knot theory, and to scientists who study the topology and geometry of biopolymers. Features Provides material that is useful and accessible to undergraduates, postgraduates, and full-time researchers Topics discussed provide an excellent catalyst for students to explore meaningful research and gain confidence and commitment to pursuing advanced degrees Edited and contributed by top researchers in the field of knot theory

Introduction to Mathematical Modeling helps students master the processes used by scientists and engineers to model real-world problems, including the challenges posed by space exploration, climate change, energy sustainability, chaotic dynamical systems and random processes. Primarily intended for students with a working knowledge of calculus but minimal training in computer programming in a first course on modeling, the more advanced topics in the book are also useful for advanced undergraduate and graduate students seeking to get to grips with the analytical, numerical, and visual aspects of mathematical modeling, as well as the approximations and abstractions needed for the creation of a viable model.

The present volume of the "Handbook of the History of Logic" is designed to establish 19th century Britain as a substantial force in logic, developing new ideas, some of which would be overtaken by, and other that would anticipate, the century's later capitulation to the mathematization of logic.
"British Logic in the Nineteenth Century" is indispensable reading and a definitive research resource for anyone with an interest in the history of logic.
- Detailed and comprehensive chapters covering the entire range of modal logic
- Contains the latest scholarly discoveries and interpretative insights that answer many questions in the field of logic

While in classical (abelian) homological algebra additive functors from abelian (or additive) categories to abelian categories are investigated , non- abelian homological algebra deals with non-additive functors and their homological properties , in particular with functors having values in non-abelian categories. Such functors haveimportant applications in algebra, algebraic topology, functional analysis, algebraic geometry and other principal areas of mathematics. To study homological properties of non-additive functors it is necessary to define and investigate their derived functors and satellites. It will be the aim of this book based on the results of researchers of A. Razmadze Mathematical Institute of the Georgian Academy of Sciences devoted to non-abelian homological algebra. The most important considered cases will be functors from arbitrary categories to the category of modules, group valued functors and commutative semigroup valued functors. In Chapter I universal sequences of functors are defined and in- vestigated with respect to (co)presheaves of categories, extending in a natural way the satellites of additive functors to the non-additive case and generalizing the classical relative homological algebra in additive categories to arbitrary categories. Applications are given in the furth- coming chapters. Chapter II is devoted to the non-abelian derived functors of group valued functors with respect to projective classes using projective pseu- dosimplicial resolutions. Their functorial properties (exactness, Milnor exact sequence, relationship with cotriple derived functors, satellites and Grothendieck cohomology, spectral sequence of an epimorphism, degree of an arbitrary functor) are established and applications to ho- mology and cohomology of groups are given.

The concept of infinity is one of the most important, and at the same time, one of the most mysterious concepts of science. Already in antiquity many philosophers and mathematicians pondered over its contradictory nature. In mathematics, the contradictions connected with infinity intensified after the creation, at the end of the 19th century, of the theory of infinite sets and the subsequent discovery, soon after, of paradoxes in this theory. At the time, many scientists ignored the paradoxes and used set theory extensively in their work, while others subjected set-theoretic methods in mathematics to harsh criticism. The debate intensified when a group of French mathematicians, who wrote under the pseudonym of Nicolas Bourbaki, tried to erect the whole edifice of mathematics on the single notion of a set. Some mathematicians greeted this attempt enthusiastically while others regarded it as an unnecessary formalization, an attempt to tear mathematics away from life-giving practical applications that sustain it. These differences notwithstanding, Bourbaki has had a significant influence on the evolution of mathematics in the twentieth century. In this book we try to tell the reader how the idea of the infinite arose and developed in physics and in mathematics, how the theory of infinite sets was constructed, what paradoxes it has led to, what significant efforts have been made to eliminate the resulting contradictions, and what routes scientists are trying to find that would provide a way out of the many difficulties.

The author provides an introduction to automated reasoning, and in particular to resolution theorem proving using the prover OTTER. He presents a new clausal version of von Neumann-Bernays-Goedel set theory, and lists over 400 theorems proved semiautomatically in elementary set theory. He presents a semiautomated proof that the composition of homomorphisms is a homomorphism, thus solving a challenge problem. The author next develops Peano's arithmetic, and gives more than 1200 definitions and theorems in elementary number theory. He gives part of the proof of the fundamental theorem of arithmetic (unique factorization), and gives and OTTER-generated proof of Euler's generalization of Fermat's theorem. Next he develops Tarski's geometry within OTTER. He obtains proofs of most of the challenge problems appearing in the literature, and offers further challenges. He then formalizes the modal logic calculus K4, in order to obtain very high level automated proofs of Loeb's theorem, and of Goedel's two incompleteness theorems. Finally he offers thirty-one unsolved problems in elementary number theory as challenge problems.

Robert A. Rankin, one of the world's foremost authorities on modular forms and a founding editor of The Ramanujan Journal, died on January 27, 2001, at the age of 85. Rankin had broad interests and contributed fundamental papers in a wide variety of areas within number theory, geometry, analysis, and algebra. To commemorate Rankin's life and work, the editors have collected together 25 papers by several eminent mathematicians reflecting Rankin's extensive range of interests within number theory. Many of these papers reflect Rankin's primary focus in modular forms. It is the editors' fervent hope that mathematicians will be stimulated by these papers and gain a greater appreciation for Rankin's contributions to mathematics.
This volume would be an inspiration to students and researchers in the areas of number theory and modular forms.

We dedicate this volume to Professor Parimala on the occasion of her 60th birthday. It contains a variety of papers related to the themes of her research. Parimala's rst striking result was a counterexample to a quadratic analogue of Serre's conjecture (Bulletin of the American Mathematical Society, 1976). Her in uence has cont- ued through her tenure at the Tata Institute of Fundamental Research in Mumbai (1976-2006),and now her time at Emory University in Atlanta (2005-present). A conference was held from 30 December 2008 to 4 January 2009, at the U- versity of Hyderabad, India, to celebrate Parimala's 60th birthday (see the conf- ence's Web site at http://mathstat.uohyd.ernet.in/conf/quadforms2008). The or- nizing committee consisted of J.-L. Colliot-Thel ' en ' e, Skip Garibaldi, R. Sujatha, and V. Suresh. The present volume is an outcome of this event. We would like to thank all the participants of the conference, the authors who have contributed to this volume, and the referees who carefully examined the s- mitted papers. We would also like to thank Springer-Verlag for readily accepting to publish the volume. In addition, the other three editors of the volume would like to place on record their deep appreciation of Skip Garibaldi's untiring efforts toward the nal publication.

This book deals with two important branches of mathematics, namely, logic and set theory. Logic and set theory are closely related and play very crucial roles in the foundation of mathematics, and together produce several results in all of mathematics. The topics of logic and set theory are required in many areas of physical sciences, engineering, and technology. The book offers solved examples and exercises, and provides reasonable details to each topic discussed, for easy understanding. The book is designed for readers from various disciplines where mathematical logic and set theory play a crucial role. The book will be of interested to students and instructors in engineering, mathematics, computer science, and technology.

This book discusses the theory of triangular norms and surveys several applied fields in which triangular norms play a significant part: probabilistic metric spaces, aggregation operators, many-valued logics, fuzzy logics, sets and control, and non-additive measures together with their corresponding integrals. It includes many graphical illustrations and gives a well-balanced picture of theory and applications. It is for mathematicians, computer scientists, applied computer scientists and engineers.

Steiner's Problem concerns finding a shortest interconnecting network for a finite set of points in a metric space. A solution must be a tree, which is called a Steiner Minimal Tree (SMT), and may contain vertices different from the points which are to be connected. Steiner's Problem is one of the most famous combinatorial-geometrical problems, but unfortunately it is very difficult in terms of combinatorial structure as well as computational complexity. However, if only a Minimum Spanning Tree (MST) without additional vertices in the interconnecting network is sought, then it is simple to solve. So it is of interest to know what the error is if an MST is constructed instead of an SMT. The worst case for this ratio running over all finite sets is called the Steiner ratio of the space. The book concentrates on investigating the Steiner ratio. The goal is to determine, or at least estimate, the Steiner ratio for many different metric spaces. The author shows that the description of the Steiner ratio contains many questions from geometry, optimization, and graph theory. Audience: Researchers in network design, applied optimization, and design of algorithms.

Harish-Chandra¿s general Plancherel inversion theorem admits a much shorter presentation for spherical functions. Previous expositions have dealt with a general, wide class of Lie groups. This has made access to the subject difficult for outsiders, who may wish to connect some aspects with several if not all other parts of mathematics. In this book, the essential features of Harish-Chandra theory are exhibited on SLn(R), but hundreds of pages of background are replaced by short direct verifications. The material is accessible to graduate students with no background in Lie groups and representation theory.