Mathematics of Discrete Structures for Computer Science

Communication complexity is the mathematical study of scenarios where several parties need to communicate to achieve a common goal, a situation that naturally appears during computation. This introduction presents the most recent developments in an accessible form, providing the language to unify several disjoint research subareas. Written as a guide for a graduate course on communication complexity, it will interest a broad audience in computer science, from advanced undergraduates to researchers in areas ranging from theory to algorithm design to distributed computing. The first part presents basic theory in a clear and illustrative way, offering beginners an entry into the field. The second part describes applications including circuit complexity, proof complexity, streaming algorithms, extension complexity of polytopes, and distributed computing. Proofs throughout the text use ideas from a wide range of mathematics, including geometry, algebra, and probability. Each chapter contain

Knowledge of the structure of the cosmos, Plato suggests, is important in organizing a human community which aims at happiness. This book investigates this theme in Plato's later works, the Timaeus, Statesman, and Laws. Dominic J. O'Meara proposes fresh readings of these texts, starting from the religious festivals and technical and artistic skills in the context of which Plato elaborates his cosmological and political theories, for example the Greek architect's use of models as applied by Plato in describing the making of the world. O'Meara gives an account of the model of which Plato's world is an image; of the mathematics used in producing the world; and of the relation between the cosmic model and the political science and legislation involved in designing a model state in the Laws. Non-specialist scholars and students will be able to access and profit from the book.

Although numerical approximation and statistical inference are traditionally covered as entirely separate subjects, they are intimately connected through the common purpose of making estimations with partial information. This book explores these connections from a game and decision theoretic perspective, showing how they constitute a pathway to developing simple and general methods for solving fundamental problems in both areas. It illustrates these interplays by addressing problems related to numerical homogenization, operator adapted wavelets, fast solvers, and Gaussian processes. This perspective reveals much of their essential anatomy and greatly facilitates advances in these areas, thereby appearing to establish a general principle for guiding the process of scientific discovery. This book is designed for graduate students, researchers, and engineers in mathematics, applied mathematics, and computer science, and particularly researchers interested in drawing on and developing this

Network flow theory has been used across a number of disciplines, including theoretical computer science, operations research, and discrete math, to model not only problems in the transportation of goods and information, but also a wide range of applications from image segmentation problems in computer vision to deciding when a baseball team has been eliminated from contention. This graduate text and reference presents a succinct, unified view of a wide variety of efficient combinatorial algorithms for network flow problems, including many results not found in other books. It covers maximum flows, minimum-cost flows, generalized flows, multicommodity flows, and global minimum cuts and also presents recent work on computing electrical flows along with recent applications of these flows to classical problems in network flow theory.

This textbook is ideal for MATLAB/Introduction to Programming courses in both Engineering and Computer Science departments.Engineering Computation with MATLAB introduces the power of computing to engi

Finite-state methods are the most efficient mechanisms for analysing textual and symbolic data, providing elegant solutions for an immense number of practical problems in computational linguistics and computer science. This book for graduate students and researchers gives a complete coverage of the field, starting from a conceptual introduction and building to advanced topics and applications. The central finite-state technologies are introduced with mathematical rigour, ranging from simple finite-state automata to transducers and bimachines as 'input-output' devices. Special attention is given to the rich possibilities of simplifying, transforming and combining finite-state devices. All algorithms presented are accompanied by full correctness proofs and executable source code in a new programming language, C(M), which focuses on transparency of steps and simplicity of code. Thus, by enabling readers to obtain a deep formal understanding of the subject and to put finite-state methods t

This tutorial reference serves as a coherent overview of various statistical and mathematical approaches used in brain network analysis, where modeling the complex structures and functions of the human brain often poses many unique computational and statistical challenges. This book fills a gap as a textbook for graduate students while simultaneously articulating important and technically challenging topics. Whereas most available books are graph theory-centric, this text introduces techniques arising from graph theory and expands to include other different models in its discussion on network science, regression, and algebraic topology. Links are included to the sample data and codes used in generating the book's results and figures, helping to empower methodological understanding in a manner immediately usable to both researchers and students.

Category theory is unmatched in its ability to organize and layer abstractions and to find commonalities between structures of all sorts. No longer the exclusive preserve of pure mathematicians, it is now proving itself to be a powerful tool in science, informatics, and industry. By facilitating communication between communities and building rigorous bridges between disparate worlds, applied category theory has the potential to be a major organizing force. This book offers a self-contained tour of applied category theory. Each chapter follows a single thread motivated by a real-world application and discussed with category-theoretic tools. We see data migration as an adjoint functor, electrical circuits in terms of monoidal categories and operads, and collaborative design via enriched profunctors. All the relevant category theory, from simple to sophisticated, is introduced in an accessible way with many examples and exercises, making this an ideal guide even for those without experien

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book w

This volume contains eight survey articles based on the invited lectures given at the 27th British Combinatorial Conference, held at the University of Birmingham in July 2019. This biennial conference is a well-established international event, with speakers from around the world. The volume provides an up-to-date overview of current research in several areas of combinatorics, including graph theory, cryptography, matroids, incidence geometries and graph limits. Each article is clearly written and assumes little prior knowledge on the part of the reader. The authors are some of the world's foremost researchers in their fields, and here they summarise existing results and give a unique preview of cutting-edge developments. The book provides a valuable survey of the present state of knowledge in combinatorics, and will be useful to researchers and advanced graduate students, primarily in mathematics but also in computer science and statistics.

Category theory is unmatched in its ability to organize and layer abstractions and to find commonalities between structures of all sorts. No longer the exclusive preserve of pure mathematicians, it is now proving itself to be a powerful tool in science, informatics, and industry. By facilitating communication between communities and building rigorous bridges between disparate worlds, applied category theory has the potential to be a major organizing force. This book offers a self-contained tour of applied category theory. Each chapter follows a single thread motivated by a real-world application and discussed with category-theoretic tools. We see data migration as an adjoint functor, electrical circuits in terms of monoidal categories and operads, and collaborative design via enriched profunctors. All the relevant category theory, from simple to sophisticated, is introduced in an accessible way with many examples and exercises, making this an ideal guide even for those without experien

Python is today's fastest growing programming language. This engaging and refreshingly different guide breaks down the skills into clear step-by-step chunks and explains the theory using brief easy-to-understand language. Rather than bamboozling readers with pages of mind-numbing technical jargon, this book includes 150 practical challenges, putting the power in the reader's hands. Through creating programs to solve these challenges the reader will quickly progress from mastering the basics to confidently using subroutines, a graphical user interface, and linking to external text, csv and SQL files. This book is perfect for anyone who wants to learn how to program with Python. In particular, students starting out in computer science and teachers who want to improve their confidence in Python will find here a set of ready-made challenges for classroom use.

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book w

Proof complexity is a rich subject drawing on methods from logic, combinatorics, algebra and computer science. This self-contained book presents the basic concepts, classical results, current state of the art and possible future directions in the field. It stresses a view of proof complexity as a whole entity rather than a collection of various topics held together loosely by a few notions, and it favors more generalizable statements. Lower bounds for lengths of proofs, often regarded as the key issue in proof complexity, are of course covered in detail. However, upper bounds are not neglected: this book also explores the relations between bounded arithmetic theories and proof systems and how they can be used to prove upper bounds on lengths of proofs and simulations among proof systems. It goes on to discuss topics that transcend specific proof systems, allowing for deeper understanding of the fundamental problems of the subject.

Classical logic is concerned, loosely, with the behaviour of truths. Epistemic logic similarly is about the behaviour of known or believed truths. Justification logic is a theory of reasoning that enables the tracking of evidence for statements and therefore provides a logical framework for the reliability of assertions. This book, the first in the area, is a systematic account of the subject, progressing from modal logic through to the establishment of an arithmetic interpretation of intuitionistic logic. The presentation is mathematically rigorous but in a style that will appeal to readers from a wide variety of areas to which the theory applies. These include mathematical logic, artificial intelligence, computer science, philosophical logic and epistemology, linguistics, and game theory.

This textbook, based on the authors' fifteen years of teaching, is a complete teaching tool for turning students into logic designers in one semester. Each chapter describes new concepts, giving extensive applications and examples. Assuming no prior knowledge of discrete mathematics, the authors introduce all background in propositional logic, asymptotics, graphs, hardware and electronics. Important features of the presentation are: • All material is presented in full detail. Every designed circuit is formally specified and implemented, the correctness of the implementation is proved, and the cost and delay are analyzed • Algorithmic solutions are offered for logical simulation, computation of propagation delay and minimum clock period • Connections are drawn from the physical analog world to the digital abstraction • The language of graphs is used to describe formulas and circuits • Hundreds of figures, examples and exercises enhance understanding. The extensive website (http://www.en

Quantum theory is the soul of theoretical physics. It is not just a theory of specific physical systems, but rather a new framework with universal applicability. This book shows how we can reconstruct the theory from six information-theoretical principles, by rebuilding the quantum rules from the bottom up. Step by step, the reader will learn how to master the counterintuitive aspects of the quantum world, and how to efficiently reconstruct quantum information protocols from first principles. Using intuitive graphical notation to represent equations, and with shorter and more efficient derivations, the theory can be understood and assimilated with exceptional ease. Offering a radically new perspective on the field, the book contains an efficient course of quantum theory and quantum information for undergraduates. It is aimed at researchers, professionals, and students in physics, computer science and philosophy, as well as the curious outsider seeking a deeper understanding of the theo

The Earth and environmental sciences are becoming progressively more quantitative due to the increased use of mathematical models and new data analysis techniques. This accessible introduction presents an overview of the mathematical methods essential for understanding Earth processes, providing an invaluable resource for students and early career researchers who may have missed (or forgotten) the mathematics they need to succeed as scientists. Topics build gently from basic methods such as calculus to more advanced techniques including linear algebra and differential equations. The practical applications of the mathematical methods to a variety of topics are discussed, ranging from atmospheric science and oceanography to biogeochemistry and geophysics. Including over 530 exercises and end-of-chapter problems, as well as additional computer codes in Python and MATLAB®, this book supports readers in applying appropriate analytical or computational methods to solving real research questi

This book combines a comprehensive state-of-the-art analysis of bifurcations of discrete-time dynamical systems with concrete instruction on implementations (and example applications) in the free MATLAB® software MatContM developed by the authors. While self-contained and suitable for independent study, the book is also written with users in mind and is an invaluable reference for practitioners. Part I focuses on theory, providing a systematic presentation of bifurcations of fixed points and cycles of finite-dimensional maps, up to and including cases with two control parameters. Several complementary methods, including Lyapunov exponents, invariant manifolds and homoclinic structures, and parts of chaos theory, are presented. Part II introduces MatContM through step-by-step tutorials on how to use the general numerical methods described in Part I for simple dynamical models defined by one- and two-dimensional maps. Further examples in Part III show how MatContM can be used to analyze

Instabilities are present in all natural fluids from rivers to atmospheres. This book considers the physical processes that generate instability. Part I describes the normal mode instabilities most important in geophysical applications, including convection, shear instability and baroclinic instability. Classical analytical approaches are covered, while also emphasising numerical methods, mechanisms such as internal wave resonance, and simple `rules of thumb' that permit assessment of instability quickly and intuitively. Part II introduces the cutting edge: nonmodal instabilities, the relationship between instability and turbulence, self-organised criticality, and advanced numerical techniques. Featuring numerous exercises and projects, the book is ideal for advanced students and researchers wishing to understand flow instability and apply it to their own research. It can be used to teach courses in oceanography, atmospheric science, coastal engineering, applied mathematics and environ

Algorithmic Aspects of Graph Connectivity is the first comprehensive book on this central notion in graph and network theory, emphasizing its algorithmic aspects. Because of its wide applications in the fields of communication, transportation, and production, graph connectivity has made tremendous algorithmic progress under the influence of the theory of complexity and algorithms in modern computer science. The book contains various definitions of connectivity, including edge-connectivity and vertex-connectivity, and their ramifications, as well as related topics such as flows and cuts. The authors thoroughly discuss new concepts and algorithms that allow for quicker and more efficient computing, such as maximum adjacency ordering of vertices. Covering both basic definitions and advanced topics, this book can be used as a textbook in graduate courses in mathematical sciences, such as discrete mathematics, combinatorics, and operations research, and as a reference book for specialists i

This Handbook reviews a wealth of research in cognitive and educational psychology that investigates how to enhance learning and instruction to aid students struggling to learn and to advise teachers on how best to support student learning. The Handbook includes features that inform readers about how to improve instruction and student achievement based on scientific evidence across different domains, including science, mathematics, reading and writing. Each chapter supplies a description of the learning goal, a balanced presentation of the current evidence about the efficacy of various approaches to obtaining that learning goal, and a discussion of important future directions for research in this area. It is the ideal resource for researchers continuing their study of this field or for those only now beginning to explore how to improve student achievement.

With the increasing public interest in artificial intelligence (AI), there is also increasing interest in learning about the benefits that AI can deliver to society. This book focuses on research advances in AI that benefit the conservation of wildlife, forests, coral reefs, rivers, and other natural resources. It presents how the joint efforts of researchers in computer science, ecology, economics, and psychology help address the goals of the United Nations' 2030 Agenda for Sustainable Development. Written at a level accessible to conservation professionals and AI researchers, the book offers both an overview of the field and an in-depth view of how AI is being used to understand patterns in wildlife poaching and enhance patrol efforts in response, covering research advances, field tests and real-world deployments. The book also features efforts in other major conservation directions, including protecting natural resources, ecosystem monitoring, and bio-invasion management through the

This Handbook reviews a wealth of research in cognitive and educational psychology that investigates how to enhance learning and instruction to aid students struggling to learn and to advise teachers on how best to support student learning. The Handbook includes features that inform readers about how to improve instruction and student achievement based on scientific evidence across different domains, including science, mathematics, reading and writing. Each chapter supplies a description of the learning goal, a balanced presentation of the current evidence about the efficacy of various approaches to obtaining that learning goal, and a discussion of important future directions for research in this area. It is the ideal resource for researchers continuing their study of this field or for those only now beginning to explore how to improve student achievement.

With a focus on models and tangible applications of probability from physics, computer science, and other related disciplines, this book successfully guides readers through fundamental coverage for en

This is a modern, introductory textbook on the dynamics of the atmosphere and ocean, with a healthy dose of geophysical fluid dynamics. It will be invaluable for intermediate to advanced undergraduate and graduate students in meteorology, oceanography, mathematics, and physics. It is unique in taking the reader from very basic concepts to the forefront of research. It also forms an excellent refresher for researchers in atmospheric science and oceanography. It differs from other books at this level in both style and content: as well as very basic material it includes some elementary introductions to more advanced topics. The advanced sections can easily be omitted for a more introductory course, as they are clearly marked in the text. Readers who wish to explore these topics in more detail can refer to this book's parent, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, now in its second edition.

With the increasing public interest in artificial intelligence (AI), there is also increasing interest in learning about the benefits that AI can deliver to society. This book focuses on research advances in AI that benefit the conservation of wildlife, forests, coral reefs, rivers, and other natural resources. It presents how the joint efforts of researchers in computer science, ecology, economics, and psychology help address the goals of the United Nations' 2030 Agenda for Sustainable Development. Written at a level accessible to conservation professionals and AI researchers, the book offers both an overview of the field and an in-depth view of how AI is being used to understand patterns in wildlife poaching and enhance patrol efforts in response, covering research advances, field tests and real-world deployments. The book also features efforts in other major conservation directions, including protecting natural resources, ecosystem monitoring, and bio-invasion management through the

Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world. The first part of this book covers groups, after some preliminaries on sets, functions, relations, and induction, and features applications such as public-key cryptography, Sudoku, the finite Fourier transform, and symmetry in chemistry and physics. The second part of this book covers rings and fields, and features applications such as random number generators, error correcting codes, the Google page rank algorithm, communication networks, and elliptic curve cryptography. The book's masterful use of colorful figures and images helps illustrate the applications and concepts in the text. Real-world examples and exercises will help students contextualize the information. Intended for a year-long undergraduate course in algebra for mathematics, engineering, and computer science majors, the only prerequisites are calculus and a bit of courage when ask

Written in an easy-to-understand manner, this comprehensive textbook brings together both basic and advanced concepts of numerical methods in a single volume. Important topics including error analysis, nonlinear equations, systems of linear equations, interpolation and interpolation for Equal intervals and bivariate interpolation are discussed comprehensively. The textbook is written to cater to the needs of undergraduate students of mathematics, computer science, mechanical engineering, civil engineering and information technology for a course on numerical methods/numerical analysis. The text simplifies the understanding of the concepts through exercises and practical examples. Pedagogical features including solved examples and unsolved exercises are interspersed throughout the book for better understanding.

Written in lucid language, this valuable textbook brings together fundamental concepts of data mining and data warehousing in a single volume. Important topics including information theory, decision tree, Naïve Bayes classifier, distance metrics, partitioning clustering, associate mining, data marts and operational data store are discussed comprehensively. The textbook is written to cater to the needs of undergraduate students of computer science, engineering and information technology for a course on data mining and data warehousing. The text simplifies the understanding of the concepts through exercises and practical examples. Chapters such as classification, associate mining and cluster analysis are discussed in detail with their practical implementation using Weka and R language data mining tools. Advanced topics including big data analytics, relational data models and NoSQL are discussed in detail. Pedagogical features including unsolved problems and multiple-choice questions are

By the end of this decade, the U.S. economy will annually create hundreds of thousands of new jobs requiring a bachelor's degree in STEM fields, particularly computer science. This increasing need for

This is an introduction to the dynamics of fluids at small scales, the physical and mathematical underpinnings of Brownian motion, and the application of these subjects to the dynamics and flow of complex fluids such as colloidal suspensions and polymer solutions. It brings together continuum mechanics, statistical mechanics, polymer and colloid science, and various branches of applied mathematics, in a self-contained and integrated treatment that provides a foundation for understanding complex fluids, with a strong emphasis on fluid dynamics. Students and researchers will find that this book is extensively cross-referenced to illustrate connections between different aspects of the field. Its focus on fundamental principles and theoretical approaches provides the necessary groundwork for research in the dynamics of flowing complex fluids.

This is an introduction to the dynamics of fluids at small scales, the physical and mathematical underpinnings of Brownian motion, and the application of these subjects to the dynamics and flow of complex fluids such as colloidal suspensions and polymer solutions. It brings together continuum mechanics, statistical mechanics, polymer and colloid science, and various branches of applied mathematics, in a self-contained and integrated treatment that provides a foundation for understanding complex fluids, with a strong emphasis on fluid dynamics. Students and researchers will find that this book is extensively cross-referenced to illustrate connections between different aspects of the field. Its focus on fundamental principles and theoretical approaches provides the necessary groundwork for research in the dynamics of flowing complex fluids.

Children come into the world completely helpless, and require well-functioning families and schools to meet their needs, protect their interests and nurture their potential. This book argues that healthy child-development depends on values, ideas and structures that promote justice for children and families; in particular, checks and balances that favour: • Fairness: allowing fair distribution of resources, so that every child and family have the best possible chance to reach their potential. • Protection: resources for families, neighbourhoods and schools to help protect and encourage their children, alongside the means to intervene, should this protection fail. • Autonomy: encouraging children's voice and participation in decision-making at a level commensurate with their maturity. Authored by leading experts in the field, the book is comprised of short, highly readable chapters with an interdisciplinary appeal, for practitioners of social science, law, social work, psychology, paedi

The prediction of crystal structures from first principles has been one of the grand challenges for computational methods in chemistry and materials science. The goal of being able to reliably predict

This volume brings together scholars of Mediterranean archaeology, ancient history, and complexity science to advance theoretical approaches and analytical tools for studying maritime connectivity. For the coast-hugging populations of the ancient Mediterranean, mobility and exchange depended on a distinct environment and technological parameters that created diverse challenges and opportunities, making the modeling of maritime interaction a paramount concern for understanding cultural interaction more generally. Network-inspired metaphors have long been employed in discussions of this interaction, but increasing theoretical sophistication and advances in formal network analysis now offer opportunities to refine and test the dominant paradigm of connectivity. Extending from prehistory into the Byzantine period, the case studies here reveal the potential of such network approaches. Collectively they explore the social, economic, religious, and political structures that guided Mediterrane

The third edition of Reinforced and Prestressed Concrete continues to be the most comprehensive text for engineering students, instructors and practising engineers. Theoretical and practical aspects of analysis and design are presented in a clear, easy-to-follow manner and are complemented by numerous illustrative and design examples to aid students' comprehension of complex concepts. This edition has been fully updated to reflect recent amendments and addenda to the Australian Standard for Concrete Structures AS3600–2009 and allied standards. Two new chapters, covering T-beams, irregular-shaped sections and continuous beams, and strut-and-tie modelling have been added as discrete modules to enhance the progression of topics. Additional information is provided on fire resistance, detailing and covering, long-term deflection and design for torsion. An expanded collection of end-of-chapter tutorial problems consolidate student learning and develop problem-solving skills. Reinforced and P

Graphical models are a statistical tool used for a wide range of applications. There has been a huge amount of research in this topic across statistics, mathematics and computer science in the last fe