Mathematics of Discrete Structures for Computer Science

Using basic category theory, this Element describes all the central concepts and proves the main theorems of theoretical computer science. Category theory, which works with functions, processes, and structures, is uniquely qualified to present the fundamental results of theoretical computer science. In this Element, readers will meet some of the deepest ideas and theorems of modern computers and mathematics, such as Turing machines, unsolvable problems, the P=NP question, Kurt Gödel's incompleteness theorem, intractable problems, cryptographic protocols, Alan Turing's Halting problem, and much more. The concepts come alive with many examples and exercises.

MATCHES THE LATEST EXAM Let us supplement your AP classroom experience with this easy-to-follow study guide The immensely popular 5 Steps to a 5: AP Computer Science A guide has been updated for the 2021-22 school year and now contains: 3 full-length practice exams that reflect the latest examComprehensive overview of the AP Computer Science A exam formatHundreds of practice exercises with thorough answer explanationsTwo full sections devoted to algorithm writing and two levels of sample questions for each Java concept (basic and advanced)Proven strategies specific to each section of the test

Bringing together contributions on a diverse range of topics, this text explores the relationship between discrete and continuum mechanics as a tool to model new and complex metamaterials. Providing a comprehensive bibliography and historical review of the field, it covers mechanical, acoustic and pantographic metamaterials, discusses Naive Model Theory and Lagrangian discrete models, and their applications, and presents methods for pantographic structures and variational methods for multidisciplinary modeling and computation. The relationship between discrete and continuous models is discussed from both mathematical and engineering viewpoints, making the text ideal for those interested in the foundation of mechanics and computational applications, and innovative viewpoints on the use of discrete systems to model metamaterials are presented for those who want to go deeper into the field. An ideal text for graduate students and researchers interested in continuum approaches to the study

The fundamental mathematical tools needed to understand machine learning include linear algebra, analytic geometry, matrix decompositions, vector calculus, optimization, probability and statistics. These topics are traditionally taught in disparate courses, making it hard for data science or computer science students, or professionals, to efficiently learn the mathematics. This self-contained textbook bridges the gap between mathematical and machine learning texts, introducing the mathematical concepts with a minimum of prerequisites. It uses these concepts to derive four central machine learning methods: linear regression, principal component analysis, Gaussian mixture models and support vector machines. For students and others with a mathematical background, these derivations provide a starting point to machine learning texts. For those learning the mathematics for the first time, the methods help build intuition and practical experience with applying mathematical concepts. Every cha

The fundamental mathematical tools needed to understand machine learning include linear algebra, analytic geometry, matrix decompositions, vector calculus, optimization, probability and statistics. These topics are traditionally taught in disparate courses, making it hard for data science or computer science students, or professionals, to efficiently learn the mathematics. This self-contained textbook bridges the gap between mathematical and machine learning texts, introducing the mathematical concepts with a minimum of prerequisites. It uses these concepts to derive four central machine learning methods: linear regression, principal component analysis, Gaussian mixture models and support vector machines. For students and others with a mathematical background, these derivations provide a starting point to machine learning texts. For those learning the mathematics for the first time, the methods help build intuition and practical experience with applying mathematical concepts. Every cha

For 80 years, mathematics has driven fundamental innovation in computing and communications. This timely book provides a panorama of some recent ideas in mathematics and how they will drive continued innovation in computing, communications and AI in the coming years. It provides a unique insight into how the new techniques that are being developed can be used to provide theoretical foundations for technological progress, just as mathematics was used in earlier times by Turing, von Neumann, Shannon and others. Edited by leading researchers in the field, chapters cover the application of new mathematics in computer architecture, software verification, quantum computing, compressed sensing, networking, Bayesian inference, machine learning, reinforcement learning and many other areas.

Paul Erdős published more papers during his lifetime than any other mathematician, especially in discrete mathematics. He had a nose for beautiful, simply-stated problems with solutions that have far-reaching consequences across mathematics. This captivating book, written for students, provides an easy-to-understand introduction to discrete mathematics by presenting questions that intrigued Erdős, along with his brilliant ways of working toward their answers. It includes young Erdős's proof of Bertrand's postulate, the Erdős-Szekeres Happy End Theorem, De Bruijn-Erdős theorem, Erdős-Rado delta-systems, Erdős-Ko-Rado theorem, Erdős-Stone theorem, the Erdős-Rényi-Sós Friendship Theorem, Erdős-Rényi random graphs, the Chvátal-Erdős theorem on Hamilton cycles, and other results of Erdős, as well as results related to his work, such as Ramsey's theorem or Deza's theorem on weak delta-systems. Its appendix covers topics normally missing from introductory courses. Filled with personal anecdot

Paul Erdős published more papers during his lifetime than any other mathematician, especially in discrete mathematics. He had a nose for beautiful, simply-stated problems with solutions that have far-reaching consequences across mathematics. This captivating book, written for students, provides an easy-to-understand introduction to discrete mathematics by presenting questions that intrigued Erdős, along with his brilliant ways of working toward their answers. It includes young Erdős's proof of Bertrand's postulate, the Erdős-Szekeres Happy End Theorem, De Bruijn-Erdős theorem, Erdős-Rado delta-systems, Erdős-Ko-Rado theorem, Erdős-Stone theorem, the Erdős-Rényi-Sós Friendship Theorem, Erdős-Rényi random graphs, the Chvátal-Erdős theorem on Hamilton cycles, and other results of Erdős, as well as results related to his work, such as Ramsey's theorem or Deza's theorem on weak delta-systems. Its appendix covers topics normally missing from introductory courses. Filled with personal anecdot

Barron's AP Computer Science Principles is designed to help students prepare for exam topics, regardless of what computer language or method they learned. The book has been updated to include the cour

Mathematicians have skills that would enable them to use data to answer questions important to them and others, and report those answers in compelling ways. Data science combines parts of mathematics,

This text is geared toward students who have an undergraduate degree or extensive coursework in engineering or the physical sciences and who wish to develop their understanding of the essential topics of applied mathematics. The methods covered in the chapters form the core of analysis in engineering and the physical sciences. Readers will learn the solutions, techniques, and approaches that they will use as academic researchers or industrial R&D specialists. For example, they will be able to understand the fundamentals behind the various scientific software packages that are used to solve technical problems (such as the equations describing the solid mechanics of complex structures or the fluid mechanics of short-term weather prediction and long-term climate change), which is crucial to working with such codes successfully. Detailed and numerous worked problems help to ensure a clear and well-paced introduction to applied mathematics. Computational challenge problems at the end of eac

★ 皇家學會科學圖書獎入圍作，最有故事性的數學史鉅作★ 從阿基米德到人工智慧，一口氣看懂微積分如何改變世界★ 沒有微積分，就沒有電腦、GPS、太空旅行與現代科技文明★ 天才、競爭、爭議與革命交織，一本讀來像史詩冒險的科普傑作這是一本把微積分寫得像歷史小說般精采的科普經典。史蒂芬．史卓格茲不只解釋數學，更帶領讀者看見天才們如何用思想改寫世界。從古代幾何到AI時代，本書呈現微積分如何成為現代文明最深層、也最驚人的力量。Shortlisted for the Royal Society Science Book Prize 2019A magisterial history of calculus (and the people behind it) from one of the world's foremost mathematicians. This is the captivating story of mathematics' greatest ever idea: calculus. Without it, there would be no computers, no microwave ovens, no GPS, and no space travel.But before it gave modern man almost infinite powers, calculus was behind centuries of controversy, competition, and even death. Taking us on a thrilling journey through three millennia, professor Steven Strogatz charts the development of this seminal achievement from the days of Archimedes to today's breakthroughs in chaos theory and artificial intelligence. Filled with idiosyncratic characters from Pythagoras to Fourier, Infinite Powers is a compelling

A top-down,layered approach to computer networking. Unique among computernetworking texts, the 8th Edition, Global Edition, of thepopular Computer Networking: A Top Down Approach buildson the authors' long tradition of teaching this complex subject through alayered approach in a "top-down manner." The text works its way from theapplication layer down toward the physical layer, motivating students byexposing them to important concepts early in their study of networking.Focusing on the Internet and the fundamentally important issues of networking,this text provides an excellent foundation for students in computer science andelectrical engineering, without requiring extensive knowledge of programming ormathematics. The 8th Edition, Global Edition, has been updatedto reflect the most important and exciting recent advances in networking,including the importance of software-defined networking (SDN) and the rapidadoption of 4G/5G networks and the mobile applications they enable.Chapter 1:Comp

The controversial matters surrounding the notion of anachronism are difficult ones: they have been broached by literary and art critics, by philosophers, as well as by historians of science. This book adopts a bottom-up approach to the many problems concerning anachronism in the history of mathematics. Some of the leading scholars in the field of history of mathematics reflect on the applicability of present-day mathematical language, concepts, standards, disciplinary boundaries, indeed notions of mathematics itself, to well-chosen historical case studies belonging to the mathematics of the past, in European and non-European cultures. A detailed introduction describes the key themes and binds the various chapters together. The interdisciplinary and transcultural approach adopted allows this volume to cover topics important for history of mathematics, history of the physical sciences, history of science, philosophy of mathematics, history of philosophy, methodology of history, non-Europ

'Phase-only Fresnel holograms,' which can be displayed on a single SLM without the need for lenses or complicated optical accessories, substantially simplifies 3-D holographic display systems. Exploring essential concepts, theories, and formulations of these phase-only Fresnel holograms, this book provides comprehensive coverage of modern methods for generating such holograms, which pave the way for commercial products such as compact holographic projectors, heads-up displays, and data security enhancement. Relevant MATLAB codes are provided for readers to implement and evaluate the theories and formulations of different methods, and can be used as a quick start framework for further research and development. This is a crucial and up-to-date treatment of phase-only Fresnel holograms for students and researchers in electrical and electronic engineering, computer science/engineering, applied physics, information technology, and multimedia technology, as well as engineers and scientists

Master simple to advanced biomaterials and structures with this essential text. Featuring topics ranging from bionanoengineered materials to bio-inspired structures for spacecraft and bio-inspired robots, and covering issues such as motility, sensing, control and morphology, this highly illustrated text walks the reader through key scientific and practical engineering principles, discussing properties, applications and design. Presenting case studies for the design of materials and structures at the nano, micro, meso and macro-scales, and written by some of the leading experts on the subject, this is the ideal introduction to this emerging field for students in engineering and science as well as researchers.

Mathematical models of bond markets are of interest to researchers working in applied mathematics, especially in mathematical finance. This book concerns bond market models in which random elements are represented by Lévy processes. These are more flexible than classical models and are well suited to describing prices quoted in a discontinuous fashion. The book's key aims are to characterize bond markets that are free of arbitrage and to analyze their completeness. Nonlinear stochastic partial differential equations (SPDEs) are an important tool in the analysis. The authors begin with a relatively elementary analysis in discrete time, suitable for readers who are not familiar with finance or continuous time stochastic analysis. The book should be of interest to mathematicians, in particular to probabilists, who wish to learn the theory of the bond market and to be exposed to attractive open mathematical problems.

●ACTIONABLE AND MEASURABLE LEARNING OBJECTIVES EMPHASIZE IMPORTANT CONCEPTS. Newly added learning objectives guide students through the edition, calling attention to critical key points.●SOLID OVERVIEW PRESENTS BREADTH OF COMPUTER SCIENCE WITHOUT PROGRAMMING LANGUAGE SPECIFICS. The authors expertly present a language-independent, breadth-first introduction to computer science that includes coverage of today's hardware, software, and applications as well as a modern view of ethics.●UNIQUE APPROACH INTEGRATES TOPICS USING THE CONCEPT OF ALGORITHMS. Rather than simply present a large collection of seemingly random and unrelated topics, the authors tie the discipline of computer science together using a six-layer hierarchy that clearly demonstrates how the subjects in the book relate to each other. Student gain a clear understanding of the entire discipline of computer science that prepares them for future study in this field.●EXTENSIVE EXERCISES AND PROBLEMS PROVIDE NUMEROUS OPPORTUNITIES

THE SERIES: INTELLIGENT BIOMEDICAL DATA ANALYSISBy focusing on the methods and tools for intelligent data analysis, this series aims to narrow the increasing gap between data gathering and data comprehension. Emphasis is also given to the problems resulting from automated data collection in modern hospitals, such as analysis of computer-based patient records, data warehousing tools, intelligent alarming, effective and efficient monitoring. In medicine, overcoming this gap is crucial since medical decision making needs to be supported by arguments based on existing medical knowledge as well as information, regularities and trends extracted from big data sets.

Discover the fundamental principles of biomedical measurement design and performance evaluation with this hands-on guide. Whether you develop measurement instruments or use them in novel ways, this practical text will prepare you to be an effective generator and consumer of biomedical data. Designed for both classroom instruction and self-study, it explains how information is encoded into recorded data and can be extracted and displayed in an accessible manner. Describes and integrates experimental design, performance assessment, classification, and system modelling. Combines mathematical concepts with computational models, providing the tools needed to answer advanced biomedical questions. Includes MATLAB® scripts throughout to help readers model all types of biomedical systems, and contains numerous homework problems, with a solutions manual available online. This is an essential text for advanced undergraduate and graduate students in bioengineering, electrical and computer engineer

This is the first rigorous, self-contained treatment of the theory of deep learning. Starting with the foundations of the theory and building it up, this is essential reading for any scientists, instructors, and students interested in artificial intelligence and deep learning. It provides guidance on how to think about scientific questions, and leads readers through the history of the field and its fundamental connections to neuroscience. The author discusses many applications to beautiful problems in the natural sciences, in physics, chemistry, and biomedicine. Examples include the search for exotic particles and dark matter in experimental physics, the prediction of molecular properties and reaction outcomes in chemistry, and the prediction of protein structures and the diagnostic analysis of biomedical images in the natural sciences. The text is accompanied by a full set of exercises at different difficulty levels and encourages out-of-the-box thinking.

Part of the bestselling Little People, BIG DREAMS series, Alan Turing tells the inspiring story of the father of theoretical computer science and artificial intelligence.In this book from the critically acclaimed, multimillion-copy bestselling Little People, BIG DREAMS series, discover the life of Alan Turing, the genius code cracker and father of theoretical computer science and artificial intelligence.Alan grew up in England, where his best friends were numbers and a little boy called Christopher. When his young friend died, Alan retreated to the world of numbers and codes, where he discovered how to crack the code of the Nazi Enigma machine. This moving book features stylish and quirky illustrations and extra facts at the back, including a biographical timeline with historical photos and a detailed profile of the brilliant mathematician's life.Little People, BIG DREAMS is a bestselling biography series for kids that explores the lives of outstanding people, from designers and artist