Diskret matematik

This book systematically establishes the almost periodic theory of dynamic equations and presents applications on time scales in fuzzy mathematics and uncertainty theory. The authors introduce a new division of fuzzy vectors depending on a determinant algorithm and develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales. Several applications are studied; in particular, a new type of fuzzy dynamic systems called fuzzy q-dynamic systems (i.e. fuzzy quantum dynamic systems) is presented. The results are not only effective on classical fuzzy dynamic systems, including their continuous and discrete situations, but are also valid for other fuzzy multidimensional dynamic systems on various hybrid domains. In an effort to achieve more accurate analysis in real world applications, the authors propose a number of uncertain factors in the theory. As such, fuzzy dynamical models, interval-valued functions, differential equations, fuzzy-valued differential equations, and their applications to dynamic equations on time scales are considered.

This volume constitutes the thoroughly refereed proceedings of the 49th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2023. The 33 full papers presented in this volume were carefully reviewed and selected from a total of 116 submissions. The WG 2022 workshop aims to merge theory and practice by demonstrating how concepts from graph theory can be applied to various areas in computer science, or by extracting new graph theoretic problems from applications.

This book is intended to provide graduate students and researchers in graph theory with an overview of the elementary methods of graph Ramsey theory. It is especially targeted towards graduate students in extremal graph theory, graph Ramsey theory, and related fields, as the included contents allow the text to be used in seminars. It is structured in thirteen chapters which are application-focused and largely independent, enabling readers to target specific topics and information to focus their study. The first chapter includes a true beginner's overview of elementary examples in graph Ramsey theory mainly using combinatorial methods. The following chapters progress through topics including the probabilistic methods, algebraic construction, regularity method, but that's not all. Many related interesting topics are also included in this book, such as the disproof for a conjecture of Borsuk on geometry, intersecting hypergraphs, Turan numbers and communication channels, etc.

This book presents a collection of 36 pieces of scientific work in the areas of complexity theory and foundations of cryptography: 20 research contributions, 13 survey articles, and 3 programmatic and reflective viewpoint statements. These so far formally unpublished pieces were written by Oded Goldreich, some in collaboration with other scientists. The articles included in this book essentially reflect the topical scope of the scientific career of Oded Goldreich now spanning three decades. In particular the topics dealt with include average-case complexity, complexity of approximation, derandomization, expander graphs, hashing functions, locally testable codes, machines that take advice, NP-completeness, one-way functions, probabilistically checkable proofs, proofs of knowledge, property testing, pseudorandomness, randomness extractors, sampling, trapdoor permutations, zero-knowledge, and non-iterative zero-knowledge.All in all, this potpourri of studies in complexity and cryptography constitutes a most valuable contribution to the field of theoretical computer science centered around the personal achievements and views of one of its outstanding representatives.

Learn all of the core mathematical topics that professional software engineers need to know—in a single book!This book summarizes all the core mathematical topics a typical professional software engineer needs to know. In condensing the various concepts covered in an undergraduate computer science program into a single volume, it provides an excellent starting point for independent study, or a refresher for those who haven’t been in a classroom for years. Early chapters cover preliminary subjects like number representation systems, set theory, and Boolean algebra, followed by a dive into the field of discrete mathematics, including functions, induction proofs, number theory, combinatorics, graphs, and trees. Later sections examine essential topics in probability, statistics, linear algebra, and calculus.Rather than confine itself to abstract theory, the book focuses on practical applications and numerical methods at the level typically encountered by working software developers. In addition, hands-on code examples in Python and C make the topics concrete.

This proceedings volume gathers selected, revised papers presented at the 51st Southeastern International Conference on Combinatorics, Graph Theory and Computing (SEICCGTC 2020), held at Florida Atlantic University in Boca Raton, USA, on March 9-13, 2020. The SEICCGTC is broadly considered to be a trendsetter for other conferences around the world - many of the ideas and themes first discussed at it have subsequently been explored at other conferences and symposia.The conference has been held annually since 1970, in Baton Rouge, Louisiana and Boca Raton, Florida. Over the years, it has grown to become the major annual conference in its fields, and plays a major role in disseminating results and in fostering collaborative work.This volume is intended for the community of pure and applied mathematicians, in academia, industry and government, working in combinatorics and graph theory, as well as related areas of computer science and the interactions among these fields.

This book constitutes the refereed proceedings of the 17th International Joint Conference on Theoretical Computer Science-Frontier of Algorithmic Wisdom (IJTCS-FAW 2023), consisting of the 17th International Conference on Frontier of Algorithmic Wisdom (FAW) and the 4th International Joint Conference on Theoretical Computer Science (IJTCS), held in Macau, China, during August 14¿18, 2023.FAW started as the Frontiers of Algorithmic Workshop in 2007 at Lanzhou, China, and was held annually from 2007 to 2021 and published archival proceedings. IJTCS, the International joint theoretical Computer Science Conference, started in 2020, aimed to bring in presentations covering active topics in selected tracks in theoretical computer science. To accommodate the diversified new research directions in theoretical computer science, FAW and IJTCS joined their forces together to organize an event for information exchange of new findings and work of enduring value in the field. The 21 full papers included in this book were carefully reviewed and selected from 34 submissions. They were organized in topical sections as follows: algorithmic game theory; algorithms and data structures; combinatorial optimization; and computational economics.

Discrete mathematics refers to the study of mathematical structures which are discrete and not continuous. Logic statements, integers and graphs are some of the objects examined in discrete mathematics. It is concerned with the investigation of countable sets, which might be finite or infinite. Various topics covered under discrete mathematics include set theory, graph theory, combinations, logic, information theory, probability, geometry and algebraic structures. The notations and concepts of discrete mathematics are beneficial in analyzing and expressing objects and issues in the field of computer science, including programming languages, automated theorem proving, computer algorithms, software development and cryptography. Most of the topics introduced in this book cover new techniques and the applications of discrete mathematical structures. It will serve as a valuable source of reference for graduate and postgraduate students. Coherent flow of topics, student-friendly language and extensive use of examples make this book an invaluable source of knowledge.

Discrete geometry refers to a subfield of geometry that examines discrete and highly symmetric structures in geometry and how discrete geometric objects might be constructed. It is closely connected to computational geometry as well as other mathematical subjects including combinatorial geometry and topology, combinatorial optimization, geometric graph theory, and finite geometry. Discrete geometry focuses on the problems that aim to characterize specific discrete arrangements of geometric objects having high symmetry. Many of the techniques used to investigate discrete arrangements are based on a wide range of symetrizations. Discrete geometry is used in numerous theoretical disciplines including algebraic geometry and topology, geometry of numbers, mathematical physics, functional analysis, geometric measure theory, calculus of variations, and group theory. This book is compiled in such a manner, that it will provide an in-depth knowledge about the theory and applications of discrete geometry and symmetry. It is a resource guide for experts as well as students.

Discrete mathematics refers to the study of mathematical structures that are countable or otherwise separable or unique. Logical statements, combinations and graphs are examples of discrete structures. Symmetry is a proportionate and balanced resemblance between two halves of an object. It implies that one half is the mirror image of the other. Discrete symmetry in mathematics is a symmetry that explains non-continuous changes in a system. Symmetry is frequently used in discrete mathematics to show relationships between two or more elements of a set, where the first element relates to the second element and vice-versa. This book explores all the important aspects of discrete mathematics and symmetry in the present day scenario. It presents researches and studies performed by experts across the globe. This book, with its detailed analyses and data, will prove immensely beneficial to professionals and students involved in this area of mathematics at various levels.

This book not only presents an essential material to understand fuzzy metric fixed point theory but also enables the readers to appreciate the recent advancements made in this direction. It contains seven chapters on different topics in fuzzy metric fixed point theory.

This book constitutes the refereed proceedings of the 13th IAPR-TC-15 International Workshop on Graph-Based Representations in Pattern Recognition, GbRPR 2023, which took place in Vietri sul Mare, Italy, in September 2023.The 16 full papers included in this book were carefully reviewed and selected from 18 submissions. They were organized in topical sections on graph kernels and graph algorithms; graph neural networks; and graph-based representations and applications.

Nachdem der Autor sich in seinen Abschlussarbeiten mit Themen der Analysis befasst hatte, arbeitete er danach beruflich ausschließlich im Bereich der diskreten Mathematik: forschend im Bereich der Spezifikation und Testgenerierung für kommunizierende diskrete Systeme und lehrend in den Bereichen Petri-Netze, Logik und Theoretischer Informatik. Das bewog ihn letztlich, eine breit angelegte Übersicht über wichtige Bereiche der diskreten Mathematik zu verfassen. Es sollte ein Lehrbuch werden, wie er es sich im Studium gewünscht hätte - und hier liegt es vor. Gegenstand des Buches sind Grundlagenfächer wie Mengen, Relationen, Funktionen, Logik und Algebra, sowie diskrete Anwendungen wie Zahlen, Anzahlen, Graphen und diskrete Wahrscheinlichkeit. Damit werden etliche Grundvorlesungen für Studierende der Mathematik bzw. Informatik abgedeckt. Reizvoll sind die gelegentlich eingeflochten Hinweise auf die Grenzen des formal-mathematischen Vorgehens. Anhand von etwa 250 Übungsaufgaben werden die Leser animiert, das Gelesene aktiv umzusetzen und ihre eigenen Lösungen mit den online einsehbaren Beispiellösungen zu vergleichen. So wird das Buch zu einem wertvollen Begleiter in den ersten Semestern des Mathematik- und Informatikstudiums. Die komplett überarbeitete 2. Auflage enthält zusätzliche Definitionen und ist noch klarer strukturiert.

This book provides an elementary introduction, complete with detailed proofs, to the celebrated tilings of the plane discovered by Sir Roger Penrose in the '70s. Quasi-periodic tilings of the plane, of which Penrose tilings are the most famous example, started as recreational mathematics and soon attracted the interest of scientists for their possible application in the description of quasi-crystals. The purpose of this survey, illustrated with more than 200 figures, is to introduce the curious reader to this beautiful topic and be a reference for some proofs that are not easy to find in the literature. The volume covers many aspects of Penrose tilings, including the study, from the point of view of Connes' Noncommutative Geometry, of the space parameterizing these tilings.

This book discusses discrete geometric analysis, especially topological crystallography and discrete surface theory for trivalent discrete surfaces. Topological crystallography, based on graph theory, provides the most symmetric structure among given combinatorial structures by using the variational principle, and it can reproduce crystal structures existing in nature. In this regard, the topological crystallography founded by Kotani and Sunada is explained by using many examples. Carbon structures such as fullerenes are considered as trivalent discrete surfaces from the viewpoint of discrete geometric analysis. Discrete surface theories usually have been considered discretization of smooth surfaces. Here, consideration is given to discrete surfaces modeled by crystal/molecular structures, which are essentially discrete objects.

"This largely self-contained text introduces discrete probability and its applications, at a level suitable for beginning graduate students in mathematics, computer science, statistics and engineering. Each chapter includes exercises and pointers to the wider literature, covering a wide spectrum of essential techniques and key examples"--

Intelligent Fractal-Based Image Analysis: Application in Pattern Recognition and Machine Vision provides insights into the current strengths and weaknesses of different applications as well as research findings on fractal graphics in engineering and science applications. The book aims to improve the exchange of ideas and coherence between various core computing methods and highlights the relevance of related application areas for advanced as well as novice-user application. The book presents core concepts, methodological aspects, and advanced feature opportunities, focusing on major, real-time applications in engineering and health science. It will appeal to researchers, data scientists, industry professionals, and graduate students. Fractals are infinite, complex patterns used in modeling physical and dynamic systems. Fractal theory research has increased across different fields of applications including engineering science, health science, and social science. Recent literature shows the vital role fractals play in digital image analysis, specifically in biomedical image processing. Fractal graphics is an interdisciplinary field that deals with how computers can be used to gain high-level understanding from digital images. Integrating artificial intelligence with fractal characteristics has resulted in new interdisciplinary research in the fields of pattern recognition and image processing analysis.

As the sophistication of cyber-attacks increases, understanding how to defend critical infrastructure systems-energy production, water, gas, and other vital systems-becomes more important, and heavily mandated. Industrial Network Security, Third Edition arms you with the knowledge you need to understand the vulnerabilities of these distributed supervisory and control systems. Author Eric Knapp examines the unique protocols and applications that are the foundation of Industrial Control Systems (ICS) and provides clear guidelines for their protection. This comprehensive reference gives you thorough understanding of the challenges facing critical infrastructures, new guidelines and security measures for infrastructure protection, knowledge of new and evolving security tools, and pointers on SCADA protocols and security implementation.

This book constitutes the refereed proceedings of the 18th International Symposium on Algorithms and Data Structures, WADS 2023, held during July 31-August 2, 2023. The 47 regular papers, presented in this book, were carefully reviewed and selected from a total of 92 submissions. They present original research on the theory, design and application of algorithms and data structures.

This book is designed to meet the requirement of undergraduate and postgraduate students pursuing computer science, information technology, mathematical science, and physical science course. No formal prerequisites are needed to understand the text matter except a very reasonable background in college algebra. The text contains in-depth coverage of all major topics proposed by professional institutions and universities for a discrete mathematics course. It emphasizes on problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof technique, algorithmic development, algorithm correctness, and numeric computations. A sufficient amount of theory is included for those who enjoy the beauty in development of the subject and a wealth of applications as well as for those who enjoy the power of problem-solving techniques.Biographical sketches of nearly 25 mathematicians and computer scientists who have played a significant role in the development of the field are threaded into the text to provide a human dimension and attach a human face to major discoveries. Each section of the book contains a generous selection of carefully tailored examples to classify and illuminate various concepts and facts. Theorems are backbone of mathematics. Consequently, this book contains the various proof techniques, explained and illustrated in details. Most of the concepts, definitions, and theorems in the book are illustrated with appropriate examples. Proofs shed additional light on the topic and enable students to sharpen thin problem-solving skills. Each chapter ends with a summary of important vocabulary, formulae, properties developed in the chapter, and list of selected references for further exploration and enrichment.

This book constitutes the refereed proceedings of the 16th International Conference on Graph Transformation, ICGT 2023, held as Part of STAF 2023 in Leicester, UK, in July 2023. The 14 full papers and 2 short papers presented in this book were carefully reviewed and selected from 29 submissions. The conference focuses on describing new unpublished contributions in the theory and applications of graph transformation as well as tool presentation papers that demonstrate main new features and functionalities of graph-based tools.

This volume collects papers based on talks given at the conference "e;Geometrias'19: Polyhedra and Beyond"e;, held in the Faculty of Sciences of the University of Porto between September 5-7, 2019 in Portugal. These papers explore the conference's theme from an interdisciplinary standpoint, all the while emphasizing the relevance of polyhedral geometry in contemporary academic research and professional practice. They also investigate how this topic connects to mathematics, art, architecture, computer science, and the science of representation. Polyhedra and Beyond will help inspire scholars, researchers, professionals, and students of any of these disciplines to develop a more thorough understanding of polyhedra.

The theory developed for  4-additive codes is the starting point for much generalization about codes over mixed alphabets. They have opened a new, emergent area of research.  The techniques used for  4-linear codes are being generalized and applied to more general codes.  By example, these codes have contributed to the classification of many nonlinear codes. Moreover, they can be considered as the starting point of many different generalizations given over mixed alphabets, thereby representing a useful area of research.  Since 2010, more than 30 papers have been published about the codes considered in the book, which includes important classes of binary codes (1-perfect, Hadamard, etc.) that are not linear in general. For example, much recent research has shown the application of the techniques described for the family of cyclic  4-linear codes.   Topics and Features:Examines everything from the basic definitions to very advanced results                             Provides numerous examples, succinctly and comprehensively gathering and using the relevant information                                                                                                         Includes examples using commands from a new Magma package, developed mostly by the same authors                                                                                                          Proposes algorithms, for instance to describe coding and decoding strategiesThis unique volume will be eminently suitable for researchers working on coding theory over rings, especially over mixed alphabets. Experts will find commands and algorithms that will be useful in the generalization to codes over mixed alphabets. Additionally, by outlining the basic theory of codes over mixed alphabets and providing numerous examples, the book will be useful to researchers wanting to be introduced to the topic.The authors are all affiliated with the Dept. of Information and Communications Engineering at the Universitat Autonoma de Barcelona, Spain. Joaquim Borges and Cristina Fernandez-Cordoba are Associate Professors, Jaume Pujol is a now retired Associate Professor, Josep Rifa is Professor Emeritus, and Merce Villanueva is Associate Professor.

Tree successor algebra: A new branch in mathematics is a book about a formal theory of tree generation with an axiomatic basis for a new object called collection space. The elements of this space, in other words collections, have a clear connection to rooted trees and are treated as variables in sum form equations, the application area of tree successor algebra. With connections to different branches of mathematics such as number theory, linear algebra and algebra, tree successor algebra shows a fundamental link between rooted tree generation and partition generation, establishing a well-defined order in which rooted trees are generated. This in turn makes it possible to define a successor operator, the unit of least action in tree generation, and generalize it in order to create a concept of tree sequences. Due to this, the concept of the infinite sequence of all rooted trees can be formed, and the notion of a rooted tree line, and thus the need for tools to solve sum form equations rises. The axiomatic system answers to this need.

This book constitutes the refereed proceedings of the 22nd International Conference on Mathematical Optimization Theory and Operations Research, MOTOR 2023, held in Ekaterinburg, Russia, during July 2¿8, 2023. The 28 full papers and 1 short paper included in this book were carefully reviewed and selected from 89 submissions. They were organized in topical sections as follows: Mathematical programming and applications; discrete and combinatorial optimization; stochastic optimization; scheduling; game theory; and optimal control and mathematical economics. The book also contains one invited talk in full paper length.

Calculation in Mathematics: Logarithm and Exponential FunctionThis self-teaching solution book consist of lots of examples in Logarithm & Exponential Functions ranging from basics to advance . The book will also enhance your knowledge in mathematics. Learn more and Increase your math skills on... Rules of Logarithms Logarithms Equations Simplification of Logarithm Equations Common Logarithms Operations of Common Logarithms Antilogarithms Exponential FunctionsImprove your math skills by grabbing a copy

The present book is based on the curriculum of undergraduate and postgraduate courses of universities in India and abroad. Every effort is made to present the various topics in the theory of graphs in a logical manner with adequate historical background and include suitable figures to illustrate concepts and results ideally. The formidable exercises, neither easy nor straightforward, are bold faced and highlighted. The theory portion of each chapter is studied thoroughly as it helps solve many of the problems with comparative ease. Selected material from this book is used for a semester course on graph theory, while the entire book serves for a whole session course.

Digital Technology in Public Health and Rehabilitation Care: COVID Era provides an in-depth examination on how digital technology has impacted public health and rehabilitation during the COVID-19 pandemic. The book showcases the range of digital technology applications utilized in healthcare, including the use of mobile phones, computers, wearable and non-wearable technologies, sensors, 3D printers, robotics, Artificial Intelligence (AI), internet technologies, websites and apps, virtual and augmented realities, and computer games. With a wealth of case studies and insights, this book is an essential resource on the impact of digital technology on public health and rehabilitation services in the COVID-19 era. The COVID-19 pandemic had a major impact on the rehabilitation services available to those in need, particularly for older adults, people with physical and cognitive impairments, children with autism and ADHD, and individuals with physical and mental health disorders. In response to these challenges, the book focuses on the potential for advanced digital technologies to revolutionize public health and rehabilitation services, highlighting the need for researchers and healthcare professionals to work together to promote digital-based interventions.