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A Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups (UNITEXT for Physics)
by Carlo Alabiso Ittay WeissThis book is an introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, resides in the very high mathematical difficulty of even the simplest physical case. Within an ordinary graduate course in physics there is insufficient time to cover the theory of Hilbert spaces and operators, as well as distribution theory, with sufficient mathematical rigor. Compromises must be found between full rigor and practical use of the instruments. The book is based on the author's lessons on functional analysis for graduate students in physics. It will equip the reader to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude. With respect to the original lectures, the mathematical flavor in all subjects has been enriched. Moreover, a brief introduction to topological groups has been added in addition to exercises and solved problems throughout the text. With these improvements, the book can be used in upper undergraduate and lower graduate courses, both in Physics and in Mathematics.
A Primer on Hilbert Space Theory: Linear Spaces, Topological Spaces, Metric Spaces, Normed Spaces, and Topological Groups (UNITEXT for Physics)
by Carlo Alabiso Ittay WeissThis book offers an essential introduction to the theory of Hilbert space, a fundamental tool for non-relativistic quantum mechanics. Linear, topological, metric, and normed spaces are all addressed in detail, in a rigorous but reader-friendly fashion. The rationale for providing an introduction to the theory of Hilbert space, rather than a detailed study of Hilbert space theory itself, lies in the strenuous mathematics demands that even the simplest physical cases entail. Graduate courses in physics rarely offer enough time to cover the theory of Hilbert space and operators, as well as distribution theory, with sufficient mathematical rigor. Accordingly, compromises must be found between full rigor and the practical use of the instruments. Based on one of the authors’s lectures on functional analysis for graduate students in physics, the book will equip readers to approach Hilbert space and, subsequently, rigged Hilbert space, with a more practical attitude. It also includes a brief introduction to topological groups, and to other mathematical structures akin to Hilbert space. Exercises and solved problems accompany the main text, offering readers opportunities to deepen their understanding. The topics and their presentation have been chosen with the goal of quickly, yet rigorously and effectively, preparing readers for the intricacies of Hilbert space. Consequently, some topics, e.g., the Lebesgue integral, are treated in a somewhat unorthodox manner. The book is ideally suited for use in upper undergraduate and lower graduate courses, both in Physics and in Mathematics.
Summability Calculus: A Comprehensive Theory Of Fractional Finite Sums
by Ibrahim M. AlabdulmohsinThis book develops the foundations of "summability calculus", which is a comprehensive theory of fractional finite sums. It fills an important gap in the literature by unifying and extending disparate historical results. It also presents new material that has not been published before. Importantly, it shows how the study of fractional finite sums benefits from and contributes to many areas of mathematics, such as divergent series, numerical integration, approximation theory, asymptotic methods, special functions, series acceleration, Fourier analysis, the calculus of finite differences, and information theory. As such, it appeals to a wide audience of mathematicians whose interests include the study of special functions, summability theory, analytic number theory, series and sequences, approximation theory, asymptotic expansions, or numerical methods. Richly illustrated, it features chapter summaries, and includes numerous examples and exercises. The content is mostly developed from scratch using only undergraduate mathematics, such as calculus and linear algebra.
Trends in Control Theory and Partial Differential Equations (Springer INdAM Series #32)
by Fatiha Alabau-Boussouira Fabio Ancona Alessio Porretta Carlo SinestrariThis book presents cutting-edge contributions in the areas of control theory and partial differential equations. Over the decades, control theory has had deep and fruitful interactions with the theory of partial differential equations (PDEs). Well-known examples are the study of the generalized solutions of Hamilton-Jacobi-Bellman equations arising in deterministic and stochastic optimal control and the development of modern analytical tools to study the controllability of infinite dimensional systems governed by PDEs. In the present volume, leading experts provide an up-to-date overview of the connections between these two vast fields of mathematics. Topics addressed include regularity of the value function associated to finite dimensional control systems, controllability and observability for PDEs, and asymptotic analysis of multiagent systems. The book will be of interest for both researchers and graduate students working in these areas.
Wavelets and Wavelet Transform Systems and Their Applications: A Digital Signal Processing Approach
by Cajetan M. AkujuobiThis textbook is unique because of its in-depth treatment of the applications of wavelets and wavelet transforms to many areas, across many disciplines. The book is written to serve the needs of a one or two semester course at either the undergraduate or graduate level. The author uses a very simplified, accessible approach that de-emphasizes mathematical rigor. The presentation includes many diagrams to illustrate points being discussed and uses MATLAB for all of application code. The author reinforces concepts introduced in the book with easy to grasp review questions and problems, tailored to each specific chapter for better mastery of the subject matter. This book enables students to understand the fundamental concepts of wavelets and wavelet transforms, as well as how to use them for problem solutions in digital signal and image processing, mixed-signal testing, space applications, aerospace applications, biomedical, cyber security, homeland security and many other application areas.
Direct and Inverse Scattering for the Matrix Schrödinger Equation (Applied Mathematical Sciences #203)
by Tuncay Aktosun Ricardo WederAuthored by two experts in the field who have been long-time collaborators, this monograph treats the scattering and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition. The existence, uniqueness, construction, and characterization aspects are treated with mathematical rigor, and physical insight is provided to make the material accessible to mathematicians, physicists, engineers, and applied scientists with an interest in scattering and inverse scattering. The material presented is expected to be useful to beginners as well as experts in the field. The subject matter covered is expected to be interesting to a wide range of researchers including those working in quantum graphs and scattering on graphs. The theory presented is illustrated with various explicit examples to improve the understanding of scattering and inverse scattering problems. The monograph introduces a specific class of input data sets consisting of a potential and a boundary condition and a specific class of scattering data sets consisting of a scattering matrix and bound-state information. The important problem of the characterization is solved by establishing a one-to-one correspondence between the two aforementioned classes. The characterization result is formulated in various equivalent forms, providing insight and allowing a comparison of different techniques used to solve the inverse scattering problem. The past literature treated the type of boundary condition as a part of the scattering data used as input to recover the potential. This monograph provides a proper formulation of the inverse scattering problem where the type of boundary condition is no longer a part of the scattering data set, but rather both the potential and the type of boundary condition are recovered from the scattering data set.
Fundamentals of Real and Complex Analysis (Springer Undergraduate Mathematics Series)
by Asuman Güven AksoyThe primary aim of this text is to help transition undergraduates to study graduate level mathematics. It unites real and complex analysis after developing the basic techniques and aims at a larger readership than that of similar textbooks that have been published, as fewer mathematical requisites are required. The idea is to present analysis as a whole and emphasize the strong connections between various branches of the field. Ample examples and exercises reinforce concepts, and a helpful bibliography guides those wishing to delve deeper into particular topics. Graduate students who are studying for their qualifying exams in analysis will find use in this text, as well as those looking to advance their mathematical studies or who are moving on to explore another quantitative science.Chapter 1 contains many tools for higher mathematics; its content is easily accessible, though not elementary. Chapter 2 focuses on topics in real analysis such as p-adic completion, Banach Contraction Mapping Theorem and its applications, Fourier series, Lebesgue measure and integration. One of this chapter’s unique features is its treatment of functional equations. Chapter 3 covers the essential topics in complex analysis: it begins with a geometric introduction to the complex plane, then covers holomorphic functions, complex power series, conformal mappings, and the Riemann mapping theorem. In conjunction with the Bieberbach conjecture, the power and applications of Cauchy’s theorem through the integral formula and residue theorem are presented.
Enlargement of Filtration with Finance in View
by Anna Aksamit Monique JeanblancIn addition to presenting the basic theory of enlargement of filtrations, this book also includes new material and applications to finance. The main results are applied to give conditions which ensure that new information does not provide arbitrage opportunities, and in those cases where the arbitrages exist, they are explicitly constructed. Numerous examples are provided together with a list of recent papers on the subject. The presentation has been kept as simple as possible. In particular, sophisticated new results are given without proofs. Addressing the question of how to model and study new information flows, the theory of enlargement of filtrations is used in credit risk modeling, insider trading and in the study of asymmetric information. This book fills a gap in the literature and will be useful to students and researchers interested in the role of information, both in financial mathematics and in econometric science.
Advances in Acoustics and Vibration IV: Proceedings of the Fourth International Conference on Acoustics and Vibration (ICAV2022), December 19-21, 2022, Sousse, Tunisia (Applied Condition Monitoring #22)
by Ali Akrout Moez Abdennadher Nabih Feki Mohamed Slim Abbes Fakher Chaari Mohamed HaddarThe book provides readers with a snapshot of recent research and industrial trends in field of industrial acoustics and vibration. Each chapter, accepted after a rigorous peer-review process, reports on a selected, original piece of work presented and discussed at the Fourth International Conference on Acoustics and Vibration (ICAV2022), which was organized by the Tunisian Association of Industrial Acoustics and Vibration (ATAVI) and held in hybrid format on December 19–21, 2022, in and from Sousse, Tunisia. The contributions cover advances in both theory and practice in a variety of subfields, such as structural and machine dynamics and vibrations, fault diagnosis and prognosis, nonlinear dynamics, and vibration control of mechatronic systems. Further topics include fluid–structure interaction, computational vibro-acoustics, vibration field measurements, and dynamic behavior of materials. This book provides a valuable resource for both academics and professionals dealing with diverse issues in applied mechanics. By combining advanced theories with industrial issues, it is expected to facilitate communication and collaboration between different groups of researchers and technology users.
Topics in Nonparametric Statistics
by Michael G. Akritas S. N. Lahiri Dimitris N. PolitisThis volume is composed of peer-reviewed papers that have developed from the First Conference of the International Society for Non Parametric Statistics (ISNPS). This inaugural conference took place in Chalkidiki, Greece, June 15-19, 2012. It was organized with the co-sponsorship of the IMS, the ISI and other organizations. M. G. Akritas, S. N. Lahiri and D. N. Politis are the first executive committee members of ISNPS and the editors of this volume. ISNPS has a distinguished Advisory Committee that includes Professors R. Beran, P. Bickel, R. Carroll, D. Cook, P. Hall, R. Johnson, B. Lindsay, E. Parzen, P. Robinson, M. Rosenblatt, G. Roussas, T. SubbaRao and G. Wahba. The Charting Committee of ISNPS consists of more than 50 prominent researchers from all over the world. The chapters in this volume bring forth recent advances and trends in several areas of nonparametric statistics. In this way, the volume facilitates the exchange of research ideas, promotes collaboration among researchers from all over the world and contributes to the further development of the field. The conference program included over 250 talks, including special invited talks, plenary talks and contributed talks on all areas of nonparametric statistics. Out of these talks, some of the most pertinent ones have been refereed and developed into chapters that share both research and developments in the field.
Graphs for the Analysis of Bipolar Fuzzy Information (Studies in Fuzziness and Soft Computing #401)
by Muhammad Akram Musavarah Sarwar Wieslaw A. DudekThis monograph discusses decision making methods under bipolar fuzzy graphical models with the aim of overcoming the lack of mathematical approach towards bipolar information—positive and negative. It investigates the properties of bipolar fuzzy graphs, their distance functions, and concept of their isomorphism. It presents certain notions, including irregular bipolar fuzzy graphs, domination in bipolar fuzzy graphs, bipolar fuzzy circuits, energy in bipolar fuzzy graphs, bipolar single-valued neutrosophic competition graphs, and bipolar neutrosophic graph structures. This book also presents the applications of mentioned concepts to real-world problems in areas of product manufacturing, international relations, psychology, global terrorism and more, making it valuable for researchers, computer scientists, social scientists and alike.
Fuzzy Hypergraphs and Related Extensions (Studies in Fuzziness and Soft Computing #390)
by Muhammad Akram Anam LuqmanThis book presents the fundamental and technical concepts of fuzzy hypergraphs and explains their extensions and applications. It discusses applied generalized mathematical models of hypergraphs, including complex, intuitionistic, bipolar, m-polar fuzzy, Pythagorean, complex Pythagorean, and q-rung orthopair hypergraphs, as well as single-valued neutrosophic, complex neutrosophic and bipolar neutrosophic hypergraphs. In addition, the book also sheds light on real-world applications of these hypergraphs, making it a valuable resource for students and researchers in the field of mathematics, as well as computer and social scientists.
Fuzzy Lie Algebras (Infosys Science Foundation Series)
by Muhammad AkramThis book explores certain structures of fuzzy Lie algebras, fuzzy Lie superalgebras and fuzzy n-Lie algebras. In addition, it applies various concepts to Lie algebras and Lie superalgebras, including type-1 fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, vague sets and bipolar fuzzy sets. The book offers a valuable resource for students and researchers in mathematics, especially those interested in fuzzy Lie algebraic structures, as well as for other scientists. Divided into 10 chapters, the book begins with a concise review of fuzzy set theory, Lie algebras and Lie superalgebras. In turn, Chap. 2 discusses several properties of concepts like interval-valued fuzzy Lie ideals, characterizations of Noetherian Lie algebras, quotient Lie algebras via interval-valued fuzzy Lie ideals, and interval-valued fuzzy Lie superalgebras. Chaps. 3 and 4 focus on various concepts of fuzzy Lie algebras, while Chap. 5 presents the concept of fuzzy Lie ideals of a Lie algebra over a fuzzy field. Chapter 6 is devoted to the properties of bipolar fuzzy Lie ideals, bipolar fuzzy Lie subsuperalgebras, bipolar fuzzy bracket product, solvable bipolar fuzzy Lie ideals and nilpotent bipolar fuzzy Lie ideals. Chap. 7 deals with the properties of m-polar fuzzy Lie subalgebras and m-polar fuzzy Lie ideals, while Chap. 8 addresses concepts like soft intersection Lie algebras and fuzzy soft Lie algebras. Chap. 9 deals with rough fuzzy Lie subalgebras and rough fuzzy Lie ideals, and lastly, Chap. 10 investigates certain properties of fuzzy subalgebras and ideals of n-ary Lie algebras.
m−Polar Fuzzy Graphs: Theory, Methods & Applications (Studies in Fuzziness and Soft Computing #371)
by Muhammad AkramThis book provides readers with an introduction to m-polar fuzzy graphs and m-polar fuzzy hypergraphs, covering both theories and applications. A special emphasis is given to m-polar fuzzy graphs at the aim of filling a gap in the literature, namely the absence of a mathematical approach to analyze multi-index, multipolar, and multi-attribute data. The book describes metrics and labeling in m-polar graphs, m-polar fuzzy matroids. It also discusses in detail important applications in decision-making problems and imaging processing. The book is expected to stimulate the curiosity of mathematics, computer scientists, and social scientists alike, and to provide both students and researchers with the necessary knowledge to understand and apply m−polar fuzzy graph theory.
Theoretical Foundations of Radar Location and Radio Navigation (Springer Aerospace Technology)
by Denis Alexandrovich Akmaykin Eduard Anatolyevich Bolelov Anatoliy Ivanovich Kozlov Boris Valentinovich Lezhankin Alexander Evgenievich Svistunov Yury Grigorievich ShatrakovThe book represents a study guide reciting theoretical basics of radar location and radio navigation systems of air and sea transport. This is the distinctive feature of this study guide.The study guide states the principal physics of radar location and radio navigation, main measuring methods of proper and relative movement parameters of an object, tactical and technical characteristics of radar location and radio navigation systems, including examining issues on radiofrequency signals detection and its parameters estimation against background and interference of different type, filtering, combined detection and rating of signals, signals resolution and classification. The structural and functioning principles of the current and advanced radar location and radio navigation systems of air and sea transport are represented in the study guide with an adequate completeness. The study guide features the result of years long lecturing on radar location and radio navigation theoretical courses at the Moscow State Technical University of Civil Aviation and G.I.Nevelskiy Maritime State Technical Academy. The study guide is designated for students of radio-engineering specialties in area of air and sea transport. The study guide can be useful for radio engineers working in the field of air and maritime transport, and for graduate students and academic researchers as well.
Handbook of Formal Analysis and Verification in Cryptography (Prospects in Pure and Applied Mathematics)
by Sedat Akleylek Besik DunduaThis handbook of formal analysis in cryptography is very important for secure communication and processing of information. It introduces readers to several formal verification methods and software used to analyse cryptographic protocols. The chapters give readers general knowledge and formal methods focusing on cryptographic protocols. Handbook of Formal Analysis and Verification in Cryptography includes major formalisms and tools used for formal verification of cryptography, with a spotlight on new-generation cryptosystems such as post-quantum, and presents a connection between formal analysis and cryptographic schemes. The text offers formal methods to show whether security assumptions are valid and compares the most prominent formalism and tools as they outline common challenges and future research directions. Graduate students, researchers, and engineers worldwide will find this an exciting read.
Substitution and Tiling Dynamics: CIRM Jean-Morlet Chair, Fall 2017 (Lecture Notes in Mathematics #2273)
by Shigeki Akiyama Pierre ArnouxThis book presents a panorama of recent developments in the theory of tilings and related dynamical systems. It contains an expanded version of courses given in 2017 at the research school associated with the Jean-Morlet chair program.Tilings have been designed, used and studied for centuries in various contexts. This field grew significantly after the discovery of aperiodic self-similar tilings in the 60s, linked to the proof of the undecidability of the Domino problem, and was driven futher by Dan Shechtman's discovery of quasicrystals in 1984. Tiling problems establish a bridge between the mutually influential fields of geometry, dynamical systems, aperiodic order, computer science, number theory, algebra and logic. The main properties of tiling dynamical systems are covered, with expositions on recent results in self-similarity (and its generalizations, fusions rules and S-adic systems), algebraic developments connected to physics, games and undecidability questions, and the spectrum of substitution tilings.
Discrete and Computational Geometry and Graphs
by Jin Akiyama Hiro Ito Toshinori Sakai Yushi UnoThis book constitutes the thoroughly refereed post-conference proceedings of the 18th Japanese Conference on Discrete and Computational Geometry and Graphs, JDCDGG 2015, held in Kyoto, Japan, in September 2015. The total of 25 papers included in this volume was carefully reviewed and selected from 64 submissions. The papers feature advances made in the field of computational geometry and focus on emerging technologies, new methodology and applications, graph theory and dynamics. This proceedings are dedicated to Naoki Katoh on the occasion of his retirement from Kyoto University.
Discrete and Computational Geometry and Graphs
by Jin Akiyama Hiro Ito Toshinori SakaiThis book constitutes the thoroughly refereed post-conference proceedings of the 16th Japanese Conference on Discrete and computational Geometry and Graphs, JDCDGG 2013, held in Tokyo, Japan, in September 2013. The total of 16 papers included in this volume was carefully reviewed and selected from 58 submissions. The papers feature advances made in the field of computational geometry and focus on emerging technologies, new methodology and applications, graph theory and dynamics.
Algebraisches Denken im Arithmetikunterricht der Grundschule: Muster entdecken – Strukturen verstehen (Mathematik Primarstufe und Sekundarstufe I + II)
by Kathrin Akinwunmi Anna Susanne SteinwegDieses Buch thematisiert algebraisches Denken in der Grundschule als wesentlichen Kern der übergreifenden Leitidee „Muster, Strukturen und funktionaler Zusammenhang“ in den aktuellen KMK-Bildungsstandards.Für algebraische Lehr-Lernprozesse ist eine Unterscheidung zwischen sichtbaren Mustern und allgemeinen Strukturen wesentlich; eine solche wird hier vorgelegt und an vielen Beispielen konkretisiert: Muster machen aufmerksam und lassen neugierig werden. Die Suche nach Begründungen des Musters erwartet, die Tür zu dahinterliegenden Strukturen zu öffnen. Strukturen, d. h. die mathematischen Eigenschaften und Relationen, können so als ursächlich für die Regelmäßigkeit des Musters erkannt werden.Für die unterrichtliche Förderung und gezielte Unterstützung der algebraischen Denkentwicklung werden in diesem Buch einerseits Grundideen algebraischen Denkens für den Arithmetikunterricht ausgearbeitet und andererseits Prinzipien für Unterricht zu algebraischen Grundideen und ihr Zusammenspiel mit prozessbezogenen, allgemeinen Kompetenzen erläutert. Den vier algebraischen Grundideen folgend werden vielfältige, didaktisch aufbereitete Anregungen zur praktischen Umsetzung sowie jeweils entsprechendes Hintergrundwissen angeboten.Das Buch richtet sich an Lehramtsstudierende, an angehende und bereits praktizierende Lehrkräfte sowie an Personen, die in der Lehrerinnen- und Lehrerbildung tätig sind.
Zur Entwicklung von Variablenkonzepten beim Verallgemeinern mathematischer Muster
by Kathrin AkinwunmiDas Verallgemeinern mathematischer Muster ist eine grundlegende Tätigkeit des Mathematikunterrichts und zugleich ein zentraler Zugang zur Algebra. Dies nutzt Kathrin Akinwunmi, um sich mit der propädeutischen Entwicklung von Variablenkonzepten in der Grundschule zu beschäftigen. Sie geht der Frage nach, wie GrundschülerInnen Muster verallgemeinern und wie sich dabei Variablenkonzepte entwickeln. In einer Interviewstudie mit 30 ViertklässlerInnen untersucht die Autorin die Verallgemeinerungsprozesse der Lernenden aus epistemologischer Perspektive. In der Datenanalyse rekonstruiert sie Begriffsbildungsprozesse zu Variablenkonzepten und arbeitet sprachliche Mittel heraus, welche die Lernenden bei Verallgemeinerungen mathematischer Muster nutzen.
Proceedings of the 3rd International Conference on Experimental and Computational Mechanics in Engineering: ICECME 2021, Banda Aceh, October 11-12 (Lecture Notes in Mechanical Engineering)
by Akhyar Syifaul Huzni Mohd IqbalThis book gathers a selection of peer-reviewed papers presented at the 3rd International Conference on Experimental and Computational Mechanics in Engineering (ICECME 2021), held as a virtual conference and organized by Universitas Syiah Kuala, Banda Aceh, Indonesia, on October 11–12, 2021.This book, prepared by international scientists and engineers, covers the latest advances in computational mechanics, metallurgy and material science, energy systems, manufacturing processing systems, industrial and system engineering, biomechanics, artificial intelligence, micro-/nano-engineering, micro-electro-mechanical system, machine learning, mechatronics, and engineering design. This book is intended for academics, including graduate students and researchers, as well as industrial practitioners working in the areas of experimental and computational mechanics.
Proceedings of the 2nd International Conference on Experimental and Computational Mechanics in Engineering: ICECME 2020, Banda Aceh, October 13–14 (Lecture Notes in Mechanical Engineering)
by AkhyarThis book gathers a selection of peer-reviewed papers presented at the 2nd International Conference on Experimental and Computational Mechanics in Engineering (ICECME 2020), held as a virtual conference and organized by Universitas Syiah Kuala, Banda Aceh, Indonesia, on 13–14 October 2020. The contributions, prepared by international scientists and engineers, cover the latest advances in computational mechanics, metallurgy and material science, energy systems, manufacturing processing systems, industrial and system engineering, biomechanics, artificial intelligence, micro/nano-engineering, micro-electro-mechanical system, machine learning, mechatronics, and engineering design. The book is intended for academics, including graduate students and researchers, as well as industrial practitioners working in the areas of experimental and computational mechanics.
Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities (Nonlinear Physical Science)
by Marat Akhmet Ardak KashkynbayevThis book is devoted to bifurcation theory for autonomous and nonautonomous differential equations with discontinuities of different types. That is, those with jumps present either in the right-hand-side or in trajectories or in the arguments of solutions of equations. The results obtained in this book can be applied to various fields such as neural networks, brain dynamics, mechanical systems, weather phenomena, population dynamics, etc. Without any doubt, bifurcation theory should be further developed to different types of differential equations. In this sense, the present book will be a leading one in this field. The reader will benefit from the recent results of the theory and will learn in the very concrete way how to apply this theory to differential equations with various types of discontinuity. Moreover, the reader will learn new ways to analyze nonautonomous bifurcation scenarios in these equations. The book will be of a big interest both for beginners and experts in the field. For the former group of specialists, that is, undergraduate and graduate students, the book will be useful since it provides a strong impression that bifurcation theory can be developed not only for discrete and continuous systems, but those which combine these systems in very different ways. The latter group of specialists will find in this book several powerful instruments developed for the theory of discontinuous dynamical systems with variable moments of impacts, differential equations with piecewise constant arguments of generalized type and Filippov systems. A significant benefit of the present book is expected to be for those who consider bifurcations in systems with impulses since they are presumably nonautonomous systems.
Dynamics with Chaos and Fractals (Nonlinear Systems and Complexity #29)
by Marat Akhmet Mehmet Onur Fen Ejaily Milad AlejailyThe book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested.