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Hyperbolic Systems with Analytic Coefficients
by Tatsuo NishitaniThis monograph focuses on the well-posedness of the Cauchy problem for linear hyperbolic systems with matrix coefficients. Mainly two questions are discussed: (A) Under which conditions on lower order terms is the Cauchy problem well posed? (B) When is the Cauchy problem well posed for any lower order term? For first order two by two systems with two independent variables with real analytic coefficients, we present complete answers for both (A) and (B). For first order systems with real analytic coefficients we prove general necessary conditions for question (B) in terms of minors of the principal symbols. With regard to sufficient conditions for (B), we introduce hyperbolic systems with nondegenerate characteristics, which contain strictly hyperbolic systems, and prove that the Cauchy problem for hyperbolic systems with nondegenerate characteristics is well posed for any lower order term. We also prove that any hyperbolic system which is close to a hyperbolic system with a nondegenerate characteristic of multiple order has a nondegenerate characteristic of the same order nearby.
Hyperbolic Triangle Centers: The Special Relativistic Approach
by A. A. UngarAfter A. Ungar had introduced vector algebra and Cartesian coordinates into hyperbolic geometry in his earlier books, along with novel applications in Einstein's special theory of relativity, the purpose of his new book is to introduce hyperbolic barycentric coordinates, another important concept to embed Euclidean geometry into hyperbolic geometry. It will be demonstrated that, in full analogy to classical mechanics where barycentric coordinates are related to the Newtonian mass, barycentric coordinates are related to the Einsteinian relativistic mass in hyperbolic geometry. Contrary to general belief, Einstein's relativistic mass hence meshes up extraordinarily well with Minkowski's four-vector formalism of special relativity. In Euclidean geometry, barycentric coordinates can be used to determine various triangle centers. While there are many known Euclidean triangle centers, only few hyperbolic triangle centers are known, and none of the known hyperbolic triangle centers has been determined analytically with respect to its hyperbolic triangle vertices. In his recent research, the author set the ground for investigating hyperbolic triangle centers via hyperbolic barycentric coordinates, and one of the purposes of this book is to initiate a study of hyperbolic triangle centers in full analogy with the rich study of Euclidean triangle centers. Owing to its novelty, the book is aimed at a large audience: it can be enjoyed equally by upper-level undergraduates, graduate students, researchers and academics in geometry, abstract algebra, theoretical physics and astronomy. For a fruitful reading of this book, familiarity with Euclidean geometry is assumed. Mathematical-physicists and theoretical physicists are likely to enjoy the study of Einstein's special relativity in terms of its underlying hyperbolic geometry. Geometers may enjoy the hunt for new hyperbolic triangle centers and, finally, astronomers may use hyperbolic barycentric coordinates in the velocity space of cosmology.
Hyperbolic and Kinetic Models for Self-organised Biological Aggregations: A Modelling and Pattern Formation Approach (Lecture Notes in Mathematics #2232)
by Raluca EftimieThis book focuses on the spatio-temporal patterns generated by two classes of mathematical models (of hyperbolic and kinetic types) that have been increasingly used in the past several years to describe various biological and ecological communities. Here we combine an overview of various modelling approaches for collective behaviours displayed by individuals/cells/bacteria that interact locally and non-locally, with analytical and numerical mathematical techniques that can be used to investigate the spatio-temporal patterns produced by said individuals/cells/bacteria. Richly illustrated, the book offers a valuable guide for researchers new to the field, and is also suitable as a textbook for senior undergraduate or graduate students in mathematics or related disciplines.
Hyperbolicity of Projective Hypersurfaces
by Simone Diverio Erwan RousseauThisbook presents recent advances on Kobayashi hyperbolicity in complex geometry,especially in connection with projective hypersurfaces. This is a very activefield, not least because of the fascinating relations with complex algebraicand arithmetic geometry. Foundational works of Serge Lang and Paul A. Vojta,among others, resulted in precise conjectures regarding the interplay of theseresearch fields (e. g. existence of Zariski dense entire curves shouldcorrespond to the (potential) density of rational points). Perhapsone of the conjectures which generated most activity in Kobayashi hyperbolicitytheory is the one formed by Kobayashi himself in 1970 which predicts that avery general projective hypersurface of degree large enough does not containany (non-constant) entire curves. Since the seminal work of Green and Griffithsin 1979, later refined by J. -P. Demailly, J. Noguchi, Y. -T. Siu and others, itbecame clear that a possible general strategy to attack this problem was tolook at particular algebraic differential equations (jet differentials) thatevery entire curve must satisfy. This has led to some several spectacularresults. Describing the state of the art around this conjecture is the maingoal of this work.
Hypercomplex Analysis and Its Applications: Extended Abstracts of the International Conference Celebrating Paula Cerejeiras’ 60th Birthday (Trends in Mathematics #9)
by Uwe Kähler Nelson Faustino Milton Ferreira Nelson VieiraThis book contains a collection of short papers based on the presentations given at the international conference on Hypercomplex Analysis and its Applications celebrating Paula Cerejeiras&’ 60th birthday. These papers present the latest results as well as overviews on specific topics in the areas of hypercomplex and harmonic analysis as well as their connections with partial differential equations and spectral theory.
Hypergeometric Summation
by Wolfram KoepfModern algorithmic techniques for summation, most of which were introduced in the 1990s, are developed here and carefully implemented in the computer algebra system Maple(tm). The algorithms of Fasenmyer, Gosper, Zeilberger, Petkovsek and van Hoeij for hypergeometric summation and recurrence equations, efficient multivariate summation as well as q-analogues of the above algorithms are covered. Similar algorithms concerning differential equations are considered. An equivalent theory of hyperexponential integration due to Almkvist and Zeilberger completes the book. The combination of these results gives orthogonal polynomials and (hypergeometric and q-hypergeometric) special functions a solid algorithmic foundation. Hence, many examples from this very active field are given. The materials covered are suitable for an introductory course on algorithmic summation and will appeal to students and researchers alike.
Hypergraph Theory
by Alain BrettoThis book provides an introduction to hypergraphs, its aim being to overcome the lack of recent manuscripts on this theory. In the literature hypergraphs have many other names such as set systems and families of sets. This work presents the theory of hypergraphs in its most original aspects, while also introducing and assessing the latest concepts on hypergraphs. The variety of topics, their originality and novelty are intended to help readers better understand the hypergraphs in all their diversity in order to perceive their value and power as mathematical tools. This book will be a great asset to upper-level undergraduate and graduate students in computer science and mathematics. It has been the subject of an annual Master's course for many years, making it also ideally suited to Master's students in computer science, mathematics, bioinformatics, engineering, chemistry, and many other fields. It will also benefit scientists, engineers and anyone else who wants to understand hypergraphs theory.
Hyperidentities and Clones (Algebra, Logic and Applications)
by Klaus Denecke S L WismathTheories and results on hyperidentities have been published in various areas of the literature over the last 18 years. Hyperidentities and Clones integrates these into a coherent framework for the first time. The author also includes some applications of hyperidentities to the functional completeness problem in multiple-valued logic and extends the
Hypernumbers and Extrafunctions
by Mark Burgin"Hypernumbers and Extrafunctions" presents a rigorous mathematical approach to operate with infinite values. First, concepts of real and complex numbers are expanded to include a new universe of numbers called hypernumbers which includes infinite quantities. This brief extends classical calculus based on real functions by introducing extrafunctions, which generalize not only the concept of a conventional function but also the concept of a distribution. Extrafucntions have been also efficiently used for a rigorous mathematical definition of the Feynman path integral, as well as for solving some problems in probability theory, which is also important for contemporary physics. This book introduces a new theory that includes the theory of distributions as a subtheory, providing more powerful tools for mathematics and its applications. Specifically, it makes it possible to solve PDE for which it is proved that they do not have solutions in distributions. Also illustrated in this text is how this new theory allows the differentiation and integration of any real function. This text can be used for enhancing traditional courses of calculus for undergraduates, as well as for teaching a separate course for graduate students.
Hyperparameter Tuning for Machine and Deep Learning with R: A Practical Guide
by Thomas Bartz-Beielstein Eva Bartz Martin Zaefferer Olaf MersmannThis open access book provides a wealth of hands-on examples that illustrate how hyperparameter tuning can be applied in practice and gives deep insights into the working mechanisms of machine learning (ML) and deep learning (DL) methods. The aim of the book is to equip readers with the ability to achieve better results with significantly less time, costs, effort and resources using the methods described here. The case studies presented in this book can be run on a regular desktop or notebook computer. No high-performance computing facilities are required. The idea for the book originated in a study conducted by Bartz & Bartz GmbH for the Federal Statistical Office of Germany (Destatis). Building on that study, the book is addressed to practitioners in industry as well as researchers, teachers and students in academia. The content focuses on the hyperparameter tuning of ML and DL algorithms, and is divided into two main parts: theory (Part I) and application (Part II). Essential topics covered include: a survey of important model parameters; four parameter tuning studies and one extensive global parameter tuning study; statistical analysis of the performance of ML and DL methods based on severity; and a new, consensus-ranking-based way to aggregate and analyze results from multiple algorithms. The book presents analyses of more than 30 hyperparameters from six relevant ML and DL methods, and provides source code so that users can reproduce the results. Accordingly, it serves as a handbook and textbook alike.
Hypersingular Integral Equations and Their Applications
by I.K. Lifanov L.N. Poltavskii MG.M. VainikkoA number of new methods for solving singular and hypersingular integral equations have emerged in recent years. This volume presents some of these new methods along with classical exact, approximate, and numerical methods. The authors explore the analysis of hypersingular integral equations based on the theory of pseudodifferential operators and co
Hypersingular Integrals and Their Applications (Analytical Methods and Special Functions)
by Stefan SamkoHypersingular integrals arise as constructions inverse to potential-type operators and are realized by the methods of regularization and finite differences. This volume develops these approaches in a comprehensive treatment of hypersingular integrals and their applications. The author is a renowned expert on the topic. He explains the basics before
Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension
by Michio KakuAre there other dimensions beyond our own? Is time travel possible? Can we change the past? Are there gateways to parallel universes? All of us have pondered such questions, but there was a time when scientists dismissed these notions as outlandish speculations. Not any more. Today, they are the focus of the most intense scientific activity in recent memory. In Hyperspace, Michio Kaku offers the first book-length tour of the most exciting (and perhaps most bizarre) work in modern physics. <p><p>The theory of hyperspace (or higher dimensional space)—and its newest wrinkle, superstring theory—stand at the center of this revolution, with adherents in every major research laboratory in the world. Beginning where Hawking's Brief History of Time left off, Kaku paints a vivid portrayal of the breakthroughs now rocking the physics establishment. Why all the excitement? As the author points out, for over half a century, scientists have puzzled over why the basic forces of the cosmos—gravity, electromagnetism, and the strong and weak nuclear forces—require markedly different mathematical descriptions. But if we see these forces as vibrations in a higher dimensional space, their field equations suddenly fit together like pieces in a jigsaw puzzle, perfectly snug, in an elegant, astonishingly simple form. This may thus be our leading candidate for the Theory of Everything.
Hyperspaces: Fundamentals and Recent Advances (Chapman & Hall/CRC Pure and Applied Mathematics)
by Alejandro IllanesPresents hyperspace fundamentals, offering a basic overview and a foundation for further study. Topics include the topology for hyperspaces, examples of geometric models for hyperspaces, 2x and C(X) for Peano continua X, arcs in hyperspaces, the shape and contractability of hyperspaces, hyperspaces and the fixed point property, and Whitney maps. The text contains examples and exercises throughout, and provides proofs for most results.
Hyperspectral Imaging Remote Sensing: Physics, Sensors, and Algorithms
by Dimitris G. Manolakis Ronald B. Lockwood Thomas W. CooleyA practical and self-contained guide to the principles, techniques, models and tools of imaging spectroscopy. Bringing together material from essential physics and digital signal processing, it covers key topics such as sensor design and calibration, atmospheric inversion and model techniques, and processing and exploitation algorithms. Readers will learn how to apply the main algorithms to practical problems, how to choose the best algorithm for a particular application, and how to process and interpret hyperspectral imaging data. A wealth of additional materials accompany the book online, including example projects and data for students, and problem solutions and viewgraphs for instructors. This is an essential text for senior undergraduate and graduate students looking to learn the fundamentals of imaging spectroscopy, and an invaluable reference for scientists and engineers working in the field. A self-contained introductory text covering the principles, techniques, and tools of imaging spectroscopy. Can be used in both undergraduate and graduate settings, and also as a reference text for practitioners. Accompanied online by example projects and data for students, and problem solutions and viewgraphs for instructors.
Hypoelliptic Laplacian and Orbital Integrals (Annals of Mathematics Studies #177)
by Jean-Michel BismutThis book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.
Hypothesen Testen: Eine Einführung für Bachelorstudierende sozialwissenschaftlicher Fächer (essentials)
by Florian G. Hartmann Daniel LoisDie Sozialwissenschaftler Florian G. Hartmann und Daniel Lois erklären in diesem Essential Schritt für Schritt und auf Nachvollziehbarkeit bedacht, wie im Rahmen einer quantitativen Untersuchung Hypothesen überprüft werden. Dabei werden methodische und statistische Grundbegriffe besprochen und komplexere Sachverhalte anhand von alltagsnahen Beispielen erläutert. Die Autoren schöpfen bei den Erklärungen aus ihrer Lehr- und Forschungstätigkeit und berücksichtigen die Erfahrungen ihres eigenen Studiums.
Hypothetical Learning Trajectories: A Special Issue of Mathematical Thinking and Learning
by Douglas H. Clements Julie SaramaThe purpose of this special issue is to present several research perspectives on learning trajectories with the intention of encouraging the broader community to reflect on, better define, adopt, adapt, or challenge the concept. The issue begins by briefly introducing learning trajectories. The remaining articles provide elaboration, examples, and discussion of the construct. They purposefully are intended to be illustrative, exploratory, and provocative with regard to learning trajectories construct; they are not a set of verification studies.
Hysteresis Phenomena in Biology
by Hamid Reza NooriThe occurrence of hysteresis phenomena has been traditionally associated with mechanical and magnetic properties of materials. However, recent studies on the dynamics of biological processes suggest switch-like behavior that could be described by mathematical models of hysteresis. This book presents the milestones and perspectives of biological hysteresis and provides a comprehensive and application-oriented introduction to this subject. The target audience primarily comprises researchers but the book may also be beneficial for graduate students.
HÜTTE - Das Ingenieurwissen
by Horst Czichos Akademischer Verein Hütte e.V. Manfred HenneckeDas Werk präsentiert die mathematisch-naturwissenschaftlichen, ökonomisch-rechtlichen sowie technologischen Grundlagen des Ingenieurwissens - alles in einem Band. Für die Neuauflage wurden sämtliche Inhalte fachlich ergänzt, insbesondere die Abschnitte zu Makromolekülen, Umweltverträglichkeit, Recycling, Festigkeitslehre, Mikrosensorik, binäre Steuerungstechnik, Software-Engineering, Kommunikationstechnik, Mensch-Maschine-Interaktion sowie Normung, Recht und Patente. Neu hinzugekommen sind die Themen Management, Qualität und Personal.
Häufigkeiten, Verteilungen, Mittelwerte und Co.: Grundlagen der beschreibenden Statistik etwas anders dargestellt und erklärt (essentials)
by Rüdiger StegenIn diesem essential steht die leichte Verständlichkeit statistischer Grundbegriffe im Vordergrund, ohne dabei die mathematische Korrektheit zu beeinträchtigen. Zunächst werden Merkmale und ihr praktischer Einsatz beschrieben. Anschließend werden Häufigkeiten in Bezug auf Mengen definiert, sodass der spätere Übergang zu Wahrscheinlichkeiten naheliegend ist. Hypergeometrische Verteilung und Binomialverteilung werden mit relativen Häufigkeiten statt mit Wahrscheinlichkeiten formuliert, sodass ein direkter Bezug zur Praxis entsteht. Arithmetisches, geometrisches und harmonisches Mittel werden aus praktischen Fragestellungen abgeleitet. Bei Klassierungen werden unverbesserbare Intervalle für das arithmetische Mittel ohne die üblichen spekulativen Annahmen hergeleitet. Alle Themen des essentials werden durch Beispiele erläutert.
Höhere Mathematik 1: Lineare Algebra
by Walter Strampp Dörthe JanssenDas Buch schildert die wichtigsten Inhalte der Linearen Algebra. Durch zahlreiche Beispiele und ausführliche Übungen wird der Leser zur sicheren Beherrschung des Stoffs geführt. Gegenüber der Vorauflage "Höhere Mathematik mit MATHEMATICA -Band 1: Grundlagen, Lineare Algebra" wurden die Inhalte zugunsten eines größeren Übungsteils inklusive Lösungen gestrafft, das Buch ist damit besonders für die Bachelor-Studiengänge geeignet.
Höhere Mathematik für Dummies (Für Dummies)
by Thoralf RäschPhysik ohne Mathematik, das ist unmöglich. Aber wenn Sie Ihre liebe Mühe mit Mathe haben, dann hilft Ihnen dieses Buch, ganz gleich aus welchem Grund Sie sich mit Physik beschäftigen müssen: als Studienanfänger der Physik, als Student der Ingenieurwissenschaften oder der Medizin. Dieses Buch erklärt Ihnen, was Sie über einfache, komplexe und mehrdimensionale Analysis, Differentialgleichungen und Lineare Algebra wissen sollten. Zahlreiche Beispiele machen die Erläuterungen noch anschaulicher.
Höhere Mathematik für Naturwissenschaftler und Ingenieure
by Günter Bärwolff Akiko KatoDieses Lehrbuch wendet sich an Studierende der Ingenieur- und Naturwissenschaften und stellt die gesamte Höhere Mathematik, wie sie üblicherweise im Grundstudium behandelt wird, in einem Band zusammen.Ausgangspunkt ist dabei stets die Frage, womit Ingenieure und Naturwissenschaftler in ihrer Arbeit konfrontiert werden, wie z. B. die Modellierung und Optimierung technischer Prozesse oder die Beschreibung physikalischer Gesetzmäßigkeiten. Das Werk erschließt systematisch die zugrunde liegenden mathematischen Themen, ausgehend von der Schulmathematik über die Lineare Algebra bis hin zu partiellen Differenzialgleichungen. Den Autoren gelingt eine in sich geschlossene und didaktisch eingängige Darstellung der Höheren Mathematik, wobei Beweise nur angegeben werden, wenn sie für das Verständnis hilfreich sind. Alle neu eingeführten Begriffe werden durch Abbildungen oder Beispiele veranschaulicht. Eine Vielzahl von Übungsaufgaben (mit Lösungen im Internet)erleichtern die Vertiefung des Lernstoffs.Für die vorliegende 4. Auflage wurde das Werk vollständig durchgesehen und u.a. um das Thema mathematische Grundlagen des Deep Learning ergänzt.Plus: Zudem erhalten Sie Zugang auf ca. 150 Flashcards (Springer-Nature-Flashcards-App), mit denen Sie die Inhalte auf spielerische Weise einüben können.
Höhere Mathematik im Alltag: Vom Regenbogen bis zur digitalen Bildkompression
by Rüdiger SeydelMINT-Kompetenz (Mathematik, Informatik, Naturwissenschaften, Technik) ist der Schlüssel zur Zukunft. Aber warum Mathematik? Dieses Buch gibt die Antwort anhand von Beispielen aus Alltag, Natur und Technik. Es präsentiert und diskutiert Fallstudien aus verschiedensten Bereichen mit Mitteln der Höheren Mathematik. Dabei zeigt sich die Macht der Mathematik beim Aufspüren verborgener Zusammenhänge und beweist, dass sogar einfache Modelle eine reiche Struktur besitzen. Die Beispiele sind gut durchdacht und führen verständlich an den mathematischen Stoff heran.Das Buch eignet sich bestens zum Selbststudium, für weiterführende Schulen, Arbeitsgemeinschaften und Seminare in MINT-relevanten Studienfächern.