Calculus and mathematical analysis Books

Book SynopsisThis book concerns testing hypotheses in non-parametric models. Classical non-parametric tests (goodness-of-fit, homogeneity, randomness, independence) of complete data are considered. Most of the test results are proved and real applications are illustrated using examples. Theories and exercises are provided. The incorrect use of many tests applying most statistical software is highlighted and discussed. Table of ContentsPreface xi Terms and Notation xv Chapter 1. Introduction 1 1.1. Statistical hypotheses 1 1.2. Examples of hypotheses in non-parametric models 2 1.3. Statistical tests 5 1.4. P-value 7 1.5. Continuity correction 10 1.6. Asymptotic relative efficiency 13 Chapter 2. Chi-squared Tests 17 2.1. Introduction 17 2.2. Pearson’s goodness-of-fit test: simple hypothesis 17 2.3. Pearson’s goodness-of-fit test: composite hypothesis 26 2.4. Modified chi-squared test for composite hypotheses 34 2.5. Chi-squared test for independence 52 2.6. Chi-squared test for homogeneity 57 2.7. Bibliographic notes 64 2.8. Exercises 64 2.9. Answers 72 Chapter 3. Goodness-of-fit Tests Based on Empirical Processes 77 3.1. Test statistics based on the empirical process 77 3.2. Kolmogorov–Smirnov test 82 3.3. ω2, Cramér–von-Mises and Andersen–Darling tests 86 3.4. Modifications of Kolmogorov–Smirnov, Cramér–von-Mises and Andersen–Darling tests: composite hypotheses 91 3.5. Two-sample tests 98 3.6. Bibliographic notes 104 3.7. Exercises106 3.8. Answers 109 Chapter 4. Rank Tests 111 4.1. Introduction 111 4.2. Ranks and their properties 112 4.3. Rank tests for independence 117 4.4. Randomness tests 139 4.5. Rank homogeneity tests for two independent samples 146 4.6. Hypothesis on median value: the Wilcoxon signed ranks test 168 4.7. Wilcoxon’s signed ranks test for homogeneity of two related samples 180 4.8. Test for homogeneity of several independent samples: Kruskal–Wallis test 181 4.9. Homogeneity hypotheses for k related samples: Friedman test 191 4.10. Independence test based on Kendall’s concordance coefficient 204 4.11. Bibliographic notes 208 4.12. Exercises 209 4.13. Answers 212 Chapter 5. Other Non-parametric Tests 215 5.1. Sign test 215 5.2. Runs test 221 5.3. McNemar’s test 231 5.4. Cochran test 238 5.5. Special goodness-of-fit tests 245 5.6. Bibliographic notes 268 5.7. Exercises 269 5.8. Answers 271 APPENDICES 275 Appendix A. Parametric Maximum Likelihood 277 Appendix B. Notions from the Theory of 281 BBibliography 293 Index 305

Read more less

Book SynopsisReal Analysis is a comprehensive introduction to this core subject and is ideal for self-study or as a course textbook for first and second-year undergraduates. Combining an informal style with precision mathematics, the book covers all the key topics with fully worked examples and exercises with solutions. All the concepts and techniques are deployed in examples in the final chapter to provide the student with a thorough understanding of this challenging subject. This book offers a fresh approach to a core subject and manages to provide a gentle and clear introduction without sacrificing rigour or accuracy.Trade ReviewVol. 85 (504), 2001) "The book is a clear and structured introduction to real analysis. ... Fully worked out examples and exercises with solutions extend and illustrate the theory. Written in an easy-to-read style, combining informality and precision, the book is ideal for self-study or as a course textbook for first- and second-year undergraduates." (I. Rasa, Zentralblatt MATH, Vol. 969, 2001)Table of Contents1. Introductory Ideas.- 1.1 Foreword for the Student: Is Analysis Necessary?.- 1.2 The Concept of Number.- 1.3 The Language of Set Theory.- 1.4 Real Numbers.- 1.5 Induction.- 1.6 Inequalities.- 2. Sequences and Series.- 2.1 Sequences.- 2.2 Sums, Products and Quotients.- 2.3 Monotonie Sequences.- 2.4 Cauchy Sequences.- 2.5 Series.- 2.6 The Comparison Test.- 2.7 Series of Positive and Negative Terms.- 3. Functions and Continuity.- 3.1 Functions, Graphs.- 3.2 Sums, Products, Compositions; Polynomial and Rational Functions.- 3.3 Circular Functions.- 3.4 Limits.- 3.5 Continuity.- 3.6 Uniform Continuity.- 3.7 Inverse Functions.- 4. Differentiation.- 4.1 The Derivative.- 4.2 The Mean Value Theorems.- 4.3 Inverse Functions.- 4.4 Higher Derivatives.- 4.5 Taylor’s Theorem.- 5. Integration.- 5.1 The Riemann Integral.- 5.2 Classes of Integrable Functions.- 5.3 Properties of Integrals.- 5.4 The Fundamental Theorem.- 5.5 Techniques of Integration.- 5.6 Improper Integrals of the First Kind.- 5.7 Improper Integrals of the Second Kind.- 6. The Logarithmic and Exponential Functions.- 6.1 A Function Defined by an Integral.- 6.2 The Inverse Function.- 6.3 Further Properties of the Exponential and Logarithmic Functions.- Sequences and Series of Functions.- 7.1 Uniform Convergence.- 7.2 Uniform Convergence of Series.- 7.3 Power Series.- 8. The Circular Functions.- 8.1 Definitions and Elementary Properties.- 8.2 Length.- 9. Miscellaneous Examples.- 9.1 Wallis’s Formula.- 9.2 Stirling’s Formula.- 9.3 A Continuous, Nowhere Differentiable Function.- Solutions to Exercises.- The Greek Alphabet.

Read more less

Book SynopsisThis book serves as an introduction to the use of mathematics in describing collective phenomena in physics and biology. Derived from a course of innovative lectures, the book shows students early in their studies how many of the topics they have encountered – partial differential equations, differential equations, Fourier series, and linear algebra – are useful in constructing, analysing and interpreting phenomena present in the real world. Throughout, ideas are developed using worked examples and exercises with solution. The text does not assume a strong background in physics. Trade ReviewFrom the reviews: "This textbook … is designed for undergraduates who have followed a first ‘mathematical methods’ course and who are ready to study in more depth the mathematics underlying phenomena … . Each chapter includes a number of worked examples and a dozen or so exercises, with solutions collected at the end of the book. The book is aimed at students of applied mathematics, physics and engineering. The emphasis on techniques and the frequent references to applications make it particularly suitable for this audience." (S.C. Russen, The Mathematical Gazette, Vol. 89 (516), 2005) "Fields, Flows, and Waves, is an introduction to continuum models based on the author’s lectures … . Ample illustrations and worked examples come with the exposition, and there are several exercises with varying degrees of difficulty; detailed solutions are included at the end of the text. … I warmly recommend this book. It reads well and is in an attractive, concise format. … It makes one yearn for a course in the curriculum where this material could be regularly taught." (SIAM Review, Vol. 46 (3), 2004) "This book is an introduction to the mathematical methods in classical fields theory. It is designed for the second-year undergraduate in physics, mathematics and engineering. … The presentation is excellent, numerous examples of increasing difficulties are considered with details. … The book ends with solutions to the exercises a short bibliography and an index. In conclusion, I warmly recommend this book to any students in physics because it’s well written … interesting and very useful." (Stéphane Métens, Physicalia, Vol. 26 (1), 2004) "This book … is a first introduction to the mathematical description of fields, flows and waves. It shows students, early in their studies, how many of the topics they have encountered are useful … . Designed for second-year undergraduate students in mathematics, mathematical physics, and engineering, it presumes only a limited familiarity with several variable calculus and vector fields. … The ideas are developed through worked examples, and a range of exercises (with solutions) is provided to test understanding." (Läenseignement Mathematique, Vol. 49 (3-4), 2003) "This is another excellent readable book in the Springer Undergraduate Mathematics Series (SUMS). It is a refreshingly modern approach to Continuum Mechanics … . Indeed Professor Parker has written this book so that it might be used directly as an elementary course … . This is a carefully written, well structured book which contains a wealth of examples complete with solutions. … a carefully structured book from which a modern undergraduate applied mathematics course may be taught directly." (Sean McKee, Journal of Fluid Mechanics, Vol.504, 2004) "Introductory books … often struggle with the balance between the motivating physical problems and the formal mathematical structures. As the title suggests, Parker … manages to keep the more technical mathematical structure in clear view. Particularly impressive is how carefully the author leads readers … . the book has a completeness that makes it attractive as a self-contained resource as well as a textbook. … complete solutions (not just answers) to all of the exercises makes the book particularly effective for independent study of this material. Summing Up: Highly recommended." (J. Feroe, CHOICE, December, 2003) "The book is well-written and illustrated by interesting figures which make the text easy to read and attractive. Of course undergraduate students in physics and maybe in mathematics will surely benefit of a lecture and practice of this book. Each of the ten chapters indeed contains some lists of significant exercises. The more or less detailed solutions of these exercises are gathered at the end of the book." (Alain Brillard, Zentralblatt MATH, 2003) "Continuum models ignoring the substructures of fluids are useful and widely applied for the description of fields, flows, and waves in different research works. This book gives a first introduction to the mathematical methods necessary for the solution of the resulting equations. … Each chapter contains some examples and exercises. … The results of the exercises are listed at the end of the book. … the book is a useful introduction in this important branch of knowledge." (Bernd Platzer, www.zamm-journal.org, 2004) "David Parker’s book Fields, Flows and Waves: An Introduction to Continuum Models … is a fine addition to the Springer Undergraduate Mathematics Series. … For the subjects considered, the author provides masterly compact accounts of the physical phenomena … and solves interesting problems. Parker takes particular care to examine the physical implications of the mathematical solutions … . An adequate selection of student exercises is included, with solutions … . could be used to enrich an advanced undergraduate or beginning graduate course on continuum mechanics." (James Casey, Physics today, October, 2004)Table of Contents1. The Continuum Description.- 1.1 Densities and Fluxes.- 1.2 Conservation and Balance Laws in One Dimension.- 1.3 Heat Flow.- 1.4 Steady Radial Flow in Two Dimensions.- 1.5 Steady Radial Flow in Three Dimensions.- 2. Unsteady Heat Flow.- 2.1 Thermal Energy.- 2.1.1 Heat Balance in One-dimensional Problems.- 2.1.2 Some Special Solutions of Equation (2.3).- 2.2 Effects of Heat Supply.- 2.3 Unsteady, Spherically Symmetric Heat Flow.- 3. Fields and Potentials.- 3.1 Gradient of a Scalar.- 3.1.1 Some Applications.- 3.2 Gravitational Potential.- 3.2.1 Special Properties of the Function ?=r?1.- 3.3 Continuous Distributions of Mass.- 3.4 Electrostatics.- 3.4.1 Gauss’s Law of Flux.- 3.4.2 Charge-free Regions.- 3.4.3 Surface Charge Density.- 4. Laplace’s Equation and Poisson’s Equation.- 4.1 The Ubiquitous Laplacian.- 4.2 Separable Solutions.- 4.3 Poisson’s Equation.- 4.4 Dipole Solutions.- 4.4.1 Uses of Dipole Solutions to ?2?=0.- 4.4.2 Spherical Inclusions.- 5. Motion of an Elastic String.- 5.1 Tension and Extension; Kinematics and Dynamics.- 5.1.1 Dynamics.- 5.2 Planar Motions.- 5.2.1 Small Transverse Motions.- 5.2.2 Longitudinal Motions.- 5.3 Properties of the Wave Equation.- 5.3.1 Standing Waves.- 5.3.2 Superposition of Standing Waves.- 5.4 D’Alembert’s Solution, Travelling Waves and Wave Reflections.- 5.4.1 Wave Reflections.- 5.5 Other One-dimensional Waves.- 5.5.1 Acoustic Vibrations in a’ lUbe.- 5.5.2 Telegraphy and High-voltage Transmission.- 6. Fluid Flow.- 6.1 Kinematics and Streamlines.- 6.1.1 Some Important Examples of Steady Flow.- 6.2 Volume Flux and Mass Flux.- 6.2.1 Incompressible Fluids.- 6.2.2 Mass Conservation.- 6.3 Two-dimensional Flows of Incompressible Fluids.- 6.3.1 The Continuity Equation.- 6.3.2 Irrotational Flows and the Velocity Potential.- 6.3.3 The Stream Function.- 6.4 Pressure in a Fluid.- 6.4.1 Resultant Force.- 6.4.2 Hydrostatics and Archimedes’ Principle.- 6.4.3 Momentum Density and Momentum Flux.- 6.5 Bernoulli’s Equation.- 6.5.1 The Material (Advected) Derivative.- 6.5.2 Bernoulli’s Equation and Dynamic Pressure.- 6.5.3 The Principle of Aerodynamic Lift.- 6.6 Three-dimensional, Incompressible Flows.- 6.6.1 The Continuity Equation.- 6.6.2 Irrotational Flows, the Velocity Potential and Laplace’s Equation.- 7. Elastic Deformations.- 7.1 The Kinematics of Deformation.- 7.1.1 Deformation Gradient.- 7.1.2 Stretch and Rotation.- 7.2 Polar Decomposition.- 7.3 Stress.- 7.3.1 Traction Vectors.- 7.3.2 Components of Stress.- 7.3.3 Traction on a General Surface.- 7.4 Isotropic Linear Elasticity.- 7.4.1 The Constitutive Law.- 7.4.2 Stretching, Shear and Torsion.- 8. Vibrations and Waves.- 8.1 Wave Reflection and Refraction.- 8.1.1 Use of the Complex Exponential.- 8.1.2 Plane Waves.- 8.1.3 Reflection at a Rigid Wall.- 8.1.4 Refraction at an Interface.- 8.1.5 Total Internal Reflection.- 8.2 Guided Waves.- 8.2.1 Acoustic Waves in a Layer.- 8.2.2 Waveguides and Dispersion.- 8.3 Love Waves in Elasticity.- 8.4 Elastic Plane Waves.- 8.4.1 Elastic Shear Waves.- 8.4.2 Dilatational Waves.- 9. Electromagnetic VVaves and Light.- 9.1 Physical Background.- 9.1.1 The Origin of Maxwell’s Equations.- 9.1.2 Plane Electromagnetic Waves.- 9.1.3 Reflection and Refraction of Electromagnetic Waves.- 9.2 Waveguides.- 9.2.1 Rectangular Waveguides.- 9.2.2 Circular Cylindrical Waveguides.- 9.2.3 An Introduction to Fibre Optics.- 10. Chemical and Biological Models.- 10.1 Diffusion of Chemical Species.- 10.1.1 Fick’s Law of Diffusion.- 10.1.2 Self-similar Solutions.- 10.1.3 Travelling Wavefronts.- 10.2 Population Biology.- 10.2.1 Growth and Dispersal.- 10.2.2 Fisher’s Equation and Self-limitation.- 10.2.3 Population-dependent Dispersivity.- 10.2.4 Competing Species.- 10.2.5 Diffusive Instability.- 10.3 Biological Waves.- 10.3.1 The Logistic Wavefront.- 10.3.2 Travelling Pulses and Spiral Waves.- Solutions.

Read more less

Book Synopsis

Read more less

Book SynopsisThis book presents articles of L.V. Kantorovich on the descriptive theory of sets and function and on functional analysis in semi-ordered spaces, to demonstrate the unity of L.V. Kantorovich's creative research. It also includes two papers on the extension of Hilbert space.

Read more less

Book SynopsisDie Theorie gewöhnlicher Differentialgleichungen und dynamischer Systeme spielt eine zentrale Rolle in der Modellierung realer zeitabhängiger Prozesse. Damit gehört sie zur universitären Grundausbildung von Mathematikern, Physikern, Informatikern und Ingenieuren. Der Band liefert eine zeitgemäße Darstellung der Theorie, wobei der Schwerpunkt auf Dynamik liegt. Um die Leistungsfähigkeit der Theorie zu belegen, nehmen Beispiele viel Raum ein. Neue Anwendungen in Biologie, Chemie und Physik werden in Modellierung und Analysis detailliert behandelt.Table of ContentsProlog.- Notationen.- I Gewöhnliche Differentialgleichungen.- 1. Einführung.- 2 Existenz und Eindeutigkeit.- 3 Lineare Systeme.- 4 Stetige und differenzierbare Abhängigkeit.- 5 Elementare Stabilitätstheorie.- II Dynamische Systeme.- 6 Existenz und Eindeutigkeit II.- 7 Invarianz.- 8 Ljapunov-Funktionen und Stabilität.- 9 Ebene autonome Systeme.- 10 Linearisierung und invariante Mannigfaltigkeiten.- 11 Periodische Lösungen.- 12 Verzweigungstheorie.- 13 Differentialgleichungen auf Mannigfaltigkeiten.- Epilog.- Abbildungsverzeichnis.- Literaturverzeichnis.- Lehrbücher und Monographien.- Originalliteratur.- Index.

Read more less

Book SynopsisThis book presents a step-by-step guide to the basic theory of multivectors and spinors, with a focus on conveying to the reader the geometric understanding of these abstract objects. Following in the footsteps of M. Riesz and L. Ahlfors, the book also explains how Clifford algebra offers the ideal tool for studying spacetime isometries and Möbius maps in arbitrary dimensions.The book carefully develops the basic calculus of multivector fields and differential forms, and highlights novelties in the treatment of, e.g., pullbacks and Stokes’s theorem as compared to standard literature. It touches on recent research areas in analysis and explains how the function spaces of multivector fields are split into complementary subspaces by the natural first-order differential operators, e.g., Hodge splittings and Hardy splittings. Much of the analysis is done on bounded domains in Euclidean space, with a focus on analysis at the boundary. The book also includes a derivation of new Dirac integral equations for solving Maxwell scattering problems, which hold promise for future numerical applications. The last section presents down-to-earth proofs of index theorems for Dirac operators on compact manifolds, one of the most celebrated achievements of 20th-century mathematics.The book is primarily intended for graduate and PhD students of mathematics. It is also recommended for more advanced undergraduate students, as well as researchers in mathematics interested in an introduction to geometric analysis. Trade Review“The book is carefully prepared and well presented, and I recommend the book … for students who have just mastered vector calculus and Maxwellian electromagnetism.” (Hirokazu Nishimura, zbMATH 1433.58001, 2020)Table of ContentsPrelude: Linear algebra.- Exterior algebra.- Clifford algebra.- Mappings of inner product spaces.- Spinors in inner product spaces.- Interlude: Analysis.- Exterior calculus.- Hodge decompositions.- Hypercomplex analysis.- Dirac equations.- Multivector calculus on manifolds.- Two index theorems.

Read more less

Book SynopsisThis book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided. We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contracted space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications. After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc. The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics. Table of ContentsHolomorphic functions.- Meromorphic functions.- Riemann's theorem.- Harmonic functions.- Riemann surfaces and their modules.- Compact Riemann surfaces and algebraic curves.- Riemann-Roch theorem and theta functions.- Integrable Systems.- The formula for the conformal mapping of an arbitrary domain into the unit disk.

Read more less

Book SynopsisThis textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations.Through 65 fully solved problems, the book offers readers a fast but in-depth introduction to the field, covering advanced topics in microlocal analysis, including pseudo- and para-differential calculus, and the key classical equations, such as the Laplace, Schrödinger or Navier-Stokes equations. Essentially self-contained, the book begins with problems on the necessary tools from functional analysis, distributions, and the theory of functional spaces, and in each chapter the problems are preceded by a summary of the relevant results of the theory.Informed by the authors' extensive research experience and years of teaching, this book is for graduate students and researchers who wish to gain real working knowledge of the subject. Trade Review“Instructors teaching courses that include one or all of the above-mentioned topics will find the exercises of great help in course preparation. Researchers in partial differential equations may find this work useful as a summary of analytic theories published in this volume.” (Vicenţiu D. Rădulescu, zbMATH 1461.35001, 2021)Table of ContentsPart I Tools and Problems.- 1 Elements of functional analysis and distributions.- 2 Statements of the problems of Chapter 1.- 3 Functional spaces.- 4 Statements of the problems of Chapter 3.- 5 Microlocal analysis.- 6 Statements of the problems of Chapter 5.- 7 The classical equations.- 8 Statements of the problems of Chapter 7.- Part II Solutions of the Problems. A Classical results. Index.

Read more less

Book SynopsisThis graduate-level textbook introduces the reader to the area of inverse problems, vital to many fields including geophysical exploration, system identification, nondestructive testing, and ultrasonic tomography. It aims to expose the basic notions and difficulties encountered with ill-posed problems, analyzing basic properties of regularization methods for ill-posed problems via several simple analytical and numerical examples. The book also presents three special nonlinear inverse problems in detail: the inverse spectral problem, the inverse problem of electrical impedance tomography (EIT), and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness, and continuous dependence on parameters. Ultimately, the text discusses theoretical results as well as numerical procedures for the inverse problems, including many exercises and illustrations to complement coursework in mathematics and engineering. This updated text includes a new chapter on the theory of nonlinear inverse problems in response to the field’s growing popularity, as well as a new section on the interior transmission eigenvalue problem which complements the Sturm-Liouville problem and which has received great attention since the previous edition was published. Trade Review“This monograph is a thorough and insightful introduction to the mathematics of inverse problems and a solid improvement of the previous editions, which were used to educate many researchers in the field over the last two and a half decades. As such, the volume is already a classic and can be recommended without reservations to any reader interested in both the foundations and specific examples of inverse problems relevant to modern engineering and sciences.” (Alexander Mamonov, SIAM Review, Vol. 65 (2), 2023)Table of ContentsIntroduction and Basic Concepts.- Regularization Theory for Equations of the First Kind.- Regularization by Discretization.- Nonlinear Inverse Problems.- Inverse Eigenvalue Problems.- An Inverse Problem in Electrical Impedance Tomography.- An Inverse Scattering Problem.- Basic Facts from Functional Analysis.- References.- Index.

Read more less

Book SynopsisThis book contains the contributions resulting from the 6th Italian-Japanese workshop on Geometric Properties for Parabolic and Elliptic PDEs, which was held in Cortona (Italy) during the week of May 20–24, 2019. This book will be of great interest for the mathematical community and in particular for researchers studying parabolic and elliptic PDEs. It covers many different fields of current research as follows: convexity of solutions to PDEs, qualitative properties of solutions to parabolic equations, overdetermined problems, inverse problems, Brunn-Minkowski inequalities, Sobolev inequalities, and isoperimetric inequalities.Table of Contents- Poincaré and Hardy Inequalities on Homogeneous Trees. - Ground State Solutions for the Nonlinear Choquard Equation with Prescribed Mass. - Optimization of the Structural Performance of Non-homogeneous Partially Hinged Rectangular Plates. - Energy-Like Functional in a Quasilinear Parabolic Chemotaxis System. - Solvability of a Semilinear Heat Equation via a Quasi Scale Invariance. - Bounds for Sobolev Embedding Constants in Non-simply Connected Planar Domains. - Sharp Estimate of the Life Span of Solutions to the Heat Equation with a Nonlinear Boundary Condition. - Neutral Inclusions, Weakly Neutral Inclusions, and an Over-determined Problem for Confocal Ellipsoids. - Nonexistence of Radial Optimal Functions for the Sobolev Inequality on Cartan-Hadamard Manifolds. - Semiconvexity of Viscosity Solutions to Fully Nonlinear Evolution Equations via Discrete Games. - An Interpolating Inequality for Solutions of Uniformly Elliptic Equations. - Asymptotic Behavior of Solutions for a Fourth Order Parabolic Equation with Gradient Nonlinearity via the Galerkin Method. - A Note on Radial Solutions to the Critical Lane-Emden Equation with a Variable Coefficient. - Remark on One Dimensional Semilinear DampedWave Equation in a CriticalWeighted L2-space.

Read more less

Book SynopsisThis volume is part of collection of contributions devoted to analytical and experimental techniques of dynamical systems, presented at the 15th International Conference “Dynamical Systems: Theory and Applications”, held in Łódź, Poland on December 2-5, 2019. The wide selection of material has been divided into three volumes, each focusing on a different field of applications of dynamical systems.The broadly outlined focus of both the conference and these books includes bifurcations and chaos in dynamical systems, asymptotic methods in nonlinear dynamics, dynamics in life sciences and bioengineering, original numerical methods of vibration analysis, control in dynamical systems, optimization problems in applied sciences, stability of dynamical systems, experimental and industrial studies, vibrations of lumped and continuous systems, non-smooth systems, engineering systems and differential equations, mathematical approaches to dynamical systems, and mechatronics.Table of ContentsOn the spinning motion of a disc under the influence a gyrostatic moment (Gamiel).- Suppression of impact oscillations in a railway current collection system with an additional oscillatory system (Nishiyama).- On Qualitative Analysis of Lattice Dynamical System of Two- and Three-Dimensional Biopixels Array: Bifurcations and Transition to Chaos (Martsenyuk).- Response sensitivity of damper-connected adjacent structural systems subjected to fully non-stationary random excitations (Genovese).- Analysis of switching strategies for the optimization of periodic chemical reactions with controlled flow-rate (Zuyev).- Quaternion based free-floating space manipulator dynamics modeling using the dynamically equivalent manipulator approach (Jarzębowska).- Slosh analysis on a full car model with SDRE control and hydraulic damper (Balthazar).- A comparison of the common types of nonlinear energy sinks (Saeed).- Stability of three wheeled narrow vehicle (Weigel-Milleret).- Testing and analysis of vibrations of a tension transmission with a thermally sealed belt (Szymański).- Modeling and experimental tests on motion resistance of double-flange rollers of rubber track systems due to sliding friction between the rollers and guide lugs of rubber tracks (Chołodowski).- Structural dynamic response of the coupling between transmission lines and tower under random excitation (Machado).- Experimental assessment of the test station support structure rigidity by the vibration diagnostics method (Šmeringaiová).- Experimental dynamical analysis of a mechatronic analogy of the human circulatory system (Olejnik).- Robust design of inhibitory neuronal networks displaying rhythmic activity (Taylor).- Nonlinear dynamics of the industrial city's atmospheric ventilation: New differential equations model and chaos (Buyadzhi).- Biomechanical analysis of different foot morphology during standing on a dynamic support surface (Shu).- Comparison of various fractional order controllers on a poorly damped system (Birs).- Asymptotic analysis of submerged spring pendulum motion in liquid (Amer).- Parametric identification of non linear structures using Particle Swarm Optimization based on power flow balance criteria (Rajan).- Vibration and buckling of laminated plates of complex form under in-plane uniform and non-uniform loading (Linnik).- Dynamical systems and stability in fractional solid mechanics (Beda).- Stability of coupled systems of stochastic Cohen-Grossberg neural networks with time delays, impulses and Markovian switching (Tojtovska).- Stability of steady states with regular or chaotic behaviour in time (Mikhlin).- Modelling of torsional vibrations in a motorcycle steering system (Dębowski).- Free vibration frequencies of simply supported bars with variable cross section (Szlachetka).- On dynamics of a rigid block on visco-elastic foundation (Garziera).

Read more less

Book SynopsisThis textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning – the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series.The text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in infinite sequences and series. It is designed for the reader who has a good working knowledge of calculus. No additional prior knowledge is required.The text is divided into five chapters, which can be grouped into two parts: the first two chapters are concerned with the sequences and series of numbers, while the remaining three chapters are devoted to the sequences and series of functions, including the power series. Within each major topic, the exposition is inductive and starts with rather simple definitions and/or examples, becoming more compressed and sophisticated as the course progresses. Each key notion and result is illustrated with examples explained in detail. Some more complicated topics and results are marked as complements and can be omitted on a first reading.The text includes a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test the understanding of key concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory, or provide counterexamples to false propositions which seem to be natural at first glance. Solutions to additional problems proposed at the end of each chapter are provided as an electronic supplement to this book.Trade Review“The text contains a large number of problems and exercises, which should make it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and to test the understanding of concepts. … Many additional problems are proposed as homework tasks at the end of each chapter.” (Hüseyin Çakallı, zbMATH 1523.40001, 2023)Table of ContentsSequences of numbers.- Series of numbers.- Sequences of functions.- Series of functions.- Power series.

Read more less

Book SynopsisThis monograph presents recent developments in comparison geometry and geometric analysis on Finsler manifolds. Generalizing the weighted Ricci curvature into the Finsler setting, the author systematically derives the fundamental geometric and analytic inequalities in the Finsler context. Relying only upon knowledge of differentiable manifolds, this treatment offers an accessible entry point to Finsler geometry for readers new to the area. Divided into three parts, the book begins by establishing the fundamentals of Finsler geometry, including Jacobi fields and curvature tensors, variation formulas for arc length, and some classical comparison theorems. Part II goes on to introduce the weighted Ricci curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler manifolds. These tools allow the derivation of the Bochner–Weitzenböck formula and the corresponding Bochner inequality, gradient estimates, Bakry–Ledoux’s Gaussian isoperimetric inequality, and functional inequalities in the Finsler setting. Part III comprises advanced topics: a generalization of the classical Cheeger–Gromoll splitting theorem, the curvature-dimension condition, and the needle decomposition. Throughout, geometric descriptions illuminate the intuition behind the results, while exercises provide opportunities for active engagement. Comparison Finsler Geometry offers an ideal gateway to the study of Finsler manifolds for graduate students and researchers. Knowledge of differentiable manifold theory is assumed, along with the fundamentals of functional analysis. Familiarity with Riemannian geometry is not required, though readers with a background in the area will find their insights are readily transferrable.Trade Review“Finsler geometry is an active area of research in mathematics and has led to numerous real-world applications. This book is a comprehensive introduction to Finsler geometry and its applications. It covers the basic concepts of this geometry. More intuitively, this book provides an accessible introduction to recent developments in comparison geometry and geometric analysis on Finsler manifolds. … this book offers a valuable perspective for those familiar with comparison geometry and geometric analysis.” (Behroz Bidabad, Mathematical Reviews, May, 2023)Table of ContentsI Foundations of Finsler Geometry.- 1. Warm-up: Norms and inner products.- 2. Finsler manifolds.- 3. Properties of geodesics.- 4. Covariant derivatives.- 5. Curvature.- 6. Examples of Finsler manifolds.- 7. Variation formulas for arclength.- 8. Some comparison theorems.- II Geometry and analysis of weighted Ricci curvature.- 9. Weighted Ricci curvature.- 10. Examples of measured Finsler manifolds.- 11. The nonlinear Laplacian.- 12. The Bochner-Weitzenbock formula.- 13. Nonlinear heat flow.- 14. Gradient estimates.- 15. Bakry-Ledoux isoperimetric inequality.- 16. Functional inequalities.- III Further topics.- 17. Splitting theorems.- 18. Curvature-dimension condition.- 19. Needle decompositions.

Read more less

Book SynopsisThis book is designed to serve as a textbook for courses offered to undergraduate and graduate students enrolled in Mathematics. The first edition of this book was published in 2015. As there is a demand for the next edition, it is quite natural to take note of the several suggestions received from the users of the earlier edition over the past six years. This is the prime motivation for bringing out a revised second edition with a thorough revision of all the chapters. The book provides a clear understanding of the basic concepts of differential and integral calculus starting with the concepts of sequences and series of numbers, and also introduces slightly advanced topics such as sequences and series of functions, power series, and Fourier series which would be of use for other courses in mathematics for science and engineering programs. The salient features of the book are - precise definitions of basic concepts; several examples for understanding the concepts and for illustrating the results; includes proofs of theorems; exercises within the text; a large number of problems at the end of each chapter as home-assignments. The student-friendly approach of the exposition of the book would be of great use not only for students but also for the instructors. The detailed coverage and pedagogical tools make this an ideal textbook for students and researchers enrolled in a mathematics course. Table of ContentsSequence and Series of Real Numbers.- Limit, Continuity and Differentiability of Functions.- Definite Integral.- Improper Integrals.- Sequence and Series of Functions.- Fourier Series.- References.- Index.

Read more less

Book SynopsisThis text provides a detailed presentation of the main results for infinite products, as well as several applications. The target readership is a student familiar with the basics of real analysis of a single variable and a first course in complex analysis up to and including the calculus of residues. The book provides a detailed treatment of the main theoretical results and applications with a goal of providing the reader with a short introduction and motivation for present and future study. While the coverage does not include an exhaustive compilation of results, the reader will be armed with an understanding of infinite products within the course of more advanced studies, and, inspired by the sheer beauty of the mathematics. The book will serve as a reference for students of mathematics, physics and engineering, at the level of senior undergraduate or beginning graduate level, who want to know more about infinite products. It will also be of interest to instructors who teach courses that involve infinite products as well as mathematicians who wish to dive deeper into the subject. One could certainly design a special-topics class based on this book for undergraduates. The exercises give the reader a good opportunity to test their understanding of each section.Trade Review“This is an excellent textbook … . It must be very satisfactory for students to learn the subject from such a nicely written book.” (Marcel G. de Bruin, zbMATH 1491.40001, 2022)Table of ContentsPreface.- 1. Introduction.- 2. Infinite Products.- 3. The Gamma Function.- 4. Prime Numbers, Partitions and Products.- 5. Epilogue.- 6. Tables of Products.- References.

Read more less

Book SynopsisThis book convenes a collection of carefully selected problems in mathematical analysis, crafted to achieve maximum synergy between analytic geometry and algebra and favoring mathematical creativity in contrast to mere repetitive techniques. With eight chapters, this work guides the student through the basic principles of the subject, with a level of complexity that requires good use of imagination.In this work, all the fundamental concepts seen in a first-year Calculus course are covered. Problems touch on topics like inequalities, elementary point-set topology, limits of real-valued functions, differentiation, classical theorems of differential calculus (Rolle, Lagrange, Cauchy, and l’Hospital), graphs of functions, and Riemann integrals and antiderivatives. Every chapter starts with a theoretical background, in which relevant definitions and theorems are provided; then, related problems are presented. Formalism is kept at a minimum, and solutions can be found at the end of each chapter.Instructors and students of Mathematical Analysis, Calculus and Advanced Calculus aimed at first-year undergraduates in Mathematics, Physics and Engineering courses can greatly benefit from this book, which can also serve as a rich supplement to any traditional textbook on these subjects as well.Trade Review“This reviewer warmly welcomes the publication of this nice selection of problems and would recommend its use as a complimentary reading for analysis and calculus courses. … The book under review nicely complements existing literature on the subject; it is a stimulating reading requiring, as suggested by the authors, a bit of critical reflection.” (Svitlana P. Rogovchenko, zbMATH 1494.00003, 2022)“A sequence of a traditional textbook for an undergraduate calculus course. From this standpoint, this book can be used as a supplement to any conventional calculus text. … It is well known that students struggle with inducing abstract math concepts to gain insight into behavior in real-world phenomena to understand better those behaviors; including such examples is seen as a potential asset.” (Andrzej Sokolowski, MAA Reviews, August 1, 2022)Table of ContentsSummary of basic theory of inequalities.- Sets, sequences, functions.- Limits of functions, continuity.- Differentiation.- Classical theorems of differential calculus.- Monotonicity, concavity, minima, maxima, inflection points.- Graphs of functions.- Integrals.

Read more less

Book SynopsisConservation and balance laws on networks have been the subject of much research interest given their wide range of applications to real-world processes, particularly traffic flow. This open access monograph is the first to investigate different types of control problems for conservation laws that arise in the modeling of vehicular traffic. Four types of control problems are discussed - boundary, decentralized, distributed, and Lagrangian control - corresponding to, respectively, entrance points and tolls, traffic signals at junctions, variable speed limits, and the use of autonomy and communication. Because conservation laws are strictly connected to Hamilton-Jacobi equations, control of the latter is also considered. An appendix reviewing the general theory of initial-boundary value problems for balance laws is included, as well as an appendix illustrating the main concepts in the theory of conservation laws on networks. Table of ContentsIntroduction.- Boundary Control.- Decentralized Control.- Distributed Control.- Lagrangian Control.- Hamilton-Jacobi Equations.- Appendix A: Balance Laws with Boundary.- Conservation Laws on Networks.

Read more less

Book SynopsisThe purpose of this book is to present the current state of the art of the Virtual Element Method (VEM) by collecting contributions from many of the most active researchers in this field and covering a broad range of topics: from the mathematical foundation to real life computational applications. The book is naturally divided into three parts. The first part of the book presents recent advances in theoretical and computational aspects of VEMs, discussing the generality of the meshes suitable to the VEM, the implementation of the VEM for linear and nonlinear PDEs, and the construction of discrete hessian complexes. The second part of the volume discusses Virtual Element discretization of paradigmatic linear and non-linear partial differential problems from computational mechanics, fluid dynamics, and wave propagation phenomena. Finally, the third part contains challenging applications such as the modeling of materials with fractures, magneto-hydrodynamics phenomena and contact solid mechanics.The book is intended for graduate students and researchers in mathematics and engineering fields, interested in learning novel numerical techniques for the solution of partial differential equations. It may as well serve as useful reference material for numerical analysts practitioners of the field.Table of Contents1 Tommaso Sorgente et al., VEM and the Mesh.- 2 Dibyendu Adak et al., On the implementation of Virtual Element Method for Nonlinear problems over polygonal meshes.- 3 Long Chen and Xuehai Huang, Discrete Hessian Complexes in Three Dimensions.- 4 Edoardo Artioli et al., Some Virtual Element Methods for Infinitesimal Elasticity Problems.- 5 Lourenço Beirão da Veiga and Giuseppe Vacca, An introduction to second order divergence-free VEM for fluidodynamics.- 6 Gabriel N. Gatica et al, A virtual marriage à la mode: some recent results on the coupling of VEM and BEM.- 7 Daniele Boffi et al., Virtual element approximation of eigenvalue problems.- 8 David Mora and Alberth Silgado, Virtual element methods for a stream-function formulation of the Oseen equations.- 9 Lorenzo Mascotto et al., The nonconforming Trefftz virtual element method: general setting, applications, and dispersion analysis for the Helmholtz equation.- 10 Paola F. Antonietti et al., The conforming virtual element method for polyharmonic and elastodynamics problems: a review.- 11 Edoardo Artioli et al., The virtual element method in nonlinear and fracture solid mechanics.- 12 Sebastián Naranjo Álvarez et al., The virtual element method for the coupled system of magneto-hydrodynamics.- 13 Peter Wriggers et al., Virtual Element Methods for Engineering Applications.

Read more less

Book SynopsisThis monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems – as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis – will find this text to be a valuable addition to the mathematical literature.Table of ContentsIntroduction.- Geometric Measure Theory.- Calderon-Zygmund Theory for Boundary Layers in UR Domains.- Boundedness and Invertibility of Layer Potential Operators.- Controlling the BMO Semi-Norm of the Unit Normal.- Boundary Value Problems in Muckenhoupt Weighted Spaces.- Singular Integrals and Boundary Problems in Morrey and Block Spaces.- Singular Integrals and Boundary Problems in Weighted Banach Function Spaces.

Read more less

Book SynopsisThis textbook describes selected topics in functional analysis as powerful tools of immediate use in many fields within applied mathematics, physics and engineering. It follows a very reader-friendly structure, with the presentation and the level of exposition especially tailored to those who need functional analysis but don’t have a strong background in this branch of mathematics. For every tool, this work emphasizes the motivation, the justification for the choices made, and the right way to employ the techniques. Proofs appear only when necessary for the safe use of the results. The book gently starts with a road map to guide reading. A subsequent chapter recalls definitions and notation for abstract spaces and some function spaces, while Chapter 3 enters dual spaces. Tools from Chapters 2 and 3 find use in Chapter 4, which introduces distributions. The Linear Functional Analysis basic triplet makes up Chapter 5, followed by Chapter 6, which introduces the concept of compactness. Chapter 7 brings a generalization of the concept of derivative for functions defined in normed spaces, while Chapter 8 discusses basic results about Hilbert spaces that are paramount to numerical approximations. The last chapter brings remarks to recent bibliographical items. Elementary examples included throughout the chapters foster understanding and self-study. By making key, complex topics more accessible, this book serves as a valuable resource for researchers, students, and practitioners alike that need to rely on solid functional analysis but don’t need to delve deep into the underlying theory.Table of ContentsRoad Map.- Basic Concepts.- Dual of a Normed Space.- Sobolev Spaces, Distributions.- The Three Basic Principles.- Compactness.- Function Derivatives in Normed Spaces.- Hilbert Bases and Approximations.

Read more less

Book SynopsisThe use of difference matrices and high-level MATLAB® commands to implement finite difference algorithms is pedagogically novel. This unique and concise textbook gives the reader easy access and a general ability to use first and second difference matrices to set up and solve linear and nonlinear systems in MATLAB which approximate ordinary and partial differential equations. Prerequisites include a knowledge of basic calculus, linear algebra, and ordinary differential equations. Some knowledge of partial differential equations is a plus though the text may easily serve as a supplement for the student currently working through an introductory PDEs course. Familiarity with MATLAB is not required though a little prior experience with programming would be helpful. In addition to its special focus on solving in MATLAB, the abundance of examples and exercises make this text versatile in use. It would serve well in a graduate course in introductory scientific computing for partial differential equations. With prerequisites mentioned above plus some elementary numerical analysis, most of the material can be covered and many of the exercises assigned in a single semester course. Some of the more challenging exercises make substantial projects and relate to topics from other typical graduate mathematics courses, e.g., linear algebra, differential equations, or topics in nonlinear functional analysis. A selection of the exercises may be assigned as projects throughout the semester. The student will develop the skills to run simulations corresponding to the primarily theoretical course material covered by the instructor. The book can serve as a supplement for the instructor teaching any course in differential equations. Many of the examples can be easily implemented and the resulting simulation demonstrated by the instructor. If the course has a numerical component, a few of the more difficult exercises may be assigned as student projects. Established researchers in theoretical partial differential equations may find this book useful as well, particularly as an introductory guide for their research students. Those unfamiliar with MATLAB can use the material as a reference to quickly develop their own applications in that language. Practical assistance in implementing algorithms in MATLAB can be found in these pages. A mathematician who is new to the practical implementation of methods for scientific computation in general can learn how to implement and execute numerical simulations of differential equations in MATLAB with relative ease by working through a selection of exercises. Additionally, the book can serve as a practical guide in independent study, undergraduate or graduate research experiences, or for reference in simulating solutions to specific thesis or dissertation-related experiments.Table of Contents1. Introduction.- 2. Review of elementary numerical methods and MATLAB(R).- 3. Ordinary Differential Equations.- 4. Partial Differential Equations.- 5. Advanced topics in semilinear elliptic BVP.- References.

Read more less

Book SynopsisThis textbook provides an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, readers are introduced to techniques to obtain exact solutions of NLPDEs. The chapters include the following topics: Nonlinear PDEs are Everywhere; Differential Substitutions; Point and Contact Transformations; First Integrals; and Functional Separability. Readers are guided through these chapters and are provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. This Second Edition includes a new method of generating contact transformations and focuses on a solution method (parametric Legendre transformations) to solve a particular class of two nonlinear PDEs.Table of ContentsNonlinear PDEs are Everywhere.- Differential Substitutions.- Point and Contact Transformations.- First Integrals.- Functional Separability.

Read more less

Book SynopsisThe book addresses the rigorous foundations of mathematical analysis. The first part presents a complete discussion of the fundamental topics: a review of naive set theory, the structure of real numbers, the topology of R, sequences, series, limits, differentiation and integration according to Riemann. The second part provides a more mature return to these topics: a possible axiomatization of set theory, an introduction to general topology with a particular attention to convergence in abstract spaces, a construction of the abstract Lebesgue integral in the spirit of Daniell, and the discussion of differentiation in normed linear spaces. The book can be used for graduate courses in real and abstract analysis and can also be useful as a self-study for students who begin a Ph.D. program in Analysis. The first part of the book may also be suggested as a second reading for undergraduate students with a strong interest in mathematical analysis.Table of ContentsPart I First half of the journey.- 1 An appetizer of propositional logic.- 2 Sets, relations, functions in a naïve way.- 3 Numbers.- 4 Elementary cardinality.- 5 Distance, topology and sequences on the set of real numbers.- 6 Series.- 7 Limits: from sequences to functions of a real variable.- 8 Continuous functions of a real variable.- 9 Derivatives and differentiability- 10 Riemann’s integral.- 11 Elementary functions.- Part II Second half of the journey.- 12 Return to Set Theory.- 13 Neighbors again: topological spaces.- 14 Differentiating again: linearization in normed spaces.- 15 A functional approach to Lebesgue integration theory.- 16 Measures before integrals.

Read more less

Book SynopsisThe book is devoted to the qualitative study of differential equations defined by piecewise linear (PWL) vector fields, mainly continuous, and presenting two or three regions of linearity. The study focuses on the more common bifurcations that PWL differential systems can undergo, with emphasis on those leading to limit cycles. Similarities and differences with respect to their smooth counterparts are considered and highlighted. Regarding the dimensionality of the addressed problems, some general results in arbitrary dimensions are included. The manuscript mainly addresses specific aspects in PWL differential systems of dimensions 2 and 3, which are sufficinet for the analysis of basic electronic oscillators.The work is divided into three parts. The first part motivates the study of PWL differential systems as the natural next step towards dynamic complexity when starting from linear differential systems. The nomenclature and some general results for PWL systems in arbitrary dimensions are introduced. In particular, a minimal representation of PWL systems, called canonical form, is presented, as well as the closing equations, which are fundamental tools for the subsequent study of periodic orbits.The second part contains some results on PWL systems in dimension 2, both continuous and discontinuous, and both with two or three regions of linearity. In particular, the focus-center-limit cycle bifurcation and the Hopf-like bifurcation are completely described. The results obtained are then applied to the study of different electronic devices.In the third part, several results on PWL differential systems in dimension 3 are presented. In particular, the focus-center-limit cycle bifurcation is studied in systems with two and three linear regions, in the latter case with symmetry. Finally, the piecewise linear version of the Hopf-pitchfork bifurcation is introduced. The analysis also includes the study of degenerate situations. Again, the above results are applied to the study of different electronic oscillators.Table of ContentsPart I: Introduction1 From linear to piecewise linear differential systems1.1 Some caveats about the notation used in the book1.2 A short review on linear systems in R21.2.1 Real and distinct eigenvalues1.2.2 Complex eigenvalues1.2.3 Nonzero double eigenvalues1.2.4 A canonical form for affine systems1.2.5 The case of vanishing determinant1.3 Degenerate bifurcations in planar affine systems1.4 The one-parameter Li´enard form1.5 Computing the exponential matrix1.6 Passing from linear to piecewise linear systems1.7 Limit cycles in a continuous piecewise linear worked example1.7.1 The right half-return map1.7.2 The left half-return map1.7.3 A bifurcation analysis2 Preliminary results2.1 A unified Li´enard form for continuous planar piecewise linear systems2.2 Canonical forms for Lur´e systems in higher dimension2.3 Some generic results about equilibria2.3.1 Observable continuous piecewise linear systems with two zones2.3.2 Observable symmetric continuous PWL systems with three zones2.4 Analysis of periodic orbits through their closing equations2.4.1 Observable continuous piecewise linear systems with two zones2.4.2 Symmetric continuous PWL systems with three zones2.5 Periodic orbits and Poincar´e maps in piecewise linear systems2.5.1 Derivatives of transition maps2.5.2 Continuous piecewise linear systems with two zones2.5.3 Continuous piecewise linear systems with three zonesPart II: Planar piecewise linear differential systems3 Continuousplanar systemswithtwo zones3.1 Equilibria in continuous planar piecewise linear systems with two zones3.2 Some preliminary results on limit cycles3.3 The Massera’s method for uniqueness of limit cycles3.4 General results about limit cycles3.5 Refracting Systems3.6 The bizonal focus-center-limit cycle bifurcation4 Continuousplanar systemswiththree zones4.1 Limit cycle existence and uniqueness4.2 The focus-center-limit cycle bifurcation for symmetric systems5 Boundary equilibriumbifurcations and limit cycles5.1 Boundary Equilibrium Bifurcations in systems with two zones5.2 Boundary Equilibrium Bifurcations in systems with three zones5.3 Analysis of Wien bridge oscillators6 Algebraically computable continuousPWLnodal oscillators6.1 Preliminary results6.2 Analysis of equilibria and periodic orbits6.3 Application to a piecewise linear van der Pol oscillator7 The focus-saddle boundary bifurcation7.1 Preliminary results7.2 Main results7.3 Application to the analysis of memristor oscillatorsPart III: Three-dimensional piecewise linear differential systems8 The FCLC bifurcation in 3D systems with 2 zones8.1 The generic focus-center-limit cycle bifurcation8.2 The onset of asymmetric oscillations in Chua’s circuit8.3 The degenerate FCLC bifurcation in systems with 2 zones8.4 The degenerate FCLC bifurcation in Chua’s circuit9 The FCLC bifurcation in 3D symmetric PWL systems9.1 The symmetric focus-center-limit cycle bifurcation9.2 The degeneration of the FCLC bifurcation in symmetric systems10 The analogue of Hopf-pitchfork bifurcation10.1 A one-parameter bifurcation analysis10.2 The degenerate PWL Hopf-pitchfork bifurcation10.3 The Hopf-pitchfork bifurcation in Chua’s circuit10.4 An extended Bonhoeffer–van der Pol oscillator11 AfterwordAppendicesA The piecewise linear characteristics of Chua’s diodeB The Chua’s oscillatorC Auxiliary resultsBibliography

Read more less

Book SynopsisThis concise textbook introduces calculus students to power series through an informal and captivating narrative that avoids formal proofs but emphasizes understanding the fundamental ideas. Power series—and infinite series in general—are a fundamental tool of pure and applied mathematics. The problems focus on ideas, applications, and creative thinking instead of being repetitive and procedural. Calculus is about functions, so the book turns on two fundamental ideas: using polynomials to approximate a function and representing a function in terms of simpler functions. The derivative is reinterpreted in terms of linear approximations, which then leads to Taylor polynomials and the question of convergence. Enough of the theory of convergence is developed to allow a more complete understanding of power series and their applications. A final chapter looks at the distant horizon and discusses other kinds of series representations. SageMath, a free open-source mathematics software system, is used throughout to do computations, provide examples, and create many graphs. While most problems do not require SageMath, students are encouraged to use it where appropriate. An instructor’s guide with solutions to all the problems is available. The book is intended as a supplementary textbook for calculus courses; lecturers and instructors will find innovative and engaging ways to teach this topic. The informal and conversational tone make the book useful to any student seeking to understand this essential aspect of analysis.Table of Contents- To the reader.- Getting close with lines.- Getting closer with polynomials.- Going all the way: Convergence.- Power series.- Distant mountains.- Appendix A: SageMath: A (very) short introduction.- Appendix B: Why I do it this way.- Bibliography.

Read more less

Book SynopsisThe monograph is devoted to the use of the moduli method in mapping theory, in particular, the meaning of direct and inverse modulus inequalities and their possible applications. The main goal is the development of a modulus technique in the Euclidean space and some metric spaces (manifolds, surfaces, quotient spaces, etc.). Particular attention is paid to the local and boundary behavior of mappings, as well as to obtaining modulus inequalities for some classes. The reader is invited to familiarize himself with all the main achievements of the author, synthesized in this book. The results presented here are of a high scientific level, are new and have no analogues in the world with such a degree of generality.Table of ContentsGeneral definitions and notation.- Boundary behavior of mappings with Poletsky inequality.- Removability of singularities of generalized quasiisometries.- Normal families of generalized quasiisometries.- On boundary behavior of mappings with Poletsky inequality in terms of prime ends.- Local and boundary behavior of mappings on Riemannian manifolds.- Local and boundary behavior of maps in metric spaces.- On Sokhotski-Casorati-Weierstrass theorem on metric spaces.- On boundary extension of mappings in metric spaces in the terms of prime ends.- On the openness and discreteness of mappings with the inverse Poletsky inequality.- Equicontinuity and isolated singularities of mappings with the inverse Poletsky inequality.- Equicontinuity of families of mappings with the inverse Poletsky inequality in terms of prime ends.- Logarithmic H¨older continuous mappings and Beltrami equation.- On logarithmic H¨older continuity of mappings on the boundary.- The Poletsky and V¨ais¨al¨a inequalities for the mappings with (p;q)-distortion.- An analog of the V¨ais¨al¨a inequality for surfaces.- Modular inequalities on Riemannian surfaces.- On the local and boundary behavior of mappings of factor spaces.- References.- Index.

Read more less

Book SynopsisOrdinary Differential Equations.- Linear Systems and Stability of Nonlinear Systems.- Applications.- Hyperbolic Theory.- Continuation of Periodic Solutions.- Homoclinic Orbits, Melnikov's Method, and Chaos.- Averaging.- Local Bifurcation.

Read more less