Complex analysis, complex variables Books

Book SynopsisThis book is devoted to the phenomenon of synchronization and its application for determining the values of Lyapunov exponents. In recent years, the idea of synchronization has become an object of great interest in many areas of science, e.g., biology, communication or laser physics. Over the last decade, new types of synchronization have been identified and some interesting new ideas concerning the synchronization have also appeared.This book presents the complete synchronization problem rather than just results from the research. The problem is demonstrated in relation to a kind of coupling applied between dynamical systems, whereby a unique classification of possible couplings is introduced. Another novel feature is the connection presented between synchronization and the problem of determining the Lyapunov exponents, especially for non-differentiable systems. A detailed proposal of such an estimation method and examples of its application are included.Table of ContentsClassification of Couplings between Dynamical Systems: Diffusive and Non-diffusive, Dissipative and Conservative, Common Signal; Methods of Determining the Complete Synchronization Thresholds in Networks of Coupled Oscillators; Synchronizability of Coupled Oscillators: Single and Multiple Synchronous Ranges; Estimation of the Largest Lyapunov Exponent via Complete Synchronization: Discrete-Time Systems, Mechanical Oscillators with Discontinuities, Systems with Time Delay.

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Book SynopsisThis heavily illustrated book collects in one source most of the mathematically simple systems of differential equations whose solutions are chaotic. It includes the historically important systems of van der Pol, Duffing, Ueda, Lorenz, Rössler, and many others, but it goes on to show that there are many other systems that are simpler and more elegant. Many of these systems have been only recently discovered and are not widely known. Most cases include plots of the attractor and calculations of the spectra of Lyapunov exponents. Some important cases include graphs showing the route to chaos. The book includes many cases not previously published as well as examples of simple electronic circuits that exhibit chaos.No existing book thus far focuses on mathematically elegant chaotic systems. This book should therefore be of interest to chaos researchers looking for simple systems to use in their studies, to instructors who want examples to teach and motivate students, and to students doing independent study.Table of ContentsFundamentals; Periodically Forced Systems; Autonomous Dissipative Systems; Autonomous Conservative Systems; Low-Dimensional Systems (D < 3); High-Dimensional Systems (D > 3); Infinite-Dimensional Systems; Chaotic Electrical Circuits.

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Book SynopsisSince the birth of the Chua circuit in 1983, a considerable number of fruitful, fascinating and relevant research topics have arisen. In honor of the 25th anniversary of the invention of Chua's circuit, this book presents the 25 years of research on the implementation of Chua's circuit, and also discusses future directions and emerging applications of recent results.The purpose of the book is to provide researchers, PhD students, and undergraduate students a research monograph containing both fundamentals on the topics and advanced results that have been recently obtained. With about 60 illustrations included in the book, it also shows the detailed schematics of several different implementations that can be easily reproduced with a low-cost experimental setup and PC-based measurement instrumentation.

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Book SynopsisThe aim of the book is to give a smooth analytic continuation from calculus to complex analysis by way of plenty of practical examples and worked-out exercises. The scope ranges from applications in calculus to complex analysis in two different levels.If the reader is in a hurry, he can browse the quickest introduction to complex analysis at the beginning of Chapter 1, which explains the very basics of the theory in an extremely user-friendly way. Those who want to do self-study on complex analysis can concentrate on Chapter 1 in which the two mainstreams of the theory — the power series method due to Weierstrass and the integration method due to Cauchy — are presented in a very concrete way with rich examples. Readers who want to learn more about applied calculus can refer to Chapter 2, where numerous practical applications are provided. They will master the art of problem solving by following the step by step guidance given in the worked-out examples.This book helps the reader to acquire fundamental skills of understanding complex analysis and its applications. It also gives a smooth introduction to Fourier analysis as well as a quick prelude to thermodynamics and fluid mechanics, information theory, and control theory. One of the main features of the book is that it presents different approaches to the same topic that aids the reader to gain a deeper understanding of the subject.

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Book SynopsisThe aim of the book is to give a smooth analytic continuation from calculus to complex analysis by way of plenty of practical examples and worked-out exercises. The scope ranges from applications in calculus to complex analysis in two different levels.If the reader is in a hurry, he can browse the quickest introduction to complex analysis at the beginning of Chapter 1, which explains the very basics of the theory in an extremely user-friendly way. Those who want to do self-study on complex analysis can concentrate on Chapter 1 in which the two mainstreams of the theory — the power series method due to Weierstrass and the integration method due to Cauchy — are presented in a very concrete way with rich examples. Readers who want to learn more about applied calculus can refer to Chapter 2, where numerous practical applications are provided. They will master the art of problem solving by following the step by step guidance given in the worked-out examples.This book helps the reader to acquire fundamental skills of understanding complex analysis and its applications. It also gives a smooth introduction to Fourier analysis as well as a quick prelude to thermodynamics and fluid mechanics, information theory, and control theory. One of the main features of the book is that it presents different approaches to the same topic that aids the reader to gain a deeper understanding of the subject.

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Book SynopsisThis volume contains short courses and recent papers by several specialists in different fields of Mathematical Analysis. It offers a wide perspective of the current state of research, and new trends, in areas related to Geometric Analysis, Harmonic Analysis, Complex Analysis, Functional Analysis and History of Mathematics. The contributions are presented with a remarkable expository nature and this makes the discussed topics accessible to a more general audience.

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Book SynopsisThis heavily illustrated book collects in one source most of the mathematically simple systems of differential equations whose solutions are chaotic. It includes the historically important systems of van der Pol, Duffing, Ueda, Lorenz, Rössler, and many others, but it goes on to show that there are many other systems that are simpler and more elegant. Many of these systems have been only recently discovered and are not widely known. Most cases include plots of the attractor and calculations of the spectra of Lyapunov exponents. Some important cases include graphs showing the route to chaos. The book includes many cases not previously published as well as examples of simple electronic circuits that exhibit chaos.No existing book thus far focuses on mathematically elegant chaotic systems. This book should therefore be of interest to chaos researchers looking for simple systems to use in their studies, to instructors who want examples to teach and motivate students, and to students doing independent study.

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Book SynopsisChaos: from simple models to complex systems aims to guide science and engineering students through chaos and nonlinear dynamics from classical examples to the most recent fields of research. The first part, intended for undergraduate and graduate students, is a gentle and self-contained introduction to the concepts and main tools for the characterization of deterministic chaotic systems, with emphasis to statistical approaches. The second part can be used as a reference by researchers as it focuses on more advanced topics including the characterization of chaos with tools of information theory and applications encompassing fluid and celestial mechanics, chemistry and biology.The book is novel in devoting attention to a few topics often overlooked in introductory textbooks and which are usually found only in advanced surveys such as: information and algorithmic complexity theory applied to chaos and generalization of Lyapunov exponents to account for spatiotemporal and non-infinitesimal perturbations.The selection of topics, numerous illustrations, exercises and proposals for computer experiments make the book ideal for both introductory and advanced courses.

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Book SynopsisClassical Complex Analysis, available in two volumes, provides a clear, broad and solid introduction to one of the remarkable branches of exact science, with an emphasis on the geometric aspects of analytic functions. Volume 2 begins with analytic continuation. The Riemann mapping theorem is proved and used in solving Dirichlet's problem for an open disk and, hence, a class of general domains via Perron's method. Finally, proof of the uniformization theorem of Riemann surfaces is given.The book is rich in contents, figures, examples and exercises. It is self-contained and is designed for a variety of usages and motivations concerning advanced studies. It can be used both as a textbook for undergraduate and graduate students, and as a reference book in general.Table of ContentsFundamental Theory: Sequences, Series and Infinite Products; Conformal Mapping and Dirichlet's Problem; Riemann Surfaces.

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Book SynopsisChaos: from simple models to complex systems aims to guide science and engineering students through chaos and nonlinear dynamics from classical examples to the most recent fields of research. The first part, intended for undergraduate and graduate students, is a gentle and self-contained introduction to the concepts and main tools for the characterization of deterministic chaotic systems, with emphasis to statistical approaches. The second part can be used as a reference by researchers as it focuses on more advanced topics including the characterization of chaos with tools of information theory and applications encompassing fluid and celestial mechanics, chemistry and biology.The book is novel in devoting attention to a few topics often overlooked in introductory textbooks and which are usually found only in advanced surveys such as: information and algorithmic complexity theory applied to chaos and generalization of Lyapunov exponents to account for spatiotemporal and non-infinitesimal perturbations.The selection of topics, numerous illustrations, exercises and proposals for computer experiments make the book ideal for both introductory and advanced courses.Table of ContentsFirst Encounter with Chaos; The Language of Dynamical Systems; Examples of Chaotic Behaviors; Probabilistic Approach to Chaos; Characterization of Chaotic Dynamical Systems; From Order to Chaos in Dissipative Systems; Chaos in Hamiltonian Systems; Chaos and Information Theory; Coarse-Grained Information and Large Scale Predictability; Chaos, Numerical Computations and Experiments; Chaos in Few-Degrees of Freedom Systems; Spatiotemporal Chaos; Turbulence as a Dynamical Systems Problem; Chaos and Statistical Mechanics: Fermi-Pasta-Ulam a Case Study.

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Book SynopsisThis book contains the contributions by the participants in the nine of a series of workshops. Throughout the series of workshops, the contributors are consistently aiming at higher achievements of studies of the current topics in complex analysis, differential geometry and mathematical physics and further in any intermediate areas, with expectation of discovery of new research directions. Concerning the present one, it is worthwhile to mention that, in addition to the new developments of the traditional trends, many attractive and pioneering works were presented and their results were contributed to the present volume. The contents of this volume therefore will provide not only significant and useful information for researchers in complex analysis, differential geometry and mathematical physics (including their related areas), but also interesting mathematics for non-specialists and a broad audience. The present volume contains new developments and trends in the studies on constructions of holomorphic Cliffordian functions; the swelling constructions of minimal surfaces with higher genus in flat tori; the spectral properties of soliton equations on symmetric spaces; new types of shallow water waves described by Camassa-Holm type equations, the properties of pseudo-hermitian boson and fermion coherent states; fractals and chaos on orthorhombic lattices, and even an ambitious proposal of a graph model for Kaehler manifolds with Kaehler magnetic fields.Table of ContentsA Discrete Model for Kaehler Magnetic Fields on a Complex Hyperbolic Space; A Characterization of Clifford Minimal Hypersurfaces of a Sphere in Terms of Their Geodesics; Hyperbolic Gauss Maps and Parallel Surfaces in Hyperbolic Three-Space; Generalizations of Conjugate Connections; Heisenberg Relations in the General Case; A Short Note on the Double-Complex Laplace Operator; Monogenic, Hypermonogenic and Holomorphic; Cliffordian Functions - A Survey; On Multicomponent Evolution Equations on Symmetric Spaces with Constant Boundary Conditions; Relativistic Strain and Electromagnetic Photon-Like Objects; Cyclic Hyper-Scalar Systems; Poisson Structures of Equations Associated with Groups of Diffeomorphisms; On the Lax Pair for Two and Three Wave Interaction System; New Integrable Equations of MKdV Type; Persistence of Solutions for Some Integrable Shallow Water Equations; Mathematical Outlook of Fractals and Chaos Related to Simple Orthorhombic Ising-Onsager-Zhang Lattices.

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Book SynopsisThe monograph, as its first main goal, aims to study the overconvergence phenomenon of important classes of Bernstein-type operators of one or several complex variables, that is, to extend their quantitative convergence properties to larger sets in the complex plane rather than the real intervals. The operators studied are of the following types: Bernstein, Bernstein—Faber, Bernstein-Butzer, q-Bernstein, Bernstein-Stancu, Bernstein-Kantorovich, Favard-Szász-Mirakjan, Baskakov and Balázs-Szabados.The second main objective is to provide a study of the approximation and geometric properties of several types of complex convolutions: the de la Vallée Poussin, Fejér, Riesz-Zygmund, Jackson, Rogosinski, Picard, Poisson-Cauchy, Gauss-Weierstrass, q-Picard, q-Gauss-Weierstrass, Post-Widder, rotation-invariant, Sikkema and nonlinear. Several applications to partial differential equations (PDEs) are also presented.Many of the open problems encountered in the studies are proposed at the end of each chapter. For further research, the monograph suggests and advocates similar studies for other complex Bernstein-type operators, and for other linear and nonlinear convolutions.Table of ContentsBernstein-Type Operators of One Complex Variable: Auxiliary Results in Complex Analysis; Bernstein Polynomials; Iterates of Bernstein Polynomials; Generalized Voronovskaja Theorems for Bernstein Polynomials; Butzer's Linear Combination of Bernstein Polynomials; q-Bernstein Polynomials; Bernstein-Stancu Polynomials; Bernstein-Kantorovich Type Polynomials; Favard-Szasz-Mirakjan Operators; Baskakov Operators; Balazs-Szabados Operators; Bibliographical Notes and Open Problems; Bernstein-Type Operators of Several Complex Variables: Introduction; Bernstein Polynomials; Favard-Szasz-Mirakjan Operators; Baskakov Operators; Bibliographical Notes and Open Problems; Complex Convolutions: Linear Polynomial Convolutions; Linear Non-Polynomial Convolutions; Nonlinear Complex Convolutions; Bibliographical Notes and Open Problems; Appendices: Related Topics: Bernstein Polynomials of Quaternion Variable; Approximation of Vector-Valued Functions; Strong Approximation by Complex Taylor Series; Bibliographical Notes and Open Problems.

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Book SynopsisThe subject matter in this volume is Schwarz's Lemma which has become a crucial theme in many branches of research in mathematics for more than a hundred years to date. This volume of lecture notes focuses on its differential geometric developments by several excellent authors including, but not limited to, L Ahlfors, S S Chern, Y C Lu, S T Yau and H L Royden.This volume can be approached by a reader who has basic knowledge on complex analysis and Riemannian geometry. It contains major historic differential geometric generalizations on Schwarz's Lemma and provides the necessary information while making the whole volume as concise as ever.Table of ContentsMean-Value Properties; Maximum Principles; A Quick Introduction to Hermitian Geometry; Classical Schwarz's Lemma; Poincare' Distance and Metric; General Schwarz's Lemma by Ahlfors, Chern-Lu, Yau, Royden and Others; Almost Maximum Principle; Chern-Lu Formula; Very Recent Developments on Differential Geometric Schwarz's Lemma.

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Book SynopsisThis proceedings is a collection of articles by front-line researchers in Mathematical Analysis, giving the reader a wide perspective of the current research in several areas like Functional Analysis, Complex Analysis and Measure Theory.The works are a fundamental source for current and future developments in these research fields. The articles and surveys have been collected as well as reference results scattered in the corresponding literature and thus, are highly useful to researchers.Table of ContentsWeyl Type Theorems in the Theory of Bounded Linear Operators; Forward Shifts Between Spaces of Continuous Functions; Algebras of Holomorphic Functions; Finitely Additive Measures; Linear Preserver Problems; Diameter Preserving Mappings; The E Property and Many Other Geometric Properties in Banach Spaces.

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Book SynopsisThe book provides a comprehensive account of particle physics linking various aspects of particle physics in a coherent manner. This self-contained book not only cover basic concepts and recent developments but also overlaps between Astrophysics, Cosmology and Particle Physics, known as astroparticle physics. Several appendices are included to make the book self-contained.Table of ContentsIntroduction; Scattering and Particle Interactions; Space-Time Symmetries; Internal Symmetries; Unitary Groups and SU(3); SU(6) and Quark Model; Colors, Gauge Principle and Quantum Chromodynamics; Heavy Flavor Physics; Heavy Quark Effective Theory; Neutrinos; Weak Interactions; Properties of Weak Hadronic Currents and Chiral Symmetry; Electroweak Unification; Deep Inelastic Scattering; Weak Decays of Heavy Flavor; Particle Mixing and CP-Violation; Grand Unification; Supersymmetry and Strings; Astroparticle Physics.

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Book SynopsisThe scope of this book is the field of evolutionary genetics. The book contains new methods for simulating evolution at the genomic level. It sets out applications using up to date Monte Carlo simulation methods applied in classical population genetics, and sets out new fields of quantifying mutation and selection at the Mendelian level. A serious limitation of Wright-Fisher process, the assumption that population size is constant, motivated the introduction of self regulating branching processes in this book. While providing a short review of the principles of probability and its application and using computer intensive methods whilst applying these principles, this book explains how it is possible to derive new formulas expressed in terms of matrix algebra providing new insights into the classical Wright-Fisher processes of evolutionary genetics. Also covered are the development of new methods for studying genetics and evolution, simulating nucleotide substitutions of a DNA molecule and on self regulating branching processes. Components of natural selection are studied in terms of reproductive success of each genotype whilst also studying the differential ability of genotypes to compete for resources and sexual selection. The concept of the gene is also reviewed in this book, and it provides a current definition of a gene based on very recent experiments with micro-array technologies. A development of stochastic models for simulating the evolution of model genomes concludes the studies in this book. Deserving of a place on the book shelves of workers in biomathematics, applied probability, stochastic processes and statistics, as well as in bioinformatics and phylogenetics, it will also be relevant to those interested in computer simulation, and evolutionary biologists interested in quantitative methods.Table of ContentsIntroduction of Mathematical Probability; Linkage and Recombination at Multiple Linker Loci; Nucleotide Substitution Models Formulated as Markov Processes in Continuous Time; Genealogies, Coalescence and Self Regulating Branching Processes; Two Sex Multi-type Self Regulating Branching Processes in Evolutionary Genetics; Selected Topics from Molecular Genetics; Detecting Genomic Signal of Selection; Suggestions of Further Research, Reading and Viewing; and other papers.

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Book SynopsisThis volume contains contributions of principal speakers of the symposium on geometry and analysis of automorphic forms of several variables, held in September 2009 at Tokyo, Japan, in honor of Takayuki Oda's 60th birthday. It presents both research and survey articles in the fields that are the main themes of his work. The volume may serve as a guide to developing areas as well as a resource for researchers who seek a broader view and for students who are beginning to explore automorphic form.Table of ContentsAutomorphic Form; Birch and Swinnerton-Dyer Conjecture; Borcherds Lift; Discrete Series; Fourier Coefficient; Fourier Transform; Hermite's Constant; Newton Polygon; Oda's Period Relation; Orbital Integral; p-Adic Group; p-Divisible Group; Real Symplectic Group; Siegel Modular Form; Siegel Modular Variety; Siegel Series; Spherical Function; Voronoi's Theorem; Whittaker Function.

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Book SynopsisThis review volume, co-edited by Nobel laureate G Ertl, provides a broad overview on current studies in the understanding of design and control of complex chemical systems of various origins, on scales ranging from single molecules and nano-phenomena to macroscopic chemical reactors. Self-organizational behavior and the emergence of coherent collective dynamics in reaction-diffusion systems, reactive soft matter and chemical networks are covered. Special attention is paid to the applications in molecular cell biology and to the problems of biological evolution, synthetic biology and design of artificial living cells. Starting with a detailed introduction on the history of research on complex chemical systems, its current state of the art and perspectives, the book comprises 19 chapters that survey the current progress in particular research fields. The reviews, prepared by leading international experts, yield together a fascinating picture of a rapidly developing research discipline that brings chemical engineering to new frontiers.Table of ContentsIntroduction: Analysis, Design and Control of Complex Chemical Systems (Alexander S Mikhailov and Gerhard Ertl); Single Molecules, Nanoscale Phenomena and Active Particles: Imaging and Manipulation of Single Molecules by Scanning Tunneling Microscopy (Leonhard Grill); Self-Organization at the Nanoscale in Far-From-Equilibrium Surface Reactions and Copolymerizations (Pierre Gaspard); Single Molecule and Collective Dynamics of Motor Protein Coupled with Mechano-Sensitive Chemical Reaction (Mitsuhiro Iwaki, Lorenzo Marcucci, Yuichi Togashi and Toshio Yanagida); Nanomotors Propelled by Chemical Reactions (Raymond Kapral); Biology of Nanobots (Wentao Duan, Ryan Pavlick and Ayusman Sen); Reaction - Diffusion Systems and Nonequilibrium Soft Matter: Wave Phenomena in Reaction - Diffusion Systems (Oliver Steinbock and Harald Engel); Self-Oscillating Polymer Gels as Smart Materials (Ryo Yoshida); Stochastic Fluctuations and Spontaneous Symmetry Breaking in the Chemotaxis Signaling System of Dicyostelium Cells (Tatsuo Shibata); Mechanochemical Pattern Formation in the Polarization of the One-Cell C. Elegans Embryo (Justin S Bois and Stephan W Grill); Oscillations and Synchronization: Synchronization of Electrochemical Oscillators (Mahesh Wickramasinghe and Istvan Z Kiss); Turbulence and Synchrony in Spatially Extended Electrochemical Oscillators (Vladimir Garcia-Morales and Katharina Krischer); Quorum Sensing and Synchronization in Populations of Coupled Chemical Oscillators (Annette F Taylor, Mark R Tinsley and Kenneth Showalter); Collective Decision-Making and Oscillatory Behaviors in Cell Populations (Koichi Fujimoto and Satoshi Sawai); Synchronization via Hydrodynamic Interactions (Franziska Kendelbacher and Holger Stark); Evolution, Synthetic Biology, and Protocells: Emergence and Selection of Biomodules: Steps in the Assembly of a Protocell (Susanna C Manrubia and Carlos Briones); From Catalytic Reaction Networks to Protocells (Kunihiko Kaneko); Constructive Approach Towards Protocells (Tadashi Sugawara, Kensuke Kurihara and Kentaro Suzuki); Network Reverse Engineering Approach in Synthetic Biology (Haoqian Zhang, Ao Liu, Yuheng Lu, Ying Sheng, Qianzhu Wu, Zhenzhen Yin, Yiwei Chen, Zairan Liu, Heng Pan and Qi Ouyang).

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Book SynopsisThis book presents the relationship between classical theta functions and knots. It is based on a novel idea of Răzvan Gelca and Alejandro Uribe, which converts Weil's representation of the Heisenberg group on theta functions to a knot theoretical framework, by giving a topological interpretation to a certain induced representation. It also explains how the discrete Fourier transform can be related to 3- and 4-dimensional topology.Theta Functions and Knots can be read in two perspectives. Readers with an interest in theta functions or knot theory can learn how the two are related. Those interested in Chern-Simons theory will find here an introduction using the simplest case, that of abelian Chern-Simons theory. Moreover, the construction of abelian Chern-Simons theory is based entirely on quantum mechanics and not on quantum field theory as it is usually done.Both the theory of theta functions and low dimensional topology are presented in detail, in order to underline how deep the connection between these two fundamental mathematical subjects is. Hence the book is self-contained with a unified presentation. It is suitable for an advanced graduate course, as well as for self-study.Table of ContentsSome Historical Facts; A Quantum Mechanical Prototype; Surfaces and Curves; The Theta Functions Associated to a Riemann Surface; From Theta Functions to Knots; Some Results About 3- and 4-Dimensional Manifolds; The Discrete Fourier Transform and Topological Quantum Field Theory; Theta Functions and Quantum Groups; An Epilogue - Abelian Chern - Simons Theory.

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Book SynopsisThe book constitutes a basic, concise, yet rigorous course in complex analysis, for students who have studied calculus in one and several variables, but have not previously been exposed to complex analysis. The textbook should be particularly useful and relevant for undergraduate students in joint programmes with mathematics, as well as engineering students. The aim of the book is to cover the bare bones of the subject with minimal prerequisites. The core content of the book is the three main pillars of complex analysis: the Cauchy-Riemann equations, the Cauchy Integral Theorem, and Taylor and Laurent series expansions.Each section contains several problems, which are not purely drill exercises, but are rather meant to reinforce the fundamental concepts. Detailed solutions to all the exercises appear at the end of the book, making the book ideal also for self-study. There are many figures illustrating the text.Table of ContentsComplex Numbers; Elementary Functions; Complex Differentiability and the Cauchy - Riemann Equations; Contour Integration; Cauchy Integral Theorem and Its Consequences; Taylor and Laurent Series; Harmonic Functions.

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Book SynopsisThis volume is an enlarged edition of a classic textbook on complex analysis. In addition to the classical material of the first edition it provides a concise and accessible treatment of Loewner theory, both in the disc and in the half-plane. Some of the new material has been described in research papers only or appears here for the first time. Each chapter ends with exercises.Table of ContentsThe Residue Theorem; Picard's Two Theorems; The Riemann Mapping Theorem; Runge's Approximation Theorems; The Prime Number Theorem; Harmonic and Subharmonic Functions; Poisson's Formula; Jordan's Curve Theorem; Caratheodory's Extension Theorem; Compact Hulls, Half-plane Hulls and their Capacities; Caratheodory's Convergence Theorems; Loewner Theory in the Disc and in the Half-plane.

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Book SynopsisThis volume is an enlarged edition of a classic textbook on complex analysis. In addition to the classical material of the first edition it provides a concise and accessible treatment of Loewner theory, both in the disc and in the half-plane. Some of the new material has been described in research papers only or appears here for the first time. Each chapter ends with exercises.Table of ContentsThe Residue Theorem; Picard's Two Theorems; The Riemann Mapping Theorem; Runge's Approximation Theorems; The Prime Number Theorem; Harmonic and Subharmonic Functions; Poisson's Formula; Jordan's Curve Theorem; Caratheodory's Extension Theorem; Compact Hulls, Half-plane Hulls and their Capacities; Caratheodory's Convergence Theorems; Loewner Theory in the Disc and in the Half-plane.

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Book SynopsisThis book investigates several classes of partial differential equations of real time variable and complex spatial variables, including the heat, Laplace, wave, telegraph, Burgers, Black-Merton-Scholes, Schrödinger and Korteweg-de Vries equations.The complexification of the spatial variable is done by two different methods. The first method is that of complexifying the spatial variable in the corresponding semigroups of operators. In this case, the solutions are studied within the context of the theory of semigroups of linear operators. It is also interesting to observe that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. The second method is that of complexifying the spatial variable directly in the corresponding evolution equation from the real case. More precisely, the real spatial variable is replaced by a complex spatial variable in the corresponding evolution equation and then analytic and non-analytic solutions are sought.For the first time in the book literature, we aim to give a comprehensive study of the most important evolution equations of real time variable and complex spatial variables. In some cases, potential physical interpretations are presented. The generality of the methods used allows the study of evolution equations of spatial variables in general domains of the complex plane.

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Book SynopsisThis second review volume is a follow-up to the book “Engineering of Chemical Complexity” that appeared in 2013. Co-edited by the Nobel laureate Gerhard Ertl, this book provides a broad perspective over the current research aimed at understanding, the design and control of complex chemical systems of various origins, on the scales ranging from single molecules and nano-phenomena to macroscopic chemical reactors. Self-organization behavior and emergence of coherent collective dynamics in reaction-diffusion systems, in active soft matter and biochemical networks are discussed. Special attention is paid to applications in cell biology, to molecular motors and microfluidics effects. The reviews, prepared by leading international experts from the EU, USA, Russia and Japan, together yield a fascinating picture of a rapidly developing research discipline that brings chemical engineering to new frontiers.Table of ContentsOscillatory Behavior in Cellular Regulatory Networks; Chemically-Driven Biological Brownian Machines; Modeling Substrate-Based Motility of Eukaryotic Cells; Stimuli-Induced Reconfiguration and Directed Motion of Responsive Gels; From Colloid Thermophoresis to Thermophoretic Machines; Collective Motions in Populations of Microdroplets; Dissipative Patterns in Systems of Coupled Reactive Nano- and Microdroplets; Dynamics of Filaments of Scroll Waves; Control of Chemical Wave Propagation; Synchronization Phenomena during Elecrodissolution of Silicon; and other papers;

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Book SynopsisThis is a book of a series on interdisciplinary topics on the Mathematical and Biological Sciences. The chapters correspond to selected papers on special research themes, which have been presented at BIOMAT 2014 International Symposium on Mathematical and Computational Biology which was held in the Stefan Banach International Mathematical Centre at Bedlewo near Poznan, Poland on November 03 - 07, 2014. The treatment is both pedagogical yet advanced in order to motivate research students as well as to fulfill the requirements of professional practitioners. As in the other volumes of this series, there are new important results on the interdisciplinary fields of mathematical and biological sciences and comprehensive reviews written by prominent scientific leaders of famous research groups.Table of ContentsPopulation Dynamics: Modelling Sustainable Development for Decision Making (D Angulo, G Olivar and F Angulo); Wild Herbivores in Forests: Four Case Studies (Giorgio Sabetta and Ezio Venturino); Effect of Viral Disease in a Diffusive Plankton System (Nanda Das and Samares Pal); Pattern Recognition of Biological Phenomena: Exploration of Different Wave Patterns in a Model of the Bovine Estrous Cycle by Fourier Analysis (C Stotzel, R Ehrig, H M T Boer, J Plontzke and S Roblitz); A Spectral Similarity Measure between Time Series Applied to Identify Protein-Protein Interactions (G E Salcedo, A M Montoya and A F Arenas); Mathematical Modelling of Infectious Diseases: An Agent-based Modelling Framework to Study the Burden of Pertussis and the Impact of Preventive Measures (J-E Poirrier, C Philemotte and D Curran); Zoonotic Visceral Leishmaniasis: A Novel Model involving Dynamic Interactions of Humans, Dogs and Sandflies (H J Shimozako, Jianhong Wu and E Massad); Mathematical Models for Vaccination, Waning Immunity and Immune System Boosting: A General Framework (M V Barbarossa and G Rost); What is the Optimal Level of Information Dissemination during an Epidemic? (Marek Laskowski, Preeti Dubey, Murray E Alexander, Shannon Collinson, Jane M Heffernan and Seyed M Moghadas); Rich Dynamics of Hepatitis C Viral Infection with Logistic Proliferation (Sandip Banerjee); Computational Biology: In Silico Manipulation of Single DNA Molecules (P Cortini, M Barbi and P Carrivain); Very High Synchrony in Evolution of Organelles and Host Genomes (A I Chernyshova, Yu A Putintzeva and M G Sadovsky); A Method for Clustering Hemagglutinin Influenza Protein Sequences (X Li, H Jankowski, X Wang and Jane M Heffernan); Comparison of the Robustness of Entropy Measures for Protein Domain Families (R P Mondaini, Cecilia Mondaini and Edgar A G G do Amaral); Dynamic and Geometric Modelling of Biomolecular Structures: Dynamics of Z-Ring Formation in Liposomes (R A Barrio, C Varea, T Alarcon, C B Picallo and A Hernandez-Machado); Optimal Control of Protein Langevin Dynamics (R P Mondaini and S C de Albuquerque Neto); Time Asymmetry of Cross-correlation Functions as a Signature of Non-equilibrium Steady States (A Lemarchand and C Bianca); Modelling Physiological Disorders: Entropy of Protein Networks in Malignant Cells of Breast Cancer and Therapeutic Treatment (Jack A Tuszynski, N Carels and T M Tilli); Molecular Dynamics and Computational Drug Discovery: Molecular Dynamics and related Computational Methods with Applications to Drug Discovery (Jack A Tuszynski, T Luchko, P Vinter, Cassandra Churchill, K Sahu, F Gentile, Sara I Omar, N Nayebi, G Hu, K Wang and J Ruan); Multi-scale Models in Biological Sciences: On a Multi-scale Analysis of a Micro-model of Heat Transfer in Biological Tissues (Abdelhamid Ainouz); Multi-scale Modelling in Cell Dynamics (M Banerjee, M Benmir and V Volpert); Mathematical Morphology of Biological Structures: Topology of Cell Membranes (E I Kats and M I Monastyrsky); Geometry of Morphogenesis (N Morozova and R Penner); Comparing Shape Trajectories of Biological Soft Tissues in the Size-and-shape Space (V Varano, S Gabriele, L Teresi, I Dryden, P E Puddu, C Torromeo and P Piras);

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