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This book presents counterexamples to false statements typically found within the study of mathematical analysis, real analysis, and calculus, all of which are related to infinite sequences, series of functions, and functions and integrals depending on a parameter.
Mathematical analysis --- Calculus --- Sequences (Mathematics) --- Functions --- Integrals
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Numerical analysis. --- Integrals. --- Entire functions. --- Vacuum systems. --- Problem solving.
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This book presents fractional difference, integral, differential, evolution equations and inclusions, and discusses existence and asymptotic behavior of their solutions. Controllability and relaxed control results are obtained. Combining rigorous deduction with abundant examples, it is of interest to nonlinear science researchers using fractional equations as a tool, and physicists, mechanics researchers and engineers studying relevant topics. ContentsFractional Difference EquationsFractional Integral EquationsFractional Differential EquationsFractional Evolution Equations: ContinuedFractional Differential Inclusions
Fractional calculus. --- Derivatives and integrals, Fractional --- Differentiation of arbitrary order, Integration and --- Differintegration, Generalized --- Fractional derivatives and integrals --- Generalized calculus --- Generalized differintegration --- Integrals, Fractional derivatives and --- Integration and differentiation of arbitrary order --- Calculus
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"Prime numbers are beautiful, mysterious, and beguiling mathematical objects. The mathematician Bernhard Riemann made a celebrated conjecture about primes in 1859, the so-called Riemann Hypothesis, which remains to be one of the most important unsolved problems in mathematics. Through the deep insights of the authors, this book introduces primes and explains the Riemann Hypothesis. Students with minimal mathematical background and scholars alike will enjoy this comprehensive discussion of primes. The first part of the book will inspire the curiosity of a general reader with an accessible explanation of the key ideas. The exposition of these ideas is generously illuminated by computational graphics that exhibit the key concepts and phenomena in enticing detail. Readers with more mathematical experience will then go deeper into the structure of primes and see how the Riemann Hypothesis relates to Fourier analysis using the vocabulary of spectra. Readers with a strong mathematical background will be able to connect these ideas to historical formulations of the Riemann Hypothesis." --Publisher description.
Riemann hypothesis --- Numbers, Prime --- Prime numbers --- Numbers, Natural --- Riemann's hypothesis --- 517.58 --- 517.58 Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials. --- Special functions. Hyperbolic functions. Euler integrals. Gamma functions. Elliptic functions and integrals. Bessel functions. Other cylindrical functions. Spherical functions. Legendre polynomials. Orthogonal polynomials. Chebyshev polynomials.
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Complexity increases with increasing system size in everything from organisms to organizations. The nonlinear dependence of a system's functionality on its size, by means of an allometry relation, is argued to be a consequence of their joint dependency on complexity (information). In turn, complexity is proven to be the source of allometry and to provide a new kind of force entailed by a system's information gradient. Based on first principles, the scaling behavior of the probability density function is determined by the exact solution to a set of fractional differential equations. The resulting lowest order moments in system size and functionality gives rise to the empirical allometry relations. Taking examples from various topics in nature, the book is of interest to researchers in applied mathematics, as well as, investigators in the natural, social, physical and life sciences. ContentsComplexityEmpirical allometryStatistics, scaling and simulationAllometry theoriesStrange kineticsFractional probability calculus
Fractional calculus. --- Fractional differential equations. --- Mathematical models. --- Models, Mathematical --- Simulation methods --- Extraordinary differential equations --- Differential equations --- Fractional calculus --- Derivatives and integrals, Fractional --- Differentiation of arbitrary order, Integration and --- Differintegration, Generalized --- Fractional derivatives and integrals --- Generalized calculus --- Generalized differintegration --- Integrals, Fractional derivatives and --- Integration and differentiation of arbitrary order --- Calculus
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This book focuses on two specific areas related to fractional order systems – the realization of physical devices characterized by non-integer order impedance, usually called fractional-order elements (FOEs); and the characterization of vegetable tissues via electrical impedance spectroscopy (EIS) – and provides readers with new tools for designing new types of integrated circuits. The majority of the book addresses FOEs. The interest in these topics is related to the need to produce “analogue” electronic devices characterized by non-integer order impedance, and to the characterization of natural phenomena, which are systems with memory or aftereffects and for which the fractional-order calculus tool is the ideal choice for analysis. FOEs represent the building blocks for designing and realizing analogue integrated electronic circuits, which the authors believe hold the potential for a wealth of mass-market applications. The freedom to choose either an integer- or non-integer-order analogue integrator/derivator is a new one for electronic circuit designers. The book shows how specific non-integer-order impedance elements can be created using materials with specific structural properties. EIS measures the electrical impedance of a specimen across a given range of frequencies, producing a spectrum that represents the variation of the impedance versus frequency – a technique that has the advantage of avoiding aggressive examinations. Biological tissues are complex systems characterized by dynamic processes that occur at different lengths and time scales; this book proposes a model for vegetable tissues that describes the behavior of such materials by considering the interactions among various relaxing phenomena and memory effects.
Engineering. --- Information theory. --- Electronic circuits. --- Circuits and Systems. --- Information and Communication, Circuits. --- Electronic Circuits and Devices. --- Fractional calculus. --- Derivatives and integrals, Fractional --- Differentiation of arbitrary order, Integration and --- Differintegration, Generalized --- Fractional derivatives and integrals --- Generalized calculus --- Generalized differintegration --- Integrals, Fractional derivatives and --- Integration and differentiation of arbitrary order --- Calculus --- Systems engineering. --- Mathematics. --- Math --- Science --- Engineering systems --- System engineering --- Engineering --- Industrial engineering --- System analysis --- Design and construction --- Communication theory --- Communication --- Cybernetics --- Electron-tube circuits --- Electric circuits --- Electron tubes --- Electronics
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This advanced monograph is concerned with modern treatments of central problems in harmonic analysis. The main theme of the book is the interplay between ideas used to study the propagation of singularities for the wave equation and their counterparts in classical analysis. In particular, the author uses microlocal analysis to study problems involving maximal functions and Riesz means using the so-called half-wave operator. To keep the treatment self-contained, the author begins with a rapid review of Fourier analysis and also develops the necessary tools from microlocal analysis. This second edition includes two new chapters. The first presents Hörmander's propagation of singularities theorem and uses this to prove the Duistermaat-Guillemin theorem. The second concerns newer results related to the Kakeya conjecture, including the maximal Kakeya estimates obtained by Bourgain and Wolff.
Fourier series. --- Fourier integral operators. --- Fourier analysis. --- Analysis, Fourier --- Mathematical analysis --- Fourier analysis --- Integral operators --- Fourier integrals --- Series, Fourier --- Series, Trigonometric --- Trigonometric series --- Calculus --- Harmonic analysis --- Harmonic functions
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This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schrödinger equation at least in the short term if the potential is differentiable sufficiently many times and its derivatives of order equal to or higher than two are bounded. The semi-classical asymptotic formula up to the second term of the fundamental solution is also proved by a method different from that of Birkhoff. A bound of the remainder term is also proved. The Feynman path integral is a method of quantization using the Lagrangian function, whereas Schrödinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrödinger's method, there is still much to be done concerning rigorous mathematical treatment of Feynman's method. Feynman himself defined a path integral as the limit of a sequence of integrals over finite-dimensional spaces which is obtained by dividing the time interval into small pieces. This method is called the time slicing approximation method or the time slicing method. This book consists of two parts. Part I is the main part. The time slicing method is performed step by step in detail in Part I. The time interval is divided into small pieces. Corresponding to each division a finite-dimensional integral is constructed following Feynman's famous paper. This finite-dimensional integral is not absolutely convergent. Owing to the assumption of the potential, it is an oscillatory integral. The oscillatory integral techniques developed in the theory of partial differential equations are applied to it. It turns out that the finite-dimensional integral gives a finite definite value. The stationary phase method is applied to it. Basic properties of oscillatory integrals and the stationary phase method are explained in the book in detail. Those finite-dimensional integrals form a sequence of approximation of the Feynman path integral when the division goes finer and finer. A careful discussion is required to prove the convergence of the approximate sequence as the length of each of the small subintervals tends to 0. For that purpose the book uses the stationary phase method of oscillatory integrals over a space of large dimension, of which the detailed proof is given in Part II of the book. By virtue of this method, the approximate sequence converges to the limit. This proves that the Feynman path integral converges. It turns out that the convergence occurs in a very strong topology. The fact that the limit is the fundamental solution of the Schrödinger equation is proved also by the stationary phase method. The semi-classical asymptotic formula naturally follows from the above discussion. A prerequisite for readers of this book is standard knowledge of functional analysis. Mathematical techniques required here are explained and proved from scratch in Part II, which occupies a large part of the book, because they are considerably different from techniques usually used in treating the Schrödinger equation.
Mathematics. --- Fourier analysis. --- Functional analysis. --- Partial differential equations. --- Mathematical physics. --- Mathematical Physics. --- Functional Analysis. --- Partial Differential Equations. --- Fourier Analysis. --- Differential equations, partial. --- Partial differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analysis, Fourier --- Mathematical analysis --- Feynman integrals. --- Feynman diagrams --- Multiple integrals --- Physical mathematics --- Physics --- Mathematics
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The first part of the book is devoted to the transport equation for a given vector field, exploiting the lagrangian structure of solutions. It also treats the regularity of solutions of some degenerate elliptic equations, which appear in the eulerian counterpart of some transport models with congestion. The second part of the book deals with the lagrangian structure of solutions of the Vlasov-Poisson system, which describes the evolution of a system of particles under the self-induced gravitational/electrostatic field, and the existence of solutions of the semigeostrophic system, used in meteorology to describe the motion of large-scale oceanic/atmospheric flows.
Mathematics. --- Partial differential equations. --- Calculus of variations. --- Geophysics. --- Partial Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Geophysics and Environmental Physics. --- Vector fields. --- Elliptic functions. --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Direction fields (Mathematics) --- Fields, Direction (Mathematics) --- Fields, Slope (Mathematics) --- Fields, Vector --- Slope fields (Mathematics) --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Vector analysis --- Differential equations, partial. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Partial differential equations --- Isoperimetrical problems --- Variations, Calculus of --- Geological physics --- Terrestrial physics --- Earth sciences --- Physics
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This textbook presents a rigorous approach to multivariable calculus in the context of model building and optimization problems. This comprehensive overview is based on lectures given at five SERC Schools from 2008 to 2012 and covers a broad range of topics that will enable readers to understand and create deterministic and nondeterministic models. Researchers, advanced undergraduate, and graduate students in mathematics, statistics, physics, engineering, and biological sciences will find this book to be a valuable resource for finding appropriate models to describe real-life situations. The first chapter begins with an introduction to fractional calculus moving on to discuss fractional integrals, fractional derivatives, fractional differential equations and their solutions. Multivariable calculus is covered in the second chapter and introduces the fundamentals of multivariable calculus (multivariable functions, limits and continuity, differentiability, directional derivatives and expansions of multivariable functions). Illustrative examples, input-output process, optimal recovery of functions and approximations are given; each section lists an ample number of exercises to heighten understanding of the material. Chapter three discusses deterministic/mathematical and optimization models evolving from differential equations, difference equations, algebraic models, power function models, input-output models and pathway models. Fractional integral and derivative models are examined. Chapter four covers non-deterministic/stochastic models. The random walk model, branching process model, birth and death process model, time series models, and regression type models are examined. The fifth chapter covers optimal design. General linear models from a statistical point of view are introduced; the Gauss–Markov theorem, quadratic forms, and generalized inverses of matrices are covered. Pathway, symmetric, and asymmetric models are covered in chapter six, the concepts are illustrated with graphs. .
Mathematics. --- Integral transforms. --- Operational calculus. --- Special functions. --- Mathematical models. --- Mathematical optimization. --- Mathematical Modeling and Industrial Mathematics. --- Optimization. --- Special Functions. --- Integral Transforms, Operational Calculus. --- Functions, special. --- Integral Transforms. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Transform calculus --- Integral equations --- Transformations (Mathematics) --- Special functions --- Fractional calculus. --- Derivatives and integrals, Fractional --- Differentiation of arbitrary order, Integration and --- Differintegration, Generalized --- Fractional derivatives and integrals --- Generalized calculus --- Generalized differintegration --- Integrals, Fractional derivatives and --- Integration and differentiation of arbitrary order --- Calculus --- Operational calculus --- Differential equations --- Electric circuits --- Models, Mathematical
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