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The asymptotic analysis has obtained new impulses with the general development of various branches of mathematical analysis and their applications. In this book, such impulses originate from the use of slowly varying functions and the asymptotic behavior of generalized functions. The most developed approaches related to generalized functions are those of Vladimirov, Drozhinov and Zavyalov, and that of Kanwal and Estrada. The first approach is followed by the authors of this book and extended in the direction of the S-asymptotics. The second approach - of Estrada, Kanwal and Vindas - is related

Asymptotic expansions. --- Asymptotic developments --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis

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This book addresses the task of computation from the standpoint of asymptotic analysis and multiple scales that may be inherent in the system dynamics being studied. This is in contrast to the usual methods of numerical analysis and computation. The technical literature is replete with numerical methods such as Runge-Kutta approach and its variations, finite element methods, and so on. However, not much attention has been given to asymptotic methods for computation, although such approaches have been widely applied with great success in the analysis of dynamic systems. The presence of different scales in a dynamic phenomenon enable us to make judicious use of them in developing computational approaches which are highly efficient. Many such applications have been developed in such areas as astrodynamics, fluid mechanics and so on. This book presents a novel approach to make use of the different time constants inherent in the system to develop rapid computational methods. First, the fundamental notions of asymptotic analysis are presented with classical examples. Next, the novel systematic and rigorous approaches of system decomposition and reduced order models are presented. Next, the technique of multiple scales is discussed. Finally application to rapid computation of several aerospace systems is discussed, demonstrating the high efficiency of such methods.

Mathematical physics -- Asymptotic theory. --- Multiscale modeling. --- System theory -- Asymptotic theory. --- Mathematical physics --- System theory --- Multiscale modeling --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Civil Engineering --- Applied Mathematics --- Applied Physics --- Asymptotic theory --- Asymptotic expansions. --- Asymptotic efficiencies (Statistics) --- Efficiencies, Asymptotic (Statistics) --- Asymptotic developments --- Engineering. --- Computer mathematics. --- Mechanics. --- Mechanics, Applied. --- Aerospace engineering. --- Astronautics. --- Theoretical and Applied Mechanics. --- Computational Mathematics and Numerical Analysis. --- Aerospace Technology and Astronautics. --- Estimation theory --- Mathematical statistics --- Statistical hypothesis testing --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- Mechanics, applied. --- Computer science --- Mathematics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Space sciences --- Aeronautics --- Astrodynamics --- Space flight --- Space vehicles --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematics --- Aeronautical engineering --- Astronautics --- Engineering --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory

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The book concerns the notion of association in probability and statistics. Association and some other positive dependence notions were introduced in 1966 and 1967 but received little attention from the probabilistic and statistics community. The interest in these dependence notions increased in the last 15 to 20 years, and many asymptotic results were proved and improved. Despite this increased interest, characterizations and results remained essentially scattered in the literature published in different journals. The goal of this book is to bring together the bulk of these results, presenting the theory in a unified way, explaining relations and implications of the results. It will present basic definitions and characterizations, followed by a collection of relevant inequalities. These are then applied to characterize almost sure and weak convergence of sequences of associated variables. It will also cover applications of positive dependence to the characterization of asymptotic results in nonparametric statistics. The book is directed towards researchers in probability and statistics, with particular emphasis on people interested in nonparametric methods. It will also be of interest to graduate students in those areas. The book could also be used as a reference on association in a course covering dependent variables and their asymptotics. As prerequisite, readers should have knowledge of basic probability on the reals and on metric spaces. Some acquaintance with the asymptotics of random functions, such us empirical processes and partial sums processes, is useful but not essential.

Asymptotes. --- Distribution (Probability theory. --- Mathematical statistics. --- Random variables. --- Statistics. --- Random variables --- Asymptotes --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Asymptotic expansions. --- Asymptotic developments --- Probabilities. --- Statistics, general. --- Statistical Theory and Methods. --- Probability Theory and Stochastic Processes. --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Distribution functions --- Frequency distribution --- Characteristic functions --- Statistics . --- Probability --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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The approximation of a continuous function by either an algebraic polynomial, a trigonometric polynomial, or a spline, is an important issue in application areas like computer-aided geometric design and signal analysis. This book is an introduction to the mathematical analysis of such approximation, and, with the prerequisites of only calculus and linear algebra, the material is targeted at senior undergraduate level, with a treatment that is both rigorous and self-contained. The topics include polynomial interpolation; Bernstein polynomials and the Weierstrass theorem; best approximations in the general setting of normed linear spaces and inner product spaces; best uniform polynomial approximation; orthogonal polynomials; Newton-Cotes , Gauss and Clenshaw-Curtis quadrature; the Euler-Maclaurin formula ; approximation of periodic functions; the uniform convergence of Fourier series; spline approximation,with an extensive treatment of local spline interpolation,and its application in quadrature. Exercises are provided at the end of each chapter.

Asymptotic expansions. --- Transformations (Mathematics). --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Theory --- Approximation theory. --- Numerical analysis. --- Theory of approximation --- Mathematics. --- Computer science --- Computer mathematics. --- Applied mathematics. --- Engineering mathematics. --- Mathematics, general. --- Computational Mathematics and Numerical Analysis. --- Approximations and Expansions. --- Mathematics of Computing. --- Appl.Mathematics/Computational Methods of Engineering. --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Mathematical analysis --- Computer science. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Informatics --- Science --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Math --- Approximation theory --- Data processing. --- Study and teaching. --- Computer science—Mathematics.

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Hybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. Filled with a wealth of examples to illustrate concepts, this book presents a complete theory of robust asymptotic stability for hybrid dynamical systems that is applicable to the design of hybrid control algorithms--algorithms that feature logic, timers, or combinations of digital and analog components. With the tools of modern mathematical analysis, Hybrid Dynamical Systems unifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems. It presents hybrid system versions of the necessary and sufficient Lyapunov conditions for asymptotic stability, invariance principles, and approximation techniques, and examines the robustness of asymptotic stability, motivated by the goal of designing robust hybrid control algorithms. This self-contained and classroom-tested book requires standard background in mathematical analysis and differential equations or nonlinear systems. It will interest graduate students in engineering as well as students and researchers in control, computer science, and mathematics.

Automatic control. --- Control theory. --- Dynamics. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Dynamics --- Machine theory --- Control engineering --- Control equipment --- Control theory --- Engineering instruments --- Automation --- Programmable controllers --- Hermes solutions. --- Krasovskii regularization. --- Krasovskii solutions. --- Lyapunov conditions. --- Lyapunov functions. --- Lyapunov-like functions. --- asymptotic stability. --- closed sets. --- compact sets. --- conical approximation. --- conical hybrid system. --- continuity properties. --- continuous time. --- continuous-time systems. --- data structure. --- differential equations. --- differential inclusions. --- discrete time. --- discrete-time systems. --- dynamical systems. --- equilibrium points. --- flow map. --- flow set. --- generalized solutions. --- graphical convergence. --- hybrid arcs. --- hybrid control algorithms. --- hybrid dynamical systems. --- hybrid feedback control. --- hybrid models. --- hybrid system. --- hybrid time domains. --- invariance principles. --- jump map. --- jump set. --- modeling frameworks. --- modeling. --- nonlinear systems. --- numerical simulations. --- output function. --- pre-asymptotic stability. --- pre-asymptotically stable sets. --- precompact solutions. --- regularity properties. --- set convergence. --- set-valued analysis. --- set-valued mappings. --- smooth Lyapunov function. --- solution concept. --- stability theory. --- state measurement error. --- state perturbations. --- switching signals. --- switching systems. --- uniform asymptotic stability. --- well-posed hybrid systems. --- well-posed problems. --- well-posedness. --- ω-limit sets. --- Nonlinear control theory.