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Informationen zum Autor Dr. A.V. Kim, PhD, is the head of the research group of the Institute of Mathematics and Mechanics of the Russian Academy of Sciences (Ural Branch). He graduated from the mathematics department at Ural State University in 1980, received his doctorate from Ural State University in 1987, and received his Doctor of Science degree with his research monograph, "Some problems of Functional Differential Equations theory," at the Institute of Mathematics and Mechanics in 2001. Before doing his research at the Russian Academy of Natural Science, he taught at the Seoul National University in the School of Electrical Engineering. Klappentext A totally new direction in mathematics, this revolutionary new study introduces a new class of invariant derivatives of functions and establishes relations with other derivatives, such as the Sobolev generalized derivative and the generalized derivative of the distribution theory. i -smooth analysis is the branch of functional analysis that considers the theory and applications of the invariant derivatives of functions and functionals. The important direction of i -smooth analysis is the investigation of the relation of invariant derivatives with the Sobolev generalized derivative and the generalized derivative of distribution theory. Until now, i -smooth analysis has been developed mainly to apply to the theory of functional differential equations, and the goal of this book is to present i -smooth analysis as a branch of functional analysis. The notion of the invariant derivative ( i -derivative) of nonlinear functionals has been introduced in mathematics, and this in turn developed the corresponding i -smooth calculus of functionals and showed that for linear continuous functionals the invariant derivative coincides with the generalized derivative of the distribution theory. This book intends to introduce this theory to the general mathematics, engineering, and physicist communities. i-Smooth Analysis: Theory and Applications Introduces a new class of derivatives of functions and functionals, a revolutionary new approach Establishes a relationship with the generalized Sobolev derivative and the generalized derivative of the distribution theory Presents the complete theory of i-smooth analysis Contains the theory of FDE numerical method, based on i-smooth analysis Explores a new direction of i-smooth analysis, the theory of the invariant derivative of functions Is of interest to all mathematicians, engineers studying processes with delays, and physicists who study hereditary phenomena in nature. AUDIENCE Mathematicians, applied mathematicians, engineers, physicists, students in mathematics Inhaltsverzeichnis Preface xi Part I Invariant derivatives of functionals and numerical methods for functional differential equations 1 1 The invariant derivative of functionals 3 1 Functional derivatives 3 1.1 The Frechet derivative 4 1.2 The Gateaux derivative 4 2 Classification of functionals on C[a, b] 5 2.1 Regular functionals 5 2.2 Singular functionals 6 3 Calculation of a functional along a line 6 3.1 Shift operators 6 3.2 Superposition of a functional and a function 7 3.3 Dini derivatives 8 4 Discussion of two examples 8 4.1 Derivative of a function along a curve 8 4.2 Derivative of a functional along a curve 9 5 The invariant derivative 11 5.1 The invariant derivative 11 5.2 The invariant derivative in the class B[a, b] 12 5.3 Examples 13 6 Properties of the invariant derivative 16 6.1 Principles of calculating invariant derivatives 16 6.2 The invariant differentiability and invariant continuity 19 6.3 High order invariant derivatives 20 ...
