Modern achievements in the intensively developing field of applied mathematics are presented in this monograph. In particular, it proposes a new approach to extremal problem theory for nonlinear operators, differential-operator equations and inclusions, and for variational inequalities in Banach spaces. An axiomatic study of nonlinear maps (including multi-valued ones) is given, and the properties of resolving operators for systems, consisting of operator and differential-operator equations, are stated in nonlinear-map terms. The solvability conditions and the properties of extremal problem solutions are obtained, while their weak expansions and necessary conditions of optimality in variational inequality form are formulated. In addition. the monograph proposes regularization methods and approximation schemes. This book is adressed to scientists, graduates and undergraduates who are interested in nonlinear analysis, control theory, system analysis and differential equations.
Here, the authors present modern mathematical methods to solve problems of differential-operator inclusions and evolution variation inequalities which may occur in fields such as geophysics, aerohydrodynamics, or fluid dynamics. For the first time, they describe the detailed generalization of various approaches to the analysis of fundamentally nonlinear models and provide a toolbox of mathematical equations. These new mathematical methods can be applied to a broad spectrum of problems. Examples of these are phase changes, diffusion of electromagnetic, acoustic, vibro-, hydro- and seismoacoustic waves, or quantum mechanical effects. This is the second of two volumes dealing with the subject.
In this sequel to two earlier volumes, the authors now focus on the long-time behavior of evolution inclusions, based on the theory of extremal solutions to differential-operator problems. This approach is used to solve problems in climate research, geophysics, aerohydrodynamics, chemical kinetics or fluid dynamics. As in the previous volumes, the authors present a toolbox of mathematical equations. The book is based on seminars and lecture courses on multi-valued and non-linear analysis and their geophysical application.
This monograph is dedicated to the systematic presentation of main trends, technologies and methods of computational intelligence (CI). The book pays big attention to novel important CI technology- fuzzy logic (FL) systems and fuzzy neural networks (FNN). Different FNN including new class of FNN- cascade neo-fuzzy neural networks are considered and their training algorithms are described and analyzed. The applications of FNN to the forecast in macroeconomics and at stock markets are examined. The book presents the problem of portfolio optimization under uncertainty, the novel theory of fuzzy portfolio optimization free of drawbacks of classical model of Markovitz as well as an application for portfolios optimization at Ukrainian, Russian and American stock exchanges. The book also presents the problem of corporations bankruptcy risk forecasting under incomplete and fuzzy information, as well as new methods based on fuzzy sets theory and fuzzy neural networks and results of their application for bankruptcy risk forecasting are presented and compared with Altman method. This monograph also focuses on an inductive modeling method of self-organization – the so-called Group Method of Data Handling (GMDH) which enables to construct the structure of forecasting models almost automatically. The results of experimental investigations of GMDH for forecasting at stock exchanges are presented. The final chapters are devoted to theory and applications of evolutionary modeling (EM) and genetic algorithms. The distinguishing feature of this monograph is a great number of practical examples of CI technologies and methods application for solution of real problems in technology, economy and financial sphere, in particular forecasting, classification, pattern recognition, portfolio optimization, bankruptcy risk prediction under uncertainty which were developed by authors and published in this book for the first time. All CI methods and algorithms are presented from the general system approach and analysis of their properties, advantages and drawbacks that enables practitioners to choose the most adequate method for their own problems solution.
Here, the authors present modern mathematical methods to solve problems of differential-operator inclusions and evolution variation inequalities which may occur in fields such as geophysics, aerohydrodynamics, or fluid dynamics. For the first time, they describe the detailed generalization of various approaches to the analysis of fundamentally nonlinear models and provide a toolbox of mathematical equations. These new mathematical methods can be applied to a broad spectrum of problems. Examples of these are phase changes, diffusion of electromagnetic, acoustic, vibro-, hydro- and seismoacoustic waves, or quantum mechanical effects. This is the first of two volumes dealing with the subject.
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