Fresh, lively text serves as a modern introduction to the subject, with applications to the mechanics of systems with a finite number of degrees of freedom. Ideal for math and physics students.

The non-local functional is an integral with the integrand depending on the unknown function at different values of the argument. These types of functionals have different applications in physics, engineering and sciences. The Euler type equations that arise as necessary conditions of extrema of non-local functionals are the functional differential equations. The book is dedicated to systematic study of variational calculus for non-local functionals and to theory of boundary value problems for functional differential equations. There are described different necessary and some sufficient conditions for extrema of non-local functionals. Theorems of existence and uniqueness of solutions to many kinds of boundary value problems for functional differential equations are proved. The spaces of solutions to these problems are, as a rule, Sobolev spaces and it is not often possible to apply the analytical methods for solution of these problems. Therefore it is important to have approximate methods for their solution. Different approximate methods of solution of boundary value problems for functional differential equations and direct methods of variational calculus for non-local functionals are described in the book. The non-local functional is an integral with the integrand depending on the unknown function at different values of the argument. These types of functionals have different applications in physics, engineering and sciences. The Euler type equations that arise as necessary conditions of extrema of non-local functionals are the functional differential equations. The book is dedicated to systematic study of variational calculus for non-local functionals and to theory of boundary value problems for functional differential equations. There are described different necessary and some sufficient conditions for extrema of non-local functionals. Theorems of existence and uniqueness of solutions to many kinds of boundary value problems for functional differential equations are proved. The spaces of solutions to these problems are, as a rule, Sobolev spaces and it is not often possible to apply the analytical methods for solution of these problems. Therefore it is important to have approximate methods for their solution. Different approximate methods of solution of boundary value problems for functional differential equations and direct methods of variational calculus for non-local functionals are described in the book.

This overview provides a single-volume treatment of key algorithms and theories. Begins with the derivation of optimality conditions and discussions of convex programming, duality, generalized convexity, and analysis of selected nonlinear programs, and then explores techniques for numerical solutions and unconstrained optimization methods. 1976 edition. Includes 58 figures and 7 tables.

This book explores the major techniques involved in optimization, control theory, and calculus of variations. The book serves as a concise contemporary guide to optimal control theory, optimization, numerical methods and beyond. As such, it is a valuable source to learn mathematical modeling and the mathematical nature of optimization and optimal control. The presence of a variety of exercises solved down to numerical values is one of the main characteristic features of the book. Another one is its compactness, and the material’s usefulness in preparing and teaching several different university courses. The investigation of trends and their formation undertaken in the book leads seamlessly into extrapolation techniques and rigorous methods of scientific prediction. The research for this book was accomplished at the Russian Technological University (RTU) MIREA, based on the courses which have been taught at the RTU for many years.

Theory of Extremal Problems

This work by Zorich on Mathematical Analysis constitutes a thorough first course in real analysis, leading from the most elementary facts about real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, Fourier, Laplace, and Legendre transforms, and elliptic functions.

There is an ever-growing interest in control problems today, con nected with the urgent problems of the effective use of natural resources, manpower, materials, and technology. When referring to the most important achievements of science and technology in the 20th Century, one usually mentions the splitting of the atom, the exploration of space, and computer engineering. Achievements in control theory seem less spectacular when viewed against this background, but the applications of control theory are playing an important role in the development of modern civilization, and there is every reason to believe that this role will be even more signifi cant in the future. Wherever there is active human participation, the problem arises of finding the best, or optimal, means of control. The demands of economics and technology have given birth to optimization problems which, in turn, have created new branches of mathematics. In the Forties, the investigation of problems of economics gave rise to a new branch of mathematical analysis called linear and convex program ming. At that time, problems of controlling flying vehicles and technolog ical processes of complex structures became important. A mathematical theory was formulated in the mid-Fifties known as optimal control theory. Here the maximum principle of L. S. Pontryagin played a pivotal role. Op timal control theory synthesized the concepts and methods of investigation using the classical methods of the calculus of variations and the methods of contemporary mathematics, for which Soviet mathematicians made valuable contributions.

This book presents a theory of necessary conditions for an extremum, including formal conditions for an extremum and computational methods. It states the general results of the theory and shows how these results can be particularized to specific problems.

This book is intended for those having only a moderate background in mathematics, who need to increase their mathematical knowledge for development in their areas of work and to read the related mathematical literature. The material covered, which includes practically all the information on functional analysis that may be necessary for those working in various areas of applications of mathematics, as well as the simplicity of presentation, differentiates this book from others. About 300 examples and more than 500 problems are provided to help readers understand and master the theories presented. The list of references enables readers to explore those topics in which they are interested, and gather further information about applications used as examples in the book. Applications: Probability Theory and Statistics, Signal and Image Processing, Systems Analysis and Design.

This book introduces the basic principles and implementation process of deep learning in a simple way, and uses python's numpy library to build its own deep learning library from scratch instead of using existing deep learning libraries. On the basis of introducing basic knowledge of Python programming, calculus, and probability statistics, the core basic knowledge of deep learning such as regression model, neural network, convolutional neural network, recurrent neural network, and generative network is introduced in sequence according to the development of deep learning. While analyzing the principle in a simple way, it provides a detailed code implementation process. It is like not teaching you how to use weapons and mobile phones, but teaching you how to make weapons and mobile phones by yourself. This book is not a tutorial on the use of existing deep learning libraries, but an analysis of how to develop deep learning libraries from 0. This method of combining the principle from 0 with code implementation can enable readers to better understand the basic principles of deep learning and the design ideas of popular deep learning libraries.