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Review Essays of Academic, Professional & Technical Books in the Humanities & Sciences


Analytic Inequalities by Nicholas D. Kazarinoff (Dover) unabridged republication of the edition published by Holt, Rinehart and Winston, New York, 1961. 28 figures.

Mathematical analysis is largely a systematic study and exploitation of inequalities—but for students the study of inequalities often remains a foreign country, difficult of access. This book is a passport to that country, offering a background on inequalities that will prepare undergraduates (and even high school students) to cope with the concepts of continuity, derivative, and integral. 

Beginning with explanations of the algebra of inequalities and conditional inequalities, the text introduces a pair of ancient theorems and their applications. Explorations of inequalities and calculus cover the number e, examples from the calculus, and approximations by polynomials. The final section presents modern theorems, including Bernstein's proof of the Weierstrass approximation theorem and the Cauchy, Bunyakovskii, Milder, and Minkowski inequalities. Numerous figures, problems, and examples appear throughout the book, offering students an excellent foundation for further studies of calculus.

Asymptotic Methods in Resonance Analytical Dynamics by E.A. Grebenikov, Yu. A. Mitropolskiy, Yu. A. Ryabov (CRC Press) presents asymptotic methods for analysis and construction of solutions (mainly periodic and quasiperiodic) of differential equations with small parameter considered in many problems in nonlinear mechanics and oscillation theory. These methods are based on the generalized averaging technique of Krylov-Bogolubov and the numeric-analytical iterations of Lyapunov-Poincare. The volume should provide a useful source of reference for postgraduates and researchers working in this area of applied mathematics.

Asymptotic Methods in Resonance Analytical Dynamics introduces the fundamental ideas of linear stochastic (optimal) estimation. The authors provide rigorous theoretical derivations with qualitative discussion, judgments, and applications while making it accessible to the beginner. They use dynamic models from mechanical and aerospace engineering to provide immediate results of estimation concepts with only a minimum reliance on mathematical skills.

Many important systems in analytical dynamics are described by nonlinear mathe­matical models, and the latter as a rule are represented by differential or integrodifferential equations. The absence of exact universal methods for the investigation of nonlinear systems has driven the development of a wide range of approximate analytic and numerical—analytic methods that can be implemented in effective com­puter algorithms.

Trying to give them a general description, one can assert that practically all approximate methods are constructed by the iteration principle. This means that first an initial approximation is somehow chosen for the problem, and then, by means of iterations, the addition of terms of different infinitesimal order to the initial approximation are found. The term "iteration" is understood here as either successive approximations, or a chain of successive transformations of phase variables, or a functional series with terms decreasing in value. From the historical aspect, the first approximation was considered as a solution of some linear problem (hence the term linearization method), to which some small functions (propor­tional to the small parameter) determined within the framework of one or another perturbation theory were added. For example, while studying the oscillations of the mathematical pendulum Newton considered the small oscillations of the pendulum that describes oscillations of a mathematical pendulum, not necessarily small in value. Lagrange did the same when considering secular perturbations in the planetary three-body problem under small perturbations.

Another approach to nonlinear problem solving is when, as the initial ap­proximation, the solution of a system is taken that is a nonlinear but essentially "simpler" system than the initial one. This simplified system can be obtained by different methods, but since the times of Lagrange and Gauss, in oscillatory models of celestial mechanics the method of averaging (or smoothing) of periodic or quasi-periodic functions being a part of the analytic structure of the equations was most often used. Later, and particularly nowadays, averaging methods in combination with asymptotic representations (in the sense of Poincaré) were used as the basic constructive means of solving intricate problems of analytical dynamics, formalized in the language of differential equations. This became possible thanks to the work of N.N. Bogolyubov in the 1930s, where the problem of nonlinear problem solving was formulated as a problem of transformation of the initial differential equations into new simplified, so-called comparison equations. Since the choice of comparison equations is arbitrary, one can make an optimal choice (e.g. from the standpoint of their solvability, or the proximity of solutions of comparison equations to those of the initial system, or the efficiency of numerical methods applied to them) that determines the corresponding transformation of phase variables. It is easy to show that if the differential equations are written in a normal form in the Cauchy sense, then the functions that perform the change of variables satisfy some system of quasi-linear systems in first-order partial derivatives.

From the aspect of geometric interpretation, the transformation of initial equations into comparison equations can be interpreted as the choice of a phase space most optimal for the given problem, of its metrics and norm. The transformation of equations means the search for the geometry that gives the simplest description to the considered problem. But the simplest description of a problem does not mean that the method of solution has been found. Therefore, as a rule, it is only a reasonable combination of iteration procedures with a successful choice of the initial approximation and with the use of fast algorithms and compact programs that can guarantee the efficiency of the construction of an approximate solution with prescribed accuracy.

One of the interesting new directions is the KAM (Kolmogorov—Arnold—Moser) theory allowing us to construct exact (in the sense of convergence) solutions of the Hamiltonian dynamics regular with respect to a small parameter, in spite of the negative impact of the small denominators on the conditions of convergence of an infinite chain of canonical transformations giving exact solutions in the limit. KAM theory made it possible to solve in an exact wording the problem of stability of a hypothetical planetary system (or rather its configuration), which should be recognized as a remarkable achievement of modern mathematics. Unfortunately, these results are so far inapplicable to our solar system because its dynamical parameters (planetary masses, above all) do not satisfy the estimations of KAM theory.

However it would be incorrect to contend that asymptotic methods in resonance analytical dynamics are in a state of completeness. This is especially noticeable in the construction of solutions of differential equations of analytical dynamics, i.e. in constructive equation theory. A modern researcher dreams of such a computer realization of the asymptotic or qualitative theory of differential equations that would allow us to use analytical operations and graphical tools fully. However it turns out that these issues are closely associated with the so-called problem of asymptotic theory bifurcation in the neighborhood of low—order frequency reso­nances, and with the problem of recalculation of the initial conditions at each step of the iterations. In other words, in resonance analytical dynamics it is possible to formulate some problems that seem to us most urgent:

Problem 1. Let a multifrequency system of differential equations be specified on a torus, as well as the corresponding initial conditions. Is it possible to construct a variant of asymptotic theory such that at each step of the transformation the iterations are minimized by changing the initial conditions? The answer is yes. Moreover, there exists an analytic algorithm allowing us to express the new initial conditions through the old ones, and vice versa.

This algorithm is easily implemented on computer.

Problem 2. In the construction of asymptotic theory by means of successive changes of variables it is inexpedient to specify beforehand the analytic structure of the equations at each step, but this should be determined from some conditions of each iteration norm minimization. We have developed an algorithm of obtaining such minimization conditions for typical problems of resonance analytical dynamics.

Algorithms to solve the above-mentioned problems have called forth the ap­pearance of the concept of the bifurcation of forms of the analytical theory of perturbations of differential equations with a small parameter.

Problem 3. The development of constructive numerical—analytic methods of the construction of periodic and quasi-periodic solutions to problems of resonance analytical dynamics that use converging analytic algorithms and are implemented on computer, using symbolic programming packages.

These problems are actually the subject of this monograph. We are investigat­ing the properties of the solutions of multifrequency regular systems of differential equations of analytical dynamics on the assumption of the presence of frequency resonances in the evolution process (with the change of t). Used for this purpose are the averaging principle, asymptotic representation in the sense of Poincaré, and converging iteration procedures of Lyapunov-Poincaré. A constructive asymptotic theory allows us to obtain in an explicit analytic form iterations of any order with respect to the small parameter, taking into account the above-mentioned minimization considerations.

In this monograph a number of nonlinear oscillations concerned with applications are considered. 

Nonlinear Analysis and Control of Physical Processes and Fields by M. Z. Zgurovskii, V. S. Melnik (Data and Knowledge in a Changing World: Springer Verlag) Considerable progress in the study of nonlinear boundary value problems for partial differential equations has been analyzed by extensive use of nonlinear functional analysis methods. These methods found their application in various fields of mathematics. It is natural to reduce these problems to non-linear operator or differential-operator equations in functional spaces. Such approach, which was introduced by F. Browder (as corollaries of operator systems), provides results for specific systems. Similar approach is effective in optimization and control of systems problems which are described by non-linear partial differential equations.

In this monograph, a new approach is given for the study of extremal problems in nonlinear operator and differential operator equations, as well as the variational inequalities in Banach spaces.

Presented is an axiomatic study of classes of nonlinear mappings (including the multivalued) in Cartesian product spaces. In their terms the functional-topological properties of the resolving operators of the systems which contain both operator and differential-operator equations are established. Defined are the existence conditions and properties of optimization problems solutions. Their weak expansions and necessary optimality conditions are constructed. Regularization methods are developed and the approximation schemes are proven.

The first part of the monograph "Methods of Nonlinear Analysis" presents the results obtained for nonlinear analysis problems which arise in the control and optimization theory of nonlinear infinite dimensional systems.

Chapter 1 "Preliminary Results" considers classes of monotone type non-linear mappings in product of spaces. Their relationships are identified and the relating examples are provided. The maximal semimonotone mappings which act from a reflexive Banach space into the conjugate multivalued mapping are introduced and studied.

Chapter 2 "Functionals and Forms" deals with some classes of extremal problems for functionals which are the superposition of a real function and multivalued mappings in Banach spaces. The results obtained here are original and for the first time are stated in monographic literature. The main results here are theorems 2.1.1–2.1.5 on the existence of solution of the corresponding extremal problems. Elements of functionals differential analysis on "nonlinear sets" are also developed.

Chapter 3 "Nonlinear Operator Equations, Inclusions and Variational In-equalities" considers issues connected with the existence of solutions of non-linear operator equations, operator inclusions and variational inequalities in Banach spaces. In addition to the known facts some new results are introduced here. In particular, for the first time, the notion of A-weak solution for an operator equation with densely defined mapping is introduced which is principal in theory of extremal problems expansion for operator equations. Theorems on existence and properties of A-weak solutions of operator equations and operator inclusions and also variational inequalities are proved. For variational inequalities with multivalued mappings "penalty mappings methods" are introduced and shown for the first time.

Chapter 4 "Differential-Operator Equations and Inclusions". In this chap-ter similar questions are raised for evolutional problems. This ends Part 1 of the monograph.

In the second part "Control Problems for the Distributed Parameters Objects", by making a considerable use of the results obtained in the pre-ceding chapters, methods of attack of control problems for objects described by nonlinear operator, differential-operator equations, inclusions and variational inequalities in Banach spaces with restrictions on control and phase variables are developed. Until recently it has been considered that mathematical models of the controlled systems had the property of uniqueness, i.e. each admissible set of controls corresponded to a unique state. At the same time numerous applied problems result in mathematical models, when the corresponding boundary problem may have many solutions with the fixed control function. The approach suggested in the monograph gives the possibility of embracing this class of problems.

Chapter 5 "Extremal Problems for the Nonlinear Operator Equations and Variational Inequalities" considers the class of optimizational problems for the objects described by nonlinear operator equations with inclusions and variational inequalities and constraints on inclusions and operator inequalities. Essential here is the fact that the corresponding operator equation has a multivalued resolving operator for which a number of extremal problems is established. Theorems on existence of the corresponding optimization problems solutions are proved, non-coercive extremal problems in the sense of V. Petryshyn are considered. Applying the notion of A-weak solution introduced in the previous section, considered here is the basis of the extremal problems weak expansions theory for operator equations and variational inequalities. Up to now this class of expansions has not been considered in the literature.

An original approach is also suggested by the. authors to the regularization and construction of optimization problems solutions. With the corresponding differentiability requirements the necessary conditions of optimality in the form of variational inequalities are given. Here the quasi-differential calculusof V, Pshenychnyi is extensively used and eventually the finite dimensional approximations of optimization problems and necessary optimality conditions are proposed.

In Chapter 6 "Optimal Control for Differential Operator Equations and Inclusions" similar range of questions is considered for evolutional problems with the corresponding appropriate modifications. Here for the first time systems appear which contain operator and differential-operator equations. To the best of my knowledge, the question of existence of solutions has not been raised (also for extremal problems). Also, for the first time, is presented the systemic analysis of the optimal control for modified equations of hydrodynamic type , in particular for the Navier-Stokes' equations.

In Chapter 7 "Some Problems of Synthesis in Distributed Parameters Systems" a rather novel idea is proposed regarding the synthesis of attracting sets for infinite dimensional systems. The results presented here are not only of theoretical, but also of practical value.

Chapter 8 "Control of Heat Transfer and Diffusion Processes". Total methodology presented in the monograph is demonstrated on the example of nonlinear heat transfer. Mathematical model of the process has been studied, theorems on existence of the generalized solutions (generally speaking, without uniqueness) have been proved, optimization problems have been set and the existence of their solutions has been proved, the necessary conditions of optimality have been identified and finite dimensional approximations have been constructed.

In the following two chapters the experience of practical application of mathematical methods for control of distributed parameters processes is described. Considerable attention is given to the numerical modeling and the development and implementation of optimization algorithms.

This monograph provides novel and up-to-date research on the theory of control of nonlinear infinite dimensional systems which has been carried out in Ukraine during past 20 years. Such material appears for the first time in monographic literature. It is for the first time that this research is presented and it is of considerable interest to specialists in the field of nonlinear analysis and its applications, control theory and dynamics systems theory, as well as optimization methods. 

Applied Analysis by Takasi Senba, Takashi Suzuki (Imperial College Press, World Scientific) provides a general introduction to applied mathematics, such as mathematical modeling of random motion of particles, chemotaxis in biology, and their theoretical study. Several tools in linear and nonlinear PDE theory and spectral theory of eigenfunction expansion are described. The book also presents the fundamental ideas in theoretical and applied analysis and discusses recent developments in nonlinear science.

Applied Analysis is intended to be an introduction to mathematical science, particularly the theoretical study from the viewpoint of applied analysis. Applied Analysis provides a general introduction to applied analysis; vector analysis with physical motivation, calculus of variation, Fourier analysis, eigenfunction expansion, distribution, and so forth, including a catalogue of mathematical theories, such as basic analysis, topological spaces, complex function theory, real analysis, and abstract analysis. This book also gives fundamental ideas of applied mathematics to discuss recent developments in nonlinear science, such as mathematical modeling of reinforced random motion of particles, semi-conductor device equation in applied physics, and chemotaxis in biology. Several tools in linear PDE theory, such as fundamental solutions, Perron's method, layer potentials, iteration scheme, are described, as well as systematic descriptions on the recent study of blowup of the solution. Contents: Geometric Objects, Calculus of Variation, Infinite-Dimensional Analysis, Random Motion of Particles, Linear PDE Theory, Nonlinear PDE Theory, System of Chemotaxis.

As basic materials, vector analysis and calculus of variation are taken, and then Fourier analysis is introduced for the eigenfunction expansion to justify. After that, statistical method is presented to control the mean field of many particles, and the mathematical theory to linear and nonlinear partial differential equations is accessed. System of chemotaxis is a special topic in this book, and well-posedness of the model is established. The authors summarize several mathematical theories and give some references for the advanced study. They also picked up some materials from classical mechanics, geometry, mathematical programming, and numerical schemes. Therefore, Applied Analysis covers some parts of undergraduate courses for mathematical study. It is also suitable for the first degree of graduate course to learn the basic ideas, mathematical techniques, systematic logic, physical and biological motivations of applied analysis.

The presentation is fully mathematical with minimal explanations. Will work well as basic reference to the mathematics.   

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