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Because magnetically confined plasmas are generally not found in a state of thermodynamic equilibrium, they have been studied extensively with methods of applied kinetic theory. In closed magnetic field line confinement devices such as the tokamak, non-Maxwellian distortions usually occur as a result of auxiliary heating and transport. In magnetic mirror configurations even the intended steady state plasma is far from local thermodynamic equilibrium because of losses along open magnetic field lines. In both of these major fusion devices, kinetic models based on the Boltzmann equation with Fokker-Planck collision terms have been successful in representing plasma behavior. The heating of plasmas by energetic neutral beams or microwaves, the production and thermalization of a-particles in thermonuclear reactor plasmas, the study of runaway electrons in tokamaks, and the performance of two-energy compo nent fusion reactors are some examples of processes in which the solution of kinetic equations is appropriate and, moreover, generally necessary for an understanding of the plasma dynamics. Ultimately, the problem is to solve a nonlinear partial differential equation for the distribution function of each charged plasma species in terms of six phase space variables and time. The dimensionality of the problem may be reduced through imposing certain symmetry conditions. For example, fewer spatial dimensions are needed if either the magnetic field is taken to be uniform or the magnetic field inhomogeneity enters principally through its variation along the direction of the field.

The long-term operation of rails has been studied with focus on (1) the formation and behavior of structural-phase states and nanoscale structures, (2) the modelling of the processes occurring in the surface layers of rails under severe plastic deformation and (3) the methods and techniques for assessing the structural and phase states of rails, internal stresses, and their evolution during the life cycle. The book references 264 original resources and includes their direct web link for in-depth reading. Keywords: Long Rails, Long-term Operation, Transmission Electron Microscopy, Steel, Differentiated Hardening, Structural Phase States, Nanoscale Structures, Wear, Deformation Effects, Recrystallization, Segregation, Homogenization, Relaxation, Phase Transitions, Phase Decomposition, Amorphization, Sintering, Filling of Micro- and Nanopores, Nanocapillaries, Severe Plastic Deformation, Megaplastic Deformation.

Various applications of the homogenization theory of partial differential equations resulted in the further development of this branch of mathematics, attracting an increasing interest of both mathematicians and experts in other fields. In general, the theory deals with the following: Let Ak be a sequence of differential operators, linear or nonlinepr. We want to examine the asymptotic behaviour of solutions uk to the equation Auk = f, as k ~ =, provided coefficients of Ak contain rapid oscillations. This is the case, e. g. when the coefficients are of the form a(e/x), where the function a(y) is periodic and ek ~ 0 ask~=. Of course, of oscillation, like almost periodic or random homogeneous, are of many other kinds interest as well. It seems a good idea to find a differential operator A such that uk ~ u, where u is a solution of the limit equation Au = f Such a limit operator is usually called the homogenized operator for the sequence Ak . Sometimes, the term "averaged" is used instead of "homogenized". Let us look more closely what kind of convergence one can expect for uk. Usually, we have some a priori bound for the solutions. However, due to the rapid oscillations of the coefficients, such a bound may be uniform with respect to k in the corresponding energy norm only. Therefore, we may have convergence of solutions only in the weak topology of the energy space.

'Et moi, ...* si j'avait su comment en revcnir. One service mathematics has rendered the je n'y scrais point aile.' human race. It has put common sense back where it belongs, on the topmost shclf next Jules Verne to the dusty canister labdlcd 'discarded non· The series is divergent; therefore we may be sense'. able to do something with it Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com­ puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

The greatest reward for an author is the feeling of satisfaction he gets when it becomes clear to him that readers find his work useful. After my book appeared in the USSR in 1975 I received many letters from fellow physicists including colleagues from Western European countries and the USA. Some of those letters, as well as official reviews of the book, made specific sug gestions for improving the book. The satisfaction I derived from all those kind and warm responses gave me the determination to continue work on the book in order to fulfill these wishes in the next edition. This possibility arose when one of the scientific editors from Springer-Verlag, Heidelberg, H. Latsch, who is the founder of the well-known series of quasi-monographs "Topics in Applied Physics", visited our Institute and suggested an English edition of my book. For all this, and for his subsequent help, I am sincerely thankful. I consider it my pleasant duty also to express my gratitude to the American physicist H. F. Ivey, who served as scientific editor of the trans lation. The English version of the book retains the structure of the Russian edition, though it is supplemented with many new data in the tables and figures. It reflects trends in the development of the physics and spectroscopy of laser crystals in recent years.

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were. thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various Isciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geom. eJry interacts with I physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and rpathematical programminglprofit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

Improvement in the methods of analysis of structures, machines, aircrafts and ships is one of the most important problems in engineering today. The computational aspects of this problem are being tackled successfully due to developments in computer science. However, for an adequate description of the physical properties of structures, especially those made of newer, non- traditional materials, it is essential to further study their behaviour under different load and kinematic conditions and to develop appropriate physical models that provide a comprehensive and correct description of the actual state of deformation. The objective of this book is to adopt a unified approach for describing the large number of models of internal friction and to offer recommendations regarding the methods of taking it into account at the time of dynamic analysis. It is also intended to provide a comprehensive analysis of the various models, accompanied by detailed solutions of specific problems, which could serve as examples for dynamic analysis of real structures taking into account the effect of internal friction.