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Since the publication of Banach's treatise on the theory of linear operators, the literature on the theory of bases in topological vector spaces has grown enormously. Much of this literature has for its origin a question raised in Banach's book, the question whether every sepa­ rable Banach space possesses a basis or not. The notion of a basis employed here is a generalization of that of a Hamel basis for a finite dimensional vector space. For a vector space X of infinite dimension, the concept of a basis is closely related to the convergence of the series which uniquely correspond to each point of X. Thus there are different types of bases for X, according to the topology imposed on X and the chosen type of convergence for the series. Although almost four decades have elapsed since Banach's query, the conjectured existence of a basis for every separable Banach space is not yet proved. On the other hand, no counter examples have been found to show the existence of a special Banach space having no basis. However, as a result of the apparent overconfidence of a group of mathematicians, who it is assumed tried to solve the problem, we have many elegant works which show the tight connection between the theory of bases and structure of linear spaces.

Historically, a major problem for the study of the large deformation of crystalline solids has been the apparent lack of unity in experimentally determined stress-strain functions. The writer's discovery in 1949 of the unexpectedly high velocity of incremental loading waves in pre-stressed large deformation fields emphasized to him the pressing need for the independent, systematic experimental study of the subject, to provide a firm foundation upon which physically plausible theories for the finite deformation of crystalline solids could be constructed. Such a study undertaken by the writer at that time and continued uninterruptedly to the present, led in 1956 to the development of the diffraction grating experiment which permitted, for the first time, the optically accurate determination of the strain-time detail of non-linear finite amplitude wave fronts propagating into crystalline solids whose prior history was precisely known. These experimental diffraction grating studies during the past decade have led to the discovery that the uniaxial stress-strain functions of 27 crystalline solids are unified in a single, generalized stress-strain function which is described, much of it hitherto unpublished, in the present monograph. The detailed study of over 2,000 polycrystal and single crystal uni­ axial stress experiments in 27 crystalline solids, in terms of the variation of a large number of pertinent parameters, has provided new unified pat­ terns of understanding which, it is hoped, will be of interest and value to theorists and experimentalists alike.

Non-linear stability problems formulated in terms of non-linear partial differential equations have only recently begun to attract attention and it will probably take some time before our understanding of those problems reaches some degree of maturity.

It is sufficient to mention that these include, among others, the logarithmic integral, the integral sine and cosine, the error integral, the Fresnel integral, the cylinder functions and the cylinder function in parabolic cylindrical coordinates.

. . . . . . . . . . . Plan and Scope of this Monograph . . . . . . . . . . . . . . . . . . . . The Dynamical Equations . . . . . . . . . . . The Principle of Material Objectivity . . . . . . The Definition of an Incompressible Simple Fluid . Static Behavior of Simple Fluids . . . . . . . . The Viscometric Functions . . . Kinematical Description .

for example, the so-called Lp approximation, the Bernstein approxima tion problem (approximation on the real line by certain entire functions), and the highly interesting studies of J.

Theoretical researches in general relativity and observational data from galactic astronomy combine in this volume in contributions to one of the oldest questions of natural philosophy: Is the structure of the physical world more adequately described by a continuous or a discrete mode of representation?

This book is an introduction to the study of methods of obtaining generating functions. Weisner's group-theoretic method is explained in detail in Chapter II and is further illustrated in Chapter III. His method of obtaining generating functions is only one of these results.

It is now stylish to teach elementary linear algebra at the sophomore college level, and we feel that the require ment that the reader should be familiar with elementary linear algebra is no more demanding than the usual requirement in most electrical engineering texts that the reader should be familiar with calculus.

The classical result for uniqueness in elasticity theory is due to Kirchhoff.

However, general theories having a thermodynamic basis, including heat conduction and constitutive equations, did not appear in general relativity until about 1955 for fluids and appeared only after 1960 for elastic or more general finitely deformed materials.

It was to be a short work on the energy theory of stability and if I had stuck to that I would have finished the writing many years ago. I am grateful to Truesdell, not so much for the invitation to spend endless hours of writing and erasing, but for the generous way he has supported my efforts and encouraged me to higher standards of good work.

In order to keep the book slender, whole fields of basic physical research had to be excluded - chiefly many-body physics, quantum field theories, and elementary particle theories.

It was to be a short work on the energy theory of stability and if I had stuck to that I would have finished the writing many years ago. I am grateful to Truesdell, not so much for the invitation to spend endless hours of writing and erasing, but for the generous way he has supported my efforts and encouraged me to higher standards of good work.

This monograph concerns the development, analysis, and application of the theory of pseudo-rigid bodies. The fact that this variety of aspects, each examined in considerable detail, can be collected together in a single, unified treat ment gives this theory an elegance that we feel sets it apart from many others.

In this book I present, in a systematic form, some local theorems on existence, uniqueness, and analytic dependence on the load, which I have recently obtained for some types of boundary value problems of finite elasticity.

This theoretical approach can actually complement and guide the computational approach by narrow- ing down trial and error procedures, pinpointing singularities and suggesting methods for handling them.

A rational study of kinematics is a treatment of the subject based on invariants, i.e., quantities that remain essentially unchanged under a change of observer. Although the kinematics of rigid bodies can be regarded as a particular case of the kinematics of continua, the former deserves attention on its own merits for several reasons.