This book comprehensively covers all aspects of distribution, taxonomy, life cycle, cultivation, application in traditional medicine, their secondary metabolites, and their biological properties, along with various parameters to yield improvement in selected species of Cordyceps. It covers the history, diversity, ecology, taxonomy, phylogeny, genomics, proteomics, metabolomics, various techniques of cultivation, preservation, significance in human health (traditional remedies, modern therapeutics, bioactive potential and curative potential of lifestyle diseases), industrial significance (pigments, nutraceuticals, cosmeceuticals, bioactive macromolecules, fermentation products, metabolic pathways and industrial products), biocontrol potential, market potential and conservation. This book will also highlight the future directions of application-oriented research and the market potential of this economically important genus.

This treatment of differential geometry and the mathematics required for general relativity makes the subject of this book accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra.

The recent most impressive developments in theoretical physics, i.e. gauge and string theories with their far- reaching relations to basic concepts of present day mathematics brought a renewed and intense exchange between mathematicians and physicists. In a way the past few years might be compared to the great period of the mid-twenties with its really decisive mathematical contributions to the foundations of both quantum theory as well as general relativity .. Among others, this far-reaching exchange led H.Weyl in 1929 to the formulation of his (second) gauge principle (concerning the phase of the quantum mechanical wave function) which led, in turn (through the most natural generalization by C.N.Yang and R.Mills in 1954) to present day gauge theory which brought, first of all, a real breakthrough with respect to a theoretical inter- pretation of an enormous realm of data in modern particle physics. It led, in fact, to the well-known unification of electromagnetism with weak interactions (Weinberg-Salam theory) and, at the same time, to a most natural unifica- tion of particle and nuclear physics through QCD which deals with the basic quark-gluon structure of heavy matter, e.g. hadrons.

Scale is a concept the antiquity of which can hardly be traced. Certainly the familiar phenomena that accompany sc ale changes in optical patterns are mentioned in the earliest written records. The most obvious topological changes such as the creation or annihilation of details have been a topic to philosophers, artists and later scientists. This appears to of fascination be the case for all cultures from which extensive written records exist. For th instance, chinese 17 c artist manuals remark that "distant faces have no eyes" . The merging of details is also obvious to many authors, e. g. , Lucretius mentions the fact that distant islands look like a single one. The one topo­ logical event that is (to the best of my knowledge) mentioned only late (by th John Ruskin in his "Elements of drawing" of the mid 19 c) is the splitting of a blob on blurring. The change of images on a gradual increase of resolu­ tion has been a recurring theme in the arts (e. g. , the poetic description of the distant armada in Calderon's The Constant Prince) and this "mystery" (as Ruskin calls it) is constantly exploited by painters.

For a number of years, beginning with the early 70's, the authors have been delivering lectures on the fundamentals of geometry and topology in the Faculty of Mechanics and Mathematics of Moscow State University. This text-book is the result of this work. We shall recall that for a long period of time the basic elements of modern geometry and topology were not included, even by departments and faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in classical differential geometry have gradually become outdated, and there has been, hitherto, no unanimous standpoint as to which parts of modern geometry should be viewed as abolutely essential to a modern mathematical education. In view of the necessity of using a large number of geometric concepts and methods, a modernized course in geometry was begun in 1971 in the Mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. In addition to the traditional geometry of curves and surfaces, the course included the fundamental priniciples of tensor analysis, Riemannian geometry and topology. Some time later this course was also introduced in the division of mathematics. On the basis of these lecture courses, the following text-books appeared:

The problems being solved by invariant theory are far-reaching generalizations and extensions of problems on the "reduction to canonical form" of various is almost the same thing, projective geometry. objects of linear algebra or, what Invariant theory has a ISO-year history, which has seen alternating periods of growth and stagnation, and changes in the formulation of problems, methods of solution, and fields of application. In the last two decades invariant theory has experienced a period of growth, stimulated by a previous development of the theory of algebraic groups and commutative algebra. It is now viewed as a branch of the theory of algebraic transformation groups (and under a broader interpretation can be identified with this theory). We will freely use the theory of algebraic groups, an exposition of which can be found, for example, in the first article of the present volume. We will also assume the reader is familiar with the basic concepts and simplest theorems of commutative algebra and algebraic geometry; when deeper results are needed, we will cite them in the text or provide suitable references.