This monograph attempts to present the results known today on bases in Banach spaces and some unsolved problems concerning them. Although this important part of the theory of Banach spaces has been studied for more than forty years by numerous mathematicians, the existing books on functional analysis (e. g. M. M. Day [43], A. Wilansky [263], R. E. Edwards [54]) contain only a few results on bases. A survey of the theory of bases in Banach spaces, up to 1963, has been presented in the expository papers [241], [242] and [243], which contain no proofs; although in the meantime the theory has...

In recent years, the study of the theory of Brownian motion has become a powerful tool in the solution of problems in mathematical physics. This self-contained and readable exposition by leading authors, provides a rigorous account of the subject, emphasizing the "explicit" rather than the "concise" where necessary, and addressed to readers interested in probability theory as applied to analysis and mathematical physics.
A distinctive feature of the methods used is the ubiquitous appearance of stopping time. The book contains much original research by the authors (some of which published...

The small book by Shimura-Taniyama on the subject of complex multi is a classic. It gives the results obtained by them (and some by Weil) plication in the higher dimensional case, generalizing in a non-trivial way the method of Deuring for elliptic curves, by reduction mod p. Partly through the work of Shimura himself (cf. Sh 1] Sh 2], and Sh 5]), and some others (Serre, Tate, Kubota, Ribet, Deligne etc.) it is possible today to make a more snappy and extensive presentation of the fundamental results than was possible in 1961. Several persons have found my lecture notes on this subject...

Every positive integer m has a product representation of the form where v, k and the ni are positive integers, and each Ei = I. A value can be given for v which is uniform in the m. A representation can be computed so that no ni exceeds a certain fixed power of 2m, and the number k of terms needed does not exceed a fixed power of log 2m. Consider next the collection of finite probability spaces whose associated measures assume only rational values. Let hex) be a real-valued function which measures the information in an event, depending only upon the probability x with which that event occurs....

Capillarity phenomena are all about us; anyone who has seen a drop of dew on a plant leaf or the spray from a waterfall has observed them. Apart from their frequently remarked poetic qualities, phenomena of this sort are so familiar as to escape special notice. In this sense the rise of liquid in a narrow tube is a more dramatic event that demands and at first defied explanation; recorded observations of this and similar occur rences can be traced back to times of antiquity, and for lack of expla nation came to be described by words deriving from the Latin word "capillus," meaning hair. It...

The aim of" the present monograph is two-fold: (a) to give a short account of the main results concerning the theory of random systems with complete connections, and (b) to describe the general learning model by means of random systems with complete connections. The notion of chain with complete connections has been introduced in probability theory by ONICESCU and MIHOC (1935a). These authors have set themselves the aim to define a very broad type of dependence which takes into account the whole history of the evolution and thus includes as a special case the Markovian one. In a sequel of...

Das vorliegende Buch ist aus einer langjahrigen Beschaftigung mit den praktischen Anwendungen hervorgegangen. Seit meiner Studenten zeit sind immer wieder Volkswirtschaftler, Mediziner, Physiologen, Biologen und Ingenieure mit statistischen Fragen zu mir gekommen. Durch N achdenken und Literaturstudium habe ich immer bessere Methoden kennengelernt. Diese Methoden sollen hier begrundet und auf moglichst lehrreiche Beispiele aus den N atur- und Sozialwissen schaften angewandt werden. So hoffe ich, dem Leser manche Irrwege, die ich anfangs gegangen bin, zu ersparen. Die Beispiele sind nicht aus...

Als ich im Jahre 1958 mit den Vorarbeiten zu diesem Buch begann, schien es noch moglich, eine einigermal3en vollstandige Darstellung der Strukturtheorie endlicher Gruppen in einem Bande zu geben. Die stiir mische Entwicklung, welche die Theorie seitdem erlebt hat (das Literatur verzeichnis gibt einen Eindruck davon), hat diese Zielsetzung unmoglich gemacht. Der vorliegende erste Band enthalt neben den Grundbegriffen die Theorie der nilpotenten, p-nilpotenten und auflosbaren Gruppen sowie die gewohnliche Darstellungstheorie. Da die Entwicklung der letzten Jahre nicht in diesen Gebieten ihren...

Our presentation of some fundamental domains of celestial mechanics requires no special preliminary knowledge; however, the chosen mathe matical method is new in so far as the pure two-body motion is described by linear differential equations, which even have constant coefficients. In other words an equivalence between the Keplerian motion and a harmonic oscillation is established; this approach to celestial mechanics will be referred to as the linear theory. Besides the possibility of the mutual fruitful interaction between celestial and oscillatory mechanics which is thereby created, our...

When I first considered writing a book about multipliers, it was my intention to produce a moderate sized monograph which covered the theory as a whole and which would be accessible and readable to anyone with a basic knowledge of functional and harmonic analysis. I soon realized, however, that such a goal could not be attained. This realization is apparent in the preface to the preliminary version of the present work which was published in the Springer Lecture Notes in Mathematics, Volume 105, and is even more acute now, after the revision, expansion and emendation of that manuscript needed...