Book Reviews

Psriodica Mathemat~ca Hungarica V ol. 20 (2), (1989), pp. 173--175 BOOK REVIEWS Ismael Herrera, Boundary methods: An algebraic theory, 136 pages, Pitman, Bos- ton, 1984. Abstract formulations of boundary value problems have been so far known only for ordinary differential equations (see e.g. in the monograph of Dunford and Schwartz). I. Herrera develope~t in the last twenty five years an abstract method applicable for linear partial differential equations too. This book is an integrated presentation of this method. Herrera's theory is based upon an algebraic structure which occurs in linear boundary value problems. In Part 1 of the book the abstract theory is developed. The algebraic theory uses functional valued operators on vector spaces without inner product. The crucial point in the algebraic development is the introduction of abstract Green formulas. The author gives a characterization of Green formulas by means of some pairs of subspaces and shows a method to obtain abstract Green formulas for the union Of two domains using that of the parts. This material is accompanied by several examples. In Part 2 the author investigates the Trefftz method and shows that every Green formula corresponds to an immersion in some Hilbert space. Finally some algorithms of the solution of boundary value and initial-boundary value problems are discussed. I. Jo6 (Budapest) H. Attouch, Variational convergence for functions and operators, 423 pages, Pitman, Boston, 1984. In the last twenty years new types of convergence appeared in the mathematical literature for sequences of functions and operators. The aim of these notions is to approach the limit of sequences of va~tional problems, hence they are called variational con- vergences. The book is an integrated presentation of the results of this type appeared in. a large number of papers. The book contains the case of a special variation~fl problem, the so called minimization problems. The corresponding convergence of functions is called in this case epi-eonvergenee. The book is divided into three chapters. In Chapter 1 the definition of epi-convergence is given in a general topological setting and there are given interesting physical applications. Chapter 2 contains a complete description of the topological properties of epi-convergence and, at last, the third and final chapter is devoted to the study of epi-convergence of convex functions. The author indicates many new extensions and new fields of application of variation convergence appeared in the literature in the last few years. I. Jo5 (Budapest) Wilbur Knott, The ancient tradition of geometric problems, ix~-411 pages, Birk- h~user, Boston, 1986. Looking through the history of famous geometric problems such as the cube duplication, angle trisection, and circle quadrature, it is possible to follow the main course of the geometry from the pre-Euclidean period through the Hellenistic and Middle Akad~iat Kiadd, ~udapest Kluwcr Ac~,mic Publishers, Dordred~ 174 BOOK RE¥II~WS Age period into the nineteen century, The ancient efforts for solving these "classical" problems are of a special interest of the present book. However, what was understood under the "problem" in the Greek geometry Of ancient time ? "Within the ancient geometry, a geometric ~problem~ seeks the construc- tion of a figure corresponding to a specific description. The solution to any problem requires for its completion an appeal to the constructions in other problems already solved, and in turn will be applied to the solutions of yet others. In effect, then, the corpus of solved problems forn~s an ordered sequence in which each problem can be reduced to those preceding." This book is conceived "as an exploratory essay, intended to reveal the opportuni- ties which the evidence available to us provides for an interpretation of the ancient field". To begin with the table of contents, we note that the book is divided into eight chapters, with the short Chapter 1 where three major recommendations bearing on the present study of the ancient problem-solving efforts are given. First, "to establish the simple chronological sequence of the various solutions". Second, "to devote separate treatment to several important metamathematieal issues: how the ancient divided the geometric field according to the types of problems and solving methods; what they viewed the special role of problems to be, especially in relation to that of theorems, and how they associated these with the important methods of analysis and synthesis; what conditions they imposed on the techniques admissible for the solution of problems, and wheather they judged that satisfactory solutions for the three special problems had actually been found". Third, "to take up the textual issues bearing on our use of authors from the later antiquity". Chapters 2--7 are dedicated to investigations "on special problems as a cross section, suitable for filling out one's portrait of the wider field". The pre-Euelldean history of Delian problem of the cube duplication, and the problem of the quadrature of the circle is given in the Chapters 2 and 3. The special attention is paid to the geometers (Archytas, Eudoxus, lYienaechmus, ... ) in Plato's Academy in Athens when it became the center for geometric studies. Chapter 4 treats the work of Euclid of Alexandria, and of geometers from his generation that witnessed important advances in the field of geometric problem solving. Archimedean problem-solving methods via conic sections and neu~e8 (~sva~, equivalent to a eublc equation) are described in the Chapter 5. Moreover, one section of this chapter is dedicated to the solution of the problem of circle quadrature by number ~, and to the Arehimedean spirals. Chapter 6 discusses the "golden age" of interest in the three classical problems; that is the later part of the 3rd century B. C. From that time survived fragments from the work of a series of Archimedes' successors: Eratosthenes, Nicomedes, Hippias, ]:)iocles, Dionysodorus, Perseus, and Zenodorus. In the Chapter 7 we are treated to the work of Appollonius of Perga. His "Gon/cs, Solid Loci and other works reported by Pappus represented the high point of the ancient classical tradition. Their later editions, even in fragmentary form, continued to challenge geometers until well into modern times". The goal of the Chapter 8 is to discover what the ancient views were by means of a consideration of the relevant historical evidence. This Chapter is concerned with the second problem mentioned in the Chapter 1. In the list of references at the end of the book author endeavored to include only those contributions which seemed to him "historically and technically stimulating". The immense secondary literature on the history of ancient problems and solving methods could not be attempted here. Book is to be recommended to the readers interested in the early history of geometric constructions. Z. ~ (Beograd) ProbabiHty theory and harmonie analysis, edited by J.-A. CHAO and W. A. WOYCZYI~TSKI (Pure and Applied Mathematics, 98), vlii~-291 pages, M. Dekker, New York, 1986. This volume of the series "Monographs and Textbooks in Pure and Applied Mathematics" is a collection of fifteen papers in probability theory and harmonic analysis. The papers are concerned with the interaction between these areas which is one of the most successful field of mathematics over the past two decades. Among the authors .

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