Projective Geometry - Volume I
Publisher: Potter Press | 2007-03-15 | ISBN:1406747173 | Pages:352 | DJVU | 4 MB
PEOJECTIVE GEOMETRY BY OSWALD VEBLEN PROFESSOR OP MATHEMATICS, PRINCETON UNIVERSITY AND JOHN WESLEY YOUNG PROFESSOR OF MATHEMATICS, UNIVERSITY OF KANSAS VOLUME I GINN AND COMPANY BOSTON NEW YORK CHICAGO LONDON PREFACE Geometry, which had been for centuries the most perfect example of a deductive science, during the creative period of the nineteenth century outgrew its old logical forms. The most recent period has however brought a clearer understanding of the logical foundations of mathematics and thus has made it possible for the exposition of geometry to resume the purely deductive form. But the treatment in the books which have hitherto appeared makes the work of lay ing the foundations seem so formidable as either to require for itself a separate treatise, or to be passed over without attention to more than the outlines. This is partly due to the fact that in giving the complete foundation for ordinary real or complex geometry, it is necessary to make a study of linear order and continuity, a study which is not only extremely delicate, but whose methods are those of the theory of functions of a real variable rather than of elemen tary geometry. The present work, which is to consist of two volumes and is in tended to be available as a text in courses offered in American uni versities to upper-class and graduate students, seeks to avoid this difficulty by deferring the study of order and continuity to the sec ond volume. The more elementary part of the subject rests on a very simple set of assumptions which characterize what may be called general protective geometry. It will be found that the theorems selected on this basis of logical simplicity are also elemen tary in the sense of being easily comprehended and often used. Even the limited space devoted in this volume to the foundations may seem a drawback from the pedagogical point of view of some mathematicians. To this we can only reply that, in our opinion, an adequate knowledge of geometry cannot be obtained without attention to the foundations. We believe, moreover, that the abstract treatment is peculiarly desirable in protective geometry, because it is through the latter that the other geometric disciplines are most readily coordinated. Read More »
Ian Graham, Gabriela Kohr, “Geometric Function Theory in One and Higher Dimensions (Pure and Applied Mathematics, Vol 255) (Pure and Applied Mathematics)”
Publisher: CRC | Pages: 528 | ISBN: 0824709764 | PDF | 16.6 MB
This reference details valuable results that lead to improvements in existence theorems for the Loewner differential equation in higher dimensions, discusses the compactness of the analog of the Caratheodory class in several variables, and studies various classes of univalent mappings according to their geometrical definitions. It introduces the infinite-dimensional theory and provides numerous exercises in each chapter for further study. The authors present such topics as linear invariance in the unit disc, Bloch functions and the Bloch constant, and growth, covering and distortion results for starlike and convex mappings in Cn and complex Banach spaces.
Steven G. Krantz, “Geometric Function Theory: Explorations in Complex Analysis”
Birkhäuser Boston; 1 edition (September 20, 2005) | ISBN: 0817643397 | 314 pages | PDF | 2 Mb
Complex variables is a precise, elegant, and captivating subject. Presented from the point of view of modern work in the field, this new book addresses advanced topics in complex analysis that verge on current areas of research, including invariant geometry, the Bergman metric, the automorphism groups of domains, harmonic measure, boundary regularity of conformal maps, the Poisson kernel, the Hilbert transform, the boundary behavior of harmonic and holomorphic functions, the inhomogeneous CauchyRiemann equations, and the corona problem.
The author adroitly weaves these varied topics to reveal a number of delightful interactions. Perhaps more importantly, the topics are presented with an understanding and explanation of their interrelations with other important parts of mathematics: harmonic analysis, differential geometry, partial differential equations, potential theory, abstract algebra, and invariant theory. Although the book examines complex analysis from many different points of view, it uses geometric analysis as its unifying theme.