Stability theorems in geometry and analysis
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Bibliographic Information
Stability theorems in geometry and analysis
（Mathematics and its applications, v. 304）
Kluwer Academic Publishers, c1994
- Other Title
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Teoremy ustoĭchivosti v geometrii i analize
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Note
Revised and updated translation of Teoremy ustoĭchivosti v geometrii i analize, translated from the Russian by N.S. Dairbekov and V.N. Dyatlov, and edited by S.S. Kutateladze
Includes bibliographical references (p. [383]-391) and index
Description and Table of Contents
Description
1. Preliminaries, Notation, and Terminology n n 1.1. Sets and functions in lR. * Throughout the book, lR. stands for the n-dimensional arithmetic space of points x = (X},X2,'" ,xn)j Ixl is the length of n n a vector x E lR. and (x, y) is the scalar product of vectors x and y in lR. , i.e., for x = (Xl, X2, *.* , xn) and y = (y}, Y2,**., Yn), Ixl = Jx~ + x~ + ...+ x~, (x, y) = XIYl + X2Y2 + ...+ XnYn. n Given arbitrary points a and b in lR. , we denote by [a, b] the segment that joins n them, i.e. the collection of points x E lR. of the form x = >.a + I'b, where>. + I' = 1 and >. ~ 0, I' ~ O. n We denote by ei, i = 1,2, ...,n, the vector in lR. whose ith coordinate is equal to 1 and the others vanish. The vectors el, e2, ...,en form a basis for the space n lR. , which is called canonical. If P( x) is some proposition in a variable x and A is a set, then {x E A I P(x)} denotes the collection of all the elements of A for which the proposition P( x) is true.
Table of Contents
Foreword to the English Translation. Preface to the First Russian Edition. 1. Introduction. 2. Moebius Transformations. 3. Integral Representations and Estimates for Differentiable Functions. 4. Stability in Liouville's Theorem on Conformal Mappings in Space. 5. Stability of Isometric Transformations of the Space Rn. 6. Stability in Darboux's Theorem. 7. Differential Properties of Mappings with Bounded Distortion and Conformal Mappings of Riemannian Spaces. References. Subject Index.
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