Nonsmooth analysis refers to differential analysis in the absence of differentiability. It can be regarded as a subfield of that vast subject known as nonlinear analysis. While nonsmooth analysis has classical roots （we claim to have traced its lineage hack to Dini）， it is only in the last decades that the subject has grown rapidly. To the point， in fact， that further development has sometimes appeared in danger of being stymied， due to the plethora of definitions and unclearly related theories.

This is the third edition of Principles of Real Alysis， first published in 1981. The aim of this edition is to accommodate the current needs for the traditional real analysis course that is usually taken by the senior undergraduate or by the first year graduate student in mathematics. This edition differs substantially from the second edition. Each chapter has been greatly improved by incorporating new material and by rearranging the old material. Moreover， a new chapter （Chapter 6） on Hilbert spaces and Fourier analysis has been added.

The present book comes from the first part of the lecture notes I used for a first-yeargraduate algebra course at the University of Minnesota，Purdue University，and PekingUniversity．The Chinese versions of these notes were published by The Peking UniversitvPress in 1986，and by Linking Publishing Co of Taiwan in 1987．

General Background I first became involved in the teaching of geometry about twenty years ago，when my department introduced an optional second year course on the geometry of plane curves，partly to redress the imbalance in the teaching of the subject。It Was mildly revolutionary，since it went back to an earlier sct of precepts where the differential and algebraic geometry of cuwes were pursued simultaneously，to their mutua！advantage.

This book consists of two parts. The first is devoted to the theory of curves， which are treated from both the analytic and algebraic points of view. Starting with the basic notions of the theory of Riemann surfaces the reader is lead into an exposition covering the Riemann-Roch theorem， Riemann's fundamental existence theorem.uniformization and automorphic functions. The algebraic material also treats algebraic curves over an arbitrary field and the connection between algebraic curves and Abelian varieties. The second part is an introduction to higher- dimensional algebraic geometry. The author deals with algebraic varieties， the corresponding morphisms，the theory of coherent sheaves and， finally， The theory of schemes.This book is a very readable introduction to algebraic geometry and will be immensely useful to mathematicians working in algebraic geometry and complex analysis and especially to graduate students in these fields.

本书是作者在英国留学期间完成的自编教材基础上，结合国内双语课教学的实际而编写成的，是一本概率统计的入门教材。全书共分八章，内容包括概率公理、随机变量及其分布、多元随机变量、期望与方差、大数定律与中心极限定理、随机抽样、估计问题和假设检验。各章取材注重实际，力求叙述清晰易懂，书中配有适量的例题和习题，书末附有习题答案，便于教学和学生自学。本书可以作为高等院校工科各专业、理科非数学专业以及管理与经济类等专业本科生的概率统计双语课程教材，也可以供相关科技人员参考。

This volume of the Encyclopaedia contains two articles which give a survey of modern research into non-regular Riemannian geometry，carried out mostly by Russian mathematicians．The first article written by Reshetnyak is devoted to the theory of two—dimensional Riemannian manifolds of bounded curvature．Concepts of Riemannian geometry such as the area and integral curvature of a set and the length and integral curvature of a curve are also defined for these manifolds．Some fundamental results of Riemannian geometry like the Gauss．Bonnet formula are true in the more general case considered in the book． The second article by Berestovskij and Nikolaev is devoted to the theory of metric spaces whose curvature lies between two giyen constants．The main result iS that these spaces are in fact Riemannian． This result has important applications in global Riemannian geometry． Both parts cover topics which have not yet been treated in monograph form．Hence the book will be immensely useful to graduate students and researchers in geometry，in particular Riemannian geometry．

I acknowledge and thank many people for their help with this book. In particular I thank Alan Sloan， who has unceasingly encouraged me， who wrote the first Collage software， and who so clearly envisioned the application of iterated function systems to image compression and communications that he founded a company named Iterated Systems Incorporated. Edward Vrscay， who taught the first course in deterministic fractal geometry at Georgia Tech， shared his ideas about how the course could be taught， and suggested some subjects for inclusion in this text. Steven Demko， who collaborated with me on the discovery of iterated function systems， made early detailed proposals on how the subject could be presented to students and scientists， and provided comments on several chapters. Andrew Harrington and Jeffrey Geronimo， who discovered with me orthogonal polynomials on Julia sets. My collaborations with them over five years formed for me the foundation on which iterated function systems are built. Watch for more papers from us！Les Karlovitz， who encouraged and supported my research over the last nine years， obtained the time for me to write this book and provided specific help， advice， and direction. His words can be found in some of the sentences in the text. Gunter Meyer， who has encouraged and supported my research over the last nine years. He has often given me good advice. Robert Kasriel， who taught me some topology over the last two years， corrected and rewrote my proof of Theorem 7.1 in Chapter II and contributed other help and warm encouragement. Nathanial Chafee， who read and corrected Chapter II and early drafts of Chapters III and IV. His apt constructive comments have increased substantially the precision of the writing. John Elton， who taught me some ergodic theory， continues to collaborate on exciting research into iterated function systems， and helped me with many parts of the book. Daniel Bessis and Pierre Moussa， who are filled with the wonder and mystery of science， and taught me to look for mathematical events that are so astonishing that they may be called miracles. Research work with Bessis and Moussa at Saclay during 1978， on the Diophantine Moment Problem and Ising Models， was the seed that grew into this book. Warren Stahle， who provided some of his experimental research results.

This book treats that part of Riemannian geometry related to more classical topics in a very original，clear and solid style.Before going to Riemannian geometry，the author presents a more general theory of manifolds with a linear connection.Having in mind different generalizations of Riemannian manifolds，it is clearly stressed which notions and theorems belong to Riemannian geometry and which of them are of a more general nature.Much attention is paid to transformation groups of smooth manifolds.Throughout the book，different aspects of symmetric spaces are treated.The author successfully combines the co-ordinate and invariant approaches to differential geometry，which give the reader tools for practical calculations as well as a theoretical understanding of the subject.The book contains a very useful large appendix on foundations of differentiable manifolds and basic structures on them which makes it self-contained and practically independent from other sources.The results are well presented and useful for students in mathematics and theoretical physics，and for experts in these fields.The book can serve as a textbook for students doing geometry，as well as a reference book for professional mathematicians and physicists.

《拟微分算子和Nash-Moser定理》以精练的篇幅在第一章中讲述了这一理论的核心内容。Nash-Moser定理是20世纪50年代末、60年代初的一个重要数学成果，直到今天，它仍然在微分几何、动力系统和非线性偏微分方程中有着重要的地位。它是《拟微分算子和Nash-Moser定理》第三章的论题。拟微分算子理论是20世纪50年代开始发展的一套分析工具，在偏微分方程和微分几何等领域的许多问题的研究中都有着广泛应用。这两套理论在数学文献中基本上都是分开单独处理的，而《拟微分算子和Nash-Moser定理》则在介绍这两个各自本身都有着非常重要意义的理论的同时，还阐明了它们是如何关联在一起的。通过大量的例子和习题，作者们给出了几乎所有结论的简洁而完整的证明。通过循序渐进地引进微局部分析、Littlewood-Paley理论、二进分析、仿微分算子及其在插值不等式中的应用、双曲方程（组）的能量不等式、隐函数定理等内容，作者们建立了上述两套理论之间的一座清晰的桥梁。《拟微分算子和Nash-Moser定理》可作为高等院校数学类专业的研究生学习非线性偏微分方程或几何学的教学用书，也可供对微局部分析、偏微分方程以及几何学感兴趣的数学工作者使用参考。《拟微分算子和Nash-Moser定理》对于有志打好分析基础的研究生来说是一本非常有价值的教学用书。对于从事分析或者几何方面研究的数学工作者来说，《拟微分算子和Nash-Moser定理》也是了解另一个领域的快速有效的途径。