A large amount of the capacity of today's computers is used for computations that can be described as computations involving real numbers. In this book, the focus is on a problem arising particularly in real number computations: the problem of veri?edor reliablecomputations. Since real numbersare objects c- taining an in?nite amount of information, they cannot be represented precisely on a computer. This leads to the well-known problems caused by unveri?ed - plementations of real number algorithms using ?nite precision. While this is t- ditionally seen to be a problem in numerical mathematics, there are also several scienti?c communities in computer science that are dealing with this problem. This book is a follow-up of the Dagstuhl Seminar 06021 on "Reliable Imp- mentation of Real Number Algorithms: Theory and Practice," which took place January 8-13, 2006. It was intended to stimulate an exchange of ideas between the di?erent communities that deal with the problem of reliable implementation of real number algorithms either from a theoretical or from a practical point of view. Forty-eight researchers from many di?erent countries and many di?erent disciplines gathered in the castle of Dagstuhl to exchange views and ideas, in a relaxed atmosphere. The program consisted of 35 talks of 30 minutes each, and of three evening sessions with additional presentations and discussions. There were also lively discussions about di?erent theoretical models and practical - proaches for reliable real number computations.

This book constitutes the thoroughly refereed post-proceedings of the 4th International Workshop on Computability and Complexity in Analysis, CCA 2000, held in Swansea, UK, in September 2000.The 23 revised full papers presented together with a report on a competition of exact real arithmetic systems were carefully selected from 28 papers accepted for presentation during the workshop. The book presents a cross-section on current research on computability and complexity in analysis.

Nine year old Herbie learns to cope with his world in spite of all the obstacles he has to overcome. He was born with special problems other kids don't have-terrible headaches, a hard time communicating, and a temper he can't always control. Herbie has learned to distrust everyone, but a teacher he gets to know while living in a group home is finally able to win his confidence. Through all the problems he faces, he learns who he can really trust, and how to communicate critical information when it is most needed. He also finds clever ways to defend himself when bullied. Throughout the story, Herbie draws on his own inner strength to meet the challenges life presents to him.