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The importance of mathematics competitions has been widely recognised for three reasons: they help to develop imaginative capacity and thinking skills whose value far transcends mathematics; they constitute the most effective way of discovering and nurturing mathematical talent; and they provide a means to combat the prevalent false image of mathematics held by high school students, as either a fearsomely difficult or a dull and uncreative subject. This book provides a comprehensive training resource for competitions from local and provincial to national Olympiad level, containing hundreds of diagrams, and graced by many light-hearted cartoons. It features a large collection of what mathematicians call "beautiful" problems - non-routine, provocative, fascinating, and challenging problems, often with elegant solutions. It features careful, systematic exposition of a selection of the most important topics encountered in mathematics competitions, assuming little prior knowledge. Geometry, trigonometry, mathematical induction, inequalities, Diophantine equations, number theory, sequences and series, the binomial theorem, and combinatorics - are all developed in a gentle but lively manner, liberally illustrated with examples, and consistently motivated by attractive "appetiser" problems, whose solution appears after the relevant theory has been expounded. Each chapter is presented as a "toolchest" of instruments designed for cracking the problems collected at the end of the chapter. Other topics, such as algebra, co-ordinate geometry, functional equations and probability, are introduced and elucidated in the posing and solving of the large collection of miscellaneous problems in the final toolchest. An unusual feature of this book is the attention paid throughout to the history of mathematics - the origins of the ideas, the terminology and some of the problems, and the celebration of mathematics as a multicultural, cooperative human achievement. As a bonus the aspiring "mathlete" may encounter, in the most enjoyable way possible, many of the topics that form the core of the standard school curriculum.
Digital Imaging targets anyone with an interest in digital imaging, professional or private, who uses even quite modest equipment such as a PC, digital camera and scanner, a graphics editor such as PAINT, and an inkjet printer. Uniquely, it is intended to fill the gap between the highly technical texts for academics (with access to expensive equipment), and the superficial introductions for amateurs. The four-part treatment spans theory, technology, programs and practice. Theory covers integer arithmetic, additive and subtractive color, greyscales, computational geometry, and a new presentation of discrete Fourier analysis; Technology considers bitmap file structures, scanners, digital cameras, graphic editors, and inkjet printers; Programs develops several processing tools for use in conjunction with a standard Paint graphics editor and supplementary processing tools; Practice discusses 1-bit, greyscale, 4-bit, 8-bit, and 24-bit images for the practice section. Relevant QBASIC code is supplied an accompanying CD and algorithms are listed in the appendix. Readers can attain a level of understanding and the practical insights to obtain optimal use and satisfaction from even the most basic digital-imaging equipment.
Cryptography has experienced rapid development, with major advances recently in both secret and public key ciphers, cryptographic hash functions, cryptographic algorithms and multiparty protocols, including their software engineering correctness verification, and various methods of cryptanalysis. This textbook introduces the reader to these areas, offering an understanding of the essential, most important, and most interesting ideas, based on the authors' teaching and research experience. After introducing the basic mathematical and computational complexity concepts, and some historical context, including the story of Enigma, the authors explain symmetric and asymmetric cryptography, electronic signatures and hash functions, PGP systems, public key infrastructures, cryptographic protocols, and applications in network security. In each case the text presents the key technologies, algorithms, and protocols, along with methods of design and analysis, while the content is characterized by a visual style and all algorithms are presented in readable pseudocode or using simple graphics and diagrams. The book is suitable for undergraduate and graduate courses in computer science and engineering, particularly in the area of networking, and it is also a suitable reference text for self-study by practitioners and researchers. The authors assume only basic elementary mathematical experience, the text covers the foundational mathematics and computational complexity theory.
The purpose of this book is to prepare the reader for coping with abstract mathematics. The intended audience is both students taking a first course in abstract algebra who feel the need to strengthen their background, and those from a more applied background who need some experience in dealing with abstract ideas. Learning any area of abstract mathematics requires not only ability to write formally but also to think intuitively about what is going on and to describe that process clearly and cogently in ordinary English. Ash tries to aid intuition by keeping proofs short and as informal as possible, and using concrete examples as illustration. Thus it is an ideal textbook for an audience with limited experience in formalism and abstraction. A number of expository innovations are included, for example, an informal development of set theory which teaches students all the basic results for algebra in one chapter.