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Normally, mathematical research has been divided into “pure” and “applied,” and only within the past decade has this distinction become blurred. However, differential geometry is one area of mathematics that has not made this distinction and has consistently played a vital role in both general areas. The papers in this volume represent the proceedings of a conference entitled “Differential Geometry: The Interface Between Pure and Applied Mathematics,” which was held in San Antonio, Texas, in April 1986. The purpose of the conference was to explore recent exciting applications and challenging classical problems in differential geometry. The papers represent a tremendous range of applications and techniques in such diverse areas as ordinary differential equations, Lie groups, algebra, numerical analysis, and control theory.

This concise textbook presents the mathematics that is foundational to multimedia applications. Featuring a rigorous survey of selected results from algebra and analysis, the work examines tools used to create application software for multimedia signal processing and communication. Key features include: • Over 100 exercises with complete solutions; • Useful algorithms presented in pseudocode and Standard C to help readers with programming, experimentation, and the solution of exercises; • Numerous illustrations based on data from real studies; • Suggestions for further reading at the end of each chapter; • A companion website—maintained by the author—providing computer programs described in the book as well as additional references and data files, such as images and sounds, to enhance the reader's understanding of key topics; • A supplementary manual—containing several hundred exercises, solutions, and sample programs not included in the book—available to instructors upon request; • Minimal prerequisites—only an undergraduate-level knowledge of mathematics, not including statistics, is required. Mathematics for Multimedia is an ideal textbook for upper undergraduate and beginning graduate students in pure and applied mathematics, engineering, and computer science seeking an innovative approach to contemporary mathematics with practical applications. The work may also serve as a useful reference for multimedia applications developers and other researchers and practitioners interested in the mathematics underlying multimedia software design and implementation.

The attraction of quantum computation and quantum communica­ tion theory and experiments hes in the fact that we engineer both them themselves and the quantum systems they treat. This approach has turned out to be very resiUent. Driven by the final goal of calculating exponentially faster and communicating infinitely more securely than we do today, as soon as we encounter a limitation in either a theory or experiment, a new idea around the no-go emerges. As soon as the decoherence'demon'threatened the first computation models, quan­ tum error correction theory was formulated and applied not only to computation theory but also to communication theory to make it un­ conditionally secure. As soon as liquid-state nuclear magnetic resonance experiments started to approach their limits, solid-based nuclear spin experiments—the Kane computer—came in. As soon as it was proved that it is theoretically impossible to completely distinguish photon Bell states, three new approaches appeared: hyperentanglement, the use of continuous variables, and the Knill-Laflamme-Milburn proposal. There are many more such examples. What facilitated all these breakthroughs is the fact that at the present stage of development of quantum computation and communication, we deal with elementary quantum systems consisting of several two-level systems. The complexity of handling and controlHng such simple sys­ tems in a laboratory has turned out to be tremendous, but the basic physical models we follow and calculate for the systems themselves are not equally intricate.

This detail-oriented text is intended for engineers and applied mathematicians who must write computer programs to perform wavelet and related analysis on real data. It contains an overview of mathematical prerequisites and proceeds to describe hands-on programming techniques to implement special programs for signal analysis and other applications.