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With the continued advance of computing power and accessibility, the view that "real mathematicians don't compute" no longer has any traction for a newer generation of mathematicians. The goal in this book is to present a coherent variety of accessible examples of modern mathematics where intelligent computing plays a significant role and in so doi
New mathematical insights and rigorous results are often gained through extensive experimentation using numerical examples or graphical images and analyzing them. Today computer experiments are an integral part of doing mathematics. This allows for a more systematic approach to conducting and replicating experiments. The authors address the role of
Keith Devlin and Jonathan Borwein, two well-known mathematicians with expertise in different mathematical specialties but with a common interest in experimentation in mathematics, have joined forces to create this introduction to experimental mathematics. They cover a variety of topics and examples to give the reader a good sense of the current sta
A quiet revolution in mathematical computing and scientific visualization took place in the latter half of the 20th century. These developments have dramatically enhanced modes of mathematical insight and opportunities for "exploratory" computational experimentation. This volume collects the experimental and computational contributions of Jonathan and Peter Borwein over the past quarter century.
One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).
This new approach to mathematics---the utilization of advanced computing technology in mathematical research---is often called experimental mathematics. The computer provides the mathematician with a "laboratory" in which she can perform experiments---analyzing examples, testing out new ideas, or searching for patterns. This book presents the rationale and historical context of experimental mathematics, and includes a series of examples that best portray the experimental methodology. For more examples and insights, the book, "Experimentation in Mathematics: Computational Paths to Discovery" is a highly recommended companion.
This revised and updated second edition maintains the content and spirit of the first edition and includes a new chapter, "Recent Experiences", that provides examples of experimental mathematics that have come to light since the publication of the first edition in 2003. For more examples and insights, Experimentation in Mathematics: Computational P
This revised and updated second edition maintains the content and spirit of the first edition and includes a new chapter, "Recent Experiences", that provides examples of experimental mathematics that have come to light since the publication of the first edition in 2003. For more examples and insights, Experimentation in Mathematics: Computational Paths to Discovery is a highly recommended companion.
As discrete mathematics rapidly becomes a required element of undergraduate mathematics programs, algebraic software systems replace compiled languages and are now most often the computational tool of choice. Newcomers to university level mathematics, therefore, must not only grasp the fundamentals of discrete mathematics, they must also learn to use an algebraic manipulator and develop skills in abstract reasoning. Experimental Mathematics with MAPLE uniquely responds to these needs. Following an emerging trend in research, it places abstraction and axiomatization at the end of a learning process that begins with computer experimentation. It introduces the foundations of discrete mathematics and, assuming no previous knowledge of computing, gradually develops basic computational skills using the latest version of the powerful MAPLE® software. The author's approach is to expose readers to a large number of concrete computational examples and encourage them to isolate the general from the particular, to synthesize computational results, formulate conjectures, and attempt rigorous proofs. Using this approach, Experimental Mathematics with MAPLE enables readers to build a foundation in discrete mathematics, gain valuable experience with algebraic computing, and develop a familiarity with basic abstract concepts, notation, and jargon. Its engaging style, numerous exercises and examples, and Internet posting of selected solutions and MAPLE worksheets make this text ideal for use both in the classroom and for self-study.
Borwein is an authority in the area of mathematical optimization, and his book makes an important contribution to variational analysis Provides a good introduction to the topic