Die Sätze von D. Klarner und N. G. de Bruijn als Exponate (German Edition)

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I am an astronomer by training, but have picked up a fair bit of mathematics along the way. It never ceases to amaze me how mathematical connections weave their way through the physical sciences. It frequently transpires that some piece of recently acquired knowledge turns out, to be just what I need to solve some apparently unrelated problem. I have therefore developed the habit of picking up and storing away odd bits of information for future use. This work has provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics.

I have also found it very difficult to find clear yet accessible explanations of technical mathematics unless I already have some familiarity with the subject. I hope this encyclopedia will provide jumping-off points for people who are interested in the subjects listed here but who, like me, are not necessarily experts. The encyclopedia has been compiled over the last 11 years or so, beginning in my college years and continuing during graduate school.

Crc concise encyclopedia of mathematics [part 1 of 4] e weisstein

The initial document was written in Microsoj? Word on a Mac Plus computer, and had reached about pages by the time I started graduate school in The selection of topics included in this compendium is not based on any fixed set of criteria, but rather reflects my own random walk through mathematics. In truth, there is no good way of selecting topics in such a work. The mathematician James Sylvester may have summed up the situation most aptly.

As he points out, mathematics itself cannot be confined to the pages of a book. The results of mathematics, however, are shared and passed on primarily through the printed and now electronic medium.


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While there is no danger of mathematical results being lost through lack of dissemination, many people miss out on fascinating and useful mathematical results simply because they are not aware of them. Not only does collecting many results in one place provide a single starting point for mathematical exploration, but it should also lessen the aggravation of encountering explanations for new concepts which themselves use unfamiliar terminology.

In this work, the reader is only a cross-reference or a mouse click away from the necessary background material. Strangely, except for relatively small works usually on particular subjects, there do not appear to have been any substantial attempts to collect and display in a place of prominence the treasure trove of mathematical results that have been discovered invented?

Finally, a few words about logistics. Because of the alphabetical listing of entries in the encyclopedia, neither table of contents nor index are included. The alphabetization of entries contains a few peculiarities which need mentioning. In this encyclopedia, I have endeavored to use the most, widely accepted form.

I have also tried to consistently give entry titles in the singular e. In cases where the same word is applied in different contexts, the context is indicated in parentheses or appended to the end. In an endeavor of this magnitude, errors and typographical mistakes are inevitable. Although the current length makes extensive additions in a printed I plan to continue updating, correcting, and improving the work,.

Although I alone have compiled and typeset this work, many people have contributed indirectly and directly to its creation. I have not yet had the good fortune to meet Donald Knuth of Stanford University, but he is unquestionably the person most directly responsible for making this work possible.

Before his mathematical typesetting program TEX, it would have been impossible for a single individual to compile such a work as this. Had Prof. Bateman owned a personal computer equipped with T , perhaps his shoe box of notes would not have had to await the labors of Erdelyi, Magnus, and Oberhettinger to become a three-volume work on mathematical functions. I hope that Steven will someday publish his detailed essays in book form.

Thanks also to Neil Sloane and Simon Plouffe for compiling and making available the printed and on-line www research.


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Thanks to Robert Dickau, Simon Plouffe, and Richard Schroeppel for reading portions of the manuscript and providing a number of helpful suggestions and additions. Thanks also to algebraic topologist Ryan Budney for sharing some of his expertise, to Charles Walkden for his helpful comments about dynamical systems theory, and to Lambros Lambrou for his contributions.


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Thanks to David W. Wilson for a number of helpful comments and corrections. Thanks to Dale Rolfsen, compiler James Bailey, and artist Ali Roth for permission to reproduce their beautiful knot and link diagrams.

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Thanks to Gavin Theobald for providing diagrams of his masterful polygonal dissections. Sincere thanks to Judy Schroeder for her skill and diligence in the monumental task of proofreading the entire document for syntax. Many thanks to the library staff at the University of Virginia, who have provided invaluable assistance in tracking down many an obscure citation. Finally, I would like to thank the hundreds of people who took the time to e-mail me comments and suggestions while this work was in its formative stages.

Your continued comments and feedback are very welcome. Although the number I used to be considered a PRIME NUMBER, it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own. Ifonly one root, solution, etc. For example, the number The largest known such number. Madachy Cook , no. Madachy, J. New York: Dover, pp. Pickover, C. A, Keys to Infinity. New York: W. Sloane, N. References RecmBall, W. Mathematical ations and Essays, 13th ed. Bogomolny, A. Johnson, W. Kraitchik, M.

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Story, W. It consists of 15 squares numbered from 1 to 15 which are placed in a 4 x 4 box leaving one position out of the 16 empty. The goal is to rearrange the squares from a given arbitrary starting arrangement by sliding them one at a time into the configuration shown above. For some initial arrangements, this rearrangement is possible, but for others, it is not. To address the solubility of a given initial arrangement, proceed as follows.

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Then define For example, in the following arrangement. References Carr, M. Fischer, R. Lefevre, V.

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Shell Centre for Mathematical Education. Formally, for a given N, does there exist a sequence of real numbers xl, x2,. Steinhaus gives a point solution 0. Surprisingly, possible to place 17 points in this manner and Graham , Warmus Warmus, M. Number Th. Now repeat the procedure with the SUM so obtained.

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For example, starting with the number produces , , , The end results of applying the algorithm to 1, 2, 3,. The value for 89 is especially large, being Lattices J. H, and Sloane, N. Sphere-Packings, and Groups, 2nd ed. New York: Springer-Verlag, Coxeter, H. Introduction to Geometry, 2nd ed. New York: Wiley, p. Coxeter, I-l. Regular Polytopes, 3rd ed.

New York: Dover, Extended entry in The Encyclopedia of Integer Sequences. San Diego: Academic Press,. The smallest numbers l. Reterences Adams, D. References Berlekamp, E. Gardner, M. Steinhaus, H. New York: Knopf, pp. Eruenberger, F. Some interesting properties as well as a few arcane ones not reiterated here of the number are discussed in Beeler et al.