Alhazen'S Billiard Problem Extended Essay

Alhazen'S Billiard Problem Extended Essay-89
This work has provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics. Ackermann Number A number of the form n t • • • T™ where Arrow Nota- n TION has been used.

This work has provided a mechanism for organizing what has turned out to be a fairly large collection of mathematics. Ackermann Number A number of the form n t • • • T™ where Arrow Nota- n TION has been used.

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First, the entire text and all the equations and figures are available in searchable electronic form on CD-ROM. see also Crunode, Spinode, Tacnode Acoptic Polyhedron A term invented by B. see also Addition, Augend, Plus, Radicand Addition i i - 15 8- J- 249 * 407- Y carries (-addend 1 Y addend 2 hsum The combining of two or more quantities using the PLUS operator.

Second, the entries are extensively cross-linked and cross-referenced, not only to related entries but also to many external sites on the Internet. It also provides a highly efficient way to "navigate" from one related concept to another, a feature that is especially powerful in the electronic version. Griinbaum in an attempt to pro- mote concrete and precise POLYHEDRON terminology. The individual numbers being combined are called ADDENDS, and the total is called the Sum. Addition Chain An addition chain for a number n is a SEQUENCE 1 = ao 2 15 contain a in base 3, which would imply that the maximum persistence in base 3 is 3 (Guy, 1994).

These formulas are comprehensible to readers with only minimal mathematical background, and are interesting to both those with and without formal mathematics training. n l X n (i) (2) be the step interval, and consider the Maclaurin Se- ries of y about x n , y n i = y n ( -T-) ( x ~ x n) (x - x n ) 2 .

However, they have not previously been collected in a single convenient location. V dx J n 1 \dxj n \ dx 2 J (3) (4) Here, the Derivatives of y are given by the Backward Differences \dx/ n Xn i ~ X 3/n i - y n h (5) (6) (7) etc.

While undertaking the Word to T^}K conversion, I also began cross-referencing entries, anticipating that eventually I would be able to convert the entire document to hypertext. 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.

This hope was realized beginning in 1995, when the Internet explosion was in full swing and I learned of Nikos Drakos's excellent I^X to HTML converter, I£Tg X2HTML. In truth, there is no good way of selecting topics in such a work. CRC Concise Encyclopedia MAm EMAf JCS CRC Concise Encyclopedia MAffl EMAff G Eric W. This problem is often further compounded by the difficulty in locating concrete and easily understood examples. ../ (y 2 Cyi ~yi£y 2 )dx = — {yi 2/2 - 2/13/2 ) Po (3) see a/50 Self-Adjoint Operator, Sturm-Liouville Theory Adjugate Matrix see Adjoint Matrix Adjunction If a is an element of a Field F over the PRIME Field P, then the set of all RATIONAL FUNCTIONS of a with Coefficients in P is a Field derived from P by ad- junction of a. Weisstein CRC Press Boca Raton London New York Washington, D. Library of Congress Cataloging-in-Publication Data Weisstein, Eric W. To give perspective to a subject, I find it helpful to learn why it is useful, how it is connected to other areas of mathematics and science, and how it is actually implemented. Adleman-Pomerance-Rumely Primality Test A modified Miller's Primality Test which gives a guarantee of Primality or COMPOSITENESS. "On Distinguishing Prime Numbers from Composite Number." Ann. This work attempts to provide enough details to give the reader a flavor for a subject without getting lost in minutiae. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. It was simplified by Cohen and Lenstra (1984), implemented by Cohen and Lenstra (1987), and subsequently optimized by Bosma and van der Hulst (1990). While absolute rigor may suffer somewhat, I hope the improvement in usefulness and readability will more than make up for the deficiencies of this approach. Cambridge, England: Cam- bridge University Press, p. Addend A quantity to be Added to another, also called a Sum- MAND. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Adele An element of an Adele GROUP, sometimes called a Repartition in older literature. For zero argument, Ai(0) 3 -2/3 (13) where T(z) is the GAMMA FUNCTION. W45 1998 510'.3— DC21 98-22385 CIP This book contains information obtained from authentic and highly regarded sources. Using the identity K n (x) TV I-n(x) - I n (x) 2 sin(n7r) (5) where K{z) is a MODIFIED BESSEL FUNCTION OF THE Second Kind, the second case can be re-expressed (8) 1 /Fir f 2 * 3/2 ^ A more commonly used definition of Airy functions is given by Abramowitz and Stegun (1972, pp. (9) 22 Airy Functions The solutions are then written y(z) = AAi(z) BBi(z) 7 where (10) Ai(*) = -*(-l,z) = |^[/_ 1/3 (I^ /2 )-/ 1/ 3(Iz 3/2 )] = ^^/3(I^ /2 ) ("J Bi(z) = y|[7_ a/ 3(fz 3/2 ) / 1 /3(! (12) In the above plot, Ai(z) is the solid curve and Bi(z) is dashed. The CRC concise encyclopedia of mathematics / Eric W. While a picture may be worth a thousand words, explicit examples are worth at least a few hundred! The Al- gorithm's running time for a number N has been provedtobeas O((lni V) clnlnln JV ) for some c 0. Although mathematics is a fascinating subject, it all too frequently is clothed in specialized jargon and dry formal exposition that make many interesting and useful mathematical results inaccessible to laypeople. Sequences A031286 and A006050/M4683 in "An On-Line Version of the Encyclopedia of Integer Se- quences." Sloane, N. 'dx 2 dx (2) 18 Adjugate Matrix Affine Hull Write the two Linearly Independent solutions as t/i (x) and 2/2 (#)■ Then the adjoint operator can also be written ? In this work, the most useful, interesting, and entertaining (at least to my mind) aspects of topics are discussed in addition to their technical definitions. A Square can be dissected into as few as 8 acute triangles. For example, in my entry for pi (71-), the definition in terms of the diameter and circumference of a circle is supplemented by a great many formulas and series for pi, including some of the amazing discoveries of Ramanujan. see also Obtuse Triangle, Right Triangle Adams-Bashforth-Moulton Method Addition Chain 15 Adams-Bashforth-Moulton Method see Adams' Method Adams' Method Adams' method is a numerical METHOD for solving linear First-Order Ordinary Differential Equa- tions of the form dy dx f{xy)- Let : 3?

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