Alhazen'S Billiard Problem Extended Essay

Alhazen'S Billiard Problem Extended Essay-85
Standard mathematical references, combined with a few popular ones, are also given at the end of most entries to facilitate additional reading and exploration. The word "coptic" derives from the Greek for "to cut," and acoptic polyhedra are defined as POLYHEDRA for which the FACES do not intersect (cut) themselves, mak- ing them 2-Manifolds. Mathematical Methods in Engineering: An Introduction to the Mathematical Treat- ment of Engineering Problems. The first of several ADDENDS, or "the one to which the oth- ers are added," is sometimes called the AUGEND. While the usual form of adding two n-digit INTEGERS (which consists of summing over the columns right to left and "Carrying" a 1 to the next column if the sum exceeds 9) requires n operations (plus carries), two n- digit INTEGERS can be added in about 21gn steps by n processors using carry-lookahead addition (Mc Geoch 1993). see also Digitadition, Digital Root, Multiplica- tive Persistence, Narcissistic Number, Recur- ring Digital Invariant References Guy, R. "The Persistence of a Number." §F25 in Unsolved Problems in Number Theory, 2nd ed.

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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.

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. The first few Ackermann numbers .3 are 1 t 1 = 1, 2 tt 2 = 4, and 3 ttt 3 = 3 3 7,625,597,484,987 see also Ackermann Function, Arrow Notation, Power Tower References Ackermann, W.

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. "Zum hilbertschen Aufbau der reellen Zahlen." Math.

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 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. The format of this work is somewhere between a handbook, a dictionary, and an encyclopedia. For example, in the expression a 6 c, a, 6, and c are all addends. It differs from existing dictionaries of mathematics in a number of important ways. The first of several addends, or "the one to which the others are added" (a in the previous example), is sometimes called the AUGEND. The initial document was written in Microsoft Word® on a Mac Plus® computer, and had reached about 200 pages by the time I started graduate school in 1990. When Andrew Treverrow made his Oz Tg X program available for the Mac, I began the task of converting all my documents to T^X, resulting in a vast improvement in readability. 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. After some additional effort, I was able to post an HTML version of my encyclopedia to the World Wide Web, currently located at www . The mathematician James Sylvester may have summed up the situation most aptly. According to Sylvester (as quoted in the introduction to Ian Stewart's book From Here to Infinity), "Mathematics is not a book confined within a cover and bound between brazen clasps, whose contents it needs only patience to ransack; it is not a mine, whose treasures may take long to reduce into possession, but which fill only a limited number of veins and lodes; it is not a soil, whose fertility can be exhausted by the yield of successive harvests; it is not a continent or an ocean, whose area can be mapped out and its "contour defined; it is as limitless as that space which it finds too narrow for its aspiration; its possibilities are as infinite as the worlds which are forever crowding in and multiplying upon the astronomer's gaze; it is as incapable of being restricted within assigned boundaries or being reduced to definitions of permanent validity, as the consciousness of life." Several of Sylvester's points apply particularly to this undertaking. The selection of topics in this work is more extensive than in most mathematical dictionaries (e.g., Borowski and Borwein's Harper Collins Dictionary of Mathematics and Jeans and Jeans' Mathematics Dictio- nary). see also ANGLE, OBTUSE ANGLE, RIGHT ANGLE, Straight Angle Acute Triangle A Triangle in which all three Angles are Acute An- gles. see also Addend, Amenable Number, Augend, Carry, Difference, Division, Multiplication, Plus, Subtraction, Sum References Mc Geoch, C. At the same time, the descriptions are more accessible than in "technical" mathematical encyclopedias (e.g., Hazewinkel's Encyclopaedia of Mathematics and Iyanaga's Encyclopedic Dictionary of Mathematics), While the latter remain models of accuracy and rigor, they are not terribly useful to the undergraduate, research scientist, or recreational mathematician. A Triangle which is neither acute nor a RIGHT Triangle (i.e., it has an Obtuse Angle) is called an Obtuse Triangle.


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