Case 6:12-cv-00799-JRG Document 143-6 Filed 04/11/14 Page 1 of 3 PageID #: 4281
`Case 6:12—cv—00799—JRG Document 143-6 Filed 04/11/14 Page 1 of 3 Page|D #: 4281
`
`EXHIBIT F
`
`EXHIBIT F
`
`

`
`Case 6:12-cv-00799-JRG Document 143-6 Filed 04/11/14 Page 2 of 3 PageID #: 4282
`
`Library of Congress Cataloging-in-Publication Data
`
`Weisstein, Eric W.
`The CRC concise encyclopedia of mathematics I Eric W. Weisstein.
`p. cm.
`Includes bibliographical references and index.
`ISBN 0-8493-9640-9 (alk. paper)
`I. Mathematics-Encyclopedias. I. Title.
`QA5.W45 1998
`510'.3-dc21
`
`98-22385
`CIP
`
`This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and
`sources are indicated. A wide variety of references are listed. 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.
`
`Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including
`photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the
`publisher.
`
`The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale.
`Specific permission must be obtained in writing from CRC Press LLC for such copying.
`
`Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
`
`Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and
`explanation, without intent to infringe.
`
`Visit the CRC Press Web site at www.crcpress.com
`
`© 1999 by CRC Press LLC
`
`No claim to original U.S. Government works
`International Standard Book Number 0-8493-9640-9
`Library of Congress Card Number 98-22385
`Printed in the United States of America
`3 4 5 6 7 8 9 0
`Printed on acid-free paper
`
`

`
`Case 6:12-cv-00799-JRG Document 143-6 Filed 04/11/14 Page 3 of 3 PageID #: 4283
`Domino Problem
`Dot Product
`489
`
`fl![JJ ~ rE/J
`~~
`ll\i'J 'J1Ed ~[['li\J~
`.. ~ . .
`~ n .. C~ c -··i L? f l }
`..
`.
`.
`LJ
`::J
`::J -=--~
`·~LI U
`_LL1::1J~··
`BE E=»c::IJ
`
`see also FIBONACCI NUMBER, GOMORY'S THEOREM,
`HEXOMINO, PENTOMINO, POLYOMINO, TETROMINO,
`TRIO MINO
`
`http://www.
`
`References
`Dickau, R. M. "Fibonacci Numbers."
`prairienet.org/-pops/fibboard.html.
`Gardner, M. "Polyominoes." Ch. 13 in The Scientific Amer(cid:173)
`ican Book of Mathematical Puzzles 8 Diversions. New
`York: Simon and Schuster, pp. 124-140, 1959.
`Kraitchik, M. "Dominoes." §12.1.22 in Mathematical Recre(cid:173)
`ations. New York: W . W. Norton, pp. 298-302, 1942.
`Lei, A. "Domino." http://www. cs. ust .hk/-philipl/omino/
`domino . html.
`Madachy, J . S. "Domino Recreations." Madachy's Mathe(cid:173)
`matical Recreations. New York: Dover, pp. 209-219, 1979.
`
`Domino Problem
`see WANG'S CONJECTURE
`
`Donaldson Invariants
`Distinguish between smooth MANIFOLDS in 4-D.
`
`Donkin's Theorem
`The product of three translations along the directed
`sides of a TRIANGLE through twice the lengths of these
`sides is the identity.
`
`Donut
`see TORUS
`
`where y is the MEAN, CTy the STANDARD DEVIATION,
`and Tr the relaxation time.
`
`References
`Doob, J. L. "Topics in the Theory of Markov Chains." Trans .
`Amer. Math. Soc. 52, 37-64, 1942.
`
`Dot
`The "dot" · has several meanings in mathematics, in(cid:173)
`cluding MULTIPLICATION (a · b is pronounced "a times
`b"), computation of a DOT PRODUCT (a· b is pronounced
`"a dot b"), or computation of a time DERIVATIVE (a is
`pronounced "a dot") .
`see also DERIVATIVE, DOT PRODUCT, TIMES
`
`Dot Product
`The dot product can be defined by
`
`X · Y = IXI IYI cosB,
`where e is the angle between the vectors. It follows
`immediately that x . y = 0 if x is PERPENDICULAR to
`Y. The dot product is also called the INNER PRODUCT
`and written (a, b). By writing
`
`(1)
`
`A,,= A cos BA
`Ay =A sin BA
`
`B,, = B cos BB
`By= B sin BB,
`
`(2)
`(3)
`
`it follows that (1) yields
`
`A· B = ABcos(BA - BB)
`= AB (cos e A cos e B + sin e A sin e B)
`= AcosBABcosBB + AsinBABsinBB
`= A,,B,, + AyBy.
`
`So, in general,
`
`X · Y = X1Y1 + · · · + XnYn·
`
`The dot product is COMMUTATIVE
`
`X·Y=Y·X,
`
`ASSOCIATIVE
`
`(rX) · Y = r(X · Y),
`
`and DISTRIBUTIVE
`
`X · (Y + Z) = X · Y + X · Z.
`
`(4)
`
`(5)
`
`(6)
`
`(7)
`
`(8)
`
`Doob's Theorem
`A theorem proved by Doob (1942) which states that any
`random process which is both GAUSSIAN and MARKOV
`has the following forms for its correlation function, spec(cid:173)
`tral density, and probability densities:
`Cy(r) = CTy 2e- r / rr
`G (j) =
`y
`
`2
`4Tr -lCTy
`(211' f
`1
`-(y-jj)2 / 2uy2
`P1 Y = ~e
`( )
`y 27!'CTy2
`1
`P2(Y1IY2 , r) = -r======
`J211'(1 - e-2-r / rr )cry2
`[(y2 - y) _ e-r / rr(yi - y)]2}
`x exp -~~--,-~~-.--.,-~-,--~~
`{
`2(1 - e - 2r / r r )cry2
`
`The DERIVATIVE of a dot product of VECTORS is
`
`dr1
`dr2
`d
`dt[r1(t) · r2(t)] = ri(t) · dt + dt · r2(t) .
`
`(9)
`
`,