H.W.GOULD West Virginia University, Morgan town, West Virginia 26506

Size: px
Start display at page:

Download "H.W.GOULD West Virginia University, Morgan town, West Virginia 26506"

Transcription

1 A F I B O N A C C I F O R M U L A OF LUCAS A N D ITS SUBSEQUENT M A N I F E S T A T I O N S A N D R E D I S C O V E R I E S H.W.GOULD West Viginia Univesity, Mogan town, West Viginia Almost eveyone who woks with Fibonacci numbes knows that diagonal sums in the Pascal tiangle give ise to the fomula [?±] (D F. ( " - * " ' ), a>1. but not many ealize that (2) F 2n = f-/i*("-*-')'- a, o that \i\ (3) F 3n = 2 { " ' I ' 1 )4"- 1-2k, that these ae special cases of a vey geneal fomula given in 1878 by Edouad Lucas [5, Eqs ], [6, pp ]. As fa as I can detemine, fomula (2) fist appeaed in ou Fibonacci Quately as a poblem posed by Luline Squie [10] when she was studying numbe theoy at West Viginia Univesity. M. N. S. Swamy's solution invoked the use of Chebyshev polynomials. I was eminded of the fomula ecently when Leon Benstein [1] found the fomula again asked me about it. He used a new technique involving algebaic numbe fields. Fomulas (2) (3) genealize in a cuious manne. On the one h we have fo even positive integes F [T] (4) f 1 = H (~V k { n ~ k k' 1 )L"' 1 ' 2k. 2\. but on the othe h fo odd positive integes we get the same tems but with all positive signs (5), IT] F f = J2 ( n ~ k k- 1 )L n - 1-2k, k=1) ' 2\, whee L is the usual Lucas numbe defined by L n+1 = L n + L n -f, with LQ = 2, Lj = 7, this of couse in contast with F n +i = F n + F n --j FQ = O f F-J = 1. Fomulas (4) (5) may be witten as a single fomula in a cleve way as noted by Hoggatt Lind [4] who would wite n-u (6) T =11 (-V k( ' 1 > {"- k k- 1 )L" - 1-2k, 25

2 26 A FIBONACCI FORMULA OF LUCAS AND [FEB. valid now fo any positive integes nj > 1. Fomula (6) of Hoggatt Lind was posed as a poblem by James E. Desmond [11] solved by him using a esult of Joseph A. Raab [7]. The pecise same poblem was posed again by David Englund [12] Douglas Lind pointed out that it was just the same fomula. Fomulas (4) (5) wee obtained by Hoggatt Lind [4] by calculations using compositions geneating functions. Although they cite Lucas [5] fo a numbe of items they wee evidently unawae that the fomulas appea in Lucas in a fa moe geneal fom. Since L = F2 /F, fomulas (4)-(5) can be witten entiely in tems of f ' s. Lucas intoduced the geneal functions U, V defined by (7) U n = a =-f> V n = a n +b n, a b whee a b ae the oots of the quadatic equation (8) x 2 -Px + Q= 0, so that a + b = P ab = Q. When we have x 2 - x ft we gets /? as (1 + ^Js)/2 (1 -yjs)/2 then U n = F n, V n = L n. One of the geneal fomulas Lucas gave is [6, pp , note mispint in fomula] l"f] (9) ^ = (~V k ( n ~ k k- 1 ) Vf- 1-2 *!]*, which unifies (4) (5) is moe geneal than (6). Cuiously, as we have intimated, Hoggatt Lind do not cite this geneal fomula. Now of couse, thee ae many othe such fomulas in Lucas' wok. Two special cases should be paaded hee fo compaison. These ae (10) L n = (-1) k --V-- ( n - k ) L n ~ 2k f o m e, f] mi L ("-*) c - 2k «' - These can be united in the same manne as (4)-(5) in (6). Thus [I] (11.1) L n = (-7) k( ~ 1) -JLj {"- k ) L n ~ 2k. Thee is nothing eally mysteious about why such fomulas exist Thee ae pefectly good fomulas fo the sums of powes of oots of algebaic equations tacing back to Lagange ealie. The two types of fomulas we ae discussing aise because of [ ] (12) (- 1) k (" ~ *) (xy) k (x + Y 2k = X -^X-, fomula (1.60) in [3], X V

3 1977] ITS SUBSEQUENT MANIFESTATIONS AND REDISCOVERIES 27 f - 1 L2J (13) ^ ( - j ) * " { n - k )(xy) k (x+y 2k = x n +y n, n * fomula (1.64) in [ 3], familia fomulas that say the same thing Lucas was saying. The eason it is not mysteious that (2) holds tue, e.g., is that Z^/? satisfies the second-ode ecuence elation F 2n+2 = 3F 2n ~ F2n-2 with which we associate the chaacteistic quadatic equation x 2 = 3x - / so that a fomula like (2) must be tue. Fo fomula (4) with = 4 we note that F4 n+ 4 = 7F4 n - F4 n _4. In geneal in fact, (14) F n+ = L F n -F m. fo even, o F m+ + F m - = L F m, (15) F m+ = L F n + F n - ioodd, o F m+ - F m. = L F m. Regulaly spaced tems in a ecuent sequence of ode two themselves satisfy such a ecuence. Set u n = F n to see this fo then we have (16) u n +i = L u n ±u n -i, with z 2 = L z± 7, so we expect a pioi that u n must satisfy a fomula athe like (1). Fomulas like (12) (13) give the sums of powes of the oots of the chaacteistic equation, whence the geneal fomulas. Fomula (12) coesponds to (B.1) (13) coesponds to (A.1) in Daim's pape [2] which the eade may also consult. Anothe inteesting fact is that these fomulas ae elated to the Fibonacci polynomials intoduced in a poblem [9] discussed at length by Hoggatt othes in late issues of the Quately. These ae defined by f n (x) = xf n -f(x) + f n - 2 (x), n > 2, with fi(x)= 1 f2(x) = x. In geneal m (17) f n (x)= {"- k k- 1 )x"- 2k - 1. whence fo odd we have by (5) that (18) f {L ) = F -P. ' Many othe such elations can be deduced. Finally we want to note two sets of invese pais given by Riodan [8] which he classifies as Chebyshev invese pais: i] (19) «W = ( - 1>k ^k ( n k k )9( n ~ 2k ) if only if [I] (20) g(n) = (" k )f(n-2k);

4 28 A FIBONACCI FORMULA OF LUCAS AND [FEB. tfi (211 IM- X. TlfH» ' " - 2kl if only if [i] (22) g(n) = (-if ( n ~ K ) f(n-2k). Applying (19) (20) to (10) we get the paticulaly nice fomula (23) L n = (" k ) L (n-2k), even. Using (21) (22) on (4) we get the slightly moe complicated fomula u i] (24), ^ i t n ^ f o ^ a n-k+1 \kl F k^o I do not ecall seeing (23) o (24) in any accessible location in ou Quately. If we let /--> 0 in (4) we can obtain the fomula (1.72) in [3] of Lucas, which is also pat of Desmond's poblem [11] who does not cite Lucas, I'l (25) n= (~1) k ( n - k k- 1 )2"- 2k - 1, n>1. It is abundantly clea that the techniques we have discussed apply to many of the genealized sequences that have been intoduced, e.g., Hoadam's genealized Fibonacci sequence, but we shall not take the space to develop the obvious fomulas. It is hoped that we have shed a little moe light on a set of athe inteesting fomulas all due to Lucas. REFERENCES 1. L. Benstein, "Units in Algebaic Numbe Fields Combinatoial Identities," Invited pape, Special Session on Combinatoial Identities, Ame. Math. Soc. Meeting, Aug. 1976, Toonto, Notices of Ame. Math. Soc, 23 (1976). p. A-408, Abstact No N. A. Daim, "Sums of n f Powes of Roots of a Given Quadatic Equation," The Fibonacci Quately, Vol. 4, No. 2 (Apil, 1966), pp H. W. Gould, "Combinatoial Identities," Revised Edition, Published by the autho, Mogantown, W. Va., V. E. Hoggatt, J., D. A. Lind, '"Compositions Fibonacci Numbes," The Fibonacci Quately, Vol. 7, No. 3 (Oct., 1969), pp E. Lucas,"The'oiedes FonctionsNume'iquesSimplementPe'iodiques/'/l/77e. J. Math., 1 (1878), pp ; E. Lucas, "The Theoy of Simply Peiodic Numeical Functions," The Fibonacci Association, Tanslated by Sidney Kavitz Edited by Douglas Lind. 7. J. A. Raab, "A Genealization of the Connection between the Fibonacci Sequence Pascal's Tiangle," The Fibonacci Quately, Vol. 1, No. 3 (Oct. 1963), pp J. Riodan, Combinatoial Identities, John Wiley Sons, New Yok, 1968.

5 1977] AND ITS SUBSEQUENT MANIFESTATIONS AND REDISCOVERIES Poblem B-74, Posed by M. N. S. Swamy, The Fibonacci Quately, Vol. 3, No. 3 (Oct., 1965), p. 236; Solved by D. Zeitlin, ibid.. Vol. 4, No. 1 (Feb. 1966), pp Poblem H-83, Posed by Ms. W. Squie, The Fibonacci Quately, Vol. 4, No. 1 (Feb., 1966), p. 57; Solved by M. N. S. Swamy, ibid., Vol. 6, No. 1 (Feb., 1968), pp Poblem H-135, Posed by J. E. Desmond, The Fibonacci Quately, Vol. 6, No. 2 (Apil, 1968), pp ; Solved by the Popose,/M/ Vol. 7, No. 5 (Dec. 1969), pp Poblem H-172, Posed by David Engl, The Fibonacci Quately, Vol. 8, No. 4 (Dec, 1970), p. 383; Solved by Douglas Lind, ibid., Vol. 9, No. 5 (Dec, 1971), p Poblem B-285, Posed by Bay Wolk, The Fibonacci Quately, Vol, 12, No. 2 (Apil 1974), p. 221; Solved by C. B. A. Peck, ibid., Vol. 13, No. 2 (Apil 1975), p ******* [Continued fom page 24.] (iii) ( ^ - 7 ) G 2 + G 2 H = G 2n +1 (n > 1) (iv) G 2 n+2- (P-flf'G 2 = G 2n+2 (n > 1) =0 (vi) { P -J- L )T, = G n+2-1 (n > V. =1 The poofs of the above esults, which ely essentially on equations (2), (3) (5), togethe with a - j 3 = V p, a+p.= 1 a(s =-(&- ), ae faily staightfowad left to the eade. Of couse, esults such as these ae not new. Fo example, (ii) was poved in a slightly moe geneal fom by E. Lucas as ealy as 1876 (see [1] page 396). Finally, tuning to the vetical sequences in the table given ealie, it follows fom (v) that the sequence unde G n (n > 1) is given by is'c-;- )'*-"'} (6) \ X [ n 'l- )(k-u \ <k> 1), t 1 =0 =0 J so that fo example the sequences unde G4 G5 ae {2k - 7} {k + k- / }, espectively. Altenatively, instead of using (6), we can apply the Binomial Theoem to (2) obtain the geneal vetical sequence in the fom ~~n~l 2 { n )(4k-3) ( - 1J/2 } (k > 1).. 2 =1 J odd REFERENCE 1. L. E. Dickson, Histoy of the Theoy of Numbes, Vol. 1, Canegie Institution (Washington 1919).

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0}, ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability

More information

On a generalization of Eulerian numbers

On a generalization of Eulerian numbers Notes on Numbe Theoy and Discete Mathematics Pint ISSN 1310 513, Online ISSN 367 875 Vol, 018, No 1, 16 DOI: 10756/nntdm018116- On a genealization of Euleian numbes Claudio Pita-Ruiz Facultad de Ingenieía,

More information

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version

Auchmuty High School Mathematics Department Advanced Higher Notes Teacher Version The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!

More information

Miskolc Mathematical Notes HU e-issn Tribonacci numbers with indices in arithmetic progression and their sums. Nurettin Irmak and Murat Alp

Miskolc Mathematical Notes HU e-issn Tribonacci numbers with indices in arithmetic progression and their sums. Nurettin Irmak and Murat Alp Miskolc Mathematical Notes HU e-issn 8- Vol. (0), No, pp. 5- DOI 0.85/MMN.0.5 Tibonacci numbes with indices in aithmetic pogession and thei sums Nuettin Imak and Muat Alp Miskolc Mathematical Notes HU

More information

6 PROBABILITY GENERATING FUNCTIONS

6 PROBABILITY GENERATING FUNCTIONS 6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to

More information

(received April 9, 1967) Let p denote a prime number and let k P

(received April 9, 1967) Let p denote a prime number and let k P ON EXTREMAL OLYNOMIALS Kenneth S. Williams (eceived Apil 9, 1967) Let p denote a pime numbe and let k denote the finite field of p elements. Let f(x) E k [x] be of fixed degee d 2 2. We suppose that p

More information

A generalization of the Bernstein polynomials

A generalization of the Bernstein polynomials A genealization of the Benstein polynomials Halil Ouç and Geoge M Phillips Mathematical Institute, Univesity of St Andews, Noth Haugh, St Andews, Fife KY16 9SS, Scotland Dedicated to Philip J Davis This

More information

(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2.

(n 1)n(n + 1)(n + 2) + 1 = (n 1)(n + 2)n(n + 1) + 1 = ( (n 2 + n 1) 1 )( (n 2 + n 1) + 1 ) + 1 = (n 2 + n 1) 2. Paabola Volume 5, Issue (017) Solutions 151 1540 Q151 Take any fou consecutive whole numbes, multiply them togethe and add 1. Make a conjectue and pove it! The esulting numbe can, fo instance, be expessed

More information

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic. Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this

More information

New problems in universal algebraic geometry illustrated by boolean equations

New problems in universal algebraic geometry illustrated by boolean equations New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic

More information

A POWER IDENTITY FOR SECOND-ORDER RECURRENT SEQUENCES

A POWER IDENTITY FOR SECOND-ORDER RECURRENT SEQUENCES A OWER IDENTITY FOR SECOND-ORDER RECURRENT SEQUENCES V. E. Hoggatt,., San ose State College, San ose, Calif. and D. A. Lind, Univesity of Viginia, Chalottesville, V a. 1. INTRODUCTION The following hold

More information

A Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction

A Short Combinatorial Proof of Derangement Identity arxiv: v1 [math.co] 13 Nov Introduction A Shot Combinatoial Poof of Deangement Identity axiv:1711.04537v1 [math.co] 13 Nov 2017 Ivica Matinjak Faculty of Science, Univesity of Zageb Bijenička cesta 32, HR-10000 Zageb, Coatia and Dajana Stanić

More information

TANTON S TAKE ON CONTINUOUS COMPOUND INTEREST

TANTON S TAKE ON CONTINUOUS COMPOUND INTEREST CURRICULUM ISPIRATIOS: www.maa.og/ci www.theglobalmathpoject.og IOVATIVE CURRICULUM OLIE EXPERIECES: www.gdaymath.com TATO TIDBITS: www.jamestanton.com TATO S TAKE O COTIUOUS COMPOUD ITEREST DECEMBER 208

More information

On a quantity that is analogous to potential and a theorem that relates to it

On a quantity that is analogous to potential and a theorem that relates to it Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich

More information

MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form

MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and

More information

Semicanonical basis generators of the cluster algebra of type A (1)

Semicanonical basis generators of the cluster algebra of type A (1) Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published:

More information

THE EVALUATION OF CERTAIN ARITHMETIC SUMS

THE EVALUATION OF CERTAIN ARITHMETIC SUMS THE EVALUATION OF CERTAIN ARITHMETIC SUMS B.C. GRIIVISON Dept. of Biostatistfes, Univesity of Woth Caolina, Chapel Hill, Woth Caolina 27514 1. In this pape we evaluate cetain cases of the expession (1.1)

More information

Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs

Math 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let

More information

Then the number of elements of S of weight n is exactly the number of compositions of n into k parts.

Then the number of elements of S of weight n is exactly the number of compositions of n into k parts. Geneating Function In a geneal combinatoial poblem, we have a univee S of object, and we want to count the numbe of object with a cetain popety. Fo example, if S i the et of all gaph, we might want to

More information

2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8

2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8 5 CHAPTER Fundamentals When solving equations that involve absolute values, we usually take cases. EXAMPLE An Absolute Value Equation Solve the equation 0 x 5 0 3. SOLUTION By the definition of absolute

More information

Numerical approximation to ζ(2n+1)

Numerical approximation to ζ(2n+1) Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at: https://woks.bepess.com/tian_xiao_he/6/

More information

A Bijective Approach to the Permutational Power of a Priority Queue

A Bijective Approach to the Permutational Power of a Priority Queue A Bijective Appoach to the Pemutational Powe of a Pioity Queue Ia M. Gessel Kuang-Yeh Wang Depatment of Mathematics Bandeis Univesity Waltham, MA 02254-9110 Abstact A pioity queue tansfoms an input pemutation

More information

OLYMON. Produced by the Canadian Mathematical Society and the Department of Mathematics of the University of Toronto. Issue 9:2.

OLYMON. Produced by the Canadian Mathematical Society and the Department of Mathematics of the University of Toronto. Issue 9:2. OLYMON Poduced by the Canadian Mathematical Society and the Depatment of Mathematics of the Univesity of Toonto Please send you solution to Pofesso EJ Babeau Depatment of Mathematics Univesity of Toonto

More information

arxiv: v1 [physics.pop-ph] 3 Jun 2013

arxiv: v1 [physics.pop-ph] 3 Jun 2013 A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,

More information

of the contestants play as Falco, and 1 6

of the contestants play as Falco, and 1 6 JHMT 05 Algeba Test Solutions 4 Febuay 05. In a Supe Smash Bothes tounament, of the contestants play as Fox, 3 of the contestants play as Falco, and 6 of the contestants play as Peach. Given that thee

More information

f h = u, h g = v, we have u + v = f g. So, we wish

f h = u, h g = v, we have u + v = f g. So, we wish Answes to Homewok 4, Math 4111 (1) Pove that the following examples fom class ae indeed metic spaces. You only need to veify the tiangle inequality. (a) Let C be the set of continuous functions fom [0,

More information

Pascal s Triangle (mod 8)

Pascal s Triangle (mod 8) Euop. J. Combinatoics (998) 9, 45 62 Pascal s Tiangle (mod 8) JAMES G. HUARD, BLAIR K. SPEARMAN AND KENNETH S. WILLIAMS Lucas theoem gives a conguence fo a binomial coefficient modulo a pime. Davis and

More information

Bounds for Codimensions of Fitting Ideals

Bounds for Codimensions of Fitting Ideals Ž. JOUNAL OF ALGEBA 194, 378 382 1997 ATICLE NO. JA966999 Bounds fo Coensions of Fitting Ideals Michał Kwiecinski* Uniwesytet Jagiellonski, Instytut Matematyki, ul. eymonta 4, 30-059, Kakow, Poland Communicated

More information

Berkeley Math Circle AIME Preparation March 5, 2013

Berkeley Math Circle AIME Preparation March 5, 2013 Algeba Toolkit Rules of Thumb. Make sue that you can pove all fomulas you use. This is even bette than memoizing the fomulas. Although it is best to memoize, as well. Stive fo elegant, economical methods.

More information

The r-bell Numbers. 1 Introduction

The r-bell Numbers. 1 Introduction 3 47 6 3 Jounal of Intege Sequences, Vol. 4 (, Aticle.. The -Bell Numbes István Meő Depatment of Applied Mathematics and Pobability Theoy Faculty of Infomatics Univesity of Debecen P. O. Box H-4 Debecen

More information

Enumerating permutation polynomials

Enumerating permutation polynomials Enumeating pemutation polynomials Theodoulos Gaefalakis a,1, Giogos Kapetanakis a,, a Depatment of Mathematics and Applied Mathematics, Univesity of Cete, 70013 Heaklion, Geece Abstact We conside thoblem

More information

Asymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function

Asymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function "Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics

More information

MODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...

MODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ... MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto

More information

Solutions to Problem Set 8

Solutions to Problem Set 8 Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics fo Compute Science Novembe 21 Pof. Albet R. Meye and Pof. Ronitt Rubinfeld evised Novembe 27, 2005, 858 minutes Solutions to Poblem

More information

I. CONSTRUCTION OF THE GREEN S FUNCTION

I. CONSTRUCTION OF THE GREEN S FUNCTION I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,

More information

Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments

Chaos and bifurcation of discontinuous dynamical systems with piecewise constant arguments Malaya Jounal of Matematik ()(22) 4 8 Chaos and bifucation of discontinuous dynamical systems with piecewise constant aguments A.M.A. El-Sayed, a, and S. M. Salman b a Faculty of Science, Aleandia Univesity,

More information

Analytical Solutions for Confined Aquifers with non constant Pumping using Computer Algebra

Analytical Solutions for Confined Aquifers with non constant Pumping using Computer Algebra Poceedings of the 006 IASME/SEAS Int. Conf. on ate Resouces, Hydaulics & Hydology, Chalkida, Geece, May -3, 006 (pp7-) Analytical Solutions fo Confined Aquifes with non constant Pumping using Compute Algeba

More information

Section 8.2 Polar Coordinates

Section 8.2 Polar Coordinates Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal

More information

PROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.

PROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr. POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and

More information

arxiv: v1 [math.nt] 28 Oct 2017

arxiv: v1 [math.nt] 28 Oct 2017 ON th COEFFICIENT OF DIVISORS OF x n axiv:70049v [mathnt] 28 Oct 207 SAI TEJA SOMU Abstact Let,n be two natual numbes and let H(,n denote the maximal absolute value of th coefficient of divisos of x n

More information

18.06 Problem Set 4 Solution

18.06 Problem Set 4 Solution 8.6 Poblem Set 4 Solution Total: points Section 3.5. Poblem 2: (Recommended) Find the lagest possible numbe of independent vectos among ) ) ) v = v 4 = v 5 = v 6 = v 2 = v 3 =. Solution (4 points): Since

More information

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function Abstact and Applied Analysis Volume 011, Aticle ID 697547, 7 pages doi:10.1155/011/697547 Reseach Aticle On Alze and Qiu s Conjectue fo Complete Elliptic Integal and Invese Hypebolic Tangent Function Yu-Ming

More information

Compactly Supported Radial Basis Functions

Compactly Supported Radial Basis Functions Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically

More information

Chapter Eight Notes N P U1C8S4-6

Chapter Eight Notes N P U1C8S4-6 Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that

More information

Geometry of the homogeneous and isotropic spaces

Geometry of the homogeneous and isotropic spaces Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant

More information

When two numbers are written as the product of their prime factors, they are in factored form.

When two numbers are written as the product of their prime factors, they are in factored form. 10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The

More information

arxiv: v1 [math.co] 6 Mar 2008

arxiv: v1 [math.co] 6 Mar 2008 An uppe bound fo the numbe of pefect matchings in gaphs Shmuel Fiedland axiv:0803.0864v [math.co] 6 Ma 2008 Depatment of Mathematics, Statistics, and Compute Science, Univesity of Illinois at Chicago Chicago,

More information

The Substring Search Problem

The Substring Search Problem The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is

More information

SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER

SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Discussiones Mathematicae Gaph Theoy 39 (019) 567 573 doi:10.7151/dmgt.096 SUFFICIENT CONDITIONS FOR MAXIMALLY EDGE-CONNECTED AND SUPER-EDGE-CONNECTED GRAPHS DEPENDING ON THE CLIQUE NUMBER Lutz Volkmann

More information

Review of the H-O model. Problem 1. Assume that the production functions in the standard H-O model are the following:

Review of the H-O model. Problem 1. Assume that the production functions in the standard H-O model are the following: Revie of the H-O model Poblem 1 Assume that the poduction functions in the standad H-O model ae the folloing: f 1 L 1 1 ) L 1/ 1 1/ 1 f L ) L 1/3 /3 In addition e assume that the consume pefeences ae given

More information

Several new identities involving Euler and Bernoulli polynomials

Several new identities involving Euler and Bernoulli polynomials Bull. Math. Soc. Sci. Math. Roumanie Tome 9107 No. 1, 016, 101 108 Seveal new identitie involving Eule and Benoulli polynomial by Wang Xiaoying and Zhang Wenpeng Abtact The main pupoe of thi pape i uing

More information

Quasi-Randomness and the Distribution of Copies of a Fixed Graph

Quasi-Randomness and the Distribution of Copies of a Fixed Graph Quasi-Randomness and the Distibution of Copies of a Fixed Gaph Asaf Shapia Abstact We show that if a gaph G has the popety that all subsets of vetices of size n/4 contain the coect numbe of tiangles one

More information

Brief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis

Brief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis Bief summay of functional analysis APPM 5440 Fall 014 Applied Analysis Stephen Becke, stephen.becke@coloado.edu Standad theoems. When necessay, I used Royden s and Keyzsig s books as a efeence. Vesion

More information

PHYS 301 HOMEWORK #10 (Optional HW)

PHYS 301 HOMEWORK #10 (Optional HW) PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

Journal of Number Theory

Journal of Number Theory Jounal of umbe Theoy 3 2 2259 227 Contents lists available at ScienceDiect Jounal of umbe Theoy www.elsevie.com/locate/jnt Sums of poducts of hypegeometic Benoulli numbes Ken Kamano Depatment of Geneal

More information

The Strain Compatibility Equations in Polar Coordinates RAWB, Last Update 27/12/07

The Strain Compatibility Equations in Polar Coordinates RAWB, Last Update 27/12/07 The Stain Compatibility Equations in Pola Coodinates RAWB Last Update 7//7 In D thee is just one compatibility equation. In D polas it is (Equ.) whee denotes the enineein shea (twice the tensoial shea)

More information

KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS

KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS Jounal of Applied Analysis Vol. 14, No. 1 2008), pp. 43 52 KOEBE DOMAINS FOR THE CLASSES OF FUNCTIONS WITH RANGES INCLUDED IN GIVEN SETS L. KOCZAN and P. ZAPRAWA Received Mach 12, 2007 and, in evised fom,

More information

Question 1: The dipole

Question 1: The dipole Septembe, 08 Conell Univesity, Depatment of Physics PHYS 337, Advance E&M, HW #, due: 9/5/08, :5 AM Question : The dipole Conside a system as discussed in class and shown in Fig.. in Heald & Maion.. Wite

More information

CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS OF COMPOSITE ENTIRE FUNCTIONS FROM THE VIEW POINT OF L -ORDER. Tanmay Biswas

CENTRAL INDEX BASED SOME COMPARATIVE GROWTH ANALYSIS OF COMPOSITE ENTIRE FUNCTIONS FROM THE VIEW POINT OF L -ORDER. Tanmay Biswas J Koean Soc Math Educ Se B: Pue Appl Math ISSNPint 16-0657 https://doiog/107468/jksmeb01853193 ISSNOnline 87-6081 Volume 5, Numbe 3 August 018, Pages 193 01 CENTRAL INDEX BASED SOME COMPARATIVE GROWTH

More information

arxiv: v2 [math.ag] 4 Jul 2012

arxiv: v2 [math.ag] 4 Jul 2012 SOME EXAMPLES OF VECTOR BUNDLES IN THE BASE LOCUS OF THE GENERALIZED THETA DIVISOR axiv:0707.2326v2 [math.ag] 4 Jul 2012 SEBASTIAN CASALAINA-MARTIN, TAWANDA GWENA, AND MONTSERRAT TEIXIDOR I BIGAS Abstact.

More information

Solving Some Definite Integrals Using Parseval s Theorem

Solving Some Definite Integrals Using Parseval s Theorem Ameican Jounal of Numeical Analysis 4 Vol. No. 6-64 Available online at http://pubs.sciepub.com/ajna///5 Science and Education Publishing DOI:.69/ajna---5 Solving Some Definite Integals Using Paseval s

More information

Chapter 5 Linear Equations: Basic Theory and Practice

Chapter 5 Linear Equations: Basic Theory and Practice Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and

More information

SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES

SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF 2 2 OPERATOR MATRICES italian jounal of pue and applied mathematics n. 35 015 (433 44) 433 SOME GENERAL NUMERICAL RADIUS INEQUALITIES FOR THE OFF-DIAGONAL PARTS OF OPERATOR MATRICES Watheq Bani-Domi Depatment of Mathematics

More information

F-IF Logistic Growth Model, Abstract Version

F-IF Logistic Growth Model, Abstract Version F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth

More information

Exploration of the three-person duel

Exploration of the three-person duel Exploation of the thee-peson duel Andy Paish 15 August 2006 1 The duel Pictue a duel: two shootes facing one anothe, taking tuns fiing at one anothe, each with a fixed pobability of hitting his opponent.

More information

Complex Eigenvalues. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Complex Eigenvalues. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Pepaed by Vince Zaccone Fo ampus Leaning ssistance Sevices at USB omplex Numbes When solving fo the oots of a quadatic equation, eal solutions can not be found when the disciminant is negative. In these

More information

arxiv: v1 [math.co] 4 May 2017

arxiv: v1 [math.co] 4 May 2017 On The Numbe Of Unlabeled Bipatite Gaphs Abdullah Atmaca and A Yavuz Ouç axiv:7050800v [mathco] 4 May 207 Abstact This pape solves a poblem that was stated by M A Haison in 973 [] This poblem, that has

More information

Surveillance Points in High Dimensional Spaces

Surveillance Points in High Dimensional Spaces Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage

More information

Application of Parseval s Theorem on Evaluating Some Definite Integrals

Application of Parseval s Theorem on Evaluating Some Definite Integrals Tukish Jounal of Analysis and Numbe Theoy, 4, Vol., No., -5 Available online at http://pubs.sciepub.com/tjant/// Science and Education Publishing DOI:.69/tjant--- Application of Paseval s Theoem on Evaluating

More information

Pearson s Chi-Square Test Modifications for Comparison of Unweighted and Weighted Histograms and Two Weighted Histograms

Pearson s Chi-Square Test Modifications for Comparison of Unweighted and Weighted Histograms and Two Weighted Histograms Peason s Chi-Squae Test Modifications fo Compaison of Unweighted and Weighted Histogams and Two Weighted Histogams Univesity of Akueyi, Bogi, v/noduslód, IS-6 Akueyi, Iceland E-mail: nikolai@unak.is Two

More information

Measure Estimates of Nodal Sets of Polyharmonic Functions

Measure Estimates of Nodal Sets of Polyharmonic Functions Chin. Ann. Math. Se. B 39(5), 08, 97 93 DOI: 0.007/s40-08-004-6 Chinese Annals of Mathematics, Seies B c The Editoial Office of CAM and Spinge-Velag Belin Heidelbeg 08 Measue Estimates of Nodal Sets of

More information

5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS

5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS 5.6 Physical Chemisty Lectue #3 page MAY ELECTRO ATOMS At this point, we see that quantum mechanics allows us to undestand the helium atom, at least qualitatively. What about atoms with moe than two electons,

More information

CALCULUS II Vectors. Paul Dawkins

CALCULUS II Vectors. Paul Dawkins CALCULUS II Vectos Paul Dawkins Table of Contents Peface... ii Vectos... 3 Intoduction... 3 Vectos The Basics... 4 Vecto Aithmetic... 8 Dot Poduct... 13 Coss Poduct... 21 2007 Paul Dawkins i http://tutoial.math.lama.edu/tems.aspx

More information

On the Poisson Approximation to the Negative Hypergeometric Distribution

On the Poisson Approximation to the Negative Hypergeometric Distribution BULLETIN of the Malaysian Mathematical Sciences Society http://mathusmmy/bulletin Bull Malays Math Sci Soc (2) 34(2) (2011), 331 336 On the Poisson Appoximation to the Negative Hypegeometic Distibution

More information

A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM

A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM DIEGO AVERNA AND GABRIELE BONANNO Abstact. The aim of this pape is twofold. On one hand we establish a thee citical

More information

3.6 Applied Optimization

3.6 Applied Optimization .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the

More information

On the integration of the equations of hydrodynamics

On the integration of the equations of hydrodynamics Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious

More information

ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi

ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS. D.A. Mojdeh and B. Samadi Opuscula Math. 37, no. 3 (017), 447 456 http://dx.doi.og/10.7494/opmath.017.37.3.447 Opuscula Mathematica ON THE INVERSE SIGNED TOTAL DOMINATION NUMBER IN GRAPHS D.A. Mojdeh and B. Samadi Communicated

More information

Physics 121 Hour Exam #5 Solution

Physics 121 Hour Exam #5 Solution Physics 2 Hou xam # Solution This exam consists of a five poblems on five pages. Point values ae given with each poblem. They add up to 99 points; you will get fee point to make a total of. In any given

More information

-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r.

-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r. The Laplace opeato in pola coodinates We now conside the Laplace opeato with Diichlet bounday conditions on a cicula egion Ω {(x,y) x + y A }. Ou goal is to compute eigenvalues and eigenfunctions of the

More information

arxiv: v1 [math.nt] 12 May 2017

arxiv: v1 [math.nt] 12 May 2017 SEQUENCES OF CONSECUTIVE HAPPY NUMBERS IN NEGATIVE BASES HELEN G. GRUNDMAN AND PAMELA E. HARRIS axiv:1705.04648v1 [math.nt] 12 May 2017 ABSTRACT. Fo b 2 and e 2, let S e,b : Z Z 0 be the function taking

More information

3.1 Random variables

3.1 Random variables 3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated

More information

Probablistically Checkable Proofs

Probablistically Checkable Proofs Lectue 12 Pobablistically Checkable Poofs May 13, 2004 Lectue: Paul Beame Notes: Chis Re 12.1 Pobablisitically Checkable Poofs Oveview We know that IP = PSPACE. This means thee is an inteactive potocol

More information

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50 woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,

More information

Chapter 3: Theory of Modular Arithmetic 38

Chapter 3: Theory of Modular Arithmetic 38 Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences

More information

The Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr

The Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez y L.H. Harper z M. Rottger U.-P. Schroeder Abstract We consider the pr The Congestion of n-cube Layout on a Rectangula Gid S.L. Bezukov J.D. Chavez y L.H. Hape z M. Rottge U.-P. Schoede Abstact We conside the poblem of embedding the n-dimensional cube into a ectangula gid

More information

Syntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland)

Syntactical content of nite approximations of partial algebras 1 Wiktor Bartol Inst. Matematyki, Uniw. Warszawski, Warszawa (Poland) Syntactical content of nite appoximations of patial algebas 1 Wikto Batol Inst. Matematyki, Uniw. Waszawski, 02-097 Waszawa (Poland) batol@mimuw.edu.pl Xavie Caicedo Dep. Matematicas, Univ. de los Andes,

More information

Using Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu

Using Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu Available at https://edupediapublicationsog/jounals Volume 3 Issue 4 Febuay 216 Using Laplace Tansfom to Evaluate Impope Integals Chii-Huei Yu Depatment of Infomation Technology, Nan Jeon Univesity of

More information

On Polynomials Construction

On Polynomials Construction Intenational Jounal of Mathematical Analysis Vol., 08, no. 6, 5-57 HIKARI Ltd, www.m-hikai.com https://doi.og/0.988/ima.08.843 On Polynomials Constuction E. O. Adeyefa Depatment of Mathematics, Fedeal

More information

Solution to HW 3, Ma 1a Fall 2016

Solution to HW 3, Ma 1a Fall 2016 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.

More information

Vanishing lines in generalized Adams spectral sequences are generic

Vanishing lines in generalized Adams spectral sequences are generic ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal

More information

C/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22

C/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22 C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.

More information

Quadratic Harmonic Number Sums

Quadratic Harmonic Number Sums Applied Matheatics E-Notes, (), -7 c ISSN 67-5 Available fee at io sites of http//www.ath.nthu.edu.tw/aen/ Quadatic Haonic Nube Sus Anthony Sofo y and Mehdi Hassani z Received July Abstact In this pape,

More information

COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT

COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit

More information

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating

More information

Practice Integration Math 120 Calculus I Fall 2015

Practice Integration Math 120 Calculus I Fall 2015 Pactice Integation Math 0 Calculus I Fall 05 Hee s a list of pactice eecises. Thee s a hint fo each one as well as an answe with intemediate steps... ( + d. Hint. Answe. ( 8 t + t + This fist set of indefinite

More information

Galois points on quartic surfaces

Galois points on quartic surfaces J. Math. Soc. Japan Vol. 53, No. 3, 2001 Galois points on quatic sufaces By Hisao Yoshihaa (Received Nov. 29, 1999) (Revised Ma. 30, 2000) Abstact. Let S be a smooth hypesuface in the pojective thee space

More information

ONE-POINT CODES USING PLACES OF HIGHER DEGREE

ONE-POINT CODES USING PLACES OF HIGHER DEGREE ONE-POINT CODES USING PLACES OF HIGHER DEGREE GRETCHEN L. MATTHEWS AND TODD W. MICHEL DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634-0975 U.S.A. E-MAIL: GMATTHE@CLEMSON.EDU, TMICHEL@CLEMSON.EDU

More information