THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES
|
|
|
- Stephanie Perkins
- 6 years ago
- Views:
Transcription
1 Joural of Mathematical Aalysis ISSN: , URL: Volume 7 Issue 4(16, Pages THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES ABEDALLAH RABABAH, AYMAN AL SHBOOL Abstract I this paper, the basis trasformatio of the Chebyshev polyomials of fourth id ad the Berstei polyomials is cosidered ad the trasformatio matrices are derived 1 Itroductio Amog all possible bases forms, choosig the appropriate basis gives us the features we wat ad maes it easier to solve the problem tha whe usig other bases Polyomials i Berstei form give valuable isight of the geometric behavior of the polyomial But the Berstei basis is ot orthogoal; therefore, the least-squares approximatio problem ca ot be reduced O the other had, the Chebyshev polyomial basis is orthogoal; so the least-squares approximatio problem has a explicit solutio Explicit forms of bases trasformatios have bee derived; betwee Legedre ad Berstei bases i [19], betwee Chebyshev polyomials of the first id ad the Berstei bases i [8], betwee Jacobi ad Berstei polyomial bases i [6], betwee the Chebyshev of the secod id ad the Berstei bases i [16], ad betwee the Chebyshev of the Third id ad the Berstei bases i [] Applicatios to bases trasformatios ca be foud i [3], [7], [5], [4] I this paper we costruct the trasformatio matrix M betwee the Chebyshev polyomials of the fourth id ad the Berstei polyomials ad its iverse The Berstei polyomials of degree are defied o [,1] by: B i (u ( (1 u i u i, i (!, i, 1,,, (11 i i!( i! We will recall some properties of the Berstei polyomials that will help us gettig our results, see [15] for details We ca write each Berstei polyomial Bi (u of degree i terms of the Berstei basis Bm (u of degree m, where < m, usig the followig degree elevatio formula: Mathematics Subect Classificatio 35A7, 35Q53 Key words ad phrases Berstei polyomials, Chebyshev polyomials of fourth id, Trasformatio matrix c 16 Uiversiteti i Prishtiës, Prishtië, Kosovë Submitted July, 16 Published August 15, 16 13
2 14 ABEDALLAH RABABAH, AYMAN AL SHBOOL B i (u m +i i m i( i ( m B m (u, i, 1,, (1 The Chebyshev polyomials of the fourth id are orthogoal o the iterval [, 1] with respect to the weight fuctio w(u (u 1 (1 u 1 They are give i explicit form as follows: ( W (x ( (x 1 (x (13 Prelimiaries I this sectio, we state ad prove some importat relatios ad idetities i order to be used later i the mai results The double factorial of a iteger is give by: (( (4(, : eve!! (( (3(1, : odd!, : eve! 1 1!, : odd The defiitio of the factorial of a iteger plus oe half is give by (1 ( + 1! ( + 1 ( 1 ( 3 (3 (1 ( This factorial (ad the mius oe half ca be further simplified to get: ( + 1 ( + 1!!! +1, ( 1 ( 1!!! (3 Usig the defiitio of the double factorial, here are other relatios that will be used (!! ( + 1!!, ( + 1 (!! ( 1!! (4 The followig lemmas are eeded for the proof of Theorems 1 ad Lemma 1: For every > ad, 1,,, we have ( + 1!! ( + 1!!( 1!! Proof ( + 1!!(! (!( + 1!! ( + 1!!!(!(!! (!!( 1!!( + 1!!!(!! ( + 1!!!(!(! (! ( 1!!! ( + 1!! ( + 1!! ( + 1!!( 1!!
3 15 Lemma : For every > ad, 1,,, we have + 1( Proof Usig combiatorial properties, we ca reach the ed of the proof as follows: ( + 1!( 1! ( + 1!(!( 1!! ( + 1!!( 1!! ( + 1!!(!( 1!!! ( + 1!( 1!! 1 ( 1! (! 1 ( 1!( + 1!(!( 1!! ( +1 ( + 1!(!!(!!!( + 1!(!(!! +1 Lemma 3: The Beta fuctio, β, satisfies the followig equality: β(z + 1, + 1 π (z 1!!( 1!! z+ (z +! Proof Usig properties of the Gamma ad Beta fuctios, we ca get the result as follows: β(z+ 1, +1 Γ(z + 1 Γ( + 1 Γ(z (z! π(! π 4 z z!4!(z +! π ( z 4 z+ ( ( z z+ z Lemma 4: The Chebyshev IV polyomial W (u is expressed i the Berstei basis as follows: W (u ( + 1!! ( 1 ( + 1!!( 1!! B (u Proof Usig relatio (13 ad lemmas (, (3 with further simplificatios, we get W (u + 1( 1 (u 1 u ( ( + 1 ( 1 (1 u u + 1 ( 1 ( B (u ( 1 ( + 1!! ( + 1!!( 1!! B (u (+1!! π (z 1!!( 1!! z+ (z +! ( 1 (1 u u ( B (u ( 1 ( + 1!!( 1!! B (u
4 16 ABEDALLAH RABABAH, AYMAN AL SHBOOL Lemma 5: The itegral of the weighted product of the Berstei polyomial of degree ad the Chebyshev IV polyomial of degree is give by u 1 u B π (uw (udu ++1 ( 1 i( ( +1 +i+1 + i i+1 +i+1 + i ++1 +i+1 (5 Proof To complete the proof, previous relatios with lemma (3, lemma (4, ad beta fuctio properties are used u I 1 u B (uw (udu Usig lemma (4 leads to B (uw (uu 1 (1 u 1 du I I I ( ( + 1!! B (u(1 u 1 1 ( 1 i u ( + 1!! (i + 1!!( i 1!! B i (udu ( 1 i( i (i + 1!!( i 1!! By the defiitio of the beta fuctio, we get ( ( + 1!! Usig lemma (3 brigs I π ++1 Usig lemma (1 I π ++1 (1 u (+ (i+ 1 u +i+ 1 du ( 1 i( i (i + 1!!( i 1!! β( + i + 1, + i ( 1 i( i ( + i + 1!!( + i 1!!( + 1!! ( + + 1!(i + 1!!( i 1!! ( 1 i( +1 Usig the relatios i (4, we have π ( 1 i( +1 i+1 i+1 ( + i + 1!!( + i 1!! ( + + 1! ( +i +i + i + i ( + i!!( + i!!( + i + 1 ( + + 1! From the defiitio of the double factorial of the eve iteger, the equatio reduced to I π ++1 ( 1 i( +1 i+1 ( +i + i +i + i + +i Doig further simplificatios completes the proof ( + i + 1 ( + + 1
5 17 More iformatio about factorials, combiatorial, beta ad gamma fuctios ca be foud i [17, 1] 3 Basis Coversio Matrices The polyomial P (u is expressed i the Berstei basis ad the Chebyshev IV basis: P (u c B (u t W (u (31 To mae use of the advatages of the Berstei basis properties ad the Chebyshev IV basis properties, we fid the ( + 1 ( + 1 trasformatio matrix M that coverts the Chebyshev IV basis coefficiets t, t 1,, t to the Berstei basis coefficiets c, c 1,, c ad its iverse, M 1, that trasforms the Berstei to the Chebyshev IV basis Hece, M ad M 1 satisfy: c Mt ad t M 1 c, where c (c, c 1,, c T, ad t (t, t 1,, t T The matrices M ad M 1 are called the trasformatio matrices betwee the Chebyshev IV ad the Berstei basis The Chebyshev IV polyomial W (u ca be writte i terms of the Berstei basis as follows: W (u N B (u, (3 where N is the ( + 1 ( + 1 basis coversio matrix Multiplyig both sides with t ad taig the summatio over we have t W (u t N B (u Compare this relatio with the equatio (31 to get c Sice c Mt ad t M 1 c, we get c M t, ad t t N B (u t N (33 M 1 c,,, 1,, Comparig with (33 we fid that M N ; thus M N T The elemets of M are give i the followig theorem Theorem 1: The elemets of the matrix M that satisfies W B M which trasforms from the Chebyshev IV polyomial basis ito the Berstei polyomial basis for, are give by: M ( + 1!! mi(, imax(,+ ( 1 i( ( i i (i + 1!!( i 1!! (34
6 18 ABEDALLAH RABABAH, AYMAN AL SHBOOL Proof By applyig the degree elevatio (1 of the Berstei polyomials for Berstei polyomials of degree ad such that < we have: B i (u +i i ( i( i B (u, i, 1,, By substitutig the degree elevatio i Lemma 4 for the Chebyshev IV polyomials of degree we get: W (x ( + 1!! B (u ( 1 i +i (i + 1!!( i 1!! i mi(, imax(,+ ( i( i B (u ( 1 i ( + 1!! ( ( i i (i + 1!!( i 1!! The elemets N of the matrix ca be costructed after solvig the liear trasformatio W (u N B (u, ad hece we get the matrix M by trasposig the matrix N Theorem : The elemets of the matrix M 1 that satisfies B W M 1 which trasforms from the Berstei polyomial basis ito the Chebyshev IV polyomial basis for, are give by: M 1 Proof We ow that 4 + ( 1 i( +1 i+1 B (u ( +i+1 + i +i+1 + i ++1 +i+1 N 1 W i(u Multiply the previous equatio by W (u( u 1 u 1, the itegrate o [, 1], ad use the orthogoality property of the Chebyshev IV polyomial basis to get: N 1 B u (uw (u( π 1 u 1 du After applyig Lemma 5, we have: N 1 + ( 1 i( i ( + i + 1!!( + i 1!!( + 1!! ( + + 1!(i + 1!!( i 1!! After some calculatios for N 1 ad traspositio of N 1 we obtai M 1 as desired Acowledgmets The authors would lie to tha the reviewers for their valuable commets that lead to better represetatio of the paper
7 19 Refereces [1] A H Bhrawy, EH Doha, D Baleau, MA Saer, Modified Jacobi-Berstei basis trasformatio ad its applicatio to multi-degree reductio of Bezier curves,joural of Computatioal ad Applied Mathematics 3,16 [] A Rababah ad A Al Shbool, The trasformatio matrix of Chebyshev III Berstei polyomial basis, Iteratioal Coferece of Numarical Aalysis ad Applied Mathematics 15 (ICNAAM 15, 1738, Issue 1, [3] A Rababah, B G Lee, ad J Yoo, Multiple degree reductio ad elevatio of Bézier curves usig Jacobi-Berstei basis trasformatio, Numerical Fuctioal Aalysis ad Optimizatio 8, Issue 9-1, ( [4] A Rababah, B G Lee, ad J Yoo, A simple matrix form for degree reductio of Bézier curves usig Chebyshev-Berstei basis trasformatios Applied Mathematics ad Computatio 181, ( [5] A Rababah, Itegratio of Jacobi ad Weighted Berstei Polyomials usig Bases Trasformatios Computatioal Methods i Applied Mathematics 7(3, (7 1-6 [6] A Rababah,Jacobi-Berstei basis trasformatio, Comput Meth Appl Math (4, 4:6-14 [7] A Rababah, M Al-Refai, ad R Al-Jarrah, Computig Derivatives of Jacobi Polyomials Usig Berstei Trasformatio ad Differetiatio Matrix Numerical Fuctioal Aalysis ad Optimizatio 9, Issue 5-6, ( [8] A Rababah, Trasformatio of Chebyshev-Berstei polyomial basis, Comput Meth Appl Math, 3 (3, pp 68-6 [9] AS Olaguu, FL Joseph,Third-id Chebyshev Polyomials V r(x i Collocatio Methods of Solvig Boudary value Problems, IOSR Joural of Maths 13; (8:4-47 [1] EDRaiville, Special Fuctios, Chelsea Publ Co, Brox, New Yor, 1971 [11] G G Lorez, Berstei polyomials, Mathematical Expositios No 8, Toroto Press, 1953 [1] G Szego, Orthogoal Polyomials, 4th ed, America Mathematical Society, Providece, RI, 1975 [13] JCMaso ad DC Hadscomb, Chebyshev Polyomials, CRC Press Compay, 3 [14] J Rice, The Approximatio of Fuctios, Vol 1, Liear Theory, Addiso Wesley, 1964 [15] K Höllig ad J Hörer (13 Approximatio ad Modelig with B-Splies SIAM Titles i Applied Mathematics 13 [16] L Lu, G Wag, Applicatio of Chebyshev II-Berstei basis trasformatios to degree reductio of Bézier curves, J Comput Appl Math 1 ( [17] RLGraham, DEKuth, ad OPatashi, Cocrete Mathematics, Addiso-Wesley, Readig, MA,1989 [18] R T Faroui, O the stability of trasformatios betwee power ad Berstei polyomial forms, Comput Aided Geom Des, 8 (1991, pp 9-36 [19] R T Faroui, Legedre Berstei basis trasformatios, J Comput Appl Math, 119 (, pp [] R T Faroui ad T N T Goodma, O the optimal stability of Berstei basis, Math Comput, 65 (1996, pp [1] R T Faroui ad V T Raa, Algorithms for polyomials i Berstei form, Comput Aided Geom Des, 5 (1988, pp 1-6 [] T S Chihara, A Itroductio to Orthogoal Polyomials, Gordo ad Breach, New Yor, 1978 [3] T Herma, O the stability of polyomial trasformatios betwee Taylor, Berstei ad Hermite forms, Numer Algorithms, 13 (1996, pp 37-3 Departmet of Mathematics ad Statistics, Jorda Uiversity of Sciece ad Techology, Irbid 11, Jorda address: rababah@usteduo Departmet of Mathematics ad Statistics, Jorda Uiversity of Sciece ad Techology, Irbid 11, Jorda address: dragool@hotmailcom
Distance for degree raising and reduction of triangular Bezier surfaces
Joural of Computatioal ad Applied Mathematics 158 (003) 33 41 www.elsevier.com/locate/cam Distace for degree raisig ad reductio of triagular Bezier surfaces Abedallah Rababah Departmet of Mathematics ad
Sequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
Solution of Differential Equation from the Transform Technique
Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from
GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION
J Korea Math Soc 44 (2007), No 2, pp 487 498 GENERALIZED HARMONIC NUMBER IDENTITIES AND A RELATED MATRIX REPRESENTATION Gi-Sag Cheo ad Moawwad E A El-Miawy Reprited from the Joural of the Korea Mathematical
Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 15), PP 1-11 www.iosrjourals.org Numerical Solutios of Secod Order Boudary Value Problems
Taylor polynomial solution of difference equation with constant coefficients via time scales calculus
TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu
SOME RELATIONS ON HERMITE MATRIX POLYNOMIALS. Levent Kargin and Veli Kurt
Mathematical ad Computatioal Applicatios, Vol. 18, No. 3, pp. 33-39, 013 SOME RELATIONS ON HERMITE MATRIX POLYNOMIALS Levet Kargi ad Veli Kurt Departmet of Mathematics, Faculty Sciece, Uiversity of Adeiz
Generating Functions for Laguerre Type Polynomials. Group Theoretic method
It. Joural of Math. Aalysis, Vol. 4, 2010, o. 48, 257-266 Geeratig Fuctios for Laguerre Type Polyomials α of Two Variables L ( xy, ) by Usig Group Theoretic method Ajay K. Shula* ad Sriata K. Meher** *Departmet
Series with Central Binomial Coefficients, Catalan Numbers, and Harmonic Numbers
3 47 6 3 Joural of Iteger Sequeces, Vol. 5 (0), Article..7 Series with Cetral Biomial Coefficiets, Catala Numbers, ad Harmoic Numbers Khristo N. Boyadzhiev Departmet of Mathematics ad Statistics Ohio Norther
Harmonic Number Identities Via Euler s Transform
1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810
The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
1 6 = 1 6 = + Factorials and Euler s Gamma function
Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio
DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan
Mathematical ad Computatioal Applicatios, Vol. 9, No. 3, pp. 30-40, 04 DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS Muhammad Aslam Noor, Khalida Iayat Noor ad Asif Waheed
CALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
CertainSequencesanditsIntegralRepresentationsinTermsofLaguerrePolynomials
Global Joural of Sciece Frotier Research: F Mathematics ad Decisio Scieces Volume 5 Issue 5 Versio. Year 5 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic. USA
Numerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
COMMON FIXED POINT THEOREMS VIA w-distance
Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3, Pages 182-189 COMMON FIXED POINT THEOREMS VIA w-distance (COMMUNICATED BY DENNY H. LEUNG) SUSHANTA
A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
Fibonacci numbers and orthogonal polynomials
Fiboacci umbers ad orthogoal polyomials Christia Berg April 10, 2006 Abstract We prove that the sequece (1/F +2 0 of reciprocals of the Fiboacci umbers is a momet sequece of a certai discrete probability,
A Note On The Exponential Of A Matrix Whose Elements Are All 1
Applied Mathematics E-Notes, 8(208), 92-99 c ISSN 607-250 Available free at mirror sites of http://wwwmaththuedutw/ ame/ A Note O The Expoetial Of A Matrix Whose Elemets Are All Reza Farhadia Received
On the Inverse of a Certain Matrix Involving Binomial Coefficients
It. J. Cotemp. Math. Scieces, Vol. 3, 008, o. 3, 5-56 O the Iverse of a Certai Matrix Ivolvig Biomial Coefficiets Yoshiari Iaba Kitakuwada Seior High School Keihokushimoyuge, Ukyo-ku, Kyoto, 60-0534, Japa
wavelet collocation method for solving integro-differential equation.
IOSR Joural of Egieerig (IOSRJEN) ISSN (e): 5-3, ISSN (p): 78-879 Vol. 5, Issue 3 (arch. 5), V3 PP -7 www.iosrje.org wavelet collocatio method for solvig itegro-differetial equatio. Asmaa Abdalelah Abdalrehma
Bijective Proofs of Gould s and Rothe s Identities
ESI The Erwi Schrödiger Iteratioal Boltzmagasse 9 Istitute for Mathematical Physics A-1090 Wie, Austria Bijective Proofs of Gould s ad Rothe s Idetities Victor J. W. Guo Viea, Preprit ESI 2072 (2008 November
ON SOME TRIGONOMETRIC POWER SUMS
IJMMS 0: 2002 185 191 PII. S016117120200771 http://ijmms.hidawi.com Hidawi Publishig Corp. ON SOME TRIGONOMETRIC POWER SUMS HONGWEI CHEN Received 17 Jue 2001 Usig the geeratig fuctio method, the closed
Bangi 43600, Selangor Darul Ehsan, Malaysia (Received 12 February 2010, accepted 21 April 2010)
O Cesáro Meas of Order μ for Outer Fuctios ISSN 1749-3889 (prit), 1749-3897 (olie) Iteratioal Joural of Noliear Sciece Vol9(2010) No4,pp455-460 Maslia Darus 1, Rabha W Ibrahim 2 1,2 School of Mathematical
Discrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
Recursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION
NEW NEWTON-TYPE METHOD WITH k -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds West Yorkshire, LS7 JS, ENGLAND ABSTRACT The objective
Gamma Distribution and Gamma Approximation
Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 456-46, USA cheg@cs.uky.edu Abstract
PAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu
MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
Computation of Error Bounds for P-matrix Linear Complementarity Problems
Mathematical Programmig mauscript No. (will be iserted by the editor) Xiaoju Che Shuhuag Xiag Computatio of Error Bouds for P-matrix Liear Complemetarity Problems Received: date / Accepted: date Abstract
Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
Problem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient
Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case
Numerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets
Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN 99-878 Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie
CALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
AMS Mathematics Subject Classification : 40A05, 40A99, 42A10. Key words and phrases : Harmonic series, Fourier series. 1.
J. Appl. Math. & Computig Vol. x 00y), No. z, pp. A RECURSION FOR ALERNAING HARMONIC SERIES ÁRPÁD BÉNYI Abstract. We preset a coveiet recursive formula for the sums of alteratig harmoic series of odd order.
ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS
Joural of Algebra, Number Theory: Advaces ad Applicatios Volume, Number, 00, Pages 7-89 ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS OLCAY KARAATLI ad REFİK KESKİN Departmet
Modified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations
Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-
Zeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
A NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS
Jural Karya Asli Loreka Ahli Matematik Vol. No. (010) page 6-9. Jural Karya Asli Loreka Ahli Matematik A NEW CLASS OF -STEP RATIONAL MULTISTEP METHODS 1 Nazeeruddi Yaacob Teh Yua Yig Norma Alias 1 Departmet
An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions
A Asymptotic Expasio for the Number of Permutatios with a Certai Number of Iversios Lae Clark Departmet of Mathematics Souther Illiois Uiversity Carbodale Carbodale, IL 691-448 USA lclark@math.siu.edu
q-lucas polynomials and associated Rogers-Ramanujan type identities
-Lucas polyomials associated Rogers-Ramauja type idetities Joha Cigler Faultät für Mathemati, Uiversität Wie johacigler@uivieacat Abstract We prove some properties of aalogues of the Fiboacci Lucas polyomials,
Roger Apéry's proof that zeta(3) is irrational
Cliff Bott cliffbott@hotmail.com 11 October 2011 Roger Apéry's proof that zeta(3) is irratioal Roger Apéry developed a method for searchig for cotiued fractio represetatios of umbers that have a form such
TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction
Math Appl 6 2017, 143 150 DOI: 1013164/ma201709 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES PANKAJ KUMAR DAS ad LALIT K VASHISHT Abstract We preset some iequality/equality for traces of Hadamard
Proc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS
Proc. Amer. Math. Soc. 139(2011, o. 5, 1569 1577. BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Zhi-Wei Su* ad Wei Zhag Departmet of Mathematics, Naig Uiversity Naig 210093, People s Republic of
Chapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
arxiv: v1 [math.nt] 28 Apr 2014
![arxiv: v1 [math.nt] 28 Apr 2014](/thumbs/88/114694424.jpg "arxiv: v1 [math.nt] 28 Apr 2014")
Proof of a supercogruece cojectured by Z.-H. Su Victor J. W. Guo Departmet of Mathematics, Shaghai Key Laboratory of PMMP, East Chia Normal Uiversity, 500 Dogchua Rd., Shaghai 0041, People s Republic of
f(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
Computation for Jacobi-Gauss Lobatto Quadrature Based on Derivative Relation
Computatio for acobi-auss obatto Quadrature Based o Derivative Relatio Z.S. Zheg uaghui Huag Abstract. The three-term recurrece relatio for derivatives of acobi-type polyomial is derived ad the auss-obatto
A MASTER THEOREM OF SERIES AND AN EVALUATION OF A CUBIC HARMONIC SERIES. 1. Introduction
Joural of Classical Aalysis Volume 0, umber 07, 97 07 doi:0.753/jca-0-0 A MASTER THEOREM OF SERIES AD A EVALUATIO OF A CUBIC HARMOIC SERIES COREL IOA VĂLEA Abstract. I the actual paper we preset ad prove
NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES
Publ. Math. Debrece 8504, o. 3-4, 85 95. ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES QING-HU HOU*, ZHI-WEI SUN** AND HAOMIN WEN Abstract. We cofirm Su s cojecture that F / F 4 is strictly decreasig
62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
A q-analogue of some binomial coefficient identities of Y. Sun
A -aalogue of some biomial coefficiet idetities of Y. Su arxiv:008.469v2 [math.co] 5 Apr 20 Victor J. W. Guo ad Da-Mei Yag 2 Departmet of Mathematics, East Chia Normal Uiversity Shaghai 200062, People
SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS
Folia Mathematica Vol. 5, No., pp. 4 6 Acta Uiversitatis Lodziesis c 008 for Uiversity of Lódź Press SOME TRIGONOMETRIC IDENTITIES RELATED TO POWERS OF COSINE AND SINE FUNCTIONS ROMAN WITU LA, DAMIAN S
Euler-type formulas. Badih Ghusayni. Department of Mathematics Faculty of Science-1 Lebanese University Hadath, Lebanon
Iteratioal Joural of Mathematics ad Computer Sciece, 7(), o., 85 9 M CS Euler-type formulas Badih Ghusayi Departmet of Mathematics Faculty of Sciece- Lebaese Uiversity Hadath, Lebao email: badih@future-i-tech.et
De la Vallée Poussin Summability, the Combinatorial Sum 2n 1
J o u r a l of Mathematics ad Applicatios JMA No 40, pp 5-20 (2017 De la Vallée Poussi Summability, the Combiatorial Sum 1 ( 2 ad the de la Vallée Poussi Meas Expasio Ziad S. Ali Abstract: I this paper
Physics 116A Solutions to Homework Set #9 Winter 2012
Physics 116A Solutios to Homework Set #9 Witer 1 1. Boas, problem 11.3 5. Simplify Γ( 1 )Γ(4)/Γ( 9 ). Usig xγ(x) Γ(x + 1) repeatedly, oe obtais Γ( 9) 7 Γ( 7) 7 5 Γ( 5 ), etc. util fially obtaiig Γ( 9)
A New Method to Order Functions by Asymptotic Growth Rates Charlie Obimbo Dept. of Computing and Information Science University of Guelph
A New Method to Order Fuctios by Asymptotic Growth Rates Charlie Obimbo Dept. of Computig ad Iformatio Sciece Uiversity of Guelph ABSTRACT A ew method is described to determie the complexity classes of
FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS. H. W. Gould Department of Mathematics, West Virginia University, Morgantown, WV 26506, USA
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A58 FLOOR AND ROOF FUNCTION ANALOGS OF THE BELL NUMBERS H. W. Gould Departmet of Mathematics, West Virgiia Uiversity, Morgatow, WV
Question 1: The magnetic case
September 6, 018 Corell Uiversity, Departmet of Physics PHYS 337, Advace E&M, HW # 4, due: 9/19/018, 11:15 AM Questio 1: The magetic case I class, we skipped over some details, so here you are asked to
Analytical solutions for multi-wave transfer matrices in layered structures
Joural of Physics: Coferece Series PAPER OPEN ACCESS Aalytical solutios for multi-wave trasfer matrices i layered structures To cite this article: Yu N Belyayev 018 J Phys: Cof Ser 109 01008 View the article
Chapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
Q-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received:
ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS
STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume XLVII, Number 4, December 2002 ON BLEIMANN, BUTZER AND HAHN TYPE GENERALIZATION OF BALÁZS OPERATORS OGÜN DOĞRU Dedicated to Professor D.D. Stacu o his 75
PROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1.
Math 7 Sprig 06 PROBLEM SET 5 SOLUTIONS Notatios. Give a real umber x, we will defie sequeces (a k ), (x k ), (p k ), (q k ) as i lecture.. (a) (5 pts) Fid the simple cotiued fractio represetatios of 6
Seed and Sieve of Odd Composite Numbers with Applications in Factorization of Integers
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-75X. Volume 1, Issue 5 Ver. VIII (Sep. - Oct.01), PP 01-07 www.iosrjourals.org Seed ad Sieve of Odd Composite Numbers with Applicatios i
Solving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods
Applied ad Computatioal Mathematics 07; 6(6): 38-4 http://www.sciecepublishiggroup.com/j/acm doi: 0.648/j.acm.070606. ISSN: 38-5605 (Prit); ISSN: 38-563 (Olie) Solvig a Noliear Equatio Usig a New Two-Step
COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun
Korea J. Math. 23 2015) No. 3 pp. 371 377 http://dx.doi.org/10.11568/kjm.2015.23.3.371 COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q } Sag Pyo Ju Abstract. I this ote we cosider a geeralized
(p, q)-type BETA FUNCTIONS OF SECOND KIND
Adv. Oper. Theory 6, o., 34 46 http://doi.org/.34/aot.69. ISSN: 538-5X electroic http://aot-math.org p, q-type BETA FUNCTIONS OF SECOND KIND ALI ARAL ad VIJAY GUPTA Commuicated by A. Kamisa Abstract. I
Optimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
Dominating Sets and Domination Polynomials of Square Of Cycles
IOSR Joural of Mathematics IOSR-JM) ISSN: 78-78. Volume 3, Issue 4 Sep-Oct. 01), PP 04-14 www.iosrjourals.org Domiatig Sets ad Domiatio Polyomials of Square Of Cycles A. Vijaya 1, K. Lal Gipso 1 Assistat
Recurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
ECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
Bertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
PHY4905: Nearly-Free Electron Model (NFE)
PHY4905: Nearly-Free Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. No-degeerate eigestates
MAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
Approximation properties of (p, q)-bernstein type operators
Acta Uiv. Saietiae, Mathematica, 8, 2 2016 222 232 DOI: 10.1515/ausm-2016-0014 Aroximatio roerties of, -Berstei tye oerators Zoltá Fita Deartmet of Mathematics, Babeş-Bolyai Uiversity, Romaia email: fzolta@math.ubbcluj.ro
Generalization of Samuelson s inequality and location of eigenvalues
Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,
CLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS
Submitted to the Bulleti of the Australia Mathematical Society doi:10.1017/s... CLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS GAŠPER JAKLIČ, VITO VITRIH ad EMIL ŽAGAR Abstract I this paper,
3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
Chapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
CHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
EVALUATION OF A CUBIC EULER SUM RAMYA DUTTA. H n
Joural of Classical Aalysis Volume 9, Number 6, 5 59 doi:.753/jca-9-5 EVALUATION OF A CUBIC EULER SUM RAMYA DUTTA Abstract. I this paper we calculate the cubic series 3 H ad two related Euler Sums of weight
Let A(x) and B(x) be two polynomials of degree n 1:
MI-EVY (2011/2012) J. Holub: 4. DFT, FFT ad Patter Matchig p. 2/42 Operatios o polyomials MI-EVY (2011/2012) J. Holub: 4. DFT, FFT ad Patter Matchig p. 4/42 Efficiet Patter Matchig (MI-EVY) 4. DFT, FFT
Determinants of order 2 and 3 were defined in Chapter 2 by the formulae (5.1)
5. Determiats 5.. Itroductio 5.2. Motivatio for the Choice of Axioms for a Determiat Fuctios 5.3. A Set of Axioms for a Determiat Fuctio 5.4. The Determiat of a Diagoal Matrix 5.5. The Determiat of a Upper
An enumeration of flags in finite vector spaces
A eumeratio of flags i fiite vector spaces C Rya Viroot Departmet of Mathematics College of William ad Mary P O Box 8795 Williamsburg VA 23187 viroot@mathwmedu Submitted: Feb 2 2012; Accepted: Ju 27 2012;
Rational Bounds for the Logarithm Function with Applications
Ratioal Bouds for the Logarithm Fuctio with Applicatios Robert Bosch Abstract We fid ratioal bouds for the logarithm fuctio ad we show applicatios to problem-solvig. Itroductio Let a = + solvig the problem
Some integrals related to the Basel problem
November, 6 Some itegrals related to the Basel problem Khristo N Boyadzhiev Departmet of Mathematics ad Statistics, Ohio Norther Uiversity, Ada, OH 458, USA k-boyadzhiev@ouedu Abstract We evaluate several
Chimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
Some properties of Boubaker polynomials and applications
Some properties of Boubaker polyomials ad applicatios Gradimir V. Milovaović ad Duša Joksimović Citatio: AIP Cof. Proc. 179, 1050 (2012); doi: 10.1063/1.756326 View olie: http://dx.doi.org/10.1063/1.756326
THE REPRESENTATION OF THE REMAINDER IN CLASSICAL BERNSTEIN APPROXIMATION FORMULA
Global Joural of Advaced Research o Classical ad Moder Geometries ISSN: 2284-5569, Vol.6, 2017, Issue 2, pp.119-125 THE REPRESENTATION OF THE REMAINDER IN CLASSICAL BERNSTEIN APPROXIMATION FORMULA DAN
On the Determinants and Inverses of Skew Circulant and Skew Left Circulant Matrices with Fibonacci and Lucas Numbers
WSEAS TRANSACTIONS o MATHEMATICS Yu Gao Zhaoli Jiag Yapeg Gog O the Determiats ad Iverses of Skew Circulat ad Skew Left Circulat Matrices with Fiboacci ad Lucas Numbers YUN GAO Liyi Uiversity Departmet
On a class of convergent sequences defined by integrals 1
Geeral Mathematics Vol. 4, No. 2 (26, 43 54 O a class of coverget sequeces defied by itegrals Dori Adrica ad Mihai Piticari Abstract The mai result shows that if g : [, ] R is a cotiuous fuctio such that