International Journal of Multidisciplinary Research and Development. M.A. Gopalan, A. Kavitha, G. Thamaraiselvi
|
|
|
- Primrose Irene O’Neal’
- 5 years ago
- Views:
Transcription
1 Volume:, Issue: 6, -7 Jue 015.allsubjectjoural.com e-issn: p-issn: Impact Factor:.76 M.A. Gopala Professor, Departmet of Idira Gadhi College, Trichy-6000, Tamiladu, Idia. A. Kavitha Lecturer, Departmet of Idira Gadhi College, Trichy-6000, Tamiladu, Idia. G. Thamaraiselvi M. Phil., Scholar, Departmet of Mathematics, Shrimati Idira Gadhi College, Trichy-6000, Tamiladu, Idia. Observatio o the Biquadratic Equatio ith Five Ukos (x-y(x y (k k ( X Y M.A. Gopala, A. Kavith G. Thamaraiselvi Abstract (x, y,x, Y, We obtai ifiitely may o-zero iteger quituples satisfyig the Bi-Quadratic (x - y (x y (k k ( X Y equatio. Various iterestig relatios betee the solutios ad special umbers, amely, Polygoal umbers, Proic umbers, Pyramidal umbers, Stella Octagular umbers, ad Octahedral umbers are exhibited. Keyords: Biquadratic equatio ith five ukos, Itegral solutios, Polygoal ad Pyramidal umbers. 010 Mathematics subject classificatio: 11D5. Notatios Used t m, : Polygoal umber of rak ith sides m t m, m P P m ( 1(m 1 : Pyramidal umber of rak ith sides m ( 1 6 ( m (5 m SO : Stella Octagular umber of rak SO ( 1 Pr : Proic umber of rak Pr OH ( 1 : Octahedral umber of rak OH 1 ( 1 Correspodece: M.A. Gopala Professor, Departmet of Idira Gadhi College, Trichy-6000, Tamiladu, Idia. 1. Itroductio The theory of Diophatie equatios offers a rich variety of fasciatig problems. I particular biquadratic Diophatie equatios, homogeeous ad o-homogeeous have aroused the iterest of umerous mathematicias sice atiquity [1-4] [5-10, 1,. I this cotext oe may refer 1] for various problems o the biquadratic Diophatie equatios ith four ad five variables [11]. have discussed the problem o biquadratic Diophatie equatio ith five ukos. This paper cocers ith yet aother problem of determiig o-trivial itegral solutios of the o-homogeeous biquadratic equatio ith five ukos give by (x - y (x y (k k ( X Y. A fe relatios amog the solutios are preseted. ~ ~
2 . Method of Aalysis The Diophatie equatio represetig the biquadratic equatio ith five ukos uder cosideratio is (x - y (x y (k k ( X Y Itroducig the liear trasformatios x u v, X u v, y u v Y u v i (1, it simplifies to u v (4k 4k 4 The above equatio ( is solved through differet choices ad thus, oe obtais distict patters of iteger solutios to (1.1 Patter: 1 Let a b (4 Substitutig (4 i ( ad usig the method of factorizatio, defie ( u i v ( a i ((k 1 i Equatig real ad imagiary parts, e have u a b ka 6kb 6ab v a b ab 4kab Substitutig the values of u ad v i (, the o-zero distict itegral solutios of (1 are give by a 6b ka 6kb 4ab 4kab ka 6kb 8ab 4kab a 9b 4ka 1kb 10ab 4kab (5 Y ( a b 4ka 1kb 14ab 4kab ( a b. Properties a 1- a 1 4t 4,a 4Pra ( 1 1a 6 0 a -1 a -1 t46, a 4t6, a (1 -a 6 0 (1 ( ( Write 1 as 1 1 i 1 i 4 Usig (7 i (6 ad employig the method of factorizatio, defie ( u i v ((k 1 i ( a i Equatig real ad imagiary parts i (8, e get 1 i * u a b 6ab ka kb 6kab v a b ab ka kb kab Usig (9 ad (, e get the itegral solutio of (1 as 8ab ka 6kb 4kab a 6b 4ab 8kab a b 14ab ka 9kb 10kab (10 Y ( a 9b 10ab ka kb 14kab ( a b.4 Properties a 1 a 1 16Pr a( k(7t4, a 0a15 0 a 1 ( a 1 4t 4,a 1OH a (1 0.5 Note It is to be oted that i additio to (7, 1 may also represeted as the product of complex cojugates as give belo: Choice: 1 1 i 1 i 1 4 Choice: 11 i4 11 i (7 (8 (9 Choice: 11 i4 11 i Patter: ( ca be ritte as u v (4k 4k 4 *1 (6 ~ 4 ~ Choice: 4 1 i4 1 i4 1 49
3 Choice: 5 1 i4 1 i Choice: 6 11 i5 11 i5 1 Choice: 7 11 i5 11 i5 1 Choice: 8 1 i 1 i 1 Choice: 9 1 i 1 i 1 Employig the procedure preseted i patter:, the correspodig itegral solutios to (1 for the above choices of 1 are obtaied as belo; Solutio for choice: 1 a 6b 4ab 8kab a 6b 4ab ka 6kb 4kab X ( 4a 1b 4ab ka kb 14kab Y ( 4a 1b 4ab ka 9kb 10kab ( a b 17a 51b 8ab 4ka 6kb 9kab 7a 81b 8ab1ka 1kb 100kab ( 7a 1b Solutio for choice: 5 10a 0b 44ab 6ka 18kb 5kab 16a 48b 8ab 10ka 0kb 44kab a 69b 6ab 4ka 1kb 100kab Y ( 9a 87b 10ab 1ka 6kb 9kab ( 7a 1b Solutio for choice: 6 1a 6b 104ab 1ka 96kb 16kab 0a 60b 88abka 6kb 104kab 8a 4b 00ab4ka 16kb 76kab 4a 7b 184ab 54ka 10kb 164kab ( 14a 4b Solutio for choice: 7 a 96b 16ab1ka 6kb 104kab 0a 60b 88ab1ka 96kb 16kab 58a 174b 0ab4ka 10kb 164kab 46a 18b 14ab4ka 16kb 76kab ( 14a 4b Solutio for choice: 14a 4b 48ab 0ka 90kb 48kab 16a 48b 1ab14ka 4kb 48kab 1a 9b 18ab5ka 156kb 96kab 17a 51b ab6ka 108kb 96kab Solutio for choice: 0a 90b 4ab 14ka 4kb 9kab 16a 48b 88ab0ka 90kb 4kab 5a 159b 8ab6ka 108kb 140kab Y ( 9a 117b 10ab 5ka 156kb 5kab ( 1a 9b Solutio for choice: 4 6a 18b 5ab 6ka 0kb 44kab 16a 48b 8ab 10ka 18kb 5kab ~ 5 ~ Solutio for choice: 8 0a 60b 88ab ka 96kb 16kab 1a 6b 104ab 0ka 60kb 88kab X ( 4a 7b 184ab 58ka 174kb 0kab 8a 4b 00ab46ka 18kb 14kab ( 14a 4b Solutio for choice: 9 a 96b 16ab 0ka 60kb 88kab 1a 6b 104abka 96kb 16kab X ( 54a 16b 76ab 46ka 18kb 14kab Y ( 4a 10b 164ab 58ka 174kb 0kab (.6 Patter: Rerite ( as u 14a 4b (k 1 ( v (11
4 (11 is ritte i the form of ratio as u (k 1 ( v, 0 v u (k 1 Which is equivalet to the system of double equatios 0 (1 0 Solvig the above system (1 by applyig the method of cross multiplicatio, e get u k 6k 6 v 4k (1 Usig (1 ad (, e get the correspodig o-zero iteger solutios of (1 to be, k 6k 8 4k, k 6 6k 4 4k X (, 4k 1k 14 4k, 4k 9 1k 10 4k (,.7 Properties a -1 a -1 6t4, a 4t 4SO a(1 60 4, a 8a (1 0 1 ( a1 t4, a(1 4 Pra (7-4ka-1 0 a - a a 5 a1 (a1 t4, a( 8 1(a k 1 4P a(5 0 a-1 ( a-1 t (11 4t (1 6k(4a Remark 4, a 6, a (11 may also be expressed i the form of ratios i three more differet ays that are preseted belo: Way 1 u (k 1 ( v ( v u (k 1 Way u (k 1 ( v ( v u (k 1 Way u (k 1 ( v ( v u (k 1 ~ 6 ~ Solvig each of the above system of equatios by folloig the procedure as preseted i Patter:, the correspodig iteger solutios to (1 are foud to be as give belo: Solutio for ay 1,, 8 6k k 4k,, 6 4 6k k 4k,, 14 1k 4k 4k,, k 4k 4k (, Solutio for ay,, 6 4 k 6k 4k,, k 6k 8 4k,, k 1k 4k,, 14 4k 1k 4k (, Solutio for ay,, 6 6k k 4k,, 6k k 4k X (,, 9 1k 4k 4k,, 1k 4k 4k (,. Coclusio I this paper, e have preseted differet choices of iteger solutios to the homogeeous biquadratic equatio ith five ukos (x - y (x y (k k ( X Y. It is orth metioig here that i (, the liear trasformatios for X ad Y may also be cosidered as I X=uv+, Y=uv- ad II X=uv+1, Y=uv-1. Employig the above to forms of trasformatios for X,Y differet values for X ad Y are obtaied. To coclude, as biquadratic equatios are rich i variety, oe may cosider other forms of biquadratic equatios ad search for correspodig properties. 4. Ackoledgemet The fiical support from the UGC, Ne Delhi (F.MRP- 51/14(SERO/UGC dated march 014 for a part of this ork is gratefully ackoledged. 5. Refereces 1. L.E. Dickso, History of Theory of Numbers, Vol.11, Chelsea Publishig Compay, Ne York (195.. L.J. Mordell, Diophatie equatios, Academic Press, Lodo (9.. Carmichael, R.D., The theory of umbers ad Diophatie Aalysis, Dover Publicatios, Ne York (1959
5 4. Telag, S.G., Number theory, Tata McGra Hill publishig compay, Ne Delhi ( M.A. Gopala & P. Shamugaadham, O the biquadratic equatio x y z, Impact J.Sci tech; vol.4, No, 4, , ( M.A. Gopala & G. Sageeth Itegral Solutios of No-Homogeeous Quartic equatio 4 4 x y ( 1( z, Impact J.Sci Tech; Vol.4 No., 15-1, M.A. Gopala & Padm Itegral Solutio of No- 4 4 Homogeeous Quartic equatio x y z, Atarctica J. Math., 7(4,71-77, M.A. Gopala & V. Padichelvi o the solutios of the 4 Biquadratic Equatio ( x y ( z 1 Paper preseted i the iteratioal coferece o Mceathematical Methods ad Computatio, Jamal Mohamed college, Tiruchirapalli, July 4-5, M.A.Gopala, S. Vidhyalakshmi ad K. Lakshmi, O The Biquadratic Equatio ith Four ukos x xy y ( z z IJPAM (RIP vol.5, No.1, 7-77, M.A. Gopala & J. Kaligarai, Quartic equato i five 4 4 ukos x y ( z p Bulleti of Pure ad Applied Scieces Vol8E, No., 05-11, M.A. Gopala, J. Kaligarai, Quartic equato i five 4 4 ukos x y ( z p. Impact J.Sci Tech; Vol.4 No., 15-1, M.A. Gopala, S. Vidhyalakshmi, A.Kavith observatios o the biquadratic ith five ukos 4 4 x y x x y Z( X Y IJESM, Vol, Issue, 19-00, Jue M.A. Gopala, S. Vidhyalakshmi, E. Premalath o the homogeeous biquadratic ith five ukos 4 4 x y 8( z p IJSRP, Vol 4, issue 1, 1-5, Ja 014. ~ 7 ~
GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES
GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES A STUDY ON THE HYPERBOLA x 9 0 S. Vidhyalakshmi, M.A. Gopala & T. Mahalakshmi*, Professor, Departmet of Mathematics, Shrimati Idira Gadhi College, Trichy-60
- Pyramidal number of rank n with size m. - Polygonal number of rank n with size m.
Scholars Joural of Egieerig ad Techology SJET Sch. J. Eg. Tech., 0; :9-3 Scholars cademic ad Scietific Publisher Iteratioal Publisher for cademic ad Scietific Resources www.saspublisher.com ISSN 3-35X
Abstract We obtain infinitely many non-zero integer sextuples ( x, y, z, w, p, T )
IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY O the No-Homogeeous Equatio of the Eighth Degree with Six Ukows x 5 -y 5 +(x 3 -y 3 xy = p(z -w T 3 S.Vidhyalakshmi *1, K.Lakshmi,
ONTERNARY QUADRATIC DIOPHANTINE EQUATION 2x 2 + 3y 2 = 4z
ONTERNARY QUADRATIC DIOPHANTINE EQUATION x + 3y = 4z M.A.Gopala 1, K.Geetha ad Maju Somaath 3 1 Professor,Dept.of Mathematics,Shrimati Idira Gadhi College,Trichy- 0 Tamiladu, Asst Professor,Dept.of Mathematics,Cauvery
OBSERVATIONS ON TERNARY QUADRATIC DIOPHANTINE EQUATION - x 63y
OBSERVATIONS ON TERNARY QUADRATIC DIOPHANTINE EQUATION - x 63 K.Lakshmi R.Someshwari Asst.Professor, Departmet of Maematics, Shrimati Idira Gadhi College,Tamil adu,idia, M.Phil Scholar, Departmet of Maematics,
Research Article. ISSN (Print) *Corresponding author C.Nithya Abstract: The binary quadratic equation x 5xy
Scholars Joural o Egieerig ad Techology (SJET) Sch. J. Eg. Tech., 04; (B):6-30 Scholars Academic ad Scietiic Publisher (A Iteratioal Publisher or Academic ad Scietiic Resources) www.saspublisher.com ISSN
Pythagorean Triangle with Area/ Perimeter as a special polygonal number
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578p-ISSN: 319-765X Volume 7 Issue 3 (Jul. - Aug. 013) PP 5-6 Pythagorea Triagle with Area/ Perimeter as a special polygoal umber M.A.Gopala 1 Maju Somaath
ON THE BINARY QUADRATIC DIOPHANTINE EQUATION
Iteratioal Joural of Scietific Egieerig ad Applied Sciece (IJSEAS) - Volume-, Issue-4, Jue 0 ISSN: 9-470 ON THE BINARY QUADRATIC DIOPHANTINE EQUATION - 6 0 M.A.Gopala * V.Geetha, D.Priaka. Professor, Departmet
International Research Journal of Engineering and Technology (IRJET) e-issn: ON THE BINARY QUADRATIC DIOPHANTINE EQUATION
Iteratioal Research Joural of Egieerig ad Techolog (IRJET) e-issn: 9-006 Volume: 0 Issue: 04 Jul-0 www.irjet.et p-issn: 9-007 ON THE BINARY QUADRATIC DIOPHANTINE EQUATION - + 8 = 0 S.Vidhalakshmi, M.A.Gopala,
On Homogeneous Ternary Quadratic Diophantine Equation
IOSR Joural of Matheatics (IOSR-JM) e-issn:78-578, p-issn: 9-765X.Volue,Issue Ver. III (Ja.-Feb.6)PP 78-8 www.iosrjourals.org O Hoogeeous Terar Quadratic Diophatie Equatio 5 x R.buselvi ad N.hila Departet
THREE CURIOUS DIOPHANTINE PROBLEMS
JP Joural of Mathematical Scieces Volume 8, Issues &, 0, Pages -0 0 Ishaa Publishig House This paper is available olie at http://www.iphsci.com THREE CRIOS DIOPHATIE PROBLEMS A. VIJAYASAKAR, M. A. GOPALA
(3) (4) The least positive integer solution of (3) is
Meea K et al.; Sch. J. Phs. Math. Stat., 06; Vol-; Issue- (Feb); pp-5-9 Scholars Joural o Phsics, Mathematics ad Statistics Sch. J. Phs. Math. Stat. 06; ():5-9 Scholars Academic ad Scietiic Publishers
SOME NON-EXTENDABLE DIOPHANTINE TRIPLES IN SPECIAL NUMBER PATTERNS
Iteratioal Joural of Iovative Research ad Review ISSN: 347 444 (Olie A Olie Iteratioal Joural Available at http://www.cibtech.org/jirr.htm 014 Vol. (3 July-September, pp.1-7/gopala et al. SOME NON-ETENDABLE
ON THE DIOPHANTINE EQUATION
Fudametal Joural of Mathematics ad Mathematical Scieces Vol., Issue 1, 01, Pages -1 This paper is available olie at http://www.frdit.com/ Published olie July, 01 ON THE DIOPHANTINE EQUATION SARITA 1, HARI
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
On The Homogeneous Quintic Equation with Five Unknowns
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-78,p-ISSN: 319-76X, Volume 7, Issue 3 (Jul. - Aug. 013), PP 7-76 www.iosrjourls.org O The Homogeeous Quitic Equtio with Five Ukows y y 3 3 ( y ) 3(( y)( z w
Some Basic Diophantine Equations
Some Basic iophatie Equatios R.Maikada, epartmet of Mathematics, M.I.E.T. Egieerig College, Tiruchirappalli-7. Email: maimaths78@gmail.com bstract- - I this paper we preset a method for solvig the iophatie
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
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
Accepted in Fibonacci Quarterly (2007) Archived in SEQUENCE BALANCING AND COBALANCING NUMBERS
Accepted i Fiboacci Quarterly (007) Archived i http://dspace.itrl.ac.i/dspace SEQUENCE BALANCING AND COBALANCING NUMBERS G. K. Pada Departmet of Mathematics Natioal Istitute of Techology Rourela 769 008
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
Polynomials with Rational Roots that Differ by a Non-zero Constant. Generalities
Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by
POWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS
Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 3-8 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity
Observations on Derived K-Fibonacci and Derived K- Lucas Sequences
ISSN(Olie): 9-875 ISSN (Prit): 7-670 Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August 06 Observatios o Derived K-iboacci
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
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
LINEAR RECURSION RELATIONS - LESSON FOUR SECOND-ORDER LINEAR RECURSION RELATIONS
LINEAR RECURSION RELATIONS - LESSON FOUR SECOND-ORDER LINEAR RECURSION RELATIONS BROTHER ALFRED BROUSSEAU St. Mary's College, Califoria Give a secod-order liear recursio relatio (.1) T. 1 = a T + b T 1,
INTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION
Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: 9-006 Volume: 04 Issue: Mr -017 www.irjet.et p-issn: 9-007 INTEGRL SOLUTIONS OF THE TERNRY CUBIC EQUTION y ) 4y y ) 97z G.Jki 1, C.Sry,* ssistt
Unsaturated Solutions of A Nonlinear Delay Partial Difference. Equation with Variable Coefficients
Europea Joural of Mathematics ad Computer Sciece Vol. 5 No. 1 18 ISSN 59-9951 Usaturated Solutios of A Noliear Delay Partial Differece Euatio with Variable Coefficiets Xiagyu Zhu Yuahog Tao* Departmet
degree non-homogeneous Diophantine equation in six unknowns represented by x y 2z
Scholrs Jourl of Egieerig d Techology (SJET Sch. J. Eg. Tech., ; (A:97- Scholrs Acdeic d Scietific Pulisher (A Itertiol Pulisher for Acdeic d Scietific Resources www.sspulisher.co ISSN -X (Olie ISSN 7-9
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-
Sigma notation. 2.1 Introduction
Sigma otatio. Itroductio We use sigma otatio to idicate the summatio process whe we have several (or ifiitely may) terms to add up. You may have see sigma otatio i earlier courses. It is used to idicate
De Moivre s Theorem - ALL
De Moivre s Theorem - ALL. Let x ad y be real umbers, ad be oe of the complex solutios of the equatio =. Evaluate: (a) + + ; (b) ( x + y)( x + y). [6]. (a) Sice is a complex umber which satisfies = 0,.
You may work in pairs or purely individually for this assignment.
CS 04 Problem Solvig i Computer Sciece OOC Assigmet 6: Recurreces You may work i pairs or purely idividually for this assigmet. Prepare your aswers to the followig questios i a plai ASCII text file or
Research Article Some E-J Generalized Hausdorff Matrices Not of Type M
Abstract ad Applied Aalysis Volume 2011, Article ID 527360, 5 pages doi:10.1155/2011/527360 Research Article Some E-J Geeralized Hausdorff Matrices Not of Type M T. Selmaogullari, 1 E. Savaş, 2 ad B. E.
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
Similarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall
Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig
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
BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH
Vol..Issue.1.14 BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH A eer Reviewed Itertiol Reserch Jourl http://www.bomsr.com RESEARCH ARTICLE INTEGRAL SOLUTIONS OF TERNARY QUADRATIC DIOHANTINE EQUATIONS
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
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
We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
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
Some remarks for codes and lattices over imaginary quadratic
Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. shaska@oaklad.edu Abstract
On Some Properties of Digital Roots
Advaces i Pure Mathematics, 04, 4, 95-30 Published Olie Jue 04 i SciRes. http://www.scirp.org/joural/apm http://dx.doi.org/0.436/apm.04.46039 O Some Properties of Digital Roots Ilha M. Izmirli Departmet
A Block Cipher Using Linear Congruences
Joural of Computer Sciece 3 (7): 556-560, 2007 ISSN 1549-3636 2007 Sciece Publicatios A Block Cipher Usig Liear Cogrueces 1 V.U.K. Sastry ad 2 V. Jaaki 1 Academic Affairs, Sreeidhi Istitute of Sciece &
Average Number of Real Zeros of Random Fractional Polynomial-II
Average Number of Real Zeros of Radom Fractioal Polyomial-II K Kadambavaam, PG ad Research Departmet of Mathematics, Sri Vasavi College, Erode, Tamiladu, Idia M Sudharai, Departmet of Mathematics, Velalar
THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES
Joural of Mathematical Aalysis ISSN: 17-341, URL: http://iliriascom/ma Volume 7 Issue 4(16, Pages 13-19 THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES ABEDALLAH RABABAH, AYMAN AL
Complex Number Theory without Imaginary Number (i)
Ope Access Library Joural Complex Number Theory without Imagiary Number (i Deepak Bhalchadra Gode Directorate of Cesus Operatios, Mumbai, Idia Email: deepakm_4@rediffmail.com Received 6 July 04; revised
A METHOD TO SOLVE THE DIOPHANTINE EQUATION ax 2 by 2 c 0
A METHOD TO SOLVE THE DIOPHANTINE EQUATION ax by c Floreti Smaradache, Ph D Associate Professor Chair of Departmet of Math & Scieces Uiversity of New Mexico College Road Gallup, NM 87, USA E-mail:smarad@um.edu
Proof of Fermat s Last Theorem by Algebra Identities and Linear Algebra
Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,
ON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
Patterns in Complex Numbers An analytical paper on the roots of a complex numbers and its geometry
IB MATHS HL POTFOLIO TYPE Patters i Complex Numbers A aalytical paper o the roots of a complex umbers ad its geometry i Syed Tousif Ahmed Cadidate Sessio Number: 0066-009 School Code: 0066 Sessio: May
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
A 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
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
Chapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
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
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
Hoggatt and King [lo] defined a complete sequence of natural numbers
![Hoggatt and King [lo] defined a complete sequence of natural numbers](/thumbs/96/128001016.jpg "Hoggatt and King [lo] defined a complete sequence of natural numbers")
REPRESENTATIONS OF N AS A SUM OF DISTINCT ELEMENTS FROM SPECIAL SEQUENCES DAVID A. KLARNER, Uiversity of Alberta, Edmoto, Caada 1. INTRODUCTION Let a, I deote a sequece of atural umbers which satisfies
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,
Some New Iterative Methods for Solving Nonlinear Equations
World Applied Scieces Joural 0 (6): 870-874, 01 ISSN 1818-495 IDOSI Publicatios, 01 DOI: 10.589/idosi.wasj.01.0.06.830 Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida
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
Chapter 2. Periodic points of toral. automorphisms. 2.1 General introduction
Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad
The Local Harmonious Chromatic Problem
The 7th Workshop o Combiatorial Mathematics ad Computatio Theory The Local Harmoious Chromatic Problem Yue Li Wag 1,, Tsog Wuu Li ad Li Yua Wag 1 Departmet of Iformatio Maagemet, Natioal Taiwa Uiversity
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
Fourier series and the Lubkin W-transform
Fourier series ad the Lubki W-trasform Jaso Boggess, Departmet of Mathematics, Iowa State Uiversity Eric Buch, Departmet of Mathematics, Baylor Uiversity Charles N. Moore, Departmet of Mathematics, Kasas
On Homogeneous Ternary Quadratic Diophantine Equation 4 x 2 + y 2 7xy =
Scholars Journal of Engineering and Technology (SJET) Sch. J. Eng. Tech., 2014; 2(5A):676-680 Scholars Academic and Scientific Publisher (An International Publisher for Academic and Scientific Resources)
APPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
Math 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
Section 10.3 The Complex Plane; De Moivre's Theorem. abi
Sectio 03 The Complex Plae; De Moivre's Theorem REVIEW OF COMPLEX NUMBERS FROM COLLEGE ALGEBRA You leared about complex umbers of the form a + bi i your college algebra class You should remember that "i"
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,
Addition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c
Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity
Exact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method
Exact Solutios for a Class of Noliear Sigular Two-Poit Boudary Value Problems: The Decompositio Method Abd Elhalim Ebaid Departmet of Mathematics, Faculty of Sciece, Tabuk Uiversity, P O Box 741, Tabuki
PAijpam.eu IRREGULAR SET COLORINGS OF GRAPHS
Iteratioal Joural of Pure ad Applied Mathematics Volume 109 No. 7 016, 143-150 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://www.ijpam.eu doi: 10.173/ijpam.v109i7.18 PAijpam.eu
-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
Stochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
ON THE HADAMARD PRODUCT OF BALANCING Q n B AND BALANCING Q n
TWMS J App Eg Math V5, N, 015, pp 01-07 ON THE HADAMARD PRODUCT OF ALANCING Q AND ALANCING Q MATRIX MATRIX PRASANTA KUMAR RAY 1, SUJATA SWAIN, Abstract I this paper, the matrix Q Q which is the Hadamard
k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
A Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations
Copyright, Darbose Iteratioal Joural o Applied Mathematics ad Computatio Volume (), pp -6, 9 http//: ijamc.darbose.com Some Variats o Newto's Method with Fith-Order ad Fourth-Order Covergece or Solvig
Pell and Lucas primes
Notes o Number Theory ad Discrete Mathematics ISSN 30 532 Vol. 2, 205, No. 3, 64 69 Pell ad Lucas primes J. V. Leyedekkers ad A. G. Shao 2 Faculty of Sciece, The Uiversity of Sydey NSW 2006, Australia
Research Article Two Expanding Integrable Models of the Geng-Cao Hierarchy
Abstract ad Applied Aalysis Volume 214, Article ID 86935, 7 pages http://d.doi.org/1.1155/214/86935 Research Article Two Epadig Itegrable Models of the Geg-Cao Hierarchy Xiurog Guo, 1 Yufeg Zhag, 2 ad
Appendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
ANOTHER GENERALIZED FIBONACCI SEQUENCE 1. INTRODUCTION
ANOTHER GENERALIZED FIBONACCI SEQUENCE MARCELLUS E. WADDILL A N D LOUIS SACKS Wake Forest College, Wisto Salem, N. C., ad Uiversity of ittsburgh, ittsburgh, a. 1. INTRODUCTION Recet issues of umerous periodicals
THE DOUBLE ANGLE FORMULA
05 HAWAII UNIVERSITY INTERNATIONAL CONERENCES S.T.E.A.. & EDUCATION JUNE 3-5, 05 ALA OANA HOTEL, HONOLULU, HAWAII S.T.E.A. & EDUCATION PUBLICATION: ISSN 333-496 (CD-RO ISSN 333-4908 (ONLINE THE DOUBLE
CSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
Computation of Hahn Moments for Large Size Images
Joural of Computer Sciece 6 (9): 37-4, ISSN 549-3636 Sciece Publicatios Computatio of Ha Momets for Large Size Images A. Vekataramaa ad P. Aat Raj Departmet of Electroics ad Commuicatio Egieerig, Quli
On the number of sums of three unit fractions
Notes o Number Theory ad Discrete Mathematics Vol. 9, 0, No., 8 O the umber of sums of three uit fractios Simo Brow School of Huma Life Scieces, Uiversity of Tasmaia, Locked Bag 0, Laucesto, Tasmaia 70,
The Sumudu transform and its application to fractional differential equations
ISSN : 30-97 (Olie) Iteratioal e-joural for Educatio ad Mathematics www.iejem.org vol. 0, No. 05, (Oct. 03), 9-40 The Sumudu trasform ad its alicatio to fractioal differetial equatios I.A. Salehbhai, M.G.
On the Fibonacci-like Sequences of Higher Order
Article Iteratioal Joural of oder atheatical Scieces, 05, 3(): 5-59 Iteratioal Joural of oder atheatical Scieces Joural hoepage: wwwoderscietificpressco/jourals/ijsaspx O the Fiboacci-like Sequeces of
MAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
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
Fuzzy Shortest Path with α- Cuts
Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) Volume 58 Issue 3 Jue 2018 Fuzzy Shortest Path with α- Cuts P. Sadhya Assistat Professor, Deptt. Of Mathematics, AIMAN College of Arts ad Sciece
Linear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy
Liear Differetial Equatios of Higher Order Basic Theory: Iitial-Value Problems d y d y dy Solve: a( ) + a ( )... a ( ) a0( ) y g( ) + + + = d d d ( ) Subject to: y( 0) = y0, y ( 0) = y,..., y ( 0) = y
An Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
NTMSCI 5, No. 1, (2017) 26
NTMSCI 5, No. 1, - (17) New Treds i Mathematical Scieces http://dx.doi.org/1.85/tmsci.17.1 The geeralized successive approximatio ad Padé approximats method for solvig a elasticity problem of based o the
Mathematics review for CSCI 303 Spring Department of Computer Science College of William & Mary Robert Michael Lewis
Mathematics review for CSCI 303 Sprig 019 Departmet of Computer Sciece College of William & Mary Robert Michael Lewis Copyright 018 019 Robert Michael Lewis Versio geerated: 13 : 00 Jauary 17, 019 Cotets
Pellian sequence relationships among π, e, 2
otes o umber Theory ad Discrete Mathematics Vol. 8, 0, o., 58 6 Pellia sequece relatioships amog π, e, J. V. Leyedekkers ad A. G. Shao Faculty of Sciece, The Uiversity of Sydey Sydey, SW 006, Australia
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