IJESMR International Journal OF Engineering Sciences & Management Research
|
|
- Shana Hill
- 5 years ago
- Views:
Transcription
1 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research CONSTRUCTION OF DIOPHANTINE QUADRUPLES WITH PROPERTY D(A PERFECT SQUARE) V.Geetha *, M.A.Gopala * Assistat Professor, Departmet of Mathematics, Cauvery Collage For wome, Trichy ,Tamiladu, Idia DProfessor, Departmet of Mathematics, Shrimathi Idira Gadhi Collage, Trichy , Tamiladu, Idia Keywords: Diophatie quadruples, System of equatios. MSC 000 Mathematics Subject Classificatio:11D99 ABSTRACT This paper cocers with the study of costructio of Diophatie quadruples such that the product of ay two elemets of the set added by a perfect square is a perfect square. INTRODUCTION Let q be a o-zero umber. A set a1, a,... a m of o-zero ratioal is called a - mtuple, if is a square for all 1 i j m. The mathematicia Diophatus of Alexadria cosidered a variety of problems Dq ( ) o idetermiat equatios with ratioal or itegers solutios. I particular, oe of the problems was to fid the sets of distict positive ratioal umbers such that the product of ay two umbers is oe less tha a ratioal square [1] ad Diophatus foud four positive ratioals,,, [,5]. The first set of four positive itegers with the same property, the set 1,3,8,10 was foud by Fermat. It was proved i 1969 by Baer ad Daveport [3] that a fifth positive iteger caot be added to this set ad oe may refer [6, 7,11] for geeralizatio. However, Euler discovered that a fifth ratioal umber ca be added to give the followig ratioal Diophatie quituple 1,3,8,10,. Ratioal sextuples with two equal elemets have bee give i []. I this 1999, Gibs [13] foud several examples of ratioal Diophatie sextuples, eg., ,,,,, , ,,,5,, All ow Diophatie quadruples are regular ad it has bee cojectured that there are o irregular Diophatie quadruples [1,13] (this is ow to be true for polyomials with iteger co-efficiets [8]). If so the there are o Diophatie quituples. However there are ifiitely may irregular ratioal Diophatie quadruples. The smallest is,5,,. May of these irregular quadruples are examples of aother commo type for which two of the subtriples are regular i.e., a, b, c, d is a irregular ratioal Diophatie quadruple, while abc,, ad,,d ab a a i j q are regular Diophatie triples. These are ow as semi regular ratioal Diophatie quadruples. These are oly fiitely may of these for ay give commo deomiator l ad they ca readily foud. Moreover i [1], it has bee proved that D( ) - triple, 1, 1 caot be exteded to a D( ) - quituple. I [10], it has bee proved that to a D( ) - quadruple if 5. - triple 1, 1, D( ) caot be exteded http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[10]
2 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research METHOD OF ANALYSIS SECTION I: I this sectio we search the diophatie quadruple (,,, ) a b c d such that product of ay two of them added with is a perfect square. Also,the fourth tuple is either iteger or ratioal umber. Cosider a (6 ) 6,& b (6 ) 6 ab Note that is a perfect square. Let c be ay o-zero iteger such that ac (1) bc () From (1), we have c a (3) Assume X ((6 ) 6) T () X ((6 ) 6) T (5) O substitutig the value of (3) i () ad by usig () ad (5), we get X ((6 ) 6)((6 ) 6) T whose iitial solutio is T0 1, X0 6( 1) Thus 6( 1) (6 ) 6 6( 1) (6 ) 6 Therefore from (3) c( 1) Let d be ay o-zero iteger such that ad A (6) bd B (7) cd C (8) Solvig (6), (7) ad (8) we get the value of d ( 1) d 1( 1) Substitutig the value of d i (6),(7) & (8) the 1( 1) 1 ( 1) ad 1( 1) 1 ( 1) bd 88( 1) cd Therefore ( a, b, c, d ) is a mixed diophatie quadruple with property D( ) as the fourth tuple may ot always be iteger. I what follows, a few examples of diophatie quadruple iteger are preseted. http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[11]
3 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research Table ( a, b, c, d) (,8,,80) 3 (9,15,8,30) (1,,7,5760) 6 (,36,10,11880) 8 (3,50,168,1835) 1 (5,78,6,31680) (,10,,19) 3 (6,18,8,70) (10,6,7,1386) 6 (18,,10,880) 1 (,90,6,77) (0,1,,7) (6,30,7,576) 8 (18,66,168,1890) (-,1,,30) 3 (0,,8,1) Sectio II: I this sectio we search the diophatie quadruple ( a, b, c, d ) such that product of ay two of them added with is a perfect square. Assume a ad 1 b. ab is a perfect square. Let c be ay o-zero iteger such that ac (9) bc (10) From (9), we have Assume c (11) a X T (1) 1 X (. ) T (13) O substitutig the value of (11) i (10) ad by usig (1) ad (13), we get Whose iitial solutio is T0 1, Thus Therefore from (11) X (. ) T 1 0 X ( ) ( ) 1 ( ) (. ) c ( 1) ( 1) Let d be ay o-zero iteger such that ad A (1) bd B (15) http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[1]
4 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research cd C (16) Solvig (1), (15) ad (16) we get the value of d ( 1) ( ) 8 ( 1) 8 (3.. d ) Substitutig the value of d i (1),(15) & (16) the ( ) (. 1) ad (. 1) (3 ) ( 1) bd (. 3) (3.. 1) ( 1) cd Therefore ( a, b, c, d) is a mixed diophatie quadruple with property D ( ) as the fourth tuple may ot always be iteger, a few umerical examples of diophatie quadruple iteger are preseted i the followig table. Table. Sectio III: I this sectio we search the diophatie quadruple ( a, b, c, d ) such that product of ay two of them added with 1 3 ( a, b, c, d ) 1 (1,8,15,58) (1,1,1,30) 3 (1,16,7,80) (1,0,33,73) 6 (1,8,5,88) 1 (,7,110,1709) (,80,10,38808) (,96,10,1390) 5 (,10,150,10500) 8 (,18,180,638) 10 (,1,00,530) 1 (9,59,77,159760) (9,608,765,1888) 3 (9,6,783,195700) (9,60,801,115630) 6 (9,67,837,565500) 8 (9,70,873,38880) 9 (9,70,891,88360) 1 (16,18,658, ) (16,160,69, ). is a perfect square. Assume a Carl 1 1 ad ab. is a perfect square. b Ky 1 1 http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[13]
5 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research Let c be ay o-zero iteger such that ac. (17) bc. (18) From (17), we have. c (19) a 1 Assume X ( 1) T (0) 1 X ( 1) T (1) O substitutig the value of (19) i (18) ad by usig (0) ad (1), we get Thus X whose iitial solutio is Therefore from (19) ( 1)( 1) T. 1 1 T0 1, 0 1 X ( 1). 1. c ( 1) Let d be ay o-zero iteger such that ad A () bd B (3) cd C () Solvig (), (3) ad () we get the value of d d Substitutig the value of d i (),(3) & () the 3 ad bd cd Remar: It is see that the fourth tuple d is iteger oly whe 1 (-1,7,1,15) with the property D ( ). ad the correspodig quadruple is CONCLUSION To coclude oe may costruct a Diophatie quadruples with suitable properties. http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[1]
6 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research REFERENCES [1]..G.Bashmaova (ed), Diophatus of Alexadria, Arithmetics ad the Boo of Polygoal Numbers, Naua, Moscow,197. [].A.F.Beardo ad M.N.Deshpade, Diophatie triples, The mathematical Gazette, 86, 00, [3].Bo He, A.Togbe, O the family of Diophatie triples 1,,9 3,Period Math Hugar,58,009, , A A,( A 1) ( A 1) with two []. Bo He, A.Togbe, O the family of Diophatie triples, parameters, Acta Math, Huger, 1,009, [5]. Bo He, A.Togbe, O the family of Diophatie triples,, A A,( A 1) ( A 1) with two parameters, Period Math Huger, 6,01, [6].Y.Bugeaud, A.Dujella ad M.migotte, O the family of Diophatie triples 1, 1,16 3, Glassgow Math J, 9, 007, [7].M.N.Deshpade ad E.Brow, Diophatie triples ad the Pell sequece, Fibaacci Quart, 39,001,-9. [8]. M.N.Deshpade, Oe iterestig family of Diophatie triples, Iterat. J.Math Ed.Sci.Tech., 33,00, [9]. M.N.Deshpade, Families of Diophatie triplets, Bulleti of the Marathwada mathematical society,,003,19-1. [10].A.Dujella ad C.Fuchs, Complete solutio of the polyomial versio of a problem of Diophatus, J.NumberTheory,106,00,36-3. [11].A.Dujella ad F.Luca, O a problem of Dioiphatus with polyomials, Rocy Moutai J. Math, 37,007, [1].A.Dujella ad V.Petricevic, Strog Diophatie triples, Experimet Math 17,008, F F, F, Fibaacci Quart, 8,01,19-7., 6 [13]. A.Filipi, Bo He, A.Togbe, O the D()-triple [1]. A.Filipi, Bo He, A.Togbe, O the family of two parameters D()-triples, Glas. Mat.ser.III,7,01, [15]. A.Filipi, No-extedability of D(-1) triples of the form {1,10,c}, Iterat J Math. Sci., 35,005,17-6. [16].M.A.Gopala ad G.Srividhya, Two special Diophatie Triples Diophatus J Math., 1(1) 01,3-7. [17].M.A.Gopala,V.Sageetha admaju Somaath, Costructio of the Diophatie Triple ivolvig polygoal umbers, Sch. J.Eg.Tech., (1),01,19- [ [18].M.A.Gopala, S.vidhyalashmi, S.Mallia, Special family of Diophatie Triples,Sch. J.Eg.Tech., (A), 01, [19].V.Padichelvi, Costructio of the Diophatie Triple ivolvig Polygoal umbers Impact J.Sci.Tech., 5(1), 011,7-11. http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[15]
New Results on Energy of Graphs of Small Order
Global Joural of Pure ad Applied Mathematics. ISSN 0973-1768 Volume 13, Number 7 (2017), pp. 2837-2848 Research Idia Publicatios http://www.ripublicatio.com New Results o Eergy of Graphs of Small Order
More informationSolving Fuzzy Assignment Problem Using Fourier Elimination Method
Global Joural of Pure ad Applied Mathematics. ISSN 0973-768 Volume 3, Number 2 (207), pp. 453-462 Research Idia Publicatios http://www.ripublicatio.com Solvig Fuzzy Assigmet Problem Usig Fourier Elimiatio
More informationOnes Assignment Method for Solving Traveling Salesman Problem
Joural of mathematics ad computer sciece 0 (0), 58-65 Oes Assigmet Method for Solvig Travelig Salesma Problem Hadi Basirzadeh Departmet of Mathematics, Shahid Chamra Uiversity, Ahvaz, Ira Article history:
More informationSuper Vertex Magic and E-Super Vertex Magic. Total Labelling
Proceedigs of the Iteratioal Coferece o Applied Mathematics ad Theoretical Computer Sciece - 03 6 Super Vertex Magic ad E-Super Vertex Magic Total Labellig C.J. Deei ad D. Atoy Xavier Abstract--- For a
More informationHigher-order iterative methods free from second derivative for solving nonlinear equations
Iteratioal Joural of the Phsical Scieces Vol 6(8, pp 887-89, 8 April, Available olie at http://wwwacademicjouralsorg/ijps DOI: 5897/IJPS45 ISSN 99-95 Academic Jourals Full Legth Research Paper Higher-order
More informationComputer Science Foundation Exam. August 12, Computer Science. Section 1A. No Calculators! KEY. Solutions and Grading Criteria.
Computer Sciece Foudatio Exam August, 005 Computer Sciece Sectio A No Calculators! Name: SSN: KEY Solutios ad Gradig Criteria Score: 50 I this sectio of the exam, there are four (4) problems. You must
More informationRelationship between augmented eccentric connectivity index and some other graph invariants
Iteratioal Joural of Advaced Mathematical Scieces, () (03) 6-3 Sciece Publishig Corporatio wwwsciecepubcocom/idexphp/ijams Relatioship betwee augmeted eccetric coectivity idex ad some other graph ivariats
More informationOn Characteristic Polynomial of Directed Divisor Graphs
Iter. J. Fuzzy Mathematical Archive Vol. 4, No., 04, 47-5 ISSN: 30 34 (P), 30 350 (olie) Published o April 04 www.researchmathsci.org Iteratioal Joural of V. Maimozhi a ad V. Kaladevi b a Departmet of
More informationAlpha Individual Solutions MAΘ National Convention 2013
Alpha Idividual Solutios MAΘ Natioal Covetio 0 Aswers:. D. A. C 4. D 5. C 6. B 7. A 8. C 9. D 0. B. B. A. D 4. C 5. A 6. C 7. B 8. A 9. A 0. C. E. B. D 4. C 5. A 6. D 7. B 8. C 9. D 0. B TB. 570 TB. 5
More informationEVALUATION OF TRIGONOMETRIC FUNCTIONS
EVALUATION OF TRIGONOMETRIC FUNCTIONS Whe first exposed to trigoometric fuctios i high school studets are expected to memorize the values of the trigoometric fuctios of sie cosie taget for the special
More informationOn Spectral Theory Of K-n- Arithmetic Mean Idempotent Matrices On Posets
Iteratioal Joural of Sciece, Egieerig ad echology Research (IJSER), Volume 5, Issue, February 016 O Spectral heory Of -- Arithmetic Mea Idempotet Matrices O Posets 1 Dr N Elumalai, ProfRMaikada, 3 Sythiya
More informationModule 8-7: Pascal s Triangle and the Binomial Theorem
Module 8-7: Pascal s Triagle ad the Biomial Theorem Gregory V. Bard April 5, 017 A Note about Notatio Just to recall, all of the followig mea the same thig: ( 7 7C 4 C4 7 7C4 5 4 ad they are (all proouced
More informationLU Decomposition Method
SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS LU Decompositio Method Jamie Traha, Autar Kaw, Kevi Marti Uiversity of South Florida Uited States of America kaw@eg.usf.edu http://umericalmethods.eg.usf.edu Itroductio
More informationAn Algorithm to Solve Multi-Objective Assignment. Problem Using Interactive Fuzzy. Goal Programming Approach
It. J. Cotemp. Math. Scieces, Vol. 6, 0, o. 34, 65-66 A Algorm to Solve Multi-Objective Assigmet Problem Usig Iteractive Fuzzy Goal Programmig Approach P. K. De ad Bharti Yadav Departmet of Mathematics
More informationMean cordiality of some snake graphs
Palestie Joural of Mathematics Vol. 4() (015), 49 445 Palestie Polytechic Uiversity-PPU 015 Mea cordiality of some sake graphs R. Poraj ad S. Sathish Narayaa Commuicated by Ayma Badawi MSC 010 Classificatios:
More informationarxiv: v2 [cs.ds] 24 Mar 2018
Similar Elemets ad Metric Labelig o Complete Graphs arxiv:1803.08037v [cs.ds] 4 Mar 018 Pedro F. Felzeszwalb Brow Uiversity Providece, RI, USA pff@brow.edu March 8, 018 We cosider a problem that ivolves
More informationOctahedral Graph Scaling
Octahedral Graph Scalig Peter Russell Jauary 1, 2015 Abstract There is presetly o strog iterpretatio for the otio of -vertex graph scalig. This paper presets a ew defiitio for the term i the cotext of
More information9.1. Sequences and Series. Sequences. What you should learn. Why you should learn it. Definition of Sequence
_9.qxd // : AM Page Chapter 9 Sequeces, Series, ad Probability 9. Sequeces ad Series What you should lear Use sequece otatio to write the terms of sequeces. Use factorial otatio. Use summatio otatio to
More informationA Comparative Study of Positive and Negative Factorials
A Comparative Study of Positive ad Negative Factorials A. M. Ibrahim, A. E. Ezugwu, M. Isa Departmet of Mathematics, Ahmadu Bello Uiversity, Zaria Abstract. This paper preset a comparative study of the
More informationCh 9.3 Geometric Sequences and Series Lessons
Ch 9.3 Geometric Sequeces ad Series Lessos SKILLS OBJECTIVES Recogize a geometric sequece. Fid the geeral, th term of a geometric sequece. Evaluate a fiite geometric series. Evaluate a ifiite geometric
More informationMath 3201 Notes Chapter 4: Rational Expressions & Equations
Learig Goals: See p. tet.. Equivalet Ratioal Epressios ( classes) Read Goal p. 6 tet. Math 0 Notes Chapter : Ratioal Epressios & Equatios. Defie ad give a eample of a ratioal epressio. p. 6. Defie o-permissible
More informationRecursion. Recursion. Mathematical induction: example. Recursion. The sum of the first n odd numbers is n 2 : Informal proof: Principle:
Recursio Recursio Jordi Cortadella Departmet of Computer Sciece Priciple: Reduce a complex problem ito a simpler istace of the same problem Recursio Itroductio to Programmig Dept. CS, UPC 2 Mathematical
More informationFuzzy Minimal Solution of Dual Fully Fuzzy Matrix Equations
Iteratioal Coferece o Applied Mathematics, Simulatio ad Modellig (AMSM 2016) Fuzzy Miimal Solutio of Dual Fully Fuzzy Matrix Equatios Dequa Shag1 ad Xiaobi Guo2,* 1 Sciece Courses eachig Departmet, Gasu
More informationCombination Labelings Of Graphs
Applied Mathematics E-Notes, (0), - c ISSN 0-0 Available free at mirror sites of http://wwwmaththuedutw/ame/ Combiatio Labeligs Of Graphs Pak Chig Li y Received February 0 Abstract Suppose G = (V; E) is
More informationCIS 121 Data Structures and Algorithms with Java Fall Big-Oh Notation Tuesday, September 5 (Make-up Friday, September 8)
CIS 11 Data Structures ad Algorithms with Java Fall 017 Big-Oh Notatio Tuesday, September 5 (Make-up Friday, September 8) Learig Goals Review Big-Oh ad lear big/small omega/theta otatios Practice solvig
More informationn n B. How many subsets of C are there of cardinality n. We are selecting elements for such a
4. [10] Usig a combiatorial argumet, prove that for 1: = 0 = Let A ad B be disjoit sets of cardiality each ad C = A B. How may subsets of C are there of cardiality. We are selectig elemets for such a subset
More informationCOSC 1P03. Ch 7 Recursion. Introduction to Data Structures 8.1
COSC 1P03 Ch 7 Recursio Itroductio to Data Structures 8.1 COSC 1P03 Recursio Recursio I Mathematics factorial Fiboacci umbers defie ifiite set with fiite defiitio I Computer Sciece sytax rules fiite defiitio,
More informationAn (or ) is a sequence in which each term after the first differs from the preceding term by a fixed constant, called the.
Sectio.2 Arithmetic Sequeces ad Series -.2 Arithmetic Sequeces ad Series Arithmetic Sequeces Arithmetic Series Key Terms: arithmetic sequece (arithmetic progressio), commo differece, arithmetic series
More informationA Note on Chromatic Transversal Weak Domination in Graphs
Iteratioal Joural of Mathematics Treds ad Techology Volume 17 Number 2 Ja 2015 A Note o Chromatic Trasversal Weak Domiatio i Graphs S Balamuruga 1, P Selvalakshmi 2 ad A Arivalaga 1 Assistat Professor,
More informationOptimum Solution of Quadratic Programming Problem: By Wolfe s Modified Simplex Method
Volume VI, Issue III, March 7 ISSN 78-5 Optimum Solutio of Quadratic Programmig Problem: By Wolfe s Modified Simple Method Kalpaa Lokhade, P. G. Khot & N. W. Khobragade, Departmet of Mathematics, MJP Educatioal
More informationCS 683: Advanced Design and Analysis of Algorithms
CS 683: Advaced Desig ad Aalysis of Algorithms Lecture 6, February 1, 2008 Lecturer: Joh Hopcroft Scribes: Shaomei Wu, Etha Feldma February 7, 2008 1 Threshold for k CNF Satisfiability I the previous lecture,
More informationLecture 1: Introduction and Strassen s Algorithm
5-750: Graduate Algorithms Jauary 7, 08 Lecture : Itroductio ad Strasse s Algorithm Lecturer: Gary Miller Scribe: Robert Parker Itroductio Machie models I this class, we will primarily use the Radom Access
More informationProject 2.5 Improved Euler Implementation
Project 2.5 Improved Euler Implemetatio Figure 2.5.10 i the text lists TI-85 ad BASIC programs implemetig the improved Euler method to approximate the solutio of the iitial value problem dy dx = x+ y,
More informationABOUT A CONSTRUCTION PROBLEM
INTERNATIONAL JOURNAL OF GEOMETRY Vol 3 (014), No, 14 19 ABOUT A CONSTRUCTION PROBLEM OVIDIU T POP ad SÁNDOR N KISS Abstract I this paper, we study the costructio of a polygo if we kow the midpoits of
More informationBOOLEAN MATHEMATICS: GENERAL THEORY
CHAPTER 3 BOOLEAN MATHEMATICS: GENERAL THEORY 3.1 ISOMORPHIC PROPERTIES The ame Boolea Arithmetic was chose because it was discovered that literal Boolea Algebra could have a isomorphic umerical aspect.
More informationCHAPTER IV: GRAPH THEORY. Section 1: Introduction to Graphs
CHAPTER IV: GRAPH THEORY Sectio : Itroductio to Graphs Sice this class is called Number-Theoretic ad Discrete Structures, it would be a crime to oly focus o umber theory regardless how woderful those topics
More informationArithmetic Sequences
. Arithmetic Sequeces COMMON CORE Learig Stadards HSF-IF.A. HSF-BF.A.1a HSF-BF.A. HSF-LE.A. Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered
More informationCreating Exact Bezier Representations of CST Shapes. David D. Marshall. California Polytechnic State University, San Luis Obispo, CA , USA
Creatig Exact Bezier Represetatios of CST Shapes David D. Marshall Califoria Polytechic State Uiversity, Sa Luis Obispo, CA 93407-035, USA The paper presets a method of expressig CST shapes pioeered by
More information5.3 Recursive definitions and structural induction
/8/05 5.3 Recursive defiitios ad structural iductio CSE03 Discrete Computatioal Structures Lecture 6 A recursively defied picture Recursive defiitios e sequece of powers of is give by a = for =0,,, Ca
More informationRedundancy Allocation for Series Parallel Systems with Multiple Constraints and Sensitivity Analysis
IOSR Joural of Egieerig Redudacy Allocatio for Series Parallel Systems with Multiple Costraits ad Sesitivity Aalysis S. V. Suresh Babu, D.Maheswar 2, G. Ragaath 3 Y.Viaya Kumar d G.Sakaraiah e (Mechaical
More informationPseudocode ( 1.1) Analysis of Algorithms. Primitive Operations. Pseudocode Details. Running Time ( 1.1) Estimating performance
Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Pseudocode ( 1.1) High-level descriptio of a algorithm More structured
More informationAlgorithms Chapter 3 Growth of Functions
Algorithms Chapter 3 Growth of Fuctios Istructor: Chig Chi Li 林清池助理教授 chigchi.li@gmail.com Departmet of Computer Sciece ad Egieerig Natioal Taiwa Ocea Uiversity Outlie Asymptotic otatio Stadard otatios
More informationMathematics and Art Activity - Basic Plane Tessellation with GeoGebra
1 Mathematics ad Art Activity - Basic Plae Tessellatio with GeoGebra Worksheet: Explorig Regular Edge-Edge Tessellatios of the Cartesia Plae ad the Mathematics behid it. Goal: To eable Maths educators
More informationCounting the Number of Minimum Roman Dominating Functions of a Graph
Coutig the Number of Miimum Roma Domiatig Fuctios of a Graph SHI ZHENG ad KOH KHEE MENG, Natioal Uiversity of Sigapore We provide two algorithms coutig the umber of miimum Roma domiatig fuctios of a graph
More informationOn (K t e)-saturated Graphs
Noame mauscript No. (will be iserted by the editor O (K t e-saturated Graphs Jessica Fuller Roald J. Gould the date of receipt ad acceptace should be iserted later Abstract Give a graph H, we say a graph
More informationSome non-existence results on Leech trees
Some o-existece results o Leech trees László A.Székely Hua Wag Yog Zhag Uiversity of South Carolia This paper is dedicated to the memory of Domiique de Cae, who itroduced LAS to Leech trees.. Abstract
More informationHomework 1 Solutions MA 522 Fall 2017
Homework 1 Solutios MA 5 Fall 017 1. Cosider the searchig problem: Iput A sequece of umbers A = [a 1,..., a ] ad a value v. Output A idex i such that v = A[i] or the special value NIL if v does ot appear
More informationRecursion. Computer Science S-111 Harvard University David G. Sullivan, Ph.D. Review: Method Frames
Uit 4, Part 3 Recursio Computer Sciece S-111 Harvard Uiversity David G. Sulliva, Ph.D. Review: Method Frames Whe you make a method call, the Java rutime sets aside a block of memory kow as the frame of
More informationBASED ON ITERATIVE ERROR-CORRECTION
A COHPARISO OF CRYPTAALYTIC PRICIPLES BASED O ITERATIVE ERROR-CORRECTIO Miodrag J. MihaljeviC ad Jova Dj. GoliC Istitute of Applied Mathematics ad Electroics. Belgrade School of Electrical Egieerig. Uiversity
More informationOn Infinite Groups that are Isomorphic to its Proper Infinite Subgroup. Jaymar Talledo Balihon. Abstract
O Ifiite Groups that are Isomorphic to its Proper Ifiite Subgroup Jaymar Talledo Baliho Abstract Two groups are isomorphic if there exists a isomorphism betwee them Lagrage Theorem states that the order
More informationRecurrent Formulas of the Generalized Fibonacci Sequences of Third & Fourth Order
Natioal Coferece o 'Advaces i Computatioal Mathematics' 7-8 Sept.03 :- 49 Recurret Formulas of the Geeralized Fiboacci Sequeces of hird & Fourth Order A. D.Godase Departmet of Mathematics V.P.College Vaijapur
More informationInternational Journal of Mathematical Archive-7(9), 2016, Available online through ISSN
Iteratioal Joural of Mathematical Archive-7(9), 06, 7- Available olie through www.ijma.ifo IN 9 5046 ON ECCENTRIC CONNECTIVITY INDEX OF F D AND F D GRAPH [ [ U. MARY*,. HAMILA *Departmet of Mathematics,
More information3D Model Retrieval Method Based on Sample Prediction
20 Iteratioal Coferece o Computer Commuicatio ad Maagemet Proc.of CSIT vol.5 (20) (20) IACSIT Press, Sigapore 3D Model Retrieval Method Based o Sample Predictio Qigche Zhag, Ya Tag* School of Computer
More informationExamples and Applications of Binary Search
Toy Gog ITEE Uiersity of Queeslad I the secod lecture last week we studied the biary search algorithm that soles the problem of determiig if a particular alue appears i a sorted list of iteger or ot. We
More informationINTERSECTION CORDIAL LABELING OF GRAPHS
INTERSECTION CORDIAL LABELING OF GRAPHS G Meea, K Nagaraja Departmet of Mathematics, PSR Egieerig College, Sivakasi- 66 4, Virudhuagar(Dist) Tamil Nadu, INDIA meeag9@yahoocoi Departmet of Mathematics,
More informationAn Algorithm to Solve Fuzzy Trapezoidal Transshipment Problem
Iteratioal Joural of Systems Sciece ad Applied Mathematics 206; (4): 58-62 http://www.sciecepublishiggroup.com/j/ssam doi: 0.648/j.ssam.206004.4 A Algorithm to Solve Fuzzy Trapezoidal Trasshipmet Problem
More informationMajor CSL Write your name and entry no on every sheet of the answer script. Time 2 Hrs Max Marks 70
NOTE:. Attempt all seve questios. Major CSL 02 2. Write your ame ad etry o o every sheet of the aswer script. Time 2 Hrs Max Marks 70 Q No Q Q 2 Q 3 Q 4 Q 5 Q 6 Q 7 Total MM 6 2 4 0 8 4 6 70 Q. Write a
More informationMath Section 2.2 Polynomial Functions
Math 1330 - Sectio. Polyomial Fuctios Our objectives i workig with polyomial fuctios will be, first, to gather iformatio about the graph of the fuctio ad, secod, to use that iformatio to geerate a reasoably
More informationSOME ALGEBRAIC IDENTITIES IN RINGS AND RINGS WITH INVOLUTION
Palestie Joural of Mathematics Vol. 607, 38 46 Palestie Polytechic Uiversity-PPU 07 SOME ALGEBRAIC IDENTITIES IN RINGS AND RINGS WITH INVOLUTION Chirag Garg ad R. K. Sharma Commuicated by Ayma Badawi MSC
More informationAn Efficient Algorithm for Graph Bisection of Triangularizations
A Efficiet Algorithm for Graph Bisectio of Triagularizatios Gerold Jäger Departmet of Computer Sciece Washigto Uiversity Campus Box 1045 Oe Brookigs Drive St. Louis, Missouri 63130-4899, USA jaegerg@cse.wustl.edu
More information3. b. Present a combinatorial argument that for all positive integers n : : 2 n
. b. Preset a combiatorial argumet that for all positive itegers : : Cosider two distict sets A ad B each of size. Sice they are distict, the cardiality of A B is. The umber of ways of choosig a pair of
More informationA Parallel DFA Minimization Algorithm
A Parallel DFA Miimizatio Algorithm Ambuj Tewari, Utkarsh Srivastava, ad P. Gupta Departmet of Computer Sciece & Egieerig Idia Istitute of Techology Kapur Kapur 208 016,INDIA pg@iitk.ac.i Abstract. I this
More informationA study on Interior Domination in Graphs
IOSR Joural of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 219-765X. Volume 12, Issue 2 Ver. VI (Mar. - Apr. 2016), PP 55-59 www.iosrjourals.org A study o Iterior Domiatio i Graphs A. Ato Kisley 1,
More informationENGI 4421 Probability and Statistics Faculty of Engineering and Applied Science Problem Set 1 Descriptive Statistics
ENGI 44 Probability ad Statistics Faculty of Egieerig ad Applied Sciece Problem Set Descriptive Statistics. If, i the set of values {,, 3, 4, 5, 6, 7 } a error causes the value 5 to be replaced by 50,
More informationPerhaps the method will give that for every e > U f() > p - 3/+e There is o o-trivial upper boud for f() ad ot eve f() < Z - e. seems to be kow, where
ON MAXIMUM CHORDAL SUBGRAPH * Paul Erdos Mathematical Istitute of the Hugaria Academy of Scieces ad Reu Laskar Clemso Uiversity 1. Let G() deote a udirected graph, with vertices ad V(G) deote the vertex
More informationImproving Information Retrieval System Security via an Optimal Maximal Coding Scheme
Improvig Iformatio Retrieval System Security via a Optimal Maximal Codig Scheme Dogyag Log Departmet of Computer Sciece, City Uiversity of Hog Kog, 8 Tat Chee Aveue Kowloo, Hog Kog SAR, PRC dylog@cs.cityu.edu.hk
More informationNTH, GEOMETRIC, AND TELESCOPING TEST
NTH, GEOMETRIC, AND TELESCOPING TEST Sectio 9. Calculus BC AP/Dual, Revised 08 viet.dag@humbleisd.et /4/08 0:0 PM 9.: th, Geometric, ad Telescopig Test SUMMARY OF TESTS FOR SERIES Lookig at the first few
More information4-PRIME CORDIAL LABELING OF SOME DEGREE SPLITTING GRAPHS
Iteratioal Joural of Maagemet, IT & Egieerig Vol. 8 Issue 7, July 018, ISSN: 49-0558 Impact Factor: 7.119 Joural Homepage: Double-Blid Peer Reviewed Refereed Ope Access Iteratioal Joural - Icluded i the
More informationIntro to Scientific Computing: Solutions
Itro to Scietific Computig: Solutios Dr. David M. Goulet. How may steps does it take to separate 3 objects ito groups of 4? We start with 5 objects ad apply 3 steps of the algorithm to reduce the pile
More informationCompactness of Fuzzy Sets
Compactess of uzzy Sets Amai E. Kadhm Departmet of Egieerig Programs, Uiversity College of Madeat Al-Elem, Baghdad, Iraq. Abstract The objective of this paper is to study the compactess of fuzzy sets i
More informationAn Improvement of the Basic El-Gamal Public Key Cryptosystem
Iteratioal Joural of Computer Applicatios Techology ad Research A Improvemet of the Basic El-Gamal Public Key Cryptosystem W.D.M.G.M. Dissaayake (PG/MPhil/2015/09 Departmet of Computer Egieerig Faculty
More informationfound that now considerable work has been done in this started with some example, which motivates the later results.
8 Iteratioal Joural of Comuter Sciece & Emergig Techologies (E-ISSN: 44-64) Volume, Issue 4, December A Study o Adjacecy Matrix for Zero-Divisor Grahs over Fiite Rig of Gaussia Iteger Prajali, Amit Sharma
More informationMath 10C Long Range Plans
Math 10C Log Rage Plas Uits: Evaluatio: Homework, projects ad assigmets 10% Uit Tests. 70% Fial Examiatio.. 20% Ay Uit Test may be rewritte for a higher mark. If the retest mark is higher, that mark will
More informationThe Platonic solids The five regular polyhedra
The Platoic solids The five regular polyhedra Ole Witt-Hase jauary 7 www.olewitthase.dk Cotets. Polygos.... Topologically cosideratios.... Euler s polyhedro theorem.... Regular ets o a sphere.... The dihedral
More information6.854J / J Advanced Algorithms Fall 2008
MIT OpeCourseWare http://ocw.mit.edu 6.854J / 18.415J Advaced Algorithms Fall 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.415/6.854 Advaced Algorithms
More informationSome cycle and path related strongly -graphs
Some cycle ad path related strogly -graphs I. I. Jadav, G. V. Ghodasara Research Scholar, R. K. Uiversity, Rajkot, Idia. H. & H. B. Kotak Istitute of Sciece,Rajkot, Idia. jadaviram@gmail.com gaurag ejoy@yahoo.co.i
More informationWebAssign Lesson 6-1b Geometric Series (Homework)
WebAssig Lesso 6-b Geometric Series (Homework) Curret Score : / 49 Due : Wedesday, July 30 204 :0 AM MDT Jaimos Skriletz Math 75, sectio 3, Summer 2 204 Istructor: Jaimos Skriletz. /2 poitsrogac alcet2
More informationCubic Polynomial Curves with a Shape Parameter
roceedigs of the th WSEAS Iteratioal Coferece o Robotics Cotrol ad Maufacturig Techology Hagzhou Chia April -8 00 (pp5-70) Cubic olyomial Curves with a Shape arameter MO GUOLIANG ZHAO YANAN Iformatio ad
More informationPartitions of a Convex Polygon
HAPTER 6 Partitios of a ovex Polygo 6 INTRODUTION I our survey we have come across some results o eumeratio of ocrossig cofiguratios o the set of vertices of a covex polygo, such as triagulatios ad trees
More informationMatrix Partitions of Split Graphs
Matrix Partitios of Split Graphs Tomás Feder, Pavol Hell, Ore Shklarsky Abstract arxiv:1306.1967v2 [cs.dm] 20 Ju 2013 Matrix partitio problems geeralize a umber of atural graph partitio problems, ad have
More informationAn Efficient Algorithm for Graph Bisection of Triangularizations
Applied Mathematical Scieces, Vol. 1, 2007, o. 25, 1203-1215 A Efficiet Algorithm for Graph Bisectio of Triagularizatios Gerold Jäger Departmet of Computer Sciece Washigto Uiversity Campus Box 1045, Oe
More informationHash Tables. Presentation for use with the textbook Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015.
Presetatio for use with the textbook Algorithm Desig ad Applicatios, by M. T. Goodrich ad R. Tamassia, Wiley, 2015 Hash Tables xkcd. http://xkcd.com/221/. Radom Number. Used with permissio uder Creative
More information1.8 What Comes Next? What Comes Later?
35 1.8 What Comes Next? What Comes Later? A Practice Uderstadig Task For each of the followig tables, CC BY Hiroaki Maeda https://flic.kr/p/6r8odk describe how to fid the ext term i the sequece, write
More informationLecture 2: Spectra of Graphs
Spectral Graph Theory ad Applicatios WS 20/202 Lecture 2: Spectra of Graphs Lecturer: Thomas Sauerwald & He Su Our goal is to use the properties of the adjacecy/laplacia matrix of graphs to first uderstad
More informationAlgorithms for Disk Covering Problems with the Most Points
Algorithms for Disk Coverig Problems with the Most Poits Bi Xiao Departmet of Computig Hog Kog Polytechic Uiversity Hug Hom, Kowloo, Hog Kog csbxiao@comp.polyu.edu.hk Qigfeg Zhuge, Yi He, Zili Shao, Edwi
More informationRecursive Procedures. How can you model the relationship between consecutive terms of a sequence?
6. Recursive Procedures I Sectio 6.1, you used fuctio otatio to write a explicit formula to determie the value of ay term i a Sometimes it is easier to calculate oe term i a sequece usig the previous terms.
More informationSEQUENCES AND SERIES
SEQUENCES AND SERIES U N I The umber of gifts set i the popular Christmas Carol days of Christmas form a sequece. A part of the sog goes this way O the th day of Christmas my true love gave to me drummers
More informationBig-O Analysis. Asymptotics
Big-O Aalysis 1 Defiitio: Suppose that f() ad g() are oegative fuctios of. The we say that f() is O(g()) provided that there are costats C > 0 ad N > 0 such that for all > N, f() Cg(). Big-O expresses
More informationComputing Vertex PI, Omega and Sadhana Polynomials of F 12(2n+1) Fullerenes
Iraia Joural of Mathematical Chemistry, Vol. 1, No. 1, April 010, pp. 105 110 IJMC Computig Vertex PI, Omega ad Sadhaa Polyomials of F 1(+1) Fullerees MODJTABA GHORBANI Departmet of Mathematics, Faculty
More informationPrime Cordial Labeling on Graphs
World Academy of Sciece, Egieerig ad Techology Iteratioal Joural of Mathematical ad Computatioal Scieces Vol:7, No:5, 013 Prime Cordial Labelig o Graphs S. Babitha ad J. Baskar Babujee, Iteratioal Sciece
More informationName of the Student: Unit I (Logic and Proofs) 1) Truth Table: Conjunction Disjunction Conditional Biconditional
SUBJECT NAME : Discrete Mathematics SUBJECT CODE : MA 2265 MATERIAL NAME : Formula Material MATERIAL CODE : JM08ADM009 (Sca the above QR code for the direct dowload of this material) Name of the Studet:
More information. Written in factored form it is easy to see that the roots are 2, 2, i,
CMPS A Itroductio to Programmig Programmig Assigmet 4 I this assigmet you will write a java program that determies the real roots of a polyomial that lie withi a specified rage. Recall that the roots (or
More informationChapter 3 Classification of FFT Processor Algorithms
Chapter Classificatio of FFT Processor Algorithms The computatioal complexity of the Discrete Fourier trasform (DFT) is very high. It requires () 2 complex multiplicatios ad () complex additios [5]. As
More informationExercise 6 (Week 42) For the foreign students only.
These are the last exercises of the course. Please, remember that to pass exercises, the sum of the poits gathered by solvig the questios ad attedig the exercise groups must be at least 4% ( poits) of
More informationAnalysis of Algorithms
Aalysis of Algorithms Ruig Time of a algorithm Ruig Time Upper Bouds Lower Bouds Examples Mathematical facts Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite
More informationMatrix representation of a solution of a combinatorial problem of the group theory
Matrix represetatio of a solutio of a combiatorial problem of the group theory Krasimir Yordzhev, Lilyaa Totia Faculty of Mathematics ad Natural Scieces South-West Uiversity 66 Iva Mihailov Str, 2700 Blagoevgrad,
More informationLecture Notes 6 Introduction to algorithm analysis CSS 501 Data Structures and Object-Oriented Programming
Lecture Notes 6 Itroductio to algorithm aalysis CSS 501 Data Structures ad Object-Orieted Programmig Readig for this lecture: Carrao, Chapter 10 To be covered i this lecture: Itroductio to algorithm aalysis
More informationA Method for Solving Balanced Intuitionistic Fuzzy Assignment Problem
P. Sethil Kumar et al t. Joural of Egieerig Research ad Applicatios SSN : 2248-9622, Vol. 4, ssue 3( Versio 1), March 2014, pp.897-903 RESEARCH ARTCLE OPEN ACCESS A Method for Solvig Balaced tuitioistic
More informationSouth Slave Divisional Education Council. Math 10C
South Slave Divisioal Educatio Coucil Math 10C Curriculum Package February 2012 12 Strad: Measuremet Geeral Outcome: Develop spatial sese ad proportioal reasoig It is expected that studets will: 1. Solve
More informationA RELATIONSHIP BETWEEN BOUNDS ON THE SUM OF SQUARES OF DEGREES OF A GRAPH
J. Appl. Math. & Computig Vol. 21(2006), No. 1-2, pp. 233-238 Website: http://jamc.et A RELATIONSHIP BETWEEN BOUNDS ON THE SUM OF SQUARES OF DEGREES OF A GRAPH YEON SOO YOON AND JU KYUNG KIM Abstract.
More information