Al-Zangana Iraqi Journal of Science, 2016, Vol. 57, No.2A, pp:
|
|
- Sara Merritt
- 5 years ago
- Views:
Transcription
1 Results n Projetve Geometry PG( r,), r, Emd Bkr Al-Zngn* Deprtment of Mthemts, College of Sene, Al-Mustnsryh Unversty, Bghdd, Ir Abstrt In projetve plne over fnte feld F, on s the unue omplete ( ) r nd ny rs on on re nomplete r of degree less thn These rs orrespond to sets n the projetve lne over the sme feld In ths pper, The number of neuvlent nomplete k rs; k 5,,,, on the on n PG (, ) nd stblzer group types re found Also, the projetve lne PG (, ) hs been splttng nto two -sets nd prttoned nto sx dsjont tetrds Keywords: Projetve plne, Projetve lne, k Ar, Complete rs ISSN: GIF: 085 r, نتائج في الهندسة االسقاطية (,r PG( الخالصة عماد بكر عبد الكريم الزنكنة * قسم الرياضيات, كلية العلوم, الجامعة المستنصرية, بغداد, الع ارق في المستوي االسقاطي على الحقل المنتهي F واي قوس اخر على المخروط هو غير تام من الدرجة اقل من االسقاطي على نفس الحقل في هذا البحث عدد االقواس الغير على المخروط في المخروط هوالقوس الوحيد التام من الدرجة هذه االقواس تقابل مجاميع في الخط تامة من الدرجة k حيث والزمر المثبتة لها قد وجد كذلك الخط االسقاطي PG(, ) قد جزء الى مجموعتين من الدرجة و قسم ايضا الى ستة مجاميع من الدرجة ال اربعة k 5,,, PG(, ) - Introduton Let PG( r, ) be projetve geometry of dmenson r over the Glos feld F of elements If r, PG(, ) s lled projetve lne nd f r, PG(, ) s lled projetve plne Defnton []: A k r n projetve plne s set of ponts, no three of them re ollner A k r s omplete f t s not ontned n ( k ) r A k set n projetve lne PG(, ) s set of dstnt ponts Defnton []: In PG( r, ), frme s set of n ponts, no n n hyperplne; tht s, every subset of n ponts s lnerly ndependent The set { U0, U, U, U} n projetve plne nd the set {, 0, } n projetve lne PG(, ) re lled the stndrd frmes, where 9
2 U [,0,0], U [0,,0], U [0,0,], U [,,] 0 Defnton []: Let F be form of degree two; tht s, F X X, 0 j j j Wth not ll j 0 n F, then the set C v( F) { P( X ) PG(, ) F( X ) 0} s lled udr plne The set v( F) s lled non-sngulr f F rreduble over F A nonsngulr plne udr C s lled on whh s formed unue omplete ( ) r Lemm : Any on form through the stndrd frme hs the followng form F X 0X bx 0X XX Theorem 5[]: In wth, there s unue on through 5-r Defnton []: The ross-rto T { P, P, P, P } of four ordered ponts P, P, P, P PG(, ) wth oordntes t, t, t, t s ( t t)( t t) { P, P ; P, P } { t, t; t, t,} CR( T) ( t t )( t t ) Defnton 7[]: Let T be tetrd(-set) wth ross-rto Then T s lled () rmon, denoted by, f or ( ) or ( ) ; () Eunhrmon, denoted by E, f ( ) or euvlently, ( ) ; () Nether hrmon nor eunhrmon, denoted by N, f the ross-rto s nother vlue Remrk 8: () The ross-rto of ny hrmon tetrd hs the vlues,, () The ross-rto of tetrd of type E stsfes the euton 0 In wth 5 odd, n r not ontned n on n hs t most ( ) ponts n ommon wth on [] Therefore, ny nomplete r n on s t most of degree ( ) ; ere, there re two uestons () Wht s the mxmum sze of omplete r other thn the on hs ( ) ponts n ommon wth on? () Wht s the number of nomplete r n the on? In [], the frst ueston hs been nswer for some In [], ueston two hs been nswered for 9 The m of ths pper s nswered ueston two for, before tht, the ons formed through the stndrd frme hve been reprmetrzed Also, the projetve lne over F hs been splts nto two -sets nd prttoned nto sx dfferent tetrds For the group types whh pper n ths pper see [] The mn omputng tool s the mthemtl progrmmng lnguge GAP [5] - Con Representton Through 5-r Aordng to Lemm nd Theorem 5, to gve on wth dfferent form through the stndrd frme, t hs to be fndng the neuvlent 5-rs 95
3 Theorem : In PG (, ), there re sx projetvely neuvlent 5-rs through the stndrd frme s gven n Tble- Tble -Ineuvlent 5-rs n PG (,) A A A A A A 5 A 5-r {7} {8} {0} {} {} {8} In the followng, the on form through eh 5-r hs been gven C X X 9X X 0 X X {P(9( t t),9( t), t) t F } A A A * 0 0 Stblzer A tht lsted n Tble nd ts prmetrzon C C X X X X X X {P(0( t t),( t), t) t F } A * 0 0 C X X X X 5 X X {P(( t t),( t), t) t F } () A5 * 0 0 C X X 0X X X X {P(( t t),0( t), t) t F } A * 0 0 C X X 9X X 8 X X {P(( t t),( t), t) t F } Where * 0 0 F * F { } Sne there s unue on up to projetvty, so t s enough to fxed one on form to fned the number of k rs on the on Theorem : The number of neuvlent nomplete k rs; k 5,,,, on the on n PG (, ) nd stblzer group types re gven n Tble- Tble -Ineuvlent, nomplete k rs on the on -r 5-r -r 7-r 8-r 9-r 0-r -r -r N = N 5 = N = N 7 = N 8 =8 N 9 =5 N 0 =9 N =7 N =8 : I : I 5:I : I : I 9: I : I 85: I 90: I :V 5:Z 9:Z 5: Z 9: Z 7 : Z 5: Z : Z 57 : Z :V 7:V :Z 0 :V : D :Z :S :D :S 0 :V : D : D :Z - Projetve Lne PG (,) 8 I Z Z Z Z Z :S : D :Z : A : D : D 8 9
4 Eh pont P( xy, ) wth y 0 n PG(, ) s determned by the non-homogeneous oordnte xy; the oordnte for P(,0) s So, the ponts of PG(, ) n be represented by the set F { } {, t, t,, t t F } () On PG (, ), the projetve lne over Glos feld of order, there re ponts The ponts of PG (, ) re the elements of the set F { } {, 0,,,,, 5,, 7, 8, 9, 0, } A tetrd s of type f the ross-rto s, or Sne the euton 0 hs no soluton n F, so there s no tetrd of type E Therefore, there re three types of tetrds of type N Let tetrds of ross rto,8,,, 7, 0 denote by N, tetrds of ross-rto,,9,, 5, 8 denote by N nd tetrds of ross-rto 5,7,0,,, 9 denote by N Let ( k, ) be the set of ll neuvlent k sets through the stndrd frme n PG(, ) nd C ( k, ) be the set of ll neuvlent k rs on the on through the stndrd frme n PG(, ) It s ler tht from () nd (), there s one to one orrespondng between projetve lne nd on s gven below * PG(, ) F C [ x, y] t P( t) Therefore; there s one to one orrespondng between the neuvlent k sets through the stndrd frme n PG(, ) nd nomplete k rs on the on through the stndrd frme up to projetvty, where k ( ) Let denote these bjetvty by the mp k : ( k,) C( k,), then 5 : ( k,) C( k,) s defned s follows: ( A ) P; 5( A) P ; 5 5( A) P ; 5 A5 P 5( A) P5 ; 5( A) P ( ) ; Corollry : The number of neuvlent k sets n PG(,) nd ts stblzer group types s the sme s gven n Tble- Exmple : In Tble- nd Tble-, the neuvlent pentds ( 5 sets) nd hexds ( sets) through the stndrd frme {, 0, } nd ts prtton to s tetrds (pentds) wth stblzer group types hve been gven Tble - Ineuvlent pentds n PG (,) 97
5 P P P P P P 5 P The pentd P {,} P {,} P {,5} P {,} P5 {,7} P {,} Tble - Ineuvlent hexd n PG (,) Type of Tetrds N NN N N N N N N N N N N N N N N N N N N N N N N N Stblzer The hexd Types of pentds Stblzer {, 0,,,, } PPPP PP V {, 0,,,, } PP PP P5 P I {, 0,,,, 5 } PP P P PP Z {, 0,,,, } PP P P P P 5 I {, 0,,,, 7 } 5 {, 0,,,, 9 } 7 {, 0,,,, 0 } 8 {, 0,,,, } {, 0,,,, } Z Z I Z Z Z PP P P P P I PP PP P P Z 5 5 PP P P P P 5 PPP P P P Z PPPPPP D 9 0 {, 0,,,, 5 } P P P P P P Z P P P P P P V {, 0,,,, } {, 0,,,, 9 } Z {, 0,,,, 0 } P P P P5 P5 P I P P P P P P Z {, 0,,,, } 5 {, 0,,,, 7 } {, 0,,, 5, 7 } 7 P P P P P P V P P P P P P Z {, 0,,, 5, 9 } PPPPPP S P P P P P P Z 8 {, 0,,, 5, } {, 0,,, 5, 8 } P P P P P P V {, 0,,,, 9 } PPPPPP S {, 0,,,, } P P5 P P P5 P Z {, 0,,, 7, 0 } PPPPPP I S 98
6 - Splttng PG (,) Eh -set, nd ts omplement lso fxes the omplement then the stblzer group of the prtton s: () If projetvely neuvlent to ts omplement group of the prtton s lso G () If projetvely euvlent to ts omplement prtton PG (,) Clerly, the stblzer group G of So, f PG (,) s prtton nto two -sets { ; },, then G s G therefore; the stblzer, then the stblzer group of the prtton s And n ths se, the stblzer group of G unon of ll lner trnsformtons between nd the prtton generted lwys by two elements one of them belong to the projetvty between G nd Theorem: The projetve lne PG (,) hs () 90 projetvely dstnt prttons nto two euvlent -sets (EQ); () 78 projetvely dstnt prttons nto two neuvlent -sets (NEQ) Tble 5- Prtton of PG(,) nto two -sets NEQ:{ ; } EQ:{ ; } Totl:78 : I : Z 8:V :S Exmple: () The unue -set type D, nd ts omplement lne suh tht j j j = P Totl:90 8: Z :Z 0 :V :S : D : D : D : D : D :S : D 8 G nd the other s {, 5,,,, 9, 0}, whh hs stblzer group of j ={, 7, 8, 9, 0,,, 5,, 7, 8, } prtton the projetve The stblzer group of the prtton s of type D s gven bellow: D t, b (8t 0) (t 9) b, b b () The -set k = P {,, 5, 7, 8, 7, 9}, whh hs stblzer group of type S, nd ts omplement k ={, 9, 0,,,,, 5,, 8, 0, }re prtton the projetve lne suh tht The stblzer group of the prtton s lso S s gven bellow: k k S (5 t) (t ), b (9t 8) b, b b 99
7 Theorem: The projetve lne PG(,) splt nto sx dsjont hrmon tetrds nd sx dsjont tetrds of type N,,, These prttons re not unue Proof : The GAP progrmmng hs been used to splttng the projetve lne nto sx dsjont tetrds () Prttons nto rmon tetrds; {, 0,, - }, CR( ) ; {,,, }, CR( ) ; {,, 5, 5 }, CR( ) ; {,, 7, 8 }, CR( ) ; 5 { 7, 9, 0, }, CR( 5 ) ; { 8, 9, 0, }, CR( ) () Prttons nto tetrds of types N ; {, 0,, }, CR( ) ; {,,, }, CR( ) 8; {,, 5, 5 }, CR( ) 0; {,, 7, 7 }, CR( ) ; { 8, 8, 9, }, CR( ) 0; 5 5 { 9, 0, 0, }, CR( ) 8 () Prttons nto tetrds of types N ; {, 0,, }, CR( )= ; {,,, 7 }, CR( )=9; {,,, 5 }, CR( )= 8; { 5,,, 8 }, CR( )=; 5 { 7, 8, 9, 0 }, CR 5 ( )=9; { 9, 0,, }, CR( )= 5; (v) Prttons nto tetrds of types N ; {, 0,, }, CR( )= ; {,,, }, CR( )=7; {,, 5, 7 }, CR( )=7; { 5,,, 8 }, CR( )= ; {7, 0, 0, }, CR( )=0; 5 5 {8, 9, CR 9, }, ( )=0 Referenes rshfeld, J W P 998 Projetve geometres over fnte felds, nd Edton, Oxford Mthemtl Monogrphs, The Clrendon Press, Oxford Unversty Press, New York Brtol, D, Dvydov, A A, Mrugn S nd Pmbno, F 0 A -yle onstruton of omplete rs shrng (+)/ ponts wth on, Advnes n Mthemts of Communtons, 7(), pp: 9-970
8 Al-Zngn, E B 0 The geometry of the plne of order nneteen nd ts pplton to errororretng odes, PhD Thess, Unversty of Sussex, Unted Kngdom Thoms, A D nd Wood, G V 980 Group tbles Shv Mthemts Seres, Seres 5 GAP Group 0 GAP Referene mnul, URL 97
VECTORS VECTORS VECTORS VECTORS. 2. Vector Representation. 1. Definition. 3. Types of Vectors. 5. Vector Operations I. 4. Equal and Opposite Vectors
1. Defnton A vetor s n entt tht m represent phsl quntt tht hs mgntude nd dreton s opposed to slr tht ls dreton.. Vetor Representton A vetor n e represented grphll n rrow. The length of the rrow s the mgntude
More informationLearning Enhancement Team
Lernng Enhnement Tem Worsheet: The Cross Produt These re the model nswers for the worsheet tht hs questons on the ross produt etween vetors. The Cross Produt study gude. z x y. Loong t mge, you n see tht
More informationPrinciple Component Analysis
Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors
More informationAbhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no
hlsh Clsses Clss- XII Dte: 0- - SOLUTION Chp - 9,0, MM 50 Mo no-996 If nd re poston vets of nd B respetvel, fnd the poston vet of pont C n B produed suh tht C B vet r C B = where = hs length nd dreton
More informationThe Number of Rows which Equal Certain Row
Interntonl Journl of Algebr, Vol 5, 011, no 30, 1481-1488 he Number of Rows whch Equl Certn Row Ahmd Hbl Deprtment of mthemtcs Fcult of Scences Dmscus unverst Dmscus, Sr hblhmd1@gmlcom Abstrct Let be X
More informationJens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers
Jens Sebel (Unversty of Appled Scences Kserslutern) An Interctve Introducton to Complex Numbers 1. Introducton We know tht some polynoml equtons do not hve ny solutons on R/. Exmple 1.1: Solve x + 1= for
More informationMATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER
MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER I One M Queston Fnd the unt veto n the deton of Let ˆ ˆ 9 Let & If Ae the vetos & equl? But vetos e not equl sne the oespondng omponents e dstnt e detons
More informationMath 497C Sep 17, Curves and Surfaces Fall 2004, PSU
Mth 497C Sep 17, 004 1 Curves nd Surfces Fll 004, PSU Lecture Notes 3 1.8 The generl defnton of curvture; Fox-Mlnor s Theorem Let α: [, b] R n be curve nd P = {t 0,...,t n } be prtton of [, b], then the
More informationOn Right α-centralizers and Commutativity of Prime Rings
On Right α-centralizers and Commutativity of Prime Rings Amira A. Abduljaleel*, Abdulrahman H. Majeed Department of Mathematics, College of Science, Baghdad University, Baghdad, Iraq Abstract: Let R be
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More information7.2 Volume. A cross section is the shape we get when cutting straight through an object.
7. Volume Let s revew the volume of smple sold, cylnder frst. Cylnder s volume=se re heght. As llustrted n Fgure (). Fgure ( nd (c) re specl cylnders. Fgure () s rght crculr cylnder. Fgure (c) s ox. A
More informationLeast squares. Václav Hlaváč. Czech Technical University in Prague
Lest squres Václv Hlváč Czech echncl Unversty n Prgue hlvc@fel.cvut.cz http://cmp.felk.cvut.cz/~hlvc Courtesy: Fred Pghn nd J.P. Lews, SIGGRAPH 2007 Course; Outlne 2 Lner regresson Geometry of lest-squres
More informationVariable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia
Vrble tme mpltude mplfcton nd quntum lgorthms for lner lgebr Andrs Ambns Unversty of Ltv Tlk outlne. ew verson of mpltude mplfcton;. Quntum lgorthm for testng f A s sngulr; 3. Quntum lgorthm for solvng
More informationUNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II
Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )
More informationComplement of an Extended Fuzzy Set
Internatonal Journal of Computer pplatons (0975 8887) Complement of an Extended Fuzzy Set Trdv Jyot Neog Researh Sholar epartment of Mathemats CMJ Unversty, Shllong, Meghalaya usmanta Kumar Sut ssstant
More informationRank One Update And the Google Matrix by Al Bernstein Signal Science, LLC
Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses
More informationTrigonometry. Trigonometry. Solutions. Curriculum Ready ACMMG: 223, 224, 245.
Trgonometry Trgonometry Solutons Currulum Redy CMMG:, 4, 4 www.mthlets.om Trgonometry Solutons Bss Pge questons. Identfy f the followng trngles re rght ngled or not. Trngles,, d, e re rght ngled ndted
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More information1/4/13. Outline. Markov Models. Frequency & profile model. A DNA profile (matrix) Markov chain model. Markov chains
/4/3 I529: Mhne Lernng n onformts (Sprng 23 Mrkov Models Yuzhen Ye Shool of Informts nd omputng Indn Unversty, loomngton Sprng 23 Outlne Smple model (frequeny & profle revew Mrkov hn pg slnd queston Model
More informationSection 1.3 Triangles
Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior
More informationAn Introduction to Support Vector Machines
An Introducton to Support Vector Mchnes Wht s good Decson Boundry? Consder two-clss, lnerly seprble clssfcton problem Clss How to fnd the lne (or hyperplne n n-dmensons, n>)? Any de? Clss Per Lug Mrtell
More informationProblem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?
Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2
More information- Primary Submodules
- Primary Submodules Nuhad Salim Al-Mothafar*, Ali Talal Husain Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq Abstract Let is a commutative ring with identity and
More informationRules of Probability
( ) ( ) = for all Corollary: Rules of robablty The probablty of the unon of any two events and B s roof: ( Φ) = 0. F. ( B) = ( ) + ( B) ( B) If B then, ( ) ( B). roof: week 2 week 2 2 Incluson / Excluson
More informationChapter 2 Introduction to Algebra. Dr. Chih-Peng Li ( 李 )
Chpter Introducton to Algebr Dr. Chh-Peng L 李 Outlne Groups Felds Bnry Feld Arthetc Constructon of Glos Feld Bsc Propertes of Glos Feld Coputtons Usng Glos Feld Arthetc Vector Spces Groups 3 Let G be set
More informationMultiple view geometry
EECS 442 Computer vson Multple vew geometry Perspectve Structure from Moton - Perspectve structure from moton prolem - mgutes - lgerc methods - Fctorzton methods - Bundle djustment - Self-clrton Redng:
More informationModule 3: Element Properties Lecture 5: Solid Elements
Modue : Eement Propertes eture 5: Sod Eements There re two s fmes of three-dmenson eements smr to two-dmenson se. Etenson of trngur eements w produe tetrhedrons n three dmensons. Smr retngur preeppeds
More informationREGULAR CUBIC LANGUAGE AND REGULAR CUBIC EXPRESSION
Advnes n Fuzzy ets nd ystems 05 Pushp Pulshng House Allhd nd Pulshed Onlne: Novemer 05 http://dx.do.org/0.7654/afde05_097_3 Volume 0 Numer 05 Pges 97-3 N: 0973-4X REGUAR CUBC ANGUAGE AND REGUAR CUBC EXPREON
More informationInterval Valued Neutrosophic Soft Topological Spaces
8 Interval Valued Neutrosoph Soft Topologal njan Mukherjee Mthun Datta Florentn Smarandah Department of Mathemats Trpura Unversty Suryamannagar gartala-7990 Trpura Indamal: anjan00_m@yahooon Department
More informationDCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
More informationINTRODUCTION TO COMPLEX NUMBERS
INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced
More informationApplied Statistics Qualifier Examination
Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng
More informationMCA-205: Mathematics II (Discrete Mathematical Structures)
MCA-05: Mthemts II (Dsrete Mthemtl Strutures) Lesson No: I Wrtten y Pnkj Kumr Lesson: Group theory - I Vette y Prof. Kulp Sngh STRUCTURE.0 OBJECTIVE. INTRODUCTION. SOME DEFINITIONS. GROUP.4 PERMUTATION
More informationPartially Observable Systems. 1 Partially Observable Markov Decision Process (POMDP) Formalism
CS294-40 Lernng for Rootcs nd Control Lecture 10-9/30/2008 Lecturer: Peter Aeel Prtlly Oservle Systems Scre: Dvd Nchum Lecture outlne POMDP formlsm Pont-sed vlue terton Glol methods: polytree, enumerton,
More informationRegular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15
Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll
More informationLecture 4: Piecewise Cubic Interpolation
Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationSolutions for HW9. Bipartite: put the red vertices in V 1 and the black in V 2. Not bipartite!
Solutions for HW9 Exerise 28. () Drw C 6, W 6 K 6, n K 5,3. C 6 : W 6 : K 6 : K 5,3 : () Whih of the following re iprtite? Justify your nswer. Biprtite: put the re verties in V 1 n the lk in V 2. Biprtite:
More informationMyhill-Nerode Theorem
Overview Myhill-Nerode Theorem Correspondence etween DA s nd MN reltions Cnonicl DA for L Computing cnonicl DFA Myhill-Nerode Theorem Deepk D Souz Deprtment of Computer Science nd Automtion Indin Institute
More informationThe Schur-Cohn Algorithm
Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for
More informationR- Annihilator Small Submodules المقاسات الجزئية الصغيرة من النمط تالف R
ISSN: 0067-2904 R- Annihilator Small Submodules Hala K. Al-Hurmuzy*, Bahar H. Al-Bahrany Department of Mathematics, College of Science, Baghdad University, Baghdad, Iraq Abstract Let R be an associative
More informationRead section 3.3, 3.4 Announcements:
Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f
More informationUNIT 31 Angles and Symmetry: Data Sheets
UNIT 31 Angles nd Symmetry Dt Sheets Dt Sheets 31.1 Line nd Rottionl Symmetry 31.2 Angle Properties 31.3 Angles in Tringles 31.4 Angles nd Prllel Lines: Results 31.5 Angles nd Prllel Lines: Exmple 31.6
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationFuzzy Rings and Anti Fuzzy Rings With Operators
OSR Journal of Mathemats (OSR-JM) e-ssn: 2278-5728, p-ssn: 2319-765X. Volume 11, ssue 4 Ver. V (Jul - ug. 2015), PP 48-54 www.osrjournals.org Fuzzy Rngs and nt Fuzzy Rngs Wth Operators M.Z.lam Department
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationFunctions. mjarrar Watch this lecture and download the slides
9/6/7 Mustf Jrrr: Leture Notes in Disrete Mthemtis. Birzeit University Plestine 05 Funtions 7.. Introdution to Funtions 7. One-to-One Onto Inverse funtions mjrrr 05 Wth this leture nd downlod the slides
More informationLecture 2: Numerical Methods for Differentiations and Integrations
Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton
More informationMore Properties of the Riemann Integral
More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl
More informationOn Solving Singular Multi Point Boundary Value Problems with Nonlocal Condition حول حل مسائل القيم الحدودية متعددة النقاط الشاذة مع شروط
ISSN: 0067-2904 GIF: 0.851 On Solving Singular Multi Point Boundary Value Problems with Nonlocal Condition Heba. A. Abd Al-Razak* Department of Mathematics, College of Science for Women, Baghdad University,
More informationAnalysis of Variance and Design of Experiments-II
Anly of Vrne Degn of Experment-II MODULE VI LECTURE - 8 SPLIT-PLOT AND STRIP-PLOT DESIGNS Dr Shlbh Deprtment of Mthemt & Sttt Indn Inttute of Tehnology Knpur Tretment ontrt: Mn effet The uefulne of hvng
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationLecture 36. Finite Element Methods
CE 60: Numercl Methods Lecture 36 Fnte Element Methods Course Coordntor: Dr. Suresh A. Krth, Assocte Professor, Deprtment of Cvl Engneerng, IIT Guwht. In the lst clss, we dscussed on the ppromte methods
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationProof that if Voting is Perfect in One Dimension, then the First. Eigenvector Extracted from the Double-Centered Transformed
Proof tht f Votng s Perfect n One Dmenson, then the Frst Egenvector Extrcted from the Doule-Centered Trnsformed Agreement Score Mtrx hs the Sme Rn Orderng s the True Dt Keth T Poole Unversty of Houston
More information1.3 SCALARS AND VECTORS
Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd
More informationFirst Midterm Examination
Çnky University Deprtment of Computer Engineering 203-204 Fll Semester First Midterm Exmintion ) Design DFA for ll strings over the lphet Σ = {,, c} in which there is no, no nd no cc. 2) Wht lnguge does
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationTrigonometry Revision Sheet Q5 of Paper 2
Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others.
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationMatrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n.
Mtrx Defto () s lled order of m mtrx, umer of rows ( 橫行 ) umer of olums ( 直列 ) m m m where j re rel umers () B j j for,,, m d j =,,, () s lled squre mtrx f m (v) s lled zero mtrx f (v) s lled detty mtrx
More informationSequences and summations
Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course
More informationFactorising FACTORISING.
Ftorising FACTORISING www.mthletis.om.u Ftorising FACTORISING Ftorising is the opposite of expning. It is the proess of putting expressions into rkets rther thn expning them out. In this setion you will
More information1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the
More informationLecture 7 Circuits Ch. 27
Leture 7 Cruts Ch. 7 Crtoon -Krhhoff's Lws Tops Dret Current Cruts Krhhoff's Two ules Anlyss of Cruts Exmples Ammeter nd voltmeter C ruts Demos Three uls n rut Power loss n trnsmsson lnes esstvty of penl
More informationI. Theory of Automata II. Theory of Formal Languages III. Theory of Turing Machines
CI 3104 /Winter 2011: Introduction to Forml Lnguges Chpter 16: Non-Context-Free Lnguges Chpter 16: Non-Context-Free Lnguges I. Theory of utomt II. Theory of Forml Lnguges III. Theory of Turing Mchines
More informationArithmetic & Algebra. NCTM National Conference, 2017
NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible
More information4. Eccentric axial loading, cross-section core
. Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line. - When grphing first-degree eqution, solve for the vrible. The grph of this solution will be single point
More informationEffects of polarization on the reflected wave
Lecture Notes. L Ros PPLIED OPTICS Effects of polrzton on the reflected wve Ref: The Feynmn Lectures on Physcs, Vol-I, Secton 33-6 Plne of ncdence Z Plne of nterfce Fg. 1 Y Y r 1 Glss r 1 Glss Fg. Reflecton
More informationEXPANSIVE MAPPINGS. by W. R. Utz
Volume 3, 978 Pages 6 http://topology.auburn.edu/tp/ EXPANSIVE MAPPINGS by W. R. Utz Topology Proceedngs Web: http://topology.auburn.edu/tp/ Mal: Topology Proceedngs Department of Mathematcs & Statstcs
More informationNon-Deterministic Finite Automata. Fall 2018 Costas Busch - RPI 1
Non-Deterministic Finite Automt Fll 2018 Costs Busch - RPI 1 Nondeterministic Finite Automton (NFA) Alphbet ={} q q2 1 q 0 q 3 Fll 2018 Costs Busch - RPI 2 Nondeterministic Finite Automton (NFA) Alphbet
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More informationON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs
ON THE INEQUALITY OF THE DIFFERENCE OF TWO INTEGRAL MEANS AND APPLICATIONS FOR PDFs A.I. KECHRINIOTIS AND N.D. ASSIMAKIS Deprtment of Eletronis Tehnologil Edutionl Institute of Lmi, Greee EMil: {kehrin,
More informationREDUCTION MODULO p. We will prove the reduction modulo p theorem in the general form as given by exercise 4.12, p. 143, of [1].
REDUCTION MODULO p. IAN KIMING We wll prove the reducton modulo p theorem n the general form as gven by exercse 4.12, p. 143, of [1]. We consder an ellptc curve E defned over Q and gven by a Weerstraß
More informationIntroduction To Matrices MCV 4UI Assignment #1
Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be
More informationJupiter Elevation Angle Determination at Baghdad Location
Jupiter Elevation Angle Determination at Baghdad Location Kamal M. Abood 1, Mourtada J. Kahdom 2, and Zeina F. Kahdome 1* 1 Department of Astronomy and Space, College of Science, University of Baghdad,
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationROB EBY Blinn College Mathematics Department
ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob Eby-Fll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationTrigonometry. Trigonometry. Curriculum Ready ACMMG: 223, 224, 245.
Trgonometry Trgonometry Currulum Rey ACMMG: 223, 22, 2 www.mthlets.om Trgonometry TRIGONOMETRY Bslly, mny stutons n the rel worl n e relte to rght ngle trngle. Trgonometry souns ffult, ut t s relly just
More information3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.
3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)
More informationInstructions. An 8.5 x 11 Cheat Sheet may also be used as an aid for this test. MUST be original handwriting.
ID: B CSE 2021 Computer Orgniztion Midterm Test (Fll 2009) Instrutions This is losed ook, 80 minutes exm. The MIPS referene sheet my e used s n id for this test. An 8.5 x 11 Chet Sheet my lso e used s
More information" = #N d$ B. Electromagnetic Induction. v ) $ d v % l. Electromagnetic Induction and Faraday s Law. Faraday s Law of Induction
Eletromgnet Induton nd Frdy s w Eletromgnet Induton Mhel Frdy (1791-1867) dsoered tht hngng mgnet feld ould produe n eletr urrent n ondutor pled n the mgnet feld. uh urrent s lled n ndued urrent. The phenomenon
More informationGraph Theory. Simple Graph G = (V, E). V={a,b,c,d,e,f,g,h,k} E={(a,b),(a,g),( a,h),(a,k),(b,c),(b,k),...,(h,k)}
Grph Theory Simple Grph G = (V, E). V ={verties}, E={eges}. h k g f e V={,,,,e,f,g,h,k} E={(,),(,g),(,h),(,k),(,),(,k),...,(h,k)} E =16. 1 Grph or Multi-Grph We llow loops n multiple eges. G = (V, E.ψ)
More informationExpectation and Variance
Expecttion nd Vrince : sum of two die rolls P(= P(= = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 P(=2) = 1/36 P(=3) = 1/18 P(=4) = 1/12 P(=5) = 1/9 P(=7) = 1/6 P(=13) =? 2 1/36 3 1/18 4 1/12 5 1/9 6 5/36 7 1/6
More informationTheory of Computation Regular Languages. (NTU EE) Regular Languages Fall / 38
Theory of Computtion Regulr Lnguges (NTU EE) Regulr Lnguges Fll 2017 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of Finite Automt A finite utomton hs finite set of control
More informationA Study on Root Properties of Super Hyperbolic GKM algebra
Stuy on Root Popetes o Supe Hypebol GKM lgeb G.Uth n M.Pyn Deptment o Mthemts Phypp s College Chenn Tmlnu In. bstt: In ths ppe the Supe hypebol genelze K-Mooy lgebs o nente type s ene n the mly s lso elte.
More informationLecture 1 - Introduction and Basic Facts about PDEs
* 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationState Minimization for DFAs
Stte Minimiztion for DFAs Red K & S 2.7 Do Homework 10. Consider: Stte Minimiztion 4 5 Is this miniml mchine? Step (1): Get rid of unrechle sttes. Stte Minimiztion 6, Stte is unrechle. Step (2): Get rid
More information