Difference Sets of Null Density Subsets of

Size: px
Start display at page:

Download "Difference Sets of Null Density Subsets of"

Transcription

1 dvces Pue Mthetcs Pulshed Ole M ( Dffeece Sets of Null Dest Susets of Dwoud hd Dsted M Hosse Deptet of Mthetcs Uvest of Gul Rsht I El: hd@gulc h@googlelco Receved Decee 9 ; evsed Feu 9 ; ccepted Feu 6 BSTRCT Let D B : BB : B d fo d B B B lsup s postve the B s cosdeed s lge set D B hs oth hgh dest d ch stuctue The set D B D B D B If B Its dffeece set s lso eltvel lge d t s log stdg coectue tht le sets postve uppe dest the hve thetc pogesso of t legth Hee we show the dffeece set ot e susttl; fo thee ests such tht d d D Kewods: Dffeece Set; Dest; -Set Itoducto suset B of hs ull dest f d B l B D B B Pehps the ost poet geel esult ths cse s theoe coectued Edős d poved Rusz [] It sttes tht f d B the l whee D B : BB : B Ou coce hee s to cosde the ull dest susets the fl : These sets c e cosdeed lge sets og ull dest sets To hve copso sets of hghe destes ote tht f d : d hs Rse popet tht s f d s pttoed to ftel sets the t lest oe of the eleets of the ptto les These two popetes eg vt ude tslto d hvg Rse popet hold fo B : d Blsup () s well Howeve f B the D B s -set [] Ths es D B s oth lge d stuctued set Fo stce t s sdetc: thee ests p such tht pd B fo B defto -set tesects the dffeece set of fte tul sequece d ogst stuctues t s IP whch es thee s sequece of tul ues such tht ll of ts fte sus e B [3] Fo lgeess pot of vew e cosdeed et to fo [4] Note tht thee s lso fl : d lsup whch d s clled the uppe Bch dest of We hve tht d s vt ude ts- lto d stsfes Rse popet [5] lso f B the D B s [5] d B hs thetc pogesso of t legth [6] Howeve B ot e -lge Fo stce B : ut B Edös coectued tht eleets of hve thetc pogesso of t legth [7] pott suset of tul ues whch s the set of pe ues les d the coectue ws poved postvel Copght ScRes

2 96 D DSTJERDI M HOSSEINI Gee-To [8] I ode to peset susttl cotst of sets those we wll show tht fo thee s such tht D D D ; the th dffeece set of eve does ot le Thoughout ths ote uless othewse stted tevl we e tevl of teges So fo stce : Refeece Set Hee we toduce d vestgte the popetes of suset of o-egtve teges sutl defed fo ou lte use Let e set of o-egtve teges the popet tht fo Fo let l e the lgest tege stctl less th Set c : fo ll those l such tht lso set c c f d : othewse Defto Let e s ove The set of oegtve teges R l l l c d s clled the efeece set ssocted to The efeece set c lso e see s follows Set : R the 3 l l 3 l whee : Let us ow ee whe R \ fo gve e- logs to : Le Suppose d let R e ts ssocted efeece set Suppose fo suffcetl lge fo soe The R \ Poof Fst ote tht () ples d lso f we set l : l the l f Now let fo But fo l ; () (3) l l I R l (4) : cd The cd R I cd R cd R cd R I I I So esttg whee R (5) e (6) e : I I 3 3 Note tht (3) fo suffcetl lge we hve 3 Now e I I 3 3 e I I 3 3 I I I lso (5) I I I Theefoe Copght ScRes

3 D DSTJERDI M HOSSEINI e e 3 d 3 d B cosdeg (3) ths ples l e lf lf e l l f d d Now the poof s coplete the to test Let the eple fo stsfg the ove le s Fo ths eple lf l Re 3 Sl guets s the poof of the ove le shows tht f ethe (3) does ot hold o f (3) holds ut thee s soe such tht fo ftel (8) the We gve setch of poof fo the R ltte Suppose s defed s (4) The (8) we hve l sup Let I e s ove d set e : I I Usg the 4 l left eqult (5) we hve Now R I e d l d e l e (7) l Fo (3) we hve Ths d the fct tht ples d we e doe 3 M Result e l sup l sup e We wll show tht copg to set of postve uppe Bch dest the dffeece set D of set d d c e ve spse The et theoe gves clss of eples ths popet Theoe Let the thee ests d D such tht s ot I ptcul d D Poof Let d Cosde the followg setc susets of teges: F (9) F fo d Set : I F F F whee : d I s the tevl Hece d s uo of sutevls of o-zeo teges the fst sutevl of legth d ll othes of legth To hve pctue of s cosde whch s pototpe fo othes The e cosdeed s whch R R R = = = Copght ScRes

4 98 D DSTJERDI M HOSSEINI Tht s hs s hlf eleets s whch e shfted to the left ppoptel So R R B Le oe ots tht R \ \ d hece (Lte we wll pove tht So dvegece of wll e \ cosequece of tht s well) Now we cl tht s ot Fst ote tht : The the cl s estlshed otg tht the ove defto fo whch ples does ot tesect D \ : d so t s ot The sets F s d s e defed such w tht D fo To see ths suppose d e two eleets of d The thee ests such tht d lso whee F So d e o equvletl Theefoe To coplete the poof t es to show tht fo ech We led hve poved ths fct \ fo d we wll pove tht fo d sce we e doe So cosde F (9) d let F : F F : F lso set The : F F Now usg the se gu- ets s Le we pove tht veget If we let the I : cd I : cd s d I I I I Let l e s the defto of R ; the whee \ e e : II 3 l 3 So I I 3 5 e I I l f l e II 5 5 l f 5 d l f 5 d Let e the set defed the poof of the ove theoe d let S e sdetc set the lg- Copght ScRes

5 D DSTJERDI M HOSSEINI 99 est gp slle th The the cocluso of the ove theoe pples fo S Ths cocluso lso holds f we let e p whee e s eve tege d p s cesg tege vlued fucto p lso ecll tht the theoe of Rusz [] fo sets of ull dest sttes tht the dffeece set s cosdelel lge th the set tself tht s f db the () holds Howeve eples such s those the ove theoe oe tes to hve dffeece set whch s s sll s possle Ou ppoch ws to hve the thetc pogesso of log possle legths Theefoe such eples ot ol do ot cotdct the Edös coectue ut stogl e the fvo of t REFERENCES [] I Z Rusz O Dffeece-Sequeces ct thetc Vol pp 5-57 [] V Begelso d N Hd ddtve d Multplctve Rse Theoes -Soe Eleet Results Cotocs Polt d Coputg Vol 993 pp -4 do:7/s [3] V Begelso Ptto Regul Stuctues Coted Lge Sets e udt Joul of Cotol Theo Sees Vol 93 No pp 8-36 do:6/ct36 [4] T C Bow d R Feed thetc Pogessos Lcu Sets Roc Mout Joul of Mthetcs Vol 7 No pp do:6/rmj [5] V Begelso N Hd d R McCutche Notos of Sze d Cotol Popetes of Quotet Sets Segoups Topolog Poceedgs Vol pp 3-6 [6] E Szeed O Sets of Iteges Cotg No Eleets thetc Pogesso ct thetc Vol pp [7] P Edös Poles d Results Cotol Nue Theo stsque 975 pp 95-3 [8] B Gee d T To The Pes Cot tl Log thetc Pogessos ls of Mthetcs Vol 67 No 4 pp do:47/ls86748 Copght ScRes

SOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE

SOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE D I D A C T I C S O F A T H E A T I C S No (4) 3 SOE REARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOIAL ASYPTOTE Tdeusz Jszk Abstct I the techg o clculus, we cosde hozotl d slt symptote I ths ppe the

More information

= y and Normed Linear Spaces

= y and Normed Linear Spaces 304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads

More information

Chapter Linear Regression

Chapter Linear Regression Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use

More information

«A first lesson on Mathematical Induction»

«A first lesson on Mathematical Induction» Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio,

More information

FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL SEQUENCES

FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL SEQUENCES Joual of Appled Matheatcs ad Coputatoal Mechacs 7, 6(), 59-7 www.ac.pcz.pl p-issn 99-9965 DOI:.75/jac.7..3 e-issn 353-588 FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL

More information

Professor Wei Zhu. 1. Sampling from the Normal Population

Professor Wei Zhu. 1. Sampling from the Normal Population AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple

More information

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,

More information

On The Circulant K Fibonacci Matrices

On The Circulant K Fibonacci Matrices IOSR Jou of Mthetcs (IOSR-JM) e-issn: 78-578 p-issn: 39-765X. Voue 3 Issue Ve. II (M. - Ap. 07) PP 38-4 www.osous.og O he Ccut K bocc Mtces Sego co (Deptet of Mthetcs Uvesty of Ls Ps de G C Sp) Abstct:

More information

A convex hull characterization

A convex hull characterization Pue d ppled Mthets Joul 4; (: 4-48 Pulshed ole My 4 (http://www.seepulshggoup.o//p do:.648/.p.4. ove hull htezto Fo Fesh Gov Qut Deptet DISG Uvesty of Se Itly El ddess: fesh@us.t (F. Fesh qut@us.t (G.

More information

On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx

On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx Iteatoal Joual of Mathematcs ad Statstcs Iveto (IJMSI) E-ISSN: 3 4767 P-ISSN: 3-4759 www.jms.og Volume Issue 5 May. 4 PP-44-5 O EP matces.ramesh, N.baas ssocate Pofesso of Mathematcs, ovt. ts College(utoomous),Kumbakoam.

More information

CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.

CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane. CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.

More information

The z-transform. LTI System description. Prof. Siripong Potisuk

The z-transform. LTI System description. Prof. Siripong Potisuk The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put

More information

Generalisation on the Zeros of a Family of Complex Polynomials

Generalisation on the Zeros of a Family of Complex Polynomials Ieol Joul of hemcs esech. ISSN 976-584 Volume 6 Numbe 4. 93-97 Ieol esech Publco House h://www.house.com Geelso o he Zeos of Fmly of Comlex Polyomls Aee sgh Neh d S.K.Shu Deme of hemcs Lgys Uvesy Fdbd-

More information

are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures.

are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures. Lectue 4 8. MRAC Desg fo Affe--Cotol MIMO Systes I ths secto, we cosde MRAC desg fo a class of ult-ut-ult-outut (MIMO) olea systes, whose lat dyacs ae lealy aaetezed, the ucetates satsfy the so-called

More information

SUBSEQUENCE CHARACTERIZAT ION OF UNIFORM STATISTICAL CONVERGENCE OF DOUBLE SEQUENCE

SUBSEQUENCE CHARACTERIZAT ION OF UNIFORM STATISTICAL CONVERGENCE OF DOUBLE SEQUENCE Reseach ad Coucatos atheatcs ad atheatcal ceces Vol 9 Issue 7 Pages 37-5 IN 39-6939 Publshed Ole o Novebe 9 7 7 Jyot cadec Pess htt//yotacadecessog UBEQUENCE CHRCTERIZT ION OF UNIFOR TTITIC CONVERGENCE

More information

Some Equivalent Forms of Bernoulli s Inequality: A Survey *

Some Equivalent Forms of Bernoulli s Inequality: A Survey * Ale Mthets 3 4 7-93 htt://oog/436/34746 Pulshe Ole Jul 3 (htt://wwwsog/joul/) Soe Euvlet Fos of Beoull s Ieult: A Suve * u-chu L Cheh-Chh eh 3 Detet of Ale Mthets Ntol Chug-Hsg Uvest Tw Detet of Mthets

More information

COMPLEX NUMBERS AND DE MOIVRE S THEOREM

COMPLEX NUMBERS AND DE MOIVRE S THEOREM COMPLEX NUMBERS AND DE MOIVRE S THEOREM OBJECTIVE PROBLEMS. s equl to b d. 9 9 b 9 9 d. The mgr prt of s 5 5 b 5. If m, the the lest tegrl vlue of m s b 8 5. The vlue of 5... s f s eve, f s odd b f s eve,

More information

ME 501A Seminar in Engineering Analysis Page 1

ME 501A Seminar in Engineering Analysis Page 1 Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt

More information

such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1

such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1 Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9

More information

Chapter #2 EEE State Space Analysis and Controller Design

Chapter #2 EEE State Space Analysis and Controller Design Chpte EEE8- Chpte # EEE8- Stte Spce Al d Cotolle Deg Itodcto to tte pce Obevblt/Cotollblt Modle ede: D D Go - d.go@cl.c.k /4 Chpte EEE8-. Itodcto Ae tht we hve th ode te: f, ', '',.... Ve dffclt to td

More information

Fredholm Type Integral Equations with Aleph-Function. and General Polynomials

Fredholm Type Integral Equations with Aleph-Function. and General Polynomials Iteto Mthetc Fou Vo. 8 3 o. 989-999 HIKI Ltd.-h.co Fedho Te Iteg uto th eh-fucto d Gee Poo u J K.J. o Ittute o Mgeet tude & eech Mu Id u5@g.co Kt e K.J. o Ittute o Mgeet tude & eech Mu Id dehuh_3@hoo.co

More information

COMP 465: Data Mining More on PageRank

COMP 465: Data Mining More on PageRank COMP 465: Dt Mnng Moe on PgeRnk Sldes Adpted Fo: www.ds.og (Mnng Mssve Dtsets) Powe Iteton: Set = 1/ 1: = 2: = Goto 1 Exple: d 1/3 1/3 5/12 9/24 6/15 = 1/3 3/6 1/3 11/24 6/15 1/3 1/6 3/12 1/6 3/15 Iteton

More information

5 - Determinants. r r. r r. r r. r s r = + det det det

5 - Determinants. r r. r r. r r. r s r = + det det det 5 - Detemts Assote wth y sque mtx A thee s ume lle the etemt of A eote A o et A. Oe wy to efe the etemt, ths futo fom the set of ll mtes to the set of el umes, s y the followg thee popetes. All mtes elow

More information

Available online through

Available online through Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo

More information

PROGRESSION AND SERIES

PROGRESSION AND SERIES INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of

More information

2. Elementary Linear Algebra Problems

2. Elementary Linear Algebra Problems . Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N 3 4 5 3 6 4 7 8 Full y tee Depth te tep log()n Veto updte the f- poe wth N : ) ( ) ( ) ( )

More information

Complete Classification of BKM Lie Superalgebras Possessing Strictly Imaginary Property

Complete Classification of BKM Lie Superalgebras Possessing Strictly Imaginary Property Appled Mthemtcs 4: -5 DOI: 59/m4 Complete Clssfcto of BKM Le Supelges Possessg Stctly Imgy Popety N Sthumoothy K Pydhs Rmu Isttute fo Advced study Mthemtcs Uvesty of Mds Che 6 5 Id Astct I ths ppe complete

More information

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s: Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,

More information

XII. Addition of many identical spins

XII. Addition of many identical spins XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.

More information

MTH 146 Class 7 Notes

MTH 146 Class 7 Notes 7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg

More information

Lecture 3 summary. C4 Lecture 3 - Jim Libby 1

Lecture 3 summary. C4 Lecture 3 - Jim Libby 1 Lecue su Fes of efeece Ivce ude sfoos oo of H wve fuco: d-fucos Eple: e e - µ µ - Agul oeu s oo geeo Eule gles Geec slos cosevo lws d Noehe s heoe C4 Lecue - Lbb Fes of efeece Cosde fe of efeece O whch

More information

Fairing of Parametric Quintic Splines

Fairing of Parametric Quintic Splines ISSN 46-69 Eglad UK Joual of Ifomato ad omputg Scece Vol No 6 pp -8 Fag of Paametc Qutc Sples Yuau Wag Shagha Isttute of Spots Shagha 48 ha School of Mathematcal Scece Fuda Uvesty Shagha 4 ha { P t )}

More information

RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S

RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, +91-9650 350 480] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets

More information

ON NILPOTENCY IN NONASSOCIATIVE ALGEBRAS

ON NILPOTENCY IN NONASSOCIATIVE ALGEBRAS Jourl of Algebr Nuber Theory: Advces d Applctos Volue 6 Nuber 6 ges 85- Avlble t http://scetfcdvces.co. DOI: http://dx.do.org/.864/t_779 ON NILOTENCY IN NONASSOCIATIVE ALGERAS C. J. A. ÉRÉ M. F. OUEDRAOGO

More information

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013 Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo

More information

this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]

this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ] Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow

More information

Spectral Continuity: (p, r) - Α P And (p, k) - Q

Spectral Continuity: (p, r) - Α P And (p, k) - Q IOSR Joul of Mthemtcs (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X Volume 11, Issue 1 Ve 1 (J - Feb 215), PP 13-18 wwwosjoulsog Spectl Cotuty: (p, ) - Α P Ad (p, k) - Q D Sethl Kum 1 d P Mhesw Nk 2 1

More information

Technical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005.

Technical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005. Techc Appedx fo Iveoy geme fo Assemy Sysem wh Poduc o Compoe eus ecox d Zp geme Scece 2005 Lemm µ µ s c Poof If J d µ > µ he ˆ 0 µ µ µ µ µ µ µ µ Sm gumes essh he esu f µ ˆ > µ > µ > µ o K ˆ If J he so

More information

CURVE FITTING LEAST SQUARES METHOD

CURVE FITTING LEAST SQUARES METHOD Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued

More information

Exponential Generating Functions - J. T. Butler

Exponential Generating Functions - J. T. Butler Epoetal Geeatg Fuctos - J. T. Butle Epoetal Geeatg Fuctos Geeatg fuctos fo pemutatos. Defto: a +a +a 2 2 + a + s the oday geeatg fucto fo the sequece of teges (a, a, a 2, a, ). Ep. Ge. Fuc.- J. T. Butle

More information

Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants

Fibonacci and Lucas Numbers as Tridiagonal Matrix Determinants Rochester Isttute of echology RI Scholr Wors Artcles 8-00 bocc d ucs Nubers s rdgol trx Deterts Nth D. Chll Est Kod Copy Drre Nry Rochester Isttute of echology ollow ths d ddtol wors t: http://scholrwors.rt.edu/rtcle

More information

A Deterministic Model for Channel Capacity with Utility

A Deterministic Model for Channel Capacity with Utility CAPTER 6 A Detestc Model fo Chel Cct wth tlt 6. todcto Chel cct s tl oeto ssocted wth elble cocto d defed s the hghest te t whch foto c be set ove the chel wth btl sll obblt of eo. Chel codg theoes d the

More information

Lecture 10: Condensed matter systems

Lecture 10: Condensed matter systems Lectue 0: Codesed matte systems Itoducg matte ts codesed state.! Ams: " Idstgushable patcles ad the quatum atue of matte: # Cosequeces # Revew of deal gas etopy # Femos ad Bosos " Quatum statstcs. # Occupato

More information

14.2 Line Integrals. determines a partition P of the curve by points Pi ( xi, y

14.2 Line Integrals. determines a partition P of the curve by points Pi ( xi, y 4. Le Itegrls I ths secto we defe tegrl tht s smlr to sgle tegrl except tht sted of tegrtg over tervl [ ] we tegrte over curve. Such tegrls re clled le tegrls lthough curve tegrls would e etter termology.

More information

Studying the Problems of Multiple Integrals with Maple Chii-Huei Yu

Studying the Problems of Multiple Integrals with Maple Chii-Huei Yu Itetol Joul of Resech (IJR) e-issn: 2348-6848, - ISSN: 2348-795X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog Studyg the Poblems of Multle Itegls wth Mle Ch-Hue Yu Detmet of Ifomto Techology,

More information

χ be any function of X and Y then

χ be any function of X and Y then We have show that whe we ae gve Y g(), the [ ] [ g() ] g() f () Y o all g ()() f d fo dscete case Ths ca be eteded to clude fuctos of ay ube of ado vaables. Fo eaple, suppose ad Y ae.v. wth jot desty fucto,

More information

2.Decision Theory of Dependence

2.Decision Theory of Dependence .Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give

More information

6.6 Moments and Centers of Mass

6.6 Moments and Centers of Mass th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder

More information

Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek

Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek Optmlt of Strteges for Collpsg Expe Rom Vrles Smple Rom Smple E Stek troucto We revew the propertes of prectors of ler comtos of rom vrles se o rom vrles su-spce of the orgl rom vrles prtculr, we ttempt

More information

Certain Expansion Formulae Involving a Basic Analogue of Fox s H-Function

Certain Expansion Formulae Involving a Basic Analogue of Fox s H-Function vlle t htt:vu.edu l. l. Mth. ISSN: 93-9466 Vol. 3 Iue Jue 8. 8 36 Pevouly Vol. 3 No. lcto d led Mthetc: Itetol Joul M Cet Exo Foule Ivolvg c logue o Fox -Fucto S.. Puoht etet o c-scece Mthetc College o

More information

Chapter 2: Descriptive Statistics

Chapter 2: Descriptive Statistics Chapte : Decptve Stattc Peequte: Chapte. Revew of Uvaate Stattc The cetal teecy of a oe o le yetc tbuto of a et of teval, o hghe, cale coe, ofte uaze by the athetc ea, whch efe a We ca ue the ea to ceate

More information

21(2007) Adílson J. V. Brandão 1, João L. Martins 2

21(2007) Adílson J. V. Brandão 1, João L. Martins 2 (007) 30-34 Recuece Foulas fo Fboacc Sus Adílso J. V. Badão, João L. Mats Ceto de Mateátca, Coputa cão e Cog cão, Uvesdade Fedeal do ABC, Bazl.adlso.badao@ufabc.edu.b Depataeto de Mateátca, Uvesdade Fedeal

More information

Parametric Methods. Autoregressive (AR) Moving Average (MA) Autoregressive - Moving Average (ARMA) LO-2.5, P-13.3 to 13.4 (skip

Parametric Methods. Autoregressive (AR) Moving Average (MA) Autoregressive - Moving Average (ARMA) LO-2.5, P-13.3 to 13.4 (skip Pmeti Methods Autoegessive AR) Movig Avege MA) Autoegessive - Movig Avege ARMA) LO-.5, P-3.3 to 3.4 si 3.4.3 3.4.5) / Time Seies Modes Time Seies DT Rdom Sig / Motivtio fo Time Seies Modes Re the esut

More information

ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE

ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE O The Covegece Theoems... (Muslm Aso) ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE Muslm Aso, Yosephus D. Sumato, Nov Rustaa Dew 3 ) Mathematcs

More information

Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi SOME PROPERTIES CONCERNING THE HYPERSURFACES OF A WEYL SPACE

Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi SOME PROPERTIES CONCERNING THE HYPERSURFACES OF A WEYL SPACE Jou of Eee d Ntu Scece Mühed e Fe Be De S 5/4 SOME PROPERTIES CONCERNING THE HYPERSURFACES OF A EYL SPACE N KOFOĞLU M S Güze St Üete, Fe-Edeyt Füte, Mtet Böüü, Beştş-İSTANBUL Geş/Receed:..4 Ku/Accepted:

More information

Mathematical Statistics

Mathematical Statistics 7 75 Ode Sttistics The ode sttistics e the items o the dom smple ed o odeed i mitude om the smllest to the lest Recetl the impotce o ode sttistics hs icesed owi to the moe equet use o opmetic ieeces d

More information

M5. LTI Systems Described by Linear Constant Coefficient Difference Equations

M5. LTI Systems Described by Linear Constant Coefficient Difference Equations 5. LTI Systes Descied y Lie Costt Coefficiet Diffeece Equtios Redig teil: p.34-4, 245-253 3/22/2 I. Discete-Tie Sigls d Systes Up til ow we itoduced the Fouie d -tsfos d thei popeties with oly ief peview

More information

Fractional Integrals Involving Generalized Polynomials And Multivariable Function

Fractional Integrals Involving Generalized Polynomials And Multivariable Function IOSR Joual of ateatcs (IOSRJ) ISSN: 78-578 Volue, Issue 5 (Jul-Aug 0), PP 05- wwwosoualsog Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto D Neela Pade ad Resa Ka Deatet of ateatcs APS uvest

More information

On Almost Increasing Sequences For Generalized Absolute Summability

On Almost Increasing Sequences For Generalized Absolute Summability Joul of Applied Mthetic & Bioifotic, ol., o., 0, 43-50 ISSN: 79-660 (pit), 79-6939 (olie) Itetiol Scietific Pe, 0 O Alot Iceig Sequece Fo Geelized Abolute Subility W.. Suli Abtct A geel eult coceig bolute

More information

Chapter 17. Least Square Regression

Chapter 17. Least Square Regression The Islmc Uvest of Gz Fcult of Egeeg Cvl Egeeg Deptmet Numecl Alss ECIV 336 Chpte 7 Lest que Regesso Assocte Pof. Mze Abultef Cvl Egeeg Deptmet, The Islmc Uvest of Gz Pt 5 - CURVE FITTING Descbes techques

More information

On the Trivariate Polynomial Interpolation

On the Trivariate Polynomial Interpolation WSEAS RANSACIONS o MAHEMAICS Sle Sf O the vte Polol Itepolto SULEYMAN SAFAK Dvso of Mthetcs Fclt of Eee Do Elül Uvest 56 ıtepe c İ URKEY. sle.sf@de.ed.t Abstct: hs ppe s coceed wth the fole fo copt the

More information

Regularization of the Divergent Integrals I. General Consideration

Regularization of the Divergent Integrals I. General Consideration Zozuly / Electoc Joul o Bouy Eleets ol 4 No pp 49-57 6 Reulzto o the Dveet Itels I Geel Coseto Zozuly Ceto e Ivestco Cetc e Yuct AC Clle 43 No 3 Colo Chubuá e Hlo C 97 Mé Yuctá Méco E-l: zozuly@ccy Abstct

More information

The Linear Probability Density Function of Continuous Random Variables in the Real Number Field and Its Existence Proof

The Linear Probability Density Function of Continuous Random Variables in the Real Number Field and Its Existence Proof MATEC Web of Cofeeces ICIEA 06 600 (06) DOI: 0.05/mateccof/0668600 The ea Pobablty Desty Fucto of Cotuous Radom Vaables the Real Numbe Feld ad Its Estece Poof Yya Che ad Ye Collee of Softwae, Taj Uvesty,

More information

Summary: Binomial Expansion...! r. where

Summary: Binomial Expansion...! r. where Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly

More information

EXERCISE - 01 CHECK YOUR GRASP

EXERCISE - 01 CHECK YOUR GRASP EXERISE - 0 HEK YOUR GRASP 3. ( + Fo sum of coefficiets put ( + 4 ( + Fo sum of coefficiets put ; ( + ( 4. Give epessio c e ewitte s 7 4 7 7 3 7 7 ( 4 3( 4... 7( 4 7 7 7 3 ( 4... 7( 4 Lst tem ecomes (4

More information

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)

More information

The use of linear parametric approximation in numerical solving of nonlinear non-smooth Fuzzy equations

The use of linear parametric approximation in numerical solving of nonlinear non-smooth Fuzzy equations vlle ole t www.choleechl.co chve o ppled Scece Reech 5 6:49-6 http://choleechl.co/chve.htl ISSN 975-58X CODEN US SRC9 The ue o le petc ppoto uecl olvg o ole o-ooth uzz equto Mjd Hllj e Ze d l Vhd Kd Nehu

More information

Super-Mixed Multiple Attribute Group Decision Making Method Based on Hybrid Fuzzy Grey Relation Approach Degree *

Super-Mixed Multiple Attribute Group Decision Making Method Based on Hybrid Fuzzy Grey Relation Approach Degree * Supe-Med Multple Attbute Goup Decso Mkg Method Bsed o Hybd Fuzzy Gey Relto Appoch Degee Gol K Fe Ye b Cete of Ntul Scece vesty of Sceces Pyogyg DPR Koe b School of Busess Adstto South Ch vesty of Techology

More information

Permutations that Decompose in Cycles of Length 2 and are Given by Monomials

Permutations that Decompose in Cycles of Length 2 and are Given by Monomials Poceedgs of The Natoa Cofeece O Udegaduate Reseach (NCUR) 00 The Uvesty of Noth Caoa at Asheve Asheve, Noth Caoa Ap -, 00 Pemutatos that Decompose Cyces of Legth ad ae Gve y Moomas Lous J Cuz Depatmet

More information

Moments of Generalized Order Statistics from a General Class of Distributions

Moments of Generalized Order Statistics from a General Class of Distributions ISSN 684-843 Jol of Sttt Vole 5 28. 36-43 Moet of Geelzed Ode Sttt fo Geel l of Dtto Att Mhd Fz d Hee Ath Ode ttt eod le d eel othe odel of odeed do le e ewed el e of geelzed ode ttt go K 995. I th e exlt

More information

ECONOMETRIC ANALYSIS ON EFFICIENCY OF ESTIMATOR ABSTRACT

ECONOMETRIC ANALYSIS ON EFFICIENCY OF ESTIMATOR ABSTRACT ECOOMETRIC LYSIS O EFFICIECY OF ESTIMTOR M. Khohev, Lectue, Gffth Uvet, School of ccoutg d Fce, utl F. K, tt Pofeo, Mchuett Ittute of Techolog, Deptet of Mechcl Egeeg, US; cuetl t Shf Uvet, I. Houl P.

More information

Union, Intersection, Product and Direct Product of Prime Ideals

Union, Intersection, Product and Direct Product of Prime Ideals Globl Jourl of Pure d Appled Mthemtcs. ISSN 0973-1768 Volume 11, Number 3 (2015), pp. 1663-1667 Reserch Id Publctos http://www.rpublcto.com Uo, Itersecto, Product d Drect Product of Prme Idels Bdu.P (1),

More information

X-Ray Notes, Part III

X-Ray Notes, Part III oll 6 X-y oe 3: Pe X-Ry oe, P III oe Deeo Coe oupu o x-y ye h look lke h: We efe ue of que lhly ffee efo h ue y ovk: Co: C ΔS S Sl o oe Ro: SR S Co o oe Ro: CR ΔS C SR Pevouly, we ee he SR fo ye hv pxel

More information

SEPTIC B-SPLINE COLLOCATION METHOD FOR SIXTH ORDER BOUNDARY VALUE PROBLEMS

SEPTIC B-SPLINE COLLOCATION METHOD FOR SIXTH ORDER BOUNDARY VALUE PROBLEMS VOL. 5 NO. JULY ISSN 89-8 RN Joul of Egeeg d ppled Sceces - s Resech ulshg Netok RN. ll ghts eseved..pouls.com SETIC -SLINE COLLOCTION METHOD FOR SIXTH ORDER OUNDRY VLUE ROLEMS K.N.S. Ks Vsdhm d. Mul Ksh

More information

Exercise # 2.1 3, 7, , 3, , -9, 1, Solution: natural numbers are 3, , -9, 1, 2.5, 3, , , -9, 1, 2 2.5, 3, , -9, 1, , -9, 1, 2.

Exercise # 2.1 3, 7, , 3, , -9, 1, Solution: natural numbers are 3, , -9, 1, 2.5, 3, , , -9, 1, 2 2.5, 3, , -9, 1, , -9, 1, 2. Chter Chter Syste of Rel uers Tertg Del frto: The del frto whh Gve fte uers of dgts ts del rt s lled tertg del frto. Reurrg ( o-tertg )Del frto: The del frto (No tertg) whh soe dgts re reeted g d g the

More information

Maximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002

Maximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002 Mmm lkelhood eme of phylogey BIO 9S/ S 90B/ MH 90B/ S 90B Iodco o Bofomc pl 00 Ovevew of he pobblc ppoch o phylogey o k ee ccodg o he lkelhood d ee whee d e e of eqece d ee by ee wh leve fo he eqece. he

More information

Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2007] Direct Method; Newton s Divided Difference; Lagrangian Interpolation; Spline Interpolation.

Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2007] Direct Method; Newton s Divided Difference; Lagrangian Interpolation; Spline Interpolation. Nuecl Methods o Eg [ENGR 39 [Les KADEM 7 CHAPTER V Itepolto d Regesso Topcs Itepolto Regesso Dect Method; Newto s Dvded Deece; Lgg Itepolto; ple Itepolto Le d o-le Wht s tepolto? A ucto s ote gve ol t

More information

For this purpose, we need the following result:

For this purpose, we need the following result: 9 Lectue Sigulities of omplex Fuctio A poit is clled sigulity of fuctio f ( z ) if f ( z ) is ot lytic t the poit. A sigulity is clled isolted sigulity of f ( z ), if f ( z ) is lytic i some puctued disk

More information

Strategies for the AP Calculus Exam

Strategies for the AP Calculus Exam Strteges for the AP Clculus Em Strteges for the AP Clculus Em Strtegy : Kow Your Stuff Ths my seem ovous ut t ees to e metoe. No mout of cochg wll help you o the em f you o t kow the mterl. Here s lst

More information

2. Sample Space: The set of all possible outcomes of a random experiment is called the sample space. It is usually denoted by S or Ω.

2. Sample Space: The set of all possible outcomes of a random experiment is called the sample space. It is usually denoted by S or Ω. Ut: Rado expeet saple space evets classcal defto of pobablty ad the theoes of total ad copoud pobablty based o ths defto axoatc appoach to the oto of pobablty potat theoes based o ths appoach codtoal pobablty

More information

The formulae in this booklet have been arranged according to the unit in which they are first

The formulae in this booklet have been arranged according to the unit in which they are first Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge og to the ut whh the e fst toue. Thus te sttg ut m e eque to use the fomule tht wee toue peeg ut e.g. tes sttg C mght e epete to use fomule

More information

Chapter 7. Bounds for weighted sums of Random Variables

Chapter 7. Bounds for weighted sums of Random Variables Chpter 7. Bouds for weghted sums of Rdom Vrbles 7. Itroducto Let d 2 be two depedet rdom vrbles hvg commo dstrbuto fucto. Htczeko (998 d Hu d L (2000 vestgted the Rylegh dstrbuto d obted some results bout

More information

African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS

African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS Af Joul of See Tehology (AJST) See Egeeg See Vol. 4, No.,. 7-79 GENERALISED DELETION DESIGNS Mhel Ku Gh Joh Wylff Ohbo Dee of Mhe, Uvey of Nob, P. O. Bo 3097, Nob, Key ABSTRACT:- I h e yel gle ele fol

More information

ME 501A Seminar in Engineering Analysis Page 1

ME 501A Seminar in Engineering Analysis Page 1 Fobeius ethod pplied to Bessel s Equtio Octobe, 7 Fobeius ethod pplied to Bessel s Equtio L Cetto Mechicl Egieeig 5B Sei i Egieeig lsis Octobe, 7 Outlie Review idte Review lst lectue Powe seies solutios/fobeius

More information

SOLUTIONS ( ) ( )! ( ) ( ) ( ) ( )! ( ) ( ) ( ) ( ) n r. r ( Pascal s equation ). n 1. Stepanov Dalpiaz

SOLUTIONS ( ) ( )! ( ) ( ) ( ) ( )! ( ) ( ) ( ) ( ) n r. r ( Pascal s equation ). n 1. Stepanov Dalpiaz STAT UIU Pctice Poblems # SOLUTIONS Stepov Dlpiz The followig e umbe of pctice poblems tht my be helpful fo completig the homewo, d will liely be vey useful fo studyig fo ems...-.-.- Pove (show) tht. (

More information

The Mathematical Appendix

The Mathematical Appendix The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.

More information

On the k-lucas Numbers of Arithmetic Indexes

On the k-lucas Numbers of Arithmetic Indexes Alied Mthetics 0 3 0-06 htt://d.doi.og/0.436/.0.307 Published Olie Octobe 0 (htt://www.scirp.og/oul/) O the -ucs Nubes of Aithetic Idees Segio lco Detet of Mthetics d Istitute fo Alied Micoelectoics (IUMA)

More information

The formulae in this booklet have been arranged according to the unit in which they are first

The formulae in this booklet have been arranged according to the unit in which they are first Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge ccog to the ut whch the e fst touce. Thus cte sttg ut m e eque to use the fomule tht wee touce peceg ut e.g. ctes sttg C mght e epecte to use

More information

A GENERAL CLASS OF ESTIMATORS UNDER MULTI PHASE SAMPLING

A GENERAL CLASS OF ESTIMATORS UNDER MULTI PHASE SAMPLING TATITIC IN TRANITION-ew sees Octobe 9 83 TATITIC IN TRANITION-ew sees Octobe 9 Vol. No. pp. 83 9 A GENERAL CLA OF ETIMATOR UNDER MULTI PHAE AMPLING M.. Ahed & Atsu.. Dovlo ABTRACT Ths pape deves the geeal

More information

7.5-Determinants in Two Variables

7.5-Determinants in Two Variables 7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt

More information

GREEN S FUNCTION FOR HEAT CONDUCTION PROBLEMS IN A MULTI-LAYERED HOLLOW CYLINDER

GREEN S FUNCTION FOR HEAT CONDUCTION PROBLEMS IN A MULTI-LAYERED HOLLOW CYLINDER Joual of ppled Mathematcs ad Computatoal Mechacs 4, 3(3), 5- GREE S FUCTIO FOR HET CODUCTIO PROBLEMS I MULTI-LYERED HOLLOW CYLIDER Stasław Kukla, Uszula Sedlecka Isttute of Mathematcs, Czestochowa Uvesty

More information

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The

More information

Compactness in Multiset Topology

Compactness in Multiset Topology opatess ultset Topolog Sougata ahata S K Saata Depatet of atheats Vsva-haat Satketa-7335 Ida Abstat The pupose of ths pape s to todue the oept of opatess ultset topologal spae e vestgate soe bas esults

More information

Appropriate Realisation of MIMO Gain-Scheduled Controllers. D.J.Leith W.E. Leithead

Appropriate Realisation of MIMO Gain-Scheduled Controllers. D.J.Leith W.E. Leithead Appopte Relsto of MIMO GScheduled Cotolles D.J.Leth W.E. Lethed Deptet of Electoc & Electcl Egeeg, Uvest of Stthclde, GLASGOW G QE, U.K. Astct The dc chctestcs of cotolle desged the gschedulg ppoch c e

More information

ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY)

ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY) ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY) Floet Smdche, Ph D Aocte Pofeo Ch of Deptmet of Mth & Scece Uvety of New Mexco 2 College Rod Gllup, NM 873, USA E-ml: md@um.edu

More information

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1. SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,

More information

A Dynamical Quasi-Boolean System

A Dynamical Quasi-Boolean System ULETNUL Uestăţ Petol Gze Ploeşt Vol LX No / - 9 Se Mtetă - otă - Fză l Qs-oole Sste Gel Mose Petole-Gs Uest o Ploest ots etet est 39 Ploest 68 o el: ose@-loesto stt Ths e oes the esto o ol theoetl oet:

More information

University of Pavia, Pavia, Italy. North Andover MA 01845, USA

University of Pavia, Pavia, Italy. North Andover MA 01845, USA Iteatoal Joual of Optmzato: heoy, Method ad Applcato 27-5565(Pt) 27-6839(Ole) wwwgph/otma 29 Global Ifomato Publhe (HK) Co, Ltd 29, Vol, No 2, 55-59 η -Peudoleaty ad Effcecy Gogo Gog, Noma G Rueda 2 *

More information

Preliminary Examinations: Upper V Mathematics Paper 1

Preliminary Examinations: Upper V Mathematics Paper 1 relmr Emtos: Upper V Mthemtcs per Jul 03 Emer: G Evs Tme: 3 hrs Modertor: D Grgortos Mrks: 50 INSTRUCTIONS ND INFORMTION Ths questo pper sts of 0 pges, cludg swer Sheet pge 8 d Iformto Sheet pges 9 d 0

More information

A Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares

A Technique for Constructing Odd-order Magic Squares Using Basic Latin Squares Itertol Jourl of Scetfc d Reserch Publctos, Volume, Issue, My 0 ISSN 0- A Techque for Costructg Odd-order Mgc Squres Usg Bsc Lt Squres Tomb I. Deprtmet of Mthemtcs, Mpur Uversty, Imphl, Mpur (INDIA) tombrom@gml.com

More information