Strong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA

Size: px
Start display at page:

Download "Strong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA"

Transcription

1 Iteatioal Joual of Reseach i Egieeig ad aageet Techology (IJRET) olue Issue July 5 Available at Stog Result fo Level Cossigs of Rado olyoials Dipty Rai Dhal D K isha Depatet of atheatics CET BUT BBSR ODISHA INDIA Abstact: 99 atheatics subject classificatio (Ae ath Soc): 6 B 99 Keywods ad phases: Idepedet idetically distibuted ado vaiables ado algebaic polyoial ado algebaic equatio eal oots I INTRODUCTION Let N be the ube of eal oots of the algebaic equatio f () k ξ k whee ae idepedet ado vaiables assuig eal values oly Seveal authos have estiated bouds fo N whe the ado vaiables satisfy diffeet distibutio laws Littlewood ad Offod [] ade the fist attept i this diectio They cosideed the cases whe the ae oally distibuted o uifoly distibuted i (- +) o assue oly the values + ad with equal pobability They obtaied i each case that N μlog logloglog A log Saal [] has cosideed the geeal case whe the have idetical distibutio with eceptio zeo vaiace ad thid absolute oet fiite ad o-zeo He has show that N s log outside a eceptioal set whose easue teds to zeo as teds to ifiity whee s teds to zeo but s log teds to ifiity Saal ad isha [4] have cosideed the case the have a coo chaacteistics fuctio ep C t whee C is a positive costat ad They have show that N μ log loglog outside a eceptioal set easue at ost ' μ (log log )(log ) μ loglog log if if I all the above cases the eceptioal set depeds upo Evas [] was the fist to obtai stog esult fo these bouds I such case the eceptioal set is idepedet of the degee of the polyoial We use the te stog esult i the followig sese: All the above esults ae of the fo N μ as i teds to ifiity wheeas the theoe of Evas is of the fo sup N μ as teds to ifiity Evas [] has show i case of oally distibuted coefficiets that thee eists a itege such that fo N μ log loglog μloglog ecept fo a set of easue at ost log Saal ad isha [5] have show i the case of C t chaacteistic fuctio ep that fo N μ log loglog outside a eceptioal set of easue at ost whee μ log log log log I [7] they have cosideed the stog esult i the geeal case Assuig that the ado vaiables (ot ecessaily idetically distibuted) have eceptio zeo 6 Iteatioal Joual of Reseach i Egieeig ad aageet Techology ISSN: ol Issue July 5

2 Dipty Rai Dhal et al vaiace ad thid absolute oet o-zeo fiite they have show that fo N (μlog)/log (K outside a set of easue at ost μ log log K log log t /t )log K li povided t is fiite ad log t = a (log) whee K = σ t aσ ad a τ σ τ beig the vaiace ad thid absolute ξ oet espectively of log Ou object is to ipove the stog esult fo lowe C t boud i case of chaacteistic fuctio ep We have show that fo N log Outside a eceptioal set of easue at ost μ log whee but log The esult of Evas [] is a special case of ous ad is (loglog ) obtaied by takig = ad i ou theoe The esult of Saal ad isha [5] is also a special case of ou theoe O the othe had ou eceptioal set is salle All authos who have estiated bouds fo N have used oe kid of basic techique oigially used by Littlewood ad Offod [] We shall deote μ fo positive costats which ay have diffeet values i diffeet occueces We suppose always that is lage so that ay iequalities tue whe is lage ay be take as satisfied Thoughout the pape [] will deote the geatest itege ot eceedig It ay be oted that although Evas [] is a special case of ous a uch bette estiate fo the lowe boud with salle eceptioal set ca be deived fo ou theoe Fo eaple if we take = <p< the fo p (loglog ) whee Stog Result fo Level Cossigs of Rado olyoials N log (log log ) p outside a eceptioal set of easue at ost μ(log log ) log p II THEORE Let f(w) be a polyoial of degee whose coefficiets ae idepedet ado vaiables with a C t coo chaacteistics fuctio ep whee = ad C is a positive costat Take {e }to be ay sequece tedig to zeo such that e log teds to ifiity as teds to ifiity The thee eists a itege such that fo each the ube of eal oots of ost of the equatios f()= is at least e log ecept fo a set of μ log easue at ost Lea If a ado vaiable ζ has chaacteistic fuctio C t ep the fo evey e C ξ This lea is due to Saal ad isha [4] ROOF OF THE THEORE Take costat A ad B such that <B< ad A Choose such that ad as teds to ifiity Let () So μ / log μ We defie (X) log Let k be the itege deteied by υ(8k 7) () 8k7 both ted to ifiity Ae B υ(8k ) 8k 7 Iteatioal Reseach i Egieeig ad aageet Techology (IJRET) ISSN: ol Issue July 5

3 Dipty Rai Dhal et al The fist iequality gives k iequality gives llog μk log log 8k log (8k ) (8k ) (8k ) log So Thus μ k μ log log log The secod (8k ) log(8k ) (8k ) log μ k μ log log () By the coditio iposed o it follows that k teds to ifiity as teds to ifiity We have f () U R at the poits fo X υ(4 ) 4 / (4) = k / k / k w hee U ξxr ξx the ide v agig fo υ(4 ) υ(4 ) υ(4 f ( ) i ad υ(4 ) 4 to fo to to i We also have ) UR f ( ) U R (5) fo Obviously U ad U + ae idepedet ado vaiables Agai it follows fo () that k+< fo lage Also the aiu ide i U + fo =k is υ(8k 7) (5) Let 8k7 which by () is cosistet with / The Stog Result fo Level Cossigs of Rado olyoials φ(4 ) 4 φ(4 ) υ(4) φ(4 ) φ(4 ) φ(4 (e / A) ) φ(4 ) 4 φ(4 ) φ(4 ) 4 (B / A)e (6) whe is lage Now we estiate U U U U U ( U U ) ( U ξ Sice the chaacteistic fuctio of is ep C t the chaacteistic fuctio of U is theefoe C t ep C t ep 4 whee the ide ages fo 8 υ(8 ) to υ(8 ) i 4 Thee foe the chaacteistic fuctio of U / is ep C t which is siilaly also the chaacteistic fuctio U / Thus the chaacteistic fuctio is depedet o Let F() be the coo distibutio fuctio Hece U ) ( U / ( U / ) F() Thus = F() F( ) F( ) F() F( ) F() δ(say Obviously δ ) We shall eed the followig leas Lea ost lage ξ / CAe ep - (4 ) B( ) ecept fo a set of easue at fo sufficietly 8 Iteatioal Reseach i Egieeig ad aageet Techology (IJRET) ISSN: ol Issue July 5

4 Dipty Rai Dhal et al oof The chaacteistics fuctio of But ξ ( ) is ep C t C X 4 υ(4) X 4 υ(4) Stog Result fo Level Cossigs of Rado olyoials / / / ξ / υ(4 ) (4 ) 4 υ(4 ) 4 (4 ) (4 ) υ(4 ) / [LOG (4) (4) [LOG (4) (4) 4 υ(4 ) 4 4 / / 4 / / υ(4 ) Sice υ(4 (4 ) (4 ) 4 υ(4 ) ) 4 log(4 ) (4) 4 υ(4 )(4 We have log(4 ) (4) 4 υ(4 ) 4 ) 4 Hece usig (6) we obtai 4 ep (4 ) CAe ep (4 ) as equied Lea B( ) ξ set of easue at ost ξ C ( ) / This follows diectly fo lea Now 4 υ(4) ecept fo a Ae B 6 6 υ (4 ) 6 Ae B / / / Ae B The last two steps above follow fo () ad (6) Hece by usig leas ad we have R < fo evey sufficietly lage ecept fo a set of easue at ost μep μ' (4 ) μep ( ) Thus we have R adr fo = + k whee =[k/]+ μ ep The easue of the eceptioal set is at ost μ ep μ' (4 μ ep ( ) ) / μ' μ ( ) (7) 4 We defie the evets E ad F as follows: E ={U U + <- + } F ={U < U } μ' 9 Iteatioal Reseach i Egieeig ad aageet Techology (IJRET) ISSN: ol Issue July 5

5 Dipty Rai Dhal et al Stog Result fo Level Cossigs of Rado olyoials We have show ealie that (E F ) δ Let η be a ado vaiable such that it takes value o E Uf ad zeo elsewhee I othe wods η with pobabilityδ with pobability - δ Let η ae thus idepedet ado vaiables with E (η )=δ ad (η )= δ δ < We wite S if R ad R othewise REFERENCES [] JE Littlewood ad AC Offod O the ube of eal oots of a ado algebaic equatio II oc Cab hilos Soc 5(99) -48 [] J Hajek ad A Reye A geealizatio of a iequality of Kologoov Acta ath Acad Sci Hugay 6(955) 8-8 [] GSaal O the ube of a ado algebaicequatio oc Cab hilos Soc 58(96) 4-44 [4] EA Evas O the ube of a ado algebaicequatio oc Lodo ath Soc (5)(965) [5] GSaal ad D atihai Stog esult fo eal zeos of ado polyoials JIdia ath Soc 4(976) -4 [6] GSaal ad D atihai Stog esult fo eal zeos of ado polyoials II JIdia ath Soc 4(977) 95-4 [7] N Ragaatha ad Sabadha O the lowe boud of the ube of eal oots of a ado algebaic equatio Idia ue Appl ath (98) [8] NNNayak ad S ohaty O the lowe boud of the ube of eal zeos of a ado algebaic polyoial J Idia ath Soc 49(985) 7-5 Iteatioal Reseach i Egieeig ad aageet Techology (IJRET) ISSN: ol Issue July 5

Strong Result for Level Crossings of Random Polynomials

Strong Result for Level Crossings of Random Polynomials IOSR Joual of haacy ad Biological Scieces (IOSR-JBS) e-issn:78-8, p-issn:19-7676 Volue 11, Issue Ve III (ay - Ju16), 1-18 wwwiosjoualsog Stog Result fo Level Cossigs of Rado olyoials 1 DKisha, AK asigh

More information

Greatest term (numerically) in the expansion of (1 + x) Method 1 Let T

Greatest term (numerically) in the expansion of (1 + x) Method 1 Let T BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value

More information

A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS

A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Discussioes Mathematicae Gaph Theoy 28 (2008 335 343 A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 e-mail: aboato@oges.com

More information

Some Integral Mean Estimates for Polynomials

Some Integral Mean Estimates for Polynomials Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi

More information

On ARMA(1,q) models with bounded and periodically correlated solutions

On ARMA(1,q) models with bounded and periodically correlated solutions Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,

More information

Using Difference Equations to Generalize Results for Periodic Nested Radicals

Using Difference Equations to Generalize Results for Periodic Nested Radicals Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =

More information

Structure and Some Geometric Properties of Nakano Difference Sequence Space

Structure and Some Geometric Properties of Nakano Difference Sequence Space Stuctue ad Soe Geoetic Poeties of Naao Diffeece Sequece Sace N Faied ad AA Baey Deatet of Matheatics, Faculty of Sciece, Ai Shas Uivesity, Caio, Egyt awad_baey@yahooco Abstact: I this ae, we exted the

More information

Some Properties of the K-Jacobsthal Lucas Sequence

Some Properties of the K-Jacobsthal Lucas Sequence Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Some Popeties of the K-Jacobsthal Lucas

More information

International Journal of Mathematics Trends and Technology (IJMTT) Volume 47 Number 1 July 2017

International Journal of Mathematics Trends and Technology (IJMTT) Volume 47 Number 1 July 2017 Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 Coe Metic Saces, Coe Rectagula Metic Saces ad Coo Fixed Poit Theoes M. Sivastava; S.C. Ghosh Deatet of Matheatics, D.A.V. College

More information

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +

More information

On randomly generated non-trivially intersecting hypergraphs

On randomly generated non-trivially intersecting hypergraphs O adomly geeated o-tivially itesectig hypegaphs Balázs Patkós Submitted: May 5, 009; Accepted: Feb, 010; Published: Feb 8, 010 Mathematics Subject Classificatio: 05C65, 05D05, 05D40 Abstact We popose two

More information

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics

More information

Conditional Convergence of Infinite Products

Conditional Convergence of Infinite Products Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this

More information

FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES

FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity

More information

MATH Midterm Solutions

MATH Midterm Solutions MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca

More information

Generalization of Horadam s Sequence

Generalization of Horadam s Sequence Tuish Joual of Aalysis ad Nube Theoy 6 Vol No 3-7 Available olie at http://pubssciepubco/tjat///5 Sciece ad Educatio Publishig DOI:69/tjat---5 Geealizatio of Hoada s Sequece CN Phadte * YS Valaulia Depatet

More information

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If

More information

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae

More information

Generalized Fibonacci-Lucas Sequence

Generalized Fibonacci-Lucas Sequence Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash

More information

Lacunary Almost Summability in Certain Linear Topological Spaces

Lacunary Almost Summability in Certain Linear Topological Spaces BULLETIN of te MLYSİN MTHEMTİCL SCİENCES SOCİETY Bull. Malays. Mat. Sci. Soc. (2) 27 (2004), 27 223 Lacuay lost Suability i Cetai Liea Topological Spaces BÜNYMIN YDIN Cuuiyet Uivesity, Facutly of Educatio,

More information

KEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow

KEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you

More information

Lecture 6: October 16, 2017

Lecture 6: October 16, 2017 Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of

More information

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3

More information

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special

More information

Math 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as

Math 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1

More information

EVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS

EVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal

More information

Multivector Functions

Multivector Functions I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed

More information

A New Result On A,p n,δ k -Summabilty

A New Result On A,p n,δ k -Summabilty OSR Joual of Matheatics (OSR-JM) e-ssn: 2278-5728, p-ssn:239-765x. Volue 0, ssue Ve. V. (Feb. 204), PP 56-62 www.iosjouals.og A New Result O A,p,δ -Suabilty Ripeda Kua &Aditya Kua Raghuashi Depatet of

More information

Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces

Lecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such

More information

A note on random minimum length spanning trees

A note on random minimum length spanning trees A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu

More information

International Journal of Mathematical Archive-3(5), 2012, Available online through ISSN

International Journal of Mathematical Archive-3(5), 2012, Available online through   ISSN Iteatioal Joual of Matheatical Achive-3(5,, 8-8 Available olie though www.ija.ifo ISSN 9 546 CERTAIN NEW CONTINUED FRACTIONS FOR THE RATIO OF TWO 3 ψ 3 SERIES Maheshwa Pathak* & Pakaj Sivastava** *Depatet

More information

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P. Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a

More information

On composite conformal mapping of an annulus to a plane with two holes

On composite conformal mapping of an annulus to a plane with two holes O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy

More information

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES #A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet

More information

Mapping Radius of Regular Function and Center of Convex Region. Duan Wenxi

Mapping Radius of Regular Function and Center of Convex Region. Duan Wenxi d Iteatioal Cofeece o Electical Compute Egieeig ad Electoics (ICECEE 5 Mappig adius of egula Fuctio ad Cete of Covex egio Dua Wexi School of Applied Mathematics Beijig Nomal Uivesity Zhuhai Chia 363463@qqcom

More information

ON CERTAIN CLASS OF ANALYTIC FUNCTIONS

ON CERTAIN CLASS OF ANALYTIC FUNCTIONS ON CERTAIN CLASS OF ANALYTIC FUNCTIONS Nailah Abdul Rahma Al Diha Mathematics Depatmet Gils College of Educatio PO Box 60 Riyadh 567 Saudi Aabia Received Febuay 005 accepted Septembe 005 Commuicated by

More information

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.

More information

Modular Spaces Topology

Modular Spaces Topology Applied Matheatics 23 4 296-3 http://ddoiog/4236/a234975 Published Olie Septebe 23 (http://wwwscipog/joual/a) Modula Spaces Topology Ahed Hajji Laboatoy of Matheatics Coputig ad Applicatio Depatet of Matheatics

More information

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig

More information

REVIEW ARTICLE ABSTRACT. Interpolation of generalized Biaxisymmetric potentials. D. Kumar* G.L. `Reddy**

REVIEW ARTICLE ABSTRACT. Interpolation of generalized Biaxisymmetric potentials. D. Kumar* G.L. `Reddy** Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 9 REVIEW ARTICLE Itepolatio of geealized Biaxisyetic potetials D Kua* GL `Reddy** ABSTRACT I this pape we study the chebyshev ad itepolatio

More information

Global asymptotic stability in a rational dynamic equation on discrete time scales

Global asymptotic stability in a rational dynamic equation on discrete time scales Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [395-699] [Vol-, Issue-, Decebe- 6] Global asyptotic stability i a atioal dyaic euatio o discete tie scales a( t) b( ( t)) ( ( t)), t T c ( ( (

More information

SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE

SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE Faulty of Siees ad Matheatis, Uivesity of Niš, Sebia Available at: http://www.pf.i.a.yu/filoat Filoat 22:2 (28), 59 64 SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE Saee Ahad Gupai Abstat. The sequee

More information

Lower Bounds for Cover-Free Families

Lower Bounds for Cover-Free Families Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set

More information

INVERSE CAUCHY PROBLEMS FOR NONLINEAR FRACTIONAL PARABOLIC EQUATIONS IN HILBERT SPACE

INVERSE CAUCHY PROBLEMS FOR NONLINEAR FRACTIONAL PARABOLIC EQUATIONS IN HILBERT SPACE IJAS 6 (3 Febuay www.apapess.com/volumes/vol6issue3/ijas_6_3_.pdf INVESE CAUCH POBLEMS FO NONLINEA FACTIONAL PAABOLIC EQUATIONS IN HILBET SPACE Mahmoud M. El-Boai Faculty of Sciece Aleadia Uivesit Aleadia

More information

On the Circulant Matrices with. Arithmetic Sequence

On the Circulant Matrices with. Arithmetic Sequence It J Cotep Math Scieces Vol 5 o 5 3 - O the Ciculat Matices with Aithetic Sequece Mustafa Bahsi ad Süleya Solak * Depatet of Matheatics Educatio Selçuk Uivesity Mea Yeiyol 499 Koya-Tukey Ftly we have defied

More information

ECE Spring Prof. David R. Jackson ECE Dept. Notes 20

ECE Spring Prof. David R. Jackson ECE Dept. Notes 20 ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we

More information

The Pigeonhole Principle 3.4 Binomial Coefficients

The Pigeonhole Principle 3.4 Binomial Coefficients Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple

More information

Using Counting Techniques to Determine Probabilities

Using Counting Techniques to Determine Probabilities Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe

More information

= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!

= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n! 0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet

More information

( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.

( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8. Mathematics. Solved Eamples JEE Mai/Boads Eample : Fid the coefficiet of y i c y y Sol: By usig fomula of fidig geeal tem we ca easily get coefficiet of y. I the biomial epasio, ( ) th tem is c T ( y )

More information

Al Lehnen Madison Area Technical College 10/5/2014

Al Lehnen Madison Area Technical College 10/5/2014 The Correlatio of Two Rado Variables Page Preliiary: The Cauchy-Schwarz-Buyakovsky Iequality For ay two sequeces of real ubers { a } ad = { b } =, the followig iequality is always true. Furtherore, equality

More information

Advanced Physical Geodesy

Advanced Physical Geodesy Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig

More information

2.4.2 A Theorem About Absolutely Convergent Series

2.4.2 A Theorem About Absolutely Convergent Series 0 Versio of August 27, 200 CHAPTER 2. INFINITE SERIES Add these two series: + 3 2 + 5 + 7 4 + 9 + 6 +... = 3 l 2. (2.20) 2 Sice the reciprocal of each iteger occurs exactly oce i the last series, we would

More information

On Some Fractional Integral Operators Involving Generalized Gauss Hypergeometric Functions

On Some Fractional Integral Operators Involving Generalized Gauss Hypergeometric Functions Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 5, Issue (Decembe ), pp. 3 33 (Peviously, Vol. 5, Issue, pp. 48 47) Applicatios ad Applied Mathematics: A Iteatioal Joual (AAM) O

More information

THE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES

THE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES Please cite this atle as: Mhal Matalyck Tacaa Romaiuk The aalysis of some models fo claim pocessig i isuace compaies Scietif Reseach of the Istitute of Mathemats ad Compute Sciece 004 Volume 3 Issue pages

More information

Asymptotic distribution of products of sums of independent random variables

Asymptotic distribution of products of sums of independent random variables Proc. Idia Acad. Sci. Math. Sci. Vol. 3, No., May 03, pp. 83 9. c Idia Academy of Scieces Asymptotic distributio of products of sums of idepedet radom variables YANLING WANG, SUXIA YAO ad HONGXIA DU ollege

More information

Technical Report: Bessel Filter Analysis

Technical Report: Bessel Filter Analysis Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we

More information

Convergence of random variables. (telegram style notes) P.J.C. Spreij

Convergence of random variables. (telegram style notes) P.J.C. Spreij Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space

More information

Taylor Transformations into G 2

Taylor Transformations into G 2 Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that

More information

Minimal order perfect functional observers for singular linear systems

Minimal order perfect functional observers for singular linear systems Miimal ode efect fuctioal obseves fo sigula liea systems Tadeusz aczoek Istitute of Cotol Idustial lectoics Wasaw Uivesity of Techology, -66 Waszawa, oszykowa 75, POLAND Abstact. A ew method fo desigig

More information

THE ANALYTIC LARGE SIEVE

THE ANALYTIC LARGE SIEVE THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig

More information

Ch 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology

Ch 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a

More information

On a Problem of Littlewood

On a Problem of Littlewood Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995

More information

Chapter 2 Sampling distribution

Chapter 2 Sampling distribution [ 05 STAT] Chapte Samplig distibutio. The Paamete ad the Statistic Whe we have collected the data, we have a whole set of umbes o desciptios witte dow o a pape o stoed o a compute file. We ty to summaize

More information

DANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD

DANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito

More information

Consider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample

Consider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample Uodeed Samples without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied

More information

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r. Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p

More information

I PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of

I PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of Two Maks Questios I PU MATHEMATIS HAPTER - 08 Biomial Theoem. Epad + usig biomial theoem ad hece fid the coefficiet of y y. Epad usig biomial theoem. Hece fid the costat tem of the epasio.. Simplify +

More information

International Journal of Mathematical Archive-5(3), 2014, Available online through ISSN

International Journal of Mathematical Archive-5(3), 2014, Available online through   ISSN Iteatioal Joual of Mathematical Achive-5(3, 04, 7-75 Available olie though www.ijma.ifo ISSN 9 5046 ON THE OSCILLATOY BEHAVIO FO A CETAIN CLASS OF SECOND ODE DELAY DIFFEENCE EQUATIONS P. Mohakuma ad A.

More information

RECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia.

RECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia. #A39 INTEGERS () RECIPROCAL POWER SUMS Athoy Sofo Victoia Uivesity, Melboue City, Austalia. athoy.sofo@vu.edu.au Received: /8/, Acceted: 6//, Published: 6/5/ Abstact I this ae we give a alteative oof ad

More information

Sums of Involving the Harmonic Numbers and the Binomial Coefficients

Sums of Involving the Harmonic Numbers and the Binomial Coefficients Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa

More information

Lecture 24: Observability and Constructibility

Lecture 24: Observability and Constructibility ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio

More information

Steiner Hyper Wiener Index A. Babu 1, J. Baskar Babujee 2 Department of mathematics, Anna University MIT Campus, Chennai-44, India.

Steiner Hyper Wiener Index A. Babu 1, J. Baskar Babujee 2 Department of mathematics, Anna University MIT Campus, Chennai-44, India. Steie Hype Wiee Idex A. Babu 1, J. Baska Babujee Depatmet of mathematics, Aa Uivesity MIT Campus, Cheai-44, Idia. Abstact Fo a coected gaph G Hype Wiee Idex is defied as WW G = 1 {u,v} V(G) d u, v + d

More information

Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)].

Probability 2 - Notes 10. Lemma. If X is a random variable and g(x) 0 for all x in the support of f X, then P(g(X) 1) E[g(X)]. Probability 2 - Notes 0 Some Useful Iequalities. Lemma. If X is a radom variable ad g(x 0 for all x i the support of f X, the P(g(X E[g(X]. Proof. (cotiuous case P(g(X Corollaries x:g(x f X (xdx x:g(x

More information

On the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers

On the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers O the Explicit Detemiats Sigulaities of -ciculat Left -ciculat Matices with Some Famous Numbes ZHAOLIN JIANG Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA jzh08@siacom JUAN LI Depatmet

More information

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence

Chapter 3. Strong convergence. 3.1 Definition of almost sure convergence Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

S. K. VAISH AND R. CHANKANYAL. = ρ(f), b λ(f) ρ(f) (1.1)

S. K. VAISH AND R. CHANKANYAL. = ρ(f), b λ(f) ρ(f) (1.1) TAMKANG JOURNAL OF MATHEMATICS Volume 35, Number, Witer 00 ON THE MAXIMUM MODULUS AND MAXIMUM TERM OF COMPOSITION OF ENTIRE FUNCTIONS S. K. VAISH AND R. CHANKANYAL Abstract. We study some growth properties

More information

The Multivariate-t distribution and the Simes Inequality. Abstract. Sarkar (1998) showed that certain positively dependent (MTP 2 ) random variables

The Multivariate-t distribution and the Simes Inequality. Abstract. Sarkar (1998) showed that certain positively dependent (MTP 2 ) random variables The Multivaiate-t distibutio ad the Simes Iequality by Hey W. Block 1, Saat K. Saka 2, Thomas H. Savits 1 ad Jie Wag 3 Uivesity of ittsbugh 1,Temple Uivesity 2,Gad Valley State Uivesity 3 Abstact. Saka

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

INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS

More information

p-adic Invariant Integral on Z p Associated with the Changhee s q-bernoulli Polynomials

p-adic Invariant Integral on Z p Associated with the Changhee s q-bernoulli Polynomials It. Joual of Math. Aalysis, Vol. 7, 2013, o. 43, 2117-2128 HIKARI Ltd, www.m-hiai.com htt://dx.doi.og/10.12988/ima.2013.36166 -Adic Ivaiat Itegal o Z Associated with the Chaghee s -Beoulli Polyomials J.

More information

b i u x i U a i j u x i u x j

b i u x i U a i j u x i u x j M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here

More information

Dynamic Programming for Estimating Acceptance Probability of Credit Card Products

Dynamic Programming for Estimating Acceptance Probability of Credit Card Products Joual of Copute ad Couicatios, 07, 5, 56-75 http://wwwscipog/joual/jcc ISSN Olie: 7-57 ISSN Pit: 7-59 Dyaic Pogaig fo Estiatig Acceptace Pobability of Cedit Cad Poducts Lai Soo Lee,, Ya Mei Tee, Hsi Vo

More information

lim za n n = z lim a n n.

lim za n n = z lim a n n. Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget

More information

Bertrand s postulate Chapter 2

Bertrand s postulate Chapter 2 Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are

More information

A PROBABILITY PROBLEM

A PROBABILITY PROBLEM A PROBABILITY PROBLEM A big superarket chai has the followig policy: For every Euros you sped per buy, you ear oe poit (suppose, e.g., that = 3; i this case, if you sped 8.45 Euros, you get two poits,

More information

THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION

THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia

More information

On Order of a Function of Several Complex Variables Analytic in the Unit Polydisc

On Order of a Function of Several Complex Variables Analytic in the Unit Polydisc ISSN 746-7659, Eglad, UK Joural of Iforatio ad Coutig Sciece Vol 6, No 3, 0, 95-06 O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Rata Kuar Dutta + Deartet of Matheatics, Siliguri

More information

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio

More information

Probability theory and mathematical statistics:

Probability theory and mathematical statistics: N.I. Lobachevsky State Uiversity of Nizhi Novgorod Probability theory ad mathematical statistics: Law of Total Probability. Associate Professor A.V. Zorie Law of Total Probability. 1 / 14 Theorem Let H

More information

FRACTIONAL CALCULUS OF GENERALIZED K-MITTAG-LEFFLER FUNCTION

FRACTIONAL CALCULUS OF GENERALIZED K-MITTAG-LEFFLER FUNCTION Joual of Rajastha Academy of Physical Scieces ISSN : 972-636; URL : htt://aos.og.i Vol.5, No.&2, Mach-Jue, 26, 89-96 FRACTIONAL CALCULUS OF GENERALIZED K-MITTAG-LEFFLER FUNCTION Jiteda Daiya ad Jeta Ram

More information

EXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI

EXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI avtei EXAMPLES Solutios of AVTE by AVTE (avtei) lass XI Leade i BSE oachig 1 avtei SHORT ANSWER TYPE 1 Fid the th tem i the epasio of 1 We have T 1 1 1 1 1 1 1 1 1 1 Epad the followig (1 + ) 4 Put 1 y

More information

MATH /19: problems for supervision in week 08 SOLUTIONS

MATH /19: problems for supervision in week 08 SOLUTIONS MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each

More information

1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND

1.2 AXIOMATIC APPROACH TO PROBABILITY AND PROPERTIES OF PROBABILITY MEASURE 1.2 AXIOMATIC APPROACH TO PROBABILITY AND NTEL- robability ad Distributios MODULE 1 ROBABILITY LECTURE 2 Topics 1.2 AXIOMATIC AROACH TO ROBABILITY AND ROERTIES OF ROBABILITY MEASURE 1.2.1 Iclusio-Exclusio Forula I the followig sectio we will discuss

More information

Applications of the Dirac Sequences in Electrodynamics

Applications of the Dirac Sequences in Electrodynamics Poc of the 8th WSEAS It Cof o Mathematical Methods ad Computatioal Techiques i Electical Egieeig Buchaest Octobe 6-7 6 45 Applicatios of the Diac Sequeces i Electodyamics WILHELM W KECS Depatmet of Mathematics

More information

The Application of a Maximum Likelihood Approach to an Accelerated Life Testing with an Underlying Three- Parameter Weibull Model

The Application of a Maximum Likelihood Approach to an Accelerated Life Testing with an Underlying Three- Parameter Weibull Model Iteatioal Joual of Pefomability Egieeig Vol. 4, No. 3, July 28, pp. 233-24. RAMS Cosultats Pited i Idia The Applicatio of a Maximum Likelihood Appoach to a Acceleated Life Testig with a Udelyig Thee- Paamete

More information

The degree sequences and spectra of scale-free random graphs

The degree sequences and spectra of scale-free random graphs The degee sequeces ad specta of scale-fee ado gaphs Joatha Joda Uivesity of Sheffield 30th July 004 Abstact We ivestigate the degee sequeces of scale-fee ado gaphs. We obtai a foula fo the liitig popotio

More information

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial. Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable

More information

Gamma Distribution and Gamma Approximation

Gamma Distribution and Gamma Approximation Gamma Distributio ad Gamma Approimatio Xiaomig Zeg a Fuhua (Frak Cheg b a Xiame Uiversity, Xiame 365, Chia mzeg@jigia.mu.edu.c b Uiversity of Ketucky, Leigto, Ketucky 456-46, USA cheg@cs.uky.edu Abstract

More information