PRIMARY DECOMPOSITION, ASSOCIATED PRIME IDEALS AND GABRIEL TOPOLOGY

Size: px
Start display at page:

Download "PRIMARY DECOMPOSITION, ASSOCIATED PRIME IDEALS AND GABRIEL TOPOLOGY"

Transcription

1 Orietal J. ath., Volue 1, Nuber, 009, Page Orietal Acadeic Publiher PRIARY DECOPOSITION, ASSOCIATED PRIE IDEALS AND GABRIEL TOPOLOGY. EL HAJOUI, A. IRI ad A. ZOGLAT Uiverité ohaed V aculté de Sciece Départeet de athéatique B.P 1014 Rabat, aroc e-ail: hajoui4@yahoo.fr Abtract Let R be a coutative rig with a uit, N be a R-ubodule of a give R- odule ad be a o-trivial Gabriel topology o R. I thi paper, we give ufficiet coditio for the cloure Cl ( N ) of N i to be priary R- ubodule ad we how that it reduce to or N if R i Noetheria. oreover, thi will be ued to deduce a priary decopoitio of Cl ( N ) fro the oe of N whe i of fiite type (ad R i Noetheria). I uch cae, we tudy A ( ) ad we give eceary ad ufficiet coditio to have A ( N ), where A ( ) deote the et of the aociated prie ideal. 1. Itroductio ad Preliiarie Let R be a coutative rig R with uit. A prie ideal P of R i aid to be aociated to a give R-odule if there exit x \ ( 0) uch that P = A( x). The et of all thee ideal i deoted by A ( ). Throughout thi paper, we will ue ubodule N to ea that N i a ubodule of. The, a N i aid to be priary i if N ad for all a R ad x \ N with ax N there exit N uch that a N. I 010 atheatic Subject Claificatio: Priary: 16D10; Secodary: 13C05, 13C11. Keyword ad phrae: priary ubodule, priary decopoitio, aociated prie ideal, Gabriel topology, ijective odule. Received Deceber 0, 009

2 . EL HAJOUI, A. IRI ad A. ZOGLAT 10 thi cae the ideal P = A( N ) i prie ad N i aid to be P -priary (ee [5]). I a give Gabriel topology o R, we deote by Cl ( N ) the cloure of N i, i.e., Cl ( N ) = { x : I Ix N}. (or ore detail o Gabriel topology ee [1,, 3, 4]). Hece, the ubodule N i called -cloed if N = Cl ( N ). Alo, we ay that N ha a uppleetary i if there exit L uch that L N =. Defiitio 1.1. A R-odule i ijective if for all R-odule hooorphi f : 1 ad θ : 1 where f i oe-to-oe, the there exit a hooorphi ρ : atifyig θ = ρof. Defiitio 1.. Let be a R-ubodule of E. We ay that E i a eetial exteio of (or iply i eetial i E ) if N ( 0) for ay ozero R-ubodule N of E.. Priary Decopoitio of Cl ( N ) I thi ectio, we give ufficiet coditio for the R-ubodule Cl ( N ) to be priary. If R i Noetheria, a eceary coditio i alo give. Thee reult will be ued to obtai a priary decopoitio of Cl ( N ) fro a priary decopoitio of N. Propoitio.1. Let P -priary. The we have: (i) If P, the Cl ( N ) = N. (ii) If Cl ( N ) =, the P. Proof. (i) Suppoe that N be two R-odule ad aue that N i P ad Cl ( N ) N. The N Cl ( N ). Now, let x Cl ( N )\ N. The there exit I uch that I x N. Hece, for all λ I we have λ x N. Sice x N ad N i P-priary,

3 PRIARY DECOPOSITION, ASSOCIATED PRIE IDEALS 103 there exit Coequetly N uch that λ N. Therefore λ P = A( N ). I P ad P, a cotradictio. (ii) N i P -priary, thu N, let x ice Cl ( N ) =, the x Cl ( N ), therefore there exit I uch that I x N. Next, proceedig i a iilar way a i (i) above we deduce that P. Reark.. The covere of (ii) i the above propoitio i alo true whe R i aued to be Noetheria. See the ext ectio for the proof. or itace, ote that if P i a prie ideal of R, the P = { I ideal of R : I P} i a topology of Gabriel defied o R. Therefore, we have the followig Corollary.3. If N i P -priary, the Cl ( N ) = N. Theore.4. Let P N be two R-odule. If N i P-priary, the Cl ( N ) = or Cl ( N ) i Q -priary with P Q. Proof. Suppoe that N i P -priary ad ad Cl ( N ). Let x \ Cl ( N ) ad a R with ax Cl ( N ). The there exit I uch that I ax N. But I x N for x Cl ( N ). Therefore we ca fid λ I uch that λ x N ad aλ x N. Hece there i a N Cl ( N ) which iplie that Cl ( N ) i priary. N with Let Q = A( Cl ( N )). The Cl ( N ) i Q -priary ad becaue N Cl ( N ). P Q Theore.5. Let R a Noetheria rig, ad N be a P -priary ubodule of. The Cl ( N ) = if ad oly if P. Proof. Accordig to Propoitio.1, we eed to prove oly that P iplie Cl ( N ) =. Let P ad aue the exitece of x \ Cl ( ). Sice R i Noetheria, P i of fiite type ad therefore 0 N

4 there exit. EL HAJOUI, A. IRI ad A. ZOGLAT 104 1,,..., R uch that P = R1 + R + + R. Sice,,..., P A( ), there exit ( 1,,..., ) ( N ) uch 1 = N k that k N, k = 1,,...,. Thu, we have k 0 N for every k = 1,,...,. Now, et = k = 1 k ad let λ P. Hece, we ca write λ = Therefore, we have k x ( λk11 + λk + + λk ) fiitek= 1 = β( ) 1,,..., 1 1. fiite = x N with =. 1 Ideed, there exit k { 1,,..., } uch that k k (becaue if ot k < k for ay 1,,...,, k k, a k= 1 k= 1 k = ad o = < = cotradictio). Next, uig k x 0 N k, we obtai k k x0 N ad coequetly 1 1 x0 N. Uig thi fact, it follow fially that λx0 = β( ) 1 1,,..., 1 fiite = ad therefore P x 0 N. But, ice P x0 Cl ( N ), a cotradictio. Thi coplete the proof. x0 N for P, we coclude that Corollary.6. If R i Noetheria ad N i a P -priary ubodule of, the Cl ( N ) = N or Cl ( N ) =. ro ow o, we uppoe that R i Noetheria ad i of fiite type (i.e., there exit x 1, x,..., x eleet i uch that = Rx 1 + Rx + + Rx ). Hece, accordig to Theore 6.8 i [5], we ee that ay R- ubodule N of ha a priary decopoitio. Let N = Q i, where Q i i i= 1 ad J = { j : 1 j adp j }. Pi -priary, a priary decopoitio of N,

5 PRIARY DECOPOSITION, ASSOCIATED PRIE IDEALS 105 Theore.7. (i) If J =, the Cl ( N ) =. (ii) If J, the Cl ( N ) = Q j i a priary decopoitio of j J Cl ( N ) extracted fro that of. Proof. ollow fro Theore.5 ad Corollary.6. Exaple.8. (1) Let p, q be two prie uber i N, ad = { Z : Z pz}. We have: Z p (i) q Cl ( q Z) = q Z. Z p (ii) = q Cl ( q Z) = Z. () Let = p p p r 1 1 r be a priary decopoitio for N \{ 0, 1}, be a Gabriel topology defied o Z, ad J = { j : 1 j ad p j Z }. Z (i) If J =, the Cl ( Z) = Z. Z j (ii) If J, the Cl ( Z) = Z, with = pj. j J (3) Let R = K[ X, Y ], with K a coutative field. ( X, XY ) = ( X ) ( ax + Y, X ) i a priary decopoitio of ( X, XY ) ( for ay a K ), becaue ( X ) i a prie ad the i priary, ad the radical of ( ax + Y, X ) i ( X, Y ) which i axial, the ( ax + Y, X ) i priary. Next, let = { I ideal of R : I ( X )}. The Cl R R R (( X, XY )) = Cl (( X )) Cl (( ax + Y, X )) = ( X ) R = ( X ). 3. Aociated Prie Ideal ad Gabriel Topology I thi ectio, we uppoe agai that R i Noetheria ad i of fiite type ad we will reproduce oe of our reult obtaied i the previou

6 . EL HAJOUI, A. IRI ad A. ZOGLAT 106 ectio uig the otio of the aociated prie ideal. Theore 3.1. Let N be a P -priary R-ubodule of. The, we have: (i) P if ad oly if Cl ( N ) = N. (ii) If Cl ( N ), the Cl ( N ) i P -priary. Proof. (i) Suppoe P. Theore 6.6 i [5] give rie to the followig equivalece: N i P -priary if ad oly if A ( N ) = { P}. However, ice N i P -priary, P A( N ) ad o we ca fid x \ N uch that P = A( ) which iplie P x. Thu, we have x Cl ( N )\ N ad x 0 0 N 0 0 therefore Cl ( N ) N. Thi how that P wheever Cl ( N ) = N. The covere i give by Propoitio.1. (ii) I view of Theore.4, we have Cl ( N ) i a Q -priary ad P Q. Alo, by Theore 6.6 i [5], we ee that A( / Cl ( N )) = { Q}. The there exit x \ Cl ( N ) uch that Q = A( ) ad hece 0 Qx0 Cl ( N ). or every Q, there exit I uch that I x 0 Cl ( N ). But I x0 N ( becaue x0 Cl ( N )), the there exit β I with βx 0 N ad β x. O the other had, for N beig P-priary, we ca fid 0 N N atifyig N. Therefore P = A ( N ). Thi prove that Q P ad hece P = Q. Theore 3.. Let Cl (( 0 )) = ( 0) the Cl ( N ) i ijective. N be two R-odule. If N i ijective ad Proof. Suppoe the cotrary, i.e., Cl ( N ) i ot ijective. The, uig Theore B.4 i [5], we ee Cl ( N ) ha a proper eetial exteio L. But ice N i ijective, N doe ot have ay eetial exteio ad the N i ot eetial i L. Therefore, there i L L atifyig L ( 0), L N = ( 0) ad L Cl ( N ) ( 0). Now, for L0 = L we have: ( 0) L Cl ( N ) L = L0 ad L0 N = ( L ) N = ( L N ) = ( 0). Next, becaue L 0 i a ubodule of L ditict of (0), the x 0

7 L PRIARY DECOPOSITION, ASSOCIATED PRIE IDEALS 107 Cl ( N ) ( 0). urtherore, ice Cl (( 0) ) = ( 0 ) we have ( 0) L 0 0 Cl ( N ) Cl ( L0 ) Cl ( N ) = Cl ( L0 N ) = Cl (( 0) ), a cotradictio. Propoitio 3.3. If i a R-odule, the A ( ) = A( Cl (( 0) )). Proof. If P A( ), there exit x0 \ ( 0) uch that P = A ( x 0 ) ad o P x 0 ( 0). x0 Cl (( ) ad P A( Cl (( 0) )). Coverely, let P A( Cl (( 0) )), the there exit x Cl (( 0) )\ ( 0) uch that P = A( ). Thu there exit I uch that I x 0 ( 0). Hece I P = A ( ) ad therefore, P A( ). x 0 x 0 Propoitio 3.4. Let N be a R-ubodule of. The, we have A ( N ) = if ad oly if Cl ( N ) = N. Proof. Cobiatio of Theore 6.1 i [5] ad Propoitio 3.3 yield the ollowig N A( N ) = A( Cl (( 0 ))) = Cl (( 0)) = ( 0) N 0 Cl ( N ) N = ( 0) Cl ( N ) = N. Corollary 3.5. If Cl (( 0 )) = ( 0) ad N i a R-ubodule of havig a uppleetary, the N i -cloed. Proof. If N ha a uppleetary i, the by Theore 6.8 of [5], it follow A( ) = A( N ) A( N ). Next, ice Cl (( 0 )) = ( 0), A ( ) = A( Cl (( 0) )) = A( ( 0) ) = ad alo A ( N ) =. Therefore Cl ( N ) = N. Propoitio 3.6. If N i a ijective R-ubodule of ad A ( ) =, the Cl ( N ) = N. Proof. Suppoe that Cl ( N ) N ad N i ijective. The, N doe ot

8 . EL HAJOUI, A. IRI ad A. ZOGLAT 108 have ay proper eetial exteio. Thi follow by applyig Theore B.4 of [5]. Therefore, there exit L Cl ( N ) atifyig L ( 0) ad L N = ( 0). Let x0 L \ ( 0). The x0 Cl ( N ). urther, we ca fid I uch hat I x0 N. Hece, I x0 Ix0 N L N = ( 0) ad the I x 0 = ( 0). O the other had, ice Rx 0 ( 0) ad R i Noetheria, the A ( Rx 0 ) ad there exit R uch that Q = A( x0 ) A( Rx 0 ). Coequetly, we have Q ( becauei A ( x0 ) Q) ad Q A ( ) =, a cotradictio. Referece [1] J. Ecoriza ad B. Torrecilla, ultiplicatio odule relative to torio theorie, Co. Algebra 3(11) (1995), [] J. Ecoriza ad B. Torrecilla, Relative ultiplicatio ad ditributive odule, Coet. ath. Uiv. Carolia 3() (1997), [3] J. Ecoriza ad B. Torrecilla, Divioriel ultiplicatio rig, Note i ath. 63, arcel-dekker, 004. [4] B. Stetrö, Rig of Quotiet, Spriger, Berli, [5] H. atuura, Coitative Rig Theory, Cabridge Uiverity Pre, 1986.

Left Quasi- ArtinianModules

Left Quasi- ArtinianModules Aerica Joural of Matheatic ad Statitic 03, 3(): 6-3 DO: 0.593/j.aj.03030.04 Left Quai- ArtiiaModule Falih A. M. Aldoray *, Oaia M. M. Alhekiti Departet of Matheatic, U Al-Qura Uiverity, Makkah,P.O.Box

More information

On Some Properties of Tensor Product of Operators

On Some Properties of Tensor Product of Operators Global Joural of Pure ad Applied Matheatics. ISSN 0973-1768 Volue 12, Nuber 6 (2016), pp. 5139-5147 Research Idia Publicatios http://www.ripublicatio.co/gjpa.ht O Soe Properties of Tesor Product of Operators

More information

Applied Mathematical Sciences, Vol. 9, 2015, no. 3, HIKARI Ltd,

Applied Mathematical Sciences, Vol. 9, 2015, no. 3, HIKARI Ltd, Applied Mathematical Sciece Vol 9 5 o 3 7 - HIKARI Ltd wwwm-hiaricom http://dxdoiorg/988/am54884 O Poitive Defiite Solutio of the Noliear Matrix Equatio * A A I Saa'a A Zarea* Mathematical Sciece Departmet

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 16 11/04/2013. Ito integral. Properties

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 16 11/04/2013. Ito integral. Properties MASSACHUSES INSIUE OF ECHNOLOGY 6.65/15.7J Fall 13 Lecture 16 11/4/13 Ito itegral. Propertie Cotet. 1. Defiitio of Ito itegral. Propertie of Ito itegral 1 Ito itegral. Exitece We cotiue with the cotructio

More information

STRONG DEVIATION THEOREMS FOR THE SEQUENCE OF CONTINUOUS RANDOM VARIABLES AND THE APPROACH OF LAPLACE TRANSFORM

STRONG DEVIATION THEOREMS FOR THE SEQUENCE OF CONTINUOUS RANDOM VARIABLES AND THE APPROACH OF LAPLACE TRANSFORM Joural of Statitic: Advace i Theory ad Applicatio Volume, Number, 9, Page 35-47 STRONG DEVIATION THEORES FOR THE SEQUENCE OF CONTINUOUS RANDO VARIABLES AND THE APPROACH OF LAPLACE TRANSFOR School of athematic

More information

MORE COMMUTATOR INEQUALITIES FOR HILBERT SPACE OPERATORS

MORE COMMUTATOR INEQUALITIES FOR HILBERT SPACE OPERATORS terat. J. Fuctioal alyi Operator Theory ad pplicatio 04 Puhpa Publihig Houe llahabad dia vailable olie at http://pph.co/oural/ifaota.ht Volue Nuber 04 Page MORE COMMUTTOR NEQULTES FOR HLERT SPCE OPERTORS

More information

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi

More information

42 Dependence and Bases

42 Dependence and Bases 42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V

More information

SOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER

SOME FINITE SIMPLE GROUPS OF LIE TYPE C n ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER Joural of Algebra, Nuber Theory: Advaces ad Applicatios Volue, Nuber, 010, Pages 57-69 SOME FINITE SIMPLE GROUPS OF LIE TYPE C ( q) ARE UNIQUELY DETERMINED BY THEIR ELEMENT ORDERS AND THEIR ORDER School

More information

A Tail Bound For Sums Of Independent Random Variables And Application To The Pareto Distribution

A Tail Bound For Sums Of Independent Random Variables And Application To The Pareto Distribution Applied Mathematic E-Note, 9009, 300-306 c ISSN 1607-510 Available free at mirror ite of http://wwwmaththuedutw/ ame/ A Tail Boud For Sum Of Idepedet Radom Variable Ad Applicatio To The Pareto Ditributio

More information

Generalized Likelihood Functions and Random Measures

Generalized Likelihood Functions and Random Measures Pure Mathematical Sciece, Vol. 3, 2014, o. 2, 87-95 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/pm.2014.437 Geeralized Likelihood Fuctio ad Radom Meaure Chrito E. Koutzaki Departmet of Mathematic

More information

On the Signed Domination Number of the Cartesian Product of Two Directed Cycles

On the Signed Domination Number of the Cartesian Product of Two Directed Cycles Ope Joural of Dicrete Mathematic, 205, 5, 54-64 Publihed Olie July 205 i SciRe http://wwwcirporg/oural/odm http://dxdoiorg/0426/odm2055005 O the Siged Domiatio Number of the Carteia Product of Two Directed

More information

A tail bound for sums of independent random variables : application to the symmetric Pareto distribution

A tail bound for sums of independent random variables : application to the symmetric Pareto distribution A tail boud for um of idepedet radom variable : applicatio to the ymmetric Pareto ditributio Chritophe Cheeau To cite thi verio: Chritophe Cheeau. A tail boud for um of idepedet radom variable : applicatio

More information

u t u 0 ( 7) Intuitively, the maximum principles can be explained by the following observation. Recall

u t u 0 ( 7) Intuitively, the maximum principles can be explained by the following observation. Recall Oct. Heat Equatio M aximum priciple I thi lecture we will dicu the maximum priciple ad uiquee of olutio for the heat equatio.. Maximum priciple. The heat equatio alo ejoy maximum priciple a the Laplace

More information

Domination Number of Square of Cartesian Products of Cycles

Domination Number of Square of Cartesian Products of Cycles Ope Joural of Discrete Matheatics, 01,, 88-94 Published Olie October 01 i SciRes http://wwwscirporg/joural/ojd http://dxdoiorg/10436/ojd014008 Doiatio Nuber of Square of artesia Products of ycles Morteza

More information

Jordan Chevalley Decomposition and Invariants for Locally Finite Actions of Commutative Hopf Algebras

Jordan Chevalley Decomposition and Invariants for Locally Finite Actions of Commutative Hopf Algebras JOURNAL OF ALGEBRA 182, 123139 1996 ARTICLE NO. 0164 JordaChevalley ecopoitio ad Ivariat for Locally Fiite Actio of Coutative Hopf Algebra Adrzej Tyc* N. Copericu Uierity, Ititute of Matheatic, ul. Chopia

More information

On the Positive Definite Solutions of the Matrix Equation X S + A * X S A = Q

On the Positive Definite Solutions of the Matrix Equation X S + A * X S A = Q The Ope Applied Mathematic Joural 011 5 19-5 19 Ope Acce O the Poitive Defiite Solutio of the Matrix Equatio X S + A * X S A = Q Maria Adam * Departmet of Computer Sciece ad Biomedical Iformatic Uiverity

More information

Bernoulli Numbers and a New Binomial Transform Identity

Bernoulli Numbers and a New Binomial Transform Identity 1 2 3 47 6 23 11 Joural of Iteger Sequece, Vol. 17 2014, Article 14.2.2 Beroulli Nuber ad a New Bioial Trafor Idetity H. W. Gould Departet of Matheatic Wet Virgiia Uiverity Morgatow, WV 26506 USA gould@ath.wvu.edu

More information

On Certain Sums Extended over Prime Factors

On Certain Sums Extended over Prime Factors Iteratioal Mathematical Forum, Vol. 9, 014, o. 17, 797-801 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/imf.014.4478 O Certai Sum Exteded over Prime Factor Rafael Jakimczuk Diviió Matemática,

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

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)

distinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k) THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject

More information

BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear

BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios

More information

The Maximum Number of Subset Divisors of a Given Size

The Maximum Number of Subset Divisors of a Given Size The Maxiu Nuber of Subet Divior of a Give Size arxiv:407.470v [ath.co] 0 May 05 Abtract Sauel Zbary Caregie Mello Uiverity a zbary@yahoo.co Matheatic Subject Claificatio: 05A5, 05D05 If i a poitive iteger

More information

Some remarks on the paper Some elementary inequalities of G. Bennett

Some remarks on the paper Some elementary inequalities of G. Bennett Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity - Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries

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

Zeta-reciprocal Extended reciprocal zeta function and an alternate formulation of the Riemann hypothesis By M. Aslam Chaudhry

Zeta-reciprocal Extended reciprocal zeta function and an alternate formulation of the Riemann hypothesis By M. Aslam Chaudhry Zeta-reciprocal Eteded reciprocal zeta fuctio ad a alterate formulatio of the Riema hypothei By. Alam Chaudhry Departmet of athematical Sciece, Kig Fahd Uiverity of Petroleum ad ieral Dhahra 36, Saudi

More information

Heat Equation: Maximum Principles

Heat Equation: Maximum Principles Heat Equatio: Maximum Priciple Nov. 9, 0 I thi lecture we will dicu the maximum priciple ad uiquee of olutio for the heat equatio.. Maximum priciple. The heat equatio alo ejoy maximum priciple a the Laplace

More information

A New Type of q-szász-mirakjan Operators

A New Type of q-szász-mirakjan Operators Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators

More information

Boundedness for multilinear commutator of singular integral operator with weighted Lipschitz functions

Boundedness for multilinear commutator of singular integral operator with weighted Lipschitz functions Aal of the Uiverity of raiova, Matheatic ad oputer Sciece Serie Volue 40), 203, Page 84 94 ISSN: 223-6934 Boudede for ultiliear coutator of igular itegral operator with weighted Lipchitz fuctio Guo Sheg,

More information

On the 2-Domination Number of Complete Grid Graphs

On the 2-Domination Number of Complete Grid Graphs Ope Joural of Dicrete Mathematic, 0,, -0 http://wwwcirporg/oural/odm ISSN Olie: - ISSN Prit: - O the -Domiatio Number of Complete Grid Graph Ramy Shahee, Suhail Mahfud, Khame Almaea Departmet of Mathematic,

More information

1. (a) If u (I : R J), there exists c 0 in R such that for all q 0, cu q (I : R J) [q] I [q] : R J [q]. Hence, if j J, for all q 0, j q (cu q ) =

1. (a) If u (I : R J), there exists c 0 in R such that for all q 0, cu q (I : R J) [q] I [q] : R J [q]. Hence, if j J, for all q 0, j q (cu q ) = Math 615, Witer 2016 Problem Set #5 Solutio 1. (a) If u (I : R J), there exit c 0 i R uch that for all q 0, cu q (I : R J) [q] I [q] : R J [q]. Hece, if j J, for all q 0, j q (cu q ) = c(ju) q I [q], o

More information

Math 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]

Math 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version] Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths

More information

THE CONCEPT OF THE ROOT LOCUS. H(s) THE CONCEPT OF THE ROOT LOCUS

THE CONCEPT OF THE ROOT LOCUS. H(s) THE CONCEPT OF THE ROOT LOCUS So far i the tudie of cotrol yte the role of the characteritic equatio polyoial i deteriig the behavior of the yte ha bee highlighted. The root of that polyoial are the pole of the cotrol yte, ad their

More information

Société de Calcul Mathématique, S. A. Algorithmes et Optimisation

Société de Calcul Mathématique, S. A. Algorithmes et Optimisation Société de Calcul Mathématique S A Algorithme et Optimiatio Radom amplig of proportio Berard Beauzamy Jue 2008 From time to time we fid a problem i which we do ot deal with value but with proportio For

More information

Math Solutions to homework 6

Math Solutions to homework 6 Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there

More information

A NOTE ON AN R- MODULE WITH APPROXIMATELY-PURE INTERSECTION PROPERTY

A NOTE ON AN R- MODULE WITH APPROXIMATELY-PURE INTERSECTION PROPERTY Joural of Al-ahrai Uiversity Vol.13 (3), September, 2010, pp.170-174 Sciece A OTE O A R- ODULE WIT APPROXIATELY-PURE ITERSECTIO PROPERTY Uhood S. Al-assai Departmet of Computer Sciece, College of Sciece,

More information

Fractional Integral Operator and Olsen Inequality in the Non-Homogeneous Classic Morrey Space

Fractional Integral Operator and Olsen Inequality in the Non-Homogeneous Classic Morrey Space It Joural of Math Aalyi, Vol 6, 202, o 3, 50-5 Fractioal Itegral Oerator ad Ole Ieuality i the No-Homogeeou Claic Morrey Sace Mohammad Imam Utoyo Deartmet of Mathematic Airlagga Uiverity, Camu C, Mulyorejo

More information

Chapter 2. Periodic points of toral. automorphisms. 2.1 General introduction

Chapter 2. Periodic points of toral. automorphisms. 2.1 General introduction Chapter 2 Periodic poits of toral automorphisms 2.1 Geeral itroductio The automorphisms of the two-dimesioal torus are rich mathematical objects possessig iterestig geometric, algebraic, topological ad

More information

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.

[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms. [ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural

More information

Certain Properties of an Operator Involving the Generalized Hypergeometric Functions

Certain Properties of an Operator Involving the Generalized Hypergeometric Functions Proceedig of the Pakita Acadey of Sciece 5 (3): 7 3 (5) Copyright Pakita Acadey of Sciece ISSN: 377-969 (prit), 36-448 (olie) Pakita Acadey of Sciece Reearch Article Certai Propertie of a Operator Ivolvig

More information

f(1), and so, if f is continuous, f(x) = f(1)x.

f(1), and so, if f is continuous, f(x) = f(1)x. 2.2.35: Let f be a additive fuctio. i Clearly fx = fx ad therefore f x = fx for all Z+ ad x R. Hece, for ay, Z +, f = f, ad so, if f is cotiuous, fx = fx. ii Suppose that f is bouded o soe o-epty ope set.

More information

On Nonsingularity of Saddle Point Matrices. with Vectors of Ones

On Nonsingularity of Saddle Point Matrices. with Vectors of Ones Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad

More information

ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS

ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS ON THE LEHMER CONSTANT OF FINITE CYCLIC GROUPS NORBERT KAIBLINGER Abstract. Results of Lid o Lehmer s problem iclude the value of the Lehmer costat of the fiite cyclic group Z/Z, for 5 ad all odd. By complemetary

More information

Multistep Runge-Kutta Methods for solving DAEs

Multistep Runge-Kutta Methods for solving DAEs Multitep Ruge-Kutta Method for olvig DAE Heru Suhartato Faculty of Coputer Sciece, Uiverita Idoeia Kapu UI, Depok 6424, Idoeia Phoe: +62-2-786 349 E-ail: heru@c.ui.ac.id Kevi Burrage Advaced Coputatioal

More information

Inverse Matrix. A meaning that matrix B is an inverse of matrix A.

Inverse Matrix. A meaning that matrix B is an inverse of matrix A. Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix

More information

FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a =

FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS. Dedicated to the memory of Paul Erdős. 1. Introduction. n k. f n,a = FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS NEIL J. CALKIN Abstract. We prove divisibility properties for sus of powers of bioial coefficiets ad of -bioial coefficiets. Dedicated to the eory of

More information

Ma/CS 6a Class 22: Power Series

Ma/CS 6a Class 22: Power Series Ma/CS 6a Class 22: Power Series By Ada Sheffer Power Series Mooial: ax i. Polyoial: a 0 + a 1 x + a 2 x 2 + + a x. Power series: A x = a 0 + a 1 x + a 2 x 2 + Also called foral power series, because we

More information

q-apery Irrationality Proofs by q-wz Pairs

q-apery Irrationality Proofs by q-wz Pairs ADVANCES IN APPLIED MATHEMATICS 0, 7583 1998 ARTICLE NO AM970565 -Apery Irratioality Proof by -WZ Pair Tewodro Adeberha ad Doro Zeilberger Departet of Matheatic, Teple Uierity, Philadelphia, Peylaia 191,

More information

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number

Integrable Functions. { f n } is called a determining sequence for f. If f is integrable with respect to, then f d does exist as a finite real number MATH 532 Itegrable Fuctios Dr. Neal, WKU We ow shall defie what it meas for a measurable fuctio to be itegrable, show that all itegral properties of simple fuctios still hold, ad the give some coditios

More information

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?

If a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero? 2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a

More information

Comments on Discussion Sheet 18 and Worksheet 18 ( ) An Introduction to Hypothesis Testing

Comments on Discussion Sheet 18 and Worksheet 18 ( ) An Introduction to Hypothesis Testing Commet o Dicuio Sheet 18 ad Workheet 18 ( 9.5-9.7) A Itroductio to Hypothei Tetig Dicuio Sheet 18 A Itroductio to Hypothei Tetig We have tudied cofidece iterval for a while ow. Thee are method that allow

More information

New integral representations. . The polylogarithm function

New integral representations. . The polylogarithm function New itegral repreetatio of the polylogarithm fuctio Djurdje Cvijović Atomic Phyic Laboratory Viča Ititute of Nuclear Sciece P.O. Box 5 Belgrade Serbia. Abtract. Maximo ha recetly give a excellet ummary

More information

Chain conditions. 1. Artinian and noetherian modules. ALGBOOK CHAINS 1.1

Chain conditions. 1. Artinian and noetherian modules. ALGBOOK CHAINS 1.1 CHAINS 1.1 Chai coditios 1. Artiia ad oetheria modules. (1.1) Defiitio. Let A be a rig ad M a A-module. The module M is oetheria if every ascedig chai!!m 1 M 2 of submodules M of M is stable, that is,

More information

PROBLEM SET I (Suggested Solutions)

PROBLEM SET I (Suggested Solutions) Eco3-Fall3 PROBLE SET I (Suggested Solutios). a) Cosider the followig: x x = x The quadratic form = T x x is the required oe i matrix form. Similarly, for the followig parts: x 5 b) x = = x c) x x x x

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

Commutativity in Permutation Groups

Commutativity in Permutation Groups Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are

More information

AN APPLICATION OF HYPERHARMONIC NUMBERS IN MATRICES

AN APPLICATION OF HYPERHARMONIC NUMBERS IN MATRICES Hacettepe Joural of Mathematic ad Statitic Volume 4 4 03, 387 393 AN APPLICATION OF HYPERHARMONIC NUMBERS IN MATRICES Mutafa Bahşi ad Süleyma Solak Received 9 : 06 : 0 : Accepted 8 : 0 : 03 Abtract I thi

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

6.4 Binomial Coefficients

6.4 Binomial Coefficients 64 Bioial Coefficiets Pascal s Forula Pascal s forula, aed after the seveteeth-cetury Frech atheaticia ad philosopher Blaise Pascal, is oe of the ost faous ad useful i cobiatorics (which is the foral ter

More information

Maximal sets of integers not containing k + 1 pairwise coprimes and having divisors from a specified set of primes

Maximal sets of integers not containing k + 1 pairwise coprimes and having divisors from a specified set of primes EuroComb 2005 DMTCS proc. AE, 2005, 335 340 Maximal sets of itegers ot cotaiig k + 1 pairwise coprimes ad havig divisors from a specified set of primes Vladimir Bliovsky 1 Bielefeld Uiversity, Math. Dept.,

More information

The Random Walk For Dummies

The Random Walk For Dummies The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli

More information

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.

Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014. Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the

More information

Measure and Measurable Functions

Measure and Measurable Functions 3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies

More information

LECTURE 13 SIMULTANEOUS EQUATIONS

LECTURE 13 SIMULTANEOUS EQUATIONS NOVEMBER 5, 26 Demad-upply ytem LETURE 3 SIMULTNEOUS EQUTIONS I thi lecture, we dicu edogeeity problem that arie due to imultaeity, i.e. the left-had ide variable ad ome of the right-had ide variable are

More information

Refinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane

Refinements of Jensen s Inequality for Convex Functions on the Co-Ordinates in a Rectangle from the Plane Filoat 30:3 (206, 803 84 DOI 0.2298/FIL603803A Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat Refieets of Jese s Iequality for Covex

More information

TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction

TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction Math Appl 6 2017, 143 150 DOI: 1013164/ma201709 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES PANKAJ KUMAR DAS ad LALIT K VASHISHT Abstract We preset some iequality/equality for traces of Hadamard

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

Inferences of Type II Extreme Value. Distribution Based on Record Values

Inferences of Type II Extreme Value. Distribution Based on Record Values Applied Matheatical Sciece, Vol 7, 3, o 7, 3569-3578 IKARI td, www-hikarico http://doiorg/988/a33365 Ierece o Tpe II tree Value Ditributio Baed o Record Value M Ahaullah Rider Uiverit, awreceville, NJ,

More information

Math 61CM - Solutions to homework 3

Math 61CM - Solutions to homework 3 Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig

More information

Seunghee Ye Ma 8: Week 5 Oct 28

Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

More information

Matrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.

Matrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc. 2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a

More information

ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS

ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS EDUARD KONTOROVICH Abstract. I this work we uify ad geeralize some results about chaos ad sesitivity. Date: March 1, 005. 1 1. Symbolic Dyamics Defiitio

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

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009

Lecture Outline. 2 Separating Hyperplanes. 3 Banach Mazur Distance An Algorithmist s Toolkit October 22, 2009 18.409 A Algorithist s Toolkit October, 009 Lecture 1 Lecturer: Joatha Keler Scribes: Alex Levi (009) 1 Outlie Today we ll go over soe of the details fro last class ad ake precise ay details that were

More information

Analysis of Analytical and Numerical Methods of Epidemic Models

Analysis of Analytical and Numerical Methods of Epidemic Models Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN 09-70 Aalyi of Aalytical ad Nuerical Metod of Epideic Model Pooa Kuari Aitat Profeor, Departet of Mateatic Magad

More information

EIGENVALUES OF SEVERAL TRIDIAGONAL MATRICES

EIGENVALUES OF SEVERAL TRIDIAGONAL MATRICES Applied Mathematics E-Notes, 5(005), 66-74 c ISSN 607-50 Available free at mirror sites of http://www.math.thu.edu.tw/ ame/ EIGENVALUES OF SEVERAL TRIDIAGONAL MATRICES We-Chyua Yueh Received 4 September

More information

SOLVED EXAMPLES

SOLVED EXAMPLES Prelimiaries Chapter PELIMINAIES Cocept of Divisibility: A o-zero iteger t is said to be a divisor of a iteger s if there is a iteger u such that s tu I this case we write t s (i) 6 as ca be writte as

More information

The Structure of Z p when p is Prime

The Structure of Z p when p is Prime LECTURE 13 The Structure of Z p whe p is Prime Theorem 131 If p > 1 is a iteger, the the followig properties are equivalet (1) p is prime (2) For ay [0] p i Z p, the equatio X = [1] p has a solutio i Z

More information

Explicit scheme. Fully implicit scheme Notes. Fully implicit scheme Notes. Fully implicit scheme Notes. Notes

Explicit scheme. Fully implicit scheme Notes. Fully implicit scheme Notes. Fully implicit scheme Notes. Notes Explicit cheme So far coidered a fully explicit cheme to umerically olve the diffuio equatio: T + = ( )T + (T+ + T ) () with = κ ( x) Oly table for < / Thi cheme i ometime referred to a FTCS (forward time

More information

Metric Dimension of Some Graphs under Join Operation

Metric Dimension of Some Graphs under Join Operation Global Joural of Pure ad Applied Matheatics ISSN 0973-768 Volue 3, Nuber 7 (07), pp 333-3348 Research Idia Publicatios http://wwwripublicatioco Metric Diesio of Soe Graphs uder Joi Operatio B S Rawat ad

More information

Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan

Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan Arkasas Tech Uiversity MATH 94: Calculus II Dr Marcel B Fia 85 Power Series Let {a } =0 be a sequece of umbers The a power series about x = a is a series of the form a (x a) = a 0 + a (x a) + a (x a) +

More information

The log-behavior of n p(n) and n p(n)/n

The log-behavior of n p(n) and n p(n)/n Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity

More information

Compositions of Fuzzy T -Ideals in Ternary -Semi ring

Compositions of Fuzzy T -Ideals in Ternary -Semi ring Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces Copositios of Fuy T -Ideals i Terary -Sei rig RevathiK, 2, SudarayyaP 3, Madhusudhaa RaoD 4, Siva PrasadP 5 Research Scholar, Departet

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

Fuzzy n-normed Space and Fuzzy n-inner Product Space

Fuzzy n-normed Space and Fuzzy n-inner Product Space Global Joural o Pure ad Applied Matheatics. ISSN 0973-768 Volue 3, Nuber 9 (07), pp. 4795-48 Research Idia Publicatios http://www.ripublicatio.co Fuzzy -Nored Space ad Fuzzy -Ier Product Space Mashadi

More information

Riesz-Fischer Sequences and Lower Frame Bounds

Riesz-Fischer Sequences and Lower Frame Bounds Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.

More information

The Coupon Collector Problem in Statistical Quality Control

The Coupon Collector Problem in Statistical Quality Control The Coupo Collector Proble i Statitical Quality Cotrol Taar Gadrich, ad Rachel Ravid Abtract I the paper, the author have exteded the claical coupo collector proble to the cae of group drawig with iditiguihable

More information

6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.

6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer. 6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio

More information

Name Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions

Name Period ALGEBRA II Chapter 1B and 2A Notes Solving Inequalities and Absolute Value / Numbers and Functions Nae Period ALGEBRA II Chapter B ad A Notes Solvig Iequalities ad Absolute Value / Nubers ad Fuctios SECTION.6 Itroductio to Solvig Equatios Objectives: Write ad solve a liear equatio i oe variable. Solve

More information

MATH10212 Linear Algebra B Proof Problems

MATH10212 Linear Algebra B Proof Problems MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix

More information

Proc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS

Proc. Amer. Math. Soc. 139(2011), no. 5, BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Proc. Amer. Math. Soc. 139(2011, o. 5, 1569 1577. BINOMIAL COEFFICIENTS AND THE RING OF p-adic INTEGERS Zhi-Wei Su* ad Wei Zhag Departmet of Mathematics, Naig Uiversity Naig 210093, People s Republic of

More information

Generalized Fibonacci Like Sequence Associated with Fibonacci and Lucas Sequences

Generalized Fibonacci Like Sequence Associated with Fibonacci and Lucas Sequences Turkih Joural of Aalyi ad Number Theory, 4, Vol., No. 6, 33-38 Available olie at http://pub.ciepub.com/tjat//6/9 Sciece ad Educatio Publihig DOI:.69/tjat--6-9 Geeralized Fiboacci Like Sequece Aociated

More information

SOME TRIBONACCI IDENTITIES

SOME TRIBONACCI IDENTITIES Mathematics Today Vol.7(Dec-011) 1-9 ISSN 0976-38 Abstract: SOME TRIBONACCI IDENTITIES Shah Devbhadra V. Sir P.T.Sarvajaik College of Sciece, Athwalies, Surat 395001. e-mail : drdvshah@yahoo.com The sequece

More information

Exponential Functions and Taylor Series

Exponential Functions and Taylor Series MATH 4530: Aalysis Oe Expoetial Fuctios ad Taylor Series James K. Peterso Departmet of Biological Scieces ad Departmet of Mathematical Scieces Clemso Uiversity March 29, 2017 MATH 4530: Aalysis Oe Outlie

More information

PROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1.

PROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1. Math 7 Sprig 06 PROBLEM SET 5 SOLUTIONS Notatios. Give a real umber x, we will defie sequeces (a k ), (x k ), (p k ), (q k ) as i lecture.. (a) (5 pts) Fid the simple cotiued fractio represetatios of 6

More information

Types Ideals on IS-Algebras

Types Ideals on IS-Algebras Ieraioal Joural of Maheaical Aalyi Vol. 07 o. 3 635-646 IARI Ld www.-hikari.co hp://doi.org/0.988/ija.07.7466 Type Ideal o IS-Algebra Sudu Najah Jabir Faculy of Educaio ufa Uiveriy Iraq Copyrigh 07 Sudu

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

Properties and Tests of Zeros of Polynomial Functions

Properties and Tests of Zeros of Polynomial Functions Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by

More information

k-equitable mean labeling

k-equitable mean labeling Joural of Algorithms ad Comutatio joural homeage: htt://jac.ut.ac.ir k-euitable mea labelig P.Jeyathi 1 1 Deartmet of Mathematics, Govidammal Aditaar College for Wome, Tiruchedur- 628 215,Idia ABSTRACT

More information