IRREDUCIBLE COVARIANT REPRESENTATIONS ASSOCIATED TO AN R-DISCRETE GROUPOID
|
|
- Britton Andrews
- 6 years ago
- Views:
Transcription
1 UPB Sc Bull Sere A Vol 69 No 7 ISSN 3-77 IRREDUCIBLE COVARIANT REPRESENTATIONS ASSOCIATED TO AN R-DISCRETE GROUPOID Roxaa VIDICAN Ue perech covarate poztv defte ( T ) relatv la u grupod r-dcret G e poate aoca prtr-o teoremă de tp Steprg o reprezetare covartă ( U ) Vom tabl o codţe eceară ş ufcetă de reductbltate petru aceata d urmă Let ( T ) be a potve defte covarat par wth repect to a r- dcrete groupod G Ug a theorem of type Steprg we ca aocate to the par ( T ) a covarat repreetato ( U ) I th artcle we provde a eceary ad uffcet codto of rreducblty for the repreetato ( U ) Keyword: r-dcrete groupod potve defte covarat par covarat repreetato Itroducto For a utable ameable r-dcrete prcpal groupod G wth the ut pace G ad the emgroup I of t compact ad ope G-et we defe a covarat repreetato of I to be a par ( T ) where () a -repreetato of C ( G ) ( the C -algebra of complexvalued fucto o G that are cotuou ad vahg at fty) o a (complex ad eparable) Hlbert pace H; () T { T ( ) I} a famly of operator o H uch that T ( ) ad T ( ) T T ( for t I; () T ( ) ( ( a ) T ( ) for a C ( G ) ad I (ee Ob (v; ad (v) T ( ) ( ℵ ) for I uch that G (where ℵ deote the charactertc fucto of ) At Departmet of Mathematc II Uverty Poltehca of Bucharet ROMANIA
2 8 Roxaa Vdca Defto If T a fucto from I to B (H ) ad a - repreetato of C ( G ) o H the ( T ) a potve defte covarat par f the followg codto are atfed: () T ( ) ( ( a ) T ( ) for a C ( G ) I; () T ( ) T T ( for t I; () T ( ) ( ℵ ) for I wth G ; ad (v) for each fte collecto of pot I the operator matrx ( T ( actg o H H o-egatve (th mea that < T ( ) ξ ξ > H for ay ( ξ ξ ) H H ) Theorem Let ( T ) be a potve defte covarat par wth value B (H ) The there are a Hlbert pace H a covarat repreetato ( U ) of I o H ad a Hlbert pace omorphm V mappg H to H uch that ( a ) V ( V for a C ( G ) ad T V U V for t I e hall how a eceary ad uffcet codto of rreducblty for the covarat repreetato ( U ) from the above theorem term of the par ( T ) Prelmare The defto for the oto of: ameable r-dcrete prcpal groupod G- et C -algebra aocated wth a locally compact groupod ca be foud [] [] [3] [4] or [5] Now to udertadg more ealy the cotet of th paper we hall preet frt of all the mot mportat tage of the proof of the theorem from Itroducto Let H be the et of all fucto f : I H uch that () there a compact et K f (depedg of f ) uch that K f G ad f for t K f Ø ; ad () ft t I uch that t t the f t ) ( ) f ( ) (where d t ( ℵ d ( t ) t
3 Irreducble covarat repreetato aocated to a r-dcrete groupod 9 e defe o H a equlear fuctoal < > by the formula < f g > < T ( f g( ) > H where the um over ay two fte et { t }{ } I uch that K f t K g ad uch that t t k l Ø f k l Let N { f H < f f > } The equlear fuctoal < > from above pae to a er product o H / N ( For th er product we keep the otato < > ) Let H be the completo of H / N the aocated orm Next we defe V : H H by V ξ ξ + N where ξ ( t ) ( E( ℵ ξ ( E beg the codtoal expectato from C ( G) to t C ( G ) C ( G ) ; ee [4] p4) Th operator V a ometry Note that for f H ad t I f we deote f ( ) f ( t t ) I we obta f t H Now for q I the operator U o H / N defed by U ( q)( f + N) f q + N f H atfe the properte U ( q ) U ( q) ad U ( U ( ) U for q t I Fally for a C ( G ) we defe ( o H by the formula ( ( f ) ( a t ) f Deote ( a )( f + N) ( f + N a -repreetato of C ( G ) o H The par ( U ) ad the operator V have the properte aked the theorem Obervato () ( U ) ut a potve defte covarat par: let ( ξ ξ ) H H ad I; the T ( )ξ ξ U ( ) Vξ Vξ From here ug V ( H) H we deduce that ( U( a potve operator matrx
4 Roxaa Vdca () A V a ometry t reult V V I H Sce V a omorphm there ext V ad V V V V I hece H ( V V ) V The VH H mple V V Coequetly T V U V VT V U t I () If a o-degeerate repreetato of C ( G ) the the famly { T ( )} I o-degeerate ( e Sp{ T ( ) ξ Iξ H} H ) ad o { U ( )} alo: let η H ad { a } C ( G ){ ξ } H uch that I η lm ( a ) ξ ; fx { } I uch that : upp a The η lm ( ℵ a ) ξ lm ( ℵ ) ( a ) ξ lmt( c ) ( a ) ξ For the ecod part of the aerto : ce V a ometry V V are cotuou operator Therefore H V ( H ) V ( Sp( T ( I) H Sp( VT ( I)( H Sp( VT ( I) V ( H Sp( U ( I) H ) (v) If ( T ) a potve defte covarat par the T ( ) T ( ) I: from the proof of the theorem we kow that U U ( t ) t I hece T V U ( t ) V T ( t ) t I (v) If ( T ) a potve defte covarat par the T () I : let I ; the we have T () T ( ) T( ) T ( ) T ( ) T ( ) ( ℵ ) ℵ (v) For a C ( G ) ad I we have a ℵaℵ ( where from the rght de a G-et whle from the left de a otato for the fucto u d( u ) ; u the elemet x wth r ( x) u ) hece a ( a ) ( ℵ aℵ ) ( ℵℵ ) a( ℵℵ ) ℵ aℵ a Now: ( T ( ) ( T( ) T ( ) ( ( ℵ ) T ( ) ( a ( T ( ) ( T( ) ( a ) T ( ) T ( ) ( a 3 Irreducble repreetato of C ( ) ) Lema If ( T )( T ) are two potve defte covarat par of I o B (H ) wth o-degeerate repreetato of ( C G ) uch that N ad I: ( T ( ( T ( G
5 Irreducble covarat repreetato aocated to a r-dcrete groupod whle ( U H) repectvely ( U H ) are the correpodg covarat repreetato the there ext a cotracto B( H H) ( e ) atfyg: () V V ; () U ( ) U( ) I; ad () a ( ) ( a C ( G ) T ( a t ) T ( a t ) t I a C ( G ) Proof: () Let h h H ad I The: U ( ) Vh V U ( ) V h h T ( ) h h T ( ) h h U ( ) Vh Defg : H H by U Vh U( Vh we remark that a cotracto wth V V () For t I ad h H we have U ( ) U Vh U ( Vh U( Vh U( ) U( Vh U( ) U Vh () If a C ( G ) t I ad h H t follow a U t V h U t ( ) ( ) ( ) ( a t ) Vh U V ( a t ) h U( V ( a t O the other had a U t V h a U t V h U t ( ) ( ) ( ) ( ) ( ) ( a t ) V h U V ( a t ) h Hece a ( ) ( V U V ( a t ) h V U V ( a t ) h T ( a t ) h T ( a t ) h Defto Let H be a Hlbert pace ad U B(H ) U a potve operator ( ad we hall deote th by U ) f U U ad U ( x) x x H Notato 3 For a Hlbert pace H ad a et M uch that M B(H ) we wrte M for the commutat of M e M { U B( H ) V M : UV VU} ) h
6 Roxaa Vdca Lemma 4 Let ( T ) a covarat potve defte par o B (H ) wth a o-degeerate repreetato of C ( G ) ( U ) the correpodg covarat repreetato o B (H ) ad V the omorphm betwee H ad H uch that ( a ) V ( V a C ( G ) ad T V U V t I For U ( I) ad S ( C ( G we defe the applcato Φ : I B( H ) by Φ V U V t I ad Ψ S : C ( G ) B( H ) by S ( V S Ψ ( V a C ( G ) Uder thee codto the followg aerto are true: () the map Φ ad S ΨS are lear ad ectve; () f U ( I) ad the for N ad I the operator matrx ( Φ ( o-egatve; ad () f U ( I) uch that ad N I the operator matrx Φ ( o-egatve the ( Proof: () It uffce to how that the lear map Φ ectve For the ectvty of S ΨS the proof aalogou e aume that Φ Let t I ad h k H The: U Vh U ( ) Vk U ( ) U Vh Vk V U ( Vh k Φ ( h k hece (Oe apple Ob ( () e ue the obervato () from Prelmare ad the followg theorem: If A a volutve algebra H a Hlbert pace π a repreetato of A o H ad T B(H ) uch that T ad T π ( A) the there ext K B(H ) wth K K K T ad K π ( A) Thu for U ( I) wth there K B(H ) uch that K K K ad K U ( I) Let I ad ( ξ ξ ) H H The requred cocluo follow by Φ ( ) ξ ξ V U ( ) Vξ ξ K U ( ) Vξ Vξ U ( ) KVξ KVξ ( Oe ue ob ( () Let a the tatemet of the lemma For N I h h H ad k U( ) Vh + + U ( ) Vh we get
7 Irreducble covarat repreetato aocated to a r-dcrete groupod 3 hece k k V U ( ) U ( ) Vh h V U ( ) Vh h Φ ( ) h h Lemma 5 Let ( T ) ad ( T ) two potve defte covarat par wth o-degeerate repreetato of C ( G ) uch that N I ( T ( ( T ( ad : t I a C ( G ) : T ( a t ) T ( a t ) If ( U ) the covarat repreetato wth repect to ( T ) the there U ( I) ( C ( G wth the properte: I ( I : H H the detty operator I ( h) h h H ) T Φ ad Ψ Proof: Let V : H H the omorphm correpodg to the par ( T ) ( U ) the covarat repreetato ad V : H H the omorphm aocated wth the par ( T ) The lemma aure the extece of a cotracto : H H uch that V V U ( ) U ( ) I ad ( ( a C ( G ) e hall deote The ad for h H we have: h h h h h h hece I Chooe I The U ( ) U( ) ( U ( ) ) ( U( ) ) ( U ( U ( ) U ( ) mple U ( I) Aalogouly ( C ( G Suppoe that t I ad h k H Sce Φ h k U( Vh Vk U Vh Vk U Vh Vk ( U Vh Vk V U( Vh k T h k T Φ Smlarly Ψ Obervato 6 () If a G-et the are alo G-et ad at the ame tme they are ubet of G Moreover ( If x the x yu wth y ad u hece x y Coverely for x we have x xx x )
8 4 Roxaa Vdca () The operator from lemma 5 atfe a follow from T Φ T ( Φ ( t I Φ ( ) Φ T ( ) T T ( Φ ( V U ( ) VV U Vh V U ( Vh t I h H V U ( Vh V U ( Vh t I h H U ( Vh U ( Vh t I h H U ( ) h U ( ) h h H U h U h I ( ) ( ) I h H Propoto 7 Let ( T ) a potve defte covarat par The there a becto betwee the et A { U ( I ) ( C ( G I } ad the famly B of the potve defte covarat par ( S θ ) o B (H ) whch have the property N I : ( ( T ( ad ( S S ( a t ) S( θ ( a t ) a C ( G ) t I Proof: Aume A By lemma 4 for N ad t t I the operator matrx ( Φ ( t t o-egatve Becaue I we ca clam that N ad t t I : ( Φ I ( t t a o-egatve matrx hece Φ T Ug t clear that Φ ( Φ ( ) Φ t I ad that Ψ a repreetato of C ( G ) For I uch that G we have Φ ( ) V U ( ) V V ( ℵ ) V Ψ ( ℵ ) Next for a C ( G ) ad I t follow Φ Ψ a V U VV ( ) ( ) ( ) ( V V U ( ) ( V V ( a ) U ( ) V V ( a ) VV U ( ) V Ψ ( a ) Φ ( ) Fally f t I ad a C ( G ) the Φ ( a t ) V U V( a t ) ad Φ Ψ ( a t ) V U V V ( a t ) V V U( ( a t ) V V U( V ( a t ) Thu Φ ( a t ) Φ Ψ ( a t ) All thee mply ( Φ Ψ ) B By the lemma 4 the map ( Φ Ψ ) ectve It urectvty follow from lemma 5 ad obervato 6 ()
9 Irreducble covarat repreetato aocated to a r-dcrete groupod 5 Propoto 8 If ( T ) a o-ull potve defte covarat par uch that for every ( S θ ) B there λ C wth S λt the the covarat repreetato ( U ) aocated wth ( T ) rreducble ( the ee that oly the ubpace ad H are cloed ad varat wth repect to U (I) ad ( C ( G Proof: Let ( T ) ad ( S θ ) B a the tatemet of the propoto By lemma 5 we ca fd A uch that S Φ ad θ Ψ Coequetly there λ C uch that Φ λt It reult: V U( V λ T t I U λvt V t I U λ U t I λi Let K a cloed ubpace of H uch that K varat wth repect to U (I) ad ( C ( G For h H we coder the wrttg h h + h where h K ad h K e hall deote wth P K the proecto o K P K : H H P K h h It rema to how that P K A ( A from propoto 7) The t wll ext λ C uch that P K λi Th equvalet wth K or K H For k K h K ad I we have U ( ) k K hece U ( ) h k h U ( ) k from where U ( ) h K Thu f h H uch that h h + h wth h K ad h K we ca wrte PK U( ) h PK ( U( ) h + U( ) h U( ) h U( ) PK h e P K U( I) Aalogouly t reult that P ( ( K C G For h H wth the ame decompoto a above we fd PK h PK h h PK h P K h h h h + h h h ad ( P K I) h h h h h h h h h Hece P P ad P K I K K Propoto 9 If ( T ) a o-ull potve defte covarat par uch that a o-degeerate repreetato of C ( G ) ad ( U ) rreducble the for ( S θ ) B there λ C uch that S λt Proof: Chooe ( T ) ad ( U ) wth the properte from the tatemet ad ( S θ ) B By lemma 5 there a operator A uch that S θ ) ( Φ Ψ ) (
10 6 Roxaa Vdca Let K (H ) The ( H ) { h H k H aî h k} { h H h k} a lear cloed ubpace of H Moreover t varat wth repect to U (I) ad ( C ( G Becaue ( U ) rreducble we deduce that (H ) or H hece λi for ome λ C Fally t I : S( Φ V λ U V λt R E F E R E N C E S [] PMuhly BSolel Subalgebra of groupod C -algebra J ree ud agew Math 4 (989) pp4-75 [] P Muhly Coordate Operator Algebra 997 preprt [3] A Patero Groupod Ivere Semgroup ad ther Operator Algebra Progre Mathematc vol7 Brkhäuer 999 [4] J Reault A groupod Approach to C -algebra Lecture Note Mathematc vol793 Sprger Verlag 98 [5] Roxaa Vdca C -ubalgebra of thec -algebra aocated wth a r-dcrete groupod UPB Sc Bull Sere A vol 64 pp3-38 [6] m B Arveo Subalgebra of C -algebra Acta Math 3 (969) pp4-4 Th reearch wa upported by grat CNCSIS (Romaa Natoal Coucl for Reearch Hgh Educato) code A 65/6
International Journal of Pure and Applied Sciences and Technology
It J Pure Appl Sc Techol, () (00), pp 79-86 Iteratoal Joural of Pure ad Appled Scece ad Techology ISSN 9-607 Avalable ole at wwwjopaaat Reearch Paper Some Stroger Chaotc Feature of the Geeralzed Shft Map
More informationStrong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity
BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sc. Soc. () 7 (004), 5 35 Strog Covergece of Weghted Averaged Appromats of Asymptotcally Noepasve Mappgs Baach Spaces wthout
More informationROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K
ROOT-LOCUS ANALYSIS Coder a geeral feedback cotrol yte wth a varable ga. R( Y( G( + H( Root-Locu a plot of the loc of the pole of the cloed-loop trafer fucto whe oe of the yte paraeter ( vared. Root locu
More informationMATH 247/Winter Notes on the adjoint and on normal operators.
MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say
More informationLinear Approximating to Integer Addition
Lear Approxmatg to Iteger Addto L A-Pg Bejg 00085, P.R. Cha apl000@a.com Abtract The teger addto ofte appled cpher a a cryptographc mea. I th paper we wll preet ome reult about the lear approxmatg for
More informationSome distances and sequences in a weighted graph
IOSR Joural of Mathematc (IOSR-JM) e-issn: 78-578 p-issn: 19 765X PP 7-15 wwworjouralorg Some dtace ad equece a weghted graph Jll K Mathew 1, Sul Mathew Departmet of Mathematc Federal Ittute of Scece ad
More information10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n
0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More informationInternational Journal of Mathematical Archive-5(8), 2014, Available online through ISSN
Iteratoal Joural of Mathematcal Archve-5(8) 204 25-29 Avalable ole through www.jma.fo ISSN 2229 5046 COMMON FIXED POINT OF GENERALIZED CONTRACTION MAPPING IN FUZZY METRIC SPACES Hamd Mottagh Golsha* ad
More informationSimple Linear Regression Analysis
LINEAR REGREION ANALYSIS MODULE II Lecture - 5 Smple Lear Regreo Aaly Dr Shalabh Departmet of Mathematc Stattc Ida Ittute of Techology Kapur Jot cofdece rego for A jot cofdece rego for ca alo be foud Such
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationA note on testing the covariance matrix for large dimension
A ote o tetg the covarace matrx for large dmeo Melae Brke Ruhr-Uvertät Bochum Fakultät für Mathematk 44780 Bochum, Germay e-mal: melae.brke@ruhr-u-bochum.de Holger ette Ruhr-Uvertät Bochum Fakultät für
More informationOn the energy of complement of regular line graphs
MATCH Coucato Matheatcal ad Coputer Chetry MATCH Cou Math Coput Che 60 008) 47-434 ISSN 0340-653 O the eergy of copleet of regular le graph Fateeh Alaghpour a, Baha Ahad b a Uverty of Tehra, Tehra, Ira
More informationSome Wgh Inequalities for Univalent Harmonic Analytic Functions
ppled Mathematc 464-469 do:436/am66 Publhed Ole December (http://wwwscrporg/joural/am Some Wgh Ieualte for Uvalet Harmoc alytc Fucto btract Pooam Sharma Departmet of Mathematc ad troomy Uverty of Lucow
More informationCollapsing to Sample and Remainder Means. Ed Stanek. In order to collapse the expanded random variables to weighted sample and remainder
Collapg to Saple ad Reader Mea Ed Staek Collapg to Saple ad Reader Average order to collape the expaded rado varable to weghted aple ad reader average, we pre-ultpled by ( M C C ( ( M C ( M M M ( M M M,
More informationα1 α2 Simplex and Rectangle Elements Multi-index Notation of polynomials of degree Definition: The set P k will be the set of all functions:
Smplex ad Rectagle Elemets Mult-dex Notato = (,..., ), o-egatve tegers = = β = ( β,..., β ) the + β = ( + β,..., + β ) + x = x x x x = x x β β + D = D = D D x x x β β Defto: The set P of polyomals of degree
More informationQ-analogue of a Linear Transformation Preserving Log-concavity
Iteratoal Joural of Algebra, Vol. 1, 2007, o. 2, 87-94 Q-aalogue of a Lear Trasformato Preservg Log-cocavty Daozhog Luo Departmet of Mathematcs, Huaqao Uversty Quazhou, Fua 362021, P. R. Cha ldzblue@163.com
More informationLebesgue Measure of Generalized Cantor Set
Aals of Pure ad Appled Mathematcs Vol., No.,, -8 ISSN: -8X P), -888ole) Publshed o 8 May www.researchmathsc.org Aals of Lebesgue Measure of Geeralzed ator Set Md. Jahurul Islam ad Md. Shahdul Islam Departmet
More informationINEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS
Joural of Mathematcal Scece: Advace ad Alcato Volume 24, 23, Page 29-46 INEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS ZLATKO PAVIĆ Mechacal Egeerg Faculty Slavok Brod Uverty of Ojek
More informationMaps on Triangular Matrix Algebras
Maps o ragular Matrx lgebras HMED RMZI SOUROUR Departmet of Mathematcs ad Statstcs Uversty of Vctora Vctora, BC V8W 3P4 CND sourour@mathuvcca bstract We surveys results about somorphsms, Jorda somorphsms,
More informationOn Submanifolds of an Almost r-paracontact Riemannian Manifold Endowed with a Quarter Symmetric Metric Connection
Theoretcal Mathematcs & Applcatos vol. 4 o. 4 04-7 ISS: 79-9687 prt 79-9709 ole Scepress Ltd 04 O Submafolds of a Almost r-paracotact emaa Mafold Edowed wth a Quarter Symmetrc Metrc Coecto Mob Ahmad Abdullah.
More informationDIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS
DIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS Course Project: Classcal Mechacs (PHY 40) Suja Dabholkar (Y430) Sul Yeshwath (Y444). Itroducto Hamltoa mechacs s geometry phase space. It deals
More informationResearch Article A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings
Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Volume 009, Artcle ID 391839, 9 pages do:10.1155/009/391839 Research Artcle A New Iteratve Method for Commo Fxed Pots of a Fte
More informationRademacher Complexity. Examples
Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed
More informationUniversity 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 informationReaction Time VS. Drug Percentage Subject Amount of Drug Times % Reaction Time in Seconds 1 Mary John Carl Sara William 5 4
CHAPTER Smple Lear Regreo EXAMPLE A expermet volvg fve ubject coducted to determe the relatohp betwee the percetage of a certa drug the bloodtream ad the legth of tme t take the ubject to react to a tmulu.
More informationREVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION
REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I lear regreo, we coder the frequecy dtrbuto of oe varable (Y) at each of everal level of a ecod varable (X). Y kow a the depedet varable. The
More informationBasic Structures: Sets, Functions, Sequences, and Sums
ac Structure: Set, Fucto, Sequece, ad Sum CSC-9 Dcrete Structure Kotat uch - LSU Set et a uordered collecto o object Eglh alphabet vowel: V { a, e,, o, u} a V b V Odd potve teger le tha : elemet o et member
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More informationJournal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi FACTORIZATION PROPERTIES IN POLYNOMIAL EXTENSION OF UFR S
Joural of Egeerg ad Natural Scece Mühedl ve Fe Bller Derg Sga 25/2 FACTORIZATION PROPERTIES IN POLYNOMIAL EXTENSION OF UFR S Murat ALAN* Yıldız Te Üverte, Fe-Edebyat Faülte, Mateat Bölüü, Davutpaşa-İSTANBUL
More informationMu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
More informationJohns Hopkins University Department of Biostatistics Math Review for Introductory Courses
Johs Hopks Uverst Departmet of Bostatstcs Math Revew for Itroductor Courses Ratoale Bostatstcs courses wll rel o some fudametal mathematcal relatoshps, fuctos ad otato. The purpose of ths Math Revew s
More informationEntropy ISSN by MDPI
Etropy 2003, 5, 233-238 Etropy ISSN 1099-4300 2003 by MDPI www.mdp.org/etropy O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100,
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationJohns Hopkins University Department of Biostatistics Math Review for Introductory Courses
Johs Hopks Uverst Departmet of Bostatstcs Math Revew for Itroductor Courses Ratoale Bostatstcs courses wll rel o some fudametal mathematcal relatoshps, fuctos ad otato. The purpose of ths Math Revew s
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More information9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d
9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,
More informationThe 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 informationCS473-Algorithms I. Lecture 12b. Dynamic Tables. CS 473 Lecture X 1
CS473-Algorthm I Lecture b Dyamc Table CS 473 Lecture X Why Dyamc Table? I ome applcato: We do't kow how may object wll be tored a table. We may allocate pace for a table But, later we may fd out that
More informationChapter 4 Multiple Random Variables
Revew for the prevous lecture: Theorems ad Examples: How to obta the pmf (pdf) of U = g (, Y) ad V = g (, Y) Chapter 4 Multple Radom Varables Chapter 44 Herarchcal Models ad Mxture Dstrbutos Examples:
More informationQuiz 1- Linear Regression Analysis (Based on Lectures 1-14)
Quz - Lear Regreo Aaly (Baed o Lecture -4). I the mple lear regreo model y = β + βx + ε, wth Tme: Hour Ε ε = Ε ε = ( ) 3, ( ), =,,...,, the ubaed drect leat quare etmator ˆβ ad ˆβ of β ad β repectvely,
More informationPacking of graphs with small product of sizes
Joural of Combatoral Theory, Seres B 98 (008) 4 45 www.elsever.com/locate/jctb Note Packg of graphs wth small product of szes Alexadr V. Kostochka a,b,,gexyu c, a Departmet of Mathematcs, Uversty of Illos,
More informationThe Arithmetic-Geometric mean inequality in an external formula. Yuki Seo. October 23, 2012
Sc. Math. Japocae Vol. 00, No. 0 0000, 000 000 1 The Arthmetc-Geometrc mea equalty a exteral formula Yuk Seo October 23, 2012 Abstract. The classcal Jese equalty ad ts reverse are dscussed by meas of terally
More informationHarmonic Curvatures in Lorentzian Space
BULLETIN of the Bull Malaya Math Sc Soc Secod See 7-79 MALAYSIAN MATEMATICAL SCIENCES SOCIETY amoc Cuvatue Loetza Space NEJAT EKMEKÇI ILMI ACISALIOĞLU AND KĀZIM İLARSLAN Aaa Uvety Faculty of Scece Depatmet
More informationTHE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/2008, pp
THE PUBLISHIN HOUSE PROCEEDINS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/8, THE UNITS IN Stela Corelu ANDRONESCU Uversty of Pteşt, Deartmet of Mathematcs, Târgu Vale
More informationThe Primitive Idempotents in
Iteratoal Joural of Algebra, Vol, 00, o 5, 3 - The Prmtve Idempotets FC - I Kulvr gh Departmet of Mathematcs, H College r Jwa Nagar (rsa)-5075, Ida kulvrsheora@yahoocom K Arora Departmet of Mathematcs,
More information02/15/04 INTERESTING FINITE AND INFINITE PRODUCTS FROM SIMPLE ALGEBRAIC IDENTITIES
0/5/04 ITERESTIG FIITE AD IFIITE PRODUCTS FROM SIMPLE ALGEBRAIC IDETITIES Thomas J Osler Mathematcs Departmet Rowa Uversty Glassboro J 0808 Osler@rowaedu Itroducto The dfferece of two squares, y = + y
More informationTrignometric Inequations and Fuzzy Information Theory
Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Volume, Iue, Jauary - 0, PP 00-07 ISSN 7-07X (Prt) & ISSN 7- (Ole) www.arcjoural.org Trgometrc Iequato ad Fuzzy Iformato Theory P.K. Sharma,
More informationIII-16 G. Brief Review of Grand Orthogonality Theorem and impact on Representations (Γ i ) l i = h n = number of irreducible representations.
III- G. Bref evew of Grad Orthogoalty Theorem ad mpact o epresetatos ( ) GOT: h [ () m ] [ () m ] δδ δmm ll GOT puts great restrcto o form of rreducble represetato also o umber: l h umber of rreducble
More informationX ε ) = 0, or equivalently, lim
Revew for the prevous lecture Cocepts: order statstcs Theorems: Dstrbutos of order statstcs Examples: How to get the dstrbuto of order statstcs Chapter 5 Propertes of a Radom Sample Secto 55 Covergece
More informationAssignment 5/MATH 247/Winter Due: Friday, February 19 in class (!) (answers will be posted right after class)
Assgmet 5/MATH 7/Wter 00 Due: Frday, February 9 class (!) (aswers wll be posted rght after class) As usual, there are peces of text, before the questos [], [], themselves. Recall: For the quadratc form
More informationTHE PROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION
Joural of Scece ad Arts Year 12, No. 3(2), pp. 297-32, 212 ORIGINAL AER THE ROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION DOREL MIHET 1, CLAUDIA ZAHARIA 1 Mauscrpt receved: 3.6.212; Accepted
More informationDouble Dominating Energy of Some Graphs
Iter. J. Fuzzy Mathematcal Archve Vol. 4, No., 04, -7 ISSN: 30 34 (P), 30 350 (ole) Publshed o 5 March 04 www.researchmathsc.org Iteratoal Joural of V.Kaladev ad G.Sharmla Dev P.G & Research Departmet
More informationOn the convergence of derivatives of Bernstein approximation
O the covergece of dervatves of Berste approxmato Mchael S. Floater Abstract: By dfferetatg a remader formula of Stacu, we derve both a error boud ad a asymptotc formula for the dervatves of Berste approxmato.
More informationSTRONG CONSISTENCY FOR SIMPLE LINEAR EV MODEL WITH v/ -MIXING
Joural of tatstcs: Advaces Theory ad Alcatos Volume 5, Number, 6, Pages 3- Avalable at htt://scetfcadvaces.co. DOI: htt://d.do.org/.864/jsata_7678 TRONG CONITENCY FOR IMPLE LINEAR EV MODEL WITH v/ -MIXING
More informationmeans the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.
9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc,
More informationComplete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables
Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl
More informationPart 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))
art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the
More informationA Result of Convergence about Weighted Sum for Exchangeable Random Variable Sequence in the Errors-in-Variables Model
AMSE JOURNALS-AMSE IIETA publcato-17-sere: Advace A; Vol. 54; N ; pp 3-33 Submtted Mar. 31, 17; Reved Ju. 11, 17, Accepted Ju. 18, 17 A Reult of Covergece about Weghted Sum for Exchageable Radom Varable
More informationNEW RESULTS IN TRAJECTORY-BASED SMALL-GAIN WITH APPLICATION TO THE STABILIZATION OF A CHEMOSTAT
NEW RESULTS IN TRAJECTORY-BASED SMALL-GAIN WITH APPLICATION TO THE STABILIZATION OF A CHEMOSTAT Iao Karafyll * ad Zhog-Pg Jag ** * Dept. of Evrometal Eg., Techcal Uverty of Crete, 73, Chaa, Greece, emal:
More informationHypersurfaces with Constant Scalar Curvature in a Hyperbolic Space Form
Hypersurfaces wth Costat Scalar Curvature a Hyperbolc Space Form Lu Xm ad Su Wehog Abstract Let M be a complete hypersurface wth costat ormalzed scalar curvature R a hyperbolc space form H +1. We prove
More informationCOMPUTERISED ALGEBRA USED TO CALCULATE X n COST AND SOME COSTS FROM CONVERSIONS OF P-BASE SYSTEM WITH REFERENCES OF P-ADIC NUMBERS FROM
U.P.B. Sc. Bull., Seres A, Vol. 68, No. 3, 6 COMPUTERISED ALGEBRA USED TO CALCULATE X COST AND SOME COSTS FROM CONVERSIONS OF P-BASE SYSTEM WITH REFERENCES OF P-ADIC NUMBERS FROM Z AND Q C.A. MURESAN Autorul
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationLecture 07: Poles and Zeros
Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationExtend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables
ppled Mathematcal Sceces, Vol 4, 00, o 3, 637-64 xted the Borel-Catell Lemma to Sequeces of No-Idepedet Radom Varables olah Der Departmet of Statstc, Scece ad Research Campus zad Uversty of Tehra-Ira der53@gmalcom
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More informationHarley Flanders Differential Forms with Applications to the Physical Sciences. Dover, 1989 (1962) Contents FOREWORD
Harley Fladers Dfferetal Forms wth Applcatos to the Physcal Sceces FORWORD Dover, 989 (962) Cotets PRFAC TO TH DOVR DITION PRFAC TO TH FIRST DITION.. xteror Dfferetal Forms.2. Comparso wth Tesors 2.. The
More informationChapter 5 Properties of a Random Sample
Lecture 6 o BST 63: Statstcal Theory I Ku Zhag, /0/008 Revew for the prevous lecture Cocepts: t-dstrbuto, F-dstrbuto Theorems: Dstrbutos of sample mea ad sample varace, relatoshp betwee sample mea ad sample
More informationOn the construction of symmetric nonnegative matrix with prescribed Ritz values
Joural of Lear ad Topologcal Algebra Vol. 3, No., 14, 61-66 O the costructo of symmetrc oegatve matrx wth prescrbed Rtz values A. M. Nazar a, E. Afshar b a Departmet of Mathematcs, Arak Uversty, P.O. Box
More informationJournal of Mathematical Analysis and Applications
J. Math. Aal. Appl. 365 200) 358 362 Cotets lsts avalable at SceceDrect Joural of Mathematcal Aalyss ad Applcatos www.elsever.com/locate/maa Asymptotc behavor of termedate pots the dfferetal mea value
More information18.413: Error Correcting Codes Lab March 2, Lecture 8
18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationChapter 9 Jordan Block Matrices
Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.
More information8 The independence problem
Noparam Stat 46/55 Jame Kwo 8 The depedece problem 8.. Example (Tua qualty) ## Hollader & Wolfe (973), p. 87f. ## Aemet of tua qualty. We compare the Huter L meaure of ## lghte to the average of coumer
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postpoed exam: ECON430 Statstcs Date of exam: Jauary 0, 0 Tme for exam: 09:00 a.m. :00 oo The problem set covers 5 pages Resources allowed: All wrtte ad prted
More informationMA 524 Homework 6 Solutions
MA 524 Homework 6 Solutos. Sce S(, s the umber of ways to partto [] to k oempty blocks, ad c(, s the umber of ways to partto to k oempty blocks ad also the arrage each block to a cycle, we must have S(,
More informationEuropean Journal of Mathematics and Computer Science Vol. 5 No. 2, 2018 ISSN
Europea Joural of Mathematc ad Computer Scece Vol. 5 o., 018 ISS 059-9951 APPLICATIO OF ASYMPTOTIC DISTRIBUTIO OF MA-HITEY STATISTIC TO DETERMIE THE DIFFERECE BETEE THE SYSTOLIC BLOOD PRESSURE OF ME AD
More informationBeam Warming Second-Order Upwind Method
Beam Warmg Secod-Order Upwd Method Petr Valeta Jauary 6, 015 Ths documet s a part of the assessmet work for the subject 1DRP Dfferetal Equatos o Computer lectured o FNSPE CTU Prague. Abstract Ths documet
More informationMinimal Surfaces and Gauss Curvature of Conoid in Finsler Spaces with (α, β)-metrics *
Advace Pre Mathematc -5 http://ddoorg/6/apm Plhed Ole Jly (http://wwwscrporg/joral/apm) Mmal Srface ad Ga Crvatre of Cood Fler Space wth (α β)-metrc Dghe Xe Q He Departmet of Mathematc Togj Uverty Shagha
More informationChapter 2 - Free Vibration of Multi-Degree-of-Freedom Systems - II
CEE49b Chapter - Free Vbrato of Mult-Degree-of-Freedom Systems - II We ca obta a approxmate soluto to the fudametal atural frequecy through a approxmate formula developed usg eergy prcples by Lord Raylegh
More informationTHE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 6, Number 1/2005, pp
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROANIAN ACADEY, Sere A, OF THE ROANIAN ACADEY Volume 6, Number /005,. 000-000 ON THE TRANSCENDENCE OF THE TRACE FUNCTION Vctor ALEXANDRU Faculty o athematc, Uverty
More informationIdeal multigrades with trigonometric coefficients
Ideal multgrades wth trgoometrc coeffcets Zarathustra Brady December 13, 010 1 The problem A (, k) multgrade s defed as a par of dstct sets of tegers such that (a 1,..., a ; b 1,..., b ) a j = =1 for all
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationDimensionality Reduction and Learning
CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that
More informationAitken delta-squared generalized Juncgk-type iterative procedure
Atke delta-squared geeralzed Jucgk-type teratve procedure M. De la Se Isttute of Research ad Developmet of Processes. Uversty of Basque Coutry Campus of Leoa (Bzkaa) PO Box. 644- Blbao, 488- Blbao. SPAIN
More informationFactorization of Finite Abelian Groups
Iteratoal Joural of Algebra, Vol 6, 0, o 3, 0-07 Factorzato of Fte Abela Grous Khald Am Uversty of Bahra Deartmet of Mathematcs PO Box 3038 Sakhr, Bahra kamee@uobedubh Abstract If G s a fte abela grou
More informationSimulation Output Analysis
Smulato Output Aalyss Summary Examples Parameter Estmato Sample Mea ad Varace Pot ad Iterval Estmato ermatg ad o-ermatg Smulato Mea Square Errors Example: Sgle Server Queueg System x(t) S 4 S 4 S 3 S 5
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationThe Lie Algebra of Smooth Sections of a T-bundle
IST Iteratoal Joural of Egeerg Scece, Vol 7, No3-4, 6, Page 8-85 The Le Algera of Smooth Sectos of a T-udle Nadafhah ad H R Salm oghaddam Astract: I ths artcle, we geeralze the cocept of the Le algera
More informationAbout k-perfect numbers
DOI: 0.47/auom-04-0005 A. Şt. Uv. Ovdus Costaţa Vol.,04, 45 50 About k-perfect umbers Mhály Becze Abstract ABSTRACT. I ths paper we preset some results about k-perfect umbers, ad geeralze two equaltes
More informationSemi-Riemann Metric on. the Tangent Bundle and its Index
t J Cotem Math Sceces ol 5 o 3 33-44 Sem-Rema Metrc o the Taet Budle ad ts dex smet Ayha Pamuale Uversty Educato Faculty Dezl Turey ayha@auedutr Erol asar Mers Uversty Art ad Scece Faculty 33343 Mers Turey
More informationA unified matrix representation for degree reduction of Bézier curves
Computer Aded Geometrc Desg 21 2004 151 164 wwwelsevercom/locate/cagd A ufed matrx represetato for degree reducto of Bézer curves Hask Suwoo a,,1, Namyog Lee b a Departmet of Mathematcs, Kokuk Uversty,
More informationE be a set of parameters. A pair FE, is called a soft. A and GB, over X is the soft set HC,, and GB, over X is the soft set HC,, where.
The Exteso of Sgular Homology o the Category of Soft Topologcal Spaces Sad Bayramov Leoard Mdzarshvl Cgdem Guduz (Aras) Departmet of Mathematcs Kafkas Uversty Kars 3600-Turkey Departmet of Mathematcs Georga
More informationLecture 4 Sep 9, 2015
CS 388R: Radomzed Algorthms Fall 205 Prof. Erc Prce Lecture 4 Sep 9, 205 Scrbe: Xagru Huag & Chad Voegele Overvew I prevous lectures, we troduced some basc probablty, the Cheroff boud, the coupo collector
More informationCIS 800/002 The Algorithmic Foundations of Data Privacy October 13, Lecture 9. Database Update Algorithms: Multiplicative Weights
CIS 800/002 The Algorthmc Foudatos of Data Prvacy October 13, 2011 Lecturer: Aaro Roth Lecture 9 Scrbe: Aaro Roth Database Update Algorthms: Multplcatve Weghts We ll recall aga) some deftos from last tme:
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More informationSome properties of symmetry classes of tensors
The d Aual Meetg Mathematcs (AMM 07) Departmet of Mathematcs, Faculty of Scece Chag Ma Uversty, Chag Ma Thalad Some propertes of symmetry classes of tesors Kulathda Chmla, ad Kjt Rodtes Departmet of Mathematcs,
More informationScheduling Jobs with a Common Due Date via Cooperative Game Theory
Amerca Joural of Operato Reearch, 203, 3, 439-443 http://dx.do.org/0.4236/ajor.203.35042 Publhed Ole eptember 203 (http://www.crp.org/joural/ajor) chedulg Job wth a Commo Due Date va Cooperatve Game Theory
More information