Convergence Rates of Density Estimation in Besov Spaces

Size: px
Start display at page:

Download "Convergence Rates of Density Estimation in Besov Spaces"

Transcription

1 Aed Mthemtc 58-6 do:436/m75 Pubhed Oe October (htt://wwwscrpor/our/m) Coverece Rte of Dety Etmto Beov Sce Abtrct Huy W Dertmet of Aed Mthemtc Be Uverty of Techooy Be Ch E-m: b865@embuteduc Receved Juy 9 ; reved Auut 3 ; cceted Auut 3 The otmty of dety etmto o Beov ce B R for the r w etbhed by Dooho Johtoe Kerychr d Pcrd ( Dety etmto by wveet threhod The A of Stttc Vo 4 No ) To how the ower boud of otm rte of coverece R Br they ue Koroteev d Aoud emm However the codto of thoe two emm re dffcut to be verfed Th er m to ve other roof for tht boud by u Fo Lemm whch oo tte mer I ddto our method c be ued my other tttc mode for ower boud of etmto Keyword: Otm Rte of Coverece Dety Etmto Beov Sce Wveet r L Itroducto Wveet y h my cto oe of whch to etmte uow dety fucto bed o deedet d detcy dtrbuted (d) rdom me Let ( P) be robbty meurbe ce d X X be d rdom vrbe wth uow dety fucto f We ue EX to deote the exectto of rdom vrbe X The euece R V : f u E f f ced otm rte f fv of coverece o the fucto c V for the L r Here f rbtrry etmtor of f wth d rdom me Kerychr d Pcrd [] tudy R V whe V Beov ce wth mtched ce Dooho Johtoe Kerychr d Pcrd [] coder umtched ce I fct they how the otm coverece rte for L r Beov c B r d L r / r/ / r r RB r () r To how the ower boud of () uthor of [3] ue Koroteev d Aoud emm However the codto of thoe two emm re dffcut to be verfed I th m er we ve other roof for the ower boud of () by u Fo emm [4] It houd be oted out tht Fo emm c be ued to vrety of tttc mode ee [5-7] A uu L R deote the cc Lebeue ce o the re e R I rtcur L R td for the Hbert ce whch cot of ure terbe fucto A ubce of L R the Soboev ce wth teer exoet me m W R : f f L R m The correod orm f : f f W Moreover the Beov ce B R [3] ( d (]) c be defed by B R f W R f wth the octed orm where B Z ( ) Z f : f f W f t f x h f xh f x : u ht I eer t c be how tht comcty uorted d B R for tme dfferetbe fucto beo to Coyrht ScRe

2 H Y WANG 59 d The Beov ce c be dcretzed by the euece orm of wveet coeffcet My uefu wveet re eerted by c fucto More recey f c fucto wth x h x the : x h x defe wveet [3] Cery whe comcty uorted d cotuou the correod wveet h the me roerte A orthoorm wveet b of L R eerted from dto d trto of c fucto d t correod wveet e x x : : x x Z Athouh wveet b re cotructed for L R mot of them cottute ucodto be for L R A c fucto ced t reur f h cotuou dervtve of order t d t correod wveet h vh momet of order t e x x d x t The foow emm [3] y mortt roe th er Lemm Let be comcty uorted t reur orthoorm c fucto wth the correod wveet d t If f L R : f d : f d the the foow two codto re euvet: ) f B R ; ) Furthermore d f d B Before troduc Fo Lemm we eed the otto of Kubc-Leber dtce [4] Let P d Q wth P be boutey cotuou wth reect to Q (deoted by P Q) The the Kubc-Leber dtce defed by x K PQ : x dx x Where d re dety fucto of P Q reectvey Lemm (Fo Lemm [4]) Let ( P ) be robbty meurbe ce d A m If A Av for v the wth A c td for the comemet of A d m : f K P Pv vmm u P m v ex 3 c A m e m m By Lemm d we c how the foow reut: Theorem Let f B r R L wth r d r If f etmtor of f wth d rdom me the / r/ / r u E f f mx fb r R L where Br R L : f Br R f L d f B r h comct uort}; The otto x y me x Cy wth cott C Remr Note tht for mx r / r/ / r/ / r / r d for r / r/ / r mx The theorem reformute of the ower boud () By u the de of referece [5] we how th theorem the ext two ecto Proof of Theorem Frty we rove u f B R L E f f / r/ / r r uch tht B r R L Oe eed cotruct d u E f / r/ / r Let be comcty uorted t t reur d orthoorm c fucto be the correod wv et wth u N Here d fter e Coyrht ScRe

3 6 H Y WANG N deote the et of otve teer The there ext comcty uorted dety fucto (e x dx ) tfy x d x Br R d x c Let : The the umber of eemet deoted by # / / Motvted by [ 5] oe defe : r d wth I : Obv- ouy x : x x I : f I ee x x d d / r xc for re whch me tht dety fucto for ech By the umto of the wveet comcty uorted d t tme dfferetbe Therefore B r Rt d B r R Bec ue // r C d o Br due to Lemm Hece x B r For / / r : B r R L Cery // r () due to : Fur- thermore A : f tfe A A for Rec tht # By Lemm c 3 u P A m ex Here d e fter P f td for the robbty meure correod- to the dety fucto f x: f x f x f x It ey to ee tht P P from the cotructo of Sce f etmtor of dety wth d rdom me c E f P f P A The c u E f u P A Next oe how 3 m ex e () c : Rec tht f x x f f f f x d f x fx f x K P P : f x dx The x x f x d f x K P P f x x K P P f K P P f x x d : d u u f Note tht for Hece u The f x f x d f x f x K P P f x dx f x x f x f x f x d x : f K P P K P P Moreover v v v x x x dx (3) Accord to the defto of u d x c o Thu x x x dx c x dx c x c by the orthoormty of The (3) reduce to / r c (4) r Te The Now oe c chooe C uch tht C d C4 r c Therefore 4/ r c Cc e e d () reduce to the dered foow from d / r Now we rove E f u C u fb r The / / r by () R L roof deed o other emm [4] E f f Our Coyrht ScRe

4 H Y WANG 6 Lemm (Vrhmov-Gbert) Let : m The there ext ubet M of wth uch tht m m/8 m M d M 8 It uffcet to cotruct M uch tht B R L d r u E f (5) A roved b ove et be comcty uorted tt reur d orthoorm c fuct o be the correod wveet wth u N Aume B r R L d [ ] c Defe / : : d x: x x wth (ote tht ) Sce r oe ow tht d By Lemm B r / r / / r r Hece B R L r Br C d o Note th t the uort of for re mutuy dot The x c c me for b Th wth x d x x d x tht dety fucto for ech Accord to Lemm there ext uch M 3 tht M d 3 (6) Becue u u for oe ow tht Th wth (6) d 3 d ed to (7) / 8 : Cery the et A f M tfy A A for The Fo Lemm yed M c M u P A m M ex 3 e (8) O the other hd t foow M c from the mr rumet to the roof of (4) Te The ( ) chooe cott C uch tht Hece oe c 4 4 c c C M Me e e c Therefore (8) reduce to u M P A C d u f ME u P C f M Th wth (7) d 3 Acowedemet yed (5) The uthor Huy W rtefu to the referee for ther vube commet d th her dvor Profeor Youm Lu for h hefu udce Th wor uorted by the Nto Ntur Scece Foudto of Ch (No 87) d Ntur Scece Foudto of Be (No 83) 4 Referece [] G Kerychr d D Pcrd Dety Etmto Beov Sce Stttc & Probbty Letter Vo 3 No do:6/67-75(9)93-s [] D L Dooho I M Johtoe G Kerychr d D Pcrd Dety Etmto by Wveet Threhod The A of Stttc Vo 4 No do:4/o/ [3] W Härde G Kerychr D Pcrd d A B Ty- Aroxmto d Stttc A- bov Wveet cto Srer-Ver New Yor 997 [4] A B Tybov Itroducto to Normetrc Etmto (Eh) Reved d Exteded from the 4 Frech Or Trted by Vdmr Zt Srer Sere Stttc Srer New Yor 9 [5] P Bd G Kerychr D Mrucc d D Pcrd Adtve Dety Etmto for Drecto Dt U Needet The A of Stttc Vo 37 No 6A Coyrht ScRe

5 6 H Y WANG do:4/ 9-AOS68 [6] C Chrtohe Rereo wth Rdom De: A Mmx Study Stttc & Probbty Letter Vo 77 No do:6/6 5 [7] A B Tybov Otm Rte of Areto COLT/ Kere 3 Lecture Note Artfc Iteece 777 Srer Hedeber Coyrht ScRe

European Journal of Mathematics and Computer Science Vol. 3 No. 1, 2016 ISSN ISSN

European Journal of Mathematics and Computer Science Vol. 3 No. 1, 2016 ISSN ISSN Euroe Jour of Mthemtcs d omuter Scece Vo. No. 6 ISSN 59-995 ISSN 59-995 ON AN INVESTIGATION O THE MATRIX O THE SEOND PARTIA DERIVATIVE IN ONE EONOMI DYNAMIS MODE S. I. Hmdov Bu Stte Uverst ABSTRAT The

More information

The Computation of Common Infinity-norm Lyapunov Functions for Linear Switched Systems

The Computation of Common Infinity-norm Lyapunov Functions for Linear Switched Systems ISS 746-7659 Egd UK Jour of Iformto d Comutg Scece Vo. 6 o. 4. 6-68 The Comutto of Commo Ifty-orm yuov Fuctos for er Swtched Systems Zheg Che Y Go Busess Schoo Uversty of Shgh for Scece d Techoogy Shgh

More information

3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS

3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS . REVIEW OF PROPERTIES OF EIGENVLUES ND EIGENVECTORS. EIGENVLUES ND EIGENVECTORS We hll ow revew ome bc fct from mtr theory. Let be mtr. clr clled egevlue of f there et ozero vector uch tht Emle: Let 9

More information

THE 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 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 information

Further Results on Pair Sum Labeling of Trees

Further Results on Pair Sum Labeling of Trees Appled Mathematcs 0 70-7 do:046/am0077 Publshed Ole October 0 (http://wwwscrporg/joural/am) Further Results o Par Sum Labelg of Trees Abstract Raja Poraj Jeyaraj Vjaya Xaver Parthpa Departmet of Mathematcs

More information

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n

10.2 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 information

International Journal of Pure and Applied Sciences and Technology

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 information

Lecture 4 Sep 9, 2015

Lecture 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 information

Abstract. 1. Introduction

Abstract. 1. Introduction Joura of Mathematca Sceces: Advaces ad Appcatos Voume 4 umber 2 2 Pages 33-34 COVERGECE OF HE PROJECO YPE SHKAWA ERAO PROCESS WH ERRORS FOR A FE FAMY OF OSEF -ASYMPOCAY QUAS-OEXPASVE MAPPGS HUA QU ad S-SHEG

More information

Complete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables

Complete 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 information

A FUZZY MULTI-ATTRIBUTE DECISION MAKING ALGORITM BASED ON INTUITIONISTIC FUZZY SETS

A FUZZY MULTI-ATTRIBUTE DECISION MAKING ALGORITM BASED ON INTUITIONISTIC FUZZY SETS FUZZY MULTI-TTRIBUTE DECISION MKING LGORITM BSED ON INTUITIONISTIC FUZZY SETS I Crme Bărbăcor, Cott Brâcş Uverty, Tg. J btrct: The rooed method ebe deco mker to chooe the mot mortt of the crter mkg deco

More information

Available online through

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

More information

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

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

More information

INEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS

INEQUALITIES 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 information

Order Statistics from Exponentiated Gamma. Distribution and Associated Inference

Order Statistics from Exponentiated Gamma. Distribution and Associated Inference It J otm Mth Scc Vo 4 9 o 7-9 Od Stttc fom Eottd Gmm Dtto d Aoctd Ifc A I Shw * d R A Bo G og of Edcto PO Bo 369 Jddh 438 Sd A G og of Edcto Dtmt of mthmtc PO Bo 469 Jddh 49 Sd A Atct Od tttc fom ottd

More information

Continuous Random Variables: Conditioning, Expectation and Independence

Continuous Random Variables: Conditioning, Expectation and Independence Cotuous Radom Varables: Codtog, xectato ad Ideedece Berl Che Deartmet o Comuter cece & Iormato geerg atoal Tawa ormal Uverst Reerece: - D.. Bertsekas, J.. Tstskls, Itroducto to robablt, ectos 3.4-3.5 Codtog

More information

D KL (P Q) := p i ln p i q i

D KL (P Q) := p i ln p i q i Cheroff-Bouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The Kullback-Lebler dvergece or relatve etroy of P ad Q s defed as m D KL

More information

= lim. (x 1 x 2... x n ) 1 n. = log. x i. = M, n

= lim. (x 1 x 2... x n ) 1 n. = log. x i. = M, n .. Soluto of Problem. M s obvously cotuous o ], [ ad ], [. Observe that M x,..., x ) M x,..., x ) )..) We ext show that M s odecreasg o ], [. Of course.) mles that M s odecreasg o ], [ as well. To show

More information

On the Rational Valued Characters Table of the

On the Rational Valued Characters Table of the Aled Mathematcal Sceces, Vol., 7, o. 9, 95-9 HIKARI Ltd, www.m-hkar.com htts://do.or/.9/ams.7.7576 O the Ratoal Valued Characters Table of the Grou (Q m C Whe m s a Eve Number Raaa Hassa Abass Deartmet

More information

STRONG CONSISTENCY FOR SIMPLE LINEAR EV MODEL WITH v/ -MIXING

STRONG 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 information

Packing of graphs with small product of sizes

Packing 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 information

Lecture 3 Probability review (cont d)

Lecture 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 information

Hypercyclic Functions for Backward and Bilateral Shift Operators. Faculty of Science, Ain Shams University, Cairo, Egypt 2 Department of Mathematics,

Hypercyclic Functions for Backward and Bilateral Shift Operators. Faculty of Science, Ain Shams University, Cairo, Egypt 2 Department of Mathematics, Jourl of themtcs d Sttstcs 5 (3):78-82, 29 ISSN 549-3644 29 Scece ublctos Hyercyclc Fuctos for Bcwrd d Blterl Shft Oertors N Fred, 2 ZA Hss d 3 A orsy Dertmet of themtcs, Fculty of Scece, A Shms Uversty,

More information

The Mathematical Appendix

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

More information

Semi-Riemann Metric on. the Tangent Bundle and its Index

Semi-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 information

A Family of Non-Self Maps Satisfying i -Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces *

A Family of Non-Self Maps Satisfying i -Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces * Advaces Pure Matheatcs 0 80-84 htt://dxdoorg/0436/a04036 Publshed Ole July 0 (htt://wwwscrporg/oural/a) A Faly of No-Self Mas Satsfyg -Cotractve Codto ad Havg Uque Coo Fxed Pot Metrcally Covex Saces *

More information

Complex Variables. Chapter 19 Series and Residues. March 26, 2013 Lecturer: Shih-Yuan Chen

Complex Variables. Chapter 19 Series and Residues. March 26, 2013 Lecturer: Shih-Yuan Chen omplex Vrble hpter 9 Sere d Redue Mrch 6, Lecturer: Shh-Yu he Except where otherwe oted, cotet lceed uder BY-N-SA. TW Lcee. otet Sequece & ere Tylor ere Luret ere Zero & pole Redue & redue theorem Evluto

More information

STA261H1.doc. i 1 X n be a random sample. The sample mean is defined by i= 1 X 1 + ( ) X has a N ( σ ) 1 n. N distribution. Then n. distribution.

STA261H1.doc. i 1 X n be a random sample. The sample mean is defined by i= 1 X 1 + ( ) X has a N ( σ ) 1 n. N distribution. Then n. distribution. TA6Hdoc tttcl Iferece RADOM AMPLE Defto: Rdom mple clled rdom mple from dtruto wth pdf f ( (or pf ( depedet d hve detcl dtruto wth pdf f (or pf P (depedet-detcll-dtruted P f re It ofte deoted d ote Let

More information

SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE. Sayali S. Joshi

SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE. Sayali S. Joshi Faculty of Sceces ad Matheatcs, Uversty of Nš, Serba Avalable at: http://wwwpfacyu/float Float 3:3 (009), 303 309 DOI:098/FIL0903303J SUBCLASS OF ARMONIC UNIVALENT FUNCTIONS ASSOCIATED WIT SALAGEAN DERIVATIVE

More information

MATH 247/Winter Notes on the adjoint and on normal operators.

MATH 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 information

Research Article A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings

Research 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 information

Chapter 5 Properties of a Random Sample

Chapter 5 Properties of a Random Sample Lecture 3 o BST 63: Statstcal Theory I Ku Zhag, /6/006 Revew for the revous lecture Cocets: radom samle, samle mea, samle varace Theorems: roertes of a radom samle, samle mea, samle varace Examles: how

More information

Theory study about quarter-wave-stack dielectric mirrors

Theory study about quarter-wave-stack dielectric mirrors Theor tud about quarter-wave-tack delectrc rror Stratfed edu tratted reflected reflected Stratfed edu tratted cdet cdet T T Frt, coder a wave roagato a tratfed edu. A we kow, a arbtrarl olared lae wave

More information

Non-uniform Turán-type problems

Non-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 information

Collapsing to Sample and Remainder Means. Ed Stanek. In order to collapse the expanded random variables to weighted sample and remainder

Collapsing 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 information

. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1)

. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1) Chapter 7 Fuctos o Bouded Varato. Subject: Real Aalyss Level: M.Sc. Source: Syed Gul Shah (Charma, Departmet o Mathematcs, US Sargodha Collected & Composed by: Atq ur Rehma (atq@mathcty.org, http://www.mathcty.org

More information

Algorithms Theory, Solution for Assignment 2

Algorithms Theory, Solution for Assignment 2 Juor-Prof. Dr. Robert Elsässer, Marco Muñz, Phllp Hedegger WS 2009/200 Algorthms Theory, Soluto for Assgmet 2 http://lak.formatk.u-freburg.de/lak_teachg/ws09_0/algo090.php Exercse 2. - Fast Fourer Trasform

More information

Unique Common Fixed Point of Sequences of Mappings in G-Metric Space M. Akram *, Nosheen

Unique Common Fixed Point of Sequences of Mappings in G-Metric Space M. Akram *, Nosheen Vol No : Joural of Facult of Egeerg & echolog JFE Pages 9- Uque Coo Fed Pot of Sequeces of Mags -Metrc Sace M. Ara * Noshee * Deartet of Matheatcs C Uverst Lahore Pasta. Eal: ara7@ahoo.co Deartet of Matheatcs

More information

CS473-Algorithms I. Lecture 12b. Dynamic Tables. CS 473 Lecture X 1

CS473-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 information

On Several Inequalities Deduced Using a Power Series Approach

On Several Inequalities Deduced Using a Power Series Approach It J Cotemp Mth Sceces, Vol 8, 203, o 8, 855-864 HIKARI Ltd, wwwm-hrcom http://dxdoorg/02988/jcms2033896 O Severl Iequltes Deduced Usg Power Seres Approch Lored Curdru Deprtmet of Mthemtcs Poltehc Uversty

More information

Random Variables. ECE 313 Probability with Engineering Applications Lecture 8 Professor Ravi K. Iyer University of Illinois

Random Variables. ECE 313 Probability with Engineering Applications Lecture 8 Professor Ravi K. Iyer University of Illinois Radom Varables ECE 313 Probablty wth Egeerg Alcatos Lecture 8 Professor Rav K. Iyer Uversty of Illos Iyer - Lecture 8 ECE 313 Fall 013 Today s Tocs Revew o Radom Varables Cumulatve Dstrbuto Fucto (CDF

More information

Chapter 5 Properties of a Random Sample

Chapter 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 information

NONDIFFERENTIABLE MATHEMATICAL PROGRAMS. OPTIMALITY AND HIGHER-ORDER DUALITY RESULTS

NONDIFFERENTIABLE MATHEMATICAL PROGRAMS. OPTIMALITY AND HIGHER-ORDER DUALITY RESULTS HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, See A, OF HE ROMANIAN ACADEMY Volue 9, Nube 3/8,. NONDIFFERENIABLE MAHEMAICAL PROGRAMS. OPIMALIY AND HIGHER-ORDER DUALIY RESULS Vale PREDA Uvety

More information

Unit 9. The Tangent Bundle

Unit 9. The Tangent Bundle Ut 9. The Taget Budle ========================================================================================== ---------- The taget sace of a submafold of R, detfcato of taget vectors wth dervatos at

More information

On Bounded Second Variation

On Bounded Second Variation Advces Pure Mthemtcs, 0,, -6 htt://dxdoor/0436/m0005 Pulshed Ole Jury 0 (htt://wwwscrpor/ourl/m) O Bouded Secod Vrto José Gméez, Lore Lóez, N Meretes, J L Sáchez Dertmeto de Mtemátcs, Uversdd de los Ades,

More information

X ε ) = 0, or equivalently, lim

X 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 information

Linear Open Loop Systems

Linear Open Loop Systems Colordo School of Me CHEN43 Trfer Fucto Ler Ope Loop Sytem Ler Ope Loop Sytem... Trfer Fucto for Smple Proce... Exmple Trfer Fucto Mercury Thermometer... 2 Derblty of Devto Vrble... 3 Trfer Fucto for Proce

More information

Methods for solving the radiative transfer equation. Part 3: Discreteordinate. 1. Discrete-ordinate method for the case of isotropic scattering.

Methods for solving the radiative transfer equation. Part 3: Discreteordinate. 1. Discrete-ordinate method for the case of isotropic scattering. ecture Metos for sov te rtve trsfer equto. rt 3: Dscreteorte eto. Obectves:. Dscrete-orte eto for te cse of sotropc sctter..geerzto of te screte-orte eto for ooeeous tospere. 3. uerc peetto of te screte-orte

More information

On a class of analytic functions defined by Ruscheweyh derivative

On a class of analytic functions defined by Ruscheweyh derivative Lfe Scece Jourl ;9( http://wwwlfescecestecom O clss of lytc fuctos defed by Ruscheweyh dervtve S N Ml M Arf K I Noor 3 d M Rz Deprtmet of Mthemtcs GC Uversty Fslbd Pujb Pst Deprtmet of Mthemtcs Abdul Wl

More information

PROJECTION PROBLEM FOR REGULAR POLYGONS

PROJECTION 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 information

A Remark on the Uniform Convergence of Some Sequences of Functions

A Remark on the Uniform Convergence of Some Sequences of Functions Advaces Pure Mathematcs 05 5 57-533 Publshed Ole July 05 ScRes. http://www.scrp.org/joural/apm http://dx.do.org/0.436/apm.05.59048 A Remark o the Uform Covergece of Some Sequeces of Fuctos Guy Degla Isttut

More information

Strong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity

Strong 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 information

Union, Intersection, Product and Direct Product of Prime Ideals

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

More information

ρ < 1 be five real numbers. The

ρ < 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 information

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

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

More information

POWERS OF COMPLEX PERSYMMETRIC ANTI-TRIDIAGONAL MATRICES WITH CONSTANT ANTI-DIAGONALS

POWERS OF COMPLEX PERSYMMETRIC ANTI-TRIDIAGONAL MATRICES WITH CONSTANT ANTI-DIAGONALS IRRS 9 y 04 wwwrppresscom/volumes/vol9issue/irrs_9 05pdf OWERS OF COLE ERSERIC I-RIIGOL RICES WIH COS I-IGOLS Wg usu * Q e Wg Hbo & ue College of Scece versty of Shgh for Scece d echology Shgh Ch 00093

More information

Chapter #3 EEE Subsea Control and Communication Systems

Chapter #3 EEE Subsea Control and Communication Systems EEE 87 Chter #3 EEE 87 Sube Cotrol d Commuictio Sytem Cloed loo ytem Stedy tte error PID cotrol Other cotroller Chter 3 /3 EEE 87 Itroductio The geerl form for CL ytem: C R ', where ' c ' H or Oe Loo (OL)

More information

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

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

More information

Chapter 4 Multiple Random Variables

Chapter 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 information

On Eccentricity Sum Eigenvalue and Eccentricity Sum Energy of a Graph

On Eccentricity Sum Eigenvalue and Eccentricity Sum Energy of a Graph Aals of Pure ad Appled Mathematcs Vol. 3, No., 7, -3 ISSN: 79-87X (P, 79-888(ole Publshed o 3 March 7 www.researchmathsc.org DOI: http://dx.do.org/.7/apam.3a Aals of O Eccetrcty Sum Egealue ad Eccetrcty

More information

Fibonacci Identities as Binomial Sums

Fibonacci Identities as Binomial Sums It. J. Cotemp. Math. Sceces, Vol. 7, 1, o. 38, 1871-1876 Fboacc Idettes as Bomal Sums Mohammad K. Azara Departmet of Mathematcs, Uversty of Evasvlle 18 Lcol Aveue, Evasvlle, IN 477, USA E-mal: azara@evasvlle.edu

More information

On Signed Product Cordial Labeling

On Signed Product Cordial Labeling Appled Mathematcs 55-53 do:.436/am..6 Publshed Ole December (http://www.scrp.or/joural/am) O Sed Product Cordal Label Abstract Jayapal Baskar Babujee Shobaa Loaatha Departmet o Mathematcs Aa Uversty Chea

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed:

More information

THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/2008, pp

THE 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 information

Problem Set 3: Model Solutions

Problem Set 3: Model Solutions Ecoomc 73 Adaced Mcroecoomc Problem et 3: Model oluto. Coder a -bdder aucto wth aluato deedetly ad detcally dtrbuted accordg to F( ) o uort [,]. Let the hghet bdder ay the rce ( - k)b f + kb to the eller,

More information

KR20 & Coefficient Alpha Their equivalence for binary scored items

KR20 & Coefficient Alpha Their equivalence for binary scored items KR0 & Coeffcet Alpha Ther equvalece for bary cored tem Jue, 007 http://www.pbarrett.et/techpaper/r0.pdf f of 7 Iteral Cotecy Relablty for Dchotomou Item KR 0 & Alpha There apparet cofuo wth ome dvdual

More information

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY 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 information

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

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

More information

4 Inner Product Spaces

4 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 information

Integration by Parts for D K

Integration by Parts for D K Itertol OPEN ACCESS Jourl Of Moder Egeerg Reserc IJMER Itegrto y Prts for D K Itegrl T K Gr, S Ry 2 Deprtmet of Mtemtcs, Rgutpur College, Rgutpur-72333, Purul, West Begl, Id 2 Deprtmet of Mtemtcs, Ss Bv,

More information

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

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

More information

Stats & Summary

Stats & Summary Stts 443.3 & 85.3 Summr The Woodbur Theorem BCD B C D B D where the verses C C D B, d est. Block Mtrces Let the m mtr m q q m be rttoed to sub-mtrces,,,, Smlrl rtto the m k mtr B B B mk m B B l kl Product

More information

NP!= P. By Liu Ran. Table of Contents. The P versus NP problem is a major unsolved problem in computer

NP!= P. By Liu Ran. Table of Contents. The P versus NP problem is a major unsolved problem in computer NP!= P By Lu Ra Table of Cotets. Itroduce 2. Prelmary theorem 3. Proof 4. Expla 5. Cocluso. Itroduce The P versus NP problem s a major usolved problem computer scece. Iformally, t asks whether a computer

More information

Chapter 7. Bounds for weighted sums of Random Variables

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

More information

Extend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables

Extend 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 information

Entropy ISSN by MDPI

Entropy 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 information

Bounds for the Connective Eccentric Index

Bounds for the Connective Eccentric Index It. J. Cotemp. Math. Sceces, Vol. 7, 0, o. 44, 6-66 Bouds for the Coectve Eccetrc Idex Nlaja De Departmet of Basc Scece, Humates ad Socal Scece (Mathematcs Calcutta Isttute of Egeerg ad Maagemet Kolkata,

More information

Mu Sequences/Series Solutions National Convention 2014

Mu 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 information

On the energy of complement of regular line graphs

On 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 information

Solutions Manual for Polymer Science and Technology Third Edition

Solutions Manual for Polymer Science and Technology Third Edition Solutos ul for Polymer Scece d Techology Thrd Edto Joel R. Fred Uer Sddle Rver, NJ Bosto Idols S Frcsco New York Toroto otrel Lodo uch Prs drd Cetow Sydey Tokyo Sgore exco Cty Ths text s ssocted wth Fred/Polymer

More information

ONE APPROACH FOR THE OPTIMIZATION OF ESTIMATES CALCULATING ALGORITHMS A.A. Dokukin

ONE APPROACH FOR THE OPTIMIZATION OF ESTIMATES CALCULATING ALGORITHMS A.A. Dokukin Iero Jor "Iforo Theore & co" Vo 463 ONE PPROH FOR THE OPTIIZTION OF ETITE UTING GORITH Do rc: I h rce he ew roch for ozo of eo ccg gorh ggeed I c e ed for fdg he correc gorh of coexy he coex of gerc roch

More information

PubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS

PubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS PubH 745: REGRESSION ANALSIS REGRESSION IN MATRIX TERMS A mtr s dspl of umbers or umercl quttes ld out rectgulr rr of rows d colums. The rr, or two-w tble of umbers, could be rectgulr or squre could be

More information

PART ONE. Solutions to Exercises

PART ONE. Solutions to Exercises PART ONE Soutos to Exercses Chapter Revew of Probabty Soutos to Exercses 1. (a) Probabty dstrbuto fucto for Outcome (umber of heads) 0 1 probabty 0.5 0.50 0.5 Cumuatve probabty dstrbuto fucto for Outcome

More information

IRREDUCIBLE COVARIANT REPRESENTATIONS ASSOCIATED TO AN R-DISCRETE GROUPOID

IRREDUCIBLE COVARIANT REPRESENTATIONS ASSOCIATED TO AN R-DISCRETE GROUPOID 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

More information

MA 524 Homework 6 Solutions

MA 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 information

ON NILPOTENCY IN NONASSOCIATIVE ALGEBRAS

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

More information

Digital Design of Coefficient Diagram Method

Digital Design of Coefficient Diagram Method 9 erc Cotro Coferece Hytt Regecy Rverfrot, St Lou, MO, US Jue -, 9 h76 gt eg of Coeffcet gr Method Ö Öc, r d e trct Coeffcet gr Method CM the oe of the ot effectve cotro deg ethod terture It gve cotro

More information

Solving Constrained Flow-Shop Scheduling. Problems with Three Machines

Solving 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 information

Matricial Potentiation

Matricial Potentiation Matrcal Potetato By Ezo March* ad Mart Mates** Abstract I ths short ote we troduce the potetato of matrces of the same sze. We study some smple propertes ad some example. * Emertus Professor UNSL, Sa Lus

More information

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b

Discrete Mathematics and Probability Theory Fall 2016 Seshia and Walrand DIS 10b CS 70 Dscrete Mathematcs ad Probablty Theory Fall 206 Sesha ad Walrad DIS 0b. Wll I Get My Package? Seaky delvery guy of some compay s out delverg packages to customers. Not oly does he had a radom package

More information

Applying the condition for equilibrium to this equilibrium, we get (1) n i i =, r G and 5 i

Applying the condition for equilibrium to this equilibrium, we get (1) n i i =, r G and 5 i CHEMICAL EQUILIBRIA The Thermodyamc Equlbrum Costat Cosder a reversble reacto of the type 1 A 1 + 2 A 2 + W m A m + m+1 A m+1 + Assgg postve values to the stochometrc coeffcets o the rght had sde ad egatve

More information

Assignment 5/MATH 247/Winter Due: Friday, February 19 in class (!) (answers will be posted right after class)

Assignment 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 information

Lecture 9: Tolerant Testing

Lecture 9: Tolerant Testing Lecture 9: Tolerat Testg Dael Kae Scrbe: Sakeerth Rao Aprl 4, 07 Abstract I ths lecture we prove a quas lear lower boud o the umber of samples eeded to do tolerat testg for L dstace. Tolerat Testg We have

More information

Preliminary Examinations: Upper V Mathematics Paper 1

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

More information

Patterns of Continued Fractions with a Positive Integer as a Gap

Patterns of Continued Fractions with a Positive Integer as a Gap IOSR Jourl of Mthemtcs (IOSR-JM) e-issn: 78-578, -ISSN: 39-765X Volume, Issue 3 Ver III (My - Ju 6), PP -5 wwwosrjourlsorg Ptters of Cotued Frctos wth Postve Iteger s G A Gm, S Krth (Mthemtcs, Govermet

More information

Chapter 9 Jordan Block Matrices

Chapter 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 information

Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients

Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients Cuch tpe proes of er Re-ouve frcto dfferet equtos wth vre coeffcets Mo-H K u-cho R u-so Choe Ho Cho O* Fcut of thetcs K Su Uverst Po PRK Correspod uthor E: ro@hooco Astrct: The estece of soutos to Cuch

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll

More information

Application of Generating Functions to the Theory of Success Runs

Application of Generating Functions to the Theory of Success Runs Aled Mathematcal Sceces, Vol. 10, 2016, o. 50, 2491-2495 HIKARI Ltd, www.m-hkar.com htt://dx.do.org/10.12988/ams.2016.66197 Alcato of Geeratg Fuctos to the Theory of Success Rus B.M. Bekker, O.A. Ivaov

More information

CS 4758 Robot Kinematics. Ashutosh Saxena

CS 4758 Robot Kinematics. Ashutosh Saxena CS 4758 Rt Kemt Ahuth Se Kemt tude the mt f de e re tereted tw emt tp Frwrd Kemt (ge t pt ht u re gve: he egth f eh he ge f eh t ht u fd: he pt f pt (.e. t (,, rdte Ivere Kemt (pt t ge ht u re gve: he

More information