Integration by Parts for D K
|
|
|
- Gloria Simon
- 5 years ago
- Views:
Transcription
1 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, Vsv-Brt, Stet West Begl,Id I PRELIMINERIES Let f e rel vlued fucto defed o set E Let c, d E, c d, d 2 te oscllto of f o [ c, E of order s defed to e O f,[ c, E sup d c[ f, c, x, x2, x, were te sup s te over ll pots x,x,x o 2 - [c, E d [f,c,x,,x, represets te - t order dvded dfferece of f t te + pots c,x,,x -,d Te we vrto of f of order s defed s follows, V f, E sup O f,[ c, d ] E were te sup s te over ll sequeces {c,d } of o overlpg tervls wt ed pots o E Te f s sd to e of t vrto te wde sese f V f, E d t s wrtte s f BV E Te fucto f s sd to e t solutely cotuous o E f for y 0 tere s 0 suc tt for every sequece of o overlpg tervls c,d wt ed pots o E d wt d c we ve ABSTRACT: I ts per we ve defed D tegrl d proved te tegrto y prts formul Key Words d prses: Asolutely Cotuous fucto, Geerlsed solutely cotuous fucto, Dejoy tegrto 2000 Mtemtcs suject Clssfcto: Prmry 26A24 Secodry 26A2, 26A48, 44A0 O f,[ c, d ] E d we wrte t s f AC E Te fucto F s sd to e geerlsed solutely t cotuous resp of geerlsed ouded t vrto o E f E E were ec E s closed d f AC E resp f BV E for ec d we wrte t s f AC GE resp f BV GE II AUXILIARY RESULT Followg result wll e eeded wc re proved [2] Lemm 2 Let E e closed set Te f ACGE resp f BVGE f d oly f every closed suset of E s porto o wc f s AC respbv Lemm 22 Te clsses of fuctos AC E,BV E,AC GE,BV GE re ll ler spces Teorem 23 If f BV E of f Teorem 24 Let 2 d te f f :[, ] R e suc tt exsts ftely e o E, were f s repeted proxmte dervtve Te reserc wor of frst utor s fclly supported y UGC, MRP 2 Correspodg uter IJMER ISSN: wwwjmercom Vol 5 Iss J 205 6
2 Itegrto By Prts For D K Itegrl f ACG[, ] D[, ] f exsts [,] 2 f for r=0,,,-2 d x, oe of f r + 'x d fr - 'x exsts te fr 'x exststs codto s weer t te smootess codto of f r v f 0 e Te f - exsts d s odecresg [,] d f AC [, ] Te followg teorem s proved for =2 Teorem-4 of [2], d smlrly c e proved y Teorem 25 Let F d G e AC respbv o E Te FG s AC respbv o E Te FG s AC respbv o E Corollry 26 If F d G re AC G respbv G o [,] te FG s so [,] Te proof s smlr of te corollry of Teorem-4 of [2] A fucto :[, ] R suc tt f AC G[, ] 2 exsts [,] III THE D INTEGRAL f :[, ] R s sd to e D tegrle o [,] f tere exsts cotuous fucto r s smoot, for r=0,,,-2, v,, exsts d,, f e [,] Were d deote rgt d proxmte dervtve of t of order - d left d proxmte dervtve of t of order - Te exstece of e [,] s gurteed y Teorem-23 Te fucto f exsts s clled t prmtve of f o [,] If D tegrl of f d,, F F s clled te defte D tegrl of f over [,]d s deoted y F we cll F to e defte D ftdt Te defte D tegrl s uque y Teorem-24 Te defte D tegrl s uque upto ddtve costt d te t prmtve s uque upto ddto of polyoml of degree - Teorem 3 Let f e D tegrle o [,] d o [,c] d let d e t prmtves of f [,], d [,c] respectvely If, d exsts te f s D tegrle [,c] d c c D f+d f=d f,, Proof: Sce d exsts, te prevous dervtve exsts some left d rgt egourood of Let IJMER ISSN: wwwjmercom Vol 5 Iss J
![Itegrto By Prts For D K Itegrl Te H s cotuous [,c] We sow tt H s te t prmtve of f [,c] Clerly H AC G[, c] Also for 0 r 2, H r x Ad ece H exsts [,c] We re to sow tt 2 H s smoot t for 0 r 2 Let 0 d 0 r](/docs-images/84/90565800/images/3-0.jpg "2 Te r r r r r r r H H 2H H H H H r r, r, r r, r, O r, r, Te frst term teds to 0 So r r, r, d te secod term teds to s r H s smoot t for 0 r 2 Hece H r x s smoot o,c for 0 r 2 Te proof of oter")
![propertes of t prmtve re esy Hece H s te t prmtve of f o [,c] So f s tegrle o [,c] Also D c f H { {,,, c H D f D,,, c f c} { } {,, c,, } } Teorem 32 Let f e D tegrle [,] d let <c< Let te t prmtve of](/docs-images/84/90565800/images/3-1.jpg "f [,] If c exsts te f s D tegrle o [,c] d o [c,] d c c D f+d f=d f Te proof s mmedte Teorem 33 Let f d g e D tegrle [,] d, re costts te f g s D tegrle [,] d D f g D f D g D IJMER ISSN: 2249 6645")
3 Itegrto By Prts For D K Itegrl Te H s cotuous [,c] We sow tt H s te t prmtve of f [,c] Clerly H AC G[, c] Also for 0 r 2, H r x Ad ece H exsts [,c] We re to sow tt 2 H s smoot t for 0 r 2 Let 0 d 0 r 2 Te r r r r r r r H H 2H H H H H r r, r, r r, r, O r, r, Te frst term teds to 0 So r r, r, d te secod term teds to s r H s smoot t for 0 r 2 Hece H r x s smoot o,c for 0 r 2 Te proof of oter propertes of t prmtve re esy Hece H s te t prmtve of f o [,c] So f s tegrle o [,c] Also D c f H { {,,, c H D f D,,, c f c} { } {,, c,, } } Teorem 32 Let f e D tegrle [,] d let <c< Let te t prmtve of f [,] If c exsts te f s D tegrle o [,c] d o [c,] d c c D f+d f=d f Te proof s mmedte Teorem 33 Let f d g e D tegrle [,] d, re costts te f g s D tegrle [,] d D f g D f D g D IJMER ISSN: wwwjmercom Vol 5 Iss J
4 Itegrto By Prts For D K Itegrl Te proof follows from te defto of te tegrl d Lemm 22 Teorem 34 If f e D tegrle o [,], te f s mesurle d fte e o [,] Proof: Sce f s r D tegrle o [,], tere s cotuous fucto AC G[, ] suc tt r s smoot for r,2, 2 d f, e o [,] Let [,]= E, were E s closed d s AC o E for ec Let E P D were P s perfect d D s coutle Sce AC G E, ACG D d so y Teorem-6 of [3], s AC o P were te dervtve s te wt respect to P Let x x for x P d e ler te closure of ll tervls cotguous to P Te BV [, ] d so exsts e o [,] Also f x x e E Sce dervtve of fucto of ouded vrto s fte e d mesurle, f s fte e d s mesurle o E for ec Te rest s cler Teorem 35 If f s D tegrle d f 0 [,], te f s Leesgue tegrle d te tegrls re equl Proof: Let e te t prmtve of f So f 0 e [,] Te from Teorem-24, exsts d s odecresg [,] d [ AC, ] Hece exsts e [,] d s Leesgue tegrle [,] Hece f s Leesgue tegrle [,] Teorem 36 Itegrto y prts Let f :[, ] R s D tegrle [,], F e ts defte D tegrl d F s D tegrle Let G - e solutely cotuous [,], te fg s D tegrle [,] d D fg [ FG] D FG Proof: Let e te t prmtve of f [,] Let x xgx, x [, ] Te sce for r=0,,,-2 d G - r exsts ftely so s smoot for r=0,,,-2 Sce G - s solutely cotuous,g s AC [,] d sce s AC G [,] Ad e [,] r So G Let G G c G c G c G G FG fg FG fg s D 2 H x G c G G 2 tegrle [,] Te smlr wy s tt of Teorem-4 of [] t c e proved tt H ACG [, ] Ag sce d G exsts e [,],we ve, H x G c c G G 0 = G c G FG G r s smoot So G c G FG s D tegrle [,] d so D tegrle d H s defte D tegrl Sce F s D tegrle d G s solutely cotuous [,], FG s D tegrle y Teorem25p246 of [4] d ece FGs D tegrle [,] Ag sce, G c G FG fg, G c G FG d FG re D tegrle, y Teorem33, fg s D tegrle [,] d IJMER ISSN: wwwjmercom Vol 5 Iss J
5 Itegrto By Prts For D K Itegrl D fg D = = G c G FG fg D G c G FG D FG [ D FG ] [ H] 2 2 G c G G FG] [ 2 2 c G G [ G ] D FG = [ FG] D FG REFERENCES [] SNMuopdyy d SKMuopdyy,"A geerlsed tegrl wt plcto to trgometrc seres" Alyss Mt, [2] S N Muopdyy d S Ry, "Geerlsed Asolutely t Cotuous Fucto" Id Jourl of Mtemtcs, Vol 53, No [3] SKMuopdyy d SNMuopdyy,"Fuctos of ouded t vrto d solutely t cotuous fuctos" BullAustrlMtSoc 46992,9-06 [4] SSs,"Teory of te tegrl",dover,newyor937 IJMER ISSN: wwwjmercom Vol 5 Iss J
Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
![Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]](/thumbs/83/87614710.jpg "Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]")
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
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
this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]
![this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]](/thumbs/78/76749440.jpg "this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]")
Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow
UNIT I. Definition and existence of Riemann-Stieltjes
1 UNIT I Defto d exstece of Rem-Steltjes Itroducto: The reder wll recll from elemetry clculus tht to fd the re of the rego uder the grph of postve fucto f defed o [, ], we sudvde the tervl [, ] to fte
Mathematically, integration is just finding the area under a curve from one point to another. It is b
Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom
MTH 146 Class 7 Notes
7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg
CS321. Numerical Analysis
CS Numercl Alyss Lecture 4 Numercl Itegrto Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 456 6 Octoer 6, 5 Dete Itegrl A dete tegrl s tervl or tegrto. For ed tegrto tervl, te result
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
under the curve in the first quadrant.
NOTES 5: INTEGRALS Nme: Dte: Perod: LESSON 5. AREAS AND DISTANCES Are uder the curve Are uder f( ), ove the -s, o the dom., Prctce Prolems:. f ( ). Fd the re uder the fucto, ove the - s, etwee,.. f ( )
Numerical differentiation (Finite difference)
MA umercl Alyss Week 4 To tk d wrte portle progrms we eed strct model of ter computtol evromet. Ftful models do est ut tey revel tt evromet to e too dverse forcg portle progrmmers to lot eve te smplest
Chapter Simpson s 1/3 Rule of Integration. ( x)
Cpter 7. Smpso s / Rule o Itegrto Ater redg ts pter, you sould e le to. derve te ormul or Smpso s / rule o tegrto,. use Smpso s / rule t to solve tegrls,. develop te ormul or multple-segmet Smpso s / rule
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.
The z-transform. LTI System description. Prof. Siripong Potisuk
The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put
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
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
Oriented Measures* Raphael Cerf. and. Carlo Mariconda. Submitted by Dorothy Maharam Stone. Received June 15, 1994
Ž JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 197, 9259 1996 ARTICLE NO 0062 Oreted Mesures* Rphel Cerf CNRS, Uerste Prs Sud, Mthemtque, Btmet ˆ 25, 9105 Orsy Cedex, Frce d Crlo Mrcod Dprtmeto d Mtemtc
On Solution of Min-Max Composition Fuzzy Relational Equation
U-Sl Scece Jourl Vol.4()7 O Soluto of M-Mx Coposto Fuzzy eltol Equto N.M. N* Dte of cceptce /5/7 Abstrct I ths pper, M-Mx coposto fuzzy relto equto re studed. hs study s geerlzto of the works of Ohsto
Sequences and summations
Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course
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
Differential Entropy 吳家麟教授
Deretl Etropy 吳家麟教授 Deto Let be rdom vrble wt cumultve dstrbuto ucto I F s cotuous te r.v. s sd to be cotuous. Let = F we te dervtve s deed. I te s clled te pd or. Te set were > 0 s clled te support set
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.
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
General Method for Calculating Chemical Equilibrium Composition
AE 6766/Setzma Sprg 004 Geeral Metod for Calculatg Cemcal Equlbrum Composto For gve tal codtos (e.g., for gve reactats, coose te speces to be cluded te products. As a example, for combusto of ydroge wt
means 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,
. 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)](/thumbs/85/92239142.jpg ". 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
Numerical Analysis Formulae Booklet
Numercal Aalyss Formulae Booklet. Iteratve Scemes for Systems of Lear Algebrac Equatos:.... Taylor Seres... 3. Fte Dfferece Approxmatos... 3 4. Egevalues ad Egevectors of Matrces.... 3 5. Vector ad Matrx
SUM PROPERTIES FOR THE K-LUCAS NUMBERS WITH ARITHMETIC INDEXES
Avlble ole t http://sc.org J. Mth. Comput. Sc. 4 (04) No. 05-7 ISSN: 97-507 SUM PROPERTIES OR THE K-UCAS NUMBERS WITH ARITHMETIC INDEXES BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA * School of
Numerical Differentiation and Integration
Numerl Deretto d Itegrto Overvew Numerl Deretto Newto-Cotes Itegrto Formuls Trpezodl rule Smpso s Rules Guss Qudrture Cheyshev s ormul Numerl Deretto Forwrd te dvded deree Bkwrd te dvded deree Ceter te
Supplementary Material for Limits on Sparse Support Recovery via Linear Sketching with Random Expander Matrices
Joata Scarlett ad Volka Cever Supplemetary Materal for Lmts o Sparse Support Recovery va Lear Sketcg wt Radom Expader Matrces (AISTATS 26, Joata Scarlett ad Volka Cever) Note tat all ctatos ere are to
CS321. Introduction to Numerical Methods
CS Itroducto to Numercl Metods Lecture Revew Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 6 6 Mrc 7, Number Coverso A geerl umber sould be coverted teger prt d rctol prt seprtely
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
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
Johns 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
Outline. Finite Difference Grids. Numerical Analysis. Finite Difference Grids II. Finite Difference Grids III
Itrodcto to Nmercl Alyss Mrc, 9 Nmercl Metods or PDEs Lrry Cretto Meccl Egeerg 5B Semr Egeerg Alyss Mrc, 9 Otle Revew mdterm soltos Revew bsc mterl o mercl clcls Expressos or dervtves, error d error order
Johns 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
Chapter 2 Intro to Math Techniques for Quantum Mechanics
Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
On Stability of a Class of Fractional Differential Equations
Proceedgs of the Pkst Acdey of Sceces 49 (: 39-43 (0 Coyrght Pkst Acdey of Sceces SSN: 0377-969 Pkst Acdey of Sceces Orgl Artcle O Stblty of Clss of Frctol Dfferetl Equtos Rbh W. brh* sttute of Mthetcl
Chapter 12-b Integral Calculus - Extra
C - Itegrl Clulus Cpter - Itegrl Clulus - Etr Is Newto Toms Smpso BONUS Itroduto to Numerl Itegrto C - Itegrl Clulus Numerl Itegrto Ide s to do tegrl smll prts, lke te wy we preseted tegrto: summto. Numerl
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
Numerical Solution of Second order Integro- Differential Equations(Ides) with Different Four Polynomials Bases Functions
Numercl Soluto of Secod order Itegro- Dfferetl Equtos(Ides) wth Dfferet Four Polyomls Bses Fuctos wo O A R M Deprtmet of Mthemtcs Uersty of Ilor Deprtmet of Mthemtcs d Sttstcs he Poly Id Astrct: - I ths
Nonlinear Piecewise-Defined Difference Equations with Reciprocal Quadratic Terms
Joural of Matematcs ad Statstcs Orgal Researc Paper Nolear Pecewse-Defed Dfferece Equatos wt Recprocal Quadratc Terms Ramada Sabra ad Saleem Safq Al-Asab Departmet of Matematcs, Faculty of Scece, Jaza
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
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
Lecture 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
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
( ) ( ) ( ( )) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) + ( ) ( ) = ( ( )) ( ) + ( ( )) ( ) Review. Second Derivatives for f : y R. Let A be an m n matrix.
Revew + v, + y = v, + v, + y, + y, Cato! v, + y, + v, + y geeral Let A be a atr Let f,g : Ω R ( ) ( ) R y R Ω R h( ) f ( ) g ( ) ( ) ( ) ( ( )) ( ) dh = f dg + g df A, y y A Ay = = r= c= =, : Ω R he Proof
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
DATA FITTING. Intensive Computation 2013/2014. Annalisa Massini
DATA FITTING Itesve Computto 3/4 Als Mss Dt fttg Dt fttg cocers the problem of fttg dscrete dt to obt termedte estmtes. There re two geerl pproches two curve fttg: Iterpolto Dt s ver precse. The strteg
CHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
Square Difference Labeling Of Some Path, Fan and Gear Graphs
Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 ISSN 9-558 Square Dfferece Labelg Of Some Path, Fa ad Gear Graphs J.Shama Assstat Professor Departmet of Mathematcs CMS College of
Some Unbiased Classes of Estimators of Finite Population Mean
Itertol Jourl O Mtemtcs Ad ttstcs Iveto (IJMI) E-IN: 3 4767 P-IN: 3-4759 Www.Ijms.Org Volume Issue 09 etember. 04 PP-3-37 ome Ubsed lsses o Estmtors o Fte Poulto Me Prvee Kumr Msr d s Bus. Dertmet o ttstcs,
CHAPTER 5 Vectors and Vector Space
HAPTE 5 Vetors d Vetor Spe 5. Alger d eometry of Vetors. Vetor A ordered trple,,, where,, re rel umers. Symol:, B,, A mgtude d dreto.. Norm of vetor,, Norm =,, = = mgtude. Slr multplto Produt of slr d
Strategies for the AP Calculus Exam
Strteges for the AP Clculus Em Strteges for the AP Clculus Em Strtegy : Kow Your Stuff Ths my seem ovous ut t ees to e metoe. No mout of cochg wll help you o the em f you o t kow the mterl. Here s lst
19 22 Evaluate the integral by interpreting it in terms of areas. (1 x ) dx. y Write the given sum or difference as a single integral in
SECTION. THE DEFINITE INTEGRAL. THE DEFINITE INTEGRAL A Clck here for swers. S Clck here for solutos. Use the Mdpot Rule wth the gve vlue of to pproxmte the tegrl. Roud the swer to four decml plces. 9
20 23 Evaluate the integral by interpreting it in terms of areas. (1 x ) dx. y 2 2
SECTION 5. THE DEFINITE INTEGRAL 5. THE DEFINITE INTEGRAL A Clck here for swers. S Clck here for solutos. 7 Use the Mdpot Rule wth the gve vlue of to pproxmte the tegrl. Roud the swer to four decml plces.
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
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:
Itō Calculus (An Abridged Overview)
Itō Clculus (A Abrdged Overvew) Arturo Ferdez Uversty of Clfor, Berkeley Sttstcs 157: Topcs I Stochstc Processes Semr Aprl 14, 211 1 Itroducto I my prevous set of otes, I troduced the cocept of Stochstc
i+1 by A and imposes Ax
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 09.9 NUMERICAL FLUID MECHANICS FALL 009 Mody, October 9, 009 QUIZ : SOLUTIONS Notes: ) Multple solutos
Lecture 12 APPROXIMATION OF FIRST ORDER DERIVATIVES
FDM: Appromato of Frst Order Dervatves Lecture APPROXIMATION OF FIRST ORDER DERIVATIVES. INTRODUCTION Covectve term coservato equatos volve frst order dervatves. The smplest possble approach for dscretzato
6.6 Moments and Centers of Mass
th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder
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,
9 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((,
Third handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
Second Mean Value Theorem for Integrals By Ng Tze Beng. The Second Mean Value Theorem for Integrals (SMVT) Statement of the Theorem
Secod Me Vlue Theorem for Itegrls By Ng Tze Beg This rticle is out the Secod Me Vlue Theorem for Itegrls. This theorem, first proved y Hoso i its most geerlity d with extesio y ixo, is very useful d lmost
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),
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.
Summary 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
MATH2999 Directed Studies in Mathematics Matrix Theory and Its Applications
MATH999 Drected Studes Mthemtcs Mtr Theory d Its Applctos Reserch Topc Sttory Probblty Vector of Hgher-order Mrkov Ch By Zhg Sho Supervsors: Prof. L Ch-Kwog d Dr. Ch Jor-Tg Cotets Abstrct. Itroducto: Bckgroud.
Linear Algebra Concepts
Ler Algebr Cocepts Nuo Vscocelos (Ke Kreutz-Delgdo) UCSD Vector spces Defto: vector spce s set H where ddto d sclr multplcto re defed d stsf: ) +( + ) = (+ )+ 5) H 2) + = + H 6) = 3) H, + = 7) ( ) = (
MATRIX AND VECTOR NORMS
Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the
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
SOME SERIES IDENTITIES FOR SOME SPECIAL CLASSES OF APOSTOL-BERNOULLI AND APOSTOL-EULER POLYNOMIALS RELATED TO GENERALIZED POWER AND ALTERNATING SUMS
Bullet of Mthemtcl Alyss d Applctos ISSN: 181-191, URL: http://www.mth.org Volume 4 Issue 4 01, Pges 76-90. SOME SERIES IDENTITIES FOR SOME SPECIAL CLASSES OF APOSTOL-BERNOULLI AND APOSTOL-EULER POLYNOMIALS
A tighter lower bound on the circuit size of the hardest Boolean functions
Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the
Asymptotic Formulas Composite Numbers II
Iteratoal Matematcal Forum, Vol. 8, 203, o. 34, 65-662 HIKARI Ltd, www.m-kar.com ttp://d.do.org/0.2988/mf.203.3854 Asymptotc Formulas Composte Numbers II Rafael Jakmczuk Dvsó Matemátca, Uversdad Nacoal
Point 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.
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,
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
Analytical Approach for the Solution of Thermodynamic Identities with Relativistic General Equation of State in a Mixture of Gases
Itertol Jourl of Advced Reserch Physcl Scece (IJARPS) Volume, Issue 5, September 204, PP 6-0 ISSN 2349-7874 (Prt) & ISSN 2349-7882 (Ole) www.rcourls.org Alytcl Approch for the Soluto of Thermodymc Idettes
A 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,
International 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
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
ME 501A Seminar in Engineering Analysis Page 1
Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt
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
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
L5 Polynomial / Spline Curves
L5 Polyomal / Sple Curves Cotets Coc sectos Polyomal Curves Hermte Curves Bezer Curves B-Sples No-Uform Ratoal B-Sples (NURBS) Mapulato ad Represetato of Curves Types of Curve Equatos Implct: Descrbe a
Chapter 3. Differentiation 3.3 Differentiation Rules
3.3 Dfferetato Rules 1 Capter 3. Dfferetato 3.3 Dfferetato Rules Dervatve of a Costat Fucto. If f as te costat value f(x) = c, te f x = [c] = 0. x Proof. From te efto: f (x) f(x + ) f(x) o c c 0 = 0. QED
Roberto s Notes on Integral Calculus Chapter 4: Definite integrals and the FTC Section 2. Riemann sums
Roerto s Notes o Itegrl Clculus Chpter 4: Defte tegrls d the FTC Secto 2 Rem sums Wht you eed to kow lredy: The defto of re for rectgle. Rememer tht our curret prolem s how to compute the re of ple rego
CURVE FITTING LEAST SQUARES METHOD
Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued
Chapter 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
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
ON THE LOGARITHMIC INTEGRAL
Hacettepe Joural of Mathematcs ad Statstcs Volume 39(3) (21), 393 41 ON THE LOGARITHMIC INTEGRAL Bra Fsher ad Bljaa Jolevska-Tueska Receved 29:9 :29 : Accepted 2 :3 :21 Abstract The logarthmc tegral l(x)
Numerical Solution of Higher Order Linear Fredholm Integro Differential Equations.
Amerc Jorl of Egeerg Reserch (AJER) 04 Amerc Jorl of Egeerg Reserch (AJER) e-iss : 30-0847 p-iss : 30-0936 Volme-03, Isse-08, pp-43-47 www.jer.org Reserch Pper Ope Access mercl Solto of Hgher Order Ler
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
Chapter 3. Differentiation 3.2 Differentiation Rules for Polynomials, Exponentials, Products and Quotients
3.2 Dfferetato Rules 1 Capter 3. Dfferetato 3.2 Dfferetato Rules for Polyomals, Expoetals, Proucts a Quotets Rule 1. Dervatve of a Costat Fucto. If f as te costat value f(x) = c, te f x = [c] = 0. x Proof.
5 Short Proofs of Simplified Stirling s Approximation
5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is
Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I
A New Method for Decision Making Based on Soft Matrix Theory
Joural of Scetfc esearch & eports 3(5): 0-7, 04; rtcle o. JS.04.5.00 SCIENCEDOMIN teratoal www.scecedoma.org New Method for Decso Mag Based o Soft Matrx Theory Zhmg Zhag * College of Mathematcs ad Computer
Metric Spaces: Basic Properties and Examples
1 Metrc Spces: Bsc Propertes d Exmples 1.1 NTODUCTON Metrc spce s dspesble termedte course of evoluto of the geerl topologcl spces. Metrc spces re geerlstos of Euclde spce wth ts vector spce structure