# 2. Elementary Linear Algebra Problems

Size: px
Start display at page:

Transcription

1 . Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N Full y tee Depth te tep log()n

2 Veto updte the f- poe wth N : ) ( ) ( ) ( ) ( ) ( ) ( ) ( N N N N ( ) ( ) [ ] ( ) ( ) [ ] ed ed fo N fo N N ; ) ; ; ( ) ; ; ( log()n te tep pllel fo veto of legth

3 evel- BS BS oute wth () pole (veto oly y ε IR ). Ft eple: DT-podut y f-: y T y y y 3 y 3 4 y 4 5 y 5 6 y y 7 8 y 8 3

4 Te tep pllel of the DT-podut ot ette th log(). Evey oputto volvg f- wll te log() te tep pllel! ddedu: Coputg the DT-podut o pel htetue: Dtute dt o -deol y of poeo wth / poeo 3 (-) P P P 3 P / P - P Ptto veto d y lo of veto wth legth. Eh poeo h to opute ptl DT-podut: ( ) y( ) y... 4

5 Te tep pllel oputto ed o th odel: ( dd ult ) te fo ddto d ultplto pllel poeo p fo ptl DT-podut fte oputg the ptl DT-podut eh poeo pllel teedte eult hve to e olleted d dded up to the fl eult. To th the poeo P (P ) ed t eult to the eghog poeo P (P - ) tht dd th ue to t ow ptl DT-podut. The the eult ed to the et egho tht dd two ptl eult g d o o utl the fl eult oputed the poeo P /. ( ) 3 (-) P P P 3 P / P - P 5

6 Totl te depedg o d : f ( ) ( dd ( dd ult) ( ed dd) dd ed ult) ze the totl te f() :! f '( ) ( dd ult) dd ed Ο ( ) ptu: ( ) te tep dt dtuted o ( ) poeo. f ( ) / Ο ( ) Cope f-: poeo log() te tep! 6

7 Futhe evel- BS Pole BS-Notto: defe SXPY: S --- gle peo (D fo doule C fo ople) --- α l X --- veto P --- plu opeto Y --- veto y α y Y α X Y Vetozto of SXPY y ppelg: α α y : α ultplto ddto α y y 7

8 8 SXPY Pllelzto y Pttog { } R I I I > < R R Y Y y y y y X X wth hot veto X d Y of legth /R. Eh poeo P get ptl veto X d Y d opute Y α X Y R. Reult: SXPY vey good vetozle d pllelzle X X X X X

9 Futhe evel-bs Route SCPY: y o y Cope SXPY No: T Cope DT-podut 9

10 evel- BS t-veto opeto wth ( ) opeto (equetlly) BS-Notto: S --- gle peo G E geel t V --- veto defe SGEV t-veto podut: y α y the evel- BS: Solvg tgul yte wth tgul t

11 evel-3 BS t-t opeto wth ( 3 ) opeto (equetlly) BS-Notto: S --- gle peo G E geel t --- t defe SGE t-t podut: C α B C PCK uoute fo olvg le equto let que pole QR-deopoto egevlue gul vlue ed o BS

12 . ly of t-veto- Podut ( ) IR IR IR Vetozto DT-podut of legth SXPY of legth (GXPY)

13 Peudoode: ()-fo: ; fo fo ed ed DT podut Dot podut of -th ow of wth veto 3

14 Peudoode: ()-fo: ; fo fo ed ed SXPY GXPY SXPY updtg veto wth -th olu of GXPY: Sequee of SXPY elted to the e veto dvtge: veto tht updted e ept ft eoy! No ddtol dt tfe. 4

15 .. Pllelzto y uldg lo Ide: Redue t-veto podut o lle t-veto podut. < > < > {... } I I I {... } J J J S R I dut: J I J fo fo Ue -deol y of poeo P. P get t lo :(I J ) :(J ) :(I ). S S ( ) I J I J 5

16 Peudoode fo R fo S () ; ed ed fo R ; fo S () ; ed ed Sll depedet t-veto podut. No outo eey! Blowe olleto d ddto of veto. Rowwe outo! F-. 6

17 7 Spel e: S P P.. No outo eey etwee poeo P P R The oputto of vetozle y GXPY. R : ( ) e depedet. The olleto of ptl eult fo poeo P P. F-. Fl u oe poeo: vetozle y GXPY.

18 Rule () Ie loop of pog hould e ple vetozle () ute loop of pog hould e uttl depedet pllelzle fo uttl d pllelzle fo. ed ple vetozle ed (3) Reue of dt (Che l dt tfe log) 8

19 9..3 fo Bded t Bdwdth (yet) dgol: dg. udg. upedg. : tdgol

20 N N N Stog ete dgolwe () t ted of. ow fo... d d ow dgol d d [ ] { } { } [ ] l [ ] { } { } [ ] l

21 Coputto of the t-veto podut ed o th toge hee o veto poeo: Fo : l l lgoth: Fo - : Fo {-} : {-} ed ed Geel tde No SXPY o Fo : Fo {--} : {-} ed ed Ptl Dot podut Spty le opeto ut lo lo of effey thee opeto

22 Pllel Pttog: R < > U fo ed I I dut l ; Poeo P get ow to de et I :[ ] ode to opute t pt of the fl veto. Wht pt of veto doe poeo P eed ode to opute t pt of?

23 Neey fo I : : l { } { } { } { } Poeo P wth de et I eed fo the de [ { } { } ] 3

24 4.3 ly of t-t Podut ( ) ( ) ( ) q q B C B ed ed q Fo Fo : :

25 ()-Fo: lgoth Fo : Fo : q Fo : ; ed ed ed Dot podut of legth B fo ll ll ete e fully oputed oe fte othe. e to d C owwe to B oluwe. 5

26 6 Dffeet vew o the t-t podut: ( ) ( ) ( ) T T e e e e t odeed oto of olu o ow ( ) T T e e e e B u of full te y oute podut of the -th olu of d the -th ow of B Full q - te

27 ()-Fo lgoth Fo : q Fo : Fo : ; ed ed ed SXPY Veto updte. GXPY Sequee of SXPY fo the e veto. C oputed oluwe; e to oluwe. e to B oluwe ut delyed. 7

28 ()-Fo lgoth 3 Fo : Fo : q Fo : ; ed ed ed SXPY No GXPY Sequee of SXPY fo dffeet veto. Veto updte. e to oluwe. e to B delyed. C oputed wth teedte vlue () whh e oputed oluwe. 8

29 vevew ove dffeet Fo e to y e to B y Coput -to of C ow ow olu olu olu ow ow olu ow ow ow olu olu olu Coput det delyed delyed det delyed delyed -to Veto opeto Veto leght DT GXPY SXPY DT GXPY SXPY q q Bette: GXPY loge veto legth. e to te odg to toge hee (owwe o oluwe) 9

30 t-t Pllel R < > U I < > U K < q > U J S T t t. Dtute the lo eltve to de et I K d J t y P t : to poeo K J t J t K. I () t I B t () Poeo P t opute ll t-t podut ll Poeo pllel. ( ) t B t () Copute u y f- : S ( t t ) 3

31 Spel Ce S J t J t. I t I B t I th e eh poeo P t opute t pt of t depedetly wthout y outo. Eh poeo eed full lo of ow of eltve to de et I d full lo of olu of B eltve to de et J t ode to opute t eltve to ow I d olu J t. Epelly wth q poeo eh poeo h to opute oe DT podut t t wth () pllel te tep. F- y q ddtol poeo fo ll thee Dot podut edue the ue of pllel te tep to (log()). 3

### Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector

evel-2 BS trx-vector opertos wth 2 opertos sequetlly BS-Notto: S --- sgle precso G E geerl mtrx V --- vector defes SGEV, mtrx-vector product: r y r α x β r y ther evel-2 BS: Solvg trgulr system x wth trgulr

### 2.Decision Theory of Dependence

.Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give

### Current Programmed Control (i.e. Peak Current-Mode Control) Lecture slides part 2 More Accurate Models

Curret Progred Cotrol.e. Pek Curret-Mode Cotrol eture lde prt More Aurte Model ECEN 5807 Drg Mkovć Sple Frt-Order CPM Model: Sury Aupto: CPM otroller operte delly, Ueful reult t low frequee, well uted

### SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS

ELM Numecl Alyss D Muhem Mecmek SOLVING SYSTEMS OF EQUATIONS DIRECT METHODS ELM Numecl Alyss Some of the cotets e dopted fom Luee V. Fusett Appled Numecl Alyss usg MATLAB. Petce Hll Ic. 999 ELM Numecl

### The formulae in this booklet have been arranged according to the unit in which they are first

Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge og to the ut whh the e fst toue. Thus te sttg ut m e eque to use the fomule tht wee toue peeg ut e.g. tes sttg C mght e epete to use fomule

### Moments of Generalized Order Statistics from a General Class of Distributions

ISSN 684-843 Jol of Sttt Vole 5 28. 36-43 Moet of Geelzed Ode Sttt fo Geel l of Dtto Att Mhd Fz d Hee Ath Ode ttt eod le d eel othe odel of odeed do le e ewed el e of geelzed ode ttt go K 995. I th e exlt

### 5 - Determinants. r r. r r. r r. r s r = + det det det

5 - Detemts Assote wth y sque mtx A thee s ume lle the etemt of A eote A o et A. Oe wy to efe the etemt, ths futo fom the set of ll mtes to the set of el umes, s y the followg thee popetes. All mtes elow

### X-Ray Notes, Part III

oll 6 X-y oe 3: Pe X-Ry oe, P III oe Deeo Coe oupu o x-y ye h look lke h: We efe ue of que lhly ffee efo h ue y ovk: Co: C ΔS S Sl o oe Ro: SR S Co o oe Ro: CR ΔS C SR Pevouly, we ee he SR fo ye hv pxel

### On Almost Increasing Sequences For Generalized Absolute Summability

Joul of Applied Mthetic & Bioifotic, ol., o., 0, 43-50 ISSN: 79-660 (pit), 79-6939 (olie) Itetiol Scietific Pe, 0 O Alot Iceig Sequece Fo Geelized Abolute Subility W.. Suli Abtct A geel eult coceig bolute

### Chapter #2 EEE State Space Analysis and Controller Design

Chpte EEE8- Chpte # EEE8- Stte Spce Al d Cotolle Deg Itodcto to tte pce Obevblt/Cotollblt Modle ede: D D Go - d.go@cl.c.k /4 Chpte EEE8-. Itodcto Ae tht we hve th ode te: f, ', '',.... Ve dffclt to td

### 3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4

// Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,,.... - - - Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples

### Matrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n.

Mtrx Defto () s lled order of m mtrx, umer of rows ( 橫行 ) umer of olums ( 直列 ) m m m where j re rel umers () B j j for,,, m d j =,,, () s lled squre mtrx f m (v) s lled zero mtrx f (v) s lled detty mtrx

### The linear system. The problem: solve

The ler syste The prole: solve Suppose A s vertle, the there ests uue soluto How to effetly opute the soluto uerlly??? A A A evew of dret ethods Guss elto wth pvotg Meory ost: O^ Coputtol ost: O^ C oly

### 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

### Chapter Linear Regression

Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use

### = y and Normed Linear Spaces

304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads

### 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

### 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

### Rendering Equation. Linear equation Spatial homogeneous Both ray tracing and radiosity can be considered special case of this general eq.

Rederg quto Ler equto Sptl homogeeous oth ry trcg d rdosty c be cosdered specl cse of ths geerl eq. Relty ctul photogrph Rdosty Mus Rdosty Rederg quls the dfferece or error mge http://www.grphcs.corell.edu/ole/box/compre.html

### Review of Linear Algebra

PGE 30: Forulto d Soluto Geosstes Egeerg Dr. Blhoff Sprg 0 Revew of Ler Alger Chpter 7 of Nuercl Methods wth MATLAB, Gerld Recktewld Vector s ordered set of rel (or cople) uers rrged s row or colu sclr

### 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

### CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find

BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,

### 1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.

SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,

### AN ALGEBRAIC APPROACH TO M-BAND WAVELETS CONSTRUCTION

AN ALGEBRAIC APPROACH TO -BAN WAELETS CONSTRUCTION Toy L Qy S Pewe Ho Ntol Lotoy o e Peeto Pe Uety Be 8 P. R. C Att T e eet le o to ott - otool welet e. A yte of ott eto ote fo - otool flte te olto e o

### Chapter 17. Least Square Regression

The Islmc Uvest of Gz Fcult of Egeeg Cvl Egeeg Deptmet Numecl Alss ECIV 336 Chpte 7 Lest que Regesso Assocte Pof. Mze Abultef Cvl Egeeg Deptmet, The Islmc Uvest of Gz Pt 5 - CURVE FITTING Descbes techques

### A convex hull characterization

Pue d ppled Mthets Joul 4; (: 4-48 Pulshed ole My 4 (http://www.seepulshggoup.o//p do:.648/.p.4. ove hull htezto Fo Fesh Gov Qut Deptet DISG Uvesty of Se Itly El ddess: fesh@us.t (F. Fesh qut@us.t (G.

### 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 9-3/8/10-14 Spatial Description and Transformation

Letue 9-8- tl Deton nd nfomton Homewo No. Due 9. Fme ngement onl. Do not lulte...8..7.8 Otonl et edt hot oof tht = - Homewo No. egned due 9 tud eton.-.. olve oblem:.....7.8. ee lde 6 7. e Mtlb on. f oble.

### COMP 465: Data Mining More on PageRank

COMP 465: Dt Mnng Moe on PgeRnk Sldes Adpted Fo: www.ds.og (Mnng Mssve Dtsets) Powe Iteton: Set = 1/ 1: = 2: = Goto 1 Exple: d 1/3 1/3 5/12 9/24 6/15 = 1/3 3/6 1/3 11/24 6/15 1/3 1/6 3/12 1/6 3/15 Iteton

### ECONOMETRIC ANALYSIS ON EFFICIENCY OF ESTIMATOR ABSTRACT

ECOOMETRIC LYSIS O EFFICIECY OF ESTIMTOR M. Khohev, Lectue, Gffth Uvet, School of ccoutg d Fce, utl F. K, tt Pofeo, Mchuett Ittute of Techolog, Deptet of Mechcl Egeeg, US; cuetl t Shf Uvet, I. Houl P.

### Chapter Gauss-Seidel Method

Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos

### Week 10: DTMC Applications Ranking Web Pages & Slotted ALOHA. Network Performance 10-1

Week : DTMC Alictions Rnking Web ges & Slotted ALOHA etwok efonce - Outline Aly the theoy of discete tie Mkov chins: Google s nking of web-ges Wht ge is the use ost likely seching fo? Foulte web-gh s Mkov

### Difference Sets of Null Density Subsets of

dvces Pue Mthetcs 95-99 http://ddoog/436/p37 Pulshed Ole M (http://wwwscrpog/oul/p) Dffeece Sets of Null Dest Susets of Dwoud hd Dsted M Hosse Deptet of Mthetcs Uvest of Gul Rsht I El: hd@gulc h@googlelco

### 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

### Analele Universităţii din Oradea, Fascicula: Protecţia Mediului, Vol. XIII, 2008

Alele Uverstăţ d Orde Fsul: Proteţ Medulu Vol. XIII 00 THEORETICAL AND COMPARATIVE STUDY REGARDING THE MECHANICS DISPLASCEMENTS UNDER THE STATIC LOADINGS FOR THE SQUARE PLATE MADE BY WOOD REFUSE AND MASSIF

### African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS

Af Joul of See Tehology (AJST) See Egeeg See Vol. 4, No.,. 7-79 GENERALISED DELETION DESIGNS Mhel Ku Gh Joh Wylff Ohbo Dee of Mhe, Uvey of Nob, P. O. Bo 3097, Nob, Key ABSTRACT:- I h e yel gle ele fol

### MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER

MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER I One M Queston Fnd the unt veto n the deton of Let ˆ ˆ 9 Let & If Ae the vetos & equl? But vetos e not equl sne the oespondng omponents e dstnt e detons

Advced Highe Mths: Fomule Advced Highe Mthemtics Gee (G): Fomule you solutely must memoise i ode to pss Advced Highe mths. Rememe you get o fomul sheet t ll i the em! Ame (A): You do t hve to memoise these

### 148 CIVIL ENGINEERING

STRUTUR NYSS fluee es fo Bems d Tusses fluee le sows te vto of effet (eto, se d momet ems, foe tuss) used movg ut lod oss te stutue. fluee le s used to deteme te posto of movele set of lods tt uses te

### Introduction to Modern Control Theory

Itroductio to Moder Cotrol Theory MM : Itroductio to Stte-Spce Method MM : Cotrol Deig for Full Stte Feedck MM 3: Etitor Deig MM 4: Itroductio of the Referece Iput MM 5: Itegrl Cotrol d Rout Trckig //4

### Exercise # 2.1 3, 7, , 3, , -9, 1, Solution: natural numbers are 3, , -9, 1, 2.5, 3, , , -9, 1, 2 2.5, 3, , -9, 1, , -9, 1, 2.

Chter Chter Syste of Rel uers Tertg Del frto: The del frto whh Gve fte uers of dgts ts del rt s lled tertg del frto. Reurrg ( o-tertg )Del frto: The del frto (No tertg) whh soe dgts re reeted g d g the

### xl yl m n m n r m r m r r! The inner sum in the last term simplifies because it is a binomial expansion of ( x + y) r : e +.

Ler Trsfortos d Group Represettos Hoework #3 (06-07, Aswers Q-Q re further exerses oer dots, self-dot trsfortos, d utry trsfortos Q3-6 volve roup represettos Of these, Q3 d Q4 should e quk Q5 s espelly

### Spherical refracting surface. Here, the outgoing rays are on the opposite side of the surface from the Incoming rays.

Sphericl refrctig urfce Here, the outgoig ry re o the oppoite ide of the urfce from the Icomig ry. The oject i t P. Icomig ry PB d PV form imge t P. All prxil ry from P which trike the phericl urfce will

### E-Companion: Mathematical Proofs

E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth

### G8-11 Congruence Rules

G8-11 ogruee Rules If two polgos re ogruet, ou ple the oe o top of the other so tht the th etl. The verties tht th re lled orrespodig verties. The gles tht th re lled orrespodig gles. The sides tht th

### SYSTEMS OF NON-LINEAR EQUATIONS. Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations

SYSTEMS OF NON-LINEAR EQUATIONS Itoduto Gaphal Method Cloe Method Ope Method Polomal Root Stem o Multvaale Equato Chapte Stem o No-Lea Equato /. Itoduto Polem volvg o-lea equato egeeg lude optmato olvg

### The formulae in this booklet have been arranged according to the unit in which they are first

Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge ccog to the ut whch the e fst touce. Thus cte sttg ut m e eque to use the fomule tht wee touce peceg ut e.g. ctes sttg C mght e epecte to use

### Chapter Unary Matrix Operations

Chpter 04.04 Ury trx Opertos After redg ths chpter, you should be ble to:. kow wht ury opertos mes,. fd the trspose of squre mtrx d t s reltoshp to symmetrc mtrces,. fd the trce of mtrx, d 4. fd the ermt

### Parametric Methods. Autoregressive (AR) Moving Average (MA) Autoregressive - Moving Average (ARMA) LO-2.5, P-13.3 to 13.4 (skip

Pmeti Methods Autoegessive AR) Movig Avege MA) Autoegessive - Movig Avege ARMA) LO-.5, P-3.3 to 3.4 si 3.4.3 3.4.5) / Time Seies Modes Time Seies DT Rdom Sig / Motivtio fo Time Seies Modes Re the esut

### Project 3: Using Identities to Rewrite Expressions

MAT 5 Projet 3: Usig Idetities to Rewrite Expressios Wldis I lger, equtios tht desrie properties or ptters re ofte lled idetities. Idetities desrie expressio e repled with equl or equivlet expressio tht

### Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates.

CURVE FITTING Descbes techques to ft cuves (cuve fttg) to dscete dt to obt temedte estmtes. Thee e two geel ppoches fo cuve fttg: Regesso: Dt ehbt sgfct degee of sctte. The stteg s to deve sgle cuve tht

### 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

### DETAIL MEASURE EVALUATE

MEASURE EVALUATE B I M E q u i t y BIM Workflow Guide MEASURE EVALUATE Introduction We o e to ook 2 i t e BIM Workflow Guide i uide wi tr i you i re ti ore det i ed ode d do u e t tio u i r i d riou dd

### minimize c'x subject to subject to subject to

z ' sut to ' M ' M N uostrd N z ' sut to ' z ' sut to ' sl vrls vtor of : vrls surplus vtor of : uostrd s s s s s s z sut to whr : ut ost of :out of : out of ( ' gr of h food ( utrt : rqurt for h utrt

### Noise in electronic components.

No lto opot5098, JDS No lto opot Th PN juto Th ut thouh a PN juto ha fou opot t: two ffuo ut (hol fo th paa to th aa a lto th oppot to) a thal at oty ha a (hol fo th aa to th paa a lto th oppot to, laka

### D. Bertsekas and R. Gallager, "Data networks." Q: What are the labels for the x-axis and y-axis of Fig. 4.2?

pd by J. Succ ECE 543 Octob 22 2002 Outl Slottd Aloh Dft Stblzd Slottd Aloh Uslottd Aloh Splttg Algoths Rfc D. Btsks d R. llg "Dt twoks." Rvw (Slottd Aloh): : Wht th lbls fo th x-xs d y-xs of Fg. 4.2?

### Lectures # He-like systems. October 31 November 4,6

Lectue #5-7 7 Octoe 3 oveme 4,6 Self-conitent field Htee-Foc eqution: He-lie ytem Htee-Foc eqution: cloed-hell hell ytem Chpte 3, pge 6-77, Lectue on Atomic Phyic He-lie ytem H (, h ( + h ( + h ( Z Z:

### STUDY PACKAGE. Subject : Mathematics Topic : DETERMINANTS & MATRICES Available Online :

o/u opkj Hkh# tu] ugh vkjehks dke] oi s[k NksM+s qj e/;e eu dj ';kea iq#"k lg ldyi dj] lgs oi vusd] ^uk^ u NksM+s /;s; dks] j?kqj jk[ks VsdAA jp% ekuo /kez iz.ksk l~q# Jh j.knksm+klth egkjkt STUDY PAKAGE

### Some Different Perspectives on Linear Least Squares

Soe Dfferet Perspectves o Lear Least Squares A stadard proble statstcs s to easure a respose or depedet varable, y, at fed values of oe or ore depedet varables. Soetes there ests a deterstc odel y f (,,

### Lecture 3 summary. C4 Lecture 3 - Jim Libby 1

Lecue su Fes of efeece Ivce ude sfoos oo of H wve fuco: d-fucos Eple: e e - µ µ - Agul oeu s oo geeo Eule gles Geec slos cosevo lws d Noehe s heoe C4 Lecue - Lbb Fes of efeece Cosde fe of efeece O whch

### XII. Addition of many identical spins

XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.

### Numerical Analysis Topic 4: Least Squares Curve Fitting

Numerl Alss Top 4: Lest Squres Curve Fttg Red Chpter 7 of the tetook Alss_Numerk Motvto Gve set of epermetl dt: 3 5. 5.9 6.3 The reltoshp etwee d m ot e ler. Fd futo f tht est ft the dt 3 Alss_Numerk Motvto

### Multi-Electron Atoms-Helium

Multi-lecto Atos-Heliu He - se s H but with Z He - electos. No exct solutio of.. but c use H wve fuctios d eegy levels s sttig poit ucleus sceeed d so Zeffective is < sceeig is ~se s e-e epulsio fo He,

### ELECTROPHORESIS IN STRUCTURED COLLOIDS

ELKIN 4 ELECTROPHORESIS IN STRUCTURE COLLOIS José M. Médez A. Cvestv Mexo I ollboto wt O. Aló-Wess ULA d J. J. Beel-Mstett Cvestv. V µ E; µ 6πη ε ζ ; ζ 3 ε ζ ζ 4 THE GENERATION OF ONE PARTICLE EFFECTIVE

### M5. LTI Systems Described by Linear Constant Coefficient Difference Equations

5. LTI Systes Descied y Lie Costt Coefficiet Diffeece Equtios Redig teil: p.34-4, 245-253 3/22/2 I. Discete-Tie Sigls d Systes Up til ow we itoduced the Fouie d -tsfos d thei popeties with oly ief peview

### Born-Oppenheimer Approximation. Kaito Takahashi

o-oppehee ppoato Kato Takahah toc Ut Fo quatu yte uch a ecto ad olecule t eae to ue ut that ft the=tomc UNT Ue a of ecto (ot kg) Ue chage of ecto (ot coulob) Ue hba fo agula oetu (ot kg - ) Ue 4pe 0 fo

### ( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x

SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.

### Maximize: x (1.1) Where s is slack variable vector of size m 1. This is a maximization problem. Or (1.2)

A ew Algorth for er Progrg Dhy P. ehedle Deprtet of Eletro See, Sr Prhurhu College, lk Rod, Pue-00, d dhy.p.ehedle@gl.o Atrt- th pper we propoe ew lgorth for ler progrg. h ew lgorth ed o tretg the oetve

### Note 7 Root-Locus Techniques

Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 Note 7 Root-Locu Techique Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd Lecture Note

### Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:

Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the

### SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is

SPH3UW Uit 7.5 Sell s Lw Pge 1 of 7 Notes Physis Tool ox Refrtio is the hge i diretio of wve due to hge i its speed. This is most ommoly see whe wve psses from oe medium to other. Idex of refrtio lso lled

### Summary: Binomial Expansion...! r. where

Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly

### ROOT-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

### 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

### 20.2. The Transform and its Inverse. Introduction. Prerequisites. Learning Outcomes

The Trnform nd it Invere 2.2 Introduction In thi Section we formlly introduce the Lplce trnform. The trnform i only pplied to cul function which were introduced in Section 2.1. We find the Lplce trnform

### Induction. Induction and Recursion. Induction is a very useful proof technique

Iductio Iductio is vey useul poo techique Iductio d Recusio CSC-59 Discete Stuctues I compute sciece, iductio is used to pove popeties o lgoithms Iductio d ecusio e closely elted Recusio is desciptio method

### Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x

Setio Notes Pge Prtil Frtio Deompositio Suppose we were sked to write the followig s sigle frtio: We would eed to get ommo deomitors: You will get: Distributig o top will give you: 8 This simplifies to:

### Empirical equations for electrical parameters of asymmetrical coupled microstrip lines

Epl equons fo elel petes of syel ouple osp lnes I.M. Bsee Eletons eseh Instute El-h steet, Dokk, o, Egypt Abstt: Epl equons e eve fo the self n utul nutne n ptne fo two syel ouple osp lnes. he obne ptne

### ( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi

Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)

### 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

### BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.)

BINOMIAL THEOREM SOLUTION. (D) ( + + +... + ) (+ + +.) The coefficiet of + + + +... + fo. Moeove coefficiet of is + + + +... + if >. So. (B)... e!!!! The equied coefficiet coefficiet of i e -.!...!. (A),

### CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.

CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.

### Mathematical Notation Math Calculus & Analytic Geometry I

Mthemticl Nottio Mth - Clculus & Alytic Geometry I Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits shoul e emile to the istructor t jmes@richl.eu. Type your me t the top

### 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

### Artificial Intelligence Markov Decision Problems

rtificil Intelligence Mrkov eciion Problem ilon - briefly mentioned in hpter Ruell nd orvig - hpter 7 Mrkov eciion Problem; pge of Mrkov eciion Problem; pge of exmple: probbilitic blockworld ction outcome

### 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

### Lattice planes. Lattice planes are usually specified by giving their Miller indices in parentheses: (h,k,l)

Ltte ples Se the epol ltte of smple u ltte s g smple u ltte d the Mlle des e the oodtes of eto oml to the ples, the use s ey smple lttes wth u symmety. Ltte ples e usully spefed y gg the Mlle des petheses:

### Chapter #2 EEE Subsea Control and Communication Systems

EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8 EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio

### Stochastic simulation methods

tot ulto etod oue oepe: oltá éd U-lu lu Ro ed@p.ubblu.o o: ote-lo etod ttp://www.p.ubblu.o/~ed/.tl obetve o te oue: - letue ote - opute ode - ee poet - ppe to du - pott oueet - teet l o te web o ve PRAIA

### 12781 Velp Avenue. West County B Rural Residential Development

U PL & EET E 28 Vel ee eded 2 P.. ) LL EET T E 2) PPVE E ) ET E ) e e e e eded eebe 2 Plg & g eeg b) Bldg Pe e: eebe ) PUBL FU ( -E TE): g be bg bee e Plg & g eel ll be de ll be e. 5) UEETFEEBK: ) be ll

### BEM with Linear Boundary Elements for Solving the Problem of the 3D Compressible Fluid Flow around Obstacles

EM wth L ou Elts o olvg th Pol o th D opssl Flu Flow ou Ostls Lut Gu o Vlsu stt hs pp psts soluto o th sgul ou tgl quto o th D opssl lu low ou ostl whh uss sopt l ou lts o Lgg tp. h sgul ou tgl quto oult

### Course Updates. Reminders: 1) Assignment #8 available. 2) Chapter 28 this week.

Couse Updtes http://www.phys.hwii.edu/~vne/phys7-sp1/physics7.html Remindes: 1) Assignment #8 vilble ) Chpte 8 this week Lectue 3 iot-svt s Lw (Continued) θ d θ P R R θ R d θ d Mgnetic Fields fom long

### Asymptotic Dominance Problems. is not constant but for n 0, f ( n) 11. 0, so that for n N f

Asymptotc Domce Prolems Dsply ucto : N R tht s Ο( ) ut s ot costt 0 = 0 The ucto ( ) = > 0 s ot costt ut or 0, ( ) Dee the relto " " o uctos rom N to R y g d oly = Ο( g) Prove tht s relexve d trstve (Recll:

### Accuplacer Elementary Algebra Study Guide

Testig Ceter Studet Suess Ceter Aupler Elemetry Alger Study Guide The followig smple questios re similr to the formt d otet of questios o the Aupler Elemetry Alger test. Reviewig these smples will give

### The use of linear parametric approximation in numerical solving of nonlinear non-smooth Fuzzy equations

vlle ole t www.choleechl.co chve o ppled Scece Reech 5 6:49-6 http://choleechl.co/chve.htl ISSN 975-58X CODEN US SRC9 The ue o le petc ppoto uecl olvg o ole o-ooth uzz equto Mjd Hllj e Ze d l Vhd Kd Nehu

### Physics 604 Problem Set 1 Due Sept 16, 2010

Physics 64 Polem et 1 Due ept 16 1 1) ) Inside good conducto the electic field is eo (electons in the conducto ecuse they e fee to move move in wy to cncel ny electic field impessed on the conducto inside

### 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

### The limit comparison test

Roerto s Notes o Ifiite Series Chpter : Covergece tests Sectio 4 The limit compriso test Wht you eed to kow lredy: Bsics of series d direct compriso test. Wht you c ler here: Aother compriso test tht does