CHAPTER 38 MARKOV MODELLING CONTENTS
|
|
- Deborah Richards
- 5 years ago
- Views:
Transcription
1 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu CHAPTER 8 MARKOV MODELLING CONTENTS INTRODUCTION MARKOV ANALYSIS EXAMPLE OF MARKOV METHODS FOR ANALYSING THE RELIABILITY OF COMPLEX SYSTEMS 6 SOLUTION OF THE DIFFERENTIAL EQUATIONS DESCRIBING THE STATE PROBABILITIES, USING LAPLACE TRANSFORMS 5 rlad documn Error! Bookmark no dfind. Iu Pag
2 Char 8 Markov Modlling INTRODUCTION. Gnral.. Par C Char 0 of hi Manual dcrib Rliabiliy Block Diagram analyi, which i rha h mo familiar mhod of analying h funcional rlaionhi of a ym from a rliabiliy andoin. Par C Char 5 dcrib analyical man of rforming calculaion uing RBD, bu h calculaion ar only valid undr crain rriciv aumion.g. indndnc of block, no quuing for rair, c.... Thr ar modlling chniqu ha may b ud a ool o ovrcom om of h rricion. Thy may b groud ino wo gnral y, h Markov modl and h imulaion or Mon Carlo modl Par D Char. Boh modl u h RBD a a man of rrning h funcional lmn of a ym and hir inrrlaionhi. Boh ar alo ochaic modlling chniqu, maning ha hy can dal wih vn having an lmn of chanc and hnc a of oibl oucom a ood o drminiic modlling, whr a ingl oucom i drivd from a dfind of circumanc. In R&M nginring ochaic modlling i ud o dcrib a ym oraion wih rc o im. Th ubym failur and rair im yically bcom h random variabl... For many yar h iz and co of comur caabl of running all bu h iml imulaion modl limid hir u di h advanag ou in Tabl. In rcn yar, howvr, incra in comuing owr and aociad co dcra hav mad Mon Carlo imulaion radily accibl, and hnc h oulariy of Markov ha dclind. Howvr i i ill uful for h analyi of mulil a ym and ho ha xhibi rong dndncy bwn comonn and i ud in commrcial AR&M modlling ool ha u a raniion diagram o calcula rliabiliy and mainainabiliy valu for comlx ym... Ohr raon for h lack of oulariy of Markov ar: Th fac ha i i no an ay ool for nginr o aly and o xlain o ohr. Mon Carlo imulaion rogram xi ha no only modl comlx ym bu alo h u of h ym in comlx oraional cnario. Pag
3 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu MARKOV ANALYSIS MONTE CARLO SIMULATION ADVANTAGES DISADVANTAGES SOURCE A. Simliic Modlling Aroach. Th modl ar iml o gnra alhough hy do rquir a mor comlicad mahmaical aroach. D. Va incra in numbr of a a h iz of h ym incra. Th ruling diagram for larg ym ar gnrally vry xniv and comlx, difficul o conruc and comuaionally xniv. A: Rfrnc D: Rfrnc Vry flxibl. Thr i virually no limi o h analyi. Can gnrally b aily xndd and dvlod a rquird. A. Rdundancy Managmn Tchniqu. Sym rconfiguraion rquird by failur i aily incororad in h modl. D. Markov modlling of rdundan rairabl ym wih auomaic faul dcion and on rair crw i flawd. Thi i bcau alhough random failur i a Markov roc, rair of mulil failur i no a Markov Proc. Th mahmaical dicrancy may b ovrcom by uing a ddicad rair crw r quimn, bu hi do no normally corrond o ral lif uor ragi. A: Rfrnc D: Rfrnc Rdundancy managmn aily handld. A. Covrag. Covrd dcd and iolad and uncovrd undcd failur of comonn ar muually xcluiv vn, no aily modlld uing claical chniqu bu radily handld by Markov mahmaic. A: Rfrnc Covrd and uncovrd failur of comonn radily handld. A. Comlx mainnanc oion can radily b modlld. D. Can only dal wih conan failur ra and conan rair ra - h lar bing unraliic in ral, oraional ym for many raon including, for xaml, changing hyical condiion and variaion in mainnanc kill. Howvr, if h MTTR i vry much horr han h MTTF, hn hi horcoming rarly inroduc riou inaccuracy in h final comud ym aramr. A: Rfrnc D: Rfrnc Comlx mainnanc oion can radily b modlld. A wid rang of diribuion including mirical diribuion can b handld. D5. Fuur a of h ym ar indndn of all a a xc h immdialy rcding on, which imli ha a rair rurn h ym o an a nw condiion. 5D: Rfrnc Can incorora diribuion ha mbrac war-ou condiion. A6. Comlx Sym. Many imlifying chniqu xi which allow h modlling of comlx ym. A6: Rfrnc Comlx ym radily handld. A7. Squncd Evn. Markov modlling aily handl h comuaion of h robabiliy of an vn ruling from a qunc of ub-vn. Thi y of roblm do no lnd ilf wll o claical chniqu. A7: Rfrnc Rfrnc Squncd vn radily handld. Tabl : Advanag and limiaion of Markov Analyi comard o Mon Carlo Simulaion Iu Pag
4 Char 8 Markov Modlling MARKOV ANALYSIS. Gnral.. Markov analyi i a comlx ubjc wih many alicaion ouid h fild of R&M nginring. Mo chnical librari will hav vral book on h ubjc. I i covrd in hi manual inc i i an analyi mhod ha can b alid o crain rliabiliy roblm. Th mhod i bad on an analyi of h raniion bwn ym a. Markov analyi i illurad by xaml in Scion of hi Char... Th bai of a Markov modl i h aumion ha h fuur i indndn of h a, givn h rn. Thi ari from h udy of Markov chain qunc of random variabl in which h fuur variabl i drmind by h rn variabl bu i indndn of h way in which h rn a aro from i rdcor. Markov analyi look a a qunc of vn and analy h ndncy of on vn o b followd by anohr. Uing hi analyi, i i oibl o gnra a nw qunc of random bu rlad vn, which aar imilar o h original... A Markov chain may b dcribd a homognou or non-homognou. A homognou chain i characrid by conan raniion im bwn a. A nonhomognou chain i characrid by raniion ra bwn h a ha ar funcion of a global clock, for xaml, lad miion im. In R&M analyi a Markov modl may b ud whr vn, uch a h failur or rair of an im can occur a any oin in im. Th modl valua h robabiliy of moving from a known a o h nx logical a, i.. from vryhing working o h fir im faild, from h fir im faild o h cond im faild and o on unil h ym ha rachd h final or oally faild a.. Sym Sa and Truh Tabl.. A SYSTEM STATE i a aricular combinaion of h a of h lmn comriing h ym. For xaml, for a ym comriing wo lmn x and y, ach lmn caabl of aking on of wo a u or down, hr ar oibl ym a: a x u b x u c x down d x down y u y down y u y down.. In gnral, if lmn can b in on of m a, h numbr of oibl ym a for an n lmn ym i m n... Th li of all oibl ym a in rm of h lmn a i calld h TRUTH TABLE for h ym; a o d abov comri a ruh abl. Each lin of h ruh abl can b idnifid wih a ym condiion, u or down or dgradd. For xaml, if lmn x and y wr in ri, hn a would b a ym u a, and b, c and d would b down a. If x and y wr in a rdundan configuraion, hn a a, b and c would rrn ym u a and d h ym down a. Pag
5 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu. Markov Analyi and Traniion Sa Diagram.. Markov analyi comu h ra a which raniion occur bwn ym a from uch aramr a h lmn failur ra and/or rair ra. Thi i hn ud o comu ym aramr uch a MTBF, rliabiliy, availabiliy, c. Th mahmaic i illurad in Scion of hi Char... Gnrally Markov analyi in rliabiliy alicaion i confind o h iuaion whr h diribuion of lmn failur and rair im i ngaiv xonnial. I i alo gnrally aumd ha wo lmn canno chang hir a imulanouly. Thu, for xaml, h lmn ym of aragrah.. canno chang from a a o a d a on im, inc hi would rquir x and y o fail imulanouly. Poibl raniion bwn ym a and h ra a which hy occur ar indicad in Traniion Sa Diagram, a hown in Scion... Di h limiaion h chniqu i of valu inc uch aumion ar frqunly mad in rliabiliy work, and i can handl iuaion whr h failur and rair im diribuion of h lmn ar no indndn a i h ca wih andby ym. Iu Pag 5
6 Char 8 Markov Modlling EXAMPLE OF MARKOV METHODS FOR ANALYSING THE RELIABILITY OF COMPLEX SYSTEMS. Inroducion.. Thi Scion dcrib, by man of an xaml, a mhod of analying h rliabiliy of comlx ym. Alhough h xaml chon i of a non-rairabl andby ym, i i rlaivly raighforward o ada h mhod o modl rairabl ym... Th analyi chniqu mloy om of h ida ud in h analyi of Markov Proc. An imoran aumion rquird by and limiaion of h mhod i ha h failur ra of h lmn comriing h ym ar conan for rairabl ym, h rair ra mu alo b conan, i.. h failur im and rair im diribuion mu b ngaiv xonnial. In ral im oraional iuaion hi may b unraliic and car mu b akn whn alying hi chniqu.. Th Analyi Mhod.. Conidr h ym hown in, comriing lmn in andby rdundancy. L h failur ra of lmn b, and h failur ra of lmn b in h oraional a and in h andby a. Figur : A Two Elmn Sandby Rdundancy Sym.. Now hi ym can occuy on of four a: and u, down, u, u, down, down, down, Th ym a may b rrnd diagrammaically a: Pag 6
7 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu,,,, Tabl : Th Sym Sa.. L h robabiliy of h ym bing in a a im Conidr h robabiliy of h ym bing in a a im Δ. Th following rlaionhi hold auming ha Δ i ufficinly mall ha h robabiliy of wo or mor raniion aking lac in h im inrval Δ i ngligibl. Δ [ ] inc h robabiliy of bing in a a Δ i h roduc of h robabiliy ha h ym wa in a a im and h robabiliy ha nihr of lmn and faild in Δ. Similarly: Th diagram can b illurad by a diagram calld h TRANSITION STATE DIAGRAM for h ym. Iu Pag 7
8 Char 8 Markov Modlling,,,, Figur : Traniion Sa Diagram No: i ii h arrow indica h dircion of raniion bwn ym a. i an aborbing a, i.. onc nrd, h ym canno lav i... Equaion can b r-arrangd a follow: Pag 8
9 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu Δ Δ Δ Δ Δ Δ Δ Δ Taking h limi a yild: 0 & & & & Whr d d &..5 Th diffrnial quaion can b olvd by aking Lalac Tranform, a hown in Andix. Th oluion ar: Iu Pag 9
10 Char 8 Markov Modlling..6 Th rliabiliy of h ym a im i h robabiliy ha h ym i in a, or : i.. R..7 Thr ar ohr mhod of analying h ym dcribd abov. On advanag of hi mhod howvr i ha givn ha h ym can b dcribd in a raniion a diagram uch a ha hown in Fig, h mhod i comlly gnral and can b incororad in a comur rogram. Thorically, a ym of any comlxiy may b analyd in hi way, alhough in racic hr will b limiaion imod by h iz of h ym and h comur rogram o analy i i.. for a ym comriing n lmn hr will b n ym a if, howvr, om of h lmn ar idnical, hi numbr may b rducd...8 A ad in h inroducion, h mhod can b adad aily o h analyi of rairabl ym. Conidr h am lmn andby rdundancy ym whr lmn and hav rair ra μ and μ rcivly. Th raniion a diagram i MTBF now: Pag 0
11 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu μ,, μ μ, μ μ, μ Figur : Traniion Sa Diagram for a Rairabl Sandby Sym No: Thr i no raniion allowd from o. Thi ari from h aumion ha a aiv failur of lmn i.. a raniion from o will no b dcd unil h lmn i rquird for oraion i.. whn lmn fail. Hnc no rair acion i akn on h aiv failur of lmn, and hnc hr can b no raniion from a o...9 Th raniional robabiliy quaion imilar o can now b u and olvd a bfor. I hould b nod ha, in hi ca, bcau w ar daling wih a rairabl ym, h um givn by rrn h availabiliy and no h rliabiliy of h ym a im. Th rliabiliy of h ym i.. i robabiliy of urvival o im R may b calculad by modifying h raniion a diagram o ha hown in Fig 5. Th raon for raing h calculaion of rliabiliy R in hi way ari from h fac ha rliabiliy i h robabiliy ha a ym will no fail in a givn riod of im. Iu Pag
12 Char 8 Markov Modlling μ,, μ,, Figur 5: Traniion Sa Diagram for a Rairabl Sandby Sym o Calcula i Rliabiliy R A gnral rul for h conrucion of raniion a diagram i ha for availabiliy calculaion, all allowabl rair raniion hould b includd, whra for rliabiliy calculaion h ym down a,.g. a in Fig 5, hould b rad a aborbing a, i.. onc nrd, h ym canno lav hm. Pag
13 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu SOLUTION OF THE DIFFERENTIAL EQUATIONS DESCRIBING THE STATE PROBABILITIES, USING LAPLACE TRANSFORMS. Inroducion.. Thi aragrah dcrib how o olv h diffrnial quaion lablld in aragrah, bu hr. Th quaion wr: & & & & whr & d d.. Th mhod of oluion adod hr i ha uing Lalac Tranform. Th bai of h mhod i no dcribd hr, xc o ay ha i convr diffrnial quaion o algbraic form by man of a ranformaion dicovrd by Lalac. Th algbraic quaion may hn b aily olvd, h invr of h ranformaion alid o obain h final oluion. Som Lalac Tranform ar givn in Tabl.. Soluion.. L L i b h Lalac Tranform of & i * and ak Lalac Tranform of quaion. Now h Lalac Tranform of & i i givn by Li i 0. If, a 0, boh lmn and ar u hn h ym will b in a aragrah.., i.. 0, * L τ τ i i dτ 0 Iu Pag
14 Char 8 Markov Modlling L L L L L L L L L L L L L L L L L L L For h objc of h xrci i o olv h quaion algbraically for h L, hn ak invr Lalac Tranform o obain oluion for h, i o in hi ca. In roducing h xrion of L i i ncary o u i ino a form uiabl for h invr ranformaion. Such form can b obaind from abl of Lalac Tranform.g. Tabl. In hi ca h uiabl form i: whr A, B,.. Thu from : A B L i c α β α, β do no involv. L * i L L i u ino h form of quaion a follow: A B Sinc hi i an idniy w hav: Pag
15 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu Iu Pag 5 B B A A Equaing cofficin of on boh id yild: A B 0 i.. B A Inring i yil h in 6 d: A A A Thrfor from 5 L * Similarly L L * Th corrc form for i mo aily obaind from h xrion: L L L L L which ilf driv from h fac ha L L *.. Th oluion now com from h quaion lablld *. Th invr Lalac Tranform of L i by dfiniion, ha of / α A i A α, and ha of / i Tabl. Hnc:
16 Char 8 Markov Modlling Pag Thu h xrion lablld 7 abov ar h oluion of h diffrnial quaion lablld in aragrah...
17 Alid R&M Manual for Dfnc Sym Par C R&M Rlad Tchniqu x n n a oiiv ingr n n! a in a a coa inh a > a a a a coh a > a a a a [ in a a.co ] in a a a a a a Lalac Tranform of x d x.lx0 whn L i Lalac d Tranform of x d d n n x n n n n dx d x. L x0... n d 0 d 0 d i x whr i h valu of h i h d 0 drivaiv of x a 0 Tabl : Tabl of Common Lalac Tranform No: Th Lalac Tranform of x i dfind by: L 0 x d Sciali book will rovid mor comrhniv abl of ranform. Iu Pag 7
18 Char 8 Markov Modlling 5 RELATED DOCUMENTS. Alicabiliy of Markov Analyi Mhod o Rliabiliy, Mainainabiliy and Safy. by Norman B. Fuqua. Publihd in h Rliabiliy Analyi Cnr - Slcd Toic in Auranc Rlad Tchnologi START Volum 0 No.. Rairabl Rdundan Sym and h Markov Fallacy by W G Gulland and Rliabiliy Amn of Rairabl Sym I Markov Modlling Corrc? by KGL Simon and M Klly. Publihd in Safy and Rliabiliy Sociy Journal Volum No.. Briih Sandard Rliabiliy of ym, quimn and comonn. Par :99. Scion. Calculaing robabiliy of Failur of Elcronic and Elcrical Sym Markov v. FTA by Vio Faraci Jr. Publihd in h Journal of h Rliabiliy Analyi Cnr, Third Quarr 00. Pag 8
Estimation of Mean Time between Failures in Two Unit Parallel Repairable System
Inrnaional Journal on Rcn Innovaion rnd in Comuing Communicaion ISSN: -869 Volum: Iu: 6 Eimaion of Man im bwn Failur in wo Uni Paralll Rairabl Sym Sma Sahu V.K. Paha Kamal Mha hih Namdo 4 ian Profor D.
More informationTransfer function and the Laplace transformation
Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and
More informationChapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System
EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +
More informationPoisson process Markov process
E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc
More informationPFC Predictive Functional Control
PFC Prdiciv Funcional Conrol Prof. Car d Prada D. of Sm Enginring and Auomaic Conrol Univri of Valladolid, Sain rada@auom.uva. Oulin A iml a oibl Moivaion PFC main ida An inroducor xaml Moivaion Prdiciv
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion
More informationInstructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems
Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi
More informationLaPlace Transform in Circuit Analysis
LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz
More informationBoyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors
Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar
More informationFinal Exam : Solutions
Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b
More information2. The Laplace Transform
Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin
More informationElementary Differential Equations and Boundary Value Problems
Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationREPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.
Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[
More informationLaplace Transforms recap for ccts
Lalac Tranform rca for cc Wha h big ida?. Loo a iniial condiion ron of cc du o caacior volag and inducor currn a im Mh or nodal analyi wih -domain imdanc rianc or admianc conducanc Soluion of ODE drivn
More informationMa/CS 6a Class 15: Flows and Bipartite Graphs
//206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d
More informationWhy Laplace transforms?
MAE4 Linar ircui Why Lalac ranform? Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A R V A _ v
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More information2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35
MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h
More informationA Method of Performance Assessment of PID Controller with Actuator Saturation
Inrnaional Confrnc on Mcharonic, Elcronic, Indurial and Conrol Enginring (MEIC 5 A Mhod of rformanc Amn of ID Conrollr wih Acuaor Sauraion Xia Hao Faculy of Elcronic and Elcrical Enginring Dalian Univriy
More informationLecture 26: Leapers and Creepers
Lcur 6: Lapr and Crpr Scrib: Grain Jon (and Marin Z. Bazan) Dparmn of Economic, MIT May, 005 Inroducion Thi lcur conidr h analyi of h non-parabl CTRW in which h diribuion of p iz and im bwn p ar dpndn.
More informationMidterm exam 2, April 7, 2009 (solutions)
Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions
More informationUsing Degradation Models to Assess Pipeline Life
Saiic Prrin Saiic 10-2014 Uing Dgradaion Modl o A Pilin Lif Shiyao Liu Iowa Sa Univriy William Q. Mkr Iowa Sa Univriy, wqmkr@iaa.du Follow hi and addiional work a: h://lib.dr.iaa.du/a_la_rrin Par of h
More information14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions
4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway
More informationChapter 12 Introduction To The Laplace Transform
Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and
More informationChap.3 Laplace Transform
Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl
More informationVoltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!
Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr
More informationInstitute of Actuaries of India
Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6
More informationCSE 245: Computer Aided Circuit Simulation and Verification
CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy
More informationwhere: u: input y: output x: state vector A, B, C, D are const matrices
Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &
More informationDecline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline.
Dlin Curvs Dlin Curvs ha lo flow ra vs. im ar h mos ommon ools for forasing roduion and monioring wll rforman in h fild. Ths urvs uikly show by grahi mans whih wlls or filds ar roduing as xd or undr roduing.
More informationPhys463.nb Conductivity. Another equivalent definition of the Fermi velocity is
39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h
More informationa dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system:
Undrdamd Sysms Undrdamd Sysms nd Ordr Sysms Ouu modld wih a nd ordr ODE: d y dy a a1 a0 y b f If a 0 0, hn: whr: a d y a1 dy b d y dy y f y f a a a 0 0 0 is h naural riod of oscillaion. is h daming facor.
More informationLecture 4: Laplace Transforms
Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions
More informationUNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o
More informationRatio-Product Type Exponential Estimator For Estimating Finite Population Mean Using Information On Auxiliary Attribute
Raio-Produc T Exonnial Esimaor For Esimaing Fini Poulaion Man Using Informaion On Auxiliar Aribu Rajsh Singh, Pankaj hauhan, and Nirmala Sawan, School of Saisics, DAVV, Indor (M.P., India (rsinghsa@ahoo.com
More informationOn the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument
Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn
More informationLecture 2: Current in RC circuit D.K.Pandey
Lcur 2: urrn in circui harging of apacior hrough Rsisr L us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R and a ky K in sris. Whn h ky K is swichd on, h charging
More informationCharging of capacitor through inductor and resistor
cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.
More informationBoyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues
Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm
More informationApplied Statistics and Probability for Engineers, 6 th edition October 17, 2016
Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...
More informationMath 266, Practice Midterm Exam 2
Mh 66, Prcic Midrm Exm Nm: Ground Rul. Clculor i NOT llowd.. Show your work for vry problm unl ohrwi d (pril crdi r vilbl). 3. You my u on 4-by-6 indx crd, boh id. 4. Th bl of Lplc rnform i vilbl h l pg.
More informationH is equal to the surface current J S
Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationSpring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an
More informationLet s look again at the first order linear differential equation we are attempting to solve, in its standard form:
Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,
More information1 Finite Automata and Regular Expressions
1 Fini Auom nd Rgulr Exprion Moivion: Givn prn (rgulr xprion) for ring rching, w migh wn o convr i ino drminiic fini uomon or nondrminiic fini uomon o mk ring rching mor fficin; drminiic uomon only h o
More informationA THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER
A THREE COPARTENT ATHEATICAL ODEL OF LIVER V. An N. Ch. Paabhi Ramacharyulu Faculy of ahmaics, R D collgs, Hanamonda, Warangal, India Dparmn of ahmaics, Naional Insiu of Tchnology, Warangal, India E-ail:
More informationCHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano
Expcaions: Th Basic Prpard by: Frnando Quijano and Yvonn Quijano CHAPTER CHAPTER14 2006 Prnic Hall Businss Publishing Macroconomics, 4/ Olivir Blanchard 14-1 Today s Lcur Chapr 14:Expcaions: Th Basic Th
More informationGeneral Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract
Applicaion of Diffrnial... Gnral Aricl Applicaion of diffrnial uaion in - and C- circui analysis by classical mhod. ajndra Prasad gmi curr, Dparmn of Mahmaics, P.N. Campus, Pokhara Email: rajndraprasadrgmi@yahoo.com
More informationCopyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin
More informationDouble Slits in Space and Time
Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an
More informationCPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees
CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs
More informationPWM-Scheme and Current ripple of Switching Power Amplifiers
axon oor PWM-Sch and Currn rippl of Swiching Powr Aplifir Abrac In hi work currn rippl caud by wiching powr aplifir i analyd for h convnional PWM (pulwidh odulaion) ch and hr-lvl PWM-ch. Siplifid odl for
More informationLecture 2: Bayesian inference - Discrete probability models
cu : Baysian infnc - Disc obabiliy modls Many hings abou Baysian infnc fo disc obabiliy modls a simila o fqunis infnc Disc obabiliy modls: Binomial samling Samling a fix numb of ials fom a Bnoulli ocss
More information5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t
AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =
More informationAR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )
AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc
More informationFrequency Response. Response of an LTI System to Eigenfunction
Frquncy Rsons Las m w Rvsd formal dfnons of lnary and m-nvaranc Found an gnfuncon for lnar m-nvaran sysms Found h frquncy rsons of a lnar sysm o gnfuncon nu Found h frquncy rsons for cascad, fdbac, dffrnc
More informationEXERCISE - 01 CHECK YOUR GRASP
DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc
More informationFirst Lecture of Machine Learning. Hung-yi Lee
Firs Lcur of Machin Larning Hung-yi L Larning o say ys/no Binary Classificaion Larning o say ys/no Sam filring Is an -mail sam or no? Rcommndaion sysms rcommnd h roduc o h cusomr or no? Malwar dcion Is
More informationWave Equation (2 Week)
Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris
More informationPartial Fraction Expansion
Paial Facion Expanion Whn ying o find h inv Laplac anfom o inv z anfom i i hlpfl o b abl o bak a complicad aio of wo polynomial ino fom ha a on h Laplac Tanfom o z anfom abl. W will illa h ing Laplac anfom.
More information3(8 ) (8 x x ) 3x x (8 )
Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 =.7 8 -. a d.6 6 6 6 6 8 8 8 b 9 d 6 6 6 8 c d.8 6 6 6 6 8 8 7 7 d 6 d.6 6 6 6 6 6 6 8 u u u u du.9 6 6 6 6 6
More informationTo become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship
Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,
More informationChapter 6. PID Control
Char 6 PID Conrol PID Conrol Mo ommon onrollr in h CPI. Cam ino u in 930 wih h inroduion of numai onrollr. Exrmly flxibl and owrful onrol algorihm whn alid rorly. Gnral Fdbak Conrol Loo D G d Y E C U +
More informationMath 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2
Mah 0 Homwork S 6 Soluions 0 oins. ( ps) I ll lav i o you o vrify ha y os sin = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add s ono h las wo rms sin hos ar xaly h omplimnary
More informationCIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 1/8 ISOPARAMERIC ELEMENS h inar rianguar mn and h iinar rcanguar mn hav vra imporan diadvanag. 1. Boh mn ar una o accura rprn curvd oundari, and. h provid
More informationReview Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )
Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division
More information4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b
4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs
More information6.8 Laplace Transform: General Formulas
48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy
More informationMinistry of Education and Science of Ukraine National Technical University Ukraine "Igor Sikorsky Kiev Polytechnic Institute"
Minisry of Educaion and Scinc of Ukrain Naional Tchnical Univrsiy Ukrain "Igor Sikorsky Kiv Polychnic Insiu" OPERATION CALCULATION Didacic marial for a modal rfrnc work on mahmaical analysis for sudns
More informationINTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS
adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC
More informationAn Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT
[Typ x] [Typ x] [Typ x] ISSN : 974-7435 Volum 1 Issu 24 BioTchnology 214 An Indian Journal FULL PAPE BTAIJ, 1(24), 214 [15197-1521] A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss LI
More informationMEM 355 Performance Enhancement of Dynamical Systems A First Control Problem - Cruise Control
MEM 355 Prformanc Enhancmn of Dynamical Sysms A Firs Conrol Problm - Cruis Conrol Harry G. Kwany Darmn of Mchanical Enginring & Mchanics Drxl Univrsiy Cruis Conrol ( ) mv = F mg sinθ cv v +.2v= u 9.8θ
More informationwhereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas
Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically
More informationLaplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)
Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful
More information( ) ( ) + = ( ) + ( )
Mah 0 Homwork S 6 Soluions 0 oins. ( ps I ll lav i o you vrify ha h omplimnary soluion is : y ( os( sin ( Th guss for h pariular soluion and is drivaivs ar, +. ( os( sin ( ( os( ( sin ( Y ( D 6B os( +
More informationB) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of
Sampl Final 00 1. Suppos z = (, y), ( a, b ) = 0, y ( a, b ) = 0, ( a, b ) = 1, ( a, b ) = 1, and y ( a, b ) =. Thn (a, b) is h s is inconclusiv a saddl poin a rlaiv minimum a rlaiv maimum. * (Classiy
More informationI) Title: Rational Expectations and Adaptive Learning. II) Contents: Introduction to Adaptive Learning
I) Til: Raional Expcaions and Adapiv Larning II) Conns: Inroducion o Adapiv Larning III) Documnaion: - Basdvan, Olivir. (2003). Larning procss and raional xpcaions: an analysis using a small macroconomic
More informationLecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey
cur : Growh and dcay of currn in circui Growh of currn in Circui us considr an inducor of slf inducanc is conncd o a DC sourc of.m.f. E hrough a rsisr of rsisanc and a ky K in sris. Whn h ky K is swichd
More informationEngineering Differential Equations Practice Final Exam Solutions Fall 2011
9.6 Enginring Diffrntial Equation Practic Final Exam Solution Fall 0 Problm. (0 pt.) Solv th following initial valu problm: x y = xy, y() = 4. Thi i a linar d.. bcau y and y appar only to th firt powr.
More informationChapter 9 - The Laplace Transform
Chaper 9 - The Laplace Tranform Selece Soluion. Skech he pole-zero plo an region of convergence (if i exi) for hee ignal. ω [] () 8 (a) x e u = 8 ROC σ ( ) 3 (b) x e co π u ω [] ( ) () (c) x e u e u ROC
More information[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then
SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd
More informationFailure Load of Plane Steel Frames Using the Yield Surface Method
ISBN 978-93-84422-22-6 Procdings of 2015Inrnaional Confrnc on Innovaions in Civil and Srucural Enginring (ICICSE'15) Isanbul (Turky), Jun 3-4, 2015. 206-212 Failur Load of Plan Sl Frams Using h Yild Surfac
More information1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to:
Rfrncs Brnank, B. and I. Mihov (1998). Masuring monary policy, Quarrly Journal of Economics CXIII, 315-34. Blanchard, O. R. Proi (00). An mpirical characrizaion of h dynamic ffcs of changs in govrnmn spnding
More information14.02 Principles of Macroeconomics Problem Set 5 Fall 2005
40 Principls of Macroconomics Problm S 5 Fall 005 Posd: Wdnsday, Novmbr 6, 005 Du: Wdnsday, Novmbr 3, 005 Plas wri your nam AND your TA s nam on your problm s Thanks! Exrcis I Tru/Fals? Explain Dpnding
More informationConsider a system of 2 simultaneous first order linear equations
Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm
More informationOn the Speed of Heat Wave. Mihály Makai
On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.
More informationPERIODICAL SOLUTION OF SOME DIFFERENTIAL EQUATIONS UDC 517.9(045)=20. Julka Knežević-Miljanović
FCT UNIVESITTIS Sri: chanic uomaic Conrol and oboic Vol. N o 7 5. 87-9 PEIODICL SOLUTION OF SOE DIFFEENTIL EQUTIONS UDC 57.95= Julka Knžvić-iljanović Facul o ahmaic Univri o lgrad E-mail: knzvic@oincar.ma.bg.ac.u
More informationLecture 4: Parsing. Administrivia
Adminitrivia Lctur 4: Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More information5.2 GRAPHICAL VELOCITY ANALYSIS Polygon Method
ME 352 GRHICL VELCITY NLYSIS 52 GRHICL VELCITY NLYSIS olygon Mehod Velociy analyi form he hear of kinemaic and dynamic of mechanical yem Velociy analyi i uually performed following a poiion analyi; ie,
More informationMixing Real-Time and Non-Real-Time. CSCE 990: Real-Time Systems. Steve Goddard.
CSCE 990: Ral-Tim Sym Mixing Ral-Tim and Non-Ral-Tim goddard@c.unl.du hp://www.c.unl.du/~goddard/cour/raltimsym 1 Ral-Tim Sym Mixd Job - 1 Mixing Ral-Tim and Non-Ral-Tim in Prioriy-Drivn Sym (Chapr 7 of
More informationThe University of Alabama in Huntsville Electrical and Computer Engineering Homework #4 Solution CPE Spring 2008
Th Univrsity of Alabama in Huntsvill Elctrical and Comutr Enginring Homwork # Solution CE 6 Sring 8 Chatr : roblms ( oints, ( oints, ( oints, 8( oints, ( oints. You hav a RAID systm whr failurs occur at
More informationfiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are
MTEMTICL PHYSICS SOLUTIONS GTE- Q. Considr an ani-symmric nsor P ij wih indics i and j running from o 5. Th numbr of indpndn componns of h nsor is 9 6 ns: Soluion: Th numbr of indpndn componns of h nsor
More informationS.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]
S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:
More informationCS4445/9544 Analysis of Algorithms II Solution for Assignment 1
Conider he following flow nework CS444/944 Analyi of Algorihm II Soluion for Aignmen (0 mark) In he following nework a minimum cu ha capaciy 0 Eiher prove ha hi aemen i rue, or how ha i i fale Uing he
More informationThe transition:transversion rate ratio vs. the T-ratio.
PhyloMah Lcur 8 by Dan Vandrpool March, 00 opics of Discussion ransiion:ransvrsion ra raio Kappa vs. ransiion:ransvrsion raio raio alculaing h xpcd numbr of subsiuions using marix algbra Why h nral im
More informationTHE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES *
Iranian Journal of Scinc & Tchnology, Tranacion A, Vol, No A Prind in h Ilamic Rpublic of Iran, 009 Shiraz Univriy THE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES E KASAP, S YUCE AND N KURUOGLU Ondokuz
More informationAn Improved Anti-windup Control Using a PI Controller
05 Third nernaional Conference on Arificial nelligence, Modelling and Simulaion An mroved Ani-windu Conrol Uing a P Conroller Kyohei Saai Graduae School of Science and Technology Meiji univeriy Kanagawa,
More informationMidterm Examination (100 pts)
Econ 509 Spring 2012 S.L. Parn Midrm Examinaion (100 ps) Par I. 30 poins 1. Dfin h Law of Diminishing Rurns (5 ps.) Incrasing on inpu, call i inpu x, holding all ohr inpus fixd, on vnuall runs ino h siuaion
More information10.5 Linear Viscoelasticity and the Laplace Transform
Scn.5.5 Lnar Vclacy and h Lalac ranfrm h Lalac ranfrm vry uful n cnrucng and analyng lnar vclac mdl..5. h Lalac ranfrm h frmula fr h Lalac ranfrm f h drvav f a funcn : L f f L f f f f f c..5. whr h ranfrm
More information