# Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach to solving systms. W thn continu on th xponntial matrix.. Variation of Paramtrs 2. Eulr s Mthod 3. Exponntial Matrix 4. Diagonalization Non-Homognous Systms A nonhomognous first-ordr linar systm of diffrntial quations has th form y = P(ty + g(t, whr g(t is a nonzro matrix. P(t is a squar matrix, but th othr symbols y, y, and g(t dnot column matrics. (This is in kping with th notation introducd at th bginning of th chaptr that column matrics would appar in bold lowrcas lttrs, but squar matrics would appar in ITALIC CAPITAL lttrs. In our study of nonhomognous quations from prvious chaptrs, w notd that th solution to a nonhomognous quation took th form y = y C + y P, whr y C rprsntd th complmntary solution th gnral solution to th rlatd homognous quation, and y P rprsntd a particular solution to th nonhomognous part of th quation.

2 Exampl: Considr th systm of diffrntial quations: 5 3t + y = y + t 3 2t t 2 Show that y = 3t 5 t t 3 + t is a solution to th systm. Find th gnral solution to th abov systm: 2 Solving Nonhomognous Equations In sctions 3.8 and 3., w discussd two tchniqus for solving nonhomognous highr-ordrd quations: th Mthod of Undtrmind Cofficints, and Variation of Paramtrs. Both tchniqus xtnd to nonhomognous systms, although th Mthod of Undtrmind Cofficints is a much wakr mthod of solution in th cas of homognous systms. (Exampl, p. 278, and th discussion immdiatly following this xampl point out th waknsss in th mthod. Thrfor, w ll focus our attntion on th mthod of Variation of Paramtrs. 2

3 2. Varation of Paramtrs W know that th complmntary solution has th form y C = Ψ(t c, whr Ψ(t dnots th solution matrix a squar matrix whos columns form a fundamntal st and c dnots a column of constants In th Variation of Paramtrs mthod, w rplac th constants c,..., c n, with functions u,...,u n, and solv th rsulting quation. Whn w usd this mthod in sction \$3., w discovrd that it was asir to solv first for thir drivativs, and thn intgrat; this continus to b tru. Howvr, solving first for th drivativs is complicatd by th fact that w ll hav to find th invrs of th squar matrix Ψ(t in th procss. c. c n. Exampl 3 Using variation of paramtrs, solv th initial valu problm: Solution: y = 2 2 2t y + 2t, y( = First, w find th complmntary solution to th homognous quation:. Find th ignvalus: λ = and λ = Find an ignvctor for ach ignvalu. Whn λ =, th ignvctor is, and whn λ = 3, th ignvctor is. 3. Writ th complmntary solution: y C (t = t t 3t 3t c c 2 To find th particular solution to th nonhomognous quation:. Comput th invrs of th solution matrix: 3

4 t 3t t 3t 2. Multiply th invrs of th solution matrix by th matrix g(t: 2 t 2 t 2t = 2t 2 3t 2 3t 3. Find th antidrivativ of Stp 2: (W usd intgration by parts hr. 2 3t + t t 2 t t 3t dt = 3t 6 t + t t t 2 + 3t + t 3t 3 4. Multiply th complmntary solution matrix Ψ(t by th rsult of stp 3: 3t y P (t = t 3t t 3t 6 t + t t = 2t 3 + 4t 3 8 t 2 + 3t + t 3t 22t 3 3 2t Combin th complmntary solution with th particular solution: y(t = y C (t + y P (t = t 3t c 2t t 3t t 3 8 c 2 22t 3 2t Us th initial condition to solv for c and c 2 : 4

5 At tim t =, y( = c c = Solving this systm of quations, w gt c = 5 and c 6 2 = 7, so: 8 5 y(t = t 3t 6 2t t 3t t t 8 3 2t 3 + 5

6 Exampl 3: A shortcut If w did not hav an initial condition, stps 5 abov would find th gnral solution. Howvr, w can us th initial condition as a limit of intgration in Stp 3 to sav us som work: 3. Intgrat th product ovr th intrval t, t. Notic that sinc t is on of our ndpoints, w ll hav to us a diffrnt variabl of intgration: t 2 3s + s s 3t ds = 6 t + t t s s 3s t 2 + 3t + t 3t Multiply th invrs of th solution matrix Ψ(t (w r valuating th invrs at t by th initial condition. Add this rsult to Stp = 2 2 3t 6 t + t t t 2 + 3t + t 3t = 3t 6 t + t t t 2 + 3t + t 3t Multiply th complmntary solution matrix Ψ(t by th rsult of stp 4. This complts th procss w don t hav to solv for c and c 2. 3t y(t = t 3t t 3t 6 t + t t t 2 + 3t + t 3t + 7 = 8 3t 2t t + 4t t 2 3 2t 5 6 t 2t 3 + (Chck that this is th sam solution w obtaind on th prvious pag. 6

7 p. 286, #7 Solv th initial valu problm: y = y + y( = 3 Eulr s Mthod Rcall that Eulr s mthod givs an approximat solution to an initial valu problm givn in th form y = f(t, y, y(t = y. W choos a stp siz h, and w will approximat th solution y at a sris of qually spacd points t = t + h, t 2 = t + 2h,..., t n = t + nh. W will us y i to dnot th approximation at t i, so y(t i y i. W gt from th approximation at t i to th approximation at t i+ using th formula y i+ = y i + hf(t i, y i which uss th linarization of y to approximat th chang in going from t i to t i+. Exampl: Suppos w wish to approximat th solution to th first ordr non-linar quation y = cos(y + t, y( =, using a stp siz h =.. W know that thr xists a uniqu solution for all t, sinc both f and f y ar continuous vrywhr. Thus w start with our initial condition, y( =. Thn w can approximat y(t = y(. y = y( + hf(, W can thn us this to approximat y(.2: = +.(cos( +.46 y(.2 y 2 =.46 + (.(cos( W can continu for as many stps as w nd, approximating y(t i+ using t i and y(t i. Eulr s mthod can b gnralizd asily to systms of first ordr quations. Eulr s mthod uss th linarization for y at ach point t i to prdict th rsult at t i+. If 7

8 w now considr a function x (which is a vctor valud function, w can still us a linarization at th point t i to prdict th valu of x at t i+. W will still us th sam formula, but it will now b basd on th vctor valud function x: x(t i+ x(t i + hx (t i Exampl: Suppos w wish to solv th following systm: x = 3 2 x x 2 x 2 = 3 + x 5 x 2 with initial conditions x ( = 25 and x 2 ( = 5. (Although w will larn how to solv homognous systms shortly, w will not b studying how to find xact solutions to non-homognous systms such as this. W will approximat th solution numrically using Eulr s mthod. In this cas, our initial condition for our vctor valud function x is ( 25 x( =, 5 and w know that x = ( 3 2 x x x 5 x 2 Thrfor, if w choos h =., w gt th following approximations: ( ( 3 25 x(. + ( ( = ( ( x(.2 + ( ( If w continu this procss, w gt th following tabl of approximations for th valus of x and x 2 : t x x

9 W can of cours us this tchniqu with a non-linar systm as wll: Exampl: Suppos w hav th initial valu problm givn by th systm x = x x 2 + t x 2 = 2x x 2 with initial conditions x ( =, x 2 ( =. Lt s us Eulr s mthod with stp siz h =.2. W will dnot th approximation of x i at stp j by x ij. So our initial conditions ar x = and x 2 =, and x will b th approximation of x (t at t =.2. Thus, for our first stp w gt th following: and x = x + (.2( x x 2 + t = + (.2( + =.2 x 2 = x 2 + (.2(2x x 2 = + (.2(2 =.2 Not that w ndd both x and x 2 to calculat th approximation at th nxt stp for ithr function. W can thn procd to x 2 x (.4 and x 22 x 2 (.4: whil x 2 = x + (.2( x x 2 + t =.2 + (.2( x 22 = x 2 + (.2(2x x 2 =.2 + (.2 (2(.2(.2.46 W can continu as w nd to, using our prvious approximations for x and x 2 to gnrat nw approximations. W can now of cours also solv highr ordr diffrntial quations using Eulr s mthod by first rwriting th quation as a systm of first ordr quations. Exampl: Approximat th solution to y + y = y + t with initial conditions y( = and y ( =, using Eulr s mthod with a stp siz of h =.25. W bgin by rwriting th quation as a systm. W lt u = y, and u 2 = y. Thn w hav first th quation u = u 2

10 and scond th quation u 2 + u 2 = u + t or u 2 = u u 2 + t In matrix form our quation is ( u = u 2 ( ( u u 2 + ( t Our initial conditions ar thn u ( = and u 2 ( =, or ( u =. Eulr s mthod bcoms u k+ = u k +.25 So w st out to find u and u 2 : ( ( u k + tk u = ( ( +.25 ( + ( = ( ( +.25 = ( u 2 = ( ( +.25 ( (.25 = ( ( = ( u = u 2 = (initial conditions W tak th first stp: u = u + (.25(u 2 = +.25 =.25 u 2 = u 2 + (.25(u u 2 + t = + (.25( + =.75 W thn tak a scond stp: u 2 = u + (.25(u 2 =.25 + (.25(.75 =.4375 u 22 = u 2 + (.25(u u 2 + t =.75 + (.25( =.6875

11 Th actual solution to y + y = y + t with initial conditions y( = and y ( = can b found by finding th complmntary solution y c (t = c xp ( + 5t/2 + c 2 xp ( 5t/2, and thn using th mthod of undtrmind cofficints to find th non-homognous trm. Th solution is 5 2 xp ( 5 t xp ( + 5 t t 2 Evaluatd at t =.5, w gt.42863, so our approximation of u 2 =.4375 is off by about.88. W also not that w happn to hav an approximation u 22 =.6875 for y (.5. Th actual valu of y (.5 is about.8738, which is not quit as good. Of cours, thr ar mor sophisticatd tchniqus than Eulr s mthod for stimating solutions (although thy ar basd loosly on Eulr s mthod, but w will not study ths mor complicatd tchniqus. 4 Th Exponntial Matrix Th xponntial function y = at has turnd up ovr and ovr in our study of diffrntial quations. On of th rasons this function has bn ubiquitous is that th function rtains th sam form whn diffrntiating th only thing that changs is th constant cofficint. Is it possibl to dfin a function on matrics that bhavs th sam way? In othr words, is thr som matrix-valud function f(a whos drivativ f (A = Af(A is a multipl of itslf? Th answr is ys, and w ll us th xact sam notation for this function: Lt A b a squar matrix. W dfin th xponntial matrix At to b th matrix whos drivativ is A tims itslf. 4.. What th xponntial matrix is NOT: Lt A = 3 2. Find At. Unfortunatly, th asy and obvious thing to try 3t t 2t

12 dos not work. 3 3t W can chck that it dosn t work by taking th drivativ: t 2 2t and multiplying A tims th matrix: 3t 3 3 3t + 2 t 2 2t. Ths arn t th sam matrix, which mans that w hav to look for anothr mthod of finding th xponntial matrix. 4. Th Maclaurin sris for at Instad, w mak us of th Maclaurin sris for at : at = n= a n t n. n! If w rplac th scalar a with th matrix A, w gt: At = n= t n n! An. In gnral, this summation is difficult to comput by hand, bcaus of th tdiousnss of calculating rpatd powrs of A n, howvr, thr ar som shortcuts that w can us: Exampl A matrix is in diagonal form (or simply a diagonal matrix if vry ntry not on th main diagonal of th matrix is zro. 3 For xampl, is a diagonal matrix. 5 2

13 λ Lt A = λ 2 b a diagonal matrix. Find At. 5 Diagonalization In th last sction, as on of th stps in prforming th Mthod of Variations on a systm, w multiplid by a matrix, and thn latr multiplid by th invrs of that matrix. A similar procss, calld diagonalization can b usd to find th xponntial of a matrix. Exampl 2 Lt A = λ λ. Find At. 3

14 Exrcis 7 Lt A = 2. Find At. If matrix A has a full st of ignvctors x, x 2,, x n, thn A is diagonalizabl: T AT = D, whr λ T = λ 2 x x 2 x n and D =... λ n Not th positions of T and T : th ignmatrix T appars aftr A and T appars bfor. λ t Also not that: At = T Dt T, Dt λ 2t =.... λnt 5. Dcoupling transformation Considr th linar systm y = Ay + g(t. 4

15 If A is diagonalizabl (i.. T AT = D, thn th abov systm can b rwrittn as T y = T AT(T y + T g(t. Us th substitution z = T y, w hav a dcoupld linar systm whr ĝ = T g. z = Dz + ĝ(t, In th dcoupld systm, ach row only concrns on unknown scalar and thrfor can b solvd as a first ordr linar diffrntial quation: Finally, w can transform z back to y by z = λ z + ĝ (t z 2 = λ 2 z 2 + ĝ 2 (t y = Tz Exampl: Find th gnral solution of th following linar systm by dcoupling transformation. y = 3y + 2y y 2 = y + 4y 2 + 5

### The Matrix Exponential

Th Matrix Exponntial (with xrciss) 92.222 - Linar Algbra II - Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial

### New Basis Functions. Section 8. Complex Fourier Series

Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting

### Question 3: How do you find the relative extrema of a function?

ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

### 5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This

### 14.3 Area Between Curves

14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and

### Section 7.4: Exponential Growth and Decay

1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

### A Derivation of Bill James Pythagorean Won-Loss Formula

A Drivation of Bill Jams Pythagoran Won-Loss Formula Ths nots wr compild by John Paul Cook from a papr by Dr. Stphn J. Millr, an Assistant Profssor of Mathmatics at Williams Collg, for a talk givn to th

### Statistical Machine Translation

Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm

### The Normal Distribution: A derivation from basic principles

Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn

### QUANTITATIVE METHODS CLASSES WEEK SEVEN

QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.

### Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)

con 37: Answr Ky for Problm St (Chaptr 2-3) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc

### by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

### Differential Equations (MTH401) Lecture That a non-homogeneous linear differential equation of order n is an equation of the form n

Diffrntial Equations (MTH40) Ltur 7 Mthod of Undtrmind Coffiints-Surosition Aroah Rall. That a non-homognous linar diffrntial quation of ordr n is an quation of th form n n d d d an + a a a0 g( ) n n +

### 5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

### Factorials! Stirling s formula

Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

### Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails

### A Note on Approximating. the Normal Distribution Function

Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

### Section 5-5 Inverse of a Square Matrix

- Invrs of a Squar Matrix 9 (D) Rank th playrs from strongst to wakst. Explain th rasoning hind your ranking. 68. Dominan Rlation. Eah mmr of a hss tam plays on math with vry othr playr. Th rsults ar givn

### Lecture 3: Diffusion: Fick s first law

Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th

### CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:

### Genetic Drift and Gene Flow Illustration

Gntic Drift and Gn Flow Illustration This is a mor dtaild dscription of Activity Ida 4, Chaptr 3, If Not Rac, How do W Explain Biological Diffrncs? in: How Ral is Rac? A Sourcbook on Rac, Cultur, and Biology.

### Lecture 20: Emitter Follower and Differential Amplifiers

Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.

### The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

### 7 Timetable test 1 The Combing Chart

7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing

### Chapter 10 Function of a Matrix

EE448/58 Vrsion. John Stnsby Chatr Function of a atrix t f(z) b a comlx-valud function of a comlx variabl z. t A b an n n comlxvalud matrix. In this chatr, w giv a dfinition for th n n matrix f(a). Also,

### Simulated Radioactive Decay Using Dice Nuclei

Purpos: In a radioactiv sourc containing a vry larg numbr of radioactiv nucli, it is not possibl to prdict whn any on of th nucli will dcay. Although th dcay tim for any on particular nuclus cannot b prdictd,

### Modelling and Solving Two-Step Equations: ax + b = c

Modlling and Solving To-Stp Equations: a + b c Focus on Aftr this lsson, you ill b abl to φ modl problms φ ith to-stp linar quations solv to-stp linar quations and sho ho you ord out th ansr Cali borrod

### Deer: Predation or Starvation

: Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,

### Solutions to Homework 8 chem 344 Sp 2014

1. Solutions to Homwork 8 chm 44 Sp 14 .. 4. All diffrnt orbitals mans thy could all b paralll spins 5. Sinc lctrons ar in diffrnt orbitals any combination is possibl paird or unpaird spins 6. Equivalnt

### CHAPTER EIGHT. Making use of the limit formula developed in Chapter 1, it can be shown that

CONTINUOUS COMPOUNDING CHAPTER EIGHT In prvious stions, th as of m ompoundings pr yar was disussd In th as of ontinuous ompounding, m is allowd to tnd to infinity Th mathmatial rlationships ar fairly asy

### Principles of Humidity Dalton s law

Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

### Sigmoid Functions and Their Usage in Artificial Neural Networks

Sigmoid Functions and Thir Usag in Artificial Nural Ntworks Taskin Kocak School of Elctrical Enginring and Computr Scinc Applications of Calculus II: Invrs Functions Eampl problm Calculus Topic: Invrs

### AP Calculus AB 2008 Scoring Guidelines

AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.

### (Analytic Formula for the European Normal Black Scholes Formula)

(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually

### SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* Rostov-on-Don. Russia

SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* Rostov-on-Don. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs

### SPECIAL VOWEL SOUNDS

SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)

### Adverse Selection and Moral Hazard in a Model With 2 States of the World

Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,

### e = C / electron Q = Ne

Physics 0 Modul 01 Homwork 1. A glass rod that has bn chargd to +15.0 nc touchs a mtal sphr. Aftrword, th rod's charg is +8.00 nc. What kind of chargd particl was transfrrd btwn th rod and th sphr, and

### Examples. Epipoles. Epipolar geometry and the fundamental matrix

Epipoar gomtry and th fundamnta matrix Epipoar ins Lt b a point in P 3. Lt x and x b its mapping in two imags through th camra cntrs C and C. Th point, th camra cntrs C and C and th (3D points corrspon

### CHAPTER 4c. ROOTS OF EQUATIONS

CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil

### A Theoretical Model of Public Response to the Homeland Security Advisory System

A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL 60607 wxding@uicdu Using a diffrntial

### CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.

Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by

### http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force

ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd

### Finite Elements from the early beginning to the very end

Finit Elmnts from th arly bginning to th vry nd A(x), E(x) g b(x) h x =. x = L An Introduction to Elasticity and Hat Transfr Applications x Prliminary dition LiU-IEI-S--8/535--SE Bo Torstnflt Contnts

### Introduction to Finite Element Modeling

Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation

### Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,

### Chapter 3: Capacitors, Inductors, and Complex Impedance

haptr 3: apacitors, Inductors, and omplx Impdanc In this chaptr w introduc th concpt of complx rsistanc, or impdanc, by studying two ractiv circuit lmnts, th capacitor and th inductor. W will study capacitors

### Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing

Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif

### Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY

Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl

### EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS

25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut

### Gas Radiation. MEL 725 Power-Plant Steam Generators (3-0-0) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi

Gas Radiation ME 725 Powr-Plant Stam Gnrators (3-0-0) Dr. Prabal Talukdar Assistant Profssor Dpartmnt of Mchanical Enginring T Dlhi Radiation in absorbing-mitting mdia Whn a mdium is transparnt to radiation,

### C H A P T E R 1 Writing Reports with SAS

C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd

### Vibrational Spectroscopy

Vibrational Spctroscopy armonic scillator Potntial Enrgy Slction Ruls V( ) = k = R R whr R quilibrium bond lngth Th dipol momnt of a molcul can b pandd as a function of = R R. µ ( ) =µ ( ) + + + + 6 3

### Policies for Simultaneous Estimation and Optimization

Policis for Simultanous Estimation and Optimization Migul Sousa Lobo Stphn Boyd Abstract Policis for th joint idntification and control of uncrtain systms ar prsntd h discussion focuss on th cas of a multipl

### SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM

RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs

### Basis risk. When speaking about forward or futures contracts, basis risk is the market

Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also

### Current and Resistance

Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt...6-6.1.1 Currnt Dnsity...6-6. Ohm s Law...6-4 6.3 Elctrical Enrgy and Powr...6-7 6.4 Summary...6-8 6.5 Solvd Problms...6-9 6.5.1 Rsistivity of a Cabl...6-9

### Incomplete 2-Port Vector Network Analyzer Calibration Methods

Incomplt -Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar

### DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN W will assum ha h radr is familiar wih h calculaor s kyboard and h basic opraions. In paricular w hav assumd ha h radr knows h funcions of

### ME 612 Metal Forming and Theory of Plasticity. 6. Strain

Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.

### Production Costing (Chapter 8 of W&W)

Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary

### Parallel and Distributed Programming. Performance Metrics

Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

### 3. Yes. You can put 20 of the 6-V lights in series, or you can put several of the 6-V lights in series with a large resistance.

CHAPTE 6: DC Circuits sponss to Qustions. Evn though th bird s ft ar at high potntial with rspct to th ground, thr is vry littl potntial diffrnc btwn thm, bcaus thy ar clos togthr on th wir. Th rsistanc

### Foreign Exchange Markets and Exchange Rates

Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

### Traffic Flow Analysis (2)

Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

### AP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com

AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos

### Budget Optimization in Search-Based Advertising Auctions

Budgt Optimization in Sarch-Basd Advrtising Auctions ABSTRACT Jon Fldman Googl, Inc. Nw York, NY jonfld@googl.com Martin Pál Googl, Inc. Nw York, NY mpal@googl.com Intrnt sarch companis sll advrtismnt

### Rent, Lease or Buy: Randomized Algorithms for Multislope Ski Rental

Rnt, Las or Buy: Randomizd Algorithms for Multislop Ski Rntal Zvi Lotkr zvilo@cs.bgu.ac.il Dpt. of Comm. Systms Enginring Bn Gurion Univrsity Br Shva Isral Boaz Patt-Shamir Dror Rawitz {boaz, rawitz}@ng.tau.ac.il

### Finite Dimensional Vector Spaces.

Lctur 5. Ft Dmsoal Vctor Spacs. To b rad to th musc of th group Spac by D.Maruay DEFINITION OF A LINEAR SPACE Dfto: a vctor spac s a st R togthr wth a oprato calld vctor addto ad aothr oprato calld scalar

### Category 7: Employee Commuting

7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil

### On the moments of the aggregate discounted claims with dependence introduced by a FGM copula

On th momnts of th aggrgat discountd claims with dpndnc introducd by a FGM copula - Mathiu BARGES Univrsité Lyon, Laboratoir SAF, Univrsité Laval - Hélèn COSSETTE Ecol Actuariat, Univrsité Laval, Québc,

### Dynamic Analysis of Policy Lag in a Keynes-Goodwin Model : Stability, Instability, Cycles and Chaos. Hiroyuki Yoshida* and Toichiro Asada**

Dynamic Analysis of Policy Lag in a Kyns-Goodwin Modl : Stability, Instability, Cycls and Chaos Hiroyuki Yoshida* and Toichiro Asada** * Faculty of Economics, Nagoya Gakuin Univrsity, Aichi, Japan ** Faculty

### Analyzing the Economic Efficiency of ebaylike Online Reputation Reporting Mechanisms

A rsarch and ducation initiativ at th MIT Sloan School of Managmnt Analyzing th Economic Efficincy of Baylik Onlin Rputation Rporting Mchanisms Papr Chrysanthos Dllarocas July For mor information, plas

### Simulation of the electric field generated by a brown ghost knife fish

C H A P T R 2 7 Simulation of th lctric fild gnratd by a brown ghost knif fish lctric fild CONCPTS 27.1 Th fild modl 27.2 lctric fild diagrams 27.3 Suprposition of lctric filds 27.4 lctric filds and forcs

### Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D

24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd

### SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY

1 SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY ALEXA Vasil ABSTRACT Th prsnt papr has as targt to crat a programm in th Matlab ara, in ordr to solv, didactically

### Math 161 Solutions To Sample Final Exam Problems

Solutions To Sampl Final Eam Problms Mat 6 Mat 6 Solutions To Sampl Final Eam Problms. Find dy in parts a - blow. a y = + b y = c y = arctan d y +lny = y = cos + f y = +ln ln g y = +t3 dt y = g a y = b

### Important Information Call Through... 8 Internet Telephony... 6 two PBX systems... 10 Internet Calls... 3 Internet Telephony... 2

Installation and Opration Intrnt Tlphony Adaptr Aurswald Box Indx C I R 884264 03 02/05 Call Duration, maximum...10 Call Through...7 Call Transportation...7 Calls Call Through...7 Intrnt Tlphony...3 two

### Chapter 5 Capacitance and Dielectrics

5 5 5- Chaptr 5 Capacitanc and Dilctrics 5.1 Introduction... 5-3 5. Calculation of Capacitanc... 5-4 Exampl 5.1: Paralll-Plat Capacitor... 5-4 Exampl 5.: Cylindrical Capacitor... 5-6 Exampl 5.3: Sphrical

### Financial Mathematics

Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution

### Improving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost

Economy Transdisciplinarity Cognition www.ugb.ro/tc Vol. 16, Issu 1/2013 50-54 Improving Managrial Accounting and Calculation of Labor Costs in th Contxt of Using Standard Cost Lucian OCNEANU, Constantin

### 81-1-ISD Economic Considerations of Heat Transfer on Sheet Metal Duct

Air Handling Systms Enginring & chnical Bulltin 81-1-ISD Economic Considrations of Hat ransfr on Sht Mtal Duct Othr bulltins hav dmonstratd th nd to add insulation to cooling/hating ducts in ordr to achiv

### Optical Modulation Amplitude (OMA) and Extinction Ratio

Application Not: HFAN-.. Rv; 4/8 Optical Modulation Amplitud (OMA) and Extinction Ratio AVAILABLE Optical Modulation Amplitud (OMA) and Extinction Ratio Introduction Th optical modulation amplitud (OMA)

### Free ACA SOLUTION (IRS 1094&1095 Reporting)

Fr ACA SOLUTION (IRS 1094&1095 Rporting) Th Insuranc Exchang (301) 279-1062 ACA Srvics Transmit IRS Form 1094 -C for mployrs Print & mail IRS Form 1095-C to mploys HR Assist 360 will gnrat th 1095 s for

### Constraint-Based Analysis of Gene Deletion in a Metabolic Network

Constraint-Basd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFG-Rsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany

### Economic Analysis of Floating Exchange Rate Systems

Economic Analysis of Floating Rat Systms Th businss sction of any nwspapr will hav a tabl of spot s. Ths ar th s at which a prson could hav bought othr currncis or forign, such as th English Pound, Frnch

### Fetch. Decode. Execute. Memory. PC update

nwpc PC Nw PC valm Mmory Mm. control rad writ Data mmory data out rmmovl ra, D(rB) Excut Bch CC ALU A vale ALU Addr ALU B Data vala ALU fun. valb dste dstm srca srcb dste dstm srca srcb Ftch Dcod Excut

### Graph Theory. 1 Graphs and Subgraphs

1 Graphs and Subgraphs Graph Thory Dfinition 1.1. A multigraph or just graph is an ordrd pair G = (V, E) consisting of a nonmpty vrtx st V of vrtics and an dg st E of dgs such that ach dg E is assignd

### ACTIVITES INCLUDED IN THE PACK:

This rsourc pack is to hlp you ﬁnd your way through th World War On displays at Nwark Houss Musum and hopfully, to mak larning about a harrowing part of history a littl mor fun. Includd in th pack ar activitis

### CIRCUITS AND ELECTRONICS. Basic Circuit Analysis Method (KVL and KCL method)

6. CIRCUITS AND ELECTRONICS Basic Circuit Analysis Mthod (KVL and KCL mthod) Cit as: Anant Agarwal and Jffry Lang, cours matrials for 6. Circuits and Elctronics, Spring 7. MIT 6. Fall Lctur Rviw Lumpd

### Logo Design/Development 1-on-1

Logo Dsign/Dvlopmnt 1-on-1 If your company is looking to mak an imprssion and grow in th marktplac, you ll nd a logo. Fortunatly, a good graphic dsignr can crat on for you. Whil th pric tags for thos famous

### The Fourier Transform

Th Fourir Transfor Larning outcos Us th Discrt Fourir Transfor to prfor frquncy analysis on a discrt (digital) signal Eplain th significanc of th Fast Fourir Transfor algorith; Eplain why windowing is

### SPECIFIC HEAT AND HEAT OF FUSION

PURPOSE This laboratory consists of to sparat xprimnts. Th purpos of th first xprimnt is to masur th spcific hat of to solids, coppr and rock, ith a tchniqu knon as th mthod of mixturs. Th purpos of th

### Global Financial Management

Global Financial Managmnt Valuation of Stocks Copyright 999 by Alon Brav, Stphn Gray, Campbll R Harvy and Ernst Maug. All rights rsrvd. No part of this lctur may b rproducd without th prmission of th authors.

### Intermediate Macroeconomic Theory / Macroeconomic Analysis (ECON 3560/5040) Final Exam (Answers)

Intrmdiat Macroconomic Thory / Macroconomic Analysis (ECON 3560/5040) Final Exam (Answrs) Part A (5 points) Stat whthr you think ach of th following qustions is tru (T), fals (F), or uncrtain (U) and brifly

### MEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM. Hans G. Jonasson

MEASUREMENT AND ASSESSMENT OF IMPACT SOUND IN THE SAME ROOM Hans G. Jonasson SP Tchnical Rsarch Institut of Swdn Box 857, SE-501 15 Borås, Swdn hans.jonasson@sp.s ABSTRACT Drum sound, that is th walking

### Systems of Equations and Inequalities

CHAPTER CONNECTIONS Systms of Equations and Inqualitis CHAPTER OUTLINE.1 Linar Systms in Two Variabls with Applications 04. Linar Systms in Thr Variabls with Applications 16.3 Nonlinar Systms of Equations

### Performance Evaluation

Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Bay-lik rputation systms: Analysis, charactrization and insuranc mchanism