New Basis Functions. Section 8. Complex Fourier Series


 Caitlin Sanders
 1 years ago
 Views:
Transcription
1 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 ralvalud Fourir sris is xplaind and formula ar givn for convrting btwn th two typs of rprsntation. Exampls ar givn of computing th complx Fourir sris and convrting btwn complx and ral sriss. Rcall that th Fourir sris builds a rprsntation composd of a wightd sum of th following basis functions. (i.. a constant trm) cos(t) cos(2t) cos(3t) cos(4t)... sin(t) sin(2t) sin(3t) sin(4t)... Computing th wights a n, b n and c oftn involvs som nasty intgration. W now prsnt an altrnativ rprsntation basd on a diffrnt st of basis functions: (i.. a constant trm) it 2it 3it 4it... it 2it 3it 4it... Ths can all b rprsntd by th trm int with n taking intgr valus from to +. Not that th constant trm is providd by th cas whn n = 0. 2
2 Sris of Complx Exponntials A rprsntation basd on this family of functions is calld th complx Fourir sris. c n int Th cofficints, c n, ar normally complx numbrs. It is oftn asir to calculat than th sin/cos Fourir sris bcaus intgrals with xponntials in ar usually asy to valuat. W will now driv th complx Fourir sris quations, as shown abov, from th sin/cos Fourir sris using th xprssions for sin() and cos() in trms of complx xponntials. d + = d + = d + = whr c n = Complx Fourir Sris a n cos(nt) + b n sin(nt)] a n ( int + int 2 (a n ib n ) int + 2 c n int ) ( int int )] + b n 2i (a n + ib n ) int 2 d, n = 0 (a n ib n ) /2, n =,2,3,... (a n + ib n ) /2, n =, 2, 3,... Not that a n and b n ar only dfind whn n is ngativ. 3 4
3 a n = cos(nt)f(t) dt b n = sin(nt)f(t) dt d = 2 f(t) dt thus for n positiv c n = 2 (a n ib n ) = cos(nt) isin(nt)] f(t) dt 2 = 2 int f(t) dt for n ngativ c n = 2 (a n + ib n ) = cos( nt) + isin( nt)] f(t) dt 2 = 2 int f(t) dt Complx Fourir Sris Summary c n = 2 int f(t) dt c n int and for n = 0 c 0 = d = 2 0 f(t) dt 5 6
4 Complx Sris Exampl Find th complx Fourir sris to modl sin(t). c n = 2 int f(t) dt = 2 int sin(t) dt = in in ] 2 n 2 Which is zro whn n dos not qual or. For ths two spcial cass w hav to st n = + ǫ and calculat th limit of c n as ǫ tnds to zro. This givs us c = 2i c = 2i Which mans th complx Fourir sris for sin(t) is = it it 2i c n int 7 Finding th limit as n tnds to c n = 2 in in ] n 2 St n = + ǫ and lt ǫ tnd to zro. c = i(+ǫ) i(+ǫ) 2 ( + ǫ) 2 = iǫ + iǫ ] 2 ( + ǫ) 2 ] iǫ + iǫ 2 + 2ǫ ] 2iǫ 2 2ǫ i 2 2i 8
5 Complx Sris Exampl 2 Find th complx Fourir sris to modl f(x) that has a priod of 2 and is whn 0 < x < T and zro whn T < x < 2. f(x) c n = 2 int f(t) dt = i 2n T int ], whn n 0 = 2 ara = T 2, whn n = 0 So th Fourir sris is c n int = 2 T + + i n i n int ] int int ] int 2 9 x Convrting c to a, b and d From our xampl on th prvious pag. i 2n int ], whn n 0 c n = 2 ara = 2 T, whn n = 0 W wish to calculat th cofficints for th quivalnt Fourir sris in trms of sin() and cos(). Clarly d = c 0 = T 2. For n > 0 c n = (a n ib n )/2 a n = 2 R{c n } and b n = 2 Im{c n } convrting our xprssion for c n into sin() and cos(): so 2c n = i cos(nt) isin(nt) ] n = sin(nt) + i(cos(nt) )] n a n = sin(nt) n and b n = cos(nt). n 20
6 Complx Fourir Sris 2 T + Ral Fourir Sris T i n int ] int i ] int int n sin(nt) cos(nt) n cos(nt) + sin(nt) n Both sriss convrg as /n. 2 Convrting from Ral to Complx Convrt th ral Fourir sris of th squar wav f(t) to a complx sris. 2 f(t) For th ral sris, w know that d = a n = 0 and b n = 4 sin(nt)f(t) dt = n, n odd giving 4 sin(t) + sin(3t) 3 + sin(5t) ] To convrt to a complx sris, us d, n = 0 c n = (a n ib n ) /2, n =,2,3,... (a n + ib n ) /2, n =, 2, 3,... so w hav c 0 = 0 c n = 2i/(n), n positiv and odd c n = 2i/( n), n ngativ and n odd 2i it 5 + 3it 3 + it t 2 + it + 3it 3 + 5it ]
7 Gnral Complx Sris For priod of 2 c n = 2 int f(t) dt 2 0 Similarly, for priod L c n = L f(x) = L c n int 0 inx2 L f(x) dx c n inx2 L Th fraction 2 L is oftn writtn as ω 0 and calld th fundamntal angular frquncy. Exampl A vn function f(t) is priodic with priod L = 2, and cosh(t ) for 0 t. Find a complx Fourir sris rprsntation for f(t). f(t) c n = L = 2 L 0 int2 L f(t) dt 2 0 int cosh(t ) dt = sinh() + n 2 2 t 23 24
8 Hnc th complx Fourir sris is = c n int2 L sinh() int + n 2 2 W can chck this answr by computing th quivalnt ral Fourir sris which w calculatd at th start of sction 7. a n = 2 R{c n }, n =,2,3,... b n = 2 Im{c n }, n =,2,3,... d = c 0 In this cas, as c n is ntirly ral, a n = 2c n = 2sinh() + n 2 2, n =,2,3,... b n = 0 d = sinh() Exampl 2 Find th complx Fourir sris of th th squar wav f(x). L f(x) Not that th man of th function is zro, so c 0 = 0. c n = L L 0 inx2 L f(x) dx = L/2 ] L inx2 L dx L 0 L/2 inx2 L dx = 2in + 2 in ] 2in in ] f(x) = inx2 L in n = n 0 f(x) = ix2 L i 5 + ix2 L + 3ix2 L 3 + 3ix 2 L 3 + 5ix2 L 5 + ix2 L L +... x 25 26
9 Convrting to a Ral Sris W wish to convrt th complx gnral rang squar wav sris into a sris with ral cofficints. { 2/(in), n odd c n = 0, n vn Clarly d = c 0 = 0. For a and b us: c n = (a n ib n )/2 a n = 2 R{c n } = 0 and b n = 2 Im{c n } = 4 n, n odd Which givs us th ral sris: 4 sin ( x 2 L ) + sin ( 3x 2 ) L 3 + sin ( 5x 2 L 5 ) +... For priod L Sction 8: Summary c n = L f(x) = L 0 inx2 L f(x) dx c n inx2 L Rlationship with th cos/sin Fourir sris. d, n = 0 c n = (a n ib n ) /2, n =,2,3,... (a n + ib n ) /2, n =, 2, 3,... a n = 2 R{c n }, n =,2,3,... b n = 2 Im{c n }, n =,2,3,... d = c
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
More informationNonHomogeneous Systems, Euler s Method, and Exponential Matrix
NonHomognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous firstordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach
More information14.3 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
More informationSection 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 # 117 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart
More informationby 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
More informationSolutions 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
More informationQuestion 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
More informationMathematics. 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
More informationCHAPTER 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
More informationLecture 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.
More informationSigmoid 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
More informationLecture 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
More informationA Derivation of Bill James Pythagorean WonLoss Formula
A Drivation of Bill Jams Pythagoran WonLoss 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
More informationCPS 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:
More informationFinancial 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
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 1213)
con 37: Answr Ky for Problm St (Chaptr 23) 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
More informatione = 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
More informationCHAPTER 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
More informationPhysics 106 Lecture 12. Oscillations II. Recap: SHM using phasors (uniform circular motion) music structural and mechanical engineering waves
Physics 6 Lctur Oscillations II SJ 7 th Ed.: Chap 5.4, Rad only 5.6 & 5.7 Rcap: SHM using phasors (unifor circular otion) Physical pndulu xapl apd haronic oscillations Forcd oscillations and rsonanc. Rsonanc
More informationBasis 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
More informationQUANTITATIVE 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.
More informationGas Radiation. MEL 725 PowerPlant Steam Generators (300) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi
Gas Radiation ME 725 PowrPlant Stam Gnrators (300) Dr. Prabal Talukdar Assistant Profssor Dpartmnt of Mchanical Enginring T Dlhi Radiation in absorbingmitting mdia Whn a mdium is transparnt to radiation,
More informationA Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 4549 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and
More informationThe 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
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More information5.4 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
More informationFactorials! 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
More informationThe 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
More informationStatistical 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
More informationMath 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
More informationThe 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 wllsuitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of Cnts. Hnc, it can b rad by popl
More informationTN Calculating Radiated Power and Field Strength for Conducted Power Measurements
Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts TN100.04 Calculating adiatd owr and Fild Strngth for Conductd owr Masurmnts Copyright Smtch 007 1 of 9 www.smtch.com Calculating adiatd
More informationChapter 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
More informationIncomplete 2Port 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
More informationPrinciples 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
More informationOptical 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)
More informationAP Calculus MultipleChoice Question Collection 1969 1998. connect to college success www.collegeboard.com
AP Calculus MultiplChoic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos
More informationDifferential Equations (MTH401) Lecture That a nonhomogeneous linear differential equation of order n is an equation of the form n
Diffrntial Equations (MTH40) Ltur 7 Mthod of Undtrmind CoffiintsSurosition Aroah Rall. That a nonhomognous linar diffrntial quation of ordr n is an quation of th form n n d d d an + a a a0 g( ) n n +
More informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. GangLn Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationModelling and Solving TwoStep Equations: ax + b = c
Modlling and Solving ToStp Equations: a + b c Focus on Aftr this lsson, you ill b abl to φ modl problms φ ith tostp linar quations solv tostp linar quations and sho ho you ord out th ansr Cali borrod
More informationExamples. 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
More informationCPU. 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
More informationGenetic 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.
More informationSUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* RostovonDon. Russia
SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* RostovonDon. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs
More informationIntroduction 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
More informationME 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.
More informationFundamentals: 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
More informationHigher. Exponentials and Logarithms 160
hsn uknt Highr Mthmtics UNIT UTCME Eponntils nd Logrithms Contnts Eponntils nd Logrithms 6 Eponntils 6 Logrithms 6 Lws of Logrithms 6 Eponntils nd Logrithms to th Bs 65 5 Eponntil nd Logrithmic Equtions
More information7 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
More informationCurrent and Resistance
Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt...66.1.1 Currnt Dnsity...66. Ohm s Law...64 6.3 Elctrical Enrgy and Powr...67 6.4 Summary...68 6.5 Solvd Problms...69 6.5.1 Rsistivity of a Cabl...69
More informationCALCULATING MARGINAL PROBABILITIES IN PROC PROBIT Guy Pascale, Memorial Health Alliance
CALCULATING MARGINAL PROBABILITIES IN PROC PROBIT Guy Pascal, Mmorial Halth Allianc Introduction Th PROBIT procdur within th SAS systm provids a simpl mthod for stimating discrt choic variabls (i.. dichotomous
More informationSimulated 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,
More informationVibrational 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
More information(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
More informationSIMULATION 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
More informationAccurate Doppler Prediction Scheme for Satellite Orbits
Accurat Dopplr Prdiction Schm for Satllit Orbits NASER AYAT, MOHAMAD MEHDIPOUR Computr nginring group Payam noor univrsity Lashgarak st., Nakhl st., Thran IRAN Abstract:  In satllit communications particular
More informationGold versus stock investment: An econometric analysis
Intrnational Journal of Dvlopmnt and Sustainability Onlin ISSN: 2688662 www.isdsnt.com/ijds Volum Numbr, Jun 202, Pag 7 ISDS Articl ID: IJDS20300 Gold vrsus stock invstmnt: An conomtric analysis Martin
More information5 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
More informationhttp://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
More informationChapter 10 Function of a Matrix
EE448/58 Vrsion. John Stnsby Chatr Function of a atrix t f(z) b a comlxvalud 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,
More informationE X C H A N G E R U L E S A N D C L E A R I N G R U L E S O F N A S D A Q O M X D E R I V A T I V E S M A R K E T S
E X C H A N G E R U L E S A N D C L E A R I N G R U L E S O F N A S D A Q O M X D E R I V A T I V E S M A R K E T S Fair Valu 1 Valuation Variabls Tabl 1 blow shows th variabls us in th rspctiv valuation
More informationProblem Solving Session 1: Electric Dipoles and Torque
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Dpatmnt of Physics 8.02 Poblm Solving Sssion 1: Elctic Dipols and Toqu Sction Tabl (if applicabl) Goup Mmbs Intoduction: In th fist poblm you will lan to apply Coulomb
More informationConstraintBased Analysis of Gene Deletion in a Metabolic Network
ConstraintBasd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFGRsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany
More informationSPREAD 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
More informationVersion 1.0. General Certificate of Education (Alevel) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.
Vrsion.0 Gnral Crtificat of Education (Alvl) 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,
More information[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lbinsec^2)
MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (voltssc/rad Motor torqu constant (lbin/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris
More informationExternalities in Online Advertising
Extrnalitis in Onlin Advrtising Arpita Ghosh Yahoo! Rsarch 2821 Mission Collg Blvd. Santa Clara, CA 95054 arpita@yahooinc.com Mohammad Mahdian Yahoo! Rsarch 2821 Mission Collg Blvd. Santa Clara, CA 95054
More informationEpipolar Geometry and the Fundamental Matrix
9 Epipolar Gomtry and th Fundamntal Matrix Th pipolar gomtry is th intrinsic projctiv gomtry btwn two viws. It is indpndnt of scn structur, and only dpnds on th camras intrnal paramtrs and rlativ pos.
More informationAnalyzing 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
More informationVan der Waals Forces Between Atoms
Van dr Waals Forcs twn tos Michal Fowlr /8/7 Introduction Th prfct gas quation of stat PV = NkT is anifstly incapabl of dscribing actual gass at low tpraturs, sinc thy undrgo a discontinuous chang of volu
More informationFar Field Estimations and Simulation Model Creation from Cable Bundle Scans
Far Fild Estimations and Simulation Modl Cration from Cabl Bundl Scans D. Rinas, S. Nidzwidz, S. Fri Dortmund Univrsity of Tchnology Dortmund, Grmany dnis.rinas@tudortmund.d stphan.fri@tudortmund.d Abstract
More informationDept. of Materials Science and Engineering. Problem Set 8 Solutions
MSE 30/ECE 30 Elctrical Prortis Of Matrials Dt. of Matrials Scinc and Enginring Fall 0/Bill Knowlton Problm St 8 Solutions. Using th rlationshi of n i n i i that is a function of E g, rcrat th lot shown
More informationArchitecture of the proposed standard
Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th
More information2.2.C Analogy between electronic excitations in an atom and the mechanical motion of a forced harmonic oscillator"
..C Analgy btwn lctrnic xcitatins in an atm and th mchanical mtin f a frcd harmnic scillatr" Hw t chs th valu f th crrspnding spring cnstant k? Rsnant Absrptin Mchanical rsnanc W idntify th mchanical rsnanc
More informationHOMEWORK FOR UNIT 51: FORCE AND MOTION
Nam Dat Partnrs HOMEWORK FOR UNIT 51: FORCE AND MOTION 1. You ar givn tn idntial springs. Dsrib how you would dvlop a sal of for (i., a mans of produing rpatabl fors of a varity of sizs) using ths springs.
More informationSimulation of a Solar Cell considering SingleDiode Equivalent Circuit Model
Simulation of a Solar Cll considring SinglDiod Equivalnt Circuit Modl E.M.G. Rodrigus, R. Mlício,, V.M.F. Mnds and J.P.S. Catalão, Univrsity of Bira Intrior R. Font do Lamiro,  Covilhã (Portugal) Phon:
More informationDeer: 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,
More informationIrrigating With High Salinity Water 1
BUL322 Irrigating With High Watr 1 Dorota Z. Haman 2 In humid aras such as Florida, salinity concrns ar diffrnt than in arid aras sinc larg amounts of rainfall will wash out salts concntrating in th soil
More informationModern Portfolio Theory (MPT) Statistics
Modrn Portfolio Thory (MPT) Statistics Morningstar Mthodology Papr May 9, 009 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription
More informationRural and Remote Broadband Access: Issues and Solutions in Australia
Rural and Rmot Broadband Accss: Issus and Solutions in Australia Dr Tony Warrn Group Managr Rgulatory Stratgy Tlstra Corp Pag 1 Tlstra in confidnc Ovrviw Australia s gographical siz and population dnsity
More informationReverberation Times Obtained Using a Numerical Model Versus Those Given by Simplified Formulas and Measurements
ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 88 () 6 Rvrbration Tims Obtaind Using a Numrical Modl Vrsus Thos Givn by Simplifid Formulas and Masurmnts J. António, L. Godinho, A. Tadu Civil Enginring Dpartmnt,
More informationAdverse 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,
More informationSection 55 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
More informationForeign Trade, Devaluation and Elasticities: A Model Approach. prof. Ing. Jaroslav Husar, CSc. Mgr. ek. University of Economics, Bratislava
Forign Trad, Dvaluation and Elaticiti: A Modl Approach prof. Ing. Jarolav Huar, CSc. Mgr.. Univrity of Econoic, Bratilava Introduction Fro th thory of th intrnational conoic it i nowwn that th pnditurwitching
More informationBusiness rules FATCA V. 02/11/2015
Elmnt Attribut Siz InputTyp Rquirmnt BUSINESS RULES TYPE ERROR ACK Xpath I.Mssag Hadr FATCA_OECD Vrsion xsd: string = Validation WrongVrsion ftc:fatca_oecd/vrsion SndingCompanyIN Unlimit d xsd: string
More informationNoise Power Ratio (NPR) A 65Year Old Telephone System Specification Finds New Life in Modern Wireless Applications.
TUTORIL ois Powr Ratio (PR) 65Yar Old Tlphon Systm Spcification Finds w Lif in Modrn Wirlss pplications ITRODUTIO by Walt Kstr Th concpt of ois Powr Ratio (PR) has bn around sinc th arly days of frquncy
More informationSeries FOURIER SERIES. Graham S McDonald. A selfcontained Tutorial Module for learning the technique of Fourier series analysis
Series FOURIER SERIES Graham S McDonald A selfcontained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 004 g.s.mcdonald@salford.ac.uk 1. Theory.
More informationGateway 125,126,130 Fall 2006 Exam 1 KEY p1
Gatway 125,126,130 Fall 2006 Exam 1 KEY p1 Q16 (1 point ach) Plas plac th corrct lttr/s in th box. 1) How many lctrons can th third principal quantum lvl (n = 3) hold? a. 2 b. 8 c. 16 d. 18. 32 2) Arrang
More informationForeign 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
More informationOn 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,
More informationNoble gas configuration. Atoms of other elements seek to attain a noble gas electron configuration. Electron configuration of ions
Valnc lctron configuration dtrmins th charactristics of lmnts in a group Nobl gas configuration Th nobl gass (last column in th priodic tabl) ar charactrizd by compltly filld s and p orbitals this is a
More informationSimulation 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
More informationCircuits with Transistors
ircuits with Transistors ontnts 1 Transistors 1 2 Amplifirs 2 2.1 h paramtrs.................................... 3 3 Bipolar Junction Transistor (BJT) 3 3.1 BJT as a switch...................................
More informationA 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
More informationCategory 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
More informationRent, 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 PattShamir Dror Rawitz {boaz, rawitz}@ng.tau.ac.il
More informationUpper 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
More informationC 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
More informationChapter 5 Capacitance and Dielectrics
5 5 5 Chaptr 5 Capacitanc and Dilctrics 5.1 Introduction... 53 5. Calculation of Capacitanc... 54 Exampl 5.1: ParalllPlat Capacitor... 54 Exampl 5.: Cylindrical Capacitor... 56 Exampl 5.3: Sphrical
More informationLong run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange
Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity
More information