8. The Normalized Least-Squares Estimator with Exponential Forgetting
|
|
- Φωτινή Θεοδοσίου
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Lecure 5 8. he Normalized Leas-Squares Esimaor wih Expoeial Forgeig his secio is devoed o he mehod of Leas-Squares wih expoeial forgeig ad ormalizaio. Expoeial forgeig of daa is a very useful echique i dealig wih ime-varyig ukow parameers. Is iuiive moivaio is ha pas daa are geeraed by pas parameers ad hus should be discoued whe beig used for he esimaio of he curre parameers. Isead of pealizig all of he pas errors from τ = o τ = ha are due o ˆ θ ( θ (as i is doe whe derivig he pure Leas-Squares Esimaor, he miimizaio problem is posed for he cos fucio wih he so-called expoeial forgeig facor. he cos fucio is chose i he form: ( ˆ θ ( ( x( τ y( τ ( ˆ J θ ( = exp λ( r dr d τ τ ( x( τ P( x( τ β(, τ =Expoeial Forgeig Facor = Normalizig Sigal β (, ( ( ( + exp λ( r dr ˆ θ ˆ θ ˆ ˆ Q θ θ (8. he esimaed value ˆ θ ( will be foud such ha: ( ( ( τ ( τ ( ( ( ˆ e J θ =, d, ˆ ˆ Q ˆ ˆ β τ τ + β θ θ θ θ mi (8. x θ y ˆ I (8. (8., P P ˆ θ ˆ = θ is he iiial esimae of he rue ukow cosa parameer vecor θ, Q= Q > is he = > is he ormalizig sigal weighig marix, ermial sae weighig marix, λ : is he so-called ime varyig forgeig facor, + + r R R, ad β(, τ = exp λ( r dr (8.3 τ 5
2 Remark 8. If λ( r = λ is a cosa he β(, τ o: ( θ ( x τ y( τ ˆ ( ( ( ( x( τ ( = e λ τ, ad he cos fucio (8. simplifies ( ( ( J ˆ θ = e dτ + e ˆ θ ˆ θ Q ˆ θ ˆ θ λ τ λ (8.4 Moreover, seig λ = ad P= Q= resuls i he iegral cos fucio ( ˆ( ˆ ( ( J θ = θ x τ y τ dτ (8.5 ha was used o derive he pure Leas-Squares Esimaor. Goig back o solvig he miimizaio problem i (8., suppose ha P > ad Q >. he cos fucio J ˆ θ ( is covex wih respec o ˆ θ (, for all. hus, ay local miimum is also global ad saisfies ( θ ( J ˆ =, (8.6 Calculaig he gradie of he cos fucio J ˆ θ ( wih respec o ˆ θ (, resuls i: ( ˆ θ ( ( ( τ ( τ x( τ x y J( ˆ θ ( = ( x( τ β(, τ dτ + β(, Q( ˆ θ ˆ θ (8.7 Equaig he gradie o zero while rearragig erms, furher yields: (, ˆ ( ( (, τ ( x( τ β β (, Q+ ( x( τ ( x( τ dτ ˆ θ Γ f (, τ ( x( τ β = β Qθ + x τ y τ dτ ( (8.8 Immediaely oe ha sice Q > ad ( x ( x, Γ exiss a each ime. Moreover, by defiiio ad because of (8.3, his marix saisfies he followig IVP: 5
3 ( x ( x ( ( d Γ ( =λ ( Γ ( + d x Γ = Q (8.9 Similarly, he fucio f i (8.8 saisfies f& f x f Q ˆ = θ ( = λ ( ( +( ( y ( x( (8. Usig (8.8, resuls i: ˆ θ ( =Γ ( f ( (8. Based o he ideiy d d ( ( ( ( = Γ Γ =Γ& Γ ( +Γ( Γ ( (8. d d oe ca wrie: d ( ( Γ & =Γ Γ ( Γ( d (8.3 Subsiuig (8.9 io (8.3, yields: ( x( x( x( Γ & ( =Γ( λ ( Γ ( Γ Γ = Q ( (8.4 or, equivalely: ( x ( x ( x( Γ & ( = λ ( Γ( Γ( Γ Γ =Γ = Q ( (8.5 Moreover, differeiaig (8. while usig (8. ad (8.4, oe ges: 5
4 & ˆ θ ( =Γ & ( f ( +Γ( f& ( x( λ ( ( ( ( λ ( f ( x( ( x( ( x( = Γ Γ Γ f +Γ + ( x( y ( ( (8.6 Collecig erms, yields: ey ( ( x( ˆ & θ ( =Γ( ( x( Γ( f ( y( ( x( 443 ˆ θ ( y ˆ( ˆ θ f {{ ( =Γ = ˆ θ Q Q ˆ θ (8.7 Fially, combiig (8.7 wih (8.5, resuls i he coiuous ime recursive versio of he Normalized Leas-Squares Esimaor wih Expoeial Forgeig (NLSEEF: ( x( x( ˆ& θ ( =Γ( e y ( ˆ ˆ θ = θ ( x( ( x( Γ & ( = λ ( Γ ( Γ ( ( x( Γ Γ =Γ = Q ( (8.8 where y ( ( ˆ = ˆ = θ ( θ ( ( ˆ θ( θ ( x( θ ( x( e y y x x = =Δ (8.9 is he oupu predicio error. Remark 8. Seig λ ( = ad P = resuls i he pure Leas-Squares Esimaor. 53
5 ˆ& θ =Γ ey Γ =Γ & Γ ( (8. O he oher had, wih ( Squares Esimaor. λ = ad P= P >, oe ges he ormalized pure Leas- ˆ& θ =Γ e y Γ & =Γ Γ ( (8. Nex, sabiliy ad covergece properies of he NLSEEF (8.8 will be aalyzed. Usig (8.3 ad (8.8, he esimaor gai ca be explicily wrie as: Cosider ( ( exp ( x( τ ( x( τ λ exp λ τ ( x( τ (8. τ r dr r dr d Γ =Γ + d d & & (8.3 ( ( ( Γ Δ θ =Γ ( Δ θ( +Γ ( ˆ θ( Subsiuig (8.9 ad (8. io (8.3, yields: d ( ( ( ( θ λ θ Γ Δ = Γ + θ λ( θ Δ +Γ Γ Δ = Γ Δ d (8.4 Iegraig boh sides of (8.4, resuls i: Γ Δ θ = exp λ( r dr Γ Δθ (8.5 Cosequely: Δ θ( = λ( r dr Γ( Γ Δ θ =Τ Γ Δθ exp Τ ( (8.6 Subsiuig for Γ ( from (8.8, oe ges 54
6 ( x( τ x( τ x( τ ( x( τ x( τ x( τ Δ θ = β(, β(, Q+ β, τ dτ Γ Δθ τ = Q+ exp λ τ Γ Δθ ( r dr d (8.7 As see from (8.6 (8.7, he sabiliy properies of he NLSEEF deped o he ime- λ r. varyig forgeig facor From (8.6 i follows ha i order o show boudedess of he esimaed parameer ˆ θ, i is sufficie o esablish boudedess of ( Τ ( = exp λ ( r dr Γ( (8.8 owards ha ed, differeiaig (8.8 ad subsiuig for he dyamics of Γ ( from (8.8, yields: =exp λ ( r dr Γ Γ d Τ ( =λ( exp λ( r dr Γ ( + exp λ( r dr λ( Γ Γ Γ d (8.9 From (8.9 i follows ha ad herefore ( Τ Τ =Γ = Q <, (8.3 Τ Q <, (8.3 Moreover, sice Τ ( is o-icreasig ad bouded from below, i has a limi: Remark 8.3 lim Τ =Τ < (8.3 55
7 Alhough ( Τ is uiformly bouded ad has a fiie limi, he esimaio gai Γ ( may grow ubouded. I fac, for a scalar liear i parameers saic model wih a sigle ukow parameer, oe ca show ha if he forgeig facor is cosa, λ( r = λ >, ad if he regressor vecor ( x( = e he Γ, as. O he oher had, if he regressor vecor ( x( is PE he o oly he esimaio gai remais bouded bu also he parameer errors coverge o zero expoeially wih he rae λ. Usig (8.3, a upper boud for he parameer esimaio error i (8.6 ca be calculaed: Q = Q Δθ Τ Γ Δ < } ( ( ( θ ( (8.33 his proves ha he esimaed parameer errors are uiformly ulimaely bouded, ad hece ˆ θ L. Also because of (8.3, akig he limi i (8.6 gives: Q lim Δ θ = lim Τ Γ Δ θ =Τ Δ θ < (8.34 ad hus ˆ Q limθ = θ +Τ Δ θ = θ < (8.35 Nex, we show ha he ormalized predicio error is uiformly ulimaely bouded ad e y square-iegrable, ha is L L. owards ha ed, cosider he fucio (, V Differeiaig (8.36 alog he rajecories of Δ θ =Δθ Γ Δ θ (8.36 ˆ& θ =Γ Δθ d ( Γ Δ θ =λ( Γ Δθ d (8.37 yields: 56
8 ˆ d V& &, Δ θ = θ Γ Δ θ +Δθ ( Γ ( Δθ d = Γ Δ Γ ( Δ ( Δ Γ Δ V(, Δθ θ θ λ θ θ ey Δθ Δ θ= (8.38 or, equivalely ( θ λ V ( θ e y V &, Δ =, Δ (8.39 Sice V ad V& he V is a o-icreasig fucio of ime ad cosequely i has a fiie limi: ( θ lim V, Δ = V < (8.4 Usig (8.39 ad iegraig boh sides of e y V& (8.4 proves ha he ormalized predicio error e y is square-iegrable: ( τ x( τ e y ey d V V < L (8.4 he uiform ulimae boudedess of he ormalized predicio error follows direcly from: y θ θ θ e Δ Δ Δ = = + P λ P mi ( P λmi (8.43 ad from he fac ha e y Δθ L. hus, L L. Γ Γ for all, (sufficie codiios for his will be esablished i Secio 9. he usig (8.8 yields Moreover, suppose ha ( 57
9 ˆ& ey θ ( =Γ e y Γ e y Γ {{ L L L (8.44 herefore ˆ& θ L L (8.45 Summarizig, yields: heorem 8. Recursive versio of he NLSEEF is give by (8.8 ad he esimaor has he followig properies: ˆ θ L e y L L lim ˆ θ( = θ < If, i addiio Γ( ˆ& L L θ Γ he Remark 8.4 Followig he lies of Secio 6 (NGE, a se of sufficie codiios ca be foud ha guaraee asympoic covergece of he predicio error e y. 9. he Normalized Leas-Squares Esimaor wih Bouded Gai Forgeig Daa forgeig feaure of he NLSEEF resuls i he esimaor abiliy o rack slowly- Γ may grow ubouded if varyig parameers. However, he esimaor gai marix he regressor vecor ( x is o PE, (read Remark 8.3. hus, i is desirable o ue λ e τ variaio of he forgeig facor β(, τ =, (i.e., choosig ( r forgeig is acivaed whe ( x is PE ad suspeded whe ( x r dr λ, so ha daa is o PE. owards 58
10 ha ed, oe ha he magiude of he gai marix Γ is a idicaor of he exciaio level of x(. hus, i is reasoable o correlae he forgeig facor variaio wih Γ (. A specific echique, (bouded gai forgeig, for achievig his purpose is o choose: λ( λ Γ = k (9. I (9., λ is he maximum forgeig rae, ad k is he pre-specified boud for gai marix magiude. Basically, if he orm of Γ is small, (idicaig srog PE, he λ. As Γ becomes larger forgeig facor i (9. discards he daa a maximum rae he forgeig speed is reduced ad whe he orm reaches he pre-specified upper boud k he forgeig is suspeded. Noe ha iiially he gai marix i (8.8 mus be chose such ha Γ k. Nex, covergece properies of he Normalized Leas-Squares Esimaor wih Bouded Gai Forgeig (NLSEBGF will be aalyzed. We show ha he form (9. guaraees x. ha he resulig gai marix Γ is upper bouded regardless of he PE of his is i coras o a cosa forgeig facor ad cosiues he mai beefi of he variable forgeig mehod. Subsiuig he bouded forgeig relaio (9. io he gai updae dyamics (8.9, resuls i: ( d Γ ( Γ =λ Γ + d k Γ = Q IN N k (9. his leads o Sice ( ( τ x x λ λ τ τ Γ ( = Qe + e Γ Γ + d k x λ ( τ τ τ τ (9.3 59
11 is posiive semi-defiie he Usig (9.5 i (9.3, oe ges: ( ( IN N ( IN N Γ Γ =Γ Γ Γ Γ (9.4 Γ Γ I N N (9.5 λ λ ( x τ λ ( τ ( x( τ ( ( τ ( x τ ( x( τ ( ( τ Γ Qe + e IN N+ dτ k x = + + λ λ Q IN Ne IN N e d k k x ( τ τ (9.6 he esimaio gai iiial codiio from (9. guaraees ha: Q I N N (9.7 k Cosequely, for all he followig lower boud akes place: or, equivalely: ( Γ I N N (9.8 k Γ k I N N (9.9 he upper boud (9.9 implies ha he bouded forgeig relaio (9. always resuls i: λ ( r (9. his proves uiform boudedess of he esimaio gai Γ ad he o-egaive aure of he fucio λ ( r. Furhermore, suppose ha he regressor vecor ( x is PE. he by Defiiio 6., here mus exis cosas α, > such ha for all + I ( x ( x d I α τ τ τ α (9. 6
12 I addiio, suppose ha x bouded, ha is x R : he for all : ad ( x is coiuous. he ( x L ( x C is uiformly (9. ( x( τ x( τ x( τ P x ( ( τ + + dτ ( x( τ ( x( τ dτ ( + λ max P C α I = λ I + λmax P C λ (9.3 his PI codiio ca equivalely be wrie as: d I (9.4 τ λ, Based o (9.4, a ew lower boud for Γ ca be compued. I fac, usig (9.6 i follows ha for all : λ( τ λ( τ Γ ( N N+ τ N N k + k I e d I e dτ + + k k λ λ IN N e dτ e λ I (9.5 herefore, for all : k Γ( I k I λ < + e λ k (9.6 his, i ur, leads o uiform lower boudedess of he ime-varyig forgeig facor λ by a posiive cosa: ( λ Γ λe λ k λ( = λ = λmi > λ k + e λ k (9.7 6
13 Recall from (8.6 ha ( λ( r dr ( θ Δ θ = e Γ QΔ (9.8 ad ha he esimaio gai Γ ( is uiformly bouded as i (9.9. he he orm of he parameer esimaio error Δ θ ( saisfies: λ( r dr e k Q e k Q ( λ mi τ Δθ Δθ Δ θ (9.9 which immediaely implies he expoeial covergece of he esimaed parameers o heir rue ukow cosa values, ha is: lim Δ θ = (9. he properies of he NLSEBGF are summarized below: heorem 9. I addiio o he properies of he NLSEEF (8.8 saed i heorem 8., usig he bouded forgeig facor (9. resuls i he uiformly bouded esimaio gai marix: Γ L If he regressor vecor ( x is PE ad bouded he he esimaed parameers coverge expoeially o heir rue ukow cosa values: limθ = θ ( 6
) 2. δ δ. β β. β β β β. r k k. tll. m n Λ + +
Techical Appedix o Hamig eposis ad Helpig Bowes: The ispaae Impac of Ba Cosolidaio (o o be published bu o be made available upo eques. eails of Poofs of Poposiios 1 ad To deive Poposiio 1 s exac ad sufficie
Διαβάστε περισσότεραAPPENDIX A DERIVATION OF JOINT FAILURE DENSITIES
APPENDIX A DERIVAION OF JOIN FAILRE DENSIIES I his Appedi we prese he derivaio o he eample ailre models as show i Chaper 3. Assme ha he ime ad se o ailre are relaed by he cio g ad he sochasic are o his
Διαβάστε περισσότερα1. For each of the following power series, find the interval of convergence and the radius of convergence:
Math 6 Practice Problems Solutios Power Series ad Taylor Series 1. For each of the followig power series, fid the iterval of covergece ad the radius of covergece: (a ( 1 x Notice that = ( 1 +1 ( x +1.
Διαβάστε περισσότεραHomework for 1/27 Due 2/5
Name: ID: Homework for /7 Due /5. [ 8-3] I Example D of Sectio 8.4, the pdf of the populatio distributio is + αx x f(x α) =, α, otherwise ad the method of momets estimate was foud to be ˆα = 3X (where
Διαβάστε περισσότεραFourier Series. Fourier Series
ECE 37 Z. Aliyazicioglu Elecrical & Compuer Egieerig Dep. Cal Poly Pomoa Periodic sigal is a fucio ha repeas iself every secods. x() x( ± ) : period of a fucio, : ieger,,3, x() 3 x() x() Periodic sigal
Διαβάστε περισσότεραOn Quasi - f -Power Increasing Sequences
Ieaioal Maheaical Fou Vol 8 203 o 8 377-386 Quasi - f -owe Iceasig Sequeces Maheda Misa G Deae of Maheaics NC College (Auooous) Jaju disha Mahedaisa2007@gailco B adhy Rolad Isiue of echoy Golahaa-76008
Διαβάστε περισσότεραThe Estimates of the Upper Bounds of Hausdorff Dimensions for the Global Attractor for a Class of Nonlinear
Advaces i Pure Mahemaics 8 8 - hp://wwwscirporg/oural/apm ISSN Olie: 6-384 ISSN Pri: 6-368 The Esimaes of he Upper Bouds of Hausdorff Dimesios for he Global Aracor for a Class of Noliear Coupled Kirchhoff-Type
Διαβάστε περισσότεραLast Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing.
Last Lecture Biostatistics 602 - Statistical Iferece Lecture 19 Likelihood Ratio Test Hyu Mi Kag March 26th, 2013 Describe the followig cocepts i your ow words Hypothesis Null Hypothesis Alterative Hypothesis
Διαβάστε περισσότεραIntrinsic Geometry of the NLS Equation and Heat System in 3-Dimensional Minkowski Space
Adv. Sudies Theor. Phys., Vol. 4, 2010, o. 11, 557-564 Irisic Geomery of he NLS Equaio ad Hea Sysem i 3-Dimesioal Mikowski Space Nevi Gürüz Osmagazi Uiversiy, Mahemaics Deparme 26480 Eskişehir, Turkey
Διαβάστε περισσότεραOSCILLATION CRITERIA FOR SECOND ORDER HALF-LINEAR DIFFERENTIAL EQUATIONS WITH DAMPING TERM
DIFFERENIAL EQUAIONS AND CONROL PROCESSES 4, 8 Elecroic Joural, reg. P375 a 7.3.97 ISSN 87-7 hp://www.ewa.ru/joural hp://www.mah.spbu.ru/user/diffjoural e-mail: jodiff@mail.ru Oscillaio, Secod order, Half-liear
Διαβάστε περισσότεραL.K.Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 4677 + {JEE Mai 04} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks:
Διαβάστε περισσότεραSUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6
SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES Readig: QM course packet Ch 5 up to 5. 1 ϕ (x) = E = π m( a) =1,,3,4,5 for xa (x) = πx si L L * = πx L si L.5 ϕ' -.5 z 1 (x) = L si
Διαβάστε περισσότεραRG Tutorial xlc3.doc 1/10. To apply the R-G method, the differential equation must be represented in the form:
G Tuorial xlc3.oc / iear roblem i e C i e C ( ie ( Differeial equaio for C (3 Thi fir orer iffereial equaio ca eaily be ole bu he uroe of hi uorial i o how how o ue he iz-galerki meho o fi ou he oluio.
Διαβάστε περισσότερα16. 17. r t te 2t i t 1. 18 19 Find the derivative of the vector function. 19. r t e t cos t i e t sin t j ln t k. 31 33 Evaluate the integral.
SECTION.7 VECTOR FUNCTIONS AND SPACE CURVES.7 VECTOR FUNCTIONS AND SPACE CURVES A Click here for answers. S Click here for soluions. Copyrigh Cengage Learning. All righs reserved.. Find he domain of he
Διαβάστε περισσότεραOscillations CHAPTER 3. ν = = 3-1. gram cm 4 E= = sec. or, (1) or, 0.63 sec (2) so that (3)
CHAPTER 3 Oscillaios 3-. a) gram cm 4 k dye/cm sec cm ν sec π m π gram π gram π or, ν.6 Hz () or, π τ sec ν τ.63 sec () b) so ha 4 3 ka dye-cm E 4 E 4.5 erg c) The maximum velociy is aaied whe he oal eergy
Διαβάστε περισσότεραΣτα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ.
Στα πόμνα θωρούμ ότι όλα συμβαίνουν σ ένα χώρο πιθανότητας ( Ω,,). Modes of covergece: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. { } ίναι οι ξής: σ μια τ.μ.. Ισχυρή σύγκλιση strog covergece { } lim = =.
Διαβάστε περισσότεραRandom Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains
Alied Maheaics 5 6 79-87 Published Olie Seeber 5 i SciRes h://wwwscirorg/oural/a h://dxdoiorg/436/a5659 Rado Aracors for Sochasic Reacio-Diffusio Equaios wih Disribuio Derivaives o Ubouded Doais Eshag
Διαβάστε περισσότεραVidyalankar. Vidyalankar S.E. Sem. III [BIOM] Applied Mathematics - III Prelim Question Paper Solution. 1 e = 1 1. f(t) =
. (a). (b). (c) f() L L e i e Vidyalakar S.E. Sem. III [BIOM] Applied Mahemaic - III Prelim Queio Paper Soluio L el e () i ( ) H( ) u e co y + 3 3y u e co y + 6 uy e i y 6y uyy e co y 6 u + u yy e co y
Διαβάστε περισσότεραIIT JEE (2013) (Trigonomtery 1) Solutions
L.K. Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 677 (+) PAPER B IIT JEE (0) (Trigoomtery ) Solutios TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE
Διαβάστε περισσότερα9.1 Introduction 9.2 Lags in the Error Term: Autocorrelation 9.3 Estimating an AR(1) Error Model 9.4 Testing for Autocorrelation 9.
9.1 Inroducion 9.2 Lags in he Error Term: Auocorrelaion 9.3 Esimaing an AR(1) Error Model 9.4 Tesing for Auocorrelaion 9.5 An Inroducion o Forecasing: Auoregressive Models 9.6 Finie Disribued Lags 9.7
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραSolutions: Homework 3
Solutios: Homework 3 Suppose that the radom variables Y,, Y satisfy Y i = βx i + ε i : i,, where x,, x R are fixed values ad ε,, ε Normal0, σ ) with σ R + kow Fid ˆβ = MLEβ) IND Solutio: Observe that Y
Διαβάστε περισσότεραAppendix. The solution begins with Eq. (2.15) from the text, which we repeat here for 1, (A.1)
Aenix Aenix A: The equaion o he sock rice. The soluion egins wih Eq..5 rom he ex, which we reea here or convenience as Eq.A.: [ [ E E X, A. c α where X u ε, α γ, an c α y AR. Take execaions o Eq. A. as
Διαβάστε περισσότεραDegenerate Perturbation Theory
R.G. Griffi BioNMR School page 1 Degeerate Perturbatio Theory 1.1 Geeral Whe cosiderig the CROSS EFFECT it is ecessary to deal with degeerate eergy levels ad therefore degeerate perturbatio theory. The
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραTime Series Analysis Final Examination
Dr. Sevap Kesel Time Series Aalysis Fial Examiaio Quesio ( pois): Assume you have a sample of ime series wih observaios yields followig values for sample auocorrelaio Lag (m) ˆ( ρ m) -0. 0.09 0. Par a.
Διαβάστε περισσότεραΜια εισαγωγή στα Μαθηματικά για Οικονομολόγους
Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους Μαθηματικά Ικανές και αναγκαίες συνθήκες Έστω δυο προτάσεις Α και Β «Α είναι αναγκαία συνθήκη για την Β» «Α είναι ικανή συνθήκη για την Β» Α is ecessary for
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραLecture 12 Modulation and Sampling
EE 2 spring 2-22 Handou #25 Lecure 2 Modulaion and Sampling The Fourier ransform of he produc of wo signals Modulaion of a signal wih a sinusoid Sampling wih an impulse rain The sampling heorem 2 Convoluion
Διαβάστε περισσότεραα ]0,1[ of Trigonometric Fourier Series and its Conjugate
aqartvelo mecierebata erovuli aademii moambe 3 # 9 BULLETIN OF THE GEORGIN NTIONL CDEMY OF SCIENCES vol 3 o 9 Mahemaic Some pproimae Properie o he Cezàro Mea o Order ][ o Trigoomeric Fourier Serie ad i
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραINDIRECT ADAPTIVE CONTROL
INDIREC ADAPIVE CONROL OULINE. Inroducion a. Main properies b. Running example. Adapive parameer esimaion a. Parameerized sysem model b. Linear parameric model c. Normalized gradien algorihm d. Normalized
Διαβάστε περισσότεραThe Heisenberg Uncertainty Principle
Chemistry 460 Sprig 015 Dr. Jea M. Stadard March, 015 The Heiseberg Ucertaity Priciple A policema pulls Werer Heiseberg over o the Autobah for speedig. Policema: Sir, do you kow how fast you were goig?
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραMATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Matrix Algebra
MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutios to Poblems o Matix Algeba 1 Let A be a squae diagoal matix takig the fom a 11 0 0 0 a 22 0 A 0 0 a pp The ad So, log det A t log A t log
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραn r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1)
8 Higher Derivative of the Product of Two Fuctios 8. Leibiz Rule about the Higher Order Differetiatio Theorem 8.. (Leibiz) Whe fuctios f ad g f g are times differetiable, the followig epressio holds. r
Διαβάστε περισσότεραLecture 17: Minimum Variance Unbiased (MVUB) Estimators
ECE 830 Fall 2011 Statistical Sigal Processig istructor: R. Nowak, scribe: Iseok Heo Lecture 17: Miimum Variace Ubiased (MVUB Estimators Ultimately, we would like to be able to argue that a give estimator
Διαβάστε περισσότεραCHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES
CHAPTER 3 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES EXERCISE 364 Page 76. Determie the Fourier series for the fuctio defied by: f(x), x, x, x which is periodic outside of this rage of period.
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραA Note on Saigo s Fractional Integral Inequalities
Turkish Joural of Aalysis ad Number Theory, 214, Vol 2, No 3, 65-69 Available olie a hp://pubssciepubcom/ja/2/3/2 Sciece ad Educaio Publishig DOI:112691/ja-2-3-2 A Noe o Saigo s Fracioal Iegral Iequaliies
Διαβάστε περισσότεραOscillation Criteria for Nonlinear Damped Dynamic Equations on Time Scales
Oscillaion Crieria for Nonlinear Damped Dynamic Equaions on ime Scales Lynn Erbe, aher S Hassan, and Allan Peerson Absrac We presen new oscillaion crieria for he second order nonlinear damped delay dynamic
Διαβάστε περισσότερα( ) ( t) ( 0) ( ) dw w. = = β. Then the solution of (1.1) is easily found to. wt = t+ t. We generalize this to the following nonlinear differential
Periodic oluion of van der Pol differenial equaion. by A. Arimoo Deparmen of Mahemaic Muahi Iniue of Technology Tokyo Japan in Seminar a Kiami Iniue of Technology January 8 9. Inroducion Le u conider a
Διαβάστε περισσότερα( ) ( ) ( ) Fourier series. ; m is an integer. r(t) is periodic (T>0), r(t+t) = r(t), t Fundamental period T 0 = smallest T. Fundamental frequency ω
Fourier series e jm when m d when m ; m is an ineger. jm jm jm jm e d e e e jm jm jm jm r( is periodi (>, r(+ r(, Fundamenal period smalles Fundamenal frequeny r ( + r ( is periodi hen M M e j M, e j,
Διαβάστε περισσότεραarxiv: v1 [math.ap] 5 Apr 2018
Large-ime Behavior ad Far Field Asympoics of Soluios o he Navier-Sokes Equaios Masakazu Yamamoo 1 arxiv:184.1746v1 [mah.ap] 5 Apr 218 Absrac. Asympoic expasios of global soluios o he icompressible Navier-Sokes
Διαβάστε περισσότεραUniversity of Washington Department of Chemistry Chemistry 553 Spring Quarter 2010 Homework Assignment 3 Due 04/26/10
Universiy of Washingon Deparmen of Chemisry Chemisry 553 Spring Quarer 1 Homework Assignmen 3 Due 4/6/1 v e v e A s ds: a) Show ha for large 1 and, (i.e. 1 >> and >>) he velociy auocorrelaion funcion 1)
Διαβάστε περισσότεραω = radians per sec, t = 3 sec
Secion. Linear and Angular Speed 7. From exercise, =. A= r A = ( 00 ) (. ) = 7,00 in 7. Since 7 is in quadran IV, he reference 7 8 7 angle is = =. In quadran IV, he cosine is posiive. Thus, 7 cos = cos
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραErrata (Includes critical corrections only for the 1 st & 2 nd reprint)
Wedesday, May 5, 3 Erraa (Icludes criical correcios oly for he s & d repri) Advaced Egieerig Mahemaics, 7e Peer V O eil ISB: 978474 Page # Descripio 38 ie 4: chage "w v a v " "w v a v " 46 ie : chage "y
Διαβάστε περισσότεραSolve the difference equation
Solve the differece equatio Solutio: y + 3 3y + + y 0 give tat y 0 4, y 0 ad y 8. Let Z{y()} F() Taig Z-trasform o both sides i (), we get y + 3 3y + + y 0 () Z y + 3 3y + + y Z 0 Z y + 3 3Z y + + Z y
Διαβάστε περισσότεραΨηφιακή Επεξεργασία Εικόνας
ΠΑΝΕΠΙΣΤΗΜΙΟ ΙΩΑΝΝΙΝΩΝ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΪΚΑ ΜΑΘΗΜΑΤΑ Ψηφιακή Επεξεργασία Εικόνας Φιλτράρισμα στο πεδίο των συχνοτήτων Διδάσκων : Αναπληρωτής Καθηγητής Νίκου Χριστόφορος Άδειες Χρήσης Το παρόν εκπαιδευτικό
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραFourier Series. constant. The ;east value of T>0 is called the period of f(x). f(x) is well defined and single valued periodic function
Fourier Series Periodic uctio A uctio is sid to hve period T i, T where T is ve costt. The ;est vlue o T> is clled the period o. Eg:- Cosider we kow tht, si si si si si... Etc > si hs the periods,,6,..
Διαβάστε περισσότεραEN40: Dynamics and Vibrations
EN40: Dyamics a Vibratios School of Egieerig Brow Uiversity Solutios to Differetial Equatios of Motio for Vibratig Systems Here, we summarize the solutios to the most importat ifferetial equatios of motio
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραINTEGRATION OF THE NORMAL DISTRIBUTION CURVE
INTEGRATION OF THE NORMAL DISTRIBUTION CURVE By Tom Irvie Email: tomirvie@aol.com March 3, 999 Itroductio May processes have a ormal probability distributio. Broadbad radom vibratio is a example. The purpose
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότεραThe Euler Equations! λ 1. λ 2. λ 3. ρ ρu. E = e + u 2 /2. E + p ρ. = de /dt. = dh / dt; h = h( T ); c p. / c v. ; γ = c p. p = ( γ 1)ρe. c v.
hp://www.nd.ed/~gryggva/cfd-corse/ The Eler Eqaions The Eler Eqaions The Eler eqaions for D flow: + + p = x E E + p where Define E = e + / H = h + /; h = e + p/ Gréar Tryggvason Spring 3 Ideal Gas: p =
Διαβάστε περισσότεραLAD Estimation for Time Series Models With Finite and Infinite Variance
LAD Estimatio for Time Series Moels With Fiite a Ifiite Variace Richar A. Davis Colorao State Uiversity William Dusmuir Uiversity of New South Wales 1 LAD Estimatio for ARMA Moels fiite variace ifiite
Διαβάστε περισσότεραhp-bem for Contact Problems and Extended Ms-FEM in Linear Elasticity
hp-bem for Coac Problems ad Exeded Ms-FEM i Liear Elasiciy Vo der Fakulä für Mahemaik ud Physik der Gofried Wilhelm Leibiz Uiversiä aover zur Erlagug des Grades Dokor der Naurwisseschafe Dr. rer. a. geehmige
Διαβάστε περισσότεραPresentation of complex number in Cartesian and polar coordinate system
1 a + bi, aεr, bεr i = 1 z = a + bi a = Re(z), b = Im(z) give z = a + bi & w = c + di, a + bi = c + di a = c & b = d The complex cojugate of z = a + bi is z = a bi The sum of complex cojugates is real:
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότερα9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Διαβάστε περισσότεραOutline. Detection Theory. Background. Background (Cont.)
Outlie etectio heory Chapter7. etermiistic Sigals with Ukow Parameters afiseh S. Mazloum ov. 3th Backgroud Importace of sigal iformatio Ukow amplitude Ukow arrival time Siusoidal detectio Classical liear
Διαβάστε περισσότεραProblem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.
Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +
Διαβάστε περισσότεραBessel function for complex variable
Besse fuctio for compex variabe Kauhito Miuyama May 4, 7 Besse fuctio The Besse fuctio Z ν () is the fuctio wich satisfies + ) ( + ν Z ν () =. () Three kids of the soutios of this equatio are give by {
Διαβάστε περισσότεραΥπόδειγµα Προεξόφλησης
Αρτίκης Γ. Παναγιώτης Υπόδειγµα Προεξόφλησης Μερισµάτων Γενικό Υπόδειγµα (Geeral Model) Ταµειακές ροές από αγορά µετοχών: Μερίσµατα κατά την διάρκεια κατοχής των µετοχών Μια αναµενόµενη τιµή στο τέλος
Διαβάστε περισσότεραα β
6. Eerg, Mometum coefficiets for differet velocit distributios Rehbock obtaied ) For Liear Velocit Distributio α + ε Vmax { } Vmax ε β +, i which ε v V o Give: α + ε > ε ( α ) Liear velocit distributio
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραd dt S = (t)si d dt R = (t)i d dt I = (t)si (t)i
d d S = ()SI d d I = ()SI ()I d d R = ()I d d S = ()SI μs + fi + hr d d I = + ()SI (μ + + f + ())I d d R = ()I (μ + h)r d d P(S,I,) = ()(S +1)(I 1)P(S +1, I 1, ) +()(I +1)P(S,I +1, ) (()SI + ()I)P(S,I,)
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραManaging Production-Inventory Systems with Scarce Resources
Managing Producion-Invenory Sysems wih Scarce Resources Online Supplemen Proof of Lemma 1: Consider he following dynamic program: where ḡ (x, z) = max { cy + E f (y, z, D)}, (7) x y min(x+u,z) f (y, z,
Διαβάστε περισσότεραDiane Hu LDA for Audio Music April 12, 2010
Diae Hu LDA for Audio Music April, 00 Terms Model Terms (per sog: Variatioal Terms: p( α Γ( i α i i Γ(α i p( p(, β p(c, A j Σ i α i i i ( V / ep β (i j ij (3 q( γ Γ( i γ i i Γ(γ i q( φ q( ω { } (c A T
Διαβάστε περισσότεραStrain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραThe choice of an optimal LCSCR contract involves the choice of an x L. such that the supplier chooses the LCS option when x xl
EHNIA APPENDIX AMPANY SIMPE S SHARIN NRAS Proof of emma. he choice of an opimal SR conrac involves he choice of an such ha he supplier chooses he S opion hen and he R opion hen >. When he selecs he S opion
Διαβάστε περισσότεραFREE VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM Revision B
FREE VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM Revisio B By Tom Irvie Email: tomirvie@aol.com February, 005 Derivatio of the Equatio of Motio Cosier a sigle-egree-of-freeom system. m x k c where m
Διαβάστε περισσότεραQuadratic Expressions
Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots
Διαβάστε περισσότεραJ. of Math. (PRC) u(t k ) = I k (u(t k )), k = 1, 2,, (1.6) , [3, 4] (1.1), (1.2), (1.3), [6 8]
Vol 36 ( 216 ) No 3 J of Mah (PR) 1, 2, 3 (1, 4335) (2, 4365) (3, 431) :,,,, : ; ; ; MR(21) : 35A1; 35A2 : O17529 : A : 255-7797(216)3-591-7 1 d d [x() g(, x )] = f(, x ),, (11) x = ϕ(), [ r, ], (12) x(
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραHomework 4.1 Solutions Math 5110/6830
Homework 4. Solutios Math 5/683. a) For p + = αp γ α)p γ α)p + γ b) Let Equilibria poits satisfy: p = p = OR = γ α)p ) γ α)p + γ = α γ α)p ) γ α)p + γ α = p ) p + = p ) = The, we have equilibria poits
Διαβάστε περισσότεραDOCUMENTOS DE ECONOMÍA Y FINANZAS INTERNACIONALES. Working Papers on International Economics and Finance
DOCUMENTOS DE ECONOMÍA Y FINANZAS INTERNACIONALES Workig Papers o Ieraioal Ecoomics ad Fiace DEFI 8- Julio 8 Time-varyig coiegraig regressio aalysis wih a applicaio o he log-ru ieres rae pass-hrough i
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότερα