Keldysh Formalism: Nonequilibrium Green s Function


 Iris Barber
 1 years ago
 Views:
Transcription
1 Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium Green s Funcion (NEGF), also named as Keldysh Formalism, is presened here. Perurbaion heory is also builded up sep by sep on he basis of nonineracing NEGF, and hen leads o he Dyson s Equaion. Finally, applicaion of NEGF on ighbinding model wih random impuriy is explicily calculaed as an example. I. DEFINITION, FREE FIELD AND PERTURBATION When only he properies of ground sae is concerned, Zeroemperaure (single and manyparicle) Green s funcion ogeher wih is perurbaion heory principally gives all he informaion. Also if he equilibrium hermal disribuion is concerned, we can urn o Masubara Green s funcion and is perurbaion heory[2]. However, we may need o consider some more general saes, such as a saionary nonequilibrium sae for example a sae wih nonzero curren, or even an arbirary sae ρ, how o ge correlaion funcion for such sysem and hen how o do he corresponding perurbaion expansion? In experimens of conduciviy, here are such nonequilibrium sysem, which has nonzero curren and includes ineracion beween paricles. So in his noe I will shorly review he idea and echnics of Nonequilibrium Green s funcion, also known as Keldysh Formalism. All he formulas defined in his noe is only for fermions, alhough i is no hard o make hem o also cover bosons. Generally we need o calculae he Green s funcion G (x, ; x, ) = i } ) (T T r ψ H (x, ) ψ H (x, ) ρ H, (1) where ψ H (x, ) is he annihilaion operaor in Heisenberg picure, and ρ is he densiy marix also in Heisenberg picure. We will se = 1 for a while. We also assume ρ is diagonal under paricle number basis, [N, ρ] = 0. More general densiy marix could be possible, bu no used anywhere ye. For free field, his can be calculaed quie srai forward, ha ψh (x, ) = k e iω(k)+ikx c k ψ H (x, ) = k eiω(k) ikx c, (2) k hen G 0 (x, ; x, ) = iθ ( ) k 1 n k e iω(k)( )+ik(x x ) iθ ( ) k n k e iω(k)( )+ik(x x ) (3) Following he roadmap of zeroemperaure Green s funcion, nex sep would be o urn ψ H (x, ) ino ineracion picure ψ I (x, ) and o somehow replace he real sae ρ H wih sae of nonineracing sysem ρ 0 H. Le s firs organize he above mapping for zeroemperaure Green s funcion and hen see if he same procedure can be applied ono NEGF. Zeroemperaure Green s funcion is defined as a special case of eq(1), ρ H = Ω Ω, } G (x, ; x, ) = i Ω T ψ H (x, ) ψ H (x, ) Ω, (4) Firs, express T A H ( 1 ) B H ( 2 )} in ineracion picure. and H = H 0 + e ɛ 0 H 1, (5) A H () = e ih( 0) e ih0( 0) A I () e ih0( 0) e ih( 0). (6)
2 2 FIG. 1: The imeloop inegraion pah. Define uniary operaor U (, ) = e ih0( 0) e ih( 0) e ih0( 0), (7) which is he formal soluion of where H I () = e ih0( 0) H 1 e ih0( 0). Or he explici form is Then for 1 > 2 he imeordered operaors, i U (, ) = H I () U (, ), (8) U (, ) = T exp i dh I (). (9) T A H ( 1 ) B H ( 2 )} = A H ( 1 ) B H ( 2 ) = U ( 1, 0 ) A I ( 1 ) U ( 1, 0 ) U ( 2, 0 ) B I ( 2 ) U ( 2, 0 ) = U ( 0, 1 ) A I ( 1 ) U ( 1, 2 ) B I ( 2 ) U ( 2, 0 ). (10) = U ( 0, ) U (, 1 ) A I ( 1 ) U ( 1, 2 ) B I ( 2 ) U ( 2, ) U (, 0 ) Nex for rue ground sae Ω, we can relae i wih 0, he ground sae of nonineracing field, U ( 0, ) 0 = lim T eih( T 0) e ih0( T 0) 0 = lim T e ih(t +0) 0. (11) According o GellMann and Low Theorem, for adiabaic coupling, back in Schrödinger picure, ground sae is sable, φ 0 ( f ) U ( f, i ) φ 0 ( i ). In our case, Ω ( 0 ) U ( 0, T ) 0 ( T ), so up o some consan, U ( 0, ) 0 Ω U (, 0 ) Ω 0, (12) and similarly, U (, 0 ) Ω 0. Therefore, when 1 > 2 } Ω T ψ H (x, ) ψ H (x, ) Ω 0 U (, 1 ) A I () U ( 1, 2 ) B I () U ( 2, ) 0. (13) Finally, he consan can be eliminaed as 0 T G (x, ; x, ) = i } ψ I (x, ) ψ I (x, ) U (, ) 0 U (, ) 0 0, (14) where now all he quaniies are from free field, and herefore can be calculaed order by order by perurbaion. However, when we apply he same procedure o general ρ H, we require he GellMann and Low heorem holds for excied saes for boh = ±. This condiion is oo srong. The idea o solve above problem is o remove some special poins from he hree special poins = (, 0, ), for example by seing 0. In his way we require ha only φ 0 () = 0 insead of boh φ 0 (± ) = 0. The pah is o sar along = o = 0 and hen come back along = 0 o =. And finally le 0, a picured in fig(1). As inhe case of usual Green s funcion, le firsly reorganize T A ( 1 ) B ( 2 )}, and hen he relaion beween ρ H and ρ 0 H. For 1 > 2, inser U (, ) ino eq(10), T A H ( 1 ) B H ( 2 )} = U ( 0, ) U (, ) U (, 1 ) A I ( 1 ) U ( 1, 2 ) B I ( 2 ) U ( 2, )} U (, 0 ). (15) Then boh sides relaes only wih U (, 0 ), which sill link he rue eigensaes n wih freefield eigensaes n 0, ha U (, 0 ) n n 0. (16)
3 3 (a) G + (, ) and G (, ) (b)g c (, ) (c) G c (, ) s (d) Impuriy Scaering FIG. 2: One possible Feynman rules for NEGF. Therefore, we arrive G (x, ; x, ) = i T r(u(, )Tψ I (x,)ψ I(x, )U(,)}ρ 0 H) T r(u(,)ρ 0 H) = i T r(tcψ I (x,)ψ I(x, )U(,)}ρ 0 H) T r(u(,)ρ 0 H), (17) where T c is he loop order as shown in fig(1) o order he ime along he loop from o. For example, a he above branch i s he same wih ime order T c = T, while a he lower branch i s aniime order T c = T. And U (, ) = T c exp i dhi (). (18) One hing need o be noiced ha in he final expression of Green s funcion, all he ime spos should be reaed as poins a above branch (also call posiive branch). However, laer on, we will see ha when we ry o calculae such Green s funcion, we will need o do inegral also over he lower branch (also call negaive branch). This means generally no all he ime spos are a he posiive branch, and he second line of eq(17) could no be separaed as U (, ) and U (, ) ouside and inside usual ime order T. Therefore, generally we need o define Green s funcion according o he branches of and. There are four differen configuraions of hem, as picured in fig(2). They are less and greaer Green s funcion[1], G + (x, ; x, ) = it r (ρ H T c ψ H (x, + ) ψ ( ) ( ) H x, = it r ρ H ψ H (x, ) ψ H (x, ) G (x, ; x, ) = it r (ρ H T c ψ H (x, ) ψ ( ) ( ), (19) H x, + = it r ρ H ψ H (x, ) ψ H (x, ) where ± means he ime is on posiive/negaive branch; ime ordered and aniime ordered Green s funcion, G c (x, ; x, ) = it r (ρ H T c ψ H (x, + ) ψ ( ) ( H x, + = it r ρ H T ψ H (x, ) ψ H (x, ) G c (x, ; x, ) = it r (ρ H T c ψ H (x, ) ψ ( ) ( ; (20) H x, = it r ρ H T ψ H (x, ) ψ H (x, )
4 and rearded and advanced Green s funcion, G r (x, ; x, ) = iθ ( ) T r (ρ H ψ H (x, ), ψ H (x, ). (21) G a (x, ; x, ) = iθ ( ) T r (ρ H ψ H (x, ), ψ H (x, ) Only hree of hem are independen, G r = G c G + = G G c, G a = G c G = G + G c. (22) This can be seen easily by separaing G ± as four pars, θ ( ) G ± and θ ( + ) G ±. Then G c = θ ( + ) G + + θ ( ) G G c = θ ( ) G + + θ ( + ) G (23) So G c + G c = G + + G. Noice for greaer/less Green s funcion, differen definiions and noaions are used in lieraure, for example, G > G and G < G + in [2]. Before we go furher o perurbaion calculaion of ineracing field, le s firs lis all he freefield Nonequilibrium Green s funcions. 4 G 0,+ (x, ; x, ) = i k n k e iω(k)( )+ik(x x ) G 0, (x, ; x, ) = i k 1 n k e iω(k)( )+ik(x x ) G 0,c (x, ; x, ) = i k θ ( ) n k e iω(k)( )+ik(x x ) G 0, c (x, ; x, ) = i k θ ( ) n k e iω(k)( )+ik(x x ) G 0,r (x, ; x, ) = iθ ( ) k e iω(k)( )+ik(x x ) G 0,a (x, ; x, ) = iθ ( ) k e iω(k)( )+ik(x x ) (24) Abou Eq(16), he expression similar wih of GellMann and Low Theorem, i should be undersood in a kind inverse logic order. For an ineracing field H = H 0 + H 1, he nonequilibrium densiy marix ρ H is no clearly defined, while ρ 0 H of he corresponding free filed H 0 could be well defined. Then he meaning of eq(16) is ha we sar from ρ 0 H a = and arrive a sae a = 0, and hen jus use his new sae as ρ H. I s no guaraneed ha his reamen will correspond o he rue nonequilibrium disribuion ρ H. So for NEGF, even concepually insead of saring from he rue ρ H, we build up he heory from ρ 0 H. This could be a disadvanage of NEGF. Also several advanages need o be noiced. Firs, his NEGF can be used for sysem a hermal equilibrium sae, which is usually he subjec of Masubara Green s funcion. We will use as an example in III. Second, even H 1 explicily depends on, his NEFG is sill applicable, because we only se one end as H 1 () = 0, H 1 ( ) could be arbirary. A las, for nonequilibrium sysem as long as a unambiguous freefield corresponding nonequilibrium sae could be defined, we could always do he perurbaion calculaion order by order. II. SELFENERGY AND DYSON S EQUATION In order o arrive a he Dyson s equaion, we would firsly go o a lile pracice of firs and second order calculaion. Consider an example wih V = αβ M αβc αc β [2]. Then he firs order perurbaion gives, G + (µ, ; ν, ) = G 0,+ (µ, ; ν, ) dsm αβ T r ( T c Cµ () C ν ( ) C α (s) C β (s). (25) αβ The second erm has one disconneced erm and one nonzero conneced erm, which includes αβ dsmαβ Tc Cµ () C α (s) } T c C ν ( ) C β (s) } = [ αβ dsg0,c (µ, ; α, s) M αβ G 0,+ (β, s; ν, ) + ] dsg0,+ (µ, ; α, s) M αβ G 0, c (β, s; ν, ) (26) A he nex order, more erms will be coupled ino his expansion series. Similarly G also couples wih all oher Gs. We will work ou explicily a special case of his ineracion in he nex secion III. Skipping some deailed calculaion here, overall hese expansion series could be represened by a marix equaion, in he form of ( ) Ĝ = Ĝ0ΣĜ0 + Ĝ0ΣĜ0ΣĜ0 + = Ĝ0 1 + ΣĜ (27)
5 5 Like he example abou disordered sysem in lecure noes[3], concep of selfenergy Σ can be inroduced. Then he Dyson s equaion of NEGF could be organized as Ĝ (, ) = Ĝ0 (, ) + d 1 d 2 Ĝ 0 (, 1 ) ˆΣ ( 1, 2 ) Ĝ ( 2, ), (28) where ( Ĝc Ĝ Ĝ = Ĝ + Ĝ c ), (29) and Ĝ (, ) is he operaor form of G (x, ; x, ) = x Ĝ (, ) x ; Ĝ 0 = ˆD is he freefield NEGF; and ˆΣ is he selfenergy erm coming from ineracion, ( ) ˆΣc ˆΣ ˆΣ = ˆΣ +. (30) ˆΣ c We will explain boh heoreically and by examples how o ge he selfenergy erm from ineracion laer. Le s emporally suppose hey are known. Because only hree are independen of hese six NEGFs, he above formula can be simplified furher by choosing he righ hree independen ones. Le s use Ĝa, Ĝr and From eq(22) i is easy o see ha his ransformaion is uniary, ( Ĝc Ĝ U Ĝ + Ĝ c ˆF = Ĝc + Ĝ c. (31) ) ( ) U 0 Ĝ = a Ĝ r Ǧ, (32) ˆF where U = 1 ( ) 1 1. (33) Induced by his ransformaion, he selfenergy erm will ransform as ( ) ˆΣc ˆΣ ˇΣ U ˆΣ + U, (34) ˆΣ c And finally, as we will see from he example in secion III, because generally Σ c + Σ c = (Σ + + Σ ), ( ) Ω Σ r ˇΣ Σ a, (35) 0 Therefore, eq(28) sill holds, bu in a new form, or simply denoed as, Ǧ = Ǧ0 + Ǧ0 ˇΣǦ. (36) Then he calculaion of Green s funcion becomes calculaion of selfenergy, where differen approximaions, such as Born or SelfConsisen Born approximaion, could be used. III. EXAMPLE Le s apply he general procedure above ono he following problem, he random impuriy, a special case of he example in secion II, H = k ɛ (k) C k C k + k,q,x M (q) e iq X C k+q V C k, (37)
6 6 s s FIG. 3: Firs order expansion on loop pah, LHS for s on posiive branch, RHS for s on negaive branch. Boh of hem are zero if M (0) = 0. s 1 s 2 s 1 s 2 FIG. 4: Second order expansion of G + on loop pah, for s 1, s 2 boh on posiive branch. wih M (0) = 0 means average effec of impuriy is zero. X is uniformly disribued so ha k is sill a good quanum number, so G 0 (k, ; k, ) = G 0 (k, ; k, ) δ (k k ) and G (k, ; k, ) = G (k, ; k, ) δ (k k ). Le s firs work in ime domain, laer on we will do he Fourier ransformaion o work in he frequency domain. Firs order expansion of G +, eq(26) for our special case will be X 1 V [ dsg 0,c (k, ; k, s) M (0) G 0,+ (k, s; k, ) + ] dsg 0,+ (k, ; k, s) M (0) G 0, c (k, s; k, ) = 0. (38) Or equivalenly by Feynman Diagram, as in fig(3). The second order expansion of G + is, i 2V 2 ds 1 ds 2 e iq1 X1 iq2 X2 M (q 1 ) M (q 2 ) C k 1,q 1,X 1 k 2,q 2,X 2 T r (T, (39) c C k () C k ( ) C k 1+q 1 (s 1 ) C k1 (s 1 ) C k 2+q 2 (s 2 ) C k2 (s 2 ) where by Wick s Theorem, = T r +T r C k () C k 2+q 2 (s 2 ) T r T r C k () C k 1+q 1 (s 1 ) T r C k () C k ( ) C k 1+q 1 (s 1 ) C k1 (s 1 ) C k 2+q 2 (s 2 ) C k2 (s 2 ) C k1 (s 1 ) C k ( ) T r T r C k2 (s 2 ) C k ( ) C k2 (s 2 ) C k 1+q 1 (s 1 ). (40) C k1 (s 1 ) C k 2+q 2 (s 2 ) This has o be calculaed for he four configuraions of s 1, s 2. For example, here le s work ou for case of boh s 1, s 2 on posiive branch. i 2V 2 ds 1 ds 2 M (q 1 ) 2 e iq1 X1+iq1 X2 G 0,c (k, ; k, s 2 ) G 0,+ (k, s 1 ; k, ) G 0,c (k + q 1, s 2 ; k + q 1, s 1 ) q 1,X 1,X 2 + i 2V 2 ds 1 ds 2 M (q 2 ) 2 e iq2 X1 iq2 X2 G 0,c (k, ; k, s 1 ) G 0,+ (k, s 2 ; k, ) G 0,c (k + q 2, s 1 ; k + q 2, s 2 ). (41) q 2,X 1,X 2 = i V 2 ds 1 ds 2 M (q) 2 e iq X1 iq X2 G 0,c (k, ; k, s 1 ) G 0,+ (k, s 2 ; k, ) G 0,c (k + q, s 1 ; k + q, s 2 ) q,x 1,X 2 Those wo erm give he same conribuion because of he inegraion and summaion over s 1, s 2, q 1,2. Excep ha here G + couples wih G 0,c and ec, his expression has exacly he same form wih he second order of he usual
7 7 G + G G c G c NonZero Impuriy Line = Σ 1,c Σ 1, Σ 1, Σ 1, c FIG. 5: Anoher se of Feynman rules for NEGF, second order expansion is shown as example o express in hese new rules. Furher perurbaion could be done in his diagram form. Green s funcion wih impuriy. Similarly, G + also coupled wih G 0,, G 0, c and obviously G 0,+ iself. So par of he firs order of selfenergy erm could be defined as Σ c (k; s 2, s 1 ) i 2V 2 M (q 1 ) 2 e iq1 X1+iq1 X2 G 0,c (k + q, s 2 ; k + q, s 1 ). (42) q 1,X 1,X 2 So we ge a erm of G + a he second order in he form of If we check he erm abou G + in eq(28), we will ge G 2,+ = G 0,+ Σ 1,c G 0,c +... (43) G 2,+ = G 0,+ Σ 1,c G 0,c + G 0, c Σ 1,+ G 0,c + G 0,+ Σ 1, G 0,+ + G 0, c Σ 1, c G 0,+, (44) which does include he above erm we already calculaed. The diagram expression of all hose four erms are shown in he follow fig(5). A deail calculaion include all he configuraion of s 1, s 2 will give us he lef hree erms. Similarly we can do he second order expansion of G,c c. Then he hird order expansion will give us more selfenergy erms. Finally, we can ge he Dyson s equaion, which afer Fourier ransformaion will lead us o frequency domain Dyson s equaion. In he Feynman diagrams in fig(2), where he ime loop is explicily down o be an indicaor of he ype of Green s funcion. Besides his, considering he characer of our specific case, M (0) = 0, we may jus use differen lines o represen differen Green s funcions, so ha he diagrams could be simpler, Here we skip he boring derivaion of eq(36), he marix form Dyson s equaion, hopefully his will no sop one undersanding he general picure of NEGF. IV. SUMMARY Keldysh NEGF Formula is a generalizaion of he usual zero and nonzero emperaure Green s funcion. I can be applied ono he usual equilibrium cases and nonequilibrium case. The way we derive such formula here is o consruc a loop from o and back o, and inroducing Green s funcions G (, ) corresponding o he differen configuraion of (, ) over branches. Formally he srucure of Green s funcion and heir Dyson s Equaion
8 looks equivalen wih he one for usual Green s Funcion. So all he perurbaion calculaion is quie sraighforward. However, in order o do he perurbaion, a welldefined nonineracing sae and Green s funcion is a necessary saring poin. Wheher such saring poin leads o he real nonequilibrium sae is quesionable, and relies on he comparison beween calculaion and experimens. There is anoher way o ge his formula in [4], where he auhors use conour pah over complex plan o replace above loop pah. The derivaion is more elegan in a sense, bu he problem of picking up he righ saring poin sill exiss, alhough he auhors claim heir reamen is exac more or less. 8 [1] C. Caroli and R. Combresco and P. Nozieres and D. SainJames, Direc calculaion of he unneling curren, J. Phys. C: Solid S. Phys, 4(1971), [2] G. D. Manhan, ManyParicle Physics(1990 New York), Plenum Press. [3] M. Berciu, Lecure noes for Phys503, 2005, hp://www.physics.ubc.ca/ berciu/. [4] H. Haug and A.P. Jauho, Quanum Kineics in Transpor and Opics of Semiconducors(1996 Berlin Heidelberg), Springer Verlag.
Random Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More information1.Blowdown of a Pressurized Tank
.Blowdown of a ressurized Tank This experimen consiss of saring wih a ank of known iniial condiions (i.e. pressure, emperaure ec.) and exiing he gas hrough a choked nozzle. The objecive is o es he heory
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationPhysics 2001 Problem Set 2 Solutions
Physics 2001 Problem Se 2 Soluions Jeff Kissel Sepember 12, 2006 1. An objec moves from one poin in space o anoher. Afer i arrives a is desinaion, is displacemen is (a) greaer han or equal o he oal disance
More informationNewton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO
Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationINTRODUCTORY MATHEMATICS FOR ECONOMICS MSC S. LECTURE 5: DIFFERENCE EQUATIONS. HUW DAVID DIXON CARDIFF BUSINESS SCHOOL. SEPTEMEBER 2009.
INTRODUCTORY MATHEMATICS FOR ECONOMICS MSC S. LECTURE 5: DIFFERENCE EQUATIONS. HUW DAVID DIXON CARDIFF BUSINESS SCHOOL. SEPTEMEBER 9. 5.1 Difference Equaions. Much of economics, paricularly macroeconomics
More informationRenewal processes and Poisson process
CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More information24.2. Properties of the Fourier Transform. Introduction. Prerequisites. Learning Outcomes
Properies of he Fourier Transform 4. Inroducion In his Secion we shall learn abou some useful properies of he Fourier ransform which enable us o calculae easily furher ransforms of funcions and also in
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More information24.3. Some Special Fourier Transform Pairs. Introduction. Prerequisites. Learning Outcomes
Some Special Fourier Transform Pairs 24.3 Inroducion In his final Secion on Fourier ransforms we shall sudy briefly a number of opics such as Parseval s heorem and he relaionship beween Fourier ransform
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationChapter 7 Response of Firstorder RL and RC Circuits
Chaper 7 Response of Firsorder RL and RC Circuis 7.1 The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationPosition, Velocity, and Acceleration
rev 12/2016 Posiion, Velociy, and Acceleraion Equipmen Qy Equipmen Par Number 1 Dynamic Track ME9493 1 Car ME9454 1 Fan Accessory ME9491 1 Moion Sensor II CI6742A 1 Track Barrier Purpose The purpose
More informationTime variant processes in failure probability calculations
Time varian processes in failure probabiliy calculaions A. Vrouwenvelder (TUDelf/TNO, The Neherlands) 1. Inroducion Acions on srucures as well as srucural properies are usually no consan, bu will vary
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationDERIVATIVES ALONG VECTORS AND DIRECTIONAL DERIVATIVES. Math 225
DERIVATIVES ALONG VECTORS AND DIRECTIONAL DERIVATIVES Mah 225 Derivaives Along Vecors Suppose ha f is a funcion of wo variables, ha is, f : R 2 R, or, if we are hinking wihou coordinaes, f : E 2 R. The
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More information2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity
.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationVelocity and Acceleration 4.6
Velociy and Acceleraion 4.6 To undersand he world around us, we mus undersand as much abou moion as possible. The sudy of moion is fundamenal o he principles of physics, and i is applied o a wide range
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationEmergence of FokkerPlanck Dynamics within a Closed Finite Spin System
Emergence of FokkerPlanck Dynamics wihin a Closed Finie Spin Sysem H. Niemeyer(*), D. Schmidke(*), J. Gemmer(*), K. Michielsen(**), H. de Raed(**) (*)Universiy of Osnabrück, (**) Supercompuing Cener Juelich
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationCh. 2 1Dimensional Motion conceptual question #6, problems # 2, 9, 15, 17, 29, 31, 37, 46, 54
Ch. 2 1Dimensional Moion concepual quesion #6, problems # 2, 9, 15, 17, 29, 31, 37, 46, 54 Quaniies Scalars disance speed Vecors displacemen velociy acceleraion Defininion Disance = how far apar wo poins
More informationPractical 1 RC Circuits
Objecives Pracical 1 Circuis 1) Observe and qualiaively describe he charging and discharging (decay) of he volage on a capacior. 2) Graphically deermine he ime consan for he decay, τ =. Apparaus DC Power
More information24.3. Some Special Fourier Transform Pairs. Introduction. Prerequisites. Learning Outcomes. Before starting this Section you should...
Some Special Fourier Transform Pairs 4.3 Inroducion Prerequisies Before saring his Secion you should... Learning Oucomes Afer compleing his Secion you should be able o... . Parseval s Theorem Recall from
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More information1. The graph shows the variation with time t of the velocity v of an object.
1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially
More informationDisplacement  Time (dt) Graphs
Lesson 9: d & v Graphs Graphing he moion of objecs gives us a way o inerpre he moion ha would oherwise be difficul. Graphs will also allow you o show a large amoun of informaion in a compac way. Essenially
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationScattering and Decays from Fermi s Golden Rule including all the h s and c s
PHY 362L Supplemenary Noe Scaering and Decays from Fermi s Golden Rule including all he h s and c s (originally by Dirac & Fermi) References: Griffins, Inroducion o Quanum Mechanics, Prenice Hall, 1995.
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationLab 2 Position and Velocity
b Lab 2 Posiion and Velociy Wha You Need To Know: Working Wih Slope In las week s lab you deal wih many graphing ideas. You will coninue o use one of hese ideas in his week s lab, specifically slope. Howeer,
More informationConcepTests and Answers and Comments for Section 2.5
90 CHAPTER TWO For Problems 9, le A = f() be he deph of read, in cenimeers, on a radial ire as a funcion of he ime elapsed, in monhs, since he purchase of he ire. 9. Inerpre he following in pracical erms,
More informationTopic 5. Energy & Power Spectra, and Correlation. 5.1 Review of Discrete Parseval for the Complex Fourier Series
Topic 5 Energy & Power Specra, and Correlaion In Lecure we reviewed he noion of average signal power in a periodic signal and relaed i o he A n and B n coeciens of a Fourier series, giving a mehod of calculaing
More information24.2. Properties of the Fourier Transform. Introduction. Prerequisites. Learning Outcomes. Before starting this Section you should...
Properies of he Fourier Transform 4. Inroducion Prerequisies Before saring his Secion you should... Learning Oucomes Afer compleing his Secion you should be able o... . Lineariy Properies of he Fourier
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationINTERSPECIFIC COMPETITION AND COMPETITIVE EXCLUSION
9 INTERSPECIFIC COMPETITION AND COMPETITIVE EXCLUSION Objecives Program he LokaVolerra model of inerspecific compeiion in a spreadshee. Undersand he compeiive exclusion principle and how i relaes o he
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationPES 1120 Spring 2014, Spendier Lecture 24/Page 1
PES 1120 Spring 2014, Spendier Lecure 24/Page 1 Today:  circuis (imevarying currens)  Elecric Eel Circuis Thus far, we deal only wih circuis in which he currens did no vary wih ime. Here we begin a
More information( ) in the following way. ( ) < 2
Sraigh Line Moion  Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and manmade. Wrie down several of hem. Horizonal cars waer
More informationMaking Use of Gate Charge Information in MOSFET and IGBT Data Sheets
Making Use of ae Charge Informaion in MOSFET and IBT Daa Shees Ralph McArhur Senior Applicaions Engineer Advanced Power Technology 405 S.W. Columbia Sree Bend, Oregon 97702 Power MOSFETs and IBTs have
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationAP CALCULUS BC 2010 SCORING GUIDELINES. Question 1
AP CALCULUS BC 2010 SCORING GUIDELINES Quesion 1 There is no snow on Jane s driveway when snow begins o fall a midnigh. From midnigh o 9 A.M., snow cos accumulaes on he driveway a a rae modeled by f()
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More information13 Solving nonhomogeneous equations: Variation of the constants method
13 Solving nonhomogeneous equaions: Variaion of he consans meho We are sill solving Ly = f, (1 where L is a linear ifferenial operaor wih consan coefficiens an f is a given funcion Togeher (1 is a linear
More informationAcceleration Lab Teacher s Guide
Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More information4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F
efore you begin: Use black pencil. Wrie and bubble your SU ID Number a boom lef. Fill bubbles fully and erase cleanly if you wish o change! 20 Quesions, each quesion is 10 poins. Each quesion has a mos
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More information1. The reaction rate is defined as the change in concentration of a reactant or product per unit time. Consider the general reaction:
CHAPTER TWELVE CHEMICAL KINETICS For Review. The reacion rae is defined as he change in concenraion of a reacan or produc per uni ime. Consider he general reacion: A Producs where rae If we graph vs.,
More information11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.
11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationPhysics 111 Fall 2007 Electric Currents and DC Circuits
Physics 111 Fall 007 Elecric Currens and DC Circuis 1 Wha is he average curren when all he sodium channels on a 100 µm pach of muscle membrane open ogeher for 1 ms? Assume a densiy of 0 sodium channels
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More informationMaking a Faster Cryptanalytic TimeMemory TradeOff
Making a Faser Crypanalyic TimeMemory TradeOff Philippe Oechslin Laboraoire de Securié e de Crypographie (LASEC) Ecole Polyechnique Fédérale de Lausanne Faculé I&C, 1015 Lausanne, Swizerland philippe.oechslin@epfl.ch
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationUsing RCtime to Measure Resistance
Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a builin ADC (Analog o Digial Converer)
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationOutline. Outline. 1 Maximum Likelihood Estimation in a Nutshell. 2 MLE of Independent Data
Ouline Ouline Ouline Maximum Likelihood Esimaion in a Nushell Maximum Likelihood: An Inroducion 2 of Independen Daa Example: esimaing mean and variance Example: OLS as Chrisian Julliard Deparmen of Economics
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More informationRC Circuit and Time Constant
ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisorcapacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Wholerange Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More information