Keldysh Formalism: Non-equilibrium Green s Function

Size: px
Start display at page:

Download "Keldysh Formalism: Non-equilibrium Green s Function"

Transcription

1 Keldysh Formalism: Non-equilibrium 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 Non-equilibrium 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 non-ineracing NEGF, and hen leads o he Dyson s Equaion. Finally, applicaion of NEGF on igh-binding 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, Zero-emperaure (single- and many-paricle) 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 non-equilibrium 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 non-equilibrium sysem, which has non-zero curren and includes ineracion beween paricles. So in his noe I will shorly review he idea and echnics of Non-equilibrium 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 zero-emperaure 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 non-ineracing sysem ρ 0 H. Le s firs organize he above mapping for zero-emperaure Green s funcion and hen see if he same procedure can be applied ono NEGF. Zero-emperaure 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 ime-loop 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 ime-ordered 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 non-ineracing field, U ( 0, ) 0 = lim T eih( T 0) e ih0( T 0) 0 = lim T e ih(t +0) 0. (11) According o Gell-Mann 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 Gell-Mann 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 free-field 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 ani-ime 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 ani-ime 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 free-field Non-equilibrium 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 Gell-Mann and Low Theorem, i should be undersood in a kind inverse logic order. For an ineracing field H = H 0 + H 1, he non-equilibrium 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 non-equilibrium 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 non-equilibrium sysem as long as a unambiguous free-field corresponding non-equilibrium sae could be defined, we could always do he perurbaion calculaion order by order. II. SELF-ENERGY 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 non-zero 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 self-energy 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 free-field NEGF; and ˆΣ is he self-energy erm coming from ineracion, ( ) ˆΣc ˆΣ ˆΣ = ˆΣ +. (30) ˆΣ c We will explain boh heoreically and by examples how o ge he self-energy 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 self-energy 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 self-energy, where differen approximaions, such as Born or Self-Consisen 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 Non-Zero 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 self-energy 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 self-energy 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 non-zero emperaure Green s funcion. I can be applied ono he usual equilibrium cases and non-equilibrium 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 well-defined non-ineracing sae and Green s funcion is a necessary saring poin. Wheher such saring poin leads o he real non-equilibrium 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. Sain-James, Direc calculaion of he unneling curren, J. Phys. C: Solid S. Phys, 4(1971), [2] G. D. Manhan, Many-Paricle 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 1-D. 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 1-D. 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 information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 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 information

The Transport Equation

The 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 information

Inductance and Transient Circuits

Inductance 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 information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT 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 information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order 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 information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 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 information

Duration 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.

Duration 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 UVA-F-38 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 information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S 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 cuing-edge

More information

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Entropy: 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 information

Signal Processing and Linear Systems I

Signal Processing and Linear Systems I Sanford Universiy Summer 214-215 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 14-15, Gibbons

More information

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS 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 information

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabot 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 information

Term Structure of Prices of Asian Options

Term 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 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

More information

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chapter 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 information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics 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 information

Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal 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 information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

TEMPORAL 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 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 information

Circuit Types. () i( t) ( )

Circuit 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 information

Lectures # 5 and 6: The Prime Number Theorem.

Lectures # 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 ζ-funcion o skech an argumen which would give an acual formula for π( and sugges

More information

Why Did the Demand for Cash Decrease Recently in Korea?

Why 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 information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE 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 information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 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 information

Emergence of Fokker-Planck Dynamics within a Closed Finite Spin System

Emergence of Fokker-Planck Dynamics within a Closed Finite Spin System Emergence of Fokker-Planck 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 information

2.5 Life tables, force of mortality and standard life insurance products

2.5 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 information

1 HALF-LIFE EQUATIONS

1 HALF-LIFE EQUATIONS R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life)

More information

Cointegration: The Engle and Granger approach

Cointegration: The Engle and Granger approach Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

More information

Individual Health Insurance April 30, 2008 Pages 167-170

Individual Health Insurance April 30, 2008 Pages 167-170 Individual Healh Insurance April 30, 2008 Pages 167-170 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 information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking 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 information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal 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 information

9. Capacitor and Resistor Circuits

9. 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 information

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1 Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

More information

Present Value Methodology

Present 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 information

A Re-examination of the Joint Mortality Functions

A Re-examination of the Joint Mortality Functions Norh merican cuarial Journal Volume 6, Number 1, p.166-170 (2002) Re-eaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali

More information

Making a Faster Cryptanalytic Time-Memory Trade-Off

Making a Faster Cryptanalytic Time-Memory Trade-Off Making a Faser Crypanalyic Time-Memory Trade-Off 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 information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase 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 information

Physics 111 Fall 2007 Electric Currents and DC Circuits

Physics 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 information

On the degrees of irreducible factors of higher order Bernoulli polynomials

On 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 information

Acceleration Lab Teacher s Guide

Acceleration 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 information

Life insurance cash flows with policyholder behaviour

Life insurance cash flows with policyholder behaviour Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK-2100 Copenhagen Ø, Denmark PFA Pension,

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential 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 information

Using RCtime to Measure Resistance

Using 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 buil-in ADC (Analog o Digial Converer)

More information

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

Measuring macroeconomic volatility Applications to export revenue data, 1970-2005 FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

More information

Making Use of Gate Charge Information in MOSFET and IGBT Data Sheets

Making 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 information

RC Circuit and Time Constant

RC Circuit and Time Constant ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

More information

A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A 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 information

Chapter 2 Kinematics in One Dimension

Chapter 2 Kinematics in One Dimension Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

More information

Capacitors and inductors

Capacitors and inductors Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear

More information

Supplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect Risk-Taking?

Supplementary Appendix for Depression Babies: Do Macroeconomic Experiences Affect Risk-Taking? Supplemenary Appendix for Depression Babies: Do Macroeconomic Experiences Affec Risk-Taking? Ulrike Malmendier UC Berkeley and NBER Sefan Nagel Sanford Universiy and NBER Sepember 2009 A. Deails on SCF

More information

Dependent Interest and Transition Rates in Life Insurance

Dependent Interest and Transition Rates in Life Insurance Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies

More information

Statistical Analysis with Little s Law. Supplementary Material: More on the Call Center Data. by Song-Hee Kim and Ward Whitt

Statistical Analysis with Little s Law. Supplementary Material: More on the Call Center Data. by Song-Hee Kim and Ward Whitt Saisical Analysis wih Lile s Law Supplemenary Maerial: More on he Call Cener Daa by Song-Hee Kim and Ward Whi Deparmen of Indusrial Engineering and Operaions Research Columbia Universiy, New York, NY 17-99

More information

Forecasting and Information Sharing in Supply Chains Under Quasi-ARMA Demand

Forecasting and Information Sharing in Supply Chains Under Quasi-ARMA Demand Forecasing and Informaion Sharing in Supply Chains Under Quasi-ARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in

More information

Stochastic Optimal Control Problem for Life Insurance

Stochastic Optimal Control Problem for Life Insurance Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian

More information

Return Calculation of U.S. Treasury Constant Maturity Indices

Return Calculation of U.S. Treasury Constant Maturity Indices Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

Morningstar Investor Return

Morningstar Investor Return Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

More information

Applied Intertemporal Optimization

Applied Intertemporal Optimization . Applied Ineremporal Opimizaion Klaus Wälde Universiy of Mainz CESifo, Universiy of Brisol, UCL Louvain la Neuve www.waelde.com These lecure noes can freely be downloaded from www.waelde.com/aio. A prin

More information

Economics Honors Exam 2008 Solutions Question 5

Economics 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

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior 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 information

A Probability Density Function for Google s stocks

A Probability Density Function for Google s stocks A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

Fourier Series & The Fourier Transform

Fourier Series & The Fourier Transform Fourier Series & The Fourier Transform Wha is he Fourier Transform? Fourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous limi: he Fourier ransform (and is inverse) The

More information

Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds

Inventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp. 1079 1097 issn 0030-364X eissn 1526-5463 06 5406 1079 informs doi 10.1287/opre.1060.0338 2006 INFORMS Invenory Planning wih Forecas Updaes:

More information

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convolution. Recommended Problems. x2[n] 1 2[n] 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

More information

AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 2010 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

More information

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

More information

LLC Resonant Converter Reference Design using the dspic DSC

LLC Resonant Converter Reference Design using the dspic DSC LLC Resonan Converer Reference Design using he dspic DSC 2010 Microchip Technology Incorporaed. All Righs Reserved. LLC Resonan Converer Webinar Slide 1 Hello, and welcome o his web seminar on Microchip

More information

Motion Along a Straight Line

Motion Along a Straight Line Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his

More information

Newton s Laws of Motion

Newton s Laws of Motion Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

More information

TSG-RAN Working Group 1 (Radio Layer 1) meeting #3 Nynashamn, Sweden 22 nd 26 th March 1999

TSG-RAN Working Group 1 (Radio Layer 1) meeting #3 Nynashamn, Sweden 22 nd 26 th March 1999 TSG-RAN Working Group 1 (Radio Layer 1) meeing #3 Nynashamn, Sweden 22 nd 26 h March 1999 RAN TSGW1#3(99)196 Agenda Iem: 9.1 Source: Tile: Documen for: Moorola Macro-diversiy for he PRACH Discussion/Decision

More information

Chapter 1.6 Financial Management

Chapter 1.6 Financial Management Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1

More information

Niche Market or Mass Market?

Niche Market or Mass Market? Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.

More information

Differential Equations and Linear Superposition

Differential Equations and Linear Superposition Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100... Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The Black-Scholes Shl Ml Moel... pricing opions an calculaing

More information

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling

Modeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se

More information

Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Journal Of Business & Economics Research September 2005 Volume 3, Number 9 Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

More information

The Torsion of Thin, Open Sections

The Torsion of Thin, Open Sections EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such

More information

Nikkei Stock Average Volatility Index Real-time Version Index Guidebook

Nikkei Stock Average Volatility Index Real-time Version Index Guidebook Nikkei Sock Average Volailiy Index Real-ime Version Index Guidebook Nikkei Inc. Wih he modificaion of he mehodology of he Nikkei Sock Average Volailiy Index as Nikkei Inc. (Nikkei) sars calculaing and

More information

Two Compartment Body Model and V d Terms by Jeff Stark

Two Compartment Body Model and V d Terms by Jeff Stark Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

More information

Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark E-mail: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal

II.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.

More information

Fourier Series and Fourier Transform

Fourier Series and Fourier Transform Fourier Series and Fourier ransform Complex exponenials Complex version of Fourier Series ime Shifing, Magniude, Phase Fourier ransform Copyrigh 2007 by M.H. Perro All righs reserved. 6.082 Spring 2007

More information

Section 7.1 Angles and Their Measure

Section 7.1 Angles and Their Measure Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

More information

µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ

µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high

More information

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS

DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper

More information

Modeling Stock Price Dynamics with Fuzzy Opinion Networks

Modeling Stock Price Dynamics with Fuzzy Opinion Networks Modeling Sock Price Dynamics wih Fuzzy Opinion Neworks Li-Xin Wang Deparmen of Auomaion Science and Technology Xian Jiaoong Universiy, Xian, P.R. China Email: lxwang@mail.xju.edu.cn Key words: Sock price

More information

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619

Working Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619 econsor www.econsor.eu Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;

More information

The option pricing framework

The option pricing framework Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.

More information

Chapter 4: Exponential and Logarithmic Functions

Chapter 4: Exponential and Logarithmic Functions Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion

More information

DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR

DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios

More information

OPERATION MANUAL. Indoor unit for air to water heat pump system and options EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1

OPERATION MANUAL. Indoor unit for air to water heat pump system and options EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1 OPERAION MANUAL Indoor uni for air o waer hea pump sysem and opions EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1 EKHBRD011ABY1 EKHBRD014ABY1 EKHBRD016ABY1 EKHBRD011ACV1 EKHBRD014ACV1 EKHBRD016ACV1 EKHBRD011ACY1

More information

Section 5.1 The Unit Circle

Section 5.1 The Unit Circle Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P

More information

Hedging with Forwards and Futures

Hedging with Forwards and Futures Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

Multiprocessor Systems-on-Chips

Multiprocessor Systems-on-Chips Par of: Muliprocessor Sysems-on-Chips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,

More information