Definition of the spin current: The angular spin current and its physical consequences

Size: px
Start display at page:

Download "Definition of the spin current: The angular spin current and its physical consequences"

Transcription

1 Definition of the spin current: The angular spin current an its physical consequences Qing-feng Sun 1, * an X. C. Xie 2,3 1 Beijing National Lab for Conense Matter Physics an Institute of Physics, Chinese Acaemy of Sciences, Beijing , China 2 Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA 3 International Center for Quantum Structures, Chinese Acaemy of Sciences, Beijing , China Receive 14 June 2005; revise manuscript receive 22 September 2005; publishe 2 December 2005 We fin that in orer to completely escribe the spin transport, apart from spin current or linear spin current, one has to introuce the angular spin current. The two spin currents, respectively, escribe the translational an rotational motion precession of a spin. The efinitions of these spin current ensities are given an their physical properties are iscusse. Both spin current ensities appear naturally in the spin continuity equation. In particular, we preict that the angular spin current or the spin torque as calle in previous works, similar to the linear spin current, can also inuce an electric fiel E. The formula for the inuce electric fiel E by the angular spin current element is erive, playing the role of Biot-Savart law or Ampere law. When at large istance r, this inuce electric fiel E scales as 1/r 2, whereas the E fiel generate from the linear spin current goes as 1/r 3. DOI: /PhysRevB PACS number s : b, b, I. INTRODUCTION Recently, a new subiscipline of conense matter physics, spintronics, is emerging rapily an generating great interests. 1,2 The spin current, the most important physical quantity in spintronics, has been extensively stuie. Many interesting an funamental phenomena, such as the spin Hall effect 3 6 an the spin precession 7,8 in systems with spinorbit coupling, have been iscovere an are uner further stuy. As for the charge current, the efinition of the local charge current ensity j e r,t =Re r,t ev r,t an its continuity equation /t e r,t + j e r,t =0 is well known in physics. Here r,t is the electronic wave function, v =ṙ is the velocity operator, an e r,t =e is the charge ensity. This continuity equation is the consequence of charge invariance, i.e., when an electron moves from one place to another, its charge remains the same. However, in the spin transport, there are still a lot of ebates over what is the correct efinition for spin current. 9,10 The problem stems from that the spin s is no invariant quantity in the spin transport, so that the conventional efining of the spin current v s is no conservative. Recently, some stuies have begun investigation in this irection, e.g., a semiclassical escription of the spin continuity equation has been propose, 11,12 as well as stuies introucing a conserve spin current uner special circumstances. 9 In this paper, we stuy the efinition of local spin current ensity. We fin that ue to the spin is vector an it has the translational an rotational motion, one has to use two quantities, the linear spin current an the angular spin current, to completely escribe the spin transport. Here the linear spin current escribe the translational motion of a spin, an the angular spin current is for the rotational motion. The efinition of two spin current ensities are given an they appear naturally in the quantum spin continuity equation. Moreover, we preict that the angular spin current can generate an electric fiel similar as with the linear spin current, an thus contains physical consequences. The paper is organize as follows. In Sec. II, we first iscuss the flow of a classical vector. The flow of a quantum spin is investigate in Sec. III. In Sec. IV an Sec. V, we stuy the problem of an inuce electric fiel an the heat prouce by spin currents, respectively. Finally, a brief summary is given in Sec. VI. II. THE FLOW OF A CLASSICAL VECTOR Before stuying the spin current in a quantum system, we first consier the classical case. Consier a classical particle having a vector m e.g., the classical magnetic moment, etc. with its magnitue m fixe uner the particle motion. To completely escribe this vector flow see Fig. 1 c, in aition to the local vector ensity M r,t = r,t m r,t, one nees two quantities: the linear velocity v r,t an the angular velocity r,t. Here r,t is the particle ensity, an v an escribe the translational an rotational motions, re- FIG. 1. Color online a an b are the schematic iagram for the translational motion an the rotational motion of the classic vector m, respectively. c Schematic iagram for a classic vector flow /2005/72 24 / /$ The American Physical Society

2 Q.-F. SUN AND X. C. XIE spectively see Figs. 1 a an 1 b. In contrast with the flow of a scalar quantity in which one only nees one quantity, namely, the local velocity v r,t, to escribe the translational motion, it is essential to use two quantities v r,t an r,t for the vector flow. We emphasize that it is impossible to use one vector to escribe both translational an rotational motions altogether. Since m is a constant, the change of the local vector M r,t in the volume element V= x y z with the time from t to t+t can be obtaine see Fig. 1 c : M r,t V = i=x,y,z V i v i r,t tm r,t V i v i r + î,t tm r + î,t + r,t M r,t Vt. 1 The first an the secon terms on the right escribe the classical vector flowing in or out the volume element V an its rotational motion respectively, an both can cause a change in the local vector ensity. When V goes to zero, we have the vector continuity equation t M r,t = v r,t M r,t + r,t M r,t, 2 where v M is a tensor, an its element v M ij =v i M j. Note that this vector continuity equation is from the kinematics an the invariance of m, an it is inepenent of the ynamic laws. It is well known that the scalar e.g., charge e continuity equation /t e + j e =0 is from the kinematics an the invariance of the charge e. It is inepenent of the external force F as well ynamic laws. In other wors, even if the acceleration a F/m, the continuity equation still survives. It is complete same with the vector continuity Eqs. 2 or 3. It is also from the kinematics an the invariance of m. In particular, it is inepenent of the external force an the torque acting on the vector, as well as the ynamic laws. Introucing j s r,t =v r,t M r,t an j r,t = r,t M r,t, then Eq. 2 reuces to t M r,t = j s r,t + j r,t. 3 Here j s =v M is from the translational motion of the classical vector m, an j = M escribes its rotational motion. Since v an are calle as the linear velocity an the angular velocity respectively, it is natural to name j s an j as the linear an the angular current ensities. Notice although the unit of j is ifferent from that of the linear spin current. However, j is inee the current in the angular space. It is worth to mention the following two points. 1 If to consier that there are many particles in the volume element V i.e., the volume element V is very small macroscopically but very large microscopically, the vector irection m i, the velocity v i, an the angular velocity i for each particle may be ifferent, however, the vector continuity equation 3 is still vali: /t M = j s + j, with j s r,t v M =lim V 0 i v i m i / V an j r,t M =lim V 0 i i m i / V. Here j s r,t an j r,t still escribe the translational an the rotational motions of the classical vector. 2 If one as an arbitrary curl A arb to the current j s =v M, the continuity equation oes not change. Does this imply that the linear spin current ensity can be efine as j s =v M + A arb C with a constant vector C?In our opinion, this oes not because the local spin current ensity has physical meanings. This reason is completely same with the charge current ensity that cannot be reefine as j e =ev + A arb. In orer to escribe a scalar e.g., charge flow, one local current j e r,t is sufficient. Why is it require to introuce two quantities instea of one to escribe a vector flow? The reason is that the scalar quantity only has the translational motion, but the vector quantity has two kins of motion, the translational an the rotational. So one has to use two quantities, the velocity v an the angular velocity, to escribe the motion of a single vector. Corresponingly, two quantities j s =v M an j = M are necessary to escribe the vector flow. In the steay state case, the scalar e.g., charge continuity equation reuces into j e =0, so the scalar current j e is a conserve quantity. But for a vector flow, the linear vector current j s is not conserve since j s = j. Whether it is possible to have a conserve vector current through reefinition? Of course, this reefine vector current shoul have a clear physical meaning an is measurable. In our opinion, this is almost impossible in the three-imensional space. The reasons are as follows. i One cannot use a three-imensional vector to combine both v an. Therefore one can also not use a three-imensional tensor to combine both j s =v M an j = M. ii Consier an example, as shown in Fig. 3 a, a one-imensional classical vector flowing along the x axis. When x 0, the vector s irection is in the +x axis. At 0 x L, the vector rotates in accompany with its translational motion. When x L, its irection is along the +y axis. Since for x 0 an x L the vector has no rotational motion, the efinition of the vector current is unambiguous, an the nonzero element is j xx for x 0, an j xy for x L. Therefore, the vector current is obviously ifferent for x 0 an x L, an the vector current is nonconservative. In our opinion, j s =v M an j = M alreay have clear physical meanings. They also completely an sufficiently escribe a vector flow, an they can etermine any physical effects cause by the vector flow see Secs. IV an V. One may not nee to enforce a conserve current. In particular, as shown in the example of Fig. 3 a, sometimes it is impossible to introuce a conserve current. 14 III. THE FLOW OF A QUANTUM SPIN Now we stuy the electronic spin s in the quantum case. Consier an arbitrary wave function r,t. The local spin ensity s at the position r an time t is s r,t

3 DEFINITION OF THE SPIN CURRENT: THE ANGULAR = r,t s ˆ r,t, where s ˆ = /2 ˆ with ˆ being the Pauli matrices. The time-erivative of s r,t is t 2 s r,t = ˆ + ˆ t t. From the Schröinger equation, we have /t r,t = 1/i H r,t an /t r,t = 1/ i H r,t. Notice here the transposition in the symbol only acts on the spin inexes. By using the above two equations, Eq. 4 changes into 4 are vali in general. They are inepenent of the special choice of Hamiltonian 6. For example, in the case with a vector potential A, the general spin-orbit coupling ˆ p V r, an so on, 17 the results still hol. Notice that for the Hamiltonian 6, one has Re vˆ iŝ j =Re ŝ j vˆ i an Re ˆ s ˆ = Re s ˆ ˆ. If Re vˆ iŝ j Re ŝ j vˆ i an Re ˆ s ˆ Re s ˆ ˆ for certain Hamiltonians, Eqs. 12 an 13 will change to /t s r,t = ˆ H H ˆ /2i. 5 In the erivation below, we use the following Hamiltonian: j s r,t =Re 1 2 v ˆs ˆ + s ˆv ˆ T, 14 H = p 2 2m + V r + ˆ B + ẑ ˆ p. 6 Note that our results are inepenent of this specific choice of the Hamiltonian. In Eq. 6 the first an secon terms are the kinetic energy an potential energy. The thir term is the Zeeman energy ue to a magnetic fiel, an the last term is the Rashba spin-orbit coupling, 15,16 which has been extensively stuie recently. 4,7,8 Next we substitute the Hamiltonian 6 into Eq. 5, an Eq. 5 reuces to t s = 2 Re p m + ẑ ˆ ˆ +Re B + p ẑ ˆ. Introucing a tensor j s r,t an a vector j r,t : j s r,t =Re p m + ẑ ˆ s ˆ, j r,t =Re 2 B + p ẑ s ˆ, then Eq. 7 reuces to t s r,t = j s r,t + j r,t, or it can also be rewritten in the integral form s V S t V = S j s j V. + V Due to the fact that v ˆ = /t r= p /m+ / ẑ ˆ an /t ˆ = 1/i ˆ,H = 2/ B + / p ẑ ˆ, Eqs. 8 an 9 become j s r,t =Re r,t v ˆs ˆ r,t, j r,t =Re s ˆ/t =Re ˆ s ˆ, where ˆ 2/ B + / p ẑ is the angular velocity operator. We emphasize that those results, 10, 12, an 13, j r,t =Re s ˆ/t =Re 1 2 ˆ s ˆ s ˆ ˆ. 15 Equation 10 is the quantum spin continuity equation, which is the same with the classic vector continuity equation 3 although the erivation process is very ifferent. In some previous works, this equation has also been obtaine in the semiclassical case. 11,12 Here we emphasize that this spin continuity equation 10 is the consequence of invariance of the spin magnitue s, i.e., when an electron makes a motion, either translation or rotation, its spin magnitue s = /2 remains a constant. Equation 10 shoul also be inepenent with the force i.e., the potential an the torque, as well the the ynamic law. The two quantities j s r,t an j r,t in Eq. 10, which are efine in Eqs. 12 an 13 respectively, escribe the translational an rotational motion precession of a spin at the location r an the time t. They will be name the linear an the angular spin current ensities accoringly, similar as v an are calle the linear an the angular velocities. In fact, j is calle the spin torque in a recent work. 11 We consier both j an j s escribing the motion of a spin, capable of inucing an electric fiel see Sec. IV, an so on. Namely, both of them play a parallel role in contributing to a physical quantity. Therefore, it is better to name j s an j both as spin currents. Otherwise, when one calculates the contribution of spin current to a given physical quantity, one may forget to inclue the contribution by the angular spin current. The linear spin current j s r,t is ientical with the conventional spin current investigate in recent stuies. 4 From j s r,t, the total linear spin current along i irection i=x,y,z is I si i,t = Sî j s r,t. To assume I si i,t inepenent on t an i e.g., in the case of the steay state an without spin flip, one has I si= 1/L Vî j s r,t = 1/L VRe vˆ is ˆ = 1/L V 1 2 vˆ is ˆ +s ˆvˆ i = 1 2 vˆ is ˆ +s ˆvˆ i, where L is sample length in the i irection. This efinition is the same as in recent publications. 4 Next, we iscuss certain properties of j s r,t an j r,t. Notice that j r,t which escribe the rotational motion precession of the spin plays a parallel role in comparison with

4 Q.-F. SUN AND X. C. XIE FIG. 2. Color online a The linear spin current element j s,xy. b an c The angular spin current element j,x. The spin current in a quasi 1D quantum wire. e The currents of two magnetic charges that are equivalent to a angular MM current. the conventional linear spin current j s r,t for the spin transport. 1 Similar to the classical case, it is necessary to introuce the two quantities j s r,t an j r,t to completely escribe the motion of a quantum spin. 2 The linear spin current is a tensor. Its element, e.g., j s,xy, represents an electron moving along the x irection with its spin in the y irection see Fig. 2 a. The angular spin current j is a vector. In Fig. 2 b, its element j,x escribes the rotational motion of the spin in the y irection an the angular velocity in the z irection. 3 From the linear spin current ensity j s r,t, one can calculate or say how much the linear spin current I s flowing through a surface S see Fig. 2 : I s S = S S j s. However, the behavior for the angular spin current is ifferent. From the ensity j r,t, it is meaningless to etermine how much the angular spin current flowing through a surface S because j is the current in the angular space. On the other han, one can calculate the total angular spin current I V in a volume V from j : I V = V j r,t V. 4 It is easy to prove that the spin currents in the present efinitions of Eqs. 12 an 13 are invariant uner a space coorinate transformation as well the gauge transformation. 5 If the system is in a steay state, j s an j are inepenent of the time t, an /t s r,t =0. Then the spin continuity equation 10 reuces to j s = j or S S j s = V j V. This means that the total linear spin current flowing out of a close surface is equal to the total angular spin current enclose. If to further consier a quasi-one-imensional 1D system see Fig. 2, then one has I S s I s S =I V. 6 The linear spin current ensity j s =Re v ˆs ˆ gives both the spin irection an the irection of spin movement, so it completely escribes the translational motion. However, the angular spin current ensity, j =Re s ˆ /t =Re ˆ s ˆ involves the vector prouct of ˆ s ˆ, not the tensor ˆ s ˆ. Is it correct or sufficient to escribe the rotational motion? For example, the rotational motion of Fig. 2 b with the spin s in the y irection an the angular velocity in the z irection is ifferent from the one in Fig. 2 c in which s is in the z irection an FIG. 3. Color online a Schematic iagram for the spin moving along the x axis, with the spin precession rotational motion in the x-y plane while 0 x L. b A 1D wire of electric ipole moment p e. This configuration will generate an electric fiel equivalent to the fiel from the spin currents in a. is in the y irection, but their angular spin currents are completely the same. Shall we istinguish them? It turns out that the physical results prouce by the above two rotational motions Figs. 2 b an 2 c are inee the same. For instance, the inuce electric fiel by them is ientical since a spin s has only the irection but no size see etail iscussion below. Thus, the vector j is sufficient to escribe the rotational motion, an no tensor is necessary. Now we give an example of applying the above formulas, 12 an 13, to calculate the spin currents. Let us consier a quasi-1d quantum wire having the Rashba spin orbit coupling, an its Hamiltonian is H = p 2 2m + V y,z + z 2 x p x + p x x k R 2m, 16 where k R x x m/ 2. x =0 for x 0 an x L, an x 0 while 0 x L. The other Rashba term / x p z is neglecte because the z irection is quantize. 7 Let be a stationary wave function r = 2 2 x e i 0 eikx kr x x y,z, e i 0 x k R x x 17 where y,z is the boun state wave function in the confine y an z irections. r represents the spin motion as shown in Fig. 3 a, in which the spin moves along the x axis, as well the spin precession in the x-y plane in the region 0 x L. 7,8 Using Eqs. 12 an 13, the spin current ensities of the wave function r are easily obtaine. There are only two nonzero elements of j s r : j sxx r = 2 k 2m y,z 2 cos 2 x, j sxy r = 2 k 2m y,z 2 sin 2 x. The nonzero elements of j r are

5 DEFINITION OF THE SPIN CURRENT: THE ANGULAR j x r = 2 kk R x y,z 2 sin 2 x, m 20 j y r = 2 kk R m y,z 2 cos 2 x, 21 where x = x 0 k R x x. Those spin current ensities confirm with the intuitive picture of an electron motion, precession in the x-y plane in 0 x L an movement in the x irection see Fig. 3 a. In particular, in the region of x 0 an x L, x =k R x =0, an ˆ is a goo quantum number, hence, j =0. In this case, the efinition of the spin current j s is unambiguous. However, the spin currents are ifferent in x 0 an x L except for L =n n=0,±1,±2,.... This is clearly seen from Fig. 3 a. Therefore, through this example, one can conclue that it is sometimes impossible to efine a conserve spin current. 14 The example of Fig. 3 a inee exists an has been stuie before. 7,8 Above iscussion shows that the linear spin current j s =Re v ˆs ˆ an the angular spin current j =Re ˆ s ˆ have clear physical meanings, representing the translational motion an the rotational motion precession respectively. They completely escribe the flow of a quantum spin. Any physical effects of the spin currents, such as the inuce electric fiel, can be expresse by j s an j. IV. SPIN CURRENTS INDUCED ELECTRIC FIELDS Recently, theoretic stuies have suggeste that the linear spin current can inuce an electric fiel E Can the angular spin current also inuce an electric fiel? If so, this gives a way of etecting the angular spin current. Following, we stuy this question by using the metho of equivalent magnetic charge. 21 Let us consier a steay-state angular spin current element j V at the origin. Associate with the spin s, there is a magnetic moment MM m =g B = 2g B / s where B is the Bohr magneton. Thus, corresponing to j, there is also a angular MM current j m V = 2g B / j V. From above iscussions, we alreay know that j m or j comes from the rotational motion of a MM m or s see Figs. 2 b an 2 c, an j m = m or j = s. Uner the metho of equivalent magnetic charge, the MM m is equivalent to two magnetic charges: one with magnetic charge +q locate at nˆ m an the other with q at nˆ m see Fig. 2 e. nˆ m is the unit vector of m an is a tiny length. The angular MM current j m is equivalent to two magnetic charge currents: one is j +q =nˆ jq sin at the location nˆ m, the other is j q =nˆ jq sin at nˆ m see Fig. 2 e, with nˆ j being the unit vector of j m an the angle between an m. In our previous work, 20 by using the relativistic theory, we have arrive at the formulae for the inuce electric fiel by a magnetic charge current. The electric fiel inuce by j m V can be calculate by aing the contributions from the two magnetic charge currents. Let 0, an note that 2q m an m sin = j m,weobtain the electric fiel E generate by an element of the angular spin current j V: E = 0 j m V r 4 r 3 = 0g B j V r h r We also rewrite the electric fiel E s generate by an element of the linear spin current using the tensor j s : 20 E s = 0g B h j s V r r Below we emphasize three points. i In the large r case, the electric fiel E ecays as 1/r 2. Note that the fiel from a linear spin current E s goes as 1/r 3. In fact, in terms of generating an electric fiel, the angular spin current is as effective as a magnetic charge current. ii In the steay-state case, the total electric fiel E T=E +E s contains the property C E T l =0, where C is an arbitrary close contour not passing through the region of spin current. However, for each E or E s, C E l or C E s l can be nonzero. iii As mentione above, an angular spin current j may consist of ifferent an s see Figs. 2 b an 2 c. However, the resulting electric fiel only epens on j = s. This is because a spin vector contains only a irection an a magnitue, but not a spatial size i.e., the istance approaches to zero. Inthe limit 0, both magnetic charge currents j ±q reuce to m /2 at the origin. Therefore, the overall effect of the rotational motion is only relate to m, not separately on an m. Hence it is enough to escribe the spin rotational motion by using a vector s, instea of a tensor s. Due to the fact that the irection of s can change uring the particle motion, the linear spin current ensity j s is not a conserve quantity. It is always interesting to uncover a conserve physical quantity from both theoretical an experimental points of view. Let us apply acting on two sies of Eq. 10, we have t s + j s j =0, 24 where j s means that acts on the secon inex of j s, i.e., j s i = j /j j s,ij with i, j x,y,z. To introuce j s = j s j. Note it is ifferent from j j. In a steay state, j j =0, however j s = j j is usually nonzero. By using j s, the above equation reuces to t s + j s =0. 25 This means that the current j s of the spin ivergence is a conserve quantity in the steay state case. In fact, s r,t represents an equivalent magnetic charge, so j s can also be name the magnetic charge current ensity

6 Q.-F. SUN AND X. C. XIE Moreover, the total electric fiel prouce by j s an j can be rewritten as E T = E s + E = 0g B h j s j V r r 3 = 0g B h j s V r r So the total electric fiel E T only epens on the current j s of the spin ivergence. Note that E T can be measure experimentally in principle. Through the measurement of E T r, j s can be uniquely obtaine. In the following, let us calculate the inuce electric fiels at the location r= x,y,z by the spin currents in the example of Fig. 3 a. Substituting the spin currents of Eqs into Eqs. 22 an 23 an assuming the transverse sizes of the 1D wire are much smaller than y 2 +z 2, the inuce fiels E an E s can be obtaine straightforwarly. Then the total electric fiel E T=E +E s is E T = a k m = a V S z sin 2 x x x 2 + y 2 + z 2 3/2x r r r r 3x, 27 where V = k/m,0,0, S =(cos 2 x,sin 2 x,0), r = x,0,0, the constant a= 0 g B s /4, an s is the linear ensity of moving electrons uner the bias of an external voltage. The total electric fiel E T represents the one generate by a 1D wire of electric ipole moment p e =(0,0,c sin 2 x ) at the x axis see Fig. 3 b, where c is a constant. It is obvious that E T=0, i.e., C E T l =0. However, in general C E l an C E s l are separately nonzero. Finally, we estimate the magnitue of E T. We use parameters consistent with realistic experimental samples. Take the Rashba parameter = ev m corresponing to k R =1/100 nm for m=0.036m e, s =10 6 /m i.e., one moving electron per 1000 nm in length, an k=k F =10 8 /m. The electric potential ifference between the two points A an B see Fig. 3 b is about 0.01 V, where the positions of A an B are 1/2k R /2,0,0.01 an 1/2k R /2,0, This value of the potential is measurable with toay s technology. 19 Furthermore, with the above parameters the electric fiel E T at A or B is about 5 V/m which is rather large. V. HEAT PRODUCED BY SPIN CURRENTS Here we consier another physical effect, the heat prouce by the spin currents. Assume a uniform isotropic conuctor having a linear spin current j s an a charge current j e, an consiering the simple case that there exists no spin flip process i.e., s is conserve so that j =0. Then the prouce Joule heat Q in unit volume an in unit time is 2 Q = i=x,y,z 4 j si + j ei 2 + j si j ei 2 = ij 2 j s,ij + j ei i 2 j 2 s + j 2 e, where is the resistivity. The term j e 2 is the Joule heat from the charge current which is well known, an the other term j s 2 is the heat prouce by the linear spin current. So the prouce heat can inee be expresse by j s for the case of j =0. Also note the prouce heat Q epens on j s 2, whereas the inuce electric fiel epens on j s j. VI. CONCLUSION In summary, we fin that in orer to completely escribe the spin flow incluing both classic an quantum flows, apart from the conventional spin current or linear spin current, one has to introuce another quantity, the angular spin current. The angular spin current escribes the rotational motion of the spin, an it plays a parallel role in comparison with the conventional linear spin current for the spin translational motion. In particular, we point out that the angular spin current or the spin torque as being calle in other works can also inuce an electric fiel. The formula for the generate electric fiel E is erive an E scales as 1/r 2 at large r. In aition, a conserve quantity, the current j s of the spin ivergence, is iscovere, an the total electric fiel only epens on j s. ACKNOWLEDGMENTS We gratefully acknowlege financial support from the Chinese Acaemy of Sciences an NSFC uner Grant Nos , , an X.C.X. was supporte by the US-DOE uner Grant No. DE-FG02-04ER46124, NSF CCF , an NSF-MRSEC uner Grant No. DMR *Electronic aress: sunqf@aphy.iphy.ac.cn 1 S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. V. Molnar, M. L. Roukes, A. Y. Chtchelkanova, an D. M. Treger, Science 294, ; G. A. Prinz, ibi. 282, I. Zutic, J. Fabian, an S. Das Sarma, Rev. Mo. Phys. 76, S. Murakami, N. Nagaosa, an S.-C. Zhang, Science 301,

7 DEFINITION OF THE SPIN CURRENT: THE ANGULAR 2003 ; Phys. Rev. B 69, ; Z. F. Jiang, R. D. Li, S.-C. Zhang, an W. M. Liu, ibi. 72, J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, an A. H. MacDonal, Phys. Rev. Lett. 92, ; B.K.Nikolic, S. Souma, L. P. Zarbo, an J. Sinova, ibi. 95, S. Q. Shen, M. Ma, X. C. Xie, an F. C. Zhang, Phys. Rev. Lett. 92, ; S.-Q. Shen, Y.-J. Bao, M. Ma, X. C. Xie, an F. C. Zhang, Phys. Rev. B 71, ; S. Murakami, N. Nagaosa, an S.-C. Zhang, Phys. Rev. Lett. 93, ; S.-P. Kou, X.-L. Qi, an Z.-Y. Weng, Phys. Rev. B 72, G. Y. Guo, Y. Yao, an Q. Niu, Phys. Rev. Lett. 94, ; Y. Yao an Z. Fang, ibi. 95, ; X. Dai, Z. Fang, Y.-G. Yao, an F.-C. Zhang, con-mat/ unpublishe. 7 S. Datta an B. Das, Appl. Phys. Lett. 56, T. Matsuyama, C.-M. Hu, D. Grunler, G. Meier, an U. Merkt, Phys. Rev. B 65, ; F. Mireles an G. Kirczenow, ibi. 64, ; J. Wang, H. B. Sun, an D. Y. Xing, ibi. 69, P. Zhang, J. R. Shi, D. Xiao, an Q. Niu, con-mat/ unpublishe ; J. Wang, B. G. Wang, W. Ren, an H. Guo, conmat/ unpublishe. 10 F. Schätz, P. Kopietz, an M. Kollar, Eur. Phys. J. B 41, D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth, A. H. Mac- Donal, an Q. Niu, Phys. Rev. Lett. 93, S. Zhang an Z. Yang, Phys. Rev. Lett. 94, A. A. Burkov, A. S. Nunez, an A. H. MacDonal, Phys. Rev. B 70, Of course, in some special cases, one may be able to introuce a conserve current, for example see Janin Splettstoesser, M. Governale, an U. Zälicke, Phys. Rev. B 68, E. I. Rashba, Fiz. Tver. Tela Leningra 2, Sov. Phys. Soli State 2, Y. A. Bychkov an E. I. Rashba, J. Phys. C 17, Recently, Ref. 10 has investigate the efinition of the spin current in the insulating Heisenberg moel. In our opinion, our efinition is vali in the Heisenberg moel, for which the linear spin current j s is always zero since particles o not move in this insulating moel. 18 J. E. Hirsch, Phys. Rev. B 42, ; 60, ; F. Meier an D. Loss, Phys. Rev. Lett. 90, F. Schütz, M. Kollar, an P. Kopietz, Phys. Rev. Lett. 91, ; Phys. Rev. B 69, Q.-F. Sun, H. Guo, an J. Wang, Phys. Rev. B 69, For example, see D. J. Griffiths, Introuction to Electroynamics Prentice-Hall, Englewoo Cliffs, NJ,

Lagrangian and Hamiltonian Mechanics

Lagrangian and Hamiltonian Mechanics Lagrangian an Hamiltonian Mechanics D.G. Simpson, Ph.D. Department of Physical Sciences an Engineering Prince George s Community College December 5, 007 Introuction In this course we have been stuying

More information

Lecture L25-3D Rigid Body Kinematics

Lecture L25-3D Rigid Body Kinematics J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L25-3D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general three-imensional

More information

Given three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B);

Given three vectors A, B, andc. We list three products with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); 1.1.4. Prouct of three vectors. Given three vectors A, B, anc. We list three proucts with formula (A B) C = B(A C) A(B C); A (B C) =B(A C) C(A B); a 1 a 2 a 3 (A B) C = b 1 b 2 b 3 c 1 c 2 c 3 where the

More information

Scalar : Vector : Equal vectors : Negative vectors : Proper vector : Null Vector (Zero Vector): Parallel vectors : Antiparallel vectors :

Scalar : Vector : Equal vectors : Negative vectors : Proper vector : Null Vector (Zero Vector): Parallel vectors : Antiparallel vectors : ELEMENTS OF VECTOS 1 Scalar : physical quantity having only magnitue but not associate with any irection is calle a scalar eg: time, mass, istance, spee, work, energy, power, pressure, temperature, electric

More information

An intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations

An intertemporal model of the real exchange rate, stock market, and international debt dynamics: policy simulations This page may be remove to conceal the ientities of the authors An intertemporal moel of the real exchange rate, stock market, an international ebt ynamics: policy simulations Saziye Gazioglu an W. Davi

More information

Proper Definition of Spin Current in Spin-Orbit Coupled Systems

Proper Definition of Spin Current in Spin-Orbit Coupled Systems Proper Definition of Spin Current in Spin-Orbit Coupled Systems Junren Shi ddd Institute of Physics Chinese Academy of Sciences March 25, 2006, Sanya Collaborators: Ping Zhang (dd) Di Xiao, Qian Niu(UT-Austin

More information

The Classical Particle Coupled to External Electromagnetic Field Symmetries and Conserved Quantities. Resumo. Abstract

The Classical Particle Coupled to External Electromagnetic Field Symmetries and Conserved Quantities. Resumo. Abstract The Classical Particle Couple to External Electromagnetic Fiel Symmetries an Conserve Quantities G. D. Barbosa R. Thibes, Universiae Estaual o Suoeste a Bahia Departamento e Estuos Básicos e Instrumentais

More information

arxiv:1309.1857v3 [gr-qc] 7 Mar 2014

arxiv:1309.1857v3 [gr-qc] 7 Mar 2014 Generalize holographic equipartition for Friemann-Robertson-Walker universes Wen-Yuan Ai, Hua Chen, Xian-Ru Hu, an Jian-Bo Deng Institute of Theoretical Physics, LanZhou University, Lanzhou 730000, P.

More information

Answers to the Practice Problems for Test 2

Answers to the Practice Problems for Test 2 Answers to the Practice Problems for Test 2 Davi Murphy. Fin f (x) if it is known that x [f(2x)] = x2. By the chain rule, x [f(2x)] = f (2x) 2, so 2f (2x) = x 2. Hence f (2x) = x 2 /2, but the lefthan

More information

Factoring Dickson polynomials over finite fields

Factoring Dickson polynomials over finite fields Factoring Dickson polynomials over finite fiels Manjul Bhargava Department of Mathematics, Princeton University. Princeton NJ 08544 manjul@math.princeton.eu Michael Zieve Department of Mathematics, University

More information

15.2. First-Order Linear Differential Equations. First-Order Linear Differential Equations Bernoulli Equations Applications

15.2. First-Order Linear Differential Equations. First-Order Linear Differential Equations Bernoulli Equations Applications 00 CHAPTER 5 Differential Equations SECTION 5. First-Orer Linear Differential Equations First-Orer Linear Differential Equations Bernoulli Equations Applications First-Orer Linear Differential Equations

More information

Unsteady Flow Visualization by Animating Evenly-Spaced Streamlines

Unsteady Flow Visualization by Animating Evenly-Spaced Streamlines EUROGRAPHICS 2000 / M. Gross an F.R.A. Hopgoo Volume 19, (2000), Number 3 (Guest Eitors) Unsteay Flow Visualization by Animating Evenly-Space Bruno Jobar an Wilfri Lefer Université u Littoral Côte Opale,

More information

Reading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14.

Reading: Ryden chs. 3 & 4, Shu chs. 15 & 16. For the enthusiasts, Shu chs. 13 & 14. 7 Shocks Reaing: Ryen chs 3 & 4, Shu chs 5 & 6 For the enthusiasts, Shu chs 3 & 4 A goo article for further reaing: Shull & Draine, The physics of interstellar shock waves, in Interstellar processes; Proceeings

More information

The Quick Calculus Tutorial

The Quick Calculus Tutorial The Quick Calculus Tutorial This text is a quick introuction into Calculus ieas an techniques. It is esigne to help you if you take the Calculus base course Physics 211 at the same time with Calculus I,

More information

i( t) L i( t) 56mH 1.1A t = τ ln 1 = ln 1 ln 1 6.67ms

i( t) L i( t) 56mH 1.1A t = τ ln 1 = ln 1 ln 1 6.67ms Exam III PHY 49 Summer C July 16, 8 1. In the circuit shown, L = 56 mh, R = 4.6 Ω an V = 1. V. The switch S has been open for a long time then is suenly close at t =. At what value of t (in msec) will

More information

JON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT

JON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT OPTIMAL INSURANCE COVERAGE UNDER BONUS-MALUS CONTRACTS BY JON HOLTAN if P&C Insurance Lt., Oslo, Norway ABSTRACT The paper analyses the questions: Shoul or shoul not an iniviual buy insurance? An if so,

More information

5 Isotope effects on vibrational relaxation and hydrogen-bond dynamics in water

5 Isotope effects on vibrational relaxation and hydrogen-bond dynamics in water 5 Isotope effects on vibrational relaxation an hyrogen-bon ynamics in water Pump probe experiments HDO issolve in liqui H O show the spectral ynamics an the vibrational relaxation of the OD stretch vibration.

More information

10.2 Systems of Linear Equations: Matrices

10.2 Systems of Linear Equations: Matrices SECTION 0.2 Systems of Linear Equations: Matrices 7 0.2 Systems of Linear Equations: Matrices OBJECTIVES Write the Augmente Matrix of a System of Linear Equations 2 Write the System from the Augmente Matrix

More information

Math 230.01, Fall 2012: HW 1 Solutions

Math 230.01, Fall 2012: HW 1 Solutions Math 3., Fall : HW Solutions Problem (p.9 #). Suppose a wor is picke at ranom from this sentence. Fin: a) the chance the wor has at least letters; SOLUTION: All wors are equally likely to be chosen. The

More information

As customary, choice (a) is the correct answer in all the following problems.

As customary, choice (a) is the correct answer in all the following problems. PHY2049 Summer 2012 Instructor: Francisco Rojas Exam 1 As customary, choice (a) is the correct answer in all the following problems. Problem 1 A uniformly charge (thin) non-conucting ro is locate on the

More information

Sensitivity Analysis of Non-linear Performance with Probability Distortion

Sensitivity Analysis of Non-linear Performance with Probability Distortion Preprints of the 19th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August 24-29, 214 Sensitivity Analysis of Non-linear Performance with Probability Distortion

More information

Homework 8. problems: 10.40, 10.73, 11.55, 12.43

Homework 8. problems: 10.40, 10.73, 11.55, 12.43 Hoework 8 probles: 0.0, 0.7,.55,. Proble 0.0 A block of ass kg an a block of ass 6 kg are connecte by a assless strint over a pulley in the shape of a soli isk having raius R0.5 an ass M0 kg. These blocks

More information

Optimal Energy Commitments with Storage and Intermittent Supply

Optimal Energy Commitments with Storage and Intermittent Supply Submitte to Operations Research manuscript OPRE-2009-09-406 Optimal Energy Commitments with Storage an Intermittent Supply Jae Ho Kim Department of Electrical Engineering, Princeton University, Princeton,

More information

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and

Here the units used are radians and sin x = sin(x radians). Recall that sin x and cos x are defined and continuous everywhere and Lecture 9 : Derivatives of Trigonometric Functions (Please review Trigonometry uner Algebra/Precalculus Review on the class webpage.) In this section we will look at the erivatives of the trigonometric

More information

INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES

INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES 1 st Logistics International Conference Belgrae, Serbia 28-30 November 2013 INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES Goran N. Raoičić * University of Niš, Faculty of Mechanical

More information

Maxwell equations in Lorentz covariant integral form

Maxwell equations in Lorentz covariant integral form ENEÑANZA REVITA MEXIANA DE FÍIA E 52 (1 84 89 JUNIO 2006 Maxwell equations in Lorentz covariant integral form E. Ley Koo Instituto e Física, Universia Nacional Autónoma e México, Apartao Postal 20-364,

More information

Lagrange s equations of motion for oscillating central-force field

Lagrange s equations of motion for oscillating central-force field Theoretical Mathematics & Applications, vol.3, no., 013, 99-115 ISSN: 179-9687 (print), 179-9709 (online) Scienpress Lt, 013 Lagrange s equations of motion for oscillating central-force fiel A.E. Eison

More information

Detecting Possibly Fraudulent or Error-Prone Survey Data Using Benford s Law

Detecting Possibly Fraudulent or Error-Prone Survey Data Using Benford s Law Detecting Possibly Frauulent or Error-Prone Survey Data Using Benfor s Law Davi Swanson, Moon Jung Cho, John Eltinge U.S. Bureau of Labor Statistics 2 Massachusetts Ave., NE, Room 3650, Washington, DC

More information

DIFFRACTION AND INTERFERENCE

DIFFRACTION AND INTERFERENCE DIFFRACTION AND INTERFERENCE In this experiment you will emonstrate the wave nature of light by investigating how it bens aroun eges an how it interferes constructively an estructively. You will observe

More information

On Adaboost and Optimal Betting Strategies

On Adaboost and Optimal Betting Strategies On Aaboost an Optimal Betting Strategies Pasquale Malacaria 1 an Fabrizio Smerali 1 1 School of Electronic Engineering an Computer Science, Queen Mary University of Lonon, Lonon, UK Abstract We explore

More information

Heat-And-Mass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Ratkovich, Nicolas Rios; Nopens, Ingmar

Heat-And-Mass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Ratkovich, Nicolas Rios; Nopens, Ingmar Aalborg Universitet Heat-An-Mass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Ratkovich, Nicolas Rios; Nopens, Ingmar Publishe in: International Journal of Heat an Mass Transfer

More information

Mannheim curves in the three-dimensional sphere

Mannheim curves in the three-dimensional sphere Mannheim curves in the three-imensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey.

More information

Elliptic Functions sn, cn, dn, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota

Elliptic Functions sn, cn, dn, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota Elliptic Functions sn, cn, n, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota Backgroun: Jacobi iscovere that rather than stuying elliptic integrals themselves, it is simpler to think of them as inverses

More information

11 CHAPTER 11: FOOTINGS

11 CHAPTER 11: FOOTINGS CHAPTER ELEVEN FOOTINGS 1 11 CHAPTER 11: FOOTINGS 11.1 Introuction Footings are structural elements that transmit column or wall loas to the unerlying soil below the structure. Footings are esigne to transmit

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 14 10/27/2008 MOMENT GENERATING FUNCTIONS Contents 1. Moment generating functions 2. Sum of a ranom number of ranom variables 3. Transforms

More information

A Universal Sensor Control Architecture Considering Robot Dynamics

A Universal Sensor Control Architecture Considering Robot Dynamics International Conference on Multisensor Fusion an Integration for Intelligent Systems (MFI2001) Baen-Baen, Germany, August 2001 A Universal Sensor Control Architecture Consiering Robot Dynamics Frierich

More information

An Introduction to Event-triggered and Self-triggered Control

An Introduction to Event-triggered and Self-triggered Control An Introuction to Event-triggere an Self-triggere Control W.P.M.H. Heemels K.H. Johansson P. Tabuaa Abstract Recent evelopments in computer an communication technologies have le to a new type of large-scale

More information

Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm

Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm Sensor Network Localization from Local Connectivity : Performance Analysis for the MDS-MAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor,

More information

Tracking Control of a Class of Hamiltonian Mechanical Systems with Disturbances

Tracking Control of a Class of Hamiltonian Mechanical Systems with Disturbances Proceeings of Australasian Conference on Robotics an Automation, -4 Dec 4, The University of Melbourne, Melbourne, Australia Tracking Control of a Class of Hamiltonian Mechanical Systems with Disturbances

More information

SOLUTIONS TO CONCEPTS CHAPTER 17

SOLUTIONS TO CONCEPTS CHAPTER 17 1. Given that, 400 m < < 700 nm. 1 1 1 700nm 400nm SOLUTIONS TO CONCETS CHATER 17 1 1 1 3 10 c 3 10 (Where, c = spee of light = 3 10 m/s) 7 7 7 7 7 10 4 10 7 10 4 10 4.3 10 14 < c/ < 7.5 10 14 4.3 10 14

More information

Chapter 2 Kinematics of Fluid Flow

Chapter 2 Kinematics of Fluid Flow Chapter 2 Kinematics of Flui Flow The stuy of kinematics has flourishe as a subject where one may consier isplacements an motions without imposing any restrictions on them; that is, there is no nee to

More information

Optimal Control Policy of a Production and Inventory System for multi-product in Segmented Market

Optimal Control Policy of a Production and Inventory System for multi-product in Segmented Market RATIO MATHEMATICA 25 (2013), 29 46 ISSN:1592-7415 Optimal Control Policy of a Prouction an Inventory System for multi-prouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational

More information

The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens

The influence of anti-viral drug therapy on the evolution of HIV-1 pathogens DIMACS Series in Discrete Mathematics an Theoretical Computer Science Volume 7, 26 The influence of anti-viral rug therapy on the evolution of HIV- pathogens Zhilan Feng an Libin Rong Abstract. An age-structure

More information

Option Pricing for Inventory Management and Control

Option Pricing for Inventory Management and Control Option Pricing for Inventory Management an Control Bryant Angelos, McKay Heasley, an Jeffrey Humpherys Abstract We explore the use of option contracts as a means of managing an controlling inventories

More information

Ch 10. Arithmetic Average Options and Asian Opitons

Ch 10. Arithmetic Average Options and Asian Opitons Ch 10. Arithmetic Average Options an Asian Opitons I. Asian Option an the Analytic Pricing Formula II. Binomial Tree Moel to Price Average Options III. Combination of Arithmetic Average an Reset Options

More information

Optimal Control Of Production Inventory Systems With Deteriorating Items And Dynamic Costs

Optimal Control Of Production Inventory Systems With Deteriorating Items And Dynamic Costs Applie Mathematics E-Notes, 8(2008), 194-202 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.eu.tw/ amen/ Optimal Control Of Prouction Inventory Systems With Deteriorating Items

More information

Firewall Design: Consistency, Completeness, and Compactness

Firewall Design: Consistency, Completeness, and Compactness C IS COS YS TE MS Firewall Design: Consistency, Completeness, an Compactness Mohame G. Goua an Xiang-Yang Alex Liu Department of Computer Sciences The University of Texas at Austin Austin, Texas 78712-1188,

More information

Force on Moving Charges in a Magnetic Field

Force on Moving Charges in a Magnetic Field [ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after

More information

Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5

Hull, Chapter 11 + Sections 17.1 and 17.2 Additional reference: John Cox and Mark Rubinstein, Options Markets, Chapter 5 Binomial Moel Hull, Chapter 11 + ections 17.1 an 17.2 Aitional reference: John Cox an Mark Rubinstein, Options Markets, Chapter 5 1. One-Perio Binomial Moel Creating synthetic options (replicating options)

More information

Data Center Power System Reliability Beyond the 9 s: A Practical Approach

Data Center Power System Reliability Beyond the 9 s: A Practical Approach Data Center Power System Reliability Beyon the 9 s: A Practical Approach Bill Brown, P.E., Square D Critical Power Competency Center. Abstract Reliability has always been the focus of mission-critical

More information

MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION

MSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUM-LIKELIHOOD ESTIMATION MAXIMUM-LIKELIHOOD ESTIMATION The General Theory of M-L Estimation In orer to erive an M-L estimator, we are boun to make an assumption about the functional form of the istribution which generates the

More information

FAST JOINING AND REPAIRING OF SANDWICH MATERIALS WITH DETACHABLE MECHANICAL CONNECTION TECHNOLOGY

FAST JOINING AND REPAIRING OF SANDWICH MATERIALS WITH DETACHABLE MECHANICAL CONNECTION TECHNOLOGY FAST JOINING AND REPAIRING OF SANDWICH MATERIALS WITH DETACHABLE MECHANICAL CONNECTION TECHNOLOGY Jörg Felhusen an Sivakumara K. Krishnamoorthy RWTH Aachen University, Chair an Insitute for Engineering

More information

Mathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition

Mathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition Mathematical Moels of Therapeutical Actions Relate to Tumour an Immune System Competition Elena De Angelis (1 an Pierre-Emmanuel Jabin (2 (1 Dipartimento i Matematica, Politecnico i Torino Corso Duca egli

More information

The one-year non-life insurance risk

The one-year non-life insurance risk The one-year non-life insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on non-life insurance reserve risk has been evote to the ultimo risk, the risk in the

More information

20. Product rule, Quotient rule

20. Product rule, Quotient rule 20. Prouct rule, 20.1. Prouct rule Prouct rule, Prouct rule We have seen that the erivative of a sum is the sum of the erivatives: [f(x) + g(x)] = x x [f(x)] + x [(g(x)]. One might expect from this that

More information

A Comparison of Performance Measures for Online Algorithms

A Comparison of Performance Measures for Online Algorithms A Comparison of Performance Measures for Online Algorithms Joan Boyar 1, Sany Irani 2, an Kim S. Larsen 1 1 Department of Mathematics an Computer Science, University of Southern Denmark, Campusvej 55,

More information

Mathematics Review for Economists

Mathematics Review for Economists Mathematics Review for Economists by John E. Floy University of Toronto May 9, 2013 This ocument presents a review of very basic mathematics for use by stuents who plan to stuy economics in grauate school

More information

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions

A Generalization of Sauer s Lemma to Classes of Large-Margin Functions A Generalization of Sauer s Lemma to Classes of Large-Margin Functions Joel Ratsaby University College Lonon Gower Street, Lonon WC1E 6BT, Unite Kingom J.Ratsaby@cs.ucl.ac.uk, WWW home page: http://www.cs.ucl.ac.uk/staff/j.ratsaby/

More information

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes

Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes previous inex next Calculating Viscous Flow: Velocity Profiles in Rivers an Pipes Michael Fowler, UVa 9/8/1 Introuction In this lecture, we ll erive the velocity istribution for two examples of laminar

More information

Modeling and Predicting Popularity Dynamics via Reinforced Poisson Processes

Modeling and Predicting Popularity Dynamics via Reinforced Poisson Processes Proceeings of the Twenty-Eighth AAAI Conference on Artificial Intelligence Moeling an Preicting Popularity Dynamics via Reinforce Poisson Processes Huawei Shen 1, Dashun Wang 2, Chaoming Song 3, Albert-László

More information

Modelling and Resolving Software Dependencies

Modelling and Resolving Software Dependencies June 15, 2005 Abstract Many Linux istributions an other moern operating systems feature the explicit eclaration of (often complex) epenency relationships between the pieces of software

More information

Minimizing Makespan in Flow Shop Scheduling Using a Network Approach

Minimizing Makespan in Flow Shop Scheduling Using a Network Approach Minimizing Makespan in Flow Shop Scheuling Using a Network Approach Amin Sahraeian Department of Inustrial Engineering, Payame Noor University, Asaluyeh, Iran 1 Introuction Prouction systems can be ivie

More information

Game Theoretic Modeling of Cooperation among Service Providers in Mobile Cloud Computing Environments

Game Theoretic Modeling of Cooperation among Service Providers in Mobile Cloud Computing Environments 2012 IEEE Wireless Communications an Networking Conference: Services, Applications, an Business Game Theoretic Moeling of Cooperation among Service Proviers in Mobile Clou Computing Environments Dusit

More information

View Synthesis by Image Mapping and Interpolation

View Synthesis by Image Mapping and Interpolation View Synthesis by Image Mapping an Interpolation Farris J. Halim Jesse S. Jin, School of Computer Science & Engineering, University of New South Wales Syney, NSW 05, Australia Basser epartment of Computer

More information

Parameterized Algorithms for d-hitting Set: the Weighted Case Henning Fernau. Univ. Trier, FB 4 Abteilung Informatik 54286 Trier, Germany

Parameterized Algorithms for d-hitting Set: the Weighted Case Henning Fernau. Univ. Trier, FB 4 Abteilung Informatik 54286 Trier, Germany Parameterize Algorithms for -Hitting Set: the Weighte Case Henning Fernau Trierer Forschungsberichte; Trier: Technical Reports Informatik / Mathematik No. 08-6, July 2008 Univ. Trier, FB 4 Abteilung Informatik

More information

The Concept of the Effective Mass Tensor in GR. The Equation of Motion

The Concept of the Effective Mass Tensor in GR. The Equation of Motion The Concept of the Effective Mass Tensor in GR The Equation of Motion Mirosław J. Kubiak Zespół Szkół Technicznych, Gruziąz, Polan Abstract: In the papers [, ] we presente the concept of the effective

More information

Measures of distance between samples: Euclidean

Measures of distance between samples: Euclidean 4- Chapter 4 Measures of istance between samples: Eucliean We will be talking a lot about istances in this book. The concept of istance between two samples or between two variables is funamental in multivariate

More information

Exponential Functions: Differentiation and Integration. The Natural Exponential Function

Exponential Functions: Differentiation and Integration. The Natural Exponential Function 46_54.q //4 :59 PM Page 5 5 CHAPTER 5 Logarithmic, Eponential, an Other Transcenental Functions Section 5.4 f () = e f() = ln The inverse function of the natural logarithmic function is the natural eponential

More information

How To Find Out How To Calculate Volume Of A Sphere

How To Find Out How To Calculate Volume Of A Sphere Contents High-Dimensional Space. Properties of High-Dimensional Space..................... 4. The High-Dimensional Sphere......................... 5.. The Sphere an the Cube in Higher Dimensions...........

More information

1. the acceleration of the body decreases by. 2. the acceleration of the body increases by. 3. the body falls 9.8 m during each second.

1. the acceleration of the body decreases by. 2. the acceleration of the body increases by. 3. the body falls 9.8 m during each second. Answer, Key Homework 3 Davi McIntyre 45123 Mar 25, 2004 1 This print-out shoul have 21 questions. Multiple-choice questions may continue on the next column or pae fin all choices before makin your selection.

More information

Net Neutrality, Network Capacity, and Innovation at the Edges

Net Neutrality, Network Capacity, and Innovation at the Edges Net Neutrality, Network Capacity, an Innovation at the Eges Jay Pil Choi Doh-Shin Jeon Byung-Cheol Kim May 22, 2015 Abstract We stuy how net neutrality regulations affect a high-banwith content provier(cp)

More information

Stochastic Theory of the Classical Molecular Dynamics Method

Stochastic Theory of the Classical Molecular Dynamics Method ISSN 070-048, Mathematical Moels an Computer Simulations, 0, Vol. 5, No. 4, pp. 05. Pleiaes Publishing, Lt., 0. Original Russian Text G.E. Norman, V.V. Stegailov, 0, publishe in Matematicheskoe Moelirovanie,

More information

Electrostatics I. Potential due to Prescribed Charge Distribution, Dielectric Properties, Electric Energy and Force

Electrostatics I. Potential due to Prescribed Charge Distribution, Dielectric Properties, Electric Energy and Force Chapter Electrostatics I. Potential ue to Prescribe Charge Distribution, Dielectric Properties, Electric Energy an Force. Introuction In electrostatics, charges are assume to be stationary. Electric charges

More information

The mean-field computation in a supermarket model with server multiple vacations

The mean-field computation in a supermarket model with server multiple vacations DOI.7/s66-3-7-5 The mean-fiel computation in a supermaret moel with server multiple vacations Quan-Lin Li Guirong Dai John C. S. Lui Yang Wang Receive: November / Accepte: 8 October 3 SpringerScienceBusinessMeiaNewYor3

More information

Notes on tangents to parabolas

Notes on tangents to parabolas Notes on tangents to parabolas (These are notes for a talk I gave on 2007 March 30.) The point of this talk is not to publicize new results. The most recent material in it is the concept of Bézier curves,

More information

Unbalanced Power Flow Analysis in a Micro Grid

Unbalanced Power Flow Analysis in a Micro Grid International Journal of Emerging Technology an Avance Engineering Unbalance Power Flow Analysis in a Micro Gri Thai Hau Vo 1, Mingyu Liao 2, Tianhui Liu 3, Anushree 4, Jayashri Ravishankar 5, Toan Phung

More information

Search Advertising Based Promotion Strategies for Online Retailers

Search Advertising Based Promotion Strategies for Online Retailers Search Avertising Base Promotion Strategies for Online Retailers Amit Mehra The Inian School of Business yeraba, Inia Amit Mehra@isb.eu ABSTRACT Web site aresses of small on line retailers are often unknown

More information

Calibration of the broad band UV Radiometer

Calibration of the broad band UV Radiometer Calibration of the broa ban UV Raiometer Marian Morys an Daniel Berger Solar Light Co., Philaelphia, PA 19126 ABSTRACT Mounting concern about the ozone layer epletion an the potential ultraviolet exposure

More information

The most common model to support workforce management of telephone call centers is

The most common model to support workforce management of telephone call centers is Designing a Call Center with Impatient Customers O. Garnett A. Manelbaum M. Reiman Davison Faculty of Inustrial Engineering an Management, Technion, Haifa 32000, Israel Davison Faculty of Inustrial Engineering

More information

The wave equation is an important tool to study the relation between spectral theory and geometry on manifolds. Let U R n be an open set and let

The wave equation is an important tool to study the relation between spectral theory and geometry on manifolds. Let U R n be an open set and let 1. The wave equation The wave equation is an important tool to stuy the relation between spectral theory an geometry on manifols. Let U R n be an open set an let = n j=1 be the Eucliean Laplace operator.

More information

Security Vulnerabilities and Solutions for Packet Sampling

Security Vulnerabilities and Solutions for Packet Sampling Security Vulnerabilities an Solutions for Packet Sampling Sharon Golberg an Jennifer Rexfor Princeton University, Princeton, NJ, USA 08544 {golbe, jrex}@princeton.eu Abstract Packet sampling supports a

More information

An Alternative Approach of Operating a Passive RFID Device Embedded on Metallic Implants

An Alternative Approach of Operating a Passive RFID Device Embedded on Metallic Implants An Alternative Approach of Operating a Passive RFID Device Embee on Metallic Implants Xiaoyu Liu, Ravi Yalamanchili, Ajay Ogirala an Marlin Mickle RFID Center of Excellence, Department of Electrical an

More information

Inductors and Capacitors Energy Storage Devices

Inductors and Capacitors Energy Storage Devices Inuctors an Capacitors Energy Storage Devices Aims: To know: Basics of energy storage evices. Storage leas to time elays. Basic equations for inuctors an capacitors. To be able to o escribe: Energy storage

More information

A Data Placement Strategy in Scientific Cloud Workflows

A Data Placement Strategy in Scientific Cloud Workflows A Data Placement Strategy in Scientific Clou Workflows Dong Yuan, Yun Yang, Xiao Liu, Jinjun Chen Faculty of Information an Communication Technologies, Swinburne University of Technology Hawthorn, Melbourne,

More information

Stock Market Value Prediction Using Neural Networks

Stock Market Value Prediction Using Neural Networks Stock Market Value Preiction Using Neural Networks Mahi Pakaman Naeini IT & Computer Engineering Department Islamic Aza University Paran Branch e-mail: m.pakaman@ece.ut.ac.ir Hamireza Taremian Engineering

More information

CALCULATION INSTRUCTIONS

CALCULATION INSTRUCTIONS Energy Saving Guarantee Contract ppenix 8 CLCULTION INSTRUCTIONS Calculation Instructions for the Determination of the Energy Costs aseline, the nnual mounts of Savings an the Remuneration 1 asics ll prices

More information

Isothermal quantum dynamics: Investigations for the harmonic oscillator

Isothermal quantum dynamics: Investigations for the harmonic oscillator Isothermal quantum ynamics: Investigations for the harmonic oscillator Dem Fachbereich Physik er Universität Osnabrück zur Erlangung es Graes eines Doktors er Naturwissenschaften vorgelegte Dissertation

More information

Differentiability of Exponential Functions

Differentiability of Exponential Functions Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an

More information

Achieving quality audio testing for mobile phones

Achieving quality audio testing for mobile phones Test & Measurement Achieving quality auio testing for mobile phones The auio capabilities of a cellular hanset provie the funamental interface between the user an the raio transceiver. Just as RF testing

More information

Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

More information

Chapter 11: Feedback and PID Control Theory

Chapter 11: Feedback and PID Control Theory Chapter 11: Feeback an ID Control Theory Chapter 11: Feeback an ID Control Theory I. Introuction Feeback is a mechanism for regulating a physical system so that it maintains a certain state. Feeback works

More information

Oberwolfach Preprints

Oberwolfach Preprints Oberwolfach Preprints OWP 2007-01 Gerhar Huisken Geometric Flows an 3-Manifols Mathematisches Forschungsinstitut Oberwolfach ggmbh Oberwolfach Preprints (OWP) ISSN 1864-7596 Oberwolfach Preprints (OWP)

More information

ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014

ISSN: 2277-3754 ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 3, Issue 12, June 2014 ISSN: 77-754 ISO 900:008 Certifie International Journal of Engineering an Innovative echnology (IJEI) Volume, Issue, June 04 Manufacturing process with isruption uner Quaratic Deman for Deteriorating Inventory

More information

A New Evaluation Measure for Information Retrieval Systems

A New Evaluation Measure for Information Retrieval Systems A New Evaluation Measure for Information Retrieval Systems Martin Mehlitz martin.mehlitz@ai-labor.e Christian Bauckhage Deutsche Telekom Laboratories christian.bauckhage@telekom.e Jérôme Kunegis jerome.kunegis@ai-labor.e

More information

Inverse Trig Functions

Inverse Trig Functions Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that

More information

Dynamics of Iain M. Banks Orbitals. Richard Kennaway. 12 October 2005

Dynamics of Iain M. Banks Orbitals. Richard Kennaway. 12 October 2005 Dynamics of Iain M. Banks Orbitals Richard Kennaway 12 October 2005 Note This is a draft in progress, and as such may contain errors. Please do not cite this without permission. 1 The problem An Orbital

More information

Optimizing Multiple Stock Trading Rules using Genetic Algorithms

Optimizing Multiple Stock Trading Rules using Genetic Algorithms Optimizing Multiple Stock Traing Rules using Genetic Algorithms Ariano Simões, Rui Neves, Nuno Horta Instituto as Telecomunicações, Instituto Superior Técnico Av. Rovisco Pais, 040-00 Lisboa, Portugal.

More information

ThroughputScheduler: Learning to Schedule on Heterogeneous Hadoop Clusters

ThroughputScheduler: Learning to Schedule on Heterogeneous Hadoop Clusters ThroughputScheuler: Learning to Scheule on Heterogeneous Haoop Clusters Shehar Gupta, Christian Fritz, Bob Price, Roger Hoover, an Johan e Kleer Palo Alto Research Center, Palo Alto, CA, USA {sgupta, cfritz,

More information

Which Networks Are Least Susceptible to Cascading Failures?

Which Networks Are Least Susceptible to Cascading Failures? Which Networks Are Least Susceptible to Cascaing Failures? Larry Blume Davi Easley Jon Kleinberg Robert Kleinberg Éva Taros July 011 Abstract. The resilience of networks to various types of failures is

More information

Seeing the Unseen: Revealing Mobile Malware Hidden Communications via Energy Consumption and Artificial Intelligence

Seeing the Unseen: Revealing Mobile Malware Hidden Communications via Energy Consumption and Artificial Intelligence Seeing the Unseen: Revealing Mobile Malware Hien Communications via Energy Consumption an Artificial Intelligence Luca Caviglione, Mauro Gaggero, Jean-François Lalane, Wojciech Mazurczyk, Marcin Urbanski

More information