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


 Jeffery Melton
 2 years ago
 Views:
Transcription
1 Definition of the spin current: The angular spin current an its physical consequences Qingfeng 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 BiotSavart 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 threeimensional space. The reasons are as follows. i One cannot use a threeimensional vector to combine both v an. Therefore one can also not use a threeimensional tensor to combine both j s =v M an j = M. ii Consier an example, as shown in Fig. 3 a, a oneimensional 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 timeerivative 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 spinorbit 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 spinorbit 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 quasioneimensional 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 xy 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 quasi1d 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 xy 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 xy 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 steaystate 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 steaystate 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 USDOE uner Grant No. DEFG0204ER46124, NSF CCF , an NSFMRSEC uner Grant No. DMR *Electronic aress: 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, conmat/ 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, conmat/ 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 PrenticeHall, Englewoo Cliffs, NJ,
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 informationLecture L253D Rigid Body Kinematics
J. Peraire, S. Winall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L253D Rigi Boy Kinematics In this lecture, we consier the motion of a 3D rigi boy. We shall see that in the general threeimensional
More informationGiven 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 information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk  Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking threeimensional potentials in the next chapter, we shall in chapter 4 of this
More informationPurpose of the Experiments. Principles and Error Analysis. ε 0 is the dielectric constant,ε 0. ε r. = 8.854 10 12 F/m is the permittivity of
Experiments with Parallel Plate Capacitors to Evaluate the Capacitance Calculation an Gauss Law in Electricity, an to Measure the Dielectric Constants of a Few Soli an Liqui Samples Table of Contents Purpose
More informationScalar : 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 informationarxiv:1309.1857v3 [grqc] 7 Mar 2014
Generalize holographic equipartition for FriemannRobertsonWalker universes WenYuan Ai, Hua Chen, XianRu Hu, an JianBo Deng Institute of Theoretical Physics, LanZhou University, Lanzhou 730000, P.
More informationAn 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 informationProper Definition of Spin Current in SpinOrbit Coupled Systems
Proper Definition of Spin Current in SpinOrbit Coupled Systems Junren Shi ddd Institute of Physics Chinese Academy of Sciences March 25, 2006, Sanya Collaborators: Ping Zhang (dd) Di Xiao, Qian Niu(UTAustin
More informationThe 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 informationAnswers 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 informationCHAPTER 5 : CALCULUS
Dr Roger Ni (Queen Mary, University of Lonon)  5. CHAPTER 5 : CALCULUS Differentiation Introuction to Differentiation Calculus is a branch of mathematics which concerns itself with change. Irrespective
More informationPHY101 Electricity and Magnetism I Course Summary
TOPIC 1 ELECTROSTTICS PHY11 Electricity an Magnetism I Course Summary Coulomb s Law The magnitue of the force between two point charges is irectly proportional to the prouct of the charges an inversely
More informationFactoring 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 informationi( 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 information15.2. FirstOrder Linear Differential Equations. FirstOrder Linear Differential Equations Bernoulli Equations Applications
00 CHAPTER 5 Differential Equations SECTION 5. FirstOrer Linear Differential Equations FirstOrer Linear Differential Equations Bernoulli Equations Applications FirstOrer Linear Differential Equations
More informationUnsteady Flow Visualization by Animating EvenlySpaced Streamlines
EUROGRAPHICS 2000 / M. Gross an F.R.A. Hopgoo Volume 19, (2000), Number 3 (Guest Eitors) Unsteay Flow Visualization by Animating EvenlySpace Bruno Jobar an Wilfri Lefer Université u Littoral Côte Opale,
More informationReading: 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 informationThe 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 informationJON HOLTAN. if P&C Insurance Ltd., Oslo, Norway ABSTRACT
OPTIMAL INSURANCE COVERAGE UNDER BONUSMALUS 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 information5 Isotope effects on vibrational relaxation and hydrogenbond dynamics in water
5 Isotope effects on vibrational relaxation an hyrogenbon 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 information10.2 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 informationMath 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 informationAs 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) nonconucting ro is locate on the
More informationOptimal Energy Commitments with Storage and Intermittent Supply
Submitte to Operations Research manuscript OPRE200909406 Optimal Energy Commitments with Storage an Intermittent Supply Jae Ho Kim Department of Electrical Engineering, Princeton University, Princeton,
More informationHomework 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 informationSensitivity Analysis of Nonlinear Performance with Probability Distortion
Preprints of the 19th Worl Congress The International Feeration of Automatic Control Cape Town, South Africa. August 2429, 214 Sensitivity Analysis of Nonlinear Performance with Probability Distortion
More informationHere 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 informationINFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES
1 st Logistics International Conference Belgrae, Serbia 2830 November 2013 INFLUENCE OF GPS TECHNOLOGY ON COST CONTROL AND MAINTENANCE OF VEHICLES Goran N. Raoičić * University of Niš, Faculty of Mechanical
More informationMaxwell 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 20364,
More informationMannheim curves in the threedimensional sphere
Mannheim curves in the threeimensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey.
More informationDIFFRACTION 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 informationLagrange s equations of motion for oscillating centralforce field
Theoretical Mathematics & Applications, vol.3, no., 013, 99115 ISSN: 1799687 (print), 1799709 (online) Scienpress Lt, 013 Lagrange s equations of motion for oscillating centralforce fiel A.E. Eison
More informationDetecting Possibly Fraudulent or ErrorProne Survey Data Using Benford s Law
Detecting Possibly Frauulent or ErrorProne 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 informationHeatAndMass Transfer Relationship to Determine Shear Stress in Tubular Membrane Systems Ratkovich, Nicolas Rios; Nopens, Ingmar
Aalborg Universitet HeatAnMass 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 informationOn 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 informationElliptic 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 information11 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 informationMASSACHUSETTS 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(We assume that x 2 IR n with n > m f g are twice continuously ierentiable functions with Lipschitz secon erivatives. The Lagrangian function `(x y) i
An Analysis of Newton's Metho for Equivalent Karush{Kuhn{Tucker Systems Lus N. Vicente January 25, 999 Abstract In this paper we analyze the application of Newton's metho to the solution of systems of
More informationA Universal Sensor Control Architecture Considering Robot Dynamics
International Conference on Multisensor Fusion an Integration for Intelligent Systems (MFI2001) BaenBaen, Germany, August 2001 A Universal Sensor Control Architecture Consiering Robot Dynamics Frierich
More informationDEVELOPMENT OF A BRAKING MODEL FOR SPEED SUPERVISION SYSTEMS
DEVELOPMENT OF A BRAKING MODEL FOR SPEED SUPERVISION SYSTEMS Paolo Presciani*, Monica Malvezzi #, Giuseppe Luigi Bonacci +, Monica Balli + * FS Trenitalia Unità Tecnologie Materiale Rotabile Direzione
More informationSensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm
Sensor Network Localization from Local Connectivity : Performance Analysis for the MDSMAP Algorithm Sewoong Oh an Anrea Montanari Electrical Engineering an Statistics Department Stanfor University, Stanfor,
More information9.3. Diffraction and Interference of Water Waves
Diffraction an Interference of Water Waves 9.3 Have you ever notice how people relaxing at the seashore spen so much of their time watching the ocean waves moving over the water, as they break repeately
More informationTracking 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 informationSOLUTIONS 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 informationAn Introduction to Eventtriggered and Selftriggered Control
An Introuction to Eventtriggere an Selftriggere 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 largescale
More informationData 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 missioncritical
More informationMSc. Econ: MATHEMATICAL STATISTICS, 1995 MAXIMUMLIKELIHOOD ESTIMATION
MAXIMUMLIKELIHOOD ESTIMATION The General Theory of ML Estimation In orer to erive an ML estimator, we are boun to make an assumption about the functional form of the istribution which generates the
More informationChapter 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 informationForce 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 informationThe influence of antiviral drug therapy on the evolution of HIV1 pathogens
DIMACS Series in Discrete Mathematics an Theoretical Computer Science Volume 7, 26 The influence of antiviral rug therapy on the evolution of HIV pathogens Zhilan Feng an Libin Rong Abstract. An agestructure
More informationOptimal Control Policy of a Production and Inventory System for multiproduct in Segmented Market
RATIO MATHEMATICA 25 (2013), 29 46 ISSN:15927415 Optimal Control Policy of a Prouction an Inventory System for multiprouct in Segmente Market Kuleep Chauhary, Yogener Singh, P. C. Jha Department of Operational
More informationOptimal Control Of Production Inventory Systems With Deteriorating Items And Dynamic Costs
Applie Mathematics ENotes, 8(2008), 194202 c ISSN 16072510 Available free at mirror sites of http://www.math.nthu.eu.tw/ amen/ Optimal Control Of Prouction Inventory Systems With Deteriorating Items
More informationOption 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 informationFirewall Design: Consistency, Completeness, and Compactness
C IS COS YS TE MS Firewall Design: Consistency, Completeness, an Compactness Mohame G. Goua an XiangYang Alex Liu Department of Computer Sciences The University of Texas at Austin Austin, Texas 787121188,
More informationS 4 Flavor Symmetry Embedded into SU(3) and Lepton Masses and Mixing
S 4 Flavor Symmetry Embee into SU() an Lepton Masses an Mixing Yoshio Koie Institute for Higher Eucation esearch an Practice, Osaka University, 11 Machikaneyama, Toyonaka, Osaka 50004, Japan Email aress:
More informationMathematics 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 informationTheoretical Considerations on Compensation of the Accommodation Vergence Mismatch by Refractive Power of FocusAdjustable 3D Glasses
Theoretical Consierations on Compensation of the Accommoation Vergence Mismatch by Refractive Power of FocusAjustable 3D Glasses DalYoung Kim* Department of Optometry, Seoul National University of Science
More informationCh 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 informationFAST 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 information20. 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 informationThe oneyear nonlife insurance risk
The oneyear nonlife insurance risk Ohlsson, Esbjörn & Lauzeningks, Jan Abstract With few exceptions, the literature on nonlife insurance reserve risk has been evote to the ultimo risk, the risk in the
More informationHull, 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. OnePerio Binomial Moel Creating synthetic options (replicating options)
More informationComparative study regarding the methods of interpolation
Recent Avances in Geoesy an Geomatics Engineering Comparative stuy regaring the methos of interpolation PAUL DANIEL DUMITRU, MARIN PLOPEANU, DRAGOS BADEA Department of Geoesy an Photogrammetry Faculty
More informationA 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 informationCalculating 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 informationStudy on the Price Elasticity of Demand of Beijing Subway
Journal of Traffic an Logistics Engineering, Vol, 1, No. 1 June 2013 Stuy on the Price Elasticity of Deman of Beijing Subway Yanan Miao an Liang Gao MOE Key Laboratory for Urban Transportation Complex
More informationMathematical 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 PierreEmmanuel Jabin (2 (1 Dipartimento i Matematica, Politecnico i Torino Corso Duca egli
More informationA Generalization of Sauer s Lemma to Classes of LargeMargin Functions
A Generalization of Sauer s Lemma to Classes of LargeMargin 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 informationMeasures 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 informationModeling and Predicting Popularity Dynamics via Reinforced Poisson Processes
Proceeings of the TwentyEighth AAAI Conference on Artificial Intelligence Moeling an Preicting Popularity Dynamics via Reinforce Poisson Processes Huawei Shen 1, Dashun Wang 2, Chaoming Song 3, AlbertLászló
More informationStress Concentration Factors of Various Adjacent Holes Configurations in a Spherical Pressure Vessel
5 th Australasian Congress on Applie Mechanics, ACAM 2007 1012 December 2007, Brisbane, Australia Stress Concentration Factors of Various Ajacent Holes Configurations in a Spherical Pressure Vessel Kh.
More informationPROBLEMS. A.1 Implement the COINCIDENCE function in sumofproducts form, where COINCIDENCE = XOR.
724 APPENDIX A LOGIC CIRCUITS (Corrispone al cap. 2  Elementi i logica) PROBLEMS A. Implement the COINCIDENCE function in sumofproucts form, where COINCIDENCE = XOR. A.2 Prove the following ientities
More informationMinimizing 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 informationModelling 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 informationView 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 informationThe 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 informationParameterized Algorithms for dhitting 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. 086, July 2008 Univ. Trier, FB 4 Abteilung Informatik
More informationGame 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 informationLecture 17: Implicit differentiation
Lecture 7: Implicit ifferentiation Nathan Pflueger 8 October 203 Introuction Toay we iscuss a technique calle implicit ifferentiation, which provies a quicker an easier way to compute many erivatives we
More informationKater Pendulum. Introduction. It is wellknown result that the period T of a simple pendulum is given by. T = 2π
Kater Penulum ntrouction t is wellknown result that the perio of a simple penulum is given by π L g where L is the length. n principle, then, a penulum coul be use to measure g, the acceleration of gravity.
More informationFall 12 PHY 122 Homework Solutions #8
Fall 12 PHY 122 Homework Solutions #8 Chapter 27 Problem 22 An electron moves with velocity v= (7.0i  6.0j)10 4 m/s in a magnetic field B= (0.80i + 0.60j)T. Determine the magnitude and direction of the
More informationExponential 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 information1. 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 printout shoul have 21 questions. Multiplechoice questions may continue on the next column or pae fin all choices before makin your selection.
More information1 HighDimensional Space
Contents HighDimensional Space. Properties of HighDimensional Space..................... 4. The HighDimensional Sphere......................... 5.. The Sphere an the Cube in Higher Dimensions...........
More informationJitter effects on Analog to Digital and Digital to Analog Converters
Jitter effects on Analog to Digital an Digital to Analog Converters Jitter effects copyright 1999, 2000 Troisi Design Limite Jitter One of the significant problems in igital auio is clock jitter an its
More informationAn 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 informationStochastic Theory of the Classical Molecular Dynamics Method
ISSN 070048, 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 informationElectrostatics 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 informationNotes 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 informationNet Neutrality, Network Capacity, and Innovation at the Edges
Net Neutrality, Network Capacity, an Innovation at the Eges Jay Pil Choi DohShin Jeon ByungCheol Kim May 22, 2015 Abstract We stuy how net neutrality regulations affect a highbanwith content provier(cp)
More informationCalibration 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 information1 Derivatives of Piecewise Defined Functions
MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.
More informationThe 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 informationUnbalanced 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 informationIsothermal 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 informationThe 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 informationInductors 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 informationSearch 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