Derivation of Mean Field Equations for Classical Systems


 Agatha Greene
 1 years ago
 Views:
Transcription
1 Derivation of Mean Field Equations for Classical Systems Master Thesis iklas Boers
2
3 Derivation of Mean Field Equations for Classical Systems Master Thesis iklas Boers Fakultät für Mathematik, Informatik und Statistik LudwigMaximiliansUniversität München München, January 2011 Supervisor: Prof. Dr. Detlef Dürr Second Supervisor: Prof. Dr. Peter Pickl
4
5 Acknowledgements This thesis would have never been possible without the great support of Detlef Dürr and Peter Pickl. Their ingenuity as well as their patience are deeply appreciated. 3
6
7 COTETS COTETS Contents 1 Introduction 7 2 The Counting Measure Construction of the Counting Measure Derivation of the Vlasov equation 27 4 Conclusion 37 5
8
9 1 ITRODUCTIO 1 Introduction Consider a ewtonian system of identical particles interacting via a spherically symmetric pair potential φ C 2 b (Rd in d dimensions, which might be either attractive or repulsive. Here, C 2 b (Rd shall denote the set of functions on R d with bounded and continuous derivatives up to order 2. The Hamiltonian of such a system has the form H (X = i=1 p 2 i 2m + i<j φ(q i q j (1 with X = (x 1,..., x R 6, x i = (q i, p i R 6, leading to the equations of motion q i = H p i = ṗ i = H q i p i m = qi φ(q i q j. (2 ote that, since φ is symmetric and thus φ(0 = 0, we can omit the restriction i j in the sum. Substituting φ = 1 φ for the potential and setting m = 1, one obtains j=1 dq i dt dp i dt = p i = d2 q i dt 2 = 1 qi φ(q i q j. (3 j=1 These are the equations of motion for classical particles interacting 1 through the potential φ. It is this dynamics from which the Vlasov equation can be derived. ote that the coupling becomes weaker with growing. In a more general setting the Hamiltonian looks like where H β (X = i=1 p 2 i 2m + φ β (q i q j, (4 i<j φ β (q = 1+β φ( β/d q (5 7
10 1 ITRODUCTIO in dimension d for spherically symmetric φ Cb 2(Rd. Hence the support of φ β shrinks with for positive β. The case described above obviously corresponds to β = 0 and m = 1. Usually the potential energy is expected to scale in the same way in as the kinetic energy. This motivates the factor of 1 in the expression for φ β. In general, the ewtonian equations of motion for interacting particles are practically impossible to solve. One way to circumvent this problem and obtain at least some kind of solution is to change the level of description. Instead of the microscopic body problem, one considers an equation for a continuous mass density which effectively still describes the same situation but from a macroscopic point of view. It is intuitively clear that this coarse grained description gets more appropriate as the number of particles increases, becoming exact in the limit. The crucial observation in the case where the potential scales with 1/, i.e. the β = 0 case, is that the force q i = 1 qi φ(q i q j, i = 1,..., (6 j=1 looks like the empirical mean of the continuous function φ(q i of the random variables q j. In the limit, one might expect this to be equal to the expectation value of φ, given by the convolution 1 f t φ(q i = qi φ(q i qf t (q, pdqdp, (7 where f t (q, p for q, p R 3 denotes the mass density at q with momentum p at time t. In replacing the empirical mean by its expectation value, we switch from the body problem to an effective external force problem: Instead of summing up all interaction terms, one expects a single particle to feel only the mean field produced by all particles together. But if there is only an external force ṗ acting, then the time evolution of f t is dictated by the continuity equation on R 6 : t f t + q f t q + p f t ṗ = 0 (8 Inserting for ṗ the expectation value (7 yields t f t + q f t q p f t f t φ = 0. (9 1 Young s inequality assures that the convolution f φ exists for f L 1 (R 6 if φ is bounded: f t φ f t 1 φ 8
11 2 THE COUTIG MEASURE This equation is known as the Vlasov equation, a partial nonlinear differential equation which has a unique global solution in the space of probability densities if φ is bounded and Lipschitz continuous (see [3]. The question is how the replacement of (6 by (7 can be rigorously justified from microscopic dynamics. To put it differently, how well does a solution of the Vlasov equation approximate an actual configuration evolving in time according to ewton s equation of motion? 2 The Counting Measure We can add to the Hamiltonian an external potential V t (q: H β (X = i=1 p 2 i 2m + i<j φ β (q i q j + V t (q i (10 i=1 Since an external force has the same effect on all particles, independently of their distribution, this should not affect the derivation of a hydrodynamic equation. Since we will in the following only consider differences between exact ewtonian and mean field dynamics, this external potential will not appear anymore. For the mass density f t, one has the continuity equation on 1particle phase space Γ 1 = R 6 containing points x = (q, p t f t (x + q f t (x dq dt + pf t (x dp dt = 0. (11 From this one obtains the approximating mean field equation by replacing the interaction part of the force with its expectation value. Hence, instead of summing up all interaction terms, one expects a single particle to feel only the mean field produced by all particles together. It is clear that, for different scaling parameters 2 β, this procedure leads to different hydrodynamic equations. In the next section we will investigate the case β = 0 as an example of the method that we shall present in the following. If one looks at the mean field approximation from the perspective of a law of large numbers, one possibility to justify the replacement in the continuity equation would be to argue that the random variables x i R 6 are 2 Recall: φ β (q = 1+β φ( β/d q 9
12 2 THE COUTIG MEASURE distributed identically and independently (i.i.d.. Formally, this means that the particle density 3 F is a product of the 1particle densities: F(X = f(x i (12 for X = (x 1,..., x R 6 and x i = (q i, p i R 6. This situation is called molecular chaos. The question of justifying the mean field approximation can now be reformulated: If, initially, the random variables x i R 6 were distributed according to F 0 (X = i=1 f 0 (x i, (13 will this product structure survive the timeevolution? Moreover, in which sense will it survive, i.e. in which sense i=1 F t (X i=1 f t (x i (14 for solutions f t of the effective equation? Remark 2.1. It is crucial that the L 1 norm is not a suitable notion of distance in this context: Think of a situation in which some m out of particles are not i.i.d.. Then the distributing function would be of the form F t = g t (x 1,..., x m i=m+1 f t (x i. We would say that this is close to a product if m was small. The L 1  distance from a product distribution would be given by = Ft (x 1,..., x f t (x i dx 1... dx gt (x 1,..., x m i=1 m i=1 f t (x i dx 1... dx m, 3 Here, we will exclusively deal with situations where the distributing measure is absolutely continuous. 10
13 2 THE COUTIG MEASURE which does not tell us anything about the value of m. The m bad particles might be distributed according to a function whose support is disjoint m or has very little intersection with the support of f t : then supp(g t supp( f m t 0, Ft (x 1,..., x i=1 f t (x i dx 1... dx = 2. independently of m. Therefore the L 1 distance does not tell us what we want to know and we have to find a for our purposes more appropriate notion of distance. The strategy for answering the above questions consists of two steps: First, look for a measure α that tells us how many particles are nice, i.e. i.i.d., in a way that α 0 if most particles are nice and α 1 if most of them are bad. This will represent the desired notion of distance from product distribution. Then, secondly, try to prove a statement of the kind: If that measure was small in the beginning, then it will remain small for all times. This second step will most likely involve an application of Gronwall s lemma, which we will now state and prove: Theorem 2.1. (Gronwall s lemma Let f and c denote realvalued functions defined on [0, and let f be differentiable on (0,. If f satisfies then d f(t c(tf(t, dt Proof. Define a function g by for t [0,. Then f(t e t 0 c(sds f(0. g(t = e t 0 c(sds It holds d g(t = c(tg(t. dt 11
14 2 THE COUTIG MEASURE on (0, and therefore d f dt g = f g fg g 2 cfg cfg g 2 = 0 f(t g(t f(0 g(0 = f(0 since g(0 = 1 and g(t > 1 for all t (0,. For finite particle numbers, all our results will contain error terms depending on. We therefore need the following Corollary 2.1. Let f denote a continuous realvalued function on [0, and let f be differentiable on (0,. If f satisfies for positive constants c 1 and c 2 then d dt f(t c 1f(t + c 2, f(t e c 1t f(0 + ( e c 1t 1 c 2 c 1. Proof. Define then g(t := f(t + c 2 c 1, d dt g(t = d dt f(t c 1f(t + c 2 = c 1 g(t and thus by Gronwall s lemma which implies g(t e c 1t g(0, f(t = g(t c 2 c 1 e c 1t f(0 + ( e c 1t 1 c 2 c 1. 12
15 2 THE COUTIG MEASURE If a given physical system admits for a mean field approximation, then it is clear that a relation of the form α t cα t + o( (15 must holds for a constant c: At a given time t, the growth of α t will be caused by two different processes: There will be interactions within the i.i.d. part of the particles and there will be interactions of not i.i.d. particles with ones that were i.i.d., thereby infecting some of the good particles. Interactions among independently distributed particles occur statistically and are controllable by the law of large numbers. Interactions of bad particles with good ones on the other hand are what really causes the growth of α t beyond statistical fluctuations. To control them we need an inequality like (15. The measure α t, if reasonably defined, shall tell us the amount of particles which are not i.i.d. at time t. Then the subsequent growth of this amount has to be bounded by a multiple of this amount, if a mean field description is to make any sense. By Gronwall s lemma, it follows from (15 that α t e ct α 0 + o(. This is intuitively clear, since the amount of particles which are not i.i.d. should grow exponentially in time. Starting with a product distribution at time t = 0 corresponds to α 0 = 0. Then the timeevolved measure will be α t o( and after carrying out the limit, it would hold α t = 0 for all times. Then we would say that the product structure of F survives the timeevolution. In other words, we would have shown the propagation of molecular chaos. Still, there remains the technical difficulty to find the right measure α such that the constant c is as small as possible. ote that it is one advantage of the method to be able to give an error estimate of the mean field approximation, depending on the particle number. Of course, due to the pair interaction, the independence will be destroyed through timeevolution. More precisely, during each small time interval t, because we consider only pair interactions, not more than two particles will fall out of molecular chaos. 4 onetheless, for appropriate scaling β one hopes for this effect to be weak compared to the mixing, such that molecular chaos survives the better the larger the number of particles is. 4 The probability that two particles interact is of order t, thus the probability of having two interacting pairs is of order ( t 2 and is therefore negligible to first order in t 13
16 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE The proximity of F t to a product density can, of course, only become exact in the limit. In fact, F t itself cannot be expected to converge, since there will always be some few particles which fell out of the product distribution. Instead, one can hope that its sparticle correlation functions Ft s will converge to an sfold product of solutions of the mean field equation f t. As will turn out, their L 1 distance will be bounded by a multiple of α, thus tending to zero as. To obtain this result, the symmetry of F t under exchange of particles will be absolutely crucial. Another way of formulating the desired result is that the following diagram commutes: For f 0 L 1 (R 6, f t L 1 (R 6 and F t L 1 (R 6 f 0 MeanF ield f t s f s t L 1 (16 f 0 = F 0 ewton F treducing F s t 2.1 Construction of the Counting Measure At a given time t, the particle density F t can be decomposed in the following way 5 : Definition 2.1. For a solution f t : R 6 R of the Vlasov equation, an particle density F t : R 6 R and k {0,..., }, we define functions g k t : R 6 R by the following conditions (C: and F t = and g k t (X = g k t (x 1,..., x = χ k t (x 1,..., x ( g k t sym := 1! σ S n χ k t,σ(x 1,..., x i=k+1 i=k+1 f t (x i f t (x σ(i, (17 5 Such a decomposition always exists: Any function F can be written in such a way because the functions χ can be chosen to be such that (17 is true. 14
17 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure χ k t,σ(x 1,..., x dx σ(1... dx σ(k = c k > 0 such that c k = 1 The sets { } M k f t := gt k : R 6 R conditions (C are fulfilled we call different sectors for distinct k s. ote that M k f t M l f t for k < l, which shows that this decomposition of F t is not unique. Remark 2.2. In the following, we will make the symmetry property of F t explicit only where it is needed and ignore it in the rest of the cases, writing F t = gt k with gt k = χ k t (x 1,..., x f t (x i i=k+1 such that χ k t (x 1,..., x dx 1... dx k = c k and c k = 1 evertheless, F t will always be understood to be symmetric. The c k s are independent of x k+1,..., x by definition and independent of time due to Liouville s theorem. The function χ k t itself depends on x k+1,..., x, which reflects the idea that those particles which are not i.i.d. depend on the positions and momenta of those particles which are, but in such a way that integrating over the noni.i.d.coordinates gives a constant. The measure 6 α t is supposed to tell us the relative amount of particles that are not distributed independently. For this purpose, we take a weighted sum of the integral over the functions g k t : α = inf F t= g k t k g k t dx (18 6 Again we remark that we mean measure in the sense of a counting device and not in the sense of measure theory. 15
18 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE with the abbreviation dx = dx 1... dx. The decomposition of the  particle density F t = gk t is not unique, one could always put F t = χ t, yielding α = 1 although it should possibly be much smaller. We want to give F t as much product structure as possible. Therefore we have to put as much of its integral mass as possible into functions gt k with low kvalues. To achieve this, we take for α the infimum of the weighted sum over all possible decompositions. We take the integral over the gt k s because we want to benefit from the volume conservation of the Hamiltonian flow on phase space and from linearity of the integral in our derivation. This will become explicit when computing the time derivative of α below. For technical reasons, we will have to allow for negative parts of the functions χ k t : χ k t = (χ k t + + (χ k t in such a way that still c k = χ k t dx > 0. This is possible since we can always put F t = χ t with χ t dx = c = 1 > 0. But it is crucial that all functions over which we integrate in (18 are positive, otherwise we could get α = 0 for a nonproduct state: e.g. then we would find ft F t = f t + F t, } {{ } } {{ } g 0 g α = 0 + (F t f t dx = 0, the positive and negative parts of (F t f t exactly compensate due to Liouville s theorem. Obviously (gt k + dx gt k dx = c k. But, since we are dealing with probability densities, we want to have just the integral mass of the gt k s, given by the c k s, in the positive parts of the gt k s. Therefore we shift some integral mass of (gt k + over to the negative part, thereby constructing new functions: Definition 2.2. Let (gt k g be defined in such a way that it fulfills (gt k g 0 and (gt k g dx 1... dx = c k In addition 16
19 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure (g k t n := g k t (g k t g (19 such that (g k t n dx 1... dx = (g k t (g k t g dx = 0. (20 These conditions are realized by 7 (χ k t g := (χ k t + c k, (21 (χ k t + dx 1... dx k where the coordinates x k+1,..., x are kept fixed. From this we get which is obviously positive. (gt k g := (gt k + c k, (22 (χ k t + dx 1... dx k We are now in a position to define the counting device which shall tell us how good F t can be approximated by a product f t : Definition 2.3. Let F t L 1 (R 6 be the particle density and let f t L 1 (R 6 be a solution of the mean field equation in question. Then we define α(f t, f t := inf F t= g k t,gk t Mk f t [ k (gt k g dx + } {{ } =:c k ] (gt k n 1 (23 Suppose that at some time t the particle density is given by F t = g k t = (g k t g + (g k t n. (24 Then, developing these g k t s a time interval t further, we get F t+ t = F t Φ t = (g k t g Φ t + (g k t n Φ t. (25 The whole construction is such that one has to find a new decomposition at each time t. We will construct a decomposition of F t+ t by timeevolving the single g k t s and decomposing in such a way that the corresponding α is not too large: 7 We could have started with this, but we wanted to keep the definition as general as possible and what we really need is that (g k t g = c k. 17
20 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE Let Φ and Φ denote the Hamiltonian flows of the microscopic and the mean field dynamics. The exact dynamics develops (gt k g M k f t into (gt k g Φ t M k+2 f t to first order in t: (g k t g Φ t = (g k t g + i=1 qi (gt k g q i t + i=1 pi (gt k g p i t (26 and with p i = 1 qi φ(q i q j j=1 we see that pi (gt k g is multiplied with qi φ(q i q j, resulting in a loss of two i.i.d. particles for i, j {k + 1,..., }. But giving (gt k g Φ t a prefactor of k+2 in the counting device would be too coarse since there still remain parts for which only k coordinates are not i.i.d.. If we look at the exact time evolution as mean field evolution plus a perturbation, i.e. (g k t g Φ t = (g k t g Φ t + ( (g k t g Φ t, where the perturbation is defined by this equation, then it is clear that all parts of (g k t g Φ t for which k+2 particles are bad are in the perturbation ( (g k t g Φ t. Liouville s theorem implies that (gt k g Φ t dx = (gt k g Φ t dx = (gt k g dx = c k (27 and hence ( (g k t g Φ t dx = 0. This means that (g k t g Φ t has as much negative as positive integral mass. But in the definition of α we demand positivity of each term. Therefore we put for 0 < θ k < 1 (gt k g Φ t = ( (gt k g Φ t + θk (gt k g Φ t } {{ } M k+2 f t + (1 θ k (g k t g Φ t } {{ } M k f t (28 with θ k such that the first term, which will get a prefactor of k+2 in α, is positive for typical fluctuations of the exact dynamics around the mean field orbit. For stronger fluctuations this term can still be negative, which will be 18
21 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure dealt with by writing ( (g k t g Φ t + θk (g k t g Φ t = + ( ( (g kt g Φ t + θk (g kt g Φ t g ( ( (g kt g Φ t + θk (g kt g Φ t n and absorbing the ( n part into the second sum in α (together with (g k t n Φ t. We could have shifted all the negative parts of the perturbation over to the second sum in α by writing ( (g k t g Φ t = ( (g k t g Φ t g + ( (g k t g Φ t n, but then the estimates for this term would become too bad. Heuristically, the introduction of θ k allows us to draw a line between typical and untypical fluctuations. Remark 2.3. At this point we can already identify a constraint of the new strategy: If (g k t g Φ t = 0 on some set, we cannot take anything from it by writing a convex combination as in (28 and the procedure fails. Eventually, this will translate into the condition that p f t c(tf t for some function c(t which is bounded on compact time intervals, as will become more explicit in due course. At time t, α is given by α(f t, f t = k inf F t= gt k,gk t Mk f t (g k t g dx + (g k t n 1 (29 Suppose the decomposition is such that 8 ( t 2 + α(f t, f t = k (g k t g dx + (g k t n 1 (30 for some (small t. At time t + t, because it includes the infimum over the decompositions at this time, α(f t+ t, f t+ t will be smaller than the α we constructed by timeevolving and suitably decomposing the right hand side of (30, which is given by 8 We do not know if the infimum is actually attained. The decomposition here is chosen to be such that its corresponding α (without an infimum in front is by ( t 2 larger than the infimum of it over all decompositions. 19
22 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE α(f t+ t, f t+ t := + + = + + k + 2 ( ( (gt k g Φ t + θk (gt k g Φ g t dx k (1 θ k (gt k g Φ t dx ( ( (gt k g Φ t + θk (gt k g Φ n t + (g k t n 1 Φ t ( k + 2 θ kc k + k (1 θ kc k (31 ( ( (gt k g Φ t + θk (gt k g Φ n t + (g k t n 1 Φ t k c k + (gt k n θ k c k ( ( (gt k g Φ t + θk (gt k g Φ n t + (g k t n 1 Φ t It follows that α(f t+ t, f t+ t α(f t, f t (32 [ t t θ k c k + t ( + ( ] (gt k g Φ t + θk (gt k g Φ n 1 t and hence we have α(t + t α(t α(t lim (33 t 0 [ t 1 2 lim θ k c k + t 0 t + ( ( (gt k g Φ t + θk (gt k g Φ n 1 t ]. 20
23 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure Remark 2.4. Observe that this derivation heavily relies on volume conservation: Throughout the computation we use line (27, implying the timeindependence of the c k s. In a way, also the introduction of θ reflects the idea that volume is preserved by the Hamiltonian flow: θ k (gt k g dx = θ k c k stands for that amount of integral mass which typically moves from a sector M k f t to M k+2 f t, see line (31 above. This is as far as we can get without specifying θ k which depends on the scaling behavior of the interaction. Later we will give the right expression for θ k for an interaction potential with scaling property φ = 1 φ and then derive the Vlasov equation. But first we will show that propagation of molecular chaos implies convergence of the marginals of the particle density towards products of solutions of the mean field equation. From that, we will also derive convergence of empirical densities towards solutions of the mean field equation. But for the sake of argument, assume ɛ k was such that for α(t c 1 α(t + c 2, (34 where c 2 stands for the error term that should go to zero as goes to infinity. Then an application of Gronwall s lemma would imply α(t e c 1t α(0 + (e c 1t 1 c 2 c 1 (35 and since α(0 = 0 if we assume all particles to be i.i.d. initially, it would follow α(t o( for all t, leading to α(t = 0 in the limit. For finite, the estimation for α will include an error term decreasing with. One advantage of the presented method is thus that the result is not just a statement for the limit, but it is actually possible to establish the speed of convergence in the particle number. If one succeeds in proving an equation of the type (15, one can show that the smarginal of the symmetric particle density F t, defined as Ft s := F t dx s+1... dx (36 converges in L 1 norm to an sfold product of solutions f t of the mean field equation: Theorem 2.2. Let F t : R 6 R + denote the timedependent probability density of particles evolving according to ewton s equations of motion with interaction potential φ β as above and let f t : R 6 R + be a solution of the mean field equation in question. If for α as defined above holds 21
24 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE then the marginals of F t converge: α(f t, f t 0, (37 as. F s t L 1 s f t (x i t R + s (38 i=1 Remark 2.5. By definition it is possible to choose for each a decomposition of F t such that α(f t, f t + 1 = k c k + (g k t n 1. To see this, observe that the integral mass of F t can be freely distributed over the g k t s (yielding the values of the c k s as long as the right hand side stays larger than α, which carries the infimum over all decompositions of F t. Since lim α = 0, sending on both sides yields lim ( k c k + (gt k n 1 = 0. This means that in the limit there exists a decomposition of F t which minimizes k c k + (gt k n 1 if the counting measure vanishes. We will need this for the following proof. Proof. Pick a decomposition of F t such that α(f t, f t + 1 = k c k + (g k t n 1. We make use of the symmetry of F t and reorder the permutations σ into permutations σ for which the first s particles are distributed according to f t and permutations σ for which this is not the case: 22
25 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure F s t := = + 1! + F t dx s+1... dx [ s 1 f t (x 1 f t (x s χ k t,σ! (x 1,..., x σ f(x σ (idx s+1... dx i=k+1,σ (i 1,...,s σ k= s+1 1! χ k t,σ (x 1,..., x σ i=k+1 χ k t,σ(x 1,..., x f t (x i dx s+1... dx ] i=k+1 The relative frequency for the permutations σ is k k s + 1 s + 1 = ( s!! f t (x i dx s+1... dx. ( k! ( k s! < 1, whereas for the permutations σ the relative frequency can be estimated from above as 9 ( s!! s 1 i=0 ( s i k! ( k! (k s + i! ( k i! for c(s = s 1 ( s i=0 i. We therefore have s 1 ( ( s i ( s k k i s i=0 ( s k c(s s i Ft s f t (x 1 f t (x s (40 ( s k + c(s χ k t,σ s (x 1,..., x f t (x i dx s+1... dx i=k χ k! t,σ(x 1,..., x f t (x i dx s+1... dx k= s+1 σ 9 ote that they sum up to one: ( s! s ( s k!! i (k s + i! i=0 i=k+1 ( k! ( k i! = 1 (39 23
26 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE With k= s+1 s + 1 k k 1! σ k 1! σ χ k t,σ(x 1,..., x χ k t,σ(x 1,..., x in the last line above it follows that i=k+1 i=k+1 f t (x i dx s+1... dx f t (x i dx s+1... dx (41 Ft s f t (x 1 f t (x s 1 by assumption and hence ( s k c(s s c k + s + 1 [( s ] c(s + s s + 1 } {{ } c(s+1 0 k c k ( α(f t, f t + 1 F s t L 1 f t (x 1 f t (x s. Remark 2.6. Under the conditions of the theorem, the L 1 convergence F s t L 1 f t (x 1 f t (x s (42 for all t and s implies w convergence of the corresponding measures: Ft s dx w s f t (x i dx i (43 i=1 Weak convergence of the empirical density µ X(t := 1 i=1 δ xi (t towards weak solutions of the Vlasov equation µ(t was shown, among others, by eunzert and Braun and Hepp (see [1],[2],[3]. The key equation in eunzert s proof is d BL (µ X(t, µ(t ect d BL (µ X(0, µ(0 (44 24
27 2 THE COUTIG MEASURE 2.1 Construction of the Counting Measure for the bounded Lipschitz distance d BL (see e.g. [3]. We will show that for absolutely continuous measures, this result follows from the L 1  convergence of the marginals towards products of solutions of the Vlasov equation: Corollary 2.2. Let µ X(t describe the positions and momenta of particles and let µ ft (dx = f t (xdx for f : R R 6 R. Then the L 1 convergence implies F 1 t f t and F 2 t f t f t (45 in mean with respect to P Ft. µ X(t d BL µ f t (46 Proof. The crucial observation is that the metric d BL is defined for functions g which are Lipschitz continuous with constant 1: D := {g : R [0, 1], g(x g(y x y }. A partition of phase space Γ 1 = R 6 into cells i of sidelength δ is very useful, because the test functions g vary at most by δ on such a cell: d BL (µ X(t, µft := sup g D = sup g D i i sup g D µ X(t (dxg(x i µ X(t f t (xg(xdx (dxg(x f t (xg(xdx i (dxg(x f t (xg(xdx i i µ X(t ow g D and hence, as already explained, there exists δ > 0 such that g(x g(y δ for all x, y i and, in addition, g(x 1 for all x: d BL (µ X(t, µft i (1 + δ 2δ + i (dx f t (xdx i i µ X(t µ X(t ( i µ ft ( i Integrating this with respect to P F and noting that 25
28 2.1 Construction of the Counting Measure 2 THE COUTIG MEASURE F(Xd BL (µ X(t, µft dx = F t (Xd BL (µ X, µ ft dx since F(X = F t (X(t by definition, we obtain by dominated convergence F t (Xd BL (µ X, µ ft dx i F t (X µ X ( i µ ft ( i dx + 2δ. (47 } {{ } =:λ Using CauchySchwarz s inequality, we get λ ( F t (XdX ( µ X ( i µ ft ( i 1/2 2 ( = F t (XdX ( 1 = Ft 1 (xdx + i 2 Ft 1 (xdx i ( + ( (µ X ( i 2 2µ X ( i µ ft ( i + ( µ ft ( i 2 1/2 ( 1 2 Ft 2 (x, ydxdy (48 i i ( 2 1/2 f t (xdx + f t (xdx i i 1 Ft 1 (xdx 1 Ft 2 (x, ydxdy i i i Ft 2 (x, y f t (xf t (y dxdy i + 2 i i Ft 1 (x f t (x dx f t (xdx i 1/2 0 and since the left hand side of (47 is independent of δ, sending δ to zero implies lim F t (Xd BL (µ X, µ ft dx = 0. (49 ote that to get line (48, we used the symmetry of F t under particle exchange. 26
29 3 DERIVATIO OF THE VLASOV EQUATIO Remark 2.7. This obviously implies for all κ > 0 since P ({X = (x 1,..., x d BL (µ X(t, µft > κ } 0 (50 F t χ { d BL (µ X(t,µf t κ >1} dx1... dx 1 κ which tends to zero for all κ. F t d BL (µ X(t, µft dx 1... dx, (51 3 Derivation of the Vlasov equation In the last section we explained how one can in principle derive an effective mean field equation from microscopic dynamics using the counting measure α. However, we did this in very general terms and stopped at the point where a specification of the parameters θ k was inevitable. θ k was introduced to compensate for the negative parts in the difference (gt k g Φ t (gt k g Φ t at least as long as the dynamics stays within typical fluctuations. At this stage it is quite clear that we expect these typical fluctuations to be of order in total, i.e. 1/ per particle, since we have the law of large numbers at our disposal and the fluctuations from the mean field orbit are the deviations from the expectation value. Hence, θ k (gt k g Φ t should be of the typical size of the difference between 10 and (g k t g Φ t = (g k t g + qi (gt k g q i + i=1 pi (gt k g p i (52 i=1 (g k t g Φ t = (g k t g + where we defined qi (gt k g q i + i=1 pi (gt k g π i, (53 i=1 10 ote that the following equations are only true to first order in t. 27
30 3 DERIVATIO OF THE VLASOV EQUATIO and π i = p i = 1 qi φ(q i q j t for i = 1,..., (54 j=1 { 1 j=1 q i φ(q i q j t, for i = 1,..., k f t (q, p qi φ(q i qdqdp t, for i = k + 1,..., (55 for a solution f t of the Vlasov equation. ote that we ignored the part of the force coming from an external potential, namely qi V t (q i, because it appears in both p i and π i and we only consider the difference p π. The splitting of the mean field momenta π i is for technical reasons. We could have let the k bad particles evolve according to the expectation value of the force, but it doesn t matter so much since they are not i.i.d. and can thus not be expected to act according to mean field dynamics. We will in the following use the abbreviation δk(q i q j := qi φ(q i q j qi φ(q i qf t (q, pdqdp. (56 For the difference between the two time evolutions we then have p (g k t g ( p π = i=k+1 [ k pi (gt k g 1 ( δk(q i q j (57 j=1 ] + ( δk(q i q j t, j=k+1 where one should keep in mind that the first k particles are not distributed according to a product, whereas the last k particles are, therefore the splitting into two sums. We need a bound for the set in particle phase space Γ on which (g k t g Φ t (g k t g Φ t +θ k (g k t g Φ t is still negative. We will now define θ k and then show that this set is in fact exponentially small in. Definition 3.1. For an interaction potential scaling as φ (q = 1 φ(q with φ such that φ is bounded, the parameter θ k shall be defined for k {0,..., } and 0 < κ < 1 as 2 θ k := pf t φ k t + pf t ( k 1/2+κ t (58 f t f t 28
Stochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationRate of convergence towards Hartree dynamics
Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationQuasistatic evolution and congested transport
Quasistatic evolution and congested transport Inwon Kim Joint with Damon Alexander, Katy Craig and Yao Yao UCLA, UW Madison Hard congestion in crowd motion The following crowd motion model is proposed
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationSome Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.
Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationt := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).
1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More information10. Proximal point method
L. Vandenberghe EE236C Spring 201314) 10. Proximal point method proximal point method augmented Lagrangian method MoreauYosida smoothing 101 Proximal point method a conceptual algorithm for minimizing
More informationON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
More informationCritical Thresholds in EulerPoisson Equations. Shlomo Engelberg Jerusalem College of Technology Machon Lev
Critical Thresholds in EulerPoisson Equations Shlomo Engelberg Jerusalem College of Technology Machon Lev 1 Publication Information This work was performed with Hailiang Liu & Eitan Tadmor. These results
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:3705:00 Copyright 2003 Dan
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationInformation Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay
Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture  17 ShannonFanoElias Coding and Introduction to Arithmetic Coding
More informationIdeal Class Group and Units
Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals
More informationChapter 21: The Discounted Utility Model
Chapter 21: The Discounted Utility Model 21.1: Introduction This is an important chapter in that it introduces, and explores the implications of, an empirically relevant utility function representing intertemporal
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
More informationNonparametric adaptive age replacement with a onecycle criterion
Nonparametric adaptive age replacement with a onecycle criterion P. CoolenSchrijner, F.P.A. Coolen Department of Mathematical Sciences University of Durham, Durham, DH1 3LE, UK email: Pauline.Schrijner@durham.ac.uk
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More information15.062 Data Mining: Algorithms and Applications Matrix Math Review
.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationCHAPTER 7 APPLICATIONS TO MARKETING. Chapter 7 p. 1/54
CHAPTER 7 APPLICATIONS TO MARKETING Chapter 7 p. 1/54 APPLICATIONS TO MARKETING State Equation: Rate of sales expressed in terms of advertising, which is a control variable Objective: Profit maximization
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationBX in ( u, v) basis in two ways. On the one hand, AN = u+
1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x
More informationChapter 7. Continuity
Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this
More informationWeierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems
Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems Wolfgang Wagner Mohrenstrasse 39 10117 Berlin Germany Tel. +49 30 20372 0 www.wiasberlin.de WIAS workshop,
More informationStationary random graphs on Z with prescribed iid degrees and finite mean connections
Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the nonnegative
More informationRotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve
QUALITATIVE THEORY OF DYAMICAL SYSTEMS 2, 61 66 (2001) ARTICLE O. 11 Rotation Rate of a Trajectory of an Algebraic Vector Field Around an Algebraic Curve Alexei Grigoriev Department of Mathematics, The
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationOPTIMAl PREMIUM CONTROl IN A NONliFE INSURANCE BUSINESS
ONDERZOEKSRAPPORT NR 8904 OPTIMAl PREMIUM CONTROl IN A NONliFE INSURANCE BUSINESS BY M. VANDEBROEK & J. DHAENE D/1989/2376/5 1 IN A OPTIMAl PREMIUM CONTROl NONliFE INSURANCE BUSINESS By Martina Vandebroek
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationOn a comparison result for Markov processes
On a comparison result for Markov processes Ludger Rüschendorf University of Freiburg Abstract A comparison theorem is stated for Markov processes in polish state spaces. We consider a general class of
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationVector and Matrix Norms
Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a nonempty
More informationThe Method of Lagrange Multipliers
The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,
More information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationA Note on Di erential Calculus in R n by James Hebda August 2010.
A Note on Di erential Calculus in n by James Hebda August 2010 I. Partial Derivatives o Functions Let : U! be a real valued unction deined in an open neighborhood U o the point a =(a 1,...,a n ) in the
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationThe BanachTarski Paradox
University of Oslo MAT2 Project The BanachTarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the BanachTarski paradox states that for any ball in R, it is possible
More information4: SINGLEPERIOD MARKET MODELS
4: SINGLEPERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: SinglePeriod Market
More informationand s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space
RAL ANALYSIS A survey of MA 641643, UAB 19992000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σalgebras. A σalgebra in X is a nonempty collection of subsets
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More information1 Lecture 3: Operators in Quantum Mechanics
1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators
More informationRandom graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to
More informationChapter 4: NonParametric Classification
Chapter 4: NonParametric Classification Introduction Density Estimation Parzen Windows KnNearest Neighbor Density Estimation KNearest Neighbor (KNN) Decision Rule Gaussian Mixture Model A weighted combination
More informationDifferentiating under an integral sign
CALIFORNIA INSTITUTE OF TECHNOLOGY Ma 2b KC Border Introduction to Probability and Statistics February 213 Differentiating under an integral sign In the derivation of Maximum Likelihood Estimators, or
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationSeveral Views of Support Vector Machines
Several Views of Support Vector Machines Ryan M. Rifkin Honda Research Institute USA, Inc. Human Intention Understanding Group 2007 Tikhonov Regularization We are considering algorithms of the form min
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More informationDate: April 12, 2001. Contents
2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........
More informationAn optimal transportation problem with import/export taxes on the boundary
An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint
More informationarxiv:1112.0829v1 [math.pr] 5 Dec 2011
How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly
More informationM2S1 Lecture Notes. G. A. Young http://www2.imperial.ac.uk/ ayoung
M2S1 Lecture Notes G. A. Young http://www2.imperial.ac.uk/ ayoung September 2011 ii Contents 1 DEFINITIONS, TERMINOLOGY, NOTATION 1 1.1 EVENTS AND THE SAMPLE SPACE......................... 1 1.1.1 OPERATIONS
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationOffline sorting buffers on Line
Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7)
Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following
More informationExercises with solutions (1)
Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal
More informationDiscussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski.
Discussion on the paper Hypotheses testing by convex optimization by A. Goldenschluger, A. Juditsky and A. Nemirovski. Fabienne Comte, Celine Duval, Valentine GenonCatalot To cite this version: Fabienne
More informationPayment streams and variable interest rates
Chapter 4 Payment streams and variable interest rates In this chapter we consider two extensions of the theory Firstly, we look at payment streams A payment stream is a payment that occurs continuously,
More informationThe Basics of Interest Theory
Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special DistributionsVI Today, I am going to introduce
More information