On Adaboost and Optimal Betting Strategies


 Ezra Kelley
 2 years ago
 Views:
Transcription
1 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 the relation between the Aaboost weight upate proceure an Kelly s theory of betting. Specifically, we show that Aaboost can be seen as an intuitive optimal betting strategy in the sense of Kelly. Technically this is achieve as the solution of the ual of the classical formulation of the Aaboost minimisation problem. Using this equivalence with betting strategies we erive a substantial simplification of Aaboost an of a relate family of boosting algorithms. Besies allowing a natural erivation of the simplification, our approach establishes a connection between Aaboost an Information Theory that we believe may cast a new light on some boosting concepts. Keywors: Ensembles methos, Statistical Methos, Regression/Classification, Aaboost, Information theory, Optimal betting strategies 1. Introuction Suppose you want to bet on a series of races of six horses, on which no information is initially available (the same horses participate in all races. The only information available after each race is in the form of the names of a subset of the losers for that race. This implies that you are not informe about the changes to your capital after each race, so your next bet can only be expresse as a percentage of your (unknown capital. Let us furthermore assume that you are require to bet all of your capital on each race. Initially, as nothing is known, it seems reasonable to sprea one s bets evenly, so that your initial bet will allocate 1/6th of your capital on each horse. Suppose now that you learn that horses 1,3, 4, 5 an 6 lost the first race. As your previous bet was optimal 1 (w.r.t. your information at the time you want the new bet to be the closest possible to the previous bet, taking into account the outcome of the race. You coul for instance ecie to bet a fraction ρ of your capital on the latest possible" winners (horse 1 an 1 ρ on the latest losers (horses 1,3,4,5,6. This ρ is a measure of how much you believe the result of the last race to be a preictor of the result of the next race. Notice that, as you have to invest all your capital at each time, ρ = 1 (i.e. betting everything on the set of horses 1 Optimal betting is here unerstoo in the information theoretical sense. In probability terms betting all capital on the horse most likely to win will give the highest expecte return, but will make you bankrupt with probability 1 in the long term. Maximizing the return has to be balance with the risk of bankruptcy; see Section 3. containing the previous winner woul be a ba choice: you woul then lose all your capital if the following race is won by any other horse. Table 1 presents an algorithm implementing these ieas: bet t + 1 on horse i is obtaine from the previous bet t by multiplying it by ρ ɛ t (if the horse was a possible winner an by 1 ρ (1 ɛ (if it was a loser; ɛ t t is the total amount bet on the latest possible winners in the previous race t. If you nee to choose a value of ρ before all races, as no information is available, you may well opt for ρ = 0.5. This is a noncommittal" choice: you believe that the subset of losers of the each race reveale to you each time will not be a very goo preictor for the result of the next race. In our example, where 1,3,4,5 an 6 lost the first race, choosing ρ = 0.5 will lea to the bets (0.1, 0.5, 0.1, 0.1, 0.1, 0.1 on the secon race. Table 2 shows the bets generate by applying the betting algorithm with ρ = 0.5 to a series of 4 races. In Section 4 we show that the Aaboost weight upate cycle is equivalent to this betting strategy with ρ = 0.5, moulo the trivial ientification of the horses with training examples an of the possible winners with misclassifie training ata: this equivalence hols uner an assumption of minimal luck, assumption which we will make more precise in section 5. As will be shown, this erives straightforwarly from a irect solution of the ual of the Aaboost minimisation problem. This result, that is the main contribution of our work, establishes the relation between Kelly s theory of optimal betting an Aaboost sketche in Figure 1. It also leas irectly to the simplifie formulation of Aaboost shown in Table 4. The rest of this paper is organise as follows: Section 2 gives a brief introuction to the Aaboost algorithm; in Section 3 we outline the theory of optimal betting unerlying the strategy isplaye in Table 1. The equivalence between this betting strategy an Aaboost is prove in Section 4. In Section 5 we procee in the opposite irection, an show how convergence theorems for Aaboost can be use to investigate the asymptotic behaviour of our betting strategy. Finally in Section 6 we generalize the corresponence between betting strategies an boosting an show that in the case of arbitrary choice of ρ we recover a known variant of Aaboost [1]. 2. Backgroun on Aaboost Boosting concerns itself with the problem of combining several preiction rules, with poor iniviual performance,
2 Table 1: The betting algorithm Betting Algorithm, ρ is the fraction of the capital allocate to the latest possible winners: Choose the initial bets b 1,i = 1 m, where m is the number of horses; For race number t: 1 Let W be the set of possible winners in the previous race. Compute ɛ t = i W b t,i; 2 Upate the bets for race t + 1: b t+1,i = ρ ɛ t b t,i if i W, b t+1,i = 1 ρ (1 ɛ b t t,i otherwise. into a highly accurate preictor. This is commonly referre to as combining a set of weak learners into a strong classifier. Aaboost, introuce by Yoav Freun an Robert Schapire [2], iffers from previous algorithms in that it oes not require previous knowlege of the performance of the weak hypotheses, but it rather aapts accoringly. This aaptation is achieve by maintaining a istribution of weights over the elements of the training set. Aaboost is structure as an iterative algorithm, that works by maintaining a istribution of weights over the training examples. At each iteration a new weak learner (classifier is traine, an the istribution of weights is upate accoring to the examples it misclassifies. The iea is to increase the importance of the training examples that are ifficult to classify. The weights also etermine the contribution of the each weak learner to the final strong classifier. In the following, we will inicate the m training examples as {(x 1, y 1,..., (x m, y m }, where the x i are vectors in a feature space X an the y i { 1, +1} are the respective class labels. Given a family of weak learners H = {h t : X { 1, +1}}, it is convenient to efine u t,i = y i h t (x i, so that u t,i is +1 if h t classifies x i correctly, an 1 otherwise. If we now let t,i be the istribution of weights at time t, with i t,i = 1, the cycle of iterations can be escribe as in Table 3. Each iteration of the Aaboost algorithm solves the following minimisation problem [3], [4]: where α t = arg min α log Z t (α, (1 Z t (α = m t,i exp( αu t,i (2 i=1 is the partition function that appears in point 3 of Table 3. In other wors, the choice of the upate weights t+1,i is such that α t will minimise Z t. 3. The gambler s problem: optimal betting strategies Before investigating the formal relation between Aaboost an betting strategies, we review in this Section the basics of the theory of optimal betting, that motivate the heuristic reasoning of Section 1. Our presentation is a summary of the ieas in Chapter 6.1 of [5], base on Kelly s theory of gambling [6]. Let us consier the problem of etermining the optimal betting strategy for a race run by m horses x 1,..., x m. For the ith horse, let p i, o i represents respectively the probability of x i winning an the os, intene as o i forone (e.g., at 3for1 the gambler will get 3 times his bet in case of victory. Assuming that the gambler always bets all his wealth, let b i be the fraction that is bet on horse x i. Then if horse X wins the race, the gambler s capital grows by the wealth relative S(X = b(xo(x. Assuming all races are inepenent, the gambler s wealth after n races is therefore S n = Π n j=1 S(X j. It is at this point convenient to introuce the oubling rate m W (b, p = E(log S(X = p i log(b i o i. (3 i=1 (in this information theoretical context, the logarithm shoul be unerstoo to be in base 2. From the weak law of large numbers an the efinition of S n, we have that 1 n log S n = 1 n n log S(X j E(log S(X (4 j=1 in probability, which allows us to express the gambler s wealth in terms of the oubling rate: S n = 2 n 1 n log Sn. = 2 ne(log S(X = 2 nw (p,b (5 where by =. we inicate equality to the first orer in the exponent. As can be shown the optimal betting scheme, i.e. the choice of the b i that maximises W (p, b, is given by Table 2: Bets b i generate using the betting algorithm (Table 1 with ρ = 1/2 by information O i over four races (w enotes a possible winner. x 1 x 2 x 3 x 4 x 5 x 6 b O 0 l w l l l l b O 1 w l l l w l b O 2 l l w w l l b O 3 l w l l l l b
3 Figure 1: Relation between Kelly s theory of gambling an the stanar an ual formulations of Aaboost. The Aaboost algorithm: Table 3: The stanar formulation of the Aaboost algorithm Initialise 1,i = 1 m. 1 Select the weak learner h t that minimises the training error ɛ t, weighte by current istribution of weights t,i : ɛ t = i u t,i= 1 t,i. Check that ɛ t < Set α t = 1 1 ɛt 2log ɛ t 3 Upate the weights accoring to t+1,i = 1 Z t(α t t,i exp( α t u t,i, where Z t (α t = m i=1 t,i exp( α t u t,i is a normalisation factor which ensures that the t+1,i will be a istribution. Output the final ( classifier: T H(x = sign t=1 α th t (x b = p, that is to say proportional betting. Note that this is inepenent of the os o i. Incientally, it is crucial to this concept of optimality that (4 is true with probability one in the limit of large n. Consier for example the case of 3 horses that win with probability 1/2, 1/4 an 1/4 respectively an that are given at the same os (say 3. In this case, it woul be tempting to bet all our capital on the 1st horse, as this woul lea to the highest possible expecte gain of 3 (1/2 = 3/2. However, it is easy to see that such a strategy woul eventually make us bankrupt with probability one, as the probability that horse 1 wins the first n races is vanishingly small for large n, so that the expecte value is not a useful criterion to guie our betting. Equation 4, instea, assures us that the capital will asymptotically grow like 2 nw. For the proportional betting strategy, in this case, W = (1/2 log(3 1/2+(1/4 log(3 1/4 + (1/4 log(3 1/6 0.08, so that the capital will grow to infinity with probability one [5]. In the special case of fair os with o i = 1 p i (fair os are efine as os such that 1/o i = 1, the oubling rate becomes W (p, b = ( bi p i log = D(p b/ log e 2 (6 p i where D(p b is the KullbackLeibler ivergence (also known as the relative entropy of the probability of victory p an the betting strategy b. Note that, since the os are fair, the gambler can at best conserve his wealth, that is max b W (p, b = 0. In this case, it follows irectly from the properties of the Kullback Leibler ivergence that the maximum occurs for b i = p i, i.e. that the optimal strategy is proportional betting.
4 In the following, for convenience, we will refer to Equation 6 as if the base of the logarithm were e instea than 2. This moification is inessential, an for the sake of convenience we will continue to refer to the new quantity as the oubling rate. 4. Equivalence of Aaboost an betting strategy We are now in a position to prove the equivalence of the betting algorithm introuce in Section 1 with the Aaboost weight upate strategy, points 2 an 3 in Table 3. Moulo the ientification of the horses with training examples an of the possible winners of each race with misclassifie examples, this leas to the simplification of Aaboost shown in Table 4. Figure 1 illustrates schematically the relationships between these algorithms. Our proof is base on the uality relations iscusse in a number of excellent papers that cast Aaboost in terms of the minimisation of KullbackLeibler or Bregman ivergences [7], [8], [3]. In these, the Aaboost weight upate equations are shown to solve the following problem: t+1 = arg min D( t subject to u t = 0. (7 In information theoretical terms, this can be interprete as the Minimum Discrimination Information principle applie to the estimation of the istribution t+1 given the initial estimate t an the linear constraint etermine by the statistics u t [9]. The key observation is now that the KullbackLeibler ivergence of the i an the t,i, D( t = i log i t,i, (8 bears a strong formal analogy to Equation 6 above. For the sake of reaability, we iscar the inex t an simply write i for t,i an i for t+1,i. The Aaboost ual problem Equation 7 then becomes = arg min D( subject to u = 0. (9 The constraint u = 0 can be rewritten as i = i = 1/2. (10 i u i=+1 With these notations, we are reay to prove the following Theorem 1: The simplifie weight upate strategy in table 4 leas to the same weights as the original algorithm in Table 3. Proof: we nee to show that the weight upate equations in table 4, that are the formal transcription of the betting strategy in Table 2, actually solve the optimisation problem Equation 9. The constraint Equation 10 allows us to split the objective function in Equation 9 into the sum of two terms: D( = = i log i + i i u i=+1 = D (, + D + (,, i log i i (11 each of which can be maximise separately. Notice that neither of these terms taken separately represents a KullbackLeibler ivergence, as the partial sums over the i an i are not over istributions. However, this is easily remeie by normalising these subsequences separately. To this purpose, we introuce the Aaboost weighte error ɛ t (see Table 3, point 1, that omitting the t inex is equal to ɛ = i. The first term of the summation can then be rewritten as D (, = = 1 2 i log 2 i 2 i /ɛ 1 log 2ɛ 2 = 1 2 D (2 /ɛ 1 log 2ɛ, 2 (12 where by D we mean the KullbackLeibler ivergence compute over the subset of inexes {i u i = 1}. The minimum point is now immeiately evient: 2 i = i/ɛ or, in the notation of Table 4, t+1,i = 1 2ɛ t t,i if u t,i = 1. (13 Similarly, the minimization of D + leas to 2 i = i/(1 ɛ, or t+1,i = 1 2(1 ɛ t t,i if u t,i = +1. (14 These are precisely the upate equations given in Table 4, which proves the theorem. We have thus shown that the simplifie algorithm in Table 4 is completely equivalent to the original Aaboost algorithm in Table 3. Inee, the two algorithms generate step by step the same sequence of weights, as well as the same ecision function. A similar simplification of Aaboost, although not so wiely use, is alreay known to researchers in the fiel (see for instance [10]. However, to the best of our knowlege it has always been erive in a purely computational way by algebraic manipulation of the equations in Table 3. We believe that our erivation as a irect solution of the ual problem Equation 7 is both more straightforwar an more illuminating, in that it brings out the irect corresponence with betting strategies represente in Figure 1. Finally, we note that a formally analogous weight upate strategy has been employe in online variants of the algorithm [11], [12].
5 Table 4: Applying the proportional betting strategy to the solution of the Aaboost ual problem leas to a simpler formulation of the algorithm, equivalent to the original Aaboost (simplifie: Initialise 1,i = 1 m. 1 Select the weak learner h t that minimises the training error ɛ t, weighte by current istribution of weights t,i : ɛ t = i u t,i= 1 t,i. Check that ɛ t < Upate the weights: t+1,i = 1 2ɛ t t,i if u t,i = 1, t+1,i = 1 2(1 ɛ t t,i otherwise. Output the final ( classifier: T H(x = sign t=1 α th t (x, with α t = 1 2 log 1 ɛt ɛ t 5. Aaboost as a betting strategy The consierations above suggest an interpretation of the betting algorithm in Table 1 as an iterative ensity estimation proceure in which an initial estimate, represente by the current bets b, is refine using the information on the previous race. The upate estimate is then use to place new bets (in the notations of Table 1, b t+1,i = p i, in accorance to the proportional betting strategy. This interpretation is supporte by convergence results for the Aaboost weight upate cycle, in either its stanar form (Table 3 or its simplifie form (Table 4. The application to our betting scenario with partial information is straightforwar, moulo the the simple formal ientifications b i = t,i an p i = t+1,i. To prove this, we will first state the following convergence theorem, which has been prove in [7], [8], [3]: Theorem 2: If the system of the linear constraints has nontrivial solutions, i.e. there is some such that u t = 0 t, then the Aaboost weight istribution converges to the istribution satisfying all the linear constraints for which D ( 1/m is minimal (here by 1 we inicate the vector of all ones. In other wors, the weight upate algorithm converges to a probability ensity that is, among all those that satisfy all the constraints, the closest to the original guess of a uniform istribution. This approximation property is best expresse by the following Pythagorean theorem for KullbackLeibler ivergences, that was erive in [13] in the context of an application to iterative ensity estimation (a generalisation to Bregman ivergences can be foun in [8]. For all solutions of the system of linear constraints, we have D ( 1/m = D ( + D ( 1/m. (15 In terms of our gambling problem, theorem 2 above means that the betting algorithm in Table 1 converges to an estimate p = b of the probability of winning of each single horse. Among all the ensities that satisfy the constraints impose by the information on the previous races, p is the one that woul maximise the oubling rate of the initial uniform bet, W (p, 1/m (strictly speaking, theorem 2 proves this result only for the case that ρ = 1/2; however, generalisation to the case of a generic ρ is straightforwar, see [7]. However, for convergence to occur the information u t on the various races must be provie in the orer etermine by Point 1 in either Table 3 or Table 4, corresponing to the Aaboost requirement that the weak learner shoul minimise the error with respect to the weights. In the betting case, of course, this appears problematic as the gambler has no influence on the orer in which the results occur. The corresponing assumption corresponing to the minimal error requirement woul be, in a sense, one of minimal luck : that is, once we ecie on a betting strategy, the next race is the one for which the set of the possible winners has, accoring to our estimate, the lowest total probability. When the conitions for convergence are met, the Pythagorean equality Equation 15 quantifies the avantage of the estimate probability p over any other ensity p compatible with the constraint, if the initial uniform istribution (or inee any other ensity obtaine in an intermeiate iteration are use for betting: W (p, 1/m = W (p, 1/m D (p p. (16 6. Extension of the simplifie approach to Aaboostρ In the following, we generalize the simplifie approach outline in Table 4. In terms of the betting strategy escribe in the introuction we are now consiering a gambler using a value of ρ ifferent from 1/2. For simplicity we will consier 0 < ρ 1/2; it is an easy observation that the case 1 > ρ 1/2 is obtaine by symmetry. This generalise constraint prouces an algorithm known as Aaboostρ, that has been introuce in [1]. This variant of Aaboost has an intuitive geometric an statistical interpretation in terms of hanling a egree of correlation between the new weights an the output of the weak learner. It also is an ieal testbe to show how our simplifie approach can lea to
6 straightforwar implementations of generalise versions of Aaboost. The question of whether an extension of our approach to the entire class of algorithms escribe in [7] is feasible is left for future work. Similarly to what has been one for Aaboost, we irectly solve the generalise ual optimisation problem t+1 = arg min D( t subject to u t = 1 2ρ. (17 where 0 < ρ 1/2. The solution to this problem will provie the simplifie weight upate rules (that this is inee the ual of the Aaboostρ problem will be clarifie in Section 6.1. Proceeing as in Section 4, we simplify the notation by omitting the inex t in t,i. Equation (17 implies the constraints i = ρ, i = 1 ρ, (18 i u i=+1 The objective function Equation 17 can again be ecompose as one in Equation 11: D(, ρ = D (, ρ + D + (, ρ, (19 where D an D + account for the misclassifie an the correctly classifie training examples respectively. Writing out the right han sie explicitly yiels D (, ρ = ρ i ρ log i /ρ i /ɛ ρ log ɛ ρ = ( = ρd ρ ρ log ɛ ɛ ρ an similarly ( D + (, ρ = (1 ρd + 1 ρ (1 ρ log 1 ɛ 1 ɛ 1 ρ where again D an D + stan for the Kullback Leibler ivergences calculate over the subsets of inexes {i u i = 1}. It follows straightforwarly that the minimum is achieve for i = ρ ɛ i if u i = 1, an i = 1 ρ 1 ɛ i if u i = +1. These effectively are the weight upate rules for the generalise algorithm, as shown in Table Stanar solution of the Aaboostρ problem For comparison, we erive here the stanar version of the Aaboostρ algorithm in the notations use in this paper (apart for trivial changes of notation, this is the algorithm originally publishe in [1]. We first nee to obtain the objective function for the minimisation problem associate to Aaboostρ, of which Equation 17 is the ual. In other wors, we nee to etermine a suitable function Z t ρ (α which generalises the function in Equation 2, that is to say, min D( t = ρ log ɛ u t=0 ρ + (1 ρ log 1 ɛ 1 ρ (20 = min α log Z t ρ (α. (21 Following the construction outline in [7] an essentially setting M ij = u j,i + 2ρ 1 in the Lagrange transform, we fin m Z t ρ (α = t,i exp ( α(u t,i + 2ρ 1, (22 i=1 that is minimise by α t ρ = 1 ( ρ 2 log 1 ρ 1 ɛ t. (23 ɛ t By substituting these values in the equation below it is easy to verify irectly that t+1,i = exp ( α t ρ (u t,i + 2ρ 1 Z t ρ ( αt ρ t,i (24 yiels the same weight upate rules as given in Table 5, an is inee the irect generalisation of the upate rules in the stanar formulation of Aaboost given in Table 3. The complete stanar Aaboostρ algorithm is shown in Table 6. Again, we note how the ual solution we propose in Table 5 presents a consierable simplification of the stanar formulation of the algorithm. 7. Conclusions We iscusse the relation between Aaboost an Kelly s theory of gambling. Specifically, we showe how the ual formulation of Aaboost formally correspons to maximising the oubling rate for a gambler s capital, in a partial information situation. The applicable moification of the optimal proportional betting strategy therefore maps to a irect, intuitive solution of the ual of the Aaboost problem. This leas naturally to a substantial formal simplification of the Aaboost weight upate cycle. Previous erivation of similar simplifications are, to the best of our knowlege, purely computational, an therefore, besies being more cumbersome, lack any meaningful interpretation. The implications of the corresponence between Aaboost an gambling theory works in both irections; as we have shown, it is for instance possible to use convergence results obtaine for Aaboost for investigating the asymptotic behaviour of the betting strategy we introuce in Table 1. We believe that a further investigation of the relation between Aaboost an Kelly s theory, or inee information theory in general, may lea to a eeper insight into some boosting concepts, an possibly a crossfertilization between these two omains.
7 Aaboostρ simplifie: Initialise 1,i = 1 m. 1 Select the weak learner h t that minimises the training error ɛ t, weighte by current istribution of weights t,i : ɛ t = i u t,i= 1 t,i. Check that ɛ t < ρ. 2 Upate the weights: t+1,i = ρ ɛ t t,i if u t,i = 1, t+1,i = 1 ρ 1 ɛ t t,i otherwise. Output the final classifier: H ρ (x = sign ( T t=1 α t ρh t (x, with α t ρ = 1 2 log ( ρ 1 ρ 1 ɛ t ɛ t Table 5: Aaboostρ generalises the Aaboost weight upate constraint to the case u = 1 2ρ, with 0 < ρ 1/2 The Aaboostρ algorithm: Initialise 1,i = 1 m. 1 Select the weak learner h t that minimises the training error ɛ t, weighte by current istribution of weights t,i : ɛ t = i u t,i= 1 t,i. Check that ɛ t < ρ. ( 2 Set α t ρ = 1 2 log ρ 1 ɛ t 1 ρ ɛ t. 3 Upate the weights accoring to t+1,i = exp( α t ρ(u t,i+2ρ 1 Z t ρ(α t ρ t,i, where Z t ρ (α t ρ = m i=1 t,i exp ( α t ρ (u t,i + 2ρ 1, is a normalisation factor which ensures that the t+1,i will be a istribution. Output the final ( classifier: T H ρ (x = sign t=1 α t ρh t (x Table 6: The stanar formulation of the Aaboostρ algorithm References [1] G. Rätsch, T. Onoa, an K.R. Muller, Soft margins for aaboost, Machine Learning, vol. 42, no. 3, pp , [Online]. Available: citeseer.ist.psu.eu/ html [2] Y. Freun an R. Schapire, A ecisiontheoretic generalization of online learning an an application to boosting, Journal of Computer an System Science, vol. 55, no. 1, [3] M. W. J. Kivinen, Boosting as entropy projection, in Proceeings of COLT 99. ACM, [4] R. Schapire an Y. Singer, Improve boosting algorithms using confiencerate preictions, Machine Learning, vol. 37, no. 3, [5] T. M. Cover an J. A. Thomas, Elements of Information Theory. Wiley Interscience, [6] J. J. L. Kelly, A new interpretation of information rate, Bell System Technical Journal, vol. 35, pp , July [7] M. Collins, R. E. Schapire, an Y. Singer, Logistic regression, aaboost an bregman istances, in Proceeings of the Thirteenth Annual Conference on Computational Learning Theory, [8] S. D. Pietra, V. D. Pietra, an H. Lafferty, Inucing features of ranom fiel, IEEE Transactions Pattern Analysis an Machine Intelligence, vol. 19, no. 4, April [9] S. Kullback, Information Theory an Statistics. Wiley, [10] C. Ruin, R. E. Schapire, an I. Daubechies, On the ynamics of boosting, in Avances in Neural Information Processing Systems, vol. 16, [11] N. C. Oza an S. Russell, Online bagging an boosting, in Proceeings of the IEEE International Conference on Systems, Man an Cybernetics, vol. 3, 2005, pp [12] H. Grabner an H. Bischof, Online boosting an vision, in Proceeings of CVPR, vol. 1, 2006, pp [13] S. Kullback, Probability ensities with given marginals, The Annals of Mathematical Statistics, vol. 39, no. 4, 1968.
On Adaboost and Optimal Betting Strategies
On Adaboost and Optimal Betting Strategies Pasquale Malacaria School of Electronic Engineering and Computer Science Queen Mary, University of London Email: pm@dcs.qmul.ac.uk Fabrizio Smeraldi School of
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 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 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 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 informationRisk Adjustment for Poker Players
Risk Ajustment for Poker Players William Chin DePaul University, Chicago, Illinois Marc Ingenoso Conger Asset Management LLC, Chicago, Illinois September, 2006 Introuction In this article we consier risk
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 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 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 informationLecture 8: Expanders and Applications
Lecture 8: Expaners an Applications Topics in Complexity Theory an Pseuoranomness (Spring 013) Rutgers University Swastik Kopparty Scribes: Amey Bhangale, Mrinal Kumar 1 Overview In this lecture, we will
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 informationInverse Trig Functions
Inverse Trig Functions c A Math Support Center Capsule February, 009 Introuction Just as trig functions arise in many applications, so o the inverse trig functions. What may be most surprising is that
More informationLagrangian 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 information19.2. First Order Differential Equations. Introduction. Prerequisites. Learning Outcomes
First Orer Differential Equations 19.2 Introuction Separation of variables is a technique commonly use to solve first orer orinary ifferential equations. It is socalle because we rearrange the equation
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 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 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 informationnparameter families of curves
1 nparameter families of curves For purposes of this iscussion, a curve will mean any equation involving x, y, an no other variables. Some examples of curves are x 2 + (y 3) 2 = 9 circle with raius 3,
More informationChapter 2 Review of Classical Action Principles
Chapter Review of Classical Action Principles This section grew out of lectures given by Schwinger at UCLA aroun 1974, which were substantially transforme into Chap. 8 of Classical Electroynamics (Schwinger
More informationReview Horse Race Gambling and Side Information Dependent horse races and the entropy rate. Gambling. Besma Smida. ES250: Lecture 9.
Gambling Besma Smida ES250: Lecture 9 Fall 200809 B. Smida (ES250) Gambling Fall 200809 1 / 23 Today s outline Review of Huffman Code and Arithmetic Coding Horse Race Gambling and Side Information Dependent
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 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 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 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 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 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 information2r 1. Definition (Degree Measure). Let G be a rgraph of order n and average degree d. Let S V (G). The degree measure µ(s) of S is defined by,
Theorem Simple Containers Theorem) Let G be a simple, rgraph of average egree an of orer n Let 0 < δ < If is large enough, then there exists a collection of sets C PV G)) satisfying: i) for every inepenent
More informationM147 Practice Problems for Exam 2
M47 Practice Problems for Exam Exam will cover sections 4., 4.4, 4.5, 4.6, 4.7, 4.8, 5., an 5.. Calculators will not be allowe on the exam. The first ten problems on the exam will be multiple choice. Work
More informationOpen World Face Recognition with Credibility and Confidence Measures
Open Worl Face Recognition with Creibility an Confience Measures Fayin Li an Harry Wechsler Department of Computer Science George Mason University Fairfax, VA 22030 {fli, wechsler}@cs.gmu.eu Abstract.
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 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 informationLecture 13: Differentiation Derivatives of Trigonometric Functions
Lecture 13: Differentiation Derivatives of Trigonometric Functions Derivatives of the Basic Trigonometric Functions Derivative of sin Derivative of cos Using the Chain Rule Derivative of tan Using the
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 informationWeek 4  Linear Demand and Supply Curves
Week 4  Linear Deman an Supply Curves November 26, 2007 1 Suppose that we have a linear eman curve efine by the expression X D = A bp X an a linear supply curve given by X S = C + P X, where b > 0 an
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 informationWeb Appendices of Selling to Overcon dent Consumers
Web Appenices of Selling to Overcon ent Consumers Michael D. Grubb A Option Pricing Intuition This appenix provies aitional intuition base on option pricing for the result in Proposition 2. Consier the
More informationA BlameBased Approach to Generating Proposals for Handling Inconsistency in Software Requirements
International Journal of nowlege an Systems Science, 3(), 7, JanuaryMarch 0 A lamease Approach to Generating Proposals for Hanling Inconsistency in Software Requirements eian Mu, Peking University,
More informationA New Evaluation Measure for Information Retrieval Systems
A New Evaluation Measure for Information Retrieval Systems Martin Mehlitz martin.mehlitz@ailabor.e Christian Bauckhage Deutsche Telekom Laboratories christian.bauckhage@telekom.e Jérôme Kunegis jerome.kunegis@ailabor.e
More informationDifferentiability of Exponential Functions
Differentiability of Exponential Functions Philip M. Anselone an John W. Lee Philip Anselone (panselone@actionnet.net) receive his Ph.D. from Oregon State in 1957. After a few years at Johns Hopkins an
More informationFOURIER TRANSFORM TERENCE TAO
FOURIER TRANSFORM TERENCE TAO Very broaly speaking, the Fourier transform is a systematic way to ecompose generic functions into a superposition of symmetric functions. These symmetric functions are usually
More informationMODELLING OF TWO STRATEGIES IN INVENTORY CONTROL SYSTEM WITH RANDOM LEAD TIME AND DEMAND
art I. robobabilystic Moels Computer Moelling an New echnologies 27 Vol. No. 23 ransport an elecommunication Institute omonosova iga V9 atvia MOEING OF WO AEGIE IN INVENOY CONO YEM WIH ANOM EA IME AN
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 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 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 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 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 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 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 informationThe Inefficiency of Marginal cost pricing on roads
The Inefficiency of Marginal cost pricing on roas Sofia GrahnVoornevel Sweish National Roa an Transport Research Institute VTI CTS Working Paper 4:6 stract The economic principle of roa pricing is that
More informationState of Louisiana Office of Information Technology. Change Management Plan
State of Louisiana Office of Information Technology Change Management Plan Table of Contents Change Management Overview Change Management Plan Key Consierations Organizational Transition Stages Change
More information20122013 Enhanced Instructional Transition Guide Mathematics Algebra I Unit 08
01013 Enhance Instructional Transition Guie Unit 08: Exponents an Polynomial Operations (18 ays) Possible Lesson 01 (4 ays) Possible Lesson 0 (7 ays) Possible Lesson 03 (7 ays) POSSIBLE LESSON 0 (7 ays)
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 informationCURRENCY OPTION PRICING II
Jones Grauate School Rice University Masa Watanabe INTERNATIONAL FINANCE MGMT 657 Calibrating the Binomial Tree to Volatility BlackScholes Moel for Currency Options Properties of the BS Moel Option Sensitivity
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 informationChapter 10 Market Power: Monopoly and Monosony Monopoly: market with only one seller. Demand function, TR, AR, and MR
ECON191 (Spring 2011) 27 & 29.4.2011 (Tutorial 9) Chapter 10 Market Power: Monopoly an Monosony Monopoly: market with only one seller a Deman function, TR, AR, an P p=a bq ( q) AR( q) for all q except
More informationarcsine (inverse sine) function
Inverse Trigonometric Functions c 00 Donal Kreier an Dwight Lahr We will introuce inverse functions for the sine, cosine, an tangent. In efining them, we will point out the issues that must be consiere
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 informationThe Kelly criterion for spread bets
IMA Journal of Applied Mathematics 2007 72,43 51 doi:10.1093/imamat/hxl027 Advance Access publication on December 5, 2006 The Kelly criterion for spread bets S. J. CHAPMAN Oxford Centre for Industrial
More informationImproved division by invariant integers
1 Improve ivision by invariant integers Niels Möller an Torbjörn Granlun Abstract This paper consiers the problem of iviing a twowor integer by a singlewor integer, together with a few extensions an
More informationCrossOver Analysis Using TTests
Chapter 35 CrossOver Analysis Using ests Introuction his proceure analyzes ata from a twotreatment, twoperio (x) crossover esign. he response is assume to be a continuous ranom variable that follows
More informationNational Sun YatSen University CSE Course: Information Theory. Gambling And Entropy
Gambling And Entropy 1 Outline There is a strong relationship between the growth rate of investment in a horse race and the entropy of the horse race. The value of side information is related to the mutual
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 informationIntegral Regular Truncated Pyramids with Rectangular Bases
Integral Regular Truncate Pyramis with Rectangular Bases Konstantine Zelator Department of Mathematics 301 Thackeray Hall University of Pittsburgh Pittsburgh, PA 1560, U.S.A. Also: Konstantine Zelator
More informationy or f (x) to determine their nature.
Level C5 of challenge: D C5 Fining stationar points of cubic functions functions Mathematical goals Starting points Materials require Time neee To enable learners to: fin the stationar points of a cubic
More informationPythagorean Triples Over Gaussian Integers
International Journal of Algebra, Vol. 6, 01, no., 5564 Pythagorean Triples Over Gaussian Integers Cheranoot Somboonkulavui 1 Department of Mathematics, Faculty of Science Chulalongkorn University Bangkok
More informationLowComplexity and Distributed Energy Minimization in Multihop Wireless Networks
LowComplexity an Distribute Energy inimization in ultihop Wireless Networks Longbi Lin, Xiaojun Lin, an Ness B. Shroff Center for Wireless Systems an Applications (CWSA) School of Electrical an Computer
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 informationImplementing IP Traceback in the Internet An ISP Perspective
Implementing IP Traceback in the Internet An ISP Perspective Dong Wei, Stuent Member, IEEE, an Nirwan Ansari, Senior Member, IEEE AbstractDenialofService (DoS) attacks consume the resources of remote
More informationWeb Appendices to Selling to Overcon dent Consumers
Web Appenices to Selling to Overcon ent Consumers Michael D. Grubb MIT Sloan School of Management Cambrige, MA 02142 mgrubbmit.eu www.mit.eu/~mgrubb May 2, 2008 B Option Pricing Intuition This appenix
More informationImage compression predicated on recurrent iterated function systems **
1 Image compression preicate on recurrent iterate function systems ** W. Metzler a, *, C.H. Yun b, M. Barski a a Faculty of Mathematics University of Kassel, Kassel, F. R. Germany b Faculty of Mathematics
More informationSprings, Shocks and your Suspension
rings, Shocks an your Suspension y Doc Hathaway, H&S Prototype an Design, C. Unerstaning how your springs an shocks move as your race car moves through its range of motions is one of the basics you must
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 informationPart I. Gambling and Information Theory. Information Theory and Networks. Section 1. Horse Racing. Lecture 16: Gambling and Information Theory
and Networks Lecture 16: Gambling and Paul Tune http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/InformationTheory/ Part I Gambling and School of Mathematical
More informationExample Optimization Problems selected from Section 4.7
Example Optimization Problems selecte from Section 4.7 19) We are aske to fin the points ( X, Y ) on the ellipse 4x 2 + y 2 = 4 that are farthest away from the point ( 1, 0 ) ; as it happens, this point
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 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 informationIntroduction to Integration Part 1: AntiDifferentiation
Mathematics Learning Centre Introuction to Integration Part : AntiDifferentiation Mary Barnes c 999 University of Syney Contents For Reference. Table of erivatives......2 New notation.... 2 Introuction
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 informationOptimizing Multiple Stock Trading Rules using Genetic Algorithms
Optimizing Multiple Stock Traing Rules using Genetic Algorithms Ariano Simões, Rui Neves, Nuno Horta Instituto as Telecomunicações, Instituto Superior Técnico Av. Rovisco Pais, 04000 Lisboa, Portugal.
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 informationGambling and Data Compression
Gambling and Data Compression Gambling. Horse Race Definition The wealth relative S(X) = b(x)o(x) is the factor by which the gambler s wealth grows if horse X wins the race, where b(x) is the fraction
More informationApplications of Global Positioning System in Traffic Studies. Yi Jiang 1
Applications of Global Positioning System in Traffic Stuies Yi Jiang 1 Introuction A Global Positioning System (GPS) evice was use in this stuy to measure traffic characteristics at highway intersections
More informationMeanValue Theorem (Several Variables)
MeanValue Theorem (Several Variables) 1 MeanValue Theorem (Several Variables) THEOREM THE MEANVALUE THEOREM (SEVERAL VARIABLES) If f is ifferentiable at each point of the line segment ab, then there
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 informationMinimumEnergy Broadcast in AllWireless Networks: NPCompleteness and Distribution Issues
MinimumEnergy Broacast in AllWireless Networks: NPCompleteness an Distribution Issues Mario Čagal LCAEPFL CH05 Lausanne Switzerlan mario.cagal@epfl.ch JeanPierre Hubaux LCAEPFL CH05 Lausanne Switzerlan
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 informationNEARFIELD TO FARFIELD TRANSFORMATION WITH PLANAR SPIRAL SCANNING
Progress In Electromagnetics Research, PIER 73, 49 59, 27 NEARFIELD TO FARFIELD TRANSFORMATION WITH PLANAR SPIRAL SCANNING S. Costanzo an G. Di Massa Dipartimento i Elettronica Informatica e Sistemistica
More information6.3 Microbial growth in a chemostat
6.3 Microbial growth in a chemostat The chemostat is a wielyuse apparatus use in the stuy of microbial physiology an ecology. In such a chemostat also known as continuousflow culture), microbes such
More informationPractical Lab 2 The Diffraction Grating
Practical Lab 2 The Diffraction Grating OBJECTIVES: 1) Observe the interference pattern prouce when laser light passes through multipleslit grating (a iffraction grating). 2) Graphically verify the wavelength
More informationIf you have ever spoken with your grandparents about what their lives were like
CHAPTER 7 Economic Growth I: Capital Accumulation an Population Growth The question of growth is nothing new but a new isguise for an ageol issue, one which has always intrigue an preoccupie economics:
More informationUniversal Gravity Based on the Electric Universe Model
Universal Gravity Base on the Electric Universe Moel By Frerik Nygaar frerik_nygaar@hotmail.com Nov 015, Porto Portugal. Introuction While most people are aware of both Newton's Universal Law of Gravity
More information_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 189 PART III. Growth Theory: The Economy in the Very Long Run
189220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 189 PART III Growth Theory: The Economy in the Very Long Run 189220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page 190 189220_Mankiw7e_CH07.qxp 3/2/09 9:40 PM Page
More information2 HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS Chapter Hyperbolic Functions Objectives After stuying this chapter you shoul unerstan what is meant by a hyperbolic function; be able to fin erivatives an integrals of hyperbolic functions;
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 information5. DEPOSITION PROCESSES. 5.1 Dry deposition
1 5. DEPOSITION PROCESSES 5.1 Dry eposition Dry eposition escribes the uptake of atmospheric species (gases or aerosol particles) at the surface of the Earth. It provies a surface bounary conition for
More informationA New Interpretation of Information Rate
A New Interpretation of Information Rate reproduced with permission of AT&T By J. L. Kelly, jr. (Manuscript received March 2, 956) If the input symbols to a communication channel represent the outcomes
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 information1 HighDimensional Space
Contents HighDimensional Space. Properties of HighDimensional Space..................... 4. The HighDimensional Sphere......................... 5.. The Sphere an the Cube in Higher Dimensions...........
More informationIntroduction to Support Vector Machines. Colin Campbell, Bristol University
Introduction to Support Vector Machines Colin Campbell, Bristol University 1 Outline of talk. Part 1. An Introduction to SVMs 1.1. SVMs for binary classification. 1.2. Soft margins and multiclass classification.
More informationLearningBased Summarisation of XML Documents
LearningBase Summarisation of XML Documents Massih R. Amini Anastasios Tombros Nicolas Usunier Mounia Lalmas {name}@poleia.lip6.fr {first name}@cs.qmul.ac.uk University Pierre an Marie Curie Queen Mary,
More information