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It should be noted that a factor r has been inserted in the electric force term because this force is applied for a fraction of the time only whereas the diffusion and pressure gradient force terms are present at all times. Further, the time increment t can be larger than or equal to one pulse repetition period .

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The German/French input document was based on these assumptions and proposed a rich data services portfolio, which respected the overall target that GSM must be optimised for telephony and offer attractive data services for the mobile of ce A range of circuit switched data services with rates of up to 96 kbit/s was envisaged, since this was the maximum possible on a single traf c channel No packet switched services were proposed, since packet switching was not possible on the traf c channels of ISDN switches Instead the concept of short message transmission was proposed The application envisaged was the following: GSM was seen at that time as a car telephone system (the discussion on the viability of hand portables in the same network was still ongoing).

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(b) Find the optimal gambling scheme b (ie, the bet b that maximizes the exponent) (c) Assuming that b is chosen as in part (b), what distribution p causes Sn to go to zero at the fastest rate 67 Horse race Consider a horse race with four horses Assume that each horse pays 4-for-1 if it wins Let the probabilities of winning of the horses be { 1 , 1 , 1 , 1 } If you started with $100 and 2 4 8 8 bet optimally to maximize your long-term growth rate, what are your optimal bets on each horse Approximately how much money would you have after 20 races with this strategy Lotto The following analysis is a crude approximation to the games of Lotto conducted by various states.

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Assume that the player of the game is required to pay $1 to play and is asked to choose one number from a range 1 to 8 At the end of every day, the state lottery commission picks a number uniformly over the same range The jackpot (ie, all the money collected that day) is split among all the people who chose the same number as the one chosen by the state For example, if 100 people played today, 10 of them chose the number 2, and the drawing at the end of the day picked 2, the $100 collected is split among the 10 people (ie, each person who picked 2 will receive $10, and the others will receive nothing) The general population does not choose numbers uniformly numbers such as 3 and 7 are supposedly lucky and are more popular than 4 or 8.

So a typical application scenario was a plumber or other technician doing some repair work in the customers home could receive short messages in his car waiting in front of the home Another scenario was to enable a user to receive a short notice while he was engaged in a call It was envisaged to carry all short messages on signalling links of the system with low priority This signalling network in an ISDN-type GSM core network and the signalling links on the radio network form essentially an embedded packet switching network in an ISDN Therefore SMS can be seen as the rst packet switched service in GSM The length of the signalling packets on the GSM radio interface is shorter than in the ISDN in order to allow an ef cient transmission over the radio channel with its dif cult transmission quality.

Assume that the fraction of people choosing the various numbers 1, 2, , 8 is (f1 , f2 , , f8 ), and assume that n people play every day Also assume that n is very large, so that any single person s choice does not change the proportion of people betting on any number (a) What is the optimal strategy to divide your money among the various possible tickets so as to maximize your long-term growth rate (Ignore the fact that you cannot buy fractional tickets) (b) What is the optimal growth rate that you can achieve in this game 1 1 1 1 (c) If (f1 , f2 , .

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