Mathematical Journeys - Saint Ann's School

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  25  stocks in the New York Stock Exchange. This inverse correlation between stock movement in the 90s and the stock movement in the 00s was not evident. The lack of correlation, however, may not refute our original hypothesis as Alex’s data consisted of the extremes of the market, the absolute best performers and the absolute worst while our correlation analysis included all of the stocks in between. Alex’s picks included only stocks that performed abnormally and we believe that there could still be a correlation present in the relationship of the movements of these abnormal stocks. Further Research to Pursue Our aspirations are tremendous. Possibilities on the horizon include the application of Sharpe Ratio standards to our portfolios, an enormously increased data pool, Mat-Lab based simulations of teenage options trading, correlative analysis of extremes in stock returns, and, summarily, the development of our own trading model.                                                                   
26    THE REALITY GAME JOSHUA GLASSER ABSTRACT. Focusing on the intricacies of variants of the Reality game, this paper explores the different mathematical constructs used to interpret such a game. Through MatLab simulations and algebraic equations, this paper attempts to explain some of the phenomena that occur as a result of the Reality game. The reality Game, although less than a decade old, has already been the focal point of a few major studies, Including Farmar’s ubiquitous paper describing it. Though there have been many variations as to what in particular constitutes the reality game, many agree that it is a game in which any number of players bet money on a coin whose probability of landing on either heads or tails are as follows: 1.N players bet X amount of money each on heads or tails We varied this eventually, but it was true in the beginning. 2. The probability of the coin landing on either heads or tails is a result of X-plus over X total bet. X + is defined as … x + / Xtotalbet 3. The winners receive an equal take of the pot. We then proceeded to have the players employ any of the following 8 strategies: S1. Bet always on the same outcome. ! ! S2. Bet on the outcome that came up in the last round. S3. Bet on the outcome that didn't come up in the last round. S4. Bet according to the monetary majority in the last round. S5. Bet according to the monetary minority in the last round. S6. Bet according to the numerical majority in the last round. S7. Bet according to the numerical minority in the last round. S8. Bet randomly, H or T with probability 1/2 each. This is closer to the original paper (Farmer et.al.), except that they allowed probabilities other than !. To see the pure results of these strategies, we simplified the number bet to a constant 1 unit throughout runs. While some advantages seemed to appear, they were contained in limited runs of 100 or 1000 tosses. More even distributions of bets often led to more even results as well, with much more varied results regarding cumulative P&L’s. After building some experience with the behavior of these strategies, we decided to embark on an analytical investigation of the expected winnings of players following different strategies in a setting where everyone bets the same amount, in order to disaggregate the effects of the amount and the direction of each bet. To begin with, we studied the "pure strategies", i.e. the situations where all N players follow the same strategy. The only pure strategies that make sense are 1 and 8. For the case of pure strategy 1, let there be N + players betting +1 and N - N + players betting -1. Then the expected winnings of a +1 player is E[P + ] = (N + /N)(N/N + - 1) - (1 - N + /N) = 0.                        
Page 1 and 2: MATHEMATICAL JOURNEYS Saint Ann’s
Page 3 and 4: 2    LINEAR ALGEBRA AND MULTIVA
Page 5 and 6: 4    EXPLORATIONS INTO EUCLIDEA
Page 11 and 12: 10    EXPLORATIONS INTO EUCLIDE
Page 13 and 14: 12    € € Let p be a prime
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Page 17 and 18: 16    ! ! ! ! "#$%&! ! ! ! ! !
Page 19 and 20: 18    ! ! ! ! ! !      
Page 21 and 22: 20    ! ! ! ! ! ! ! ! ! ! ! ! !
Page 23 and 24: 22    ! ! ! ! "#!$%&'()*$!+(),!
Page 25: 24    The average return of all
Page 29 and 30: 28    investigate qualitative o
Page 31 and 32:   30  WALMART  SAVE MONEY?
Page 33 and 34:   32             
Page 35 and 36: 34     Pull Factor  Lumbe
Page 37 and 38: 36    TETRAHEDRA ON SQUARE LATT
Page 39 and 40: 38    So there are an infinite
Page 41 and 42: 40    Another mathematical conc
Page 43 and 44: 42    geometric properties that
Page 45 and 46: 44    We both have learned so m
Page 47 and 48: 46    hypothesizing, so we
Page 49 and 50: 48    2 LUCAS NEVILLE This also
Page 51 and 52: 50    4 LUCAS NEVILLE Figure 2.
Page 53 and 54: 52    2 HARRY BOYER AND SEAMUS
Page 55 and 56: 54    4 HARRY BOYER AND SEAMUS
Page 57 and 58:   56  EXPLORATIONS INTO DIS
Page 59 and 60:   58  the above case woul
Page 61 and 62:   60  But over the course

Mathematical Journeys - Saint Ann's School

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