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Math Accessibility Project (A-MAP)

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Miscellaneous Information, Examples and Resources:

In order to view and listen to the examples below you must do the following:

  1. Install the Windows Math Fonts above. Follow the instructions presented to you.

  2. Click on the "CLiCk, Speak" link above. Click on the "Run" button twice. A blank DOS window will appear. Do not close it. 

  3. After 1-2 minutes our "FireFox CLiCk, Speak" application will launch. Use this browser to re-connect to this page [http://ideal-group.org/math/].  You can then click on any of the MathML links below, click your mouse pointer in front of any formula and then click the Green "CLC Go" button. You will hear the formulae being read using correct math notation. Make sure that your PC speakers are turned on and the volume of your PC speakers is turned up.

  4. Warning: Do not try accessing the MAthML links below without first following the above process. If you do, all you will see is garbage and incorrectly presented math notations.

Author: Hammett, Adam

Year: 2007

Advisor: Pittel, Boris

Abstract: Two permutations of [n]:={1,2,...,n} are comparable in Bruhat order if one can be obtained from the other by a sequence of transpositions decreasing the number of inversions. Let P(n) be the probability that two independent and uniformly random permutations are comparable in Bruhat order. We demonstrate that P(n) is of order n^{-2} at most, and (0.708)^n at least. We also extend this result to r-tuples of permutations. Namely, if P(n,r) denotes the probability that r independent and uniformly random permutations form an r-long chain in Bruhat order, we demonstrate that P(n,r) is of order n^{-r(r-1)} at most, an exact extension of the case P(n,2)=P(n). For the related "weak order" - when only adjacent transpositions are admissible - we show that P^*(n) is of order (0.362)^n at most, and (H(1)/2)*(H(2)/2)*...*(H(n)/n) at least. Here H(i)=1/1+1/2+....+1/i, and P^*(n) is defined analogously to P(n), but for weak order. Finally, the weak order poset is a lattice, and we study Q(n,r), the probability that r independent and uniformly random permutations have trivial infimum, 12...n. We prove that [Q(n,r)]^{1/n}-->1/q(r), as n tends to infinity. Here, q(r) is the unique (positive) root of the equation 1-z+z^2/(2!)^r+...+(-z)^j/(j!)^r=0, lying in the disk |z|<2. 

Thesis: Hammett, Adam Joseph.pdf (131 Pages)

Transcoded version of Adam Joseph Hammett's Thesis

  • Example 29

    • Some results on recurrence and entropy

Author: Pavlov, Ronald

Year: 2007

Advisor: Bergelson, Vitaly

Abstract: This thesis is comprised primarily of two separate potions. In the first portion, we exhibit, for any sparse enough increasing sequence {p_n} of integers, a totally minimal, totally uniquely ergodic, and topologically mixing system (X,T) and a continuous function f on X for which ergodic averages of f along {p_n} fail to converge on a residual set in X, answering negatively an open question of Bergelson. We also construct here a totally minimal, totally uniquely ergodic, and topologically mixing system (X',T') and x' a point in X' so that x' is not a limit point of {T^(p_n}(x')}.

In the second portion, we study perturbations of multidimensional shifts of finite type. Given any Z^d shift of finite type X for d>1 and any word w in the language of X, denote by X_w the set of elements of X in which w does not appear. If X satisfies a uniform mixing condition called strong irreducibility, we obtain exponential upper and lower bounds on the difference of the topological entropies of X and X_w dependent only on the size of w. This result generalizes a result of Lind about Z shifts of finite type.

Thesis: Ronald Lee Pavlov, Jr.pdf (175 Pages)

Transcoded version of Ronald Lee Pavlov, Jr's. Thesis

  • Example 30

    • Risk analysis and hedging in incomplete markets

Author: George Argesanu, M.Sc.

Year: 2004

Advisor: Prof Bostwick Wyman, Ph. D.

Abstract: Variable annuities are in the spotlight in today’s insurance market. The tax deferral feature and the absence of the investment risk for the insurer (while keeping the possibility of investment benefits) boosted their popularity. They represent the sensible way found by the insurance industry to compete with other stock market and financial intermediaries. A variable annuity is an investment wrapped with a life insurance contract. An insurer who sells variable annuities bears two different types of risk. On one hand, he deals with a financial risk on the investment. On the other hand there exists an actuarial (mortality) risk, given by the lifetime of the insured.

Should the insured die, the insurer has to pay a possible claim, depending on the options elected (return of premium, reset, ratchet, roll-up). In the Black-Scholes model, the share price is a continuous function of time. Some rare events (which are rather frequent lately), can accompany jumps in the share price. In this case the market model is incomplete and hence there is no perfect hedging of options. I considered a simple market model with one riskless asset and one risky asset, whose price jumps in different proportions at some random times which correspond to the jump times of a Poisson process. Between the jumps the risky asset follows the Black-Scholes model. The mathematical model consists of a probability space, a Brownian motion and a Poisson process. The jumps are independent and identically distributed. The approach consists of defining a notion of risk and choosing a price and a hedge in order to minimize the risk. In the dual market (insurance and financial) the riskminimizing strategies defined by Follmer and Sondermann and the work of Moller with equity-linked insurance products are reviewed and used for variable annuities, with death or living benefits.

The theory of incomplete markets is complex and intriguing. There are many interesting connections between such models and game theory, while the newest and maybe the most powerful research tool comes from economics, the utility function (tastes and preferences).

Thesis: George Argesanu.pdf (97 Pages)

Transcoded version of George Argesanu's Thesis

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