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Math Accessibility Project (A-MAP)
"rendering STEM documents into fully accessible formats"
Food for Thought
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A Little New Math Humor
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Miscellaneous Information, Examples and Resources:
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In order to view and listen to the examples below
you must do the following:
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Install the Windows Math Fonts above. Follow the
instructions presented to you.
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Click on the "CLiCk, Speak" link above. Click on the
"Run" button twice. A blank DOS window will appear. Do not close it.
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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.
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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.
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Example 27
Example 28
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
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Example 29
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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
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
Worksheet 1:
Worksheet 2:
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Worksheet 4:
Worksheet 5:
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