There are N urinals; men prefer to stand away from others and they come in following a Poisson process and occupy one urinal for a fixed amount of time. How many flushes each of the urinals would get for each day?
COMMENT: this problem gets interesting when N is larger than 3 and the restroom is relatively crowded (say always slightly more than half occupied). Say N=2^n+1, the # of flushes is not a monotonous function from one side to the other; instead as n goes to infinity, it could very well have an asymptotic form like the Weierstrass function (not proved)...
Friday, January 16, 2009
Wednesday, January 14, 2009
Optimal trading strategy for raising silkworms
Jingles mentioned that when she was a child, she used to trade silkworms for Mulberry leaves because otherwise her worms would starve. Here is the question: what is the optimal trading strategy to have the most living silkworms by day T?
Some basic assumptions:
1, trade is one way only.
2, each silkworm consumes 1 unit of leaf every day.
3, leaves have a half life of k days (2 leaves from k days ago is equivalent to 1 leaf from today).
4, the price formula (# of leaves traded for one worm) is P(t) on day t, for starters, assume P(t) is a constant function.
5, silkworm will starve to death if not fed for one day, however, one can choose which worms to feed (so if leaves are insufficient, at least some of the worms can survive).
Some basic assumptions:
1, trade is one way only.
2, each silkworm consumes 1 unit of leaf every day.
3, leaves have a half life of k days (2 leaves from k days ago is equivalent to 1 leaf from today).
4, the price formula (# of leaves traded for one worm) is P(t) on day t, for starters, assume P(t) is a constant function.
5, silkworm will starve to death if not fed for one day, however, one can choose which worms to feed (so if leaves are insufficient, at least some of the worms can survive).
Monday, January 12, 2009
Stock price a chaotic revelation?
For a process x(t), define survivability ratio P(t) = chance of finding x(t) within a predetermined range of x(0); Brownian motion has P(t) = t^(-1/2). For chaotic systems with Cantori trapping mechanisms, P(t) also follows a power law P(t) = t^(-p), p=1+α, with 0<α<1.
Now, assume s(t) is the drift-adjusted part of the stock price while S(t) is the unadjusted (original) stock price. In other words:
d(lnS(t)) = μ dt + ds(t),
we might be able to check the index α for the process s(t), by sampling a lot of (t=0).
Also Re: self-criticality; people have discussed (with mixed opinions) the relationship between financial markets and earthquakes (Gutenberg-Richter law). It might be worthwhile to check the survivability function of fBm. See also 1/f noises.
Now, assume s(t) is the drift-adjusted part of the stock price while S(t) is the unadjusted (original) stock price. In other words:
d(lnS(t)) = μ dt + ds(t),
we might be able to check the index α for the process s(t), by sampling a lot of (t=0).
Also Re: self-criticality; people have discussed (with mixed opinions) the relationship between financial markets and earthquakes (Gutenberg-Richter law). It might be worthwhile to check the survivability function of fBm. See also 1/f noises.
