Re: [math-fun] Miscounting errors
Henry's last comment reminds me of a project I was involved in that required optimizing a function over a certain rectangular box in some N-space. All the algorithms I could find (knowing nothing of optimization theory) were for optimizing on all of Euclidean space. No problem. (Or as they used to say, "You're welcome.") Just tessellate all of N-space by copies of that box, with the objective function extended by reflection (repeated if necessary) in the walls of the boxes. Now it's continuous on all of R^N with the same extrema as before. —Dan ----- Consider a plot of distance v. time. This is a long strip 25' wide and 30 minutes long. But you can *unfold* it -- e.g., make it 50' wide and 30 minutes long, which means that the wall doesn't exist and the swimmer keeps going. The elastic bounce has become a straight line. ...
Henry Baker writes:
I recall a question from my high school AP math or math SAT exam circa 1964:
Two swimmers are swimming laps of a 25' pool, but they swim at different (constant) speeds, say S1 & S2. (Assume that they bounce elastically off the pool ends!)
How many times do they pass one another in 30 minutes?
That's like the old spider vs. fly problem: http://mathworld.wolfram.com/SpiderandFlyProblem.html Tom Dan Asimov writes:
Henry's last comment reminds me of a project I was involved in that required optimizing a function over a certain rectangular box in some N-space. All the algorithms I could find (knowing nothing of optimization theory) were for optimizing on all of Euclidean space.
No problem. (Or as they used to say, "You're welcome.") Just tessellate all of N-space by copies of that box, with the objective function extended by reflection (repeated if necessary) in the walls of the boxes. Now it's continuous on all of R^N with the same extrema as before.
—Dan
----- Consider a plot of distance v. time.
This is a long strip 25' wide and 30 minutes long.
But you can *unfold* it -- e.g., make it 50' wide and 30 minutes long, which means that the wall doesn't exist and the swimmer keeps going.
The elastic bounce has become a straight line.
...
Henry Baker writes:
I recall a question from my high school AP math or math SAT exam circa 1964:
Two swimmers are swimming laps of a 25' pool, but they swim at different (constant) speeds, say S1 & S2. (Assume that they bounce elastically off the pool ends!)
How many times do they pass one another in 30 minutes?
participants (2)
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Dan Asimov -
Tom Karzes