[math-fun] ALSEP lunar retroreflectors
A story I've been telling about the Apollo 11 array is apparently completely apocryphal: The McDonald Observatory was dismayed to receive no return signal, and feared the departing ascent stage blew dust on the array. (Supposedly they were enjoined by NASA from plinking the landing site before the astronauts departed.) Becoming even more dismayed by assurances that the corner reflectors *exceeded* spec, someone finally calculated that Earth rotated several thousand feet during the laser pulse transit time, duly separated the transmitter from the receiver, and became joyful. But https://www.youtube.com/watch?v=VmVxSFnjYCA (poor video) has Mythbusters visit the Apache Point Observatory and supposedly put to rest the Moon landing hoax hoax by getting a paltry few photons back from gigawatt pulses, apparently with receiver and transmitter in the same building. Perhaps even the same instrument? Could the observatory possibly be unable to afford separate transmitter and receiver? Or is the spread of the reflected signal so huge as to dwarf the calculated separation? Anyway, from these few photons Mythbusters triumphantly conclude we walked on the moon, conveniently forgetting the retroreflector arrays on the Lunakhod rovers. Refuting conspiracy theories is a waste of time, especially when the refuters do a bad job. --rwg
Puzzle: ------- Let D in R^2 denote the open unit disk {(x,y) | x^2 + y^2 < 1} Let J denote the interval [0,1) x {0} in R^2. Find a rational mapping f(x,y) = P(x,y) / Q(x,y) taking D onto its subset D - J and prove that this holds. —Dan
On 17/10/2015 22:25, Dan Asimov wrote:
Puzzle: -------
Let D in R^2 denote the open unit disk {(x,y) | x^2 + y^2 < 1}
Let J denote the interval [0,1) x {0} in R^2. ... f(x,y) = P(x,y) / Q(x,y)
Just to clarify: do you mean f(x,y) = (g(x,y),h(x,y)) where g(x,y) = P(x,y) / Q(x,y) h(x,y) = R(x,y) / S(x,y) with P,Q,R,S being polynomials? (Rather than, e.g., anything explicitly to do with complex numbers.) -- g
More dumb carelessness on my part — sorry. I mean a rational mapping f: D —> D carrying D to D - J (where D = {{x,y) in R^2 | x^2+y^2 < 1} and J = {(x,0) in R^2 | 0 <= x < 1}) of the form f(x,y) = (P_1(x,y)/Q_1(x,y), P_2(x,y)/Q_2(x,y)). Arrgh. —Dan
On Oct 17, 2015, at 4:35 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 17/10/2015 22:25, Dan Asimov wrote:
Puzzle: -------
Let D in R^2 denote the open unit disk {(x,y) | x^2 + y^2 < 1}
Let J denote the interval [0,1) x {0} in R^2. ... f(x,y) = P(x,y) / Q(x,y)
Just to clarify: do you mean
f(x,y) = (g(x,y),h(x,y))
where
g(x,y) = P(x,y) / Q(x,y) h(x,y) = R(x,y) / S(x,y)
with P,Q,R,S being polynomials? (Rather than, e.g., anything explicitly to do with complex numbers.)
Add 'nontrivial', because f(x,y)=-1/2 is a trivial solution. On Oct 17, 2015 5:26 PM, "Dan Asimov" <asimov@msri.org> wrote:
Puzzle: -------
Let D in R^2 denote the open unit disk {(x,y) | x^2 + y^2 < 1}
Let J denote the interval [0,1) x {0} in R^2.
Find a rational mapping
f(x,y) = P(x,y) / Q(x,y)
taking D onto its subset D - J
and prove that this holds.
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
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Allan Wechsler -
Bill Gosper -
Dan Asimov -
Gareth McCaughan