[math-fun] Salamin polarization puzzle
ES is correct that mirror+sugar does not solve his problem (sorry). More generally mirror plus anything will not solve the problem because Maxwell eqns are time-reversible and the mirror converts a beam into its time reverse. Hence, any incoming beam that gets frobulated on the way to the mirror, will get unfrobulated on the way back. So how about, (1) sugar, (2) mirror, (3) now beam is both helicity reversed and 90degree plane of polarization rotated, (4) beam now re-enters sugar, which ruins it. But, regarding (3) as the "output" it seems like succeeded in some sense.
If you increase the separation between the mirror and sugar (to a few million miles), it becomes reasonable to remove the container of sugar whilst the light is en route to the mirror.
Sent: Tuesday, June 02, 2015 at 6:04 PM From: "Warren D Smith" <warren.wds@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Salamin polarization puzzle
ES is correct that mirror+sugar does not solve his problem (sorry). More generally mirror plus anything will not solve the problem because Maxwell eqns are time-reversible and the mirror converts a beam into its time reverse. Hence, any incoming beam that gets frobulated on the way to the mirror, will get unfrobulated on the way back.
So how about, (1) sugar, (2) mirror, (3) now beam is both helicity reversed and 90degree plane of polarization rotated, (4) beam now re-enters sugar, which ruins it. But, regarding (3) as the "output" it seems like succeeded in some sense.
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Or more simply, just tilt the mirror. But then my earlier reply as to why the Faraday rotator fails to provide a solution also applies here. -- Gene From: Adam P. Goucher <apgoucher@gmx.com> To: math-fun@mailman.xmission.com Sent: Tuesday, June 2, 2015 10:15 AM Subject: Re: [math-fun] Salamin polarization puzzle If you increase the separation between the mirror and sugar (to a few million miles), it becomes reasonable to remove the container of sugar whilst the light is en route to the mirror.
Sent: Tuesday, June 02, 2015 at 6:04 PM From: "Warren D Smith" <warren.wds@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Salamin polarization puzzle
ES is correct that mirror+sugar does not solve his problem (sorry). More generally mirror plus anything will not solve the problem because Maxwell eqns are time-reversible and the mirror converts a beam into its time reverse. Hence, any incoming beam that gets frobulated on the way to the mirror, will get unfrobulated on the way back.
So how about, (1) sugar, (2) mirror, (3) now beam is both helicity reversed and 90degree plane of polarization rotated, (4) beam now re-enters sugar, which ruins it. But, regarding (3) as the "output" it seems like succeeded in some sense.
In[1]:= num[x_,y_,z_] := (x y z - (1-x)(1-y)(1-z))^2 In[2]:= den(x_,y_,z_] := (1-x(1-z))(1-y(1-x))(1-z(1-y)) In[3]:= f[x_,y_,z_] := num[x,y,z]/den[x,y,z] In[4]:= NIntegrate[f[x,y,z],{x,0,1},{y,0,1},{z,0,1}] Out[4]= 0.130396 but: In[5]:= Integrate[f[x,y,z],{x,0,1},{y,0,1},{z,0,1}] Out[5]= 10 10 ??? Huh??? (Especially since the maximum value of f[x,y,z] over [0,1]^3 is f[0,0,0] = 1.) Rational explanations, if any, welcomed. ——Dan
exploiting symmetry: Integrate[(x y z - (1 - x) (1 - y) (1 - z))^2/((1 - y (1 - z)) (1 - z (1 - x)) (1 - x (1 - y))), {x, 0, 1}, {y, 0, 1}, {z, 0, 1}] gives 10 - π^2 = 0.130396 So clearly a Mma bug. Report it and maybe you’ll get a free license!
On Jun 2, 2015, at 5:45 PM, Dan Asimov <asimov@msri.org> wrote:
In[1]:= num[x_,y_,z_] := (x y z - (1-x)(1-y)(1-z))^2
In[2]:= den(x_,y_,z_] := (1-x(1-z))(1-y(1-x))(1-z(1-y))
In[3]:= f[x_,y_,z_] := num[x,y,z]/den[x,y,z]
In[4]:= NIntegrate[f[x,y,z],{x,0,1},{y,0,1},{z,0,1}]
Out[4]= 0.130396
but:
In[5]:= Integrate[f[x,y,z],{x,0,1},{y,0,1},{z,0,1}]
Out[5]= 10
10 ??? Huh??? (Especially since the maximum value of f[x,y,z] over [0,1]^3 is f[0,0,0] = 1.)
Rational explanations, if any, welcomed.
——Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
<< In[2]:= den(x_,y_,z_] := ... >> Mismatched bracket/parenthesis ? WFL On 6/3/15, Veit Elser <ve10@cornell.edu> wrote:
exploiting symmetry:
Integrate[(x y z - (1 - x) (1 - y) (1 - z))^2/((1 - y (1 - z)) (1 - z (1 - x)) (1 - x (1 - y))), {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]
gives
10 - π^2 = 0.130396
So clearly a Mma bug. Report it and maybe you’ll get a free license!
On Jun 2, 2015, at 5:45 PM, Dan Asimov <asimov@msri.org> wrote:
In[1]:= num[x_,y_,z_] := (x y z - (1-x)(1-y)(1-z))^2
In[2]:= den(x_,y_,z_] := (1-x(1-z))(1-y(1-x))(1-z(1-y))
In[3]:= f[x_,y_,z_] := num[x,y,z]/den[x,y,z]
In[4]:= NIntegrate[f[x,y,z],{x,0,1},{y,0,1},{z,0,1}]
Out[4]= 0.130396
but:
In[5]:= Integrate[f[x,y,z],{x,0,1},{y,0,1},{z,0,1}]
Out[5]= 10
10 ??? Huh??? (Especially since the maximum value of f[x,y,z] over [0,1]^3 is f[0,0,0] = 1.)
Rational explanations, if any, welcomed.
——Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Actually that was a copying error; I used a square bracket in the calculation. It was a genuine Mma error. ——Dan
On Jun 3, 2015, at 5:16 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
<< In[2]:= den(x_,y_,z_] := ... >>
Mismatched bracket/parenthesis ?
WFL
On 6/3/15, Veit Elser <ve10@cornell.edu> wrote:
exploiting symmetry:
Integrate[(x y z - (1 - x) (1 - y) (1 - z))^2/((1 - y (1 - z)) (1 - z (1 - x)) (1 - x (1 - y))), {x, 0, 1}, {y, 0, 1}, {z, 0, 1}]
gives
10 - π^2 = 0.130396
So clearly a Mma bug. Report it and maybe you’ll get a free license!
On Jun 2, 2015, at 5:45 PM, Dan Asimov <asimov@msri.org> wrote:
In[1]:= num[x_,y_,z_] := (x y z - (1-x)(1-y)(1-z))^2
In[2]:= den(x_,y_,z_] := (1-x(1-z))(1-y(1-x))(1-z(1-y))
In[3]:= f[x_,y_,z_] := num[x,y,z]/den[x,y,z]
In[4]:= NIntegrate[f[x,y,z],{x,0,1},{y,0,1},{z,0,1}]
Out[4]= 0.130396
but:
In[5]:= Integrate[f[x,y,z],{x,0,1},{y,0,1},{z,0,1}]
Out[5]= 10
10 ??? Huh??? (Especially since the maximum value of f[x,y,z] over [0,1]^3 is f[0,0,0] = 1.)
Rational explanations, if any, welcomed.
——Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (6)
-
Adam P. Goucher -
Dan Asimov -
Eugene Salamin -
Fred Lunnon -
Veit Elser -
Warren D Smith