Pensieve header: $50 offered for solutions.

Date: Tue, 23 Aug 2011 11:28:18 -0400 (EDT)
From: Dror Bar-Natan <drorbn@math.toronto.edu>
To: LazyKnots
Subject: Invitation / warning.


Dear All,

Within an hour I will post on my pensieve a couple of equations (for unknowns that are functions of two variables) and offer $50 for the first closed-form solution thereof. Within a short time later some hints will be posted, and perhaps some simplifications. If I will deem the simplifications major, I reserve the right to lower the sum offered. If offers for simplifications will be made by you, I may partially pay out the bounty. If no progress will be made for a while, I may raise the prize.

I will send a further notification with a URL when the equations/hints are posted. Shall anybody be interested, I will gladly describe what the equations are about at a LazyKnots meeting today at 4:30PM.

Best,

Dror.

Date: Thu, 25 Aug 2011 10:10:42 -0400 (EDT)
From: Dror Bar-Natan <drorbn@math.toronto.edu>
To: LazyKnots
Subject: Partial credits.


Dear All,

Some very useful ways to earn partial credit:

1. ($5) Prove that indeed a solution exists and is unique.

2. ($10) Prove that indeed f_21(x,y)=1/2+f_12(y,x).

3. ($30) Find a solution which is a composition of elementary functions (+, -, *, /, log, exp) and one single-variable “special” function. That special function may be anything provided it is a function of a single variable, though you can get a $5 extra credit if you can identify that special function in terms of known ones, perhaps in terms of the Lambert W function of, say, http://en.wikipedia.org/wiki/Lambert_W_function.

Best,

Dror.

The Equations.

Ok, here we go. For unknown functions (Subscript[f, 12])(x,y) and (Subscript[f, 21])(x,y), solve the equations:

((E^(-x (1+(Subscript[f, 12]))-y (1+(Subscript[f, 21])))) ((-y+(E^(x (Subscript[f, 12])+y (Subscript[f, 21]))) (x+y)) (Subscript[f, 12])+y (Subscript[f, 21])))/((x+y) (x (Subscript[f, 12])+y (Subscript[f, 21])))==(((E^(-x-y)) (-1+(E^x)))/((-1+(E^(x+y))) x))

and

((E^(-x-x (Subscript[f, 12])-y (Subscript[f, 21]))) (x y ((Subscript[f, 12])-(Subscript[f, 21]))+(E^(x (Subscript[f, 12])+y (Subscript[f, 21]))) y (x+y) (Subscript[f, 21])+(E^((x+y)/2)) x (x (Subscript[f, 12])+y (Subscript[f, 21]))))/((x+y) (x (Subscript[f, 12])+y (Subscript[f, 21])))==(((E^(-x (Subscript[g, 12])-y (Subscript[g, 21]))) x (Subscript[g, 12])+y (Subscript[g, 21]))/(x (Subscript[g, 12])+y (Subscript[g, 21])))

where in the second equation, (Subscript[g, 12]) stands for (Subscript[f, 21])(y,x) and likewise (Subscript[g, 21]) stands for (Subscript[f, 12])(y,x). I have good reasons to believe (see the “perturbative hint” below) that a solution exists and is unique.

An Analytical Hint

One solution of only the first equation is (Subscript[f, 21])=0 and (Subscript[f, 12])=((Log[(((e^(-x)) (-1+(e^(x+y))) y)/((-1+(e^y)) (x+y)))])/x).

A Perturbative Hint

Up to degree 10 there exists a unique power-series solution; it is given by:

(Subscript[f, 12]) = -(1/4)+((x/32)+((5 y)/96)) h+(((-57 (x^3)-169 (x^2) y-187 x (y^2)-99 (y^3)) (h^3))/138240)+(1/69672960)(555 (x^5)+2558 (x^4) y+5000 (x^3) (y^2)+5078 (x^2) (y^3)+2669 x (y^4)+780 (y^5)) (h^5)+(1/200658124800)(-31989 (x^7)-205357 (x^6) y-592097 (x^5) (y^2)-964985 (x^4) (y^3)-949967 (x^3) (y^4)-565943 (x^2) (y^5)-186139 x (y^6)-34275 (y^7)) (h^7)+(1/1589212348416000)(5104911 (x^9)+42750073 (x^8) y+165124408 (x^7) (y^2)+376113312 (x^6) (y^3)+552802446 (x^5) (y^4)+542218746 (x^4) (y^5)+352835712 (x^3) (y^6)+146191048 (x^2) (y^7)+33851563 x (y^8)+4104741 (y^9)) (h^9)+((O[h])^11)

(Subscript[f, 21]) = (1/4)+(((5 x)/96)+(y/32)) h+(((-99 (x^3)-187 (x^2) y-169 x (y^2)-57 (y^3)) (h^3))/138240)+(((780 (x^5)+2669 (x^4) y+5078 (x^3) (y^2)+5000 (x^2) (y^3)+2558 x (y^4)+555 (y^5)) (h^5))/69672960)+(1/200658124800)(-34275 (x^7)-186139 (x^6) y-565943 (x^5) (y^2)-949967 (x^4) (y^3)-964985 (x^3) (y^4)-592097 (x^2) (y^5)-205357 x (y^6)-31989 (y^7)) (h^7)+(1/1589212348416000)(4104741 (x^9)+33851563 (x^8) y+146191048 (x^7) (y^2)+352835712 (x^6) (y^3)+542218746 (x^5) (y^4)+552802446 (x^4) (y^5)+376113312 (x^3) (y^6)+165124408 (x^2) (y^7)+42750073 x (y^8)+5104911 (y^9)) (h^9)+((O[h])^11)

(Here the variable “h” is only used to organize the power series by degree. Feel free to substitute h=1.)

It seems that (Subscript[f, 12])(x,y)=-(1/2)+(Subscript[f, 21])(y,x) and that both functions are odd except for their constant term.

Scratch Work.

Simplify[(Subscript[f, 12])/.
Solve[(((E^(-x (1+(Subscript[f, 12]))-y (1+(Subscript[f, 21])))) ((-y+(E^(x (Subscript[f, 12])+y (Subscript[f, 21]))) (x+y)) (Subscript[f, 12])+y (Subscript[f, 21])))/((x+y) (x (Subscript[f, 12])+y (Subscript[f, 21]))))==(((E^(-x-y)) (-1+(E^x)))/((-1+(E^(x+y))) x)) /. (Subscript[f, 21]) -> 0, (Subscript[f, 12])]
]

Simplify[(Subscript[f, 21])/.
Solve[(((E^(-x (1+(Subscript[f, 12]))-y (1+(Subscript[f, 21])))) ((-y+(E^(x (Subscript[f, 12])+y (Subscript[f, 21]))) (x+y)) (Subscript[f, 12])+y (Subscript[f, 21])))/((x+y) (x (Subscript[f, 12])+y (Subscript[f, 21]))))==(((E^(-x-y)) (-1+(E^x)))/((-1+(E^(x+y))) x)) /. (Subscript[f, 12]) -> 0, (Subscript[f, 21])]
]

Simplify[(#/(-1+(E^(x+y))))& /@ (((E^(-(1+f12) h[1]-(1+f21) h[2])) (-1+(E^(h[1]+h[2]))) ((-f12+f21) h[2]+(E^(f12 h[1]+f21 h[2])) f12 (h[1]+h[2])) h[3])/((h[1]+h[2]) (f12 h[1]+f21 h[2])) == (((E^(-h[1]-h[2])) (-1+(E^(h[1]))) h[3])/(h[1])) /. {h[1]->x, h[2]->y, h[3]->1, f12 -> (Subscript[f, 12]), f21 -> (Subscript[f, 21])})
]

Simplify[((E^(-(1+f12) x-f21 y)) ((f12-f21) x y+(E^(f12 x+f21 y)) f21 y (x+y)+(E^((x+y)/2)) x (f12 x+f21 y)))/((x+y) (f12 x+f21 y))==(((E^(-g12 x-g21 y)) g12 x+g21 y)/(g12 x+g21 y)) /. {f12 -> (Subscript[f, 12]), f21 -> (Subscript[f, 21]), g12 -> (Subscript[g, 12]), g21 -> (Subscript[g, 21])}
]

pf12 = -(1/4)+((x/32)+((5 y)/96)) h+(((-57 (x^3)-169 (x^2) y-187 x (y^2)-99 (y^3)) (h^3))/138240)+(1/69672960)(555 (x^5)+2558 (x^4) y+5000 (x^3) (y^2)+5078 (x^2) (y^3)+2669 x (y^4)+780 (y^5)) (h^5)+(1/200658124800)(-31989 (x^7)-205357 (x^6) y-592097 (x^5) (y^2)-964985 (x^4) (y^3)-949967 (x^3) (y^4)-565943 (x^2) (y^5)-186139 x (y^6)-34275 (y^7)) (h^7)+(1/1589212348416000)(5104911 (x^9)+42750073 (x^8) y+165124408 (x^7) (y^2)+376113312 (x^6) (y^3)+552802446 (x^5) (y^4)+542218746 (x^4) (y^5)+352835712 (x^3) (y^6)+146191048 (x^2) (y^7)+33851563 x (y^8)+4104741 (y^9)) (h^9)+((O[h])^11);

pf21 = (1/4)+(((5 x)/96)+(y/32)) h+(((-99 (x^3)-187 (x^2) y-169 x (y^2)-57 (y^3)) (h^3))/138240)+(1/69672960)(780 (x^5)+2669 (x^4) y+5078 (x^3) (y^2)+5000 (x^2) (y^3)+2558 x (y^4)+555 (y^5)) (h^5)+(1/200658124800)(-34275 (x^7)-186139 (x^6) y-565943 (x^5) (y^2)-949967 (x^4) (y^3)-964985 (x^3) (y^4)-592097 (x^2) (y^5)-205357 x (y^6)-31989 (y^7)) (h^7)+(1/1589212348416000)(4104741 (x^9)+33851563 (x^8) y+146191048 (x^7) (y^2)+352835712 (x^6) (y^3)+542218746 (x^5) (y^4)+552802446 (x^4) (y^5)+376113312 (x^3) (y^6)+165124408 (x^2) (y^7)+42750073 x (y^8)+5104911 (y^9)) (h^9)+((O[h])^11);

pf12+(pf21 /. {x->-y, y->-x})

Simplify[pf12 - (-1/2+pf21 /. {x->y, y->x})]

((((E^(-x-y-x (Subscript[f, 12])-y (Subscript[f, 21]))) ((-y+(E^(x (Subscript[f, 12])+y (Subscript[f, 21]))) (x+y)) (Subscript[f, 12])+y (Subscript[f, 21])))/((x+y) (x (Subscript[f, 12])+y (Subscript[f, 21]))))==(((E^(-x-y)) (-1+(E^x)))/((-1+(E^(x+y))) x))) /. y (Subscript[f, 21]) -> -x (Subscript[f, 12]) + g

x pf12+y pf21 // Simplify

(x pf12+y pf21 /. {x->a+b, y-> a-b} )// Simplify

Simplify[
((((E^(-x-x (Subscript[f, 12])-y (Subscript[f, 21]))) (x y ((Subscript[f, 12])-(Subscript[f, 21]))+(E^(x (Subscript[f, 12])+y (Subscript[f, 21]))) y (x+y) (Subscript[f, 21])+(E^((x+y)/2)) x (x (Subscript[f, 12])+y (Subscript[f, 21]))))/((x+y) (x (Subscript[f, 12])+y (Subscript[f, 21]))))/((((E^(-x (Subscript[g, 12])-y (Subscript[g, 21]))) x (Subscript[g, 12])+y (Subscript[g, 21]))/(x (Subscript[g, 12])+y (Subscript[g, 21]))))) /. {(Subscript[g, 12]) -> 1/2+(Subscript[f, 12]), (Subscript[g, 21]) -> -1/2+(Subscript[f, 21])}
]

Simplify[((E^(-x-x (Subscript[f, 12])-y (Subscript[f, 21]))) (x ((1/2)+(Subscript[f, 12]))+y (-(1/2)+(Subscript[f, 21]))) (x y ((Subscript[f, 12])-(Subscript[f, 21]))+(E^(x (Subscript[f, 12])+y (Subscript[f, 21]))) y (x+y) (Subscript[f, 21])+(E^((x+y)/2)) x (x (Subscript[f, 12])+y (Subscript[f, 21]))))/((x+y) ((E^((1/2) (-x+y-2 x (Subscript[f, 12])-2 y (Subscript[f, 21])))) x ((1/2)+(Subscript[f, 12]))+y (-(1/2)+(Subscript[f, 21]))) (x (Subscript[f, 12])+y (Subscript[f, 21]))) /. {x-> h x, y-> h y, (Subscript[f, 12])-> pf12, (Subscript[f, 21])->pf21}]