Science > Physics > Independent/Dependent Phases 26: A Beautiful Mind Via Saddle Points
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Science > Physics |
| User: |
"OsherD" |
| Date: |
16 Oct 2005 01:07:55 PM |
| Object: |
Independent/Dependent Phases 26: A Beautiful Mind Via Saddle Points |
From Osher Doctorow
Nobel Laureate John Nash's Nash Equilibrium is related to saddle
points, for example by the theorem that for a matrix A, a_(ij) (or aij
for short) is a saddle point of A iff (i, j) is a Nash Equilibrium.
Let's return to quadric surfaces and see why saddles have unusual
properties regarding 1/2 even though they didn't show up in
intersection of quadrics.
It turns out that the only quadrics for which partial differentiation
yields 1/2 directly are saddles (hyperbolic paraboloids) and
paraboloids, respectively given by the equations:
1) x^2/a^2 + y^2/b^2 = cz
2) y^2/a^2 - x^2/b^2 = cz
Let's compare (2) with a typical quadric like an ellipsoid:
3) x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
Let's take Dz(y) in the respective cases of (2) and (3) by implicit
differentiation:
4) 2yDz(y)/a^2 = c
5) 2yDz(y)/b^2 = -2z/c^2
Since the 2s cancel in (5) for the ellipsoid but not in (4) for the
saddle, only (4) yields (1/2):
6) yDz(y)/a^2 = (1/2)c
It turns out that every quadric except the hyperbolic paraboloid
(saddle) and paraboloid has the difficulty of (3) because of quadratic
terms in x, y, and z, including the hyperboloid of one sheet, the
hyperboloid of two sheets, the cone, and the ellipsoid (including the
sphere).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Independent/Dependent Phases 26: A Beautiful Mind Via Saddle Points |
16 Oct 2005 01:19:48 PM |
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From Osher Doctorow
From this viewpoint, let's take another look at the Riccati
Differential equation:
1) dy/dt = A(t) + B(t)y + C(t)y^2
Notice the similarity of the right hand side to the right hand side of
the saddle, but also notice the similarity of dy/dt to Dz(y) in the
previous posting! We have a similar 1/2 scenario, and this is
precisely what I formulated earlier in this thread with the Dyt(y)
scenario, since with the right hand side formally regarded as a
function of (y, t) with y and t "independent", its y partial is B(t) +
2yC(t).
Osher Doctorow
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