Saturday, August 21, 2010

How do I solve for the steady-state value of y using the Solow model?

I've been given the equation y=sqrt k. I need to solve for the steady-state value of y as a function of s, n, g, and d (little delta). I'm having A LOT of trouble trying to figure out which equations to use and which variables to substitute, so any help would be really appreciated!How do I solve for the steady-state value of y using the Solow model?
A Variant on Waring's Problem











Legendre showed that every positive integer can be written as the sum of four squares in 1770. Subsequently, in answer to a question of Waring, it was shown that every positive integer is the sum of 9 (non-negative) cubes, the sum of 19 fourth powers, etc.





What if we extend this investigation to rational numbers? It is not difficult to see that Legendre's result actually shows that every positive rational number is the sum of four rational squares and that this number cannot be decreased. Passing to the case of cubes, we claim that every positive rational number is the sum of three non-negative rational cubes.





Given a rational number, r, we are looking for rational solutions to x^3 + y^3 + z^3 = r. Clearing denominators, we are looking for integer solutions to a^3 + b^3 + c^3 = r d^3. Dividing by c^3 and letting s=a/c, t=b/c, and u=d/c we are looking for rational solutions to s^3 + t^3 + 1 = r u^3. But this has the trivial solution s= -1, t=0, u=0. If we take an arbitrary line through this point (note that there is a two-parameter family of these) and intersect it with our cubic surface, in general we will not get rational points but points whose coordinates lie in the field Q(Sqrt[k]) = {x+y Sqrt[k]| x and y are rational}, where k is some rational number. If we take one of these points, find the equation of the tangent plane to the cubic surace at this point and intersect the surface with this plane, we will get a singular cubic curve with coefficients in Q(Sqrt[k]).

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