Quantum Gravity 250.1: Arithmetic Partial Differential Equations May Supercede GR Equations

From Osher Doctorow

Alexandru Buium and Santiago R. Simanca of U. New Mexico USA have just
come out with "Arithmetic partial differential equations, II: Modular
curves," arXiv: 0804.4856 v1 [math.NT], 30 April 2008, 26 pages. It
is a sequel to their earlier: "Arithmetic partial differential
equations," 82 pages, arXiv: math/0605107 v2 [math.AP] 10 May 2006.

Although their papers are oriented toward linear PDEs, it probably
will only be a matter of time before the nonlinear cases are
developed. They develop arithmetic analogs of linear PDEs in two
independent "spacetime" (space, time) variables with "flow" of
integers or more general points on algebraic groups, including
canonical flows that are arithmetic analogs of heat and wave equations
and convection equations.

This ties in with Analytic Number Theory, Arithmetic Derivatives (see
for example Wikipedia under the last title).

This is not simply "discrete approximation" but a whole new
direction. Take a look at "Computational methods and experiments in
analytic number theory," by Michael Rubinstein, arXiv: math/0412181 v1
[math.NT] 8 Dec 2004, 75 pages, for some value basic methods including
summation by parts (analogous to integration by parts), Euler-
MacLaurin summation, and Mobius inversion.

Osher Doctorow
.

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