Exact code scaling

Abstract

A new possibility of code scaling is introduced. We show that there are notextreme volumes over which some equations of mathematical physicshave the same eigenvalues and there exists a simple transplantation rule toget the eigenfunction of the first volume once that of the second volume isknown. We present two techniques. In the first, the domain of thenumerical method is a discretized volume. Congruent elements are gluedtogether to get the domain over which the solution is sought. We associatea group and a graph to that volume. When the group is a symmetry of theboundary value problem, one can specify the structure of the solution, andpredict the existance of other equispectral volumes. The second techniqueuses a complex mapping to transplant the solution from volume V1 tovolume V2 and a correction function. Equation for the correction function isgiven. A simple example demonstrates the feasibility of the suggestedmethod. We show that a measurement associated with the fundamentaleigenfunction of a linear operator on a volume is sufficient to predict resultsof a measurement on another volume by a computer program.