The 2-circle and 2-disk problems on trees
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Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σ n f(x) be the sum Σ d(x, y)=n f(y) and letB n f(x)=Σ d(x, y)≦n f(y). The purpose is to study to what extent Σ n f andB n f determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σ n f orB n f vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σ n andB n respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.
KeywordsCommon Factor Elementary Function Nontrivial Solution Leading Edge Homogeneous Tree
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