Manny Scarowsky

On a class of transformations which have unique absolutely continuous invariant measures

A class of piecewise C2 transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.

A Note on the Diophantine Equation: x n + y n + z n = 3

In this note solutions for the Diophantine equation 3 v3 + 9 = 3 are sought along planes x + v + z = 3m, m E Z. This was done for Iml < 50000, and no new solutions were found.

A bound on the number of periodic orbits of certain piecewise linear maps

Abstract Let f: [0, 1] → [0, 1] be a piecewise-linear continuous map. Let the slopes mi, i = 1, …, n, satisfy the condition that miϵ {±mj}j = 0∞, MϵN, ¦m¦ >1 . Let θ N = { a bp N : (a, p) = 1, p is prime a bp N ϵ [0, 1]} , N ⩾ 1 , where we assume (p, m) = 1. The main result states that f¦ θ N has a bounded number of periodic orbits independent of N.

Piecewise monotonic funcitons that commute

Let 0=a0 1,Let ⋀n be the continuous piecewise linear function with f(0)=0, f(1/n)=1, f(2/n)=0,…. Then τ is topologically conjugate to ⋀n.

On the computation of the class numbers of some cubic fields

Class numbers are calculated for cubic fields of the form x3+12Ax-12 0, A > 0, for ! a ! 17 and for some other values of A. These fields have a known unit, which under certain conditions is the fundamental unit, and are important in