Almost periodic functions and an analytical method of solving the number partitioning problem
In the present paper, we study the limit sets of the almost periodic functions f(x). It is interesting that the values r=inf|f(x)| and R=sup|f(x)| may be expressed in the exact form. We show that the ring r≤ |z|≤ R is the limit set of the almost periodic function f(x) (under some natural conditions on f). The exact expression for r coincides with the well known partition problem formula and gives a new analytical method of solving the corresponding partition problem. Several interesting examples are considered. For instance, in the case of the five numbers, the well-known Karmarkar–Karp algorithm gives the value m=2 as the solution of the partition problem in our example, and our method gives the correct answer m=0. The figures presented in Appendix illustrate our results.
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