Almost periodic functions and an analytical method of solving the number partitioning problem
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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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