| Abstract: | Let h : ? → ? be a computable function. A real number x is called h‐monotonically computable (h‐mc, for short) if there is a computable sequence (xs) of rational numbers which converges to x h‐monotonically in the sense that h(n)|x – xn| ≥ |x – xm| for all n andm > n. In this paper we investigate classes h‐ MC of h‐mc real numbers for different computable functions h. Especially, for computable functions h : ? → (0, 1)?, we show that the class h‐ MC coincides with the classes of computable and semi‐computable real numbers if and only if Σi∈?(1 – h(i)) = ∞and the sum Σi∈?(1 – h(i)) is a computable real number, respectively. On the other hand, if h(n) ≥ 1 and h converges to 1, then h‐ MC = SC (the class of semi‐computable reals) no matter how fast h converges to 1. Furthermore, for any constant c > 1, if h is increasing and converges to c, then h‐ MC = c‐ MC . Finally, if h is monotone and unbounded, then h‐ MC contains all ω‐mc real numbers which are g‐mc for some computable function g. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |
