The Hausdorff and box dimensions for measures associated with recurrent self-similar sets generated by similitudes is explicitly given. The box dimension of the attractor associated with a class of two-dimensional affine maps is also computed.

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We estimate the Hausdorff measure and dimension of Cantor sets in terms of a sequence given by the lengths of the bounded complementary intervals. The results provide the relation between the decay rate of this sequence and the dimension of the associated Cantor set.It is well-known that not every Cantor set on the line is an s-set for some 0 ≤ s ≤ 1. However, if the sequence associated to the Cantor set C is nonincreasing, we show that C is an h-set for some continuous, concave dimension function h. We construct the function h from the sequence associated to the set C.

IntroductionIn this article we will be concerned with the optimization of a scalar-valued function u = f (x), x ∈ D, defined on some set D. Often D is a subset of n-dimensional Euclidean space R n . Further we impose the condition that D be a finite set although we do not restrict the size of its cardinality. Thus D might be the set of all n-tuples of computer floating point numbers. In this way our assumption on D is not computationally restrictive. We make no assumption about the smoothness or even the continuity of f since we use only the values u of f . In the event that f has at least first order derivatives, much studied gradient methods efficiently find the optimal local value of f for the basin corresponding to any given starting value x 0 . However should f have a large number of local optima, finding a global optimum is then a matter of chance depending on the x 0 selected. And so it is that when f has no smoothness as well as when f has a very large number of local optima, one can benefit from an understanding of how to optimize by means of function values alone.Two widely known Monte Carlo methods are Simulated Annealing, [8], [9], and Simulated Evolution (or Genetic Algorithms), [3], [5]. The former is meant to model thermodynamic systems and their ability to achieve near minimal energy configurations through Boltzmann kinetics as environmental temperature is lowered. The latter is the effort to find optimal states in an abstract setting in a manner similar to the way biological systems optimize survival using abstract analogs of the mechanisms of mating, mutation and enhanced reproduction of the fittest. Because of the generality and importance of these two methods and because our results apply directly to them, we will briefly describe and compare them below.An optimization process proceeds in steps indexed by "time" t = 1, 2, . . .. On the t th step one or more domain points x are selected to constitute the current state X t of the process X t = {x t 1 , . . . , x t n(t) } ⊂ D,