Dimensions of level sets related to tangential dimensions

Integrative Dimensions of Concept Sets

Explores the problem of minimizing loss of significance when a concept set is considered too complex to be communicated comprehensibly to a particular audience. In such a case the set must be transformed, possibly by collapsing some distinctions, so as to reduce the number of conceptual elements in the set by a factor of 2, 3 or more. If collapsed in an orderly manner, maintaining integrative d...

Dimensions of subsets of cantor-type sets

Let I = [0,1] be the unit interval on the real line andm> 1 be an integer. Let J = {0,1, . . . , m− 1}. For every point x ∈ I , there is a unique base-m representation x = Σk=1 jkm−k with jk ∈ J except for countable many points. Since countable sets do not interfere with our work, we neglect them here. For each j ∈ J , x ∈ [0,1], and n ∈ N, let τj(x,n) = {k : ik = j, 1 ≤ k ≤ n}, then the limit ...

Dimensions of Julia Sets of Meromorphic Functions

It is shown that for any meromorphic function f the Julia set J(f) has constant local upper and lower box dimensions, d(J(f)) and d(J(f)) respectively, near all points of J(f) with at most two exceptions. Further, the packing dimension of the Julia set is equal to d(J(f)). Using this result it is shown that, for any transcendental entire function f in the class B (that is, the class of function...

Symmetries of Julia sets in higher dimensions

Dimensions of Sums with Self-similar Sets

For some self-similar sets K ⇢ R we obtain certain lower bounds for the lower Minkowski dimension of K + E in terms of the lower Minkowski dimension of E.