The notion of a fractal has been widely used to describe self-similar structures [1]. The simplest way to construct a fractal is to repeat a given operation over and over again. The classical example of such a repetitive consruction is the Cantor’s “middle-third erasing” set [1]. Recall the definition of this set: One divides an interval into three equal intervals and then removes the middle in...

It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-α Cantor set and α ∈ ( 1 3 , 1 2 ). We show that for any compact set K and for a.e. α ∈ (0, 1), the arithmetic sum of K and the middle-α Cantor set does indeed have positive...

The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor-Bendixson rank of any member of this space. For instance, the Cantor-Bendixson rank of the free commutative ring on n generators is ωn, where ω is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.

The articles [10], [4], [12], [11], [7], [13], [2], [3], [6], [8], [5], [1], and [9] provide the notation and terminology for this paper. Let Y be a set and let x be a non empty set. Note that Y 7−→ x is non-empty. Let X be a set and let A be a family of subsets of X . The functor UniCl(A) yields a family of subsets of X and is defined as follows: (Def. 1) For every subset x of X holds x ∈ UniC...

Cantor Digitalis is a performative singing synthesizer that is composed of two main parts: a chironomic control interface and a parametric voice synthesizer. The control interface is based on a pen/touch graphic tablet equipped with a template representing vocalic and melodic spaces. Hand and pen positions, pen pressure, and a graphical user interface are assigned to specific vocal controls. Th...

Diffractals are electromagnetic waves diffracted by a fractal aperture. In an earlier paper, we reported an important property of Cantor diffractals, that of redundancy [R. Verma et. al., Opt. Express 20, 8250 (2012)]. In this paper, we report another important property, that of robustness. The question we address is: How much disorder in the Cantor grating can be accommodated by diffractals to...

A well known theorem of Cantor asserts that the cardinal of the power-set of a given set always exceeds the cardinal of the original set. An analogous result for sets having additional structure is the well known theorem that the set of initial segments of a well ordered set always has order type greater than the original set. These two theorems suggest that there should be a similar result for...

Abstract . To any pair (M , θ ) , where M is a family of finite subsets of N , compact in the pointwise topology, and 0 < θ < 1 , we associate a Tsirelson-type Banach space T θ M . It is shown that if the Cantor-Bendixson index of M is greater than n and θ > 1 n then T θ M is reflexive. Moreover, if the Cantor-Bendixson index of M is greater than ω then T θ M does not contain any l , while if t...