Abstract:
The ternary Cantor set $C$, constructed by George Cantor in 1883, is probably the best known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article we will explore the richness, the peculiarities and the generalities that has $C$ and explore some variants and generalizations of it. For a more systematic treatment the Cantor like sets we refer to our previous paper.

Abstract:
In this paper we discuss several variations and generalizations of the Cantor set and study some of their properties. Also for each of those generalizations a Cantor-like function can be constructed from the set. We will discuss briefly the possible construction of those functions.

Abstract:
The fractal tree-like structures can be divided into three classes, according to the value of the similarity dimension Ds:DsD, where D is the topological dimension of the embedding space. It is argued that most of the physiological tree-like structures have Ds ￠ ‰ ￥D. The notion of the self-overlapping exponent is introduced to characterise the trees with Ds>D. A model of the human blood-vessel system is proposed. The model is consistent with the processes governing the growth of the blood-vessels and yields Ds=3.4. The model is used to analyse the transport of passive component by blood.

Abstract:
We study the tunneling through an arbitrary number of finite rectangular opaque barriers and generalize earlier results by showing that the total tunneling phase time depends neither on the barrier thickness nor on the inter-barrier separation. We also predict two novel peculiar features of the system considered, namely the independence of the transit time (for non resonant tunneling) and the resonant frequency on the number of barriers crossed, which can be directly tested in photonic experiments. A thorough analysis of the role played by inter-barrier multiple reflections and a physical interpretation of the results obtained is reported, showing that multibarrier tunneling is a highly non-local phenomenon.

Abstract:
In 1984, Kurt Mahler posed the following fundamental question: How well can irrationals in the Cantor set be approximated by rationals in the Cantor set? Towards development of such a theory, we prove a Dirichlet-type theorem for this intrinsic diophantine approximation on Cantor-like sets, and discuss related possible theorems/conjectures. The resulting approximation function is analogous to that for R^d, but with d being the Hausdorff dimension of the set, and logarithmic dependence on the denominator instead.

Abstract:
We obtain the Assouad dimensions of Moran sets under suitable condition. Using the homogeneous set, we also study the Assouad dimensions of Cantor-like sets.

Abstract:
Composite films with an array of needle-like TiO2 particles in urethane resin matrix were fabricated by applying a 1 kHz AC bias with a square wave. In the resulting film, needle-like TiO2 particles were arrayed in the composite films in a direction normal to the film surface. The composite films showed angular dependence of transmittance in the visible-NIR range. For the composite film with arrayed 0.1 vol% TiO2 needle-like particles, the transmittance changed by 16.6% between 0° and 60° at a wavelength of 500 nm.

Abstract:
A fractal-like (Cantor-like) stratified structure of chiral and convenient isotropic layers is considered. Peculiarities of the wave localization, self-similarity, scalability and sequential splitting in the reflected field of both the co-polarized and cross-polarized components are studied. The appearing of the additional peak multiplets in stopbands is revealed, and a correlation of their properties with chirality parameter is established.

Abstract:
One of the best systems for the study of quantum chaos is the atomic nucleus. A confined particle with general boundary conditions can present chaos and the eigenvalue problem can exhibit this fact. We study a toy model in which the potential has a Cantor-like form. The eigenvalue spectrum presents a Devil's staircase ordering in the semi-classical limit.

Abstract:
In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.