Symmetric Cantor Research Articles

On the union of homogeneous symmetric Cantor set with its translations

On the union of homogeneous symmetric Cantor set with its translations

NUMERICAL MODELLING OF ELECTROMAGNETIC WAVE SCATTERING ON GRATINGS WITH ONE-DIMENSIONAL QUASI-FRACTAL PERIOD STRUCTURE

Numerical modelling of the properties of E-polarised and H-polarised waves scattered on periodic screened quasi-fractal gratings is carried out. The location of the band system at each period is determined by the principle of constructing a generalized symmetric Cantor set at a certain step of the algorithm. A mathematical model of the problems based on systems of boundary singular integral equations of the first kind was used in the study. These systems of equations were obtained using the method of parametric representations of singular and hypersingular integral operators. The systems of singular integral equations were solved numerically using the computational schemes of the method of discrete singularities. The solutions of these equations are used to obtain the main characteristics of the electric and magnetic fields. This experiment proved the possibility of using the MDS computational scheme to analyse systems containing 8 – 16 bands at different distances from each other. Graphs of the dependence of harmonic amplitudes on the wavenumber, point plots of absolute values of all non-zero harmonics at resonant wavenumber values, and maps of electric and magnetic field components in the region above the grating were obtained. It is confirmed that the overall field structure in the case of normal incidence is significantly influenced by all harmonics with absolute numbers from 0 to 50. The harmonics had a large number of resonances that were observed at different values of the wavenumber. This led to a complex structure of the isolines of absolute values of the scattered electric and magnetic field amplitudes in the region above the structure, and a significant difference in amplitude values with small changes in coordinates. In the future, it is planned to carry out computer simulations for imperfectly conducting structures and compare the results with the numerical results for the ideal case considered in this paper. The proposed structure may be of interest for the design of multimode broadband antennas.

Tunable and giant spatial Goos–Hänchen shifts in a parity-time symmetric Cantor photonic crystals incorporated with a centered graphene layer

In this study, we present the spectral features of a one-dimensional parity-time symmetric layered structure was composed of two quasi-photonic crystals which submit to the Cantor sequence and a graphene layer is embedded in the center of the quasi-crystals. Exceptional points, reflection and transmission spectra and the spatial Goos-Hänchen (GH) shifts are investigated at two distinct terahertz regions in the presence and absence of the graphene layer and compared them. The effect of the modification of imaginary part of refractive index of constituting gain and loss media are also examined. Our results show that, the proposed structure display giant enhanced GS shifts which are tunable with the chemical potential of embedded graphene layer, while GH shifts are weak in the absence of graphene layer. Results display different value and sign of GH shifts for the zero and nonzero chemical potentials. Very extreme GH shifts are obtained by judicious choice of chemical potential and imaginary value of the refractive index of constituting materials. Our results display that not only the photonic bandgap edge modes, but also bandgap modes can support giant GH shifts at Terahertz frequencies. Functionally, these types of structures are very desirable for designing optoelectronic devices that can be adjusted by the amount of chemical potential.

Multiple Exceptional Points in APT–Symmetric Cantor Multilayers

In this study, we explore the anisotropic reflection of light waves around the exceptional points (EPs) in anti-parity-time−symmetric (APT−symmetric) Cantor dielectric multilayers. This one-dimensional fractal structure governed by the Cantor substitution law is modulated to satisfy APT symmetry. The Cantor multilayers are aperiodic and support optical fractal resonances. The optical fractal effect combined with APT symmetry can induce multiple exceptional points (EPs) in the parameter space by modulating the loss coefficient of materials and optical frequency. Reflection anisotropy for light waves incident from two opposite directions presents unidirectional suppression and enhancement around EPs. This study can be utilized for multiple wavelengths of photonic suppressors and reflectors.

Daubechies\u2019 Time\u2013Frequency Localization Operator on Cantor Type Sets I

We study Daubechies’ time–frequency localization operator, which is characterized by a window and weight function. We consider a Gaussian window and a spherically symmetric weight as this choice yields explicit formulas for the eigenvalues, with the Hermite functions as the associated eigenfunctions. Inspired by the fractal uncertainty principle in the separate time–frequency representation, we define the n-iterate mid-third spherically symmetric Cantor set in the joint representation. For the n-iterate Cantor set, precise asymptotic estimates for the operator norm are then derived up to a multiplicative constant.

A continuous function with universal Fourier series on a given closed set of Lebesgue measure zero

Given a closed set E of Lebesgue measure zero on the unit circle T there is a continuous function f on T such that for every continuous function g on E there is a subsequence of partial Fourier sums Sn+(f,ζ)=∑k=0nfˆ(k)ζkof f, which converges to g uniformly on E. This result completes one result in a recent paper by C. Papachristodoulos and M. Papadimitrakis (2019), see Papachristodoulos and Papadimitrakis (2019). They proved that for a classical one third Cantor set C there is no universal function in the disk algebra. They also proved that for a symmetric Cantor set C∗ on T there is no universal continuous function for the classical symmetric Fourier sums. See also [2].

Families of symmetric Cantor sets from the category and measure viewpoints

Abstract We consider the family 𝒞 ⁢ 𝒮 {\mathcal{CS}} of symmetric Cantor subsets of [ 0 , 1 ] {[0,1]} . Each set in 𝒞 ⁢ 𝒮 {\mathcal{CS}} is uniquely determined by a sequence a = ( a n ) {a=(a_{n})} belonging to the Polish space X : = ( 0 , 1 ) ℕ {X\mathrel{\mathop{:}}=(0,1)^{\mathbb{N}}} equipped with probability product measure μ. This yields a one-to-one correspondence between sets in 𝒞 ⁢ 𝒮 {\mathcal{CS}} and sequences in X. If 𝒜 ⊂ 𝒞 ⁢ 𝒮 {\mathcal{A}\subset\mathcal{CS}} , the corresponding subset of X is denoted by 𝒜 ∗ {\mathcal{A}^{\ast}} . We study the subfamilies ℋ 0 {\mathcal{H}_{0}} , 𝒮 ⁢ 𝒫 {\mathcal{SP}} and ℳ {\mathcal{M}} of 𝒞 ⁢ 𝒮 {\mathcal{CS}} , consisting (respectively) of sets with Haudsdorff dimension 0, and of strongly porous and microscopic sets. We have ℳ ⊂ ℋ 0 ⊂ 𝒮 ⁢ 𝒫 {\mathcal{M}\subset\mathcal{H}_{0}\subset\mathcal{SP}} , and these inclusions are proper. We prove that the sets ℳ ∗ {\mathcal{M}^{\ast}} , ℋ 0 ∗ {\mathcal{H}_{0}^{\ast}} , 𝒮 ⁢ 𝒫 ∗ {\mathcal{SP}^{\ast}} are residual in X, and μ ⁢ ( ℋ 0 ∗ ) = 0 {\mu(\mathcal{H}_{0}^{\ast})=0} , μ ⁢ ( 𝒮 ⁢ 𝒫 ∗ ) = 1 {\mu(\mathcal{SP}^{\ast})=1} .

SELF-SIMILAR SUBSETS OF SYMMETRIC CANTOR SETS

This paper concerns the affine embeddings of general symmetric Cantor sets. Under certain condition, we show that if a self-similar set [Formula: see text] can be affinely embedded into a symmetric Cantor set [Formula: see text], then their contractions are rationally commensurable. Our result supports Conjecture 1.2 in [D. J. Feng, W. Huang and H. Rao, Affine embeddings and intersections of Cantor sets, J. Math. Pures Appl. 102 (2014) 1062–1079].

Exact packing measure of central Cantor sets in the line

In this paper we consider a class of symmetric Cantor sets in R . Under certain separation condition we determine the exact packing measure of such a Cantor set through the computation of the lower density of the uniform probability measure supported on the set. With an additional restriction on the dimension we give also the exact centered Hausdorff measure by computing the upper density.

Self‐similar structure on intersection of homogeneous symmetric Cantor sets

AbstractFor a homogeneous symmetric Cantor set C, we consider all real numbers tsuch that the intersection C∩(C + t)is a self‐similar set and investigate the form of the corresponding iterated function systems. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Symmetric Cantor measure, coin-tossing and sum sets

Construct a probability measure $\mu$ on the circle by successive removal of middle third intervals with redistributions of the existing mass at the $n$th stage being determined by probability $p_n$ applied uniformly across that level. Assume that the sequence $\{p_n\}$ is bounded away from both $0$ and $1$. Then, for sufficiently large $N$, (estimates are given) the Lebesgue measure of any algebraic sum of Borel sets $E_1,E_2,\ldots,E_N$ exceeds the product of the corresponding $\mu(E_i)^\alpha$, where $\alpha$ is determined by $N$ and $\{p_n\}$. It is possible to replace 3 by any integer $M\geq 2$ and to work with distinct measures $\mu_1,\mu_2,\ldots,\mu_N$. This substantially generalizes work of Williamson and the author (for powers of single-coin coin-tossing measures in the case $M=2$) and is motivated by the extension to $M=3$. We give also a simple proof of a result of Yin and the author for random variables whose binary digits are determined by coin-tossing.

THE POINTWISE DENSITIES OF NON-SYMMETRIC CANTOR SETS

We consider a class of non-symmetric Cantor sets C determined by C = a0C ∪ (a1C + 1 - a1), a0, a1 ∈ (0, 1). Under certain conditions on a0 and a1, the upper and lower densities are obtained explicitly for each x ∈ C. It extends the results for symmetric Cantor sets obtained by Feng, Hua and Wen (The pointwise densities of the Cantor measure, J. Math. Anal. Appl.250 (2000) 692–705).

DIMENSIONS OF A DERANGED CANTOR SET WITH SPECIFIC CONTRACTION RATIOS

We investigate a deranged Cantor set (a generalized Cantor set) using the similar method to find the dimensions of cookie-cutter repeller. That is, we will use a Gibbs measure which is a weak limit of a subsequence of discrete Borel measures to find the dimensions. The deranged Cantor set that will be considered is a generalized form of a perturbed Cantor set (a variation of the symmetric Cantor set) and a cookie-cutter repeller.

Differentiable structures of central Cantor sets

Central Cantor sets form a class of symmetric Cantor sets of the real line. Here we give a complete characterization of the $C^{k + \alpha}$ regularity of these Cantor sets. We also give a classification of central Cantor sets up to global and local diffeomorphisms. Examples of central Cantor sets with special dynamical and measure-theoretical properties are also provided. Finally, we calculate the fractal dimensions of an arbitrary central Cantor set.

Dimensions and gauges for symmetric Cantor sets

Dimensions and gauges for symmetric Cantor sets

A Characterization of σ-Symmetrically Porous Symmetric Cantor Sets

A Characterization of σ-Symmetrically Porous Symmetric Cantor Sets

A characterization of 𝜎-symmetrically porous symmetric Cantor sets

The purpose of this paper is to characterize those symmetric Cantor sets which are σ \sigma -symmetrically porous in terms of a defining sequence of deleted proportions. In contrast to other notions of porosity, a symmetric Cantor set can be σ \sigma -symmetrically porous without being symmetrically porous.

Symmetric porosity of symmetric Cantor sets

Symmetric porosity of symmetric Cantor sets

Symmetric Porosity and Symmetric Cantor Sets

Symmetric Porosity and Symmetric Cantor Sets