Non-integer Dimensional Space Research Articles

Cloaking and magnifying using radial anisotropy in non-integer dimensional space

This manuscript studies the impact of non-integer dimensional (NID) space on cloaking and magnification introduced through polarly radially anisotropic (PRA) cylindrical shell/spherically radially anisotropic (SRA) spherical shell. By selecting appropriate value of ratio, between components of tensor permittivity, these structures in ordinary space (α=3) can yield cloaking and magnification under the quasi-static limit. In case of NID space (α≠3), PRA cylindrical shell can only yield cloaking. It has been shown that bound on anisotropy ratio get effected by the NID parameter in case of PRA shell. It has also been noted that magnification cannot be achieved using PRA shell in NID space. Further, cloaking and magnification of SRA shell remains independent of parameter describing NID space.

Analysis of reflection and transmission from a NID-interface/NID-dielectric interface in the presence of losses

The propagation of time harmonic linearly polarized wave in unbounded lossy/lossless noninteger dimensional space (NID) dielectric medium is investigated. It has been shown how the NID parameter changes the behavior of the propagating wave in the presence of losses. Then, it has been used as a source of excitation for an interface. A planar NID interface located in lossy dielectric medium is treated and its comparison has been made when the host medium is lossless or free-space. Furthermore, NID dielectric interface due to two lossy NID dielectric media is also investigated in this regard. The impact of different parameters, incident angle, real/imaginary part of the permittivity on reflection and transmission characteristics of a planar NID-interface/NID-dielectric interface is noted. Differences between the two cases are highlighted.

The hydrogen atom in α -dimensional space

The hydrogen atom system can be solved strictly by the eigenvalue equation and get analytical solutions, which is uncommon as a quantum mechanics system. In this study, we obtained wave functions and bound state energies of the hydrogen atom by the Schrödinger equation of the hydrogen atom in the noninteger-dimensional space. And we discussed dimensional behavior in binding energy, angular momentum, and radial density. The dimension α is given by the degree of anisotropy of the actual physical system. The excitons in solids are similar to the hydrogen atoms, except that the mass of their center positive charge is different, so the results obtained in this study were also applicable to excitons in anisotropic solids and hydrogenic impurities, which have application value.

Examples of reflection positive field theories

The requirement of reflection positivity (RP) for Euclidean field theories is considered. This is done for the cases of a scalar field, a higher derivative scalar field theory and the scalar field theory defined on a non-integer dimensional space (NIDS). It is shown that regarding RP, the analytical structure of the corresponding Schwinger functions plays an important role. For the higher derivative scalar field theory, RP does not hold. However for the scalar field theory on a NIDS, RP holds in a certain range of dimensions where the corresponding Minkowskian field is defined on a Hilbert space with a positive definite scalar product that provides a unitary representation of the Poincaré group. In addition, and motivated by the last example, it is shown that, under certain conditions, one can construct non-local reflection positive Euclidean field theories starting from the corrected two point functions of interacting local field theories.

TWO DIMENSIONAL GREEN'S FUNCTION FOR A HALF SPACE GEOMETRY DUE TO TWO DIFFERENT NON-INTEGER DIMENSIONAL SPACES

TWO DIMENSIONAL GREEN'S FUNCTION FOR A HALF SPACE GEOMETRY DUE TO TWO DIFFERENT NON-INTEGER DIMENSIONAL SPACES

Circuit elements at optical frequency in non-integer dimensional space

In present discussion, the idea of non-integer dimensional (NID) optoelectronics has been investigated. For this purpose, a nano-size sphere of any material (plasmonic or non-plasmonic) placed in NID space has been excited by an optical signal. Fields in the presence of sphere have been used to derive impedances. Reactance for the region inside the sphere is independent of NID parameter. Whereas, the capacitive and inductive behavior of impedance for the region outside the sphere can be controlled through material properties and NID parameter. This gives the existence of fringing capacitance and inductance.

Propagation of transverse magnetic mode in a non-integer dimensional dielectric slab waveguide

In present paper, solutions for the transverse magnetic modes propagating in a non-integer dimensional (NID) dielectric slab waveguide have been derived. Medium within the slab is NID in direction normal to the dielectric interfaces of the slab. Transcendental equations in terms of non-integer order Hankel functions have been derived. The order of non-integer order Hankel functions depends on parameter describing dimension of the NID space. Using graphical treatment, it has been shown that behavior of propagating mode changes substantially when value of the parameter describing dimension of the NID space changes. It has also been noted that NID parameter significantly effects the value of propagation constant, attenuation constant and cutoff frequency. The classical results have also been recovered when dimension of the NID slab has been taken equal to the dimension of the medium surrounding the slab.

Electromagnetic behavior of a planar interface of non-integer dimensional spaces

Reflection and transmission properties of a planar interface of two non-integer dimensional (NID) half spaces are studied. It is assumed that dimensions of both half spaces normal to the interface are non-integer. Both half spaces are distinguished from each other by non integer dimensions rather than wavenumbers. A plane wave has been taken as source of excitation. It is observed that non integer dimensions are also the factors controlling the reflection and transmission properties of the interface. Special cases are also considered when one or neither of the two half-spaces are NID.

Two dimensional Green's function for planar grounded dielectric layer in non-integer dimensional space

Two dimensional Green's function for grounded dielectric half space geometry in non-integer dimensional space has been derived. It has been assumed that both half spaces are of non-integer dimension. Radiated fields are written in terms of unknown plane wave spectrum. Unknown spectrum functions are determined using the boundary conditions. Special cases have also been addressed considering one or both the spaces are of integer dimensional space. The solution for both the spaces are of integer dimension is used to make a comparison with previously reported results. Results are reported for different values of physical and geometrical parameters.

Scattering of electromagnetic plane wave from a low contrast circular cylinder buried in non-integer dimensional Half Space

The objective of this paper is to study ‘scattering of electromagnetic waves from a dielectric circular cylinder buried in non-integer dimensional dielectric half-space’. To achieve this objective first two auxiliary problems are solved. The problem of reflection and transmission of electromagnetic waves for an interface between two dielectric mediums is studied first. The dimension normal to the interface is non-integer. Secondly, radiated fields from a buried current carrying line source in an non-integer dimensional half-space have been derived. By using the results of these auxiliary problems, scattering from a low contrast dielectric cylinder buried in non-integer dimensional half-space is studied using perturbation technique. Special cases are also considered when one of the two or both half-spaces are taken to be of integer dimension

Poiseuille equation for steady flow of fractal fluid

Fractal fluid is considered in the framework of continuous models with noninteger dimensional spaces (NIDS). A recently proposed vector calculus in NIDS is used to get a description of fractal fluid flow in pipes with circular cross-sections. The Navier–Stokes equations of fractal incompressible viscous fluids are used to derive a generalization of the Poiseuille equation of steady flow of fractal media in pipe.

Transient acoustic wave in self-similar porous material having rigid frame: Low frequency domain

This study concerns the acoustic wave diffusion in fractional dimensional rigid porous media in the low frequency domain. An equivalent fluid model with a non-integer dimensional space is developed using the Stillinger–Palmer–Stavrinou formalism. A generalized lossy diffusive equation is derived and solved analytically in the time domain. The coefficients of the diffusion equation are not constant and depend on the fractional dimension, static permeability and porosity of the porous material. The dynamic response of the material is obtained using the Laplace transform method. Numerical simulations of a diffusive wave in porous material with a non-integer dimensional space are given, showing the sensitivity of the waveform to the fractional dimension. It is found that the attenuation of the acoustic wave is more important for the highest value of the fractional dimension d. This result is especially true for resistive porous materials (low permeability value) and for large thicknesses.

Two dimensional Green's function for non-integer dimensional dielectric half space geometry

Two dimensional Green's function for non-integer dimensional dielectric half space geometry has been derived. It has been assumed that both half spaces are of non-integer dimension. Radiated fields are written as unknown spectrum of plane waves for non-integer dimensional space. Unknown spectrum functions are derived using the boundary conditions. Special cases with one of the two or both half spaces taken as integer dimensional have also been addressed.

Acoustic waves in fractal media: Non-integer dimensional spaces approach

Acoustic waves in fractal media are considered in the framework of continuum models with non-integer dimensional spaces. Using recently suggested vector calculus for non-integer dimensional space, we consider waves in isotropic fractal media. The wave equation for non-integer dimensional space is similar to the equation of waves in non-fractal medium with power-law heterogeneity. We discuss some properties of speed of acoustic waves in fractal materials.

Exciton binding energies in GaAs films on AlxGa1−xAs substrates

We use the fractional–dimensional approach (FDA) to study exciton binding energies in GaAs films on [Formula: see text] substrates. In this approach, the Schrödinger equation for a given anisotropic system is solved in a noninteger-dimensional space where the interactions are assumed to occur in an isotropic effective environment. The heavy-hole and light-hole exciton binding energies are calculated as functions of the film thickness and substrate thickness. The numerical results show that both the heavy-hole and light-hole exciton binding energies decrease monotonously as the film thickness increases. When the film thickness and the substrate thickness is relatively small, the change of substrate thickness has comparatively remarkable influence on both heavy-hole and light-hole exciton binding energies. As the substrate thickness increases, both the heavy-hole and light-hole exciton binding energies increase gradually. When the film thickness or the substrate thickness is relatively large, the change of substrate thickness has no significant influence on both heavy-hole and light-hole exciton binding energies.

Heat transfer in fractal materials

Heat transfer in fractal materials is considered in the framework of continuous models with non-integer dimensional spaces. We use a recently proposed vector calculus in non-integer dimensional spaces to describe heat flow in fractal materials. Solutions of the steady heat flow in fractal pipe and rod are derived.

Electromagnetic waves in non-integer dimensional spaces and fractals

Electromagnetic waves in non-integer dimensional spaces are considered in the framework of continuous models of fractal media and fields. Using the recently suggested vector calculus for non-integer dimensional space, we consider electromagnetic fields in isotropic case. This D-dimensional calculus allows us to describe fractal properties by continuous models with non-integer dimensional spaces. We prove that the wave equation for non-integer dimensional space is similar to equation of waves in non-fractal medium with heterogeneity of power-law type. The speed of electromagnetic waves and the effective refractive index of non-integer dimensional spaces and fractals are discussed.

Fractal electrodynamics via non-integer dimensional space approach

Using the recently suggested vector calculus for non-integer dimensional space, we consider electrodynamics problems in isotropic case. This calculus allows us to describe fractal media in the framework of continuum models with non-integer dimensional space. We consider electric and magnetic fields of fractal media with charges and currents in the framework of continuum models with non-integer dimensional spaces. An application of the fractal Gauss's law, the fractal Ampere's circuital law, the fractal Poisson equation for electric potential, and equation for fractal stream of charges are suggested. Lorentz invariance and speed of light in fractal electrodynamics are discussed. An expression for effective refractive index of non-integer dimensional space is suggested.

Elasticity of fractal materials using the continuum model with non-integer dimensional space

Using a generalization of vector calculus for space with non-integer dimension, we consider elastic properties of fractal materials. Fractal materials are described by continuum models with non-integer dimensional space. A generalization of elasticity equations for non-integer dimensional space, and its solutions for the equilibrium case of fractal materials are suggested. Elasticity problems for fractal hollow ball and cylindrical fractal elastic pipe with inside and outside pressures, for rotating cylindrical fractal pipe, for gradient elasticity and thermoelasticity of fractal materials are solved.

Anisotropic fractal media by vector calculus in non-integer dimensional space

A review of different approaches to describe anisotropic fractal media is proposed. In this paper, differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of vector calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of vector calculus for anisotropic fractal materials and media.