Self-similarity Symmetry Research Articles

Anisotropy Induced by Electric Charge: A Computational Analytical Approach

This paper presents a novel class of interior solutions for anisotropic stars under the imposition of a self-similar symmetry. This means proposing exact solutions to the Einstein field equations to describe charged matter distribution with radiation flow. The Einstein–Maxwell system by employing specific choices of mass function is formulated to describe the gravitational collapse of charged, anisotropic, spherically symmetric distributions using the Schwarzschild metric. Two ordinary differential equations governing the dynamics are derived by matching a straightforward solution of the symmetry equations to the charged exterior (Reissner–Nordström–Vaidya). Models with satisfactory physical behavior are constructed by extensively exploring self-similar solutions for a set of parameters and initial conditions. Finally, the paper presents the evolution of physical variables and the collapsing radius, demonstrating the inevitable collapse of the matter distribution.

Self-Similar Solutions of a Bianchi Type-III Model with a Perfect Fluid and Cosmic String Cloud in Riemannian Geometry

A Bianchi type-III cosmological model with self-similarity symmetry is investigated in cases with perfect fluid distribution and a cosmic string cloud. We show that this model admits a non-trivial homothetic vector field which possess non-null homothetic bivectors. This homothetic vector field is not parallel to its source vector (current vector). We discussed cases where the homothetic vector is either orthogonal or parallel to the 4-velocity of a fluid element. We solved Einstein’s field equations without making any assumptions on the geometry of space-time, only assuming that it admits self-similarity symmetry. We obtained new exact self-similar solutions for the Bianchi type-III model. We show that all obtained solutions are shear, and space-time is shear free in only the case where the homothetic vector field is parallel to the 4-velocity vector. The kinematical and physical properties of the obtained solutions are discussed.

Defect mode properties of self-similar symmetry order in triadic Cantor photonic crystals

The discovery of quasi-crystal tells us that not only the translational symmetry order (TSO) but also other orders can produce Bragg diffraction. As far as we know, in one dimensional (1D) photonic crystals (PCs) there are two symmetry orders, TSO and self-similar symmetry order (SSO). There has been a lot of researches exploring the properties of defect of TSO in the 1D PCs, but until now we do not find any studies related to the defect of SSO. In this work we consider the triadic Cantor set (TCS) PCs to stage 6, and attempt to create the defect of SSO by changing the scale factor of stage 2 to 3 and stage 3 to 4, where for the high-refractive-index layer only the thickness of stage 3 changes. It's well known that the defect of TSO governs the frequency of a kind of transmission peaks, called defect mode. Our results show that the defect of SSO influences the background intensity of the transmittance spectrum rather than the transmission peaks. We note that the defect mode frequency and the transmittance background intensity interchange their roles for the two symmetry orders, and it follows that the long neglected transmittance background intensity should be a significant optical property.

Optical properties of symmetry breaking of the self-similar order in triadic Cantor photonic crystals.

In general, the photonic crystal (PC) is a periodical optical structure, but there are some studies considering aperiodic structures. If we insert a defect layer into a one-dimensional periodic PC to break its translational symmetry order (TSO), some peaks, called defect modes, appear in the transmittance spectrum. The defect layer thickness governs the frequencies of these defect modes but almost does not affect the other part of the spectrum. The discovery of quasi-crystals tells us that not only the TSO but also other orders can produce Bragg diffraction. It is well known that triadic Cantor set (TCS) PCs, which lack TSO but have a self-similar symmetry order (SSO), still exhibit narrow transmission peaks. In this work, we try to break the SSO in TCS PCs and find the resulting optical phenomena, where single-negative materials and dielectrics are chosen as the constituents of PCs. The study method is the transfer matrix method, and the calculation results show that the background intensity of the transmittance spectrum rather than the frequency of peaks obviously periodically changes with the break of SSO. It follows that the SSO does have physical meaning, and not only the transmission peaks but also the background should be treated as a significant optical property.

PORE FLOW FRACTAL PROPERTIES OF POROUS LiOH ABSORBENTS

The flow law of carbon dioxide in the pores of the novel porous LiOH adsorbent is studied, and the intrinsic relationship between the pore structure of the absorbent and the fractal dimension is analyzed. In this paper, the fractal structure — Sierpinski–Menger sponge structure — is used to establish a three-dimensional fractal pore model to achieve different porosities and pore structures. The flow process of the working fluid is numerically simulated, and the flow law under different flow velocities (1, 0.1, and 0.01 mm/s) is analyzed. The simulation results show that the flow of gas in the Sierpinski–Menger sponge structure has fractal characteristics. The streamlines and velocity clouds of different pore structures and porosities reflect the distribution characteristics of the flow velocity in the porous media. The velocity field and pressure field distribution of sponge models of different series are consistent, which has obvious self-similarity symmetry. It provides a numerical reference for the study of the pore adsorption law of real LiOH absorbent.

Derivation of a Unified Theory from the Holographic Principle

Holographic principle and the related self-similarity symmetry are observed widely in nature. Research in black hole, string theory and quantum theory of gravity suggests that holographic principle can be an important symmetry for solving challenging problems in theoretical physics, such as finding the grand unification theory (GUT). In this paper, we derive a quantum theory incorporating the holographic principle by introducing the new concept and elementary information. The derived quantum action incorporating holographic principle, holographic action, turns out to be the generalized action encompassing string theory, general relativity and thermodynamics. This holographic quantum theory indicates that phenomena and laws of physics emerge from the holograms represented by the holographic action. Specifically, it predicts the following: (1) Elementary particles, gravity and gauge interactions and the classical equations of motion are the emergence of the hologram due to Poincaré symmetry, diffeomorphic symmetry and Weyl symmetry, respectively. (2) Dark matter and dark energy are the vibrations on the horizon scale of the universe. (3) Cosmological constant is calculated to be [Formula: see text] in Planck unit, in agreement with the cosmological constant deduced from astrophysical observation (4) The observed space-time is negatively curved if its dimension is greater than 4, positively curved if its dimension is less than 4, and flat if its dimension is 4. (5) It gives the mathematical formula to derive the entropy of black hole and study the internal dynamics of black hole. (6) It provides the mathematical framework to study the dynamics of spacetime compactification and the large hierarchy between Planck scale and electroweak scale. One may conclude that the holographic quantum theory based on holographic principle may be not only a GUT but also able to tackle some of the problems impossible to be addressed before.

Unconventional self-similar Hofstadter superconductivity from repulsive interactions

Fractal Hofstadter bands have become widely accessible with the advent of moiré superlattices, opening the door to studies of the effect of interactions in these systems. In this work we employ a renormalization group (RG) analysis to demonstrate that the combination of repulsive interactions with the presence of a tunable manifold of Van Hove singularities provides a new mechanism for driving unconventional superconductivity in Hofstadter bands. Specifically, the number of Van Hove singularities at the Fermi energy can be controlled by varying the flux per unit cell and the electronic filling, leading to instabilities toward nodal superconductivity and chiral topological superconductivity with Chern number {{{{{{{mathcal{C}}}}}}}}=pm 6. The latter is characterized by a self-similar fixed trajectory of the RG flow and an emerging self-similarity symmetry of the order parameter. Our results establish Hofstadter quantum materials such as moiré heterostructures as promising platforms for realizing novel reentrant Hofstadter superconductors.

Computational Algorithms for the Study of Distributions with Electric Charge and Radiation Flux in General Relativity

The following research contributes to the understanding of how electric charge influences the unwrapping of spherically symmetric distributions using a relativistic approach. With the recent detection of gravitational waves and the location of black holes, where a strong emission of electromagnetic radiation prevails, interest has arisen to consider the existence of compact charged objects. Thus, the appearance of charge in small quantities during gravitational collapse, the process by which black holes originate, is not ruled out; this also includes the emission of electromagnetic waves from them. This article intends to establish algorithms and write field equations for a charged fluid as those corresponding to an anisotropic fluid with radiation flux. Using an appropriate definition of the mass function, considering self-similar symmetry and Bondian observers, dynamical results are obtained for the Einstein–Maxwell electromagnetic system with added gravity. Imposing an additional homothetic symmetry, the field equations are solved, and the most relevant conclusions are drawn about the influence of the electric charge during the collapse and subsequent changes in the physical variables.

Hox Gene Collinearity May Be Related to Noether Theory on Symmetry and Its Linked Conserved Quantity

Hox Gene Collinearity (HGC) is a fundamental property that controls the development of many animal species, including vertebrates. In the Hox gene clusters, the genes are located in a sequential order Hox1, Hox2, Hox3, etc., along the 3’ to 5’ direction of the cluster in the chromosome. During Hox cluster activation, the Hox genes are expressed sequentially in the ontogenetic units D1, D2, D3, etc., along the anterior–posterior axis (A-P) of the early embryo. This collinearity, first observed by E.B. Lewis, is surprising because the spatial collinearity of these structures (Hox clusters and embryos) correlates entities that differ by about four orders of magnitude. Biomolecular mechanisms alone cannot explain such correlations. Long-range physical interactions, such as diffusion or electric attractions, should be involved. A biophysical model (BM) was formulated, which, in alignment with the biomolecular processes, successfully describes the existing vertebrate genetic engineering data. One hundred years ago, Emmy Noether made a fundamental discovery in mathematics and physics. She proved, rigorously, that a physical system obeying a symmetry law (e.g., rotations or self-similarity) is followed by a conserved physical quantity. It is argued here that HGC obeys a ‘primitive’ self-similarity symmetry. In this case, the associated primitive conserved quantity is the irreversibly increasing ‘ratchet’-like Hoxgene ordering where some genes may be missing. The genes of a vertebrate Hox clusterare located along a finite straight line. The same order follows the ontogenetic unitsof the vertebrate embryo. Therefore, HGC is a manifestation of a primitive Noether Theory (NT). NT may be applied to other than the vertebrate case, for instance, to animals with a circular topological symmetry. For example, the observed abnormal Hox gene ordering of the echinoderm Hox clusters may be reproduced by a double-strand break of the circular Hox gene ordering and its subsequent incorporation in the flanking chromosome.

Aperiodic-Order-Induced Multimode Effects and Their Applications in Optoelectronic Devices

Unlike periodic and random structures, many aperiodic structures exhibit unique hierarchical natures. Aperiodic photonic micro/nanostructures usually support optical multimodes due to either the rich variety of unit cells or their hierarchical structure. Mainly based on our recent studies on this topic, here we review some developments of aperiodic-order-induced multimode effects and their applications in optoelectronic devices. It is shown that self-similarity or mirror symmetry in aperiodic micro/nanostructures can lead to optical or plasmonic multimodes in a series of one-dimensional/two-dimensional (1D/2D) photonic or plasmonic systems. These multimode effects have been employed to achieve optical filters for the wavelength division multiplex, open cavities for light–matter strong coupling, multiband waveguides for trapping “rainbow”, high-efficiency plasmonic solar cells, and transmission-enhanced plasmonic arrays, etc. We expect that these investigations will be beneficial to the development of integrated photonic and plasmonic devices for optical communication, energy harvesting, nanoantennas, and photonic chips.

Cosmological constant decaying with CMB temperature

Recent observations of the dark energy density have demonstrated the fine-tuning problem and the challenges faced by theoretical modeling. In this study, we apply the self-similar symmetry (SSS) model, describing the hierarchical structure of the universe based on the Dirac large numbers hypothesis, to Einstein’s cosmological term. We introduce a new similarity dimension, [Formula: see text], in the SSS model. Using the [Formula: see text] SSS model, the cosmological constant [Formula: see text] is simply expressed as a function of the cosmic microwave background (CMB) temperature. The result shows that both the gravitational constant [Formula: see text] and [Formula: see text] are coupled with the CMB temperature, which simplifies the solution of Einstein’s field equations for the variable [Formula: see text]–[Formula: see text] model.

Chemical Bonding and Physical Properties in Quasicrystals and Their Related Approximant Phases: Known Facts and Current Perspectives

Quasicrystals are a class of ordered solids made of typical metallic atoms but they do not exhibit the physical properties that usually signal the presence of metallic bonding, and their electrical and thermal transport properties resemble a more semiconductor-like than metallic character. In this paper I first review a number of experimental results and numerical simulations suggesting that the origin of the unusual properties of these compounds can be traced back to two main features. For one thing, we have the formation of covalent bonds among certain atoms grouped into clusters at a local scale. Thus, the nature of chemical bonding among certain constituent atoms should play a significant role in the onset of non-metallic physical properties of quasicrystals bearing transition-metal elements. On the other hand, the self-similar symmetry of the underlying structure gives rise to the presence of an extended chemical bonding network due to a hierarchical nesting of clusters. This novel structural design leads to the existence of quite diverse wave functions, whose transmission characteristics range from extended to almost localized ones. Finally, the potential of quasicrystals as thermoelectric materials is discussed on the basis of their specific transport properties.

Self-Similar Symmetry Model and Cosmic Microwave Background

In this paper, we present the self-similar symmetry (SSS) model that describes the hierarchical structure of the universe. The model is based on the concept of self-similarity, which explains the symmetry of the cosmic microwave background (CMB). The approximate length and time scales of the six hierarchies of the universe---grand unification, electroweak unification, the atom, the pulsar, the solar system, and the galactic system---are derived from the SSS model. In addition, the model implies that the electron mass and gravitational constant could vary with the CMB radiation temperature.

Scaling in the optical characteristics of nanocluster structures

The possibility of identifying nanoparticle systems by comparing the scaling parameters of their geometry and the diffraction patterns of the probing radiation is considered. This technique is applied to fractal objects with different types of self-similarity symmetry. The relationship between the scaling in different spatial distributions of nanoparticles and their Fourier spectra is considered.

Influence of strong motion characteristics on permanent displacement of slopes

Landslides triggered by moderate to major earthquakes are a recognized seismic hazard. Arias Intensity (I a) is a key intensity measure of the ground motion, but significant duration is widely used to define strong motion duration. We calculate Newmark’s displacements using earthquake records bracketing a broad range of Arias Intensity and significant duration employing Newmark’s rigid block method and a number of yield accelerations. Total landslide displacement increases with the increase in the energy content of the ground motion (I a) above a threshold. Such threshold may be expressed as a function of yield acceleration of the slope regardless of the ground motion characteristics. Newmark displacement decreases with increasing duration for earthquakes with similar energy content. The wide scatter in the results converges when using formal dimensional analysis. Self-similar symmetry may facilitate the assessment of the performance of slopes during earthquakes. The mathematical framework for probabilistic determination of landslides displacement may be a useful aid to estimate the likelihood of landslide hazards provided that the geotechnical properties of the slopes are known.

Symmetry transformations in ontogeny and evolution

Symmetry transformations are fundamental phenomena in the development and evolution of multicellular animals. Symmetry transformations at the cellular level during oogenesis and early development determine the basic axes of the future body, whereas the scale of these transformations is reduced in further development. In addition to the classical symmetry types, such as rotational (radial), mirror (bilateral), and translational symmetries, characteristic of biological processes is scale symmetry (self-similarity symmetry). The chaos is growing during fractal morphogenesis and other manifestations of fluctuating asymmetry. Biological symmetry and other variants of morphofunctional iterations are an effective way of morphogenesis via iteration of genetic programs.

3D simulations of disc winds extending radially self-similar MHD models

Disc-winds originating from the inner parts of accretion discs are considered as the basic component of magnetically collimated outflows. The only available analytical MHD solutions to describe disc-driven jets are those characterized by the symmetry of radial self-similarity. However, radially self-similar MHD jet models, in general, have three geometrical shortcomings, (i) a singularity at the jet axis, (ii) the necessary assumption of axisymmetry, and (iii) the non-existence of an intrinsic radial scale, i.e. the jets formally extend to radial infinity. Hence, numerical simulations are necessary to extend the analytical solutions towards the axis, by solving the full three-dimensional equations of MHD and impose a termination radius at finite radial distance. We focus here on studying the effects of relaxing the (ii) assumption of axisymmetry, i.e. of performing full 3D numerical simulations of a disc-wind crossing all magnetohydrodynamic critical surfaces. We compare the results of these runs with previous axisymmetric 2.5D simulations. The structure of the flow in all simulations shows strong similarities. The 3D runs reach a steady state and stay close to axisymmetry for most of the physical quantities, except for the poloidal magnetic field and the toroidal velocity which slightly deviate from axisymmetry. The latter quantities show signs of instabilities, which, however, are confined to the region inside the fast magnetosonic separatrix surface. The forces present in the flow, both of collimating and accelerating nature, are in good agreement in both the 2.5D and the 3D runs. We conclude that the analytical solution behaves well also after relaxing the basic assumption of axisymmetry.

Pattern stability in diffraction of light from structures with self-similarity symmetry

A method for determining the structural features of 1D–3D aperiodic diffraction gratings with self-similarity symmetry has been proposed and substantiated. This method is based on the analysis of local patterns in scaling characteristics of diffracted waves. The stability of patterns to changes in grating illumination conditions and the presence of defects in gratings is estimated.

The self-similar field and its application to a diffusion problem

We introduce a continuum approach which accounts for self-similarity as a symmetry property of an infinite medium. A self-similar Laplacian operator is introduced which is the source of self-similar continuous fields. In this way ‘self-similar symmetry’ appears in an analogous manner as transverse isotropy or cubic symmetry of a medium. As a consequence of the self-similarity the Laplacian is a non-local fractional operator obtained as the continuum limit of the discrete self-similar Laplacian introduced recently by Michelitsch et al (2009 Phys. Rev. E 80 011135). The dispersion relation of the Laplacian and its Green’s function is deduced in closed forms. As a physical application of the approach we analyze a self-similar diffusion problem. The statistical distributions, which constitute the solutions of this problem, turn out to be Lévi-stable distributions with infinite variances characterizing the statistics of one-dimensional Lévi flights. The self-similar continuum approach introduced in this paper has the potential to be applied on a variety of scale invariant and fractal problems in physics such as in continuum mechanics, electrodynamics and in other fields.

Scaling in the Optical Characteristics of Aperiodic Structures with Self-Similarity Symmetry

The properties of diffraction gratings and multilayered systems constructed using 1D models of quasicrystals are considered based on numerical simulation. It is shown that there is a direct relationship between the self-similarity symmetry of quasicrystals and scaling in the characteristics of the above-mentioned optical devices. The degree of structural correspondence between the graphical representations of the geometric properties of crystals, light diffraction patterns of gratings, and the transmission spectra of multilayered systems is estimated. It is shown that certain types of self-similarity symmetry make the characteristics of aperiodic diffraction gratings highly stable to a change in the size ratio of forming elements.