Self-similar Structure Research Articles

PERMUTATION-BASED PRESENTATIONS FOR BRIN’S HIGHER-DIMENSIONAL THOMPSON GROUPS

AbstractThe higher-dimensional Thompson groups $nV$ , for $n \geqslant 2$ , were introduced by Brin [‘Presentations of higher dimensional Thompson groups’, J. Algebra284 (2005), 520–558]. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of n-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.

Large-scale motions and self-similar structures in compressible turbulent channel flows

The present study investigates the scale characteristics of the log- and outer-region motions and structures in subsonic and supersonic wall turbulence. The energy distribution among the multiscale structures in the outer region is found to be dominated by the semilocal friction-Reynolds-number effects rather than the Mach-number effects. The geometrical characteristics of the self-similar structures populating the logarithmic region are also revealed by adopting a linear coherence spectrum.

Seaward expansion of salt marshes maintains morphological self-similarity of tidal channel networks

Tidal channel networks (TCNs) dissect ecologically and economically valuable salt marsh ecosystems. These networks evolve in response to complex interactions between hydrological, sedimentological, and ecological processes that act in tidal landscapes. Thus, improving current knowledge of the evolution of salt-marsh TCNs is critical to providing a better understanding of bio-morphodynamic processes in coastal environments. Existing studies of coastal TCNs have typically focussed on marshes with either laterally stable or eroding edges, and suggested that TCN morphology evolves primarily through the progressive landward erosion of channel tips, that is, via channel headward growth. In this study, we analyze for the first time the morphological evolution of TCNs found within salt marshes that are characterized by active lateral expansion along their seaward edges and anthropogenically-fixed landward boundaries. We use remote-sensing and numerical-modeling analyses to show that marsh seaward expansion effectively limits headward channel growth and prompts the evolution of TCNs that maintain self-similar morphological structures. In particular, we demonstrate that the overall TCN length increases proportionally to the rate at which marshes expand laterally and that these morphological changes do not significantly alter the drainage properties of the coupled marsh-TCN system. Such behavior is not observed in marshes that are not expanding laterally. Our results allow for elucidating the mechanisms of TCN formation and evolution in tidal wetlands, and are therefore critical to improving our current understanding of coastal-landscape ecomorphodynamics, as well as to developing sustainable strategies for the conservation and restoration of these environments.

A Multiscale Picture of the Magnetic Field and Gravity from a Large-scale Filamentary Envelope to Core-accreting Dust Lanes in the High-mass Star-forming Region W51

We present 230 GHz continuum polarization observations with the Atacama Large Millimeter/Submillimeter Array at a resolution of 0.″1 (∼540 au) in the high-mass star-forming regions W51 e2 and e8. These observations resolve a network of core-connecting dust lanes, marking a departure from earlier coarser, more spherical continuum structures. At the same time, the cores do not appear to fragment further. Polarized dust emission is clearly detected. The inferred magnetic field orientations are prevailingly parallel to dust lanes. This key structural feature is analyzed together with the local gravitational vector field. The direction of local gravity is found to typically align with dust lanes. With these findings, we derive a stability criterion that defines a maximum magnetic field strength that can be overcome by an observed magnetic field–gravity configuration. Equivalently, this defines a minimum field strength that can stabilize dust lanes against a radial collapse. We find that the detected dust lanes in W51 e2 and e8 are stable, hence possibly making them a fundamental component in the accretion onto central sources, providing support for massive star formation models without the need of large accretion disks. When comparing to coarser resolutions, covering the scales of envelope, global, and local collapse, we find recurring similarities in the magnetic field structures and their corresponding gravitational vector fields. These self-similar structures point at a multiscale collapse-within-collapse scenario until finally the scale of core-accreting dust lanes is reached where gravity is entraining the magnetic field and aligning it with the dust lanes.

A Frame Theory of Energetic Life: A Twisting Energy Solidified on the Holographic Fractal Structure

Life, as the most mysterious and unique phenomenon on the Earth, has confused humans since time began. Why does life exist as it does and how has the diversity of life developed? We, herein, propose a new theory of energetic life, based on existing energy laws, to interpret the evolution and categorization of physical life forms, from microscopic life to macroscopic life. According to this theory, life is a process in which a mass of energy flows and diffuses in the environment. This energy takes DNA as the three-dimensional blueprint, protein as the basic material unit, and fractal network structure as the framework, so as to solidify from energy and form a semi-solid structure. DNA base pairs simultaneously have dual properties as protein pointers and spatial coordinates, and the multi-level self-similar fractal helix structure ultimately guides the formation of different levels of the fractal structure of organisms. This theory organically links the life phenomenon from microscopic to macroscopic levels, from gene, cell and organ to organism, and it provides a new perspective on life, which may inspire biologists to better reveal the mystery of life.

Energy absorption of self-similar inspired multi-cell tubes under quasi-static and dynamic loading

Self-similar multi-cell structures have been widely applied in engineering fields for their excellent mechanical properties and multi-functional characteristics. In this paper, self-similar inspired multi-cell tubes are constructed by replacing the substructures according to different strategies. The hybrid tubes are built by the space-filling and tessellation of these self-similar inspired multi-cell tubes. The effect of the diameter, layout and hierarchical order are experimentally investigated. The tessellation and space-filling of these multi-cell tubes are numerically and theoretically investigated. The results show that the specific energy absorption (SEA) increases with the hierarchical fractal order. The specific energy absorption of the 2nd order self-similar inspired multi-cell tubes increased by 342.5% and 116.5% compared to the 0th and 1st order multi-cell tubes. It is also found that the increase of the side length is not an effective method to improve the crashworthiness of the lower-order multi-cell tubes. Finally, a theoretical model of the angle element was constructed to predict the mean crushing force of these multi-cell tubes. In general, our research provides an effective method for the design of self-similar inspired multi-cell structures with excellent crashworthiness.

A NARRATIVE REVIEW OF THE USE OF FRACTAL GEOMETRY IN VARIOUS ASPECTS OF URBAN PLANNING

Urban structure is one of the complex geometry due to its formation process and structural elements distribution over the space. Generally, its formation represents a degree of irregularity, spatial hierarchy, unevenness. But, at different observation scales, cities spatial arrangement represents the important characteristic of fractal, which is self-similarity (a small part of an object is exactly similar to the whole). Therefore, to characterize cities formation process and measure its structural complexity quantitatively, different researchers have introduced the concept of fractal geometry. Fractal geometry is used to explain the hierarchical arrangement of objects, structural self-similarity over different viewing scales, and heterogeneity within it. In order to plan a better city, a detailed study of its formation process is essential. The purpose of this study is to gather existing knowledge about how fractal geometry has been widely used in different domains of urban planning and synthesized existing knowledge. Most of the previous studies have carried out fractal dimension to measure fractal properties within objects. Box-counting method is one of the most common methods to calculate fractal dimension. Different studies concluded that if the fractal dimension of a city increases, the complexity of the city's physical arrangement also increases. Finally, this study will give future scholars with vital information regarding the use of fractal theory in urban planning and may provide profound insight in this area.

Flow of a Self-Similar Non-Newtonian Fluid Using Fractal Dimensions

In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid is depicted as a continuum in a noninteger dimensional space. Involving vector calculus for the instance of a noninteger dimensional space, we determine an analytical solution of the Cauchy equation for the instance of a non-Newtonian self-similar fluid flow in a cylindrical pipe. The plot of the velocity profile obtained shows that the rheological behavior of a non-Newtonian power-law fluid is essentially impacted by its self-similar structure. A self-similar shear thinning fluid and a self-similar Newtonian fluid take on a shear-thickening way of behaving, and a self-similar shear-thickening fluid becomes more shear thickening. This approach has many useful applications in industry, for the investigation of blood flow and fractal fluid hydrology.

Aharonov-Bohm cages, flat bands, and gap labeling in hyperbolic tilings

Aharonov-Bohm caging is a localization mechanism stemming from the competition between the geometry and the magnetic field. Originally described for a tight-binding model in the dice lattice, this destructive interference phenomenon prevents any wavepacket spreading away from a strictly confined region. Accordingly, for the peculiar values of the field responsible for this effect, the energy spectrum consists of a discrete set of highly degenerate flat bands. In the present work, we show that Aharonov-Bohm cages are also found in an infinite set of hyperbolic dice tilings defined on a negatively curved hyperbolic plane. We detail the construction of these tilings and compute their Hofstadter butterflies by considering periodic boundary conditions on high-genus surfaces. As recently observed for some regular hyperbolic tilings, these butterflies do not manifest the self-similar structure of their Euclidean counterparts but still contain some gaps. We also consider the energy spectrum of hyperbolic kagome tilings (which are the dual of hyperbolic dice tilings), which displays interesting features, such as highly degenerate states arising for some particular values of the magnetic field. For these two families of hyperbolic tilings, we compute the Chern number in the main gaps of the Hofstadter butterfly and propose a gap labeling inspired by the Euclidean case. Finally, we also study the triangular Husimi cactus, which is a limiting case in the family of hyperbolic kagome tilings, and we derive an exact expression for its spectrum versus magnetic flux.

The small-world effect of fractal networks modeled on ‘dust-like’ cubes

In this paper, the sequence of evolving networks is generated from some ‘dust-like’ cubes by applying the encoding methods in fractal and symbolic dynamical systems. Based on the self-similar structures of fractals, we study the mean clustering coefficient, the mean geodesic distance and the mean Fermat distance. The relevant results show the small-world effect of our evolving networks.

Enhanced Sparse Low-Rank Representation via Nonconvex Regularization for Rotating Machinery Early Fault Feature Extraction

The weak fault feature extraction is key to early fault diagnosis of rotary machinery. However, the existing sparse low-rank fault feature extraction methods have the deficiency of underestimation and low peak signal-to-noise ratio. To solve these problems, this article presents an enhanced sparse low-rank (ESL) representation approach for weak fault feature extraction. Considering the periodic self-similarity and shift invariance of fault feature, a weighted dual approximation regularization is proposed for noise and fault irrelevant harmonic suppression, which provides a cornerstone for rotating machinery weak fault feature extraction. To be specific, the truncated nuclear norm (TNN) and weighted generalized minimax-concave (WGMC) penalty are leveraged to form the weighted dual approximation regularization so that it can inherit their superior properties. The TNN can capture the periodic self-similarity and shift-invariance structure of fault impulses while restraining the noise; the WGMC penalty can enhance the sparsity, restrain the fault irrelevant harmonics, and overcome the deficiency of underestimating the large amplitude components. Therefore, the proposed model can effectively extract the weak fault feature. The proposed approach is applied to bearing and planet gear fault diagnosis to evaluate its effectiveness. Comparison results show the significant improvements of the proposed ESL method, indicating that it has great potentials in fault diagnosis of rotating machinery.

Numerical modeling of imposed magnetohydrodynamic effects in hypersonic flows

Weakly ionized plasmas, formed in high enthalpy hypersonic flows, can be actively manipulated via imposed magnetic fields—a concept termed magnetohydrodynamic (MHD) flow control. Imposed MHD effects, within flows that exhibit multiple shock interactions, are consequential for emerging aerospace technologies, including the possibility of replacing mechanical control surfaces with magnetic actuation. However, numerical modeling of this flow type remains challenging due to the sensitivity of feature formation and the real gas modeling of weakly ionized, electrically conductive, air plasma. In this work, numerical simulation capabilities have been developed for the study of MHD affected, hypersonic flows, around two-dimensional axisymmetric non-simple geometries. The validated numerical methodology, combined with an advanced 19 species equation of state for air plasma, permits the realistic and efficient simulation of air plasmas in the equilibrium regime. Quantitative agreement is achieved between simulation and experiment for a Mach 5.6 double cone geometry with applied magnetic field. In the context of the magnetic actuation concept, numerical studies are conducted for varied conical surface angle and magnetic field configuration. For simple geometries with an elemental shock type, the MHD enhancement effect produces a self-similar shock structure. This paper demonstrates how, for hypersonic flows with complex shock interactions, the MHD affected flow is not only augmented in terms of shock position but may exhibit topological adaptations in the fundamental flow structure. A classification system is introduced for the emergent flow topologies identified in this work. Fluid-magnetic interactions are explored and explained in terms of the coupled mechanisms leading to (1) differences in magnitude of MHD enhancement effect and (2) structural adaptations of the flow topology. The applied numerical studies examine why increased conical surface angle does not amplify the MHD enhancement effect as expected from the base flow conditions, and the mechanisms by which the magnetic field configuration influences the MHD augmented shock structure. Most critically, classes of conditions are identified that produce topological equivalence between the magnetic interaction effects and a generalized mechanical control surface.

Wheat Flour-Based Edible Films: Effect of Gluten on the Rheological Properties, Structure, and Film Characteristics.

This work investigates the structure, rheological properties, and film performance of wheat flour hydrocolloids and their comparison with that of a wheat starch (WS)–gluten blend system. The incorporation of gluten could decrease inter-chain hydrogen bonding of starch, thereby reducing the viscosity and solid-like behavior of the film-forming solution and improving the frequency-dependence, but reducing the surface smoothness, compactness, water vapor barrier performance, and mechanical properties of the films. However, good compatibility between starch and gluten could improve the density of self-similar structure, the processability of the film-forming solution, and film performance. The films based on wheat flours showed a denser film structure, better mechanical properties, and thermal stability that was no worse than that based on WS–gluten blends. The knowledge gained from this study could provide guidance to the development of other flour-based edible packaging materials, thereby promoting energy conservation and environmental protection.

AVERAGE TRAPPING TIME ON THE 3-DIMENSIONAL 3-LEVEL SIERPINSKI GASKET NETWORK WITH A SET OF TRAP NODES

As a basic dynamic feature on complex networks, the property of random walk has received a lot of attention in recent years. In this paper, we first studied the analytical expression of the mean global first passage time (MGFPT) on the 3-dimensional 3-level Sierpinski gasket network. Based on the self-similar structure of the network, the correlation between the MGFPT and the average trapping time (ATT) is found, and then the analytical expression of the ATT is obtained. Finally, by establishing a joint network model, we further give the standard process of solving the analytical expression of the ATT when there is a set of trap nodes in the network. By illustrating examples and numerical simulations, it can be proved that when the trap node sets are different, the ATT will be quite different, but the super-linear relationship with the number of iterations will not be changed.

Success of social inequality measures in predicting critical or failure points in some models of physical systems

Statistical physicists and social scientists both extensively study some characteristic features of the unequal distributions of energy, cluster, or avalanche sizes and of income, wealth, etc., among the particles (or sites) and population, respectively. While physicists concentrate on the self-similar (fractal) structure (and the characteristic exponents) of the largest (percolating) cluster or avalanche, social scientists study the inequality indices such as Gini and Kolkata, given by the non-linearity of the Lorenz function representing the cumulative fraction of the wealth possessed by different fractions of the population. Here, using results from earlier publications and some new numerical and analytical results, we reviewed how the above-mentioned social inequality indices, when extracted from the unequal distributions of energy (in kinetic exchange models), cluster sizes (in percolation models), or avalanche sizes (in self-organized critical or fiber bundle models) can help in a major way in providing precursor signals for an approaching critical point or imminent failure point. Extensive numerical and some analytical results have been discussed.

Holographic bubbles with Jecco: expanding, collapsing and critical

Cosmological phase transitions can proceed via the nucleation of bubbles that subsequently expand and collide. The resulting gravitational wave spectrum depends crucially on the properties of these bubbles. We extend our previous holographic work on planar bubbles to cylindrical bubbles in a strongly-coupled, non-Abelian, four-dimensional gauge theory. This extension brings about two new physical properties. First, the existence of a critical bubble, which we determine. Second, the bubble profile at late times exhibits a richer self-similar structure, which we verify. These results require a new 3+1 evolution code called Jecco that solves the Einstein equations in the characteristic formulation in asymptotically AdS spaces. Jecco is written in the Julia programming language and is freely available. We present an outline of the code and the tests performed to assess its robustness and performance.

Research on out-of-plane impact characteristics of self-similar gradient hierarchical quasi-honeycomb structure

From the perspective of bionics, this paper proposes a new self-similar hierarchical quasi-honeycomb (SHQ) structure. It specifically analyzes the hierarchy and gradient distribution’s influence on the impact characteristics under different impact loads by using a combination of numerical simulation, theoretical analysis and experimental validation. Research found that the hierarchical design can improve the out-of-plane impact performance of the honeycomb, and a reasonable out-of-plane gradient distribution can greatly reduce the peak crushing force (PCF) of the out-of-plane impact. In addition, based on the simplified super folding element theory, an out-of-plane impact theoretical model of the SHQ was established, which can better predict the mean crushing force (MCF) and energy absorption (EA).

Scaling laws of statistics of wall-bounded turbulence at supercritical pressure: Evaluation and mechanism

A growing body of studies support that the real fluid effects related to the abrupt density changes in supercritical fluids significantly affect statistical properties of turbulence, yet developing appropriate scaling laws for wall-bounded turbulence at supercritical state is still difficult. In the present study, we conduct direct numerical simulations on channel flows of supercritical fluids to evaluate the usefulness of classical scaling developed for variable-property flows. We find that the expressions based on semi-local scaling [ϕ=f(y*,Reτ*) and ϕ=f(y*,Reτ*,Pr*)] fail to collapse the statistical profiles at supercritical pressure. We analyze the mechanism of the failure of semi-local scaling by quantifying the modulations of turbulent structures of supercritical fluids due to changes in fluid properties. The intensified ejection and sweep of low-speed streaks destabilize the stream-wise streaks and reduce the stream-wise coherence, changing the statistics and affecting the usefulness of semi-local scaling. To shed light on the scaling laws of fluctuating velocities, we finally examine the hypotheses in Townsend wall-attached eddy theory in the context of flows at a supercritical state. It is found that the attached eddies are self-similar near-wall structures, which result in the logarithmic profiles of stream-wise and span-wise velocity fluctuations; the population density of the attached eddies can be well approximated by an exponential scaling.

On self-similar chemical structures, once again

On self-similar chemical structures, once again

Reduced cerebral vascular fractal dimension among asymptomatic individuals as a potential biomarker for cerebral small vessel disease

Cerebral small vessel disease is a neurological disease frequently found in the elderly and detected on neuroimaging, often as an incidental finding. White matter hyperintensity is one of the most commonly reported neuroimaging markers of CSVD and is linked with an increased risk of future stroke and vascular dementia. Recent attention has focused on the search of CSVD biomarkers. The objective of this study is to explore the potential of fractal dimension as a vascular neuroimaging marker in asymptomatic CSVD with low WMH burden. Df is an index that measures the complexity of a self-similar and irregular structure such as circle of Willis and its tributaries. This exploratory cross-sectional study involved 22 neurologically asymptomatic adult subjects (42 ± 12 years old; 68% female) with low to moderate 10-year cardiovascular disease risk prediction score (QRISK2 score) who underwent magnetic resonance imaging/angiography (MRI/MRA) brain scan. Based on the MRI findings, subjects were divided into two groups: subjects with low WMH burden and no WMH burden, (WMH+; n = 8) and (WMH−; n = 14) respectively. Maximum intensity projection image was constructed from the 3D time-of-flight (TOF) MRA. The complexity of the CoW and its tributaries observed in the MIP image was characterised using Df. The Df of the CoW and its tributaries, i.e., Df (w) was significantly lower in the WMH+ group (1.5172 ± 0.0248) as compared to WMH− (1.5653 ± 0.0304, p = 0.001). There was a significant inverse relationship between the QRISK2 risk score and Df (w), (rs = − .656, p = 0.001). Df (w) is a promising, non-invasive vascular neuroimaging marker for asymptomatic CSVD with WMH. Further study with multi-centre and long-term follow-up is warranted to explore its potential as a biomarker in CSVD and correlation with clinical sequalae of CSVD.