Geometrical Exponents Research Articles

Kosambi-Cartan-Chern perspective on chaos: Unveiling hidden attractors in nonlinear autonomous systems.

This article confronts the formidable task of exploring chaos within hidden attractors in nonlinear three-dimensional autonomous systems, highlighting the lack of established analytical and numerical methodologies for such investigations. As the basin of attraction does not touch the unstable manifold, there are no straightforward numerical processes to detect those attractors and one has to implement special numerical and analytical strategies. In this article we present an alternative approach that allows us to predict the basin of attraction associated with hidden attractors, overcoming the existing limitations. The method discussed here is based on the Kosambi-Cartan-Chern theory which enables us to conduct a comprehensive theoretical analysis by means of evaluating geometric invariants and instability exponents, thereby delineating the regions encompassing chaotic and periodic zones. Our analytical predictions are thoroughly validated by numerical results.

Study on rheological, adsorption and hydration properties of cement slurries incorporated with EPEG-based polycarboxylate superplasticizers

A series of polycarboxylate superplasticizers (PCEs) with different side-chain densities, main chain polymerization degrees, and side-chain lengths were designed and synthesized using a novel highly active ethylene glycol mono vinyl ether polyethylene glycol as the ether monomer. The influence of polycarboxylate ether on the rheological properties, interface adsorption, and hydration characteristics in cement paste was investigated through characterization of charge density, rheological properties, adsorption behavior, and hydration heat. The results indicate that the adsorption process of PCE on cement particles is spontaneous physical adsorption, and the hydration kinetics fitting reveals that the geometric crystal growth exponent n is in the range of 1–2, with needle-like and lamellar hydration products formed. With a decrease in side-chain density, the broadening of molecular weight distribution and the increase of charge density accelerate the flow of slurry, reduces saturation adsorption, and delays cement hydration. A decrease in main chain polymerization degree results in a downward trend in molecular weight and charge density, leading to larger molecular conformations, reduced slurry flow, decreased saturation adsorption, and delayed cement hydration. As the side-chain length of PCE (molecular weight) increases, the charge density decreases, and the molecular conformation exhibits a compact structure with reduced slurry flow, decreased saturation adsorption, and delayed cement hydration. In cases of low side-chain density, short side chains, and low molecular weight, enhanced adsorption capacity and faster adsorption rates are observed, resulting in the lower viscosity and a delay in the cement hydration process.

Geometrical exponents of contour loops on ballistic deposition model with power-law distributed noise

In this paper, we study geometrical properties of contour loops to characterize the morphology of the porous ballistic deposition model with power-law distributed noise. In this model, rod-like particles with variable lengths instead of homogeneous particles are deposited, the length of each rod, l, is determined by a power-law distribution, P(l)∼l−(μ+1), and μ indicates the power-law strength exponent. The accumulation of rods with different lengths leads to a porous structure. The porous structure is converted into contour loops by using the Hoshen–Kopelman algorithm and the complexity and fractal features of the loops are investigated. On the other hand, the distributions of loops indicating the porosity distribution of the growth rough surfaces are investigated. The fractal dimension of the contour set, d, the fractal dimension of each loop, Df, cumulative distributions exponent of areas, ξ, and the perimeter distribution exponent, τ, are calculated for the investigated porous structures. Our results show that enhancement of μ exponent and appearance of the Gaussian ballistic deposition model leads to reduction of structure porosity and enhancement of contour loops area and perimeter. The fractal dimension of the contour set, and the fractal dimension of each loop increases by enhancement of μ exponent. Distributions of loops and geometrical exponents of contour loops in different μ exponent are determined. The results indicate that the hyperscaling relation and Zipf’s law hold for μ≥μc=3 which power-law distributed noise approaches to Gaussian one.

Connecting Complex Electronic Pattern Formation to Critical Exponents

Scanning probes reveal complex, inhomogeneous patterns on the surface of many condensed matter systems. In some cases, the patterns form self-similar, fractal geometric clusters. In this paper, we advance the theory of criticality as it pertains to those geometric clusters (defined as connected sets of nearest-neighbor aligned spins) in the context of Ising models. We show how data from surface probes can be used to distinguish whether electronic patterns observed at the surface of a material are confined to the surface, or whether the patterns originate in the bulk. Whereas thermodynamic critical exponents are derived from the behavior of Fortuin–Kasteleyn (FK) clusters, critical exponents can be similarly defined for geometric clusters. We find that these geometric critical exponents are not only distinct numerically from the thermodynamic and uncorrelated percolation exponents, but that they separately satisfy scaling relations at the critical fixed points discussed in the text. We furthermore find that the two-dimensional (2D) cross-sections of geometric clusters in the three-dimensional (3D) Ising model display critical scaling behavior at the bulk phase transition temperature. In particular, we show that when considered on a 2D slice of a 3D system, the pair connectivity function familiar from percolation theory displays more robust critical behavior than the spin-spin correlation function, and we calculate the corresponding critical exponent. We discuss the implications of these two distinct length scales in Ising models. We also calculate the pair connectivity exponent in the clean 2D case. These results extend the theory of geometric criticality in the clean Ising universality classes, and facilitate the broad application of geometric cluster analysis techniques to maximize the information that can be extracted from scanning image probe data in condensed matter systems.

Inferring the Subsurface Geometry and Strength of Slow‐Moving Landslides Using 3‐D Velocity Measurements From the NASA/JPL UAVSAR

AbstractThe hazardous impact and erosive potential of slow‐moving landslides depends on landslide properties including velocity, size, and frequency of occurrence. However, constraints on size, in particular, subsurface geometry, are lacking because these types of landslides rarely fully evacuate material to create measurable hillslope scars. Here, we use pixel offset tracking with data from the NASA/JPL Uninhabited Aerial Vehicle Synthetic Aperture Radar to measure the three‐dimensional surface deformation of 134 slow‐moving landslides in the northern California Coast Ranges. We apply volume conservation to infer the actively deforming thickness, volume, geometric scaling, and frictional strength of each landslide. These landslides move at average rates between ∼0.1–2 m/yr and have active areas of ∼6.10 × 103–2.35 × 106 m2, inferred mean thicknesses of ∼1.1–25 m, and volumes of ∼7.01 × 103–9.75 × 106 m3. The best fit volume‐area geometric scaling exponent is γ ∼ 1.2–1.5, indicating that these landslides fall between typical soil and bedrock landslide scaling. A rollover in the scaling relationship suggests that the largest landslide complexes in our data set become large primarily by increasing in area rather than thickness. In addition, the slow‐moving landslides display scale‐dependent frictional strength, such that large landslide tend to be weaker than small landslides. This decrease in frictional strength with landslide size is likely because larger landslides are composed of higher proportions of weak material. Our work shows how state of the art remote sensing techniques can be used to better understand landslide processes and quantify their contribution to landscape evolution and hazards to human safety.

On topological properties of hierarchical hypercube network based on Ve and Ev degree

Abstract Grid implementation is a principal unit in electrical and electronic engineering but it depends on the domain of these projects. For example, depending on the grid and the signal processing in that fields of electronic and electrical engineering, such as more abstract mathematics in signal conversion and e-transmission theory griding, etc. Provides transmission through grid nodes. Graph theory is very useful in research fields. As topological indices, there are more actual numbers associated with chemical composition complaints connected to the chemical grid with physical and chemical properties and reactions. In this paper, we expand the work to interconnected grid and examine the first Zagreb, the second Zagreb, Randic, sum-connectivity, harmonic, geometric, and atom bond connectivity exponents of hierarchical hypercube network based on vertex-edge and edge-vertex degree.

Two-dimensional super-roughening in the three-dimensional Ising model.

We present a random-interface representation of the three-dimensional (3D) Ising model based on thermal fluctuations of a uniquely defined geometric spin cluster in the 3D model and its 2D cross section. Extensive simulations have been carried out to measure the global interfacial width as a function of temperature for different lattice sizes which is shown to signal the criticality of the model at T_{c} by forming a size-independent cusp in 3D, along with an emergent super-roughening at its 2D cross section. We find that the super-rough state is accompanied by an intrinsic anomalous scaling behavior in the local properties characterized by a set of geometric exponents which are the same as those for a pure 2D Ising model.

An electronic avalanche model for metal–insulator transition in two dimensional electron gas

In this paper, we present an electronic avalanche model for the transport of electrons in the disordered two-dimensional (2D) electron gas which has the potential to describe the 2D metal–insulator transition (MIT) in the zero electron–electron interaction limit. The disorder is considered to be uncorrelated-Coulomb noise with a uniform distribution. In this model we sub-divide the system to some virtual cells each of which has a linear size of the order of phase coherence length of the system. Using Thomas-Fermi-Dirac theory we propose some simple energy functions for the cells and using the thermodynamics of 2DEG we develop some rules for the charge transfer between the cells. A second order transition line arises from our model with some similarities with the experiments. The compressibility of the system also diverges on this line. We characterize this (disorder-driven) phase transition which is between the non-percolating phase and the percolating phase (in which the system shows metallic behavior) and obtain some geometrical critical exponents. The fractal dimension of the exterior frontier of the electronic avalanches on the transition line is compatible with the percolation theory, whereas the other exponents are different. The exponents are robust against disorder in the low disordered 2DEGs and change considerably in the high disordered ones.

Geometric properties of the Fortuin-Kasteleyn representation of the Ising model.

We present a Monte Carlo study of the geometric properties of Fortuin-Kasteleyn (FK) clusters of the Ising model on square [two-dimensional (2D)] and simple-cubic [three-dimensional (3D)] lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio and yields a high-precision critical coupling as K_{c}(3D)=0.221654631(8). We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size C_{1} follows a single-variable function as P(C_{1},L)dC_{1}=P[over ̃](x)dx with x≡C_{1}/L^{d_{F}} (L is the linear size), where the fractal dimension d_{F} is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution P[over ̃](x) in three dimensions, and attributed to the different approaching behaviors for K→K_{c}+0^{±}. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as d_{min}(3D)=1.25936(12) and d_{min}(2D)=1.0940(2). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as d_{B}(3D)=2.1673(15) and d_{B}(2D)=1.7321(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents d_{min} and d_{B} either improve over the existing results or have not been reported yet. We believe that these numerical results would provide a testing ground in the development of further theoretical treatments of the 3D Ising model.

Iso-height lines of multifractal etched silicon rough surfaces

Morphology of etched silicon wafers during anisotropic etching process is investigated utilizing the geometrical exponents of iso-height contour lines. The images obtained from field emission scanning electron microscopy show pyramidal microstructures which disappear in large time scales and porous silicon structures are formed. The morphology of these microstructures is investigated utilizing the data obtained from atomic force microscopy and contour loop analysis. The results show that the anisotropic etching breaks the up-down symmetry in the height fluctuations of etched silicon surfaces. Area of the contour loops increases by enhancement of pyramids sizes. Fractal dimensions of the loops decrease at the beginning of etching process. Distributions of loops and geometrical exponents of iso-height lines in different etching times are determined. The results indicate that in spite of asymmetry in the height fluctuations, the hyperscaling relation and Zipf's law are verified for the multi-affine silicon surfaces during etching process.

Field verification of analytical at-a-station hydraulic-geometry relations

We used data from 109 surveyed cross sections at 34 reaches of New Zealand rivers ranging in bankfull width from 8 to 234 m, each with 4–13 measured flows, to compare regression-derived at-a-station hydraulic geometry (AHG) coefficients and exponents to the coefficients and exponents predicted by Dingman’s (2007) analytical AHG relations. The analytical geometrical relations depend on measured values of 1) bankfull width, 2) bankfull maximum depth, and 3) a geometric exponent fit to the measured cross section. Because of data limitations and the lack of a generally applicable analytical hydraulic relation, the hydraulic relation is here represented by an empirical power-law relation between depth and velocity developed at each section. The fits between the quasi-analytical and empirical values of the AHG coefficients and exponents are highly statistically significant, with intercepts near 0 and slopes near 1, validating Dingman’s original analytical relations. This confirmation of the physical basis underlying empirically-observed AHG relations, along with some statistically-significant relations among the quantities that determine AHG coefficients and exponents that are not predicted by theory, may be useful for estimating AHG relations from remotely-sensed or partial information.

Near-source attenuation of high-frequency body waves beneath the New Madrid Seismic Zone

Attenuation characteristics in the New Madrid Seismic Zone (NMSZ) are estimated from 157 local seismograph recordings out of 46 earthquakes of 2.6 ≤ M ≤ 4.1 with hypocentral distances up to 60 km and focal depths down to 25 km. Digital waveform seismograms were obtained from local earthquakes in the NMSZ recorded by the Center for Earthquake Research and Information (CERI) at the University of Memphis. Using the coda normalization method, we tried to determine Q values and geometrical spreading exponents at 13 center frequencies. The scatter of the data and trade-off between the geometrical spreading and the quality factor did not allow us to simultaneously derive both these parameters from inversion. Assuming 1/R 1.0 as the geometrical spreading function in the NMSZ, the Q P and Q S estimates increase with increasing frequency from 354 and 426 at 4 Hz to 729 and 1091 at 24 Hz, respectively. Fitting a power law equation to the Q estimates, we found the attenuation models for the P waves and S waves in the frequency range of 4 to 24 Hz as Q P = (115.80 ± 1.36) f (0.495 ± 0.129) and Q S = (161.34 ± 1.73) f (0.613 ± 0.067), respectively. We did not consider Q estimates from the coda normalization method for frequencies less than 4 Hz in the regression analysis since the decay of coda amplitude was not observed at most bandpass filtered seismograms for these frequencies. Q S/Q P > 1, for 4 ≤ f ≤ 24 Hz as well as strong intrinsic attenuation, suggest that the crust beneath the NMSZ is partially fluid-saturated. Further, high scattering attenuation indicates the presence of a high level of small-scale heterogeneities inside the crust in this region.

Bak-Tang-Wiesenfeld model in the upper critical dimension: Induced criticality in lower-dimensional subsystems.

We present extensive numerical simulations of Bak-Tang-Wiesenfeld (BTW) sandpile model on the hypercubic lattice in the upper critical dimension D_{u}=4. After re-extracting the critical exponents of avalanches, we concentrate on the three- and two-dimensional (2D) cross sections seeking for the induced criticality which are reflected in the geometrical and local exponents. Various features of finite-size scaling (FSS) theory have been tested and confirmed for all dimensions. The hyperscaling relations between the exponents of the distribution functions and the fractal dimensions are shown to be valid for all dimensions. We found that the exponent of the distribution function of avalanche mass is the same for the d-dimensional cross sections and the d-dimensional BTW model for d=2 and 3. The geometrical quantities, however, have completely different behaviors with respect to the same-dimensional BTW model. By analyzing the FSS theory for the geometrical exponents of the two-dimensional cross sections, we propose that the 2D induced models have degrees of similarity with the Gaussian free field (GFF). Although some local exponents are slightly different, this similarity is excellent for the fractal dimensions. The most important one showing this feature is the fractal dimension of loops d_{f}, which is found to be 1.50±0.02≈3/2=d_{f}^{GFF}.

Attenuation of Lg waves in the New Madrid seismic zone of the central United States using the coda normalization method

Unique properties of coda waves are employed to evaluate the frequency dependent quality factor of Lg waves using the coda normalization method in the New Madrid seismic zone of the central United States. Instrument and site responses are eliminated and source functions are isolated to construct the inversion problem. For this purpose, we used 121 seismograms from 37 events with moment magnitudes, M, ranging from 2.5 to 5.2 and hypocentral distances from 120 to 440km recorded by 11 broadband stations.A singular value decomposition (SVD) algorithm is used to extract Q values from the data, while the geometric spreading exponent is assumed to be a constant. Inversion results are then fitted with a power law equation from 3 to 12Hz to derive the frequency dependent quality factor function. The final results of the analysis are QVLg (f)=(410±38) f0.49±0.05 for the vertical component and QHLg (f)=(390±26) f0.56±0.04 for the horizontal component, where the term after ± sign represents one standard error. For stations within the Mississippi embayment with an average sediment depth of 1 km around the Memphis metropolitan area, estimation of quality factor using the coda normalization method is not well-constrained at low frequencies (f<3Hz). There may be several reasons contributing to this issue, such as low frequency surface wave contamination, site effects, or even a change in coda wave scattering regime which can exacerbate the scatter of the data.

Roughening transition and universality of single step growth models in (2+1)-dimensions

We study (2+1)-dimensional single step model for crystal growth including both deposition and evaporation processes parametrized by a single control parameter p. Using extensive numerical simulations with a relatively high statistics, we estimate various interface exponents such as roughness, growth and dynamic exponents as well as various geometric and distribution exponents of height clusters and their boundaries (or iso-height lines) as function of p. We find that, in contrary to the general belief, there exists a critical value at which the model undergoes a roughening transition from a rough phase with in the Kardar–Parisi–Zhang universality to a smooth phase with , asymptotically in the Edwards–Wilkinson class. We validate our conclusion by estimating the effective roughness exponents and their extrapolation to the infinite-size limit.

Criticality and turbulence in a resistive magnetohydrodynamic current sheet

Scaling properties of a two-dimensional (2d) plasma physical current-sheet simulation model involving a full set of magnetohydrodynamic (MHD) equations with current-dependent resistivity are investigated. The current sheet supports a spatial magnetic field reversal that is forced through loading of magnetic flux containing plasma at boundaries of the simulation domain. A balance is reached between loading and annihilation of the magnetic flux through reconnection at the current sheet; the transport of magnetic flux from boundaries to current sheet is realized in the form of spatiotemporal avalanches exhibiting power-law statistics of lifetimes and sizes. We identify this dynamics as self-organized criticality (SOC) by verifying an extended set of scaling laws related to both global and local properties of the current sheet (critical susceptibility, finite-size scaling of probability distributions, geometric exponents). The critical exponents obtained from this analysis suggest that the model operates in a slowly driven SOC state similar to the mean-field state of the directed stochastic sandpile model. We also investigate multiscale correlations in the velocity field and find them numerically indistinguishable from certain intermittent turbulence (IT) theories. The results provide clues on physical conditions for SOC behavior in a broad class of plasma systems with propagating instabilities, and suggest that SOC and IT may coexist in driven current sheets which occur ubiquitously in astrophysical and space plasmas.

Point to point continuum percolation in two dimensions

The outcome of the classic percolation approach is several power-law curves with some universal (critical) exponents. Here, the universality means that these power laws as well as their critical exponents, which control the global properties of a system, are independent of its details. Classic percolation considers the connectivity between two lines and two faces at opposite sides of a system in 2- and 3D problems, respectively; whereas, in practice (e.g. hydrocarbon formations), production and injection wells are represented by points (in 2D areal models) and lines (in 3D models). This study presents the results of Monte Carlo simulations of a 2D percolation model wherein the connection locations (i.e. wells) are represented by two points, called point-to-point (P2P) connectivity. The main contribution is to find the percolation threshold as well as the geometrical and dynamical critical exponents of a continuum percolation system with a P2P connection, which is closer to reality in some applications. The result shows that in comparison to classical percolation, some critical exponents definitely changes in the P2P connection.

Observation of Schramm–Loewner evolution on the geometrical clusters of the Ising model

Schramm–Loewner Evolution (SLE) is a stochastic process that, by focusing on the geometrical features of the two-dimensional (2D) conformal invariant models, classifies them using one real parameter κ. In this work we apply the SLE formalism to the exterior frontiers of the geometrical clusters (interfaces) of the two-dimensional critical Ising model on the triangular lattice. We first analyze the critical curves going from the real axis to the real axis in the upper half plane geometry and show numerically that SLE(κ, κ − 6) works well to extract the diffusivity parameter κ. We then analyze the conformal loops of the critical Ising model. After determining some geometrical exponents of the critical loops as the interfaces of the model in hand, we address the problem of application of SLE to conformal loops. We numerically show that SLE(κ, κ − 6) is more reliable than previous methods.

Dynamical and geometrical exponents of self-affine rough surfaces on regular and random lattices

In this paper, a wide range of rough surfaces, such as random deposition (RD) and incorporating with relaxation (RDR), ballistic deposition (BD) and restricted solid-on-solid model (RSOS), in (2 + 1)-dimension, on the regular (square, triangular, honeycomb) and random (Voronoi) lattices is simulated. We numerically show that the dynamical (growth and roughness) and geometrical (fractal dimension, loop correlation and the length distribution) exponents of these rough surfaces are independent of the underlying regular or irregular lattices. Also the universality holds at the level of statistical properties of the iso-height lines (contours) on different lattices. Finally, we indicate that the hyperscaling relations are valid for the contours of all the studied Gaussian and non-Gaussian self-affine rough surfaces.

Geometric critical exponents in classical and quantum phase transitions.

We define geometric critical exponents for systems that undergo continuous second-order classical and quantum phase transitions. These relate scalar quantities on the information theoretic parameter manifolds of such systems, near criticality. We calculate these exponents by approximating the metric and thereby solving geodesic equations analytically, near curvature singularities of two-dimensional parameter manifolds. The critical exponents are seen to be the same for both classical and quantum systems that we consider, and we provide evidence about the possible universality of our results.