Power-law Phase Research Articles

Heavy-tailed distributions of confirmed COVID-19 cases and deaths in spatiotemporal space.

This paper conducts a systematic statistical analysis of the characteristics of the geographical empirical distributions for the numbers of both cumulative and daily confirmed COVID-19 cases and deaths at county, city, and state levels over a time span from January 2020 to June 2022. The mathematical heavy-tailed distributions can be used for fitting the empirical distributions observed in different temporal stages and geographical scales. The estimations of the shape parameter of the tail distributions using the Generalized Pareto Distribution also support the observations of the heavy-tailed distributions. According to the characteristics of the heavy-tailed distributions, the evolution course of the geographical empirical distributions can be divided into three distinct phases, namely the power-law phase, the lognormal phase I, and the lognormal phase II. These three phases could serve as an indicator of the severity degree of the COVID-19 pandemic within an area. The empirical results suggest important intrinsic dynamics of a human infectious virus spread in the human interconnected physical complex network. The findings extend previous empirical studies and could provide more strict constraints for current mathematical and physical modeling studies, such as the SIR model and its variants based on the theory of complex networks.

Size Distributions Reveal Regime Transition of Lake Systems Under Different Dominant Driving Forces

AbstractPower law size distribution is found to associate with fractal, self‐organized behaviors and patterns of complex systems. Such distribution also emerges from natural lakes, with potentially important links to the dynamics of lake systems. But the driving mechanism that generates and shapes this feature in lake systems remains unclear. Moreover, the power law itself was found inadequate for fully describing the size distribution of lakes, due to deviations at the two ends of size range. Based on observed and simulated lakes in China's 11 hydro‐climatic zones, we established a conceptual model for lake systems, which covers the whole size range of lake size distribution and reveals the underlying driving mechanism. The full lake size distribution is composed of three components, with three phases featured by exponential, stretched‐exponential and power law distribution. The three phases represent system states with successively increasing degrees of heterogeneity and orderliness, and more importantly, indicate the dominance of exogenic and endogenic forces, respectively. As the dominant driving force changes from endogenic to exogenic, a phase transition occurs with lake size distribution shifted from power law to stretched‐exponential and further to exponential distribution. Apart from compressing the power law phase, exogenic force also increases its scaling exponent, driving the corresponding lake size power spectrum into the regime of “blue noise.” During this process, the autocorrelation function of the lake system diverges with a possibility of going to infinity, indicating the loss of system resilience.

Holographic realization from inflation to reheating in generalized entropic cosmology

The growing cosmological interest of different entropy functions (like the Tsallis entropy, the Rényi entropy, the Barrow entropy, the Sharma-Mittal entropy, the Kaniadakis entropy and the Loop Quantum gravity entropy) naturally raises an important question: ”Does there exist a generalized entropy that can bring all the known entropies proposed so far within a single umbrella?” In spirit of this, recently a four parameter generalized entropy has been formulated that reduces to different known entropies for suitable limits of the parameters. Based on such four parameter generalized entropy (symbolized by Sg), in the present paper, we examine the universe’s evolution during its early phase, particularly from inflation to reheating, in the context of entropic cosmology where the entropic energy density acts as the inflaton. It turns out that the entropic energy successfully drives an early inflationary phase with a graceful exit, and moreover, the theoretical expectations of the observable indices get consistent with the recent Planck data for suitable ranges of the entropic parameters. After the inflation ends, the universe enters to a reheating stage when the entropic energy decays to relativistic particles with a certain decay rate. Actually the presence of the entropic parameters in the Sg ensures a continuous evolution of the Hubble parameter from a quasi de-Sitter phase during the inflation to a power law phase during the reheating stage dominated by a constant EoS parameter. Consequently we investigate the reheating phenomenology, and scan the entropic parameters from both the inflation and reheating requirements. We further address the possibility of instantaneous reheating in the present context of generalized entropy.

Statistical Uncertainty in the Frequency Dependence of the Intensity Scintillation Index (S4)

AbstractThe intensity scintillation index (S4) exhibits a linear frequency dependence derived from a power law phase screen model, with the exponent determined by the phase spectral index (p). However, it is well known that a departure from linearity occurs as S4 increases and saturates under strong scatter conditions. Additionally, statistical errors over a finite time window can also cause the scintillation statistic to deviate from the power law dependence on frequency. This paper presents a parametric analysis on the deviation in the S4 frequency dependence from the power law form. This investigation focuses on the contribution of statistical uncertainty in S4 calculations to the deviation in the frequency dependence according to the selected parameters: S4 and Fresnel frequency (fF). The deviation is determined by comparing p estimates calculated from the power law relationship of S4 at different L‐band frequencies with those inferred from a model‐fitting method using a multi‐frequency global navigation satellite system scintillation data set. The results show the well‐known departure from linearity as S4 increases, and reveal increasing deviations with decreasing fF. The spectral analysis indicates that the variability at the lowest frequencies of intensity spectra contributes to the statistical errors in S4 calculations due to the finite time window, particularly when scintillations are dominated by low‐frequency contributions at low fF. Simulation results of scintillation realizations show much lower deviations with a 30‐min compared to 1‐min window. This suggests a 1‐min window can be too short for reliable S4 calculations due to statistical uncertainty, especially at low fF.

ГЕНЕРАЦИЯ ВНЕОСЕВЫХ ОПТИЧЕСКИХ ЛОВУШЕК НА ОСНОВЕ ПУЧКОВ СО СТЕПЕННОЙ ФАЗОВОЙ ЗАВИСИМОСТЬЮ ОТ РАДИУСА

The paper considers laser beams with a power-law phase dependence on the radius, which make it possible to form optical traps for capturing particles and molecules using a set of scattering and collecting axicons, which are complemented by a focusing lens, as well as a phase element with a deflecting wavefront in the case of off-axis trap formation. The possibility of forming off-axis optical traps of different sizes with the possibility of changing their position and using their combinations is shown.

A two-phase model of collective memory decay with a dynamical switching point

Public memories of significant events shared within societies and groups have been conceptualized and studied as collective memory since the 1920s. Thanks to the recent advancement in digitization of public-domain knowledge and online user behaviors, collective memory has now become a subject of rigorous quantitative investigation using large-scale empirical data. Earlier studies, however, typically considered only one dynamical process applied to data obtained in just one specific event category. Here we propose a two-phase mathematical model of collective memory decay that combines exponential and power-law phases, which represent fast (linear) and slow (nonlinear) decay dynamics, respectively. We applied the proposed model to the Wikipedia page view data for articles on significant events in five categories: earthquakes, deaths of notable persons, aviation accidents, mass murder incidents, and terrorist attacks. Results showed that the proposed two-phase model compared favorably with other existing models of collective memory decay in most of the event categories. The estimated model parameters were found to be similar across all the event categories. The proposed model also allowed for detection of a dynamical switching point when the dominant decay dynamics exhibit a phase shift from exponential to power-law. Such decay phase shifts typically occurred about 10 to 11 days after the peak in all of the five event categories.

Robust features of off-axis gamma-ray burst afterglow light curves

ABSTRACT The ultra-relativistic outflows powering gamma-ray bursts (GRBs) acquire angular structure through their interaction with external material. They are often characterized by a compact, nearly uniform narrow core (with half-opening angle θc,{ϵ, Γ}) surrounded by material with energy per unit solid angle ($\epsilon =\epsilon _{\rm c}\Theta _{\epsilon }^{-a}$, where $\Theta _{\lbrace \epsilon ,\Gamma \rbrace }=[1+\theta ^2/\theta _{{\rm c},\lbrace \epsilon ,\Gamma \rbrace }^2]^{1/2}$) and initial specific kinetic energy ($\Gamma _0-1=[\Gamma _{\rm c}-1]\Theta _\Gamma ^{-b}$) declining as power laws. Multiwavelength afterglow light curves of off-axis jets (with viewing angle θobs > θc) offer robust ways to constrain a, b, and the external density radial profile (ρ ∝ R−k), even while other burst parameters may remain highly degenerate. We extend our previous work on such afterglows to include more realistic angular structure profiles derived from three-dimensional hydrodynamic simulations of both long and short GRBs (addressing also jets with shallow angular energy profiles, whose emission exhibits unique evolution). We present afterglow light curves based on our parametrized power-law jet angular profiles for different viewing angles θobs and k = {0, 1, 2}. We identify a unique evolutionary power-law phase of the characteristic synchrotron frequencies (νm and νc) that manifests when the light curve is dominated by emission sensitive to the angular structure of the outflow. We calculate the criterion for obtaining single or double peaked light curves in the general case when θc,Γ ≠ θc,ϵ. We emphasize how the shape of the light curve and the temporal evolution of νm and νc can be used to constrain the outflow structure and potentially distinguish between magnetic and hydrodynamic jets.

De Sitter fractional quantum cosmology

We employ Riesz's fractional derivative into the Wheeler--DeWitt equation for a closed de Sitter geometry and obtain the no-boundary and tunneling wavefunctions. From the corresponding probability distributions, the event horizon of the nucleated universe can be a fractal surface with dimensions between $2\leq D<3$. Concretely, the tunneling wavefunction favors fractal dimensions less than $2.5$ and an accelerated power-law phase. Differently, the no-boundary proposal conveys fractal dimensions close to $3$, with the universe instead entering a decelerated phase. Subsequently, we extend our discussion towards (non-trivial compact) flat and open scenarios. Results suggest that given the probability of creation of a closed inflationary universe in the tunneling proposal is exponentially suppressed, a flat or an open universe becomes favored within fractional inflationary quantum universe.

Free-space realization of tunable pin-like optical vortex beams

We demonstrate, both analytically and experimentally, free-space pin-like optical vortex beams (POVBs). Such angular-momentum-carrying beams feature tunable peak intensity and undergo robust antidiffracting propagation, realized by judiciously modulating both the amplitude and the phase profile of a standard laser beam. Specifically, they are generated by superimposing a radially symmetric power-law phase on a helical phase structure, which allows the inclusion of an orbital angular momentum term to the POVBs. During propagation in free space, these POVBs initially exhibit autofocusing dynamics, and subsequently their amplitude patterns morph into a high-order Bessel-like profile characterized by a hollow core and an annular main lobe with a constant or tunable width during propagation. In contrast with numerous previous endeavors on Bessel beams, our work represents the first demonstration of long-distance free-space generation of optical vortex “pins” with their peak intensity evolution controlled by the impressed amplitude structure. Both the Poynting vectors and the optical radiation forces associated with these beams are also numerically analyzed, revealing novel properties that may be useful for a wide range of applications.

Lattice Boltzmann Method Simulation of Power-Law Phase Change Materials’ Solid–Liquid Phase Change

Solid–liquid phase change has been widely used in practice, but most of the existing literature focuses on Newtonian phase change materials, and there are few studies on power-law phase change materials (PLPCM). To study the heat transfer problem of PLPCM in the process of solid–liquid phase change, this paper combines the enthalpy method with Lattice Boltzmann method to study the melting process of PLPCM, and the effect of the power-law index is analyzed. The results show that, under the same conditions, the increase of the power-law index will reduce the natural convective heat transfer strength, which leads to a slower average temperature rise rate and a slower melting rate. Besides, the effect of nonlinear wall temperature on the phase change process is also investigated. The results show that the melting efficiency is higher for the enhanced temperature distribution, and the smaller the power-law index is, the higher the efficiency.

Differences in power law growth over time and indicators of COVID-19 pandemic progression worldwide

Error analysis and data visualization of positive COVID-19 cases in 27 countries have been performed up to August 8, 2020. This survey generally observes a progression from early exponential growth transitioning to an intermediate power-law growth phase, as recently suggested by Ziff and Ziff. The occurrence of logistic growth after the power-law phase with lockdowns or social distancing may be described as an effect of avoidance. A visualization of the power-law growth exponent over short time windows is qualitatively similar to the Bhatia visualization for pandemic progression. Visualizations like these can indicate the onset of second waves and may influence social policy.

COVID-19 Pandemic: Power Law Spread and Flattening of the Curve.

In this paper, we analyze the real-time infection data of COVID-19 epidemic for nine nations. Our analysis is up to May 04, 2020. South Korea, China, Italy, France, Spain, and Germany have either flattened or close to flattening their epidemic curves. USA and Japan have transitioned to a linear regime, while India is still in a power-law phase. We argue that the transition from an exponential regime to a succession of power-law regimes is a good indicator for flattening of the epidemic curve. We also argue that lockdowns, long-term community transmission, and the transmission by asymptomatic carriers traveling long distances may be inducing the power-law growth of the epidemic.

A two‐parameter multifrequency GPS signal simulator for strong equatorial ionospheric scintillation: modeling and parameter characterization

<h3>Abstract</h3> This paper presents a physics-based, multifrequency, strong scintillation simulator that requires only two input parameters: the <i>S</i>₄ index and decorrelation time <i>τ</i>₀. This simulator is developed based on the two-component power-law phase screen model, which is specified by five parameters and suitable for representation of strong equatorial scintillation. By setting three of the model parameters to representative values, numerical mappings from the user input parameter set (<i>S</i>₄ and <i>τ</i>₀) to the remaining model parameter subset of two parameters can be established through numerical evaluation. The numerical mappings are then used to drive the scintillation model to generate time series of scintillation amplitude and phase. A large group of strong scintillation data (with <i>S</i>₄ &gt; 0.6) from two equatorial sites are processed to obtain the representative values of the three model default parameters and to validate the numerical parameter mappings. A MATLAB implementation of the scintillation simulator has been made available.

Measuring the Power Law Phase Noise of an RF Oscillator with a Novel Indirect Quantitative Scheme

In conventional phase noise metrology, the phase noise of an oscillator is measured by instruments equipped with specialized and sophisticated devices. Such hardware-based testing usually requires high-performance and costly apparatuses. In this paper, we carried out a novel phase noise measurement method based on a mathematical model. The relationship between the phase noise of a radio frequency oscillator and its power spectral density (PSD) was established, different components of the power law phase noise were analyzed in the frequency domain with their characteristic parameters. Based on the complete physical model of an oscillator, we fitted and extracted the parameters for the near-carrier Gaussian and the power law PSD with Levenberg-Marquardt optimization algorithm. The fitted parameters were used to restore the power law phase noise with considerable precision. Experimental validation showed an excellent agreement between the estimation from the proposed method and the data measured by a high-performance commercial instrument. This methodology can be potentially used to realize fast and simple phase noise measurement and reduce the overall cost of hardware.

Comprehensive phase diagram for logistic populations in fluctuating environment.

Population dynamics reflects an underlying birth-death process, where the rates associated with different events may depend on external environmental conditions and on the population density. A whole family of simple and popular deterministic models (such as logistic growth) supports a transcritical bifurcation point between an extinction phase and an active phase. Here we provide a comprehensive analysis of the phases of that system, taking into account both the endogenous demographic noise (random birth and death events) and the effect of environmental stochasticity that causes variations in birth and death rates. Three phases are identified: in the inactive phase the mean time to extinction T is independent of the carrying capacity N and scales logarithmically with the initial population size. In the power-law phase T∼N^{q}, and in the exponential phase T∼exp(αN). All three phases and the transitions between them are studied in detail. The breakdown of the continuum approximation is identified inside the power-law phase, and the accompanying changes in decline modes are analyzed. The applicability of the emerging picture to the analysis of ecological time series and to the management of conservation efforts is briefly discussed.

Phase Diagram for Logistic Systems under Bounded Stochasticity.

Extinction is the ultimate absorbing state of any stochastic birth-death process; hence, the time to extinction is an important characteristic of any natural population. Here we consider logistic and logisticlike systems under the combined effect of demographic and bounded environmental stochasticity. Three phases are identified: an inactive phase where the mean time to extinction T increases logarithmically with the initial population size, an active phase where T grows exponentially with the carrying capacity N, and a temporal Griffiths phase, with a power-law relationship between T and N. The system supports an exponential phase only when the noise is bounded, in which case the continuum (diffusion) approximation breaks down within the Griffiths phase. This breakdown is associated with a crossover between qualitatively different survival statistics and decline modes. To study the power-law phase we present a new WKB scheme, which is applicable both in the diffusive and in the nondiffusive regime.

A Multifrequency GPS Signal Strong Equatorial Ionospheric Scintillation Simulator: Algorithm, Performance, and Characterization

This paper presents a physics-based, data-consistent, multifrequency GPS scintillation signal simulator. The simulator is based on the two-component power-law phase screen derived from real GPS scintillation measurements. The real data used to initialize the simulator, simulator outputs, and receiver-processed results are compared to demonstrate the simulator accuracy and the receiver processing effects. This simulator will enable better characterization of ionospheric scintillation effects and development of robust receiver algorithms during strong equatorial scintillation.

Ionospheric Scintillation Simulation on Equatorial GPS Signals for Dynamic Platforms

This paper presents a physics-based, data-consistent, multifrequency GPS scintillation signal simulator for dynamic platforms in the equatorial region. This simulator is based on a two-dimensional two-component power-law phase screen model which is described by a compact set of parameters. The time scale factor is mainly investigated in this paper, which determines the decorrelation time of the simulation signal and is affected by various factors including propagation geometry, irregularity drift velocity, and dynamics of the platform. The comparison between the initialization and simulation data for stationary receivers shows the effectiveness of the simulator. In addition, this paper examines the influence of the platform dynamics on simulation signals with respect to the geomagnetic field direction and the irregularity drift velocity. The simulator presented in this paper will provide a viable tool for dynamic GPS scintillation signal simulation and will aid the design for robust receiver processing algorithms during scintillation for dynamic platforms.

Inferring processes of cultural transmission: the critical role of rare variants in distinguishing neutrality from novelty biases.

Neutral evolution assumes that there are no selective forces distinguishing different variants in a population. Despite this striking assumption, many recent studies have sought to assess whether neutrality can provide a good description of different episodes of cultural change. One approach has been to test whether neutral predictions are consistent with observed progeny distributions, recording the number of variants that have produced a given number of new instances within a specified time interval: a classic example is the distribution of baby names. Using an overlapping generations model, we show that these distributions consist of two phases: a power-law phase with a constant exponent of [Formula: see text], followed by an exponential cut-off for variants with very large numbers of progeny. Maximum-likelihood estimations of the model parameters provide a direct way to establish whether observed empirical patterns are consistent with neutral evolution. We apply our approach to a complete dataset of baby names from Australia. Crucially, we show that analyses based on only the most popular variants, as is often the case in studies of cultural evolution, can provide misleading evidence for underlying transmission hypotheses. While neutrality provides a plausible description of progeny distributions of abundant variants, rare variants deviate from neutrality. Further, we develop a simulation framework that allows the detection of alternative cultural transmission processes. We show that anti-novelty bias is able to replicate the complete progeny distribution of the Australian dataset.This article is part of the themed issue 'Process and pattern in innovations from cells to societies'.

Complex motion of a vehicle through a series of signals controlled by power-law phase

We study the dynamic motion of a vehicle moving through the series of traffic signals controlled by the position-dependent phase of power law. All signals are controlled by both cycle time and position-dependent phase. The dynamic model of the vehicular motion is described in terms of the nonlinear map. The vehicular motion varies in a complex manner by varying cycle time for various values of the power of the position-dependent phase. The vehicle displays the periodic motion with a long cycle for the integer power of the phase, while the vehicular motion exhibits the very complex behavior for the non-integer power of the phase.