Piecewise Power Law Research Articles

Research on weight initialization of CNN student models based on knowledge distillation

Knowledge distillation is a process of weight compression where a complex teacher model trains a simplified student model, making the weights and their distribution crucial. This paper investigates the weight distribution in convolutional and fully connected layers of both teacher and student models. For convolutional layers, it was discovered that both teacher and student models exhibit a piecewise power-law distribution. A verification method based on the piecewise power law distribution of convolutional layers was proposed, and the correctness of this law was confirmed. Detailed analysis of the breakpoints and power exponents reveals that the teacher model has smaller breakpoints than the student model; for weights smaller than the breakpoint, the teacher model’s power exponent is lower than that of the student model, whereas, for weights larger than the breakpoint, the teacher model’s power exponent is higher. Based on these findings, a new weight initialization algorithm for convolutional layers was proposed. For fully connected layers, both models demonstrate a skewed distribution. A verification method based on the skewed distribution of fully connected layers was proposed, and the correctness of this law was confirmed. Analysis of the kurtosis and skewness indicates that the student model exhibits higher kurtosis and skewness than the teacher model. Based on these observations, a new weight initialization algorithm for fully connected layers was proposed. Experimental results show that both initialization methods improve the initial and final accuracy of the student model compared to the He initialization method.

A novel numerical method for mixed-frame multigroup radiation-hydrodynamics with GPU acceleration implemented in the quokka code

Abstract Mixed-frame formulations of radiation-hydrodynamics (RHD), where the radiation quantities are computed in an inertial frame but matter quantities are in a comoving frame, are advantageous because they admit algorithms that conserve energy and momentum to machine precision and combine more naturally with adaptive mesh techniques, since unlike pure comoving-frame methods they do not face the problem that radiation quantities must change frame every time a cell is refined or coarsened. However, implementing multigroup RHD in a mixed-frame formulation presents challenges due to the complexity of handling frequency-dependent interactions and the Doppler shift of radiation boundaries. In this paper, we introduce a novel method for multigroup RHD that integrates a mixed-frame formulation with a piecewise powerlaw approximation for frequency dependence within groups. This approach ensures the exact conservation of total energy and momentum while effectively managing the Lorentz transformation of group boundaries and evaluation of group-averaged opacities. Our method takes advantage of the locality of matter-radiation coupling, allowing the source term for Ng frequency groups to be handled with simple equations with a sparse Jacobian matrix of size Ng + 1, which can be inverted with O(Ng) complexity. This results in a computational complexity that scales linearly with Ng and requires no more communication than a pure hydrodynamics update, making it highly efficient for massively parallel and GPU-based systems. We implement our method in the GPU-accelerated RHD code quokka and demonstrate that it passes a wide range of numerical tests, including preserving the asymptotic diffusion limit. We demonstrate that the piecewise powerlaw method shows significant advantages over traditional opacity averaging methods for handling rapidly variable opacities with modest frequency resolution.

The goodness-of-fit of models for fracture trace length distribution: Does the power law provide a good fit?

Fracture trace length distributions are often assumed to follow a power law, which implies that the distribution is scale-independent. The present study tests this assumption by evaluating the goodness-of-fit of three statistical models—the power law, piecewise power law, and lognormal distribution—upon a dataset of 57 trace maps that cover a range of fracture modes, host rock types, network scales, and topologies. The goodness-of-fit was assessed through the unbiased Kolmogorov-Smirnov (KS) test, which accounts for the fitting procedure and the degrees of freedom of each model. The results show that the power law provides a poor fit to trace length distributions, being rejected in 24 trace maps at a significance level of 0.05. In contrast, the piecewise power law and lognormal distribution demonstrated better fits across the fracture networks, with the piecewise power law performing the best overall. The poor fit of the power law can be attributed to mechanical and chemical controls on fracture growth, mainly fracture abutment, as well as stress shadowing and cementation, which affect growth rate at different length scales and result in scale-dependent trace length distributions. The consistent poor fit of the power law across various fracture networks suggests that these controls are prevalent in natural systems. While the power law remains a simple and effective model for trace length distribution, it should be recognized that it overlooks such controls that can influence the mechanical and hydraulic properties of fracture networks. Meanwhile, the fit of the piecewise power-law suggests the existence of a characteristic length where a transition in fracture growth occurs.

Fast likelihood-free reconstruction of gravitational wave backgrounds

We apply state-of-the-art, likelihood-free statistical inference (machine-learning-based) techniques for reconstructing the spectral shape of a gravitational wave background (GWB). We focus on the reconstruction of an arbitrarily shaped signal (approximated by a piecewise power-law in many frequency bins) by the LISA detector, but the method can be easily extended to either template-dependent signals, or to other detectors, as long as a characterisation of the instrumental noise is available. As proof of the technique, we quantify the ability of LISA to reconstruct signals of arbitrary spectral shape (blind reconstruction), considering a diversity of frequency profiles, and including astrophysical backgrounds in some cases. As a teaser of how the method can reconstruct signals characterised by a parameter-dependent template (template reconstruction), we present a dedicated study for power-law signals. While our technique has several advantages with respect to traditional MCMC methods, we validate it with the latter for concrete cases. This work opens the door for both fast and accurate Bayesian parameter estimation of GWBs, with essentially no computational overhead during the inference step. Our set of tools are integrated into the package GWBackFinder, which is publicly available in GitHub.

No Top-heavy Stellar Initial Mass Function Needed: The Ionizing Radiation of GS9422 Can Be Powered by a Mixture of an Active Galactic Nucleus and Stars

JWST is producing high-quality rest-frame optical and UV spectra of faint galaxies at z > 4 for the first time, challenging models of galaxy and stellar populations. One galaxy recently observed at z = 5.943, GS9422, has nebular line and UV continuum emission that appears to require a high ionizing photon production efficiency. This has been explained with an exotic stellar initial mass function (IMF; Cameron et al. 2023a), 10–30x more top-heavy than a Salpeter IMF. Here we suggest an alternate explanation to this exotic IMF. We use a new flexible neural net emulator for CLOUDY, Cue, to infer the shape of the ionizing spectrum directly from the observed emission line fluxes. By describing the ionizing spectrum with a piecewise power law, Cue is agnostic to the source of the ionizing photons. Cue finds that the ionizing radiation from GS9422 can be approximated by a double power law characterized by QHeIIQH=−1.5 , which can be interpreted as a combination of young, metal-poor stars and a low-luminosity active galactic nucleus with F ν ∝ λ 2 in a 65%/35% ratio. This suggests a significantly lower nebular continuum contribution to the observed UV flux (24%) than a top-heavy IMF (≳80%), and hence, necessitates a damped Lyα absorber to explain the continuum turnover bluewards of ∼1400 Å. While current data cannot rule out either scenario, given the immense impact the proposed top-heavy IMF would have on models of galaxy formation, it is important to propose viable alternative explanations and to further investigate the nature of peculiar high-z nebular emitters.

Fixed-Time Cluster Optimization for Multi-Agent Systems Based on Piecewise Power-Law Design

Fixed-Time Cluster Optimization for Multi-Agent Systems Based on Piecewise Power-Law Design

Tracing curves in the plane: Geometric-invariant learning from human demonstrations.

The empirical laws governing human-curvilinear movements have been studied using various relationships, including minimum jerk, the 2/3 power law, and the piecewise power law. These laws quantify the speed-curvature relationships of human movements during curve tracing using critical speed and curvature as regressors. In this work, we provide a reservoir computing-based framework that can learn and reproduce human-like movements. Specifically, the geometric invariance of the observations, i.e., lateral distance from the closest point on the curve, instantaneous velocity, and curvature, when viewed from the moving frame of reference, are exploited to train the reservoir system. The artificially produced movements are evaluated using the power law to assess whether they are indistinguishable from their human counterparts. The generalisation capabilities of the trained reservoir to curves that have not been used during training are also shown.

Piecewise power law approximation of the interlayer relaxation curve for the long term viscoelastic fractional modeling of laminated glass

Laminated glass is composed of glass layers coupled by thin polymeric films which, being highly viscoelastic, influence the bending response over time. Here, an approximation of the polymer relaxation curve is proposed by means of continuously-connected power law branches. In the bi-logarithmic stress-time plane, this corresponds to a piecewise trilinear path, which well reproduces the experimental tests on most commercial polymers. A power law is naturally suited to be modeled via fractional calculus, as the Boltzmann convolution integral coincides with the definition of fractional derivative. However, the piecewise characterization through multiple power laws requires to properly extend the Grünwald–Letnikov integration scheme for fractional differential equations. For the paradigmatic case of a simply supported three-layer laminated beam, numerical results show an excellent correspondence with experiments under long duration loads. The comparison with the classic Prony series approach highlights, on the one hand, how calibration of material parameters is simplified; on the other hand, how the fractional approach is computationally more efficient and stable, even when the integration time step is wide enough to follow long term phenomena.

Analysis of urban turbulence intensity observed by Beijing 325-m tower and comparison with the IEC turbulence model for small wind turbines

The accurate assessment of turbulence intensity is crucial for the effective utilization of urban wind energy. However, the Normal Turbulence Model (NTM) prescribed by the IEC standard has not been adequately evaluated for its applicability in urban environments. To address this gap, we analyze one-year continuous turbulence observations collected at multiple levels on a 325-m meteorological tower in Beijing. The results show that: the turbulence intensity follows the Johnsonsb distribution instead of the Normal distribution in the IEC standard; it has significant diurnal variation and piecewise power-law vertical variation; the NTM underestimates turbulence intensity at most times and heights. A new turbulence intensity model with an exponential form is proposed, which significantly outperforms the NTM. Further, the exponential model fits the observations better than power-law model that outperforms the NTM. In terms of the fitting errors to the measurements, the full-year, monthly, and hourly RMSE (MAE) of this model are reduced on average by 51.81% (54.31%), 33.04% (34.38%) and 29.46% (28.24%) compared to that of the power-law model. These findings will contribute to the revision of the IEC standards governing the safe operation of small wind turbines in complex wind environments such as urban areas.

Distributed Fixed/Preassigned-Time Optimization Based on Piecewise Power-Law Design.

The problem of fixed-time (FXT) and preassigned-time (PAT) optimization is concerned in this article based on multiagent systems (MASs) and power-law algorithms. Under the framework of strong convexity of the cost functions, two types of piecewise algorithms are proposed, which ensure that the FXT optimization can be solved either by first achieving the FXT consensus or by first achieving local optimization. Correspondingly, the PAT optimization problem is also considered by designing several piecewise protocols, where the finished time of optimization can be arbitrary prescribed according to actual demands. Furthermore, these piecewise power-law algorithms on the weighted undirected graphs are generalized to the weighted digraphs. Finally, by providing two numerical examples, the presented algorithms are further verified.

Fast generation of reliable primary radiation damage of BCC tungsten by sampling molecular dynamics databases

Fast generation of reliable primary radiation damage plays a key role in modeling of long-term defect evolution in materials under irradiation within the multi-scale modeling framework. We have proposed a model, PradG, to fast generate reliable primary radiation damage by sampling limited molecular dynamics databases. First, MD cascade simulations for body-centered-cubic W are systematically carried out with a wide range of discrete cascade energies. Through introducing the spherical symmetry approximation of cascade volume and analyzing cascade configurations, we find that the size distribution shows a piecewise power-law formula, and the radial distribution shows the Gaussian and exponential decay types for interstitial and vacancy clusters, respectively. Second, by using a sampling method based on these fitting size and spatial distribution, regardless of the spatial correlation of interstitial and vacancy clusters, PradG is developed for generating primary defects of arbitrary cascade energies. Third, the reliability and validation of PradG are verified by comparing with cascade annealing simulations and defect evolution in fission neutron irradiated W. It provides a reliable damage source term for multi-scale modeling of long-term defect evolution with a certain PKA energy spectrum.

Exoplanets Catalogue Analysis: The Distribution of Exoplanets at FGK Stars by Mass and Orbital Period Accounting for the Observational Selection in the Radial Velocity Method

When studying the statistics of exoplanets, it is necessary to take into account the effects of observational selection and the inhomogeneity of the data in the exoplanets databases. When considering exoplanets discovered by the radial velocity technique (RV), we propose an algorithm to account for major inhomogeneities. We show that the de-biased mass distribution of the RV exoplanets approximately follows to a piecewise power law with the breaks at ~0.14 and ~1.7 MJ. FGK host stars planets group shows an additional break at 0.02 MJ. The distribution of RV planets follows the power laws of: dN/dm α m−3 (masses of 0.011–0.087 MJ), dN/dm α m−0.8…−1 (0.21–1.7 MJ), dN/dm ∝ m−1.7…−2 (0.087–0.21 MJ). There is a minimum of exoplanets in the range of 0.087–0.21 MJ. Overall, the corrected RV distribution of the planets over the minimum masses is in good agreement with the predictions of population fusion theory in the range (0.14–13 MJ) and the new population fusion theory in the range (0.02–0.14 MJ). The distributions of planets of small masses (0.011–0.14 MJ), medium masses (0.14–1.7 MJ), and large masses (1.7–13 MJ) versus orbital period indicate a preferential structure of planetary systems, in which the most massive planets are in wide orbits, as analogous to the Solar system.

An accurate treatment of scattering and diffusion in piecewise power-law models for cosmic ray and radiation/neutrino transport

ABSTRACT A popular numerical method to model the dynamics of a ‘full spectrum’ of cosmic rays (CRs), also applicable to radiation/neutrino hydrodynamics, is to discretize the spectrum at each location/cell as a piecewise power law in ‘bins’ of momentum (or frequency) space. This gives rise to a pair of conserved quantities (e.g. CR number and energy) that are exchanged between cells or bins, which in turn give the update to the normalization and slope of the spectrum in each bin. While these methods can be evolved exactly in momentum-space (e.g. considering injection, absorption, continuous losses/gains), numerical challenges arise dealing with spatial fluxes, if the scattering rates depend on momentum. This has often been treated either by neglecting variation of those rates ‘within the bin,’ or sacrificing conservation – introducing significant errors. Here, we derive a rigorous treatment of these terms, and show that the variation within the bin can be accounted for accurately with a simple set of scalar correction coefficients that can be written entirely in terms of other, explicitly evolved ‘bin-integrated’ quantities. This eliminates the relevant errors without added computational cost, has no effect on the numerical stability of the method, and retains manifest conservation. We derive correction terms both for methods that explicitly integrate flux variables (e.g. two-moment or M1-like) methods, as well as single-moment (advection-diffusion, FLD-like) methods, and approximate corrections valid in various limits.

Flame attachment and downstream heating effect of inclined line fires

Experiments were conducted to quantify downstream heating from inclined fires generated by a 25 cm wide, 5 cm deep gaseous line burner. A variety of orientation angles (θ) and fire heat-release rates (HRR) were employed simulating, at reduced scale, the dynamics of two-dimensional (2-D) inclined flame spread as found in wildland or building fires. The total heat flux q˙f″(x) to a nearly-adiabatic surface ahead of the burner was measured by a water-cooled total heat flux gage at different downstream locations (x). It was found that, with an increase in fire HRR or θ, q˙f″(x) increases. The same is true for both the flame projection length on the inclined surface (Lf,p) and the flame attachment length (La). The location of flame attachment divides a flame into an attachment region (0≤x≤La) and a liftoff region (La<x≤Lf,p). A scaling analysis was performed to correlate the modified non-dimensional heat flux q˙f*(x) and the normalized downstream distance x/La in these two regions as a piecewise power-law function. Lf,p is scaled by a non-dimensional HRR, Q˙θ*=Q˙/(ρ∞cpT∞(gcosθ)1/2D3/2L), together with the angle between a flame and an inclined surface (α). La is well represented by two derived non-dimensional quantities, L/Lf,0 and Lf,p/Lf,0 (Lf,0 is free flame height), based on the analysis of the balance between the buoyancy force of a flame (upward) and the inertial force due to air entrainment (horizontal). The quantitative comparison with previous flame spread tests over inclined PMMA samples demonstrates the effectiveness of the proposed correlations in predicting downstream heating of 2-D flame spread over inclined terrains.

Long-Term Effects of Humidity on Stainless Steel Pitting in Sea Salt Exposures

Ground 304 stainless steel (SS) samples were exposed to sea salt particles at 35 °C and two relative humidity (RH) levels for durations ranging from 1 week to 2 years. For all exposure times, pit number density and total pit volume at 40% RH were observed to be considerably greater than those at 76% RH. Statistical analysis of distributions of pit populations for both RH conditions showed that pit number density and total pit volume increased rapidly at first but slowed as exposure time increased. Cross-hatched features were observed in the 40% RH pits while ellipsoidal, faceted pits were observed at 76% RH. Optical profilometry indicated that most pits were not hemispherical. X-ray tomography provided evidence of undercutting and fissures. Piecewise curve fitting modeled the 40% RH data closely, predicting that corrosion damage would eventually plateau. However, a similar treatment of the 76% RH data suggested that corrosion damage would continuously increase, which implied that the piecewise power-law fit was limited in its ability to model atmospheric corrosion generally. Based on these observations, the operative mechanisms determining long-term corrosion behavior were hypothesized to be different depending on the RH of exposure.

Cosmological implications of the composite spectra of galactic X-ray binaries constructed using MAXI data

We have investigated the long term average spectral properties of galactic X-ray binaries in the energy range of 3-20 keV, using long term monitoring data from MAXI -Gas Slit Camera (GSC). These long term average spectra are used to construct separately the composite spectra of galactic High Mass X-ray binaries (HMXBs) and Low Mass X-ray binaries (LMXBs). These composite spectra can be described empirically with piece-wise power-law with three components. X-rays from HMXBs are considered as important contributors to heating and ionization of neutral hydrogen in the intergalactic medium during the Epoch of Reionization. Using the above empirical form of the composite HMXB spectra extrapolated to lower energies as an input, we have studied the impact of these sources on the 21-cm signal using the outputs of N-body simulation and 1D radiative transfer. The heating due to the composite spectrum is less patchy compared to power-law spectrum with a spectral index $\alpha = 1.5$, used in previous studies. The amplitude of the heating peak of large scale power spectrum, when plotted as a function of the redshift, is less for the composite spectrum.

Dynamics and scaling laws of underwater granular collapse with varying aspect ratios

We perform coupled fluid-particle simulations to understand the granular collapse in an ambient fluid (in particular, water) with a wide range of initial aspect ratios. We observe both similar and distinct features in underwater collapses compared to their dry counterparts. As aspect ratio $a$ increases, the normalized runout distance follows a piecewise power-law growth, transitioning at $a=2.5$. We associate this transition with the different growth rates of kinetic energy (with $a$) in vertical and horizontal directions. The ability of utilizing available energy for horizontal motion becomes limited when $ag2.5$. Moreover, the front propagation during underwater collapses can be well scaled by using the initial column height as length scale and considering a reduced gravity (due to buoyancy) in timescale. Under the reduced gravity, the initial fall of tall columns is found to be ballistic, consistent with dry collapses. On the other hand, underwater collapses (especially for large $a$) exhibit unique dynamics due to the presence of water. The eddies generated in water, which may carry considerable fluid inertia, tend to erode the surface of the granular layer, thus modifying the deposit morphology. The energy conversion is also affected by the ambient fluid. While water obviously consumes energy from the granular phase through fluid-particle interactions, it actually increases the efficiency of energy conversion from vertical to horizontal directions. The latter effect compensates the difference of runout distance between underwater and dry collapses.

Possible Accretion Disk Origin of the Emission Variability of a Blazar Jet

Abstract We analyze X-ray light curves of the blazar Mrk 421 obtained from the Soft X-ray Imaging Telescope (SXT) and the Large Area X-ray Proportional Counter (LAXPC) instrument on board the Indian space telescope AstroSat and archival observations from Swift. We show that the X-ray power spectral density (PSD) is a piece-wise power-law with a break; i.e., the index becomes more negative below a characteristic “break timescale.” Galactic black hole (BH) X-ray binaries and Seyfert galaxies exhibit a similar characteristic timescale in their X-ray variability that is proportional to their respective BH mass. X-rays in these objects are produced in the accretion disk or corona. Hence, such a timescale is believed to be linked to the properties of the accretion flow. Any relation observed between events in the accretion disk and those in the jet can be used to characterize the disk–jet connection. However, evidence of such a link has been scarce and indirect. Mrk 421 is a BL Lac object that has a prominent jet pointed toward us and a weak disk emission, and it is assumed that most of its X-rays are generated in the jet. Hence, the existence of the break in its X-ray PSD may indicate that changes in the accretion disk, which may be the source of the break timescale, are translating into the jet where the X-rays are produced.

Heterogeneous characters modeling of instant message services users' online behavior.

Research on temporal characteristics of human dynamics has attracted much attentions for its contribution to various areas such as communication, medical treatment, finance, etc. Existing studies show that the time intervals between two consecutive events present different non-Poisson characteristics, such as power-law, Pareto, bimodal distribution of power-law, exponential distribution, piecewise power-law, et al. With the occurrences of new services, new types of distributions may arise. In this paper, we study the distributions of the time intervals between two consecutive visits to QQ and WeChat service, the top two popular instant messaging services in China, and present a new finding that when the value of statistical unit T is set to 0.001s, the inter-event time distribution follows a piecewise distribution of exponential and power-law, indicating the heterogeneous character of IM services users’ online behavior in different time scales. We infer that the heterogeneous character is related to the communication mechanism of IM and the habits of users. Then we develop a combination model of exponential model and interest model to characterize the heterogeneity. Furthermore, we find that the exponent of the inter-event time distribution of the same service is different in two cities, which is correlated with the popularity of the services. Our research is useful for the application of information diffusion, prediction of economic development of cities, and so on.

Determination of corner frequencies of source spectra for subduction earthquakes in Avacha Gulf (Kamchatka)

The source spectra of M = 4.0-6.5 subduction earthquakes of 2011-2014 in Kamchatka are studied. The dataset comprises 1272 source spectra recovered from S waves of 372 earthquakes recorded by six digital rock-ground stations. The structure of the spectra is examined on the basis of a spectral model with three corner frequencies fc1, fc2, and fc3. It was assumed that the spectra behave as f−2 between fc2 and fc3, where fc3 denotes “source-controlled fmax” after Aki and Gusev. To determine the corner frequencies, we extracted the source spectra from S-wave spectra using a previously developed attenuation model for the study area. The spectra were first reduced to the reference hard-rock station, employing a specially determined set of spectral amplifications of stations. We approximated the recovered source spectrum by a piecewise power-law function, estimated fc1, fc2, and fc3, and examined their dependence on the seismic moment M0 (i.e., scaling). The dependence fc1(M0) does not contradict the hypothesis of source similarity when one expects fc1∝M0−1/3. For fc2 and fc3, the scaling is close to fc2∝M0−0.23 and fc3∝M0−0.13, respectively, indicating a clear violation of the similarity, especially prominent for fc3. Systematic identification of the frequency fc3, its determination, and analysis of its scaling are the main results of the study, important for understanding the physics of earthquake source processes. The use of fc3 as a source parameter in strong ground motion simulations will eliminate biases in estimating attenuation parameters, in particular, the spectral decay parameter “kappa”.