Lognormal Function Research Articles

Damage softening model and behavior of reinforced soil interface based on the Weibull distribution

A damage-softening model for a reinforced soil interface is proposed based on the trilinear model of a reinforced soil pullout interface, aiming at the softening phenomenon of the geogrid reinforced soil interface during the pullout process and considering the damage softening of the reinforced soil interface during the pullout process. The damage variable factor D is introduced, and it is assumed that D is a function of the interface displacement. The two-parameter Weibull distribution function is used to express the damage evolution law of the reinforced soil interface. Based on the basic control equation of the reinforced soil interface, the calculation model of the stress state of the geogrid at different stages under the pullout load is deduced. To verify the accuracy and applicability of the model, the prediction results of the model are compared with the test results and the ideal elastic–plastic model, hyperbolic model, elastic–exponential softening model, and the damage softening model based on a lognormal distribution function proposed in this paper. The distribution of interfacial shear stress is studied, and the influence of parameters at each stage is analyzed. The results show that in the elastic stage, with the increase of shear stiffness, the maximum shear stress at the interface increases, and the curvature of the curve also increases, while the elastic modulus is the opposite. In the softening stage, with the increase of softening length, the peak value of the curve moves to the free end. In the residual stage, the shear stress increases from the drawing end to the free end and tends to the residual stress. The research results are in accordance with the actual situation and can be applied to reinforced soil engineering.

Session V:

In this study, work on modelling uncertainty in the tensile properties of two-dimensional plain woven silicon carbide fibre-reinforced silicon carbide matrix (2D SiC/SiC) composite and B-basis values of tensile strength was carried out in the following four aspects. First, the uncertainty law and formation mechanism of the tensile properties of 2D SiC/SiC composites were researched through microstructural analysis. Subsequently, three distribution function models were established for the tensile properties of 2D SiC/SiC composites with Weibull, normal, and log-normal distribution functions. The development of three functional models was based on 30 groups of valid tensile data, linear regression methods, and Kolmogorov hypothesis testing methods. Based on the application of the model validation method, the accuracy of the three functional models was quantitatively evaluated by employing the area index of model validation. B-basis values for the tensile strength of 2D SiC/SiC composites were calculated and finally discussed for three distribution functions. The results indicate that the distribution regularity of tensile properties of 2D SiC/SiC composites could be characterized by using the three function models, and the Weibull distribution function was the most accurate. Therefore, it was preferable to calculate the B-basis values by using the Weibull distribution, with a B-basis value of 266 MPa.

Comparison of burst properties between FRB 20190520B and FRB 20121102A

ABSTRACT A comparative analysis of the individual bursts between FRB 20190520B and FRB 20121102A is presented by compiling a sample of bursts in multiple wavelengths. It is found that the peak frequency (νp) distribution of the bursts of FRB 20190520B illustrates four discrete peaks in ∼1–6 GHz and their spectral width distribution can be fitted with a lognormal function peaking at 0.35 GHz. The discrete νp distribution and the narrow-banded spectral feature are analogous to FRB 20121102A. The burst duration of FRB 20190520B in the rest frame averages to 10.72 ms, longer than that of FRB 20121102A by a factor 3. The specific energy $\left(E_{\rm\mu_{\rm c}}\right)$ at 1.25 GHz of FRB 20190520B observed with the Five-hundred-meter Aperture Spherical Radio Telescope narrowly ranges in [0.4, 1] × 1038 erg, different from the bimodal $E_{\rm \mu _{\rm c}}$ distribution of FRB 20121102A. Assuming a Gaussian spectral profile of the bursts, our Monte Carlo simulation analysis suggests that a power law (PL) or a cut-off power-law (CPL) energy function can comparably reproduce the $E_{\rm \mu _{\rm c}}$ distribution of FRB 20190520B. The derived energy function index of the PL model is 4.46 ± 0.17, much steeper than that of FRB 20121102A ($1.82^{+0.10}_{-0.30}$ ). For the CPL model, we obtain an index of 0.47 and a cut-off energy of 7.4 × 1037 erg. Regarding the predicted νp distribution in 1–2 GHz, the CPL model is more preferred than the PL model. These results indicate that FRB 20190520B and FRB 20121102A share similar spectral properties, but their energy functions are intrinsically different.

Bayesian inversion of log-normal eikonal equations

We study the Bayesian inverse problem for inferring the log-normal slowness function of the eikonal equation, given noisy observation data on its solution at a set of spatial points. We contribute rigorous proofs on the existence and well-posedness of the problem. We then study approximation of the posterior probability measure by solving the truncated eikonal equation, which contains only a finite number of terms in the Karhunen–Loeve expansion of the slowness function, by the fast marching method (FMM). The error of this approximation in the Hellinger metric is deduced in terms of the truncation level of the slowness and the grid size in the FMM resolution. It is well known that the plain Markov chain Monte Carlo (MCMC) procedure for sampling the posterior probability is highly expensive. We develop and justify the convergence of a multilevel MCMC method. Using the heap sort procedure in solving the forward eikonal equation by the FMM, our multilevel MCMC method achieves a prescribed level of accuracy for approximating the posterior expectation of quantities of interest, requiring only an essentially optimal level of complexity. Numerical examples confirm the theoretical results.

On the hosts of neutron star mergers in the nearby Universe

ABSTRACT Recently, the characterization of binary systems of neutron stars has become central in various fields such as gravitational waves, gamma-ray bursts (GRBs), and the chemical evolution of galaxies. In this work, we explore possible observational proxies that can be used to infer some characteristics of the delay time distribution (DTD) of neutron star mergers (NSMs). We construct a sample of model galaxies that fulfils the observed galaxy stellar mass function, star formation rate versus mass relation, and the cosmic star formation rate density. The star formation history of galaxies is described with a log-normal function characterized by two parameters: the position of the maximum and the width of the distribution. We assume a theoretical DTD that mainly depends on the lower limit and the slope of the distribution of the separations of the binary neutron star systems at birth. We find that the current rate of NSMs ($\mathcal {R}=320^{+490}_{-240}$ Gpc−3 yr−1) requires that ∼0.3 per cent of neutron star progenitors lives in binary systems with the right characteristics to lead to an NSM within a Hubble time. We explore the expected relations between the rate of NSMs and the properties of the host galaxy. We find that the most effective proxy for the shape of the DTD of NSMs is the current star formation activity of the typical host. At present, the fraction of short-GRBs observed in star-forming galaxies favours DTDs with at least ${\sim}40~{{\ \rm per\ cent}}$ of mergers within 100 Myr. This conclusion will be put on a stronger basis with larger samples of short-GRBs with host association (e.g. 600 events at z ≤ 1).

Bridge seismic fragility model based on support vector machine and relevance vector machine

In the area of bridge seismic fragility studies, the fragility curve is commonly defined as a log-normal cumulative distribution function, and its parameters are usually estimated through two methods. However, the fragility curves derived by one of these two methods may intersect, whereas the other imposes significant restrictions on the shape of the curves. The purpose of this study is to derive fragility curves by utilizing machine learning tools (in particular, tools that are based on multi-classification probabilistic prediction), to address the limitations that are present in existing methodologies. To be more precise, support vector machine for classification (SVC) and relevance vector machine for classification (RVC) are utilized to construct fragility curves, while two other classic methodologies are chosen for comparison and validation. The four approaches were tested on a dataset acquired using numerical analysis, and their performance was measured using Brier-Score and Log-Loss. The results show that both SVC and RVC not only overcome the limitations and outperform the classical method, but also show robustness to arbitrarily partitioned training sets, whereas the classical method sometimes performs poorly.

Bayesian Implications for the Primordial Black Holes from NANOGrav’s Pulsar-Timing Data Using the Scalar-Induced Gravitational Waves

Assuming that the common-spectrum process in the NANOGrav 12.5-year dataset has an origin of scalar-induced gravitational waves, we study the enhancement of primordial curvature perturbations and the mass function of primordial black holes, by performing the Bayesian parameter inference for the first time. We obtain lower limits on the spectral amplitude, i.e., A≳10−2 at 95% confidence level, when assuming the power spectrum of primordial curvature perturbations to follow a log-normal distribution function with width σ. In the case of σ→0, we find that the primordial black holes with 2×10−4−10−2 solar mass are allowed to compose at least a fraction 10−6 of dark matter. Such a mass range is shifted to more massive regimes for larger values of σ, e.g., to a regime of 4×10−3−0.2 solar mass in the case of σ=1. We expect the planned gravitational-wave experiments to have their best sensitivity to A in the range of 10−4 to 10−7, depending on the experimental setups. With this level of sensitivity, we can search for primordial black holes throughout the entire parameter space, especially in the mass range of 10−16 to 10−11 solar masses, where they could account for all dark matter. In addition, the importance of multi-band detector networks is emphasized to accomplish our theoretical expectation.

Stochastic Rational Method for Estimation of Flood Peak Uncertainty in Arid Basins: Comparison between Monte Carlo and First Order Second Moment Methods with a Case Study in Southwest Saudi Arabia

The flood peak is commonly estimated using the rational method for the design of hydraulic structures. The method is mainly used in a deterministic context. However, there is often uncertainty in flood predictions, which should be incorporated in the design of mitigation schemes. This research proposes a methodology to cope with uncertainty in the rational method via the application of a stochastic framework. Data from 158 storms, recorded in the period 1984–1987 in 19 subbasins in the southwestern part of Saudi Arabia, were used to implement the proposed methodology. A tri-variate log-normal probability density function was used to model the joint relationship between the rational method parameters. The model considered the parameters as random variables. The uncertainty in the rainstorms was represented by intensity or depth; the uncertainty in basin delineation (due to the use of different digital elevation model resolution) was represented by the basin area; and the uncertainty in the land use/land cover was represented by the runoff coefficient. The Monte Carlo method was used to generate realizations of the peak flow and runoff volume with 95% and 99% confidence levels from the input parameters. Although the correlation between the parameters was weak, the model was capable of simulating the rational model parameters and estimating the peak flow and runoff volume relatively well, and the generated realizations fell within the confidence levels, except for a few marginal cases. The model can be used to generate peak flows and the associated confidence limits in ungauged basins from the statistics of the input parameters using the equations developed in this study.

Constraints on primordial-black-hole population and cosmic expansion history from GWTC-3

Gravitational waves (GWs) from compact binary coalescences provide an independent probe of the cosmic expansion history other than electromagnetic waves. In this work, we assume the binary black holes (BBHs) detected by LIGO-Virgo-KAGRA (LVK) collaborations are of primordial origin and constrain the population parameters of primordial black holes (PBHs) and Hubble parameter H(z) using 42 BBHs from third LVK GW transient catalog (GWTC-3). Three PBH mass models are considered: lognormal, power-law, and critical collapse PBH mass functions. By performing a hierarchical Bayesian population analysis, the Bayes factor strongly disfavors the power-law PBH mass function against the other two in GWTC-3. The constraints on standard Λ CDM cosmological parameters are rather weak and in agreement with current results. When combining the multi-messenger standard siren measurement from GW170817, the Hubble constant H 0 is constrained to be 69-8 +19 km s-1 Mpc-1 and 70-8 +26 km s-1 Mpc-1 at 68% confidence for the lognormal and critical collapse mass models, respectively. Furthermore, we consider a mixed ABH+PBH model, in which we assume LVK BBHs can come from both the astrophysical black hole (ABH) and PBH channels. We find that the ABH+PBH model can better describe the mass distribution in GWTC-3 than any single ABH or PBH model, thus improving the precision to constrain the Hubble constant. With the increased BBH events, the mixed ABH+PBH model provides a robust statistical inference for both the population and cosmological parameters.

Evidence for a bottom-light initial mass function in massive star clusters

ABSTRACT We have determined stellar mass functions of 120 Milky Way globular clusters and massive Large Magellanic Cloud/Small Magellanic Cloud star clusters based on a comparison of archival Hubble Space Telescope photometry with a large grid of direct N-body simulations. We find a strong correlation of the global mass function slopes of star clusters with both their internal relaxation times and their lifetimes. Once dynamical effects are being accounted for, the mass functions of most star clusters are compatible with an initial mass function described by a broken power-law distribution N(m) ∼ mα with break masses at 0.4 and 1.0 M⊙ and mass function slopes of αLow = −0.3 for stars with masses m < 0.4 M⊙, αHigh = −2.30 for stars with m > 1.0 M⊙, and αMed = −1.65 for intermediate-mass stars. Alternatively, a lognormal mass function with a characteristic mass log MC = −0.36 and width σC = 0.28 for low-mass stars and a power-law mass function for stars with m > 1 M⊙ also fit our data. We do not find a significant environmental dependence of the initial mass function on cluster mass, density, global velocity dispersion, or metallicity. Our results lead to a larger fraction of high-mass stars in globular clusters compared to canonical Kroupa/Chabrier mass functions, increasing the efficiency of self-enrichment in clusters and helping to alleviate the mass budget problem of multiple stellar populations in globular clusters. By comparing our results with direct N-body simulations, we finally find that only simulations in which most black holes are ejected by natal birth kicks correctly reproduce the observed correlations.

Modelling the evolution of dual-pore structure for compacted clays along hydro-mechanical paths

The evolution of dual-pore structure for compacted clays along hydro-mechanical paths exerts considerable influences on their phenomenological behavioral properties. This study aims to propose a novel approach for modelling the evolution of dual-pore structure. Here bimodal pore size density (PSD) function is considered to be obtained by superposing two unimodal lognormal distribution functions, which are inter-aggregate PSD (inter-PSD) and intra-aggregate PSD (intra-PSD). Unimodal PSD at arbitrary state is assumed to be obtained from initial one through shifting, scaling and spreading. On the basis, six evolution parameters are defined with unambiguous physical meanings. The modelling approach for evolving bimodal PSD is proposed through relating these evolution parameters to inter-aggregate and intra-aggregate void ratios respectively, where inter-aggregate and intra-aggregate void ratios are determined based on an empirical relationship between intra-aggregate void ratio and water ratio. The empirical parameter calibration in this approach is specific to stress paths. The proposed approach is applied to model the dual-pore structure evolution for compacted clays along loading path, wetting and drying paths under free condition, as well as wetting path under isochoric condition. Comparisons between the predicted and measured data show the novel approach can effectively capture the evolution of dual-pore structure along various hydro-mechanical paths.

Quantifying diameter distributions in seasonally dry tropical forest in Bahia, Brazil

This study aims to investigate the diametric structure of a seasonally dry tropical forest in Bahia and to select the best functions for the stand and species of greater importance. Two subsets were selected for fitting and validation from pre- and post-selective harvest inventories. Normal, Log-normal functions were tested; Gamma, Weibull 2P, and Weibull 3P. The best fits were selected using the Kolmogorov-Smirnov test for Akaike information criterion. Negative exponential curves and positive skewness best described the diameter distribution. This behavior was observed for the species with the highest IV and the stand. The Log-normal function best describes the diametric structure of Commiphora leptophioeeos, Manihot catingae, and Pataggonula bahiensis. It is recommended to individually describe the diameter structures of Aspidosperma pyrifolium, Pseudobombax simplicifolium, and population by the Weibull 3P function. The Normal, Gamma, and Weibull 2P functions did not present feasible predictions for frequencies of the diameter classes in all analyzed cases.

Extended primordial black hole mass functions with a spike

ABSTRACT We introduce a modification of the Press–Schechter formalism aimed to derive general mass functions for primordial black holes (PBHs). In this case, we start from primordial power spectra (PPS) which include a monochromatic spike, typical of ultra slow-roll inflation models. We consider the PBH formation as being associated to the amplitude of the spike on top of the linear energy density fluctuations coming from a PPS with a blue index. By modelling the spike with a lognormal function, we study the properties of the resulting mass function spikes, and compare these to the underlying extended mass distributions. When the spike is at PBH masses, which are much lower than the exponential cut-off of the extended distribution, very little mass density is held by the PBHs within the spike, and it is not ideal to apply the Press–Schechter formalism in this case as the resulting characteristic overdensity is too different from the threshold for collapse. It is more appropriate to do so when the spike mass is similar to, or larger than the cut-off mass. Additionally, it can hold a similar mass density as the extended part. Such particular mass functions also contain large numbers of small PBHs, especially if stable PBH relics are considered, and they can provide ${\sim}1000\, {\rm M}_\odot$ seeds for the supermassive black holes at the centres of present-day galaxies. The constraints on the fraction of dark matter in PBHs for monochromatic mass functions are somewhat relaxed when there is an additional underlying extended distribution of masses.

Optimal power flow solutions incorporating stochastic wind power generators by moth flow optimizer

The variables that maximize an objective function, specific equality requirements, such as the power balancing and power flow equations, and inequality constraints, restrictions on the variables, are the most crucial elements of the optimal power flow (OPF). The set of parameters and constraints, as well as the shape of the objective, will vary depending on the type of OPF. Additionally, with the gradual integration of renewable energy sources into the contemporary smart grid, it is now possible to develop new optimization problems with a substantially greater number of variables. In this paper, a method for choosing the optimal power flow in a system including both conventional thermal power plants and stochastic wind turbines is presented. To anticipate wind energy output, Weibull and lognormal probability distribution functions are used, respectively. The loss from overestimating intermittent renewable sources and the penalty factor from underestimating them are both considered by the objective function. To handle the optimization problem in the IEEE 30 bus system, Moth Flow Optimizer (MFO) and Graw Wolf Optimization are used (GWO). The approach's combination and design produce the best outcomes, which meet all network limitations.

Effective permeability of fractured porous media with fracture density near the percolation threshold

This study investigates the effective permeability of fractured porous media with fracture density near the percolation threshold. A new equivalent matrix-fracture network (EMFN) model was introduced, the code was used to efficiently deal with the huge number of modeling works for steady state flow in fractured porous media. The numerical model was verified by comparing with two previous seepage methods. We conducted a series of Monte Carlo simulations to explore the relationship between permeability discontinuity and three factors, namely fracture density, contrast between fracture and matrix permeability, and the scale of research domain. It was found that the effective permeability presents a bimodal distribution with specified fracture density in the transition regime. Each bimodal function can be subdivided into two well-fitting log-normal functions according to the percolation state to obtain analyzable average permeabilities. The results revealed significant differences between percolating and non-percolating configurations in the response to the change of each of the three factors mentioned above. More importantly, the finite-size effect plays an important role when fracture density is lower than or close to the percolation threshold; but this effect disappears gradually as the increase of fracture density. A stereoscopic approach is introduced to rationalize the scale effect.

Identification of a Function to Fit the Flow Duration Curve and Parameterization of a Semi-Arid Region in North China

The discharge process has undergone major changes in many river basins throughout the world as a result of the simultaneous influences of global climate change and human activity. Flow duration curves (FDCs) are crucial indicators of river basins’ hydrological processes. However, it is challenging to compare FDCs in a quantitative way. This study will identify the best function with which to fit the flow duration curve in a semi-arid region of North China, so as to quantify the FDC, and parameterize the function of the FDC of the region in order to describe the FDCs of ungauged basins. In this work, six small- and medium-sized catchments in North China are selected as the study area, and three functions, i.e., log normal, generalized Pareto and H2018 functions, were chosen to fit the FDC at nineteen hydrological stations. The relationship between the parameters of the FDC and the basin characteristics, such as the climatic factors and geographical features, were analyzed. A regression formula of the parameters of the FDC function was established, and its spatial and temporal distributions were examined. Based on the evaluation of four indicators, the Nash–Sutcliffe efficiency, the root mean square relative error, the logarithmic Nash efficiency coefficient and the coefficient of determination, the results demonstrate that the H2018 function can match FDCs the best. Through the annual runoff, annual precipitation, precipitation in summer, potential evapotranspiration, catchment area, mean elevation, length of the main channel and maximum flow frequency, the parameters of a, b, and k in the H2018 function can be formulated. The regression formula constructed in this study can obtain a regional flow duration curve with satisfactory performance, which provides a reference for the validation of remote-sensing-based runoff data in ungauged regions.

MODELAGEM DA DISTRIBUIÇÃO DIAMÉTRICA DE ESPÉCIES ARBÓREAS EM FRAGMENTO DE FLORESTAS ESTACIONAL SEMIDECIDUAL

ABSTRACT Modeling diameter distribution in natural forests is an important tool for understanding the native woody species dynamics, supporting decision-making for degraded lands management and restoration. Therefore, this work aimed to fit probabilistic density functions to evaluate the diameter structure of three tree species with economic interest, such as Campomansesia xanthocarpa Marl. Ex. O. Berg, Piptadenia gonoacantha (Mart.) J.F. Macbr. and Zeyheria tuberculosa (Vell.) Bureau ex Verl., in a Semideciduous Seasonal Forest fragment at São Paulo State, Brazil. The data came from 83 temporary plots of 10 m x 20 m systematically distributed along 164 ha. Log-normal, Gamma, and Weibull functions were fitted to the three species using the fitdistrplus-package in the R program. Kolmogorov-Smirnov's adherence test was used to evaluate the fits at a 5% probability level. The functions were selected employing Akaike's Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC), in addition to a graphical analysis of the fitted functions. The results indicated that the three species diameter structure is positively asymmetric, representing the exponential pattern, representing continuous natural regeneration. AIC and BIC statistics indicated the Log-normal function to describe the diameter distribution of C. xanthocarpa and Z. tuberculosa, while the Gamma function was the most appropriate for P. gonoacantha. For the three species, the graphical analysis showed the Gamma function results in the best fit without tendency for estimating frequency density per diameter class.

Global dynamics and density function in a class of stochastic SVI epidemic models with Lévy jumps and nonlinear incidence

<abstract><p>The paper studies the global dynamics and probability density function for a class of stochastic SVI epidemic models with white noise, Lévy jumps and nonlinear incidence. The stability of disease-free and endemic equilibria for the corresponding deterministic model is first obtained. The threshold criteria on the stochastic extinction, persistence and stationary distribution are established. That is, the disease is extinct with probability one if the threshold value $ R_{0}^{s} < 1 $, and the disease is persistent in the mean and any positive solution is ergodic and has a unique stationary distribution if $ R_{0}^{s} > 1 $. Furthermore, the approximate expression of the log-normal probability density function around the quasi-endemic equilibrium of the stochastic model is calculated. A new technique for the calculation of the probability density function is proposed. Lastly, the numerical examples and simulations are presented to verify the main results.</p></abstract>

Micromechanics of non-embedded spruce wood: Novel polishing and indentation protocol

Since the start of nanoindentation in wood, cell walls have been supported by epoxy resin during the microtoming of surfaces. However, wood embedding in epoxy resin was initially developed to harvest thin sections destined for transmission electron microscopy. In nanoindentation, embedding can affect the identified mechanical properties and might be difficult for modified wood samples. For this reason, we propose a protocol that does not involve resin within the wood specimen, uses a more robust miller to achieve surfaces with root means squared roughness of around 35 nm, and can exclude invalid edge indents through a novel evaluation procedure based on fitting 2D histograms with a linear combination of bi-variate lognormal probability density functions. We examined twelve samples of Norway spruce (Picea abies) late wood and their counterparts soaked with a biological preservation pretreatment. A nanoindentation protocol on every sample led to a mean reduced modulus and mean hardness of around 12 % and 26 %, respectively, lower than those found in the literature for embedded Norway spruce. These results agree well with previous results found for Loblolly pine (Pinus taeda L.) when comparing embedded and non-embedded specimens.

Probability Distribution of SARS-Cov-2 (COVID) Infectivity Following Onset of Symptoms: Analysis from First Principles

The phasing out of protective measures by governments and public health agencies, despite continued seriousness of the coronavirus pandemic, leaves individuals who are concerned for their health with two basic options over which they have control: 1) minimize risk of infection by being vaccinated and by wearing a face mask when appropriate, and 2) minimize risk of transmission upon infection by self-isolating. For the latter to be effective, it is essential to have an accurate sense of the probability of infectivity as a function of time following the onset of symptoms. Epidemiological considerations suggest that the period of infectivity follows a lognormal distribution. This proposition is tested empirically by construction of the lognormal probability density function and cumulative distribution function based on quantiles of infectivity reported by several independent investigations. A comprehensive examination of a prototypical ideal clinical study, based on general statistical principles (the Principle of Maximum Entropy and the Central Limit Theorem) reveals that the probability of infectivity is a lognormal random variable. Subsequent evolution of new variants may change the parameters of the distribution, which can be updated by the methods in this paper, but the form of the probability function is expected to remain lognormal as this is the most probable distribution consistent with mathematical requirements and available information.