Logarithmic Distribution Research Articles

Benford’s law in atomic spectra and opacity databases

The intriguing law of anomalous numbers, also named Benford’s law, states that the significant digits of data follow a logarithmic distribution favoring the smallest values. In this work, we test the compliance with this law of the atomic databases developed at the National Institute of Standards and Technology (NIST) focusing on line energies, oscillator strengths, Einstein coefficients and radiative opacities. The considered databases are the Atomic Spectra Database (ASD) and the NIST-LANL (Los Alamos National Laboratory) Lanthanide/Actinide Opacity Database. The present study is not limited to the first digit and the case of multipole lines is also considered. The fact that the law is verified with a high accuracy means that the occurrence of digits reflects the constraints induced, in a given angular-momentum coupling, by the selection rules for atomic processes. As a consequence, Benford’s law may be of great interest to detect inconsistencies in atomic databases.

Effect of nano-SiO2 particles on THF hydrate induction time and its distribution

Clathrate hydrate has been studied for a few decades as a potential alternative energy source and gas storage medium. The direct engineering application of hydrate is still struggling due to the slow formation rate and the ambiguity of induction time. To understand the effect of nanoparticles on the hydrate nucleation induction time, we used different amounts and magnitude sizes of nano-SiO2 particles as promoter to observe the Tetrahydrofuran (THF) hydrate induction time under repeated experiments. We conducted a statistical analysis method based on Kolmogorov-Smirnov (K–S) test and probability-probability plot (P–P plot) test for the analysis of induction time. The results showed that the logarithmic normal distribution can be used to describe the distribution of THF hydrate induction time with addition of nano-SiO2 under repeated experiments. The expected value and variance of induction time vary with particle sizes of different magnitudes at different mass concentrations, showing different effects of nano-SiO2 on hydrate formation and dispersion of induction time. There is a critical condition (with different sizes or mass concentrations) for the influence of nano-SiO2 particles on the THF hydrate induction time, which determines the promoting or inhibiting effect of nano-SiO2 particles on THF hydrate formation.

Research on calculation model for ultimate bearing capacity of tensile type anchor cables and the shape of internal fracture surface in anchorage segment

AbstractThe generalized model and parameter equation for the internal fracture surface of tensile type anchor cable anchorage section are constructed, and the calculation model of ultimate bearing capacity is derived based on the principle of limit equilibrium. The shape of the internal fracture surface in the anchorage section and the expression of its parameter equation are experimentally studied, and the variation law of the parameter equation and fracture range with the diameter of the anchor cable and confining pressure is analyzed. The parameter equation is substituted into the calculation model to realize the calculation and verification of the model. Through theoretical derivation, a calculation model of ultimate bearing capacity of anchor cable is obtained, which comprehensively considers the average shear strength of anchorage section, the shape of fracture surface, fracture range and confining pressure. The tests show that according to the magnitude of the confining pressure on the anchorage section, the internal fracture surface takes on two shapes: cylindrical and trumpet‐shaped, the parameter equation of trumpet‐shaped fracture surface obeys logarithmic distribution approximately. The coefficient a decreases with the increase of confining pressure but increases with the increase of anchor cable diameter, the constant b increases with the increase of confining pressure but decreases with the increase of anchor cable diameter. The ranges of horizontal and vertical fracture decrease with the increase of confining pressure, and increase with the increase of anchor cable diameter. The deviations between the theoretical calculation and the experimental results are less than 6.5%.

Frequency distribution of naturally occurring arsenic leached from excavated rocks of Hokkaido Shinkansen tunnels between Shin-Hakodate-Hokuto and Oshamambe

Excavated rock produced during tunnel construction projects in volcanic areas and sedimentary rock of marine origin pose the potential hazard of introducing naturally occurring arsenic (As) into the surrounding soils and groundwater. Thus, appropriate management and/or countermeasures against As mobility is required by comprehending the leaching characteristics of the As in excavated and sedimentary rock. This study focused on the Neogene Miocene sedimentary rock of marine origin that is widely distributed between Shin-Hakodate-Hokuto and Oshamambe Stations along the Hokkaido Shinkansen in Hokkaido, Japan. The probability distribution of the leaching concentration of As from the rock excavated from the tunnels between the above stations was evaluated via batch leaching tests using crushed rock as a countermeasure against the rock containing naturally occurring toxic elements. A statistical evaluation of this probability distribution was conducted using a vast amount of test results. The results showed that the As leaching concentration followed a logarithmic normal distribution depending on the geological formation. The concentration was influenced by various factors, including the As content, key chemical composition in the rock, such as calcium (Ca), iron (Fe), and sulfur (S), as well as the pH and electrical conductivity (EC) in the leachate. In addition, the thickness of the overburden was found to impact the As leaching concentration. These results suggest that reasonable countermeasures against the risks of As leaching into the surrounding environment can be devised by considering the probability distribution of the As leaching concentration.

Water dynamics in solutions of linear poly (N-isopropyl acrylamide) studied by 2H NMR field-cycling relaxometry

We use 2H nuclear magnetic resonance to study the dynamics of deuterated water in a solution of linear poly (N-isopropyl acrylamide) (pNIPAM, 4 wt%) across its coil-to-globule transition at a lower critical solubility temperature (LCST) around 32°C. In agreement with previous studies, we find that the 2H spin-lattice (T1) and, in particular, spin-spin (T2) relaxation times abruptly decrease when heating through the LCST, indicating that the polymer collapse causes an emergence of a water fraction with strongly reduced mobility. To quantify the dynamics of this slow water fraction, we exploit the fact that 2H field-cycling relaxometry allows us to measure the spectral density of the water reorientation in a broad frequency range. We find that the slow water fraction is characterised by a broad logarithmic Gaussian distribution of correlation times (σLG = 2.3), which is centred about τLG ≈ 10–9 s near the LCST. Hence, the common assumption of a Debye spectral density does not apply. We argue that a minor water fraction, which is located inside the pNIPAM globules and shows dynamics governed by the disordered polymer matrix, accompanies a major water fraction with bulk-like dynamics above the LCST. The former fraction amounts to about 0.4 water molecules per NIPAM monomer. Several findings indicate fast exchange between these bound and free water fractions on the T1 and T2 time scales.

Erosion processes and sediment source tracing on heterogeneous slopes in the red soil region of southern China

AbstractUnderstanding erosion, sediment sources and the spatial distribution along heterogeneous slopes is essential for soil and water conservation. In this study, soils derived from nodular limestone, mud‐like limestone and limestone breccia (abbreviated as SMTL, SMDL and SLBC, respectively) were utilized to establish artificial homogeneous and heterogeneous slopes. The homogeneous slopes MTL, MDL and LBC represent SMTL‐, SMDL‐ and SLBC‐filled‐ slopes, respectively. The heterogeneous slopes were abbreviated as MTL/MDL, MDL/LBC and LBC/MTL. Rare earth elements (REEs; lanthanide, cerium and ytterbium) were used to tag soils on the upper and lower slope sections to track sediment transport and deposition. The results showed that for homogeneous slopes, the LBC slope was the most severely eroded, followed by the MDL and MTL slopes. For heterogeneous slopes, the order of erosion was LBC/MTL > MTL/MDL > MDL/LBC. Upslope sections contributed a sizable portion of the total sediment yield, ranging from 51% to 79%. Over time, the difference in the sediment contribution between the upslope and downslope regions decreased and reached equilibrium. Runoff rates on heterogeneous slopes demonstrated an inverse function (R2 > 0.75) in contrast to a logarithmic distribution (R2 > 0.87) on homogeneous slopes. Heterogeneous slopes displayed minimal intersegment variation and the absence of a consistent ordering of the magnitude of hydraulic parameters. Conversely, the hydraulic parameters of homogeneous slopes were consistently ordered across the slope sections. For sediment transport, all slopes exhibited suspension‐dominated flow with percentages greater than 89%. However, the particle size distribution of the sediment of heterogeneous slopes exhibited greater diversity and complexity because of the coexistence of distinct soils. These findings underscore the importance of customized strategies for soil and water conservation in environments characterized by diverse soil parent materials.

Spatially quantifying and mapping the riverbed roughness of a tropical river via acoustic Doppler current profiler (ADCP) measurements to develop a 2-D numerical model

Field measurement of the bed roughness of tropical tidal rivers is a long and expensive process. In this study, we quantified the roughness coefficient from the acoustic Doppler current profiler (ADCP) measurements considering flow characteristics along a 240 km length of the Vu Gia Thu Bon River system, in central Vietnam. Our approach entailed the application of the logarithmic velocity distribution trend of turbulent flows to determine the bottom roughness height. Then, we applied the equation suggested by Strickler (1923) to calculate the Manning roughness coefficient. The TELEMAC-2D hydraulic model was then used to simulate the effect of the roughness coefficient on the estimated water level. Despite its uncertainty, the estimated bed roughness was sufficient for the accurate calculation of the hydraulic variables. Compared to those in previous research, the statistical indicators showed good performance in the first simulation process during both the calibration and validation periods along the river system.

Higher-order expansions of powered extremes of logarithmic general error distribution

Higher-order expansions of powered extremes of logarithmic general error distribution

Unveiling measles transmission dynamics: Insights from a stochastic model with nonlinear incidence

AbstractIn this paper, taking into account the inevitable impact of environmental perturbations on disease transmission, we primarily investigate a stochastic model for measles infection with nonlinear incidence. The transmission rate in this model follows a logarithmic normal distribution influenced by an Ornstein–Uhlenbeck (OU) process. To analyze the dynamic properties of the stochastic model, our first step is to establish the existence and uniqueness of a global solution for the stochastic equations. Next, by constructing appropriate Lyapunov functions and utilizing the ergodicity of the OU process, we establish sufficient conditions for the existence of a stationary distribution, indicating the prevalence of the disease. Furthermore, we provide sufficient conditions for disease elimination. These conditions are derived by considering the interplay between the model parameters and the stochastic dynamics. Finally, we validate the theoretical conclusions through numerical simulations, which allow us to assess the practical implications of the established conditions and observe the dynamics of the stochastic model in action. By combining theoretical analysis and numerical simulations, we gain a comprehensive understanding of the stochastic model's behavior, contributing to the broader understanding of measles transmission dynamics and the development of effective control strategies.

Characteristics of dust plasma environment at lunar south pole

Unlike the Earth, the Moon lacks is not protected from the atmosphere and global magnetic field, and will be directly exposed to complex radiation environments such as high-energy cosmic rays, solar wind, and the Earth’s magnetotail plasma. The surface of the Moon is covered with a thick layer of lunar soil, and the particles in the soil with a diameter between 30 nm–20 μm are called lunar dust. In the complex environments such as solar wind or magnetotail plasma, lunar dust carries an electric charge and becomes charged lunar dust. Charged lunar dust is prone to migration under the action of the electric field on the lunar surface. Charged migrated lunar dust is easy to adhere to the surface of instruments and equipment, resulting in visual impairment, astronauts’ movement disorders, equipment mechanical blockage, sealing failure, and material wear, which affects the lunar exploration mission. As an important lunar exploration landing site, the lunar south pole receives special solar radiation and produces a special dust plasma environment due to its special location. In order to provide an environmental reference for lunar south pole exploration, it is necessary to explore the characteristics of the dust plasma environment in the lunar south pole and its impact. In view of the lunar south pole environment, The Spacecraft Plasma Interactions Software (SPIS) software developed by the European Space Agency is used to carry out modelling and simulation in this work. Through the simulation, the logarithmic distribution of the lunar dust space density in a range of 0–200 m at the lunar south pole, the potential distribution near the lunar surface, and the spatial distribution characteristics of plasma electrons and ions are obtained. The obtained lunar dust space density and lunar surface potential are similar to the previous theoretical derivation and field detection data, so the simulation results have high reliability. The spatial potential distribution and the spatial density distribution of electrons and ions in the lunar environment with and without lunar dust are compared. Finally, the conclusions can be drawn as follows. The space potential increases with altitude increasing. The potential at 0–10 m near the lunar south pole is about –40 V, and the space potential at 100 m is about –20 V. The density of lunar dust in an altitude range below 10 m is 10<sup>7.22</sup> m<sup>–3</sup>–10<sup>4.66</sup> m<sup>–3</sup>. The electron density in the dust plasma near the lunar surface is 10<sup>5.47</sup> m<sup>–3</sup>, and the ion density is 10<sup>6.07</sup> m<sup>–3</sup>, and both increase with altitude increasing. Charged lunar dust affects the spatial distribution of lunar dust, mainly through affecting the distribution of the space electric field, which leads to difference in electron distribution, but has little effect on ions.

Probabilistic floor spectra for fully self-centering structures with flag-shaped hysteretic behavior

Self-centering structures are emerging structural systems developed for enhancing the seismic resilience of building structures by minimizing residual displacements. Various investigations were conducted to study the design methods and seismic performance of self-centering structural systems, but the research on nonstructural performance in self-centering structures is limited. This paper aims to develop the probabilistic floor spectra that can be used for designing the acceleration-sensitive nonstructural components in self-centering building structures under near-fault ground motions. To this end, comprehensive dynamic analyses were conducted to obtain the floor spectra. 320 near-fault ground motions were selected for considering the seismic excitation's uncertainty. The Anderson-Darling test was adopted to estimate the probabilistic distribution of the floor spectra. The test results indicate that the floor spectra follow the logarithmic normal distribution under earthquakes. The effects of the nonlinear properties of primary self-centering structures (including the initial period, hysteretic features, and structural damping) and the damping of the nonstructural components on the median and dispersion of floor spectra were discussed based on the parametric dynamic analysis results. The prediction model of the probability density function of the probabilistic floor spectra for self-centering structures was developed based on the artificial neural network algorithm. The software FloorSpectraNet was developed to facilitate practical applications.

Subduction Interface Earthquake Rise-Time Scaling Relations

ABSTRACT The slip duration in a fault plane, also known as the rise time (Tr), is determined in finite-fault rupture models (FFRMs) through the analysis of seismic source inversions using strong ground-motion (SGM) records and teleseismic data. For subduction interface earthquakes (megathrust), models exist that provide estimates for Tr values. The finite-source rupture model database and National Earthquake Information Center databases include FFRMs that allow for the extension of source-scaling relations. Currently, Tr versus seismic moment (M0) scaling relations specifically derived for large megathrust earthquakes in the near-source region are scarce. The relationship between stress drop and M0 is not straightforward; therefore, the logarithmic distribution of stress drop among earthquakes of different magnitudes (Mw) appears to be constant or self-similar. This self-similarity refers to a symmetry of the time-dependent fields, which remain unchanged under certain scale transformations in space and time characterized by similarity exponents and a function of the scaled variable, called the scaling function. In this study, Tr scaling has been conducted using 45 FFRMs derived from large megathrust earthquakes (Mw≥7.3) obtained from the previously mentioned databases. The scaling relation derived from the FFRMs based on SGM records closely approximates log(Tr)=const+1/3log(M0), which agrees with the self-similarity assumption for earthquake ruptures. On the other hand, the scaling relation obtained from the teleseismic dataset exhibits a smaller slope, indicating that the teleseismic data may overestimate source time characteristics compared with SGM data from seismic stations located close to the source.

Experimental and numerical study on airflow-dust migration behavior in an underground cavern group construction for cleaner production

A large amount of dust will be generated during the construction of underground cavern group. The understanding of the dust migration in the construction ventilation of the underground cavern group can help to optimize the cavern group’s ventilation design and improve the ventilation scheme of the cavern group’s rationality. In this paper, the effects of dust particle size, dust migration path, initial dust concentration in cavern, tunnel entrance wind speed and air duct outlet wind speed on dust migration were studied by combining a model test and numerical calculation. The results show that in the process of dust ventilation migration, the larger the diameter of the dust particles, the more difficult it is to discharge. The time of dust concentration reducing to 2 mg/m3 has a logarithmic distribution with ventilation conditions. The high-speed airflow from the outlet of the air duct is an important factor affecting the migration of dust. The development of airflow is divided into vortex region, recirculation region, collision region and stability region. Vortex region and collision region have lower wind speed and higher dust concentration, which are the unfavorable positions in the ventilation process. In the actual construction process, shortening the distance of dust migration can make the dust discharge quickly. The amount of dust generated by construction has a significant impact on the peak concentration of dust migration. During the ventilation process, the wind speed at the entrance of the tunnel and at the outlet of the air duct has a significant impact on the migration time of dust, at the same time, the ventilation efficiency should be considered while selecting the wind speed.

Regression of the Rician Noise Level in 3D Magnetic Resonance Images from the Distribution of the First Significant Digit

This paper investigates the distribution characteristics of Fourier, discrete cosine, and discrete sine transform coefficients in T1 MRI images. This paper reveals their adherence to Benford’s law, characterized by a logarithmic distribution of first digits. The impact of Rician noise on the first digit distribution is examined, which causes deviations from the ideal distribution. A novel methodology is proposed for noise level estimation, employing metrics such as the Bhattacharyya distance, Kullback–Leibler divergence, total variation distance, Hellinger distance, and Jensen–Shannon divergence. Supervised learning techniques utilize these metrics as regressors. Evaluations on MRI scans from several datasets coming from a wide range of different acquisition devices of 1.5 T and 3 T, comprising hundreds of patients, validate the adherence of noiseless T1 MRI frequency domain coefficients to Benford’s law. Through rigorous experimentation, our methodology has demonstrated competitiveness with established noise estimation techniques, even surpassing them in numerous conducted experiments. This research empirically supports the application of Benford’s law in transforms, offering a reliable approach for noise estimation in denoising algorithms and advancing image quality assessment.

A simple stress field solution for conical indentation

Classical contact mechanics has revealed the theoretical conical indentation solution of a stress field for ideal elastic–plastic materials based on the expansion cavity model and elastoplastic theory. This study proposes an equivalent stress analytical equation for the representative volume element of the median energy density point under conical indentation. For materials whose constitutive relations conform to power-law hardening, the equivalent stress solution is defined as the characteristic stress factor, and a simple stress field solution characterized by the logarithmic distribution law of dimensionless stress for conical indentation is presented. The verification results of finite element analysis showed that the proposed solution can uniformly describe the distribution of equivalent stress and three principal stresses in the material deformation region near the cone tip.

Last Digit Tendency: Lucky Numbers and Psychological Rounding in Mobile Transactions

The distribution of digits in numbers obtained from different sources reveals interesting patterns. The well-known Benford’s law states that the first digits in many real-life numerical data sets have an asymmetric, logarithmic distribution in which small digits are more common; this asymmetry diminishes for subsequent digits, and the last digit tends to be uniformly distributed. In this paper, we investigate the digit distribution of numbers in a large mobile transaction data set with 835 million mobile transactions and payments made by approximately 460,000 users in more than 300 cities. Although the first digits of the numbers in these mobile transactions follow Benford’s law, the last digit has a strong tendency to be a lucky number or be influenced by psychological rounding. This lucky number tendency is more significant in transactions that are more strongly connected to social interactions, such as money sent as gifts or as “red envelopes” (a traditional method of gift-giving during Chinese holidays), and in transactions by individuals with potentially greater emotional needs, such as during COVID-19 outbreaks and natural disasters. This psychological rounding tendency is more common in online e-commerce payments, in-store purchases, and money transfers between individuals. These findings are key for understanding the last digit tendency and its psychological and emotional mechanisms, which could be used as an indicator of public sentiment or in methods of detecting fraudulent business activity.

Digital backpropagation based on binary logarithmic step size distribution for fibre non‐linearity compensation

AbstractCapacity crunch has become critical in recent years as commercial communication systems approach their theoretical data rate limits. This work presents a low‐complexity digital backpropagation (DBP) implementation approach based on step size distribution that uses a binary logarithmic step size method to achieve high data rate optical transmission. The proposed scheme shows performance improvements (∆Q) of 2.36, 1.19, and 0.71 dB over linear compensation, constant step size (CSS) DBP, and logarithmic step size DBP techniques in a 2400 km 112 Gbit/s DP‐16 quadrature amplitude modulation (QAM) system, respectively. At 13 dBm, a high performance (Q) of 10.9 dB (BER = 2.25 × 10−4) is achieved, above the 3.80 × 10−3 hard‐decision forward error correction (HD‐FEC) limit, using the proposed scheme. Also, the allowable transmission distance is extended by 960 km at the HD‐FEC limit over the linear compensation technique. The optimization achieves a 38% saving in the number of DBP calculation steps compared to the CSS DBP, which considerably reduces the computational cost since a few steps are required for effective non‐linearity compensation.

Statistical Analysis of Step Stress Acceleration Test Data

The step stress accelerated life test is the selection of a set of accelerated stress levels that are higher than normal stress levels. At the beginning of the test, a certain number of samples are placed under a certain stress level for life testing. After some time, the stress level is increased, and this continues until a certain number of samples fail. In this paper, the acceleration coefficient is used to convert the failure data under each acceleration stress level into life data. The life distribution is exponential distribution, Weibull distribution, and logarithmic normal distribution. Under a certain number of samples, the statistical analysis of step stress accelerated life test data is completed, and the estimation of each reliability index is calculated.

Certain results concerning (p, q)-parameterized beta logarithmic function and their properties

The primary object of this article is to introduce (p, q)-beta logarithmic function with extended beta function by using the logarithmic mean. We evaluate different properties and representations of beta logarithmic function. Further, it is evaluated logarithmic distribution, hypergeometric and confluent hypergeometric functions via logarithmic mean are evaluated and their essential properties are studied. Numerous formulas of (p, q)-beta logarithmic functions such as integral formula, derivative formula, transformation formula and generating function are analyzed.

Analysis of insoluble particles in hailstones in China

Abstract. Insoluble particles influence weather and climate by means of heterogeneous freezing process. Current weather and climate models face considerable uncertainties in freezing-process simulation due to limited information regarding species and number concentrations of heterogeneous ice-nucleating particles, particularly insoluble particles. Here, for the first time, the size distribution and species of insoluble particles are analyzed in 30 shells of 12 hailstones collected from China using scanning electron microscopy and energy-dispersive X-ray spectrometry. A total of 289 461 insoluble particles were detected and divided into three species – organics, dust, and bioprotein – utilizing machine learning methods. The size distribution of the insoluble particles of each species varies greatly among the different hailstones but little in their shells. Further, a classic size distribution of organics and dust followed logarithmic normal distributions, which could potentially be adapted in future weather and climate models despite the existence of uncertainties. Our findings highlight the need for atmospheric chemistry to be considered in the simulation of ice-freezing processes.