We give two examples of the appearance of heavy-tailed distributions in applications to social sciences. Among these distributions are the laws of Pareto, Lotka, and some new ones. The examples are illustrated by constructing suitable toy models.

This article presents six probability distributions from the Gamma family with three parameters, for the flood frequency analysis in hydrology. The choice of the Gamma family of statistical dis-tributions was driven by its frequent use in hydrology. In the Faculty of Hydrotechnics, the im-provement of the estimation of maximum flows and including the methodological bases for the realization of a regionalization study with the linear moments method with the corrected pa-rameters was researched, being an element of novelty. The linear moments method is better than MOM because it avoids the choice of skewness depending on the origin of the flows, practiced in Romania. The L-moments method conforms to the current trend for estimating the parameters of statistical distributions. Observed data from hydrometric stations are of relatively short length, so the statistical parameters that characterize them are of a sample that requires correction. The correction of the statistical parameters is proposed, using the method of least squares based on the inverse functions of the statistical distributions expressed with the frequency factor for L-moments. All the necessary elements for their use are presented like, quantile functions, the exact and ap-proximate relations for estimating parameters and frequency factors. A flood frequency analysis case study was carried out for the Ialomita river, to verify the proposed methodology. The per-formance of this distributions is evaluated using Kiling-Gupta and Nash-Sutcliff coefficients.

Estimating the parameters of statistical distributions generally involves solving a system of nonlinear equations or a nonlinear equation, being a technical difficulty in their usual application in hydrology. This article presents improved approximations and, in some cases, new approximations for the estimation with the method of ordinary moments and the method of linear moments, which are useful for the direct calculation of the parameters, because the errors in the approximate estimation are similar to the use of iterative numerical methods. Thirteen statistical distributions of two and three parameters frequently used in hydrology are presented, for which parameter estimation was laborious. Thus, the approximate estimation of the parameters by the two methods is simple but also precise and easily applicable by hydrology researchers. The approximate forms characterized by very small errors are the result of the research conducted within the Faculty of Hydrotechnics to update the romanian normative standards in the hydrotechnical field.

This article analyzes 6 probability distributions from the Generalized Pareto family, with 3, 4 and 5 parameters, with main purpose to identify other distributions from this family with applicability in flood frequency analysis compared to the distribution already used in the literature from this family such as Generalized Pareto Type II and Wakeby. This analysis is part of a larger and more complex research carried out in the Faculty of Hydrotechnics regarding the elaboration of a norm for flood frequency analysis using the linear moments method. In Romania, the standard method of parameter estimation is the method of ordinary moments, thus the transition from this method to the method of linear moments is desired. All the necessary elements for the distributions use are presented like, the probability density functions, the complementary cumulative distribution functions, the quantile functions, the exact and approximate relations for estimating parameters, for both methods of parameters estimation. All these elements are necessary for a proper transition between the two methods, especially since the use of the method of ordinary moments is done by choosing the skewness of the observed data depending on the origin of the maximum flows. A flood frequency analysis case study, using annual maximum and annual exceedance series, was carried out for the Prigor river, to numerically present the analyzed distributions. The performance of this distributions is evaluated using relative mean error, relative absolute error and linear moments diagram.

Most existing flexible count regression models allow only approximate inference. This work proposes a new framework to provide an exact and flexible alternative for modeling and simulating count data with various types of dispersion (equi-, under- and overdispersion). The new method, referred as “balanced discretization”, consists in discretizing continuous probability distributions while preserving expectations. It is easy to generate pseudo random variates from the resulting balanced discrete distribution since it has a simple stochastic representation in terms of the continuous distribution. For illustrative purposes, we have developed the family of balanced discrete gamma distributions which can model equi-, under- and overdispersed count data. This family of count distributions is appropriate for building flexible count regressionmodels because the expectation of the distribution has a simple expression in terms of the parameters of the distribution. Using the Jensen–Shannon divergence measure, we have shown that under equidispersion restriction, the family of balanced discrete gamma distributions is similar to the Poisson distribution. Based on this, we conjecture that while covering all types of dispersion, a count regression model based on the balanced discrete gamma distribution will allow recovering a near Poisson distribution model fit when the data is Poisson distributed.

Understanding swash zone dynamics is of crucial importance for coastal management as the swash motion, consisting of the uprush of the wave on the beach face and the subsequent downrush, is responsible for driving changes the beach morphology trough sediment exchanges between the sub-aerial and sub-aqueous beach. Improved understanding of the probabilistic characteristics of these motions has the potential to allow coastal engineers to develop improved sediment transport models which, in turn, can be further developed into coastal management tools. In this paper, novel descriptors of swash motions are obtained by combining field data and statistical modelling. Our results indicate that the probability distribution function (PDF) of shoreline height (p(ζ)) and trough-to-peak swash heights (p(ρ)) measured at a high energy, sandy beach were both inherently multimodal. Based on the observed multimodality of these PDFs, Gaussian Mixtures are shown to be the best method to statistically model them. Further, our results show that both offshore and surf zone dynamics are responsible for driving swash zone dynamics, which indicates unsaturated swash. The novel methods and results developed in this paper, both data collection and analysis, could aid coastal managers to develop improved swash zone models in the future.

White noise is fundamentally linked to many processes, it has a flat power spectral density and a delta-correlated autocorrelation. Operators acting on white noise can result in coloured noise, whether they operate in the time domain, like fractional calculus, or in the frequency domain, like spectral processing. We investigate whether any of the white noise properties remain in the coloured noises produced by the action of an operator. For a coloured noise, which drives a physical system, we provide evidence to pinpoint the mother process from which it came. We demonstrate the existence of two indices, that is, kurtosis and codifference, whose values can categorise coloured noises according to their mother process. Four different mother processes are used in this study: Gaussian, Laplace, Cauchy, and Uniform white noise distributions. The mother process determines the kurtosis value of the coloured noises that are produced. It maintains its value for Gaussian, never converges for Cauchy, and takes values for Laplace and Uniform that are within a range of its white noise value. In addition, the codifference function maintains its value for zero lag-time essentially constant around the value of the corresponding white noise.

Panel data of our interest consist of a moderate number of panels, while the panels contain a small number of observations. An estimator of common breaks in panel means without a boundary issue for this kind of scenario is proposed. In particular, the novel estimator is able to detect a common break point even when the change happens immediately after the first time point or just before the last observation period. Another advantage of the elaborated change point estimator is that it results in the last observation in situations with no structural breaks. The consistency of the change point estimator in panel data is established. The results are illustrated through a simulation study. As a by-product of the developed estimation technique, a theoretical utilization for correlation structure estimation, hypothesis testing, and bootstrapping in panel data is demonstrated. A practical application to non-life insurance is presented as well.

In this paper, four probability functions are compared with the purpose of establishing a methodology to improve the accuracy of the estimated wind energy in a desert city in northwestern Mexico. For the statistical modeling, 3 time series of wind speed data were used, corresponding to the years 2017, 2018, and 2019, recorded with a sonic anemometer at a sampling frequency of 10 Hz. From these, the analysis is performed with different stationarity periods (5, 30, 60, and 600 seconds). The estimation of the parameters characterizing the probability density functions was performed using different methods; the statistical models were evaluated by the coefficient of determination and the Nash-Sutcliffe efficiency coefficient, while their accuracy by the measured quadratic error, the mean square error, the mean absolute error, and the mean absolute percentage error. It was found that Weibull with the energy pattern factor method and Gamma using the method of moments were the probability density functions that best describe the statistical behavior of the wind speed and, therefore, better estimate the energy generated, so it is concluded that the proposed methodology will allow having greater confidence, both in the estimation of the wind speed and in the small wind energy available for its use.

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc…Using the PBN, many formulas, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman-Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential Applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.

The Rényi entropy functional was optimised for the novel findings of this investigation under various conditions. Their research's findings suggest that the complete spectrum of Lévy stable distributions is covered by generalised t-distributions. Additionally, an exposition was undertaken to prove that the Lévy distribution generalizes Tsallisian distribution, rather than the reverse. The current study is a strong generalization of an existing research work in the literature.

The philosophy of testing statistical hypothesis is a natural consequence and functional extension of mathematical analysis of Probability. Along with the concept of recurrence when applied to random sequences and functions, it leads to the analysis of a priori and posterior which implies testing statistical hypothesis. Testing statistical hypothesis also involves algebraic, functional and dimensional considerations, which are found in the works of Laplace. Aspects of mathematical analysis such as universality of solutions, Laws of Large Numbers, Entropy, Information, and various functional dependencies are the main factors explained in the five properties that lead to implication of testing statistical hypothesis. Various interesting examples with modern scientific significance from genetics, astrophysics, and other areas give methodological access to answers of different problems and phenomena which are involved in the logic of testing statistical hypothesis.

This study has developed a Matlab application for simulating statistical models project (SMp) probabilistic distributions that are similar to binomial and Poisson, which were created by mathematical procedures. The simulated distributions are graphically compared with these popular distributions. The application allows to obtain many probabilistic distributions, and shows the trend (τ ) for n trials with success probability p, i.e. the maximum probability as τ=np. While the Poisson distribution PD(x;µ) is a unique probabilistic distribution, where PD=0 in x=+∞, the application simulates many SMp(x;µ,Xmax) distributions, where µ is the Poisson parameter and value of x with generally the maximum probability, and Xmax is the upper limit of x with SMp(x;µ,Xmax) ≥ 0 and limit of the stochastic region of a random discrete variable. It is shown that by simulation via, one can get many and better probabilistic distributions than by mathematical one.

This study has developed a Matlab application for simulating statistical models project (SMp) probabilistic distributions that are similar to binomial and Poisson, which were created by mathematical procedures. The simulated distributions are graphically compared with these legendary distributions. The application allows to obtain many probabilistic distributions, and shows the trend (τ ) for n trials with success probability p, i.e. the maximum probability as τ=np. While the Poisson distribution PD(x;µ) is a unique probabilistic distribution, where PD=0 in x=+∞, the application simulates many SMp(x;µ,Xmax) distributions, where µ is the Poisson parameter and value of x with generally the maximum probability, and Xmax is upper limit of x with SMp(x;µ,Xmax) ≥ 0 and limit of the stochastic region of the random discrete variable X. It is shown that by simulation via, one can get many and better probabilistic distributions than by mathematical one.

Statistical modeling of lifetime data plays an essential role in a wide range of practical fields, such as health and engineering. There have been a lot of studies done to develop statistical models that can better describe health data than traditional models. For the first time, we pioneer a novel family of continuous probability distributions called the generalized odd beta prime generalized (GOBP-G) family of distributions. The cumulative distribution and probability density functions of the new family are presented. A new generalization of the Weibull distribution called "generalized odd beta prime-Weibull" (GOBPW) is proposed using the pioneered GOBP-G family. The mixture representations of the new distribution are defined and derived. Some formal statistical properties of the GOBPW distribution, such as the moments, moment generating function, incomplete moments, information generating function, entropies, stress-strength function, quantile function, and order statistics, are derived. The estimation of the parameters of the proposed distribution is evaluated using the maximum likelihood estimation approach. Different cancer disease data sets, such as the bladder, head and neck, acute bone, and blood cancers, are used to illustrate the applicability and usefulness of the new model and were compared using several statistical accuracy measures with that of well-established extended Weibull distributions, which are the beta modified Weibull distribution, Kumaraswamy modified Weibull distribution, gamma generalized modified Weibull distribution, gamma log-logistic Weibull distribution, and beta log-logistic Weibull distribution. The results show that the proposed model gives better results than the competitive models. This study could guide the relevant stakeholders in choosing a suitable statistical model for the health data instead of relying on traditional models to enhance decision-making.

Developing new ways to estimate probabilities can be valuable for science, statistics, and engineering. By considering the information content of different output patterns, recent work invoking algorithmic information theory has shown that a priori probability predictions based on pattern complexities can be made in a broad class of input-output maps. These algorithmic probability predictions do not depend on a detailed knowledge of how output patterns were produced, or historical statistical data. Although quantitatively fairly accurate, a main weakness of these predictions is that they are given as an upper bound on the probability of a pattern, but many low complexity, low probability patterns occur, for which the upper bound has little predictive value. Here we study this low complexity, low probability phenomenon by looking at example maps, namely a finite state transducer, natural time series data, RNA molecule structures, and polynomial curves. Some mechanisms causing low complexity, low probability behaviour are identified, and we argue this behaviour should be assumed as a default in the real world algorithmic probability studies. Additionally, we examine some applications of algorithmic probability and discuss some implications of low complexity, low probability patterns for several research areas including simplicity in physics and biology, a priori probability predictions, Solomonoff induction and Occam's razor, machine learning, and password guessing.

The development of one technology can be portrayed by common methods like Gartner’s hype cycle or S-curve, however, there is no method to characterize the feature distributions of multiple technologies within a specific domain. This study proposes a big data-based method in terms of four proposed features, namely versatility, significance, commerciality, and disruptiveness, to characterize the technologies within a given domain. The features of technologies are quantitively portrayed using the representative keywords and volumes of returned search results from Google and Google Scholar in two-dimensional analytical spaces of technique and application. We demonstrate the applicability of this method using 452 technologies in the domain of intelligent robotics. The results of our assessment indicate that the versatility values are normally distributed, while the values of significance, commerciality, and disruptiveness follow power-law distributions, in which few technologies possess higher feature values. We also show that significant technologies are more likely to be commercialized or causing potential disruption, as such technologies have higher scores in these features. Further, we validly prove the robustness of our approach via comparing historical trends with literature and characterizing technologies in reduced analytical spaces. Our method can be widely applied in analyzing feature distributions of technologies in different domains, and it can potentially be exploited in decisions like investment, trade, and science policy.

We find that the standard relative entropy and the Umegaki entropy are designed for the purpose of inferentially updating probability and density matrices respectively. From the same set of inferentially guided design criteria, both of the previously stated entropies are derived in parallel. This formulates a quantum maximum entropy method for the purpose of inferring density matrices in the absence of complete information.

This paper discusses about the quantum energy levels of muonic hydrogen considering only Coulomb interaction. The computational analysis in this paper will help the reader get a good understanding of how different a muonic hydrogen is from its electronic counterparts. Energy eigenvalues are calculated by numerical method and the probability density distribution of muon around the nucleus is plotted for different quantum states using special functions. The code is written in python. A comparative study of emission spectra between the muonic hydrogen and hydrogen has also been discussed.

In recent years, with the rapid development of technology, artificial intelligence(AI) security issues represented by adversarial sample attack have aroused widespread concern in society. Adversarial samples are often generated by surrogate models and then transfer to attack the target model, and most AI models in real-world scenarios belong to black boxes, thus transferability becomes a key factor to measure the quality of adversarial samples. The traditional method relies on the decision boundary of the classifier and takes the boundary crossing as the only judgment metric without considering the probability distribution of the sample itself, which results in an irregular way of adding perturbations to the adversarial sample, an unclear path of generation, and a lack of transferability and interpretability. In the probabilistic generative model, after learning the probability distribution of the samples, a random term can be added to the sampling to gradually transform the noise into a new independent and identically distributed sample. Inspired by this idea, we believe that by removing the random term, the adversarial sample generation process can be regarded as static sampling of the probabilistic generative model, which guides the adversarial samples out of the original probability distribution and into the target probability distribution, and helps to improve transferability and interpretability. Therefore, we propose a Score Matching-Based Attack(SMBA) method to perform the adversarial sample attacks by manipulating the probability distribution of the samples, which can show good transferability in the face of different datasets and models, and give reasonable explanations from the perspective of mathematical theory and feature space. In conclusion, our research establishes a bridge between probabilistic generative models and adversarial samples, provides a new entry angle for the study of adversarial samples, and brings new thinking to AI security.

Ranked Set Sampling (RSS) is a useful technique for improving the estimator of population mean when the sampling units in a study can be easily ranked than the actual measurement. RSS performs better than simple random sampling (SRS) when the mean of units corresponding to each rank is used. The performance of RSS can be increased further by assigning weights to the ranked observations. In this paper, we propose weighted RSS procedures to estimate the population mean of positively skew distributions. It is shown that the gain in the relative precisions of the population mean for chosen distributions are uniformly higher than those based on RSS. The gains in relative precisions are substantially high. Further, the relative precisions of our estimator are slightly higher than the ones based on Neyman’s optimal allocation model for small sample sizes. Moreover, it is shown that, the performance of the proposed estimator increases as the skewness increases by using the example of lognormal family of distributions.

This work reports measurements of calcified gallstone elemental compositions using laser-induced optical emission spectroscopy. The experimental results support the importance of the magnesium concentration in gallstone growth. Granular stones reveal an increased magnesium concentration at the periphery of the granules, suggesting the inhibition of further growth. Non-granular gallstones reveal lower overall magnesium concentrations but with higher values near the center.

For evaluating probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based upon the Gauss-Laplace law. The latter will be considered here as an element of the newly introduced family of (p,q)-spherical distributions. Based upon a suitably defined non-Euclidean arc-length measure on (p,q)-circles we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly like with elliptically contoured distributions and more general homogeneous star-shaped ones. This is demonstrated at hand of a generalization of the Box-Muller simulation method. En passant, we prove an extension of the sector and circle number functions.

This paper addresses the issues of how we can quantify structural information for nonparametric distributions and how we can detect its changes. Structural information refers to an index for a global understanding of a data distribution. When we consider the problem of clustering using a parametric model such as a Gaussian mixture model, the number of mixture components (clusters) can be thought of as structural information in the model. However, there does not exist any notion of structural information for nonparametric modeling of data. In this paper we introduce a novel notion of {\em kernel complexity} (KC) as structural information in the nonparametric setting. The key idea of KC is to combine the information bias inspired by the Gini index with the information quantity measured in terms of the normalized maximum likelihood (NML) code length. We empirically show that KC has a property similar to the number of clusters in a parametric model. We further propose a framework for structural change detection with KC in nonparametric distributions. With synthetic and real data sets we empirically demonstrate that our framework enables us to detect structural changes underlying the data and their early warning signals.

Based on recent contributions regarding the treatment of unsteady hydraulic conditions into the state-of-the-art of scour literature, the theoretically derived probability distribution of bridge scour is introduced. The model is derived assuming a rectangular hydrograph shape with a given duration, and random flood peak following a Gumbel distribution. A model extension for a more complex flood event is also presented, assuming a synthetic exponential hydrograph shape. The mathematical formulation can be extended to any flood-peak probability distribution. The aim of the manuscript is to move forward the current approaches adopted for the bridge design coupling hydrological, hydraulic, and erosional models in a mathematical closed form.

Philippines as an archipelago and tropical country, which is situated near the Pacific ocean, faces uncertain rainfall intensities. This makes environmental, agricultural and economic systems affected by precipitation difficult to manage. Time series analysis of Philippine rainfall pattern has been previously done, but there is no study investigating its probability distribution. Modeling the Philippine rainfall using probability distributions is essential, especially in managing risks and designing insurance products. Here, daily and cumulative rainfall data (January 1961 - August 2016) from 28 PAGASA weather stations are fitted to probability distributions. Moreover, the fitted distributions are examined for invariance under subsets of the rainfall data set. We observe that the Gamma distribution is a suitable fit for the daily up to the ten-day cumulative rainfall data. Our results can be used in agriculture, especially in forecasting claims in weather index-based insurance.

Four decades of snooker rankings are analyzed in terms of prize money distributions. In particular, it is investigated how much these rankings are dominated by the top players, and also by players from the UK. The main conclusions are: (1) with an increasing amount of available prize money, the rankings are less dominated by the top players, and (2) players from the UK have historically earned a significantly larger share of the available prize money than could be reasonably expected based on the actual percentage of UK players.

Congestion pricing has received lots of attention in the scientific discussion. Congestion pricing means that the operator increases prices at the time of congestion and the traffic demand is expected to decrease. In a certain sense, shadow prices are an optimal way of congestion pricing: users are charged shadow prices, i.e., the expectations of future losses because of blocked connections. The shadow prices can be calculated exactly from Howard’s equation, but this method is difficult. The paper presents simple approximations to the solution of Howard’s equation and a way to derive more exact approximations. If users do not react by lowering their demand, they will receive higher bills to pay. Many users do not react to increased prices but would want to know how the congestion pricing mechanism affects the bills. The distribution of the price of a connection follows from knowing the shadow prices and the probability of a congestion state. There is another interesting distribution. The network produces profit to the operator, or equivalently, blocked connections produce a cost to the operator. The average cost rate can be calculated from Howard’s equation, but the costs have some distribution. The distribution gives the risk that the actual costs exceed the average costs, and the operator should include this risk to the prices. The main result of this paper shows how to calculate the distribution of the costs in the future for congestion pricing by shadow prices and for congestion pricing with a more simple pricing scheme that produces the same average costs.

We propose a novel adaptive importance sampling scheme for Bayesian inversion problems where the inference of the variables of interest and the power of the data noise is split. More specifically, we consider a Bayesian analysis for the variables of interest (i.e., the parameters of the model to invert), whereas we employ a maximum likelihood approach for the estimation of the noise power. The whole technique is implemented by means of an iterative procedure, alternating sampling and optimization steps. Moreover, the noise power is also used as a tempered parameter for the posterior distribution of the the variables of interest. Therefore, a sequence of tempered posterior densities is generated, where the tempered parameter is automatically selected according to the actual estimation of the noise power. A complete Bayesian study over the model parameters and the scale parameter can be also performed. Numerical experiments show the benefits of the proposed approach.

Efficient textual data distributions (TDD) alignment and generation are open research problems in textual analytics and NLP. It is presently difficult to parsimoniously and methodologically confirm that two or more natural language datasets belong to similar distributions, and to identify the extent to which textual data possess alignment. This study focuses on addressing a segment of the broader problem described above by applying multiple supervised and unsupervised machine learning (ML) methods to explore the behavior of TDD by (i) topical alignment, and (ii) by sentiment alignment. Furthermore we use multiple text generation methods including fine-tuned GPT-2, to generate text by topic and by sentiment. Finally we develop a unique process driven variation of Kullback-Leibler divergence (KLD) application to TDD, named KL Textual Distributions Contrasts (KL-TDC) to identify the alignment of machine generated textual corpora with naturally occurring textual corpora. This study thus identifies a unique approach for generating and validating TDD by topic and sentiment, which can be used to help address sparse data problems and other research, practice and classroom situations in need of artificially generated topic or sentiment aligned textual data.

This paper investigates the randomly stopped sums, minima and maxima of heavy- and light-tailed random variables. The conditions on the primary random variables, which are independent but generally not identically distributed, and counting random variable are given in order that the randomly stopped sum, random minimum and maximum is heavy/light tailed. The results generalize some existing ones in the literature. The examples illustrating the results are provided.

This article explores the enduring mysteries of consciousness and the afterlife, two enigmatic topics that have fascinated humanity for ages. Despite extensive scientific efforts, the existence of an afterlife remains unproven, and understanding consciousness remains a significant challenge. The research introduces innovative hypotheses through simple thought experiments with empirical evidence and robust theoretical foundations. It delves into the complexities of consciousness, its relationship with the brain, and the need for interdisciplinary approaches encompassing physics, psychology, and philosophy. Boldly contemplating the probability of a continuous consciousness after death, the study argues that existing evidence strongly supports this idea. Beyond theoretical implications, it envisions practical outcomes, suggesting that insights into these mysteries could enhance human well-being and contribute to a more harmonious world. In summary, this article embarks on a quest to shed light on age-old questions about consciousness and the afterlife. Through fresh methodologies, it aims to provide new perspectives that could reshape our understanding of existence and inspire intellectuals to positive individual and societal advancements including psychological well-being and intelligence.

In public health literature the risk of death or disease associated with a dietary, environmental of behavioral factor is most commonly denoted by odds ratio (OR), hazard ratio (HR) or risk ratio (RR). The ratio indices have several desirable statistical properties. However, the most important question is whether there are some evolved innate norms of perception of risk that people use and what they are. We conducted a simple one question survey of 98 individuals with different age, sex, educational and professional backgrounds. The respondents were asked to judge the relative perceived risk of four different hypothetical habits for which data on the percentage of people affected by the disease with and without the habit was given. They were asked to rank the risks for the four habits. Results showed that the habits that had the highest difference between probability of acquiring the disease were ranked high on risk perception. The probability ratios did not affect risk perception significantly. Further age, sex, profession or formal training in statistics did not affect the response significantly. Even individuals that were formally trained to use OR and HR as risk indicators, preferred using probability differences over ratios for judging their own risk in the perceived context. This preliminary inquiry into intuitive statistical perception suggests that designing statistical indices based on people’s innate perception may be a better strategy than trying to train people to understand the indices designed by expert statisticians.

This paper presents the preservation property of the average intensity order and the preservation property of the monotonic average intensity classes under distorted distributions. Several sufficient conditions are given to get the preservation properties. It is shown that the imposed conditions are achievable as we examine in some examples. The preservation of the average intensity order and the preservation of the decreasing average intensity class under the structure of a parallel system with independent and identically distributed components' lifetime are made. A lower bound for the average intensity function of a random lifetime with an increasing failure rate in average distribution is derived. The results are applied to some semiparametric models as a particular standard family of distorted distribution.

Spatially resolved, line-of-sight measurements of aluminum monoxide emission spectra in laser ablation plasma are used with Abel inversion techniques to extract radial plasma temperatures. Contour mapping of the radially deconvolved signal intensity shows a ring of AlO formation near the plasma boundary with the ambient atmosphere. Simulations of the molecular spectra were coupled with the line profile fitting routines. Temperature results are presented with simultaneous inferences from lateral, asymmetric radial, and symmetric radial AlO spectral intensity profiles. This analysis indicates that we measured shockwave phenomena in the radial profiles, including a temperature drop behind the blast wave created during plasma initiation.

Many systems in nature and laboratories are far from equilibrium and exhibit significant fluc- tuations, invalidating the key assumptions of small fluctuations and short memory time in or near equilibrium. A full knowledge of Probability Distribution Functions (PDFs), especially time- dependent PDFs, becomes essential in understanding far-from-equilibrium processes. We consider a stochastic logistic model with multiplicative noise, which has gamma distributions as stationary PDFs. We numerically solve the transient relaxation problem, and show that as the strength of the stochastic noise increases the time-dependent PDFs increasingly deviate from gamma distributions. For sufficiently strong noise a transition occurs whereby the PDF never reaches a stationary state, but instead forms a peak that becomes ever more narrowly concentrated at the origin. The addition of an arbitrarily small amount of additive noise regularizes these solutions, and re-establishes the existence of stationary solutions. In addition to diagnostic quantities such as mean value, standard deviation, skewness and kurtosis, the transitions between different solutions are analyzed in terms of entropy and information length, the total number of statistically distinguishable states that a system passes through in time.

Identifying the SARS-CoV-2 virus has been a unique challenge for the scientific community. In this paper, we discuss a practical solution to help guide clinicians with interpretation of the probability that a positive, or negative, COVID-19 test result indicates an infected person, based on their clinical estimate of pre-test probability of infection.The authors conducted a small survey on LinkedIn to confirm that hypothesis that that the clinical pre-test probability of COVID-19 increases relative to local prevalence of disease plus patient age, known contact, and severity of symptoms. We examined results of PPA (Positive Percent Agreement, sensitivity) and NPA (Negative Percent Agreement, specificity) from 73 individual laboratory experiments for molecular tests for SARS-CoV-2 as reported to the FIND database ¹, and for selected methods in FDA EUA submissions^2,3. Authors calculated likelihood ratios to convert pre-test to post-test probability of disease and designed an online calculator to create graphics and text to report results. Despite best efforts, false positive and false negative Covid-19 test results are unavoidable^4,5. A positive or negative test result from one laboratory has a different probability for the presence of disease than the same result from another laboratory. Likelihood ratios and confidence intervals can convert the physician or other healthcare professional’s clinical estimate of pre-test probability to post-test probability of disease. Ranges of probabilities differ depending on proven method PPA and NPA in each laboratory. We recommend that laboratories verify PPA and NPA and utilize a the “Clinician’s Probability Calculator” to verify acceptable test performance and create reports to help guide clinicians with estimation of post-test probability of COVID-19.

In this work, the nearest neighbor distances and Voronoi cell features of the Cu-Ag deposits were analyzed and fitted with Lognormal, Weibull, and Gamma distributions. The nearest neighbor distance distributions of the samples were compared with those of complete spatially random points, showing spatial inhomogenity due to the nucleation exclusion effect. The radial distribution function was calculated, showing both influences from the grain size and the nucleation exclusion effect. Voronoi cells were generated based on the shape of the grains. The size, occupancy, and coordination of the Voronoi cells were examined and fitted. The results show that despite the Cu-Ag deposits seem to be governed by instantaneous nucleation mode, the spatial distribution of the nuclei is more impacted by the nucleation exclusion effect than the Cu-only samples. This behavior is also justified by the grain size distribution generated with Voronoi cell size and occupancy distributions.

Distributions with bounded support show considerable sparsity over those with unbounded support, despite the fact that there are a number of real-world contexts where observations take values from a bounded range (proportions, percentages, and fractions are typical examples). For proportion modelling, a flexible family of two-parameter distribution functions associated with the exponential distribution is proposed here. Mathematical and statistical properties of the novel distribution are examined, including quantiles, mode, moments, hazard rate function, and its characterization. The parameter estimation procedure using the maximum likelihood method was carried out, and applications to environmental and engineering data were also considered. To this end, various statistical tests are used, along with some other information criterion indicators to determine how well the model fits the data. The proposed model is found to be the most efficient plan in most cases for the datasets considered.

One of creators of quantum mechanics P. Jordan in his work on quantum biology claimed that life's missing laws were the rules of chance and probability of the quantum world. The article presents author’s results of studying frequencies (or probabilities) of nucleotides on so-called epi-chains of long DNA sequences of various eukaryotic and prokaryotic genomes. DNA epi-chains are algorithmically constructed subsequencies of DNA nucleotide sequences. According to the algorithm of construction of any epi-chain of the order n, the epi-chain is such nucleotide subsequence, in which the numerations of adjacent nucleotides differ by natural number n (n = 1, 2, 3, 4,…). Correspondingly each epi-chain of order n ≥ 2 contains n times less nucleotides than the original DNA sequence. The presented results unexpectedly discover that in long single-stranded and double-stranded DNA of any tested genome its DNA epi-chains of different orders n (values n are not too large) have practically identical frequencies (or probabilities) of each kind of nucleotides. These data allow considering DNA as a regular rich set of epi-chains, which can play a certain role in genetic and epigenetic phenomena as the author belives. Appropriate rules of nucleotide frequencies on epi-chains of long DNA sequences are formulated for further their tests on a wider set of genomes. These results testify on existence of long-range coherence in long DNA and remind the Fröhlich's theory of long-range coherence in biological systems. The phenomenological data are discussed from different standpoints: the DNA double helices and helical antennas with circular polarizations of electromagnetic waves; relations with the Fröhlich's theory; numerical analysis of DNA epi-chains under binary representations of nucleotides. Results are useful for developing quantum and algebraic biology.

This work seeks to approach the comparative analysis of the distribution of POFs (polymeric optical fibers) arranged in a random and sorted manner in the manufacture of translucent concretes. The following characteristics were analyzed: compressive strength, flexural tensile strength, capillary water absorption and light transmittance. Portland cement, water, fine aggregate (sand), silica fume, polycarboxylate ether-based superplasticizer and polymeric optical fiber were used in the production of the composites. Except the fiber content, the mixture of all other inputs was kept as a fixed parameter. Groups with the following levels of POFs were produced: 0%, 2%, 3.5% and 5%. The POF content is the sum of the areas of the fiber sections in the composite with the total area of the translucent surface. The results showed that for the analysis of mechanical behavior (flexural tensile and compressive strength) and durability (capillary water absorption), composites with random and sorted fibers within the 2% POF presented to belong to statistically distinct groups. For the other fiber contents, there was no pattern of behavior, that is, it cannot be said that there is a difference within 3.5% or 5% POF between sorted and random fibers in relation to the mechanical and durability characteristics analyzed. As for the light transmittance, it was attributed that the statistical difference between the fiber arrangements only within the 5% POF content can be a consequence of their accumulation in certain points of the material, thus influencing this characteristic measured by a light meter.

In statistics and probability theory, one of the most important statistics is the sums of random variables. After introducing a probability distribution, determining the sum of n independent and identically distributed random variables is one of the interesting topics for authors. This paper presents the probability density functions for the sum of n independent and identically distributed random variables such as Shanker, Akash, Ishita, Pranav, Rani, and Ram Awadh. In order to determine all aforementioned distributions, the problem-solving methods are applied which is based on the change-of-variables technique.

For bounded unit interval, we propose a new Kumaraswamy generalized (G) family of distributions from a new generator which could be an alternate to the Kumaraswamy-G family proposed earlier by Cordeiro and de-Castro in 2011. This new generator can also be used to develop alternate G-classes such as beta-G, McDonald-G, Topp-Leone-G, Marshall-Olkin-G and Transmuted-G for bounded unit interval. Some mathematical properties of this new family are obtained and maximum likelihood method is used for estimating the family parameters. We investigate the properties of one special model called a new Kumaraswamy-Weibull (NKwW) distribution. Parameter estimation is dealt and maximum likelihood estimators are assessed through simulation study. Two real life data sets are analyzed to illustrate the importance and flexibility of this distribution. In fact, this model outperforms some generalized Weibull models such as the Kumaraswamy-Weibull, McDonald-Weibull, beta-Weibull, exponentiated-generalized Weibull, gamma-Weibull, odd log-logistic-Weibull, Marshall-Olkin-Weibull, transmuted-Weibull, exponentiated-Weibull and Weibull distributions when applied to these data sets. The bivariate extension of the family is proposed and the estimation of parameters is given. The usefulness of the bivariate NKwW model is illustrated empirically by means of a real-life data set.

Probability is an open question in the ontological interpretation of quantum mechanics. It has been discussed in some trajectory interpretations such as Bohmian mechanics and stochastic mechanics. New questions arise when the domain of probability extends to the complex space, including the generation of complex trajectories, the definition of the complex probability, the relation of the complex probability to the real quantum probability, and so on. The complex treatment proposed here applies the optimal quantum guidance law to derive the stochastic differential (SD) equation governing the particle’s random motions in the complex plane. The ensemble of the complex quantum random trajectories (CQRTs) solved from the complex SD equation is used to construct the probability distribution ρc(t,x,y) of the particle’s position over the complex plane z=x+iy. The correctness of the obtained complex probability is confirmed by the solution of the complex Fokker-Planck equation. The significant contribution of the complex probability is that it can be used to reconstruct both quantum probability and classical probability, and to clarify their relationship. Although quantum probability and classical probability are both defined on the real axis, they are obtained by projecting complex probability onto the real axis in different ways. This difference explains why the quantum probability cannot exactly converge to the classical probability when the quantum number is large.

To generate a probability density function (PDF) for fitting probability distributions of real data, this study proposes a deep learning method which consists of two stages: (1) a training stage for estimating the cumulative distribution function (CDF) and (2) a performing stage for predicting the corresponding PDF. The CDFs of common probability distributions can be adopted as activation functions in the hidden layers of the proposed deep learning model for learning actual cumulative probabilities, and the differential equation of trained deep learning model can be used to estimate the PDF. To evaluate the proposed method, numerical experiments with single and mixed distributions are performed. The experimental results show that the values of both CDF and PDF can be precisely estimated by the proposed method.

In this paper the problem of sampling from uniform probability distributions is approached by means of space-filling curves (SFCs), a topological concept that has found a number of important applications in recent years. Departing from the theoretical fact that they are surjective but not necessarilly injective, the investigation focused upon the structure of the distributions obtained when their domain is swept in a uniform and discrete manner, and the corresponding values used to build histograms, that are approximations of their true PDFs. This work concentrates on the real interval [0,1], and the Sierpinski space-filling curve was chosen because of its favorable computational properties. In order to validate the results, the Kullback-Leibler distance is used when comparing the obtained distributions in several levels of granularity with other already established sampling methods. In truth, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. In this fashion, sequences resulting from this sort of process are not truly random.

Labelling techniques such as electron paramagnetic resonance spectroscopy and single-molecule fluorescence resonance energy transfer, allow access to distances in the range of tens of angstroms, corresponding to the size of proteins and small to medium-sized protein complexes. Such measurements do not require long-range ordering and are therefore applicable to systems with partial disorder. Data from spin-label-based measurements can be processed into distance distributions that provide information about the extent of such disorder. Using such information in modelling presents several challenges, including a small number of restraints, the influence of the label itself on the measured distance and distribution width, and balancing the fitting quality of the long-range restraints with the fitting quality of other restraint subsets. Starting with general considerations about integrative and hybrid structural modelling, this review provides an overview of recent approaches to these problems and identifies where further progress is needed.

Research for a theory of quantum gravity has recently led to the use of presheaf topos. Quantum uncertainty is linked to the truth values of intuitionistic logic. This paper proposes transposing this model into a classic probability context, that of conditional mathematical expectations. A simulation of Brownian motion is offered for illustrative purposes.

In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard Black-Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton-Jacobi-Bellman approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

In this paper, we derive the distribution of eigenvalues for a 2×2 random confusion matrix stemming from a machine learning (ML) evaluation problem. Additionally, we present the distributions of both the matrix’s trace and the difference between the two traces of random confusion matrices. We also illustrate the application of these derived distributions in calculating the superiority probability of four baseline ML models.

Increasingly measured data in the context of smart cities can be used to develop new and innovative business models to increase efficiency and the value of life. A time-series classification algorithm can support to automatize many different processes such as forecasting services. In order to ensure data security and privacy, Federated Learning trains a global model collaboratively on multiple clients. Having different data-distributions and data-quantities across participating clients, neural networks suffer from slow convergence and overfitting. Based on different data-distributions, data-quantities and number of clients, we develop and evaluate different data-clustering strategies to update global model weights in comparison to the state of the art. We use public time-series data, generate various synthetic datasets and train a Relational-Regularized Autoencoder for classification purposes. Our results show an improvement of model performance concerning generalization.

This work addresses the problem of sampling from Gaussian probability distributions by means of uniform samples obtained deterministically and directly from space-filling curves (SFCs), a purely topological concept. To that end, the well-known inverse cumulative distribution function method is used, with the help of the probit function,which is the inverse of the cumulative distribution function of the standard normal distribution. Mainly due to the central limit theorem, the Gaussian distribution plays a fundamental role in probability theory and related areas, and that is why it has been chosen to be studied in the present paper. Numerical distributions (histograms) obtained with the proposed method, and in several levels of granularity, are compared to the theoretical normal PDF, along with other already established sampling methods, all using the cited probit function. Final results are validated with the Kullback-Leibler and two other divergence measures, and it will be possible to draw conclusions about the adequacy of the presented paradigm. As is amply known, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. That said, sequences resulting from this kind of procedure are not truly random. Even so, and to be coherent with the literature, the expression ”random number” will be used along the text to mean ”pseudo-random number”.

Considering that a fair coin has two sides and a cylindrical edge, the probability that it would fall on its edge is calculated, yielding the probability of heads or tails less than 50%. Theoretical models for a static case and for five dynamic cases, without and with rebounds, show that there is a small probability that the coin does not fall on its head or tail, depending on initial toss conditions, the coin geometry and conditions of the coin and landing surfaces. For the dynamic model with rebounds, it is found that the probability that a 50 Eurocent coin thrown from a normal height with common initial velocity conditions and appropriate surface conditions will end up on its edge is in the order of one against several thousand.

This paper aims to describe the geometrical structure and explicit expressions of family of finitely parametrized probability densities over smooth manifold $M$. The geometry of family of probability densities on $M$ are inherited from probability densities on Euclidean spaces $\left\{U_\alpha \right\}$ via bundle morphisms, induced by an orientation-preserving diffeomorphisms $\rho_\alpha:U_\alpha \rightarrow M$. Current literature inherits densities on $M$ from tangent spaces via Riemannian exponential map $\exp: T_x M \rightarrow M$; densities on $M$ are defined locally on region where the exponential map is a diffeomorphism. We generalize this approach with an arbitrary orientation-preserving bundle morphism; we show that the dualistic geometry of family of densities on $U_\alpha$ can be inherited to family of densities on $M$. Furthermore, we provide explicit expressions for parametrized probability densities on $\rho_\alpha(U_\alpha) \subset M$. Finally, using the component densities on $\rho_\alpha(U_\alpha)$, we construct parametrized mixture densities on totally bounded subsets of $M$. We provide a description of inherited mixture product dualistic geometry of the family of mixture densities.

We show the tensor properties of joint probabilities on a Riemannian parametric manifold. Initially we develop a binary data matrix for parameter measurements of a large number of particles confining in a closed system in order to retrieve the joint probability densities of related parameters. By introducing a new generalized inner product as a multilinear operation on dual basis of parametric space, we extract the set of joint probabilities and prove them to meet contravariant tensor properties in a general Riemannian parametric space. We show these contravariant tensors reduce to classical ordinary partial derivatives definition in ordinary Euclidean parametric space. Finally we prove by a theorem that symmetrized iterative contravariant derivative of cumulative probability function on Riemannian manifold gives the set of joint probabilities in those manifold. We bring some examples for compatibility with physical tensors.

We start with the discussion on misapplication of classical probability theory by Feynman in his analysis of the two slit experiment (by following the critical argumentation of Koopman, Ballentine, and the author of this paper). The seed of Feynman's conclusion on the impossibility to apply the classical probabilistic description for the two slit experiment is treatment of conditional probabilities corresponding to different experimental contexts as unconditional ones. Then we move to the Bell type inequalities. Bell applied classical probability theory in the same manner as Feynman and, as can be expected, he also obtained the impossibility statement. In contrast to Feynman, he formulated his no-go statement not in the probabilistic terms, but by appealing to nonlocality. This note can be considered as a part of the author's attempts for getting rid off nonlocality from quantum physics.

We propose two methods for obtaining estimations on the minimum distance and covering radius of orthogonal arrays. Both methods are based on knowledge about the (feasible) sets of distance distributions of orthogonal arrays with given length, cardinality, factors and strength. New bounds are presented either in analytic form and as products of an ongoing project for computation and investigation of the possible distance distributions of orthogonal arrays with parameters in doable ranges.

In this work, we derive the spectral efficiency, secrecy outage probability, and bit error rate of a communication system assisted by a large intelligent surface (LIS). We consider a single-antenna user and an array of antennas at the transmitter side and the possibility of a direct link between transmitter and receiver. Additionally, there is a single-antenna eavesdropper with a direct link to the transmitter, which is modeled as a Nakagami-m distributed fading coefficient. The channels from transmitter to the LIS and from the LIS to the user may or may not have the line-of-sight (LoS) and are modeled by the Nakagami- m distribution. Moreover, we assume that the LIS elements perform non-ideal phase cancellation leading to a residual phase error that assumes a Von Mises distribution. We show that the resulting channel can be accurately approximated by a Gamma distribution whose parameters are analytically estimated using the moments of the equivalent signal-to-noise ratio. We also provide an upper bound for the error probability for M-QAM modulations. With the derived formulas, we analyze the effect of the strength of the LoS link by varying the Nakagami parameter, m.

The present paper investigates flaw strength distributions established using various flexural tests on batches of SiC bar test specimens: 4-point bending as well as 3-point bending with different span lengths. Flaw strength is given by the elemental stress operating on the critical flaw at fracture of a test specimen. Fracture inducing flaws and their locations are identified by fractography. A single population of pores was found to dominate fracture. The construction of diagrams of p-quantile vs elemental strengths was aimed at assessing the Gaussian nature of flaw strengths. Then, empirical cumulative distributions of strengths were constructed using the normal distribution function. Weibull cumulative probability distributions were derived from the mean and the standard deviation using the first moment equation. Agreement was obtained between both Weibull and normal flaw strengths distributions. The cumulative distributions for the different bending tests were predicted satisfactorily from the flaw strength density function using the elemental strength model. Flaw strength distributions that include potential weaker flaws encountered in larger test pieces are extrapolated using the p-quantile diagrams. Implications are discussed about the pertinence of an intrinsically representative flaw strength distribution, about failure predictions, about size effects, and about the significance of flaw spatial distributions.

In this paper, a novel strategy is adopted in a degradation model to affect the implied lifetime distribution. The multiplicative degradation model is utilized as a postulate in the model. It will be established that the implied lifetime distribution forms a classical mixture model. In this mixture model, time-to-failure is lying with some probabilities between two first passage times of the degradation process to reach two specified levels. Stochastic comparisons in the model under a change in the probabilities are studied. Several examples are provided to highlight the applicability of the results in the cases when typical degradation models are candidate.

In this paper we define a continuous wavelet transform of a Schwartz tempered distribution $f \in S^{'}(\mathbb R^n)$ with wavelet kernel $\psi \in S(\mathbb R^n)$ and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of $S^{'}(\mathbb R^n)$. It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution.

This paper presents the tensor properties of joint probability densities on a Riemannian manifold. Initially we develop a binary data matrix to record variables' values of a large number of particles confining in a closed system at a certain time in order to retrieve the joint probability densities of related variables. By introducing the particle oriented coordinate and generalized inner product as a multi-linear operation on the basis of this coordinate, we extract the set of joint probabilities and prove them to meet covariant tensor properties on a general Riemannian space of variables. Based on the Taylor expansion of scalar field in Riemannian manifolds, it has been shown that the symmetrized iterative covariant derivatives of cumulative probability function defined on Riemannian manifold also gives the set of related joint probability densities equivalent to the aforementioned multi-linear method. We show these covariant tensors reduce to classical ordinary partial derivatives in ordinary Euclidean space with Cartesian coordinates and gives the formal definition of joint probabilities by partial derivatives of cumulative distribution function. The equivalence between symmetrized covariant derivative and generalized inner product has been concluded. Some examples of well-known physical tensors clarify that many deterministic physical variables are presented as tensor densities with an interpretation similar to probability densities.

The evolution of even and odd Schrödinger cat states of the inverted oscillator is obtained in the center-of-mass tomographic probability description of the two-mode oscillator.The notion of entangled probability distribution is reviewed. Evolution equations describing the time-dependence of probability distributions identified with quantum system states are discussed.The connection with the Schrödinger equation and the von Neumann equation is clarified.

In this article we explore the limiting behavior of the universal prior distribution obtained when applied over multiple meta-level hierarchy of programs and output data of a computational automata model. We were motivated to alleviate the effect of Solomonoff's assumption that all computable functions or hypotheses of the same length are equally likely, by weighing each program in turn by the algorithmic probability of their description number encoding. In the limiting case, the set of all possible program strings of a fixed-length converges to a distribution of self-replicating quines and quine-relays - having the structure of a constructor. We discuss how experimental algorithmic information theory provides insights towards understanding the fundamental metrics proposed in this work and reflect on the significance of these result in digital physics and the constructor theory of life.

Soil nitrogen (N) is a common limiting factor where soil N cycling is a key component of agro-ecosystems. Soil N transformation processes are largely mediated by microbes and understanding bacteria involvement in soil N cycling in agricultural systems has both agronomic and environmental importance. This 2-yr field scale study examined the abundances and spatial distributions of total bacterial community (16s RNA), bacterial involved in nitrification (amoA), and de-nitrification (narG, nirK, and nosZ) and soil physicochemical properties of winter wheat (Triticum aestivum L.) – soybean (Glycene max L.) double cropping with 2-3 weeks of spring grazing (WGS) and without grazing (WS) and of tall fescue (Festuca arundinacea (L.) Schreb.) pasture (TF) managed to near natural conditions with similar grazing. The TF had a significantly higher abundance of 16S rRNA, amoA, narG, nirK, and nosZ genes than the WS and WGS, which had similar levels between themselves. Soil organic matter (OM) and soil pH had a stronger effect on the N-cycling bacteria gene abundance. All bacterial gene concentrations and soil pH showed non-random distribution patterns with 141-186 m range autocorrelation. These results indicate that biological N transformation processes are more important in natural agricultural systems and the abundance of N-cycling bacteria can be manipulated by field scale management strategies.

The Dual-Frequency Precipitation Radar (DPR) on the Core GPM satellite provides spaceborne 3-dimensional observations of the precipitation field and surface rainfall rate with quasi global coverage. The present study evaluates the behavior of liquid precipitation intensity, radar reflectivity factor (ZKu and ZKa) and Drop Size Distribution (DSD) parameters (weighted mean diameter Dm and intercept parameter Nw) of the GPM DPR derived products version 07 from 2014 to 2023. Observations from seven Parsivel disdrometers located in different topographic zones in the Western Mediterranean are taken as ground reference. Four matching techniques between satellite estimates and ground level observations were tested, and best results were found for the so-called optimal comparison approach. Overall, GPM DPR products capture well the variability of the observed DSD at different rainfall intensities. However, overestimation of the mean Dm and underestimation of the mean Nw are observed, being much more sensitive to errors in drop diameters larger than 1.5 mm. Moreover, lowest errors were found for radar reflectivity factor and Dm, and highest for Nw and rainfall rate. In addition, the GPM DPR convective and stratiform classification was tested and it was found a substantial overestimation of stratiform cases compared to disdrometer observations.

At the plasma membrane, transmembrane receptors are at the interface between cells and their environment. They allow sensing and transduction of chemical and mechanical extracellular signals. The spatial distribution of receptors and the specific recruitment of receptor subunits to the cell membrane is crucial for the regulation of signaling and cell behavior. However, it is challenging to define what regulates such spatial patterns for receptor localization, as cell shapes are extremely diverse when cells are maintained in standard culture conditions. Bone morphogenic protein receptors (BMPRs) are serine-threonine kinases, which build heteromeric complexes of BMPRI and II. These are especially interesting targets for receptor distribution studies, since the signaling pathways triggered by BMPR-complexes depends on their dimerization mode. They might exist as pre-formed complexes, or assemble upon binding of BMP, triggering cell signaling which leads to differentiation or migration. In this work we analyzed BMPR receptor distributions in single cells grown on micropatterns, which allows not only to control cell shape, but also the distribution of intracellular organelles and protein assemblies. We developed a script called ComRed (Center Of Mass Receptor Distribution), which uses center of mass calculations to analyze the shift and spread of receptor distributions according to the different cell shapes. ComRed was tested by simulating changes in experimental data, showing that shift and spread of distributions can be reliably detected. Our ComRed-based analysis of BMPR-complexes indicates that receptor distribution depends on cell polarization. The absence of a coordinated internalization after addition of BMP suggests that a rapid and continual recycling of BMPRs occurs. Receptor complexes formation and localization in cells induced by BMP might yield insights into the local regulation of different signaling pathways.

Neuronal timing is critical for many brain functions such as perception and learning. At the level of the chemical synapse, the synaptic delay is determined by the presynaptic release probability (Pr) and the waveform of the presynaptic action potential (AP). Depolarization-induced analogue-digital facilitation (d-ADF) is a form of context-dependent synaptic facilitation that is induced by constant depolarization of the presynaptic neuron and is mediated by voltage inactivation of presynaptic Kv1 channels. Whether synaptic delay is modulated during d-ADF remains unclear. We show here that despite Pr is elevated during d-ADF at L5-L5 synapses, the synaptic delay is surprisingly unchanged, suggesting that both Pr- and AP-dependent changes in synaptic delay compensate for each other. We conclude that in contrast with other short- or long-term modulation of presynaptic release, timing is not affected during d-ADF because of the opposite interaction of Pr- and AP-dependent modulations of synaptic delay.

$\beta$-decay is one of the key factors to understand the $r$-process and evolution of massive stars. The Gamow-Teller (GT) transitions drive the $\beta$-decay process. We employ the proton-neutron quasiparticle random phase approximation (pn-QRPA) model to calculate terrestrial and stellar $\beta$-decay rates for 50 top-ranked nuclei possessing astrophysical significance according to a recent survey \cite{Nab21}. The model parameters of the pn-QRPA model effect the predicated results of $\beta$-decay. The current study investigates the effect of nucleon-nucleon pairing gaps on charge-changing transitions and the associated $\beta$ decay rates. Three different values of pairing gaps, namely TF, 3TF and 5TF, were used in our investigation. It was concluded that both GT strength distributions and half-lives are sensitive to pairing gap values. The 3TF pairing gap scheme, in our chosen nuclear model, resulted in best prediction with around 80$\%$ of the calculated half-lives within a factor 10 of the measured ones. The 3TF pairing scheme also led to calculation of biggest $\beta$-decay rates in stellar matter.

Using a simple model for internal heat evolution and mass-internal heat relations found for rocky exo planetary objects in the inner solar system we have inferred the phases of volcanism at earth ( EA) or host star ages (SA) in 52 potentially habitable rocky exo planets. We have also calculated the internal tidal heat contributions and magnetic moments of these exo planets. Based on these results we have inferred the probability of existence of life favouring geophysical conditions in the above exo planets at EA and SA. In M star associated exo planets, occurrences of super flares and long night periods may pose problems for the development of advanced life

L\'evy walks represent important modeling tools for variety of real life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far these scaling limits were derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here we derive the corresponding limiting equations in the case of position depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo-Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalisations are analysed.

In this work, we explore the extreme searching of multidimensional functions by natural gradient descent based on Dirichlet and generalized Dirichlet distributions. The natural gradient is based on describing multidimensional surface with probability distributions, which allows us to reduce changing the accuracy of gradient and step-size. In this article, we propose an algorithm of natural gradient descent based on Dirichlet and generalized Dirichlet distributions. We demonstrate that the natural gradient descent with step-size adaptation with Dirichlet and generalized Dirichlet distributions has higher accuracy and does not take a large number of iterations for minimizing test functions than gradient descent and Adam.

Reliability analysis techniques are customary standard tools that are used for evaluating the performance of different equipment and devices in order to minimize their downtime. To predict the reliability, life data from a sample that is satisfactorily representative of the equipment should be fitted to the suitable statistical distribution. The parameterized distribution may be used to estimate essential characteristics such as failure rate; and probability at a precise time, as well as system reliability. In the current study, Weibull++/ALTA software package is used as a novel tool to fit the available data set to estimate the best fitted probability density function (PDF) using Maximum Likelihood (MLE) for parameter estimation. The determined distributions are then assessed using goodness-of-fit test to define how well it fits the available data set. There are multiple methods for determining goodness-of-fit. Weibull distributions and their special cases’ parameters have an effect on life times.

Plants grown under fluctuating light impact plant developments compared with those grown under non-fluctuating light conditions. However, our knowledge on the underlying regulatory mechanisms is still quite limited, particularly from the transcriptional perspective. In order to investigate the influence of different light intensity distributions on tomato plant development, we designed three fluctuating light intensity distributions with the non-fluctuating light intensity as control and compared the transcriptional differences after five weeks of treatment. We found plant height and aerial/root weight were significantly reduced under all fluctuating light treatments. Transcriptome analysis revealed that the number of up and down regulated genes had a distinct distribution pattern between different treatments and control. The largest difference between the numbers of down and up regulated genes was found between treatment 1 and 3, reaching to a total of 416 genes. The number and type of the top 20 enriched pathways differed between treatments and control. The largest number of genes enriched was involved in the biosynthesis of secondary metabolites. These results provide insights into the transcriptional regulations of tomato under different light intensity distributions.

Despite the availability of COVID-19 vaccines, the global pandemic remains a significant health challenge. Timely detection of the virus is crucial for effective containment efforts, prompting the exploration of machine learning (ML) models for COVID-19 detection. highlights the ongoing significance of COVID-19 as a global health crisis despite the availability of vaccines. To address the need for timely virus detection, the study explores the efficacy of machine learning (ML) models, including DNN, Voting, Bagging, SVC_rbf, SVC_linear, SVC_polynomial, and SVC_sigmoid, in predicting COVID-19 infection probability. With a dataset comprising 4000 samples split into 3200 for training and 800 for testing, the analysis reveals that DNN and Voting models demonstrate the highest performance, achieving an impressive accuracy of 89%. These results underscore the potential of ML techniques, particularly DNN and ensemble methods like Voting, in facilitating early COVID-19 detection and contributing to effective pandemic management. The study emphasizes the importance of continued research to further refine and optimize ML-based COVID-19 detection systems.

This study investigates the impact of temperature-induced quantum proton tunneling probability on DNA amplification during polymerase chain reactions (PCR). Using a simulation model based on a Gaussian wavefunction and finite-difference time-domain method, quantum tunneling of protons across square potential barriers is examined. The results unveil consistent probability distributions for quantum tunneling across various PCR temperatures, with distinct oscillation patterns emerging post-barrier crossing. Acknowledging limitations in initial conditions due to temperature-dependent proton energy, the study highlights the need for refined models and experimental validation. These findings accentuate the potential interplay between quantum mechanics and biological systems, prompting further research to understand quantum tunnelling’s comprehensive effect on genetic variations and molecular processes.

The interaction between oceanographic, meteorological, and hydrological factors can result in an extreme flooding scenario in the low-lying coastal area, called compound flooding (CF) events. For instance, rainfall and storm surge (or high river discharge) can be driven by the same meteorological, tropical or extra-tropical cyclones, resulting in a CF phenomenon. The trivariate distributional framework can significantly explain compound events' statistical behaviour reducing the associated high-impact flood risk. Resolving heterogenous dependency of the multidimensional CF events by incorporating traditional 3-D symmetric or fully nested Archimedean copula is quite complex. The main challenge is to preserve all lower-level dependencies. An approach based on decomposing the full multivariate density into simple local building blocks via conditional independence called vine or pair-copulas is a much more comprehensive way of approximating the trivariate flood dependence structure. In this study, a parametric vine copula of a drawable (D-vine) structure is introduced in the trivariate modelling of flooding events with 46 years of observations of the west Coast of Canada. This trivariate framework searches dependency by combining the joint impact of annual maximum 24-hr rainfall and the highest storm surge and river discharge observed within the time ±1 day of the highest rainfall event. The D-vine structures are constructed in three alternative ways by permutation of the conditioning variables. The most appropriate D-vine structure is selected using the fitness test statistics and estimating trivariate joint and conditional joint return periods. The investigation confirms that the D-vine copula can effectively define the compound phenomenon's dependency. The failure probability (FP) method is also adopted in assessing the trivariate hydrologic risk. It is observed that hydrologic events defined in the trivariate case produce higher FP than in the bivariate (or univariate) case. It is also concluded that hydrologic risk increases (i) with an increase in the service design life of the hydraulic facilities and (ii) with a decrease in return periods.

Both macro and microprojectiles (e.g., interplanetary, interstellar and even intergalactic material) are seen as important vehicles for the exchange of potential (bio)material within our solar system as well as between stellar systems in our Galaxy. Accordingly, this requires estimates of the impact probabilities for different source populations of projectiles, including for intergalactic meteor particles which have received relatively little attention since considered as rare events (discrete occurrences that are statistically improbable due to their very infrequent appearance). We employ the simple but comprehensive model of intergalactic microprojectile capture by the gravity of exoplanets which enables us to estimate the map of collisional probabilities for an available sample of exoplanets in habitable zones around host stars. The model includes a dynamical description of the capture adopted from Mautner model of interstellar exchange of microparticles and changed for our purposes. We use statistical and information metrics to calculate probability map of intergalactic meteorite particle capture. Moreover, by calculating the entropy index map we estimate the concentration of these rare events. We further adopted a model from immigration theory, to show that the time dependent distribution of single molecule immigration of material indicates high survivability of the immigrated material taking into account birth and death processes on our planet. At present immigration of material can not be observationally constrained but it seems reasonable to think that it will be possible in the near future, and to use it along other proposed parameters for life sustainability on some planet.

This paper introduces an approach that chooses the fault detection by calculating probabilities using probability mass function (pmf) and cumulative distribution function (CDF). This work used a method for multiple stuck-at faults by producing a new test pattern in combinational circuits. We assumed that existence of all multiple faults is only because of one single component that is faulty. A complete test set can be created by all possible single stuck-at faults in a combinational circuit using some combination of gates. The test set generation fault detection method is applied on two different 3-bit input variable and 4-bit input variable circuits. The probability of error occurrence is calculated at both 3-bit and 4-bit input variable circuits. The resulting feature is used to obtain maximum error occurrence probability to detect faults by the logic used that the complexity of the circuit is inversely proportional to the fault occurrence probability. Then again, undetectability is directly proportional to the complexity of the circuit. Therefore, finest feasible circuit should have large input variable components with less complexity to reduce the fault occurrence probability.

Cartels cause tremendous damage to the market economy and disadvantage consumers by causing higher prices and lower quality; moreover, they are difficult to detect. We need to prevent them by scientific analysis, which includes the determination of an indicator to explain antitrust enforcement. Particularly, the probability of cartel penalization is a useful indicator for the evaluation of the competition enforcement. This study is to estimate the probability of cartel penalization by using a Bayesian approach. In the empirical study, the probability of cartel penalization is estimated by Bayesian approach from cartel data of Department of Justice in United States from 1970 to 2009. The probability of cartel penalization is seen to be sensitive to change of competition law and the results shows the usefulness of higher interpretation than other research. The result of the policy simulation shows how effective the leniency program is. From this estimation, antitrust enforcement is evaluated, and thereby, can be improved.

Multicriteria decision aid requires a database as a decision matrix, in which two or more alternatives are evaluated according to two or more variables selected as decision criteria. Several problems of this nature use measures by Likert scales. Depending on the method, parameters from these data (e.g. means, modes or medians) are required for calculations. This parameterization of data in ordinal scales has fueled controversy for decades between authors who favor mathematical/statistical rigor and argue against the procedure, stating that ordinal scales should not be parameterized, and scientists from other areas who have shown gains from the process that compensate for this relaxation. The aim of this article is to demonstrate the advantages of the Composition of Probabilistic Preferences (CPP) method in multicriteria problems with data from Likert scales. The CPP is capable of allaying the protests raised and obtaining more accurate results than descriptive statistics or parametric models can bring. The proposed algorithm in R-code involves the use of CPP with empirical distributions and fitting histograms of data measured by Likert scales. Two case studies with simulated datasets having peculiar characteristics and a real case illustrate the advantages of the CPP.

In the paper, by convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the manifold of the beta distributions. Finally, the author poses several guesses and open problems related to monotonicity, complete monotonicity, and inequalities of several functions involving polygamma functions.

The probability distribution of the interevent time between two successive earthquakes has been subject of numerous studies due to its key role in seismic hazard assessment. In recent decades, many distributions have been considered and there has been a long debate about the possible universality of the shape of this distribution when the interevent times are suitably rescaled. In this work we aim to find out if there is a link between the different phases of a seismic cycle and the variations in the distribution that best fits the interevent times. To do this, we consider the seismic activity related to the Mw 6.1 L’Aquila earthquake that occurred on April 6, 2009 in central Italy by analyzing the sequence of events recorded from April 2005 to July 2009, and then the seismic activity linked to the sequence of the Amatrice-Norcia earthquakes of Mw 6 and 6.5 respectively and recorded in the period from January 2009 to June 2018. We take into account some of the most studied distributions in the literature: q-exponential, q-generalized gamma, gamma and exponential distributions and, according to the Bayesian paradigm, we compare the value of their posterior marginal likelihood in shifting time windows with a fixed number of data. The results suggest that the distribution providing the best performance changes over time and its variations may be associated with different phases of the seismic crisis.

6G is already being planned and will employ much higher frequencies, leading to a revolutionary era in communication between people as well as things. It is well known that weather, especially rain, can cause increased attenuation of signal transmission for higher frequencies. The standard methods for evaluating the effect of rain on symbol error rate are based on long-term averaging. These methods are an inaccurate, which results with an inefficient system design. This is critical regarding bandwidth scarcity and energy consumption and requires a more significant margin of effort to cope with the imprecision. Recently we have developed a new and more precise method for calculating communication system performance in case of rain, using the probability density function of rain rate. For high rain rate (above 10mm/hr), for a typical set of parameters, our method shows the symbol error rate in this range to be higher by orders of magnitude than that found by ITU standard methods. Our model also indicates that sensing and measuring the rain rate probability is important in order to provide the required bit error rate to the users. To the best knowledge of the authors, this novel analysis is unique. It can constitute a more effi-cient performance metric for the new era of 6G communication and prevent disruption due to incorrect system design. Keywords: atmospheric propagation, communication system performance, attenuation, com-munication

The development of forecasting models for the evaluation of potential slope instability after rainfall event represents an important issue for the scientific community. This topic has received considerable impetus due to climate change effect on the territory [1, 2] as several studies demonstrate that the increase in global warming can significantly influence the landslide activity and stability conditions of natural and artificial slopes [3]. A consolidated approach in evaluating rainfall induced landslide hazard is based on the integration of rainfall forecasts and physically based (PB) predictive models through deterministic laws. However, considering the complex nature of the processes and the high variability of the random quantities involved, probabilistic approaches are recommended in order to obtain reliable predictions. A crucial aspect of the stochastic approach is represented by the definition of appropriate probability density functions (pdfs) to model the uncertainty of the input variables as this may have an important effect on the evaluation of the probability of failure (PoF). The role of the pdf definition on reliability analysis is discussed through a comparison of PoF maps generated using Monte Carlo (MC) simulations performed over a study area located in the Umbria Region of central Italy.

In both managed and unmanaged forests, termites are functionally important members of the dead-wood-associated (saproxylic) insect community. However, little is known about regional-scale environmental drivers of geographic distributions of termite species, and how these environmental factors impact co-occurrence among congeneric species. Here we focus on the southern Appalachian Mountains—a well-known center of endemism for forest biota—and use Ecological Niche Modeling (ENM) to examine the distributions of three species of Reticulitermes termites (i.e., R. flavipes, R. virginicus, and R. malletei). To overcome deficiencies in public databases, ENMs were underpinned by field-collected high-resolution occurrence records coupled with molecular taxonomic species identification. Spatial overlap among areas of predicted occurrence of each species was mapped, and aspects of niche similarity were quantified. We also identified environmental factors that most strongly contribute to among-species differences in occupancy. Overall, we found that R. flavipes and R. virginicus showed significant niche divergence, which was primarily driven by dry-season precipitation. Also, all three species were most likely to co-occur in the mid-latitudes of the study area (i.e., northern Alabama and Georgia, eastern Tennessee and western North Carolina), which is an area of considerable topographic complexity. This work provides important baseline information for follow-up studies of local-scale drivers of these species’ distributions. It also identifies specific geographic areas where future assessments of the frequency of true syntopy vs. micro-allopatry, and associated interspecific competitive interactions, should be focused.

Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the negligibled effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

Bipolar disorder (BD) is a high suicide-risk mental disorder. The purpose of this study was to identify the relationship between temperament and character traits with suicide probability, suicide attempt and perceived stress level in patients with BD. A total of 39 euthymic patients with bipolar disorder who had a history of suicide attempt and 39 euthymic patients without a history of suicide attempt were included in this study. Sociodemographic and clinical data form, Hamilton Depression Rating Scale (HDRS), Young Mania Rating Scale (YMRS), The Structured Clinical Interview for DSM-5-Clinician Version (SCID-5/CV), Temperament and Character Inventory (TCI), Perceived Stress Scale (PSS), Suicide Probability Scale were used for obtaining the data. HDRS, PSS and SPS scores of the group comprised of patients who attempted suicide were higher than the other group. There was no significant difference between the group of patients who had attempted suicide and the other group in terms of temperament characteristics. In the group of patients who had attempted suicide, self-directedness (SD) and cooperativeness (CO) scores were lower and self-transcendence (ST) score was higher than the other group. HA and ST were positively and SD negatively associated with SPS scores. In the regression analysis for suicide risk, the factors most associated with suicide risk were high HDRS and low CO score. Low SD in BD, high ST with CO may be associated with suicide attempt. Alongside low SD, high HA and ST may be associated with suicidal ideation. Treating residual depressive symptoms can reduce the risk of suicide.

Over the past 150 million years, the Chilean Atacama Desert has been transformed into one of the most inhospitable landscapes by geophysical changes, which makes it an ideal Mars analog that has been explored for decades. However, two heavy rainfalls that occurred in the Atacama in 2015 and 2017 provide a unique opportunity to study the response of resident extremophiles to rapid environmental change associated with excessive water and salt shock. Here we combine mineral/ salt composition measurements, amendment cell culture experiments, and next-generation sequencing analyses to study the variations in salts and microbial communities along a latitudinal aridity gradient of the Atacama Desert. In addition, we examine the reshuffling of Atacama microbiomes after the two rainfall events by comparing with previous researches. Analysis of microbial community composition revealed that soils within the southern arid desert were consistently dominated by Actinobacteria, Proteobacteria, Acidobacteria, Planctomycetes, Chloroflexi, Bacteroidetes, Gemmatimonadetes, and Verrucomicrobia. Intriguingly, the hyperarid microbial consortia exhibited a similar pattern to the more southern desert. Salts at the shallow subsurface were dissolved and leached down to a deeper layer, challenging indigenous microorganisms with the increasing osmotic stress. Microbial viability was found to change with aridity and rainfall events. This study sheds light on the structure of xerophilic, halophilic, and radioresistant microbiomes from the hyperarid northern desert to the less arid southern transition region, as well as their response to changes in water availability. Our findings may infer similar events that happened on the wetter early Mars.

As global warming intensifies, the development of offshore wind farms is swiftly progressing, especially deep-water floating offshore wind turbines (FOWTs) capable of energy capture in deep-sea regions, which have emerged as a focal point of both academic and industrial interest. Although numerous researchers have conducted comprehensive and multifaceted studies on various components of wind turbines, less attention has been paid to the operational stage response of FOWTs to wind, waves, and currents, and the reliability of structural components. This study primarily employs theoretical analysis to establish mathematical models under a series of reasonable assumptions, examining the possibility of collisions between FOWT transport fleets and other vessels in the passage area during the towing process. Using the model, this paper takes the Wanning Offshore Floating Wind Farm (FOWF) project, which is scheduled to be deployed in the South China Sea, as the research object and calculates the probability of collisions between FOWTs and other vessels in three time periods from the total assembly pier in Hainan Wanning to the predetermined position 22 km away. The findings of the analysis suggest that the mathematical model has the capability to efficiently determine the frequency of collisions between vessels within the domain and the transport team during the transportation process. This holds substantial relevance for future risk assessment and disaster prevention and mitigation measures in the context of FOWT transportation.

For each $t\in(-1,1)$, exact values of the least upper bounds $$ H(t)=\sup_{\E X=t,\,\E X^2=1} \frac{\E\abs{X}^3}{\E \abs{X-t}^3},\quad \sup_{\E X=t,\,\E X^2=1} \frac{L_1(X)}{L_1(X-t)} $$ are obtained, where $L_1(X)=\E|X|^3/(\E X^2)^{3/2}$ is the non-cental Lyapunov ratio. It is demonstrated that these values are attained only at two-point distributions. As a corollary, S.\,Shorgin's conjecture is proved that states that the exact value is $$ \sup\frac{L_1(X)}{L_1(X-\E X)}= \frac{\sqrt{17 + 7\sqrt7}}{4} = 1.4899\ldots, $$ where the supremum is taken over all non-degenerate distributions of the random variable $X$ with the finite third moment. Also, in terms of the central Lyapunov ratio $L_1(X-\E X)$, an analog of the Berry--Esseen inequality is proved for Poisson random sums of independent identically distributed random variables with the constant $$ 0.3031\cdot H\left(\frac{\E X}{\sqrt{\E X^2}}\right) \left(1-\frac{(\E X)^2}{\E X^2}\right)^{3/2} \le 0.3031\cdot \frac{\sqrt{17 + 7\sqrt7}}{4}<0.4517.$$ where $\Law(X)$ is the common distribution of the summands.

This work is the first in literature to tackle the difficult open problem of determining the upper bound and threshold theorem for the TDCDP (time-dependent controller parameter) of the (Fokker Planck Kolmogorov) probability density function. This revolutionary exposition will put control theory and other related inter-disciplinary fields to a higher level towards contemporary control theory. Notably, based on the influential role of control theory in both engineering and industry, this paper will be of great value to all engineering and industry professionals who seek to know more about advanced trends within control theory settings. On the other remit of the spectrum, Fokker Planck Kolmogorov(FPK) equations are of high importance to physicists as well as mathematicians, based on their multiple applicability to information theory, graph theory, data science, finance, economics, and beyond. So, this by default adds more taste and credibility to this study. This leads by nature to introducing a different flavor to this ground-breaking research by highlighting the impact of Fokker Planck Kolmogorov(FPK) to revolutionize crypocurrency,which have received its name because it uses encryption to verify transactions, a new debatable digital payment system that doesn't rely on banks to verify transactions. It’s a peer-to-peer system that can enable anyone anywhere to send and receive payments. The paper ends with closing remarks combined with some challenging open problems and the next phase of research.

With the global warming and the frequent occurrences of harsh environments, mooring line failures are frequent. Response analysis of the platform structure and its residual mooring system under multiple mooring line failures is needed. Therefore, it is necessary to investigate the change law of platform motion response after mooring line failure to find out the reasons for the change. Studies for different failure combinations of mooring lines show that the variation of the steady-state equilibrium position of the platform is closely related to the stiffness, with a clear functional. The duration and amplitude of the transient response are mainly influenced by the Morison drag force. And it was found that the suspended mooring line reduced the motion responses. It is also found that the mooring line failure increased the effective tension and suspension length of the riser. Based on the improved Dezert-Smarandache theory (DSmT), it is calculated that mooring line failure has little effect on the failure probability in different zones of the riser.

The interlaminar shear behavior of a [±45°] laminated carbon fiber reinforced plastic (CFRP) specimen was investigated utilizing microscale strain mapping in a wide field of view. A three-point bending device was developed under a laser scanning microscope, and the full-field strain distributions including normal, shear and principal strains of CFRP in a three-point bending test were measured using a developed sampling Moire technique. The microscale shear strain concentrations at interfaces between each two adjacent layers were successfully detected and found to be positive-negative alternately distributed before damage occurrence. The 45° layers slipped to the right relative to the -45° layers, visualized from the revised Moire phases and shear strain distributions of the angle-ply CFRP under different loads. The absolute values of the shear strain at interfaces gradually rose with the increase of the bending load, and the sudden decrease of the shear strain peak value implied the occurrence of interlaminar damage. The evolution of the shear strain concentrations is useful in the quantitative evaluation of the potential interlaminar shear failure.

Modelling diameter distribution is an integral part of forest management. It requires the use of a suitable probability density function or cumulative distribution function with the appropriate fitting method. In this study, we compared the suitability of eight probability density functions fitted by derivative and optimization methods used for diameter distribution estimation of forest stands. The derivation and optimization were used on A Charlier, beta, generalized beta, gamma, Gumbel, Johnson’s SB, and Weibull (2 and 3-parameter). We used data from 167 permanent sample plots from Atlantic forest (Quercus robur) and 59 temporary sample plots from tropical forests (Tectona grandis). The quality of the fits was evaluated with different indices such as Kolmogorov–Smirnov, Cramer-von Mises, mean absolute error, bias and mean squared error. The results showed Johnson’s SB function was more suitable for describing the diameter distribution of the stands. Johnson’s SB, 3-parameter Weibull and generalized beta had consistent performance regardless of the fitting methods. The quality of fits produced by gamma, Gumbel and 2-parameter Weibull was poor.

For a long time, physicists always directly use mathematical tools to deal with physical problems, and few people pay attention to the difference between mathematical theory and physical theory. Just like the dilemma that physicists once faced when dealing with the problem of blackbody radiation function.By analyzing the difference between the theoretical basis of mathematics and the theoretical basis of physics, this paper draws the following conclusions: (1) The theoretical basis of mathematics and the theoretical basis of physics are different, so when we use mathematical tools for physics research, we need to be very careful. (2) Finiteness and discreteness should be the basis of the whole physical theory; This paper points out that it is not advisable to use infinite " " and infinitesimal "0" without restriction and demonstration in physics, as well as the continuity of functions, which will bring a lot of trouble to physical theory.At the same time, through the analysis of Banach-Tarski paradox and Bertrand paradox, this paper proposes that if we revise the basic assumptions of probability theory: assuming that "points" have quantized sizes, and "lines" also have quantized widths. After the correction, we can not only avoid the troubles caused by Bertrand paradox, but also make probability theory better for practical application.

In this study, we propose a radiomics-clinical probability weighted model for the prediction of prognosis for NSCLC. The model combines radiomics features extracted from RT planning images with clinical factors such as age, gender, histology, and tumor stage. CT images with radiotherapy structures of 422 NSCLC patients were retrieved from The Cancer Imaging Archive (TCIA). Radiomic features were extracted from gross tumor volume (GTV). Five machine learning algorithms, namely decision trees (DT), random forests (RF), extreme boost (EB), support vector machine (SVM) and generalized linear model (GLM), were optimized by a voted ensemble machine learning (VEML) model. A probabilistic weighted approach is used to incorporate the uncertainty associated with both radiomic and clinical features and to generate a probabilistic risk score for each patient. The performance of the model is evaluated using a receiver operating characteristic (ROC). Radiomic model, clinical factors model and combined radiomic-clinical probability weighted model demonstrated good performance in predicting NSCLC survival with AUC of 0.941, 0.856 and 0.949 respectively. The combined radiomics-clinical probability weighted enhanced model achieved significantly better performance than radiomic model in 1-year survival prediction (chi-square test, p<0.05). The proposed model has the potential to improve NSCLC prognosis and facilitate personalized treatment decisions.

Tidal stream energy is a renewable energy resource that might be developed to offset carbon emissions. A tidal energy demonstration (TED) area has been designated in Minas Passage, Bay of Fundy, for testing and installing marine hydrokinetic (MHK) turbines. Regulations require quantification of the potential for MHK turbine installations to harm local populations of marine animals. Here we use acoustic telemetry to quantify probability that post smolt Inner Bay of Fundy salmon encounter a turbine installation at the TED area. Previous work has quantified detection efficiency of Innovasea HR acoustic tags as a function of current speed and range from a moored HR2 receiver and also demonstrated that drifters carrying HR tags will be effectively detected when the drifter track crosses the array of HR2 receivers in Minas Passage. Salmon smolts were tagged and released in Gaspereau and Stewiacke Rivers, Nova Scotia, in order that the HR2 receiver array could monitor seaward migration of the post smolts through Minas Passage and particularly through the TED area. Presently we demonstrate methods by which tag signals detected by the HR2 array can be used to estimate the expected number of times that a post smolt would encounter a single near-surface MHK turbine installation during its seaward migration.

The rapid development of the Industrial Internet of Things (IIoT) has brought enormous opportunities to the industrial environment but has also posed significant challenges to network security, especially the physical layer security challenges,which is crucial for the reliable transmission of messages interconnected devices and sensors. Focus on the enhancing the physical layer security in the IIoT communication, this paper proposed a mechanism of Spatial Secrecy Outage Probability (SSOP) to ensure the secure transmission of data. The SSOP algorithm assumes that the location of legitimate device is known, while eavesdropper on the random distribution of device locations, forming the SSOP model related to device locations from the perspective of insecure regions (ISR) and secure regions (SR), and deduce the closed-form expression for its upper bound. Subsequently under the constraint of SSOP conditions, establish an optimization model with the objective of system throughput maximization based on Spatial Secrecy Outage Probability(SRM-SSOP). By optimizing the transmission rate of legitimate channels or controlling the transmission power, the maximization of system throughput can be achieved, thus reducing the consumption of communication resources. Through sophisticated modeling and simulation in the MATLAB, we verified the accuracy of the definition and derived upper bound of SSOP. The experimental results also show that SSOP algorithm gradually converges to a specific value as the number of antenna elements increases. In addition, we observed that when the receiver device is aligned with the normal direction of the transmit antenna array, the SSOP reaches a minimum value. These performance are crucial for understanding and optimizing the security performance of IIoT communication systems.

Background: Qazvin province has suffered a lot of natural disasters (earthquakes) for a long time. so, the establishment of the first industrial city in Iran, has always been a source of concern. One of these problems is the possibility of chlorine gas leaking from the chlorination system of drinking water and treatment plant the existing ones are on the health of the inhabitants of this region.Material and method: This analytical study was carried out in Novorossiysk district of Alborz in December 2012. Initially, chlorine gas leakage scenarios from these facilities were simulated by ALOHA and WISER software, and then using the Emergency Level System, according to the CCPS Consultative Center's Chemical Process Safety Advice, the Emergency Response Program was presented. Results: ALOHA software, in comparison with WISER software, has more precision to identify and address the consequences of chlorine gas leakage hazards and according to the simulation of this software in the event of a leak from the well and water resources unit at 8240 m from the area and at night 7520 meters will be affected. Due to the speed of delivering results through the WISER software, this software can be used at times when the operating speed is considered (incident time).Conclusion: According to the CCPS model, the emergency conditions were classified at levels 3 and 4, therefore, the relevant crisis management agencies (Qazvin Crisis Management Committee, Alborz City Health Center, Red Crescent and Firefighting Authority) should provide training maneuvers At Alborz city level, and informing people of cities and industrial companies, many efforts have been made to reduce human and environmental damage in case of possible leakage.