Continuous Distribution Function Research Articles

Enthalpy-based cascaded lattice Boltzmann method for conjugate heat transfer

Lattice Boltzmann (LB) methods for conjugate heat transfer are widely studied because of the simplicity and efficiency in dealing with complex boundary conditions. Recently, cascaded (or central-moments-based) LB (CLB) method has shown the potential to improve the stability of LB method. For conventional CLB method which is constructed through the continuous equilibrium distribution function, the temperature-based energy equation is recovered and solved. The continuity of temperature and heat flux at conjugate interface is unable to be accurately enforced without special treatment. To circumvent this deficiency, an enthalpy-based CLB method for conjugate heat transfer is developed in this work. In the proposed method, the enthalpy-based formulation of equilibrium moments is directly designed in the central-moment space, and the macroscopic quantity solved is converted from temperature to enthalpy. As demonstrated by the Chapman-Enskog multiscale analysis, the enthalpy-based energy equation can be recovered by the CLB equation. Compared with the conventional CLB method, the continuity condition for temperature and heat flux on the conjugate interface can be enforced automatically without special treatment. In addition, the proposed CLB method provides a new way to develop a CLB equation that can recover to a general convection-diffusion equation. Finally, the accuracy and stability of the temperature field are validated by benchmarks for the proposed enthalpy-based CLB method.

Adaptive unified gas-kinetic scheme for diatomic gases with rotational and vibrational nonequilibrium

Multiscale nonequilibrium physics at large variations of local Knudsen number are encountered in applications of aerospace engineering and micro-electro-mechanical systems, such as high-speed flying vehicles and low pressure of the encapsulation. An accurate description of flow physics in all flow regimes within a single computation requires a genuinely multiscale method. The adaptive unified gas-kinetic scheme (AUGKS) is developed for such multiscale flow simulation. The AUGKS applies discretized velocity space to accurately capture the non-equilibrium physics in the multiscale UGKS, and adaptively employs continuous distribution functions following Chapman–Enskog expansion to efficiently recover near-equilibrium flow region in GKS. The UGKS and GKS are dynamically connected at the cell interface through the fluxes from the discretized and continuous gas distribution functions, which avoids any buffer zone between them. In this study, the AUGKS with rotation and vibration non-equilibrium is developed based on a multiple temperature relaxation model. The real gas effect in different flow regimes has been properly captured. To capture aerodynamic heating accurately, the heat flux modifications from the rotation and vibration modes are also included in the current scheme. Unstructured discrete particle velocity space is adopted to further improve the computational performance of the AUGKS. Numerical tests, including Sod tube, normal shock structure, high-speed flow around the two-dimensional cylinder and three-dimensional sphere and space vehicles, and an unsteady nozzle plume flow from the continuum flow to the background vacuum, have been conducted to validate the current scheme. In comparison with the original UGKS, the current scheme speeds up the computation, reduces the memory requirement, and maintains the equivalent accuracy for multiscale flow simulation, which provides an effective tool for nonequilibrium flow simulations, especially for the flows at low and medium speed.

On the probability of a Pareto record

Abstract Given a sequence of independent random vectors taking values in ${\mathbb R}^d$ and having common continuous distribution function F, say that the $n^{\rm \scriptsize}$ th observation sets a (Pareto) record if it is not dominated (in every coordinate) by any preceding observation. Let $p_n(F) \equiv p_{n, d}(F)$ denote the probability that the $n^{\rm \scriptsize}$ th observation sets a record. There are many interesting questions to address concerning pn and multivariate records more generally, but this short paper focuses on how pn varies with F, particularly if, under F, the coordinates exhibit negative dependence or positive dependence (rather than independence, a more-studied case). We introduce new notions of negative and positive dependence ideally suited for such a study, called negative record-setting probability dependence (NRPD) and positive record-setting probability dependence (PRPD), relate these notions to existing notions of dependence, and for fixed $d \geq 2$ and $n \geq 1$ prove that the image of the mapping pn on the domain of NRPD (respectively, PRPD) distributions is $[p^*_n, 1]$ (resp., $[n^{-1}, p^*_n]$ ), where $p^*_n$ is the record-setting probability for any continuous F governing independent coordinates.

On Discrete Frechet Distribution : Estimation and Application

A discrete two-parameter distribution corresponding to the continuous distribution is introduced and derived by using the approach of discretization of continuous distribution functions . The Survival and Hazard functions of the proposed distribution were also found and examined . Two methods were discussed to estimate the parameters , Maximum Likelihood and the Bayesian method , and then compare the performance using the simulation method . In addition, the compatibility of the proposed distribution was investigated through real data , and then the estimates of the survival and Hazard functions were reached . The superiority of the Bayesian method in data modelling was shown .

Incentive Mechanism in the Sponsored Content Market With Network Effects

We propose an incentive mechanism for the sponsored content provider (CP) market in which the communication of users can be represented by a graph, and the private information of the users is assumed to have a continuous distribution function. The CP stipulates incentive rewards to encourage users to reveal their private information truthfully and increase their content demand, which leads to an increase in the advertising revenue. We prove that all users gain a nonnegative utility and disclose their private information truthfully. Moreover, we study the effectiveness and scalability of the proposed mechanism in a case study with different network structures.

Uncertainty quantification based on residual Tsallis entropy of order statistics

<abstract><p>In this study, we focused on investigating the properties of residual Tsallis entropy for order statistics. The reliability of engineering systems is highly influenced by order statistics, for example, when modeling the lifetime of a series system and the lifetime of a parallel system. The residual Tsallis entropy of the ith order statistic from a continuous distribution function and its deviation from the residual Tsallis entropy of the ith order statistics from a uniform distribution were investigated. In the mathematical framework, a method was provided to represent the residual Tsallis entropy of the ith order statistic in the continuous case with respect to the case where the distribution was uniform. This approach can provide insight into the behavior and properties of the residual Tsallis entropy for order statistics. We also investigated the monotonicity of the new uncertainty measure under different conditions. An investigation of these properties leads to a deeper understanding of the relationship between the position of the order statistics and the resulting Tsallis entropy. Finally, we presented the computational results and proposed estimators for estimating the residual Tsallis entropy of an exponential distribution. For this purpose, we derived a maximum likelihood estimator.</p></abstract>

On the Use of Lehmann’s Alternative to Capture Extreme Losses in Actuarial Science

This paper studies properties and applications related to the mixture of the class of distributions built by the Lehmann’s alternative (also referred to in the statistical literature as max-stable or exponentiated distribution) of the form [G(·)]λ, where λ>0 and G(·) is a continuous cumulative distribution function. This mixture can be useful in economics, financial, and actuarial fields, where extreme and long tails appear in the empirical data. The special case in which G(·) is the Stoppa cumulative distribution function, which is a good description of the random behaviour of large losses, is studied in detail. We provide properties of this mixture, mainly related to the analysis of the tail of the distribution that makes it a candidate for fitting actuarial data with extreme observations. Inference procedures are discussed and applications to three well-known datasets are shown.

On quantile-based dynamic survival extropy and its applications

The cumulative residual extropy is an uncertainty measure that parallels extropy in an absolutely continuous cumulative distribution function. The dynamic version of this measure is known as dynamic survival extropy. In this paper, we study some properties of the dynamic survival extropy using quantile function approach. Unlike the dynamic survival extropy, the quantile-based dynamic survival extropy determines the quantile density function uniquely through a simple relationship. We also extend the definition of quantile-based dynamic survival extropy into order statistics. Finally, an application of new quantile-based uncertainty measure as a risk measure is derived.

On asymptotic properties of spacings

In this work, we investigate spacings based on order statistics obtained from continuous distribution functions. At the beginning of the paper, we present distributional results for spacings and a method of classification of distributions according to their tails. Then we use this method to derive asymptotic results for spacings. By applying some special versions of Borel–Cantelli lemma, we obtain their strong limit results. At the end of the paper, we present some illustrative examples.

Nonequilibrium Study of the J1−J2 Ising Model with Random J2 Couplings in the Square Lattice

We studied the phase transitions of the [Formula: see text] spin-1/2 Ising model in the square lattice by considering [Formula: see text] fixed and [Formula: see text] as random interactions following discrete and continuous probability distribution functions. The configuration of [Formula: see text] in the lattice evolves in time through a competing kinetics using the Metropolis algorithm leading to a steady state without reaching the free-energy minimum. So, different phase diagrams were obtained according to the distribution of [Formula: see text]. For fixed and random values of [Formula: see text], we noted that the amplitude of the correlation length is different from the Ising value at the critical point of the superantiferromagnetic–paramagnetic frontier.

Convex Stochastic Approaches for the Optimal Allocation of Distributed Energy Resources in AC Distribution Networks with Measurements Fitted to a Continuous Probability Distribution Function

This paper addresses the optimal stochastic allocation of distributed energy resources in distribution networks. Typically, uncertain problems are analyzed in multistage formulations, including case generation routines, resulting in computationally exhaustive programs. In this article, two probabilistic approaches are proposed–range probability optimization (RPO) and value probability optimization (VPO)–resulting in a single-stage, convex, stochastic optimal power flow problem. RPO maximizes probabilities within a range of uncertainty, whilst VPO optimizes the values of random variables and maximizes their probabilities. Random variables were modeled with hourly measurements fitted to the logistic distribution. These formulations were tested on two systems and compared against the deterministic case built from expected values. The results indicate that assuming deterministic conditions ends in highly underestimated losses. RPO showed that by including ±10% uncertainty, losses can be increased up to 40% with up to −72% photovoltaic capacity, depending on the system, whereas VPO resulted in up to 85% increases in power losses despite PV installations, with 20% greater probabilities on average. By implementing any of the proposed approaches, it was possible to obtain more probable upper envelopes in the objective, avoiding case generation stages and heuristic methods.

Finite-Mixture Continuous Logit Model: Formulation and Application for Shopping Start Time Choice of Non-Commuters

As a continuous generalization of the multinomial logit (MNL) model, the continuous logit (CL) model can be used for continuous response variables (e.g., departure time and activity duration). However, the existing CL model requires the calculation of numerical integrals to obtain the choice probabilities; it thus takes a long time to estimate the model parameters, particularly when the sample size is large. In this paper, we formulate the finite-mixture CL (FMCL) model as a new continuous choice model by combining the finite-mixture method and the CL model, in which the continuous distributional function of the finite mixture is embedded in the CL model. As a result, the individual choice probability can be obtained directly by computing the probability density of the continuous distribution function; this avoids calculation of the integral but still obeys the random utility maximization (RUM) principle. Simulation experiments are conducted to demonstrate the capability of the model. In an empirical study, the proposed model is applied for non-commuters’ shopping activity start time using the expectation-maximization (EM) algorithm based on Shanghai Household Travel Survey data. The results show that the FMCL model developed in this paper can greatly reduce the model estimation time (10,048 observations requiring only 3 min) of the CL model, and the model also has a more intuitive interpretation of model coefficients, directly reflecting variable effects on time-of-day choice. These two advantages can greatly enhance the practical value of the proposed modeling method.

Review of Play and Preisach Models for Hysteresis in Magnetic Materials

This paper studies the properties of the Preisach model and the play model, and compare their similarities. Both are history-dependent hysteresis models that are used to model magnetic hysteresis. They are described as discrete sums of simple hysteresis operators but can easily be reformulated as integral equations of continuous distribution functions using either a Preisach weight distribution function or a play distribution function. The models are mostly seen as phenomenological or mathematical tools but can also be related to friction-like pinning of domain-wall motions, where Rayleigh's law of magnetic hysteresis can be seen as the simplest case on either the play model or the Preisach model. They are poor at modeling other domain behavior, such as nucleation-driven hysteresis. Yet another hysteresis model is the stop model, which can be seen as the inverted version of the play model. This type of model has advantages for expressions linked to energy and can be related to Steinmetz equation of hysteresis losses. The models share several mathematical properties, such as the congruency property and wiping-out property, and both models have a history of dependence that can be described by the series of past reversal points. More generally, it is shown that the many models can be expressed as Preisach models, showing that they can be treated as subcategories of the Preisach type models. These include the play model, the stop model and also the alternative KP-hysteron model.

On the Solvability of a System of Nonlinear Integral Equations with the Hammerstein–Stieltjes Operator on the Half-Line

We study a system of nonlinear integral equations of the Hammerstein–Stieltjes type whose pre-kernels are continuous distribution functions. A constructive existence theorem for a nontrivial, nonnegative, and bounded solution of the system is proved. The integral asymptotics of the constructed solution is studied. Examples of systems for which all the conditions of the theorem are satisfied are given.

An Impact of the Discrete Representation of the Bubble Size Distribution Function on the Flow Structure in a Bubble Column Reactor

The purpose of the present study is to analyze the effect of different discrete representations of the continuous bubble size distribution function on the flow structure in a bubble column reactor. Poly- and monodisperse media were considered, such that the mathematical expectation of the bubble size in the polydisperse case was equal to the bubble size in the monodisperse case at the same volumetric bubble contents. For these computations the normalized variances of the velocity profiles of the carrier and the disperse phases, the volume fraction of the disperse phase, and the specific area of the interfacial surface were determined. The normalized variances were calculated from a reference scenario with a detailed resolution of the bubble size distribution function with ten bubble classes. It was shown that with increase of the average bubble sizes mono- and polydisperse approaches provide converging solutions. A modified hybrid discretization of the bubble size distribution function with four classes of bubbles was shown to predict the flow structure with normalized variance less than 5% over the entire computational domain for all monitored parameters.

The Method of Probabilistic Nodes Combination in Biomechanics

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

The Method of Probabilistic Nodes Combination in Handwritten Character Reconstruction

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function.

A New Extension Form for Continuous Probability Distributions: Uniform-X Distributions

In this paper, generating extension forms for continuous probability distribution functions is studied. The considered transformer function is applied to three well-known probability distributions- Normal, Kumaraswamy, Weibull- and new extensions of these distributions are obtained. The related functions of the new extensions are defined, random samples are generated from the new extensions and the results are presented. Parameter estimation procedures of the extensions are studied, and likelihood equations are obtained. To demonstrate the modeling performance of the extensions, three different data sets are considered, separately. Each data set is modeled by both the corresponding probability distribution and its extensions. The new extensions give the best fit to the corresponding data over the well-known probability distributions.

Application of Analytical Hierarchical Process in Equipment Maintenance Scheduling and Decision-Making Process in Oil/Gas

Operational efficiency and reliability of equipment directly impact production and revenue streams. Maintenance, an essential aspect of Oil/Gas operations, is a set of activities that aim at preserving the condition of equipment to reduce the probability of failure and increase operating life cycle of equipment. The culture of maintenance evolves considerably from run to failure and corrective to preventive maintenance practices. These set of activities are expensive and significantly affect the cost of sales and operating expenditure especially in Oil and Gas sector. With the need to optimize operating cost, use of data in prioritizing equipment maintenance becomes necessary. This paper provides a novel approach for prioritizing critical equipment maintenance activities using Analytical Hierarchical Process (AH). A family of Weibull distribution function are use to define five parameters (criteria) namely the Weibull Continuous Distribution Function (CDF), Weibull Probability Density Function, Reliability function, failure rate and equipment availability. To validate the use of AHP method, data from Nine (9) critical equipment from a pump station in Port-Harcourt, Nigeria was used to prioritize maintenance activities. The slope shape parameter values of \(\gamma\) \(\epsilon\){0.5,1,2} are considered, which affects the shape of the distribution functions. The results show that multi-criterion AHP-approach supports subjective decision making by providing quantitative weighted ranking of equipment based on priority. The result indicates that equipment with best values has lowest priority ranking.

Representation of the infimum and supremum of a family of multivariate distribution functions

Sklar's theorem represents a single multivariate distribution function through its univariate marginal distributions and a copula. However, it fails in general when it comes to the representation of the envelopes of a family F of multivariate distribution functions (i.e. its point-wise infimum and supremum), even if we allow more general representing functions than copulas. In this paper we develop an alternative representation which describes the envelopes of a family F in terms of the copulas that correspond to the members of F. Our representation is reminiscent of Sklar's representation but is based on the idea of corner patches of copulas. We prove a series of four representation theorems; (1) a general one, which holds even if the envelopes of the family do not possess a Sklar type representation, (2) a version tailored for the case when the envelopes do possess a Sklar type representation, (3) a theorem for families of continuous distribution functions, and (4) a variant appropriate for modeling dependencies of absolutely continuous random variables. In addition, we demonstrate how our results can be applied to give a Sklar type theorem in the imprecise probability setting, which offers a new approach to multivariate probability boxes.