Gamma-type Distribution Research Articles

Dynamics of a two-patch logistic model with diffusion and time delay

In this paper, we proposed a two-patch logistic model connected by diffusion, where one patch includes the Gamma type distribution time delay while the other patch does not include the time delay. In general, Routh–Hurwitz criterion is applied to the derivation for the conditions of Hopf bifurcation, but the more the order of the time delay increases the more the difficulty rises. Hence we adopt the polar form method for the characteristic equation to study the stability of coexistence equilibrium. Our findings show that the diffusion prevents the instabilization of the coexistence equilibrium. Besides, we found that the coexistence equilibrium is stable when time delay is small, and becomes unstable as the delay increases. But it can be restabilized for further increasing of time delay and continues to be stable afterwards. In other words, the diffusion and the time delay are beneficial to the stability of the coexistence equilibrium.

Random networks grown by fusing edges via urns

AbstractMany classic networks grow by hooking small components via vertices. We introduce a class of networks that grows by fusing the edges of a small graph to an edge chosen uniformly at random from the network. For this random edge-hooking network, we study the local degree profile, that is, the evolution of the average degree of a vertex over time. For a special subclass, we further determine the exact distribution and an asymptotic gamma-type distribution. We also study the “core,” which consists of the well-anchored edges that experience fusing. A central limit theorem emerges for the size of the core.At the end, we look at an alternative model of randomness attained by preferential hooking, favoring edges that experience more fusing. Under preferential hooking, the core still follows a Gaussian law but with different parameters. Throughout, Pólya urns are systematically used as a method of proof.

New bivariate gamma types with MIMO application

ABSTRACTIn this paper a bivariate gamma type distribution emanating from the diagonal elements of an inverse Wishart type distribution is developed; which in turn originates from the complex matrix variate elliptical class. From this, a bivariate Weibullised gamma type distribution is also presented, of which the bivariate Nakagami-m type is a special case. The derived results may be applied as decision statistics for a MIMO (multiple input multiple output) system with two transmit antennas. It is proposed that under this elliptical umbrella some performance measures such as the outage probability of MIMO systems can be analyzed in broad generality.

Probabilistic analysis and fatigue damage assessment of offshore mooring system due to non-Gaussian bimodal tension processes

Both wave-frequency (WF) and low-frequency (LF) components of mooring tension are in principle non-Gaussian due to nonlinearities in the dynamic system. This paper conducts a comprehensive investigation of applicable probability density functions (PDFs) of mooring tension amplitudes used to assess mooring-line fatigue damage via the spectral method. Short-term statistical characteristics of mooring-line tension responses are firstly investigated, in which the discrepancy arising from Gaussian approximation is revealed by comparing kurtosis and skewness coefficients. Several distribution functions based on present analytical spectral methods are selected to express the statistical distribution of the mooring-line tension amplitudes. Results indicate that the Gamma-type distribution and a linear combination of Dirlik and Tovo-Benasciutti formulas are suitable for separate WF and LF mooring tension components. A novel parametric method based on nonlinear transformations and stochastic optimization is then proposed to increase the effectiveness of mooring-line fatigue assessment due to non-Gaussian bimodal tension responses. Using time domain simulation as a benchmark, its accuracy is further validated using a numerical case study of a moored semi-submersible platform.

Limiting Approach to Generalized Gamma Bessel Model via Fractional Calculus and Its Applications in Various Disciplines

The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral, one can list out almost all of the extended densities for the pathway parameter q < 1 and q → 1. Here, we bring out the idea of thicker- or thinner-tailed models associated with a gamma-type distribution as a limiting case of the pathway operator. Applications of this extended gamma model in statistical mechanics, input-output models, solar spectral irradiance modeling, etc., are established.

FRACTIONAL ORNSTEIN–UHLENBECK PROCESSES MIXED WITH A GAMMA DISTRIBUTION

We study the asymptotic behavior of the aggregated fractional Ornstein–Uhlenbeck processes with random coefficients that follows a Gamma-type distribution. We also analyze the properties of the limit process.

Development of the dynamic model of production quality during manufacture

The study sets and solves the problem of finding out explicit dependencies of garment products quality on the production stage by means of developing algorithms for constructing a system of determiners of the overall quality of a batch of products. We have revealed the basic types of quality indicators distribution for various stages of production. The paper describes a solution of the urgent task of developing a dynamic model that determines quality on the basis of expert research indicators of the quality of the light industry products by taking into account the actual distribution of these parameters at different stages of the process. Furthermore, the study solves the problem of finding out explicit dependencies of garment products quality on the production stage by means of developing algorithms for constructing a system of determiners of the overall quality of a batch of products. We have revealed that production quality indicators may have different distributions depending on the level of production. Different stages of manufacturing have been found determinant for normal, gamma-type, and reverse gamma distribution. A conclusion has been made on the fundamental differences of processing the results of quality indicators for different levels of production stages while using different statistical distributions.

Structure, growth and secondary production of two Tyrrhenian populations of the white gorgonian Eunicella singularis (Esper 1791)

Growth and secondary production of the shallow-water gorgonian Eunicella singularis were investigated at two infralittoral rocky stations located in the Tyrrhenian Sea (Cannitello and Cape Enfola), separated by a cline of 4° latitude. Colonies were aged by counting the number of annual growth rings at the base of the skeleton. Mean annual linear growth rates were calculated by fitting a von Bertalanffy growth equation for the species. Growth rates were 3.55 and 4.52 cm yr−1 respectively. The Enfola population exhibited an age structure reflecting a past pulse in recruitment. The Cannitello population exhibited a gamma-type distribution, dominated by younger age classes, thus suggesting a continuous recruitment rate over time. The two population differed also in colony density, which was higher at Enfola. Secondary production was estimated by means of the increment-summation method, and yielded 5.6 and 3.2 g ash-free dry weight (AFDW) m−2 yr−1 for Cannitello and Enfola populations, respectively. P/B ratio was higher at Cannitello, because of the age structure of the population. Strong currents probably cause population density-thinning mechanisms, favoring a more continuous recruitment.

Bivariate generalized gamma distributions of Kibble's type

In this paper, a new type of bivariate generalized gamma (BGG) distribution derived from the bivariate gamma distribution of Kibble [Two-variate gamma-type distribution. Sankhȳa 1941;5:137–150] by means of a power transformation is presented. The explicit expressions of statistical properties of the BGG distribution are presented. The estimation of marginal and dependence parameters using the method of moments and the method of inference functions for margins are discussed, and their performance through a Monte Carlo simulation study is assessed. Finally, an example is given to illustrate the applicability of the distributions introduced here.

A generalized gamma model associated with a Bessel function

A new probability density function associated with a Bessel function is introduced, which is the generalization of a gamma-type distribution. Some of its special cases are also mentioned. Multivariate analogue, conditional density, best predictor function, Bayesian analysis, etc., connected with this new density are also introduced. Suitability of this density as a good model in Bayesian inference and regression theory is also discussed.

Característica produtiva e estrutural de pastos mistos de aveia e azevém manejados em quatro alturas sob lotação contínua

Monitoraram-se variáveis relacionadas à produção e à estrutura de pastos de aveia-preta (Avena strigosa, Schreb) e azevém (Lolium multiflorum, Lam) manejados em quatro alturas (10, 20, 30 e 40 cm), em pastejo contínuo com lotação variável (put and take). Utilizaram-se novilhos de corte mestiços de aproximadamente 10 meses de idade com peso médio inicial de 190 kg, em delineamento de blocos completos casualizados com três repetições. Investigou-se o tipo de distribuição das frequências de altura em seis datas, bem como o potencial de predição da massa seca de forragem por meio da altura do pasto. A massa de forragem aumentou de forma linear de acordo com a altura do pasto: cada 1 cm de aumento na altura acima de 10 cm correspondeu a um acréscimo de cerca de 107 kg de MS/ha na massa de forragem. Não houve efeito da altura do pasto na taxa de acúmulo (55,8 kg de MS/ha.dia) nem na produção total de matéria seca (8.210 kg de MS/ha). A distribuição das frequências ajustou ao modelo normal em apenas uma das 96 séries analisadas. A distribuição tipo Gamma foi a que mais frequentemente se ajustou aos dados de altura do pasto, porém, uma vez iniciado o pastejo, a heterogeneidade no pasto foi tão alta que a distribuição das alturas não se ajustou a nenhum dos modelos estudados. É possível estimar a massa seca de forragem existente no pasto por meio de sua altura média, assim, sugere-se estabelecer alturas variáveis ao longo do ciclo de pastejo, com o intuito de se administrar a heterogeneidade causada pelo animal.

Description of the transport critical current density behavior of polycrystalline superconductors as a function of the applied magnetic field

Abstract A model to describe the critical current density behavior of high- T c polycrystalline superconductors is proposed for all magnetic field values. The main features of the model are as follows: the transport critical current density is controlled by the weak-link network at grain boundaries. The size distribution of weak links is well represented by a Gamma-type distribution. Finally, the tunneling critical current between grains follows a Fraunhofer diffraction pattern or a modified pattern if the applied magnetic field is lower or higher than the first critical field H c 1 .

The Suppression of Four-Wave Mixing by Random Dispersion

Pairwise interactions of optical pulses lead to the creation of four-wave mixing (FWM) products in the presence of periodic damping and amplification. In this manuscript, we examine how these FWM products grow in the presence of small to moderate random dispersion. Namely, we show that (i) the growth of the FWM products in the presence of white noise is inversely proportional to the noise strength $\bar{D}$, confirmed by both analytical and numerical results; (ii) the FWM products as a function of $\bar{D}$ obey a gamma-type probability distribution function; and (iii) the presence of either white or Ornstein–Uhlenbeck (OU) noise has a similar influence on the growth of these FWM products. This work shows that small random dispersion can effectively mitigate the deleterious effects of FWM in wavelength-division–multiplexed optical communications systems for either sech-type or Gaussian-type input pulses.

Accumulated strain in polymer-modified asphalts

One of the distress modes of asphalt pavements is the rutting that occurs at high operating temperatures, and it is believed that the accumulated strain in asphalt binder is mainly responsible for the rutting. The asphalt industry has tried to use the parameter 1/J″ as the specification parameter for the determination of the susceptibility of asphalt pavements to rutting. It was, however, recognized that this parameter is not suitable for polymer-modified asphalts, and currently, the repeated creep and recovery test (dynamic creep test) is studied as a new specification tool. In this paper, we study the general response of binders to a train of shear-stress pulses with a constant rest time between the pulses. We have found that the viscous approximation of the creep compliance function yields a suitable description of the test when the creep recovery is very weak. To extend the description to cases with significant recovery, an alternative model is proposed. Using the simple approximation of the continuous retardation spectrum, L, by the reduced loss compliance, L is modeled by a gamma-type distribution function. With this form of the shear compliance function, one can successfully describe the dynamic creep test in various polymer-modified asphalts.

Finite mixture of gamma distributions: A conjugate prior

A finite mixture of gamma distributions [Finite mixture of certain distributions. Comm. Statist. Theory Methods 31(12), 2123–2137] is used as a conjugate prior, which gives a nice form of posterior distribution. This class of conjugate priors offers a more flexible class of priors than the class of gamma prior distributions. The usefulness of a mixture gamma-type prior and the posterior of uncertain parameters λ for the Poisson distribution are illustrated by using Markov Chain Monte Carlo ( MCMC), Gibbs sampling approach, on hierarchical models. Using the generalized hypergeometric function, the method to approximate maximum likelihood estimators for the parameters of Agarwal and Al-Saleh [Generalized gamma type distribution and its hazard rate function. Comm. Statist. Theory Methods 30(2), 309–318] generalized gamma-type distribution is also suggested.

Some complex matrix-variate statistical distributions on rectangular matrices

In a recent survey, Olkin [The 70th anniversary of the distribution of random matrices: a survey, Linear Algebra Appl. 354 (2002) 231–243] has looked into 70 years of development on random matrices, giving special attention to rectangular matrices. Gupta and Richards [Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233–256] have looked into generalizations of the Dirichlet family of distributions for the multivariate case. Mathai [An Introduction to Geometrical Probability: Distributional Aspects with Applications, Gordon and Breach Scientific Publishers, New York, 1999] developed a gamma type distribution on rectangular matrices in connection with the distributional aspects of random volumes. This idea is developed further in the present article to define rectangular matrix-variate gamma type, beta type and Dirichlet type distributions with location and scale matrices; connections among these distributions and various properties are pointed out. This paper discusses the complex matrix-variate cases and the corresponding real cases are also listed along with each result. The complex rectangular matrix-variate t and Cauchy distributions are obtained as particular cases.

Coherence, power laws, and the negative polarization surge.

We develop a second-order ray-tracing model that includes the constructive interference of reciprocal rays and Fresnel reflections from an inverse gamma-type distribution function of path length to predict the negative polarization branch seen in some astronomical bodies. We expect that such a path-length distribution might resemble the path lengths undertaken by rays incident upon some astronomical bodies. The resulting negative polarization is largely wavelength independent and depends primarily on the power law in the path-length distribution, which coincides with some observations.

A theoretical analysis of crack nucleation due to grain boundary dislocation pile-ups in a random ice microstructure

Abstract The nucleation of cracks is generally recognized as the precursor to the macroscopic fracture of many brittle materials. In this paper, the existence of load-induced cracks in polycrystalline S2 ice is investigated theoretically in terms of the pile-ups of grain boundary dislocations around grain triple points. Stress concentrations due to the elastic or thermal expansion anisotropy of the ice crystals are taken into account. Considering a random microstructure, the numerical results show that: (i) both stable and unstable cracks can be nucleated during loading, (ii) the microstructural stresses due to either type of crystal anisotropy obey a Gaussian distribution, (iii) the sizes of the stable cracks obey a gamma-type distribution with a small mean and a large skewness, (iv) the orientations of the stable cracks are approximately uniformly distributed, and (v) the crack statistics is a function of the microstructure and the loading conditions.

Theory of gas-solid adsorption chromatography for heterogeneous adsorbents

Presented here is a general integral equation for describing the total net retention volume in gas-solid adsorption chromatography (GSC). This integral equation contains a distribution function that characterizes the energetic heterogeneity or a solid adsorbent. For the gamma-type distribution function, this integral generates a simple analytical equation for the total net retention volume. Application or this analytical equation for describing the retention or n-hexane and 1-hexene on controlled porosity glass demonstrated its utility for evaluating the physicochemical quantities that characterize gas-solid interactions and the energetic heterogeneity or the adsorbent

Application of the generalized Jaroniec-Choma isotherm equation for describing benzene adsorption on acttvated carbons

This equation was derived by assuming the Dubinin-Radushkevich (DR) equation to describe the gas adsorption in uniform micropores, and by applying the gamma-type distribution function to represent the structural heterogeneity of micropores