Probability Laws Derived from the Gamma Function

Several densities or probability laws of continuous random variables derive from the Euler Gamma function. These laws form the basis of sampling theory, namely hypothesis testing and estimation. Namely the gamma, beta, and Student law, through the chi-square law and the normal law are all distributions resulting from applications of Euleur functions.

COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITELY MANY GAMMA FUNCTIONS

In this paper,by deriving an inequality involving the generating function of the Bernoulli numbers,the author introduces a new ratio of finitely many gamma functions,finds complete monotonicity of the second logarithmic derivative of the ratio,and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.

SOME COMPLETELY MONOTONIC FUNCTIONS ASSOCIATED WITH THE q-GAMMA AND THE q-POLYGAMMA FUNCTIONS

In this paper, the q-analogue of the Stirling formula for the q-gamma function （Moak formula） is exploited to prove the complete monotonicity properties of some functions involving the q-gamma and the q-polygamma functions for all real number q 〉 0. The monotonicity of these functions is used to establish sharp inequalities for the q-gamma and the q-polygamma functions and the q-Harmonic number. Our results are shown to be a generalization of results which were obtained by Selvi and Batir [23].

Some Properties on the Function Involving the Gamma Function

We studied the monotonicity and Convexity properties of the new functions involving the gamma function, and get the general conclusion that Minc-Sathre and C. P. Chen-G. Wang’s inequality are extended and refined.

Gamma Functions and Siegel Integrals on Nonselfdual Cones

In this paper,we will define Gamma functions and Siegel integrals on nonselfdual cones and give some applications.

Analytical solution of modified point kinetics equations for linear reactivity variation in subcritical nuclear reactors adopting an incomplete gamma function approximation

The present work aims to achieve a fast and accurate analytical solution of the point kinetics equations applied to subcritical reactors such as ADS (Accelerator-Driven System), assuming a linear reactivity and external source variation. It was used a new set of point kinetics equations for subcritical systems based on the model proposed by Gandini & Salvatores. In this work it was employed the integrating factor method. The analytical solution for the case of interest was obtained by using only an approximation which consists of disregarding the term of the second derivative for neutron density in relation to time when compared with the other terms of the equation. And also, it is proposed an approximation for the upper incomplete gamma function found in the solution in order to make the computational processing faster. In addition, for purposes of validation and comparison a numerical solution was obtained by the finite differences method. Finally, it can be concluded that the obtained solution is accurate and has fast numerical processing time, especially when compared with the results of numerical solution by finite difference. One can also observe that the gamma approximation used achieve a high accuracy for the usual parameters. Thus we got satisfactory results when the solution is applied to practical situations, such as a reactor startup.

Investigations into Some Simple Expressions of the Gamma Function in Wind Power Theoretical Estimate by the Weibull Distribution

The Weibull distribution is a probability density function (PDF) which is widely used in the study of meteorological data. The statistical analysis of the wind speed v by using the Weibull distribution leads to the estimate of the mean wind speed , the variance of v around and the mean power density in the wind. The gamma function Γ is involved in those calculations, particularly Γ (1+1/k), Γ (1+2/k) and Γ (1+3/k). The paper reports the use of the Weibull PDF f(v) to estimate the gamma function. The study was performed by looking for the wind speeds related to the maximum values of f(v), v2 f(v) and v3 f(v). As a result, some approximate relationships were obtained for Γ (1+1/k), Γ (1+2/k) and Γ (1+3/k), that use some fitting polynomial functions. Very good agreements were found between the exact and the estimated values of Γ (1+n/k) that can be used for the estimation of the mean wind speed , the variance σ2 of the wind speed v;around the mean speed and the average wind power density.

A Distributional Representation of Gamma Function with Generalized Complex Domian

In this paper, we present a new representation of gamma function as a series of complex delta functions. We establish the convergence of this representation in the sense of distributions. It turns out that the gamma function can be defined over a space of complex test functions of slow growth denoted by Z. Some properties of gamma function are discussed by using the properties of delta function.

Difference Independence of the Euler Gamma Function

In this paper,the authors established a sharp version of the difference analogue of the celebrated Holder’s theorem concerning the differential independence of the Euler gamma functionГ.More precisely,if P is a polynomial of n+1 variables in C[X,Y0,…,Yn-1]such that P(s,Г(s+a0),…,Г(s+an-1))≡0 for some(a0,…,an-1)∈Cn and ai-aj■Z for any 0≤i<j≤n-1,then they have P≡0.Their result complements a classical result of algebraic differential independence of the Euler gamma function proved by H?lder in 1886,and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006.

The Infinite Polynomial Products of the Gamma and Zeta Functions

Starting with the binomial coefficient and using its infinite product representation, the infinite product representation of the gamma function and of the zeta function are composed of an exponential and of a trigonometric component and proved. It is proved, that all these components define imaginary roots on the critical line, if written in the form as they are in the functional equation of the zeta function.

Serum Gamma-glutamyl Transferase Levels Predict Functional Outcomes after Aneurysmal Subarachnoid Hemorrhage

Objective We aim to explore the potential association between serum gamma-glutamyl transferase levels and functional outcome after aneurysmal subarachnoid hemorrhage in a Chinese population. Methods A total of 386 aneurysmal subarachnoid hemorrhage patients were included in the study from September 2007 to February 2015. Baseline serum gamma-glutamyl transferase levels and 6-month follow-up functional outcomes were determined. A poor outcome was defined as a modified ranking scale score of ≥ 3. The multivariable logistic model was used to analyze the relationship between serum gamma-glutamyl transferase and clinical outcomes after aneurysmal subarachnoid hemorrhage. Results The adjusted poor outcome rates of patients with gamma-glutamyl transferase levels of 〈 30 U/L, 30-50 U/L and ≥ 50 U/L were 16.7%, 19.6%, and 34.4%, respectively （P 〈 0.01）. The age-sex and multivariable adjusted odds ratios （95% confidence intervals） of poor prognosis comparing the top group （≥ 50 U/L） with the lowest group （〈 30 U/L） were 5.76 （2.74-12.13）, 6.64 （2.05-21.52）, and 6.36 （1.92-21.02）. A significant linear trend existed between gamma-glutamyl transferase level and aneurysmal subarachnoid hemorrhage prognosis. This association was also observed among nondrinkers. Conclusion Patients with higher gamma-glutamyl transferase levels were more likely to have a poor prognosis. Serum gamma-glutamyl transferase can be considered to be an independent predictor of functional outcomes after aneurysmal subarachnoid hemorrhage.

Gamma-ray luminosity function of gamma-ray bright AGNs

Detection of γ-ray emissions from a class of active galactic nuclei （viz blazars）, has been one of the important findings from the Compton Gamma-Ray Observatory （CGRO）. However, their γ-ray luminosity function has not been well determined. Few attempts have been made in earlier works, where BL Lacs and Flat Spectrum Radio Quasars （FSRQs） have been considered as a single source class. In this paper, we investigated the evolution and γ-ray luminosity function of FSRQs and BL Lacs separately. Our investigation indicates no evolution for BL Lacs, however FSRQs show significant evolution. Pure luminosity evolution is assumed for FSRQs and exponential and power law evolution models are examined. Due to the small number of sources, the low luminosity end index of the luminosity function for FSRQs is constrained with an upper limit. BL Lac luminosity function shows no signature of break. As a consistency check, the model source distributions derived from these luminosity functions show no significant departure from the observed source distributions.

Point spread functions for a small gamma camera with pinhole collimator

A set of point spread functions (PSF) has been obtained by means of Monte-Carlo simulation for asmall gamma camera with a pinhole collimator of various hole diameters. The FOV (field of view) of the camera isexpended from 45 mm to 70 mm in diameter. The position dependence of the variances of PSF is presented, and theacceptance for the 140 kev gamma rays is explored. A phantom of 70 mm in diameter was experimentally imaged inthe camera with effective FOV of only 45 mm in diameter.

Females with schizophrenia have abnormal functional cortical connectivity in the gamma frequency during an auditory oddball task using magnetoencephalography

We studied differences in imaginary coherence (IC) of the gamma band between brain regions of female schizophrenia patients during the auditory oddball task using magnetoencephalography (MEG). Subjects were 12 right-handed female schizophrenia patients, who were evaluated by the Positive and Negative Syndrome Scales (PANSS). Functional connectivity during an auditory oddball task was reconstructed in low gamma (30 - 50 Hz) and high gamma (50 - 100 Hz and 100 - 150 Hz) bands, and represented by IC using seeds determined by the significant oscillatory power changes obtained by event-related synchronization (ERS) and event-related desynchronization (ERD) power measurements. Gamma ERS (30 - 50 Hz) power was decreased in the left precuneus at 500 - 750 ms and in the right precuneus at 750 - 1000 ms. IC in the gamma band (50 - 100 Hz) was decreased between the right precuneus (seed) and right paracentral lobule (target) and between the right precuneus and right hypothalamus at 0 - 250 ms. IC in the gamma band (100 - 150 Hz) was increased between the left precuneus and right cuneus (Brodmann area 7) at 250 - 500 ms, between the left precuneus and right culmen at 500 - 750 ms, and between the left precuneus and right cuneus (Brodmann area 17), between the left precuneus and right posterior cingulate cortex, and between the left precuneus and right caudate nucleus at 750 - 1000 ms. In the high gamma band (50 - 100 Hz) at 0 - 250 ms, significant positive correlations were shown between IC and conceptual disorganization in PANSS scores, between IC and unusual thought content score, and between IC and positive scale score. IC within the high gamma band in female schizophrenia patients showed two types of functional disconnection, intrahemispheric and interhemispheric. IC between the right or left precuneus and other specific cortical areas showed dysfunction, suggesting that the parietal lobe plays an important role in dysfunction in connectivity in the gamma band during the oddball task.

联系着广义gamma函数的Raabe型积分

通过渐近公式与分析方法计算了若干联系着广义gamma函数的Raabe型积分.

一般Gamma型算子在Orlicz空间内的逼近

讨论一般Gamma型算子在Orlicz空间内的逼近问题,由于Orlicz空间比连续函数空间和Lp空间“大”,范数结构也比Lp空间复杂,所以在Orlicz空间内研究Gamma算子的逼近问题具有一定的拓展性。借助Cauchy⁃Schwarz不等式、Jensen不等式、光滑模和N函数的凸性等,证明一般Gamma型算子在Orlicz空间内的有界性,给出该算子在Orlicz空间内的逼近正定理。

Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New Probability Measures Generalizing the Gamma-Type and Gaussian-Type Distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.

Shrinkage Testimator in Gamma Type-II Censored Data under LINEX Loss Function

Prakash and Singh presented the shrinkage testimators under the invariant version of LINEX loss function for the scale parameter of an exponential distribution in presence Type-II censored data. In this paper, we extend this approach to gamma distribution, as Prakash and Singh’s paper is a special case of this paper. In fact, some shrinkage testimators for the scale parameter of a gamma distribution, when Type-II censored data are available, have been suggested under the LINEX loss function assuming the shape parameter is to be known. The comparisons of the proposed testimators have been made with improved estimator. All these estimators are compared empirically using Monte Carlo simulation.

钢梁腐蚀相关性对钢-混组合梁桥体系可靠度的影响

服役环境中钢主梁的腐蚀本质上是一个随机过程,腐蚀深度随机变量既有时变性亦有空间性。现有测量方法只能获得单根钢梁腐蚀深度数据,尚不能明确结构体系腐蚀深度变量的联合分布模式,可能导致腐蚀结构体系可靠度计算模型失真。以某钢-混凝土组合梁桥为研究对象,首先建立了钢主梁腐蚀深度增长Gamma随机过程模型,并利用钢梁腐蚀深度实测数据对模型进行了验证;然后采用Kendall秩相关系数量化表征主梁腐蚀深度相关性,推导了腐蚀钢梁的时变抗力计算公式;最后基于Copula函数提出了考虑主梁腐蚀相关性的梁桥体系失效概率计算方法,对比研究了钢梁腐蚀相关性对钢-混组合简支梁桥体系可靠度的影响。研究结果表明:桥梁运营前期腐蚀相关性对梁桥受力性能影响较小,在运营中后期,中高程度的腐蚀相关性将会对梁桥体系的可靠度影响较大。研究提出了钢梁腐蚀空间相关性量化方法,获得了更精确的钢-混组合梁桥体系可靠度计算方法,可为组合梁桥体系的时变可靠度评定提供参考。

Stein方法在求解随机变量线性组合概率分布中的应用

给出一种基于改进后的Stein方程求解随机变量线性组合概率分布的证明方法.基于随机变量Z_(r)和Z_(r)的线性组合H=m^(-1 )Z_r+n^(-1),利用伽马分布Γ(r,1)的Stein特征,求得关于H的Stein方程;加入二次可微函数,并采用改进后的Stein方程,求得H_(+)和H_(-)概率密度函数及显示公式.