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].

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.

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.

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.

A Two-Sided Inequality of Gamma Function

This note shows that the inequaiity （x/e）^x√2πx（1＋1/12x）〈Г（x＋1）〈（x/e）^x√2πx（1＋1/12x-0.5） holds for all x≥1.

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.

Evaluation of Generalized Error Function via Fast-Converging Power Series

A generalized form of the error function, Gp(x)=pΓ(1/p)∫0xe−tpdt, which is directly associated with the gamma function, is evaluated for arbitrary real values of p>1and 0x≤+∞by employing a fast-converging power series expansion developed in resolving the so-called Grandi’s paradox. Comparisons with accurate tabulated values for well-known cases such as the error function are presented using the expansions truncated at various orders.

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.

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.

基于Weibull函数和Gamma函数的环境污染与经济增长的关系

环境库兹涅茨曲线是研究环境污染与经济增长之间关系的有效工具。传统的环境库兹涅茨曲线研究大多采用线性模型或对数线性模型，其在采用模型描述关系的问题上存在一定的不足。为克服这些不足，本文采用Weibull函数和Gamma函数形式的面板数据模型对中国29个省区1989～2005年四种环境污染指标人均排放量与人均收入之间的关系予以研究。结果表明Weibull函数和Gamma函数的面板数据模型拟合效果较好，且参数具有较好的解释能力；人均废水和人均SO2都随人均收入增加先上升后减少，在25000元附近出现结构转变点，而人均固体废弃物和人均废气随人均收入增加则呈现单调增加的变化趋势，潍有出现结构转变点。

非完整滑坡编目三参数反Gamma概率分布模型

中国"5.12"地震震级高,震源浅,于震区内触发了大量滑坡。为了研究地震滑坡的概率分布特征,对四川映秀极震区(约109.3 km2)地震滑坡进行系统编目,该编目属非完整滑坡编目。利用该编目建立三参数反Gamma概率分布模型,其拟合参数分别为b=1.11,a=2.17×10-3 km2及s=-3.925×10-4 km2。基于该模型对编目区内滑坡面积均值、总面积及最大滑坡面积进行计算,并引入滑坡标度及建立指数修正函数分别对"5.12"四川汶川地震触发滑坡总数及滑坡总面积进行预测,同时建议模型建立原则。研究结果表明:模型的幂律衰减参数值与已有研究结果相近;模型对编目区内滑坡面积均值、总面积及最大滑坡面积的计算结果与实测值之间的相对误差均为10%左右;模型对"5.12"汶川地震触发滑坡总数及滑坡总面积的预测结果均与已有研究结果比较接近。