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.

Cortico-striatal gamma oscillations are modulated by dopamine D3 receptors in dyskinetic rats

Long-term levodopa administration can lead to the development of levodopa-induced dyskinesia.Gamma oscillations are a widely recognized hallmark of abnormal neural electrical activity in levodopa-induced dyskinesia.Currently,studies have reported increased oscillation power in cases of levodopa-induced dyskinesia.However,little is known about how the other electrophysiological parameters of gamma oscillations are altered in levodopa-induced dyskinesia.Furthermore,the role of the dopamine D3 receptor,which is implicated in levodopa-induced dyskinesia,in movement disorder-related changes in neural oscillations is unclear.We found that the cortico-striatal functional connectivity of beta oscillations was enhanced in a model of Parkinson’s disease.Furthermore,levodopa application enhanced cortical gamma oscillations in cortico-striatal projections and cortical gamma aperiodic components,as well as bidirectional primary motor cortex(M1)↔dorsolateral striatum gamma flow.Administration of PD128907(a selective dopamine D3 receptor agonist)induced dyskinesia and excessive gamma oscillations with a bidirectional M1↔dorsolateral striatum flow.However,administration of PG01037(a selective dopamine D3 receptor antagonist)attenuated dyskinesia,suppressed gamma oscillations and cortical gamma aperiodic components,and decreased gamma causality in the M1→dorsolateral striatum direction.These findings suggest that the dopamine D3 receptor plays a role in dyskinesia-related oscillatory activity,and that it has potential as a therapeutic target for levodopa-induced dyskinesia.

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.

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.

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].

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.

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.

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.

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.

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.

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 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.

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.

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 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.

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.

Reconstruction of Probability Density Function for Gamma Distribution

The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the statistical analysis.Based on gamma distribution,four methods of probability density function(PDF)reconstruction with early failure data are proposed,and then the mean time between failures(MTBF)evaluation expressions are concluded from the reconstructed PDFs.Both theory analysis and an example show that method 2 is the best evaluation method in dealing with early-failure-small-sample data.The reconstruction methods of PDF also have certain guiding significance for other distribution types.

Analysis of Progressively Type-Ⅱ Inverted Generalized Gamma Censored Data and Its Engineering Application

A novel inverted generalized gamma(IGG)distribution,proposed for data modelling with an upside-down bathtub hazard rate,is considered.In many real-world practical situations,when a researcher wants to conduct a comparative study of the life testing of items based on cost and duration of testing,censoring strategies are frequently used.From this point of view,in the presence of censored data compiled from the most well-known progressively Type-Ⅱ censoring technique,this study examines different parameters of the IGG distribution.From a classical point of view,the likelihood and product of spacing estimation methods are considered.Observed Fisher information and the deltamethod are used to obtain the approximate confidence intervals for any unknown parametric function of the suggestedmodel.In the Bayesian paradigm,the same traditional inferential approaches are used to estimate all unknown subjects.Markov-Chain with Monte-Carlo steps are considered to approximate all Bayes’findings.Extensive numerical comparisons are presented to examine the performance of the proposed methodologies using various criteria of accuracy.Further,using several optimality criteria,the optimumprogressive censoring design is suggested.To highlight how the proposed estimators can be used in practice and to verify the flexibility of the proposed model,we analyze the failure times of twenty mechanical components of a diesel engine.

Gamma Approximation Based Multi-Antenna Covert Communication Detection

Covert communication technology makes wireless communication more secure,but it also provides more opportunities for illegal users to transmit harmful information.In order to detect the illegal covert communication of the lawbreakers in real time for subsequent processing,this paper proposes a Gamma approximation-based detection method for multi-antenna covert communication systems.Specifically,the Gamma approximation property is used to calculate the miss detection rate and false alarm rate of the monitor firstly.Then the optimization problem to minimize the sum of the missed detection rate and the false alarm rate is proposed.The optimal detection threshold and the minimum error detection probability are solved according to the properties of the Lambert W function.Finally,simulation results are given to demonstrate the effectiveness of the proposed method.

Clinical efficacy of Gamma Knife® combined with transarterial chemoembolization and immunotherapy in the treatment of primary liver cancer

BACKGROUND This study was designed to investigate the clinical efficacy and safety of Gamma Knife®combined with transarterial chemoembolization(TACE)and immunotherapy in the treatment of primary liver cancer.AIM To investigate the clinical efficacy and safety of Gamma Knife®combined with TACE and immune-targeted therapy in the treatment of primary liver cancer.METHODS Clinical data from 51 patients with primary liver cancer admitted to our hospital between May 2018 and October 2022 were retrospectively collected.All patients underwent Gamma Knife®treatment combined with TACE and immunotherapy.The clinical efficacy,changes in liver function,overall survival(OS),and progression-free survival(PFS)of patients with different treatment responses were evaluated,and adverse reactions were recorded.RESULTS The last follow-up for this study was conducted on October 31,2023.Clinical evaluation of the 51 patients with primary liver cancer revealed a partial response(PR)in 27 patients,accounting for 52.94%(27/51);stable disease(SD)in 16 patients,accounting for 31.37%(16/51);and progressive disease(PD)in 8 patients,accounting for 15.69%(8/51).The objective response rate was 52.94%,and the disease control rate was 84.31%.Alanine aminotransferase,aspartate aminotransferase,lactate dehydrogenase,and alpha-fetoprotein isoform levels decreased after treatment compared with pretreatment(all P=0.000).The median OS was 26 months[95%confidence interval(95%CI):19.946-32.054]in the PR group and 19 months(95%CI:14.156-23.125)in the SD+PD group,with a statistically significant difference(P=0.015).The median PFS was 20 months(95%CI:18.441-34.559)in the PR group and 12 months(95%CI:8.745-13.425)in the SD+PD group,with a statistically significant difference(P=0.002).Common adverse reactions during treatment included nausea and vomiting(39.22%),thrombocytopenia(27.45%),and leukopenia(25.49%),with no treatment-related deaths reported.CONCLUSION Gamma Knife®combined with TACE and immune-targeted therapy is safe and effective in the treatment of primary liver cancer and has a good effect on improving the clinical benefit rate and liver function of patients.