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...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.展开更多
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 logarithm...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.展开更多
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 fu...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].展开更多
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.
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 extern...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.展开更多
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 es...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.展开更多
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 f...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.展开更多
This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specific...This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.展开更多
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 poly...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.展开更多
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 trigonom...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.展开更多
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...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.展开更多
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 ane...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.展开更多
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 h...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.展开更多
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 ...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.展开更多
To improve image quality under low illumination conditions,a novel low-light image enhancement method is proposed in this paper based on multi-illumination estimation and multi-scale fusion(MIMS).Firstly,the illuminat...To improve image quality under low illumination conditions,a novel low-light image enhancement method is proposed in this paper based on multi-illumination estimation and multi-scale fusion(MIMS).Firstly,the illumination is processed by contrast-limited adaptive histogram equalization(CLAHE),adaptive complementary gamma function(ACG),and adaptive detail preserving S-curve(ADPS),respectively,to obtain three components.Then,the fusion-relevant features,exposure,and color contrast are selected as the weight maps.Subsequently,these components and weight maps are fused through multi-scale to generate enhanced illumination.Finally,the enhanced images are obtained by multiplying the enhanced illumination and reflectance.Compared with existing approaches,this proposed method achieves an average increase of 0.81%and 2.89%in the structural similarity index measurement(SSIM)and peak signal-to-noise ratio(PSNR),and a decrease of 6.17%and 32.61%in the natural image quality evaluator(NIQE)and gradient magnitude similarity deviation(GMSD),respectively.展开更多
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-h...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.展开更多
文摘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.
基金partially supported by the National Nature Science Foundation of China(12061033)。
文摘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.
文摘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].
文摘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.
文摘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.
文摘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.
文摘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.
文摘This paper attempts to form a bridge between a sum of the divisors function and the gamma function, proposing a novel approach that could have significant implications for classical problems in number theory, specifically the Robin inequality and the Riemann hypothesis. The exploration of using invariant properties of these functions to derive insights into twin primes and sequential primes is a potentially innovative concept that deserves careful consideration by the mathematical community.
基金supported by the National Natural Science Foundation of China(No.11901311)。
文摘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.
文摘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.
文摘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.
基金supported by the Jiangsu Provincial Medical Youth Talent of the Project of Invigorating Health Care through Science,Technology and Education(Grant No.QNRC2016694)the Six Talents Peak Project of Jiangsu Province(Grant No.2015-WSN-061)+2 种基金the fifth‘226’High Level Talent Training Project of Nantong Citythe National Natural Science Foundation of China(Grant No.81502867)the Technology Innovation Programme of Nantong University(Grant No.YKS14017)
文摘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.
文摘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.
基金Supported by the National Natural Science Foundation of China(10275063)
文摘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.
基金supported by the National Key R&D Program of China(No.2022YFB3205101)NSAF(No.U2230116)。
文摘To improve image quality under low illumination conditions,a novel low-light image enhancement method is proposed in this paper based on multi-illumination estimation and multi-scale fusion(MIMS).Firstly,the illumination is processed by contrast-limited adaptive histogram equalization(CLAHE),adaptive complementary gamma function(ACG),and adaptive detail preserving S-curve(ADPS),respectively,to obtain three components.Then,the fusion-relevant features,exposure,and color contrast are selected as the weight maps.Subsequently,these components and weight maps are fused through multi-scale to generate enhanced illumination.Finally,the enhanced images are obtained by multiplying the enhanced illumination and reflectance.Compared with existing approaches,this proposed method achieves an average increase of 0.81%and 2.89%in the structural similarity index measurement(SSIM)and peak signal-to-noise ratio(PSNR),and a decrease of 6.17%and 32.61%in the natural image quality evaluator(NIQE)and gradient magnitude similarity deviation(GMSD),respectively.
文摘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.
