Incomplete Gamma Function Research Articles

Environmental factors that influence milk production of Pantaneiro ewes and the weight gain of their lambs during the pre-weaning period.

The main objective of this research was to conduct an exploratory study of the lactation curve in order to characterize the productive potential of Pantaneiro ewes and lambs. Fifty ewes were bred using four rams in two different mating seasons. The ewes were kept with their lambs on pasture of Brachiaria brizantha. Ewe body score, ewe weight, and lamb weight were evaluated. Milk sampling was performed every week. In the morning for milk collections, the ewes were treated with 1 UI of oxytocin (intramuscular) for complete milking. Lambs were separated from the ewes for 4h and milk collections were performed. The total milk production over 24h was estimated by multiplying the production of this period (4h) by 6. The data were analyzed using the MIXED procedure (P < 0.05) in SAS. Milk production data were fitted to the curve using the incomplete gamma function of Wood, and lamb growth data were fitted using the Gompertiz equation. The average milk production of the ewes was 1.03kg/day-1. Younger ewes had the lowest milk production (18 = 798 ± 330, 24 = 1001 ± 440, 36 = 1100 ± 490, and 48 = 1106 ± 490g/day-1). Ewe body score at lambing affected initial milk production (1.0 = 816 ± 660, 1.5 = 1089 ± 105, and 2.0 = 1424 ± 1600g/day-1). Lambs were weaned with an average weight of 20.3kg. Daily weight gain from birth to weaning was 181g. Locally adapted Pantaneiro ewes showed a linear decreasing lactation curve, with reduced production from the second week of lactation. Overall, evaluation of the dairy production and lamb performance revealed great variation, denoting potential for selection.

Integral Representation for Bessel’s Functions of the First Kind and Neumann Series

A Fourier-type integral representation for Bessel’s functions of the first kind and complex order is obtained by using the Gegenbauer extension of Poisson’s integral representation for the Bessel function along with a suitable trigonometric integral representation of Gegenbauer’s polynomials. By using this representation, expansions in series of Bessel’s functions of various functions which are related to the incomplete gamma function can be obtained in a unified way. Neumann series are then considered and a new closed-form integral representation for this class of series is given. The density function of this representation is the generating function of the sequence of coefficients of the Neumann series on the unit circle. Examples of new closed-form integral representations of special functions are thus presented.

New bounds for the function involving incomplete gamma function

This paper reveals a new relation between the two functions \( A_{p}\left( x\right) =\)\(\left( \int _{0}^{x}e^{-t^{p}}dt\right) /x\) and \(B_{q}\left( x\right) =\)\(1-\left( 1-e^{-qx^{p}}\right) /\left( q\left( p+1\right) \right) \) . At the same time, we study the upper and lower bounds for the function \(A_{p}\left( x\right) /B_{\alpha }\left( x\right) \) in another sense.

An algorithm for the evaluation of the incomplete gamma function

We introduce an algorithm for the evaluation of the Incomplete Gamma Function, P(m, x), for all m, x > 0. For small m, a classical recursive scheme is used to evaluate P(m, x), whereas for large m a newly derived asymptotic expansion is used. The number of operations required for evaluation is O(1) for all x and m. Nearly full double and extended precision accuracies are achieved in their respective environments. The performance of the scheme is illustrated via several numerical examples.

Detection Capability and CFAR Loss Under Fluctuating Targets of Different Swerling Model for Various Gamma Parameters in RADAR

Target detection of RADAR deals with different and manifold problems over few decades. The detection capability is one of the most significant factors in RADAR system. The main aim of detection is to increase probability of detection while decreasing rate of false alarm. The threshold of detection is modified as a function of the receiver noise level to keep a fixed rate of false alarm. Constant False Alarm Rate (CFAR) processors are used to maintain the amount of false alarm under supervision in a diverse background of interference. In Signal to Noise Ratio (SNR) level, a loss can be occurred due to CFAR processor. Gamma function is used to determine the probability of false alarm. It is assumed in adaptive CFAR that the interference distribution is familiar here. This type of CFAR also approximates the unknown parameters connected with various interference distributions. CFAR loss depends on gamma function. Incomplete gamma function plays an important role in maintaining threshold voltage as well as probability of detection. Changing the value of gamma function can improve the probability of detection for various Swerling Models which are proposed here. This paper has also proposed a technique to compare various losses due to CFAR in terms of different gamma function in presence of different number of pulses for four Swerling Models.

Some Families of the Incomplete H-Functions and the Incomplete $$\overline H $$ H ¯ -Functions and Associated Integral Transforms and Operators of Fractional Calculus with Applications

Our present investigation is inspired by the recent interesting extensions (by Srivastava et al. [35]) of a pair of the Mellin–Barnes type contour integral representations of their incomplete generalized hypergeometric functions p γ q and p Γ q by means of the incomplete gamma functions γ(s, x) and Γ(s, x). Here, in this sequel, we introduce a family of the relatively more general incomplete H-functions γ (z) and Γ (z) as well as their such special cases as the incomplete Fox-Wright generalized hypergeometric functions p Ψ (γ) [z] and p Ψ (Γ) [z]. The main object of this paper is to study and investigate several interesting properties of these incomplete H-functions, including (for example) decomposition and reduction formulas, derivative formulas, various integral transforms, computational representations, and so on. We apply some substantially general Riemann–Liouville and Weyl type fractional integral operators to each of these incomplete H-functions. We indicate the easilyderivable extensions of the results presented here that hold for the corresponding incomplete $$\overline H $$ -functions as well. Potential applications of many of these incomplete special functions involving (for example) probability theory are also indicated.

Improved quantum efficiency models of CZTSe: GE nanolayer solar cells with a linear electric field.

We fabricated and characterized CZTSe:Ge nanolayer (<10 nm) thin film solar cells to quantitatively demonstrate an exact analytical model of quantum efficiency for Ge doped CZTSe devices. The linear electric field model is developed with the incomplete gamma function of the quantum efficiency as compared to the empirical data at forward bias conditions. This model is characterized with a consistent set of parameters from a series of measurements and the literature. Using the analytical modelling method, the carrier collection profile in the absorber is calculated and closely fitted by the developed mathematical expressions to identify the carrier dynamics during the quantum efficiency measurement of the device. The analytical calculation is compared with the measured quantum efficiency data at various bias conditions.

The exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids

Usually, the temperature distribution of nanofluids and the nanoparticles’ concentration are finally governed by second-order ordinary differential equations with polynomial coefficients. In this work, a class of second-order boundary value problems with applications on nanofluids has been theoretically solved in terms of the Kummer function. Several lemmas have been presented to relate the Kummer function with the generalized incomplete gamma function. Accordingly, the current solutions reduce to those in the literature at certain values of the coefficients as special cases. Furthermore, the present results are very useful in obtaining the solutions for any future similar problems without any need to perform further calculations.

Nonparametric estimation and testing on discontinuity of positive supported densities: a kernel truncation approach

Discontinuity in density functions is of economic importance and interest. For instance, in studies on regression discontinuity designs, discontinuity in the density of a running variable suggests violation of the no-manipulation assumption. In line with this notion, estimation and testing procedures on discontinuity in densities with positive support are developed. The proposed approach is built on splitting the asymmetric, gamma kernel into two parts at a prespecified cutoff that is suspected to be a discontinuity point and constructing two truncated kernels. The jump-size magnitude of the density at the cutoff can be estimated nonparametrically by two kernels and a multiplicative bias correction method. The estimator is easy to implement, and its convergence properties are delivered by various approximation techniques on incomplete gamma functions. Based on the jump-size estimator, two versions of test statistics for the null of continuity at a given cutoff are also proposed. Moreover, estimation theory of the entire density in the presence of a discontinuity point is explored. Monte Carlo simulations confirm nice finite-sample properties of the jump-size estimator and the test statistics.

Effects of partial slip boundary condition and radiation on the heat and mass transfer of MHD-nanofluid flow

In this paper, the effect of partial slip boundary condition on the heat and mass transfer of the Cu-water and Ag-water nanofluids over a stretching sheet in the presence of magnetic field and radiation. Such partial slip boundary condition has attracted much attention due to its wide applications in industry and chemical engineering. The flow is basically governing by a system of partial differential equations which are reduced to a system of ordinary differential equations. This system has been exactly solved, where exact analytical expression has been obtained for the fluid velocity in terms of exponential function, while the temperature distribution, and the nanoparticles concentration are expressed in terms of the generalized incomplete gamma function. In addition, explicit formulae are also derived from the rates of heat transfer and mass transfer. The effects of the permanent parameters on the skin friction, heat transfer coefficient, rate of mass transfer, velocity, the temperature profile, and concentration profile have been discussed through tables and graphs.

New Sharp Approximations Involving Incomplete Gamma Functions

In the article, we deal with the monotonicity of the function \(x\mapsto x(c+be^{-ax^{p}}){/}\int _{0}^{x}e^{-t^{p}}dt\) for \(a,b,c,p>0\) on \((0,\infty )\) and prove that the double approximating inequality $$\begin{aligned} 1-\frac{1-e^{-ax^{p}}}{a(p+1)}<\frac{1}{x}\int _{0}^{x}e^{-t^{p}}dt<1-\frac{ 1-e^{-bx^{p}}}{b(p+1)} \end{aligned}$$ holds for all \(x>0\) if and only if \(0<a\le 1/(p+1)\) and \(b\ge (p+1)/(2p+1)\). Also, some approximating properties concerning incomplete gamma function and exponential integral are established.

English

English

Incorporating big data into treatment plan evaluation: Development of statistical DVH metrics and visualization dashboards

PurposeTo develop statistical dose-volume histogram (DVH)–based metrics and a visualization method to quantify the comparison of treatment plans with historical experience and among different institutions. Methods and materialsThe descriptive statistical summary (ie, median, first and third quartiles, and 95% confidence intervals) of volume-normalized DVH curve sets of past experiences was visualized through the creation of statistical DVH plots. Detailed distribution parameters were calculated and stored in JavaScript Object Notation files to facilitate management, including transfer and potential multi-institutional comparisons. In the treatment plan evaluation, structure DVH curves were scored against computed statistical DVHs and weighted experience scores (WESs). Individual, clinically used, DVH-based metrics were integrated into a generalized evaluation metric (GEM) as a priority-weighted sum of normalized incomplete gamma functions. Historical treatment plans for 351 patients with head and neck cancer, 104 with prostate cancer who were treated with conventional fractionation, and 94 with liver cancer who were treated with stereotactic body radiation therapy were analyzed to demonstrate the usage of statistical DVH, WES, and GEM in a plan evaluation. A shareable dashboard plugin was created to display statistical DVHs and integrate GEM and WES scores into a clinical plan evaluation within the treatment planning system. Benchmarking with normal tissue complication probability scores was carried out to compare the behavior of GEM and WES scores. ResultsDVH curves from historical treatment plans were characterized and presented, with difficult-to-spare structures (ie, frequently compromised organs at risk) identified. Quantitative evaluations by GEM and/or WES compared favorably with the normal tissue complication probability Lyman-Kutcher-Burman model, transforming a set of discrete threshold-priority limits into a continuous model reflecting physician objectives and historical experience. ConclusionsStatistical DVH offers an easy-to-read, detailed, and comprehensive way to visualize the quantitative comparison with historical experiences and among institutions. WES and GEM metrics offer a flexible means of incorporating discrete threshold-prioritizations and historic context into a set of standardized scoring metrics. Together, they provide a practical approach for incorporating big data into clinical practice for treatment plan evaluations.

ON SUMS OF INDEPENDENT GENERALIZED PARETO RANDOM VARIABLES WITH APPLICATIONS TO INSURANCE AND CAT BONDS

We derive single integral representations for the exact distribution of the sum of independent generalized Pareto random variables. The integrands involve the incomplete and complementary incomplete gamma functions. Applications to insurance and catastrophe bonds are described.

A novel metaheuristic for multi-objective optimization problems: The multi-objective vortex search algorithm

This study investigates a multi-objective Vortex Search algorithm (MOVS) by modifying the single-objective Vortex Search algorithm or VS. The VS is a metaheuristic-based algorithm that uses a new adaptive step-size adjustment strategy to improve the performance of the search process.Search mechanism of the VS is inspired by the vortex pattern, so it is called a “Vortex Search” algorithm. The original VS is an improved way of solving single-objective continuous problems. To improve the MOVS algorithm, the VS algorithm is enhanced with added calculation approaches, such as fast-nondominated-sorting and crowding-distance, in order to identify the degree of non-dominance of the solutions and the densities of their occurrence.In addition, a crossover operation is added to the MOVS algorithm in order to enhance the Pareto front convergence capacity of the solutions. Finally, to spread the solutions more successfully over the Pareto front, it has been randomly produced using the inverse incomplete gamma function using a parameter between 0 and 1. The proposed MOVS algorithm is tested against 36 different benchmark problems together with NSGAII, MOCell, IBEA and MOEA/D algorithms. The test results indicate that the MOVS algorithm achieves a better performance on accuracy and convergence speed than any other algorithms when comparisons are made against several test problems, and they also show that it is a competitive algorithm.

General Laplace Integral Problems: Accuracy Improvement and Extension to Finite Upper Limits

Many problems in engineering and science involve calculation of difficult Laplace integrals of the form: In previous papers [Hanna and Davis (2011), Davis and Hanna (2013), referred to as Papers I and II, respectively], the authors introduced two new complementary analytical methods for the improved asymptotic (x → ∞) approximation of these integrals when the upper limit A equal to ∞ [the general Watson lemma (WL) problem for any A if x → ∞]. A procedure is developed here that extends application of the previous improvement methods to the difficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems having growing exponential behavior, certain infinite Fourier integrals and problems having large (tα) factors, are also considered. The main result is that, with some modifications, the exponential, expansion-point, and combination procedures developed for infinite integrals (Papers I and II) can be easily applied directly to FUL Laplace integrals. This is accomplished with the aid of a simple new “generalized incomplete gamma function (IGF)” algorithm which itself utilizes an improvement procedure. The new FUL procedure requires only F(t), F′(t), and a few terms of the Taylor expansion. A simple EXCEL program which implements the new procedure is discussed in detail in Appendix A and is freely available to users at http://www.d.umn.edu/~rdavis/CEC/. Many numerical comparisons presented here indicate that good engineering accuracy is achieved for these improved approximations at virtually all positive A values, over a very wide range in x, for various α, β, and F(t) functions. Where comparisons are possible (A = ∞), the new results are far superior to those of the best Watson’s lemma results.

On New Formulas for the Cumulative Distribution Function of the Noncentral Chi-Square Distribution

The main aim of this article is to derive three new formulas for the cumulative distribution function of the noncentral chi-square distribution. The main advantage of such formulas is that they are given in terms of modified Bessel functions, leaky aquifer function and generalized incomplete gamma function which have a wide range of applications. In addition, the computational efficiency of the newly derived formulas versus already known formulas is established.

Empirical rate formula for ion-dipolar molecule reactions

New empirical rate formula for the low temperature reactions is proposed. The formula proposed previously has been simplified by using incomplete gamma function. A few examples of temperature dependence of rate constants for the reactions of ions with dipolar molecules and that for the reaction F+H2→HF+H are demonstrated by using new rate formula.

Length distributions of nanowires: Effects of surface diffusion versus nucleation delay

It is often thought that the ensembles of semiconductor nanowires are uniform in length due to the initial organization of the growth seeds such as lithographically defined droplets or holes in the substrate. However, several recent works have already demonstrated that most nanowire length distributions are broader than Poissonian. Herein, we consider theoretically the length distributions of non-interacting nanowires that grow by the material collection from the entire length of their sidewalls and with a delay of nucleation of the very first nanowire monolayer. The obtained analytic length distribution is controlled by two parameters that describe the strength of surface diffusion and the nanowire nucleation rate. We show how the distribution changes from the symmetrical Polya shape without the nucleation delay to a much broader and asymmetrical one for longer delays. In the continuum limit (for tall enough nanowires), the length distribution is given by a power law times an incomplete gamma-function. We discuss interesting scaling properties of this solution and give a recipe for analyzing and tailoring the experimental length histograms of nanowires which should work for a wide range of material systems and growth conditions.

Medical applications for the flow of carbon-nanotubes suspended nanofluids in the presence of convective condition using Laplace transform

In nano-medicine, attempts of using the carbon-nanotubes (CNTs) as drug-carriers are undertaken especially in the treatment of cancer. These (CNTs) are first injected into the blood which then reach the tumor cite under the actions of the waves propagated by the walls of the arteries with an external force such as a magnetic field or laser beams. The flow near the boundary of the artery may be treated as a boundary layer flow only for simplification as we consider here. In applied science, the flow and heat transfer of CNTs are usually described by systems of nonlinear differential equations. Due to nonlinearities, the exact solutions of such systems cannot be obtained in most cases. In this paper, an effective analytical procedure is proposed to deduce the exact solution of a system of nonlinear differential equations describing the effect of a convective heat condition on the flow and the heat transfer of carbon-nanotube suspended nanofluids with suction/injection in the presence of a magnetic field. The heat transfer equation is solved via applying Laplace transform and the solution is expressed in terms of the generalized incomplete gamma function. Also, it is proved that the present exact solutions for the flow and the heat transfer reduce to those in literature in the absence of the suction/injection and the convective parameters. The results declare that the temperature profiles are very sensitive regarding the value assigned to the convective parameter. Moreover, the effects of other physical parameters on the studied phenomena are displayed through graphs. Besides, possible applications of the current results have been also discussed.