Incomplete Gamma Function Research Articles

Forecast of natural gas consumption in the Asia-Pacific region using a fractional-order incomplete gamma grey model

To forecast natural gas consumption more accurately, to clearly understand the future supply situation, and to optimize the allocation of resources, a new fractional-order accumulation-based incomplete gamma grey forecasting model is proposed in this paper. To further optimize the traditional grey action quantity, dynamic nonlinear action-based incomplete gamma functions are taken as the grey action quantity and combined with fractional-order accumulation. The role of new information is fully considered, and a detailed modeling process is presented, including computational steps and intelligent optimization algorithms. In this study, this new model is used to simulate and forecast natural gas consumption in the Asia-Pacific region from 2008 to 2018. First, Bangladesh and the Philippines are taken as examples to show the error changes incurred by the model under the control of two parameters. Then, a simulation and prediction of natural gas consumption are conducted and compared with those of other traditional univariate grey models. The results show that the MAPE obtained by the new model is the lowest, which indicates the prediction accuracy and effectiveness of the model. This model has good prediction performance for natural gas consumption and can be extended to more energy consumption prediction problems.

Definite Integral of Logarithmic Functions in Terms of the Incomplete Gamma Function

In this article we derive some entries and errata for the book of Gradshteyn and Ryzhik which were originally published by Bierens de Haan. We summarize our results using tables for easy reading and referencing.

Improved Asymptotic Expansions in High- and Low-Dose Ranges for Generalized Multi-Hit Model of Radiation-Induced Cell Survival.

By considering an upper bound on the number of radiation-induced potential lethal damages that can be repaired in a cell, we have proposed the generalized multi-hit (GMH) model with a closed-form solution, which can better fit various radiation-induced cell survival curves. Recent analysis shows that the asymptotic expansions that we gave before can be used to approximate the generalized single-hit single-target (GSHST) model rather than the GMH model. To illustrate the asymptotic trends of radiation-induced cell survival curves, in this study, we improve the asymptotic expansions of the GMH model in low- and high-dose ranges based on the limit formula of the incomplete gamma function in the corresponding dose ranges. When the upper limit of the number of radiation-induced potential lethal damages is one, the improved expansions of the GMH model can be reduced to the previous expansions of the GSHST model, and the improved asymptotic expansions of the GMH model also indicate that the GMH model has the generalized linear-quadratic-linear (LQL) feature. The numerical simulations indicate that the improved asymptotic expansions in high- and low-dose ranges agree well with the non-linear fitting of the GMH model in six kinds of cell lines under the corresponding dose ranges. In addition, we analyze the relative errors of the improved expansions of the GMH model in high- and low-dose ranges to demonstrate the accuracy and effectiveness of the improved expansions. Based on the error analysis, we further give the reasonable ranges of radiation dose applicable to the improved asymptotic expansions of the GMH model.

SNR Coverage Probability Analysis of RIS-Aided Communication Systems

The reconfigurable intelligent surface (RIS) technique has received many interests, thanks to its advantages of low cost, easy deployment, and high controllability. It is acknowledged that the RIS can significantly improve the quality of signal transmission, especially in the line-of-sight-blocked scenarios. Therefore, it is critical to analyze the corresponding signal-to-noise ratio (SNR) coverage probability of RIS-aided communication systems. In this correspondence, we consider many practical issues to analyze the SNR coverage probability. We employ the realistic path loss model and Rayleigh fading model as large-scale and small-scale channel models, respectively. Meanwhile, we take the number and size of the RIS elements, as well as the placement of the RIS plane into considerations. We first derive the exact closed-form SNR coverage probability for a single element. Afterward, with the moment matching method, a highly accurate approximation of SNR coverage probability is formulated as the ratio of the upper incomplete Gamma function and Gamma function, allowing an arbitrary number of elements in the RIS. Finally, we comprehensively evaluate the impacts of essential factors on the SNR coverage probability, such as the number and size of the element, the coefficients of fading channel, and the angles of incidence and RIS plane. Overall, this work provides a succinct and general expression of SNR coverage probability, which can be helpful in the performance evaluation and practical implementation of the RIS.

Exactly solvable SIR models, their extensions and their application to sensitive pandemic forecasting.

The classic SIR model of epidemic dynamics is solved completely by quadratures, including a time integral transform expanded in a series of incomplete gamma functions. The model is also generalized to arbitrary time-dependent infection rates and solved explicitly when the control parameter depends on the accumulated infections at time t. Numerical results are presented by way of comparison. Autonomous and non-autonomous generalizations of SIR for interacting regions are also considered, including non-separability for two or more interacting regions. A reduction of simple SIR models to one variable leads us to a generalized logistic model, Richards model, which we use to fit Mexico’s COVID-19 data up to day number 134. Forecasting scenarios resulting from various fittings are discussed. A critique to the applicability of these models to current pandemic outbreaks in terms of robustness is provided. Finally, we obtain the bifurcation diagram for a discretized version of Richards model, displaying period doubling bifurcation to chaos.

Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals

The cumulative distribution function of the non-central chi-square distribution χn′2(λ) of n degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin.

Some inequalities for multiplicative tempered fractional integrals involving the $ \lambda $-incomplete gamma functions

In this paper, we introduce a class of the multiplicative tempered fractional integral operators. Then, we investigate two Hermite–Hadamard type inequalities for this class. By using the established identity and the multiplicative convexity, we establish some integral inequalities for the multiplicative tempered fractional integrals involving the $ \lambda $-incomplete gamma functions. And our results obtained in the present paper generalize some results given by Budak and Tunc (2020) and Ali et al. (2019). Also, we provide three examples to demonstrate the simplicities of the calculations.

Efficient procedure to generate generalized Gaussian noise using linear spline tools

In this paper, we propose a simple method to generate generalized Gaussian noises using the inverse transform of cumulative distribution. This inverse is expressible by means of the inverse incomplete Gamma function. Since the implementation of Newton's method is rather simple, for approximating inverse incomplete Gamma function, we propose a better and new initial value exploiting the close relationship between the incomplete Gamma function and its piecewise linear interpolant. The numerical results highlight that the proposed method simulates well the univariate and bivariate generalized Gaussian noises.

On a Langevin equation involving Caputo fractional proportional derivatives with respect to another function

<abstract><p>In this work, we introduce and study a class of Langevin equation with nonlocal boundary conditions governed by a Caputo fractional order proportional derivatives of an unknown function with respect to another function. The qualitative results concerning the given problem are obtained with the aid of the lower regularized incomplete Gamma function and applying the standard fixed point theorems. In order to homologate the theoretical results we obtained, we present two examples.</p></abstract>

Unified and non-recursive formulas for moments of the normal distribution with stripe truncation

Stripe truncation in the normal distribution is introduced such that the variable is truncated when it is located on several intervals like stripes. This truncation includes single, double and elliptical truncation as special cases. Then, the moments and absolute moments of arbitrary orders for the deviation of the truncated variable from an arbitrary reference point are derived using closed-form formulas based on the incomplete gamma function. The corresponding absolute moments of non-integer valued orders are also derived employing the parabolic cylinder distribution. The formulas do not suffer from difficulties associated with the conventional recursive methods for higher-order moments.

New fractional inequalities of Hermite\u2013Hadamard type involving the incomplete gamma functions

A specific type of convex functions is discussed. By examining this, we investigate new Hermite–Hadamard type integral inequalities for the Riemann–Liouville fractional operators involving the generalized incomplete gamma functions. Finally, we expose some examples of special functions to support the usefulness and effectiveness of our results.

On existence\u2013uniqueness results for proportional fractional differential equations and incomplete gamma functions

In this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form 1Dρ=(1−ρ)+ρD,ρ∈[0,1],\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ D^{\\rho }= (1-\\rho )+ \\rho D, \\quad \\rho \\in [0,1], $$\\end{document} where D is the ordinary differential operator.

On poly-Bell numbers and polynomials

This paper aims to construct a new family of numbers and polynomials which are related to the Bell numbers and polynomials by means of the confluent hypergeometric function. We give various properties of these numbers and polynomials (generating functions, explicit formulas, integral representations, recurrence relations, probabilistic representation, …). We also derive some combinatorial sums including the generalized Bernoulli polynomials, lower incomplete gamma function, generalized Bell polynomials. Finally, by applying Cauchy formula for repeated integration, we introduce poly-Bell numbers and polynomials.

Road Load Based Model for Vehicle Repair and Maintenance Cost Estimation

A techno-economic model is developed based on road-load simulation results expressed in relation to slip energy (SE) at the tire–pavement interface and the repair and maintenance (R&M) cost obtained from published sources and data from state agencies. Tire SE allows for the consideration of aggressive acceleration and deceleration, high torque conditions (for instance, driving an upslope grade), and roadway curvature. Tire SE data used in this effort were generated using physics-based simulation models of different vehicle types for arrays of road conditions (e.g., grades, curvatures) and driving cycles (i.e., vehicle speed profiles). R&M costs were estimated for various vehicle categories and accumulated vehicle mileage. The approach is based on relating the probability density functions (PDFs) of SE and R&M costs. Asymptotic series expansion for an incomplete gamma function was used to approximate the gamma functions and to determine the gamma ratio function that is used as the coefficient to SE to estimate R&M costs. The average R&M cost per mile results from the model compared with the arithmetic mean R&M cost data from fleet operators and published data. The model can serve as a method for predicting R&M cost as a function of road load to vehicle fleet.

Performance evaluation of different computational methods to estimate Wood’s lactation curve by nonlinear mixed-effects models

Nonlinear mixed-effects models allow modeling repeated measures over time. The fixed effects of these models allow incorporating covariates, whereas the random effects reflect the multiple sources of heterogeneity and correlation between and within the units. To estimate the parameters of these models, it is necessary to use iterative processes, which can be done through different approaches, some of which are applied by the statistical software SAS. In this work, through simulation, we studied the performance of the estimators of a Wood’s incomplete gamma function, obtained through three different methods: linearization method through a Taylor series expansion on the empirical best linear unbiased predictor of random effects, applied by the NLINMIX macro, and expansion around the expected value of random effects, applied by both the NLINMIX macro and the NLMIXED procedure. We also investigated the impact of an incorrect specification of the covariance matrix for random errors. The linearization method through a Taylor series expansion on the empirical best linear unbiased predictor of random effects, applied by the NLINMIX macro, provided estimators with good performance, approximately normally distributed and with biases lower than those obtained with the other methods, even when the covariance matrix for random errors was incorrectly specified.

Signaling Pathways in Immune Responses with Incomplete Information: A Novel Model Founded on Coupled Uniform Sequential Reaction Systems

A novel mathematical model for determining the forms of signaling pathways, ranging from linear structures to networks, on the basis of single-mode and multimodal time courses of immune responses under the condition of incomplete information, was elaborated. The subcellular immune processes and systems lacking the kinetic parameters still remain, in general, undetermined by deterministic quantitative models, which has resulted in the resignation from these models in favor of the so-called kinetically agnostic models. The model proposed in this paper intends to give a deterministic description of signaling pathways implicated in immune responses with unknown kinetic parameters. To do that, uniform sequential reaction systems and coupled sequential reaction systems were constructed using systems of differential equations and the novel notions of uniform functions, transition, coupling and branch operations founded on the assumptions of a uniform distribution of unknown rate constants, mass action kinetics, and first-order kinetics. Reaction step-dependent analytical solutions to the uniform sequential systems were expressed by the means of the incomplete gamma function and unambiguously characterized using, amongst others, the Lambert W function, which were necessary to obtain strict analytical solutions to those systems. These systems were validated by comparing their solutions with those obtained for ordinary sequential systems, regarded as the systems with complete information. The numerical solutions to the uniform sequential systems obtained by the Lederberg-Marquardt method were compared with those obtained by the Lambert W function. The signaling pathways were determined by analyzing the uniform sequential and coupled sequential reaction systems that describe the time courses of multimodal immune responses. Bi- and tri-modal immune processes representing temporal expressions of the transcription factor T-bet, the IL-12Rβ2 receptor, the IL-4, Tbx21 and Twist1 genes—all in helper T cells, calcium ion in 2G12.1 T cell hybridoma, c-Fos sumoylation in serum-stimulated HeLa cells, and IFN-γ-secreting cytotoxic T cells were modeled. Application of the coupled uniform sequential systems to the above immune processes allowed accurate approximations of their time behavior and to determine the structures of the signaling pathways and to predict some. A notion of the fractional reaction step, and its meaning in revealing the coupled sequential systems behind the seeming sequential ones, was introduced and analyzed. A consistent approach to the different forms of signaling pathways, beginning from the linear sequential systems through coupled (nonlinear) sequential ones to networks, was presented together with an algebraic insight into those systems.

Full-scale analytical perturbation calculation of He-isoelectronic series with Hydrogenic bound states via Green's function expansion (monopole) of Coulomb interactions

Increasing number of carrier electrons and nuclear charge in Hydrogenic systems associated with bound and ionised (scattered) states invites long-standing divergence issue of Schrödinger's equations due to Coulomb interactions. In this regard, He-isoelectronic series represents the benchmark for atoms and molecules within non-relativistic quantum limit. Schrödinger equations for such complex systems being transformed into either Whittaker-M or Associated Laguerre Polynomial, furnish multipole operators of Green's function expansion of Coulomb interaction a robust framework of lower and upper incomplete Gamma functions. The resulting monopole integral gives terminating, finitely summed, simple and analytical forms of Lauricella functions. This novel form succinctly remedies the paradox of Energy Contribution or Correction occurring due to divergent length scales among Hydrogenic bound states. Both singly and doubly excited spherically symmetric states of two-electron systems ( ) are examined to register full-scale third-order perturbation for monopole factor within 0.2–8.2% deviation of ground-state energies from experimental values. The deviation urges for further analytical treatment to dipole factor. Moreover, exchange of electronic coordinates by virtue of exchanging quantum numbers exhibits invariant integrals of multipoles which confirms reciprocity condition of Green's function. This methodology has achieved a new paradigm of employing generic H-like orbitals for electronic structure calculations.

The Bayesian Cut.

An important task in the analysis of graphs is separating nodes into densely connected groups with little interaction between each other. Prominent methods here include flow based graph cutting procedures as well as statistical network modeling approaches. However, adequately accounting for this, the so-called community structure, in complex networks remains a major challenge. We present a novel generic Bayesian probabilistic model for graph cutting in which we derive an analytical solution to the marginalization of nuisance parameters under constraints enforcing community structure. As a part of the solution a large scale approximation for integrals involving multiple incomplete gamma functions is derived. Our multiple cluster solution presents a generic tool for Bayesian inference on Poisson weighted graphs across different domains. Applied on three real world social networks as well as three image segmentation problems our approach shows on par or better performance to existing spectral graph cutting and community detection methods, while learning the underlying parameter space. The developed procedure provides a principled statistical framework for graph cutting and the Bayesian Cut source code provided enables easy adoption of the procedure as an alternative to existing graph cutting methods.

The correct cutoff for Rutherford scattering cross-section

The correct cutoff variable for the integrals with Rutherford scattering cross-section is established in this paper. The traditional cutoff variables in plasma physics, such as scattering angle θ and impact parameter b, are incorrect or just partial cutoff variables for lacking the necessary cutoff on relative speed g. The correct cutoff variable is the same variable that correctly describing the singularity of the integrals. The difference of the partial cutoff variables and the correct one is compared through their contour lines in the b − g plane. With the correct cutoff, many physical results become more simplified and structured for both rigid-sphere interaction and Coulomb interaction. These physical results include the arbitrary higher order of Fokker-Planck coefficients, transition moments, and energy transfer rates for both rigid-sphere and Coulomb interactions. All the physical results depending on velocity are expressed by a set of functions . A useful integral formula∫0∞qn(k)(ymin,u)uk+2e−u2ξdu=ξk+3221+ξk−n2∑j=0⌊k/2⌋k!Γn+k+12−j,ymin21+ξj!(k−2j)!4ξj,which associates with incomplete gamma functions is also proved. This integral formula is the key to show that the correct cutoff constants for both rigid-sphere and Coulomb interactions are all in the form of incomplete gamma functions of different orders. In particular, the so called Coulomb logarithm should be replaced by the exact form—the zeroth order incomplete Gamma function.

On the Generalized Hermite–Hadamard Inequalities via the Tempered Fractional Integrals

Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of λ -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite–Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann–Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.