Asymptotic inversion of the incomplete beta function

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.

New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function. In each case the expansion contains the complementary error function and an asymptotic series. The expansions are uniformly valid with respect to certain domains of the parameters.

This chapter presents basic definitions and characteristics of fractional calculus. Definitions and characteristics of the Riemann–Liouville fractional integral and derivative, Caputo fractional derivative, Grünwald–Letnikov fractional derivative, Riesz fractional derivative, modified Riemann–Liouville derivative, and local fractional derivative have been primarily explored here. However, the Riemann–Liouville definitions of fractional differentiation play a significant role in developing fractional calculus. Because the derivative of constant is not zero, the Riemann–Liouville definition is not widely applicable. In such scenarios, the Caputo fractional derivative is more suited than the standard Riemann–Liouville derivative for applications in various engineering problems. It may be noted that the Riemann–Liouville technique is challenging to understand physically, while the Caputo approach employs integer-order initial conditions, which are easier to apply in real-world applications. This chapter also covers the Euler psi function, incomplete gamma function, beta function, incomplete beta function, Wright function, Mellin-Ross function, error function, hypergeometric functions such as Gauss, Kummer, and extended hypergeometric functions, and the H -function.

We construct asymptotic expansions for the normalised incomplete gamma function Q ( a , z ) = Γ ( a , z ) / Γ ( a ) Q(a,z)=\Gamma (a,z)/\Gamma (a) that are valid in the transition regions, including the case z ≈ a z\approx a , and have simple polynomial coefficients. For Bessel functions, these types of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as a → − ∞ a\to -\infty of the unique negative zero of the regularised incomplete gamma function γ ∗ ( a , x ) \gamma ^*(a,x) .

11 - The Gamma, Beta, Pi, and Psi Functions

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several ‘classical’ inverse problems for fractional differential equations involving a Djrbashian–Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm–Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of ‘fractional’ inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm–Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our findings indicate fractional diffusion inverse problems also provide an excellent case study in the differences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a finite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature.

By allowing the number of terms in an asymptotic expansion to depend on the asymptotic variable, it is possible to obtain an error term that is exponentially small as the asymptotic variable tends to its limit. This procedure is called “exponential improvement.” It is shown how to improve exponentially the well-known Poincare expansions for the generalized exponential integral (or incomplete Gamma function) of large argument. New uniform expansions are derived in terms of elementary functions, and also in terms of the error function.Inter alia, the results supply a rigorous foundation for some of the recent work of M. V. Berry on a smooth interpretation of the Stokes phenomenon.

In this paper, the discrete principal-mode propagation constant of an imperfectly-conducting stripline field applicator is investigated. Specifically, a complex correction to the ideal (i.e., perfectly-conducting) propagation constant is sought using a full-wave spectral-domain analysis. First, the electric-field dyadic Green's function for a general three-dimensional (3D) current source immersed within an imperfectly-conducting stripline background environment is determined with the aid of Hertzian-potential impedance boundary conditions. These boundary conditions, which are developed in the paper, follow from a Leontovich boundary condition and include important coupling terms that are absent in the perfectly-conducting case. Next, the stripline structure is realized by specializing the general 3D current to a 2D current flowing in an imperfectly-conducting center strip. Boundary conditions are subsequently enforced on the center strip, leading to an electric field integral equation (EFIE). The complex propagation constant for the imperfectly-conducting stripline is found through solution of the EFIE via formulation of a non-Galerkin's method-of-moments (MoM) technique. In particular, Chebyshev expansion and testing functions of the first and second kind are implemented to closely model the strip current, to accelerate convergence and to allow resulting spatial integrals to be performed in closed form. The advantage of this full-wave analysis over the standard perturbation method is discussed and various numerical results are given for the attenuation and phase constant of the principal-mode.

Fostered by advances in computing, simulation and data science have emerged as the third and fourth paradigms enabling scientific discovery. Many mathematical challenges arise when integrating these paradigms to produce new knowledge. One of the most important challenges is the inverse or calibration problem: find the inputs of a simulation such that its outputs agree with a given set of measurements. As both the cost of the complex simulations and the volume of measurement data sets increase, intuition-driven trial-and-error methods for calibration quickly become untenable; modern scientists need more computationally efficient and mathematically rigorous methods. When measurements and simulated predictions contain uncertainty, the Bayesian perspective on statistical inference provides a popular mathematical framework for developing computational methods to solve inverse problems. Markov chain Monte Carlo (MCMC) methods are a mainstay of this framework; they characterize a probability distribution on the space of model inputs that is conditioned on the given measurements. However, despite recent advances, MCMC methods remain impractical for expensive simulations with more than a handful of inputs. At its heart, MCMC searches the space of inputs for corresponding outputs that match measurements, and so it is afflicted with the curse of dimensionality. Additionally, MCMC methods become less efficient as the volume of the measurement data increases, particularly if the measurements are redundant. To make MCMC—and Bayesian methods, in general—amenable to today’s computer models with millions of inputs and terabytes of measurement data, one must devise methods to reduce the dimension of both the simulation inputs and the measurements. For many models used in practice, the outputs vary primarily on a low-dimensional manifold of the nominally high-dimensional input space. Recently developed active subspace methods discover this low-dimensional manifold and exploit it to enable analyses typically suited for low-dimensional settings (e.g., statistical inversion of complex systems). We hypothesize that active subspace methods can provide the necessary dimension reduction, enabling existing Bayesian methods to find solutions for a large class of otherwise intractable statistical inverse problems. We will test the hypothesis with a series of mathematical analyses and computational experiments that will proceed along three research veins: (i) integrating active subspace methods into methods for statistical inversion, (ii) addressing current challenges in active subspace methods, and (iii) incorporating methods for nonlinear dimensionality reduction into active subspace methods. We will apply the methods we develop to real inverse problems in chemical kinetics and turbulent flame modeling. Our team combines expertise in methods for inverse problems, dimension reduction, and reduced order modeling. With a history of active collaboration and thoughtful engagement, we bring enthusiasm and fresh perspectives to a well studied subject. We are excited by the potential opportunity to work together to extend and apply active subspace methods to data-intensive inverse problems.

PyTorch-FEA: Autograd-enabled finite element analysis methods with applications for biomechanical analysis of human aorta

We denote by and the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to , a lower bound established by Elbert and Laforgia (2000) for the function , with , only for .

در این تحقیق به منظور تعیین بهترین منحنی تولید شیر در یک دوره شیردهی از 260709 رکورد روز آزمون تولید شیر متعلق به 30903 رأس گاو هلشتاین شکم اول استان خراسان استفاده شد. اطلاعات مورد استفاده مربوط به گاو‌های 216 گله بود که طی سالهای 1367 الی 1385 زایش داشتند. توابع مورد استفاده شامل تابع گامای ناقص (وود)، تابع ویلمینک، تابع علی‌شفر و تابع لژاندر (با درجه برازش پنجم) بود. معیارهای مورد نظر برای مقایسه توابع شامل مجموع مربعات خطا، میانگین مجموع مربعات خطا، ضریب تعیین، ضریب تعیین تصحیح شده ، ضریب همبستگی و فاکتور تورم واریانس بود. براساس تمام معیارها به استثناء فاکتور تورم واریانس تابع علی- شفر و براساس فاکتور تورم واریانس تابع لژاندر به نحو مطلوب‌تری با داده‌های تولید شیر مطابقت داشتند.

Geophysical Inverse Theory and Regularization Problems

The object of this study was to test the incomplete gamma function previously developed as a descriptor for grapite disintegration. Samples of disintegrating granite were taken from several sites at three locations in a hot semiarid environment in south‐central Africa (southern Rhodesia). These samples were from surface rock, profile corestones, and disintegrating rock at the base of the soil profile. Particlesize distributions were obtained using the standard sieving and pipette method. For materials > 10 mm, Martin's diameter was measured directly. The incomplete gamma function was fitted to these data, and also to U.S. and French data from the literature, for granite in an apparently similar state of disintegration. The values for the degree‐of‐disintegration parameter were fairly uniform. The mode of disintegration appeared the same regardless of site. The two‐parameter incomplete gamma distribution was found to be a reasonably good descriptor of the particle‐size distribution of disintegrating granite. The mean size appeared to be a sensitive and useful single‐valued descriptor for these distributions.