A numerical routine for the crossed vertex diagram with a massive-particle loop

The two loop crossed ladder vertex diagram with two massive exchanges

We present analytical formulas for the calculation of the two-center two-electron integrals in the basis of Slater geminals and products of Slater orbitals. Our derivation starts with establishing a inhomogeneous fourth-order ordinary differential equation that is obeyed by the master integral, the simplest integral with inverse powers of all interparticle distances. To solve this equation it was necessary to introduce a new family of special functions which are defined through their series expansions around regular singular points of the differential equation. To increase the power of the interparticle distances under the sign of the integral we developed a family of open-ended recursion relations. A handful of special cases of the integrals is also analysed with some remarks on simplifications that occur. Additionally, we present some numerical examples of the master integral that validate the usefulness and correctness of the key equations derived in this paper. In particular, we compare our results with the calculations based on the series expansion of the exp(-\gamma r12) term in the master integral.

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions

Analysis of traffic flow signals plays an important role in traffic prediction and management. As an intrinsic property, the singular point of a traffic flow signal labels a new nonsteady status. Therefore, detecting the singular point is an effective approach to determine the moment of traffic flow prediction. In this paper, an improved wavelet transform is proposed to detect singular points of real-time traffic flow signals. The number of detected singular points is output via the heuristic selection of multiple scales. Then, a weighted similarity measurement of historical traffic flow signals is utilized to predict the next singular point. The position of the next singular point decides the duration of prediction adaptively. The detected and predicted singular points are applied to dynamically update the unequal interval prediction of traffic flow. Furthermore, a Vasicek model is used to predict the traffic flow by minimizing the sum of the relative mean standard error (RMSE) between the traffic flow increment in the predicted interval and the sampled increments of previous intervals. A decomposition method is used to solve the unequal matrix problem. Based on the scenario and traffic flow imported from the real-world map, the simulation results show that the proposed algorithm outperforms existing approaches with high prediction accuracy and much lower computing cost.

The work addresses studying the possibilities of a numerical-analytical variant of the boundary elements method (BEM) in determining the internal force factors and stresses at singular points when bending thin isotropic plates. The simplest type of singularity has been investigated, a point of application of external concentrated forces and moments. The importance of a given problem is due to the fact that at these points the internal force factors tend towards infinity and it is not possible to determine the size using elementary methods. At the same time, these singular points are the significant stress concentrators (both tangential and normal), which is why calculating the limits to which the internal forces and moments tend is essential to analyze the strength of plate structures. In order to describe an external load, it is proposed to apply the Dirac delta function of two variables. The models of external loads are presented. A given proposal makes it possible to accurately calculate the limits to which the transverse forces, as well as bending and torsional moments, tend at singular points of thin plates. We simulated plate bending using the variational Kantorovich-Vlasov method, which is fully compatible with the models of external load. The internal force factors at the singular points of plates were determined while solving the boundary value problems, formed based on the algorithm of BEM. The MATLAB environment was used for programming and computation. Results of the calculations are characterized by high accuracy and reliability, in particular the errors in determining the deflections of plates at singular points do not exceed 2.0 % and the errors for bending moments are not above 3.0 %. Recommendations have been given to solving different types of boundary problems on bending the plates with singular points based on the proposed approach. It has been established that an accurate model of the external load in the form of concentrated forces and moments fundamentally enables determining the internal forces and moments at the singular points of thin plates applying an algorithm of the variational Kantorovich-Vlasov method. Up to now, there are no data on the importance of internal forces and moments at the singular points of plates. It is also shown that when calculating the internal forces and moments of plates, it is inappropriate to apply a single term from a series of the Kantorovich-Vlasov method; the errors amount to significant magnitudes of the order of 43‒44 %

A bstract In this paper, we present analytical results for the two-loop QCD corrections to the production of two partons or a photon and a parton in hadronic collisions, mediated by loops of massive quarks. These amplitudes involve Feynman integrals defined on an elliptic curve. We compute them by generalizing our recent results for the production of two photons to include additional crossings of the corresponding master integrals, which we compute in terms of the same basis of independent iterated integrals. We discuss the analytical properties of the amplitudes, highlighting the cancellations of a large number of elliptic differential forms in their finite remainders. Finally, we elaborate on a strategy for their numerical evaluation based on generalized series expansions at singular points of the physical amplitude, through the introduction of suitable sets of variables that allow us to resolve all singularities.

It is a very demanding task to design a reliable fingerprint matching approach with high accuracy and speed. An algorithm based on the novel structure combining the singular point with its neighborhood minutiae proposed in this paper can solve this problem efficiently. The structure introduced in this paper has two novel ideas as follows: First, we give an efficient singular points detection method by the inter-relationship between the singular points and the minutiae around them. It can reject the spurious singular points detected by the Poincare index. Second, an improvement of minutiae pairing strategy is introduced, which can sharply decrease the number of candidate minutiae pairs. Experiment results show that these improvements can highly speed up the matching with a preferable accuracy. This algorithm can be used in the one-to-many matching of the on-line fingerprint identification system.KeywordsFingerprintMatchingSingular PointMinutiae Pairing

Fingerprint classification is used to assign fingerprints into five established classes, namely, Whorl, Left loop, Right loop, Arch, and Tented Arch, on the basis of ridge structures and singular points’ trait. Although some progresses have been achieved in improving accuracy rates, problems arise from ambiguous fingerprints, especially those with large intraclass and small interclass variations. Poor-quality images, such as those with blur, dry, wet, low contrast, cut, scarred and smudgy features, are equally challenging. This study proposes a new classification technique based on template matching by using fingerprint salient features as a matching tool. In classification phase, a new set of fingerprint features is created based on singular points’ occurrence and location along the symmetric axis. A set of five templates, in which each template represents a specific true class, is then generated. Finally, classification is performed by calculating similarity between the query fingerprint image and the template images by using ×2 distance measure. The performance of the current method is evaluated in terms of accuracy by using 27,000 fingerprint images acquired from the National Institute of Standard and Technology Special Database 14, which is a de facto dataset for development and testing of fingerprint classification systems. Experimental results have a high accuracy rate of 93.05%.

The main objective of this contribution is a constructive generalization of the holomorphic power and Laurent series expansions in ℂ to dimension 3 using the framework of hypercomplex function theory. This first article on hand deals with generalized Fourier and Taylor series expansions in the space of square integrable quaternion-valued functions which possess peculiar properties regarding the hypercomplex derivative and primitive. In analogy to the complex one-dimensional case, both series expansions are orthogonal series with respect to the unit ball in ℝ3 and their series coefficients can be explicitly (one-to-one) linked with each other. Finally, very compact and efficient representation formulae (recurrence, closed-form) for the elements of the orthogonal bases are presented.

In a practical computation with a structured grid around a complex body, singular points can be frequently found where an abrupt grid change exists. The grid singularity poses a troublesome problem when some finite difference scheme with high accuracy and resolution is applied. An excellent theory has been proposed, which solve the above singular problem by decomposing a computational domain into two blocks along a line or a surface which contains the singular points and by imposing accurate characteristic-based interface conditions at the block interface. However, the original theory has a limitation on the combination between the adjacent computational coordinate definitions. Concretely, these two coordinates have to be the same direction on the block interface. For a flexible coordinate arrangement without the restriction, the original characteristic-based interface treatment is extended and generalized. Consequently, the coincidence of the computational coordinate definitions becomes unnecessary, and a more flexible multi-block computation can be realized successfully. Numerical test analysis of vortex convection is performed for validation of the new theory, and excellent performance is confirmed as a result.

We compute the two-loop master integrals for leading-color QCD scattering amplitudes including a closed light-quark loop in tt¯H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t\\overline{t}H $$\\end{document} production at hadron colliders. Exploiting numerical evaluations in modular arithmetic, we construct a basis of master integrals satisfying a system of differential equations in ϵ-factorized form. We present the analytic form of the differential equations in terms of a minimal set of differential one-forms. We explore properties of the function space of analytic solutions to the differential equations in terms of iterative integrals which can be exploited for studying the analytic form of related scattering amplitudes. Finally, we solve the differential equations using generalized series expansions to numerically evaluate the master integrals in physical phase space. As the first computation of a set of two-loop seven-scale master integrals, our results provide valuable input for analytic studies of scattering amplitudes in processes involving massive particles and a large number of kinematic scales.

In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points --- two of them are regular and the third one is irregular of rank 1. The purpose of the work is to propose a procedure for numerical evaluation of the equation's solutions (confluent Heun functions). A scheme based on power series, asymptotic expansions and analytic continuation is described. Results of numerical tests are given.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

The compatible expansion in series of solutions of both the equations of P–Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P–Q pair are derived.