The fate of a new theory is to a large extent determined by the time intervening its development and the availability of a set of tools that supports practical applications. This last step is crucial if the methods developed are to become widely accepted and used. During the last two decades, considerable progress has been made in small-sample asymptotics leading to a well-established theory on...

Context. Compton scattering is involved in many astrophysical situations. It is well known and has been studied in detail for the past fifty years. Exact formulae for the different cross sections are often complex, and essentially asymptotic expressions have been used in the past. Numerical capabilities have now developed to a point where they enable the direct use of exact formulae in sophisti...

To simplify the computational process of homogeneous slope stability, improve computational accuracy, and find multiple potential slip surfaces of a complex geometric slope, this study utilized the limit equilibrium method to derive expression equations of overall and partial factors of safety. This study transformed the solution of the minimum factor of safety (FOS) to solving of a constrained...

This paper serves to prove the thesis that a computational trick can open entirely new approaches to theory. We illustrate by describing such random matrix techniques as the stochastic operator approach, the method of ghosts and shadows, and the method of “Riccatti Diffusion/Sturm Sequences,” giving new insights into the deeper mathematics underneath random matrix theory.

After the introduction basic properties of interval arithmetic are discussed and different approaches are repeated by which one can compute verified numerical approximations for a solution of a nonlinear equation.

This paper describes a numerical implementation in Maple V R. 5 of an algorithm to compute the Legendre–Fenchel conjugate, namely the Linear-time Legendre Transform algorithm. After a brief motivation on the importance of the Legendre–Fenchel transform, we illustrate the information the conjugate gives (how to test for convexity or compacity, and how to smooth a convex function), with several e...

The numerical computation of a multivariate normal probability is often a diicult problem. This article describes a transformation that simpliies the problem and places it into a form that allows eecient calculation using standard numerical multiple integration algorithms. Test results are presented that compare implementations of two algorithms that use the transformation, with currently avail...

Relative equilibria are special solutions of partial differential equations (PDEs), which are stationary in an appropriate comoving frame of reference. Such solutions occur frequently in biological and chemical models, e.g. when describing pattern formation of reaction-diffusion equations. Examples are traveling waves in 1d, planar and spiral waves in 2d and scroll waves in 3d. If the equation ...

An exact analysis of the numerical errors being generated during the computation of the Zernike moments, by using the well-known ‘q-recursive’ method, is attempted in this paper. Overflow is one kind of error, which may occur when one needs to calculate the Zernike moments up to a high order. Moreover, by applying a novel methodology it is shown that there are specific formulas, which generate ...