Related papers: NumExp: Numerical epsilon expansion of hypergeomet…
We revisit and generalize the geometric procedure of regularizing a sequence of real numbers with respect to a so-called regularizing function. This approach was studied by S. Mandelbrojt and becomes useful and necessary when working with…
This paper describes a package for calculations of expressions with Dirac matrixes. Advantages to existing similar packages are described. MatrixExp package is intended for simplification of complex expressions involving $\gamma$-matrixes,…
In this paper, we introduce a new finite expression method (FEX) to solve high-dimensional partial integro-differential equations (PIDEs). This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel…
Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator…
The Random Hypersurface Model (RHM) is introduced that allows for estimating a shape approximation of an extended object in addition to its kinematic state. An RHM represents the spatial extent by means of randomly scaled versions of the…
We extend the technique of using the Trapezoidal Rule for efficient evaluation of the Special Functions of Mathematical Physics given by integral representations. This technique was recently used for Bessel functions, and here we treat…
Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were…
This paper describes an algorithm for determining the branching geometry of algebraic functions. The graphs of these complex-valued functions have a complicated interweaving structure that can be described by analytic branches separated by…
We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
In this paper, we present a NURBS-enhanced finite element method that integrates the NURBS-based boundary representation of a geometric domain into a standard finite element framework for hexahedral meshes. We decompose an open, bounded,…
The ratio between two probability density functions is an important component of various tasks, including selection bias correction, novelty detection and classification. Recently, several estimators of this ratio have been proposed. Most…
We introduce an expansion scheme in reproducing kernel Hilbert spaces, which as a special case covers the celebrated Blaschke unwinding series expansion for analytic functions. The expansion scheme is further generalized to cover Hardy…
In many applications (hupergeometric-type) special functions like orthogonal polynomials are needed. For example in more than 50% of the published solutions for the (application-oriented) questions in the "Problems Section" of SIAM Review…
A new nonparametric estimator of a convex regression function in any dimension is proposed and its convergence properties are studied. We start by using any estimator of the regression function and we \emph{convexify} it by taking the…
In this paper we explore special values of Gaussian hypergeometric functions in terms of products of Euler $\Gamma$-functions and exponential functions of linear functions of the hypergeometric parameters. They include some classical…
In this manuscript we review some recent results about approximation of solutions of elliptic problems with high-contrast coefficients. In particular, we detail the derivation of asymptotic expansions for the solution in terms of the…
The established technique of eliminating upper or lower parameters in a general hypergeometric series is profitably exploited to create pathways among confluent hypergeometric functions, binomial functions, Bessel functions, and exponential…
We describe constructions of extended formulations that establish a certain relaxed version of the Hirsch conjecture and prove that if there is a pivot rule for the simplex algorithm for which one can bound the number of steps by a…
We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for…