Related papers: Simple derivation of basic quadrature formulas
The Linear Smoothing (LS) scheme \cite{francisa.ortiz-bernardin2017} ameliorates linear and quadratic approximations over convex polytopes by employing a three-point integration scheme. In this work, we propose a linearly consistent one…
An efficient entailment proof system is essential to compositional verification using separation logic. Unfortunately, existing decision procedures are either inexpressive or inefficient. For example, Smallfoot is an efficient procedure but…
We propose a variation of the forward--backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility…
Standard numerical integrators suffer from an order reduction when applied to nonlinear Schr\"{o}dinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$…
Given a polynomial system f associated with a simple multiple zero x of multiplicity {\mu}, we give a computable lower bound on the minimal distance between the simple multiple zero x and other zeros of f. If x is only given with limited…
We derived the formulae of central differentiation for the finding of the first and second derivatives of functions given in discrete points, with the number of points being arbitrary. The obtained formulae for the derivative calculation do…
We study modified trigonometric integrators, which generalize the popular class of trigonometric integrators for highly oscillatory Hamiltonian systems by allowing the fast frequencies to be modified. Among all methods of this class, we…
This survey paper discusses the history of approximation formulas for n-th order derivatives by integrals involving orthogonal polynomials. There is a large but rather disconnected corpus of literature on such formulas. We give some results…
In this note, we provide a simple derivation of expressions for the restricted partition function and its polynomial part. Our proof relies on elementary algebra on rational functions and a lemma that expresses the polynomial part as an…
In this paper we present a new class of cubature rules with the aim of accurately integrating weakly singular double integrals. In particular we focus on those integrals coming from the discretization of Boundary Integral Equations for 3D…
We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of…
The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the…
We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and another that has small Gowers uniformity norm…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces…
Accurate evaluation of nearly singular integrals plays an important role in many boundary integral equation based numerical methods. In this paper, we propose a variant of singularity swapping method to accurately evaluate the layer…
We consider the numerical evaluation of a class of double integrals with respect to a pair of self-similar measures over a self-similar fractal set (the attractor of an iterated function system), with a weakly singular integrand of…
A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector…
It is known that one can solve semidefinite programs to within fixed accuracy in polynomial time using the ellipsoid method (under some assumptions). In this paper it is shown that the same holds true when one uses the short-step, primal…
When considering stochastic integration and the theory of stochastic differential equations, P. Protter's textbook \cite{protter} undoubtedly is a main piece of standard literature. Not only is it well-written, but it also contains various…