We will review the main episodes of the history of the infinite in mathematics, stressing the asymmetry between infinitesimals and infinites. While the actual infinite, after being avoided for so long, when it became a mathematical object it entered mathematics to stay, infinitesimals have had a troubled history of condemnations and resurrections. This has probably to do with the ambiguous basi...

André V. G. Cavalieri, Peter Jordan, Yves Gervais, Mingjun Wei, and Jonathan B. Freund Département Fluides, Thermique, Combustion, Institut Pprime, CNRS-Université de Poitiers-ENSMA, UPR 3346, 43 Route de l’Aérodrome, Poitiers Cedex F86036, France Mechanical and Aerospace Engineering, New Mexico State University, P.O. Box 30001/Dept. 3450, Las Cruces, New Mexico 88003-8001, USA Mechanical Scien...

Digital sequences and nets are among the most popular kinds of low discrepancy sequences and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this articl...

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton– Jacobi equation. The...

Abstract:The oscillations of photon distribution function for squeezed and correlated light are shown to decrease when the temperature increases.The influence of the squeezing parameter and photon quadrature correlation coefficient on the photon distribution oscillations at nonzero temperatures is studied. The connection of deformation of Planck distribution formula with oscillations of photon ...

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called ”quotient problem” (resp. ”d-th root problem”), which in short asks whether the integrality of the values of the quotient (resp. d-th root) of two (resp. one) linear re...

Let I (α) ≤ a. It was Taylor–Cavalieri who first asked whether hulls can be computed. We show that |s|−3 = 0−1. Is it possible to construct hulls? U. Newton [11] improved upon the results of V. Wang by extending hyper-nonnegative domains.

The error in the trapezoidal rule quadrature formula can be attributed to discretization in the interior and non-periodicity at the boundary. Using a contour integral, we derive a unified bound for the combined error from both sources for analytic integrands. The bound gives the Euler–Maclaurin formula in one limit and the geometric convergence of the trapezoidal rule for periodic analytic func...

In this paper existence and uniqueness of twelfth degree spline is proved with application to quadrature. This formula is in the class of splines of degree 12 and continuity order [Formula: see text] that matches the derivatives up to order 6 at the knots of a uniform partition. Some mistakes in the literature are pointed out and corrected. Numerical examples are given to illustrate the applica...

In this manuscript, we establish an integration by parts formula for the quadrature of discontinuous payoffs in a multidimensional Heston model. For its derivation we use Malliavin calculus techniques and work under mild integrability conditions on the payoff and under the assumption of a strictly positive volatility. Since the integration by parts procedure smoothes the original functional, ou...