It is shown in this paper that Euler was first to derive the finite rotation formula which is often erroneously attributed to Rodrigues, while Rodrigues was responsible for the derivation of the composition formulae for successive finite rotations and the so-called Euler parameters of finite rotation. Therefore, based upon historical facts, the following nomenclature is suggested: Euler's finit...

by using euler-maclaurin's summation formula and the way of real analysis, a more accurate multiplehilbert-type inequality and the equivalent form are given. we also prove that the same constantfactor in the equivalent inequalities is the best possible.

We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety X and for any constructible function α with respect to a complex algebraic Whitney stratification of X . We define generalized degrees of the global polar varieties and of the MacPherson cycles and we prove a global index formula for the Euler characteristic of α. Whenever α is the Euler obstr...

Following the pioneering work of Schapira, we consider topological Radon-type integral transforms on constructible Z-valued functions using the Euler characteristic as a measure. Contributions include: (1) application of the Schapira inversion formula to target localization and classification problems in sensor networks; (2) extension and application of the inversion formula to weighted Radon t...

We prove strong convergence of order [Formula: see text] for arbitrarily small [Formula: see text] of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying...

(A, B,C, D, . . . are positive constants), where the symbol means that rn lies between any two successive partial sums of the series. Usually (as in [1, pp. 529–534]) this formula is deduced from Euler’s summation formula. In this note we show how to compute in succession the constants A, B,C, . . . by a very simple technique. Only derivatives and limits of ln and of rational functions are invo...

We study a connection between Euler-MacLaurin Summation and Boole Summation suggested in an AMM note from 1960, which explains them as two cases of a general approach to approximation. Herein we give details and extensions of this idea.

A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space V , namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in V . The paper introduces the notion of an interpolator between...