We study a connection between Euler-MacLaurin Summation and Boole Summation suggested in an AMM note from 1960, which explains them as two cases in a general approach to approximation that also encompasses Taylor sums. Here we give additional details of the construction.

Abstract. Euler–Maclaurin and Poisson analogues of the summations ∑a<n≤b χ(n) f (n), ∑a<n≤b d(n) f (n), ∑a<n≤b d(n)χ(n) f (n) have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence; d(n) is the divisor function and f (x) is a sufficiently smooth function on [a,b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s...

We give a formula of the Riemann-Hurwitz type for classes defined by multiplicative sequences as corollaries of the Chern number formula for ramified coverings. 1. 1.1. In the classical results on Riemann surfaces, we have the Riemann-Hurwitz formula which relates the topological Euler numbers of the covering space and base space. The difference of the Euler numbers can be expressed by local re...

We give an elementary derivation of the Montgomery phase formula for the motion of an Euler top, using only basic facts about the Euler equation and parallel transport on the 2-sphere (whose holonomy is seen to be responsible for the geometric phase). We also give an approximate geometric interpretation of the geometric phase for motions starting close to an unstable equilibrium point. Introduc...

In the article, we prove that the double inequality [Formula: see text] holds for all [Formula: see text], we present the best possible constants λ and μ such that [Formula: see text] for all [Formula: see text], and we find the value of [Formula: see text] in the interval [Formula: see text] such that [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], ...

In this work, we establish the maximal [Formula: see text]-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order [Formula: see text], [Formula: see text], in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler met...

The sum formula for multiple zeta values are derived, via the Mellin transform, from the Euler connection formula and the Landen connection formula for polylogarithms. These connection formulas for multiple polylogarithms will be considered in the framework of the theory of the formal Knizhnik-Zamolodchikov equation.

We formulate and prove a formula for transgressing characteristic forms in general associated bundles following a method of Chern [Che91]. As applications, we derive D. Johnson’s explicit formula in [Joh07] for such general transgression and Chern’s first transgression formula in [Che45] for the Euler class.

the complex-step derivative approximation is applied to compute numerical derivatives. in this study, we propose a new formula of fractional complex-step method utilizing jumarie definition. based on this method, we illustrated an approximate analytic solution for the fractional cauchy-euler equations. application in image denoising is imposed by introducing a new fractional mask depending on s...

In a previous paper [1] an Euler angle parameterization for SU(4) was given. Here we present the derivation of a generalized Euler angle parameterization for SU(N). The formula for the calculation of the Haar measure for SU(N) as well as its relation to Marinov’s volume formula for SU(N) [2,3] will also be derived. As an example of this parameterization’s usefulness, the density matrix paramete...