Riemann sums over polytopes

We show that the Euler-MacLaurin formula for Riemann sums has an n-dimensional analogue in which intervals on the line get replaced by convex polytopes.Comment: 13 page

A note on the Euler-Maclaurin Sum formula

In this note we give a real variable approach for calculating the constant term that arises in the application of the Euler-Maclaurin expansion for a special class of series of the form Σ =n1rf(r).asn∞→ In particular the method is used to derive the approximate summation of the expression where ℓ is a non negative integer. Σ=n1rnr,rl

Spectral Action for Robertson-Walker metrics

We use the Euler-Maclaurin formula and the Feynman-Kac formula to extend our previous method of computation of the spectral action based on the Poisson summation formula. We show how to compute directly the spectral action for the general case of Robertson-Walker metrics. We check the terms of the expansion up to a_6 against the known universal formulas of Gilkey and compute the expansion up to a_{10} using our direct method

Harish-Chandra's volume formula via Weyl's Law and Euler-Maclaurin formula

Harish-Chandra's volume formula shows that the volume of a flag manifold $G/T$, where the measure is induced by an invariant inner product on the Lie algebra of $G$, is determined up to a scalar by the algebraic properties of $G$. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing the Euler-Maclaurin formula. This approach leads to a mystery that lies under the Atiyah-Singer index theorem