Some recent ideas are generalized from four dimensions to the general dimension n. Two terms of the trace anomaly in external gravity, the Euler density Gn and 2 R, are relevant to the problem of quantum irreversibility. By adding the divergence of a gauge-invariant current, Gn can be extended to a new notion of Euler density G̃n, linear in the conformal factor. We call it pondered Euler density...

Abstract. We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich’s stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles – including any direct sum of line bundles –...

The classical Thom–Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situa...

The double zeta function was first studied by Euler in response to a letter from Goldbach in 1742. One of Euler’s results for this function is a decomposition formula, which expresses the product of two values of the Riemann zeta function as a finite sum of double zeta values involving binomial coefficients. Here, we establish a q-analog of Euler’s decomposition formula. More specifically, we s...

By using the Euler-Maclaurin’s summation formula and the weight coefficient, a pair of new inequalities is given, which is a decomposition of Hilbert’s inequality. The equivalent forms and the extended inequalities with a pair of conjugate exponents are built.

In this note we give an elementary combinatorial proof of a formula of Macris and Pul6 for the number of Euler trails in a digraph all of whose vertices have in-degree and out-degree equal to2.

The Euler characteristic of asymptotically flat, stationary vacuum spacetimes is calculated based on compactifications and Lefschetz’s fixed point formula. It turns out that it can be zero or two.

We give a Euler-Maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope. Our proof uses elementary methods.

We establish a group-invariant version of the variational bicomplex that is based on a general moving frame construction. The main application is an explicit group-invariant formula for the Euler-Lagrange equations of an invariant variational problem.

The Smith arithmetic determinant is investigated in this paper. By using two different methods, we derive the explicit formula for the Smith arithmetic determinant. Keywords—Elementary row transformation, Euler function, Matrix decomposition, Smith arithmetic determinant.