We present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection π : P(E )→ B, for B a non-singular compact complex algebraic variety of any dimension. Our formula readily produces generalizations of formulas derived by Sethi,Vafa, and Witten to compute the Euler characteristic of elliptically fibered Calabi-Yau fourfolds used for F-...

where V , E, and F are, respectively, the number of vertices, edges, and faces of p. The formula was first stated in print by Euler in 1758 [11]. The proof given here is based on Poincaré’s linear algebraic proof, stated in [17] (with a corrected proof in [18]), as adapted by Imre Lakatos in the latter’s Proofs and Refutations [15]. As is well known, Euler’s formula is not true for all polyhedr...

Because ∫ b+h2 a−h1 f(x)dx is a polynomial in h1 and h2 if f is a polynomial in x, applying the infinite order differential operator L( ∂ ∂hi ) then yields a finite sum, so the right hand side of (2) is well defined when f is a polynomial. A polytope in Rn is called integral, or a lattice polytope, if its vertices are in the lattice Zn; it is called simple if exactly n edges emanate from each v...

with respect to étale cohomology with compact supports. This relation fails in general for l = p, but holds in an important special case. Suppose f is a Galois cover and deg f is a power of p. Then the relation χc(X,Qp) = (deg f)χ(Y,Ql) holds; for complete curves, this is due to Shafarevich [2], for arbitrary curves it is equivalent to the Deuring-Shafarevich formula, and in general it is due t...

A new class of quadrature rules for the integration of both regular and singular functions is constructed and analyzed. For each rule the quadrature weights are positive and the class includes rules of arbitrarily high-order convergence. The quadratures result from alterations to the trapezoidal rule, in which a small number of nodes and weights at the ends of the integration interval are repla...

Using Hermite’s, i.e. the Euler-Maclaurin summation formula of order four, new approximations to Mathieu’s series S(x) ≡ ∑∞ k=1 2k (k2+x2) are obtained, which are more accurate than the approximations presented recently in the literature. Mathematics Subject Classification: 26D15, 33E20, 33F05, 40A05, 40A25, 65B10, 65B15, 65D20

We construct skew-adjoint operators associated to nowhere zero vector elds on manifolds with vanishing Euler number. The mod 2 indices of these operators provide potentially new invariants for such manifolds. An odd index theorem for corresponding Toeplitz operators is established. This last result may be viewed as an odd dimensional analogue of the Gauss-Bonnet-Chern theorem. It is well-known ...

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...

σ = κ1κ2...κω this decomposition is unique up to the order of the multiplicands. We will call a function f : Sn → C multiplicative if f(σ) = f(κ1)f(κ2)...f(κn). In what follows we will assume that the value of f on cycles depends only on the length of cycle, that is f(κ) = f̂(|κ|), where |κ| the order of cycle κ. Let mk(σ) be equal to the number of cycles in the decomposition σ whose order is eq...

Abstract. We show that the classical Faulhaber’s theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a + b, a + 2b, . . . , a + nb is a polynomial in na+ n(n+ 1)b/2. The coefficients of these polynomials are given in terms of the Bernoulli polynomials. Following Knuth’s approach by using the central factorial...