This is a continuation of our work [L-Q] on blowup formulae for the S-duality conjecture of Vafa and Witten. In [V-W], Vafa and Witten formulated some mathematical predictions about the Euler characteristics of instanton moduli spaces derived from the S-duality conjecture in physics. From these mathematical predictions, a blowup formula was proposed based upon the work of Yoshioka [Yos]. Roughl...

In [14], P.Schapira obtained a formula for RS in the general situation. This formula gives an inversion formula for the Radon transform of constructible functions from a real projective space to its dual in the case when the whole dimension is odd. We can, that is, reconstruct a constructible function φ on the projective space from its Radon transform RS(φ). This topological meaning is that we ...

In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta inva...

where n is even, h = (b − a)/n, xi = a + ih, and ξ ∈ (a, b). We usually derive (1) using Lagrange polynomials or making the formula exact for f(x) = 1, x, x. A standard exercise for a numerical analysis class is to use the composite Simpson’s rule to approximate an integral with n equal to successive powers of two, and verify that (as long as the fourth derivative of f is well behaved) the erro...

In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations [Formula: see text] [Formula: see text], in a neighborhood of the unique fixed point [Formula: see text] of the strictly increasing continuous function β, defined on an interval [Formula: see text]. These equations are based on the general quantum difference operator ...

When differential-algebraic equations of index 3 or higher are solved with backward differentiation formulas, the solution can have gross errors in the first few steps, even if the initial values are equal to the exact solution and even if the stepsize is kept constant. This raises the question of what are consistent initial values for the difference equations. Here we study how to change the e...

An explicit formula is given for a sequence of numbers by using the Eichler-Selberg trace formula. A criterion is obtained by using these numbers for the location of nontrivial zeros of Hecke L-functions, which are associated with a family of cusp forms of weight 2.

We relate Fubini’s theorem for Euler characteristics to Riemann-Hurwtiz formulae, and reprove a classical result of Iversen. The techniques used include algebraic geometry, complex geometry, and model theory. Possible applications to the study of wild ramification in finite characteristic are discussed. Introduction The first section of the paper reviews the concept of an Euler characteristic f...

While the Euler characteristic χ(G) = ω1(G) = ∑ x ω(x) super counts simplices, the Wu characteristics ωk(G) = ∑ x1∼x2···∼xk ω(x1) · · ·ω(xk) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial complex G. More generally, one can define the k-intersection number ωk(G1, . . . Gk), which is the same sum but where xi ∈ Gi. For every k ≥ 1 we define ...