Recall that in an earlier paper by one of the authors DonaldsonThomas type invariants were expressed as certain weighted Euler characteristics of the moduli space. The Euler characteristic is weighted by a certain canonical Z-valued constructible function on the moduli space. This constructible function associates to any point of the moduli space a certain invariant of the singularity of the sp...

In his famous paper, “On certain arithmetical functions”, Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan’s tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for...

Let M be the interior of a compact 3–manifold with boundary, and let T be an ideal triangulation of M. This paper describes necessary and sufficient conditions for the existence of angle structures, semi–angle structures and generalised angle structures on (M ;T ) respectively in terms of a generalised Euler characteristic function on the solution space of the normal surface theory of (M ; T )....

Fix a one-dimensional group variety G with Euler–characteristic χ(G) = 0, and a quasi–projective variety Y , both defined over C. For any f ∈ Hom(Y, G) and constructible sheaf F on Y , we construct an invariant cF (f) ∈ G, which provides substantial information about the topology of the fiber–structure of f and the structure of F along the fibers of f . Moreover, cF : Hom(Y,G)→ G is a group hom...

A new expansion is given for partial sums of Euler’s pentagonal number series. As a corollary we derive an infinite family of inequalities for the partition function, p(n).

We study N = 4 supersymmetric Yang-Mills theory on a Kähler manifold with b2 ≥ 3. Adding suitable perturbations we show that the partition function of the N = 4 theory is the sum of contributions from two branches: (i) instantons, (ii) a special class of SeibergWitten monopoles. We determine the partition function for the theories with gauge group SU(2) and SO(3), using S-duality. This leads us...

The paper investigates vanishing conditions on the first cohomology module of a normalized rank 2 vector bundle F on P which force F to split. The present vanishing conditions improve other conditions known in the literature and are obtained with simple computations on the Euler characteristic function, avoiding the speciality lemma and other heavy tools.

We prove that any weak space-time L vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of R satisfies the Euler equation if the solutions’ local enstrophies are uniformly bounded. We also prove that t − a.e. weak L inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they...

We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety X and for any constructible function α with respect to a complex algebraic Whitney stratification of X . We define generalized degrees of the global polar varieties and of the MacPherson cycles and we prove a global index formula for the Euler characteristic of α. Whenever α is the Euler obstr...

This is the fourth part in the series of articles [4], [5], [6] (see also [3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations is an automorphism of the algebra of valuations. Then an integration functional on valuations wi...