We present an algorithm to invert the Euler function φ(m). The algorithm, for a given integer n ≥ 1, in polynomial time “on average”, finds the set Ψ(n) of all solutions m to the equation φ(m) = n. In fact, in the worst case the set Ψ(n) is exponentially large and cannot be constructed by a polynomial time algorithm. In the opposite direction, we show, under some widely accepted number theoreti...

1 f-vector for convex polytopes Given a convex polytope P of full dimension in R, we can consider the f-vector, fk = number of k-faces, 0 ≤ k ≤ d− 1. A natural question is Question 1. Given an abstract tuple f of d integers, what are necessary and sufficient conditions for f to be the f-vector of a convex polytope? It turns out to be quite difficult to give sufficient conditions, so let’s deter...

We show that a partition function of topological twisted N = 4 Yang-Mills theory is given by Seiberg-Witten invariants on a Riemannian four manifolds under the condition that the sum of Euler number and signature of the four manifolds vanish. The partition function is the sum of Euler number of instanton moduli space when it is possible to apply the vanishing theorem. And we get a relation of E...

We compute (algebraically) the Euler characteristic of a complex of sheaves with constructible cohomology. A stratified Poincaré-Hopf formula is then a consequence of the smooth Poincaré-Hopf theorem and of additivity of the Euler-Poincaré characteristic with compact supports, once we have a suitable definition of index. AMS classification: 55N33 57R25

We define motivic Milnor fibre of cyclic L∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space ...

The aim of this paper is to introduce finite spaces and their simplicial complexes. Next, we give an application to combinatorics in the form of a relation between the Euler characteristic and the Möbius function. We begin by giving an overview of finite topological spaces. We introduce beat points and weak homotopy equivalences. Then, we show that finite spaces are weak homotopy equivalent to ...

We prove higher integrability properties of solutions to the problem of minimizing ∫ Ω L(x, u(x),∇u(x))dx, where ξ → L(x, u, ξ) is a convex function satisfying some additional conditions. As an application, we prove the validity of the Euler–Lagrange equation for a class of functionals with growth faster than exponential.

We find a quartic example of a smooth embedded negatively curved surface in R homeomorphic to a doubly punctured torus. This constitutes an explicit solution to Hadamard’s problem on constructing complete surfaces with negative curvature and Euler characteristic in R. Further we show that our solution has the optimal degree of algebraic complexity via a topological classification for smooth cub...