To each non totally real cubic extension K of Q and to each generator α of the cubic field K, we attach a family of cubic Thue equations, indexed by the units of K, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.

Morse 2-functions and trisections of 4-manifolds (Room 1.316) Robion Kirby (University of California, Berkeley) I’ll talk about work with David Gay which extends the notion of a Morse function f : M → R to a function from a 4-manifold X to a 2-dimensional surface. Just as the former gives an existence and uniqueness theorem for Heegaard splittings of a 3-manifold M , the latter gives existence ...

In this paper, the numerical solution methods of one- particale, one – dimensional time- independentSchrodinger equation are presented that allows one to obtain accurate bound state eigen values andeigen functions for an arbitrary potential energy function V(x). These methods included the FEM(Finite Element Method), Cooly, Numerov and others. Here we considered the Numerov method inmore details...

Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a non-commutative generalization of the result of Hutchings-Lee on the abelian set...

We consider the problem of segmenting triangle meshes endowed with a discrete scalar function f based on the critical points of f . The watershed transform induces a decomposition of the domain of function f into regions of influence of its minima, called catchment basins. The discrete Morse gradient induced by f allows recovering not only catchment basins but also a complete topological charac...

Many motion planning algorithms use Morse functions to characterize the free space. Speci cally, these algorithms look at the critical points of a Morse function to denote the topological changes in the free space. This paper introduces methods to sense critical points and ensure all critical points are \seen" by a coverage algorithm. Experimental results performed on a mobile robot are also pr...

Morse theory provides a powerful framework to study the topology of a manifold from a function de ned on it, but discrete constructions have remained elusive due to the di culty of translating smooth concepts to the discrete setting. Consider the problem of approximating the Morse-Smale (MS) complex of a Morse function from a point cloud and an associated nearest neighbor graph (NNG). While fol...

In this work, we design a nearly linear time discrete Morse theory based algorithm for computing homology groups of 2-manifolds, thereby establishing the fact that computing homology groups of 2-manifolds is remarkably easy. Unlike previous algorithms of similar flavor, our method works with coefficients from arbitrary abelian groups. Another advantage of our method lies in the fact that our al...

In [18, 20], Pajitnov considers the closed orbit structure of generic gradient flows of circle-valued Morse functions. It turns out that the torsion of a chain homotopy equivalence between the Novikov complex and the completed simplicial chain complex of the universal cover detects the eta function of the flow. This eta function counts the closed orbits and reduces to the logarithm of the zeta ...