A common problem of vector field topology algorithms is the large number of the resulting topological features. This paper describes a method to simplify Morse decompositions by iteratively merging pairs of Morse sets that are adjacent in the Morse Connection Graph (MCG). When Morse sets A and B are merged, they are replaced by a single Morse set, that can be thought of as the union of A, B and...

The connection matrix theory for Morse decompositions is introduced. The connection matrices are matrices of maps between the homology indices of the sets in the Morse decomposition. The connection matrices cover, in a natural way, the homology index braid of the Morse decomposition and provide information about the structure of the Morse decomposition. The existence of connection matrices of M...

This article follows the previous works [HKN] by Helffer-KleinNier and [HeNi1] by Helffer-Nier about the metastability in reversible diffusion processes via a Witten complex approach. Again, exponentially small eigenvalues of some self-adjoint realization of ∆ f,h = −h∆ + |∇f(x)| − h∆f(x) , are considered as the small parameter h > 0 goes to 0. The function f is assumed to be a Morse function o...

In this note, we consider the question of classicality for theory which is known to be effective description two-dimensional black holes - Morse quantum mechanics. We calculate Wigner function and Fisher information characterizing classicality/quantumness single-particle systems briefly discuss further directions study.

We use discrete Morse theory to provide another proof of Bernini, Ferrari, and Steingrímsson’s formula for the Möbius function of the consecutive pattern poset. In addition, we are able to determine the homotopy type of this poset. Earlier, Björner determined the Möbius function and homotopy type of factor order and the results are remarkably similar to those in the pattern case. In his thesis,...

Reeb graphs provide a method to combinatorially describe the shape of a manifold endowed with a Morse function. One question deserving attention is whether Reeb graphs are robust against function perturbations. Focusing on 1-dimensional manifolds, we define an editing distance between Reeb graphs of curves, in terms of the cost necessary to transform one graph into another through editing moves...

It is shown that the critical point relations of Morse theory, together with the maximum principle, comprise a complete set of critical point relations for harmonic functions of three variables. The proof proceeds by first constructing a simplified example and then developing techniques to modify this example to realize all admissible possibilities. Techniques used differ substantially from tho...

This article grew out of several talks that the author presented at the Banach Institute and at the University of Bialystok in Poland during November of 2001. It describes six problems from the geometry of submanifolds. Some of the problems come from the theory of constant curvature submanifolds in Euclidean space, as well as applications of Morse theory of the height function to the problem of...

By the famous theorem of Morse and Hedlund, a word is ultimately periodic if and only if it has bounded subword complexity, i.e., for sufficiently large n, the number of factors of length n is constant. In this paper we consider relational periods and relationally periodic sequences, where the relation is a similarity relation on words induced by a compatibility relation on letters. We investig...

Given a smooth closed manifold M , the Morse-Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology ofM . The geometric approach presented here was developed in [We-93] and is based on...