We define the geometric complex associated to a Morse– Bott–Smale vector field, cf. [AB95], and its associated spectral sequence. We prove an extension of the Bismut–Zhang theorem to Morse–Bott–Smale functions, see Theorem 2 below. The proof is based on the Bismut–Zhang theorem for Morse–Smale functions, see [BZ92].

Ascending and descending Morse complexes, defined by the critical points and integral lines of a scalar field f defined on a manifold domain D, induce a subdivision of D into regions of uniform gradient flow, and thus provide a compact description of the morphology of f on D. Here, we propose a dimension independent representation for the ascending and descending Morse complexes, and a data str...

This is an expository paper devoted to the Morse index of Chaperon’s generating families of Hamiltonian diffeomorphisms. After reviewing the construction of such generating families, we present Bott’s iteration theory in this setting: we study how the Morse index of a critical point corresponding to an iterated periodic orbit depends on the order of iteration of the orbit. We also investigate t...

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of...

We consider the approximability of maximization and minimization variants of the Morse matching problem, posed as open problems by Joswig and Pfetsch [12]. We establish hardness results for MaxMorse matching and Min-Morse matching. In particular, we show that, for a simplicial complex with n simplices and dimension d ≥ 3, it is NP-hard to approximate Min-Morse matching within a factor of O(n), ...

Countering the claim that brain determination (or more broadly: causation) undermines legal responsibility, Stephen Morse argued repetitively that penal law is not founded on metaphysical free will but rather the idea that people are, generally, rational agents; causation, determination, or alleged lack of free will are no excusing conditions, unlike lack of rational capacity, compulsion, or co...

The Morse-Smale complex is an important tool for global topological analysis in various problems in computational topology and data analysis. A certified algorithm for computing the Morse-Smale complexes has been presented for two-dimensional Morse-Smale systems in bounded planar domains [3]. In the current article we extend the approach in case of MorseSmale systems on two-dimensional manifold...

In this paper we consider 6-superlinear Chern–Simons–Schrödinger systems. contrast to most studies, the case where potential V is indefinite so that Schrödinger operator −Δ+V possesses a finite-dimensional negative space. We obtain nontrivial solutions for problem via Morse theory.

In this paper, we consider a discrete delay problem with negative feedback _ x(t) = f(x(t); x(t?1)) along with a certain family of time discretizations with stepsize 1=n. In the original problem, the attractor admits a Morse decomposition. We proved in G,H] that the discretized problems have global attractors. It was proved in G,M] that such attractors also admit Morse decompositions. In G,H] w...