We introduce an algorithm for construction of the Morse hierarchy, i.e. a hierarchy of Morse decompositions of a piecewise constant vector field on a surface driven by stability of the Morse sets with respect to perturbation of the vector field. Our approach builds upon earlier work on stable Morse decompositions, which can be used to obtain Morse sets of user-prescribed stability. More stable ...

The potential for communication through the kinesthetic aspect of the tactual sense was examined in a series of experiments employing Morse code signals. Experienced and inexperienced Morse code operators were trained to identify Morse code signals that were delivered as sequences of motional stimulation through up-down displacements (roughly 10 mm) of the fingertip. Performance on this task wa...

Using graph representations a new class of computable topological invariants associated with a tame real or angle valued map were recently introduced, providing a theory which can be viewed as an alternative to MorseNovicov theory for real or angle valued Morse maps. The invariants are ”barcodes” and ”Jordan cells”. From them one can derive all familiar topological invariants which can be deriv...

Let F be a flat vector bundle over a compact Riemannian manifold M and let f : M → R be a Morse function. Let g be a smooth Euclidean metric on F , let g t = e g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g t . Assuming that ∇f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log ρ(t) for t → +∞ of the form a0 + a1t +...

In the paper [N], while studying adiabatic deformations of Dirac operators on manifolds with boundary, we were led to the following nite dimensional dynamics problem. Consider (n) the grassmannian of lagrangian subspaces in the canonical symplectic space E = R2n . If A : E ! E is a selfadjoint operator anticommuting with the canonical complex structure J on E, then A belongs to the Lie algebra ...

A brief overviewof Forman’s discrete Morse theory is presented, from which analogues of the main results of classical Morse theory can be derived for discrete Morse functions, these being functions mapping the set of cells of a CW complex to the real numbers satisfying some combinatorial relations. The discrete analogue of the strong Morse inequality was proved by Forman for finite CW complexes...

We apply Morse theory in the study of sensor networks and distributed sensor data. Sensor nodes are deployed in a 2D region M with boundaries and possibly interior holes, and the sensor data samples a continuous real function f . We are interested in both the topology of the discrete sensor field in terms of the sensing holes (voids without sufficient sensors deployed), as well as the topology ...

In 1998, Forman introduced discrete Morse theory as a tool for studying CW complexes by producing smaller, simpler-to-understand complexes of critical cells with the same homotopy types as the original complexes. This paper addresses two questions: (1) under what conditions may several gradient paths in a discrete Morse function simultaneously be reversed to cancel several pairs of critical cel...

Given, in the Lagrangian torus fibration R → R, a Lagrangian submanifold L, endowed with a trivial flat connection, the corresponding mirror object is constructed on the dual fibration by means of a family of Morse homologies associated to the generating function of L, and it is provided with a holomorphic structure. Morse homology, however, is not defined along the caustic C of L or along the ...