In this paper we define and study the moduli space of metric-graphflows in a manifold M . This is a space of smooth maps from a finite graph to M , which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on M . Using the model of Gromov-Witten theory, with this moduli space replacing the space of stable holomorphic curves in a symplectic manifol...

In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph theoretical operations. We use iterated Morse decomposition, which allows us to sidetrack many problems related to the standard discrete Morse theory. In particu...

Take M , a finite-dimensional differentiable manifold, and f : M → R a smooth function. Such a function f is called a Morse function if it has no degenerate critical points. Morse theory allows us to connect the topology, in particular the homotopy type, of M with the behavior of f on M . In the following sections, we will state and prove two important theorems in Morse theory. Using these two ...

x0. Introduction. Consider a game played by 2 players, whom we call the hider and the seeker. Let S be a simplex of dimension n, with vertices x 0 , x 1 ; : : :; x n , and M a subcomplex of S, known to both the hider and the seeker. Let be a simplex of S, known only to the hider. The seeker is permitted to ask questions of the sort \Is vertex x i in ?" The seeker's goal is to determine whether ...

1. Abstract theory. Let M be a C-Riemannian manifold without boundary modeled on a separable Hubert space (see Lang [3]). For pÇzM we denote by ( , )p the inner product in the tangent space Mp and we define a function || || on the tangent bundle T(M) by ||z>|| = (v, v)J for vÇzMp. Given p and q in the same component of M we define p(p, q)==lnïfl\\<r'(t)\\dtt where the Inf is over all C 1 paths ...

In this paper, we consider non-linear Ginsburg-Pitaevski-Gross equation with the Rosen-Morse and modifiedWoods-Saxon potentials which is corresponding to the quantum vortices and has important applications in turbulence theory. We use the Runge- Kutta-Fehlberg approximation method to solve the resulting non-linear equation.    

1. I n t roduc t i on Discrete Morse theory was developed by Forman [9, 11] as a combinatorial analog to the classical smooth Morse theory. Applications to questions in combinatorial topology and related fields are numerous: e.g., Babson et al. [3], Forman [10], Batzies and Welker [4], and Jonsson [20]. I t turns out that the topologically relevant information of a discrete Morse function f on ...

In Morse Theory for Cell Complexes, we presented a discrete Morse theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlyin...

Morse theory inspired several robust and well grounded tools in discrete function analysis, geometric modeling and visualization. Such techniques need to adapt the original differential concepts of Morse theory in a discrete setting, generally using either piecewise–linear (PL) approximations or Forman’s combinatorial formulation. The former carries the intuition behind Morse critical sets, whi...