The proof follows the same outline as the previous study except that we use discrete

Morse theory to determine the homotopy type of the associated complexes of ordered set partitions.

The topics include some algebraic methods in semi-Markov chains, statistical topology through

Morse theory persistence and non-parametric estimation, structural properties of the generalized Dirichlet distributions, projections on invariant subspaces, combining statistical models, regular fractions and indicator polynomials, and some hypothesis tests for Wishart models on symmetric cones.

Under Phase I of the NSF grant, Raindrop Geomagic used

Morse Theory research to create a structured representation of a computer model by distinguishing between flat and highly curved feature surfaces.

The authors work with a new sequence of eigenvalues that employs the cohomological index to systematically develop alternative tools such as nonlinear linking and local splitting theories to apply

Morse theory to quasilinear problems.

2 is closely related to results of Jollenbeck-Welker, of Skoldberg, and of Kozlov (see (3), (8), and (4)) developing algebraic versions of discrete

Morse theory.

They then move briskly through Toponogov's theorem, homogeneous spaces,

Morse theory, closed deodesics and the cut locus, the sphere theorem and its generalizations, the differentiable space theorem, complete manifolds of non-negative curvature, and compact manifolds of non-positive curvature.

Other topics include automated reverse engineering of free form objects using

Morse theory, dense depth and color acquisition of repetitive motions, examplar-based shape from shading, and generalized MPU implicits using belief propagation.

Some 60 years later Novikov, who was working on a problem in hydrodynamics, formulated a circle-based

Morse theory.

Coverage includes the basics of various geometries in linear spaces, the geometry of two-dimensional manifolds, basics of topology of smooth manifolds, Lie groups, classical tensor algebra and tensor calculus, differential forms theory, the Riemannian theory of connections and curvature, conformal geometry, complex geometry,

Morse theory and Hamiltonian formalism, Poisson and Lagrange manifolds, multidimensional variational problems, and geometric fields in physics.

presents the basic concepts of topology and

Morse theory for nonspecialists, approaching the mathematical end first with spaces and filtration, group theory, homology,

Morse theory (including the Morse-Smale complex), and following with algorithms such as the persistence algorithms and topical simplification.

Emphasizing interactions of dynamics, geometry, and topology, they present chapters on manifolds, vector fields and dynamical systems, Riemannian metrics, Riemannian connections and geodesics, curvature, tensors and differential forms, fixed points and intersection numbers,

Morse theory, and hyperbolic systems.

Topics of the 14 papers include the maximum principle for vector fields, notes on p- harmonic analysis, dynamics in bounded domains, recent progress on the Monge-Ampere equation, and Riemann-Hurwitz formulas and

Morse theory.