Abstract Classical turnpikes correspond to optimal steady states which are attractors of infinite-horizon control problems. In this paper, motivated by mechanical systems with symmetries, we generalize concept manifold turnpikes. Specifically, the necessary optimality conditions projected onto a symmetry-induced coincide those reduced-order problem defined on under certain conditions. We also p...

Manifold Learning learns a low-dimensional embedding of the latent manifold. In this report, we give the definition of distance metric learning, provide the categorization of manifold learning, and describe the essential connection between manifold learning and distance metric learning, with special emphasis on nonlinear manifold learning, including ISOMAP, Laplacian Eigenamp (LE), and Locally ...

Even if one is interested only in finite-dimensional manifolds, the need for infinitedimensional manifolds sometimes arises. For example, one approach to study closed geodesics on a manifold is to use Morse theory on its (free) loop space; while for some purposes it is enough to work with finite-dimensional approximations, it is helpful for some finer aspects of the theory to use models of the ...

We describe the stability manifold of the bounded derived category D(P) of coherent sheaves on P, denoted Stab(D(P)). This is the first complete picture of a stability manifold for a non-Calabi-Yau manifold.

The aim of this report is to study the two nonlinear manifold learning techniques recently proposed (Isomap [12] and Locally Linear Embedding (LLE) [8]) and suggest directions for further research. First the Isomap and the LLE algorithm are discussed in detail. Some of the areas that need further work are pointed out. A few novel applications which could use these two algorithms have been discu...

We construct a nonlinear mapping from a high-dimensional sample space to a low-dimensional vector space, effectively recovering a Cartesian coordinate system for the manifold from which the data is sampled. The mapping preserves local geometric relations in the manifold and is pseudo-invertible. We show how to estimate the intrinsic dimensionality of the manifold from samples, decompose the sam...

An alternative exhaust manifold system, the Perforated Manifold, Muffler, and Catalyst (PMMC), is proposed to improve sound suppression while reducing engine pumping losses and exhaust emissions. One-dimensional predictions from acoustic theory are used to configure the initial design. Preliminary evaluation of the concept is based on bench tests, including an extended impedance tube set-up for...