This paper studies how to generalize Tukey's depth problems defined in a restricted space that may be curved or have boundaries, and with nondifferentiable objective. First, using manifold approach, we propose broad class of Riemannian
 for smooth on Riemannian manifold, showcase its applications spherical data analysis, principal component multivariate orthogonal regression. Moreover, non...

When studying a slow-fast (singularly perturbed) Hamiltonian system in two degrees of freedom one can say unexpectedly much about its dynamics near its slow manifold. We demonstrate some details of this picture assuming the phase portrait of the slow system be known and generic. Since this latter system is in one degree of freedom, the problem in question is about the structure near the annulus...

*Correspondence: Peng Gao, Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794, USA e-mail: [email protected] We conjecture that a non-flat D-real-dimensional compact Calabi-Yau manifold, such as a quintic hypersurface with D = 6, or a K3 manifold with D = 4, has locally length minimizing closed geodesics, and that the number of these with length less th...

For a large class of reaction-diiusion systems on the plane, we show rigorously that m-armed spiral waves bifurcate from a homogeneous equilibrium when the latter undergoes a Hopf bifurcation. In particular, we construct a nite-dimensional manifold which contains the set of small rotating waves close to the homogeneous equilibrium. Examining the ow on this center-manifold in a very general exam...

For a large class of reaction-diiusion systems on the plane, we show rigorously that typically m-armed spiral waves bifurcate from a homogeneous equilibrium when the latter undergoes a Hopf bifurcation. In particular, we construct a nite-dimensional manifold which contains the set of small rotating waves close to the homogeneous equilibrium. Examining the ow on this center-manifold in a very ge...

In a broad sense, model reduction means producing a low-dimensional dynamical system that replicates either approximately, or more strictly, exactly and topologically, the output of a dynamical system. Model reduction has an important role in the study of dynamical systems and also with engineering problems. In many cases, there exists a good low-dimensional model for even very high-dimensional...

The present paper deals with a toxin producing phytoplankton (TPP)-zooplankton interaction in spatial environment in thecontext of phytoplankton bloom. In the absence of diffusion the stability of the given system in terms of co-existence and hopf bifurcation has been discussed. After that TPP-zooplankton interaction is considered in spatiotemporal domain by assuming self diffusion in both popu...

Semi-supervised learning is more powerful than supervised learning by using both labeled and unlabeled data. In particular, the manifold regularization framework, together with kernel methods, leads to the Laplacian SVM (LapSVM) that has demonstrated state-of-the-art performance. However, the LapSVM solution typically involves kernel expansions of all the labeled and unlabeled examples, and is ...