Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.

Asymptotically exact unweighted particle filter for manifold-valued hidden states and point process observations.

Abstract: The filtering of a Markov diffusion process on a manifold from counting process observations leads to `large' changes in the conditional distribution upon an observed event, corresponding to a multiplication of the density by the intensity function of the observation process. If that distribution is represented by unweighted samples or particles, they need to be jointly transformed such that they sample from the modified distribution. In previous work, this transformation has been approximated by a translation of all the particles by a common vector. However, such an operation is ill-defined on a manifold, and on a vector space, a constant gain can lead to a wrong estimate of the uncertainty over the hidden state. Here, taking inspiration from the feedback particle filter (FPF), we derive an asymptotically exact filter (called ppFPF) for point process observations, whose particles evolve according to intrinsic (i.e. parametrization-invariant) dynamics that are composed of the dynamics of the hidden state plus additional control terms. While not sharing the gain-times-error structure of the FPF, the optimal control terms are expressed as solutions to partial differential equations analogous to the weighted Poisson equation for the gain of the FPF. The proposed filter can therefore make use of existing approximation algorithms for solutions of weighted Poisson equations.

Geodesics in Geometry with Constraints and Applications

Abstract: In this course we carefully define the notion of a non-holonomic manifold which is a manifold with a certain non-integrable distribution. We describe such concepts as horizontal distribution, the Ehresmann connection, bracket generating condition for a distribution, sub-Riemannian structure and sub-Riemannian metric, Hamiltonian system, normal and abnormal geodesics, principal bundle and others.

An equivariant index formula for almost-CR manifolds

Abstract: We consider a consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J(E,X) whose properties we describe. When E is equipped with a complex structure, we define a class of symbol mappings in terms of the resulting almost-CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only J(E,X) and standard equivariant characteristic classes. This formula generalizes the formula given in arXiv:0712.2431 for the case of a contact manifold.

Imposing angle boundary conditions on B-spline/NURBS surfaces

Abstract: In this paper, we study the construction of a B-spline surface satisfying prescribed angle distribution (with respect to a chosen vector) of tangent planes along its boundary curve. This problem arises e.g. in a creation of a parametric geometric model of a Pelton turbine bucket, where specific angle distributions along a splitter and an outlet curve have to be fulfilled in order to control the flow of water into and out of the bucket. We prove that for a given B-spline curve c ( t ) , t ? 0 , 1 ] , the exact solution exists only in very special cases (for a special form of an angle function f ( t ) ). Further, we formulate an algorithm for finding an approximate solution. We also derive a bound on its approximation error and give a numerical evidence that the approximation order of the proposed algorithm is four. Finally, the method is demonstrated on several examples. We study the construction of a B-spline surface satisfying prescribed angle distribution of tangent planes along its boundary curve.We prove that for a given B-spline curve, the exact solution exists only in very special cases (for a special form of an angle function).We propose an algorithm for finding an approximate solution, derive a bound on its approximation error and study the approximation order of the proposed algorithm. Display Omitted

On Semi-Invariant Submanifolds of a Generalized Kenmotsu Manifold Admitting a Semi-Symmetric Non-Metric Connection

Abstract: In this paper, semi-invariant submanifolds of a generalized Kenmotsu manifold endowed with a semi-symmetric non-metric connection are studied. Necessary and sufficient conditions are given on a submanifold of a generalized Kenmotsu manifold to be semi-invarinat submanifold with semi-symmetric non-metric connection. Morever, we studied the integrability condition of the distribution on semi-invariant submanifolds of generalized Kenmotsu manifold with semi-symmetric non-metric connection.