Abstract: We are interested in solving Liouville-type problems to explore constancy properties for maps or differential forms on Riemannian manifolds. Geometric structures on manifolds, the existence of constancy properties for maps or differential forms, and energy growth for maps or differential forms are intertwined. In this article, we concentrate on discovery of solutions to Liouville-type problems where manifolds are Euclidean spaces (i.e. flat Riemannian manifolds) and maps become real-valued functions. Liouville-type results of vanishing properties for functions are obtained. The original work in our research findings is to extend the q-energy for a function from finite in Lq space to infinite in non-Lq space by applying p-balanced technique where q = p = 2. Calculation skills such as Hölder's Inequality and Tests for Series have been used to evaluate limits and integrations for function energy. Calculation ideas and computational techniques for solving Liouville-type problems shown in this article, which are utilized in Euclidean spaces, can be universalized as a successful algorithm, which works for both maps and differential forms on Riemannian manifolds. This innovative algorithm has a far-reaching impact on research work of solving Liouville-type problems in the general settings involved with infinite energy. The p-balanced technique in this algorithm provides a clue to success on the road of q-energy extension from finite to infinite.
Abstract: In this paper, we present an application of Riemannian
geometry for processing non-Euclidean image data. We consider the
image as residing in a Riemannian manifold, for developing a new
method to brain edge detection and brain extraction. Automating this
process is a challenge due to the high diversity in appearance brain
tissue, among different patients and sequences. The main contribution, in this paper, is the use of an edge-based
anisotropic diffusion tensor for the segmentation task by integrating
both image edge geometry and Riemannian manifold (geodesic,
metric tensor) to regularize the convergence contour and extract
complex anatomical structures. We check the accuracy of the
segmentation results on simulated brain MRI scans of single
T1-weighted, T2-weighted and Proton Density sequences. We
validate our approach using two different databases: BrainWeb
database, and MRI Multiple sclerosis Database (MRI MS DB). We
have compared, qualitatively and quantitatively, our approach with
the well-known brain extraction algorithms. We show that using
a Riemannian manifolds to medical image analysis improves the
efficient results to brain extraction, in real time, outperforming the
results of the standard techniques.
Abstract: Equations on curved manifolds display interesting properties in a number of ways. In particular, the symmetries and, therefore, the conservation laws reduce depending on how curved the manifold is. Of particular interest are the wave and Gordon-type equations; we study the symmetry properties and conservation laws of these equations on the Milne and Bianchi type III metrics. Properties of reduction procedures via symmetries, variational structures and conservation laws are more involved than on the well known flat (Minkowski) manifold.