Semi-hyperbolic mappings in Banach spaces

The definition of semi-hyperbolic dynamical systems generated by Lipschitz continuous and not necessarily invertible mappings in Banach spaces is presented in this thesis. Like hyperbolic mappings, they involve a splitting into stable and unstable spaces, but a slight leakage from the strict invariance of the spaces is possible and the unstable subspaces are assumed to be finite dimensional. Bi-shadowing is a combination of the concepts of shadowing and inverse shadowing and is usually used to compare pseudo-trajectories calculated by a computer with the true trajectories. In this thesis, the concept of bi-shadowing in a Banach space is defined and proved for semi-hyperbolic dynamical systems generated by Lipschitz mappings. As an application to the concept of bishadowing, linear delay differential equations are shown to be bi-shadowing with respect to pseudo-trajectories generated by nonlinear small perturbations of the linear delay equation. This shows robustness of solutions of the linear delay equation with respect to small nonlinear perturbations. Complicated dynamical behaviour is often a consequence of the expansivity of a dynamical system. Semi-hyperbolic dynamical systems generated by Lipschitz mappings on a Banach space are shown to be exponentially expansive, and explicit rates of expansion are determined. The result is applied to a nonsmooth noninvertible system generated by delay differential equation. It is shown that semi-hyperbolic mappings are locally φ-contracting, where -0 is the Hausdorff measure of noncompactness, and that a linear operator is semi-hyperbolic if and only if it is φ-contracting and has no spectral values on the unit circle. The definition of φ-bi-shadowing is given and it is shown that semi-hyperbolic mappings in Banach spaces are φ-bi-shadowing with respect to locally condensing continuous comparison mappings. The result is applied to linear delay differential equations of neutral type with nonsmooth perturbations. Finally, it is shown that a small delay perturbation of an ordinary differential equation with a homoclinic trajectory is ‘chaotic’.

Evaluation and comparison of 3D intervertebral disc localization and segmentation methods for 3D T2 MR data: A grand challenge.

The evaluation of changes in Intervertebral Discs (IVDs) with 3D Magnetic Resonance (MR) Imaging (MRI) can be of interest for many clinical applications. This paper presents the evaluation of both IVD localization and IVD segmentation methods submitted to the Automatic 3D MRI IVD Localization and Segmentation challenge, held at the 2015 International Conference on Medical Image Computing and Computer Assisted Intervention (MICCAI2015) with an on-site competition. With the construction of a manually annotated reference data set composed of 25 3D T2-weighted MR images acquired from two different studies and the establishment of a standard validation framework, quantitative evaluation was performed to compare the results of methods submitted to the challenge. Experimental results show that overall the best localization method achieves a mean localization distance of 0.8 mm and the best segmentation method achieves a mean Dice of 91.8%, a mean average absolute distance of 1.1 mm and a mean Hausdorff distance of 4.3 mm, respectively. The strengths and drawbacks of each method are discussed, which provides insights into the performance of different IVD localization and segmentation methods.