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’.

Constitutive analysis of hot deformation behavior of a Ti6Al4V alloy using physical based model

The effect of deformation parameters on the flow behavior of a Ti6Al4V alloy has been studied to understand the deformation mechanisms during hot compression. Cylindrical samples with partially equiaxed grains were deformed in the α+β phase region at different thermo-mechanical conditions. To develop components with tailored properties, the physically based Estrin and Mecking (EM) model for the work hardening/dynamic recovery combined with the Avrami equation for dynamic recrystallization was used to predict the flow stress at varying process conditions. The EM model revealed good predictability up to the peak strain, however, at strain rates below 0.01s^-1, a higher B value was observed due to the reduced density of dislocation tangles. In contrast, the flow softening model revealed higher value of constants a and b at high strain rates due to the reduction in the volume fraction of dynamic recrystallization and larger peak strain. The predicted flow stress using the combined EM+Avrami model revealed good agreement with the measured flow stress resulted in very low average absolute relative error value. The microstructural analysis of the samples suggests the formation of coarse equiaxed grains together with the increased β phase fraction at low strain rate leads to a higher flow softening.

Links between copper and cholesterol in Alzheimer's disease

Altered copper homeostasis and hypercholesterolemia have been identified independently as risk factors for Alzheimer's disease (AD). Abnormal copper and cholesterol metabolism are implicated in the genesis of amyloid plaques and neurofibrillary tangles (NFT), which are two key pathological signatures of AD. Amyloidogenic processing of a sub-population of amyloid precursor protein (APP) that produces Aβ occurs in cholesterol-rich lipid rafts in copper deficient AD brains. Co-localization of Aβ and a paradoxical high concentration of copper in lipid rafts fosters the formation of neurotoxic Aβ:copper complexes. These complexes can catalytically oxidize cholesterol to generate H2O2, oxysterols and other lipid peroxidation products that accumulate in brains of AD cases and transgenic mouse models. Tau, the core protein component of NFTs, is sensitive to interactions with copper and cholesterol, which trigger a cascade of hyperphosphorylation and aggregation preceding the generation of NFTs. Here we present an overview of copper and cholesterol metabolism in the brain, and how their integrated failure contributes to development of AD.