Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere S3S^3 and on the Hyperbolic space H3H^3

The superintegrability of several Hamiltonian systems defined on three-dimensional configuration spaces of constant curvature is studied. We first analyze the properties of the Killing vector fields, Noether symmetries and Noether momenta. Then we study the superintegrability of the harmonic oscillator, the Smorodinsky-Winternitz system and the harmonic oscillator with ratio of frequencies 1:1:2 and additional nonlinear terms on the three-dimensional sphere S-3 (kappa > 0) and on the hyperbolic space H-3 (kappa 0) and H-3 (kappa 0, kappa = 0, or kappa < 0, the corresponding properties are obtained for the system on the sphere S-3, the Euclidean space E-3, or the hyperbolic space H-3, respectively

A Super-Integrable Two-Dimensional Non-Linear Oscillator with an Exactly Solvable Quantum Analog

Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant curvature, the deformation parameter being related with the curvature. In this sense these systems are to be considered as a harmonic oscillator and a Smorodinsky-Winternitz system in such bi-dimensional spaces of constant curvature. The quantization of the first system will be carried out and it is shown that it is super-solvable in the sense that the Schrödinger equation reduces, in three different coordinate systems, to two separate equations involving only one degree of freedom

Recent Applications of the Theory of Lie Systems in Ermakov Systems

We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found from this new perspective. We also obtain new results, such as a new superposition rule for the Pinney equation in terms of three solutions of a related Riccati equation

Stem cell bioprocessing: The bioengineering of lung epithelium in 3D from embryonic stem cells

Stem cell therapies and tissue engineering strategies are required for the clinical treatment of respiratory diseases. Previous studies have established protocols for the differentiation of airway epithelium from stem cells but have involved costly and laborious culture methods. The aim of this thesis was to achieve efficient and reproducible maintenance and differentiation of embryonic stem cells to airway epithelium, in 2D and 3D culture, by developing appropriate bioprocessing technology. Firstly, the 2D differentiation process of human and murine ES cells into pulmonary epithelial cells was addressed. The main finding in was that the proportion of type II pneumocytes, the major epithelial component of the gas-exchange area of lung, differentiated with this method was higher than that obtained in previous sudies, 33% of resultant cell expressed the specific marker surfactant protein C (SPC) compared with up to 10%. Secondly, the maintenance and differentiation was carried out in 3D. A protocol was devised that maintained undifferentiated human ES cells in culture for more than 200 days encapsulated in alginate without any feeder layer or growth factors. For ES cell differentiation in 3D, a method was devised to provide a relatively cheap and simple means of culture and use medium conditioned by a human pneumocyte tumour cell line (A549). The differentiation of human and murine ES cells into pulmonary epithelial cells, particularly type II pneumocytes, was found to be upregulated by culture in this conditioned medium, with or without embryoid body formation. The third step was to test whether this differentiation protocol was amenable to scale-up and automation in a bioreactor using cell encapsulation. It was possible to show that encapsulated murine ES cells cultured in static, co-culture or rotating wall bioreactor (HARV) systems, differentiate into endoderm and, predominantly, type I and II pneumocytes. Flow cytometry revealed that the mean yield of differentiated type II pneumocytes was around 50% at day 10 of cultivation. The final stage of the work was to design and produce a perfusion system airlift bioreactor to mimic the pulmonary microenvironment in order to achieve large scale production of biologically functional tissue. The results of these studies thus provide new protocols for the maintenance of ES cells and their differentiation towards pulmonary phenotypes that are relatively simple and cheap and can be applied in bioreactor systems that provide for the kind of scale up of differentiated cell production needed for future clinical applications