Optimal embedding of a toroidal mesh in a path

We prove that the dilation of an $m \times n$ toroidal mesh in an $mn$-vertex path equals $2\min\{m,n\}$, if $m\not= n$ and $2n-1$, if $m=n$

Aerosol loaded toroidal vortices for enhanced ocular drug delivery

Over that last few decades there has been an increase in the creation of new medicinal agents to treat various ophthalmic diseases, from small molecule drugs to highly targeted biologics. While these advances in drug discovery have enabled a new wave of therapies, many problems regarding their accurate, reproducible and safe administration still remain. The standard vehicle for topical ocular drug delivery is the eye drop, and despite its popularity, there are some significant limitations to their use. It is estimated that in many cases less than 5% of the active reaches the target tissues. While the remainder of the dose is either spilled out onto surrounding tissue or it is rapidly drained through the lacrimal ducts where it can be absorbed into the systemic circulation. Current drug delivery technologies have focused on improving bioavailability and patient compliance, and it is expected that further improvements can be made in these areas by incorporating the use of precision drug loaded aerosol vortices. Within the framework of this dissertation, two main aspects of a novel ophthalmic aerosol drug delivery device were investigated; the mechanical features that dictate the dose delivery, and the formulation aspects that control the characteristics of the aerosols that are being delivered. In early studies, investigations into the mechanism of the dose deposition were explored in order to gain knowledge into predicting the performance and to tune the delivery of a wide range of therapeutic concentrations. In later studies, after the device and dosing characteristics where established, studies were conducted on different formulation strategies in order to incorporate active pharmaceutical ingredients that would otherwise have unfavorable physicochemical characteristics for incorporation into an aqueous based system. These studies included the use of solubilizing agents and their effect on the characteristics of the aerosol generated from the device, as well as a novel particle engineering technology that could be utilized to incorporate the use of nanoparticles or colloidal particulates into the device or for other uses. The use of these formulation techniques thereby increases scope of therapeutic agents that can be incorporated for use in the device, further improving its therapeutic potential.Pharmaceutical Science

A Monte Carlo method for critical systems in infinite volume: the planar Ising model

In this paper we propose a Monte Carlo method for generating finite-domain marginals of critical distributions of statistical models in infinite volume. The algorithm corrects the problem of the long-range effects of boundaries associated to generating critical distributions on finite lattices. It uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of "holographic" boundary condition that encodes the presence of an infinite volume beyond it. We check the quality of the distribution obtained in the case of the planar Ising model by comparing various observables with their infinite-plane prediction. We accurately reproduce planar two-, three- and four-point functions of spin and energy operators. We also define a lattice stress-energy tensor, and numerically obtain the associated conformal Ward identities and the Ising central charge.Comment: 43 pages, 21 figure

Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum

A quantum version of transition state theory based on a quantum normal form (QNF) expansion about a saddle-centre-...-centre equilibrium point is presented. A general algorithm is provided which allows one to explictly compute QNF to any desired order. This leads to an efficient procedure to compute quantum reaction rates and the associated Gamov-Siegert resonances. In the classical limit the QNF reduces to the classical normal form which leads to the recently developed phase space realisation of Wigner's transition state theory. It is shown that the phase space structures that govern the classical reaction d ynamicsform a skeleton for the quantum scattering and resonance wavefunctions which can also be computed from the QNF. Several examples are worked out explicitly to illustrate the efficiency of the procedure presented.Comment: 132 pages, 31 figures, corrected version, Nonlinearity, 21 (2008) R1-R11

Computational estimation of haemodynamics and tissue stresses in abdominal aortic aneurysms

'o e Abdominal aortic aneurysm is a vascular disease involving a focal dilation of the aorta. The exact cause is unknown but possibilities include infection and weakening of the connective tissue. Risk factors include a history of atherosclerosis, current smoking and a close relative with the disease. Although abdominal aortic aneurysm can affect anyone, it is most often seen in older men, and may be present in up to 5.9 % of the population aged 80 years. Biomechanical factors such as tissue stresses and shear stresses have been shown to play a part in aneurysm progression, although the specific mechanisms are still to be determined. The growth rate of the abdominal aortic aneurysm has been found to correlate with the peak stress in the aneurysm wall and the blood flow is thought to influence disease development. In order to resolve the connections between biology and biomechanics, accurate estimations of the forces involved are required. The first part of this thesis assesses the use of computational fluid dynamics for modelling haemodynamics in abdominal aortic aneurysms. Boundary conditions from the literature o

Aspects of practical implementations of PRAM algorithms

The PRAM is a shared memory model of parallel computation which abstracts away from inessential engineering details. It provides a very simple architecture independent model and provides a good programming environment. Theoreticians of the computer science community have proved that it is possible to emulate the theoretical PRAM model using current technology. Solutions have been found for effectively interconnecting processing elements, for routing data on these networks and for distributing the data among memory modules without hotspots. This thesis reviews this emulation and the possibilities it provides for large scale general purpose parallel computation. The emulation employs a bridging model which acts as an interface between the actual hardware and the PRAM model. We review the evidence that such a scheme crn achieve scalable parallel performance and portable parallel software and that PRAM algorithms can be optimally implemented on such practical models. In the course of this review we presented the following new results: 1. Concerning parallel approximation algorithms, we describe an NC algorithm for finding an approximation to a minimum weight perfect matching in a complete weighted graph. The algorithm is conceptually very simple and it is also the first NC-approximation algorithm for the task with a sub-linear performance ratio. 2. Concerning graph embedding, we describe dense edge-disjoint embeddings of the complete binary tree with n leaves in the following n-node communication networks: the hypercube, the de Bruijn and shuffle-exchange networks and the 2-dimcnsional mesh. In the embeddings the maximum distance from a leaf to the root of the tree is asymptotically optimally short. The embeddings facilitate efficient implementation of many PRAM algorithms on networks employing these graphs as interconnection networks. 3. Concerning bulk synchronous algorithmics, we describe scalable transportable algorithms for the following three commonly required types of computation; balanced tree computations. Fast Fourier Transforms and matrix multiplications