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Density functional theory (DFT) is a popular approach to solving the many-electron Schrödinger equation, in order to investigate the properties of matter from first principles. While DFT can give the exact ground state electronic density of a system, in practice, an approximation is required for the many-body effects contained in the exchange-correlation functional. The accuracy of calculations performed using DFT is strongly related to the choice of approximation. In this thesis we will investigate and build upon a fully non-local approach to modeling exchange-correlation in the form of the weighted density approximation (WDA). Central to the WDA is the model function chosen for the coupling-constant averaged pair-correlation function (PCF). We show that a model PCF can be selected from a set to give excellent bulk properties for a particular system. However, this model is not necessarily transferable to other systems and there is no method of selecting an appropriate model from this set a priori. We suggest that the model PCF can be improved systematically by satisfying known physical constraints. One such constraint is the Kimball cusp condition, which we include in our model and implement. We demonstrate that surfaces are systems that require a non-local treatment of exchange-correlation by applying the WDA to metal surfaces and investigate the dissociative adsorption of H2 on the Cu(100) surface. A new framework for a model PCF with spin resolution is developed, providing a route for more physical constraints to be satisfied within a weighted spin density approximation (WSDA). A simple model is suggested and implemented and comparisons are made to the coupling-constant averaged PCF in the homogeneous electron gas. We then apply a selection of our new models to a number of materials and show that our model for the WSDA gives improved band gaps over the local density approximation. Application of the WSDA to spin polarised materials reveals shortcomings in our simple model. We then suggest further refinements to our implementation of the WSDA. It is expected that the inclusion of additional physical constraints will systematically improve results given in a weighted-density based approximation to exchange-correlation.

In this paper, we prove a weighted atom inequality for Hermite expansions and want to obtain similar type of inequalities for 1-dimensional Hermite expansions of weighted Hardy's inequalities.

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In a distributed system many underlying nodes or processes work in tandem to come up with a solution to a given problem. In this report, we are concerned with distributed systems where each node is given the same problem, and the system uses the solutions provided by the nodes to formulate the answer. In our case, the problem is a simple question with two possible answers, and only one answer is correct. The system is asked the question at the beginning of a round. Once the system answers the question, the round is over, and the system is given the correct answer, then another round is started. To answer the question, the system uses the answers from each node, and based on the weight of the individual nodes, it decides on its answer. In this report, we experiment with multiple ways to update the weights of the underlying nodes, and aim to study the impact of certain limitations and parameters imposed on the system; such as the maximum accuracy of the underlying nodes and the number of underlying nodes. / text

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In this thesis we use a diffeo-geometric framework based on manifolds hat are locally modeled on ``convenient'' vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold M, we construct a weak symplectic structure on each leaf I_w of a foliation of the space of compact oriented isotropic submanifolds in M equipped with top degree forms of total measure 1. These forms are called weightings and such manifolds are said to be weighted. We show that this symplectic structure on the particular leaves consisting of weighted Lagrangians is equivalent to a heuristic weak symplectic structure of Weinstein. When the weightings are positive, these symplectic spaces are symplectomorphic to reductions of a weak symplectic structure of Donaldson on the space of embeddings of a fixed compact oriented manifold into M. When M is compact, by generalizing a moment map of Weinstein we construct a symplectomorphism of each leaf I_w consisting of positive weighted isotropics onto a coadjoint orbit of the group of Hamiltonian symplectomorphisms of M equipped with the Kirillov-Kostant-Souriau symplectic structure. After defining notions of Poisson algebras and Poisson manifolds, we prove that each space I_w can also be identified with a symplectic leaf of a Poisson structure. Finally, we discuss a kinematic description of spaces of weighted submanifolds.

Let T be the unit circle,(mu) be a Borel probability measure on T and (phi) be a bounded Lebesgue measurable function on T. in this paper we consider the weighted composition operator W(phi) on L^2(T,mu) defined by W(phi)f:=(phi)*(f(circle)(tau)), f in L^2(T), where (tau) is the map (tau)(z)=z^2, z in T. We will study the von Neumann-Wold decomposition of W(phi) when W(phi) is an isometry and (mu)<< m,where m is the normalized Lebesgue measure on T.

Sleep problems are common in children with ASD. Despite this, evidence for interventions, particularly alternative approaches such as weighted blankets is limited. Aims. The aims of the study were to examine weighted blankets for: (1) their effectiveness as an intervention for sleep problems; (2) their impact on sleep state organisation; (3) their impact on night-time movement; (4) the sensory characteristics of children responding to the intervention; (5) parent’s perceptions of the intervention; (6) behavioural interventions as a second intervention if required. Method. Using a non-concurrent multiple baseline design, five children with autism aged between 7 to 13 years received a weighted blanket intervention with the option of a secondary behavioural intervention. Results. One participant rejected the weighted blanket outright. Four participants showed no substantial improvements in sleep problems or changes in sleep state organisational. Despite this, parents of four participants still perceived the weighted blanket to be moderately effective. The sensory profiles of participants were not related to their response to the intervention. Participants’ night-time movements were not found to be suppressed by the weighted blankets. Two participants went on to receive a behavioural intervention, one was successful and the other withdrew from the study before implementing the strategies. One other participant’s sleep difficulties resolved over time without further intervention. Implications. The current study neither supported weighted blankets as an intervention for sleep problems in children with autism nor supported contentions of its effectiveness for children with sensation seeking behaviours.

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