Aerodynamic devices, such as wings, are used in higher levels of motorsport (Formula-1 etc.) to increase the contact force between the road and tyres (i.e. to generate downforce). This in turn increases the performance envelope of the race car. However the extra downforce increases aerodynamic drag which (apart from when braking) is generally detrimental to lap-times. The drag acts to slow the vehicle, and hinders the effect of available drive power and reduces fuel economy. Wings, in automotive use, are not constrained by the same parameters as aircraft, and thus higher angles of attack can be safely reached, although at a higher cost in drag. Variable geometry aerodynamic devices have been used in many forms of motorsport in the past offering the ability to change the relative values of downforce and drag. These have invariably been banned, generally due to safety reasons. The use of active aerodynamics is currently legal in both Formula SAE (engineering compet ition for university students to design, build and race an open-wheel race car) and production vehicles. A number of passenger car companies are beginning to incorporate active aerodynamic devices in their designs. In this research the effect of ground proximity on the lift, drag and moment coefficients of inverted, two-dimensional aerofoils was investigated. The purpose of the study was to examine the effect ground proximity on aerofoils post stall, in an effort to evaluate the use of active aerodynamics to increase the performance of a race car. The aerofoils were tested at angles of attack ranging from 0° - 135°. The tests were performed at a Reynolds number of 2.16 x 105 based on chord length. Forces were calculated via the use of pressure taps along the centreline of the aerofoils. The RMIT Industrial Wind Tunnel (IWT) was used for the testing. Normally 3m wide and 2m high, an extra contraction was installed and the section was reduced to form a width of 295mm. The wing was mounted between walls to simulate 2-D flow. The IWT was chosen as it would allow enough height to reduce blockage effect caused by the aerofoils when at high angles of incidence. The walls of the tunnel were pressure tapped to allow monitoring of the pressure gradient along the tunnel. The results show a delay in the stall of the aerofoils tested with reduced ground clearance. Two of the aerofoils tested showed a decrease in Cl with decreasing ground clearance; the third showed an increase. The Cd of the aerofoils post-stall decreased with reduced ground clearance. Decreasing ground clearance was found to reduce pitch moment variation of the aerofoils with varied angle of attack. The results were used in a simulation of a typical Formula SAE race car.

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For elliptic problems on manifolds with edges, we construct index formulas in form of a sum of homotopy invariant contributions of the strata (the interior of the manifold and the edge). Both terms are the indices of elliptic operators, one of which acts in spaces of sections of finite-dimensional vector bundles on a compact closed manifold and the other in spaces of sections of infinite-dimensional vector bundles over the edge.

Reciprocal processes, whose concept can be traced back to E. Schrödinger, form a class of stochastic processes constructed as mixture of bridges, that satisfy a time Markov field property. We discuss here a new unifying approach to characterize several types of reciprocal processes via duality formulae on path spaces: The case of reciprocal processes with continuous paths associated to Brownian diffusions and the case of pure jump reciprocal processes associated to counting processes are treated. This presentation is based on joint works with M. Thieullen, R. Murr and C. Léonard.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

Communication networks are vulnerable to natural as well as man-made disasters. The geographical layout of the network influences the impact of these disasters. It is therefore, necessary to identify areas that could be most affected by a disaster and redesign those parts of the network so that the impact of a disaster has least effect on them. In this work, we assume that disasters which have a circular impact on the network. The work presents two new algorithms, namely the WHF-PG algorithm and the WHF-NPG algorithm, designed to solve the problem of finding the locations of disasters that would have the maximum disruptive effect on the communication infrastructure in terms of capacity.

Around 1990, Arthur proved a local (ordinary) trace formula for real or p-adic connected reductive groups. The local trace formula is a powerful tool in the local harmonic analysis of reductive groups. One of the aims of this thesis is to establish a local twisted trace formula for certain non-connected reductive groups, which is a twisted version of Arthur’s local trace formula. As an application of the local twisted trace formula, we will prove some twisted orthogonality relations, which are generalizations of Arthur’s results about orthogonality relations for tempered elliptic characters. To establish these relations, we will also give a classification of twisted elliptic representations.

In a recent paper with N. Tarkhanov, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.