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Conformal methods have proven to be very useful in the analysis global properties and stability of vacuum spacetimes in general relativity. These methods transform the physical spacetime into a different Lorentzian manifold known as the unphysical spacetime where the ideal points at infinity are located at a finite position. This thesis makes use of conformal methods and applies them to various problems involving trace-free matter models. In particular, it makes progress towards the understanding of the evolution of unphysical spacetimes perturbed by trace-free matter as well as the behaviour of the the matter itself. To this end, evolution equations (wave equations) are derived and analyzed for both the unphysical spacetime and the matter. To investigate the relation between solutions of these wave equations to the Einstein field equations, a suitable system of subsidiary evolution equations is also derived. Furthermore, this thesis looks in detail at the behaviour of an unphysical spacetime coupled to the simplest matter trace free model: the confomally invariant scalar field. Finally, the system of conformal wave equations is used to show that the deSitter spacetime is non-linearly stable under perturbations by trace-free matter.

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Embedded below the radome of a missile, existing RF-seekers use a mechanical rotating antenna to steer the radiating beam in the direction of a target. Latest research is looking at replacing the mechanical antenna components of the RF seeker with a novel 3D conformal antenna array that can steer the beam electronically. 3D antennas may oer signicant advantages, such as faster beamsteering and better coverage but, at the same time, introduce new challenges resulting from a much more complex radiation pattern than that of 2D antennas. Thanks to the mechanical system removal, the new RF-seeker has a wider available space for the design of a new 3D conformal antenna. To take best benets of this space, dierent array shapes are studied, hence the impact of the position, orientation and conformation of the elements is assessed on the antenna performance in terms of directivity, ellipticity and polarisation. To facilitate this study of 3D conformal arrays, a Matlab program has been developed to compute the polarisation pattern of a given array in all directions. One of the task of the RF-seeker consists in estimating the position of a given target to correct the missile trajectory accordingly. Thus, the impact of the array shape on the error between the measured direction of arrival of the target echo and its true value is addressed. The Cramer-Rao lower bound is used to evaluate the theoretical minimum error. The model assumes that each element receives independently and allows therefore to analyse the potential of active 3D conformal arrays. Finally, the phase monopulse estimator is studied for 3D conformal arrays whose quadrants do not have the same characteristics. A new estimator more adapted to non-identical quadrants is also proposed.

Optimal missile guidance and flight performance require accurate and continuously updated in-flight coordinate data. The Global Positioning System (GPS) is used for this positional awareness. However, due to missile rotation and orientation variations during flight, GPS signal reception using traditional antennas may be intermittent. To remain cost competitive, Stellar Exploration Inc. is developing a low-cost omnidirectional GPS antenna for guided missile prototypes. In this thesis, existing products and design techniques are examined, design constraints for supersonic missile applications are investigated, and corresponding performance goals are established. A conformal microstrip patch antenna is developed and simulated in Agilent’s Advanced Design System (ADS). The resulting antenna is constructed and characterized. Prototype testing verifies that the antenna maintains GPS signal lock regardless of orientation. The final cost is significantly lower than existing conformal products. A second revision investigates enhanced modeling, dimensional reductions (via increased dielectric constant), and radome construction. Performance is compared to first revision antenna results and differences are examined. Suggestions for further revisions are discussed.

This dissertation is in the area of complex dynamics, more specifically focused on the iteration of rational functions. Given a well-chosen family of rational functions, parameterized by a complex parameter, we are especially interested in regularity properties of the Hausdorff dimension of Julia sets of these polynomials considered as a function of the parameters. In this dissertation I deal with a family of polynomials of degree at least 3 depending in a holomorphic way on a parameter, focusing on the point where the dynamics and topology of the polynomials drastically change. In such a context proving continuity is quite challenging while real analyticity will most likely break. Our approach will, on the one hand, build on the existing methods of proving continuity of Hausdorff dimension, primarily based on proving continuity, in the weak* topology of measures on the Riemann sphere, of canonical conformal measures, but will also require methods which, up to my best knowledge, have not been implemented anywhere yet. Our main result gives a surprising example where the Hausdorff dimension of the Julia set is continuous in the parameter, but where the Julia set itself is not.

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