A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating bath, where its horizontal `walking' motion is induced by repeated impacts with its accompanying Faraday wave field. This hydrodynamic pilot-wave system exhibits many features that were previously thought to be exclusive to the quantum realm, including quantized dynamics and emergent wavelike statistics. We develop a discrete-time iterative map to analyse the pilot-wave dynamics in a number of settings, employing a sophisticated fluid model to capture the intricacies of the Faraday wave evolution neglected by previous works. We first study the stability of bouncing and walking dynamics, and elucidate further features of the droplet's wave-induced added mass. We also explore the periodic and chaotic dynamics arising when the droplet is subjected to a harmonic potential, a Coriolis force, or the interaction with a seconddroplet. Finally, we modify our fluid model to account for interactions with submerged boundaries, allowing us to rationalise the pilot-wave dynamics in a circular corral.

Abstract not available

Systems involving quantisation arise in many areas of engineering, especially when digital implementations are involved. In this thesis we consider different aspects of quantisation in feedback control systems. We study two topics of interest: (a) quantisers that quadratically stabilise a given system and are efficient in the use of their quantisation levels and (b) the derivation of ultimate bounds for perturbed systems, especially when the perturbations arise from the use of quantisers. In the first part of the thesis we address problem (a) above. We consider quadratic stabilisation of discrete-time multiple-input systems by means of quantised static feedback and we measure the efficiency of a quantiser via the concept of quantisation density. Intuitively, the lower the density of a quantiser is, the more separated its quantisation levels are. We thus deal with the problem of optimising density over all quantisers that quadratically stabilise a given system with respect to a given control Lyapunov function. Most of the available results on this problem treat single-input systems, and the ones that deal with the multiple-input case consider only two-input systems. In this thesis, we derive several new results for multiple-input systems and also provide an alternative approach to deal with the single-input case. Our new results for multiple-input systems include the derivation of the structure of optimal quantisers and the explicit design of multivariable quantisers with finite density that are able to quadratically stabilise systems having an arbitrary number of inputs. For single-input systems, we provide an alternative approach to the analysis and design of optimal quantisers by establishing a link between the separation of the quantisation levels of a quantiser and the size of its quantisation regions. In the second part of the thesis we address problem (b) above. In the presence of perturbations, asymptotic stabilisation may not be possible. However, there may exist a bounded region that contains the equilibrium point and has the property that the system trajectories converge to this bounded region. When this bounded region exists, we say that the system trajectories are ultimately bounded, and that this bounded region is an ultimate bound for the system. The size of the ultimate bound quantifies the performance of the system in steady state. Hence, it is important to derive ultimate bounds that are as tight as possible. This part of the thesis addresses the problem of ultimate bound computation in settings involving several scalar quantisers, each having different features. We consider each quantised variable in the system to be a perturbed copy of the corresponding unquantised variable. This turns the original quantised system into a perturbed system, where the perturbation has a natural \emph{componentwise} bound. Moreover, according to the type of quantiser employed, the perturbation bound may depend on the system state. Typical methods to estimate ultimate bounds are based on the use of Lyapunov functions and usually require a bound on the norm of the perturbation. Applying these methods in the setting considered here may disregard important information on the structure of the perturbation bound. We therefore derive ultimate bounds on the system states that explicitly take account of the componentwise structure of the perturbation bound. The ultimate bounds derived also have a componentwise form, and can be systematically computed without having to, e.g. select a suitable Lyapunov function for the system. The results of this part of the thesis, though motivated by quantised systems, apply to more general perturbations, not necessarily arising from quantisation. / PhD Doctorate

Représentations des groupes de Lie conformes et quantification des espaces symétriques

The image comparison operation ??sessing how well one image matches another ??rms a critical component of many image analysis systems and models of human visual processing. Two norms used commonly for this purpose are L1 and L2, which are specific instances of the Minkowski metric. However, there is often not a principled reason for selecting one norm over the other. One way to address this problem is by examining whether one metric better captures the perceptual notion of image similarity than the other. With this goal, we examined perceptual preferences for images retrieved on the basis of the L1 versus the L2 norm. These images were either small fragments without recognizable content, or larger patterns with recognizable content created via vector quantization. In both conditions the subjects showed a consistent preference for images matched using the L1 metric. These results suggest that, in the domain of natural images of the kind we have used, the L1 metric may better capture human notions of image similarity.

In this thesis we extend Lerman’s cutting construction to spin-c structures. Every spin-c structure on an even-dimensional Riemannian manifold gives rise to a Dirac operator D+ acting on sections of the associated spinor bundle. The spin-c quantization of a spin-c manifold is defined to be ker(D+)−coker(D+). It is a virtual vector space, and in the presence of a Lie group action, it is a virtual representation. In 2004, Guillemin et al defined signature quantization and showed that it is additive under cutting. We prove that the spin-c quantization of an S^1-manifold is also additive under cutting. Our proof uses the method of localization, i.e., we express the spin-c quantization of a manifold in terms of local data near connected components of the fixed point set. For a symplectic manifold (M,ω), a spin-c prequantization is a spin-c structure together with a connection compatible with ω. We explain how one can cut a spin-c prequantization and show that the choice of a spin-c structure on the complex plane (which is part of the cutting process) must be compatible with the moment level set along which the cutting is performed. Finally, we prove that the spin-c and metaplectic-c groups satisfy a universal property: Every structure that makes the construction of a spinor bundle possible must factor uniquely through a spin-c structure in the Riemannian case, or through a metaplectic-c structure in the symplectic case.

In this thesis we extend Lerman’s cutting construction to spin-c structures. Every spin-c structure on an even-dimensional Riemannian manifold gives rise to a Dirac operator D+ acting on sections of the associated spinor bundle. The spin-c quantization of a spin-c manifold is defined to be ker(D+)−coker(D+). It is a virtual vector space, and in the presence of a Lie group action, it is a virtual representation. In 2004, Guillemin et al defined signature quantization and showed that it is additive under cutting. We prove that the spin-c quantization of an S^1-manifold is also additive under cutting. Our proof uses the method of localization, i.e., we express the spin-c quantization of a manifold in terms of local data near connected components of the fixed point set. For a symplectic manifold (M,ω), a spin-c prequantization is a spin-c structure together with a connection compatible with ω. We explain how one can cut a spin-c prequantization and show that the choice of a spin-c structure on the complex plane (which is part of the cutting process) must be compatible with the moment level set along which the cutting is performed. Finally, we prove that the spin-c and metaplectic-c groups satisfy a universal property: Every structure that makes the construction of a spinor bundle possible must factor uniquely through a spin-c structure in the Riemannian case, or through a metaplectic-c structure in the symplectic case.

SAS has become an increasingly important public-health problem in recent years. It can adversely affect neurocognitive, cardiovascular, respiratory diseases and can also cause behavior disorder. Since up to 90% of these cases are obstructive sleep apnea (OSA), therefore, the study of how to diagnose, detect and treat OSA is becoming a significant issue, academically and medically. Polysomnography (PSG) can monitor the OSA with relatively fewer invasive techniques. However, PSG-based sleep studies are expansive and time-consuming because they require overnight evaluation in sleep laboratories with dedicated systems and attending personnel. This work develops a flow rate based detection method for apneas. In particular, via signal processing, feature extraction and neural network, this thesis introduces a flow rate based detective system. The goal is to detect OSA with less time and reduced financial costs.

No description available.

Designing the Nearest Neighbor Classifiers via the VQ Method