At the dawn of the Large Hadron Collider era, the Brout-Englert-Higgs mechanism remains the most appealing theoretical explanation of the electroweak symmetry breaking, despite the fact that the associated fundamental scalar boson has escaped any direct detection attempt. In this thesis, we consider a particular extension of the minimal Brout-Englert-Higgs scalar sector implemented in the Standard Model of strong and electroweak interactions. This extension, which is a specific, "twisted", realisation of the generic two-Higgs-doublet model, is motivated by a relative phase in the definition of the phenomenologically successful CP and custodial symmetries. Considering extensively various theoretical, indirect and direct constraints, this model appears as a viable alternative to more conventional scenarios like supersymmetric models, and gives grounds to largely unexplored possibilities of exotic scalar signatures at present and future collider experiments.

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.

We give an explicit construction of the trace on the algebra of quantum observables on a symplectic orbifold and propose an index formula.

The master thesis is devoted to an analysis of equilibrium or reaction-function models in illiquidity markets of derivatives. The main equation is a nonlinear equation which is a perturbation of Black-Scholes model. By using analytical methods we study invariant and scaling properties for the considered model.

Perceptual grouping is essential to manage the complexity of real world scenes. We explore bottom-up grouping at three different levels. Starting from low-level grouping, we propose a novel method for oversegmenting an image into compact superpixels, reducing the complexity of many high-level tasks. Unlike most low-level segmentation techniques, our geometric flow formulation enables us to impose additional compactness constraints, resulting in a fast method with minimal undersegmentation. Our subsequent work utilizes compact superpixels to detect two important mid-level shape regularities, closure and symmetry. Unlike the majority of closure detection approaches, we transform the closure detection problem into one of finding a subset of superpixels whose collective boundary has strong edge support in the image. Building on superpixels, we define a closure cost which is a ratio of a novel learned boundary gap measure to area, and show how it can be globally minimized to recover a small set of promising shape hypotheses. In our final contribution, motivated by the success of shape skeletons, we recover and group symmetric parts without assuming prior figure-ground segmentation. Further exploiting superpixel compactness, superpixels are this time used as an approximation to deformable maximal discs that comprise a medial axis. A learned measure of affinity between neighboring superpixels and between symmetric parts enables the purely bottom-up recovery of a skeleton-like structure, facilitating indexing and generic object recognition in complex real images.

The dynamics of a single underwater vehicle in an ideal irrotational fluid may be modeled by a Lagrangian system with configuration space the Euclidean group. If hydrodynamic coupling is ignored then two coupled vehicles may be modeled by the direct product of two single-vehicle systems. We consider this system in the case that the vehicles are coupled mechanically, with an ideal spherically symmetric joint, finding all of the relative equilibria. We demonstrate that there are relative equilibria in certain novel momentum-generator regimes identified by Patrick et.al. "<i>Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods</i>", Arch. Rational Mech. Anal., 174:301--344, 2004.

Sound source localization depends on the auditory system to identify, recognize, and segregate elements of salient sources over distracting noise. My research investigates sensory mechanisms involved in these auditory processing tasks of an insect hearing specialist, to isolate individual sound sources of interest over noise. I first developed quantitative methods to determine signal features that the acoustic parasitoid fly Ormia ochracea (Diptera: Tachinidae) evaluate for host cricket song recognition. With flies subjected to a no-choice paradigm and forced to track a switch in the broadcast location of test songs, I describe several response features (distance, steering velocity, and angular orientation) that vary with song pulse rate preferences. I incorporate these response measures in a phonotaxis performance index that is sensitive to capturing response variation that may underlie song recognition. I demonstrate that Floridian O. ochracea exhibit phonotaxis to a combination of pulse durations and interpulse intervals that combine to a range of accepted pulse periods. Under complex acoustic conditions of multiple coherent cricket songs that overlap in time and space, O. ochracea may experience a phantom source illusion and localize a direction between actual source locations. By varying the temporal overlap between competing sources, I demonstrate that O. ochracea are able to resolve this illusion via the precedence effect: exploitation of small time differences between competing sources to selectively localize the leading over lagging sources. An increase in spatial separation between cricket song and masking noise does not reduce song detection thresholds nor improve song localization accuracy. Instead, walking responses are diverted away from both song and noise. My findings support the idea that the ears of O. ochracea function as bilateral symmetry detectors to balance sound intensity, sound arrive time differences, and temporal pattern input to both sides of the auditory system. Asymmetric acoustic input result in corrective turning behaviour to re-establish balance for successful source localization.

Perceptual grouping is essential to manage the complexity of real world scenes. We explore bottom-up grouping at three different levels. Starting from low-level grouping, we propose a novel method for oversegmenting an image into compact superpixels, reducing the complexity of many high-level tasks. Unlike most low-level segmentation techniques, our geometric flow formulation enables us to impose additional compactness constraints, resulting in a fast method with minimal undersegmentation. Our subsequent work utilizes compact superpixels to detect two important mid-level shape regularities, closure and symmetry. Unlike the majority of closure detection approaches, we transform the closure detection problem into one of finding a subset of superpixels whose collective boundary has strong edge support in the image. Building on superpixels, we define a closure cost which is a ratio of a novel learned boundary gap measure to area, and show how it can be globally minimized to recover a small set of promising shape hypotheses. In our final contribution, motivated by the success of shape skeletons, we recover and group symmetric parts without assuming prior figure-ground segmentation. Further exploiting superpixel compactness, superpixels are this time used as an approximation to deformable maximal discs that comprise a medial axis. A learned measure of affinity between neighboring superpixels and between symmetric parts enables the purely bottom-up recovery of a skeleton-like structure, facilitating indexing and generic object recognition in complex real images.

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Symmetry has been a very useful tool to researchers in various scientific fields. At its most basic, symmetry refers to the invariance of an object to some transformation, or set of transformations. Usually one searches for, and uses information concerning an existing symmetry within given data, structure or concept to somehow improve algorithm performance or compress the search space. This thesis examines the effects of imposing or inducing symmetry on a search space. That is, the question being asked is whether only existing symmetries can be useful, or whether changing reference to an intuition-based definition of symmetry over the evaluation function can also be of use. Within the context of optimization, symmetry induction as defined in this thesis will have the effect of equating the evaluation of a set of given objects. Group theory is employed to explore possible symmetrical structures inherent in a search space. Additionally, conditions when the search space can have a symmetry induced on it are examined. The idea of a neighborhood structure then leads to the idea of opposition-based computing which aims to induce a symmetry of the evaluation function. In this context, the search space can be seen as having a symmetry imposed on it. To be useful, it is shown that an opposite map must be defined such that it equates elements of the search space which have a relatively large difference in their respective evaluations. Using this idea a general framework for employing opposition-based ideas is proposed. To show the efficacy of these ideas, the framework is applied to popular computational intelligence algorithms within the areas of Monte Carlo optimization, estimation of distribution and neural network learning. The first example application focuses on simulated annealing, a popular Monte Carlo optimization algorithm. At a given iteration, symmetry is induced on the system by considering opposite neighbors. Using this technique, a temporary symmetry over the neighborhood region is induced. This simple algorithm is benchmarked using common real optimization problems and compared against traditional simulated annealing as well as a randomized version. The results highlight improvements in accuracy, reliability and convergence rate. An application to image thresholding further confirms the results. Another example application, population-based incremental learning, is rooted in estimation of distribution algorithms. A major problem with these techniques is a rapid loss of diversity within the samples after a relatively low number of iterations. The opposite sample is introduced as a remedy to this problem. After proving an increased diversity, a new probability update procedure is designed. This opposition-based version of the algorithm is benchmarked using common binary optimization problems which have characteristics of deceptivity and attractive basins characteristic of difficult real world problems. Experiments reveal improvements in diversity, accuracy, reliability and convergence rate over the traditional approach. Ten instances of the traveling salesman problem and six image thresholding problems are used to further highlight the improvements. Finally, gradient-based learning for feedforward neural networks is improved using opposition-based ideas. The opposite transfer function is presented as a simple adaptive neuron which easily allows for efficiently jumping in weight space. It is shown that each possible opposite network represents a unique input-output mapping, each having an associated effect on the numerical conditioning of the network. Experiments confirm the potential of opposite networks during pre- and early training stages. A heuristic for efficiently selecting one opposite network per epoch is presented. Benchmarking focuses on common classification problems and reveals improvements in accuracy, reliability, convergence rate and generalization ability over common backpropagation variants. To further show the potential, the heuristic is applied to resilient propagation where similar improvements are also found.