Missing data is one of the challenges we are facing today in modeling valid statistical models. It reduces the representativeness of the data samples. Hence, population estimates, and model parameters estimated from such data are likely to be biased. However, the missing data problem is an area under study, and alternative better statistical procedures have been presented to mitigate its shortcomings. In this paper, we review causes of missing data, and various methods of handling missing data. Our main focus is evaluating various multiple imputation (MI) methods from the multiple imputation of chained equation (MICE) package in the statistical software R. We assess how these MI methods perform with different percentages of missing data. A multiple regression model was fit on the imputed data sets and the complete data set. Statistical comparisons of the regression coefficients are made between the models using the imputed data and the complete data.

Transcranial routing of signal (TCROS) was accomplished using completely-in-the-canal (CIC) hearing aids in 5 profoundly unilaterally hearing-impaired individuals. The functional gain realized by the participants far exceeded the gain predicted by measuring the acoustic output and real ear aided response of the hearing aids. The difference between predicted and functional gain increased with signal frequency and was attributed at least in part to mechanical vibration of the hearing aid in the external canal. Implications for fitting unilateral hearing loss using TCROS amplifying systems are discussed.

<p>The effects of open system evolution on single particle interferometry, quantum computation, and the adiabatic approximation are investigated.</p><p>Single particle interferometry: Three concepts concerning completely positive maps (CPMs) and trace preserving CPMs (channels), named subspace preserving (SP) CPMs, subspace local channels, and gluing of CPMs, are introduced. SP channels preserve probability weights on given orthogonal sum decompositions of the Hilbert space of a quantum system. Subspace locality determines what channels act locally with respect to such decompositions. Gluings are the possible total channels obtainable if two evolution devices, characterized by channels, act jointly on a superposition of a particle in their inputs. It is shown that gluings are not uniquely determined by the two channels. We determine all possible interference patterns in single particle interferometry for given channels acting in the interferometer paths. It is shown that the standard interferometric setup cannot distinguish all gluings, but a generalized setup can.</p><p>Quantum computing: The robustness of local and global adiabatic quantum search subject to decoherence in the instantaneous eigenbasis of the search Hamiltonian, is examined. In both the global and local search case the asymptotic time-complexity of the ideal closed case is preserved, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, it is shown that the local adiabatic quantum search performs as the classical search with scaling N, and that the global search scales like N^3/2 , where N is the list length. We consider success probabilities p<1 and prove bounds on the run-time with the same scaling as in the conditions for the p → 1 limit.</p><p>Adiabatic evolution: We generalize the adiabatic approximation to the case of open quantum systems in the joint limit of slow change and weak open system disturbances. </p>

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The effects of open system evolution on single particle interferometry, quantum computation, and the adiabatic approximation are investigated. Single particle interferometry: Three concepts concerning completely positive maps (CPMs) and trace preserving CPMs (channels), named subspace preserving (SP) CPMs, subspace local channels, and gluing of CPMs, are introduced. SP channels preserve probability weights on given orthogonal sum decompositions of the Hilbert space of a quantum system. Subspace locality determines what channels act locally with respect to such decompositions. Gluings are the possible total channels obtainable if two evolution devices, characterized by channels, act jointly on a superposition of a particle in their inputs. It is shown that gluings are not uniquely determined by the two channels. We determine all possible interference patterns in single particle interferometry for given channels acting in the interferometer paths. It is shown that the standard interferometric setup cannot distinguish all gluings, but a generalized setup can. Quantum computing: The robustness of local and global adiabatic quantum search subject to decoherence in the instantaneous eigenbasis of the search Hamiltonian, is examined. In both the global and local search case the asymptotic time-complexity of the ideal closed case is preserved, as long as the Hamiltonian dynamics is present. In the case of pure decoherence, where the environment monitors the search Hamiltonian, it is shown that the local adiabatic quantum search performs as the classical search with scaling N, and that the global search scales like N3/2 , where N is the list length. We consider success probabilities p&lt;1 and prove bounds on the run-time with the same scaling as in the conditions for the p → 1 limit. Adiabatic evolution: We generalize the adiabatic approximation to the case of open quantum systems in the joint limit of slow change and weak open system disturbances.

What is in common between the Kepler problem, a Hydrogen atom and a rotating black- hole? These systems are described by different physical theories, but much information about them can be obtained by separating an appropriate Hamilton-Jacobi equation. The separation of variables of the Hamilton-Jacobi equation is an old but still powerful tool for obtaining exact solutions. The goal of this thesis is to present the theory and application of a certain type of conformal Killing tensor (hereafter called concircular tensor) to the separation of variables problem. The application is to spaces of constant curvature, with special attention to spaces with Euclidean and Lorentzian signatures. The theory includes the general applicability of concircular tensors to the separation of variables problem and the application of warped products to studying Killing tensors in general and separable coordinates in particular. Our first main result shows how to use these tensors to construct a special class of separable coordinates (hereafter called Kalnins-Eisenhart-Miller (KEM) coordinates) on a given space. Conversely, the second result generalizes the Kalnins-Miller classification to show that any orthogonal separable coordinates in a space of constant curvature are KEM coordinates. A closely related recursive algorithm is defined which allows one to intrinsically (coordinate independently) search for KEM coordinates which separate a given (natural) Hamilton-Jacobi equation. This algorithm is exhaustive in spaces of constant curvature. Finally, sufficient details are worked out, so that one can apply these procedures in spaces of constant curvature using only (linear) algebraic operations. As an example, we apply the theory to study the separability of the Calogero-Moser system.

Cette thèse porte sur des propriétés d'approximation généralisant la moyennabilité pour les groupes localement compacts. Ces propriétés sont définies à partir des multiplicateurs de certaines algèbres associés aux groupes. La première partie est consacrée à l'étude de la propriété p-AP, qui est une extension de la AP de Haagerup et Kraus au cadre des opérateurs sur les espaces Lp. Le résultat principal dit que les groupes de Lie simples de rang supérieur et de centre fini ne satisfont p-AP pour aucun p entre 1 et l'infini. La deuxième partie se concentre sur les multiplicateurs de Schur radiaux sur les graphes. L'étude de ces objets est motivée par les liens avec les actions de groupes discrets et la moyennabilité faible. Les trois résultats principaux donnent des conditions nécessaires et suffisantes pour qu'une fonction sur les nombres naturels définisse un multiplicateur radial sur des différentes classes de graphes généralisant les arbres. Plus précisément, les classes de graphes étudiées sont les produits d'arbres, les produits de graphes hyperboliques et les complexes cubiques CAT(0) de dimension finie. / This thesis focusses on some approximation properties which generalise amenability for locally compact groups. These properties are defined by means of multipliers of certain algebras associated to the groups. The first part is devoted to the study of the p-AP, which is an extension of the AP of Haagerup and Kraus to the context of operators on Lp spaces. The main result asserts that simple Lie groups of higher rank and finite centre do not satisfy p-AP for any p between 1 and infinity. The second part concentrates on radial Schur multipliers on graphs. The study of these objects is motivated by some connections with actions of discrete groups and weak amenability. The three main results give necessary and sufficient conditions for a function of the natural numbers to define a radial multiplier on different classes of graphs generalising trees. More precisely, the classes of graphs considered here are products of trees, products hyperbolic graphs and finite dimensional CAT(0) cube complexes.

This thesis provides an introduction to methods for handling missing data. A thorough review of earlier methods and the development of the field of missing data is provided. The thesis present the methods suggested in today’s literature, multiple imputation and maximum likelihood estimation. A simulation study is performed to see if there are circumstances in small samples when any of the two methods are to be preferred. To show the importance of handling missing data, multiple imputation and maximum likelihood are compared to listwise deletion. The results from the simulation study does not show any crucial differences between multiple imputation and maximum likelihood when it comes to point estimates. Some differences are seen in the estimation of the confidence intervals, talking in favour of multiple imputation. The difference is decreasing with an increasing sample size and more studies are needed to draw definite conclusions. Further, the results shows that listwise deletion lead to biased estimations under a missing at random mechanism. The methods are also applied to a real dataset, the Swedish enrollment registry, to show how the methods work in a practical application.

Completely locked-in state (CLIS) patients are unable to speak and have lost all muscle movement. From the external view, the internal brain activity of such patients cannot be easily perceived, but CLIS patients are considered to still be conscious and cognitively active. Detecting the current state of consciousness of CLIS patients is non-trivial, and it is difficult to ascertain whether CLIS patients are conscious or not. Thus, it is important to find alternative ways to re-establish communication with these patients during periods of awareness, and one such alternative is through a brain–computer interface (BCI). In this study, multiscale-based methods (multiscale sample entropy, multiscale permutation entropy and multiscale Poincaré plots) were applied to analyze electrocorticogram signals from a CLIS patient to detect the underlying consciousness level. Results from these different methods converge to a specific period of awareness of the CLIS patient in question, coinciding with the period during which the CLIS patient is recorded to have communicated with an experimenter. The aim of the investigation is to propose a methodology that could be used to create reliable communication with CLIS patients.

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