This dissertation originates from a classical result that the lp-stability of the convolution operator associated with a summable sequence are equivalent to each other for different p . This dissertation is motivated by the recent result by C. E. Shin and Q. Sun (Journal ofFunctional Analysis, 256(2009), 2417-2439), where the lp-stability of infinite matrices in the Gohberg-Baskakov-Sjostrand class are proved to be equivalent to each other for different p. In the dissertation, for an infinite matrix having certain off-diagonal decay, its weighted lp-stability for different p are proved to be equivalent to each other and hence a result by Shin and Sun is generalized.

A new method of realizing linear, time-invariant digital filters is developed and demonstrated. The result is based on the convolution integral. It is assumed that the specifications of the filter are known and from these, an appropriate analog filter is chosen. The properties of this filter are then retained by digital filter after transformation. The behaviour of lowpass, highpass bandpass and bandstop digital filters is investigated in both the frequency and time domains, for both cascade and parallel structure is superior for lowpass and bandpass digital filters, and that the cascade structure is superior for high pass and bandstop digital filters. / Thesis / Master of Engineering (ME)

Many real-world engineering problems require large amounts of data in order to accurately model and predict outcomes. However, this data is often noisy, sampled and discontinuous, making the data difficult to process and giving rise to incorrect models. In order to address this issue, different interpolation techniques are commonly used to make the data continuous. This can then followed by a filtering process in order to reduce noise and further reduce discontinuities. In this report, our approach to filtering is the use of convolution kernels, which smooths out the data. By doing so, a better visual representation of the limited data available can be obtained. For instance, in the specific case of studying streamlines and vortices, filtering techniques have been used to produce more realistic plots. While the use of filters can be beneficial, it is important to note that the choice of filter and its parameters can greatly impact the results obtained. In particular, we found that, for the filters we studied, applying these to analytical functions can actually increase the error. On the other hand, when filters are applied to discontinuous functions, they can improve the accuracy of the data. Overall, when analyzing stream functions with filters, significant improvements can be seen in the quality of the data. This underscores the importance of careful selection and application of filtering techniques in engineering problems that involve large amounts of noisy and discontinuous data.

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No / In the last few years, there has been growing interest in near-real-time solar data processing, especially for space weather applications. This is due to space weather impacts on both space-borne and ground-based systems, and industries, which subsequently impacts our lives. In the current study, the deep learning approach is used to establish an automated hybrid computer system for a short-term forecast; it is achieved by using the complexity level of the sunspot group on SDO/HMI Intensitygram images. Furthermore, this suggested system can generate the forecast for solar flare occurrences within the following 24 h. The input data for the proposed system are SDO/HMI full-disk Intensitygram images and SDO/HMI full-disk magnetogram images. System outputs are the “Flare or Non-Flare” of daily flare occurrences (C, M, and X classes). This system integrates an image processing system to automatically detect sunspot groups on SDO/HMI Intensitygram images using active-region data extracted from SDO/HMI magnetogram images (presented by Colak and Qahwaji, 2008) and deep learning to generate these forecasts. Our deep learning-based system is designed to analyze sunspot groups on the solar disk to predict whether this sunspot group is capable of releasing a significant flare or not. Our system introduced in this work is called ASAP_Deep. The deep learning model used in our system is based on the integration of the Convolutional Neural Network (CNN) and Softmax classifier to extract special features from the sunspot group images detected from SDO/HMI (Intensitygram and magnetogram) images. Furthermore, a CNN training scheme based on the integration of a back-propagation algorithm and a mini-batch AdaGrad optimization method is suggested for weight updates and to modify learning rates, respectively. The images of the sunspot regions are cropped automatically by the imaging system and processed using deep learning rules to provide near real-time predictions. The major results of this study are as follows. Firstly, the ASAP_Deep system builds on the ASAP system introduced in Colak and Qahwaji (2009) but improves the system with an updated deep learning-based prediction capability. Secondly, we successfully apply CNN to the sunspot group image without any pre-processing or feature extraction. Thirdly, our system results are considerably better, especially for the false alarm ratio (FAR); this reduces the losses resulting from the protection measures applied by companies. Also, the proposed system achieves a relatively high scores for True Skill Statistics (TSS) and Heidke Skill Score (HSS).

Dans cette thèse, on étudie les sous-groupes boréliens des groupes de Lie et leur dimension de Hausdorff. Si G est un groupe de Lie nilpotent connexe, on construit dans G des sous-groupes de dimension de Hausdorff arbitraire, tandis que si G est semisimple compact, on démontre que la dimension de Hausdorff d'un sous-groupe borélien strict de G ne peut pas être arbitrairement proche de celle de G. / Given a Lie group G, we investigate the possible Hausdorff dimensions for a measurable subgroup of G. If G is a connected nilpotent Lie group, we construct measurable subgroups of G having arbitrary Hausdorff dimension, whereas if G is compact semisimple, we show that a proper measurable subgroup of G cannot have Hausdorff dimension arbitrarily close to the dimension of G.

Avec l'émergence des nouvelles technologies, les communications sans fil n'ont cessé de croître afin de permettre aux utilisateurs un accès à l'information et aux services électroniques, et ceci indépendamment de leur position géographique. Les réseaux sans fil ont aussi trouvé leur place pour des applications spécifiques telles que les transmissions radio utilisées pour l'interconnexion de capteurs. Ce type de réseau peut être considéré comme un sous-ensemble des réseaux ad hoc. Des contraintes spécifiques s'appliquent alors aux utilisateurs de ces réseaux, telles que la difficulté d'accès pour la maintenance, les problèmes liés à la miniaturisation et au nombre élevé de capteurs. L'objectif de cette thèse est d'étudier les contraintes énergétiques liées à l'utilisation des batteries à capacité limitée pour l'alimentation des capteurs. Pour atteindre cet objectif, nous avons proposé de représenter les réseaux de capteurs à travers une image à échelle de gris : les zones claires correspondant aux zones riches en énergie, alors que les zones sombres représentent des régions avec une capacité énergétique faible. Des filtres issus du monde de traitement d'image sont alors appliqués à cette image représentant l'énergie. Ainsi, nous proposons des filtres de convolution de type Sobel ou de type filtre moyen pour nos algorithmes de routage et nous construisons une matrice énergétique pour chaque capteur. Cette matrice est alors utilisée avec le produit de convolution pour guider le routage. Les différents algorithmes proposés font ensuite l'objet de simulations avec le simulateur de réseaux OMNeT++. / With the emergence of new technologies, wireless communications have been developed in order to allow users an access to information and to electronic services, independently of their geographical position. Wireless networks have also been developed for specific applications such as radio transmissions used for interconnection of sensors. This type of network can be considered as a subset of ad hoc networks. On other side, this implies specific constraints on users, such as the problem of the access for maintenance, the problems of miniaturization, and the large number of sensors. The objective of this thesis is to focus on energy constraints related to the use of batteries with limited capacity for the supply of sensors. In our work, we propose routing algorithms to route information while controlling energy consumption. To achieve this goal, we have represented the sensor network as a grayscale image: light areas represent regions rich in energy, whereas dark areas represent regions with low energy capacity. Filters used in image processing are then applied to the image representing the energy. Thus, we propose convolution filters like Sobel or mean filter in our routing algorithms and we construct an energy matrix for each sensor. This matrix will be used with the convolution to find the best path. The proposed algorithms are verified by simulations performed with the network simulator OMNeT++.