Modern array sensing systems, such as distributed fiber optic sensing, are used in many applications which may record a mixture of responses to multiple physical quantities. In these applications, it may be helpful to be able to separate this mixture of responses into the signals resulting from the individual sources. This is similar to the cocktail party problem posed with Independent Component Analysis (ICA), in which we use gradient ascent and fixed point iteration optimization algorithms to achieve this separation. We then seek to apply the problem setup from ICA to mixed signals resulting from a sensor array with the goal of maintaining coherence throughout resulting spatial arrays. We propose a new post-processing technique after separation to pair up the signals from different types of physical quantities based on the Symmetric Reverse Cuthill-McKee (SRCM) and Symmetric Approximate Minimum Degree (SAMD) permutations of the coherence matrix. / Master of Science / Some modern sensing systems are able to collect data resulting from different types of sources, such as vibrations and electromagnetic waves, at the same time. This means we have signals resulting from a mixture of sources. An example of one such modern sensing system is distributed fiber optic sensors used in geoscience applications, such as seismology and subsurface imaging, which measures strain along the fiber optic cable. In many applications, it may be helpful to obtain the signals from each of these sources separately, instead of having a mixture of these sources. We propose the use of optimization algorithms, in particular two algorithms arising from Independent Component Analysis (ICA), which seek to maximize a function in order to separate these signals. We then explore changes required to the algorithms for scenarios in which we have multiple sensors spaced some distance away from each other which record signals from two different sources. We also present a method of determining which separated signals correspond to which sensors after performing signal separation.

We study minimizers of the Landau-de Gennes energy in the exterior region around a smooth 2-manifold in R3 with a constant external magnetic field present. Uniaxial boundary data and a strong tangential anchoring are imposed on the surface of the manifold and we consider the large particle limit in a regime where the magnetic field is relatively weak. Before studying the general manifold, we analyze a more simple case in which the manifold is spherical. After deriving a lower bound for the energy in this limiting regime, we prove that a director field on the boundary which maximizes its vertical component yields a minimal lower bound. We then construct a recovery sequence to show that this lower bound is in fact the optimal energy bound. These steps are later repeated in more generality for a larger class of smooth manifolds. / Thesis / Master of Science (MSc)

La majeure partie de cette thèse est consacrée à l'inégalité de Blaschke-Santaló, qui s'énonce ainsi : parmi les ensembles symétriques, la boule euclidienne maximise le produit vol(K) vol(K°), K° désignant le polaire de K. Il existe des versions fonctionnelles de cette inégalité, découvertes par plusieurs auteurs (Ball, Artstein, Klartag, Milman, Fradelizi, Meyer. . .), mais elles sont toutes dérivées de l'inégalité ensembliste. L'objet de cette thèse est de proposer des démonstrations directes de ces inégalités fonctionnelles. On obtient ainsi de nouvelles preuves de l'inégalité de Santaló, parfois très simples. La dernière partie est un peu à part et concerne le chaos gaussien : on démontre une majoration précise des moments du chaos gaussien due à Lataªa par des arguments de chaînage à la Talagrand

This phenomenological study explored how college students’ perceptions of experiences with their secondary mathematics teachers affected their mathematical identities. The study was rooted in Wenger’s notion that learning is an experience of identity and Dewey’s theory that all experiences are inextricably linked to past and future experiences. Constructed narratives of eight college developmental mathematics students with high and low levels of mathematics anxiety were created from autobiographical essays and semistructured interviews. Analysis of the constructed narratives employed a deductive coding process using a priori themes related to experiences with secondary teachers and dimensions of mathematical identity. The study answered three research questions: What kind of experiences did students recall having with their secondary mathematics teachers? How did students perceive that those experiences influenced their mathematical identities? What common student experiences positively or negatively affecting mathematical identity emerged from the data? Two general factors that affect student mathematical identity emerged from the research: student-teacher interactions and student-mathematics interactions. Interconnectivity existed between positive student-teacher relationships, meaningful student-mathematics interactions, and strong mathematical identities. Positive student-teacher relationships were foundational to the overall connection.

The purpose of this study was twofold: (a) to investigate the prevalence of mathematics anxiety among freshman Algebra I students in an urban, Midwestern high school, and (b) to find out if a pre-quiz and quiz intervention could reduce mathematics anxiety in one specific class. The Mathematics Anxiety Rating Scale for Adolescents (MARS-A) was the primary quantitative data collection instrument. Qualitative data were collected using the Mathematicsitude Survey, student reflections, and interviews. Findings from the MARS-A showed that 50% of students experienced a significant amount of mathematics anxiety, particularly associated with testtaking. However, there was a large amount of variation among scores. In the treatment class, a strategy of pre-quiz followed by the same or similar quiz the following day was used to build student confidence and thereby lessen anxiety. The strategy did not meet this objective as many students reported greater anxiety levels after the intervention than before. Qualitative probing did show that in some isolated cases the strategy worked very well. / Thesis (M.Ed.)--Wichita State University, College of Education, Dept. of Curriculum and Instruction

In the course Discrete Deterministic Chaos, Dr. Mark Daniels introduces students to Chaos Theory and explores many topics within the field. Students prove many of the key results that are discussed in class and work through examples of each topic. Connections to the secondary mathematics curriculum are made throughout the course, and students discuss how the topics in the course could be implemented in the classroom. This paper will provide an overview of the topics covered in the course, Discrete Deterministic Chaos, and provide additional discussion on various related topics. / text

Dans les années quatre-vingts dix du siècle dernier, R. W. Thomason a démontréun théorème de concentration pour la K-théorie équivariante algébrique sur lesschémas munis d'une action d'un groupe algébrique G diagonalisable. Comme d'habitude,un tel théorème entraîne une formule des points fixes de type Lefschetz qui permetde calculer la caractéristique d'Euler-Poincaré équivariante d'un G-faisceau cohérent surun G-schéma propre en termes d'une caractéristique sur le sous-schéma des points fixes.Le but de cette thèse est de généraliser les résultats de R.W. Thomason dans le contextede la géométrie d'Arakelov. Dans ce travail, nous considérons les schémas arithmétiquesau sens de Gillet-Soulé et nous tout d'abord démontrons un analogue arithmétiquedu théorème de concentration pour les schémas arithmétiques munis d'une action duschéma en groupe diagonalisable associé à Z/nZ. La démonstration résulte du théorèmede concentration algébrique joint à des arguments analytiques. Dans le dernier chapitre,nous formulons et démontrons deux types de formules de Lefschetz arithmétiques. Cesdeux formules donnent une réponse positive à deux conjectures énoncées par K. Köhler,V. Maillot et D. Rössler.

We study in this thesis a particular machine learning approach to represent signals that that consists of modelling data as linear combinations of a few elements from a learned dictionary. It can be viewed as an extension of the classical wavelet framework, whose goal is to design such dictionaries (often orthonormal basis) that are adapted to natural signals. An important success of dictionary learning methods has been their ability to model natural image patches and the performance of image denoising algorithms that it has yielded. We address several open questions related to this framework: How to efficiently optimize the dictionary? How can the model be enriched by adding a structure to the dictionary? Can current image processing tools based on this method be further improved? How should one learn the dictionary when it is used for a different task than signal reconstruction? How can it be used for solving computer vision problems? We answer these questions with a multidisciplinarity approach, using tools from statistical machine learning, convex and stochastic optimization, image and signal processing, computer vision, but also optimization on graphs.

Many laser devices acquire directly 3D objects and reconstruct their surface. Nevertheless, the final reconstructed surface is usually smoothed out as a result of the scanner internal de-noising process and the offsets between different scans. This thesis, working on results from high precision scans, adopts the somewhat extreme conservative position, not to loose or alter any raw sample throughout the whole processing pipeline, and to attempt to visualize them. Indeed, it is the only way to discover all surface imperfections (holes, offsets). Furthermore, since high precision data can capture the slightest surface variation, any smoothing and any sub-sampling can incur in the loss of textural detail.The thesis attempts to prove that one can triangulate the raw point cloud with almost no sample loss. It solves the exact visualization problem on large data sets of up to 35 million points made of 300 different scan sweeps and more. Two major problems are addressed. The first one is the orientation of the complete raw point set, an the building of a high precision mesh. The second one is the correction of the tiny scan misalignments which can cause strong high frequency aliasing and hamper completely a direct visualization.The second development of the thesis is a general low-high frequency decomposition algorithm for any point cloud. Thus classic image analysis tools, the level set tree and the MSER representations, are extended to meshes, yielding an intrinsic mesh segmentation method.The underlying mathematical development focuses on an analysis of a half dozen discrete differential operators acting on raw point clouds which have been proposed in the literature. By considering the asymptotic behavior of these operators on a smooth surface, a classification by their underlying curvature operators is obtained.This analysis leads to the development of a discrete operator consistent with the mean curvature motion (the intrinsic heat equation) defining a remarkably simple and robust numerical scale space. By this scale space all of the above mentioned problems (point set orientation, raw point set triangulation, scan merging, segmentation), usually addressed by separated techniques, are solved in a unified framework.

Dans une première partie, on donne une condition nécessaire et suffisante à ce qu'un opérateur de Schrödinger sur une variété complète non-compacte ait un nombre fini de valeurs propres négatives. Dans une deuxième partie, on s'intéresse à la transformée de Riesz sur une classe de variétés complètes non-compactes vérifiant une inégalité de Sobolev. On montre d'abord une estimée gaussienne pour le noyau de la chaleur d'opérateurs de Schrödinger généralisés, comme par exemple le Laplacien de Hodge agissant sur les formes différentielles, puis on utilise ceci pour montrer que la transformée de Riesz est bornée sur les espaces $L^p$ si $p$ est compris entre $1$ et la dimension de Sobolev. Enfin, on montre un résultat de perturbation pour la transformée de Riesz.