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This thesis is concerned with two different problems in harmonic analysis: the multilinear Kakeya theorem, and Wolff-type inequalities for paraboloids. Chapter 1 gives an overview of both of these problems. In Chapter 2 we investigate an important special case of the multilinear Kakeya theorem, the so-called “bush example”. While the endpoint case of the multilinear Kakeya theorem was recently proved by Guth, the proof is highly abstract; our aim is to provide a more elementary proof in this special case. This is achieved for a significant part of the three-dimensional case in the main result of the chapter. Chapter 3 is a study of the endpoint case of a mixed-norm Wolff-type inequality for the paraboloid. The main result adapts an example of Bourgain to show that the endpoint inequality cannot hold with an absolute constant; there must be a dependence on the thickening of the paraboloid. The remainder of the chapter is a series of case studies, through which we establish positive endpoint results for certain classes of function, as well as indicating specific examples which need to be better understood in order to obtain the full endpoint result.

SENA, E. D. R. Técnicas multilineares em reconhecimento facial. 2014. 77 f. Dissertação (Mestrado em Engenharia de Teleinformática) - Centro de Tecnologia, Universidade Federal do Ceará, Fortaleza, 2014. / Submitted by Marlene Sousa (mmarlene@ufc.br) on 2015-02-12T17:11:46Z No. of bitstreams: 1 2014_dis_edrsena.pdf: 858896 bytes, checksum: f934617e0c123b479858e1a2ff40193a (MD5) / Approved for entry into archive by Marlene Sousa(mmarlene@ufc.br) on 2015-02-12T17:12:06Z (GMT) No. of bitstreams: 1 2014_dis_edrsena.pdf: 858896 bytes, checksum: f934617e0c123b479858e1a2ff40193a (MD5) / Made available in DSpace on 2015-02-12T17:12:06Z (GMT). No. of bitstreams: 1 2014_dis_edrsena.pdf: 858896 bytes, checksum: f934617e0c123b479858e1a2ff40193a (MD5) Previous issue date: 2014-11-07 / In this dissertation, the face recognition problem is investigated from the standpoint of multilinear algebra, more specifically the tensor decomposition, and by making use of Gabor wavelets. The feature extraction occurs in two stages: first the Gabor wavelets are applied holistically in feature selection; Secondly facial images are modeled as a higher-order tensor according to the multimodal factors present. Then, the HOSVD is applied to separate the multimodal factors of the images. The proposed facial recognition approach exhibits higher average success rate and stability when there is variation in the various multimodal factors such as facial position, lighting condition and facial expression. We also propose a systematic way to perform cross-validation on tensor models to estimate the error rate in face recognition systems that explore the nature of the multimodal ensemble. Through the random partitioning of data organized as a tensor, the mode-n cross-validation provides folds as subtensors extracted of the desired mode, featuring a stratified method and susceptible to repetition of cross-validation with different partitioning. / Nesta dissertação o problema de reconhecimento facial é investigado do ponto de vista da álgebra multilinear, mais especificamente por meio de decomposições tensoriais fazendo uso das wavelets de Gabor. A extração de características ocorre em dois estágios: primeiramente as wavelets de Gabor são aplicadas de maneira holística na seleção de características; em segundo as imagens faciais são modeladas como um tensor de ordem superior de acordo com o fatores multimodais presentes. Com isso aplicamos a decomposição tensorial Higher Order Singular Value Decomposition (HOSVD) para separar os fatores que influenciam na formação das imagens. O método de reconhecimento facial proposto possui uma alta taxa de acerto e estabilidade quando há variação nos diversos fatores multimodais, tais como, posição facial, condição de iluminação e expressão facial. Propomos ainda uma maneira sistemática para realização da validação cruzada em modelos tensoriais para estimação da taxa de erro em sistemas de reconhecimento facial que exploram a natureza multilinear do conjunto de imagens. Através do particionamento aleatório dos dados organizado como um tensor, a validação cruzada modo-n proporciona a criação de folds extraindo subtensores no modo desejado, caracterizando um método estratificado e susceptível a repetições da validação cruzada com diferentes particionamentos.

The presentation of the thesis comprises the Dissertation component (66%) along with the Practice Component and the Practice Report (33%). In this Video Chaos dissertation, through an examination of current video practices, I note an emerging trend towards disseminating audio-visual content simultaneously in the form of poly-sequential narrative structures. I argue that this is a significant development within the video medium, and that this is an effect of video new media artist-practitioners' engagement with the relationships between art and technology. Two extensive case studies are investigated and, whilst a number of issues come to the fore in this research, exploring the issue of narrative structuration is the primary focus and exploration of this dissertation. The presentation of the thesis comprises the Dissertation component (66%) along with the Practice Component and the Practice Report (33%). The Practice Report documents the nature and development of the research undertaken during the course of the study. The culmination of the Practice Component takes the form of an exhibition and archiving of video works from June 2003 to the date of submission, January 2005. The Practice Component has been based in the following locations and used resources from Central Queensland University (Bundaberg campus), and The Australian National University, Centre for New Media Arts in Canberra. The practice has examined the topic through the production of the audio-video installation Sugartown and three video works The Hazzards, Nodal Dialectics 1 and boomsplatbangwhack. While these video works exist as discrete media artworks, they also operate as a type of practice process diary for working through the ideas explored in the written dissertation. Even though the video works are not meant to literally 'illustrate' those ideas, they nevertheless explore ways of integrating the theoretical concepts into my own research practice. In this Video Chaos dissertation, through an examination of current video practices, I note an emerging trend towards disseminating audio-visual content simultaneously in the form of poly-sequential narrative structures. I argue that this is a significant development within the video medium, and that this is an effect of video new media artist-practitioners' engagement with the relationships between art and technology. Two extensive case studies are investigated and, whilst a number of issues come to the fore in this research, exploring the issue of narrative structuration is the primary focus and exploration of this dissertation.

This screenplay is a fictional Coming of Age story of a young restless woman who, on an existentialist search for her inner self, investigates into the truth behind her famous grandmother's past, an anthropologist who conducted controversial research in the Samoa of the 1920s. In the turbulent streams of her consciousness, Kat drifts in between an urban reality in Berlin, daydreams about her grandmother's journey into the exotic unknown, fantasies about the enigmatic young Samoan single mom Penei, and memories of a once loving family. The encounter with Penei and the resulting friendship and frail romance of the two women proves to be an eye-opener for Kat who finally discovers that the objective truth proves to be the ultimate myth and that only the acknowledgement of her own, subjective vision will lead her on the path to her inner happiness. Although the character of Anna König is inspired by the historic figure of anthropologist Margaret Mead, the script has no intent to refer to actual facts of Mead's life. All the characters depicted are entirely fictional.

The presentation of the thesis comprises the Dissertation component (66%) along with the Practice Component and the Practice Report (33%). In this Video Chaos dissertation, through an examination of current video practices, I note an emerging trend towards disseminating audio-visual content simultaneously in the form of poly-sequential narrative structures. I argue that this is a significant development within the video medium, and that this is an effect of video new media artist-practitioners' engagement with the relationships between art and technology. Two extensive case studies are investigated and, whilst a number of issues come to the fore in this research, exploring the issue of narrative structuration is the primary focus and exploration of this dissertation. The presentation of the thesis comprises the Dissertation component (66%) along with the Practice Component and the Practice Report (33%). The Practice Report documents the nature and development of the research undertaken during the course of the study. The culmination of the Practice Component takes the form of an exhibition and archiving of video works from June 2003 to the date of submission, January 2005. The Practice Component has been based in the following locations and used resources from Central Queensland University (Bundaberg campus), and The Australian National University, Centre for New Media Arts in Canberra. The practice has examined the topic through the production of the audio-video installation Sugartown and three video works The Hazzards, Nodal Dialectics 1 and boomsplatbangwhack. While these video works exist as discrete media artworks, they also operate as a type of practice process diary for working through the ideas explored in the written dissertation. Even though the video works are not meant to literally 'illustrate' those ideas, they nevertheless explore ways of integrating the theoretical concepts into my own research practice. In this Video Chaos dissertation, through an examination of current video practices, I note an emerging trend towards disseminating audio-visual content simultaneously in the form of poly-sequential narrative structures. I argue that this is a significant development within the video medium, and that this is an effect of video new media artist-practitioners' engagement with the relationships between art and technology. Two extensive case studies are investigated and, whilst a number of issues come to the fore in this research, exploring the issue of narrative structuration is the primary focus and exploration of this dissertation.

This screenplay is a fictional Coming of Age story of a young restless woman who, on an existentialist search for her inner self, investigates into the truth behind her famous grandmother's past, an anthropologist who conducted controversial research in the Samoa of the 1920s. In the turbulent streams of her consciousness, Kat drifts in between an urban reality in Berlin, daydreams about her grandmother's journey into the exotic unknown, fantasies about the enigmatic young Samoan single mom Penei, and memories of a once loving family. The encounter with Penei and the resulting friendship and frail romance of the two women proves to be an eye-opener for Kat who finally discovers that the objective truth proves to be the ultimate myth and that only the acknowledgement of her own, subjective vision will lead her on the path to her inner happiness. Although the character of Anna König is inspired by the historic figure of anthropologist Margaret Mead, the script has no intent to refer to actual facts of Mead's life. All the characters depicted are entirely fictional.

This thesis contains work from the author's papers Palsson and Sovine (2020); Iosevich, Palsson, and Sovine (2022); and Palsson and Sovine (2022) with coauthors Eyvindur Palsson and Alex Iosevich. These works establish new $L^p$-improving, quasi-Banach, and sparse bounds for several bilinear and multilinear operators that generalize the linear spherical average to the multilinear setting, and maximal variants of these operators, with an emphasis on the triangle averaging operator and the bilinear spherical averaging operator. / Doctor of Philosophy / This thesis establishes new regularity properties for several mathematical operations that generalize the operation of taking the average of a function over a sphere to operations that average the product of several input functions over a surface to produce a single output function. These operations include the triangle averaging operator, the $k$-simplex averaging operators for $k$ an integer greater than 1, and the bilinear spherical averaging operator, as well as maximal operators obtained by allowing the radius of the averaging surface to vary over some range of values.

One of the differentiating features of PARAFAC decompositions is that, under certain conditions, unique solutions are possible. The search for uniqueness conditions for the PARAFAC Decomposition has a limited past, spanning only three decades. The complex structure of the problem and the need for tensor algebras or other similarly abstract characterizations provided a roadblock to the development of uniqueness conditions. Theoretically, the PARAFAC decomposition surpasses its bilinear counterparts in that it is possible to obtain solutions that do not suffer from the rotational problem. However, not all PARAFAC solutions will be constrained sufficiently so that the resulting decomposition is unique. The work of Kruskal, 1977, provides the most in depth investigation into the conditions for uniqueness, so much so that many have assumed, without formal proof, that his sufficient conditions were also necessary. Aided by the introduction of Khatri-Rao products to represent the PARAFAC decomposition, ten Berge and Sidiropoulos (2002) used the column spaces of Khatri-Rao products to provide the first evidence for countering the claim of necessity, identifying PARAFAC decompositions that were unique when Kruskals condition was not met. Moreover, ten Berge and Sidiropoulos conjectured that, with additional k-rank restrictions, a class of decompositions could be formed where Kruskals condition would be necessary and sufficient. Unfortunately, the column space argument of ten Berge and Sidiropoulos was limited in its application and failed to provide an explanation of why uniqueness occurred. On the other hand, the use of orthogonal complement spaces provided an alternative approach to evaluate uniqueness that would provide a much richer return than the use of column spaces for the investigation of uniqueness. The Orthogonal Complement Space Approach (OCSA), adopted here, would provide: (1) the answers to lingering questions about the occurrence of uniqueness, (2) evidence that necessity would require more than a restriction on k-rank, and (3) an approach that could be extended to cases beyond those investigated by ten Berge and Sidiropoulos.

Oscillatory integral operators have been of interest to both mathematicians and physicists ever since the emergence of the work Theorie Analytique de la Chaleur of Joseph Fourier in 1822, in which his chief concern was to give a mathematical account of the diffusion of heat. For example, oscillatory integrals naturally arise when one studies the behaviour at infinity of the Fourier transform of a Borel measure that is supported on a certain hypersurface. One reduces the study of such a problem to that of having to obtain estimates on oscillatory integrals. However, sub-level set operators have only come to the fore at the end of the 20th Century, where it has been discovered that the decay rates of the oscillatory integral I(lambda) above may be obtainable once the measure of the associated sub-level sets are known. This discovery has been fully developed in a paper of A. Carbery, M. Christ and J.Wright. A principal goal of this thesis is to explore certain uniformity issues arising in the study of sub-level set estimates.