Accompanying the advances in computer technology is an increase collection of high dimensional data in many scientific and social studies. Sufficient dimension reduction (SDR) is a statistical method that enable us to reduce the dimension ofpredictors without loss of regression information. In this dissertation, we introduce principal asymmetric least squares (PALS) as a unified framework for linear and nonlinear sufficient dimension reduction. Classical methods such as sliced inverse regression (Li, 1991) and principal support vector machines (Li, Artemiou and Li, 2011) often do not perform well in the presence of heteroscedastic error, while our proposal addresses this limitation by synthesizing different expectile levels. Through extensive numerical studies, we demonstrate the superior performance of PALS in terms of both computation time and estimation accuracy. For the asymptotic analysis of PALS for linear sufficient dimension reduction, we develop new tools to compute the derivative of an expectation of a non-Lipschitz function. PALS is not designed to handle symmetric link function between the response and the predictors. As a remedy, we develop expectile-assisted inverse regression estimation (EA-IRE) as a unified framework for moment-based inverse regression. We propose to first estimate the expectiles through kernel expectile regression, and then carry out dimension reduction based on random projections of the regression expectiles. Several popular inverse regression methods in the literature including slice inverse regression, slice average variance estimation, and directional regression are extended under this general framework. The proposed expectile-assisted methods outperform existing moment-based dimension reduction methods in both numerical studies and an analysis of the Big Mac data. / Statistics

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Let $\bk$ be a field of characteristic zero. A Pham-Brieskorn ring is a $\bk$-algebra of the form $B_{a_0,\dots,a_n} = \bk[X_0,\dots,X_n] / \lb X_0^{a_0} + \cdots + X_n^{a_n} \rb$, where $n \geq 2$ and $a_0, \dots, a_n$ are positive integers. A ring $B$ is rigid if the only locally nilpotent derivation $D : B \to B$ is the zero derivation. Consider the following conjecture. \begin{conjnonumber}\label{PBConjectureAbstract} Let $n \geq 2$, and let $B_{a_0, \dots, a_n} = \bk[X_0, \dots, X_n] / \langle X_0^{a_0} + \cdots + X_n^{a_n} \rangle$ be a Pham-Brieskorn ring. If $\min\{a_0, \dots,a_n \} \geq 2$ and at most one element $i$ of $\{0,\dots ,n\}$ satisfies $a_i = 2$, then $B_{a_0, \dots, a_n}$ is rigid. \end{conjnonumber} The $n = 2$ case of the Conjecture is known to be true. In this thesis, we make progress towards solving the above conjecture. Our main results are: \begin{enumerate}[\rm(1)] \item For any $n \geq 3$, in order to prove the above conjecture, it suffices to prove rigidity of $B_{a_0, \dots, a_n}$ in the cases where $\bk = \Comp$ and $\cotype(a_0, \dots, a_n) = 0$. \item For any $n \geq 2$, $X = \Proj B_{a_0, \dots, a_n}$ is a well-formed quasismooth weighted complete intersection if and only if $\cotype(a_0, \dots, a_n) = 0$. \item When $n = 3$ and $\cotype(a_0, a_1, a_2, a_3) = 0$, $B_{a_0, a_1, a_2, a_3}$ is rigid, except possibly in the cases where, up to a permutation of the $a_i$, $(a_0, a_1, a_2, a_3) \in \{(2,3,4,12), (2,3,5,30)\}$. \item We summarize the list of 3-dimensional Pham-Brieskorn rings $B_{a_0, a_1, a_2, a_3}$ for which rigidity is known. It follows in particular that if $B_{2,3,4,12}$ and $B_{2,3,5,30}$ are rigid then the $n = 3$ case of the above conjecture is true. \end{enumerate} In addition to the above, we develop techniques for proving rigidity of rings in general; prove rigidity of many Pham-Brieskorn rings whose dimension is greater than 3; give simple examples of rational projective surfaces with quotient singularities that have an ample canonical divisor and prove that the members of a certain family of singular hypersurfaces are not rational.

Non-projective dependencies remain an at large issue in the field of dependency parsing. Regardless of what parsing algorithm is used, researchers run into the issue of computational speed and lower parsing performance on non-projective dependencies than on projective dependencies. Through a better understanding of non-projectivity, we may be able to address both issues. This thesis is aimed to discover what types of non-projective dependencies are prevalent in the three languages English, German, and Czech. Moreover, this thesis is aimed to define and create a linguistically informed projectivization scheme and to find out the extent to which the scheme improves upon the performance of the baseline parser. In order to achieve these aims, the eight most frequently occurring non-projective dependencies in English, German, and Czech were surveyed. This means that the causes of their non-projectivity were discovered, the structures of the non-projective dependencies were analyzed, and generalizations and comparisons between non-projective dependencies were made. After the survey, an attempt to define and create a linguistically informed projectivization scheme was made. The goals were not only to projectivize the non-projective relations but to do so by assigning the closest possible new parent in the sentence to the non-projective child and to minimize the number of projectivization transformations that needed to be made. Although the survey of the non-projective dependencies yielded good results, as we were able to identify that the causes of the more frequently occurring non-projective dependencies in German and Czech were the same and the structures of them the same as well, we reached no solid conclusion on how a linguistically informed projectivization scheme could be defined, as further research is needed. However, the novel projectivization scheme we did come up with managed to marginally outperform the baseline parser in English and German, and moderately outperform the baseline parser in Czech which is the language with the most non-projective dependencies of the group.

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[pt] A presente dissertação analisa a relação interdisciplinar entre a matemática e as artes, dando especial destaque ao Teorema de Desargues como uma ponte entre estas áreas. Destaca-se a importância atual da interdisciplinaridade na educação, embasada pela Base Nacional Comum Curricular (BNCC), que dá destaque à integração de tecnologia e conhecimento em múltiplas áreas do currículo escolar. O Teorema de Desargues é abordado como um conceito que rompe os limites da matemática, alcançando também os campos da arte e da tecnologia. A Geometria Projetiva é contextualizada historicamente, apresentando seus primeiros passos e progresso ao longo do tempo. Revela Girard Desargues como um como precursor de ideias nesse contexto, contribuindo tanto para o avanço da matemática quanto para a expressão artística. A dissertação enfatiza a aplicação prática do Teorema de Desargues no contexto educacional, propondo atividades significativas e atrativas para os alunos no contexto escolar. Apresenta o produto educacional desenvolvido pelos autores como uma fonte valiosa de sugestões para educadores que pretendem se dedicar à interdisciplinaridade. A dissertação promove uma abordagem educacional que estimula o diálogo entre disciplinas, destacando a conexão entre matemática, geometria projetiva, arte e tecnologia, para isso utiliza o Teorema de Desargues desempenhando um papel central nesse processo. / [en] The present dissertation examines the interdisciplinary relationship between mathematics and the arts, with special emphasis on Desargues Theorem as a bridge between these fields. It highlights the current importance of interdisciplinarity in education, supported by the National Common Curricular Base (BNCC), which emphasizes the integration of technology and knowledge across multiple areas of the school curriculum. Desargues Theorem is approached as a concept that transcends the boundaries of mathematics, also reaching into the realms of art and technology. Projective Geometry is historically contextualized, tracing its origins and development over time. Girard Desargues is revealed as a precursor of ideas in this context, contributing to both the advancement of mathematics and artistic expression. The dissertation emphasizes the practical application of Desargues Theorem in the educational context, proposing meaningful and engaging activities for students in the school setting. It presents the educational product developed by the authors as a valuable source of suggestions for educators looking to dedicate themselves to interdisciplinarity. The dissertation promotes an educational approach that encourages dialogue between disciplines, highlighting the connection between mathematics, projective geometry, art, and technology, utilizing Desargues Theorem as a central element in this process.

This research presents improvements to the velocity field reconstruction process achieved through computer vision. The first improvement of the velocity reconstruction process is the automation of the scanning laser Doppler vibrometer (SLDV) pose procedure. This automated process results in superior estimates of the position and orientation of the SLDV. The second improvement is the refinement of the formulation for reconstruction of the velocity field. The refined formulation permits faster computation, evaluation, and interpretation of the reconstructed structural velocity field. Taken together, these new procedures significantly improve the overall velocity reconstruction process which results in better, unbiased out-of-plane velocity estimates in the presence of noise. The automation of the SLDV pose procedure is achieved through a computer vision model of the SLDV. The SLDV is modeled as a projective camera, i.e. an imager which preserves projectivities. This projective camera model permits the precise association of object features with image features. Specifically, circular features in the object space are seen by the SLDV as ellipses in the image space. In order to extract object points, the bitangents among the circular features are constructed and the bitangent points selected. The accuracy and precision of the object points are improved through the use of a calibrated object whose circular features are measured with a coordinate measuring machine. The corresponding image points are determined by constructing the bitangents among the ellipses and selecting the tangent points. Taken together, these object/image bitangent point sets are a significantly improved data set for previously developed SLDV pose algorithms. Experimental verification of this automated pose procedure includes demonstrated repeatability, independent validation of the estimated pose parameters, and comparison of the estimated poses with previous methods. The refinement of the velocity reconstruction formulation is a direct result of the computer vision viewpoint adapted for this research. By viewing the velocity data as images of the harmonically excited structure's velocity field, analytical techniques developed for holographic interferometry are extended and applied to SLDV velocity images. Specifically, the "absolute" and "relative" fringe-order methods are used to reconstruct the velocity field with the "best" set of bases. Full and partial least squares solutions with experimental velocity data are calculated. Statistical confidence bounds of the regressed velocity coefficients are analyzed and interpreted to reveal accurate out-of-plane, but poor in-plane velocity estimates. Additionally, the reconstruction process is extended to recover the velocity field of a family of surfaces in the neighborhood of the "real" surface. This refinement relaxes the need for the exact experimental geometry. Finally, the velocity reconstruction procedure is reformulated so that independent least squares solutions are obtained for the two in-plane directions and the out-of plane direction. This formulation divides the original least squares problem into three smaller problems which can be analyzed and interpreted separately. These refinements to the velocity reconstruction process significantly improve the out-of-plane velocity solution and interpretation of the regressed velocity parameters. / Ph. D.

Let <i>R</i> be a commutative ring, <i>I</i> be an ideal in <i>R</i> and let <i>M</i> be a <i>R/ I</i> -module. In this thesis we construct a <i>R/ I</i> -projective resolution of <i>M</i> using given <i>R</i>-projective resolutions of <i>M</i> and <i>I</i>. As immediate consequences of our construction we give descriptions of the canonical maps Ext_R/I<i>(M,N)</i> -> Ext_R<i>(M,N)</i> and Tor^R_N<i>(M, N)</i> -> Tor^R/I_n<i>(M, N)</i> for a <i>R/I</i> module <i>N</i> and we give a new proof of a theorem of Gulliksen [6] which states that if <i>I</i> is generated by a regular sequence of length r then ∐∞_n=o Tor^R/I_n <i>(M, N)</i> is a graded module over the polynomial ring </i>R/ I</i> [X₁. .. X_r] with deg X_i = -2, 1 ≤ i ≤ r. If <i>I</i> is generated by a regular element and if the <i>R</i>-projective dimension of <i>M</i> is finite, we show that <i>M</i> has a <i>R/ I</i>-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when <i>R</i> = (<i>R</i>, m) is a Noetherian local ring and <i>M</i> is a finitely generated <i>R/ I</i> -module, we discuss the minimality of the constructed resolution. If it is minimal we call (<i>M, I</i>) a Golod pair over <i>R</i>. We give a direct proof of a theorem of Levin [10] which states thdt if (<i>M,I</i>) is a Golod pair over <i>R</i> then (Ωⁿ_R/IR/I(M),I) is a Golod pair over <i>R</i> where Ωⁿ_R/IR/I(M) is the nth syzygy of the constructed <i>R/ I</i> -projective resolution of <i>M</i>. We show that the converse of the last theorem is not true and if (Ω¹_R/IR/I(M),I) is a Golod pair over <i>R</i> then we give a necessary and sufficient condition for (<i>M, I</i>) to be a Golod pair over <i>R</i>. Finally we prove that if (<i>M, I</i>) is a Golod pair over <i>R</i> and if a ∈ <i>I</i> - m<i>I</i> is a regular element in </i>R</i> then (<i>M</i>, (a)) and (1/(a), (a)) are Golod pairs over <i>R</i> and (<i>M,I</i>/(a)) is a Golod pair over <i>R</i>/(a). As a corrolary of this result we show that if the natural map π : <i>R</i> → <i>R/1</i> is a Golod homomorphism ( this means (<i>R</i>/m, <i>I</i>) is a Golod pair over <i>R</i> ,Levin [8]), then the natural maps π₁ : <i>R</i> → <i>R</i>/(a) and π₂ : <i>R</i>/(a) → <i>R/1</i> are Golod homomorphisms. / Ph. D.

K3 surfaces have garnered attention across various fields, from optics and dynamics to high energy physics, making them a subject of extensive study for many decades. Recent work by mathematicians, including Brandhorst [1], has focused on non-symplectic automorphisms, aiming to categorize K3 surfaces that admit such automorphisms. Brandhorst made a list of unique K3 surfaces with purely non-symplectic automorphisms and established specific criteria for a K3 surface to be isomorphic to one on his list. This thesis aims to provide an alternative representation of select K3 surfaces from Brandhorst's list. While Brandhorst predominantly characterizes these surfaces as elliptic K3 surfaces, we offer a description of these surfaces as hypersurfaces in weighted projective space. Our approach involves verifying the criteria established by Brandhorst, thereby establishing an isomorphism between the surfaces in question. Through this study, we contribute to the understanding of K3 surfaces and their automorphisms while also demonstrating the correspondence between different spaces and methodologies for analyzing K3 surfaces. This work lays the groundwork for further investigations into K3 surfaces with purely non-symplectic automorphisms, paving the way for deeper insights into their structural properties and geometric intricacies.