The purpose of this paper is to extend, as much as possible, the modern theory of condition numbers for conic convex optimization: z* := minz ctx s.t. Ax - b Cy C Cx , to the more general non-conic format: z* := minx ctx (GPd) s.t. Ax-b E Cy X P, where P is any closed convex set, not necessarily a cone, which we call the groundset. Although any convex problem can be transformed to conic form, such transformations are neither unique nor natural given the natural description of many problems, thereby diminishing the relevance of data-based condition number theory. Herein we extend the modern theory of condition numbers to the problem format (GPd). As a byproduct, we are able to state and prove natural extensions of many theorems from the conic-based theory of condition numbers to this broader problem format.

Aim: The primary aim of this study was to find if there is any correlation between the corneal radius of curvature and its eccentricity. Method: 45 subjects participated in this study, 24 emmetropes, 18 myopes and three hyperopes. All subjects were free of ocular abnormalities and had no media opacities. All the subjects had normal ocular health and good visual acuity of 1.0 or better for both distance and near. The values for eccentricity and corneal radius of curvature were obtained by using a Topcon CA-100F Corneal Analyzer. Results: For the 4.5 mm zone the only significant correlation between corneal radius of curvature and eccentricity was obtained for the mean of the meridian (p = 0.007). On the other hand, we found no significant correlation for the average of two meridians or for meridian 1 and meridian 2 separately in the 8.0 mm zone. Conclusions: We found no correlation between the corneal radius of curvature and the eccentricity for both zones. In addition, no correlation could be found between the spherical equivalent of the refractive errors and the corneal eccentricity. The reason for not finding any significant correlation between the two entities could be due to factors such as smaller sample size and poor distribution of refractive errors in the sample. Moreover, there may be other factors that could influence the overall corneal shape like eye shape, axial length and corneal diameter, which was not evaluated in this study.

In statistics, regression analysis is a technique, used to understand and model the relationship between a dependent variable and one or more independent variables. Multiple Adaptive Regression Spline (MARS) is a form of regression analysis. It is a non-parametric regression technique and can be seen as an extension of linear models that automatically models non-linearities and interactions. MARS is very important in both classification and regression, with an increasing number of applications in many areas of science, economy and technology. In our study, we analyzed Generalized Partial Linear Models (GPLMs), which are particular semiparametric models. GPLMs separate input variables into two parts and additively integrates classical linear models with nonlinear model part. In order to smooth this nonparametric part, we use Conic Multiple Adaptive Regression Spline (CMARS), which is a modified form of MARS. MARS is very benefical for high dimensional problems and does not require any particular class of relationship between the regressor variables and outcome variable of interest. This technique offers a great advantage for fitting nonlinear multivariate functions. Also, the contribution of the basis functions can be estimated by MARS, so that both the additive and interaction effects of the regressors are allowed to determine the dependent variable. There are two steps in the MARS algorithm: the forward and backward stepwise algorithms. In the first step, the model is constructed by adding basis functions until a maximum level of complexity is reached. Conversely, in the second step, the backward stepwise algorithm reduces the complexity by throwing the least significant basis functions from the model. In this thesis, we suggest not using backward stepwise algorithm, instead, we employ a Penalized Residual Sum of Squares (PRSS). We construct PRSS for MARS as a Tikhonov Regularization Problem. We treat this problem using continuous optimization techniques which we consider to become an important complementary technology and alternative to the concept of the backward stepwise algorithm. Especially, we apply the elegant framework of Conic Quadratic Programming (CQP) an area of convex optimization that is very well-structured, hereby, resembling linear programming and, therefore, permitting the use of interior point methods. At the end of this study, we compare CQP with Tikhonov Regularization problem for two different data sets, which are with and without interaction effects. Moreover, by using two another data sets, we make a comparison between CMARS and two other classification methods which are Infinite Kernel Learning (IKL) and Tikhonov Regularization whose results are obtained from the thesis, which is on progress.

Submitted by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-11-25T13:59:24Z No. of bitstreams: 1 arquivototal.pdf: 7192997 bytes, checksum: ff89ff26a871c70f0d3a3940ed7325bc (MD5) / Made available in DSpace on 2015-11-25T13:59:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 7192997 bytes, checksum: ff89ff26a871c70f0d3a3940ed7325bc (MD5) Previous issue date: 2014-08-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / To teach classes on the conic 3rd year of high school, always observed that most students did not recall the main de nitions, and had great di culty. To improve learning this, think of a proposed intervention that did the student work in a more pleasant way this content. With the advent of computing and software GeoGebra, elaborate didactic itineraries practices, so that they could revise and deepen the main concepts of conic. This work was then developed in order to make the students could visualize and deepen the knowledge previously studied in books, aiming at a better xation of the content. It is designed with questions, so that students examine the de nitions, discuss the results, and from there to better understand the content, and know a little of the appearance, the history and evolution of knowledge on conic. / Ao lecionar cônicas nas turmas do 3a ano do Ensino Médio, sempre observava que a maioria dos alunos não se lembravam das principais de ni coes, e tinha muita di dificuldade. Visando melhorar esse aprendizado, pensamos em uma proposta de intervenção que fizesse o aluno trabalhar, de uma forma mais prazerosa, esse conte udo. Com o advento da computa c~ao e do software GeoGebra, elaboramos roteiros did áticos pr áticos, para que eles pudessem revisar e aprofundar os principais conceitos das conicas. Esse trabalho foi então elaborado de forma a fazer com que os alunos pudessem visualizar e aprofundar os conhecimentos previamente estudados nos livros, visando a uma melhor fixação do conteúdo. Os roteiros foram planejado com questões, de modo que o aluno analisassem as de ni coes, discutissem os resultados obtidos, e a partir da melhor compreendessem o conte udo, além de conhecer um pouco do surgimento, da história e da evolução do conhecimento sobre cônicas.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Neste trabalho, estudam-se de maneira sucinta as caracterÃsticas e os principais elementos algÃbricos e geomÃtricos das secÃÃes cÃnicas. Inicialmente, foram abordados os trÃs tipos gerais das secÃÃes cÃnicas, enfocando em princÃpio suas formas regulares para, em dado momento, tambÃm se discutir suas formas degeneradas. Verificou-se que, segundo um sistema cartesiano fixado, à possÃvel estabelecer uma representaÃÃo global das cÃnicas por meio de uma equaÃÃo algÃbrica do segundo grau em duas variÃveis, sendo que qualquer equaÃÃo desta forma,reciprocamente, pode ser identificada, por meio de rotaÃÃo ou translaÃÃo, com a equaÃÃo reduzida de algum tipo particular de cÃnica. Encerrando-se este trabalho, foi proposta uma abordagem inovadora para a lida no ensino mÃdio desse tÃpico de ensino-aprendizagem riquÃssimo em conceitos bÃsicos de Ãlgebra e geometria elementares, tÃo necessÃrios para uma boa formaÃÃo matemÃtica. Foi proposto que no estudo das cÃnicas se use de forma complementar a maneira alternativa de defini-las a partir de sua excentricidade, convergindo para a equaÃÃo geral unificada das cÃnicas em coordenadas polares. Essa forma generalizada e absolutamente simples de se representar as cÃnicas à tida como facilmente manipulÃvel e reflete em si mesma uma beleza singular da MatemÃtica, a Rainha das CiÃncias. Vale ressaltar que todo o enfoque dos conteÃdos aqui discutidos foi apresentado numa linguagem bastante simples e de fÃcil compreensÃo, uma vez que este trabalho se destina preferencialmente a alunos e professores da educaÃÃo bÃsica. / In this work, the features and the main algebraic and geometric elements of conic sections were succinctly studied. lnitially, the three general types of the conic sections were discussed, focusing at first on their regular forms and then, in certain moment, discussing about their degenerate forms. It was found that, according to a fixed Cartesian system, it is possible to establish a global representation of the conics through a two-variable algebraic quadratic equation, in a way that, in any equation of this form, reciprocally, can be identified by means of rotation or translation, with the reduced equation of any particular type of conic. At the end of this work, it was proposed an innovative approach of this rich teaching-leaming topic in basic concepts of elementary algebra and geometry to be applied in high school, as required for good mathematical training. lt was proposed that the study of conic would be used complementing the alternativa way to define them from their eccentricity, converging to a unified general equation of the conics in polar coordinates. This generalized way and absolutely simple to represent the conical is seen as easily manipulated and reflects on itself a singular beauty of mathematics, The Queen of Sciences. lt is worth to emphasize that the focus of the content discussed was presented in a very simple, easy to understand language, once this work is preferably intended for students and teachers of basic education.

nÃo hà / Neste trabalho, falaremos sobre cÃnicas e equaÃÃo geral do segundo grau, histÃrico e parte teÃrica, aplicando estes conhecimentos na elaboraÃÃo de planilhas eletrÃnicas para identificar qual a cÃnica (elipse, hipÃrbole ou parÃbola, bem como seus casos degenerados) e determinar seus principais elementos a partir de suas equaÃÃes, na forma canÃnica ou geral do segundo grau. Para melhor compreensÃo e fixaÃÃo do que vai ser exposto, faremos uso de atividades interativas com a utilizaÃÃo de computador. / In this paper, we will discuss general conic and quadratic equation, historical and theoretical part, applying this knowledge in designing spreadsheets to identify which conic (ellipse, parabola or hyperbola, as well as their degenerate cases) and determine its main elements from their equations in canonical or general high school.For better understanding and assessment of what will be, we will make use of interactive activities with the use of computer.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2015-04-27T13:09:17Z No. of bitstreams: 2 Dissertação - Ruimar Calaça de Menezes - 2014.pdf: 3563938 bytes, checksum: 780f3827908c962633ef10e7001a936e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-04-27T13:14:20Z (GMT) No. of bitstreams: 2 Dissertação - Ruimar Calaça de Menezes - 2014.pdf: 3563938 bytes, checksum: 780f3827908c962633ef10e7001a936e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-04-27T13:14:20Z (GMT). No. of bitstreams: 2 Dissertação - Ruimar Calaça de Menezes - 2014.pdf: 3563938 bytes, checksum: 780f3827908c962633ef10e7001a936e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-03-06 / The objective of this study is to present the Quadratic Function in a new graphic perspective. The Quadratic Function content is taught in the ninth year of elementary school and rst year of high school, the following concepts are learned: equations and inequalities involving polynomials of second degree, graphing, concavity, zeros of the quadratic function, the parable vertex, minimum or maximum and applications in other elds of knowledge. Two mathematical softwares, Winplot and Geogebra, were used in order to facilitate the understanding of graphic representations of functions. The parable was discussed from a historical point of view that also included the concept of speci c terms and de nitions involving functions. The concept of tangent lines to growth and decrease of functions was related. The parable was also exhibited in the context of Analytic Geometry and Calculus, emphasizing the in uence of each coe cient in relation to the graph. The parable was studied as one of the conic sections so studied by Apollonius. Some applications of the parable in daily life, physics and modeling problems were cited. / Este trabalho tem como objetivo apresentar a Fun c~ao quadr atica em uma nova perspectiva gr a ca. O conte udo Fun c~ao Quadr atica e ensinado no nono ano do Ensino Fundamental e primeira s erie do Ensino M edio, sendo trabalhados os seguintes conceitos: equa c~oes e inequa c~oes envolvendo polin^omios de 2o grau, representa c~ao gr a ca, concavidade, zeros da fun c~ao quadr atica, v ertice da par abola, valor m nimo ou m aximo e aplica c~oes em outras areas do conhecimento. Para desenvolver este trabalho, foram utilizados os softwares matem aticos Winplot e Geogebra, com o intuito de facilitar a compreens~ao das representa c~oes gr a cas das fun c~oes. A par abola foi comentada a partir de um contexto hist orico e abrangeu tamb em o conceito de termos espec cos e de ni c~oes envolvendo fun c~oes. Foi relacionado o conceito de retas tangentes ao crescimento e decrescimento das fun c~oes. Tamb em foi exibido a par abola no contexto da Geometria Anal tica e do c alculo diferencial, salientando a in u^encia de cada coe ciente em rela c~ao ao gr a co. A par abola foi estudada como uma das sec c~oes c^onicas t~ao pesquisadas por Apol^onio. Foi citada algumas aplica c~oes da par abola no cotidiano, na f sica e em problemas de modelagem.

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2016-08-30T20:00:18Z No. of bitstreams: 2 Dissertação - Anderson de Azevedo Gomes - 2015.pdf: 2449743 bytes, checksum: 3ff7d4c2fbea731994b5df9da8e7311d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Jaqueline Silva (jtas29@gmail.com) on 2016-08-30T20:00:30Z (GMT) No. of bitstreams: 2 Dissertação - Anderson de Azevedo Gomes - 2015.pdf: 2449743 bytes, checksum: 3ff7d4c2fbea731994b5df9da8e7311d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-08-30T20:00:30Z (GMT). No. of bitstreams: 2 Dissertação - Anderson de Azevedo Gomes - 2015.pdf: 2449743 bytes, checksum: 3ff7d4c2fbea731994b5df9da8e7311d (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2015-11-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Beginning with the concept plan and how it can be represented in a Vector Plan. Show the identity of vectors and the basic properties of the vector sum with examples and continue with the properties of the Scalar product and how to nd the Angle Between Vectors. After we nd the projection vectors and the distance between a point and a Straight. In the next chapter, I review on Conic Sections. Tract of Ellipse, after Hyperbole and analyze talking about the parable and its utilities. In sequence, show the Parametric equations of a curve. Come to a formula of how to nd a tangent vector and how to nd the length of an arc. Show the usefulness of the Frenet formulas and do the Fundamental Theorem of Plane Curves and their utilities. To end tract Importance of Mathematics in Primary Education with a little history of Dynamic Geometry Software and Software Geogebra, showing how to make some charts in GeoGebra. My goal is to show the beauty of mathematics and attract people to the area. / Começo com o conceito de Plano e como pode ser representado um Vetor no Plano. Mostro a identidade de vetores e as propriedades básicas da soma de vetores com exemplos e continuo com as propriedades do Produto Escalar e como encontrar o Ângulo entre os vetores. Depois, encontramos a projeção de vetores e a distância entre um ponto e uma reta. No próximo capítulo, faço uma revisão sobre as Seções Cônicas. Começo falando da Elipse, depois da Hipérbole e nalizo falando da Parábola e suas utilidades. Em sequência, mostro as Equações Paramétricas de uma curva. Chego a uma fórmula de como encontrar um Vetor Tangente e como encontrar o Comprimento de um Arco. Mostro a utilidade das Fórmulas de Frenet e faço o Teorema Fundamental das Curvas Planas e suas utilidades. Finalizo mostrando como fazer alguns grá cos no GeoGebra e da Importância da Matemática na Formação Básica com um pouco de História dos Softwares de Geometria Dinâmica. Meu objetivo é mostrar a beleza da matemática e atrair pessoas para essa área.

The dissertation research investigates estimating of power system static and dynamic states (e.g. rotor angle, rotor speed, mechanical power, voltage magnitude, voltage phase angle, mechanical reference point) as well as identification of synchronous generator parameters. The research has two focuses: i. Synchronous generator dynamic model states and parameters estimation using real-time PMU data. ii.Integrate PMU data and conventional measurements to carry out static state estimation. The first part of the work focuses on Phasor Measurement Unit (PMU) data-based synchronous generator states and parameters estimation. In completed work, PMU data-based synchronous generator model identification is carried out using Unscented Kalman Filter (UKF). The identification not only gives the states and parameters related to a synchronous generator swing dynamics but also gives the states and parameters related to turbine-governor and primary and secondary frequency control. PMU measurements of active power and voltage magnitude, are treated as the inputs to the system while voltage phasor angle, reactive power, and frequency measurements are treated as the outputs. UKF-based estimation can be carried out at real-time. Validation is achieved through event play back to compare the outputs of the simplified simulation model and the PMU measurements, given the same input data. Case studies are conducted not only for measurements collected from a simulation model, but also for a set of real-world PMU data. The research results have been disseminated in one published article. In the second part of the research, new state estimation algorithm is designed for static state estimation. The algorithm contains a new solving strategy together with simultaneous bad data detection. The primary challenge in state estimation solvers relates to the inherent non-linearity and non-convexity of measurement functions which requires using of Interior Point algorithm with no guarantee for a global optimum solution and higher computational time. Such inherent non-linearity and non-convexity of measurement functions come from the nature of power flow equations in power systems. The second major challenge in static state estimation relates to the bad data detection algorithm. In traditional algorithms, Largest Normalized Residue Test (LNRT) has been used to identify bad data in static state estimation. Traditional bad data detection algorithm only can be applied to state estimation. Therefore, in a case of finding any bad datum, the SE algorithm have to rerun again with eliminating found bad data. Therefore, new simultaneous and robust algorithm is designed for static state estimation and bad data identification. In the second part of the research, Second Order Cone Programming (SOCP) is used to improve solving technique for power system state estimator. However, the non-convex feasible constraints in SOCP based estimator forces the use of local solver such as IPM (interior point method) with no guarantee for quality answers. Therefore, cycle based SOCP relaxation is applied to the state estimator and a least square estimation (LSE) based method is implemented to generate positive semi-definite programming (SDP) cuts. With this approach, we are able to strengthen the state estimator (SE) with SOCP relaxation. Since SDP relaxation leads the power flow problem to the solution of higher quality, adding SDP cuts to the SOCP relaxation makes Problem’s feasible region close to the SDP feasible region while saving us from computational difficulty associated with SDP solvers. The improved solver is effective to reduce the feasible region and get rid of unwanted solutions violate cycle constraints. Different Case studies are carried out to demonstrate the effectiveness and robustness of the method. After introducing the new solving technique, a novel co-optimization algorithm for simultaneous nonlinear state estimation and bad data detection is introduced in this dissertation. ${\ell}_1$-Norm optimization of the sparse residuals is used as a constraint for the state estimation problem to make the co-optimization algorithm possible. Numerical case studies demonstrate more accurate results in SOCP relaxed state estimation, successful implementation of the algorithm for the simultaneous state estimation and bad data detection, and better state estimation recovery against single and multiple Gaussian bad data compare to the traditional LNRT algorithm.

This thesis is intended for teachers and students of high schools and universities. It consists collection of solved problems from plane analytical geometry including various solutions and their comparison. The thesis aims to increase the student knowledge of the topic and to provide different approaches to problems and working materials for lessons for teachers. Pictures for better understanding are added for more difficult problems. The practical part focusing on common mistakes and their elimination is included.