31 Reunión Latinoamericana de Matemática Educativa (Relme), evento desarrollado en la Universidad de Medellín, Colombia, del 31 Julio al 04 de Agosto de 2017. / Esta investigación presenta el análisis de cómo se generó la génesis instrumental de la noción de integral definida en estudiantes universitarios desde el Enfoque instrumental, mediada por el Geogebra. Al contrastar los resultados esperados y obtenidos por medio de la Ingeniería didáctica, se observó que los estudiantes generaron esquemas de utilización de la integral definida al emplearla en la resolución de problemas matemáticos, evidenciando la instrumentalización e instrumentación de dicha noción matemática. / This research shows the analysis of how the instrumental genesis of definite-integral notion was generated by university students from the GeoGebra-based Instrumental Approach. When contrasting the expected results to the ones obtained by using Didactic Engineering, we could observe that the students generated schemes to use the definite integral by using it in solving mathematical problems, showing the instruments used and the instrumentation of this mathematical notion.

In this dissertation we study the Lyapunov diagonal stability and its extensions through partitions of the index set {1,...,n}. This type of matrix stability plays an important role in various applied areas such as population dynamics, systems theory and complex networks. We first examine a result of Redheffer that reduces Lyapunov diagonal stability of a matrix to common diagonal Lyapunov solutions on two matrices of order one less. An enhanced statement of this result based on the Schur complement formulation is presented here along with a shorter and purely matrix-theoretic proof. We develop a number of extensions to this result, and formulate the range of feasible common diagonal Lyapunov solutions. In particular, we derive explicit algebraic conditions for a set of 2 x 2 matrices to share a common diagonal Lyapunov solution. In addition, we provide an affirmative answer to an open problem concerning two different necessary and sufficient conditions, due to Oleng, Narendra, and Shorten, for a pair of 2 x 2 matrices to share a common diagonal Lyapunov solution. Furthermore, the connection between Lyapunov diagonal stability and the P-matrix property under certain Hadamard multiplication is extended. Accordingly, we present a new characterization involving Hadamard multiplications for simultaneous Lyapunov diagonal stability on a set of matrices. In particular, the common diagonal Lyapunov solution problem is reduced to a more convenient determinantal condition. This development is based upon a new concept called P-sets and a recent result regarding simultaneous Lyapunov diagonal stability by Berman, Goldberg, and Shorten. Next, we consider various types of matrix stability involving a partition alpha of {1,..., n}. We introduce the notions of additive H(alpha)-stability and P_0(alpha)-matrices, extending those of additive D-stability and nonsingular P_0-matrices. Several new results are developed, connecting additive H(alpha)-stability and the P_0(alpha)-matrix property to the existing results on matrix stability involving alpha. We also point out some differences between these types of matrix stability and the conventional matrix stability. Besides, the extensions of results related to Lyapunov diagonal stability, D-stability, and additive D-stability are discussed. Finally, we introduce the notion of common alpha-scalar diagonal Lyapunov solutions over a set of matrices, which is a generalization of common diagonal Lyapunov solutions. We present two different characterizations of this new concept based on the well-known results for Lyapunov alpha-scalar stability [34]. The first one hinges on a general version of the theorem of the alternative, and the second one using Hadamard multiplications stems from an extension of the P-set property. Several illustrative examples and an application concerning a set of block upper triangular matrices are provided.

The thesis describes the usage of definite article with proper nouns in current Spanish language. Proper nouns distribution and their dependence of the definite article according to the grammar books and textbooks is explained in the first part. The proper nouns are divided into geographical and personal nouns and these two groups are classified into charts according to the usage of the definite article. The second part deals with the selective geographical nouns and their usage of definite article in Spanish speaking countries according to the corpus CREA. The achieved results are analysed in the last part.

Inicialmente, estudamos alguns resultados clássicos da teoria dos núcleos positivos definidos e alguns resultados pertinentes. Estudamos em seguida, o Teorema de Mercer e algumas de suas generalizações e conseqüências, incluindo a caracterização da transformada de Fourier de um núcleo positivo definido com domínio Rm£Rm, m ¸ 1. O trabalho traz um enfoque especial nos núcleos cujo domínio é um subconjunto não-compacto de Rm £ Rm, uma vez que os demais casos são considerados de maneira extensiva na literatura. Aplicamos esses estudos na análise do decaimento dos autovalores de operadores integrais gerados por núcleos positivos definidos / Firstly, we study some classical results from the theory of positive definite kernels along with some related results. Secondly, we focus on generalizations of Mercer\'s theorem and some of their implications. Special attention is given to the cases where the domain of the kernel is not compact, once the other cases are considered consistently in the literature. We include a characterization for the Fourier transform of a positive definite kernel on Rm£Rm, m ¸ 1. Finally, we apply the previous study in the analysis of decay rates for eigenvalues of integral operators generated by positive definite kernels

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.

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We investigate the ultraconnectivity condition on graphs, and provide further connections between critical and ultraconnected graphs in the positive definite partial matrix completion problem. We completely characterize when the join of graphs is ultraconnected, and prove that ultraconnectivity is preserved by Cartesian products. We completely characterize when adding a vertex to an ultraconnected graph preserves ultraconnectivity. We also derive bounds on the number of vertices which guarantee ultraconnectivity of certain classes of regular graphs. We give results from our exhaustive enumeration of ultraconnected graphs up to 11 vertices. Using techniques involving the Lovász theta parameter for graphs, we prove certain classes of graphs are critical (and hence ultraconnected) in the positive definite partial matrix completion problem.

<p>Parallel MRI, in which k-space is regularly or irregularly undersampled, is critical for imaging speed acceleration. In this thesis, we show how to optimize a regular undersampling pattern for three-dimensional Cartesian imaging in order to achieve faster data acquisition and/or higher signal to noise ratio (SNR) by using nonlinear optimization. A new sensitivity profiling approach is proposed to produce better sensitivity maps, required for the sampling optimization. This design approach is easily adapted to calculate sensitivities for arbitrary planes and volumes. The use of a semi-definite, linearly constrained model to optimize a parallel MRI undersampling pattern is novel. To solve this problem, an iterative trust-region is applied. When tested on real coil data, the optimal solution presents a significant theoretical improvement in accelerating data acquisition speed and eliminating noise.</p> / Master of Applied Science (MASc)

Let Q(√(-d)) be an imaginary quadratic field with discriminant Δ. We use the isomorphism between the ideal class groups of the field and the equivalence classes of binary quadratic forms to find the structure of the class group. We determine the structure by combining two of Shanks' algorithms [7, 8]. We utilize this method to find fields with cyclic factors that have order a large power of 2, or fields with class groups of high 5-ranks or high 7-ranks. / Master of Science

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex num hers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagona lizab le matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingutar matrices into Involutions. Chapter 5 studies factorization of a comp 1 ex matrix into Positive-( semi )definite matrices, emphasizing the least number of such factors required / Mathematical Sciences / M.Sc. (MATHEMATICS)