Flórez Gómez, Edwin
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Publication Tournament matrices: Survey and new results(2007) Flórez Gómez, Edwin; Yong, Xuerong; College of Arts and Sciences - Sciences; Urintsev, Alexander; Acar, Robert; Department of Mathematics; Macchiavell, RaúlTournament are simple and complete direct graph. In this thesis we survey and study particular casa of tournament. Since the famous Seven bridges problem, which was analyzed by Leonard Euler in 1736 and stimulated the development of graph theory, graph have been considered an important subject in mathematics and other applied science, such as physics , biology, chemistry, etc. Over the last decades, the study of graph spectra has been interesting, because is characterized the topological structure of a graph. But it turn out that this is noe easy to attack. In this thesiswe obtain new results about tournaments matrices, in particular, about Brualdi-Li matrix and r-partite tournament matrices. The original inspiration of the thesis was improve and extended the ideas introduces in Algebraix Multiplicity of the eigenvalue of a bipartite tournament matrix, by Yi-Zheng Fan and Jiong sheng Li published in SIAm Journal on Matrix Analysis and Application (SIMAX, 2002) and in upper bounf on the perron vlue of almost regular tournament matrix, by S Kirkland, in linear Algebra and its Application (2003).Publication A novel framework of structured measurement matrix for compressed sensing in wireless sensor networks(2018-05) Flórez Gómez, Edwin; Lu, Kejie; College of Engineering; Ierkic, Henrick Mario; Arzuaga, Emmanuel; Portnoy, Arturo; Seguel, Jaime; Department of Electrical and Computer Engineering; Rullán, AgustínWireless Sensor Network (WSN) is a wireless networking technology that can facilitate many important applications in our real life, from environmental monitoring to smart grid, and thus is a key component in the emerging Internet of Things (IoT). To design e cient WSNs, two major issues are (1) the throughput capacity that is the maximal data rate at which a WSN can collect data from a eld, and (2) the delay that is the duration from the time a signal is sensed to the time that is a received by the gateway of a WSN, known as the sink. To address these two issues, there are many solutions in the literature and Compressed Sensing (CS) is one of the most promising solutions because it can combine data collection and compression at the same time. By using CS, sensor nodes can collaboratively generate measurements by using a measurement matrix to linearly combine the original signals. In theory, if the original signals are sparse, then they can be reconstructed by well-known convex optimization, using a much smaller number of measurements, which leads to much higher throughput and smaller delay. In the literature, several CS based data collection schemes for WSNs have been investigated. However, the impact of the measurement matrix has not been fully investigated. On one hand, many researchers focused on the performance of WSN by simply assuming there exists a measurement matrix. On the other hand, some researchers focused on the design of measurement matrix, particularly the structured measurement matrix, without considering the feature of WSN. Therefore, there is still a signi cant gap between CS and WSN. In this dissertation, we aim at tackling this challenging issue and we propose a novel framework for structured measurement matrix to improve the performance of data collection in WSN. Speci cally, the proposed structured measurement matrix consists of rectangular blocks with non-zero elements. Each of the blocks is used to produce measurements by linearly combining original signals collected from a subset of sensors. Moreover, if two bands are adjacent in the matrix, then the corresponding subset can have intersections, and all such intersections have the same cardinality. In this manner, the measurement matrix is a circular overlapping block diagonal (COB) matrix. To evaluate the performance of the proposed COB matrix, we rst investigate a particular COB case, in which the cardinality of the intersection is one half of the cardinality of a subset of sensors. For this type of matrix, we conduct theoretical analysis to prove that it satis es the Restricted Isometric Property (RIP), which is widely used to determine the minimal number of measurements that can guarantee the reconstruction of the original signals. The theoretical analysis reveals the impacts of several important factors, including the number of blocks, the sparsity of original signals, and the total number of signals. We also conduct extensive simulation and the numerical results validate the theoretical analysis and demonstrate that the proposed COB matrix outperform existing block diagonal matrix. Based on the understandings from the speci c COB case, we generalize the COB in a way such that the size of overlapping can be an arbitrary number. For the generalized COB, we rst prove that it also satis es the RIP with a certain bound for the number of measurement. In addition to the aforementioned factors, we investigate the impact of the size of overlapping. Extensive numerical results show that our analysis again is very accurate. Finally, we conduct theoretical analysis to evaluate the throughput and delay performance of CS-based WSN with the proposed measurement matrix. In the analysis, we rst derive schemes to partition a unit area into equal-sized region, we then develop feasible time division multiple access (TDMA) schemes to facilitate two sensing scenarios in WSN. Using the theoretical analysis, we further analyze the performance of WSN using practical settings, such as the transmission range, data rate, etc. The numerical results con rm that the proposed COB scheme can improve throughout and delay performance.