En el espacio de matrices se pueden definir distintas operaciones, cada una de las cuales presenta aplicaciones diferentes. El producto usual de matrices representa la composición de transformaciones lineales, y el mismo está definido sólo entre matrices que respetan la siguiente propiedad: el número de columnas de la primera matriz coincide con el número de filas de la segunda. El producto de Kronecker se define para cualquier par de matrices, y representa el producto tensorial de las transformaciones lineales asociadas a cada una de las matrices. Este producto es asociativo, bilineal, no conmutativo, y se comporta bien con la inversa y con el cálculo de valores singulares. En el trabajo [I. Ojeda, Kronecker square roots and the block vec matrix, Amer. Math. Monthly 122 (2015), no. 1, 60–64] se estudia la existencia de las raíces cuadradas del producto de Kronecker, esto es, dada una matriz A se estudia, bajo qué condiciones, existe una matriz B tal que A=B⊗B. Estas condiciones se describen en función de la simetría y del rango de una matriz especial construida a partir de A. El propósito de este trabajo es establecer condiciones necesarias y suficientes para la existencia de raíces enésimas de Kronecker de una matriz dada. Empleando propiedades del producto de Kronecker y de la vectorización de matrices, construimos una matriz especial cuyas características nos permiten decidir cuándo una matriz es potencia de Kronecker de otra matriz dada. En caso afirmativo, describimos un algoritmo que nos permite calcular dicha matriz. En caso negativo, encontramos cotas de min┬⁡〖∥A-X^(⨂n ) ∥_2 〗 en función de los valores singulares de A. Así mismo se estudian dos problemas de Procrusto que involucran sumas y potencias de Kronecker. Los resultados teóricos desarrollados son aplicados a problemas vinculados a la identificación de grafos de Kronecker y a la resolución de ciertas ecuaciones matriciales. / Different operations can be defined in the space of matrices, each of which has different applications. The usual product of matrices represents the composition of linear transformations, and it is defined only between matrices that respect the following property: the number of columns in the first matrix coincides with the number of rows in the second one. The Kronecker product is defined for any pair of matrices, and represents the tensor product of the linear transformations associated with each of these matrices. This product is associative, bilinear, non-commutative, and behaves well with the inverse and with the calculation of singular values. The existence of square roots for the Kronecker product is studied in the paper [I. Ojeda, Kronecker square roots and the block vec matrix, Amer. Math. Monthly 122 (2015), no. 1, 60–64], that is, given a matrix A, there are certain conditions that ensures the existence of a matrix Bsuch that A=B⨂B. These conditions are described in terms of the symmetry and the rank of a special matrix associated to A. The purpose of this work is to establish necessary and sufficient conditions for the existence of n-th roots of Kronecker of a given matrix. Using properties of the Kronecker product and of the vectorization of matrices, we construct a special matrix whose characteristics allow us to decide when a matrix is the Kronecker power of another given matrix. If so, we describe an algorithm that allows us to find such matrix. If not, we find bounds of min┬⁡〖〖∥A-X^(⨂n)∥〗_2 〗 in terms of the singular values of A. Finally we study two Procrusto problems involving Kronecker sums and powers. The theoretical results developed are applied to problems related to the identification of Kronecker graphs and the resolution of certain matrix equations.

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Orientador: T.M. Viswanathan / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Científica / Made available in DSpace on 2018-07-14T18:55:37Z (GMT). No. of bitstreams: 1 Datta_Karabi_D.pdf: 1911726 bytes, checksum: 53a7f5d3f9aa2cb5a34874a76c56abb6 (MD5) Previous issue date: 1982 / Resumo: Não informado. / Abstract: Not informed. / Doutorado / Doutor em Matemática

PEREIRA, José Ilhano da Silva. Hipersuperfícies mínimas de R4 com curvatura de Gauss-Kronecker nula. 2017. 44 f. Dissertação (Mestrado em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-10-02T15:01:31Z No. of bitstreams: 1 2017_dis_jispereira.pdf: 596580 bytes, checksum: 3c2c1a16d4ce273bfb7c246f7926c01a (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Estou devolvendo a Dissertação de JOSÉ ILHANO DA SILVA PEREIRA, pois há alguns erros a serem corrigidos. Os mesmos seguem listados a seguir. 1- FOLHA DE APROVAÇÃO (substitua a folha de aprovação, por outra que não contenha as assinaturas dos membros da banca examinadora) 2- NUMERAÇÃO INDEVIDA (a numeração indevida de página que aparece na folha de aprovação deve ser retirada) 3- RESUMO (retire o recuo de parágrafo presente no resumo e no abstract) 4- PALAVRAS-CHAVE (apenas o primeiro elemento de cada palavra-chave deve começar com letra maiúscula, assim reescreva as palavras-chave como no exemplo a seguir: Hipersuperfícies mínimas) 5- SUMÁRIO (Os títulos dos capítulos principais, que aparecem no sumário e no interior do trabalho, devem estar em caixa alta (letra maiúscula). Ex.: 2 PRELIMINARES 2.1 Tensores 6 – REFERÊNCIAS (retire o conjunto de “citações” à autores que aparece no final das referências bibliográficas, pois elas fogem ao padrão ABNT para a página das referências) Atenciosamente, on 2017-10-04T17:50:58Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-10-23T19:57:28Z No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-11-01T12:35:13Z (GMT) No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) / Made available in DSpace on 2017-11-01T12:35:13Z (GMT). No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) Previous issue date: 2017-08-25 / This work does study the complete minimal hypersurfaces in the Euclidean space R4 , with Gauss-Kronecker curvature identically zero. Our main result is to prove that if f: M3 → R4 is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature boun-ded from below, then f(M3) splits as a Euclidean product L2 × R , where L2 is a complete minimal surface in R3 with Gaussian curvature bounded from below. Moreover, we show a result about the Gauss-Kronecker curvature of f, without any assumption on the scalar curvature. / Este trabalho tem como objetivo estudar as hipersuperfícies mínimas em R4, com curvatura de Gauss-Kronecker identicamente zero. Como resultado principal provamos que se f : M3 → R4 é uma hipersuperfície mínima com curvatura de Gauss-Kronecker identicamente zero, segunda forma fundamental não se anulando em nenhum ponto e curvatura escalar limitada inferiormente, então f(M3) se decompõe como um produto euclidiano do tipo L2 × R , onde L2 é uma superfície mínima de R3 com curvatura Gaussiana limitada inferiormente. Finalmente, apresentamos um resultado sobre a curvatura de Gauss-Kronecker de f sem nenhuma hipótese sobre a curvatura escalar.

Thesis advisor: Patrick Byrne / In the late 19th century, Georg Cantor opened up the mathematical field of set theory with his development of transfinite numbers. In his radical departure from previous notions of infinity espoused by both mathematicians and philosophers, Cantor created new notions of transcendence in order to clearly described infinities of different sizes. Leading the opposition against Cantor's theory was Leopold Kronecker, Cantor's former mentor and the leading contemporary German mathematician. In their lifelong dispute over the transfinite numbers emerge philosophical disagreements over mathematical existence, consistency, and freedom. This thesis presents a short summary of Cantor's controversial theories, describes Cantor and Kronecker's philosophical ideas, and attempts to state clearly their differences of opinion. In the end, the author hopes to present the shock caused by Cantor's work and an appreciation of the two very different philosophies of mathematics represented by Cantor and Kronecker. / Thesis (BA) — Boston College, 2005. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Philosophy. / Discipline: College Honors Program.

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Reductions in the cost and size of integrated circuits are allowing more and more complex functions to be included in previously simple tools such as lawn-mowers, ovens, and thermostats. Because of this, the process of synthesizing such functions from their initial representation to an optimal VLSI implementation is rarely hand-performed; instead, automated synthesis and optimization tools are a necessity. The factors such tools must take into account are numerous, including area (size), power consumption, and timing factors, to name just a few. Existing tools have traditionally focused upon optimization of two-level representations. However, new technologies such as Field Programmable Gate Arrays (FPGAs) have generated additional interest in three-level representations and structures such as Kronecker Decision Diagrams (KDDs). The reason for this is that when implementing a circuit on an FPGA, the cost of implementing exclusive-or logic is no more than that of traditional AND or OR gates. This dissertation investigates the use of the autocorrelation coefficients in logic synthesis for these types of structures; specifically, whether it is possible to pre-process a function to produce a subset of its autocorrelation coefficients and make use of this information in the choice of a three-level decomposition or of decomposition types within a KDD. This research began as a general investigation into the properties of autocorrelation coefficients of switching functions. Much work has centered around the use of a function's spectral coefficients in logic synthesis; however, very little work has used a function's autocorrelation coefficients. Their use has been investigated in the areas of testing, optimization for Programmable Logic Arrays (PLAs), identification of types of complexity measures, and in various DD-related applications, but in a limited manner. This has likely been due to the complexity in their computation. In order to investigate the uses of these coefficients, a fast computation technique was required, as well as knowledge of their basic properties. Both areas are detailed as part of this work, which demonstrates that it is feasible to quickly compute the autocorrelation coefficients. With these investigations as a foundation we further apply the autocorrelation coefficients to the development of a classification technique. The autocorrelation classes are similar to the spectral classes, but provide significantly different information. The dissertation demonstrates that some of this information highlighted by the autocorrelation classes may allow for the identification of exclusive-or logic within the function or classes of functions. In relation to this, a major contribution of this work involves the design and implementation of algorithms based on these results. The first of these algorithms is used to identify three-level decompositions for functions, and the second to determine decomposition type lists for KDD-representations. Each of these implementations compares well with existing tools, requiring on average less than one second to complete, and performing as well as the existing tools about 70% of the time. / Graduate

Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. On the other hand, the Inverse Galois Problem, given a finite group G, find a finite extension of the rational field Q whose Galois group is G, is still an open problem. We give an introduction to the Inverse Galois Problem and compare some radically different approaches to finding an extension of Q that gives a desired Galois group. In particular, a proof of the Kronecker-Weber theorem, that any finite extension of Q with an abelian Galois group is contained in a cyclotomic extension, will be discussed using an approach relying on the study of ramified prime ideals. In contrast, a different method will be explored that defines rigid groups to be groups where a selection of conjugacy classes satisfies a series of specific properties. Under the right conditions, such a group is also guaranteed to be the Galois group of an extension of Q. / Master of Science

Neste trabalho mostramos que hipersuperfícies completas da esfera Euclidiana S^4, com curvatura média constante e curvatura de Gauss-Kronecker nula são mínimas, sempre que o quadrado da norma da segunda forma fundamental for limitado superiormente. Além disso apresentamos uma descrisão local das hipersuperfícies mínimas e completas em S^5 com curvatura de Gauss- Kronecker nula e algumas hipóteses adicionais sobre as funções simétricas das curvaturas principais. / In this work we show that a complete hipersurface of the unitary sphere S^4, with constant mean curvature and zero Gauss-Kronecker curvature must be minimal, if the squared norm of the second fundamental form is bounded from above. Also, we present a local description for complete minimal hipersurfaces in S^5 with zero Gauss-Kronecker curvature, and some restrictions for the symmetric functions of the principal curvatures.

Neste trabalho mostramos que hipersuperfícies completas da esfera Euclidiana S^4, com curvatura média constante e curvatura de Gauss-Kronecker nula são mínimas, sempre que o quadrado da norma da segunda forma fundamental for limitado superiormente. Além disso apresentamos uma descrisão local das hipersuperfícies mínimas e completas em S^5 com curvatura de Gauss- Kronecker nula e algumas hipóteses adicionais sobre as funções simétricas das curvaturas principais. / In this work we show that a complete hipersurface of the unitary sphere S^4, with constant mean curvature and zero Gauss-Kronecker curvature must be minimal, if the squared norm of the second fundamental form is bounded from above. Also, we present a local description for complete minimal hipersurfaces in S^5 with zero Gauss-Kronecker curvature, and some restrictions for the symmetric functions of the principal curvatures.