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Transformações lineares no plano e aplicações / Linear transformations on the plane and applicationsNogueira, Leonardo Bernardes 15 March 2013 (has links)
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Previous issue date: 2013-03-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper begins with a brief history about the development of vector spaces and linear
transformations, then presents fundamental concepts for the study of Linear Algebra, with
greater focus on linear operators in the R2 space. Through examples it explores a wide
range of operators in R2 in order to show other applications of matrices in high school
and prepares the ground for the presentation a version of Spectral Theorem for selfadjoint
operators in R2, which says that for every operator self-adjoint T : E!E in finite
dimensional vector space with inner product, exists an orthonormal basis fu1; : : : ;ung E
formed by eigenvectors of T, and culminates with their applications on the study of conic
sections, quadratic forms and equations of second degree in x and y; on the study of
operators associated to quadratic forms, a version of Spectral Theorem could be called
as The Main Axis Theorem albeit this nomenclature is not used in this paper. Thereby
summarizing a study made by Lagrange in "Recherche d’arithmétique ", between 1773
and 1775, which he studied the property of numbers that are the sum of two squares.
Thus he was led to study the effects of linear transformation with integer coefficients in a
quadratic form in two variables. / Este trabalho inicia-se com um breve embasamento histórico sobre o desenvolvimento
de espaços vetoriais e transformações lineares. Em seguida, apresenta conceitos fundamentais
básicos, que formam uma linguagem mínima necessária para falar sobre Álgebra
Linear, com enfoque maior nos operadores lineares do plano R2. Através de exemplos,
explora-se um vasto conjunto de transformações no plano a fim de mostrar outras aplicações
de matrizes no ensino médio e prepara o terreno para a apresentação do Teorema
Espectral para operadores auto-adjuntos de R2. Este Teorema diz que para todo operador
auto-adjunto T : E!E, num espaço vetorial de dimensão finita, munido de produto
interno, existe uma base ortonormal fu1; : : : ;ung E formada por autovetores de T. O trabalho
culmina com aplicações sobre o estudo das secções cônicas, formas quadráticas e
equações do segundo grau em x e y, no qual o Teorema Espectral se traduz como Teorema
dos Eixos Principais, embora essa nomenclatura não seja usada nesse trabalho (para um
estudo mais aprofundado neste tema ver [3], [4], [5], [7]). Retomando assim um estudo
feito por Joseph Louis Lagrange em "Recherche d’Arithmétique", entre 1773 e 1775, no
qual estudou a propriedade de números que são a soma de dois quadrados. Assim, foi
levado a estudar os efeitos das transformações lineares com coeficientes inteiros numa
forma quadrática de duas variáveis.
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Tutte-Equivalent MatroidsRocha, Maria Margarita 01 September 2018 (has links)
We begin by introducing matroids in the context of finite collections of vectors from a vector space over a specified field, where the notion of independence is linear independence. Then we will introduce the concept of a matroid invariant. Specifically, we will look at the Tutte polynomial, which is a well-defined two-variable invariant that can be used to determine differences and similarities between a collection of given matroids. The Tutte polynomial can tell us certain properties of a given matroid (such as the number of bases, independent sets, etc.) without the need to manually solve for them. Although the Tutte polynomial gives us significant information about a matroid, it does not uniquely determine a matroid. This thesis will focus on non-isomorphic matroids that have the same Tutte polynomial. We call such matroids Tutte-equivalent, and we will study the characteristics needed for two matroids to be Tutte-equivalent. Finally, we will demonstrate methods to construct families of Tutte-equivalent matroids.
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Fibonacci VectorsSalter, Ena 20 July 2005 (has links)
By the n-th Fibonacci (respectively Lucas) vector of length m, we mean the vector whose components are the n-th through (n+m-1)-st Fibonacci (respectively Lucas) numbers. For arbitrary m, we express the dot product of any two Fibonacci vectors, any two Lucas vectors, and any Fibonacci vector and any Lucas vector in terms of the Fibonacci and Lucas numbers. We use these formulas to deduce a number of identities involving the Fibonacci and Lucas numbers.
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Classification of second order symmetric tensors in the Lorentz metricHjelm Andersson, Hampus January 2010 (has links)
This bachelor thesis shows a way to classify second order symmetric tensors in the Lorentz metric. Some basic prerequisite about indefinite and definite algebra is introduced, such as the Jordan form, indefinite inner products, the Segre type, and the Minkowski space. There are also some results concerning the invariant 2-spaces of a symmetric tensor and a different approach on how to classify second order symmetric tensor.
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The Minimum Witt Index of a GraphElzinga, Randall J. 17 September 2007 (has links)
An independent set in a graph G is a set of pairwise nonadjacent vertices, and the maximum size, alpha(G), of an independent set in G is called the independence number.
Given a graph G and weight matrix A of G with entries from some field F, the maximum dimension of an A-isotropic subspace, known as the Witt index of A, is an upper bound on alpha(G). Since any weight matrix can be used, it is natural to seek the minimum upper bound on the independence number of G that can be achieved by a weight matrix. This
minimum, iota_F^*(G), is called the minimum Witt index of G over F, and the resulting bound, alpha(G)<= iota_F^*(G), is called
the isotropic bound.
When F is finite, the possible values of iota_F^*(G) are determined and the graphs that attain the isotropic bound are characterized. The characterization is given in terms of graph classes CC(n,t,c)
and CK(n,t,k) constructed from certain spanning subgraphs called C(n,t,c)-graphs and K(n,t,k)-graphs. Here t is the term rank
of the adjacency matrix of G.
When F=R, the isotropic bound is known as the Cvetkovi\'c bound. It is shown that it is sufficient to consider a finite number of
weight matrices A when determining iota_R^*(G) and that, in many cases, two weight values suffice. For example, if the vertex set of G can be covered by alpha(G) cliques, then G attains the Cvetkovi\'c bound with a weight matrix with two weight values.
Inequalities on alpha and iota_F^* resulting from graph operations such as sums, products, vertex deletion, and vertex identification are examined and, in some cases, conditions that imply equality are proved. The equalities imply that the problem of determining whether or not alpha(G)=iota_F^*(G) can be reduced to that of determining iota_F^*(H) for certain crucial graphs H found from G. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-09-04 15:38:47.57
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Algorithm-Architecture Co-Design for Dense Linear Algebra ComputationsMerchant, Farhad January 2015 (has links) (PDF)
Achieving high computation efficiency, in terms of Cycles per Instruction (CPI), for high-performance computing kernels is an interesting and challenging research area. Dense Linear Algebra (DLA) computation is a representative high-performance computing ap-
plication, which is used, for example, in LU and QR factorizations. Unfortunately, mod-
ern off-the-shelf microprocessors fall significantly short of achieving theoretical lower bound in CPI for high performance computing applications. In this thesis, we perform an in-depth analysis of the available parallelisms and propose suitable algorithmic
and architectural variation to significantly improve the computation efficiency. There
are two standard approaches for improving the computation effficiency, first, to perform
application-specific architecture customization and second, to do algorithmic tuning.
In the same manner, we first perform a graph-based analysis of selected DLA kernels.
From the various forms of parallelism, thus identified, we design a custom processing
element for improving the CPI. The processing elements are used as building blocks for
a commercially available Coarse-Grained Reconfigurable Architecture (CGRA). By per-
forming detailed experiments on a synthesized CGRA implementation, we demonstrate
that our proposed algorithmic and architectural variations are able to achieve lower CPI compared to off-the-shelf microprocessors. We also benchmark against state-of-the-art custom implementations to report higher energy-performance-area product.
DLA computations are encountered in many engineering and scientific computing ap-
plications ranging from Computational Fluid Dynamics (CFD) to Eigenvalue problem.
Traditionally, these applications are written in highly tuned High Performance Comput-
ing (HPC) software packages like Linear Algebra Package (LAPACK), and/or Scalable
Linear Algebra Package (ScaLAPACK). The basic building block for these packages is Ba-
sic Linear Algebra Subprograms (BLAS). Algorithms pertaining LAPACK/ScaLAPACK
are written in-terms of BLAS to achieve high throughput. Despite extensive intellectual
efforts in development and tuning of these packages, there still exists a scope for fur-
ther tuning in this packages. In this thesis, we revisit most prominent and widely used
compute bound algorithms like GMM for further exploitation of Instruction Level Parallelism (ILP). We further look into LU and QR factorizations for generalizations and
exhibit higher ILP in these algorithms. We first accelerate sequential performance of the algorithms in BLAS and LAPACK and then focus on the parallel realization of these
algorithms. Major contributions in the algorithmic tuning in this thesis are as follows:
Algorithms:
We present graph based analysis of General Matrix Multiplication (GMM) and
discuss different types of parallelisms available in GMM
We present analysis of Givens Rotation based QR factorization where we improve
GR and derive Column-wise GR (CGR) that can annihilate multiple elements of a
column of a matrix simultaneously. We show that the multiplications in CGR are
lower than GR
We generalize CGR further and derive Generalized GR (GGR) that can annihilate
multiple elements of the columns of a matrix simultaneously. We show that the
parallelism exhibited by GGR is much higher than GR and Householder Transform
(HT)
We extend generalizations to Square root Free GR (also knows as Fast Givens
Rotation) and Square root and Division Free GR (SDFG) and derive Column-wise
Fast Givens, and Column-wise SDFG . We also extend generalization for complex
matrices and derive Complex Column-wise Givens Rotation
Coarse-grained Recon gurable Architectures (CGRAs) have gained popularity in the
last decade due to their power and area efficiency. Furthermore, CGRAs like REDEFINE also exhibit support for domain customizations. REDEFINE is an array of Tiles where each Tile consists of a Compute Element and a Router. The Routers are responsible
for on-chip communication, while Compute Elements in the REDEFINE can be domain
customized to accelerate the applications pertaining to the domain of interest. In this
thesis, we consider REDEFINE base architecture as a starting point and we design Processing Element (PE) that can execute algorithms in BLAS and LAPACK efficiently.
We perform several architectural enhancements in the PE to approach lower bound of the
CPI. For parallel realization of BLAS and LAPACK, we attach this PE to the Router of
REDEFINE. We achieve better area and power performance compared to the yesteryear
customized architecture for DLA. Major contributions in architecture in this thesis are as follows:
Architecture:
We present design of a PE for acceleration of GMM which is a Level-3 BLAS
operation
We methodically enhance the PE with different features for improvement in the
performance of GMM
For efficient realization of Linear Algebra Package (LAPACK), we use PE that can
efficiently execute GMM and show better performance
For further acceleration of LU and QR factorizations in LAPACK, we identify
macro operations encountered in LU and QR factorizations, and realize them on a
reconfigurable data-path resulting in 25-30% lower run-time
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Problemas de Programação Linear: uma proposta de resolução geométrica para o ensino médio com o uso do GeoGebra / Linear Programming Problems: a proposal for geometric resolution to high school with the use of GeoGebraZachi, Juliana Mallia [UNESP] 02 September 2016 (has links)
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Previous issue date: 2016-09-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho são apresentados os fundamentos da Programação Linear, em especial, da Programação Linear Geométrica, instrumento importante de otimização para problemas de Economia, gestão de empresas, problemas de transportes, obtenção de misturas ótimas, entre outros. Além disso, é apresentada uma proposta didática para os professores de educação básica da escola pública, utilizando o software GeoGebra como instrumento motivador para o estudo de uma situação de aprendizagem proposta no material de apoio idealizado pela Secretaria da Educação do Estado de São Paulo, abordada no caderno do aluno do 3º ano do Ensino Médio. / In this work presents the fundamentals of linear programming in particular, of geometric linear programming, important instrument of optimization for economic problems, business management, transport problems, obtaining optimal mixtures, among others. In addition, presents a didactic proposal for teachers of the basic education of public school, using the GeoGebra software as a motivating tool for the study of a learning situation proposed in the support material designed by the Education secretary of the State of São Paulo is presented, adressed in the student notebook of 3rd year of high school.
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Matrizes, determinantes e sistemas lineares: aplicações na Engenharia e Economia / Matrices, determinants and linear systems: applications in Engineering and EconomicsLevorato, Gabriela Baptistella Peres [UNESP] 18 August 2017 (has links)
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Previous issue date: 2017-08-18 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O presente trabalho mostra a importância da Álgebra Linear e em particular da Teoria de Matrizes, Determinantes e Sistemas Lineares para resolver problemas práticos e contextualizados. Mostramos aplicações em circuitos elétricos, no balanceamento de equações químicas, nos modelos aberto e fechado de Leontief, e no funcionamento do GPS. Ainda, foi aplicado um plano de aula para os alunos do segundo ano do Ensino Médio e apresentamos sugestões de exercícios de vestibulares sobre os tópicos estudados, para serem abordados em sala de aula. / The present work shows the importance of Linear Algebra and in particular of Matrix Theory, Determinants and Linear Systems to solve practical and contextualized problems. We show applications in electrical circuits, in the balancing of chemical equations, in the open and closed models of Leontief, and in the operation of GPS. Also, a lesson plan was applied to the students of the second year of high school and we presented suggestions of exercises of vestibular about the topics studied, to be approached in the classroom.
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Aplicações de algebra linear em ruidos quanticos / Applications of linear algebra in quantum noiseLima, Leandro Bezerra de, 1979- 08 August 2007 (has links)
Orientador: Carlile Campos Lavor / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-08T22:21:40Z (GMT). No. of bitstreams: 1
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Previous issue date: 2007 / Resumo: Neste trabalho, usando conceitos de álgebra linear e de operações quânticas, obtemos algumas propriedades de ruído quântico (para o caso particular de um q-bit), a fim de apresentar uma interpretação geométrica dos diferentes ruídos em canais quânticos, cujo processo é fundamental para a compreensão do processamento da informação quântica / Abstract: In this work, using concepts of linear algebra and quantum operations, we obtain some properties of quantum noise (for the one qubit case), in order to present a geometrical interpretation of different noises in quantum channels, which process is fundamental to the comprehension of the quantum information processing / Mestrado / Computação Quantica / Mestre em Matemática
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Aplicações da álgebra linear nas cadeias de Markov / Applications of linear algebra in Markov chainsSilva, Carlos Eduardo Vitória da 11 April 2013 (has links)
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Previous issue date: 2013-04-11 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The theory of linear algebra and matrices and systems particularly are linear math
topics that can be applied not only within mathematics itself, but also in various
other areas of human knowledge, such as physics, chemistry, biology, all engineering,
psychology, economy, transportation, administration, statistics and probability, etc...
The Markov chains are used to solve certain problems in the theory of probability.
Applications of Markov chains in these problems, depend directly on the theory of
matrices and linear systems.
In this work we use the techniques of Markov Chains to solve three problems of
probability, in three distinct areas. One in genetics, other in psychology and the other
in the area of mass transit in a transit system. All work is developed with the intention
that a high school student can read and understand the solutions of three problems
presented. / A teoria da álgebra linear e particularmente matrizes e sistemas lineares são
tópicos de matemática que podem ser aplicados não só dentro da própria matemática,
mas também em várias outras áreas do conhecimento humano, como física, química,
biologia, todas as engenharias, psicologia, economia, transporte, administração, estat
ística e probabilidade, etc.
As Cadeias de Markov são usadas para resolver certos problemas dentro da teoria
das probabilidades. As aplicações das Cadeias de Markov nesses problemas, dependem
diretamente da teoria das matrizes e sistemas lineares.
Neste trabalho usamos as técnicas das Cadeias de Markov para resolver três problemas
de probabilidades, em três áreas distintas. Um na área da genética, outro na
área da psicologia e o outro na área de transporte de massa em um sistema de trânsito.
Todo o trabalho é desenvolvido com a intenção de que um estudante do ensino médio
possa ler e entender as soluções dos três problemas apresentados.
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