Rastros da formação matemática na prática profissional do professor de matemática

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Um estudo introdutório da Teoria de Grafos através de matrizes

Pós-graduação em Matemática em Rede Nacional - IBILCE

Singular Value Decomposition and Ligand Binding Analysis

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Grupos wallpaper e sua relação com cohomologia de grupos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Classificação de cônicas e quádricas

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Um estudo da geometria projetiva elíptica

Pós-graduação em Matemática Universitária - IGCE

Origamic design: uma nova forma de desenvolver produtos

During a long time, origami was associated with decoration and craft production of ornaments and figures. However, in the end of 20th century, it began to be studied by mathematicians who were looking for interrelationships between this art and science. Through disciplines like geometry, trigonometry, calculation and linear algebra, they generated a set of axioms and theorems that became possible specific conversion of origami in computational geometry and the development of several softwares. Thus, origami began to be applied in engineering and design studies of innovative product and the term “origamics” was created to demonstrate its interdisciplinary nature. In this article will be presented some works exploring the constructive principles of origami to contribute with the diffusion of origamics. In this way more professionals will be able to understand the scientific and technological potential of this art.

Formação de professores de matemática da educação superior e as tecnologias digitais: aspectos do conhecimento revelados no contexto de uma comunidade de prática online

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Special identities for Bol algebras

Bol algebras appear as the tangent algebra of Bol loops. A (left) Bol algebra is a vector space equipped with a binary operation [a, b] and a ternary operation {a, b, c} that satisfy five defining identities. If A is a left or right alternative algebra then A(b) is a Bol algebra, where [a, b] := ab - ba is the commutator and {a, b, c} := < b, c, a > is the Jordan associator. A special identity is an identity satisfied by Ab for all right alternative algebras A, but not satisfied by the free Bol algebra. We show that there are no special identities of degree <= 7, but there are special identities of degree 8. We obtain all the special identities of degree 8 in partition six-two. (C) 2011 Elsevier Inc. All rights reserved.

Miniversal deformations of matrices of bilinear forms

Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29-43] constructed miniversal deformations of square complex matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct miniversal deformations of matrices under congruence. (C) 2011 Elsevier Inc. All rights reserved.

Kleiner's theorem for unitary representations of posets

A subspace representation of a poset S = {s(1), ..., S-t} is given by a system (V; V-1, ..., V-t) consisting of a vector space V and its sub-spaces V-i such that V-i subset of V-j if s(i) (sic) S-j. For each real-valued vector chi = (chi(1), ..., chi(t)) with positive components, we define a unitary chi-representation of S as a system (U: U-1, ..., U-t) that consists of a unitary space U and its subspaces U-i such that U-i subset of U-j if S-i (sic) S-j and satisfies chi 1 P-1 + ... + chi P-t(t) = 1, in which P-i is the orthogonal projection onto U-i. We prove that S has a finite number of unitarily nonequivalent indecomposable chi-representations for each weight chi if and only if S has a finite number of nonequivalent indecomposable subspace representations; that is, if and only if S contains any of Kleiner's critical posets. (c) 2012 Elsevier Inc. All rights reserved.

Remarks on the classification of a pair of commuting semilinear operators

Gelfand and Ponomarev [I.M. Gelfand, V.A. Ponomarev, Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space, Funct. Anal. Appl. 3 (1969) 325-326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tuples of linear operators for any k. We prove an analogous statement for semilinear operators. (C) 2011 Elsevier Inc. All rights reserved.

Electron Interactions with Disulfide Bridges

Because of its electronic properties, sulfur plays a major role in a variety of metabolic processes and, more in general, in the chemistry of life. In particular, S-S bridges between cysteines are present in the amino acid backbone of proteins. Protein disulfur radical anions may decay following different paths through competing intra and intermolecular routes, including bond cleavage, disproportionation, protein-protein cross linking, and electron transfer. Indeed, mass spectrometry ECD (electron capture dissociation massspectroscopy) studies have shown that capture of low-energy (<0.2 eV) electrons by multiply protonated proteins is followed by dissociation of S-S bonds holding two peptide chains together. In view of the importance of organic sulfur chemistry, we report on electron interactions with disulphide bridges. To study these interactions we used as prototypes the molecules dimethyl sulfide [(CH3)2S] and dimethyl disulfide [(H3C)S2(CH3)]. We seek to better understand the electron-induced cleavage of the disulfide bond. To explore dissociative processes we performed electron scattering calculations with the Schwinger Multichannel Method with pseudopotentials (SMCPP), recently parallelized with OpenMP directives and optimized with subroutines for linear algebra (BLAS) and LAPACK routines. Elastic cross sections obtained for different S-S bond lengths indicate stabilization of the anion formed by electron attachment to a σ*SS antibonding orbital, such that dissociation would be expected.

Weighted geometric mean of large-scale matrices: numerical analysis and algorithms

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators arising in domain decomposition of elliptic problems. In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v. Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.