Towards the Calculation of Jack Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

Towards the Calculation of Jack Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

Hierarchical Filling of N-Dimensional Spaces

The hierarchical filling of the n-dimensional space with geometric figures is studied, accompanied by a process of discrete similar changes in their sizes, that is, process of scaling. The scaling process in these fillings does not depend on time and is determined only by the geometric characteristics of the figures, which are preserved when their size is changed. Two possible ways of hierarchical filling of space are defined, under which the original figure incrementally increases its size fills the space. Investigations of the hierarchical filling of concrete geometric figures of a plane, three-dimensional space, four- and five-dimensional spaces are carried out. The denominator of geometric progressions characterizing sequences of figures in the process of scaling are determined depending on the shape of the figure and its dimension.

EDUCAÇÃO FINANCEIRA: APRENDIZAGEM DE PROGRESSÕES GEOMÉTRICAS APLICADAS AOS JUROS COMPOSTOS NA PERSPECTIVA DA EDUCAÇÃO MATEMÁTICA CRÍTICA

Resumo: O presente artigo trata-se de um recorte de dissertação de mestrado em Educação Matemática. O problema surge da análise de duas tendências que tratamos de articular, por um lado os dados indicam eventuais falhas no aprendizado da matemática e por outro os indicadores superiores a 60% de endividamentos das famílias. O objetivo principal é analisar o desenvolvimento do conteúdo de juros compostos a partir de uma sequência de ensino, tendo como base o estudo das Progressões Geométricas (PG) a partir do movimento da Educação Financeira. O referencial teórico articula as ideias de Skovsmose, referente à Educação Matemática Crítica (EMC), que enfatiza a aprendizagem da matemática de forma política e consciente. O estudo foi aplicado com estudantes do 1° ano do Ensino Médio, de uma escola na região oeste da Bahia, Brasil, em julho de 2018, com a aplicação de uma sequência de ensino. A metodologia se fundamenta na pesquisa de natureza empírica, caráter exploratório e explicativo. As análises foram organizadas em três categorias tratando-se da concepção de dinheiro, do entendimento sobre PA e PG e as propagandas. As conclusões evidenciam a necessidade de ampliação do trabalho procurando contemplar todos os elementos pertencentes à Matemática Financeira, de modo que sejam tratados criticamente.Palavras-chave: Educação Matemática Crítica; Educação Financeira; Progressões Geométricas. Abstract: This article is an excerpt of a master's thesis in Mathematics Education. The problem arises from the analysis of two trends that we tried to articulate, on the one hand, the data indicate possible flaws in the learning of mathematics and on the other, the indicators above 60% of household indebtedness. The main objective is to analyze the development of the content of compound interest based on a teaching sequence, based on the study of Geometric Progressions (PG) from the Financial Education movement. The theoretical framework articulates the ideas of Skovsmose, referring to Critical Mathematical Education (EMC), which emphasizes the learning of mathematics in a political and conscious way. The study was applied to students of the 1st year of high school, from a school in the western region of Bahia, Brazil, in July 2018, with the application of a teaching sequence. The methodology is based on research of an empirical nature, exploratory and explanatory. The analyzes were organized into three categories, dealing with the concept of money, understanding of PA and PG and advertisements. The conclusions show the need to expand the work, seeking to contemplate all elements belonging to Financial Mathematics, so that they are treated critically.Keywords: Critical Mathematics Education. Financial education. Geometric progressions.