scholarly journals Sameness between based universal algebras

2009 ◽  
Vol 42 (1) ◽  
Author(s):  
Gabriele Ricci
Keyword(s):  
Linear Algebra ◽  
Linear Transformations ◽  
Universal Algebras

AbstractThis is the continuation of the paper “Transformations between Menger systems”. To define when two universal algebras with bases “are the same”, here we propose a universal notion of transformation that comes from a triple characterization concerning three representation facets: the determinations of theHence, this notion consists of three equivalent definitions. It characterizes another technical variant and also the universal version of the very semi-linear transformationsUniversal transformations allow us to check theContrary to present beliefs, even the foundation of abstract Linear Algebra turns out to be incomplete.

scholarly journals Linear functionals on hypervector spaces

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3031-3043
Author(s):  
O.R. Dehghan
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Linear Functional ◽  
Dual Basis ◽  
Linear Functionals ◽  
Linear Transformations ◽  
Important Special Case ◽  
Finite Dimensional ◽  
Key Topics ◽  
Special Case

The study of linear functionals, as an important special case of linear transformations, is one of the key topics in linear algebra and plays a significant role in analysis. In this paper we generalize the crucial results from the classical theory and study main properties of linear functionals on hypervector spaces. In this way, we obtain the dual basis of a given basis for a finite-dimensional hypervector space. Moreover, we investigate the relation between linear functionals and subhyperspaces and conclude the dimension of the vector space of all linear functionals over a hypervector space, the dimension of sum of two subhyperspaces and the dimension of the annihilator of a subhyperspace, under special conditions. Also, we show that every superhyperspace is the kernel of a linear functional. Finally, we check out whether every basis for the vector space of all linear functionals over a hypervector space V is the dual of some basis for V.

A Critical Comparison of Reduced and Conventional EOS Algorithms

SPE Journal ◽  
2013 ◽  
Vol 18 (02) ◽  
pp. 378-388 ◽  
Author(s):  
Kjetil B. Haugen ◽  
Bret L. Beckner
Keyword(s):  
Linear Algebra ◽  
Computational Cost ◽  
Linear Transformations ◽  
Independent Variables ◽  
Reservoir Simulations ◽  
Performance Benchmarks ◽  
Speed Up ◽  
The Cost ◽  
Careful Comparison ◽  
Equilibrium Calculations

Summary Phase-equilibrium calculations can be a time-consuming part of process simulators and compositional reservoir simulations. Various authors have presented encouraging speed improvements based on reduced methods that can lower the computational cost by reducing the number of independent variables and thus generating a smaller system of equations to solve. This paper presents a careful comparison of conventional and reduced algorithms, showing that they can be expressed as linear transformations of each other. Consequently, the two sets of algorithms exhibit identical convergence behavior, and the performance gain of the reduced methods is entirely caused by reducing the cost of linear-algebra operations. Performance benchmarks show much smaller speed-up numbers than seen in previously published material. Highly optimized linear-algebra operations significantly limit the opportunity for further speed improvement from reduced methods. Only a marginal speed-up potential is observed for mixtures with 15 components or less. This suggests that reduced methods may be less attractive for reservoir simulation than previously thought.

Transformations between Menger systems

2008 ◽  
Vol 41 (4) ◽  
Author(s):  
Gabriele Ricci
Keyword(s):  
Linear Algebra ◽  
Universal Algebra ◽  
Geometric Property ◽  
Vector Spaces ◽  
Transformation Groups ◽  
Universal Algebras ◽  
General Vector

AbstractTo define transformations between based universal algebras we must introduce representations that depend on the bases, contrary to what was possible for general vector spaces and believed possible for universal algebras. In fact, a counterexample shows that by representation-free transformations alone one cannot even ascertain whether a universal algebra has any dimension or not.A transformation notion, which can do, concerns basis dependent Menger systems. It enjoys a basic geometric property of universal algebras, the preservation of reference flocks, and generalizes the transformation groups of Linear Algebra into groupoids.

scholarly journals Designing active Maths for undergraduate STEAM students

2021 ◽  
Author(s):  
Domenico Brunetto ◽  
Ana Moura Santos
Keyword(s):  
Pilot Study ◽  
Linear Algebra ◽  
Undergraduate Students ◽  
Learning Experience ◽  
Class Discussion ◽  
First Year ◽  
Linear Transformations ◽  
Learning Mathematics ◽  
Specific Goal ◽  
Teaching Learning

This work presents a set of student-centred activities that may help undergraduate students understand mathematics in their first year of a STEAM degree. In particular, the authors refer to the difficulties students meet in making connections between syntactic and semantic dimensions in learning mathematics, especially in Linear Algebra topics. The specific goal of this paper is to present and discuss how it can work in the case of linear transformations. This topic stands in the middle of every Linear Algebra standard course and is pivotal in many recent applications, such as computer graphics. The study describes the teaching-learning experience and reports the results of the first pilot study, which involves about 100 undergraduate Architecture students of Politecnico di Milano. One of the peculiarities of this work is its context since the class is composed of heterogeneous group of students, in terms of knowledge background and attitudes towards mathematics. The main findings of this paper are underlining how a student-centred strategy, based on asynchronous activities and synchronous class discussion, allows misconceptions to emerge and be appropriately addressed

scholarly journals Um panorama das dissertações e teses sobre o tema álgebra linear no Ensino Superior no Brasil nos anos de 2011 a 2018

2020 ◽  
Vol 9 (2) ◽  
Author(s):  
JULIANA MARTINS PHILOT ◽  
BARBARA LUTAIF BIANCHINI
Keyword(s):  
Higher Education ◽  
Linear Algebra ◽  
Teaching And Learning ◽  
Education Technology ◽  
Linear Transformations ◽  
Mathematics Courses ◽  
Semiotic Representations ◽  
Online Distance Learning ◽  
Online Distance ◽  
Theses And Dissertations

ResumoEste artigo, de natureza teórica, apresenta uma pesquisa bibliográfica sobre as teses e dissertações no Brasil sobre o tema Álgebra Linear no Ensino Superior publicados  entre os anos de 2011 a 2018, com o objetivo de identificar as principais tendências e/ou abordagens das pesquisas realizadas e as lacunas não investigadas. Os 20 trabalhos analisados foram divididos em 6 categorias, sendo estas: Utilização de Recursos Tecnológicos no Ensino e na Aprendizagem; Formação de Professores; Ensino a Distância - EAD; Análise de Livros Didáticos; Currículo e Sequências Didáticas. Neste artigo focamos na categoria A, com dez trabalhos, nos quais buscamos identificar os principais referencias teóricos utilizados, os cursos nos quais foram aplicados e quais os conteúdos abordados. Com essa abordagem identificamos que estes estudos foram conduzidos principalmente nos cursos de Matemática abordando o conteúdo de Transformações Lineares, sendo que o principal referencial teórico utilizado foi a Teoria dos Registros de Representação Semiótica. Palavras-chave: Álgebra Linear; Ensino e Aprendizagem de Álgebra Linear; Ensino Superior; Tecnologia.AbstractThis theoretical nature article, presents a bibliographic research on the theses and dissertations in Brazil about the theme Linear Algebra in Higher Education published between the years 2011 to 2018, in order to identify the main trends and/or approaches of the researches carried out and the gaps not investigated. The 20 works analyzed were divided into 6 categories, namely: Use of Technological Resources in Teaching and Learning; Teacher Training; Online Distance Learning (ODL); Analysis of Textbooks; Curriculum and Didactic Sequences. In this article we focus on category A, with ten works, in which we seek to identify the main theoretical references used, the courses in which they were applied and which contents were covered. With this approach we identified that these studies were conducted mainly in Mathematics courses addressing the content of Linear Transformations, and the main theoretical framework used was the Registers of Semiotic Representations Theory.Keywords: Linear Algebra; Teaching and Learning Linear Algebra; Higher Education; Technology.    

Geometric Mappings on Geometric Lattices

1971 ◽  
Vol 23 (1) ◽  
pp. 22-35 ◽  
Author(s):  
David Sachs
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Classical Result ◽  
Linear Independence ◽  
Closure Operator ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Algebraic Methods ◽  
Linear Transformations ◽  
Geometric Lattices

It is a classical result of mathematics that there is an intimate connection between linear algebra and projective or affine geometry. Thus, many algebraic results can be given a geometric interpretation, and geometric theorems can quite often be proved more easily by algebraic methods. In this paper we apply topological ideas to geometric lattices, structures which provide the framework for the study of abstract linear independence, and obtain affine geometry from the mappings that preserve the closure operator that is associated with these lattices. These mappings are closely connected with semi-linear transformations on a vector space, and thus linear algebra and affine geometry are derived from the study of a certain closure operator and mappings which preserve it, even if the “space” is finite.

scholarly journals Basis properties for semigroups

1988 ◽  
Vol 30 (1) ◽  
pp. 101-109
Author(s):  
Peter R. Jones
Keyword(s):  
Linear Algebra ◽  
Universal Algebra ◽  
Basis Property ◽  
Inverse Semigroups ◽  
Vector Spaces ◽  
Classical Theorem ◽  
Universal Algebras ◽  
Generating Sets

A universal algebra A is said to have the basis property (BP) if any two minimal generating sets (bases) for a subalgebra of A have the same cardinality. This property was studied by the author for inverse semigroups in [5, 6]. For instance free inverse semigroups have BP. When treated as universal algebras, a classical theorem of linear algebra states that vector spaces have BP. In this paper we study BP for semigroups.

scholarly journals Generalized magnification in visual optics. Part 2: Magnification as affine transformation

2010 ◽  
Vol 69 (4) ◽  
Author(s):  
W. F. Harris
Keyword(s):  
Linear Algebra ◽  
Affine Transformation ◽  
Mathematical Structure ◽  
Linear Transformations ◽  
Visual Optics

In astigmatic systems magnification may be different in different directions.  It may also be accompanied by rotation or reflection.  These changes from object to image are examples of generalized magnification.  They are represented by  2 2×  matrices.  Because they are linear transformations they can be called linear magnifications.  Linear magnifications account for a change in appearance without regard to position.  Mathematical structure suggests a natural further generalization to a magnification that is complete in the sense that it accountsfor change in appearance and position.  It is represented by a  3 3×  matrix with a dummy third row. The transformation is called affine in linear algebra which suggests that these generalized magnifica-tions be called affine magnifications.  The purpose of the paper is to define affine magnification in the context of astigmatic optics.  Several examples are presented and illustrated graphically. (S Afr Optom 2010 69(4) 166-172)

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