Algebraic Aspects Research Articles

An Investigation into Some Lexical Ambiguities in Algebra: South African Experience

Many scholars in the field of mathematics education have written a lot about algebra. They commented that the simple and consistent rules for using abstract numbers and the shorthand symbols for mathematical verbs and operators in algebra evolved very slowly. Therefore, this study has revealed some of the lexical ambiguities in algebra aspect of mathematics. These ambiguities include among others; word problem in algebra, fallacies and hitches of word problem in algebra, linguistic complexity in algebra, proficiency in language of instruction problem just to mention but a few. The model for conceptualizing algebraic activity was no left behind. The study however recommends among others that there should be a paradigm shift in the method of teaching algebra and mathematics as a subject so as to enhance students’ academic performance. The choice of methodology especially the student-centred methods (contemporary methods of teaching) are very pertinent at this stage since researches have shown that many students have phobia for mathematics. Hence, there is need for teachers to update their knowledge about the problem solving skills that can be used as a panacea to mathematics phobia and algebra ambiguities. DOI: 10.5901/mjss.2014.v5n20p1243

Algebraic aspects of gauge theories

Gauge theories are primary tools in modern elementary particle physics. The generally recognized mathematical foundations of these theories are in differential geometry, namely, in the theory of connections in a principal fiber bundle. We propose another approach to the mathematical description of gauge theories based on a combination of algebraic and geometric methods.

English

The quality of mathematics education depends largely on the quality of education in general. The main idea may be summarized as follows: in order to educate the younger generation of people to be able to meet adequately the demands of the time, it is necessary to create conditions for the high-quality mathematics education. Improving the quality of mathematics education of pupils in secondary school is one of the most pressing problems. Contents of the school course of mathematics and its teaching method has always been the subject of undammed and sometimes stormy scientific debates. There are especially true methods of teaching algebra and the analisis in the high secondary school. Still in the study process the algebraic concepts and principles of analysis are given in such an abstract and generalized form that the student may has considerable difficulties to map these general abstract concepts to the certain concrete images, they are generalizations of. Improving education quality indicators can be achieved by using the appropriate computer technology. The article deals with the use of the cloud-oriented systems of computer mathematics (SCM). The prospects of development of the Web-SCM in terms of cloud-based learning environment are considered. The pedagogical features of the SageMath Cloud use as a tool for mathematics learning are revealed. The methodological aspects of algebra and elementary analysis teaching in a high profile school using the cloud-oriented the SCM SageMath Cloud are revealed.

Sedeonic Equations of Massive Fields

Prior work on space-time sedeon algebra models relativistic quantum mechanical equation of motion with corresponding field equations, mediated by massive or massless spin-1 or spin-1/2 particles. In the massless spin-1 case, such exchange particles mediate fields in analogy to Maxwell’s equations in Lorentz gauge. This paper demonstrates fundamental aspects of massive field’s theory, such as gauge invariance, charge conservation, Poynting’s theorem, potential of a stationary scalar point source, plane wave solution, and interaction between point sources. We briefly discuss some aspects of sedeonic algebra and their potential physical applications.

Numerical tensor calculus

The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid withngrid points per direction, a spatial dimensiondyieldsndgrid points. A grid function defined on such a grid is an example of a tensor of orderd. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly ind. In this way, data of sizend= 10001000can also be treated.This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.

Hamiltonian functional integrals representing regularized traces of higher-order differential operators

In this paper, we obtain a representation for the regularized traces (1) of evenorder differential opera� tors in terms of Hamiltonian functional integrals and the corresponding Feynman formulas. We also present an example showing how such a representation can be applied to obtain formulas not containing functional integrals. A Feynman formula (2) is a representation of a solution of an evolution differential equation or an object somehow related to this equation in terms of limit integrals over Cartesian powers of some space. If the space under consideration coincides with the spa� tial domain of the solution, then the corresponding formulas are said to be Lagrangian; if this is the prod� uct of this spatial domain and the dual of the linear space containing it, then the formulas are said to be Hamiltonian. A similar terminology is used for func� tional integrals being approximated by Feynman for� mulas. In this paper, we extend results of (3) to higher� order differential operators, which requires substan� tially modifying the technique; namely, we use Hamil� tonian (rather that Lagrangian, as in (3)) Feynman formulas and functional integrals. Moreover, in those Hamiltonian functional integrals which are used in the paper, integration is over the set of functions taking values in the product of the momentum space and a domain in the configuration space not coinciding with the entire space. Main attention is paid to the algebraic aspects of the problems under consideration; some analytic assumptions are omitted.

Sage 행렬계산기와 선형대수학 Sage 콘텐츠

For over 20 years, the issue of using an adequate CAS tool in teaching and learning of linear algebra has been raised constantly. And a variety of CAS tools were introduced in many linear algebra textbooks. In Korea, however, because of some realistic problems, they have not been introduced in the class and the theoretical aspect of linear algebra has been focused on in teaching and learning of it. In this paper, we suggest Sage as an alternative for CAS tools overcoming the problems mentioned above. And, we introduce full contents for linear algebra and matrix calculator that Sage was used to develop. Taking advantage of them, almost all the concepts of linear algebra can be easily covered and the size of matrices can be expanded without difficulty.

Complexity of Tropical and Min-plus Linear Prevarieties

A tropical (or min-plus) semiring is a set $${\mathbb{Z}}$$ (or $${\mathbb{Z \cup \{\infty\}}}$$ ) endowed with two operations: $${\oplus}$$ , which is just usual minimum, and $${\odot}$$ , which is usual addition. In tropical algebra, a vector x is a solution to a polynomial $${g_1(x) \oplus g_2(x) \oplus \cdots \oplus g_k(x)}$$ , where the g i (x)s are tropical monomials, if the minimum in min i (g i (x)) is attained at least twice. In min-plus algebra solutions of systems of equations of the form $${g_1(x)\oplus \cdots \oplus g_k(x) = h_1(x)\oplus \cdots \oplus h_l(x)}$$ are studied. In this paper, we consider computational problems related to tropical linear system. We show that the solvability problem (both over $${\mathbb{Z}}$$ and $${\mathbb{Z} \cup \{\infty\}}$$ ) and the problem of deciding the equivalence of two linear systems (both over $${\mathbb{Z}}$$ and $${\mathbb{Z} \cup \{\infty\}}$$ ) are equivalent under polynomial-time reductions to mean payoff games and are also equivalent to analogous problems in min-plus algebra. In particular, all these problems belong to $${\mathsf{NP}\cap \mathsf{coNP}}$$ . Thus, we provide a tight connection of computational aspects of tropical linear algebra with mean payoff games and min-plus linear algebra. On the other hand, we show that computing the dimension of the solution space of a tropical linear system and of a min-plus linear system is $${\mathsf{NP}}$$ -complete.

Homotopical aspects of commutative algebras I: freeness conditions for crossed squares

We give an alternative description of the top algebra of the free crossed square of algebras on 2-construction data in terms of tensors and coproducts of crossed modules of commutative algebras.

Different aspects of the monotonicity of a function

In this paper, we investigate which aspects are overriding in the concept images of monotonicity of Finnish tertiary mathematics students, i.e., on which aspects of monotonicity they base their argument in different types of exercises related to that concept. Further, we examine the relationship between the quality of principal aspects and the success in solving monotonicity exercises and a few other standard problems in calculus. Our findings indicate that a mathematics student's conception about monotone functions is often restricted to continuous or differentiable functions and the algebraic aspect – the nearest one to the formal definition of monotonicity – is rare.

Autotopies and quasigroup identities: New aspects of non-associative division algebras

Abstract In this article, we explore new aspects in the classification of non-associative division algebras. By a detailed study of the representations of the Lie group of autotopies of real division algebras we show that, if the group of autotopies has a sufficiently rich structure, then the algebra is isotopic to one of the classical real division algebras. This turns out to be the case for large classes of real division algebras, including many that are defined by identities. In several cases, a classification up to isomorphism can be worked out from this information.

Geometric and algebraic properties of minimal bases of singular systems

For a general singular system with an associated pencil T(S), a complete classification of the right polynomial vector pairs , connected with the rational vector space, is given according to the proper–nonproper property, characterising the relationship of the degrees of those two vectors. An integral part of the classification of right pairs is the development of the notions of canonical and normal minimal bases for and rational vector spaces, where R(s) is the state restriction pencil of . It is shown that the notions of canonical and normal minimal bases are equivalent; the first notion characterises the pure algebraic aspect of the classification, whereas the second is intimately connected to the real geometry properties and the underlying generation mechanism of the proper and nonproper state vectors . The results describe the algebraic and geometric dimensions of the invariant partitioning of the set of reachability indices of singular systems. The classification of all proper and nonproper polynomial vectors induces a corresponding classification for the reachability spaces to proper–nonproper and results related to the possible dimensions feedback-spectra assignment properties of them are also given. The classification of minimal bases introduces new feedback invariants for singular systems, based on the real geometry of polynomial minimal bases, and provides an extension of the standard theory for proper systems (Warren, M.E., & Eckenberg, A.E. (1975).

NOETHERIAN HOPF ALGEBRAS

AbstractA brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.

Parametric quantum search algorithm by CP maps: algebraic, geometric and complexity aspects

A toy model relaxation of the unitarity of a quantum search algorithm is introduced and the resulting parametric quantum search is investigated mathematically via a one-parameter family of completely positive trace preserving maps by introducing the oracle algebra (an SU(2) isomorphic matrix algebra). The parametric search is shown to be conditionally as fast as the original algorithm. Geometrically, it is related to spherical geodesics which exhibit an azimuthal symmetry, while its search map and density matrix enjoy geometric and structural symmetries.

Tutorial on Geometric Calculus

The book Clifford Algebra to Geometric Calculus is the first and still the most complete exposition of Geometric Calculus (GC). But it is more of a reference book than a textbook, so can it be a difficult read for beginners. This tutorial is a guide for serious students who want to dig deeply into the subject. It presents helpful background and aims to clarify objectives, important results and methods in the book. It is supplemented by a hot-linked annotated bibliography of papers elaborating various aspects of Geometric Algebra and Calculus.

Upper primary school teachers’ mathematical knowledge for teaching functional thinking in algebra

This article is based on a project that investigated teachers’ knowledge in teaching an important aspect of algebra in the middle years of schooling—functions, relations and joint variation. As part of the project, 105 upper primary teachers were surveyed during their participation in Contemporary Teaching and Learning of Mathematics, a research project funded by the Catholic Education Office, Melbourne (2008–2012). Analysis of the survey responses revealed that two-thirds of teachers demonstrated content knowledge on a pattern generalisation task appropriate for upper primary levels of schooling (8- to 12-year-old students), but less than half demonstrated reasonable pedagogical content knowledge (PCK). On a paired variable (function machine) task, only one quarter of teachers demonstrated appropriate PCK. Although two-thirds of the teachers indicated that they currently taught content from the “Patterns and Algebra” strand of the new Australian Curriculum, less than half were able to provide examples of appropriate learning experiences for students. More than two-thirds of teachers expressed concern about their ability to teach this area of mathematics. Implications for the professional learning of teachers to improve their mathematics knowledge for developing students’ functional thinking are presented.

Algebraic aspects of the Hofstadter problem in graphene

We consider the Hofstadter problem on a honeycomb lattice. ita relevance to the quantum group U q (sl 2) is explicitly shown. We point out the reducibility of the corresponding characteristic polynomials and conjecture its relation to supersymmetric quantum mechanics.

Quivers, invariants and quotient correspondence

This paper studies the geometric and algebraic aspects of the moduli spaces of quivers of fence type. We first provide two quotient presentations of the quiver varieties and interpret their equivalence as a generalized Gelfand–MacPherson correspondence. Next, we introduce parabolic quivers and extend the above from the actions of reductive groups to the actions of parabolic subgroups. Interestingly, the above geometry finds its natural counterparts in the representation theory as the branching rules and transfer principle in the context of the reciprocity algebra. The last half of the paper establishes this connection.

ASSESSING CONCEPTUAL UNDERSTANDING IN MATHEMATICS: Using Derivative Function to Solve Connected Problems

Open and distance education plays an important role in the actualization of cultural goals as well as in societal developments. This is an independent teaching and learning method for mathematics which forms the dynamic of scientific thinking. Distance education is an important alternative to traditional teaching applications. These contributions brought by technology enable students to participate actively in having access to information and questioning it. Such an application increases students’ motivation and teaches how mathematics can be used in daily life. Derivative is a mathematical concept which can be used in many areas of daily life. The aim of this study is to enable the concept of derivatives to be understood well by using the derivative function in the solution of various problems. It also aims at interpreting difficulties theoretically in the solution of problems and determining mistakes in terms of teaching methods. In this study, how various aspects of derivatives are understood is emphasized. These aspects concern the explanation of concepts and process, and also their application to certain concepts in physics. Students’ depth of understanding of derivatives was analyzed based on two aspects of understanding; theoretical analysis and contextual application. Follow-up interviews were conducted with five students. The results show that the students preferred to apply an algebraic symbolic aspect instead of using logical meanings of function and its derivative. In addition, in relation to how the graph of the derivative function affects the aspect of function, it was determined that the students displayed low performance.

Orthogonal Laurent polynomials. A new algebraic approach

In this paper, some important algebraic aspects in the theory of orthogonal Laurent polynomials, such as the three-term recurrence relation, the Christoffel–Darboux identity or the Liouville–Ostrogradski formula, are revisited from the Riemann–Hilbert window. These topics are considered for general ordered Laurent polynomial sequences, and not only for the usual “balanced” cases. In addition, a connection with Szegö polynomials (orthogonal polynomials in the unit circle) is explored.