Search results for "AIC"

Algebra Without Context Is Empty, Visualizations Without Concepts Are Blind

2018

In the acquisition and formalization of mathematical concepts, the transition between algebraic and geometric representations and the use of different modes of representation contextualizes abstract algebra. Regrettably, the role of geometry is often limited to the visualization of algebraic facts and figurative memory aids. Such visualizations are blind for the underlying concepts, since transitions between concepts in different representations assume the existence of symbols, language, rules and operations in both systems. The history of mathematics offers contexts to develop geometrical language and intuition in areas currently being taught in school in a purely algebraic fashion. The ex…

Applications of the Connection between Approximation Theory and Algebra

2009

The aim of this paper is to illustrate a possibility of obtaining various theoretical results using the connection between multivariate interpolation and reduction process with respect to a H-basis of an ideal. Using this connection we can switch between interpolation theory and the theory of ideals. As a application of this connection, we found and proved an interesting identity, which is satisfied for all polynomials in d variables from an interpolation polynomial subspace.

The History of Algebra in Mathematics Education

2006

In this chapter, we analyse key issues in algebra history from which some lessons can be extracted for the future of the teaching and learning of algebra. A comparative analysis of two types of pre-Vietan languages (before 16th century), and of the corresponding methods to solve problems, leads to conjecture the presence of didactic obstacles of an epistemological origin in the transition from arithmetic to algebraic thinking. This illustrates the value of historic and critical analysis for basic research design in mathematics education. Analysing the interrelationship between different evolution stages of the sign system of symbolic algebra and vernacular language supports the inference th…

Counterexamples for unique continuation

1988

The Abel–Jacobi map for higher Chow groups

2006

We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel–Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.

Lambda substitution algebras

1993

In the paper an algebraic metatheory of type-free λ-calculus is developed. Our version is based on lambda substitution algebras (λSAs), which are just SAs introduced by Feldman (for algebraizing equational logic) enriched with a countable family of unary operations of λ-abstraction and a binary operation of application. Two representation theorems, syntactical and semantic, are proved, what directly provides completeness theorems.

An Overview on Algebraic Structures

2016

This chapter recaps and formalizes concepts used in the previous sections of this book. Furthermore, this chapter reorganizes and describes in depth the topics mentioned at the end of Chap. 1, i.e. a formal characterization of the abstract algebraic structures and their hierarchy. This chapter is thus a revisited summary of concepts previously introduced and used and provides the mathematical basis for the following chapters.

On the group of the automorphisms of some algebraic systems

1968

Within a framework of general algebra we firstly formulate a proposition on the group of the automorphisms of some irreducible algebrae (id est algebrae without proper non trivial subalgebrae). This proposition includes as particular cases the uniqueness of the automorphisms of the rational field and the Burnside theorem on the commutant of an irreducible set of operators of a finite dimensional vector space over an algebraically closed field. Afterwards we apply the general proposition to modules with irreducible sets of semilinear operators and we obtain a theorem which generalises from several points of view the Burnside theorem. Finally we derive as an application a proposition which sp…

Hodge Numbers for the Cohomology of Calabi-Yau Type Local Systems

2014

We determine the Hodge numbers of the cohomology group \(H_{L^{2}}^{1}(S, \mathbb{V}) = H^{1}(\bar{S},j_{{\ast}}\mathbb{V})\) using Higgs cohomology, where the local system \(\mathbb{V}\) is induced by a family of Calabi-Yau threefolds over a smooth, quasi-projective curve S. This generalizes previous work to the case of quasi-unipotent, but not necessarily unipotent, local monodromies at infinity. We give applications to Rohde’s families of Calabi-Yau 3-folds.

Applied Linear Algebra: Electrical Networks

2016

This chapter shows how mathematical theory is not an abstract subject which has no connection with the real world. On the contrary, this entire book is written by stating that mathematics in general, and algebra in this case, is an integrating part of every day real life and that the professional life of computational scientists and engineers requires a solid mathematical background. In order to show how the contents of the previous chapters have an immediate technical application, the last chapter of this book describes a core engineering subject, i.e. electrical networks, as an algebraic exercise. Furthermore, this chapter shows how the combination of the algebraic topics give a natural r…