Marco Antonio Rodríguez-Andrade

Instituto Politécnico Nacional | IPN · Departamento de Fisicomatemáticas

A method for generating all rational generalized matrices on indefinite real inner product spaces isomorphic to $R^{p,q}$ is presented. The proposed method is based on the proof of a weak version of the Cartan–Dieudonné theorem, handled using Clifford algebras. It is shown that all rational B-orthogonal matrices in an indefinite inner product space...

The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, we present an introduction to the main ideas of Clifford algebra, with the main goal to develop a package for Clifford algebra calculations for the com...

The problem of characterizing the coincidence site lattices obtained by superimposing a planar hexagonal lattice with its rotated version is solved using simple reflections handled with Clifford algebras. For any possible coincidence rotation, analytical expressions for the coincidence index and for a basis of the coincidence site lattices are deduce...

We have shown that the expression theta=2tan-1/ derived by Ranganathan to calculate the angles at which there exists a CSL for rotational interfaces in the cubic system can also be applied to general (oblique) two-dimensional lattices provided that the quantities 2 and /cos() are rational numbers, with =|b|/|a| and is the angle between the basis ve...

In this work simple reflections or rotations of canonical vectors are used to generate all Pythagorean vectors, i.e. vectors in \mathbbQn{\mathbb{Q}^{n}} that satisfy the Pythagoras generalized equation. By using Clifford algebra we develop a constructive method that explicitly provides an algorithm to generate generalized Pythagorean numbers. M...

In this work, the equivalence class representatives of integer solutions of the Diophantine equation of the type \({{a_1x_1^2+ .\,.\,. + a_px_p^2 = a_{p+1}x^2_{p+1} + .\,.\,. +a_{p+q}x^2_{p+q} +a_1x^2_{n+1} (a_i > 0,i=1, .\,.\,.\,,p+q,x_{n+1}\neq0)}}\) are found using simple reflections of orthogonal vectors, manipulated using the Clifford algebra...

Traditionally, equity agendas in research have not involved the education of gifted children, in general, and attention to mathematically gifted children or youth, in particular. We consider that the equity issue concerning this population can be discussed at two levels: from an individual point of view taking into account dissatisfaction, loss tal...

Éste es un libro de texto autorizado por la Secretaria de Educación Pública (México) que operacionaliza didácticamente el currículo oficial del primer grado de secundaria (séptimo año de educación básica). Esto mediante un modelo constructivista, que promueve aprendizaje significativos al partir de la problematización de contextos cotidianos de los...

Éste es un libro de texto autorizado por la Secretaria de Educación Pública (México) que operacionaliza didácticamente el currículo oficial del segundo grado de secundaria (octavo año de educación básica). Esto mediante un modelo constructivista, que promueve aprendizaje significativos al partir de la problematización de contextos cotidianos de los...

Éste es un libro de texto autorizado por la Secretaria de Educación Pública (México) que operacionaliza didácticamente el currículo oficial del tercer grado de secundaria (noveno año de educación básica). Esto mediante un modelo constructivista, que promueve aprendizaje significativos al partir de la problematización de contextos cotidianos de los...

The problem of coincidences of lattices in the space R(p,q), with p + q = 2, is analyzed using Clifford algebra. We show that, as in R(n), any coincidence isometry can be decomposed as a product of at most two reflections by vectors of the lattice. Bases and coincidence indices are constructed explicitly for several interesting lattices. Our proced...

We present an algorithmic proof of the Cartan-Dieudonn\'e theorem on generalized real scalar product spaces with arbitrary signature. We use Clifford algebras to compute the factorization of a given orthogonal transformation as a product of reflections with respect to hyperplanes. The relationship with the Cartan-Dieudonn\'e-Scherk theorem is also...

In this article, simple reflections, rotations and the Cartan theorem are handled using Clifford algebras. With this tool we provide a constructive proof of the Cartan theorem and the relationship with Pythagorean numbers is discussed.

The Clifford algebra of a n-dimensional Euclidean vector space provides a general language comprising vectors, complex numbers, quaternions, Grassman algebra, Pauli and Dirac matrices. In this work, a package for Clifford algebra calculations for the computer algebra program Mathematica is introduced through a presentation of the main ideas of Clif...

The scaling limit of the energy correlations in non-integrable Ising models In this work, an algorithm to decompose a given orthogonal transformation as a product of reflections through hyperplanes is presented. This in fact constitutes a constructive proof of a Cartan theorem, valid over any field K = Q, R or C. Clifford algebras are used to expli...

The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail....

The aim of mathematical crystallography is the classification of periodic structures by means of different equivalence relationships, yielding the well known crystallographic classes and Bravais lattices [1]. Periodicity (crys-tallinity) has been the paradigm of classical crystallography. Recently, more systematical attention has been paid to struc...

The geometric algebra as defined by D. Hestenes is compared with a constructive definition of Clifford algebras. Both approaches are discussed and the equivalence between a finite geometric algebra and the universal Clifford algebra R p, q is shown. Also an intermediate way to construct Clifford algebras is sketched. This attempt to conciliate two...