| 34/2002 |
S-Convex Fuzzy Processes Yurilev Chalco-Cano, Marko A.Rojas-Medar, R. Osuna-Gómez We introduce the notion of s-convex fuzzy processes. We study their properties and we give some applications.  rp-2002-34.pdf |
| 33/2002 |
Trees and Reflection Groups Humberto L. Talpo, Marcelo Firer We define a reflection in a tree as an involutive automorphism whose set of fixed points is exactly a complete geodesic. We consider the group $\mathrm{Aut}\left( \Gamma \right) $ of automorphism of a tree $\Gamma $ with the topology of uniform convergence in compact sets. With this topological structure, we prove that, for the case of a regular tree of degree $4k$, the group generated by reflections is dense in the group of automorphism with even translation function. It follows that the topological closure of the group generated by reflections has index $2$ in $\mathrm{Aut}\left( \Gamma \right) $. rp-2002-33.pdf |
| 32/2002 |
Almost summung mappings Daniel Pellegrino We introduce a general definition of almost $p$-summing mappings and give several concrete examples of such mappings. Some known results are considerably generalized and we present various situations in which the space of almost $p$-summing multilinear mappings coincides with the whole space of continuous multilinear mappings. rp-2002-32.pdf |
| 31/2002 |
Multivector Functionals Virginia V. Fernández, Antonio M. Moya, Waldyr A. Rodrigues Jr. In this paper we introduce the concept of multivector functionals. We study some possible kinds of derivative operators that can act in interesting ways on these objects such as, e.g., the $A$-directional derivative and the generalized concepts of curl, divergence and gradient. The derivation rules are rigorously proved. Since the subject of this paper has not been developed in previous literature, we work out in details several examples of derivation of multivector functionals. rp-2002-31.pdf |
| 30/2002 |
Multivector Functions of a Multivector Variable Virginia V. Fernández, Antonio M. Moya, Waldyr A. Rodrigues Jr. In this paper we develop with considerable details a theory of multivector functions of a $p$-vector variable. The concepts of limit, continuity and differentiability are rigorously studied. Several important types of derivatives for these multivector functions are introduced, as e.g., the $A$-directional derivative (where $A$ is a $p$-vector) and the generalized concepts of curl, divergence and gradient. The derivation rules for different types of products of multivector functions and for compositon of multivector functions are proved. rp-2002-30.pdf |
| 29/2002 |
Multivector Functions of a Real Variable Virginia V. Fernández, Antonio M. Moya, Waldyr A. Rodrigues Jr. This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar to the usual theory of vector functions of a real variable, has some subtle issues which make its presentation worhtwhile.We refer in particular to the derivative rules involving exterior and Clifford products, and also to the rule for derivation of a composition of an ordinary scalar function with a multivector function of a real variable. rp-2002-29.pdf |
| 28/2002 |
Metric Clifford Algebra Virginia V. Fernández, Antonio M. Moya, Waldyr A. Rodrigues Jr. In this paper we introduce the concept of metric Clifford algebra $\mathcal{C\ell }(V,g)$ for a $n$-dimensional real vector space $V$ endowed with a metric extensor $g$ whose signature is $(p,q)$, with $p+q=n$. The metric Clifford product on $\mathcal{C\ell }(V,g)$ appears as a well-defined deformation (induced by $g$) of an euclidean Clifford product on $\mathcal{C\ell }(V)$. Associated with the metric extensor $g,$ there is a gauge metric extensor $h$ which codifies all the geometric information just contained in $g.$ The precise form of such $h$ is here determined. Moreover, we present and give a proof of the so-called golden formula, which is important in many applications that naturally appear in ours studies of multivector functions, and differential geometry and theoretical physics. rp-2002-28.pdf |
| 27/2002 |
Metric Tensor Vs. Metric Extensor Virginia V. Fernández, Antonio M. Moya, Waldyr A. Rodrigues Jr. In this paper we give a comparison between the formulation of the concept of metric for a real vector space of finite dimension in terms of tensors and extensors. A nice property of metric extensors is that they have inverses which are also themselves metric extensors. This property is not shared by metric tensors because tensors do not have inverses. We relate the definition of determinant of a metric extensor with the classical determinant of the corresponding matrix associated to the metric tensor in a given vector basis. Previous identifications of these concepts are equivocated. The use of metric extensor permits sophisticated calculations without the introduction of matrix representations. rp-2002-27.pdf |
| 26/2002 |
Extensors Virginia V. Fernández, Antonio M. Moya, Waldyr A. Rodrigues Jr. In this paper we introduce a class of mathematical objects called extensors and develop some aspects of their theory with considerable detail. We give special names to several particular but important cases of extensors. The extension, adjoint and generalization operators are introduced and their properties studied. For the so-called $(1,1)$-extensors we define the concept of determinant, and their properties are investigated. Some preliminary applications of the theory of extensors are presented in order to show the power of the new concept in action. An useful formula for the inversion of $(1,1)$-extensors is obtained. rp-2002-26.pdf |
| 25/2002 |
Euclidean Clifford Algebra Virginia V. Fernández, Antonio M. Moya, Waldyr A. Rodrigues Jr. Let $V$ be a $n$-dimensional real vector space. In this paper we introduce the concept of euclidean Clifford algebra $\mathcal{C\ell }(V,G_{E})$ for a given euclidean structure on $V$, i.e., a pair $(V,G_{E})$ where $G_{E} $ is a euclidean metric for $V$ (also called an euclidean scalar product). Our construction of $\mathcal{C\ell }(V,G_{E})$ has been designed to produce a powerful computational tool. We start introducing the concept of multivectors over $V$. These objects are elements of a linear space over the real field, denoted by $\bigwedge V.$ We introduce moreover, the concepts of exterior and euclidean scalar product of multivectors. This permits the introduction of two contraction operators on $\bigwedge V$, and the concept of euclidean interior algebras. Equipped with these notions an euclidean Clifford product is easily introduced. We worked out with considerable details several important identities and useful formulas, to help the reader to develop a skill on the subject, preparing himself for the reading of the following papers in this series. rp-2002-25.pdf |
