A note on totally regular variables and Appell sequences in hypercomplex function theory

Series title : Lecture notes in computer science, vol. 7971, ISSN 0302-9743The aim of our contribution is to call attention to the relationship between totally regular variables, introduced by R. Delanghe in 1970, and Appell sequences with respect to the hypercomplex derivative. Under some natural normalization condition the set of all paravector valued totally regular variables defined in the three dimensional Euclidean space will be completely characterized. Together with their integer powers they constitute automatically Appell sequences, since they are isomorphic to the complex variables.Fundação para a Ciência e a Tecnologia (FCT

Matrix representations of a special polynomial sequence in arbitrary dimension

This paper provides an insight into different structures of a special polynomial sequence of binomial type in higher dimensions with values in a Clifford algebra. The elements of the special polynomial sequence are homogeneous hypercomplex differentiable (monogenic) functions of different degrees and their matrix representation allows to prove their recursive construction in analogy to the complex power functions. This property can somehow be considered as a compensation for the loss of multiplicativity caused by the non-commutativity of the underlying algebra.Fundação para a Ciência e a Tecnologia (FCT

Incorporation of a Horizontally Transferred Gene into an Operon during Cnidarian Evolution

Genome sequencing has revealed examples of horizontally transferred genes, but we still know little about how such genes are incorporated into their host genomes. We have previously reported the identification of a gene (flp) that appears to have entered the Hydra genome through horizontal transfer. Here we provide additional evidence in support of our original hypothesis that the transfer was from a unicellular organism, and we show that the transfer occurred in an ancestor of two medusozoan cnidarian species. In addition we show that the gene is part of a bicistronic operon in the Hydra genome. These findings identify a new animal phylum in which trans-spliced leader addition has led to the formation of operons, and define the requirements for evolution of an operon in Hydra. The identification of operons in Hydra also provides a tool that can be exploited in the construction of transgenic Hydra strains