7,276 research outputs found

    The set of autotopisms of partial Latin squares

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    Symmetries of a partial Latin square are determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism Θ\Theta, the cardinality of the set PLSΘ\mathcal{PLS}_{\Theta} of partial Latin squares which are invariant under Θ\Theta only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms of partial Latin squares are characterized and several related properties studied. It is also seen that the cycle structure of Θ\Theta determines the possible sizes of the elements of PLSΘ\mathcal{PLS}_{\Theta} and the number of those partial Latin squares of this set with a given size. Finally, it is generalized the traditional notion of partial Latin square completable to a Latin square.Comment: 20 pages, 4 table

    An Application of Total-Colored Graphs to Describe Mutations in Non-Mendelian Genetics

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    Any gene mutation during the mitotic cell cycle of a eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra describing the genetic pattern of the inheritance process. We identify any such pattern with a total-colored graph so that any isotopism of the former is uniquely related to an isomorphism of the latter. This enables us to develop some results on graph theory in the context of the molecular processes that occur during the S-phase of a mitotic cell cycle. In particular, each monochromatic subset of edges is identified with a mutation or regulatory mechanism that relates any two statuses of the genotypes of a pair of chromatids.Junta de Andalucía FQM-016Junta de Andalucía FQM-32

    Mutations in evolution algebras by means of isotopisms

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    Any mutation of genotypes that occurs during the mitotic cell cycle in an eukaryotic cell can be algebraically represented by an isotopism of the evolution algebra that describes the genetic pattern of the inheritance process. This talk deals with the theory of isotopisms of non-associative algebras and, particularly, with the distribution of evolution algebras into isotopism classes in order to determine the spectrum of genetic patterns, up to mutation, that describe the mentioned inheritance process of a mitotic cell cycle.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech
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