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Eigenvalues and Eigenvectors of Endomorphisms,envalue and associated eigenvector on matrices with coefficients in the dioid (R.,+,×). Indeed, it asserts that such a matrix has an eigenvalue in this dioid, with an associated eigenvector having all components in the dioid; moreover, it establishes a special property for this eigenvalue, as compar

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Dioids and Nonlinear Analysis, show that the structures of dioids lend themselves to defining, in the ., new branches of .The basic idea is to replace the classical field structure on the reals by a dioid structure. Thus, a new branch of nonlinear analysis will correspond to each type of dioid. This approach was pioneered by Mas

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https://doi.org/10.1007/978-3-476-04213-2t of .) and to the solution of equations of the fixed-point type..Various types of topologies may be introduced, depending on the nature of the ordered sets considered. The simplest cases correspond to a totally ordered set, or to a product of totally ordered sets (e.g. R. with the partial order ind

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invertible, i.e. where one cannot always solve a . x = b and a . x = b?.The key idea in the present chapter is to observe that the solution of a “fixed point” type equation such as x = a . x.b only requires the existence of the . a* of the element a, defined in the previous chapter as the “l(fā)imit” of

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https://doi.org/10.1057/9780230508866 moduloid structure) is the one which arises naturally in the properties of sets of vectors with entries in a semiring (resp. in a dioid). Thus, they turn out to be analogues for algebraic structures on semirings and dioids to the concept of a module for rings..Section 2 introduces the main basic no

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https://doi.org/10.1007/978-3-030-89058-2envalue and associated eigenvector on matrices with coefficients in the dioid (R.,+,×). Indeed, it asserts that such a matrix has an eigenvalue in this dioid, with an associated eigenvector having all components in the dioid; moreover, it establishes a special property for this eigenvalue, as compar