Abstract
We study assertion objects that constitute a particular class of symbolic objects. Symbolic objects constitute a data analysis driven formalism, which can be compared to propositional calculus, but which is oriented toward the duality intension (characteristic properties) versus extension (set of all individuals verifying a given set of properties). The set of assertion objects is endowed with a partial order and a quasi-order. We focus on the property of completeness, which precisely expresses the duality intension-extension. The order structure of complete assertion objects is studied, using notions of lattice theory and Galois connection, and extending Wille's work to multiple-valued data. Two results are then obtained for particular cases.

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Abstract
The coordinate change eigenvalues for the MacKay and van Zeijts period doubling renormalisation operator for bimodal 1D maps are derived. They are found in numerical computations of the spectrum at all the periodic orbits of renormalisation of period up to five.

Abstract
Symbolic objects (Diday (1987, 1992), Brito, Diday (1990), Brito (1991)) allow to model data on the form of descriptions by intension, thus generalizing the usual tabular model of data analysis. This modelisation allows to take into account variability within a set. The formalism of symbolic objects has some notions in common with VL1, proposed by Michalski (1980); however VL1 is mainly based on prepositional and predicate calculus, while the formalism of symbolic objects allows for an explicit interpretation within its framework, by considering the duality intension-extension. That is, given a set of observations, we consider the couple (symbolic object — extension in the given set). This results from the wish to keep a statistics point of view. The need to represent non-deterministic knowledge, that is, data for which the values for the different variables are assigned a weight, led to considering an extension of assertion objects to probabilist objects (Diday 1992). In this case, data are represented by probability distributions on the variables observation sets. The notions previously defined for assertion objects are the generalized to this new kind of symbolic objects. Other extensions can be found in Diday (1992). © Springer-Verlag Berlin Heidelberg 1993.

Abstract