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On rough definability and types of approximation of classifications

Tripathy B.K., Ojha J, Mohanty D., Kumar Ch M.S.

Foundations of Computing and Decision Sciences

2010

Vol. 35, No. 3

197-215

The study of rough definability of classifications was initiated by Busse [3]. Classifications are of great interest, as in the process of learning from examples, rules are derived from classifications generated by single decisions. It was established by him that not all concepts associated with rough definability of sets can be extended to classifications. Four propositions were established in [3] and used in defining types of classifications. In this article we extend these propositions to obtain necessary and sufficient type theorems from which several results besides the above four results of Busse could be derived. We interpret and illustrate each of these results through examples. It is found that there are 11 possible types of classifications. Out of which only 5 which are basic in nature were considered by Busse without stating any reason for doing so. However, in this paper we shall establish that the other six types reduce to these five considered by him, either directly or transitively. In this paper, we prove a general theorem which provides a complete picture of the types of elements in a classification and introduce an algorithm to generate these classifications, given the number of elements in the classification. In fact, this result establishes the statement of Pawlak [9] that the notions of complement for sets and that for classifications are different. Also, we shall establish some properties of the parameters of measures of uncertainty; the accuracy of approximation and the quality of approximation of classifications.

On the elastic energy invariance of crystal twins

Glüge R.

Archives of Mechanics

2010

Vol. 62, nr 6

441-464

An analysis by Zanzotto (1992, 1996) showed that, if crystal twinning is considered as a material symmetry operation, the symmetry groups of many crystals that form twins correspond to the unimodular group, which is the symmetry group of elastic fluids. It is therefore evident that an elastic modelling based on an elastic energy that includes any possible combination of twin system activation must fail. However, one may try to construct an elastic energy which includes only the stressfree configurations of the parent and the first order twins. It is known that, even under this restriction, many twinning modes exhibit an elastic energy invariance, which renders them indistinguishable in terms of the elastic energy. In this article, it is demonstrated that this energy invariance holds for all compound twins. The implications of the latter are discussed. It is suggested to distinguish between crystallographically equivalent and crystallographically distinct compound twins, since different implications regarding the elastic energy invariance emerge.