Incomparable element

Comparable and incomparable elements Chain and Anti-chain... 

The sensitivity coefficient dCtldkj is of limited applicability in its original form. The parameters and the various output quantities of a model may have different units, for example, rate coefficients belonging to reactions of different orders have different units. In such cases, the elements of the sensitivity matrix are incomparable. The usual solution to this problem is the introduction of normalized sensitivity coefficients. These coefficients form the normalized sensitivity matrix,... 

Anti-chain Subset of the ground set, where all elements are mutually incomparable. An example is the anti-chain ( f, a, d, q1 q2, q, ). Any other element of the ground set added to the set f, a, d would introduce a comparability. Therefore f, a, d is a maximal anti-chain. Attribute profiles being results of monotonous variations as seen in chains are not considered as essentially different. Contrary, attribute profiles through anti-chains are essentially different. Hence the width, Wd(A ), of the poset is considered as a measure of diversity. 

A rather good approximation for an element of interest, x, may be obtained, if the successors (all elements "below" x) and predecessors (all elements "above" x), respectively, are organized into a so-called S-x-P chain, all remaining elements, i.e. those incomparable to x being considered as isolated. From a combinatorial study follows that the averaged rank of an element x can be expressed as... 

The site 5, on the other hand, differs from the above discussed sites as a rather smeared out probability plot is disclosed. Thus, the eventual assignment of a rank for site 5 is uncertain. This can also be seen directly from the visualization in the Hasse diagram (Fig s. 20 and 19). The site 5 is not as strongly connected as the other three elements. In more detail the consequences are discussed in Briiggemann et al. (2001b). We can calculate the local quantity U(x), i.e. the number of incomparabilities of an element x. The larger the values of U(x) the more uncertain the rank of x is. In the case of site 5 it turns out that U(5) = 6, whereas the corresponding values for the site 91, 1, 17 are U(91) = 1, U(l) = 2, and U(17) = 1, respectively. Therefore the measure of uncertainty about the ranks is U(91) = U(17) < U(l) U(5). 

Considering the other clusters of Fig. 6a it is evident that also K2 and K4 consist of samples from almost one river section, namely FL (Fahlberg List) and HM (MciBcn harbour). However, in case of FL cluster K2 is not comparable to the remaining samples of FL four of overall ten samples from FL are assigned to K2. Moreover, most of the samples of FL are incomparable to all other samples (they are isolated elements). Reasons for this specific pattern could be the discontinuous sewage draining from an old contaminated site there. 

Anti-chain An alignment of objects, which are not comparable with one another. Elements of the same level (see chapter by Briigge-mann and Carlsen, p. 61) are incomparable. They can be considered to be similarly polluted but with different pollution patterns. As sometimes the construction of levels cannot be done uniquely, their interpretation needs some care. 

The total number of comparabilities V and incomparabilities U and their local analogues (i.e. the no of comparabilities V(x) and incomparabilities U(x) of a certain element x are useful quantities for the documentation of the Hasse diagram and for the estimation of ranking uncertainties (Bruggemann and Welzl 2002). 

Mathematicians have termed a set of elements in a poset that are all mutually incomparable an anti-chain. (See chapter by Briiggemann and Carlsen, p. 61 and for more detailed mathematics and definitions see Combinatorics and Partially Ordered Sets Dimension Theory by Trotter (Trotter 1992)). If we consider all anti-chains that contain a partition [A] as an element, the complexity of [a] is the number of elements in those antichains (i.e. the cardinality or size of the anti-chains) that have the maximum number of elements, maximum anti-chains. Clearly, this concept can be generalized to any poset, though, as we have seen, the case of the YDL is of particular interest and relevance to physics and chemistry. 

For j < k, the poset elements bj, are incomparable with the rest hence they... 

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