Anti-chain

Osmotic pressure il anti chain concentration can be expressed in terms of... 

Finite units, chains and layers bonded together by m-m bonds anti chain and layer structures. 

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

In the chapter of Briiggemann and Carlsen some concepts introduced in the chapter of El-Basil are revitalized and explained in the context of the multivariate aspect. Basic concepts, like chain, anti-chain, hierarchies, levels, etc., as well as more sophisticated ones, like sensitivity studies, dimension theory, linear extensions and some basic elements of probability concepts are at the heart of this chapter. The difficult problem of equivalent objects, which lead to the items object sets vs. quotient sets are explained and exemplified. 

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. 

Note that by construction of levels any level is to be considered as antichain. However, this anti-chain may not necessarily be a maximal one. The evaluation of sampling sites for sediment samples of the Lake Ontario is used as a further illustrative example (cf. pp. 94). More details can be found in Bruggemann Halfon (1997) and Briiggemann et al. (2001 b). 

Wd(E) The width of a Hasse diagram. It is the maximum number of elements of A /. H. which are found in an anti-chain. In the context of Young diagrams (see Seitz, p. 373) also called a breadth . 

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. 

Stability An estimator for changes in the diagram to be anticipated, if any attributes will be added or omitted. Symbol P(IB). Since P(IB) is normalised and can only take values between 0 and 1 it can easily be interpreted. If P(//i) = 1, then all objects are arranged in an anti-chain - the inclusion of additional attributes will not change the structure. If P(IB) = 0, then all objects are arranged in a chain or they are equivalent to each other - the chain (and/or the equivalence) remains, if attributes are omitted. 

In general, the Elbe sites and tributaries appeared in the upper part of the diagram, whereas the river Rhine sites appeared in the lower half, allowing a tentative assessment of the contamination for both streams. The most important substances for the partial order shown in Fig. 7 are HCH and PCB-sum (for each W= 17). The stability P(IB) is 0.73, indicating that the order achieved is still near an anti-chain. 

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. 

The discussions so far have focused on the Young Diagram poset, but the general ideas may possibly be of use more widely. Clearly, since every (non-trivial) poset contains (non-trivial) anti-chains, the complexity measure of an element in any poset may be defined in the same way as was done here for the YDL poset. However, the utility of such a quantity will obviously depend on the system being described by the partial ordering. 

Figure 4.2.6 Two effects determine the configurations of open-chain carbohydrate derivatives (a) gauche confirmations are favored in 1,2-diols over anti conformations, and (b) 1,3-syn diaxial interactions between hydroxyl substituents are repulsive and often lead to distortions of the all-anti chain.

Fig. 34. Chain (Ol) and anti-chain (05) site occupancies as a lunction of oxygen nonstoichiometry. Note the different notation of the chain site in this figure. The transition T—O takes place abruptly, in contrast to previous investigations (cf. fig. 13). After Radaelli et al. (1992).

We have previously observed [412, 664] that in a fully anti-chain, a single interconversion from anti- to syn- placement creates a bend of about 20° in the chain direction. The same bend occurs with a single anti-placement in an otherwise fully syn- chain. Very recently these observations were expanded to cover a broader series of aromatic polyamides [801]. In modeling studies of aromatic polyamide chain bending [663], we have found that two other chain... 

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