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Minimal Paths between Maximal Chains in Finite Rank Semimodular Lattices (1998)

Herscovici, David Samuel

Springer

Journal of algebraic combinatorics 7 (1998), S. 17-37

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ISSN: 1572-9192

Keywords: semimodular lattice ; maximal chain ; Jordan-Hölder permutation ; reduced decomposition

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract We study paths between maximal chains, or “flags,” in finite rank semimodular lattices. Two flags are adjacent if they differ on at most one rank. A path is a sequence of flags in which consecutive flags are adjacent. We study the union of all flags on at least one minimum length path connecting two flags in the lattice. This is a subposet of the original lattice. If the lattice is modular, the subposet is equal to the sublattice generated by the flags. It is a distributive lattice which is determined by the “Jordan-Hölder permutation” between the flags. The minimal paths correspond to all reduced decompositions of this permutation. In a semimodular lattice, the subposet is not uniquely determined by the Jordan-Hölder permutation for the flags. However, it is a join sublattice of the distributive lattice corresponding to this permutation. It is semimodular, unlike the lattice generated by the two flags, which may not be ranked. The minimal paths correspond to some reduced decompositions of the permutation, though not necessarily all. We classify the possible lattices which can arise in this way, and characterize all possibilities for the set of shortest paths between two flags in a semimodular lattice.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1023/A:1008615111070

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The mechanism of formation of stereoregular polymers (1961)

Mihail, R. ; Herscovici, J.

Hoboken, NJ : Wiley-Blackwell

Journal of Polymer Science 52 (1961), S. 45-50

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ISSN: 0022-3832

Keywords: Chemistry ; Polymer and Materials Science

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Chemistry and Pharmacology , Physics

Notes: At present various polymerization reactions are known which yield approximately identical types of eutactic polymers. This has been confirmed experimentally for a sufficient number of monomers (α-olefins, conjugated diolefins, vinyl esters, alkylene oxides, etc.) to lead to the conclusion that stereoregularity is based on a single or a few general acts. The mechanisms proposed for the formation of stereoregular polymers, i.e., radical, through alkyltitanium compounds (Nenitescu), semiconductor surface (Topchiev), or ionic (Natta, Furukawa), do not give a sufficiently general solution to the problem. Whereas some authors do not take into account the ionic character of some types of polymerization reactions and the possibility of free rotation of the macroradical around the C—C axis, others neglect the fact that radical reactions are known which leading to the formation of stereoregular units. Particularly useful is a generalization of the remarks concerning the steric control of unsymmetric induction (Cram). Stereospecific polymerization, or, to be more exact, the act or elementary acts leading to stereo-regularity of the molecules formed, should proceed just in the same manner as other stereospecific reactions. The authors of the present report attempted to find common elements between the different reaction mechanism leading to the formation of eutactic polymers.

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