Fuzzy Implications: Some Recently Solved Problems

In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference

Polymerization of propene with modified constrained geometry complexes. Double-bond isomerization in pendant alkenyl groups attached to cyclopentadienyl ligands

Polymerization of propene with dimethylsilylene-bridged (amidocyclopentadienyl) dichlorotitanium( IV) complexes [TiCl2 {eta(5)-1-(t-BuSiMe2N-kappaN)- 2,3,4- Me-3 -5- R-C-5}], where R =Me (1), H (2), Ph (3), 4-fluorophenyl (4), but-2-en-2-yl (5), and butyl (6), combined with excess methylaluminoxane revealed a moderate effect of the substituent R on the catalyst activity and the molecular weight of polypropene. The asymmetric substitution in the position adjacent to the bridging carbon atom resulted in polymer yields decreasing in the order 1 > 6 > 3 approximate to 5 > 4 > 2 while polymers with the molecular weights (M-w) close to 2.5 x 10(5) for 1, 3, and 4, 1.5 x 10(5) for 5 and 6, and 7.5 x 10(4) for 2 were obtained. The C-13 NMR analysis of the polymers has shown that atactic polypropene is slightly enriched with syndiotactic triads for all the catalysts. Investigation of the crystal structure of 5 by X-ray crystallography revealed that the double bond in but-3-en-2-yl had shifted to an internal position to give the isomeric, but-2-en-2-yl-substituted complex. Likewise, the spectroscopic data for complex 7 prepared from the ligand containing but-3-en-1-yl substituent, indicate the absence of terminal double bond