Some analytical semigroups occurring in probability theory

One-parameter semigroups occurring in operator-limit distributions are investigated. The topological-algebraic background of the relevant monoids is discussed and Lie semigroup theory is applied to the Urbanik Decomposability Semigroup.     

Dithiacrown Ether Substituted Porphyrazines: Synthesis, Single-Crystal Structure, and Control of Aggregation in Solution by Complexation of Transition-Metal Ions

The synthesis of novel magnesium, copper, and metal-free porphyrazines, peripherally substituted with dithia-7-crown-2 (MPz(7)), dithia-15-crown-5 (MPz(15)), and dithia-18-crown-6 (MPz(18)) macrocycles is reported. These compounds are prepared starting from dicyanoethylene containing crown ethers 3, 2(1), and 2(2), respectively, which contain sulfur as well as oxygen heteroatoms. The "crowned" porphyrazines bind silver(I) and mercury(II) perchlorates. UV/vis spectroscopy and electron paramagnetic resonance measurements reveal that addition of the transition-metal ions leads to dimerization of the porphyrazine complexes. In the case of the dithia-18-crown-6-substituted porphyrazines, the dimers break up to form monomeric 6:1 guest-host complexes when more than 2 equiv of the metal ion is added. The single-crystal structures of the crown ether 2(2) and the porphyrazine MgPz(18) are presented. Compound C(14)H(20)N(2)O(4)S(2) (2(2)) crystallizes in the monoclinic space group P2(1)/c with a = 10.9310(13) ?, b = 19.383(3) ?, c = 8.6976(14) ?, beta = 108.898(11) degrees, V = 1743.5(5) ?(3), and Z = 4. The structure refinement converged to R = 0.0366 and R(w) = 0.0504. Compound C(56)H(82)MgN(8)O(17)S(8) (MgPz(18)) crystallizes in the triclinic space group P&onemacr; with a = 9.584(3) ?, b = 17.672(2) ?, c = 19.620(4) ?, alpha = 84.904(14) degrees, beta = 85.21(2) degrees, gamma = 89.29(2) degrees, V = 3298.4(13) ?(3), and Z = 2. The structure refinement converged to R1 = 0.0839 and wR2 = 0.2196. The electrical properties of H(2)Pz(18) have been studied by complex impedance spectroscopy. The bulk electrical conductivity of this compound is approximately 1 order of magnitude higher than that of the corresponding 18-crown-6 phthalocyanine.