Necessary conditions for characterization of laws via mixed sums

SupposeX andY are independent and identically distributed, and independent ofU which satisfies 0U1. Recent work has centered on finding the lawsL(X) for whichX U(X+Y) where denotes equality in law. We show that this equation corresponds to a certain projective invariance property under random rotations. Implicitly or explicitly, it has been assumed that the characteristic function ofX has an expansion property near the origin. We show that solutions may be admitted in the absence of this condition when –logU has a lattice law. A continuous version of the basic problem replaces sums with a Lévy process. Instead we consider self-similar processes, showing that a solution exists only whenU is constant, and then all processes of a given order are admitted.This research was in part supported by NSERC grant A-8466.     

The impact of polystyrene resins in solid-phase organic synthesis

Summary A major objective of the DIVERSOMER^® technology is to provide pure and characterized compounds for biological testing in order to prevent false negatives in our libraries. On several occasions, analysis of the final products by¹H-NMR and MS, has revealed by-products from the polystyrene solid support. Subsequently, three alternative methods were studied to remove polystyrene by-products; (i) prewashing of the resin prior to execution of the synthesis; (ii) pretreatment of the resin with the cleavage conditions consistent with the solid-phase synthesis reaction scheme; and (iii) parallel purification.     

Synthesis and Molecular Structures of Pyridine-Containing Large-Membered Cyclic Bis(alkoxy)silanes(1)

The new cyclic silanes [(C(5)H(3)N)(CH(2)O)(2)SiMe(2)](2) (1) and (C(5)H(3)N)(CH(2)CPh(2)O)(2)SiMe(2) (2) containing 16-membered and 10-membered rings, respectively, were prepared by the condensation reaction of Me(2)SiCl(2) with an appropriate pyridine diol in the presence of Et(3)N. X-ray studies show that the dimeric formulation for 1 represents a tetracoordinate cyclic silane, whereas 2 has a geometry halfway from a tetrahedron toward a trigonal bipyramid (TBP) as a result of Si-N(ax) donor action. (29)Si and (1)H NMR indicate retention of the coordination geometry for 2 in solution that undergoes rapid Si-N cleavage and ring rearrangement. In comparison with other silanes containing five- and six-membered rings that exhibit nitrogen or oxygen coordination, the presence of larger rings, as in 2 and related silanes having sulfur coordination, indicates that retention of donor action persists, thus largely ruling out ring size as a dominant factor controlling the possibility of donor action at silicon. The dimeric silane 1 crystallizes in the triclinic space group P&onemacr; with a = 6.347(3) ?, b = 12.455(4) ?, c = 14.289(5) ?, alpha = 101.63(3) degrees, beta = 102.99(3) degrees, gamma = 104.71(3) degrees, and Z = 2. The cyclic silane 2 crystallizes in the triclinic space group P&onemacr; with a = 9.733(4) ?, b = 10.938(2) ?, c = 14.312(3) ?, alpha = 89.03(2) degrees, beta = 74.59(3) degrees, gamma = 79.24(3) degrees, and Z = 2. The final conventional unweighted residuals are 0.040 (1) and 0.039 (2).     

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.     

Total Synthesis of d,l-Isospongiadiol: An Intramolecular Radical Cascade Approach to Furanoditerpenes

A stereoselective oxidative free-radical cyclization of beta-keto ester polyenes 7 and 19 has been accomplished as a one-step entry to the tricarbocyclic synthons 8and 21 which contain five and six stereogenic centers, respectively. These key synthons possessing an axial carboethoxy group at C-4 were ultimately converted to the spongian skeleton (8--> 14 and 21 --> 25 -->14). The synthesis of d,l-isospongiadiol (3) from the common intermediate 14 was realized after introduction of the 2alpha-hydroxy group in the spongian A-ring via epoxidation of silyl enol ether 28 and subsequent desilylation.