We prove the transcendence results for the infinite product , where Ek(x), Fk(x) are polynomials, α is an algebraic number, and r?2 is an integer. As applications, we give necessary and sufficient conditions for transcendence of and , where Fn and Ln are Fibonacci numbers and Lucas numbers respectively, and {ak}k?0 is a sequence of algebraic numbers with log‖ak‖=o(rk). 相似文献
Three new sesquiterpenoids, metachromins R--T (1--3), have been isolated from an Okinawan marine sponge Spongia sp. The structures and stereochemistry of 1--3 were elucidated on the basis of the spectroscopic data. Metachromins S (2) and T (3) showed modest cytotoxicity. 相似文献
Liquid–liquid equillibria (LLE) of the tertiary system of hydroxypropylated polyrotaxane (HPPR)–poly(vinyl alcohol) (PVA)–solvent have been investigated by focusing on the internal structures of HPPR–PVA blend gels. The phase diagrams of the HPPR–PVA aqueous systems displayed two liquid phases at a high concentration and molecular weight of PVA. This result was consistent with the prediction of the Flory–Huggins lattice model. On the contrary, the HPPR–PVA–DMSO system exhibited only a single phase. The HPPR–PVA blend gels crosslinked in dimethylsulfoxide (DMSO) were highly transparent over a wide concentration range, while the gels prepared in water were opaque at high polymer concentrations. Spherical domains were observed in the opaque gels by laser scanning confocal microscopy, and the sizes of the domains were significantly dependent on the amount of cross-linking reagent utilized. These results indicated that the transparency of the HPPR–PVA blend gels was strongly affected by the competition between the liquid–liquid two-phase separation and the crosslinking HPPR and PVA polymers during the preparation of the blend gels. 相似文献
The self-organized, position-controlled and parallel growth of GaAs and InAs nanowhiskers is successfully demonstrated by using a metal–organic chemical vapour deposition method. The growth takes place preferentially along the 111 As direction with the aid of the catalytic effect of Au nanodroplets, and not along 111 Ga or In directions. The diameter and length of the whisker can be controlled artificially down to 10 nm and to over 1 μm, respectively. Doping and composition control of p- or n-type such as GaAs–InAs heterostructure formation are possible along the length direction of the whisker by changing the source gases. In order to control the growth position of the whisker, positioning of a Au nanodroplet is essential and realized by a lithographic method. By choosing the [111]B direction to the substrate surface and normal to the patterned side edges, and by positioning the Au nanodroplet on the side wall, the positioned planar nanowhisker growth and bridging are successfully demonstrated. The growth mechanism of the nanowhiskers is revealed by the scanning and transmission electron microscope observations. Nanometer-size Au-alloy droplets play an important role in the growth of the whiskers. The whisker growth process is governed by the vapor–liquid–solid growth mechanism. 相似文献
1,3‐Disubstituted chiral allenes without any heteroatoms act as chiral initiators in the addition of (i‐Pr)2Zn to pyrimidine‐5‐carbaldehyde to afford, in combination with the subsequent asymmetric autocatalysis, chiral pyrimidin‐5‐yl alkanols with up to 98% ee. The absolute configuration of the pyrimidin‐5‐yl alkanol formed depend on that of the chiral allene. 相似文献
In the rac isomer of the title compound, C34H28O2, the two C—Phdimethylphenyl bond axes make an angle of 58.7 (1)°. There is no short contact between the two 3,5‐dimethylphenyl rings, although the dihedral angle between them is 4.93 (7)°. The meso isomer has a center of symmetry at the middle of the C=C bond, and the two C—Phdimethylphenyl bond axes are antiparallel to one another. 相似文献
Porphyrin–AuIII complexes, which were partially or totally modified with C6F5 at the meso positions, were synthesized. The highly electron‐withdrawing substituents induced electron‐deficient states and Lewis acid properties. Single‐crystal X‐ray analysis of the ion pairs revealed ion‐pairing assemblies with characteristics dependent on the number and substitution pattern of the C6F5 units and the geometries of the anions. 相似文献
We carried out a small-angle neutron scattering (SANS) study of dynamically polarized polyethylene (PE) samples doped with 2,2,6,6-tetramethylpiperidine 1-oxyl (TEMPO). The transmission of the PE with almost fully polarized neutrons (98.5%) increased with increasing the proton polarization, P. The incoherent scattering cross section decreased with increasing P. The effect of P on the polarized neutrons’ transmission and the incoherent scattering cross section agreed well with the theory. The q-dependence of the coherent scattering, which reflects a two-phase structure of PE composed of crystalline and amorphous domains, was kept unchanged by the proton polarization, but the intensity increased by a factor of 3 and 6 for P=+23% and −23%, respectively. The results mean that the contrast between the two phases was successfully enhanced by a dynamic nuclear polarization (DNP) technique. However, the enhancement is only 1/13–1/16 of the enhancement calculated by assuming a homogeneous polarization through the PE sample. The discrepancy suggests that P in amorphous domains (25%) should be higher than that in crystalline domains (22%) by 3%, which in turn may suggest the partial depolarization of proton spins on the way of the spin diffusion from amorphous domains, where TEMPO radicals localize, to crystalline domains. 相似文献
Throw your hat in the ring : A highly diastereoselective synthesis of the ABC rings of (?)‐norzoanthamine has been achieved starting from the (?)‐Hajos–Parrish ketone (see scheme). Three asymmetric quaternary carbon centers on the C ring were constructed by a 1,4‐addition, and an intramolecular Diels–Alder reaction provided a trans‐decalin scaffold on the AB rings.