Transcendence of certain infinite products

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).     

Liquid-liquid equilibria of polyrotaxane and poly(vinyl alcohol)

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.     

Fabrication and Patterning of Submierometer-Thick Optical Polarizing Films for 800 nm Using Stretched Silver Island Multilayers

Stretched silver island multilayers have been investigated for patterned optical polarizers for the wavelength of 800 nm. Submicrometer-thick optical polarizing films are fabricated by stretching periodic multilayers consisting of silver island layers and Pyrex layers at the temperature of 660°C. As the optical anisotropy of the optical polarizing film is lost by heating at a temperature higher than the stretching temperature, the fine non-polarizing areas can be deliberately and easily formed on the optical polarizing film by laser irradiation with high power density. We have successfully formed various non-polarizing areas on the optical polarizing film with a 1 W-class carbon dioxide laser. The demonstrated fabricating techniques of deliberately patterned optical polarizing films should be useful for novel optical computing and sensing devices.     

New and simple method for fabricating polymer-coated silver hollow fibers with large mechanical strength

A new method is proposed for fabricating polymer-coated silver hollow glass fibers to avoid the flexibility deterioration after the curing process. Transmission properties of fibers made by the two procedures with and without curing process are compared. Little difference was observed in the transmission properties at the wavelength 2.94 μm of Er:YAG laser light and 10.6 μm of CO₂ laser light. The polymer layer is shown to be stable after 2-h, 5-W, continuous wave CO₂ laser light transmission.     

Highly Enantioselective Asymmetric Autocatalysis of Pyrimidin‐5‐yl Alkanol Induced by Chiral 1,3‐Disubstituted Hydrocarbon Allenes

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.     

rac‐ and meso‐3,3′‐Bis(3,5‐di­methyl­phenyl)‐3H,3′H‐2,2′‐biindenyl­idene‐1,1′(2H,2′H)‐dione

In the rac isomer of the title compound, C₃₄H₂₈O₂, the two C—Ph_{di­methyl­phenyl} 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—Ph_{di­methyl­phenyl} bond axes are antiparallel to one another.     

D3h‐Symmetric Porphyrin‐Based Rigid Macrocyclic Ligands for Multicofacial Multinuclear Complexes in a One‐Nanometer‐Sized Cavity

The one‐step synthesis of D_3h‐symmetric cyclic porphyrin trimers 1 composed of three 2,2′‐[4,4′‐bis(methoxycarbonyl)]bipyridyl moieties and three porphyrinatozinc moieties was achieved from a nickel‐mediated reductive coupling of meso‐5,15‐bis(6‐chloro‐4‐methoxycarbonylpyrid‐2‐yl)porphyrinatozinc. Although cyclic trimers 1 were obtained as a mixture that included other cyclic and acyclic porphyrin oligomers, an extremely specific separation was observed only for cyclic trimers 1 when using columns of silica gel modified with pyrenylethyl, cyanopropyl, and other groups. Structural analysis of cyclic trimers 1 was carried out by means of NMR spectroscopy and X‐ray crystallography. Treatment of an η³‐allylpalladium complex with a cyclic trimer gave a tris(palladium) complex containing three η³‐allylpalladium groups inside the space, which indicated that the bipyridyl moieties inside the ring could work as bidentate metalloligands.     

Total Synthesis of the 7,10‐Epimer of the Proposed Structure of Amphidinolide N,Part II: Synthesis of C17–C29 Subunit and Completion of the Synthesis

The total synthesis of 7,10‐epimer of the proposed structure of amphidinolide N was accomplished. The requisite chiral C17–C29 subunit was assembled stereoselectively via Keck allylation, Shi epoxidation, diastereoselective 1,3‐reduction, and a later oxidative synthesis of the THF framework. The C1–C13 and C17–C29 subunits were successfully coupled using a Enders RAMP “linchpin” as the C14–C16 three carbon unit, thereby controlling the chirality at C14 and C16. The labile allyl epoxy moiety was successfully constructed by Grieco–Nishizawa olefination at a final stage of the synthesis.