Time decay for nonlinear wave equations in two space dimensions

The Cauchy Problem for the equation u_tt–u+|u|^p–1u=0 (x², t>0, >1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For >5, it is shown that the maximum amplitude of such a wave decays at the expected rate t^–1/2 as t. For 1+8<5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when >_o, where _o is the root of the cubic equation ³-2²-7-8=0; thus _o4.15.Alfred P. Sloan Research Fellow     

Recurrence vs. transience for spatially inhomogeneous birth and death in a random environment

Summary We consider birth and death in a random environment, having probabilitiesp(,x) andq(,x) of a step to the right or left, respectively, from statex in environment. Under several sets of conditions involving the existence of limp(,x) and limq(,x) asx, we give conditions for certain extinction or escape to when 0 is absorbing, corresponding to recurrence or transience when 0 is reflecting. The environmental sequence is assumed to be stationary ergodic, and for some results stronger assumptions are required.     

Polymerization and polymers of silicic acid

Linear relationships among molecular weight, polymerization time, and gelation for the condensation of any monomer, including those of unknown size and functionality, are developed and applied to data on soluble silicic acids generated from tetraethyl silicate and from sodium silicate. The results suggest the formation of monomers containing ca. 12–15 ? OSi units with functionality of ca. 2.05 that condense with a rate constant of ca. 10^?4 liter/mole sec and an activation energy of 40–70 kJ/mole. One model compatible with these characteristics and the stoichiometry involved is a ladder polymer ca. 3 ? SiO units wide. Polymer isolation was achieved by replacing residual ? OH with (CH₃)₃Sio? , as well as by freezing of of aqueous solutions, which yielded fibers under special conditions. Solutions of the uncapped and capped polymers and melts of the latter had low viscosity even for fractions with M?_n ～100,000. This implies a coiled or globular nature for the polymers, which is supported by their limited propensity for film and fiber formation. Attemps to improve this situation by copolymerization, the use of other capping agents, and by the polymerization of precapped monomers were unsuccessful.     

Structure, 31P and 13C chemical shift anisotrophy of (CH3)3P, (CH3)3PO, (CH3)3PS and (CH3)3PSe as determined by liquid crystal NMR spectroscopy

The ¹³P and ¹³C spectra of the triply ¹³C labelled molecules (CH₃)₃P, (CH₃)₃PO, (CH₃)₃PS and (CH₃)₃PSe oriented in a nematic phase are reported. The CPC bond angles have been measured. The ¹³P chemical shift tensor shows a large anisotropy except in the case of (CH₃)₃P. The abnormal large value observed for the PSe bond length suggests a large anisotropy of the ¹J(PSe) spin coupling.     

Synthesis of 1,4-annulated cyclooctatetraenophanes based on a novel cubane building block approach

A general synthetic approach to strained 1,4-annulated cyclooctatetraene-based cyclophanes is described. A key feature in this approach is exploitation of the cubane core as a masked cyclooctatetraene synthon. Thus, 1,4-disubstituted cubanes 3 and 4 used as precursors to cyclooctatetraenophanes have been prepared in four steps from the readily available 1,4-cubanedicarboxaldehyde (5). The synthesis of 3 was effected by palladium/copper-mediated coupling of 1,4-bis[(Z,Z)-2-iodovinyl]cubane (6) and 1,4-bis[(Z,Z)-but-1-en-3-ynyl]cubane (8). For the synthesis of 4, on the other hand, modified Eglington-Glaser coupling was applied for the macrocyclization step. The general characteristic of Rh(I) to induce [2 + 2] cycloreversion of the cubane core to syn-tricyclo[4.2.0.0(2,5)]octa-3,7-diene followed by thermal rearrangement to cyclooctatetraene was applied as a key structural transformation toward targeted cyclooctatetraenophanes 1 and 2.     

Linear dichroism and fluorescence polarization of diphenyl polyenes in stretched polyethylene films

A study of the linear diehroism and fluorescence polarization of diphenylpolyenes (C₆H₅—(CH=CH)_n—C₆H₅) with n = 1,2,3,4,6,8 in stretched polyethylene films shows that the polyenes orient in the anisotropic matrix with their transition dipole (emission and absorption, π → π^*) aligned with the direction of stretch. The maximum dichroic ratio is observed for 1,6-diphenyl-1,3,5-hexatriene (DPH), with n = 3, whereas the values are substantially lower for n < 3 or n > 3. The high orientation of DPH and poor alignment of the higher polyenes may be accounted for in terms of the growing flexibility of the polyene chains with increasing chain length. The results justify the use of DPH as a convenient fluorescence probe for biological and synthetic membranes.     

FLUORESCENCE QUANTUM YIELDS AND LIFETIMES FOR BACTERIOCHLOROPHYLL c

Abstract— Absorption and emission spectra of bacteriochlorophyll c dissolved in a variety of solvents were measured and fluorescence quantum yields determined from the integrated emission spectra. Values for the emission maxima calculated from the positions and bandwidths of the red absorption band using the Stepanov relationship agreed closely with the experimental values. Fluorescence quantum yields varied between 0.10 in methanol and 0.36 in tetrahydrofuran and in dibutylamine. Fluorescence lifetimes were also determined for bacteriochlorophyll c in four of the solvents, and ranged from 2.7 ns in methanol to 7.6 ns in dimethylsulfoxide.     

PHOTOCHEMICAL PROPERTIES OF PORPHYRIN c: AN AGENT FOR USE IN TUMOR PHOTOTHERAPY

Abstract— The absorption and fluorescence properties of porphyrin c (P c ), the porphyrin chromophore present in cytochrome c , have been determined in several solvents and micellar environments. In aqueous buffer solutions at pH 7.5 Pc may exist in both a fluorescent monomeric form with quantum yield of fluorescence, (Φ_f,) ∼ 0.03, and fluorescence lifetime, (τ_f) ∼ 8 ns, and as a non-fluorescent aggregate. The proportion of monomeric form is higher in organic solvents and micelles but is reduced with increasing porphyrin concentrations in aqueous solutions. Porphyrin c readily complexes with Zn²⁺ to produce a fluorescent chelate (Zn-P c ) with Φ_f, ∼ 0.02 and τ_f, ∼ 2 ns at pH 7.5. The yields of singlet excited oxygen formation from Pc and the Zn-P c complex are higher than observed for hematoporphyrin derivative (HpD). Both P c and Zn-P c are effective agents in tumor phototherapy and do not induce the prolonged cutaneous photosensitivity observed with the use of HpD.     

A Remarkable Two-Step Synthesis of Diverse 1,4-Benzodiazepine-2,5-diones Using the Ugi Four-Component Condensation

A two-step, general synthesis of 1,4-benzodiazepine-2,5-diones (BZDs) is presented. This synthesis employs an Ugi four-component condensation using a convertible isocyanide (1-isocyanocyclohexene), followed by an acid-activated cyclization reaction. This synthesis represents a dramatically improved route to BZDs over those currently in the literature. In addition, since amino acids are not used as inputs, the potential for molecular diversity is much greater than that with existing syntheses. It was also found that BZDs substituted with methylenes at the C-3 and N-4 positions display conformational isomerism in the NMR spectra at room temperature. Variable-temperature NMR experiments support this observation and offer the interesting conclusion that the BZD core structure, in certain examples, might not be as rigid as previously supposed.     

Valence bond studies of AH2 molecules

Minimal basis set (STO) molecular orbital and valence-bond calculations are reported for the³ B ₁ and¹ A ₁ states of CH₂. The open-shell molecular orbital calculations used the Roothaan formulation. The valence-bond calculations used the Prosser-Hagstrom biorthogonalisation technique to evaluate the cofactors required in using Löwdin's formulae. Optimisation of geometry and orbital exponents in the molecular orbital calculation on the³ B ₁ state gave a geometry of R_C-H=2.11 a.u. and H-C-H=123.2 °. The energy obtained was ?38.8355 a.u. The molecular orbital and valencebond calculations are compared. In the valence-bond calculations the variation with bond-length and bond-angle of the configuration energies was studied. Valence bond “build-up” studies are also reported. Valence-bond calculations using hybrid orbitals instead of natural atomic orbitals showed that the perfect-pairing approximation is not as good for CH₂ as BeH₂. The nature of the lone-pair and bonding orbitals is found to be significantly different between the³ B ₁ and¹ A ₁ states. In the³ B ₁ state the 2s and 2p orbitals are fairly equally mixed between both types of orbital. However in the¹ A ₁ state the bonding orbitals have mainly 2p character and the lone pair orbitals have mainly 2s character. As was found for H₂O, the bonding hybrid orbitals do not follow the hydrogen nuclei as the bond angle varies but continue to point approximately in their equilibrium directions.