On pointwise decay rates of time‐periodic solutions to the Navier–Stokes equation

We study the existence of a time‐periodic solution with pointwise decay properties to the Navier–Stokes equation in the whole space. We show that if the time‐periodic external force is sufficiently small in an appropriate sense, then there exists a time‐periodic solution $\{u,p\}$ of the Navier–Stokes equation such that $|{?}^j{u(t,x)|=O(|x|}^{1?n?j})$ and $|{?}^j{p(t,x)|=O(|x|}^{?n?j})(j=0,1,\dots )$ uniformly in $t\in \mathbb{R}$ as $|x|\to \infty$. Our solution decays faster than the time‐periodic Stokes fundamental solution and the faster decay of its spatial derivatives of higher order is also described.     

Conjugacy relationship between M-convex and L-convex functions in continuous variables

By extracting combinatorial structures in well-solved nonlinear combinatorial optimization problems, Murota (1996,1998) introduced the concepts of M-convexity and L-convexity to functions defined over the integer lattice. Recently, Murota–Shioura (2000, 2001) extended these concepts to polyhedral convex functions and quadratic functions in continuous variables. In this paper, we consider a further extension to more general convex functions defined over the real space, and provide a proof for the conjugacy relationship between general M-convex and L-convex functions.Mathematics Subject Classification (1991): 90C10, 90C25, 90C27, 90C35This work is supported by Grant-in-Aid of the Ministry of Education, Culture, Sports, Science and Technology of Japan     

Phase Conjugation in Saturable Absorbing Dye Films by Degenerate Four-Wave Mixing and Holographic Processes Using Nanosecond Pulse and CW Lasers

We report on the simultaneous generation of phase-conjugate signals by degenerate four-wave mixing (DFWM) and holographic processes using a nanosecond pulse and a CW lasers in polyvinyl alcohol (PVA) films doped with four kinds of saturable absorbing dyes. For the pulse laser, of the four kinds of dye-doped PVA films, the erythrosine B-doped PVA and uranine doped-PVA films generate PC signals only by the DFWM process, while the other dye-doped PVA films generate PC signals simultaneously by not only the DFWM process but also the holographic process. Especially, the safranin T-doped PVA film generates strong PC signals by the holographic process. In contrast, all of the dye-doped PVA films generate the two types of PC signals for the CW laser. The fading of dye molecules is found to result in the generation of the holographic component of PC signals which governs the temporal behavior of the total PC signals.     

Fast Optical Switching Using Oriented Cyanine Dye-Doped Nematic Liquid Crystal

A cyanine dye, 2-[7-（1,3-dihydro-1,3,3-trimethyl-2H-indol-2-ylidene）-1,3,5-heptatrienyl]-1,3,3-trimethyl-3H-indolium iodide （NK-125）, is doped in 4-cyano-4＇-pentylbiphenyl （5 CB）, and the mixture is sandwiched between two pieces of rubbed glass plates. The third-order nonlinear optical properties of the oriented NK-125-SCB layers are measured by the resonant femtosecond degenerate four-wave mixing （DFWM） technique at 760 nm. The third-order nonlinear optical susceptibility of one of the present samples is 5.5×10^-8 esu. The slow DFWM response of the NK-125-SCB layers due to a population grating is accelerated by the increasing laser power because of amplified spontaneous emission （ASE）. On the other hand, we do not observe a similar phenomenon for NK-125- polyethylene glycol （PEG-400）. Oriented NK-125 molecules in nematic liquid crystals must have very high ASE efficiency. Hence the population grating in a DFWM signal disappears within about 4 ps. It is expected that NK-125-SCB can be used as a material for very fast all-optical switching.     

Constructive and Destructive Two-Pulse Excitation Investigated with a White-Light Michelson Interferometer

Linear propagation of two pulses through methanol solution of aluminum phthalocyanine chloride is investigated using a modified white-light Michelson interferometer. The observed coherence time of the white light is 6 fs, and the separation between the two-excitation pulses is set to about 10 fs. The excitation is dependent on the phase-relation between the two pulses. We have observed an enhancement of the excitation when the two pulses are in-phase and strong suppression of the excitation when the two pulses are out of phase by π.     

A SENSITIVE METHOD FOR DETERMINING CHLOROPHYLL b IN PLANT EXTRACTS

Abstract— A new method of using hydroxylamine as the reagent was developed to determine the concentrations of chlorophylls a and b in plant extracts accurately and sensitively. The method is based on the principle that hydroxylamine reacts in a short time with the aldehyde group of chlorophyll b in 95 per cent methanol at pH 5.8 to afford a derivative having a red band similar in position to the red band of chlorophyll a , while the spectrum of chlorophyll a is unaffected by the treatment with hydroxylamine. The method includes the measurements of absorbanlce for the mixture of a plant extract with hydroxylamine and for the reagent blank at a single wavelength, 666 mμ, and the concentrations of chlorophylls a and b are calculated from the absorbance value for the reagent blank and the increment of absorbance due to the reaction. By measuring this increment directly by difference photometry, one can determine a low concentration of chlorophyll b relative to chlorophyll a . The contents of chlorophylls in matured and etiolated leaves were determined by the method, and the errors included in this method were shown to be smaller than those in a commonly used technique including the measurements at two different wavelengths.     

Synthesis of pyrimidinopurinophanes

Synthesis and structures of the title compounds 2a–d are described. In spite of non-parallel orientation of the two base rings, fairly large hypochromism was observed for 2a–d.