The number of labeled graphs placeable by a given permutation

Let S be a finite set and σ a permutation on S. The permutation σ* on the set of 2-subsets of S is naturally induced by σ. Suppose G is a graph and V(G), E(G) are the vertex set, the edge set, respectively. Let V(G) = S. If E(G) and σ*(E(G)), the image of E(G) by σ*, have no common element, then G is said to be placeable by σ. This notion is generalized as follows. If any two sets of {E(G), (σ¹)*(E(G)),…,(σ^l−1)* (E(G))} have no common element, then G is said to be I-placeable by σ. In this paper, we count the number of labeled graphs which are I-placeable by a given permutation. At first, we introduce the interspaced Ith Fibonacci and Lucas numbers. When I = 2 these numbers are the ordinary Fibonacci and Lucas numbers. It is known that the Fibonacci and Lucas numbers are rounded powers. We show that the interspaced Ith Fibonacci and Lucas numbers are also rounded powers when I = 3. Next, we show the number of labeled graphs which are I-placeable by a given permutation is a product of the interspaced Ith Lucas numbers. Finally, using a property of the generalized binomial series, we count the number of labeled graphs of size k which are I-placeable by σ. © 1996 John Wiley & Sons, Inc.     

Ir(III)‐Catalyzed C7‐Position‐Selective Oxidative CH Alkenylation of Indolines with Alkenes in Air

An efficient method for C7‐position‐selective alkenylation of N‐substituted indolines with alkenes is reported. Various 7‐alkenylindolines were obtained in moderate to excellent yields in air in the presence of catalytic amounts of [Cp*IrCl₂]₂, AgOTf, and Cu(OAc)₂. The protocol relies on the use of a carbonyl or carbamoyl group on the nitrogen atom of indoline as a directing group and is potentially applicable to the synthesis of 7‐alkenylindoles and 7‐alkylindoles.     

Morphological studies of a hydrogen-bonded LC polymer obtained by photopolymerization in LC solvents

Anisotropic morphologies and the phase behaviour of a hydrogen-bonded LC polymer obtained by photopolymerization in two kinds of LC solvent are discussed. The hydrogen-bonded LC monomer, 4-(6-acryloyloxyhexyloxy) benzoic acid (A6OBA), was photopolymerized in 4-cyano-4′-hexyloxybiphenyl (6OCB) and in 4-cyano-4′-undecyloxybiphenyl (11OCB), which show a nematic phase and a smectic A phase, respectively. After photo-polymerization, the LC media were removed by extraction and the pure polymer was observed by scanning electron microscopy. SEM images showed that the polymer possessed fibrous morphology with a fibre diameter of a few micrometers, based on polymerization-induced phase separation. The overall geometries reflected typical LC characteristics such as schlieren and focal-conic fan textures. It was found that the hydrogen bond between benzoic acid groups in the monomer was rigid enough to fix the anisotropic phase-separated structure forming during the early stage of phase separation; however, it could not permanently maintain the fibre structure due to dissociation at elevated temperature. X-ray measurements revealed that a well developed layer structure of the hydrogen-bonded mesogen existed in the polymer obtained from the smectic phase of 11OCB, whereas a polymer layer structure could develop only partially from the nematic phase of 6OCB.     

Evaluating a time-delay of hydrogen production quantitatively in photosynthetic bacteria for stabilizing intermittency

Renewable energy is regarded as a clean energy source but has some problems, one of which is intermittency. To reduce this, the time-delay of hydrogen production by photosynthetic bacteria can be effective. In this study, we qualitatively evaluated the time-delay of hydrogen production by photosynthetic bacteria under various irradiation conditions, and we also quantitatively evaluated it by fitting the experimental data and the hydrogen production model with a genetic algorithm. As a result of model fitting, we found that the relationship between the lengths of the optimized time-delay of hydrogen production by photosynthetic bacteria and the amount of light irradiation is linear. And we also found that the time-delay of hydrogen production by photosynthetic bacteria had an upper limit under low light intensity. We have suggested the existence of an energy store mechanism in photosynthetic bacteria.     

Dithienylethenes with a novel photochromic performance

Dithienylethenes with low decoloration quantum yields and thermal reversibility at high temperature above 100 degrees C were prepared. Introduction of bulky alkoxy substituents at 2- and 2'-positions of the thiophene rings strongly suppressed the cycloreversion quantum yields. The quantum yields were lower than 10(-3), and the photogenerated color remained stable enough under room light. On the other hand, the bulky alkoxy substituent decreased the thermal stability of the colored closed-ring isomers at high temperature. The color of the dithienylethene with cyclohexyloxy substituents faded out in less than 1 min at 160 degrees C.     

Catalytic Pauson-Khand-type reaction using aldehydes as a CO source

[reaction: see text]. Rhodium complex-catalyzed carbonylative alkene-alkyne coupling proceeds using aldehydes as a CO source. Cinnamaldehyde is the best CO donor, and various cyclopentenones were provided in high isolated yields by a solvent-free system.     

Dalitz analysis of D0-->K(0)(S)pi(+)pi(-)

In e(+)e(-) collisions using the CLEO detector, we have studied the decay of the D0 to the final state K(0)(S)pi(+)pi(-) with the initial flavor of the D0 tagged by the decay D(*+)-->D0pi(+). We use the Dalitz technique to measure the resonant substructure in this final state and clearly observe ten different contributions by fitting for their amplitudes and relative phases. We observe a K(*)(892)(+)pi(-) component which arises from doubly Cabibbo suppressed decays or D0-D0; mixing.     

Observation of B-->K(0)(S)pi(+)pi(-) and Evidence for B-->K(*+/-)pi(-/+)

We report on a search for charmless hadronic B decays to the three-body final states K(0)(S)h(+)pi(-), K(+)h(-)pi(0), K(0)(S)h(+)pi(0) (h(+/-) denotes a charged pion or kaon), and their charge conjugates, using 13.5 fb(-1) of integrated luminosity produced near sqrt[s]=10.6 GeV, and collected with the CLEO detector. We observe the decay B-->K0pi(+)pi(-) with a branching fraction (50(+10)(-9)(stat.)+/-7(syst.))x10(-6) and the decay B-->K(*+)(892)pi(-) with a branching fraction (16(+6)(-5)(stat.)+/-2(syst.))x10(-6).