An elementary proof of the exponential blow-up for semi-linear wave equations

This paper deals with the upper bound of the life span of classical solutions to □u = ∣u∣^p, u∣_{t = 0} = εφ(x), u_t∣_t=0 = εψ(x) with the critical power of p in two or three space dimensions. Zhou has proved that the rate of the upper bound of this life span is exp(ε^−p(p−1)). But his proof, especially the two-dimensional case, requires many properties of special functions. Here we shall give simple proofs in each space dimension which are produced by pointwise estimates of the fundamental solution of □. We claim that both proofs are done in almost the same way.     

Synthesis of Trideca-O-methyl-alpha-pedunculagin. Diastereo-Favoritism Studies on Intramolecular Ester-Cyclization of Axially Chiral Diphenic Acids with Carbohydrate Core

Total synthesis of trideca-O-methyl-alpha-pedunculagin was achieved by a simple sequence. The key step is the synthesis of methyl 4,6-O-benzylidene-2,3-O-[(S)-4,4',5,5',6,6'-hexamethoxydiphenoyl]-alpha-D-glucopyranoside through intramolecular ester-cyclization of racemic hexamethoxydiphenoyl chloride with methyl 4,6-O-benzylidene-alpha-D-glucopyranoside at the 2,3-position. The diastereoselectivity results obtained in the intramolecular cyclization of hexamethoxydiphenic acid to the carbohydrate core raises a very interesting point in considering the pathway of (R)-diphenic acid biosynthesis.     

Total synthesis of gambierol

The convergent total synthesis of gambierol (1) is described. The octacyclic ether framework of 1 was constructed via the intramolecular allylation of alpha-chloroacetoxy ether followed by ring-closing metathesis. A modified Stille coupling was successfully applied to the synthesis of the triene side chain.     

Synthesis of 1,8-, 1,6- and 3,6-dichloro-9H-thioxanthen-9-ones

Cyclization of 2-chloro-6-[(3-chlorophenyl)thio]benzoic acid ( 2 ) gave a mixture of 1,8-, 3 , and 1,6-dichloro-9H-thioxanthen-9-ones 4 . The mixture was converted to 1,8-diamino- 7 , and 1-amino-6-chloro-9H-thioxanthen-9-ones 8 , from which 3 and 4 were prepared separately, respectively. From a mixture of 4 and 3,6-dichloro-9H-thioxanthen-9-one ( 11 ) obtained by cyclizing 4-chloro-2-[(3-chlorophenyl)thio]benzoic acid ( 10 ) was separated 11 by conversion of 4 to 8 .     

Total synthesis of brevetoxin B

The convergent total synthesis of brevetoxin B (1) has been achieved. The intramolecular allylation of the O,S-acetal 20, prepared from the alpha-chlorosulfide 17 and the alcohol 5, was carried out using AgOTf as a Lewis acid to give the diene 21, predominantly. Ring-closing metathesis of 21 with the Grubbs catalyst 23 afforded the hexacyclic ether 25 which was converted to the A-G ring segment 2 through several steps. The intramolecular allylation of the alpha-acetoxy ether 42, prepared from 2 and the J-K ring segment 3, followed by ring-closing metathesis provided the polycyclic ether framework 44. A series of reactions of 44, including oxidation of the A ring, deprotection of the silyl ethers, and selective oxidation of the resulting allylic alcohol, furnished 1.     

Amino acids and peptides—XIV : A simple and convenient method for preparation of α-substituted α-diazo esters

Treatment of 16 α-amino acid esters with isoamyl nitrite in the presence of a small amount of organic acid in chloroform or benzene, followed by chromatographic purification on alumina was found to afford the corresponding α-substituted-α-diazo esters (1) in fairly good yields.     

Low-voltage electroosmosis pump for stand-alone microfluidics devices

Two types of low-voltage electroosmosis pumps were developed using microfabrication technology for usage in handy or stand-alone applications of the micrototal analysis systems (micro-TAS) and the lab-on-a-chip. This was done by making a thin (< 1 microm) region in the flow path and by only applying voltages near this thin region using electrodes inserted into the flow path. The inserted electrodes must be free from bubble formation and be gas-tight in order to avoid pressure leakage. For these electrodes, Ag/AgCl or a gel salt bridge was used. For patterning the gel on the chip, a hydrophilic photopolymerization gel and a photolithographic technique were optimized for producing a gel with higher electric conductivity and higher mechanical strength. For high flow rate application, wide (33.2 mm) and thin (400 nm) pumping channels were compacted into a 1 mm x 6 mm area by folding. This pump achieves an 800 Pa static pressure and a flow of 415 nL/min at 10 V. For high-pressure application, a pump was designed with the thin and thick regions in series and positive and negative electrodes were inserted between them alternatively. This pump could increase the pumping pressure without increasing the supply voltage. A pump with 10-stage connections generated a pressure of 25 kPa at 10 V.