Global wellposedness for a one-dimensional Chern–Simons–Dirac system in Lp

The global wellposedness in L^p(?) for the Chern–Simons–Dirac equation in the 1+1 space and time dimension is discussed. We consider two types of quadratic nonlinearity: the null case and the non-null case. We show the time global wellposedness for the Chern–Simon–Dirac equation in the framework of L^p(?), where 1≤p≤∞ for the null case. For the scaling critical case, p = 1, mass concentration phenomena of the solutions may occur in considering the time global solvability. We invoke the Delgado–Candy estimate which plays a crucial role in preventing concentration phenomena of the global solution. Our method is related to the original work of Candy (2011), who showed the time global wellposedness for the single Dirac equation with cubic nonlinearity in the critical space L²(?).     

Birational rigidity of complete intersections

We prove that every smooth complete intersection \(X=X_{d_{1}, \ldots , d_{s}}\subset \mathbb {P}^{\sum _{i=1}^{s}d_{i}}\) defined by s hypersurfaces of degree \(d_{1}, \ldots , d_{s}\) is birationally superrigid if \(5s +1\le \frac{2(\sum _{i=1}^{s}d_{i}+1)}{\sqrt{\prod _{i=1}^{s}d_{i}}}\). In particular, X is non-rational and \({{\mathrm{Bir}}}(X)={{\mathrm{Aut}}}(X)\). We also prove birational superrigidity of singular complete intersections with similar numerical condition. These extend the results proved by Tommaso de Fernex.     

Calix[3]amides—bowl-shaped cyclic trimers toward building block for molecular recognition: self-complementary dimeric structure in the crystal

Bowl-shaped cyclic trimers of aromatic amides were simply synthesized in high yield by condensation reaction of meta-substituted 3-(alkylamino)benzoic acid using dichlorotriphenylphosphorane. The cyclic amides exist in syn conformation, which has a small chiral cavity, and a pair of each enantiomeric conformer formed a dimeric structure in the crystal.     

Communication: Synthesis of 3-O-Benzyl-β-L-Arabinofuranose 1,2,5-Orthopivalate as a Starting Monomer for Ring-Opening Polymerization

We have studied the ring-opening polymerizations of a variety of glucose orthoester derivatives and found that substituents on the monomer play an important role in stereo- and regioregularity of the resulting polymers.^1-3 These substituent effects open the possibility of application to ring-opening polymerizations of other sugar orthoesters to give stereo- and regioregular polysaccharides. Additionally, the ring-opening polymerization of the galactose orthoester derivative⁴ gave stereoregular (1→5)-β-D-galactofuranan.     

Studies on Phosphorus-Containing Polymers. V. Ring-Opening Polymerization of Tetrahydrofuran Catalyzed by Linear Phosphonitrilic Chloride

It was found that linear phosphonitrilic chloride could be used as a catalyst for ring-opening polymerization of tetrahydrofuran. Bulk polymerizations were carried out in a nitrogen atmosphere. After termination of polymerization, the reaction mixture was poured into water, thereby decomposing the catalyst. The product was dissolved in benzene and then subjected to lyophilization. The polymerization of tetrahydrofuran in the presence of linear phosphonitrilic chloride was found to be an equilibrium and a “living” polymerization. The polymerization product includes little phosphorus, and its infrared absorption spectrum agrees well with that of the polymer obtained with PF₅ catalyst. The results of the polymerization using epichlorohydrin as a promoter show that the number of active sites in the molecule of linear phosphonitrilic chloride is considerably smaller. Consequently it is conceivable that the catalytic activity of the linear phosphonitrilic chloride is attributed to its terminal ～～P⁺Cl₃PCl_?6 structure. Furthermore we presume that the polymerization of tetra-hydrofuran in the presence of this catalyst proceeds through a cationic ring-opening mechanism.     

Total syntheses of (+)-polygalolide a and (+)-polygalolide b: elucidation of the absolute stereochemistry and biogenetic implications

The total syntheses of (+)-polygalolide?A and (+)-polygalolide?B have been completed by using a carbonyl ylide cycloaddition strategy. Three of the four stereocenters, including two consecutive tetrasubstituted carbon atoms at C2 and C8, were incorporated through internal asymmetric induction from the stereocenter at C7 by a [Rh(2) (OAc)(4)]-catalyzed carbonyl ylide formation/intramolecular 1,3-dipolar cycloaddition sequence. The arylmethylidene moiety of these natural products was successfully installed by a Mukaiyama aldol-type reaction of a silyl enol ether with a dimethyl acetal, followed by elimination under basic conditions. We have also developed an alternative approach to the carbonyl ylide precursor based on a hetero-Michael reaction. This approach requires 18 steps, and the natural products were obtained in 9.8 and 9.3?% overall yields. Comparison of specific rotations of the synthetic materials and natural products suggests that polygalolides are biosynthesized in nearly racemic forms through a [5+2] cycloaddition between a fructose-derived oxypyrylium zwitterion with an isoprene derivative.     

Synthesis and Structure/Property Relationships of Regioselective 2‐O‐, 3‐O‐ and 6‐O‐Ethyl Celluloses

Regioselectively ethylated celluloses, 2‐O‐ ( 1 ), 3‐O‐ ( 2 ), and 6‐O‐ethyl‐ ( 3 ) celluloses were synthesized via ring‐opening polymerization of glucopyranose orthopivalate derivatives. The number‐average degrees of polymerization (DP_ns) of compounds 1 and 2 were calculated to be 10.6 and 49.4, respectively. Three kinds of compound 3 with different DP_ns were prepared: DP_ns = 12.9 ( 3‐1 ), 60.3 ( 3‐2 ), and 36.1 ( 3‐3 ). The 2‐O‐, 3‐O‐, and 6‐O‐ethylcelluloses were soluble in water, confirmed by NMR analysis. Furthermore, the 3‐O‐ ( 2 ), and 6‐O‐ethyl‐ ( 3‐2 ) celluloses showed thermo‐responsive aggregation behavior and had a lower critical solution temperature (LCST) at about 40 °C and 70 °C, respectively, based on the results from turbidity tests and DSC measurements. The 6‐O‐ethyl‐cellulose ( 3‐3 ) with DP_n = 36.1 and DP_w = 54.6 showed gelation behavior over approx 70 °C, whereas the 6‐O‐ethyl‐celluloses 3‐1 and 3‐2 with lower and higher molecular weight, such as DP_ns 12.9 and 60.3, did not show gelation behavior at this temperature. It was revealed that the position of ethyl group affected the phase transition temperature. According to our experiments, the 3‐O‐ethyl and 6‐O‐ethyl groups along the cellulose chains caused the thermo‐responsive property of their aqueous solutions. The appropriate DP of the regioselective 6‐O‐ethyl‐cellulose existed for gelation of the aqueous solution.

     

ABA- and BAB-triblock cooligomers of tri-<Emphasis Type="Italic">O</Emphasis>-methylated and unmodified cello-oligosaccharides: syntheses and structure-solubility relationship

Triblock cooligomers consisting of tri-O-methyl-glucopyranosyl and unmodified glucopyranosyl residues, methyl 2,3,4,6-tetra-O-methyl-β-d-glucopyranosyl-(1 → 4)-2,3,6-tri-O-methyl-β-d-glucopyranosyl-(1 → 4)-β-d-glucopyranosyl-(1 → 4)-β-d-glucopyranosyl-(1 → 4)-2,3,6-tri-O-methyl-β-d-glucopyranosyl-(1 → 4)-2,3,6-tri-O-methyl-α-d-glucopyranoside (1: ABA triblock cooligomer; DS = 2.1) and β-d-glucopyranosyl-(1 → 4)-2,3,6-tri-O-methyl-β-d-glucopyranosyl-(1 → 4)-2,3,6-tri-O-methyl-β-d-glucopyranosyl-(1 → 4)-2,3,6-tri-O-methyl-β-d-glucopyranosyl-(1 → 4)-2,3,6-tri-O-methyl-β-d-glucopyranosyl-(1 → 4)-d-glucopyranose (2: BAB triblock cooligomer; DS = 1.8) were prepared. Compound 1 dissolved both in distilled water and chloroform but compound 2 dissolved in distilled water not in chloroform, though compounds 1 and 2 consist of 4 tri-O-methyl-glucopyranosyl and 2 unmodified anhydro glucopyranosyl units.