On singularities arising from the contraction op the minimal section of a ruled surface

We will discuss the Gorenstein property of the singularity which is blown down from the minimal section of a ruled surface in terms of the extension class. In the case that the base field has positive characteristic, we find a new example (3.4) of Gorenstein singularity in connection with Theorem B.     

Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings.     

Application of intramolecular 1,3-dipolar cyclic addition of azide and olefin; construction of (pyrrolidine-2-ylidene)glycinate and glycinamides

Oxopropyl E-(pyrrolidine-2-ylidene)glycinamide (5c) and allyl E-(pyrrolidine-2-ylidene)glycinate (5d) were effectively synthesized from 2,3,5-tri-O-benzyl-4-O-tert-butyldimethylsilyl(TBDMS)-D-arabinal (7) using intramolecular 1,3-dipolar cyclic reaction of azide and olefin as a key reaction. These results proved this cyclic reaction should be applicable for the synthesis of various (pyrrolidine-2-ylidene)glycinate and glycinamide. In addition, the development of a synthetic route for the precursor of an unsaturated cyclic dehydro amino acid involved in azinomycins (carzinophilin) using relating glycinate, methyl E-(pyrrolidine-2-ylidene)glycinate (5a) was described.     

An intramolecular diels-alder approach to quassinoids—a stereoselective construction of a-aromatic klaineanone

Thermolysis of a benzocyclobutene derivative ($\text{20}$) prepared from norcamphor produced stereoselectively a tetracyclic compound ($\text{2}$), which was converted to a lactone ($\text{21}$) having correct stereochemistry at C₇, C₈, C₉, C₁₁, C₁₃, and C₁₄ positions of (±)-klaineanone.     

Preparation of Sr2CeO4:Eu3+,Dy3+ white luminescence phosphor particles and thin films by using an emulsion liquid membrane system

Sr(2)CeO(4) and Sr(2)CeO(4):Eu(3+),Dy(3+) phosphor particles and thin films were prepared by using an emulsion liquid membrane (ELM, water-in-oil-in-water (W/O/W) emulsion) system, containing VA-10 (2-methyl-2-ethylheptanoic acid) as extractant (cation carrier). A two-step extraction enabled efficient extraction for Sr(3+) and rare earth ions, and the resulting precursor metal oxalate particles produced in the internal water phase of the ELM system were about 60 nm in diameter. Calcination of the oxalate particles in air gave submicrometer-sized Sr(2)CeO(4) and Sr(2)CeO(4):Eu(3+),Dy(3+) particles, which showed blue and white luminescence, respectively, by UV excitation. Blue and white luminescence phosphor thin films were also prepared by soaking alumina substrates into the W/O emulsion containing precursor oxalate particles, followed by calcination in air.     

Electron spin resonance study of polymerization of butadiene with tris(acetylacetonato)titanium and triethylaluminum

ESR spectra of homogeneous catalyst derived from tris(acetylacetonato)titanium(III) and triethylaluminum were observed at several temperatures from ?78°C, to +25°C, at molar ratios of aluminum to titanium of 1–108. At ?78°C, this catalyst yields a violet complex which shows an ESR signal with a g value of 1.959 and is associated with the first intermediate. At ?40°C to ?30°C, this signal decreases, and two signals with g values of 1.947 and 1.960 are observed. The latter two signals diminish at ?5°C to +10°C, while two kinds of new signals with g values of 1.965 and 1.969 appear overlapping each other. The structures of the species corresponding to these five signals are discussed on the basis of the ESR spectra, the intensity change, and the unpaired spin distribution. A new signal with a g value of 1.978 is found in the presence of butadiene at 25°C at Al/Ti > 8 and is assigned to a growing end of polybutadiene with this catalyst. The polymer yield increases remarkedly at Al/Ti molar ratios greater than 10. The microstructure of the resulting polymer consists almost completely of 1,2 units. The structure of the growing end is proposed to be a titanium (III) species containing two 1-substituted allyl groups, by comparison with the structure ascribed to the growing end of polybutadiene with n-butyl titanate-triethylaluminum catalyst.     

Electron spin resonance study on polymerization of alkyl acrylates with n-butyl titanate(IV)–triethylaluminum catalyst

n-Butyl titanate(IV)–triethylaluminum catalyst at Al/Ti molar ratios greater than 6 polymerizes methyl and n-butyl acrylates at ?78°C. The polymerization system which includes methyl acrylate at ?78°C, gives two ESR signals with g factors of 1.958 and 1.961 that overlap each other. The absorption intensity of the latter signal is approximately proportional to the polymer chain concentration calculated from polymer yield and the molecular weight. The polymerization system at Al/Ti ratios smaller than 3 has no catalytic activity on the polymerization and shows only the ESR signal with the g factor of 1.958. On the basis of these facts the ESR signal with the g factor of 1.961 is attributed to the active growing end of poly(methyl acrylate) with this catalyst. The character of this active growing end is discussed.     

Polyiodide composition in poly(vinyl acetate)-iodine-iodide complex

Polyiodide formed by complexation of poly(vinyl acetate) (PVAc) with iodine in the presence of iodide has been investigated by chemical analysis and resonance Raman spectrophotometry. When PVAc films were immersed in iodide-iodine aqueous solutions which had different ratios of iodide to iodine concentration [I^?]/[I₂], the complex films exhibited tremendous variations of swelling degree, despite the relatively small change in the amount of bound iodine. From a quantitative chemical analysis, the composition of polyiodide bound to PVAc was found to be 1.01 ± 0.035 in the molar ratio of iodide to iodine irrespective of the composition of the iodide-iodine aqueous solution ([I^?]/[I₂] = 2–500). The polyiodide formed in PVAc-iodine-iodide complex was therefore inferred to be (I₃^?)_n. Resonance Raman spectra obtained on PVAc-iodine-iodide complexes were also identical to those of the benzamide-iodine complex, in which the polyiodide consists of (I₃^?)_n, consistent with the result from chemical analysis.