Morphology and orientational disorder of C70 single crystals

C₇₀ single crystals, free from solvent contamination, were grown via vapor phase transport technique. The (0001) face of the C₇₀ crystal was imaged with Atomic Force Microscopy (AFM) under ambient atmosphere. Low-resolution images of a freshly prepared crystal revealed well-ordered faceted regions with multiple terraces extending over a region of several hundred square nanometers. The terraced structure in the images indicates that the crystals formed as a layer growth. In the high-resolution images, C₇₀ molecules were resolved in a hexagonal close packed (hcp) structure with an average center-to-center distance of 10.5±0.5 Å. After one month's exposure to the ambient atmosphere, the molecules rearranged into a mixed structure consisting of hcp and face centered cubic (fcc) regions.     

On the presence of the S4z-term in Fe(HCOO)2 · 2H2O

The magnetic specific heat of ferrous formate dihydrate is analysed on the basis of a model, $\text{S}{}_{\mathrm{A}}\text{=}\text{1}\text{2}$ and S_B = 2. The experimental result turns out to be well explained by the introduction of a large S⁴_z-term for B site ions.     

Effect of the frustration to the ground state energy and entropy of the spin-glass in the random bond Ising model on the square lattice

The energies and the entropies of the spin-glass state and the paramagnetic state at T = 0 of the random-bond Ising mixture of the ferromagnetic bond (concentration p) and the antiferromagnetic bond (concentration 1 ? p) on the square lattice are calculated by the method of the square approximation in the simple version. A self-consistent relation that the partial trace of the normalized density matrix of the square cluster is equal to that of the vertex ($\text{tr}\text{(}{}_{\mathrm{jkl}}\text{?}\text{?}{}^{\mathrm{(4)}}\text{(i,j,k,l) =}\text{?}\text{?}{}^{\mathrm{(1)}}\text{(i)}$) leads to an integral equation for the distribution function of the effective fields, and it is solved exactly at T = 0. The symmetric solution of the integral equation contains the paramagnetic state and two spin-glass states, SG1 and SG2. The energies and the entropies of these states are obtained as functions of the concentration p. The values of the energies per spin at $\text{p =}\text{1}\text{2}$ are -0.75|E_F|, -0.72746|E_F|, -0.72543|E_F|, and correspond to a minimum, a saddle point, and a maximum, respectively, and the values of the entropies are 0, 0.082886k_B, and 0.054457k_B, respectively. The present results are compared with those of the pair approximation and discussed.     

Homological properties of balanced Cohen-Macaulay algebras

A balanced Cohen-Macaulay algebra is a connected algebra having a balanced dualizing complex in the sense of Yekutieli (1992) for some integer and some graded - bimodule . We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem:
 

As a corollary, we will have the following characterizations of AS Gorenstein algebras and AS regular algebras:
 

     

Novel catalytic hydrogenolysis of silyl enol ethers by the use of acidic ruthenium dihydrogen complexes

Treatment of 1-trimethylsilyloxy-1-cyclohexene (1a) in the presence of a catalytic amount of the acidic dihydrogen complex [RuCl(η²-H₂)(dppe)₂]OTf (4a) [dppe=1,2-bis(diphenylphosphino)ethane, OTf=OSO₂CF₃] (10 mol.%) under 1 atm of H₂ in anhydrous ClCD₂CD₂Cl at 50 °C for 8 h afforded cyclohexanone (3a) and Me₃SiH in quantitative NMR yields. Silyl enol ethers such as 1-triethylsilyloxy-1-cyclohexene (1b), 1-t-butyldimethylsilyloxy-1-cyclohexene (1c), and other trimethylsilylethers (1d, 1e, and 1f) reacted similarly with H₂ to afford the corresponding ketones and trialkylsilanes. The direct proton transfer from H₂ to the trimethylsilyl enol ethers (1a and 1d-1f) was confirmed by the experiments employing D₂ gas, where α-monodeuterated ketones (3a′ and 3d′-3f′) were obtained in high yields. The enantioselective protonation of prochiral silyl enol ethers with 1 atm of H₂ by employing [RuCl(η²-H₂)((S)-BINAP)₂]OTf (4e) [BINAP=2,2′-bis(diphenylphosphino)-1,1′-binaphthyl] and [RuCl(η²-H₂)((R, R)-CHIRAPHOS)₂]OTf (4f) [CHIRAPHOS=2,3-bis(diphenylphosphino)butane] showed that no enantioselectivity was observed in either catalytic or stoichiometric protonation reactions under various reaction conditions. The reaction of [RuHCl(dppe)₂] (5a) with one equivalent of Me₃SiOTf under 1 atm of H₂ produced rapidly 4a, concurrent with the formation of Me₃SiH. Based on these studies, the mechanism for this novel hydrogenolysis of silyl enol ethers is proposed which involves heterolytic cleavage of the coordinated H₂ on the ruthenium atom caused by the nucleophilic attack of the oxygen atom of enol ethers to give ketones and Me₃SiOTf, and the subsequent reaction of the resultant complex 5a with Me₃SiOTf under 1 atm of H₂ to regenerate the original dihydrogen complex 4a. On the other hand, the stoichiometric reaction of a lithium enolate 6e with one equivalent of 4e at −78 °C in CH₂Cl₂ under 1 atm of H₂ afforded 2-methyl-1-tetralone (3e) with 75% ee (S) in >95% yield, together with the formation of [RuHCl((S)-BINAP)₂] (5e).     

Studies on the syntheses of heterocyclic compounds—DCCLXII : Alternative synthesis of trans-3-(1'R*-hydroxyethyl)-4-(2',2'-dimethoxyethyl)-2-azetidinone,a synthetic intermediate of thienamycin

Synthesis of trans-3-(1'R^*-hydroxyethyl)-4-(2',2'-dimethoxyethyl)-2-azetidinone (5), an important intermediate for the synthesis of thienamycin (1), was investigated starting from the isoxazoline derivatives 3 and 9. The most effective method was catalytic hydrogenation of trans-4-t-butoxycarbonyl-3-(2,2'-dimethoxyethyl)-5-methyl-isoxazoline (9) with Adams catalyst in acetic acid, followed by trimethylsilylation of the resulting epimeric aminoesters 11A and B, cyclization with EtMgBr, and deblocking. Novel reductions of the isoxazolines with sodium borohydride and nickel chloride or with diborane followed by catalytic hydrogenation were also reported.     

The structure of AS-Gorenstein algebras

In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.     

Symmetry in the vanishing of Ext over stably symmetric algebras

A Frobenius algebra over a field k is called symmetric if the Nakayama automorphism is an inner automorphism. A stably symmetric algebra is defined to be a generalization of a symmetric k-algebra. In this paper we will study symmetry in the vanishing of Ext for such algebras R, namely, for all finitely generated R-modules M and N, for all i0 if and only if for all i0. We show that a certain class of noetherian stably symmetric Gorenstein algebras, such as the group algebra of a finite group and the exterior algebra Λ(kⁿ) when n is odd, have this symmetry using Serre duality. We also show that every exterior algebra Λ(kⁿ), whether n is even or odd, has this symmetry for graded modules using Koszul duality.     

The classification of 3-dimensional noetherian cubic Calabi–Yau algebras

It is known that every 3-dimensional noetherian Calabi–Yau algebra generated in degree 1 is isomorphic to a Jacobian algebra of a superpotential. Recently, S.P. Smith and the first author classified all superpotentials whose Jacobian algebras are 3-dimensional noetherian quadratic Calabi–Yau algebras. The main result of this paper is to classify all superpotentials whose Jacobian algebras are 3-dimensional noetherian cubic Calabi–Yau algebras. As an application, we show that if S is a 3-dimensional noetherian cubic Calabi–Yau algebra and σ is a graded algebra automorphism of S, then the homological determinant of σ can be calculated by the formula $\mathrm{hdet}\sigma ={(\det \sigma )}^2$ with one exception.     

Co-Point Modules Over Frobenius Koszul Algebras

Let A be a Frobenius Koszul algebra such that its Koszul dual A ^! is a quantum polynomial algebra. Co-point modules over A were defined as dual notion of point modules over A ^! with respect to the Koszul duality. In this article, we will see that various important functors between module categories over A used in representation theory of finite dimensional algebras send co-point modules to co-point modules. As a consequence, we will show that if (E, σ) is a geometric pair associated to A ^!, then the map σ:E → E is an automorphism of the point scheme E of A ^!, so that there is a bijection between isomorphism classes of left point modules over A ^! and those of right point modules over A ^!.