An example of a polarized K3 surface with elliptic free Clifford divisors

Let (S, L) be a polarized K3 surface consisting of a Kummer surface S constructed from a product abelian surface of two elliptic curves which are not mutually isogenous and a base point free line bundle L such that L ² > 0. In this paper, we give some sufficient conditions for the existence of elliptic free Clifford divisors on (S, L).     

The Effects of Side-Chain Configurations of a Retro–Inverso-Type Inhibitor on the Human T-Cell Leukemia Virus (HTLV)-1 Protease

In this study, the effects of side-chain configurations of D-Ile residues of a retro–inverso (RI)-type inhibitor on the human T-cell leukemia virus type 1 (HTLV-1) protease containing a hydroxyethylamine dipeptide isostere were clarified. Prior to evaluation using the RI-type inhibitor, the effects of side-chain configurations of Ile residues of the substrate peptide on the HTLV-1 protease were examined to estimate the influence of side-chain configurations on enzyme activity. Based on the estimation of the influence of side-chain configurations on protease affinity, the RI-type inhibitors containing a D-allo-Ile residue in the corresponding substrate sequence, instead of a D-Ile residue, were synthesized via 9-fluorenylmethoxycarbonyl-based solid-phase peptide synthesis. Refolded recombinant HTLV-1 protease (1-116, L40I) was used for the simple and short evaluation of the inhibitory activities of the synthesized RI-type inhibitors. The results clearly indicated that mimicking the whole topology, comprising both the main- and side-chain structures of the parent inhibitor, is effective for the design of potent RI-modified protease inhibitors.     

Palladium doping of magic gold cluster Au38(SC2H4Ph)24: formation of Pd2Au36(SC2H4Ph)24 with higher stability than Au38(SC2H4Ph)24

A phenylethanethiolate-protected Pd(2)Au(36)(SC(2)H(4)Ph)(24) cluster, which is a two-Pd atom-doped cluster of the well studied magic gold cluster Au(38)(SC(2)H(4)Ph)(24), was synthesized in high purity and its stability was investigated. The results demonstrate that Pd(2)Au(36)(SC(2)H(4)Ph)(24) is more stable than Au(38)(SC(2)H(4)Ph)(24) against degradation in solution and core etching by thiols.     

2‐ and 3‐factors of graphs on surfaces

It has been conjectured that any 5‐connected graph embedded in a surface Σ with sufficiently large face‐width is hamiltonian. This conjecture was verified by Yu for the triangulation case, but it is still open in general. The conjecture is not true for 4‐connected graphs. In this article, we shall study the existence of 2‐ and 3‐factors in a graph embedded in a surface Σ. A hamiltonian cycle is a special case of a 2‐factor. Thus, it is quite natural to consider the existence of these factors. We give an evidence to the conjecture in a sense of the existence of a 2‐factor. In fact, we only need the 4‐connectivity with minimum degree at least 5. In addition, our face‐width condition is not huge. Specifically, we prove the following two results. Let G be a graph embedded in a surface Σ of Euler genus g.

• (1) If G is 4‐connected and minimum degree of G is at least 5, and furthermore, face‐width of G is at least 4g?12, then G has a 2‐factor.
• (2) If G is 5‐connected and face‐width of G is at least max{44g?117, 5}, then G has a 3‐factor.

The connectivity condition for both results are best possible. In addition, the face‐width conditions are necessary too. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 67:306‐315, 2011     

Detailed Investigation of the Structural,Thermal, and Electronic Properties of Gold Isocyanide Complexes with Mechano‐Triggered Single‐Crystal‐to‐Single‐Crystal Phase Transitions

Mechano‐induced phase transitions in organic crystalline materials, which can alter their properties, have received much attention. However, most mechano‐responsive molecular crystals exhibit crystal‐to‐amorphous phase transitions, and the intermolecular interaction patterns in the daughter phase are difficult to characterize. We have investigated phenyl(phenylisocyanide)gold(I) ( 1 ) and phenyl(3,5‐dimethylphenylisocyanide)gold(I) ( 2 ) complexes, which exhibit a mechano‐triggered single‐crystal‐to‐single‐crystal phase transition. Previous reports of complexes 1 and 2 have focused on the relationships between the crystalline structures and photoluminescence properties; in this work we have focused on other aspects. The face index measurements of complexes 1 and 2 before and after the mechano‐induced phase transitions have indicated that they undergo non‐epitaxial phase transitions without a rigorous orientational relationship between the mother and daughter phases. Differential scanning calorimetry analyses revealed the phase transition of complex 1 to be enthalpically driven by the formation of new aurophilic interactions. In contrast, the phase transition of complex 2 was found to be entropically driven, with the closure of an empty void in the mother phase. Scanning electron microscopy observation showed that the degree of the charging effect of both complexes 1 and 2 was changed by the phase transitions, which suggests that the formation of the aurophilic interactions affords more effective conductive pathways. Moreover, flash‐photolysis time‐resolved microwave conductivity measurements revealed that complex 1 increased in conductivity after the phase change, whereas the conductivity of complex 2 decreased. These contrasting results were explained by the different patterns in the aurophilic interactions. Finally, an intriguing disappearing polymorphism of complex 2 has been reported, in which a polymorph form could not be obtained again after some period of time, even with repeated trials. The present studies provide us with a variety of hitherto unknown insights into mechano‐responsive molecular crystals, which help us to understand the phase transition behaviors upon mechanical stimulation and establish rational design principles.