Reflecting a triangle in the plane

We prove that if the three angles of a triangleT in the plane are different from (60°, 60°, 60°), (30°, 30°, 120°), (45°,45°,90°),(30°,60°,90°), then the set of vertices of those triangles which are obtained fromT by repeating ‘edge-reflection’ is everywhere dense in the plane.     

Arranging Solid Balls to Represent a Graph

 By solid balls, we mean a set of balls in R ³ no two of which can penetrate each other. Every finite graph G can be represented by arranging solid balls in the following way: Put red balls in R ³, one for each vertex of G, and connect two red balls by a chain when they correspond to a pair of adjacent vertices of G, where a chain means a finite sequence of blue solid balls in which each consecutive balls are tangent. (We may omit the chain if the two red balls are already tangent.) The ball number b(G) of G is the minimum number of balls (red and blue) necessary to represent G. If we put the balls and chains on a table so that all balls sit on the table, then the minimum number of balls for G is denoted by b _T (G). Among other things, we prove that b(K ₆)=8,b(K ₇)=13 and b _T (K ₅)=8,b _T (K ₆)=14. We also prove that c ₁ n ³<b(K _n )<c ₂ n ³ log n, c ₃ n ⁴ log n<b _T (K _n )<c ₄ n ⁴. Received: March 29, 1999 Final version received: January 17, 2000     

Quantum D-modules and equivariant Floer theory for free loop spaces

The objective of this paper is to clarify the relationships between the quantum D-module and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper ``Homological Geometry'. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work of Guest, we formulate the notion of ``abstract quantum D-module' which generalizes the D-module defined by the small quantum cohomology algebra. Second, we define the equivariant Floer cohomology of toric complete intersections rigorously as a D-module, using Givental's model. This is shown to satisfy the axioms of abstract quantum D-module. By Givental's mirror theorem [Giv3], it follows that equivariant Floer cohomology defined here is isomorphic to the quantum cohomology D-module.     

Rh-catalyzed π-facial selective intermolecular hydroacylation of norbornenes

Rh-catalyzed π-facial selective intermolecular hydroacylations of norbornenes with salicylaldehyde have been attained. In the reaction with norbornylene, the exo-hydroacylated product was produced because of steric hindrance. In the case of norbornadiene, the endo-product was obtained because of chelation effect. Lastly, because of chelation and remote substituent effects, the product formed in the reaction of 7-tert-butoxynorbornadiene was the endo,syn-product. Deuterium-labeling experiments revealed that the hydroacylation stereoselectively proceeded via endo- and exo-intermediates.     

Structural analysis of the c(4 × 2) reconstruction in Si(0 0 1) and Ge(0 0 1) surfaces by low-energy electron diffraction

The c(4 × 2) structures in (0 0 1) surfaces of Si and Ge have been studied by low-energy electron diffraction (LEED). Using a proper cleaning method for the Si surface, we were able to observe clear c(4 × 2) LEED patterns up to incident energy of ∼400 eV as well as the Ge surface. Extensive experimental intensity-voltage curves allowed us to optimize the asymmetric dimer model up to the eighth layer (including the dimer layer) in depth in the dynamical LEED calculation. Optimized structural parameters are almost the same for the Si and Ge except for the height of the buckled-up atom of the asymmetric dimer. For the Ge surface, the structural parameters are in excellent agreement with those obtained by a previous theoretical calculation. The tilt angle and bond length of the dimer are 18 ± 1 (19 ± 1)° and 2.4 ± 0.1 (2.5 ± 0.1) Å for the Si(0 0 1) (Ge(0 0 1)), respectively.     

Liquid-liquid extraction of silver ion with benzothiacrown ether derivatives

The extraction behavior of mono-, poly-, anal bis-(benzothiacrown ether)s is studied for alkali, alkaline-earth, heavy, and transition metal ions in the presence of picrate. These ethers show high selectivity for silver ion. Silver extractions from binary and quaternary mixtures of silver with heavy or transition metal ions were successful, except for mixtures with mercury(II) ion. Silver was extracted from a copper ore with ca. 80% recovery.     

Sharp photonic Crystal Defect Modes and Their Response to Ultrashort Optical Pulses

Single photonic crystal defects based on an air-bridge structure were fabricated. We obtained sharp defect modes with quality factors higher than 600 and observed their response to ultrashort optical pulses by utilizing two-photon absorption.