Numerical experiments of the planar equal-mass three-body problem: the effects of rotation

The planar three-body problem with angular momentum is numerically and systematically studied as a generalization of the free-fall problem (i.e., the three-body problem with zero initial velocities). The initial conditions in the configuration space exhaust all possible forms of a triangle, whereas the initial conditions in the momentum space are chosen so that position vectors and momentum vectors are orthogonal. Numerical results are organized according to the value of virial ratio k defined as the ratio of the total kinetic energy to the total potential energy. Final motions are mapped in the initial value space. Several interesting features are found. Among others, binary collision curves seem to spiral into the Lagrange point, and for large k, binary collision curves connect the Lagrange point and the Euler point. The existence of a lunar periodic orbit and a periodic orbit of petal-type is suggested. The number of escape orbits as a function of the escape time is analyzed for different k. The behavior of this number for different time and k shows most remarkably the effects of rotation of triple systems. The number of escape orbits increases exponentially for k相似文献   

Planar-chiral macrocyclic host pillar[5]arene: no rotation of units and isolation of enantiomers by introducing bulky substituents

Enantiomers of bulky percyclohexylmethyl-substituted pillar[5]arene (Cy-C1-Pillar) were able to be separated by chiral column chromatography, and the separated enantiomers did not racemize. Even though modified with the bulky cyclohexylmethyl-substituents at both rims, Cy-C1-Pillar was able to capture a guest molecule.     

Arrangements of n Points whose Incident-Line-Numbers are at most n/2

We consider a set X of n noncollinear points in the Euclidean plane, and the set of lines spanned by X, where n is an integer with n ≥ 3. Let t(X) be the maximum number of lines incident with a point of X. We consider the problem of finding a set X of n noncollinear points in the Euclidean plane with t(X) ￡ ?n/2 ?{t(X) \le \lfloor n/2 \rfloor}, for every integer n ≥ 8. In this paper, we settle the problem for every integer n except n = 12k + 11 (k ≥ 4). The latter case remains open.     

Planar-chiral pillar[5]arene: chiral switches induced by multiexternal stimulus of temperature, solvents, and addition of achiral guest molecule

We synthesized chiral-substituents modified pillar[5]arene for the first time. The chiral-substituents modified pillar[5]arene showed planar chirality and interconversion between (pS) and (pR) forms took place quickly. The planar chirality was switched by temperature, solvents, and addition of achiral guest. As the measurement temperature increased, the diastereomeric excess was decreased. The diastereomeric excesses were high in low-permittivity solvents, while a low diastereomeric excess was observed in high-permittivity solvents. Addition of achiral guest induced an increase of negative CD intensities.     

Formation and coordination modes of the C4E4 moiety (E = COOMe) bound to two palladium(II) moieties

The transmetallation of the palladacyclopentadiene complex Pd{C(COOMe)C(COOMe)C(COOMe)C(COOMe)}(bipy) with the dicationic Pd(II) complex [Pd(bipy)(CH(3)CN)(2)][BF(4)](2) afforded a terminally σ-palladated diene complex [Pd(2){μ-η(1):η(1)-C(COOMe)C(COOMe)C(COOMe)C(COOMe)}(bipy)(2)(CH(3)CN)(2)][BF(4)](2). It was revealed by X-ray crystallographic analysis that replacement of the acetonitrile ligands in a terminally σ-palladated diene complex with PPh(3) ligands resulted in the conformation change of the σ-palladated diene moiety from skewed s-cis to planar s-trans. Treatment of a bis-triphenylphosphine dipalladium complex [Pd(2)(PPh(3))(2)(CH(3)CN)(4)][PF(6)](2) with dimethoxyacetylene dicarboxylate (DMAD) (1 equiv.) in acetonitrile resulted in the insertion of DMAD to the Pd-Pd bond to afford [Pd(2){μ-η(1):η(1)-C(COOMe)C(COOMe)}(PPh(3))(2)(CH(3)CN)(4)][PF(6)](2). Addition of the second DMAD gave the ylide-type complex [Pd(2){μ-η(2):η(3)-C(COOMe)C(COOMe)C(COOMe)C(COOMe)(PPh(3))}(PPh(3))(2)(CH(3)CN)(3)][PF(6)](2) of which the structure was determined by X-ray crystallographic analysis.     

Helly Numbers of Polyominoes

We define the Helly number of a polyomino P as the smallest number h such that the h-Helly property holds for the family of symmetric and translated copies of P on the integer grid. We prove the following: (i) the only polyominoes with Helly number 2 are the rectangles, (ii) there does not exist any polyomino with Helly number 3, (iii) there exist polyominoes of Helly number k for any k ≠ 1, 3.     

Reversibly tunable lower critical solution temperature utilizing host-guest complexation of pillar[5]arene with triethylene oxide substituents

A thermoresponsive macromolecule consisting of 10 outer triethylene oxide groups and a pillar[5]arene core was prepared. The macromolecule showed lower critical solution temperature behavior. Moreover, its clouding point can be reversibly tuned based on the addition of guest and host compounds; the clouding point increased upon addition of a guest didecylviologen salt and decreased when the competitive host cucurbit[7]uril was added.