Variational and Geometric Structures of Discrete Dirac Mechanics

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.     

Quatre applications du lemme de Zalcman à la dynamique complexe

We give four applications of Zalcman’s lemma to the dynamics of rational maps on the Riemann sphere: a parameter analogue of a proof of the density of repelling cycles in the Julia sets; similarity between the Mandelbrot set and the Julia sets; a construction of the Lyubich-Minsky lamination and its variant; and a unified characterization of conical points by Lyubich and Minsky and those by Martin and Mayer.     

Comparison of ultrasonic degradation rates constants of methylene blue at 22.8 kHz, 127 kHz, and 490 kHz

Techniques such as solvent extraction, incineration, chemical dehalogenation, and biodegradation have been investigated for the degradation of hazardous organic compounds. We found ultrasound to be an attractive technology for the degradation of hazardous organic compounds in water. However, the effects of ultrasonic frequency on degradation rate constants were not investigated quantitatively. In this study, the degradation process of a model for hazardous organic compound methylene blue was investigated using ultrasonic irradiation. The study focused on the effects of ultrasonic frequency and ultrasonic power on the degradation rate constant. The apparent degradation rate constants were estimated based on time dependence of methylene blue concentration assuming pseudo-first-order kinetics for the decomposition. A linear relationship between the apparent degradation rate constant and ultrasonic power was identified. In addition, the apparent degradation rate constants at frequencies of 127 and 490 kHz were much larger than those at 22.8 kHz. A relationship between the apparent degradation rate constant and the sonochemical efficiency value (SE value) was also found. Based on these results, a simple model for estimating the apparent degradation rate constant of methylene blue based on the ultrasonic power and the SE value is proposed in this study.     

Significant Effect of the Flexibility of Bridging Alkyl Chains on the Proximity of Stacked Porphyrin and Phthalocyanine Conjugated with a Fourfold Rotaxane Linkage

Spatial distance is an important factor in controlling the functional interactions between molecular units in a conjugate; therefore, the bridging unit has been closely examined. Here, we examined the effect of the flexibility of bridging alkyl chains on the proximity of stacked porphyrin and phthalocyanine conjugated with a fourfold rotaxane linkage. We found that closely stacking two π systems requires bridging alkyl chains above a certain length, and the shorter bridges hinder stacking because of their lower flexibility. The stacking distance between porphyrin and phthalocyanine in the conjugate with decyl (C₁₀) chains was estimated to be 4.03 Å and showed a unique physical character arising from short-distance interactions. The longer alkyl chains minimized steric restriction inside the fourfold rotaxane and allowed efficient communication between the porphyrin and phthalocyanine units. This is due to the flexibility of the side chains.     

Precise nuclear moments of extremely proton-rich nuclus 23 Al

Electric quadrupole coupling constant eqQ/h of the extremely proton-rich ²³Al (I ^π ?=?5/2^?+?, T _1/2?=?0.47 s) nucleus implanted into an Al₂O₃ single crystal has been measured for the first time, using the β-ray detecting nuclear quadrupole resonance method (β-NQR) in a high magnetic field. As a preliminary result, the quadrupole coupling constant was determined as |eqQ/h(²³Al) |?=?2.66±0.77 MHz. Using the quadrupole coupling constant of ²⁷Al in Al₂O₃ as a reference, the Q moment of the ground state of ²³Al was extracted as |Q(²³Al)|?~?160 mb, which is well explained by the shell model calculation in the sd-shell model space with the USD interaction.     

Spin polarization of β-emitting fragments in high energy heavy ions on Be collisions

The degrees of nuclear spin polarization of ^8,12B, ⁹C, ¹²N, ¹³O, ^20,21F, ²³Mg, ²⁷Si and ³⁹Ca produced in the high energy ^12,13C, ¹⁶O, ²²Ne, ²⁴Mg, ²⁸Si and ⁴⁰Ca ions on Be collisions have been measured systematically, for the technical developments of the nuclear moment studies as well as the hyperfine interaction studies by means of β-NMR technique. The fragment momentum dependences were well reproduced by the simple kinematical model. Incident energy dependence and the mass dependence as well as the reaction angle dependences were qualitatively explained by the mixing of the near and far side collisions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Experimental study of critical heat flux in open two-phase thermosyphon

Critical heat flux in an open two-phase thermosyphon has been measured employing water, R113 and R22 for a wide range ofL/D=4.8 to 960 and density ratio ofρ _L /ρ _G =6.17 to 1602. The CHF data measured, are found to be in fairly good agreement with those calculated from a criterion that the CHF takes place when reaches a maximum liquid falling rate in counter-current annular flow of liquid and vapor. Normal operation of the thermosyphon cannot be obtained at any liquid of water, R113 and R22 for a small tube diameter of less than 2 mm.     

Electric Field Gradients of B in TiO2

We observed the electric quadrupole interaction of ¹²B implanted in the interstitial site of TiO₂ using β-NMR method. The electric field gradients including the direction of the principal axes were determined. The direction agreed well with the theoretical calculation.