An approximation theorem for second-order evolution equations

Summary A class of approximation schemes of arbitrary accuracy, generated by a two-step recurrence relation, is devised for evolution equations of the second order. The schemes are effected via a specially constructed family of rational approximations to cos for 0 and yield computationally efficient methods for systems of second-order ordinary differential equations and semidiscrete approximations for initial-boundary value problems for second-order hyperbolic equations.Research supported by ONR grant N00014-57-A-0298-0015Research supported by USARO grant DAAG 29-278-C-0024     

Kinetics of the X-ray induced gold nanoparticle synthesis

Time resolved small angle scattering has been employed to follow the formation kinetics of gold nanoparticles from a solution of gold hydrochlorate and tris-sodium citrate at room temperature. While UV irradiation leads to a slow kinetics as observed by optical spectroscopy, the irradiation by the X-rays led to a reduction of the gold within a few minutes. The size and morphology evolution during the radiolytic formation of particles displays the initial nucleation phase and the growth kinetics, which exhibit very nonuniform behavior. They show that the electrostatics of the particle stabilization plays a pronounced role in the repulsive forces and transient clustering of particles.     

A family of enneanuclear iron(II) single-molecule magnets

Complexes [Fe₉(X)₂(O₂CMe)₈{(2‐py)₂CO₂}₄] (X^?=OH^? ( 1 ), N₃^? ( 2 ), and NCO^? ( 3 )) have been prepared by a route previously employed for the synthesis of analogous Co₉ and Ni₉ complexes, involving hydroxide substitution by pseudohalides (N₃^?, NCO^?). As indicated by DC magnetic susceptibility measurements, this substitution induced higher ferromagnetic couplings in complexes 2 and 3 , leading to higher ground spin states compared to that of 1 . Variable‐field experiments have shown that the ground state is not well isolated from excited states, as a result of which it cannot be unambiguously determined. AC susceptometry has revealed out‐of‐phase signals, which suggests that these complexes exhibit a slow relaxation of magnetization that follows Arrhenius behavior, as observed in single‐molecule magnets, with energy barriers of 41 K for 2 (τ₀=3.4×10^?12 s) and 44 K for 3 (τ₀=2.0×10^?11 s). Slow magnetic relaxation has also been observed by zero‐field ⁵⁷Fe Mössbauer spectroscopy. Characteristic integer‐spin electron paramagnetic resonance (EPR) signals have been observed at X‐band for 1 , whereas 2 and 3 were found to be EPR‐silent at this frequency. ¹H NMR spectrometry in CD₃CN has shown that complexes 1 – 3 are stable in solution.     

Zero point energy of the Pullen–Edmonds Hamiltonian

Here we wish to apply the newly developed Generalized Moments Expansion (GMX) to the well-known potential

Stabilization and relative phase effects in a dichromatically driven diatomic Morse molecule: interpretation based on nonlinear classical dynamics

The dissociation dynamics of a dichromatically laser-driven diatomic Morse molecule vibrating in the ground state is investigated by applying tools of the nonlinear theory of classical Hamiltonian systems. Emphasis is placed on the role of the relative phase of the two fields, phi. First, it is found that, just like in quantum mechanics, there is dependence of the dissociation probability on phi. Then, it is demonstrated that addition of the second laser leads to suppression of probability (stabilization), when the intensity of the first laser is kept constant just above or below the single laser dissociation threshold. This "chemical bond hardening" diminishes as phi increases. These effects are investigated and interpreted in terms of modifications in phase space topology. Variations of phi as well as of the intensity of the second laser may cause (i) appearance/disappearance of the stability island corresponding to the common resonance with the lowest energy and (ii) deformation and movement of the region of Kolmogorov-Arnold-Moser tori that survive from the undriven system. The latter is the main origin in phase space of stabilization and phi dependence. Finally, it is shown that the use of short laser pulses enhances both effects.     

An initial- and boundary-value problem for a model equation for propagation of long waves

An initial- and boundary-value problem for a model equation for small-amplitude long waves is shown to be well-posed. The model has the form u_t + u_x + uu_x ? vu_xx ? α²u_xxt = 0, where x? [0, 1] and t ? 0. The solution u = u(x, t) is specified at t = 0 and on the two boundaries x = 0 and x = 1. Unique classical solutions are shown to exist, which depend continuously on variations of the specified data within appropriate function classes.     

Synthesis,Spectroscopy and Crystal Structure Analysis of N1,N3-dicyclohexyl-N1-(all-trans-retinoyl)urea

The title compound, C₃₃H₅₀N₂O₂, is a side product in the reaction of all-trans-retinoic acid (atRA) with N-hydroxysuccinimide, in the presence of the coupling agent N,N′-dicyclohexylcarbodiimide, which produces the ‘active’ ester succinimidyl all-trans-retinoate as the product. It crystallizes in the orthorhombic Pbca space group. The compound was characterized by ¹H-NMR, ¹³C-NMR, ESI–MS and IR spectroscopy and its structure was determined by single-crystal X-ray diffraction. For example in the ¹³C-NMR spectrum, diagnostic peaks are those of the two amide carbonyl C atoms at δ 169.5 and 154.2 ppm, the ten olefinic C atoms of the unsaturated chain of atRA moiety at δ 149.0, 139.3, 137.7, 137.3, 134.9, 130.2, 130.0, 129.4, 128.5 and 121.5 ppm and the two methine C atoms of the N,N′-dicyclohexylurea moiety at δ 57.9 and 49.5 ppm. Detailed analysis of its molecular and supramolecular structure showed that close-packing principles (elongated shape/large hydrophobic region of the molecule) together with chemical factors (N–H?O and C–H?O intermolecular interactions) direct the 3D self-assembly process in the crystalline state. Hirshfeld surface analysis was employed, a powerful approach to quickly and easily gain insight into molecular environments in the crystalline state.

Graphical Abstract

The synthesis and X-ray structure of 1-((2E, 4E, 6E, 8E)-3,7-dimethyl-9-(2,6,6-trimethylcyclohex-1-enyl)nona-2,4,6,8-tetraenoyl)-1,3-dicyclohexylurea, a side product in the synthesis of succinimidyl all-trans-retinoate, is reported; Hirshfeld surface analysis was employed to identify intermolecular interactions.

     

Analytic properties of moments matrices

Associated with each matrix element of the Generalized Moments Expansion, GMX(n,m) there is a unique expansion for the ground state energy in terms of the “connected moments” Ik of the Hamiltonian. That is, for any set {n,m} a polynomial in the Ik's may be generated to any desired order L, which is dependent upon the highest moment calculated. Here we wish to study the eigenvectors and eigenvalues of the GMX matrix itself. Furthermore we investigate the interplay between the set {n,m} and the order L of the matrix in determining which combination {n,m,L} yields the “best” (i.e. most convergent) result for the ground state energy.     

Turning points in random permutations of two kinds of distinct elements

The mean and variance of the number of turning points in random permutations of two kinds of distinct elements are evaluated. These results are applied to a Wald–Wolfowitz run test.