An iterative starting method to control parasitism for the Leapfrog method

The Leapfrog method is a time-symmetric multistep method, widely used to solve the Euler equations and other Hamiltonian systems, due to its low cost and geometric properties. A drawback with Leapfrog is that it suffers from parasitism. This paper describes an iterative starting method, which may be used to reduce to machine precision the size of the parasitic components in the numerical solution at the start of the computation. The severity of parasitic growth is also a function of the differential equation, the main method and the time-step. When the tendency to parasitic growth is relatively mild, computational results indicate that using this iterative starting method may significantly increase the time-scale over which parasitic effects remain acceptably small. Using an iterative starting method, Leapfrog is applied to the cubic Schrödinger equation. The computational results show that the Hamiltonian and soliton behaviour are well-preserved over long time-scales.     

Mean values with cubic characters

We investigate various mean value problems involving order 3 primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large sieve-type result for order 3 (and 6) Dirichlet characters.     

Cubic Diophantine Inequalities

Let λ₁, λ₂,..., λ₇ be real numbers satisfying λ _i ≥ 1. In this paper, we prove there are integers x ₁,..., x ₇ such that the inequalities |λ₁ x ³ ₁ + λ₂ x ³ ₂ + ⋯ + λ₇ x ³ ₇| < 1 and hold simultaneously. Received November 18, 1997, Accepted October 23, 1998     

Integral closure of a cubic extension and applications

In this paper, we compute the integral closure of a cubic extension over a Noetherian unique factorization domain. We also present some applications to triple coverings and to rank two reflexive sheaves over an algebraic variety.

     

Perturbation approach to elastic constitutive relations of polycrystals

Herein we obtain a formula for the effective elastic stiffness tensor C^eff of an orthorhombic aggregate of cubic crystallites by the perturbation method. The effective elastic stiffness tensor of the polycrystal gives the relationship between volume average stress and volume average strain. Under Voigt's model, Reuss’ model and Man's theory, the elastic constitutive relation accounts for the effect of the orientation distribution function (ODF) up to terms linear in the texture coefficients. However, the formula derived in this paper delineates the effect of crystallographic texture on elastic response and shows quadratic texture dependence. The formula is very simple. We also consider the influence of grain shape to elastic constitutive relations of polycrystals. Some examples are given to compare computational results of the formula with those given by Voigt's model, Reuss's model, the finite element method, and the self-consistent method. In Section 3, we also present an expression of the perturbation displacement field, in which Green's function for an orthorhombic aggregate of cubic crystallites is included.     

pH-Dependent Release from Monoolein Cubic Phase Containing Hydrophobically Modified Chitosan

Monoolein (MO) cubic phase incorporating hydrophobically modified chosan (Hm chitosan) was prepared to obtain a pH-dependent release. Following calorimetric study, Hm chitosan had little effect on the crystal structure of MO cubic phase under acidic condition where Hm chitosan is readily soluble. At a higher pH (e.g., pH 9.0), however, the crystal structure of MO cubic phase was disturbed, possibly due to the insolubilization of Hm chitosan at the alkali condition. Whether the dye included in the cubic phase is anionic (amaranth) or cationic (methylene blue), the release from the cubic phase was suppressed as the pH of release medium increased. The structural change of cubic phase caused by the insolubilization of Hm chitosan, or the blockage of the water channel of the cubic phase by precipitated Hm chitosan would be responsible for the suppressed released.     

Cubic and Columnar Supramolecular Architectures of Rod–Coil Molecules in the Melt State

The liquid crystalline behavior of compounds 1 (n = 7, 12, 15) differs significantly from that exhibited for conventional rodlike molecules. They organize into layered smectic, bicontinuous cubic or hexagonal columnar mesophases depending on the temperature or the volume fraction of coil segments.     

Synthesis and crystal structures of extremely crowded oligophenylenes as model precursors to ‘cubic graphite’

By using drastic conditions for a Diels-Alder cycloaddition reaction, it was possible to synthesize an oligophenylene with an extremely dense packing of the benzene rings. Crystallographic data could be obtained and a projection of the structure on the plane of the central phenyl ring reveals that the molecule retained its theoretical threefold symmetry with only minor deviations. Due to its dense packing of interlocked benzene rings, this oligophenylene could be furthermore used as a suitable precursor for constructing a subunit of ‘cubic graphite’.     

Phase equilibria of the ternary system vinyl acetate/(R,S)-1-phenylethanol/carbon dioxide at high pressure conditions

Phase equilibrium measurements, correlations and predictions are presented for the binary systems (R,S)-1-phenylethanol/CO₂ and vinyl acetate/CO₂ and for the ternary system vinyl acetate/(R,S)-1-phenylethanol/CO₂. Experiments for the ternary system were performed in the temperature range of 323–343 K and in the pressure range of 7–12 MPa, using a high pressure phase equilibrium apparatus with a high pressure visual variable volume cell. Phase compositions were determined by taking samples of each phase and analysing them by gas chromatography. Equilibrium data were correlated with the Peng–Robinson equation of state combined with the Mathias–Klotz–Prausnitz mixing rule. A good correlation of both phases behaviour was obtained with an average absolute deviation (AAD) of 6.80%. Predictions for the binary sub-systems and for the ternary system were performed using the Peng–Robinson and the Soave–Redlich–Kwong equation of state, with the predictive mixing rule MHV1.