Many-Polaron States in the Holstein–Hubbard Model

A variational approach is proposed to study some properties of the adiabatic Holstein–Hubbard model which describes an assembly of fermionic charges interacting with a static atomic lattice. The sum of the electronic energy and the lattice elastic energy is proved to have minima with a many-polaron structure in a certain domain of model parameters. Our analytical work consists in expanding these energy minima from the zero electronic transfer limit which remarkably holds for a finite amplitude of the onsite Hubbard repulsion and for an unbounded lattice size.     

Zum Nachweis der Salicyls?ure im Bier

Ohne Zusammenfassung     

tert‐Butyl (6S)‐4‐hydroxy‐6‐isobutyl‐2‐oxo‐1,2,5,6‐tetra­hydropyridine‐1‐carboxyl­ate and tert‐butyl (4R,6S)‐4‐hydroxy‐6‐iso­butyl‐2‐oxopiperidine‐1‐carboxyl­ate

X‐ray studies reveal that tert‐butyl (6S)‐6‐iso­butyl‐2,4‐dioxo­piperidine‐1‐carboxyl­ate occurs in the 4‐enol form, viz. tert‐butyl (6S)‐4‐hydroxy‐6‐iso­butyl‐2‐oxo‐1,2,5,6‐tetra­hydropyri­dine‐1‐carboxyl­ate, C₁₄H₂₃NO₄, when crystals are grown from a mixture of di­chloro­methane and pentane, and has an axial orientation of the iso­butyl side chain at the 6‐position of the piperidine ring. Reduction of the keto functionality leads predominantly to the corresponding β‐hydroxy­lated δ‐lactam, tert‐butyl (4R,6S)‐4‐hydroxy‐6‐iso­butyl‐2‐oxo­piperidine‐1‐car­boxyl­ate, C₁₄H₂₅NO₄, with a cis configuration of the 4‐hydroxy and 6‐iso­butyl groups. The two compounds show similar molecular packing driven by strong O—H⋯O=C hydrogen bonds, leading to infinite chains in the crystal structure.     

Pseudo-symplectic Runge-Kutta methods

Apart from specific methods amenable to specific problems, symplectic Runge-Kutta methods are necessarily implicit. The aim of this paper is to construct explicit Runge-Kutta methods which mimic symplectic ones as far as the linear growth of the global error is concerned. Such method of orderp have to bepseudo-symplectic of pseudosymplecticness order2p, i.e. to preserve the symplectic form to within ⊗(h ^2p )-terms. Pseudo-symplecticness conditions are then derived and the effective construction of methods discussed. Finally, the performances of the new methods are illustrated on several test problems.     

Formulation–composition map of a lecithin-based emulsion

Formulation–composition map is an interesting tool to predict the nature of an emulsion, stability, viscosity and nevertheless to decide the mixing protocol of its ingredients. Information based on optimum formulation (environmental conditions at which the affinity of an emulsifier for oil and for aqueous phase is same), which is depicted through hydrophilic–lipophilic deviation (HLD) concept, is necessary to make a formulation–composition map of an emulsion. In order to apply this concept in food emulsions, it is necessary to determine characteristic constants of each component of the system, i.e. the aqueous phase, the oil phase and the emulsifier at equilibrium. In this work formulation–composition map of a sunflower oil–water–lecithin system, based on the knowledge of phase behavior of lecithin at equilibrium and emulsification, was made. The shape of inversion line on formulation–composition map was not the classical stair type rather an almost vertical inversion line at water-fraction (f_w) near 0.20 was observed. It was supposed to be linked to the viscosity of oil phase which was 50 times the viscosity of aqueous phase. Additionally, emulsions were of oil-in-water (O/W) type for f_w higher than 0.20, but their viscosity and the drop size behavior with respect to salt concentration as formulation variable did not show the existence of transitional inversion line on formulation–composition map. Such map in advance can certainly facilitate the guidelines for dynamic emulsification.     

The dynamics of spatiotemporal modulations

The modulational instability of traveling waves is often thought to be a crucial point in the mechanism of transition to space-time disorder and turbulence. The aim of this paper is to study the effect of spatiotemporal modulations on some dynamics u(0)(x,t), which may occur as an instability process when a control parameter varies, for instance. We analyze the properties of the modulated dynamics of the form g(1)(x)g(2)(t)u(0)(x,t) compared to those of the reference dynamics u(0)(x,t), using operator theory. We show that, if the reference dynamics is invariant under some space-time symmetry in the sense of Ref. [J. Nonlinear Sci. 2, 183 (1992)], the modulation has the effect of either deforming this symmetry or breaking it, depending on whether the corresponding operator remains unitary or not. We also demonstrate that the smallest Euclidean space containing the modulated dynamics has a dimension smaller than or equal to the smallest Euclidean space containing u(0)(x,t). The previous results are then applied to the case of modulated uniformly traveling waves. While the spatiotemporal translation invariance of the wave never persists in the presence of a modulation, the existence of a spatiotemporal symmetry depends on the resonance of the Fourier sidebands due to the modulation. In case of nonresonance, a spatiotemporal symmetry exists and is explicitly determined. In this situation, the modulated wave and the carrier wave have the same spectrum (up to a normalization factor), the same entropy, and the spatial (resp., temporal) two-point correlation is deformed only by the spatial (resp., temporal) modulation. (c) 1995 American Institute of Physics.     

A pipecolic acid (Pip)‐containing dipeptide,Boc‐d‐Ala‐l‐Pip‐NHiPr

The title dipeptide, 1‐(tert‐butoxy­carbonyl‐d ‐alanyl)‐N‐iso­propyl‐l ‐pipecol­amide or Boc‐d ‐Ala‐l ‐Pip‐NHⁱPr (H‐Pip‐OH is pipecolic acid or piperidine‐2‐carboxylic acid), C₁₇H₃₁N₃­O₄, with a d –l heterochiral sequence, adopts a type II′β‐­turn conformation, with all‐trans amide functions, where the C‐terminal amide NH group interacts with the Boc carbonyl O atom to form a classical i+3 i intramolecular hydrogen bond. The C^α substituent takes an axial position [H^α (Pip) equatorial] and the trans pipecolamide function is nearly planar.