Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

We study the existence of solutions u:R³→R²$u:{\mathbb{R}}^3\to {\mathbb{R}}^2$ for the semilinear elliptic systems

equation(0.1)

−Δu(x,y,z)+∇W(u(x,y,z))=0,$-\mathrm{\Delta }u(x,y,z)+\mathrm{\nabla }W(u(x,y,z))=0,$

where W:R²→R$W:{\mathbb{R}}^2\to \mathbb{R}$ is a double well symmetric potential. We use variational methods to show, under generic non-degenerate properties of the set of one dimensional heteroclinic connections between the two minima _a±${\mathbf{a}}_{\pm }$ of W, that (0.1) has infinitely many geometrically distinct solutions u∈C²(R³,R²)$u\in C^2({\mathbb{R}}^3,{\mathbb{R}}^2)$ which satisfy u(x,y,z)→_a±$u(x,y,z)\to {\mathbf{a}}_{\pm }$ as x→±∞$x\to \pm \infty$ uniformly with respect to (y,z)∈R²$(y,z)\in {\mathbb{R}}^2$ and which exhibit dihedral symmetries with respect to the variables y and z  . We also characterize the asymptotic behavior of these solutions as |(y,z)|→+∞$|(y,z)|\to +\infty$.     

Synthesis of the Four Configurational Isomers of N-Benzoyl-2, 3, 6-Trtdeoxy-3-C-Methyl-3-Amino-L-Hexose from the (2S, 3R)-Diol Obtained from α-Methylcinnamaldehyde by Fermentation with Baker'S Yeast

Abstract

The erythro and threo chiral C₅ methyl ketones (4) and (5), prepared from the (2S, 3R)-methyl diel (1b), were converted into the phenylsulfenimines (6) and (7), which, in turn, on reaction with allyl-magnesiutn bromide, yielded after acid hydrolysis and benzoylation, the diastereoisomeric C₈-N-aminodiol derivatives (9) and (11), with threo stereochemistry relative to positions 4 and 5. Ozonolysis of (9) and (11) yielded the l-arabino and l-xylo 3-O-methyl branched aminodeoxysugar derivatives (13) and (15), respectively. Using diallylzinc as the reagent, the diastereoisomeric erythro products (8) and (10) were obtained. The latter materials gave the l-ribo-and l-lyxo-(lL-vancosamine) derivatives (12) and (14) upon oxonolysis. The ¹H and ¹³C NMR spectra of the four isomeric aminodeoxysugar derivatives (12)—(15) were discussed.     

Chirality transfer across length-scales in nematic liquid crystals: fundamentals and applications

When a chiral dopant is dissolved in an achiral liquid crystal medium, the whole sample organizes into a helical structure with a characteristic length-scale of the order of microns. The relation between chirality at these quite different length-scales can be rationalized by a relatively simple model, which retains the relevant factors coming into play: the molecular shape of the chiral dopant, which controls the chirality of short range intermolecular interactions, and the elastic properties of the nematic environment, which control the restoring torques opposing distortion of the director. In this tutorial review the relation between molecular and phase chirality will be reviewed and several applications of the chiral doping of nematic LCs will be discussed. These range from the exploitation of the amplified molecular chirality for stereochemical purposes (e.g., the determination of the absolute configuration or the enantiomeric excess), to newer applications in physico-chemical fields. The latter take advantage of the periodicity of the chiral field, with length-scales ranging from hundreds to thousands of nanometres, which characterise the cholesteric phase.     

Hydration kinetics of tricalcium silicate by calorimetric methods

The kinetics of the cement hydration reaction is a relevant issue in the cement research field, particularly in the presence of additional inorganic and organic components that consistently increase the complexity of the cement paste. In the present study, the hydration reaction of pure tricalcium silicate has been monitored by different calorimetric approaches: the conventional Isothermal Conduction Calorimetry (IC) and a novel Differential Scanning Calorimetry (DSC) protocol. The measured hydration curves have been modeled by using the Boundary Nucleation and Growth Model (BNGM) to extract thermodynamic parameters of the early stages of the hydration reaction. IC and DSC methods provide similar results in terms of rate constants, linear growth, and nucleation rates even though the IC accesses the total evolved heat while DSC discloses the fraction of unreacted water. The validation of the DSC approach as a reliable analytical method to the study of cement hydration kinetic is of particular importance because it allows following very long hydration processes, such as those of pastes containing organic retarders or superplasticizers. The thermodynamic and kinetic parameters for the tricalcium silicate setting has been also evaluated and discussed as a function of the surface area of the powder.