On ground states for the 2D Schrödinger equation with combined nonlinearities and harmonic potential

We consider the nonlinear Schrödinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a framework includes physical models, and ensures that finite energy solutions are global in time. We address the questions of the existence and the orbital stability of the set of standing waves. Given the mathematical features of the equation (external potential and inhomogeneous nonlinearity), the set of parameters for which standing waves exist in unclear. In the two-dimensional case, we adapt the method of fundamental frequency solutions, introduced by the second author in the higher-dimensional case without potential. This makes it possible to describe accurately the set of fundamental frequency standing waves and ground states, and to prove its orbital stability.     

Characterization of transversal homothetic solutions in the n-body problem

Homothetic solutions of the n-body problem can be seen as heteroclinic orbits when the dynamical variables are changed via the McGehee blow-up and the time is suitably scaled. Transversality of the invariant asymptotic manifolds which contain the heteroclinic orbits is related to some structural stability. We fully characterize the cases in which such transversality is obtained for the n-body problem in any dimension.     

Generalizing the ENO-DB2<Emphasis Type="Italic">p</Emphasis> transform using the inverse wavelet transform

The essentially non-oscillatory (ENO)-wavelet transform developed by Chan and Zhou (SIAM J. Numer. Anal. 40(4), 1369–1404, 2002) is based on a combination of the Daubechies-2p wavelet transform and the ENO technique. It uses extrapolation methods to compute the scaling coefficients without differencing function values across jumps and obtains a multiresolution framework (essentially) free of edge artifacts. In this work, we present a different way to compute the ENO-DB2p wavelet transform of Chan and Zhou which allows us to simplify the process and easily generalize it to other families of orthonormal wavelets.     

Development and validation of a GC-MS method for rapid determination of galanthamine in Leucojum aestivum and Narcissus ssp.: a metabolomic approach

Galanthamine, an acetylcholinesterase inhibitor marketed as a hydrobromide salt for the treatment of Alzheimer's disease, is obtained from some Amaryllidaceae plants. A new method was developed and validated for its quantification by GC-MS in different plant sources: bulbs and leaves from Narcissus confusus; bulbs from N. pseudonarcissus cv. Carlton; and leaves and in vitro cultures from L. aestivum. Samples (50 mg) were extracted with methanol (1 mL) for 2 h, then aliquots of the extracts were silylated and analyzed by GC-MS. The calibration line was linear over a range of 15-800 μg galanthamine/sample, ensuring an analysis of samples with a content of 0.03-1.54% analyte referred to dry weight. The recovery was generally more than 95%. Good inter- and intra assay precision was observed (RSD < 3%). Principal component analysis of GC-MS chromatograms allowed discrimination of the plant raw material with respect to species, organs and geographical regions. The analytical method developed in this study proved to be simple, sensitive and far more informative than the routine analytical methods (GC, HPLC, CE and NMR), so it may be useful for quality control of plant raw materials in the pharmaceutical industry.     

PtII Coordination to N1 of 9‐Methylguanine: Why it Facilitates Binding of Additional Metal Ions to the Purine Ring

The preparation and X‐ray crystal structure analysis of {trans‐[Pt(MeNH₂)₂(9‐MeG‐N1)₂]} ? {3 K₂[Pt(CN)₄]} ? 6 H₂O ( 3 a ) (with 9‐MeG being the anion of 9‐methylguanine, 9‐MeGH) are reported. The title compound was obtained by treating [Pt(dien)(9‐MeGH‐N7)]²⁺ ( 1 ; dien=diethylenetriamine) with trans‐[Pt(MeNH₂)₂(H₂O)₂]²⁺ at pH 9.6, 60 °C, and subsequent removal of the [(dien)Pt^II] entities by treatment with an excess amount of KCN, which converts the latter to [Pt(CN)₄]^2?. Cocrystallization of K₂[Pt(CN)₄] with trans‐[Pt(MeNH₂)₂(9‐MeG‐N1)₂] is a consequence of the increase in basicity of the guanine ligand following its deprotonation and Pt coordination at N1. This increase in basicity is reflected in the pK_a values of trans‐[Pt(MeNH₂)₂(9‐MeGH‐N1)₂]²⁺ (4.4±0.1 and 3.3±0.4). The crystal structure of 3 a reveals rare (N7,O6 chelate) and unconventional (N2,C2,N3) binding patterns of K⁺ to the guaninato ligands. DFT calculations confirm that K⁺ binding to the sugar edge of guanine for a N1‐platinated guanine anion is a realistic option, thus ruling against a simple packing effect in the solid‐state structure of 3 a . The linkage isomer of 3 a , trans‐[Pt(MeNH₂)₂(9‐MeG‐N7)₂] ( 6 a ) has likewise been isolated, and its acid–base properties determined. Compound 6 a is more basic than 3 a by more than 4 log units. Binding of metal entities to the N7 positions of 9‐MeG in 3 a has been studied in detail for [(NH₃)₃Pt^II], trans‐[(NH₃)₂Pt^II], and [(en)Pd^II] (en=ethylenediamine) by using ¹H NMR spectroscopy. Without exception, binding of the second metal takes place at N7, but formation of a molecular guanine square with trans‐[(Me₂NH₂)Pt^II] cross‐linking N1 positions and trans‐[(NH₃)₂Pt^II] cross‐linking N7 positions could not be confirmed unambiguously, despite the fact that calculations are fully consistent with its existence.     

Diastereoselective preparation of (S)-(1,4,4-trimethylpyrrolidin-3-yl)amine, a new chiral 1,2-diamine for thiourea-type organocatalysts

The enantioselective synthesis of the title compound, its conversion into a thiourea-type organocatalyst and the behavior of this organocatalyst in several enantioselective Michael reactions are described.     

Interaction of monovalent ions with hydrophobic and hydrophilic colloids: charge inversion and ionic specificity

Here we study experimentally and by simulations the interaction of monovalent organic and inorganic anions with hydrophobic and hydrophilic colloids. In the case of hydrophobic colloids, our experiments show that charge inversion is induced by chaotropic inorganic monovalent ions but it is not induced by kosmotropic inorganic anions. For organic anions, giant charge inversion is observed at very low electrolyte concentrations. In addition, charge inversion disappears for both organic and inorganic ions when turning to hydrophilic colloids. These results provide an experimental evidence for the hydrophobic effect as the driving force for both ion specific effects and charge inversion. In the case of organic anions, our molecular dynamics (MD) simulations with full atomic detail show explicitly how the large adsorption free energies found for hydrophobic colloids are transformed into large repulsive barriers for hydrophilic colloids. Simulations confirm that solvation free energy (and hence the hydrophobic effect) is responsible for the build up of a Stern layer of adsorbed ions and charge inversion in hydrophobic colloids and it is also the mechanism preventing charge inversion in hydrophilic colloids. Overall, our experimental and simulation results suggest that the interaction of monovalent ions with interfaces is dominated by solvation thermodynamics, that is, the chaotropic/kosmotropic character of ions and the hydrophobic/hydrophilic character of surfaces.     

Probing the acceptor substrate binding site of Trypanosoma cruzi trans-sialidase with systematically modified substrates and glycoside libraries

Systematically modified octyl galactosides and octyl N-acetyllactosamines were assessed as inhibitors of, and substrates for, T. cruzi trans-sialidase (TcTS) in the context of exploring its acceptor substrate binding site. These studies show that TcTS, which catalyses the α-(2→3)-sialylation of non-reducing terminal β-galactose residues, is largely intolerant of substitution of the galactose 2 and 4 positions whereas substitution of the galactose 6 position is well tolerated. Further studies show that even the addition of a bulky sugar residue (glucose, galactose) does not impact negatively on TcTS binding and turnover, which highlights the potential of 'internal' 6-substituted galactose residues to serve as TcTS acceptor substrates. Results from screening a 93-membered thiogalactoside library highlight a number of structural features (notably imidazoles and indoles) that are worthy of further investigation in the context of TcTS inhibitor development.     

Ostrowski type methods for solving systems of nonlinear equations

Four generalized algorithms builded up from Ostrowski’s method for solving systems of nonlinear equations are written and analyzed. A development of an inverse first-order divided difference operator for functions of several variables is presented, as well as a direct computation of the local order of convergence for these variants of Ostrowski’s method. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.     

Frozen divided difference scheme for solving systems of nonlinear equations

The development of an inverse first-order divided difference operator for functions of several variables, as well as a direct computation of the local order of convergence of an iterative method is presented. A generalized algorithm of the secant method for solving a system of nonlinear equations is studied and the maximum computational efficiency is computed. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.