Quasi-periodic stability of normally resonant tori

We study quasi-periodic tori under a normal-internal resonance, possibly with multiple eigenvalues. Two non-degeneracy conditions play a role. The first of these generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus. The second condition involves a Kolmogorov-like variation of the internal frequencies and simultaneously versality of the Floquet matrix unfolding. We focus on the reversible setting, but our results carry over to the Hamiltonian and dissipative contexts.     

EC–SPE–stripline-NMR analysis of reactive products: a feasibility study

Flow-through electrochemical conversion (EC) of drug-like molecules was hyphenated to miniaturized nuclear magnetic resonance spectroscopy (NMR) via on-line solid-phase extraction (SPE). After EC of the prominent p38α mitogen-activated protein kinase inhibitor BIRB796 into its reactive products, the SPE step provided preconcentration of the EC products and solvent exchange for NMR analysis. The acquisition of NMR spectra of the mass-limited samples was achieved in a stripline probe with a detection volume of 150 nL offering superior mass sensitivity. This hyphenated EC–SPE–stripline-NMR setup enabled the detection of the reactive products using only minute amounts of substrate. Furthermore, the integration of conversion and detection into one flow setup counteracts incorrect assessments caused by the degradation of reactive products. However, apparent interferences of the NMR magnetic field with the EC, leading to a low product yield, so far demanded relatively long signal averaging. A critical assessment of what is and what is not (yet) possible with this approach is presented, for example in terms of structure elucidation and the estimation of concentrations. Additionally, promising routes for further improvement of EC–SPE–stripline-NMR are discussed.     

Synthesis and mesogenic properties of several homologous series of aldose dialkyl dithioacetals. A model for their behaviour

Abstract

Ten homologous series (n-butyl through n-decyl) of aldose S,S-acetals (D-glucose, D-galactose, D-mannose, L-rhamnose, 2-deoxy-D-glucose, D-xylose, D-lyxose, D- or L-arabinose, D-ribose and 2-deoxy-D-ribose) have been prepared. Most of these compounds form thermotropic liquid crystals, the exceptions being the entire L-rhamnose series and some of the derivatives with the shortest alkyl chains. All of the compounds have been investigated with differential scanning calorimetry and polarization microscopy. Some temperature dependent powder X-ray data are also presented. A model is proposed which correlates the carbohydrate configuration with the melting behaviour. On the basis of now available behaviourial characteristics, visual inspection, mixing experiments and precedent, the mesophase is identified as smectic A_d, the partially overlapping carbohydrate moieties being in the centre and the aliphatic chains pointing outward at an angle of about 62°. Despite the intrinsic chirality of all the carbohydrate mesogens, no evidence for chiral mesophases was found, not even after introduction of a homochiral branched alkyl chain.     

Giant spin splitting through surface alloying

The long-range ordered surface alloy Bi/Ag(111) is found to exhibit a giant spin splitting of its surface electronic structure due to spin-orbit coupling, as is determined by angle-resolved photoelectron spectroscopy. First-principles electronic structure calculations fully confirm the experimental findings. The effect is brought about by a strong in-plane gradient of the crystal potential in the surface layer, in interplay with the structural asymmetry due to the surface-potential barrier. As a result, the spin polarization of the surface states is considerably rotated out of the surface plane.     

Surface complexation of carbonate on goethite: IR spectroscopy, structure and charge distribution

The adsorption of carbonate on goethite has been evaluated, focussing on the relation between the structure of the surface complex and corresponding adsorption characteristics, like pH dependency and proton co-adsorption. The surface structure of adsorbed CO3(-2) has been assessed with (1) a reinterpretation of IR spectroscopy data, (2) determination of the charge distribution within the carbonate complex using surface complexation modeling, and (3) evaluation of the proton co-adsorption of various oxyanions, including carbonate, in relation with structural differences. Carbonate adsorption leads to a degeneration of the nu3 IR vibration. Currently, the magnitude of the Deltanu3 band splitting is used as a criterion for metal coordination. However, the interpretation is not unambiguous, since the magnitude of Deltanu3 is influenced by polarization and additional field effects, due to, e.g., H bonding. Our evaluation shows that for goethite the magnitude of band splitting Deltanu3 falls within the range of values that is representative for bidentate complex formation, despite contrarily assignments made in literature. Determination of the charge distribution (CD), derived by modeling available carbonate adsorption data, shows that a very large part (2/3) of the carbonate charge resides in the surface. Interpretation of this result with a bond valence and a ligand charge analysis strongly favors the bidentate surface complexation option for adsorbed carbonate. This option is also supported by the proton co-adsorption of carbonate. The H co-adsorption is very high, which corresponds closely to an oxyanion surface complex in which 2/3 of the ligands are common with the surface. The high H co-adsorption is in conflict with the monodentate option for adsorbed CO3(-2). The study shows that the H co-adsorption of CO3(-2) is almost equal to the experimental H co-adsorption obtained for SeO3(-2) adsorption, which can be rationalized supposing for both XO3(-2) complexes the same ligand distribution in the interface, i.e., bidentate complex formation.     

Tiling Hamming Space with Few Spheres

We show that if the collection of all binary vectors of lengthnis partitioned intokspheres, then eitherk2 orkn+2. Moreover, such partitions withk=n+2 are essentially unique.     

The smallest length of binary 7–dimensional linear codes with prescribed minimum distance

Let n(k,d) denote the smallest value of n for which a binary (n,k,d) code exists. Then n(k, d) was known for all d, when k ?6. All values of n(7, d) will now be presented.     

A potential approach for ordinal games

We introduce the class of ordinal games with a potential, which are characterized by the absence of weak improvement cycles, the same condition used by Voorneveld and Norde (1997) for ordinal potential games.     

Characterizing properties of approximate solutions for optimization problems

Approximate solutions for optimization problems become of interest if the ‘true’ optimum cannot be found: this may happen for the simple reason that an optimum does not exist or because of the ‘bounded rationality’ (or bounded accuracy) of the optimizer. This paper characterizes several approximate solutions by means of consistency and additional requirements. In particular we consider invariance properties. We prove that, where the domain contains optimization problems without maximum, there is no non-trivial consistent solution satisfying non-emptiness, translation and multiplication invariance. Moreover, we show that the class of ‘satisficing’ solutions is obtained, if the invariance axioms are replaced with Chernoff’s Choice Axiom.     

Boundary relations and their Weyl families

The concepts of boundary relations and the corresponding Weyl families are introduced. Let be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space , let be an auxiliary Hilbert space, let

and let be defined analogously. A unitary relation from the Krein space to the Krein space is called a boundary relation for the adjoint if . The corresponding Weyl family is defined as the family of images of the defect subspaces , , under . Here need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space and the class of unitary relations , it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every -valued maximal dissipative (for ) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.