Bounded sets structure of CpX and quasi‐(DF)‐spaces

For wide classes of locally convex spaces, in particular, for the space $C_p(X)$ of continuous real‐valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of $(DF)$‐spaces have led us to introduce quasi‐ $(DF)$‐spaces, a class of locally convex spaces containing $(DF)$‐spaces that preserves subspaces, countable direct sums and countable products. Regular $(LM)$‐spaces as well as their strong duals are quasi‐ $(DF)$‐spaces. Hence the space of distributions $D^{\prime }(\mathrm{\Omega })$ provides a concrete example of a quasi‐ $(DF)$‐space not being a $(DF)$‐space. We show that $C_p(X)$ has a fundamental bounded resolution if and only if $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if the strong dual of $C_p(X)$ is a quasi‐ $(DF)$‐space if and only if X is countable. If X is metrizable, then $C_k(X)$ is a quasi‐ $(DF)$‐space if and only if X is a σ‐compact Polish space.     

On the Smith normal form of a skew‐symmetric d‐optimal design of order

We show that the Smith normal form of a skew‐symmetric D ‐optimal design of order is determined by its order. Furthermore, we show that the Smith normal form of such a design can be written explicitly in terms of the order , thereby proving a recent conjecture of Armario. We apply our result to show that certain D ‐optimal designs of order are not equivalent to any skew‐symmetric D ‐optimal design. We also provide a correction to a result in the literature on the Smith normal form of D ‐optimal designs.     

Densitometry and indirect normal‐phase HPTLC–ESI–MS for separation and quantitation of drugs and their glucuronide metabolites from plasma

The present work describes novel methods using densitometry and indirect or off‐line high performance thin‐layer chromatography–mass spectrometry (HPTLC–MS) for the simultaneous detection and quantification of asenapine, propranolol and telmisartan and their phase II glucuronide metabolites. After chromatographic separation of the drugs and their metabolites the analytes were scraped, extracted in methanol and concentrated prior to mass spectrometric analysis. Different combinations of toluene and methanol–ethanol–n‐butanol–iso‐propanol were tested for analyte separation and the best results were obtained using toluene–methanol–ammonia (6.9:3.0:0.1, v/v/v) as the elution solvent. All of the drug–metabolite pairs were separated with a homologous retardation factor difference of ≥22. The conventional densitometric approach was also studied and the method performances were compared. Both of the approaches were validated following the International Conference on Harmonization guidelines, and applied to spiked human plasma samples. The major advantage of the TLC–MS approach is that it can provide much lower limits of detection (1.98–5.83 pg/band) and limit of quantitation (5.97–17.63 pg/band) with good precision (?3.0% coefficient of variation) compared with TLC–densitometry. The proposed indirect HPTLC–MS method is simple yet effective and has tremendous potential in the separation and quantitation of drugs and their metabolites from biological samples, especially for clinical studies.     

Chiral separability of boron cluster species studied by screening approaches utilizing polysaccharide‐based chiral stationary phases

In this study, the screening steps of chiral separation strategies with polysaccharide‐based chiral stationary phases were applied on boron cluster compounds in normal‐phase liquid chromatography (NPLC) and polar organic solvents chromatography (POSC). Since the screening steps were initially developed to analyze organic compounds, their applicability for boron clusters was investigated. Overall, the screening steps in NPLC were applicable for the separation of zwitterions, while for anions mostly no elution was observed. A hypothesis for the latter behavior is precipitation of anions in the nonpolar mobile phases. Ten out of 11 compounds could be partially or baseline separated on the NPLC screening systems. In POSC, all zwitterions were separated on at least one of the screening systems, with an overall lower retention as in NPLC. Anions were detected but not separated in the majority of the experiments. Also their retention on the chiral stationary phases was very limited. This study showed that the chiral discrimination potential of chemically modified polysaccharides is meaningful for chiral separations of structurally chiral boron cluster species, but needs further systematic research, in which recognition mechanisms should be further explored. In addition, some unusual peaks also indicated that conditions with a high separation efficiency must first be searched for some of the tested systems. Copyright © 2014 John Wiley & Sons, Ltd.     

Free groups in a normal subgroup of the field of fractions of a skew polynomial ring

Let k(t) be the field of rational functions over the field k, let σ be a k-automorphism of K = k(t), let D = K(X;σ) be the ring of fractions of the skew polynomial ring K[X;σ], and let D^? be the multiplicative group of D. We show that if N is a noncentral normal subgroup of D^?, then N contains a free subgroup. We also prove that when k is algebraically closed and σ has infinite order, there exists a specialization from D to a quaternion algebra. This allows us to explicitly present free subgroups in D^?.     

Stability,periodicity, and symmetries of certain second‐order fractional difference equation with quadratic terms via KAM theory

By using the Kolmogorov–Arnold–Moser theory, we investigate the stability of the equilibrium solution of the difference equation where A ,B ,D > 0,u _?1,u ₀>0. We also use the symmetries to find effectively the periodic solutions with feasible periods. Copyright © 2016 John Wiley & Sons, Ltd.     

Pseudo‐spherical normal Darboux images of curves on a lightlike surface

We introduce the notions of the pseudospherical normal Darboux images for the curve on a lightlike surface in Minkowski 3‐space and study these Darboux images by using technics of the singularity theory. Furthermore, we give a relation between these Darboux images and Darboux frame from the viewpoint of Legendrian dualities.