Geometrically Convex Solutions of Certain Difference Equations and Generalized Bohr-Mollerup Type Theorems

Let G: (0, ∞) → (0, ∞) be logarithmically concave on a neighbourhood of ∞ and suppose lim_x→∞ G(x + δ)/G(x) = 1 for some δ > 0. Then, the functional equation $$g(x+1)=G(x)\cdot g(x),\ \ \ x\in (0,\infty),$$ admits, up to a multiplicative constant, at most one solution g: (0, ∞) → (0, ∞), geometrically convex on a neighbourhood of ∞. Sufficient conditions on G are given, for which also such a unique geometrically convex solution of (D) exists. This result improves the classical theorems of Bohr-Mollerup type and gives a new characterization of the gamma function and the q-gamma function for q ∈ (0, 1).     

An Asymptotic Formula for the Iterates of a Function

Let IK be either IR or ? and D an open set of IK containing 0 and starlike with respect to 0 (i.e. an open interval containig 0 in the case IK = IR). If f: D » IK is a continuous function with fixed point 0, then under certain conditions stated below we can prove for the kn- th iterates of f the following asymptotic formula: 1 $$f^{(kn)}\bigg({x \over n}\bigg )=\sum_{i-1}^r{1\over (nk)^i}\ f_i(kx)+o \bigg({1\over n^r}\bigg),$$ for n » ∞, k, n and r beeing positive integers and x close enough to 0. The functions f _i are continuous and uniquely determined by f. In particular (1) holds for any function holomorphic on a neighbourhood of zero, having a convergent power series expansion of the form $$f(z)=z+a_2z^2+\cdots=\sum_{j=1}^\infty\ a_jz^j,\ a_j\in {\cal C},a_1=1,$$ and for any integers k, r with r > 0.     

Separation of (Z)- and (E)-isomers of thioxanthene and dibenz[b,e]oxepin derivatives with calixarenes and resorcinarenes as additives in nonaqueous capillary electrophoresis

Five acidic calix[4]arenes with carboxylic or sulfonic groups at either the upper or lower rim of the cavity and one resorc[4]arene were investigated to separate three thioxanthenes (flupentixol, clopenthixol, chlorprothixene) and a dibenz[b,e]oxepin derivative (doxepin) with cis-/trans-isomerism by nonaqueous capillary electrophoresis (NACE). Partial filling of the capillary with the UV-absorbing selectors led to a low detection limit and an advantageous signal-to-noise ratio (S/N). A sufficient electrophoretic mobility of the calixarenes towards the anode was necessary to outweigh the oppositely directed electroosmotic flow (EOF). This depended from the functional groups, the dissociation and the hydrodynamic radius of the cyclophanes. In contrast, the resorcinarene was useable only by addition of sodium dodecyl sulfate (SDS) because only the complex of the two selectors had an anodic apparent electrophoretic mobility. p-Sulfonyl-calix[4]arene (ss-a1) was the most capable selector for all E/Z-isomers with maximal alpha-values ranging from 1.056 for doxepin to 1.224 for chlorprothixene. This was due to the sufficient migration in reversed direction to the EOF even at low pH* values of 3.0. Otherwise, electrostatic as well as hydrophobic interactions with the positively charged isomers seem to contribute to a superior recognition. Increasing the concentration up to 15 mM ss-a1 and using acidic media (pH* 5.0) led to high separation efficiency. Changing the organic solvent provides a powerful tool to improve selectivity with N,N-dimethylformamide-methanol (DMF-MeOH)-mixtures for thioxanthenes. Further electrophoretic parameters were optimized, such as the concentration of the electrolytes, the addition of SDS, the kind of electrolytes and the voltage. Distinct differences in selectivities were found between the derivatives with thioxanthene and dibenzo[b,e]oxepin ring system, respectively. Further, the different basic side chain was responsible for the different selectivity at higher pH* values. In contrast, the substitution at position 2 of the thioxanthenes played a secondary role. Based on the studies of single parameters a method for a simultaneous separation of the four pairs of isomers within 13 min was developed.     

Asymptotische LÖsungen Der Translationsgleichung

For the solutions of the translation equation special asymptotic developments are considered. It is shown that the coefficients of such developments can be determined by linear recursion formulas or by Lie-series. Examples illustrate these results.     

On maximal antichains consisting of sets and their complements

Let 1 ? k₁ ? k₂ ? … ? k_n be integers and let S denote the set of all vectors x = (x₁, x₂, …, x_n) with integral coordinates satisfying 0 ? x_i ? k_i, i = 1, 2, …, n. The complement of x is (k₁ ? x₁, k₂ ? x₂, …, k_n ? x_n) and a subset X of S is an antichain provided that for any two distinct elements x, y of X, the inequalities x_i ? y_i, i = 1, 2, …, n, do not all hold. We determine an LYM inequality and the maximal cardinality of an antichain consisting of vectors and its complements. Also a generalization of the Erdös-Ko-Rado theorem is given.