Pointwise characterization of Sobolev classes

We prove that a function f is in the Sobolev class W _loc ^m,p (ℝ ⁿ ) or W ^m,p (Q) for some cube Q ⊂ ℝ ⁿ if and only if the formal (m − 1)-Taylor remainder R ^m−1 f(x,y) of f satisfies the pointwise inequality |R ^m−1 f(x,y)| ≤ |x − y| ^m [a(x) + a(y)] for some a ε L ^p (Q) outside a set N ⊂ Q of null Lebesgue measure. This is analogous to H. Whitney’s Taylor remainder condition characterizing the traces of smooth functions on closed subsets of ℝ ⁿ . Dedicated to S.M. Nikol’skiĭ on the occasion of his 100th birthday The main results and ideas of this paper were presented in the plenary lecture of the author at the International Conference and Workshop Function Spaces, Approximation Theory and Nonlinear Analysis dedicated to the centennial of Sergei Mikhailovich Nikol’skii, Moscow, May 24–28, 2005.     

Quasi-column chromatography of barbituric acid derivatives II. Separation on Non-polar adsorbents using binary water-organic solvent mixtures as the mobile phase

Summary Twenty barbituric acid derivatives having four different types of substitution were separated in TLC-S chambers. Using non-polar adsorbents (silanized silica gel or silica gel coated with paraffin oil) and binary water-organic solvent mixtures as the mobile phase. Linear relationships between R_M and the concentration of the organic solvent were observed for the majority of the investigated compounds. The non-polar absorbents ensure a better separation than untreated silica gel especially for the therapeutically useful C₅ disubstituted barbiturates. The results can be used for the optimization of the systems for the chromatography of barbiturates. The R_f values were correlated with the number of carbon atoms of the substitutions, molecular connectivity and a parameter associated with the molecular volume. The best correlations were obtained for this last parameter.     

The influence of reverse energy transport on emission anisotropy in two-component viscous solutions

The effect of nonradiative reverse energy transport (NRET) in two donor-acceptor systems was studied experimentally. It was found that the NRET occurring in system I; rhodamine 6G (donor) and rhodamine B (acceptor), considerably lowers the emission anisotropy at medium and high concentrations. These results qualitatively confirm the predictions of the approximate theoretical approach of L. Kulak and C. Bojarski (see the preceding paper). In system II; rhodamine 6G (donor) and Nile Blue (acceptor), for which the NRET process does not occur, a good agreement with no-back-transport theory was obtained.     

Synthesis of Some Aminophosphonates Bearing N‐(Fluorophenyl)‐piperazynyl Moiety and Their Activity toward Serotonin Receptors

Syntheses of a series of compounds bearing a 1‐(fluorophenyl)piperazin‐4‐ylmethyl moiety, namely dialkyl 1‐(fluorophenyl)piperazin‐4‐ylmethyl phosphonates 3a–f , diethyl 2‐[1‐(fluorophenyl)piperazin‐4‐yl]‐ethyl phosphonates 4a,b , diethyl 3‐[1‐(4‐fluorophenyl)‐piperazin‐4‐yl]‐propyl phosphonate 5 , and di[1‐(fluorophenyl)piperazin‐4‐yl]methanes 6a,b were performed, and some of them were screened for their affinity for serotonin 5‐HT_1A, 5‐HT₆, and 5‐HT₇ receptors. Moderate interactions with these receptors were demonstrated.     

Sard's theorem for mappings in Hölder and Sobolev spaces

We prove various generalizations of classical Sard's theorem to mappings f:M ^m →N ⁿ between manifolds in Hölder and Sobolev classes. It turns out that if f ∈ C ^k,λ (M ^m ,N ⁿ ), then—for arbitrary k and λ—one can obtain estimates of the Hausdorff measure of the set of critical points in a typical level set f ^?1(y). The classical theorem of Sard holds true for f ∈ C ^k with sufficiently large k, i.e., k>max(m?n,0); our estimates contain Sard's theorem (and improvements due to Dubovitskii and Bates) as special cases. For Sobolev mappings between manifolds, we describe the structure of f ^?1(y).