Boolean functions with a simple certificate for CNF complexity

In this paper we study relationships between CNF representations of a given Boolean function $f$ and essential sets of implicates of $f$. It is known that every CNF representation and every essential set must intersect. Therefore the maximum number of pairwise disjoint essential sets of $f$ provides a lower bound on the size of any CNF representation of $f$. We are interested in functions, for which this lower bound is tight, and call such functions coverable. We prove that for every coverable function there exists a polynomially verifiable certificate (witness) for its minimum CNF size. On the other hand, we show that not all functions are coverable, and construct examples of non-coverable functions. Moreover, we prove that computing the lower bound, i.e. the maximum number of pairwise disjoint essential sets, is $NP$-hard under various restrictions on the function and on its input representation.     

Optimal quality of exceptional points for the Lebesgue density theorem

In spite of the Lebesgue density theorem, there is a positive δ such that, for every non-trivial measurable set S⊂ℝ, there is a point at which both the lower densities of S and of ℝ∖S are at least δ. The problem of determining the supremum of possible values of this δ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.     

Evaluation of extraction approaches linked to ELISA and HPLC for analyses of microcystin-LR, -RR and -YR in freshwater sediments with different organic material contents

The efficiencies of conventional extraction techniques and analytical methods (HPLC–DAD and ELISA) were investigated for analyses of microcystins (MCs) in sediments. Our results showed several limitations. First, the extraction efficiency strongly depends on the extraction solvent, and extraction with 5% acetic acid in 0.2% trifluoroacetic acid (TFA)–methanol was confirmed as being the most appropriate for three different sediments (recovery: 33.1–44.9% of total MCs according to HPLC analyses). Second, the recovery of MCs was affected by the type of sediment but did not clearly correlate with the content of organic carbon. These results suggest that the sorption of MCs onto inorganic materials such as clay minerals is probably a more important process than interactions of the MCs with organic sediment matter. Third, the structure of the MCs is another crucial factor that affects the sorption of MCs and their recovery from sediments. Hydrophilic MC-RR gave much lower recoveries (20.0–38.8%) than MC-YR (44.1–59.5%) or MC-LR (55.3–77.8%) from three different types of spiked sediments. Recovery results analysed with HPLC–DAD correlated well with ELISA analyses. Further, extraction with 5% acetic acid in 0.2% TFA–methanol was used for analyses of MCs in 34 natural sediment samples collected from Brno reservoir (Czech Republic) from April to October 2005. Concentrations of MCs in sediments ranged from 0.003 to 0.380 μg/g sediment d.m. (ELISA results) or 0.016–0.474 μg/g d.m. (HPLC results). These values are equivalent to 0.63–96.47 μg/L of sediment (ELISA) or 4.67–108.68 μg/L (HPLC), respectively. Concentrations of sediment MCs showed both temporal and spatial variability, with the highest MC contents observed in the spring (April and May) and the lowest concentrations in July and August. Our results demonstrate the suitability of the methods described here for studying the occurrence, fate and ecological role of MCs in the aquatic environment.      

Nonprincipal solutions of half-linear second order differential equations

We consider the nonoscillatory half-linear second order differential equation

(∗)     

Enantioselective complexation of carbamoylated quinine and quinidine with <Emphasis Type="Italic">N</Emphasis>-blocked amino acids: vibrational and electronic circular dichroism study

Infrared absorption (IR) and vibrational and electronic circular dichroism (VCD and ECD, respectively) spectra of tert-butylcarbamoylquinine (t-BuCQN) and pseudoenantiomeric tert-butylcarbamoylquinidine (t-BuCQD), denoted as selectors (SO), complexed with chiral (S) and (R)-3,5-dinitrobenzoylleucine (DNB-Leu) and achiral 3,5-dinitrobenzoylglycine (DNB-Gly), denoted as selectands (SA), in methanol and acetonitrile, with the spectra of pure SA and SO are reported. H–D exchange of exchangeable hydrogen atoms of SA and SO in deuterated methanol which occurs in IR and VCD experiments is exploited to identify Amide II and Amide III vibrational modes. The formation of preferentially bound complexes composed of sterically compatible combinations of DNB-Leu and SO are manifested by increased intensity of VCD bands assigned to vibrations of amide, carbamate, quinoline, and dissociated carboxylate group and also by increased ECD signals. The VCD technique revealed similarities between the strongly bound diastereomeric complex of chiral DNB-Leu and SO and the complex of achiral DNB-Gly and SO, highlighting the leading role of SO in the formation of SA–SO complex. Figure Vibrational circular dichroism study: Interaction markers typical of the binding between the quinine selector and the derivatized amino acid selectand     

Convergent Assembly of the Tricyclic Labdane Core Enables Synthesis of Diverse Forskolin-like Molecules

We report a new synthetic strategy for the flexible preparation of forskolin-like molecules. The approach is different from the previously published works and employs a convergent assembly of the tricyclic labdane-type core from pre-functionalized cyclic building blocks. Stereoselective Michael addition enabled the fragment coupling with excellent control over three newly created contiguous stereocenters, all-carbon quaternary centers included. Silyl enol ether-promoted ring-opening metathesis paired with ring closure were the other key steps enabling concise assembly of the tricyclic core. Late-stage functionalization sequences transformed the tricyclic intermediates into a set of different forskolin-like molecules. The modular nature of the synthetic scheme described herein has the potential to become a general platform for the preparation of analogs of forskolin and other complex tricyclic labdanes.