Some axioms for constructive analysis

This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and number-theoretic sequences, we propose some modified choice principles and other function existence axioms which may be of use in reverse constructive analysis. Specifically, we consider the function comprehension principles assumed by the two minimal theories EL and M, introduce an axiom schema CF_d asserting that every decidable property of numbers has a characteristic function, and use it to describe a precise relationship between the minimal theories. We show that the axiom schema AC₀₀ of countable choice can be decomposed into a monotone choice schema AC ₀₀ ^m (which guarantees that every Cauchy sequence has a modulus) and a bounded choice schema BC₀₀. We relate various (classically correct) axiom schemas of continuous choice to versions of the bar and fan theorems, suggest a constructive choice schema AC_1/2,0 (which incidentally guarantees that every continuous function has a modulus of continuity), and observe a constructive equivalence between restricted versions of the fan theorem and correspondingly restricted bounding axioms ${AB_{1/2,0}^{2^{\mathbb{N}}}}$ . We also introduce a version WKL!! of Weak K?nig’s Lemma with uniqueness which is intermediate in strength between WKL and the decidable fan theorem FT_d.     

Development of an Electrochemical Biosensor for the Rapid Detection of Carbofuran Based on Air Stable Lipid Films with Incorporated Calix[4]arene Phosphoryl Receptor

This work describes the preparation of a selective receptor for the rapid, selective and sensitive electrochemical flow injection analysis of carbofuran in foods using air stable lipid films supported on a methacrylate polymer on a glass fiber filter with incorporated artificial receptor. The selective receptor was synthesized by transformation of the ? OH groups of resorcin[4]arene receptor into phosphoryl groups. These lipid films were supported on a methylacrylate polymer (i.e., methacrylic acid was the functional monomer for the polymerization, ethylene glycol dimethacrylate was used as the crosslinker and 2,2′‐azobis‐2‐methylpropionitrile as an initiator). A minisensor device was constructed for the electrochemical flow injection analysis of toxicants based on air stabilized lipid films supported on a polymer. The device can sense the analyte in a drop (50 μL) of sample. Carbofuran was injected into flowing streams of a carrier electrolyte solution. A host‐guest complex formation between the calix[4]arene phosphoryl receptor and carbofuran takes place through hydrogen bonding. This enhances the preconcentration of carbofuran at the lipid membrane surface which in turn causes dynamic alterations of the electrostatic fields and phase structure of membranes; as a result ion current transients were obtained and the magnitude of these signals was correlated to the substrate concentration. The response times were ca. 80 s and carbofuran was determined at concentration levels of nM. The effect of potent interferences included a wide range of compounds and other insecticides. The effect of interference of proteins and lipids was also examined. The reproducibility of the method was checked by recovery experiments in fruit and vegetable samples with satisfactory results.     

Headspace solid-phase microextraction for the gas chromatographic analysis of organophosphorus insecticides in vegetables

Residues of organophosphorus insecticides (diazinon, methyl parathion, fenitrothion, malathion, and parathion) were determined in 13 different vegetable matrixes by headspace solid-phase microextraction performed with a polydimethyl-siloxane fiber (100 microm). Determination was carried out by gas chromatography with a nitrogen-phosphorus detector. Limits of detection and quantification were < 0.005 and 0.017 mg/kg, respectively; thus, the limits of maximum residue levels (MRLs) required by European regulations can be verified without difficulty. Pesticide residues were found in 38% of the 125 fresh commercial samples (imported and domestic) that were analyzed. Residues of methyl parathion and parathion, which were withdrawn in Greece in 2003, were detected in 36.8 and 4% of all samples, respectively. The MRLs were exceeded overall by 1%.     

Edge partitions of optimal 2-plane and 3-plane graphs

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for $k=1$, we focus on optimal 2-plane and on optimal 3-plane graphs, which are 2-plane and 3-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple (i.e., with neither self-loops nor parallel edges) optimal 2-plane graph into a 1-plane graph and a forest, while (ii) an edge partition formed by a 1-plane graph and two plane forests always exists and can be computed in linear time. (iii) There exist efficient algorithms to partition the edges of a simple optimal 2-plane graph into a 1-plane graph and a plane graph with maximum vertex degree at most 12, or with maximum vertex degree at most 8 if the optimal2-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) There exists an infinite family of simple optimal 2-plane graphs such that in any edge partition composed of a 1-plane graph and a plane graph, the plane graph has maximum vertex degree at least 6 and the 1-plane graph has maximum vertex degree at least 12. (v) Every optimal 3-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a 2-plane graph and two plane forests.     

One-pot synthesis of novel fused pentacyclic chromenopyrimidobenzimidazolones and benzimidazolyl-chromenyl-substituted thiazolidinones

The ultrasound promoted synthesis of a number of novel fused pentacyclic chromenopyrimido[1,2-α]benzimidazolones by the one-pot reaction of 3-formylchromones with 2-aminobenzimidazole is described. Moreover, the isolated pentacyclic chromone derivatives upon microwave irradiation with 2-mercaptocarboxylic acids afforded benzimidazolyl-chromenylthiazolidinones incorporating three pharmacophoric heterocycles; the same thiazolidinones were also formed through a multicomponent reaction under microwave irradiation involving 3-formylchromones, aminobenzimidazole and 2-mercaptocarboxylic acids. The structural elucidation of the products was accomplished by 1D and 2D NMR experiments and for thiazolidinones was also confirmed by X-ray crystallographic analysis. Full assignment of all ¹H and ¹³C NMR chemical shifts has been unambiguously achieved. The proposed reaction mechanism is also discussed.     

Preparation of a selective receptor for carbofuran for the development of a simple optical spot test for its rapid detection using stabilized in air lipid films with incorporated receptor

The present technique describes the preparation of a selective receptor for carbofuran and the development of a simple sensitive spot optical test for the rapid one-shot detection of carbofuran using stabilized lipid films supported on a methacrylate polymer on a glass fiber filter with incorporated artificial receptor. The selective receptor was synthesized by a chemical reaction using a resorcin[4]arene receptor by transforming all the –OH groups into phosphoryl groups. The lipid films without this receptor provided fluorescence under a UV lamp. The use of the receptor in these films quenched this fluorescence and the colour became similar to that of the filters without the lipid films. A drop of aqueous solution of carbofuran provided a “switching on” of the fluorescence which allows the rapid detection of this insecticide at the levels of 10⁻⁹ M concentrations. The effect of potent interferences included a wide range of compounds. The results showed no interferences from these compounds in concentration levels usually found in real samples. The effect of interference of proteins and lipids was also examined. The reproducibility of the method was checked in about 100 samples and all of them were found to provide similar results. The device was tested/evaluated in real samples of fruits, vegetables and dairy products. Note that the colours of the filters remain stable for periods of more than 2 months.