Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

Perturbative manifolds and the Noether generators of nth-order Poisson equations

A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions.     

Investigation on Ochratoxin A stability using different extraction techniques

The stability of Ochratoxin A during its extraction using different extraction techniques has been evaluated. Microwave-assisted extraction and pressurised liquid extraction, in addition to two other reference methods of extraction, i.e. ultrasound-assisted and magnetic stirring-assisted extraction, were evaluated. The effect of extraction temperature using the cited techniques was checked.The results show that Ochratoxin A can be extracted using microwave-assisted extraction at temperatures up to 150 °C without degradation. Pressurised liquid extraction can be used at temperatures up to 100 °C, for extraction times of less than 30 min. Further, both ultrasound-assisted extraction and magnetic stirring extraction can be applied at temperatures up to 65 °C.High-performance liquid chromatography combined with fluorescence detection using a Chromolith RP-18e column at a flow rate of 5 mL min⁻¹ was used to quantify the Ochratoxin A. The retention time for the Ochratoxin A was 1.3 min. The limits of detection (LOD) and of quantification (LOQ) were 0.03 and 0.10 μg L⁻¹, respectively.     

The deoxygenation kinetics of new oxovanadium(IV) complexes

Summary The deoxygenation reaction kinetics of some monomeric, dimeric and polymeric oxovanadium(IV) complexes with SOCl₂ have been studied by the stopped-flow technique in DMF. The reaction is bimolecular and obeys second-order kinetics; first-order in both SOCl₂ and complex. The polymeric halogeno product decays with time, but at a rate much slower compared to the rate of its formation. The correlation observed between the structure of these complexes and their reactivities is discussed.     

Substituted γ-lactones. Some observations on the synthesis and reactivity of 4-aroyl-3-hydroxy-2(5H)-furanones

The reaction pathway towards formation of 4-aroyl-3-hydroxy-2(5H)-furanones 1 from the base catalyzed reaction between an acetophenone, diethyl oxalate and formaldehyde was investigated. If methoxide was used as base, a transesterification was found to occur in the first step, while a side reaction, namely the retro Claisen reaction, was in competition with the desired lactone formation in the second step. The nature of the solvent and the acidic character of 1 as well as the basicity of the aminoarylene were found to have a profound influence on the course of the reaction of 1 with aminoarylenes.     

Optimization of a New Extraction Technique for Analysis of Verbenone and cis-Verbenol in Pine Seeds

Results from a systematic study of the factors affecting extraction of cis-verbenol and verbenone from pine seeds are presented. Five extraction conditions were investigated: extraction solvent, method of extraction, extraction temperature, volume of solvent, and the ratio of the mass of sample to the amount of extraction solvent. The resulting optimized method uses magnetic-stirring-assisted extraction of pine seeds (5 g) with ethyl acetate (75 mL) for 20 min, at room temperature. RSDs were less than 5% for both compounds. GC–FID was used for quantification of cis-verbenol and verbenone in the extracts.     

Substituted γ-lactones: On the structural elucidation of the reaction products from 4-aroyl-3-hydroxy-2(5H)-furanones and 1,2-diamines

The reaction pathway of 4-aroyl-3-hydroxy-2(5H)-furanones 1 with diamines depends on the nature of the amine as well as on the applied reaction conditions. Thus, the reaction of 1a-d with 5,6-diamino-1,3-dimethyluracil 5 led to the formation of two isomeric Schiff bases 7a-d and 8a-d . Conversely type 1 compounds reacted with 4,5-diaminopyrimidine 9 or 2,3-diaminopyridine 10 to form the mono acid-base adducts 11a and 11b respectively. When type 1 compounds were reacted with aliphatic diamines 13a-d or p-phenylenediamine and p-xylenediamine, respectively also an immediate formation of acid-base adducts 15a-f was observed. The reaction of a number of O-methylated type 1 compounds with 1,2-ethylenediamine afforded the novel seven-membered ring compounds 18a-d in good yields. The analogous reaction of O-alkylated 1a with o-phenylenediamine 2 or 2,3-diaminonaphthalene gave the expected tricyclic ring systems 19 or 20 .     

Synthesis of Important New Pyrrolo[3,4‐c]Pyrazoles and Pyrazolyl‐Pyrrolines from Heterocyclic β‐Ketonitriles

Treatment of heterocyclic β‐ketonitriles 1a,b with hydrazine hydrate and phenylhydrazine afforded the hydrazine derivatives 2a‐d which cyclized in PPA into pyrrolo[3,4‐c]pyrazoles 3a‐d. Reaction of 1a,b with cyanoacetohydrazide furnished the cyanoacetyl pyrrolo[3,4‐c]pyrazoles 4a,b. The hydrazine 2c reacted with β‐diketone and β‐ketoesters to afford pyrazolyl‐pyrrolines 5‐7. Also the later hydrazine reacted with some D‐aldoses and aceteophenone to give the corresponding hydrazones 10‐12 and hydrazine carboxamide derivatives 15a,b respectively.