Embedding the planar structure of 4-dimensional linear spaces into projective spaces

It is known that a linear spaces of dimensiond has at least as many hyperplanes as points with equality if it is a (possibly degenerate) projective space. If there are only a few more hyperplanes than points, then the linear space can still be embedded in a projective space of the same dimension. But even if the difference between the number of hyperplanes and points is too big to ensure an embedding, it seems likely that the linear space is closely related to a projective space. We shall demonstrate this in the cased=4.     

A discontinuous Galerkin method for the Wigner-Fokker-Planck equation with a non-polynomial approximation space

A discontinuous Galerkin approach to the Wigner-Fokker-Planck equation, a model for quantum devices including environmental effects, is proposed. Evaluation of a pseudo-differential term is the main challenge. The approach can be applied to a variety of potential functions and uses general approximation spaces. Simulations using the method are in agreement with established analytic results and produce reasonable solutions for several potentials. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tetra-2,3-Pyrazinoporphyrazines with Externally Appended Pyridine Rings 22 Synthesis,Physicochemical and Photoactivity Studies on In(III) Mono- and Heteropentanuclear Complexes

The basic macrocyclic octapyridinotetrapyrazinoporphyrazine In^III complex of formula [Py₈TPyzPzIn(OAc)]·8H₂O, prepared by reaction of the free ligand [Py₈TPyzPzH₂]·2H₂O with In(OAc)₃, is a stable-to-air species of which the structure has been studied by its X-ray powder diffraction and mass spectra and characterization operated by IR and UV-visible spectral behavior. The complex has been further examined and proven to be of potential interest for its response as an anticancer agent in the field of photodynamic therapy (PDT), the value of Φ_Δ = 0.55 (in DMF) being in the range of 0.4–0.6 at the level of similar phthalocyanine and porphyrazine analogs and qualifying the species as a highly efficient anticancer agent. Planned parallel types of investigation, including their photoactive behaviour in PDT, have been extended to the mononuclear octacation [(2-Mepy)₈TPyzPzIn(OAc)]⁸⁺ (salted by iodide ions) and the heteropentanuclear derivatives [(M’Cl₂)₄Py₈TPyzPzIn(OAc)]·xH₂O (M’ = Pd^II, x = 8; Pt^II, x = 1)) and [{(Pd(CBT)₂)₄}Py₈TPyzPzIn(OAc)]·19H₂O (CBT = m-carborane-1-thiolate anion).     

Chemical and Genetic Studies on the Formation of Pyrrolones During the Biosynthesis of Cytochalasans

A key step during the biosynthesis of cytochalasans is a proposed Knoevenagel condensation to form the pyrrolone core, enabling the subsequent 4+2 cycloaddition reaction that results in the characteristic octahydroisoindolone motif of all cytochalasans. In this work, we investigate the role of the highly conserved α,β-hydrolase enzymes PyiE and ORFZ during the biosynthesis of pyrichalasin H and the ACE1 metabolite, respectively, using gene knockout and complementation techniques. Using synthetic aldehyde models we demonstrate that the Knoevenagel condensation proceeds spontaneously but results in the 1,3-dihydro-2H-pyrrol-2-one tautomer, rather than the required 1,5-dihydro-2H-pyrrol-2-one tautomer. Taken together our results suggest that the α,β-hydrolase enzymes are essential for first ring cyclisation, but the precise nature of the intermediates remains to be determined.     

First synthesis of 4-chloro-2,2-difluoro[1,3]dioxole[4,5-c]pyridine

The 2,2-difluorobenzodioxole moiety has been introduced in medicinal chemistry research as a potential metabolically more stable derivative of the benzodioxole fragment. Herein we present, to the best of our knowledge, the first synthesis of 4-chloro-2,2-difluoro[1,3]dioxole[4,5-c]pyridine, a 5-aza-derivative of the 2,2-difluorobenzodioxole, from simple and cheap starting materials. The chlorine atom in position 4 could be useful for further functionalisation by cross coupling reactions.     

On decomposability of Multilinear sets

We consider the Multilinear set \({\mathcal {S}}\) defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form \(y_I = \prod _{i \in I} x_i\), \(I \in {\mathcal {I}}\), where \({\mathcal {I}}\) denotes a family of subsets of \(\{1,\ldots , n\}\) of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is possible when \({\mathcal {S}}\) is decomposable into simpler Multilinear sets \({\mathcal {S}}_j\), \(j \in J\); namely, the convex hull of \({\mathcal {S}}\) can be obtained by convexifying each \({\mathcal {S}}_j\), separately. In this paper, we study the decomposability properties of Multilinear sets. Utilizing an equivalent hypergraph representation for Multilinear sets, we derive necessary and sufficient conditions under which \({\mathcal {S}}\) is decomposable into \({\mathcal {S}}_j\), \(j \in J\), based on the structure of pair-wise intersection hypergraphs. Our characterizations unify and extend the existing decomposability results for the Boolean quadric polytope. Finally, we propose a polynomial-time algorithm to optimally decompose a Multilinear set into simpler subsets. Our proposed algorithm can be easily incorporated in branch-and-cut based global solvers as a preprocessing step for cut generation.     

On convergence in mixed integer programming

Let ${P \subseteq {\mathbb R}^{m+n}}$ be a rational polyhedron, and let P _I be the convex hull of ${P \cap ({\mathbb Z}^m \times {\mathbb R}^n)}$ . We define the integral lattice-free closure of P as the set obtained from P by adding all inequalities obtained from disjunctions associated with integral lattice-free polyhedra in ${{\mathbb R}^m}$ . We show that the integral lattice-free closure of P is again a polyhedron, and that repeatedly taking the integral lattice-free closure of P gives P _I after a finite number of iterations. Such results can be seen as a mixed integer analogue of theorems by Chvátal and Schrijver for the pure integer case. One ingredient of our proof is an extension of a result by Owen and Mehrotra. In fact, we prove that for each rational polyhedron P, the split closures of P yield in the limit the set P _I .     

Stereoselective cyclopropanation of chiral 5-substituted dihydro-2H-piperazines

The Simmon–Smith cyclopropanation of enantiomerically enriched dehydropiperazines is reported. The reaction is highly stereoselective and allowed us to prepare new, enantiomerically enriched 3-substituted 2,5-diazabicyclo[4.1.0]heptane cores with high diastereomeric purity and a relative anti-configuration, which was assigned by NMR analysis.     

Extraction and determination of the potato glycoalkaloid α -solanine in soil

The toxic glycoalkaloids α-solanine and α-chaconine are produced in all parts of the potato plant, and post-harvest potato tubers may represent a source of soil and water contamination. A new method was developed for extraction and purification of α-solanine in soil samples. Soil samples were extracted with THF?:?H₂O?:?ACN?:?CH₃COOH (50?:?30?:?20?:?1) and the extract purified by SPE before HPLC determination of α-solanine. The limit of detection was 2.4?mg of α-solanine?kg^?1 soil. The new procedure was used for determination of α-solanine in spiked soils with varying content of organic matter and texture. Recovery for soil samples spiked with α-solanine 1?h before extraction was 61–68% for soils low in organic carbon (<2.2% C), and to 47% for soil high in organic carbon. Similar recoveries were obtained for α-chaconine. The reproducibility of the method shown by the relative standard deviation varied from 1.7 to 10.1%, depending on the soil type. No decrease in extractable α-solanine was observed until day 17 for soil samples spiked with pure α-solanine kept at 5°C, while the content in samples spiked with potato materials showed a faster decline. This indicates that the degradation and/or ageing processes proceed relatively slowly for glycoalkaloids in soil matrices. This is the first method reported for determination of potato glycoalkaloids in soil.