Exact L 2-norm plane separation

We consider the problem of separating two sets of points in an n-dimensional real space with a (hyper)plane that minimizes the sum of L _p -norm distances to the plane of points lying on the wrong side of it. Despite recent progress, practical techniques for the exact solution of cases other than the L ₁ and L _∞-norm were unavailable. We propose and implement a new approach, based on non-convex quadratic programming, for the exact solution of the L ₂-norm case. We solve in reasonable computing times artificial problems of up to 20000 points (in 6 dimensions) and 13 dimensions (with 2000 points). We also observe that, for difficult real-life instances from the UCI Repository, computation times are substantially reduced by incorporating heuristic results in the exact solution process. Finally, we compare the classification performance of the planes obtained for the L ₁, L ₂ and L _∞ formulations. It appears that, despite the fact that L ₂ formulation is computationally more expensive, it does not give significantly better results than the L ₁ and L _∞ formulations.     

rac‐9‐Ethyl‐12a‐hydroxytetradecahydrotriphenylene‐1,5(2H,4bH)‐dione: stabilization of a new isomer of a functionalized perhydrotriphenylene through a tandem Michael addition–aldol reaction

The title compound, C₂₀H₃₀O₃, is a new functionalized perhydrotriphenylene derivative formed via a tandem Michael addition–aldol reaction. The structural study reveals that the system of fused rings approximates a C₂ point symmetry, with trans–cis–cis ring junctions, while highly symmetric all‐trans perhydrotriphenylene, previously characterized, approximates a D₃ symmetry. The perhydrotriphenylene nucleus of the title compound corresponds to the third stable stereoisomer isolated for this polycyclic system. Considering that the C_s isomer was obtained recently through a similar tandem reaction, a general strategy is proposed which may help to obtain other stable stereoisomers of perhydrotriphenylene.     

The first chemical synthesis of the core structure of the benzoylhydrazine-NAD adduct, a competitive inhibitor of the Mycobacterium tuberculosis enoyl reductase

[reaction: see text] An isoniazid-NAD adduct has been recently proposed as the ultimate metabolite responsible for the antituberculous activity of isoniazid (INH). Its structure results from binding of the isonicotinoyl radical at C4 position of the nicotinamide ring of NAD with further possible and debated cyclization to form a cyclic hemiamidal derivative. Replacing the pyridine cycle of INH in INH-NAD adduct by a phenyl cycle (BH-NAD adduct) was shown previously to still retain the activity. On these bases, the core structure (4-benzoyl-1,4-dihydronicotinamide ribonucleoside) of the BH-NAD adduct and a series of analogues have been synthesized by using 3,4-pyridinedicarboximide as starting material. Depending on the nature of the substituent (pyridine or aryl) and on the oxidized or the reduced state of the nicotinamide nucleus, they were found either in a cyclized hemiamidal or an opened form or were shown to exist in equilibrium under cyclized or opened forms. Although none of these compounds could significantly inhibit activity of the InhA or MabA reductases (two possible targets of isoniazid), they represent attractive targets to develop potential second-generation inhibitors, including the total chemical synthesis of the bioactive BH-NAD adduct.     

Periodic orbits and chain-transitive sets of C1-diffeomorphisms

We prove that the chain-transitive sets of C¹-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C¹-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C¹-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff¹(M).     

First synthesis of methyl 3-amino-4-(het)aryl-1H-pyrrole-2-carboxylates as useful scaffolds in medicinal chemistry

The preparation of new methyl 4-(het)aryl-3-amino-1H-pyrrole-2-carboxylates was achieved starting from commercial arylacetonitriles. This four steps synthesis afforded with good yields interesting building-blocks useful in the access to many nitrogen heterocycles with potential therapeutic interest.