ON THE CONVERGENCE OF THE PARABOLIC APPROXIMATION OF A CONSERVATION LAW IN SEVERAL SPACE DIMENSIONS

1.IntrodnctionWegiveaproofofthestrongconvergenceinofthesolutionoftheparabolicapproximationtowardstheentropicsolutiontothescalarconservationlawwhereuo(RN),udenotessomeapproximationofuosuchthatandthefluxfsatisfiesTheconvergenceoftheapproximatesolutions...     

A novel class of fused heterocycles,benzo­[b]­furo­[2,3‐c]­pyrroles

In the isomeric compounds 2‐benzyl‐3‐methyl‐1‐phenyl­benzo­[b]­furo­[2,3‐c]­pyrrole and 2‐benzyl‐1‐methyl‐3‐phenyl­benzo­[b]­furo­[2,3‐c]­pyrrole, both C₂₄H₁₉NO, the pyrrole ring, although presumably somewhat strained, does not differ appreciably from N‐methyl­pyrrole except for a relatively short C—C single bond in the pyrrole ring.     

ON LACUNARY STATISTICAL CONVERGENCE OF ORDER α

In this article, we introduce the concept of lacunary statistical convergence of order α of real number sequences and give some inclusion relations between the sets of lacunary statistical convergence of order α and strong Nαθ(p)-summability. Furthermore, some relations between the spaces Nαθ(p) and Sαθare examined.     

Synthesis of unsymmetrical 3,4-diaryl-3-pyrrolin-2-ones utilizing pyrrole Weinreb amides

A regiocontrolled synthesis of unsymmetrical 3,4-diaryl-3-pyrrolin-2-ones has been achieved in three steps from 1,2-diaryl-1-nitroethenes with pyrrole-2-carboxamides (pyrrole Weinreb amides) serving as the key linchpin intermediates. Two different methods for the preparation of the requisite nitroalkenes were investigated: (1) modified Henry reaction between arylnitromethanes and arylimines; and (2) Suzuki-Miyaura cross-coupling reaction of 2-aryl-1-bromo-1-nitroethenes with arylboronic acids. Some difficulty was encountered in the preparation of arylnitromethanes, thus leading to the exploration of a cross-coupling strategy that proved more useful. A Barton-Zard pyrrole cyclocondensation reaction between 1,2-diaryl-1-nitroethenes and N-methoxy-N-methyl-2-isocyanoacetamide gave the corresponding pyrrole Weinreb amides, which were then converted into the desired 3-pyrrolin-2-ones in two steps. Overall, this method allowed for the construction of 3,4-diaryl-3-pyrrolin-2-ones with complete regiocontrol of the substituents with respect to the lactam carbonyl. The utility of this synthetic methodology was demonstrated by the preparation of eight unsymmetrical and symmetrical 3,4-diaryl-3-pyrrolin-2-ones including the N-H lactam analogue of the selective COX-II inhibitor, rofecoxib.     

Regioselective 1,3-Dipolar Cycloaddition Reactions of Unsymmetrical Münchnones (1,3-Oxazolium-5-olates) with 2- and 3-Nitroindoles. A New Synthesis of Pyrrolo[3,4-b]indoles

The unsymmetrical mesoionic münchnones 13 (3-benzyl-2-methyl-4-phenyl-1,3-oxazolium-5-olate) and 14 (3-benzyl-4-methyl-2-phenyl-1,3-oxazolium-5-olate) react with the N-protected 2- and 3-nitroindoles 1 (ethyl 2-nitroindole-1-carboxylate), 6 (3-nitro-1-(phenylsulfonyl)indole), and 17 (ethyl 3-nitroindole-1-carboxylate) in refluxing THF to afford in good to excellent yields the pyrrolo[3,4-b]indoles 15 (2-benzyl-1-methyl-3-phenyl-4-carboethoxy-2,4-dihydropyrrolo[3,4-b]indole), 16 (2-benzyl-3-methyl-1-phenyl-4-carboethoxy-2,4-dihydropyrrolo[3,4-b]indole), 18 (2-benzyl-1-methyl-3-phenyl-4-(phenylsulfonyl)-2,4-dihydropyrrolo[3,4-b]indole), and 19 (2-benzyl-3-methyl-1-phenyl-4-(phenylsulfonyl)-2,4-dihydropyrrolo[3,4-b]indole). In several cases the regiochemistry, which is opposite to that predicted by FMO theory, is very high and leads essentially to a single pyrrolo[3,4-b]indole; e.g., 6+13→19 in 74% yield.     

Dendrimeric molecular transporters: synthesis and evaluation of tunable polyguanidino dendrimers that facilitate cellular uptake

[reaction: see text] Nine fluorescently labeled structurally varied polyguanidino dendrimers based on diamino acid monomeric units were individually synthesized in an efficient, scalable sequence using a trifluoroacetamide protecting group-perguanidinylation strategy. While the dendrimers varied significantly in their ability to enter a human lymphocyte cell line, the best transporters out-performed an oligoarginine reference standard.     

An efficient, scalable synthesis of the molecular transporter octaarginine via a segment doubling strategy

[reaction: see text]. Short oligomers of arginine function as remarkably efficient molecular transporters of drugs and probe molecules into cells and tissue. Currently, these compounds are prepared on resin through a unidirectional solid-phase synthesis. To extend the utility of these compounds for therapeutic and research applications, a scalable solution-phase synthesis of Arg8 (1) has been developed on the basis of a segment doubling strategy that proceeds in 13 steps and 28% overall yield from 4, including a novel one-step perdeprotection-perguanidinylation reaction.     

The fused heterocycle 2‐benzyl‐3‐methyl‐1‐phenyl‐4‐phenyl­sulfonyl‐2H,4H‐pyrrolo­[3,4‐b]­indole

Crystal structure determination of the title mol­ecule, C₃₀H₂₄N₂O₂S, reveals that the pyrrole ring in this fused heterocycle, although presumably strained and reactive in cyclo­addition reactions, does not differ appreciably from N‐methyl­pyrrole except for a shorter C—C single bond [1.409 (4) Å] in the pyrrole ring.