Approximate Radical for Clusters: A Global Approach Using Gaussian Elimination or SVD

We introduce a matrix of traces, attached to a zero dimensional ideal . We show that the matrix of traces can be a useful tool in handling systems of polynomial equations with clustered roots. We present a method based on Dickson’s lemma to compute the “approximate radical” of in which has zero clusters: the approximate radical ideal has exactly one root in each cluster for sufficiently small clusters. Our method is “global” in the sense that it works simultaneously for all clusters: the problem is reduced to the computation of the numerical nullspace of the matrix of traces, a matrix efficiently computable from the generating polynomials of . To compute the numerical nullspace of the matrix of traces we propose to use Gaussian elimination with pivoting or singular value decomposition. We prove that if has k distinct zero clusters each of radius at most ɛ in the ∞-norm, then k steps of Gaussian elimination on the matrix of traces yields a submatrix with all entries asymptotically equal to ɛ². We also show that the (k + 1)-th singular value of the matrix of traces is proportional to ɛ². The resulting approximate radical has one root in each cluster with coordinates which are the arithmetic mean of the cluster, up to an error term asymptotically equal to ɛ². In the univariate case our method gives an alternative to known approximate square-free factorization algorithms which is simpler and its accuracy is better understood. This work was completed with the support of NSF grants CCR-0306406 and CCR-0347506 and OTKA grants T42481 and T42706 and NK63066.     

An expeditious synthesis for γ-carboline analogue 4-aryl-1,3-thiazino[6,5-b]indole derivatives via the trifluoromethanesulfonic acid-promoted isomerization of 3-amidomethylthioindole intermediates to 2-indolyl sulfides

A highly efficient trifluoromethanesulfonic acid-mediated rearrangement of the benzoylaminomethylthio group of 3-benzoylaminomethylthioindoles (7a-e) to position 2 of the indole ring was developed. Thus, 2-benzoylaminomethylthioindoles (9a-e) were obtained in good yields and were involved as key intermediates in the synthesis of the new γ-carboline analogue ring system: 2,9-dihydro-4-aryl-1,3-thiazino[6,5-b]indole derivatives (11a-e). The target thiazinoindoles (11a-e) were prepared via 2-thiobenzoylaminomethylindoles (10a-e) in modified Bischler-Napieralski reactions.     

The D-property in unions of scattered spaces

We show in a direct way that a space is D if it is a finite union of subparacompact scattered spaces. This result cannot be extended to countable unions, since it is known that there is a regular space which is a countable union of paracompact scattered spaces and which is not D. Nevertheless, we show that every space which is the union of countably many regular Lindelöf C-scattered spaces has the D-property. Also, we prove that a space is D if it is a locally finite union of regular Lindelöf C-scattered spaces.     

New isomers of 4,1-benzothiazepines. The first evidence for the desmotropy of seven-membered heterocycles

A novel procedure was developed for the preparation of 2,3-disubstituted 4,1-benzothiazepines, via the ring transformation of (2R^∗,2aS^∗)-2-chloro-2a-phenyl-2,2a-dihydro-2H,4H-azeto[1,2-a][3,1]benzothiazin-1-one (1) with sodium ethoxide in ethanol. The tautomeric products (R^∗)-3-ethoxycarbonyl-2-phenyl-3,5-dihydro-4,1-benzothiazepine (4) and 3-ethoxycarbonyl-2-phenyl-1,5-dihydro-4,1-benzothiazepine (5) exhibit the rare phenomenon of desmotropy of the condensed seven-membered heterocycles. Surprisingly, these desmotropes could be separated by column chromatography. The products are unexpectedly stable in solution and their structures were proved by means of NMR and mass spectrometry.     

Single diastereomers of polyhydroxylated 9-oxa-1-azabicyclo[4.2.1]nonanes from intramolecular 1,3-dipolar cycloaddition of omega-unsaturated nitrones

8-Benzyloxymethyl-3,4,5-tribenzoyloxy-9-oxa-1-azabicyclo[4.2.1]nonane has been prepared as the single diastereoisomer 8 from an intramolecular 1,3-dipolar cycloaddition involving 2-(benzyloxy)acetaldehyde and omega-unsaturated hydroxylamine 7 derived from methyl alpha-D-glucopyranoside. The analogous 8-methoxycarbonyl 9-oxa-1-azabicyclo[4.2.1]nonane was afforded in a similar manner, from methyl D-galactopyranoside and methyl glyoxylate, as a 3:1 mixture of diastereoisomers 15 and 16. When conducted in achiral ionic liquid 17 this ratio increased to 8:1, and in chiral ionic liquid 18, compound 15 was formed exclusively.     

Electrophoretic mobility and molecular distribution studies of poly(amidoamine) dendrimers of defined charges

Generation 5 ethylenediamine (EDA)-cored poly(amidoamine) (PAMAM) dendrimers (E5, E denotes the EDA core and 5 the generation number) with different degrees of acetylation and carboxylation were synthesized and used as a model system to investigate the effect of charge and the influence of dendrimer surface modifications on electrophoretic mobility (EM) and molecular distribution. The surface-modified dendrimers were characterized by size-exclusion chromatography, 1H NMR, MALDI-TOF-MS, PAGE, and CE. The focus of our study was to determine how EM changes as a function of particle charge and molecular mass, and how the molecular distribution changes due to surface modifications. We demonstrate that partially modified dendrimers have much broader migration peaks than those of fully surface functionalized or unmodified E5 dendrimers due to variations in the substitution of individual dendrimer surfaces. EM decreased nonlinearly with increases in surface acetylation for both PAMAM acetamides and PAMAM succinamic acids, indicating a complex migration activity in CE separations that is not solely due to charge/mass ratio changes. These studies provide new insights into dendrimer properties under an electric field, as well as into the characterization of dendrimer-based materials being developed for medical applications.     

Novel indole syntheses by ring transformation of β-lactam-condensed 1,3-benzothiazines into indolo[2,3-b][1,4]benzothiazepines and indolo[3,2-c]isoquinolines

ortho-Nitrophenyl-substituted condensed 1,3-benzothiazines proved to be a useful core unit in indole syntheses under non-reductive conditions. Thus, the treatment of ortho-nitro-2-aryl-2a-chloro-4H-azeto[2,1-b][1,3]benzothiazin-1-ones with sodium methoxide in methanol provided indolo-1,4-benzothiazepines via a novel rearrangement. Through the sulfur extrusion reaction of indolo[2,3-b][1,4]benzothiazepines, further alkaloid-type indole derivatives, indolo[3,2-c]isoquinolines, were obtained. The structures of the new ring systems were determined by means of NMR spectroscopy.     

HPTLC and HPLC Determination of Alkaloids in Poppies Subjected to Stress

JPC – Journal of Planar Chromatography – Modern TLC - Papaver somniferum produces secondary metabolites which have important roles in self-defense processes, in plant biochemistry, and...     

Synthesis of alkenoic acid derivatives containing cyclopropane ring,new juvenile hormone analogs

A synthetic method is described for the preparation of new type juvenoids [$\text{2}$].