Non-central limit theorems for random selections

Selection from finite sets is a basic procedure of statistics and the partial sum behavior of selected elements is completely known under the “uniform asymptotic negligibility” condition of central limit theory. The purpose of the present paper is to determine the asymptotic behavior of partial sums when the central limit theorem fails. As an application, we describe the limiting properties of permutation and bootstrap statistics in case of infinite variance.     

Synthesis of 4-thiaharmalan analogue 4-aryl-1,3-thiazino[5,6-b]indole derivatives by prevention of rearrangements to position two of the indole moiety

An efficient and selective non-acidic protocol has been developed for the synthesis of derivatives of a new ring system: 4-aryl-1,3-thiazino[5,6-b]indole, a 4-thiaharmalan analogue. The convenient amidomethylation of indole-3-thiol (5) afforded 3-benzoylaminomethylthio-1H-indole (7a), with ortho-amidomethylated 2-benzoylamino-methyl-3-benzoylaminomethylthio-1H-indole (8) as side product. Following the Bischler-Napieralski reaction of 7a the rearranged 2-benzoylaminomethylthio-1H-indole 11 could be isolated. In order to prevent such rearrangements the target thiazinoindoles were prepared via 3-thiobenzoylaminomethylthioindole (13) via a modified Bischler-Napieralski reaction.     

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.     

Gravity-Related Wave Function Collapse

The gravity-related model of spontaneous wave function collapse, a longtime hypothesis, damps the massive Schrödinger Cat states in quantum theory. We extend the hypothesis and assume that spontaneous wave function collapses are responsible for the emergence of Newton interaction. Superfluid helium would then show significant and testable gravitational anomalies.     

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.     

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.