QR factoring to compute the GCD of univariate approximate polynomials

We present a stable and practical algorithm that uses QR factors of the Sylvester matrix to compute the greatest common divisor (GCD) of univariate approximate polynomials over /spl Ropf/[x] or /spl Copf/[x]. An approximate polynomial is a polynomial with coefficients that are not known with certainty. The algorithm of this paper improves over previously published algorithms by handling the case when common roots are near to or outside the unit circle, by splitting and reversal if necessary. The algorithm has been tested on thousands of examples, including pairs of polynomials of up to degree 1000, and is now distributed as the program QRGCD in the SNAP package of Maple 9.     

The number of ideals in a quadratic field

LetK be a quadratic Geld, and letR(N) be the number of integer ideals inK with norm at most AT. Letx with conductork be the quadratic character associated withK. Then |R(N)−NL(1,x)|⩽Bk ^50/73 N ^23/73(logN)^461/146 forN ≥Ak, whereA andB are constants. ForN ≥Ak ^c,C sufficiently large, the factork ^50/73 may be replaced by (d(k))^4/73 k ^46/73. Dedicated to the memory of Professor K G Ramanathan     

Reactions of amidoximes with orthoacetates

The reaction of two amidoximes 4 with ethyl orthoacetate in the presence of an acid catalyst gave triazine N-oxides 7 . A mechanism via a Beckmann rearrangement to a carbodiimide, followed by reaction with amidoxime and the ortho ester, is proposed.     

A new approach to the synthesis of conjugated polymer-nanocrystal composites for heterojunction optoelectronics

We report a simple one pot process for the preparation of lead sulfide (PbS) nanocrystals in the conjugated polymer poly (2-methoxy-5-(2'ethyl-hexyloxy)-p-phenylene vinylene)(MEH-PPV), and we demonstrate electronic coupling between the two components.     

Vibrational spectra of nitromethanes and the effects of internal rotation

Infrared spectra have been obtained in gas and crystalline phases for CH₃NO₂, CD₃NO₂, and CHD₂NO₂. A few Raman liquid measurements have also been made. The crystal spectra show the methyl group to be locked in conformation I,

with isolated CH stretching frequencies 3065 (ν_∥CH) and 3000 (ν_⊥CH), respectively. In the gas phase, the bands due to ν_aCH₃ and ν_aCD₃ are split, the splitting being compatible with isolated frequencies of 3065 (ν_∥CH) and 3006 (ν_⊥CH) cm^?1, respectively. These indicate a variation of 0.006 Å in bond length during internal rotation. The gas phase infrared band at 3028.7 cm^?1 and the Raman liquid band at 3024 cm^?1 in the spectra of CHD₂NO₂ represent averages of the frequencies during internal rotation. All calculations are performed with a potential function involving variations both in diagonal bond stretching, and in off-diagonal stretching interaction constants, proportional to cos2π, where π is the internal rotation angle.In the Appendix, vibrational assignments below 1600 cm^?1 are examined. Product rule considerations and intensity changes upon deuteration lead to a conflict with the infrared gas phase band contour over the assignment of the rCD₃ mode at 884 cm^?1. This and other features in the gas phase spectra may arise from the internal rotation.     

Synthesis of Substituted Trans-1,6-Dimethylbicyclo[4.3.0]-non-2-enes Using A Claisen Rearrangement of α-(Thioalkoxy) Esters

The Claisen rearrangement of α-(thioalkoxy)-esters provided access to substituted trans-1, 6-dimethylbicyclo[4.3.0]non-2-enes characteristic of the CD rings of the cucurbitanes.     

Synthetic Approaches to Taxodione. Synthesis of Methyl 12-Oxopodocarpa-5,9(11)-diene-8β-Carboxylate

In connection with studies leading to the total synthesis of taxodione¹ (1), we required the tricyclic enone 3. We have previously described² an AB→ABC approach to this intermediate which was flawed by a low yield in the penultimate closure of the C ring (2→3). We now wish to report an alternate synthesis of 3 which involves a BC→ABC approach for the construction of this tricyclic system.