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.     

Narrow band quantitative and multivariate electroencephalogram analysis of peri-adolescent period

ABSTRACT: BACKGROUND: The peri-adolescent period is a crucial developmental moment of transition from childhood to emergent adulthood. The present report analyses the differences in Power Spectrum (PS) of the Electroencephalogram (EEG) between late childhood (24 children between 8 and 13 years old) and young adulthood (24 young adults between 18 and 23 years old). RESULTS: The narrow band analysis of the Electroencephalogram was computed in the frequency range of 0--20 Hz. The analysis of mean and variance suggested that six frequency ranges presented a different rate of maturation at these ages, namely: low delta, delta-theta, low alpha, high alpha, low beta and high beta. For most of these bands the maturation seems to occur later in anterior sites than posterior sites. Correlational analysis showed a lower pattern of correlation between different frequencies in children than in young adults, suggesting a certain asynchrony in the maturation of different rhythms. The topographical analysis revealed similar topographies of the different rhythms in children and young adults. Principal Component Analysis (PCA) demonstrated the same internal structure for the Electroencephalogram of both age groups. Principal Component Analysis allowed to separate four subcomponents in the alpha range. All these subcomponents peaked at a lower frequency in children than in young adults. CONCLUSIONS: The present approaches complement and solve some of the incertitudes when the classical brain broad rhythm analysis is applied. Children have a higher absolute power than young adults for frequency ranges between 0-20 Hz, the correlation of Power Spectrum (PS) with age and the variance age comparison showed that there are six ranges of frequencies that can distinguish the level of EEG maturation in children and adults. The establishment of maturational order of different frequencies and its possible maturational interdependence would require a complete series including all the different ages.     

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.