Fonctions de partitions à parité périodique

Let be the set of positive integers and a subset of . For , let denote the number of partitions of n with parts in . In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any and , there is a unique set , such that is even for n>N. Soon after, Ben Saïd and Nicolas (Acta Arith. 106 (2003) 183) considered , and proved that for all k≥0, the sequence is periodic on n. In this paper, we generalise the above works for any formal power series f in with f(0)=1, by constructing a set such that the generating function of is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in with P(0)=Q(0)=1, then for all k≥0 the sequence is periodic on n.     

Synthesis,proton NMR stereochemical study and diels-alder reaction of (E)-7-arylidene-2H,6H-naphtho[1,8-bc]furan-2,6-diones

7-Arylidene-2H,6H-naphtho[1,8-bc]furan-2,6-diones 3 were prepared by a retro-Diels-Alder reaction of the corresponding dihydro 1,3-oxazines 2 or in a one pot synthesis from 6-hydroxy-2H-naphtho[1,8-bc]furan-2-one 1 . Their E configuration was established from the chemical shift's values of the vinylic proton and by ¹H nmr NOE difference spectroscopy. Cycloadditions of 3b upon styrene or stilbene were catalyzed by boron trifluoride etherate. The cycloadducts were obtained with the relative trans configuration.     

(m‐CH3C6H4NH3)6P6O18

In hexakis(m‐toluidinium) cyclo­hexaphosphate, 6C₇H₁₀N⁺·­P₆O₁₈^6?, the atomic arrangement is typical of a layer structure. Layers including the centrosymmetric P₆O₁₈ ring anions develop around the (100) planes at x = . The hydrogen‐bond distribution is described.     

An efficient novel synthesis of beta-(azuleno[1,2-b]benzothienyl)- and beta-(azuleno[2,1-b]benzothienyl)-alpha,beta-unsaturated ketones by the tropylium ion-mediated intramolecular furan ring-opening reaction and X-ray investigation of methyl ketone derivative

Intramolecular reaction of 2-tropylio-3-(5-substituted 2-furyl)benzothiophenes (3), prepared from the corresponding 2-cycloheptatrienyl-3-(5-substituted 2-furyl)benzothiophenes (2), afforded the beta-(azuleno[1,2-b]benzothienyl)-alpha,beta-unsaturated ketones (4), which are otherwise difficult to obtain, in moderate yields. The reaction involves a ring-opening process of the furan ring by intramolecular attack of the tropylium ion onto the 2-position of the furan ring. Similarly, beta-(azuleno[2,1-b]benzothienyl)-alpha,beta-unsaturated ketones (8) were obtained from the corresponding 3-tropylio-2-(5-substituted 2-furyl)benzothiophenes (7) albeit in lower yields. The molecular and crystal structures of the methyl ketone derivative, 8a, are discussed on the basis of X-ray structure analysis.     

Yields and average cross sections of recoil charged particle induced reactions on11B,12C,13C,14N,16O and18O

The yield and average cross section for the reactions¹¹B(p, n)¹¹C,¹²C(p, )¹³N,¹³C(p, n)¹³N, ₁₂ ¹² C(d, n)¹³N,¹⁴N(p, )¹¹C,¹⁶O(p, )¹³N,¹⁶O(d, n)¹⁷F,¹⁶O(t, n)¹⁸F, and¹⁸O(p, n)¹⁸F have been measured in different compounds. The charged particles were created in the samples themselves either through recoil by scattering of 14 MeV neutrons off hydrogen and deuterium, or by the (n, t) reaction on⁶Li using thermal neutrons. The yields of reactions¹²C(d, n);¹⁶O(p, );¹⁶O(t, n) and¹⁸O(p, n) have been measured using proton, deuteron and triton spectra generated by 14 MeV neutrons in the reactions D(n, p)2n;⁶Li(n, d);⁷Li(n, d) and¹⁰B(n, d);⁷Li(n, t) and¹⁰B(n, t), respectively.     

Synthesis and Thermal Properties of Monodispersed Telechelic Diols Prepared from Radical Telomerization of Undecylenol with Novel Dithiols

Abstract

The synthesis of novel monodispersed telechelic diols from the radical telomerization of undecylenol with new α,ω-dithiols was investigated. These diols are semi-crystalline and exhibit a thermal stability much higher than that of commercially available polydispersed diols.     

Ce-promoted Fe–Cu–ZSM-5 catalyst: SCR-NO activity and hydrothermal stability

Research on Chemical Intermediates - Fe–Cu–ZSM-5 and Ce–Fe–Cu–ZSM-5 solids prepared using solid-state ion exchange method (SSIE) were tested in the NH3–SCR of NO...     

Surface effects on Sm valence in amorphous Sm alloys

We present the results of XPS and X-ray absorption measurements performed on the amorphous Sm Au and La Sm Au compounds. The XPS Sm 3d$\text{5}\text{2}$ core level spectra in these compounds reveal that at the surface (5 ? 7 Å) the samarium ions present both the Sm²⁺ (4f⁶) and Sm³⁺ (4f⁵) configurations. When the concentration in samarium decreases it is shown that the Sm²⁺ configuration is strongly enhanced at the surface whereas the X-ray absorption measurements indicate on the contrary that in the bulk only the Sm³⁺ configuration is present. Comparison is made with similar findings in pure crystalline Sm and various crystalline rare earth compounds.     

A noncontact atomic force microscope in air using a quartz resonator and the FM detection method

Noncontact imaging was successfully performed with the quartz resonator in air by using the FM detection method. A constant frequency shift mode was used. The frequency shift was about -30 mHz, which was induced by the attractive force gradient. A noncontact image of the Si(111) atomic step was obtained with vertical and lateral resolutions of 0.8 Å and 60 Å.