K1 of Chevalley groups are nilpotent

Abstract Let F be a reduced irreducible root system and R be a commutative ring. Further, let G(F,R) be a Chevalley group of type F over R and E(F,R) be its elementary subgroup. We prove that if the rank of F is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G(F,R)/E(F,R) is nilpotent by abelian. In particular, when G(F,R) is simply connected the quotient K1(F,R)=G(F,R)/E(F,R) is nilpotent. This result was previously established by Bak for the series A1 and by Hazrat for C1 and D1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

Stable hydrogen isotope ratios of lignin methoxyl groups as a palaeoclimate proxy

• Stable isotope ratios of organic compounds are valuable tools for determining the geographical origin, identity, authenticity or history of samples from a vast range of sources such as sediments, plants and animals, including humans. • Hydrogen isotope ratios (d2H values) of methoxyl groups in lignin from wood of trees grown in different geographical areas were measured using compound-specificpyrolysis isotope ratio mass spectrometry analysis. • Lignin methoxyl groups were depleted in 2H relative to both meteoric water andwhole wood. A high correlation (r2=0.91) was observed between the d2 H valuesof the methoxyl groups and meteoric water, with a relatively uniform fractionation of –216±19 recorded with respect to meteoric water over a range of d2H values from –110 in northern Norway to + 20‰ in Yemen. Thus, woods from northernlatitudes can be clearly distinguished from those from tropical regions. By contrast, the d2H values of bulk wood were only relatively poorly correlated (r 2 = 0.47) with those of meteoric water. • Measurement of the d 2H values of lignin methoxyl groups is potentially a powerful tool that could be of use not only in the constraint of the geographical origin of lignified material but also in paleoclimate, food authenticity and forensic investigations.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.