On the measure of the spectrum for the almost Mathieu operator

We obtain partial results on the conjecture that for the almost Mathieu operator at irrational frequency, , the measure of the spectrum,S(, , )=|4–2|. For ||2 we show that if _n is rational and irrational, then .Dedicated to Res Jost and Arthur WightmanResearch partially supported by U.S. NSF grant number DMS-8801918 and by BSF under grant number 88-00325     

Air-stable liquid clathrates. 1. Crystal structure of [NBu4][Br3] and reactivity of the [NBu4][Br3]·5 C6H6 liquid clathrate

[NBu₄][Br₃] interacts with C₆H₆ to form an air-stable liquid clathrate. The tribromide anion reacts with phenol to give para-bromophenol in>95% yield. The HBr which is generated in the reaction combines with the Br^– anion to form [Br-H-Br]^–. The latter forms the basis for a new liquid clathrate, [NBu₄][Br-H-Br]·n C₆H₆. The parent salt, [NBu₄][Br₃], crystallizes in the monoclinic space group C2/c witha=12.983(5),b=10.380(7),c=16.222(6) Å,=93.93(3)°, andD _c =1.47 g cm^–3 forZ=4. The final R value is 0.068 based on 787 observed reflections.     

SO(n,1) Wigner rotation as an SL(2,R) problem

We show that the composition of not only two SO(3,1) boosts, but also that of two SO(n,1) boosts for anyn 2, is basically an SO(2,1) problem and hence can be analysed completely using SL(2,R) matrices. By computing the expression for the Thomas/Wigner angle directly using SL(2,R) matrices we show that this approach results in considerable economy of algebra.     

Approximate neutrality of large-Z ions

LetN(Z) denote the number of electrons which a nucleus of chargeZ can bind in non-relativistic quantum mechanics (assuming that electrons are fermions). We prove thatN(Z)/Z1 asZ.Research partially supported by the NSERC under Grant NA7901 and by the USNSF under Grants DMS-8416049 and PHY 85-15288-A01