On the spectrum of the sum of generators for a finitely generated group

Let Γ be a finitely generated group. In the group algebra ℂ[Γ], form the averageh of a finite setS of generators of Γ. Given a unitary representation π of Γ, we relate spectral properties of the operator π(h) to properties of Γ and π. For the universal representationπ _un of Γ, we prove in particular the following results. First, the spectrum Sp(π _un (h)) contains the complex numberz of modulus one iff Sp(π _un (h)) is invariant under multiplication byz, iff there exists a character such that η(S)={z}. Second, forS ⁻¹=S, the group Γ has Kazhdan’s property (T) if and only if 1 is isolated in Sp(π _un (h)); in this case, the distance between 1 and other points of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.     

On a new method for constructing good point sets on spheres

We use quadrature formulas with equal weights in order to constructN point sets on spheres ind-space (d 3) which are almost optimal with respect to a discrepancy concept, based on distance functions (potentials) and distance functionals (energies). By combining this approach with the probabilistic method, we obtain almost best possible approximations of balls by zonotopes, generated byN segments of equal length.Editors' note: We learned with sadness of Gerold Wagner's untimely death as a result of an avalanche in the Alps shortly after the submission of this paper. When one of the referees, Joram Lindenstrauss, suggested that Wagner's results might be extended to dimensions >6, we invited Professor Lindenstrauss to submit a paper containing that extension which we would publish alongside the Wagner paper. The result is the paper by Bourgain and Lindenstrauss that follows the present one.     

Non-linear parabolic problem associated with the penetration of a magnetic field into a substance

We study the following initial and boundary value problem: In section 1, with u₀ in L²(Ω), f continuous such that f(u) + ? non-decreasing for ? positive, we prove the existence of a unique solution on (0,T), for each T > 0. In section 2 it is proved that the unique soluition u belongs to L²(0, T; H ∩ H²) ∩ L^∞(0, T; H) if we assume u₀ in H and f in C¹(?,?). Numerical results are given for these two cases.     

Etude des conditions d'accès au triazino-1,2,4-[4,5-b] indazole

1,2,4-Triazino[4,5-b]indazol-1(2H)one and its derivatives were prepared by transposition of 3-[2-(-1,3,4-oxadiazolyl)]indazole or by ring closure of indazole ethoxymethylidenehydrazides. The synthesis of 1,2,3,4-tetrahydro-l,2,4-triazino[4,5-b]indazole-1,4-dione was achieved by cyclising the N-carbethoxyhydrazide of indazole-3-carboxylic acid and the synthesis of 1,2,4-triazino-[4,5-b]indazol-4(3H)one was made by cyclising the N-carbethoxy-hydrazone of indazole-3-carboxaldehyde. The Oxydation of 1,2,4-triazino[4,5-b]indazole-l(2H)thione gave 1,2,4-triazino-[4,5-b]indazoles. Nmr spectral data are reported.     

Radiohochspannungsionophoretische Untersuchungen der Isotopenaustauschreaktionen zwischen Polythionaten,Thiosulfat und Sulfit

The isotopic exchange of³⁵S between thiosulfate/sulfite and tetrathionate/thiosulfate is examined. The kinetic data are determined. In both cases the reactions are of second order. As the isotopic exchange between tetrathionate/thiosulfate is very fast, these reaction components are used in high dilution. Furthermore, the extent of exchange during the separation is determined.