The great theorem of A.A. Markoff and Jean Bernoulli sequences

A proof of Markoff’s Great Theorem on the Lagrange spectrum using continued fractions is sketched. Markoff’s periods and Jean Bernoulli sequence ¹ are used to obtain a simple algorithm for the computation of the Lagrange spectrum below 3.     

Syntheses and characterization of a new class of mono‐ and heterodinuclear derivatives of boron derived from Schiff bases

Reactions of 2‐isopropoxy‐1, 3, 2‐ benzodioxaborole with equivalent amounts of Schiff base ligands having two hydroxyl groups ( 1a–3a ) yield mononuclear derivatives with one residual hydroxy group. The reactions of these mononuclear derivatives with hexamethyldisilazane in a 2:1 ratio yield heterodinuclear derivatives. All these newly synthesized derivatives have been characterized by elemental analyses and molecular weight measurements. Tentative structures have been proposed on the basis of IR and NMR (¹H, ¹³C, ¹¹B,²⁹Si)spectral data and Fab‐mass studies. Schiff bases and their corresponding mono‐ and heterodinuclear derivatives of boron have also been screened for antifungal activities. Copyright © 2010 John Wiley & Sons, Ltd.     

Arithmetic families of smooth surfaces and equisingularity of embedded schemes

This article deals with the foundations of a theory of equisingularity for families of zero-dimensional sheaves of ideals on smooth algebraic surfaces, in the arithmetic context, i.e., where one works with schemes defined over Dedekind rings. Here, different equisingularity conditions are analyzed and compared, based on one of the following requirements: 1) each member of the the family has the same desingularization tree, 2) the family admits a simultaneous desingularization, 3) a naturally associated family of curves is equisingular. Similar conditions had been investigated, in the context of Complex Local Analytic Geometry, by J. J. Risler. Received: 17 November 1997 / Revised version: 19 April 1999     

A remark on gauge symmetries in Ising spin glasses

This paper studies a particular line in the parameter space of the FK random interaction random cluster model for spin glasses following Katsura ([K]) and Mazza ([M]). We show that, after averaging over the random couplings, the occupied FK bonds have exactly a Bernoulli distribution. Comparison with explicit calculations on trees confirms the marginal role of FK percolation in determining phase transitions. Received: 1 October 1997 / Revised version: 18 May 1998     

Nombres de Tamagawa des groupes réductifs quasi-connexes

We define Tamagawa numbers for quasi-connected linear algebraic groups. This category of non-connected linear algebraic groups occurs naturally in the stabilization of the Arthur-Selberg trace formula. An expression for their Tamagawa numbers, needed for the stabilization process, is given in term of cohomological invariants generalizing results of Ono, Sansuc, Kottwitz and Kottwitz & Shestad. Received: 7 March 2000