Strengthened semidefinite programming bounds for codes

We give a hierarchy of semidefinite upper bounds for the maximum size A(n,d) of a binary code of word length n and minimum distance at least d. At any fixed stage in the hierarchy, the bound can be computed (to an arbitrary precision) in time polynomial in n; this is based on a result of de Klerk et al. (Math Program, 2006) about the regular ∗-representation for matrix ∗-algebras. The Delsarte bound for A(n,d) is the first bound in the hierarchy, and the new bound of Schrijver (IEEE Trans. Inform. Theory 51:2859–2866, 2005) is located between the first and second bounds in the hierarchy. While computing the second bound involves a semidefinite program with O(n ⁷) variables and thus seems out of reach for interesting values of n, Schrijver’s bound can be computed via a semidefinite program of size O(n ³), a result which uses the explicit block-diagonalization of the Terwilliger algebra. We propose two strengthenings of Schrijver’s bound with the same computational complexity. Supported by the Netherlands Organisation for Scientific Research grant NWO 639.032.203.     

Prime Fano threefolds and integrable systems

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system. The first author is partially supported by grant MI1503/2005 of the Bulgarian Foundation for Scientific Research.     

Pauli-Fierz Hamiltonians defined as quadratic forms

We study Pauli-Fierz Hamiltonians-self-adjoint operators describing a small quantum system interacting with a bosonic field. Using quadratic form techniques, we extend the results of Dereziński-Gérard and Gérard about the self-adjointness, the location of the essential spectrum and the existence of a ground state to a large class of Pauli-Fierz Hamiltonians.     

Design of the cold atom PHARAO space clock and initial test results

In this paper we describe the cold atom clock PHARAO, designed for microgravity operation. All elements of the PHARAO engineering model have been manufactured and delivered to CNES, the French space agency. We present the clock design, its main characteristics, and initial science operation. PHARAO is one of the main components of the Atomic Clock Ensemble in Space payload that is scheduled to fly on board the International Space Station in 2010. PACS 07.87.+v; 06.30.Ft; 95.55.Sh; 32.80.Pj     

Synthesis of latonduine derivatives via intramolecular Heck reaction

The synthesis of indole ring-fused benzazepinone series as latonduine derivatives has been developed via an intramolecular Heck reaction. The scope has been enlarged not only to indole moiety but also to pyrrolo and benzo[b]thiophene nuclei. Several derivatives prepared have been evaluated in vitro for their antiproliferative activities on breast cancer cell lines. Some of them showed promising cytotoxic activities.     

Study of the spontaneous formation of organic layers on carbon and metal surfaces from diazonium salts

This study investigates the spontaneous grafting of different para-substituted phenyl groups on carbon and metallic surfaces from diazonium salts solutions. Glassy carbon, nickel, zinc and iron plates were allowed to react with an acetonitrile solution of aryldiazonium tetrafluoroborate salt by simple dipping. The surfaces were characterized before and after their immersion by XPS and AFM to evidence the formation of a coating on the different materials. The results are indicative of the presence of substituted phenyl groups on all the investigated surfaces. This study also aims at correlating grafting efficiency with metal reactivities and diazonium salt electronic properties by means of AFM and FT-IRRAS. For this purpose, zinc and nickel were chosen due to their opposite reducing properties and two diazonium salts were selected with electron-donor or -withdrawing para-substituents. The results tend to indicate that redox properties of both partners (diazonium + metal) are of prime importance for grafting to occur.     

Connections between semidefinite relaxations of the max-cut and stable set problems

We describe links between a recently introduced semidefinite relaxation for the max-cut problem and the well known semidefinite relaxation for the stable set problem underlying the Lovász’s theta function. It turns out that the connection between the convex bodies defining the semidefinite relaxations mimics the connection existing between the corresponding polyhedra. We also show how the semidefinite relaxations can be combined with the classical linear relaxations in order to obtain tighter relaxations. This work was done while the author visited CWI, Amsterdam, The Netherlands. Svata Poljak untimely deceased in April 1995. We shall both miss Svata very much. Svata was an excellent colleague, from whom we learned a lot of mathematics and with whom working was always a very enjoyable experience; above all, Svata was a very nice person and a close friend of us. The research was partly done while the author visited CWI, Amsterdam, in October 1994 with a grant fom the Stieltjes Institute, whose support is gratefully acknowledged. Partially supported also by GACR 0519. Research support by Christian Doppler Laboratorium für Diskrete Optimierung.