The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption

In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation ?Δu + g(x, u) = 0 in a smooth domain Ω ? ?^N, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities g_α,q of the form g_α,q (x, u) = d(x, ?Ω)^α |u|^q?1 u, with α > ?2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {g_α,q}, determine the critical value of the exponent q (for fixed α > ?2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for ?Δu + g_α,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.     

Control of the Morphology of Organic-Inorganic Hybrid Materials Elaborated by Reactive Processing Without Solvent

We study the synthesis of organic-inorganic hybrids in the molten state of the polymer without solvent, by reactive processing. The synthesis is based in a first step upon the crosslinking of ethylene vinyl acetate copolymer (EVA) through an exchange reaction between the pendant ester groups of EVA and tetrapropoxysilane (TPOS) in presence of dibutyltin oxide (DBTO) as an indispensable catalyst. Then in a second step the hydrolysis-condensation reactions of residual alkoxide groups results in silica network formation, which is covalently bonded with the organic network. An original approach was developed to correlate the evolution of the hydrolysis-condensation reactions through the assessment of propanol by gas chromatography and the evolution of the morphology through SAXS and TEM analysis. Silica-rich domains with a diameter close to 5 nm are obtained.     

Global expression analysis of CESA and CSL genes in Arabidopsis

We have used Affymetrix gene chips to measure the expression of 10 CESA and 29 CSL genes of Arabidopsis in different developmental stages or organs. These measurements reveal that many of the genes exhibit different levels of expression in the various organs. While several CESA genes are highly expressed in all the tissues examined, very few CSL genes approach such high levels of expression. This suggests that the CSL genes either encode enzymes for the synthesis of minor components of cell walls or are expressed only in specific cell types. The expression data also highlights the potential importance of the CESA genes for primary and secondary cell wall formation during different developmental stages and in the different organs examined.     

On the construction of 2-substituted 1,4-diacetoxybutadiene moiety: application to the synthesis of (±)-caulerpenyne

The synthesis of 2-substituted 1,4-diacetoxybutadiene derivatives with a partial control of stereochemistry is described from two potentially precursor enals by a judicious choice of experimental conditions (Ac₂O/DMAP in Et₃N). These conditions have been successfully applied in the first total synthesis of caulerpenyne.     

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2

We classify all the global phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 2. In other words we characterize all the global phase portraits of the quadratic polynomial vector fields having all their orbits contained in conics. For such a vector field there are exactly 25 different global phase portraits in the Poincaré disc, up to a reversal of sense.     

The Chaotic Mixmaster and the Suppression of Chaos in Scalar–Tensor Cosmologies

We show that there is a threshold in energy for the onset of chaos in cosmology for the Universe described as a dynamical system derived from the Einstein equations of General Relativity (GR). In the case of the mixmaster model (homogeneous and anisotropic cosmology with a Bianchi IX metric), the chaos occurs precisely at the prescribed necessary value H _vac=0 of the GR for the energy of the Universe while the system is found to be regular for H<0 and chaotic for H>0 with respect to its pure vacuum part. In the case of generalized scalar tensor theories within the Bianchi IX model, we show using the ADM formalism and a conformal transformation that the energy of the dynamical system as compared to vacuum lies below the zero energy threshold. The system is thus not exhibiting chaos and the conclusion still holds in the presence of ordinary matter as well. The suppression of chaos occurs in a similar way for stiff matter alone.     

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.