Thermodynamic Limit and Proof of Condensation for Trapped Bosons

We study condensation of trapped bosons in the limit when the number of particles tends to infinity. For the noninteracting gas we prove that there is no phase transition in any dimension, but in any dimension, at any temperature the system is 100% condensated into the one-particle ground state. In the case of an interacting gas we show that for a family of suitably scaled pair interactions, the Gross–Pitaevskii scaling included, a less-than-100% condensation into a single-particle eigenstate, which may depend on the interaction strength, persists at all temperatures.     

Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

We present here an improved version of the method introduced by the first author to derive pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering a general qualinear equation in divergence form we apply the method to the case of a nonlinear diffusion-convection equation. The conclusions are stated first for classical solutions and then for generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing effects for t > 0. Some unilateral pointwise gradient estimates are also obtained. The case of the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments showing the connections among these estimates and the study of the free boundaries associated to the solutions of the diffusion-convection equation.     

Computation of the Maslov index and the spectral flow via partial signatures

Given a smooth Lagrangian path, both in the finite and in the infinite dimensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the Maslov cycle. For real-analytic paths, we give a formula for the computation of the Maslov index using the partial signatures; a similar formula holds for the spectral flow of real-analytic paths of Fredholm self-adjoint operators on real separable Hilbert spaces. As applications of the theory, we obtain a semi-Riemannian version of the Morse index theorem for geodesics with possibly conjugate endpoints, and we prove a bifurcation result at conjugate points along semi-Riemannian geodesics. To cite this article: R. Giambò et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Separating the solution sets of analytical and polynomial systems

A linear inequality system with infinitely many constraints is polynomial (analytical) if its index set is a compact interval of the real line and all its coefficients are polynomial (analytical, respectively) functions of the index on this interval. This paper provides an example of analytical system whose solution set cannot be the solution set of any polynomial system. Research supported by DGES of Spain and FEDER of UE, Grant BFM2002-04114-C02-01. Research supported by CONACyT of Mexico, Grant 130036. Research partially supported by CONACyT of Mexico, Grant 44003.     

Geometric formulas for Smale invariants of codimension two immersions

We give three formulas expressing the Smale invariant of an immersion f of a (4k−1)-sphere into (4k+1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k−1)-space.The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k−1)-sphere into (4k+1)-space then they are also non-regularly homotopic as immersions into (6k−1)-space. Moreover, any generic homotopy in (6k−1)-space connecting f to g must have at least a_k(2k−1)! cusps, where a_k=2 if k is odd and a_k=1 if k is even.     

Second Order Representations of the Least Absolute Deviation Regression Estimator

We consider exact weak and strong Bahadur-Kiefer representations of the least absolute deviation estimator for the linear regression model. The precise behavior of these representations is obtained under minimal conditions.     

Mimicking catalase and catecholase enzymes by copper(II)-containing complexes

An imidazolate-bridged copper(II)-zinc(II) complex (Cu(II)-diethylenetriamino-μ-imidazolato-Zn(II)-tris(2-aminoethyl)amine perchlorate (denoted as “Cu,Zn complex”) and a simple copper(II) complex (Cu(II)-tris(2-aminoethyl) amine chloride (“Cu-tren”) were prepared and immobilised on silica gel (by hydrogen or covalent bonds) and montmorillonite (by ion exchange). The immobilised substances were characterised by FT-IR spectroscopy and their thermal characteristics were also studied. The obtained materials were tested in two probe reactions: catalytic oxidation of 3,5-di-tert-butyl catechol (DTBC) (catecholase activity) and the decomposition of hydrogen peroxide (catalase activity). It was found that the catecholase activity of the Cu,Zn complex increased considerably upon immobilization on silica gel via hydrogen bonds and intercalation by ion exchange among the layers of montmorillonite. The imidazolate-bridged copper(II)-zinc(II) complex and its immobilised versions were inactive in hydrogen peroxide decomposition. The Cu(II)-tris(2-aminoethyl)amine chloride complex displayed good catalase activity; however, immobilisation could not improve it.