On some stochastic coalescents

We consider infinite systems of macroscopic particles characterized by their masses. Each pair of particles with masses x and y coalesce at a given rate K(x, y). We assume that K satisfies a sort of Hölder property with index λ ∈ (0,1], and that the initial condition admits a moment of order λ. We show the existence of such infinite particle systems, as strong Markov processes enjoying a Feller property. We also show that the obtained processes are the only possible limits when making the number of particles tend to infinity in a sequence of finite particle systems with the same dynamics.     

Coherent Reflection of Electromagnetic Plane Waves from Infinite Rough Slabs

We adopt the prospect of an observer interested to optimise the signal-to-noise ratio in the reception of the backward radiation coming from a surface illuminated by an electromagnetic wave with a wavelength chosen to minimize the diffuse scattering so that he has just to point his receiver in the direction of the coherent reflection. Then, to analyse the coherent reflection for harmonic plane waves impinging on a dielectric infinite film deposited on a metallic substrate we develop a formalism generalizing the customary angular spectrum representation used to tackle this kind of problem. This new approach whose efficiency is proved in the easier situation of a dielectric film endowed with an impedance, is used to get the coherent reflection from a structured 1D-dielectric film illuminated by TE and TM electromagnetic plane waves when the rough amplitude h is small enough to justify 0(h ²) approximations. The Idemen technique is used to get the boundary conditions needed to tackle these scattering problems.     

Solution to the initial boundary-value problem for the regularized Buckley-Leverett system

We solve the initial boundary-value problem for the regularized Buckley-Leverett system, which describes the flow of two immiscible incompressible fluids through a porous medium. This is the case of the flow of water and oil in an oil reservoir. The system is formed by a hyperbolic equation and an elliptic equation coupled by a vector field which represents the total velocity of the mixture. The regularization is done by means of a filter acting on the velocity field. We consider the critical situation in which we inject pure water into the reservoir. At this critical value for the water saturation, the spatial components of the characteristics of the hyperbolic equation vanish and this motivates the use of a new technique to prove the achievement of the boundary condition for the hyperbolic equation. We treat the case of a horizontal plane reservoir. We also prove that the time averages of the saturation component of the solution converge to one, as the time interval increases indefinitely, for almost all points of the reservoir, with a rate of convergence which depends only on the flux function.     

Degenerate two-phase incompressible flow II: regularity, stability and stabilization

In this paper, we analyze a coupled system of highly degenerate elliptic-parabolic partial differential equations for two-phase incompressible flow in porous media. This system involves a saturation and a global pressure (or a total flow velocity). First, we show that the saturation is Hölder continuous both in space and time and the total velocity is Hölder continuous in space (uniformly in time). Applying this regularity result, we then establish the stability of the saturation and pressure with respect to initial and boundary data, from which uniqueness of the solution to the system follows. Finally, we establish a stabilization result on the asymptotic behavior of the saturation and pressure; we prove that the solution to the present system converges (in appropriate norms) to the solution of a stationary system as time goes to infinity. An example is given to show typical regularity of the saturation.     

A local trace formula for resonances of perturbed periodic Schrödinger operators

We give a local trace formula for the pair (P₁(h)=P₀+W(hy),P₀), where P₀ is a periodic Schrödinger operator, W is a decreasing perturbation and h is a small positive parameter. We apply this result to establish the existence of ∼h⁻ⁿ resonances near some energy λ of σ(P₀).     

Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations

We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of isolated compact invariant sets. As the frequency of the oscillations tends to infinity, we prove that every isolated compact invariant set of the averaged autonomous equation can be continued to an isolated compact invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index can be explicitly computed and can be exploited to detect the existence of recurrent dynamics in the equation.     

Galois theory for bialgebroids, depth two and normal Hopf subalgebras

We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a leftT-Galois extension for some right finite projective left bialgebroid over some algebraR if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

---

Sunto Riduciamo alcune prove di [16,11,12] a quasibasi di profondità due da un lato solo, un approccio minimalistico che conduce ad una caratterizzazione di estensioni di Galois per bialgebroidi proietivi finiti senza la proprietà di estensione di Frobenius. Dimostriamo che un'algebra che sia un'estensione propria è un'estensioneT-Galois sinistra per qualche bialgebroide finito proiettivo a sinistra su qualche algebraR se, e solo se, è un'estensione di profondità due a sinistra e bilanciata a sinistra. Scambiando destra e sinistra nell'enunciato, otteniamo una caratterizzazione di estensioni di Galois destre per bialgebroidi finiti proiettivi a destra. Guardando ad esempi di profondità due, otteniamo che una sottoalgebra di Hopf è normale se, e solo se, è un'estensione Hopf-Galois. Caratterizziamo le estensioni Hopf-Galois deboli finite usando un'applicazione canonica di Galois alternativa ottenendo parecchi corollari: queste sono di profondità due e la suriettività dell'applicazione di Galois implica la sua biiettività.

     

Effective elastic moduli and stress concentrator factors in random structure aligned fiber composites

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically uniform random set of aligned fibers. Effective elastic moduli as well as the stress concentrator factors in the components are estimated. The micromechanical approach is based on the Green’s function technique as well as on the generalization of the “multiparticle effective field method” (MEFM, see for references, Buryachenko [1]). The refined version of the MEFM takes into account the variation of the effective fields acting on each pair of fibers. The dependence of effective elastic moduli and stress concentrator factors on the radial distribution function of the fiber locations is analyzed. Received: October 20, 2004     

Neither G9（Q） nor G11（Q） Is a Subgroup of K2（Q）

It is proved that neither G9(Q) nor G11(Q) is a subgroup of K2(Q) which confirms two special cases of a conjecture proposed by Browkin, J. (Lecture Notes in Math., 966, Springer-Verlag, New York, Heidelberg, Berlin, 1982, 1-6).     

Mob families and mad families

We show the consistency of where is the size of the smallest off-branch family, and is as usual the dominating number. We also prove the consistency of with large continuum. Here, is the unbounding number, and is the almost disjointness number. Received: September 12, 1996 / Revised version received: June 16, 1997