Randomness and Nonlinear Evolution Equations

In this paper we survey some results on existence, and when possible also uniqueness, of solutions to certain evolution equations obtained by injecting randomness either on the set of initial data or as a perturbative term.     

Local well-posedness for higher order nonlinear dispersive systems

We study nonlinear dispersive systems of the form

Semilinear Schrödinger flows on hyperbolic spaces: scattering in <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>

We prove global well-posedness and scattering in H ¹ for the defocusing nonlinear Schrödinger equations

$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$

on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H ¹ is not known for any exponent \({\sigma \in (1/d, 2/d]}\) and is known to fail for \({\sigma \in (0, 1/d]}\). Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

     

Coordination properties of sulfonyl-N-aminoacids: Crystal and molecular structure of the [Zn(II) (N-(p-toluenesulfonyl)-L-glutaminate)2(H2O)2] complex

The coordination behavior of N-tosyl-L-glutamine (hereafter abbreviated as tsglnH, tosyl =toluenesulfonyl group) toward the Zn(II) ion is investigated by means of structural and spectroscopic techniques. From aqueous solution, at acidic pH the Zn(II) ion separates a compound of empirical formula Zn(tsgln)₂·2H₂O whose crystal structure is determined. The crystal is monoclinic, space group C2, with cell dimensions:a=19.775(5),b=5.3146(3),c=16.543(8) Å, =119.93(3)°,Z=2. The Zn(II) is tetrahedrally coordinated by two symmetry-related tsgln monoanions acting as monodentate carboxylate ligands and two symmetry related water molecules. The Zn–O bond distances are similar while the angles at Zn(II) ion differ significantly from those of regular tetrahedron. The side chain amide group does not participate in metal coordination and forms intermolecular hydrogen bonds.     

Thermodynamic aspects of the adsorption of hexametaphosphate on kaolinite

The adsorption of hexametaphosphate ion, an important deflocculant used in the ceramic industry, from aqueous solutions onto kaolinite has been studied at different temperatures. The adsorption isotherm follows the Langmuir model: the thermodynamic parameters DeltaG(ads)(0), DeltaH(ads)(0), and DeltaS(ads)(0) were calculated and found to be consistent with an interaction model involving the formation of an inner-sphere complex between HMP and aluminol groups. Also, the dependence of the adsorption behavior on the kaolinite volume fraction has been studied and discussed in term of association processes between the clay particles.     

A computational tool for the prediction of crystalline phases obtained from controlled crystallization of glasses

An automatic tool (named CLUSTER) for the prediction of the most probable crystal phases that can separate from glasses has been developed. The program analyzes the output of molecular dynamics simulations of glasses or glass ceramics, systematically sampling the ratios of the ions in different portions of the simulation box and comparing them to the stoichiometric ratio of compositionally equivalent crystalline phases retrieved from a crystal structure database. The efficacy of the similarity index elaborated has been judged by comparing the results obtained with the crystal phases identified by XRD analysis after thermal treatment in a series of multicomponent potential bioactive glasses and glass ceramics for which the advantages of rational-designed erosion-controlled release is straightforward.     

Nonconventional synthesis and characterization of ultrahigh-molar-mass polyacrylamides

In spite of the significant progresses in the field of replaceable sieving matrices for separating DNA in capillary electrophoresis (CE), an intense research activity is still going on to improve the separation of large size DNA sequencing fragments. There are evidences, both from experimental and theoretical sides that the resolution of these fragments, at the single base, requires the use of sieving matrices comprised of long chain linear polymers. In the separation of DNA fragments by CE are of upmost importance: (i) the complete solubility of the polymer, (ii) the linearity of the chain, (iii) the achievement of ultrahigh viscosity in dilute solutions. The aim of this work is the synthesis of ultrahigh-molecular-weight polymers which possess the three requirements mentioned above by employing a nonconventional method. We demonstrate that the sieving performance of polyacrylamide is directly correlated to its intrinsic viscosity.     

Randomization and the Gross–Pitaevskii Hierarchy

Archive for Rational Mechanics and Analysis - We study the Gross–Pitaevskii hierarchy on the spatial domain $${\mathbb{T}^3}$$ . By using an appropriate randomization of the Fourier...     

On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy

In this paper, we present a uniqueness result for solutions to the Gross–Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear Schrödinger equation. In this way, we obtain a periodic analogue of the uniqueness result on R³${\mathbb{R}}^3$ previously proved by Klainerman and Machedon [75], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H^α$H^{\alpha }$ for α>1$\alpha >1$. By constructing a specific counterexample, we show that, on T³${\mathbb{T}}^3$, the existing techniques from the work of Klainerman and Machedon approach do not apply in the endpoint case α=1$\alpha =1$. This is in contrast to the known results in the non-periodic setting, where these techniques are known to hold for all α?1$\alpha ?1$. In our analysis, we give a detailed study of the crucial spacetime estimate associated to the free evolution operator. In this step of the proof, our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice. This method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius R, depending on the number-theoretic properties of the normal vector to the plane. We are hence able to obtain a sharp range of admissible Sobolev exponents for which the spacetime estimate holds.