On the polynomial closure in a valued field

Let D be an integral domain with quotient field K. For any subset S of K, the D-polynomial closure of S is the largest subset T of K such that, for every polynomial f in K[X], if f maps S into D then f maps also T into D. When D is not local, the D-polynomial closure is not a topological closure. We prove here that, when D is any rank-one valuation domain, then there exists a topology on K such that the closed subsets for this topology are exactly the D-polynomially closed subsets of K.     

Solving Chisini’s functional equation

We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x) = F(G(x), . . . , G(x)), where F is a given function of n real variables. We provide necessary and sufficient conditions on F for the existence and uniqueness of solutions. When F is nondecreasing in each variable, we show in a constructive way that if a solution exists then a nondecreasing and idempotent solution always exists. We also provide necessary and sufficient conditions on F for the existence of continuous solutions and we show how to construct such a solution. We finally discuss a few applications of these results.     

Quasi-polynomial functions over bounded distributive lattices

In [6] the authors introduced the notion of quasi-polynomial function as being a mapping f : X ⁿ → X defined and valued on a bounded chain X and which can be factorized as ${f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))}In [6] the authors introduced the notion of quasi-polynomial function as being a mapping f : X ⁿ → X defined and valued on a bounded chain X and which can be factorized as f(x₁,?,x_n)=p(j(x₁),?,j(x_n)){f(x_1,\ldots,x_n)=p(\varphi(x_1),\ldots,\varphi(x_n))} , where p is a polynomial function (i.e., a combination of variables and constants using the chain operations ù{\wedge} and ú{\vee}) and j{\varphi} is an order-preserving map. In the current paper we study this notion in the more general setting where the underlying domain and codomain sets are, possibly different, bounded distributive lattices, and where the inner function is not necessarily order-preserving. These functions appear naturally within the scope of decision making under uncertainty since, as shown in this paper, they subsume overall preference functionals associated with Sugeno integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-polynomial functions, we propose several generalizations of well-established properties in aggregation theory, as well as show that some of the characterizations given in [6] still hold in this general setting. Moreover, we investigate the so-called transformed polynomial functions (essentially, compositions of unary mappings with polynomial functions) and show that, under certain conditions, they reduce to quasi-polynomial functions.     

Si epitaxial growth on LaAlO3(0 0 1)

We report on the growth of Si on c(2 × 2) reconstructed LaAlO₃(0 0 1) surfaces at high substrate temperature (700 °C) by molecular beam epitaxy. An initial Volmer-Weber mode is evidenced using reflection high energy electron diffraction (RHEED), X-ray photoelectron diffraction (XPD) and atomic force microscopy. After the deposition of a few monolayers, the islands coalesce. Using X-ray photoelectron spectroscopy, we demonstrate that Si islands exhibit an abrupt interface with the LaAlO₃ substrate without formation of silicate or silica. Finally, combined RHEED and XPD measurements show the epitaxial growth of Si with a unique Si(0 0 1)//LaAlO₃(0 0 1) and Si<1 0 0>//LAO<1 1 0> relationship.     

Contamination-free high capacity converging waves sonoreactors for the chemical industry

A new sonoreactor technology is presented here which should give a decisive impulse to sonochemistry in various areas of chemical processing. These exclusive systems use high power converging acoustic waves in a tube to produce a relatively large volume confined acoustic cavitation zone in flowing liquid reagents under pressure. It is well known that numerous chemical reactions are strongly accelerated when they take place inside such a zone. The new cylindrical sonoreactors do not contaminate the processed liquids with erosion products as most other devices do since the cavitation zone is maintained away from the wall of the tube. The processing conditions can be widely varied with pressure, power, temperature, and flow rate. The processing capacity of the largest models may be up to several tons per hour, depending on the required cavitation energy per unit volume to produce the desired process enhancement, using an electric power input of about 50 kW.     

Influence of the microstructure on the photooxidative degradation of poly(3-hexylthiophene)

The influence of the microstructure of poly(3-hexylthiophene) (P3HT) on its photodegradation upon exposure to UV-visible light was studied. High-regioregular (>98%) and low-regioregular (<95%) P3HTs with different molecular weights, dispersities and purities were submitted to irradiation under accelerated artificial ageing conditions. While the molecular weight had no effect on the photodegradation rate, a linear relationship was observed between the photodegradation rate and the regioregularity. The higher photo-instability of low-regioregular P3HTs, which have low crystalline phase content, could be due to the fact that the radical chain oxidation occurs predominantly in the amorphous phase of the polymer. Low-regioregular P3HTs also have a higher number of impurities, especially Fe residues, which are well known for their photocatalytic effect. The results also show that, at low concentrations, oxidation products are able to quench the singlet state of P3HT by acting as electron traps, and that the shortest wavelengths of solar light are the most harmful. Finally, the photooxidation of P3HT leads to the formation of low-molecular-weight carboxylic acids that can easily diffuse and migrate out of low-regioregular P3HT thin films. These results are therefore relevant to the preparation, storage and lifetimes of P3HT-based organic solar cells.     

Carbon Nanoparticle Production by Inductively Coupled Thermal Plasmas: Controlling the Thermal History of Particle Nucleation

The process control for reproducibility, uniformity, and achievement of desired structures for carbon black generated in thermal plasma devices is studied in this paper through modeling, and correlated with experimental results. A numerical simulation of the flow and energy fields, stream function lines and the quench rates of the plasma gas in a conical shape reactor at different pressures was made. An argon plasma is used with highly diluted methane (0.6–7%) as the carbon precursor. The quench rates were studied in order to observe the flow development and hence the thermal history of particle nucleation. Three pressure cases of 20.7, 55.2 and 101.3 kPa and two plasma powers cases of 10 and 20 kW were studied. The modeling results enabled carbon nanoflakes production in the experimental tests performed on an inductively coupled thermal plasma system. Results indicate a robust process control enabling very little particle morphology variation over this wide range of reactor pressure values and varying plasma power, and a very high reproducibility of the particle morphologies obtained.