The dynamics of a triopoly Cournot game

This paper reconsiders the Cournot oligopoly (noncooperative) game with iso-elastic demand and constant marginal costs, one of the rare cases where the reaction functions can be derived in closed form. It focuses the case of three competitors, and so also extends the critical line method for non-invertible maps to the study of critical surfaces in 3D. By this method the various bifurcations of the attractors and their basins are studied. As a special case the restriction of the map to an invariant plane when two of the three firms are identical is focused.     

Discrete compactness property for quadrilateral finite element spaces

The main purpose of the present article is to prove the discrete compactness property for Arnold‐Boffi‐Falk spaces of any order. Results of numerical experiments confirming the theory are also reported. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Self-similarity of the bandcount adding structures: Calculation by map replacement

Recently it has been demonstrated that the domain of robust chaos close to the periodic domain, which is organized by the period-adding structure, contains an infinite number of interior crisis bifurcation curves. These curves form the so-called bandcount adding scenario, which determines the occurrence of multi-band chaotic attractors. The analytical calculation of the interior crisis bifurcations represents usually a quite sophisticated and cumbersome task. In this work we demonstrate that, using the map replacement approach, the bifurcation curves can be calculated much easier. Moreover, using this approach recursively, we confirm the hypothesis regarding the self-similarity of the bandcount adding structure.     

NMR spectra of Δ3- and Δ4-pyrrolin-2-one

The spectra of Δ³- and Δ⁴-pyrrolin-2-one were analysed and the sign of all the coupling constants determined by tickling and triple resonance experiments. A positive allylic interaction (J_xz in 2 ) is reported and four-bond couplings are discussed in particular. Deuterium exchange affords evidence for the tautomeric equilibrium between 1 and 2 .     

Structurally unstable regular dynamics in 1D piecewise smooth maps,and circle maps

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X′ = aX if ∣X∣ < z, X′ = bX if ∣X∣ > z, where a and b can take any real value, and may have several applications. We show that its dynamic behaviors are those of a linear rotation: either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X′ = F(X) if ∣X∣ < z, X′ = G(X) if ∣X∣ > z is also given, proving the conditions leading to a homeomorphism of the circle.     

Synthesis of nanostructured magnetic photocatalyst by colloidal approach and spray-drying technique

Nanostructured particles with a magnetic core and a photocatalytic shell are very interesting systems for their properties to be magnetically separable (and so reusable) in photocatalytic water depuration implant. Here, a robust, low time-consuming, easily scale up method to produce Fe₃O₄/SiO₂/TiO₂ hierarchical nanostructures starting from commercial precursors (i.e. Fe₃O₄, SiO₂) by employing a colloidal approach (i.e. heterocoagulation) coupled with the spray-drying technique is presented. In particular, a self-assembled layer-by-layer methodology based on the coagulation of dissimilar colloidal particles was applied. First, a passive layer of silica (SiO₂, amorphous) was created on magnetite in order to avoid detrimental phenomena arising from the direct contact between magnetite and titania, then the deposition of titania onto silica-coated-magnetite was promoted. TiO₂, SiO₂ and Fe₃O₄ nanosols were characterized in terms of zeta potential, optimized and a self-assembled layer-by-layer approach was followed in order to promote the heterocoagulation of silica onto magnetite surface and of titania onto silica coated magnetite. Once optimized the colloidal route, the mixture was then spray-dried to obtain a granulated powder with nano-scale reactivity, easier to handle and re-disperse in comparison to starting nanopowders with the same surface properties. The nanostructured particles have been characterized by different techniques such as SEM, TEM, XDR and their magnetic properties have been investigated. Moreover, preliminary photocatalytic texts have been performed.     

Convenient Synthesis of N-Benzyl-N′-acyl-piperazines

The importance of N,N′ asymmetrically disubstituted piperazines in pharmaceutical and medicinal chemistry does not need to be emphasized. Such a basic moiety is present in a number of drugs belonging to different classes, and especially in those of the central and peripheral nervous system. In recent times the interest connected with piperazines appears to be a fast growing topic in drug research.