A reduction approach for determining generalized simple waves

A reduction approach is developed in order to construct generalized simple wave solutions to quasilinear nonhomogenous hyperbolic systems of first order PDEs. The solutions sought must possess a special ansatz which permits time-evolution of the profile of a simple wave due to a source-like term. These solutions involve a free function which can be used to fit classes of initial or boundary value problems. By means of the proposed approach two governing models of interest in a variety of applications are investigated. Model constitutive laws consistent with the full reduction process are obtained and the occurence of singularities at a finite time for the resulting solutions is analysed. Furthermore a comparison is made between the results obtained within the present theoretical framework and the standard simple wave solutions of the corresponding homogeneous (source free) governing models.      

Equivariant entire solutions to the elliptic system \Delta u=W_u(u) for general G-invariant potentials

We consider the system $\Delta u - W_u(u) = 0$ , where $u: \mathbb R ^n \rightarrow \mathbb R ^m$ , for potentials $W: \mathbb R ^m \rightarrow \mathbb R $ that possess $N$ global minima and are invariant under a finite reflection group $G$ . We prove the existence of nontrivial $G$ -equivariant entire solutions connecting the $N$ minima of $W$ . Our proof only requires the minima of $W$ to be nondegenerate and an assumption on the behavior of $W$ for large $u$ .     

Differential Constraints and Exact Solution to Riemann Problems for a Traffic Flow Model

In view of solving in a closed form initial and/or boundary value problems of interest in nonlinear hyperbolic and dissipative wave processes it is considered a reduction approach based upon appending differential constraints to quasilinear nonhomogeneous hyperbolic systems of first order PDEs. In this context a governing model of traffic flow is analyzed thoroughly and a classification of possible constraints along with sets of consistent response functions involved therein is worked out whereupon the classes of corresponding exact solutions are determined. To some extent these solutions generalize the classical simple wave solutions and may also incorporate dissipative effects. Furthermore, in order to solve a Riemann Problem, an exact rarefaction wave-like solution is built. Finally an application of the results to the so-called ??green traffic light problem?? is also illustrated.     

Pulselike Exact Solutions for a Model Describing Nerve Fibers

A quasi-linear nonhomogeneous first order hyperbolic system describing nerve pulse transmission is considered. By requiring the compatibility of the governing equations with quasi-linear differential constraints, exact solutions to the model in question are determined. Furthermore classes of material response functions amenable to the mathematical approach are characterized. Initial and/or boundary value problems of interest in nerve pulse propagation are solved. It is proved that the governing model admits solutions which describe a "space clamp" situation and the propagation of a localized action potential pulse along the nerve.     

Integrated spatial assessment: a multicriteria approach to sustainable development of cultural and environmental heritage in San Marco dei Cavoti,Italy

Starting from the concept of “territory” and from the definition of “integrated assessment”, the paper defines the operational dimensions of spatial integrated assessment (ISA), applying it to the territorial context of San Marco dei Cavoti (a rural village in Southern Italy). In particular, the paper proposes an integration between analytic hierarchy process (AHP) and geographic information system (GIS) for the strategies definition of planning choices, recognizing the relevant role of the environmental aspects in the decision-making process and alternatives selection.      

The Disproportionation of [Ni(tacn)]2+ in Ni2+ and [Ni(tacn)2]2+ Crystallographically Demonstrated (tacn=1,4,7‐Triazacyclononane)

From an EtOH/H₂O solution, 0.3 M each of Ni²⁺ and cyclic triamine tacn ( 2 ; tacn=1,4,7‐triazacyclononane), after 10 min refluxing and cooling at room temperature, copious and comparable amounts of blue crystals (containing the complex [Ni(tacn)(H₂O)₃]²⁺) and pink crystals (containing in the same cell both [Ni(H₂O)₆]²⁺ and [Ni(tacn)₂]²⁺) precipitated. This unusual behaviour is ascribed to the fact that at the refluxing temperature the three species are present at the equilibrium in similar concentrations, which are frozen on cooling, due to the inertness of the macrocyclic complexes [Ni(tacn)(H₂O)₃]²⁺ and [Ni(tacn)₂]²⁺.     

Asymmetric Catalysis on the Nanoscale: The Organocatalytic Approach to Helicenes

The first asymmetric organocatalytic synthesis of helicenes is reported. A novel SPINOL‐derived phosphoric acid, bearing extended π‐substituents, catalyzes the asymmetric synthesis of helicenes through an enantioselective Fischer indole reaction. A variety of azahelicenes and diazahelicenes could be obtained with good to excellent yields and enantioselectivities.     

The Equations of Ostwald Ripening for Dilute Systems

We consider a dilute mixture in 3D of a finite number of particles initially close to spherical, but of varying sizes, and representing one of the phases evolving according to the quasistatic dynamics. Under the scaling hypotheses that (1) typical size/typical distance and (2) deviation from sphericity/typical size are small, we associate centers and radii to each particle for the whole evolution and derive rigorously a set of ODEs fo the radii which we relate to the Lifschitz–Slyosov–Wagner theory of coarsening.