Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn–Hilliard equation.     

Exploring the Phase-Selective,Green, Hydrothermal Synthesis of Upconverting Doped Sodium Yttrium Fluoride: Effects of Temperature,Time, and Precursors

The aim of this work was i) to develop a hydrothermal, low-temperature synthesis protocol affording the upconverting hexagonal phase NaYF₄ with suitable dopants while adhering to the “green chemistry” standards and ii) to explore the effect that different parameters have on the products. In optimizing the synthesis protocol, short reaction times and low temperatures (below 150 °C) were considered. Yb³⁺ and Er³⁺ ions were chosen as dopants for the NaYF₄ material. Within the context of the second goal, parameters including nature of the precursors, treatment temperature, and treatment time were investigated to afford a pure hexagonal crystalline phase, both in the doped and undoped materials. To fully explore the synthesis results, the prepared materials were characterized from a structural (XRD), compositional (XPS, ICP-MS), and morphological (SEM) point of view. The upconverting properties of the compounds were confirmed by photoluminescence measurements.     

Molecular Size and Electronic Structure Combined Effects on the Electrogenerated Chemiluminescence of Sulfurated Pyrene‐Cored Dendrimers

The electrochemistry, photophysics, and electrochemically generated chemiluminescence (ECL) of a family of polysulfurated dendrimers with a pyrene core have been thoroughly investigated and complemented by theoretical calculations. The redox and luminescence properties of dendrimers are dependent on the generation number. From low to higher generation it is both easier to reduce and oxidize them and the emission efficiency increases along the family, with respect to the polysulfurated pyrene core. The analysis of such data evidences that the formation of the singlet excited state by cation–anion annihilation is an energy‐deficient process and, thus, the ECL has been justified through the triplet–triplet annihilation pathway. The study of the dynamics of the ECL emission was achieved both experimentally and theoretically by molecular mechanics and quantum chemical calculations. It has allowed rationalization of a possible mechanism and the experimental dependence of the transient ECL on the dendrimer generation. The theoretically calculated Marcus electron‐transfer rate constant compares very well with that obtained by the finite element simulation of the whole ECL mechanism. This highlights the role played by the thioether dendrons in modulating the redox and photophysical properties, responsible for the occurrence and dynamics of the electron transfer involved in the ECL. Thus, the combination of experimental and computational results allows understanding of the dendrimer size dependence of the ECL transient signal as a result of factors affecting the annihilation electron transfer.