Radial ground states and singular ground states for a spatial-dependent p-Laplace equation

We consider the following equation

Δpu(x)+f(u,|x|)=0,     

On the Hamiltonian structure of Hirota‐Kimura discretization of the Euler top

This paper deals with a remarkable integrable discretization of the so (3) Euler top introduced by Hirota and Kimura. Such a discretization leads to an explicit map, whose integrability has been understood by finding two independent integrals of motion and a solution in terms of elliptic functions. Our goal is the construction of its Hamiltonian formulation. After giving a simplified and streamlined presentation of their results, we provide a bi‐Hamiltonian structure for this discretization, thus proving its integrability in the standard Liouville‐Arnold sense (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Realistic quantitative descriptions of electron transfer reactions: diabatic free-energy surfaces from first-principles molecular dynamics

A general approach to calculate the diabatic surfaces for electron-transfer reactions is presented, based on first-principles molecular dynamics of the active centers and their surrounding medium. The excitation energy corresponding to the transfer of an electron at any given ionic configuration (the Marcus energy gap) is accurately assessed within ground-state density-functional theory via a novel penalty functional for oxidation-reduction reactions that appropriately acts on the electronic degrees of freedom alone. The self-interaction error intrinsic to common exchange-correlation functionals is also corrected by the same penalty functional. The diabatic free-energy surfaces are then constructed from umbrella sampling on large ensembles of configurations. As a paradigmatic case study, the self-exchange reaction between ferrous and ferric ions in water is studied in detail.     

A One-Pot Entry to a Novel 3-H-benzo[d]pyrazolo[1,5-b]isothiazole Ring System

Summary The phosphorus ylide obtained from the reaction between 2-aminobenzo[d]isothiazol-3-one and dimethyl acetylenedicarboxylate in the presence of triphenylphosphine undergoes a smooth intramolecular Wittig-type reaction to produce, in a one-pot reaction, 3-H-benzo[d]pyrazolo[1,5-b]isothiazole-2,3a-dicarboxylic acid dimethyl ester, a novel functionalized heterocyclic compound.     

Scale anomaly as the origin of time

We explore the problem of time in quantum gravity in a point-particle analogue model of scale-invariant gravity. If quantized after reduction to true degrees of freedom, it leads to a time-independent Schrödinger equation. As with the Wheeler–DeWitt equation, time disappears, and a frozen formalism that gives a static wavefunction on the space of possible shapes of the system is obtained. However, if one follows the Dirac procedure and quantizes by imposing constraints, the potential that ensures scale invariance gives rise to a conformal anomaly, and the scale invariance is broken. A behaviour closely analogous to renormalization-group (RG) flow results. The wavefunction acquires a dependence on the scale parameter of the RG flow. We interpret this as time evolution and obtain a novel solution of the problem of time in quantum gravity. We apply the general procedure to the three-body problem, showing how to fix a natural initial value condition, introducing the notion of complexity. We recover a time-dependent Schrödinger equation with a repulsive cosmological force in the ‘late-time’ physics and we analyse the role of the scale invariant Planck constant. We suggest that several mechanisms presented in this model could be exploited in more general contexts.     

A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant Coefficients

We present a construction of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. The aim is to write detailed expressions for such functions. Such expressions are then exploited to prove regularity properties in the frame of Schauder spaces and jump properties of the corresponding single layer potentials.     

There and Back Again—The Journey of LiNiO2 as a Cathode Active Material

This Review provides a comprehensive overview of LiNiO₂ (LNO), almost 30 years after its introduction as a cathode active material. We aim to highlight the physicochemical peculiarities that make LNO a complex material in every aspect. We specifically stress the effect of the Li off‐stoichiometry (Li_1?zNi_1+zO₂) on every property of LNO, especially the electrochemical ones. The key instability issues that plague the compound and the strategies that have been implemented so far to overcome them are discussed in detail. Finally, the open questions that remain to be addressed by the scientific community are summarized, and the research directions that seem the most promising to enable LNO to be fully exploited are elucidated.