Temporal exponential decay for the Stark effect in atoms

Let $\text{H}{}_1\text{= ?∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{(Δ}{}_{\mathrm{i}}\text{+ V(}\text{x}{}_{\mathrm{i}}\text{)) + ∑}{}_{\mathrm{1\; ?\; i\; <j\; ?\; N}}\text{¦}\text{x}{}_{\mathrm{i}}\text{?}\text{x}{}_{\mathrm{j}}\text{¦}{}^{\mathrm{?1}}\text{, V(}\text{x}{}_{\mathrm{i}}\text{) = N ∝ ¦}\text{x}\text{?}\text{y}\text{¦}{}^{\mathrm{?1}}\text{?(}\text{y}\text{)d}\text{y}$, with ? a normalized Gaussian. Suppose $\text{E}$ ≠ 0 and that $\text{H = H}{}_1\text{+}\text{E}\text{· (∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{N}}\text{x}{}_{\mathrm{i}}\text{)}$ has no eigenfunctions in L²($\text{R}$^3N. If H₁ψ = μψ with μ < infσ_ess(H₁), then (ψ, e^?itHψ) decays exponentially at a rate governed by the positions of the resonances of H.     

On Chelnokov-Zeitlin solutions to the vacuum Einstein equations

The equivalence of the Chelnokov-Zeitlin solutions to the vacuum Einstein equations with a special class of Lewis solutions is established in a direct way. Also, an oversight on the signature of the solutions is pointed out and corrected.     

Equidistributing principles in moving finite element methods

Recently Miller and his co-workers proposed a moving finite element method based on a least squares principle. This was followed by a similar method by the present authors using a Petrov—Galerkin approach. In this paper the two methods are compared. In particular, it is shown that both methods move their nodes according to an approximate equidistributing principle. This observation leads to a criterion for the placement of the nodes. It is also shown that the penalty function designed by Miller may also be used with the Petrov—Galerkin method. Finally, numerical examples are given, illustrating the performance of the two methods.     

Effect of Ni and Pd on the geometry, electronic properties, and active sites of copper clusters

The geometry, electronic properties, and active sites of copper clusters doped with Ni or Pd atoms, Cu(n)()(-)(1)M (n = 2-6; M = Ni, Pd) have been investigated using first-principles methods. Planar structures are energetically favorable in Cu(n)()(-)(1)Ni (n = 2-6). However, for Pd-doped clusters, three-dimensional structures are competitive in energy, and for n = 6, the most stable structure is not planar. Several properties of doped copper clusters present odd-even oscillations as the number of copper atoms grow. The different atomic ground-state configuration of Ni and Pd determines the bonding and electronic properties of doped copper clusters. The interaction between impurities and copper atoms can modify the chemical hardness and active sites of doped copper clusters markedly inducing directionality in the reactivity. This effect is relevant to the behavior of catalysts as well as in the growth of metallic films.     

Definition of a nucleophilicity scale

This work deals with exploring some empirical scales of nucleophilicity. We have started evaluating the experimental indices of nucleophilicity proposed by Legon and Millen on the basis of the measure of the force constants derived from vibrational frequencies using a probe dipole H-X (X = F,CN). The correlation among some theoretical parameters with this experimental scale has been evaluated. The theoretical parameters have been chosen as the minimum of the electrostatic potential V(min), the binding energy (BE) between the nucleophile and the H-X dipole, and the electrostatic potential measured at the position of the hydrogen atom V(H) when the complex nucleophile and dipole H-X is in the equilibrium geometry. All of them present good correlations with the experimental nucleophilicity scale. In addition, the BEs of the nucleophiles with two other Lewis acids (one hard, BF(3), and the other soft, BH(3)) have been evaluated. The results suggest that the Legon and Millen nucleophilicity scale and the electrostatic potential derived scales can describe in good approximation the reactivity order of the nucleophiles only when the interactions with a probe electrophile is of the hard-hard type. For a covalent interaction that is orbital controlled, a new nucleophilicity index using information of the frontier orbitals of both, the nucleophile and the electrophile has been proposed.     

Interior regularity results for zeroth order operators approaching the fractional Laplacian

In this article we are interested in interior regularity results for the solution \({\mu _ \in } \in C(\bar \Omega )\) of the Dirichlet problem

$$\{ _{\mu = 0in{\Omega ^c},}^{{I_ \in }(\mu ) = {f_ \in }in\Omega }$$

where Ω is a bounded, open set and \({f_ \in } \in C(\bar \Omega )\) for all ? ∈ (0, 1). For some σ ∈ (0, 2) fixed, the operator \(\mathcal{I}_{\in}\) is explicitly given by

$${I_ \in }(\mu ,x) = \int_{{R^N}} {\frac{{[\mu (x + z) - \mu (x)]dz}}{{{ \in ^{N + \sigma }} + |z{|^{N + \sigma }}}}} ,$$

which is an approximation of the well-known fractional Laplacian of order σ, as ? tends to zero. The purpose of this article is to understand how the interior regularity of u? evolves as ? approaches zero. We establish that u_? has a modulus of continuity which depends on the modulus of f_?, which becomes the expected Hölder profile for fractional problems, as ? → 0. This analysis includes the case when f? deteriorates its modulus of continuity as ? → 0.

     

A pH-tunable nanofluidic diode: electrochemical rectification in a reconstituted single ion channel

We report pH-dependent electrochemical rectification in a protein ion channel (the bacterial porin OmpF) reconstituted on a planar phospholipid membrane. The measurements performed at single-channel level show that the electric current is controlled by the protein fixed charge and it can be tuned by adjusting the local pH. Under highly asymmetric pH conditions, the channel behaves like a liquid diode. Unlike other nanofluidic devices that display also asymmetric conductance, here the microscopic charge distribution of the system can be explored by using the available high-resolution (2.4 A) channel crystallographic structure. Continuum electrostatics calculations confirm the hypothesized bipolar structure of the system. The selective titration of the channel residues is identified as the underlying physicochemical mechanism responsible for current rectification.     

On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator

In this article we study uniqueness of positive solutions for the nonlinear uniformly elliptic equation in RN, limr→∞u(r)=0, where denotes the Pucci's extremal operator with parameters 0<λ?Λ and p>1. It is known that all positive solutions of this equation are radially symmetric with respect to a point in RN, so the problem reduces to the study of a radial version of this equation. However, this is still a nontrivial question even in the case of the Laplacian (λ=Λ). The Pucci's operator is a prototype of a nonlinear operator in no-divergence form. This feature makes the uniqueness question specially challenging, since two standard tools like Pohozaev identity and global integration by parts are no longer available. The corresponding equation involving is also considered.     

Modeling the quantum evolution of the universe through classical matter

It is well known that the canonical quantization of the Friedmann–Lemaître–Robertson–Walker (FLRW) filled with a perfect fluid leads to nonsingular universes which, for later times, behave as their classical counterpart. This means that the expectation value of the scale factor $\left<a\right>(t)$ never vanishes and, as $t\rightarrow \infty $ , we recover the classical expression for the scale factor. In this paper, we show that such universes can be reproduced by classical cosmology given that the universe is filled with an exotic matter. In the case of a perfect fluid, we find an implicit equation of state (EoS). We then show that this single fluid with an implict EoS is equivalent to two non-interacting fluids, one of them representing stiff matter with negative energy density. In the case of two non-interacting scalar fields, one of them of the phantom type, we find their potential energy. In both cases we find that quantum mechanics changes completely the configuration of matter for small values of time, by adding a fluid or a scalar field with negative energy density. As time passes, the density of negative energy decreases and we recover the ordinary content of the classical universe. The more the initial wave function of the universe is concentrated around the classical big bang singularity, the more it is necessary to add negative energy, since this type of energy will be responsible for the removal of the classical singularity.