Synthesis of 2D Germanane (GeH): a New,Fast, and Facile Approach

Germanane (GeH), a germanium analogue of graphane, has recently attracted considerable interest because its remarkable combination of properties makes it an extremely suitable candidate to be used as 2D material for field effect devices, photovoltaics, and photocatalysis. Up to now, the synthesis of GeH has been conducted by substituting Ca by H in a β‐CaGe₂ layered Zintl phase through topochemical deintercalation in aqueous HCl. This reaction is generally slow and takes place over 6 to 14 days. The new and facile protocol presented here allows to synthesize GeH at room temperature in a significantly shorter time (a few minutes), which renders this method highly attractive for technological applications. The GeH produced with this method is highly pure and has a band gap (E_g) close to 1.4 eV, a lower value than that reported for germanane synthesized using HCl, which is promising for incorporation of GeH in solar cells.     

Totally non‐immune sets

Let be a countable first‐order language and be an ‐structure. “Definable set” means a subset of M which is ‐definable in with parameters. A set is said to be immune if it is infinite and does not contain any infinite definable subset. X is said to be partially immune if for some definable A, is immune. X is said to be totally non‐immune if for every definable A, and are not immune. Clearly every definable set is totally non‐immune. Here we ask whether the converse is true and prove that it is false for every countable structure whose class of definable sets satisfies a mild condition. We investigate further the possibility of an alternative construction of totally non‐immune non‐definable sets with the help of a subclass of immune sets, the class of cohesive sets, as well as with the help of a generalization of definable sets, the semi‐definable ones (the latter being naturally defined in models of arithmetic). Finally connections are found between totally non‐immune sets and generic classes in nonstandard models of arithmetic.     

The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.

     

On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces

Let be a real reflexive Banach space with dual and open and bounded and such that  Let be maximal monotone with and and with and A general and more unified eigenvalue theory is developed for the pair of operators  Further conditions are given for the existence of a pair such that

The ``implicit" eigenvalue problem, with in place of is also considered.  The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators No compactness assumptions have been made in most of the results.  The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators.  Applications to nonlinear partial differential equations are included.

     

An improved method for the computation of the Moore-Penrose inverse matrix

In this article we provide a fast computational method in order to calculate the Moore-Penrose inverse of singular square matrices and of rectangular matrices. The proposed method proves to be much faster and has significantly better accuracy than the already proposed methods, while works for full and sparse matrices.     

Sample Size for Testing Homogeneity of Two a Priori Dependent Binomial Populations Using the Bayesian Approach

This paper establishes new methodology for calculating the optimal sample size when a hypothesis test between two binomial populations is performed. The problem is addressed from the Bayesian point of view, with prior information expressed through a Dirichlet distribution. The approach of this paper sets an upper bound for the posterior risk and then chooses as “optimum ”the combined sample size for which the likelihood of the data does not satisfy this bound. The combined sample size is divided equally between the two binomials. Numerical examples are discussed for which the two proportions are equal to either a fixed or to a random value.