Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.Comment: 20 pages, 5 figures. Keywords: Charlier polynomials, Laguerre polynomials, Jacobi polynomials, asymptotic expansions, saddle point methods, two-points Taylor expansion

New Series Expansions of the Gauss Hypergeometric Function

The Gauss hypergeometric function ${}_2F_1(a,b,c;z)$ can be computed by using the power series in powers of $z, z/(z-1), 1-z, 1/z, 1/(1-z),(z-1)/z$. With these expansions ${}_2F_1(a,b,c;z)$ is not completely computable for all complex values of $z$. As pointed out in Gil, {\it et al.} [2007, \S2.3], the points $z=e^{\pm i\pi/3}$ are always excluded from the domains of convergence of these expansions. B\"uhring [1987] has given a power series expansion that allows computation at and near these points. But, when $b-a$ is an integer, the coefficients of that expansion become indeterminate and its computation requires a nontrivial limiting process. Moreover, the convergence becomes slower and slower in that case. In this paper we obtain new expansions of the Gauss hypergeometric function in terms of rational functions of $z$ for which the points $z=e^{\pm i\pi/3}$ are well inside their domains of convergence . In addition, these expansion are well defined when $b-a$ is an integer and no limits are needed in that case. Numerical computations show that these expansions converge faster than B\"uhring's expansion for $z$ in the neighborhood of the points $e^{\pm i\pi/3}$, especially when $b-a$ is close to an integer number.Comment: 18 pages, 6 figures, 4 tables. In Advances in Computational Mathematics, 2012 Second version with corrected typos in equations (18) and (19

Multi-point Taylor Expansions of Analytic Functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.Comment: 20 pages, 7 figures. Keywords: multi-point Taylor expansions, Cauchy's theorem, analytic functions, multi-point Laurent expansions, uniform asymptotic expansions of integral

Structures and Stabilities of Doubly-charged (MgO)nMg2+ (n=1-29) Cluster Ions

Ab initio perturbed ion plus polarization calculations are reported for doubly-charged nonstoichiometric (MgO)nMg2+ (n=1-29) cluster ions. We consider a large number of isomers with full relaxations of the geometries, and add the correlation correction to the Hartree-Fock energies for all cluster sizes. The polarization contribution is included at a semiempirical level also for all cluster sizes. Comparison is made with theoretical results for neutral (MgO)n clusters and singly-charged alkali-halide cluster ions. Our method is also compared to phenomenological pair potential models in order to asses their reliability for calculations on small ionic systems. The large coordination-dependent polarizabilities of oxide anions favor the formation of surface sites, and thus bulklike structures begin to dominate only after n=24. The relative stabilities of the cluster ions against evaporation of a MgO molecule show variations that are in excellent agreement with the experimental abundance spectra.Comment: Final version accepted in Journal of Chemical Physics; 8 pages plus 8 figures (6 GIFs and 2 PSs). The main difference with respect to the original submission is the inclusion of coordination-dependent polarizabilities for oxide anions. That results in substantial changes in the result

The Rothschild House business network in Spain as an example of entrepreneurial decision-taking and management structure.

In the period between 1835 and 1931 the Rothschild House established one of the most complex and influential business networks in Spanish economic history. This network controlled a wide range of activities, including financial services for the Spanish Government and the Bank of Spain, as well as the management of industrial, mining and railway firms. Instances of these firms are Rio Tinto Co., Société Minière et Metallurgique Peñarroya, MZA railway company and Deutsch & Cie, which was the main oil refining firm in Spain at the end of the 19th century.Spain 1835-1931, Rothschild, networks, Weisweiller, Bauer, foreign investments in Spain, railways, mining and refining companies, international raw material market, Public Finances, entrepreneurial function, agency problems, rent-seeking.

Consumption and habits : evidence from panel data

The purpose of this paper is to test for the presence of habit formation in consumption decisions using household panel data. We use the test proposed by Meghir and Weber (1996) and estimate the within -period marginal rate of substitution between commodities, which is robust to the presence of liquidity constraints. To that end, we use a Spanish panel data set in which households are observed up to eight consecutive quarters. This temporal dimension is crucial, since it allows us to take into account time invariant unobserved heterogeneity across households ("fixed effects") and, therefore, to investigate if the relationship between current and past consumption reflects habits or heterogeneity. Our results conf irm the importance of accounting for fixed effects when analyzing intertemporal consumption decisions allowing for time non-separabilities. Once fixed effects are controlled for and a proper set of instruments is used, the results yield supporting evidence of habit formation in the demand system of food at home, transport and services

A 2-set-up Routley-Meyer Semantics for the 4-valued Relevant Logic E4

The logic BN4 can be considered as the 4-valued logic of the relevant conditional and the logic E4, as the 4-valued logic of (relevant) entailment. The aim of this paper is to endow E4 with a 2-set-up Routley-Meyer semantics. It is proved that E4 is strongly sound and complete w.r.t. this semantics

CONSUMPTION AND HABITS: EVIDENCE FROM PANEL DATA

The purpose of this paper is to test for the presence of habit formation in consumption decisions using household panel data. We use the test proposed by Meghir and Weber (1996) and estimate the within -period marginal rate of substitution between commodities, which is robust to the presence of liquidity constraints. To that end, we use a Spanish panel data set in which households are observed up to eight consecutive quarters. This temporal dimension is crucial, since it allows us to take into account time invariant unobserved heterogeneity across households (“fixed effects”) and, therefore, to investigate if the relationship between current and past consumption reflects habits or heterogeneity. Our results conf irm the importance of accounting for fixed effects when analyzing intertemporal consumption decisions allowing for time non-separabilities. Once fixed effects are controlled for and a proper set of instruments is used, the results yield supporting evidence of habit formation in the demand system of food at home, transport and services.

Iridium-and Palladium-Based Catalysts in the Pharmaceutical Industry

Transition metal catalysts play a vital role in a wide range of industrial organic processes. The large-scale production of chemicals relying on catalyzed organic reactions represents a sustainable approach to supply society with end products for many daily life applications. Homogeneous (mainly for academic uses) and heterogeneous (crucial in industrial processes) metal-based catalysts have been developed for a plethora of organic reactions. The search for more sustainable strategies has led to the development of a countless number of metal-supported catalysts, nanosystems, and electrochemical and photochemical catalysts. In this work, although a vast number of transition metals can be used in this context, special attention is devoted to Ir-and Pd-based catalysts in the industrial manufacture of pharmaceutical drugs. Pd is by far the most widely used and versatile catalyst not only in academia but also in industry. Moreover, Ir-based complexes have emerged as attractive catalysts, particularly in asymmetric hydrogenation reactions. Ir-and Pd-based asymmetric reductions, aminations, cross-coupling reactions, and C–H activation are covered herein in the production of biologically active compounds or precursors; adaptation to bulk conditions is particularly highlighted.Gobierno de Canarias SD-19/02Ministerio de Ciencia e Innovación PID2020-116460RB-I00, 10.13039/50110001103