Discrete Entropies of Orthogonal Polynomials

Given a nontrivial Borel measure on ℝ, let p _n be the corresponding orthonormal polynomial of degree n whose zeros are λ _j ⁽ⁿ⁾, j=1,…,n. Then for each j=1,…,n,

Uniformly continuous maps between ends of $${\mathbb{R}}$$-trees

There is a well-known correspondence between infinite trees and ultrametric spaces which can be interpreted as an equivalence of categories and comes from considering the end space of the tree. In this equivalence, uniformly continuous maps between the end spaces are translated to some classes of coarse maps (or even classes of metrically proper Lipschitz maps) between the trees.     

Spatial mapping of price competition using logit-type market share models and store-level scanner-data

This paper proposes a methodology to obtain reliable spatial maps of price competition using store-level scanner data. Specifically, a procedure to obtain a symmetric matrix of similarities between brands considering their substitutability depending on price variations is proposed. The matrix is derived from a market response model where price cross-effects are split into two components. The first component accounts for the fact that price variation in one brand can have different effects to price variation in other brands (ie j → j′≠j′ → j). The second component accounts for the fact that the price of each brand can have different effects across competing brands (ie j → j′≠j → j^″). The matrix is obtained by imposing symmetry on this second component of price cross-effects. The parameterization of this symmetric matrix of similarities as the distances between the spatial representations of brands allows us to obtain the positioning maps. The proposed approach is illustrated through an empirical application.     

Improved scatter search for the global optimization of computationally expensive dynamic models

A new algorithm for global optimization of costly nonlinear continuous problems is presented in this paper. The algorithm is based on the scatter search metaheuristic, which has recently proved to be efficient for solving combinatorial and nonlinear optimization problems. A kriging-based prediction method has been coupled to the main optimization routine in order to discard the evaluation of solutions that are not likely to provide high quality function values. This makes the algorithm suitable for the optimization of computationally costly problems, as is illustrated in its application to two benchmark problems and its comparison with other algorithms.     

Hybrid heuristics for the maximum diversity problem

The maximum diversity problem presents a challenge to solution methods based on heuristic optimization. We undertake the development of hybrid procedures within the scatter search framework with the goal of uncovering the most effective designs to tackle this difficult but important problem. Our research revealed the effectiveness of adding simple memory structures (based on recency and frequency) to key scatter search mechanisms. Our extensive experiments and related statistical tests show that the most effective scatter search variant outperforms state-of-the-art methods.     

Distinct dynamic behaviors of water molecules in hydrated pores

Water molecules confined inside narrow pores are of great importance in understanding the structure, stability, and function of water channels. Here we report that besides the H-bonding water that structures the pore, the permanent presence of a significant, fast-moving fraction of incompletely H-bonded water molecules inside the pore should control the free entry and exit of water. This is achieved by means of complementary DSC and solid-state NMR studies. We also present compelling evidence from X-ray diffraction data that the cluster formed by six water molecules in the most stable cage-like structure is sufficiently hydrophobic to be stably adsorbed in a nonpolar environment.     

Dinuclear gold(I) isocyanide complexes with luminescent properties, and displaying thermotropic liquid crystalline behavior

Dinuclear gold(I) complexes [mu-(4,4'-CN-R-NC){Au(C6F4OC4H9)}2] [R = 1,4-phenylene, n = 8; R = 4,4'-biphenylene, 2,2'-dichloro-4,4'-biphenylene, 2,2'-dimethyl-4,4'-biphenylene, n = 4,6,8,10] have been prepared and their liquid crystal behavior and optical properties studied. Although the free ligands are not mesomorphic, all the gold(I) derivatives described, except the phenylisonitrilegold(I) derivative [mu-(1,4-CN-C6H4-NC){Au(C6F4OC8H17)}2], display liquid crystal behavior, giving rise to a nematic mesophase. The transition temperatures decrease in the order 4-4'-biphenylene > 2,2'-dichloro-4-4'-biphenylene > 2,2'dimethyl-4-4'-biphenylene. All compounds show photoluminescence in the solid state and in solution. The single-crystal X-ray diffraction structures of [mu-(4,4'-CN-R-NC){Au(C6F4OCnH2n+1)}2] (R = 4-4'-biphenylene and 2,2'-dichloro-4-4'-biphenylene) have been determined confirming the rodlike structure of the molecule, with a linear coordination around the gold atoms. There are Au...Au interactions in the 2,2'-dichlorobiphenyl derivative but not in the 4-4'-biphenyl compound.     

Dinuclear cyano-bridged CoIII-FeII complexes as precursors for molecular mixed-valence complexes of higher nuclearity

The preparation and characterization of a series of trinuclear mixed-valence cyano-bridged Co(III)-Fe(II)-Co(III) compounds derived from known dinuclear [[L(n)Co(III)(mu-NC)]Fe(II)(CN)(5)](-) complexes (L(n)() = N(5) or N(3)S(2) n-membered pendant amine macrocycle) are presented. All of the new trinuclear complexes were fully characterized spectroscopically (UV-vis, IR, and (13)C NMR). Complexes exhibiting a trans and cis arrangement of the Co-Fe-Co units around the [Fe(CN)(6)](4-) center are described (i.e., cis/trans-[{L(n)Co(III)(mu-NC)](2)Fe(II)(CN)(4)](2+)), and some of their structures are determined by X-ray crystallography. Electrochemical experiments revealed an expected anodic shift of the Fe(III/II) redox potential upon addition of a tripositively charged [Co(III)L(n)] moiety. The Co(III/II) redox potentials do not change greatly from the di- to the trinuclear complex, but rather behave in a fully independent and noncooperative way. In this respect, the energies and extinction coefficients of the MMCT bands agree with the formal existence of two mixed-valence Fe(II)-CN-Co(III) units per molecule. Solvatochromic experiments also indicated that the MMCT band of these compounds behaves as expected for a class II mixed-valence complex. Nevertheless, its extinction coefficient is dramatically increased upon increasing the solvent donor number.     

Isotope dilution-thermal ionisation mass spectrometric analysis for tin in a fly ash material

Isotope dilution-thermal ionisation mass spectrometry (ID-TIMS) analysis has been applied to the determination of tin in a fly ash sample supplied by the EC Joint Research Centre (Ispra, Italy). The proposed procedure includes the silica gel/phosphoric acid technique for tin thermal ionisation activation and a strict heating protocol for isotope ratio measurements. Instrumental mass discrimination factor has been previously determined measuring a natural tin standard solution. Spike solution has been prepared from ¹¹²Sn-enriched metal and quantified by reverse isotope dilution analysis. Two sample aliquots were spiked and tin was extracted with 4.5 M HCl during 25 min ultrasound exposure time. Due to the complex matrix of this fly ash material, a two-step purification stage using ion-exchange chromatography was required prior TIMS analysis. Obtained results for the two sample-spike blends (10.11 ± 0.55 and 10.50 ± 0.64 μmol g⁻¹) are comparable, both value and uncertainty. Also a good reproducibility is observed between measurements. The proposed ID-TIMS procedure, as a primary method and due to the lack of fly ash reference materials certified for tin content, can be used to validate more routine methodologies applied to tin determination in this kind of samples.