A stable Nyström interpolant for some Mellin convolution equations

We consider the numerical solution of second kind integral equations of the form $$u(y) - \int\limits_0^1 {k(y/x)\frac{{u(x)}}{x}dx = f(y), 0 \le y \le 1,} $$ for some given kernelk(t). These equations, usually indicated as of Mellin type, arise in a variety of applications. In particular, we examine a Nyström interpolant based on the following product quadrature rule: $$\int\limits_0^1 {k(y/x)\frac{{u(x)}}{x}dx \approx \sum\limits_{i = 0}^n {w_{ni} (y)u(x_{mi} ).} } $$ This rule is obtained by interpolatingu(x) by the Lagrange polynomial associated with the set of Gauss-Radau nodes {x _ni}. Under certain assumptions on the kernelk(t), we are able to prove the stability of our interpolant and derive convergence estimates.     

Lelong numbers of positive plurisubharmonic currents

A (k,k)-current T on an open subset of C^N is plurisubharmonic if is positive. Positive plurisubharmonic currents admit Lelong numbers; we prove here that they are independent on the coordinates system. Moreover, if Y is an analytic subset of pure dimension, then the Lelong numbers of T on Y are given by a non negative weakly plurisubharmonic function.     

The actions ofS n+1 andS n on the cohomology ring of a coxeter arrangement of typeA n−1

Let be the complexified Coxeter arrangement of hyperplanes of typeA _n−1. In this paper we construct anS _n+1 extension of the naturalS _n action on the complex cohomology ring of the complement ofA _n−1. Recurrence formulas connecting characters with respect to theS _n and theS _n+1 action are given.     

Di-2-pyridylketone semicarbazone as ligand in metal complexes: synthesis and X-ray crystal structure

A series of complexes of di-2-pyridylketone semicarbazone (Hdips) and Mn(II), Co(II), Co(III), Ni(II) and Cu(II) nitrates were synthesized and characterized by means of IR spectroscopy and for cobalt and nickel by X-ray crystal structures. The results are in agreement with the formulae: Mn(Hdips)₂(NO₃)₂·2H₂O, [Co(Hdips)₂](NO₃)₂·H₂O (I), [Ni(Hdips)₂](NO₃)₂·H₂O (II), Cu(Hdips)(NO₃)₂·2H₂O, [Co(dips)₂](NO₃)·2H₂O (III). The structure of I and II are monoclinic, space groupP2₁/c, with, I,a=15.980(4),b=11.531(2),c=16.170(2)Å;=104.20(2)°,Z=4,R=0.032; II,a=16.109(5),b=11.480(3),c=16.135(6)Å;=104.15(2)°,Z=4,R=0.069. Compound III is also monoclinic, space groupP2₁/c witha=12.173(5),b=15.619(5),c=15.338(8)Å;=111.40(4)°,Z=4,R=0.059. In these complexes the ligand is tridentate via carbonylic oxygen, semicarbazone and pyridine nitrogens forming each two five membered chelate rings with the metal in a distorted octahedral geometry.     

Intermolecular proton transfer in solid phase: a rare example of crystal-to-crystal transformation from hydroxo- to oxo-bridged iron(III) molecule-based magnet

Intermolecular proton transfer in solid phase from the hydroxo bridge to a water molecule occurs in a new mu-hydroxo iron(III) compound of formula {EtNH3[Fe2(ox)2Cl2(mu-OH)].2H2O}n leading to a still crystalline compound in which the mu-oxo bridge replaces the mu-hydroxo one. Both three-dimensional compounds exhibit magnetic ordering at Tc ca. 70 K due to a spin canting.     

Gadolinium(III) Complexes of dota‐Derived N‐Sulfonylacetamides (H4(dota‐NHSO2R)=10‐{2‐[(R)sulfonylamino]‐2‐oxoethyl}‐1,4,7,10‐tetraazacyclododecane‐1,4,7‐triacetic Acid): A New Class of Relaxation Agents for Magnetic Resonance Imaging Applications

Four new ligands for lanthanide ions based on the H₃do3a (=1,4,7,10‐tetraazacyclododecane‐1,4,7‐triacetic acid) structure and bearing one N‐sulfonylacetamide arm were synthesized, i.e., H₄dota‐NHSO₂R=10‐{2‐[(R)sulfonylamino]‐2‐oxoethyl}‐1,4,7,10‐tetraazacyclododecane‐1,4,7‐triacetic acids 1a – e . A ¹⁵N‐NMR study of the ¹⁵N‐labelled Eu³⁺ complex of one such ligands, 1d , showed that the coordination of the N‐sulfonylacetamide arm involves the carbonyl O‐atom rather than the N‐atom. The relaxometric properties of the corresponding Gd³⁺ complexes were investigated as a function of pH and temperature. These complexes have relaxivities in the range 4.5–5.3 mM ^?1 s^?1, at 20 MHz and 25°, and are characterized by a single H₂O molecule in their inner coordination sphere. The mean residence lifetime of this molecule is relatively long (500–700 ns) compared to other anionic complexes. The slow rate of H₂O exchange can be justified by the extensive delocalization of the negative charge on the N‐sulfonylacetamide arm. The long residence time of the coordinated H₂O allowed the observation of the effect of the prototropic exchange on the relaxivity. The study of the interaction between the complex [Gd( 1e )]‐ and HSA revealed a weak affinity constant highlighting the importance of a localized negative charge on the complex to promote a strong interaction with the protein.     

N,O-heterocyclics. 13. Isoxazolidinium salts as synthons to N,N, O-trisubstituted hydroxylamines

Substituted isoxazolidinium salts react with lithium aluminium hydride to yield open-ring products which have hydroxylamine structures. The bimolecular reaction-mechanism has been investigated by substituent effect and the structure of the products ascertained by spectroscopic methods with the aid of the MIKE technique. The overall process of the ring-opening substitution is controlled by the polarisation of the C? N bond with steric and conformational factors acting mainly at the C-5 position of the nucleus. The mechanism of isoxazolidinium ion reaction defines the use of these synthons towards the synthesis of N,N,O-trisubstituted hydroxylamines and substituted 1,3-amino-alcohols.     

Infrared active fundamental vibrations of TCNQ and chloranil monovalent anions

The interpretation of the infrared active fundamentals of TCNQ and chloranil monovalent anions is given for the first time. The rationalization of the infrared spectra was made possible by the recently achieved vibrational assignment of the corresponding neutral molecules and is mainly based on a set of judicious adjustments of their empirical valence force fields.