A $\Gamma$-convergence result for doubling metric measures and associated perimeters

Abstract. In this paper we study the notion of perimeter associated with doubling metric measures or strongly weights. We prove that the metric perimeter in the sense of L. Ambrosio and M. Miranda jr. coincides with the metric Minkowski content and can be obtained also as a -limit of Modica-Mortola type degenerate integral functionals. Received: 27 August 2001 / Accepted: 29 November 2001 / Published online: 10 June 2002 Investigation supported by University of Bologna, funds for selected research topics and by GNAMPA of INdAM, Italy. The authors are very grateful to Luigi Ambrosio and Francesco Serra Cassano for making their preprints available to them, for listening with patience and for many unvaluable suggestions.     

Error estimate for a stabilised domain decomposition method with nonmatching grids

Summary. In this paper, we analyse a stabilisation technique for the so-called three-field formulation for nonoverlapping domain decomposition methods. The stabilisation is based on boundary bubble functions in each subdomain which are then eliminated by static condensation. The discretisation grids in the subdomains can be chosen independently as well as the grid for the final interface problem. We present the analysis of the method and we construct a set of bubble functions which guarantees the optimal rate of convergence. Received May 12, 1998 / Revised version received November 21, 2000 / Published online June 7, 2001     

Synthesis,characterization and coordination chemistry of the new tetraazamacrocycle 4,10-dimethyl-1,4,7,10-tetraazacyclododecane-1,7-bis(methanephosphonic acid monoethyl ester) dipotassium salt

A new potentially hexadentate tetraazamacrocycle based on the cyclen skeleton has been synthesized and fully characterized. The macrocycle 4,10-dimethyl-1,4,7,10-tetraazacyclododecane-1,7-bis(methanephosphonic acid monoethyl ester) dipotassium salt (Me2DO2PME) contains mutually trans monoethyl ester phosphonate acid substituents on two nitrogen atoms, and trans methyl substituents on the other two nitrogen atoms. The protonation constants of this macrocycle and the stability constants of its complexes with Cu2+, Zn2+, Gd3+ and Ca2+ ions have been determined by pH potentiometric titrations. The protonation sequence of the macrocycle has been studied by 1H, 31P[1H] and 13C[1H] NMR spectroscopy: the first and second protonation steps take place at the methyl-substituted nitrogen atoms, while the third protonation involves one oxygen from a phosphonate group. Upon protonation, all the CH2 ring protons become magnetically inequivalent on the NMR time scale due to a slow conformational rearrangement, most likely occasioned by the formation of multiple hydrogen bonds within the macrocyclic ring. Me2DOPM forms neutral, mononuclear complexes with all the metals investigated. The presence of hydroxo complexes was observed for Ca2+ and Zn2+ at high pH values. Structural information on the neutral complex [Cu(Me2DO2PME)] has been obtained by a solution X-Band EPR study. It is proposed that Me2DO2PME binds Cu2+ in a distorted octahedral structure using all of its donor atoms, i.e. the four nitrogen atoms and the two phosphonate oxygen atoms. The redox chemistry of [Cu(Me2DO2PME)] in dimethyl sulfoxide and water has been studied by electrochemical measurements. Cyclic voltammetry in DMSO shows the complex to undergo a quasireversible one-electron reduction step leading to an unstable CuI species.     

Single electron traps at the surface of polycrystalline MgO: assignment of the main trapping sites

Paramagnetic centers at the surface of ionic oxides in the form of trapped electrons can be generated by exposure of the material to alkali metal or hydrogen atoms or of molecular hydrogen under UV irradiation. For many years, it has been assumed that the resulting paramagnetic centers consist of oxygen vacancies filled by one electron. High-resolution electron spin resonance spectra and ab initio quantum chemical calculations show that the paramagnetic centers consist of (H(+))(e(-)) electron pairs formed at morphological irregularities of the surface. At least three different kinds of (H(+))(e(-)) centers, [A], [B], and [C], have been identified with abundances of 80%, 10%, and 8%, respectively. In this work, we compare a wide set of measured and computed g-factors and hyperfine coupling constants of the unpaired electron with the surrounding (25)Mg, (17)O, and (1)H nuclei and we propose a general assignment of the centers. (H(+))(e(-)) pairs formed at Mg(4c) ions at steps and edges account for species [A], centers formed at Mg(4c) ions at reverse corners correspond to species [B], and species [C] originates from (H(+))(e(-)) pairs formed at Mg(3c) ions at corners and kinks.     

A new tripodal receptor for molecular recognition of monosaccharides. A paradigm for assessing glycoside binding affinities and selectivities by 1H NMR spectroscopy

A new tripodal receptor for the recognition of monosaccharides is described. The prototypical host 1 features a 1,3,5-substituted 2,4,6-triethylbenzene scaffold bearing three convergent H-bonding units. The binding ability of the t-octyl derivative 1a toward a set of octylglycosides of biologically relevant monosaccharides, including Glc, Gal, Man, and GlcNAc, was investigated by 1H NMR in CDCl3. A protocol for the correct evaluation of binding affinities was established, which can be generally applied for the recognition of monosaccharides by 1H NMR spectroscopy. A three-constant equilibrium model, including 1:1 and 2:1 host-guest association and dimerization of the receptor, was ascertained for the interaction of 1a with all the investigated glycosides. An affinity index, which we defined median binding concentration BC50 in analogy to the IC50 parameter, intended to address the general issue of comparing dimensionally heterogeneous binding data, and a limiting BC0(50)quantity describing intrinsic binding affinities were developed for evaluating the results. BC0(50) values for 1a range from 1 to 6 mM, indicating an intrinsic binding affinity in the millimolar range and a selectivity factor of 5 toward the investigated glycosides. The treatment has been extended to include any generic host-guest system involved in single or multiple binding equilibria.     

Miniquad-A general computer programme for the computation of formation constants from potentiometric data

A new computer programme for the computation of formation constants of complex species in solution is presented. The programme can be applied to all kinds of potentiometric titration data, including multi-reactant and multi-electrode systems. A statistical analysis is made of the results in order to assess their validity, and to assist in hypothesis testing.     

Die Komplexe der Hexamine «penten» und «ptetraen» Thermodynamik ihrer Bildung in wässeriger Lösung

The two hexamines (H₂N? CH₂? CH_{2? )2}N? (CH_2)n? N(? CH₂? CH₂? NH₂₎₂, «penten» (n = 2) and «ptetraen» (n=3) have been investigated as chelating agents for COIII (preparative Study) and some of the divalent metal ions (potentiometric and calorimetric studies). Both amines function as sexadentate ligands for Co^III, Co^II and Ni^II, but one of the terminal aminogroups is much easier detached from the metal in case of M (penten)^v+ than in case of M(ptetraen)^v+, thus revealing more strain in the fivemembered chelate rings of the girdle plane of the «penten» complexes. On the other hand, the sixmembered chelate ring in M(ptetraen)^v+ is more strained than the five-membered ring comprising the tertiary nitrogen atoms of M(penten)^v+ Cu^II and Zn^II coordinate with both ligands only 5 of the 6 basic nitrogen atoms present. Both hexaamines function as sexadentates again with Mn^II, but the metal is coordinated with a molecule of water in addition to the 6 nitrogen atoms in the «penten» complex in contrast to the «ptetraen» complex. The thermodynamic functions for the protonation of the hexamines and for the addition of metal ions in aqueous solution are understood in almost every detail. The dielectric shielding of the charges of the reactants exerted by the solvent has to be taken into account; it is reduced by electrostriction as well as by an increase in temperature. It is shown that the approach of charges of equal sign often is an exothermic process.     

M(R-dmet)2] bis(1,2-dithiolenes): a promising new class intermediate between [M(dmit)2] and [M(R,R'-timdt)2] (M = Ni, Pd, Pt)

We report the first examples of metal dithiolenes belonging to the class [M(R-dmet)(2)] [R-dmet = formally monoreduced N-substituted thiazolidine-2,4,5-trithione; R = Et, M = Ni (1), Pd (2), Pt (3)]. A comparative spectroscopic, electrochemical, and density functional theory theoretical investigation indicates that [M(R-dmet)(2)] complexes show features intermediate between those of the dithiolenes belonging to the previously reported classes [M(R,R'-timdt)(2)] and [M(dmit)(2)] (R,R'-timdt = formally monoreduced N,N'-disubstituted imidazolidine-2,4,5-trithione; dmit = 2-thioxo-1,3-dithiole-4,5-dithiolato). UV-vis-near-IR spectroscopy and cyclic voltammetry/differential pulsed voltammetry measurements performed on 1 and 3 proved that the new dithiolenes are stable as neutral, monoanionic, and bianionic species and feature a near-IR electrochromic absorption falling at about 1000 and 1250 nm for neutral and monoanionic species, respectively.     

Beryllium(II) complexes of the Kläui tripodal ligand cyclopentadienyltris(diethylphosphito-p)cobaltate(-)

The interactions of the beryllium(II) ion with the cyclopentadienyltris(diethylphosphito-P)cobaltate monoanion, L(-), have been investigated, in aqueous solution, by synthetic methods, potentiometry, ESMS, and (1)H, (31)P, and (9)Be NMR spectroscopy. L(-) has been found able to displace either two or three water molecules in the beryllium(II) coordination sphere, to form mononuclear, dinuclear, and trinuclear derivatives, in which the metal ion is pseudotetrahedrally coordinated. The species [BeL(H(2)O)](+) and [Be(2)L(2)(mu-OH)](+) have been identified in solution while complexes of formula BeL(2) and [Be(3)L(4)](ClO(4))(2) have been isolated as solid materials. The species [BeL(OPPh(2))](+), closely related to [BeL(H(2)O)](+), has been characterized in acetone solution and isolated as tetraphenylborate salt. The structure of the unusual trimeric complex [Be(3)L(4)](2+) has been elucidated by an unprecedented 2D (9)Be-(31)P NMR correlation spectrum showing the presence of a single central beryllium nucleus and two equivalent terminal beryllium nuclei. The three beryllium centers are held together by four cobaltate ligands, which display two different bonding modes: two ligands are terminally linked with all the three oxygen donors to one terminal beryllium, and the other two bridge two metal centers, sharing the oxygen donors between central and terminal beryllium atoms.     

Synthesis and coordination properties of new macrocyclic ligands: equilibrium studies and crystal structures

The synthesis and characterization of two new macrocyclic ligands, the bis-macrocyclic compound 2,6-bis(1,4,13-triaza-7,10-dioxacyclopentadec-1-ylmethyl)phenol (L) and 38-methoxy-1,4,13,16,19,28-hexaaza-7,10,22,25-tetraoxatricyclo[14.14.7.1(32,36)]octatriconta-32,34,Delta(36,38)-triene (L1) are reported. Equilibrium studies of basicity and coordination properties toward metal ions such as Cu(II), Zn(II), Cd(II) and Pb(II) were performed for ligand by potentiometric measurements in aqueous solution (298.1 +/- 0.1 K, I= 0.15 mol dm(-3)). L behaves as a hexaprotic base (logK(1)= 10.93, logK(2)= 9.70, logK(3)= 8.79, logK(4)= 8.05, logK(5)= 6.83, logK(6)= 2.55). All metal ions form stable mono- and dinuclear complexes: logK(MLH(-1))= 25.61 for Cu(II), 15.37 for Zn(II), 12.58 for Cd(II) and 13.79 for Pb(II); logK(M(2)LH(-1))= 31.61 for Cu(II), 23.38 for Zn(II), 24.49 for Cd(II) and 23.68 for Pb(II). All these dinuclear species show a great tendency to add the OH(-) group: the equilibrium constant for the addition reaction was found to be logK(M(2)LH(-1)OH)= 4.77 for Cu(II), 5.66 for Zn(II), 2.8 for Cd(II) and 3.18 for Pb(II). In the case of Ni(II), kinetic inertness prevents the possibility of solution studies. The dinuclear solid adducts [Ni(2)H(-1)L(N(3))(3)].EtOH and [Cu(2)H(-1)L(N(3))](ClO(4))(2) were characterized by X-ray analysis.