A New Minimum Principle for Lagrangian Mechanics

We present a novel variational view at Lagrangian mechanics based on the minimization of weighted inertia-energy functionals on trajectories. In particular, we introduce a family of parameter-dependent global-in-time minimization problems whose respective minimizers converge to solutions of the system of Lagrange’s equations. The interest in this approach is that of reformulating Lagrangian dynamics as a (class of) minimization problem(s) plus a limiting procedure. The theory may be extended in order to include dissipative effects thus providing a unified framework for both dissipative and nondissipative situations. In particular, it allows for a rigorous connection between these two regimes by means of Γ-convergence. Moreover, the variational principle may serve as a selection criterion in case of nonuniqueness of solutions. Finally, this variational approach can be localized on a finite time-horizon resulting in some sharper convergence statements and can be combined with time-discretization.     

Weighted energy-dissipation functionals for doubly nonlinear evolution

This paper is concerned with the Weighted Energy-Dissipation (WED) functional approach to doubly nonlinear evolutionary problems. This approach consists in minimizing (WED) functionals defined over entire trajectories. We present the features of the WED variational formalism and analyze the related Euler-Lagrange problems. Moreover, we check that minimizers of the WED functionals converge to the corresponding limiting doubly nonlinear evolution. Finally, we present a discussion on the functional convergence of sequences of WED functionals and present some application of the abstract theory to nonlinear PDEs.     

Amination reaction on copper and germanium β-nitrocorrolates

Copper and germanium complexes of β-substituted nitrocorroles were reacted with 4-amino-4H-1,2,4-triazole to give the corresponding β-amino-β-nitro derivatives, in moderate to good yields. This is the first successful example of a vicarious nucleophilic substitution performed on corrole derivatives, because the same reaction carried out on silver complexes afforded the corresponding 6-azahemiporphycenes by way of corrole ring expansion. The first step of this work is related to the modification of a synthetic protocol for preparation of the β-substituted nitro corroles. The nitration reaction was carried out on a copper corrole using NaNO(2) as the primary source of NO(2)(-) coupled with AgNO(2) used as oxidant. By variation of the molar ratio of the reagents it was possible to direct the product distribution toward mono- and dinitro derivatives. The reaction between mono- and dinitro derivatives of (TtBuCorrCu) with 4-amino-4H-1,2,4-triazole gave good results, leading to the isolation of 2-(NH(2))-3-(NO(2))-TtBuCorrCu and 2,18-(NH(2))(2)-3,17-(NO(2))(2)-TtBuCorrCu in moderate yields. To elucidate factors that influence the reaction, and to highlight the different behavior observed for different metal complex substrates, the electrochemistry of three copper complexes, TtBuPCorrCu, (NO(2))TtBuPCorrCu, and (NO(2))(2)TtBuPCorrCu, was studied by cyclic voltammetry and thin-layer UV-visible spectroelectrochemistry. The nitro groups on (NO(2))(x)TtBuPCorrCu are highly electron-withdrawing, which leads not only to a substantial positive shift of all redox potentials but also to a unique redox behavior and UV-vis spectrum of the singly reduced product as compared to the parent compound, TtBuPCorrCu. Finally, the amination reaction was carried out on a Ge(IV) nitrocorrolate, giving in good yield the 2-amino-3-nitroderivative, which was structurally characterized by single crystal X-ray crystallography.     

A variational view at the time-dependent minimal surface equation

We present a global variational approach to the L ²-gradient flow of the area functional of cartesian surfaces through the study of the so-called weighted energy-dissipation (WED) functional. In particular, we prove a relaxation result which allows us to show that minimizers of the WED converge in a quantitatively prescribed way to gradient-flow trajectories of the relaxed area functional. The result is then extended to general parabolic quasilinear equations arising as gradient flows of convex functionals with linear growth.     

Internal quality control as a tool for planning a robustness study regarding a multiresidue method for pesticides found in olive oil

The implementation of the internal quality assurance program allows for demonstration of the performance characteristics of a method, as well as the avoidance of erroneous results. There is increasing concern in testing laboratories to ensure that the analytical process remains stable, giving reliable results under statistical control. Robustness is tested by introducing variations in experimental conditions and examining the effects on the results. The European document SANCO/12495/2011, “Method validation and quality control procedures for pesticide residue analysis in food and feed”, defines robustness as a parameter that can be derived from ongoing method verification. Internal quality control tools, such as the recovery control chart, were employed in this study to enhance the stability of the recovery rates and to investigate the experimental conditions that have a major influence on the quantification of recovery rates. The method investigated in this study permitted us to investigate eighteen pesticides in olive oil by using a gas chromatography–mass spectrometry technique. This method has been accredited to ISO/IEC 17025:2005 standards and was applied for 1 year in routine conditions for pesticide residues at the Italian National Reference Laboratory. The recovery control chart has showed that all recovery rates for 1 year were close to the maximum limit (120 %) of recovery performance criteria. Consequently, a “positive” bias has affected all data over longer periods of time. A robustness test was planned in order to investigate the grounds that most influenced the variability of the results. The robustness test involved the following three parameters: solvent used to dissolve the final extract, internal standards, and type of olive oil.     

Γ-limits and relaxations for rate-independent evolutionary problems

This work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals and the dissipation distance . For sequences and we address the question under which conditions the limits q _∞ of solutions satisfy a suitable limit problem with limit functionals and , which are the corresponding Γ-limits. We derive a sufficient condition, called conditional upper semi-continuity of the stable sets, which is essential to guarantee that q _∞ solves the limit problem. In particular, this condition holds if certain joint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator converge if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model. Research partially supported by LC06052 (MŠMT), MSM21620839 (MŠMT), A1077402 (GAČR), by the Deutsche Forschungsgemeinschaft under MATHEON C18 and under SFB404 C7, by the European Union under HPRN-CT-2002-00284 Smart Systems, and by the Alexander von Humboldt-Stiftung. Both, TR and US gratefully acknowledge the kind hospitality of the WIAS, where this research was initiated.     

β-Nitro-5,10,15-tritolylcorroles

Functionalization of the β-pyrrolic positions of the corrole macrocycle with -NO(2) groups is limited at present to metallocorrolates due to the instability exhibited by corrole free bases under oxidizing conditions. A careful choice of the oxidant can limit the transformation of corroles into decomposition products or isocorrole species, preserving the corrole aromaticity, and thus allowing the insertion of nitro groups onto the corrole framework. Here we report results obtained by reacting 5,10,15-tritolylcorrole (TTCorrH(3)) with the AgNO(2)/NaNO(2) system, to give mono- and dinitrocorrole derivatives when stoichiometry is carefully controlled. Reactions were found to be regioselective, affording the 3-NO(2)TTCorrH(3) and 3,17-(NO(2))(2)TTCorrH(3) isomers as the main products in the case of mono- and disubstitution, in 53 and 20% yields, respectively. In both cases, traces of other mono- and disubstituted isomers were detected, which were structurally characterized by X-ray crystallography. The influence of the β-nitro substituents on the corrole properties is studied in detail by UV-visible, electrochemical, and spectroelectrochemical characterization of these functionalized corroles. Density functional theory (DFT) and time-dependent DFT (TDDFT) calculations of the ground and excited state properties of these β-nitrocorrole derivatives also afforded significant information, closely matching the experimental observations. It is found that the β-NO(2) substituents conjugate with the π-aromatic system of the macrocycle, which initiates significant changes in both the spectroscopic and redox properties of the so functionalized corroles. This effect is more pronounced when the nitro group is introduced at the 2-position, because in this case the conjugation is, for steric reasons, more efficient than in the 3-nitro isomer.