Multivariate Gini Indices

Two extensions of the univariate Gini index are considered:R_D, based on expected distance between two independent vectors from the same distribution with finite meanμ ^d; andR_V, related to the expected volume of the simplex formed fromd+1 independent such vectors. A new characterization ofR_Das proportional to a univariate Gini index for a particular linear combination of attributes relates it to the Lorenz zonoid. TheLorenz zonoidwas suggested as a multivariate generalization of the Lorenz curve.R_Vis, up to scaling, the volume of the Lorenz zonoid plus a unit cube of full dimension. Whend=1, bothR_DandR_Vequal twice the area between the usual Lorenz curve and the line of zero disparity. Whend>1, they are different, but inherit properties of the univariate Gini index and are related via the Lorenz zonoid:R_Dis proportional to the average of the areas of some two-dimensioned projections of the lift zonoid, whileR_Vis the average of the volumes of projections of the Lorenz zonoid over all coordinate subspaces.     

Extraktion von alkali- und erdalkalimetallionen mit (sym-dibenzo-14-krone-4-oxy)- und (sym-dibenzo-16-krone-5-oxy)-carbonsäuren

(Extraction of alkali on alkaline earth metal ions with (sym-dibenzo-14-crown-4-oxy)- and (sym-dibenzo-16-crown-5-oxy)-carboxylic acids.)The extraction of lithium, sodium, potassium, calcium and some other metal ions with dibenzo-4-crown-4-oxy- and dibenzo-16-crown-5-oxycarboxylic acids containing the groups -CH₂COOH, -(CH₂)₂COOH, -(CH₂)₃COOH, -CH(C₂H₅)COOH and -CH(C₄H₉)COOH was studied. The extraction increases as a function of the lipophilic character of the carboxylic acid group. Calcium, barium and strontium ions are better extracted than Li⁺, Na⁺ and K⁺; there are only small differences among the alkaline earth metal ions. Evaluated from the extraction data, the composition of the extracted species was 1:1 (metal/ligand) for Li⁺, and 1:2 for CaCa²⁺; Na⁺ and K⁺ favour the formation of 1:2 complexes with dibenzo-14-crown-4-derivatives bbut 1:1 complexes with dibenzo-16-crown-5-oxy-carboxylic acids. The dependence of the distribution ratio on pH does not provide unequivocal evidence for the composition of the extracted compounds.     

Stochastic dominance with nonadditive probabilities

Choquet expected utility which uses capacities (i.e. nonadditive probability measures) in place of-additive probability measures has been introduced to decision making under uncertainty to cope with observed effects of ambiguity aversion like the Ellsberg paradox. In this paper we present necessary and sufficient conditions for stochastic dominance between capacities (i.e. the expected utility with respect to one capacity exceeds that with respect to the other one for a given class of utility functions). One wide class of conditions refers to probability inequalities on certain families of sets. To yield another general class of conditions we present sufficient conditions for the existence of a probability measureP with f dC= f dP for all increasing functionsf whenC is a given capacity. Examples includen-th degree stochastic dominance on the reals and many cases of so-called set dominance. Finally, applications to decision making are given including anticipated utility with unknown distortion function.     

Untersuchungen zur anwendung ternärer komplexe in der photometrie—III: Die bestimmung des titanium(iv) mit brompyrogallolrot in gegenwart von nitrilotriessigsäure bzw. ethylenediamintetraessigsäure

As part of a continued investigation of the analytical applications of ternary complexes, the systems Ti(IV)/BPR, I), Ti(IV)/BPR/EDTA (II), and Ti(IV)/BPR/NTA (III), where BPR = Bromopyrogallol Red, have been investigated and compared in terms of working concentration range, molar absorptivity, standard deviation, coefficient of variation, calibration sensitivity, limit of detection and of determination. A novel way of summarizing the effects of interferences of a large group of ions (some 47 in all) is presented. The results show that the ternary systems, in particular system II, have a better selectivity than the simple binary system I of titanium with Bromopyrogallol Red.     

Generalized distortional hardening in a thermomechanically coupled framework

In many metals like aluminum the evolution of the microstructure leads to an evolution of the macroscopic yield surface. Clearly, isotropic and kinematic hardening models cannot capture this effect realistically. For that purpose, a new generalized distortional hardening model is proposed. This novel approach belongs to the class of so-called generalized standard materials and is thermodynamically consistent. Although the approach is relatively simple, it shows several advantages like yield surface convexity and yield limit saturation. This novel model is embedded into a thermomechanically coupled finite strain framework. Several numerical simulations show the predictive capabilities. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the variationally consistent modeling of material failure

Material failure associated with cracks or shear bands is frequently analyzed by utilizing so-called cohesive models. Such models are based on traction-separation laws. Within such approaches, the stress vector of the considered crack or shear band is related to its conjugate variable being the respective displacement jump (such as the material separation or the crack opening). In the present work, a framework suitable for the analysis of shear bands is discussed. All models belonging to that framework are consistently derived from thermodynamical principles. Hence, the second law of thermodynamics is automatically fulfilled. Furthermore, a variational principle strongly relying on the postulate of maximum dissipation is elaborated leading finally to a variationally consistent implementation. More precisely, all state variables, together with the unknown deformation mapping, follow naturally from minimizing an incrementally defined potential within the presented algorithmic formulation. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermodynamically consistent temperature prediction in dissipative solids

The present work deals with the thermomechanical coupling in dissipative solids and the consistent temperature prediction. One of the first works dealing with this subject was written by Taylor & Quinney (TQ) where the fraction between dissipated energy eventually transformed to heat and plastic work is assumed as constant (typically between 0.8-1.0 for metals). Although this assumption often leads to reasonable temperature predictions, it is not always in agreement with experimental observations. Furthermore, the TQ model does not comply with the first and second law of thermodynamics in general. Unfortunately, a standard thermodynamically consistent framework is not convincing either, since it usually leads to a significant overprediction of the temperature increase during dissipative processes. Within the present work, a novel framework suitable for the modeling of thermomechanically coupled processes is discussed. It will be shown that this framework is thermodynamically consistent and leads to a temperature increase, as a result of plastic deformation, in good agreement with the underlying experiments. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fully variational algorithmic formulation for wrinkling at finite strains

This contribution is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The presented approach allows to employ three–dimensional constitutive models directly, i.e., plane stress conditions characterizing membranes are variationally enforced by minimizing the potential energy with respect to the transversal strains. Since the proposed formulation for wrinkling in membranes is fully variational, it can be conveniently combined with other variational methods (based on energy minimization). As an example, a variationally consistent framework for finite strain plasticity theory is considered. More precisely, the minimization principle characterizing wrinkling in elastic membranes and that describing plasticity in inelastic solids are coupled leading to a novel variational approach for inelastic membranes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An efficient quantitative ultrasonic elastography based on variational h-refinement

Quantitative elastography is a method to visualise a stiffness distribution. It is motivated by the observation that changes in mechanical properties of soft tissue mostly include important diagnostic information. With an ultrasound–elastography system, the displacement field can be calculated from the pre– and post–deformation image. Using the assumption that the material is elastic, isotropic and nearly incompressible, the distribution of shear modulus is determined by solving an inverse problem. While common approaches use a constant mesh, a variational h–refinement is in the focus of this work. In doing so, the efficiency and accuracy of the determination can be increased. The presented results are generated by using numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)