Error estimates for approximate approximations on compact intervals

The aim of this paper is the investigation of the error which results from the method of approximate approximations applied to functions defined on compact in- tervals, only. This method, which is based on an approximate partition of unity, was introduced by V. Mazya in 1991 and has mainly been used for functions defied on the whole space up to now. For the treatment of differential equations and boundary integral equations, however, an efficient approximation procedure on compact intervals is needed. In the present paper we apply the method of approximate approximations to functions which are defined on compact intervals. In contrast to the whole space case here a truncation error has to be controlled in addition. For the resulting total error pointwise estimates and L1-estimates are given, where all the constants are determined explicitly.

Quasi-Interpolation von Funktionen und Anwendungen auf Potentialprobleme

Ausgangspunkt der Dissertation ist ein von V. Maz'ya entwickeltes Verfahren, eine gegebene Funktion f : Rn ! R durch eine Linearkombination fh radialer glatter exponentiell fallender Basisfunktionen zu approximieren, die im Gegensatz zu den Splines lediglich eine näherungsweise Zerlegung der Eins bilden und somit ein für h ! 0 nicht konvergentes Verfahren definieren. Dieses Verfahren wurde unter dem Namen Approximate Approximations bekannt. Es zeigt sich jedoch, dass diese fehlende Konvergenz für die Praxis nicht relevant ist, da der Fehler zwischen f und der Approximation fh über gewisse Parameter unterhalb der Maschinengenauigkeit heutiger Rechner eingestellt werden kann. Darüber hinaus besitzt das Verfahren große Vorteile bei der numerischen Lösung von Cauchy-Problemen der Form Lu = f mit einem geeigneten linearen partiellen Differentialoperator L im Rn. Approximiert man die rechte Seite f durch fh, so lassen sich in vielen Fällen explizite Formeln für die entsprechenden approximativen Volumenpotentiale uh angeben, die nur noch eine eindimensionale Integration (z.B. die Errorfunktion) enthalten. Zur numerischen Lösung von Randwertproblemen ist das von Maz'ya entwickelte Verfahren bisher noch nicht genutzt worden, mit Ausnahme heuristischer bzw. experimenteller Betrachtungen zur sogenannten Randpunktmethode. Hier setzt die Dissertation ein. Auf der Grundlage radialer Basisfunktionen wird ein neues Approximationsverfahren entwickelt, welches die Vorzüge der von Maz'ya für Cauchy-Probleme entwickelten Methode auf die numerische Lösung von Randwertproblemen überträgt. Dabei werden stellvertretend das innere Dirichlet-Problem für die Laplace-Gleichung und für die Stokes-Gleichungen im R2 behandelt, wobei für jeden der einzelnen Approximationsschritte Konvergenzuntersuchungen durchgeführt und Fehlerabschätzungen angegeben werden.

Approximate approximations for the Poisson and the Stokes equations

The method of approximate approximations is based on generating functions representing an approximate partition of the unity, only. In the present paper this method is used for the numerical solution of the Poisson equation and the Stokes system in R^n (n = 2, 3). The corresponding approximate volume potentials will be computed explicitly in these cases, containing a one-dimensional integral, only. Numerical simulations show the efficiency of the method and confirm the expected convergence of essentially second order, depending on the smoothness of the data.

The electronic structure of anionic halide complexes of element 105 in aqueous solutions and their extraction by aliphatic amines

To study the complex formation of group 5 elements (Nb, Ta, Ha, and pseudoanalog Pa) in aqueous HCI solutions of medium and high concentrations the electronic structures of anionic complexes of these elements [MCl_6]^-, [MOCl_4]^-, [M(OH)-2 Cl_4]^-, and [MOCl_5]^2- have been calculated using the relativistic Dirac-Slater Discrete-Variational Method. The charge density distribution analysis has shown that tantalum occupies a specific position in the group and has the highest tendency to form the pure halide complex, [TaCl_6-. This fact along with a high covalency of this complex explains its good extractability into aliphatic amines. Niobium has equal trends to form pure halide [NbCl_6]^- and oxyhalide [NbOCl_5]^2- species at medium and high acid concentrations. Protactinium has a slight preference for the [PaOCl_5]^2- form or for the pure halide complexes with coordination number higher than 6 under these conditions. Element 105 at high HCl concentrations will have a preference to form oxyhalide anionic complex [HaOCl_5]^2- rather than [HaCl_6]^-. For the same sort of anionic oxychloride complexes an estimate has been done of their partition between the organic and aqueous phases in the extraction by aliphatic amines, which shows the following succession of the partition coefficients: P_Nb < P_Ha < P_Pa.

Cleanliness versus cow comfort – an insolvable problem?

Cubicle should provide good resting comfort as well as clean udders. Dairy cows in cubicle houses often face a restrictive environment with regard to resting behaviour, whereas cleanliness may still be impaired. This study aimed to determine reliable behavioural measures regarding resting comfort applicable in on-farm welfare assessments. Furthermore, relationships between cubicle design, cow sizes, management factors and udder cleanliness (namely teats and teat tips) were investigated. Altogether 15 resting measures were examined in terms of feasibility, inter-observer reliability (IOR) and consistency of results per farm over time. They were recorded during three farm visits on farms in Germany and Austria with cubicle, deep litter and tie stall systems. Seven measures occurred to infrequently to allow reliable recording within a limited observation time. IOR was generally acceptable to excellent except for 'collisions during lying down', which only showed good IOR after improvement of the definition. Only three measures were acceptably repeatable over time: 'duration of lying down', 'percentage of collisions during lying down' and 'percentage of cows lying partly or completely outside lying area'. These measures were evaluated as suitable animal based welfare measures regarding resting behaviour in the framework of an on-farm welfare assessment protocol. The second part of the thesis comprises a cross-sectional study on resting comfort and cow cleanliness including 23 Holstein Friesian dairy herds with very low within-farm variation in cubicle measures. Height at withers, shoulder width and diagonal body length were measured in 79-100 % of the cows (herd size 30 to115 cows). Based on the 25 % largest animals, compliance with recommendations for cubicle measures was calculated. Cleanliness of different body parts, the udder, teats and teat tips was assessed for each cow in the herd prior to morning milking. No significant correlation was found between udder soiling and teat or teat tip soiling on herd level. The final model of a stepwise regression regarding the percentage of dirty teats per farm explained 58.5 % the variance and contained four factors. Teat dipping after milking which might be associated with an overall clean and accurate management style, deep bedded cubicles, increasing cubicle maintenance times and decreasing compliance concerning total cubicle length predicted lower teat soiling. The final model concerning teat tip soiling explained 46.0 % of the variance and contained three factors. Increasing litter height in the rear part of the cubicle and increased alley soiling which is difficult to explain, predicted for less soiled teat tips, whereas increasing compliance concerning resting length was associated with higher percentages of dirty teat tips. The dependent variable ‘duration of lying down’ was analysed using again stepwise regression. The final model explained 54.8 % of the total variance. Lying down duration was significantly shorter in deep bedded cubicles. Further explanatory though not significant factors in the model were neck-rail height, deep bedding or comfort mattresses versus concrete floor or rubber mats and clearance height of side partitions. In the attempt to create a more comprehensive lying down measure, another analysis was carried out with percentage of ‘impaired lying down’ (i.e. events exceeding 6.3 seconds, with collisions or being interrupted) as dependent variable. The explanatory value of this final model was 41.3 %. An increase in partition length, in compliance concerning cubicle width and the presence of straw within bedding predicted a lower proportion of impaired lying down. The effect of partition length is difficult to interpret, but partition length and height were positively correlated on the study farms, possibly leading to a bigger zone of clear space for pelvis freedom. No associations could be found between impaired lying down and teat or teat tip soiling. Altogether, in agreement with earlier studies it was found that cubicle dimensions in practice are often inadequate with regard to the body dimensions of the cows, leading to high proportions of impaired lying down behaviour, whereas teat cleanliness is still unsatisfactory. Connections between cleanliness and cow comfort are far from simplistic. Especially the relationship between cubicle characteristics and lying down behaviour apparently is very complex, so that it is difficult to identify single influential factors that are valid for all farm situations. However, based on the results of the present study the use of deep bedded cubicles can be recommended as well as improved management with special regard to cubicle and litter maintenance in order to achieve both better resting comfort and teat cleanliness.