A crack on the interface of piezo-electric bodies

Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezo-electric media under various boundary conditions on the crack surfaces. The Griffith formulae are obtained for increments of energy functionals due to growth of the crack and the notion of the energy release matrix is introduced. Normalization conditions for bases of singular solution are proposed to adapt them to the energy, stress, and deformation fracture criteria. Connections between these bases are determined and additional properties of the deformation basis related to the notion of electric surface enthalpy are established.

Statistical evaluation of texture analysis from the biocrystallization method

The consumers are becoming more concerned about food quality, especially regarding how, when and where the foods are produced (Haglund et al., 1999; Kahl et al., 2004; Alföldi, et al., 2006). Therefore, during recent years there has been a growing interest in the methods for food quality assessment, especially in the picture-development methods as a complement to traditional chemical analysis of single compounds (Kahl et al., 2006). The biocrystallization as one of the picture-developing method is based on the crystallographic phenomenon that when crystallizing aqueous solutions of dihydrate CuCl2 with adding of organic solutions, originating, e.g., from crop samples, biocrystallograms are generated with reproducible crystal patterns (Kleber & Steinike-Hartung, 1959). Its output is a crystal pattern on glass plates from which different variables (numbers) can be calculated by using image analysis. However, there is a lack of a standardized evaluation method to quantify the morphological features of the biocrystallogram image. Therefore, the main sakes of this research are (1) to optimize an existing statistical model in order to describe all the effects that contribute to the experiment, (2) to investigate the effect of image parameters on the texture analysis of the biocrystallogram images, i.e., region of interest (ROI), color transformation and histogram matching on samples from the project 020E170/F financed by the Federal Ministry of Food, Agriculture and Consumer Protection(BMELV).The samples are wheat and carrots from controlled field and farm trials, (3) to consider the strongest effect of texture parameter with the visual evaluation criteria that have been developed by a group of researcher (University of Kassel, Germany; Louis Bolk Institute (LBI), Netherlands and Biodynamic Research Association Denmark (BRAD), Denmark) in order to clarify how the relation of the texture parameter and visual characteristics on an image is. The refined statistical model was accomplished by using a lme model with repeated measurements via crossed effects, programmed in R (version 2.1.0). The validity of the F and P values is checked against the SAS program. While getting from the ANOVA the same F values, the P values are bigger in R because of the more conservative approach. The refined model is calculating more significant P values. The optimization of the image analysis is dealing with the following parameters: ROI(Region of Interest which is the area around the geometrical center), color transformation (calculation of the 1 dimensional gray level value out of the three dimensional color information of the scanned picture, which is necessary for the texture analysis), histogram matching (normalization of the histogram of the picture to enhance the contrast and to minimize the errors from lighting conditions). The samples were wheat from DOC trial with 4 field replicates for the years 2003 and 2005, “market samples”(organic and conventional neighbors with the same variety) for 2004 and 2005, carrot where the samples were obtained from the University of Kassel (2 varieties, 2 nitrogen treatments) for the years 2004, 2005, 2006 and “market samples” of carrot for the years 2004 and 2005. The criterion for the optimization was repeatability of the differentiation of the samples over the different harvest(years). For different samples different ROIs were found, which reflect the different pictures. The best color transformation that shows efficiently differentiation is relied on gray scale, i.e., equal color transformation. The second dimension of the color transformation only appeared in some years for the effect of color wavelength(hue) for carrot treated with different nitrate fertilizer levels. The best histogram matching is the Gaussian distribution. The approach was to find a connection between the variables from textural image analysis with the different visual criteria. The relation between the texture parameters and visual evaluation criteria was limited to the carrot samples, especially, as it could be well differentiated by the texture analysis. It was possible to connect groups of variables of the texture analysis with groups of criteria from the visual evaluation. These selected variables were able to differentiate the samples but not able to classify the samples according to the treatment. Contrarily, in case of visual criteria which describe the picture as a whole there is a classification in 80% of the sample cases possible. Herewith, it clearly can find the limits of the single variable approach of the image analysis (texture analysis).

Eigenschaften chromatischer Polynome

Die Berechnung des 1912 von Birkhoff eingeführten chromatischen Polynoms eines Graphen stellt bekanntlich ein NP-vollständiges Problem dar. Dieses gilt somit erst recht für die Verallgemeinerung des chromatischen Polynoms zum bivariaten chromatischen Polynom nach Dohmen, Pönitz und Tittmann aus dem Jahre 2003. Eine von Averbouch, Godlin und Makowsky 2008 vorgestellte Rekursionsformel verursacht durch wiederholte Anwendung im Allgemeinen einen exponentiellen Rechenaufwand. Daher war das Ziel der vorliegenden Dissertation, Vereinfachungen zur Berechnung des bivariaten chromatischen Polynoms spezieller Graphentypen zu finden. Hierbei wurden folgende Resultate erzielt: Für Vereinigungen von Sternen, für vollständige Graphen, aus welchen die Kanten von Sternen mit paarweise voneinander verschiedenen Ecken gelöscht wurden, für spezielle Splitgraphen und für vollständig partite Graphen konnten rekursionsfreie Gleichungen zur Berechnung des bivariaten chromatischen Polynoms mit jeweils linear beschränkter Rechenzeit gefunden werden. Weiterhin werden Möglichkeiten der Reduktion allgemeiner Splitgraphen, bestimmter bipartiter Graphen sowie vollständig partiter Graphen vorgestellt. Bei letzteren erweist sich eine hierbei gefundene Rekursionsformel durch eine polynomiell beschränkte Laufzeit als effektive Methode. Ferner konnte in einem Abschnitt zu Trennern in Graphen gezeigt werden, dass der Spezialfall der trennenden Cliquen, welcher im univariaten Fall sehr einfach ist, im bivariaten Fall sehr komplexe Methoden erfordert. Ein Zusammenhang zwischen dem bivariaten chromatischen Polynom und dem Matchingpolynom wurde für vollständige Graphen, welchen die Kanten von Sternen mit paarweise voneinander verschiedenen Ecken entnommen wurden, sowie für Bicliquen hergestellt. Die vorliegende Dissertation liefert darüber hinaus auch einige Untersuchungen zum trivariaten chromatischen Polynom, welches auf White (2011) zurückgeht und eine weitere Verallgemeinerung des bivariaten chromatischen Polynoms darstellt. Hierbei konnte gezeigt werden, dass dessen Berechnung selbst für einfache Graphentypen schon recht kompliziert ist. Dieses trifft sogar dann noch zu, wenn man die einzelnen Koeffizienten als bivariate Polynome abspaltet und einzeln berechnet. Abschließend stellt die Arbeit zu vielen Resultaten Implementierungen mit dem Computeralgebrasystem Mathematica bereit, welche zahlreiche Möglichkeiten zu eigenständigen Versuchen bieten.