Proefschrift--Letteren--Nijmegen, 1978. / Résumé en anglais. Bibliogr. p. 327-330.

Texte remanié de: Habilitationsschrift--Philosophische Fakultät für Altertumskunde und Kulturwissenschaften--München--Universität, 1986. / Notes bibliogr. Index.

Thesis (doctoral)--Westfälische Wilhelms-Universität, Münster, 1999.

Thesis (Ph. D.)--University of Delaware, 1981. / Principal faculty adviser: Barbara Marie Stafford, Dept. of Art History. Illustrations are mounted photographs. Includes bibliographical references (leaves 184-195).

Thesis (M.F.A.)--The University of North Carolina at Greensboro, 2008. / Title from PDF t.p. (viewed May 28, 2009). Advisor: Mariam Stephan; submitted to the Dept. of Art. Includes bibliographical references (p. 10).

Thesis (M.F.A.)--Washington State University, May 2010. / Title from PDF title page (viewed on July 19, 2010). "Department of Fine Arts." Includes bibliographical references (p. 9).

Magisterarbeit--Kunsthistorisches Institut--München--Ludwig-Maximilians-Universität, [1990?].

The augmented phase portrait, introduced by Sabrina Streipert and Gail Wolkowicz, is used to analyze second order rational discrete maps of the form \begin{align*} x_{n+1} = \frac{\alpha + \beta x_n + \gamma x_{n-1}}{A + Bx_n + C x_{n-1}}, \text{ for } n \in \mathbb{N}_0 =\{0,1,2,\dots, \} \end{align*} with parameters $\alpha, \, \beta, \, \gamma, \, A, \, B, \, C \geq 0$, and initial conditions, $x_{0}, \, x_{-1} > 0$. First we study the special case, \begin{align*} x_{n+1} = \frac{\alpha + \gamma x_{n-1}}{A + Bx_n}, \end{align*} with $\alpha, \, \gamma, \, B > 0$ and $A \geq 0$. Applying the change of variables, $y_n = x_{n-1}$, this equation can be rewritten as a planar system. We provide a new proof to show that oscillatory solutions have semicycles of length one, except possibly the first cycle, and that nonoscillatory solutions must converge monotonically to the equilibrium. This was originally done by Gibbons, Kulenovic, and Ladas. We also show that when the unique positive equilibrium is a saddle point, there exist nontrivial positive solutions that increase and decrease monotonically to the equilibrium, proving Conjecture 5.4.6 from the monograph by Kulenovic and Ladas. In particular, Theorem 1.2 from this monograph defines the tangent vector to the stable manifold at the equilibrium. We show that specific regions defined by the augmented phase portrait have solutions that increase and decrease monotonically to the equilibrium along the stable manifold. While Conjecture 5.4.6 was previously proven in a paper by Hoag and a paper by Sun and Xi, our proof provides a more intuitive and elementary solution. We then consider the case, \begin{equation*} x_{n+1} = \frac{\alpha + \beta (x_n + x_{n-1})}{A + B(x_n + x_{n-1})}, \end{equation*} with $\alpha, \beta, A, B > 0$. Again, using $y_n = x_{n-1}$, this system can be written as a planar system. Thus, applying the augmented phase plane, we prove global asymptotic stability of the positive equilibrium for some cases. In other cases, we show this using other theorems from the monograph by Kulenovic and Ladas as was previously done by Atawna, et al. / Thesis / Master of Science (MSc) / The augmented phase portrait, introduced by Sabrina Streipert and Gail Wolkowicz, is used to analyze second order rational discrete maps with nonnegative parameters and positive initial conditions. Using a change of variable to transform the second order rational discrete maps into planar maps, various properties of solutions were analyzed for various cases. For one case, we provide a new proof to show that oscillatory solutions have semicycles of length one, except possibly the first cycle, and that nonoscillatory solutions must converge monotonically to the equilibrium. This was originally done by Gibbons, Kulenovic, and Ladas. We also prove Conjecture 5.4.6 from the monograph by Kulenovic and Ladas, showing the existence of solutions that increase and decrease monotonically to the equilibrium. While Conjecture 5.4.6 was previously proved in a paper by Hoag and a paper by Sun and Xi, our proof provides a more intuitive and elementary solution. Finally, for another case we prove global asymptotic stability of the positive equilibrium using the augmented phase portrait. Sometimes, we show this using other theorems from the monograph by Kulenovic and Ladas as was previously done by Atawna, et al.

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The purpose of this essay is to investigate thoughts of pupils in third grade about language, language acquisition, multilingualism and mother tongue. The group is highly multicultural and multilingual which has resulted in many opinions regarding the following issues of conduct: How do the pupils describe their use of their language skills and what do they do to develop them? What thoughts do they have about multilingualism? To what and when do the pupils make use of their mother tongue? A mixed method has been used which makes the study equally quantitative and qualitative. The results show that language is a means of communication encountering people in your daily life, in example at home, school and in your free time. The children show great skills in language usage when they cross the different social arenas during a normal day and also an awareness regarding learning strategies. The pupils’ mother tongues are being used mostly with family though and although multilingualism is considered positive the children do not use all their language capacity when learning a second language.