This study aims to describe how researchers may incorporate four different generalisation methods (working hypothesis, analytic generalisation, critical case and naturalistic generalisation) to aid them in properly generaliseing from case study results. It was found that critical cases can be used in combination with working hypothesis to determine the falsifiability of the hypotheses. Working hypothesis and analytic generalisation complement each other by making it possible for researchers to generalise to both theory and new hypotheses (i.e. both inductively and deductively). Lastly, naturalistic and analytic generalisation enables a double-ended generalisation, where both the reader and the scientist themself generalise to specific situations and over-all theory, respectively.

La théorie des processus de branchement multi-type en environnement i.i.d. est considérablement moins développée que dans le cas univarié, et les questions fondamentales ne sont pas résolues en totalité à ce jour. Les réponses exigent une compréhension profonde du comportement des produits des matrices i.i.d. à coefficients positifs. Sous des hypothèses assez générales et lorsque les fonctions génératrices de probabilité des lois de reproduction sont “linéaire fractionnaires”, nous montrons que la probabilité de survie à l’instant n du processus de branchement multi-type en environnement aléatoire est proportionnelle à 1/√n lorsque n → ∞. La démonstration de ce résultat suit l’approche développée pour étudier les processus de branchement uni-variés en environnement aléatoire i. i. d. Il utilise de façon cruciale des résultats récents portant sur les fluctuations des normes de produits de matrices aléatoires i.i.d. / The theory of multi-type branching process in i.i.d. environment is considerably less developed than for the univariate case, and fundamental questions are up to date unsolved. Answers demand a solid understanding of the behavior of products of i.i.d. matrices with non-negative entries. Under mild assumptions, when the probability generating functions of the reproduction laws are fractional-linear, the survival probability of the multi-type branching process in random environment up to moment n is proportional to 1/√n as n → ∞. Techniques for univariate branching process in random environment and methods from the theory of products of i.i.d. random matrices are required.

The stability with respect to a part of variables in the critical case of pair of pure imaginary and single zero roots is researched. Also case of double zero roots with two solution groups and case of two pairs of pure imaginary roots are considered. Particular conditions of the partial asymptotic stability groups) and the partial instability were obtained. In conclusion, we give an example of solving the problem. / Исследуется устойчивость относительно части переменных в критическом случае пары чисто мнимых и одного нулевого корней. Также рассмотрены случай двойного нулевого корня с двумя группами решений и случай двух пар чисто мнимых корней. Для данных случаев получены частные условия асимптотической устойчивости по части переменных и неустойчивости. В заключение приведен пример решения задачи.

Исследуется устойчивость относительно части переменных в критическом случае двойного нулевого корня (m=2). Рассмотрены случаи двойного нулевого корня с одной и с двумя группами решений. Для данных случаев получены частные условия асимптотической устойчивости по части переменных (при двойном нулевом корне с двумя группами решений) и неустойчивости (при двойном нулевом корне с одной группой решений). / The stability with respect to a part of variables in the critical case of double zero roots (m=2) is investigated. Cases of double zero roots with one solution group or two solution groups are considered. Particular conditions of the partial asymptotic stability (case of two solution groups) and the partial instability (case of one solution group) were obtained.

This research project conducted an in-depth, qualitative assessment of vocational education and training (VET) teachers’ perceptions of pedagogic innovation, with an emphasis on obstacles and supporting factors. The main research question was: “How do teachers’ roles and perspectives shape innovation processes in VET and what does this imply for the development of teaching and learning practices?” Three clusters of subsidiary questions were derived around thematic foci: ‘perceptions and concepts’, ‘documentation of practice’, and ‘dynamics, limitations, and lessons for innovation’. Based on analytical strategies derived from grounded theory, two phases of interviews – the first with ten experts and the second with 62 VET practitioners at 20 colleges – were conducted in England, Germany, and Austria, with a focus on full-time VET (Further Education Colleges, Berufskollegs, and Berufsbildende Mittlere und Höhere Schulen) in the 16-19 age range. Classroom observation preceded semi-structured, 30 to 60 minute interviews with teachers. The study builds on previous research and existing frameworks such as Lipsky’s concept of ‘street-level bureaucracy’ and Flyvbjerg’s ‘critical cases’. However, it fills a gap in the literature by focusing on practitioner perceptions, motivations, professionalism, autonomy, work contexts, and own learning in relation to pedagogic innovation, whilst tracing relevant connections to educational policy, college management, and societal influences. Teachers are shown in multiple roles as inventors, designers, and implementers of innovation, facing nine categories of obstacles. Those include limited time and budgets, bureaucracy and lack of autonomy, problems with project planning and execution, and issues related to lack of support. In addition, this study provides a comparative investigation of practitioners’ interpretations of key terms (‘pedagogy’, ‘didactics’, ‘innovation’), revealing differences between England on the one hand, and Germany and Austria on the other, based on different degrees of autonomy and innovative focus. Based on such findings, the study details recommendations for college leaders and policy makers for facilitating pedagogic innovation, placing each in their respective national contexts.