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In this paper we establish the non--multiplicity of solutions to first order matrix dynamic equations on time scales. The new results verify and extend the notions developed in \cite{thesis} to more complex systems of $n^2$ matrices with the help of ideas developed in \cite[Chap 5]{BP}, identifying Lipschitz conditions suitable to generalised $n^2$--models on time scales., Comment: 17 pages

The theory of dynamic equations on time scales provides an important bridge between the fields of differential and difference equations. It is particularly useful in describing phenomena that possess a hybrid continuous-discrete behaviour in their growth, like many temperate--2one insect populations and crops. A dynamic equation on a time scale is a generalised. 'two-in-one' model, it serves as a differential equation for purely continuous domains and as a difference equation for purely discrete ones. The field of "dynamic equations on time scales" is about 20 years old. As such, much of the basic (yet very important) linear theory has been established, however the non-linear extensions are yet to be fully developed. This thesis aims to fill this gap by providing the foundational framework of non-linear results from which further lines of inquiry can be launched. This thesis answers several important questions regarding the qualitative and quantitative properties of solutions to non-linear dynamic equations on time scales. Namely, (i) When do solutions exist? (ii) If solutions exist, then are they unique? (iii) How can such solutions be closely approximated? (iv) How can we explicitly solve certain problems to extract their solutions? The methods employed to address the above questions include: dynamic inequalities; iterative techniques and the method of successive approximations; and the fixed point approaches of Banach and Schauder.

This study examines transformational and transactional leadership behaviours of heads of schools/departments, and the individual level cultural dimensions idiocentrism-allocentrism, as perceived by academic staff in Australian and Pakistani universities. The principal research aim was to investigate the relationships between academics’ attributions concerning leadership behaviours of heads of schools/departments and cultural norms in a collectivistic and an individualistic culture. Samples of academic staff were randomly selected from Australian and Pakistani universities. The Australian sample consisted of 117 academics, and the Pakistani sample comprised 120 academics. Analysis was carried out using exploratory factor analysis, multiple regression analysis, and discriminant analysis. Within the context of the research, the results suggested that Australian and Pakistani university academics had similar attribution patterns for entities related to transformational, transactional, and passive leadership behaviours, and the cultural dimension allocentrism. The results also suggested that the academics of both samples did not differentiate between various transformational leadership behaviours, and perceived the transactional leadership behaviour contingent reward as part of the transformational leadership behaviour proactive leadership. Further, the academics of both samples perceived the transactional leadership behaviour passive management-by-exception and laissez-faire leadership as a single dimension, passive leadership. Analyses identified some critical relationships between perceived leadership behaviours of heads of schools/departments and idiocentrism-allocentrism variables for the two samples. The transformational leadership behaviour participative leadership was positively related to allocentrism and self-reliance for the Australian sample. The transactional leadership behaviour active management-by-exception was positively related to positive-competitiveness and negatively related to negative-competitiveness for the Pakistani sample. Passive leadership was negatively related to leadership outcomes for the Pakistani sample. Moreover, active management-by-exception and positive-competitiveness discriminated between the two samples. Leadership of schools/departments in universities has been found to be a multifaceted phenomenon. The research suggests that academics’ perceptions of leadership behaviours, cultural norms, and context have important implications for school/department leadership in universities.