In this thesis, we address problems from two topics of applied mathematics: linear integer programming and polyhedral computation. Linear integer programming concerns solving optimisation problems to maximise a linear cost function over the set of integer points in a polyhedron. Polyhedral computation is concerned with algorithms for computing different properties of convex polyhedra. First, we explore the theory and computation of Gröbner bases and Markov bases for linear integer programming. Second, we investigate and improve an algorithm from polyhedral computation that converts between different representations of cones and polyhedra. A Markov basis is a set of integer vectors such that we can move between any two feasible solutions of an integer program by adding or subtracting vectors in the Markov basis while never moving outside the set of feasible solutions. Markov bases are mainly used in algebraic statistics for sampling from a set of feasible solutions. The major contribution of this thesis is a fast algorithm for computing Markov bases, which we used to solve a previously intractable computational challenge. Gröbner basis methods are exact local search approaches for solving integer programs. We present a Gröbner basis approach that can use the structure of an integer program in order to solve it more efficiently. Gröbner basis methods are interesting mainly from a purely theoretical viewpoint, but they are also interesting because they may provide insight into why some classes of integer programs are difficult to solve using standard techniques and because someday they may be able to solve these difficult problems. Computing the properties of convex polyhedra is useful for solving problems within different areas of mathematics such as linear programming, integer programming, combinatorial optimisation, and computational geometry. We investigate and improve an algorithm for converting between a generator representation of a cone or polyhedron and a constraint representation of the cone or polyhedron and vice versa. This algorithm can be extended to compute circuits of matrices, which are used in computational biology for metabolic pathway analysis.

The theoretical and methodological assumptions of this research imply a move away from a positivist empiricist approach with its reliance on the real, the measurable and the predictable towards an interactive and collaborative methodology situated in a constructivist and social epistemology. This thesis comprises a recursive intervention in the researching therapist's life. The author sets out on a voyage of self-research to investigate her "choreography of coexistence" (Maturana & Varela, 1987, p.248), due to curiosity about personal and professional impact. The purpose is to create a map of relational modes that in itself creates shifts: in the therapist. This invention-orientated research creates the context of the researcher and moves through processes: the writing of autobiography, detailed contextual description, the interpreting of feedback, and deconstruction. The contents that pour forth are many different narratives tracking the evolvement of the self in the original family, through further definition in new relationships and the expansion of roles in many professional systems both with clients and colleagues. " Extracts from conversations provide new perspectives and feedback about impact. Thus a continual 'provoking of voices' becomes a thesis theme that highlights the researcher's structure, organisation and interpersonal processes. An emancipatory and developmental process is documented through the researching therapist's positioning as actor, observer and then critic in relationship to the data. Patterns and themes emerge that facilitate both self- differentiation and connectedness and many new head-heart connections. This new knowledge could enable the professional's skilful and intuitive use of self. The self comes into being as it reflects itself, ·so a recursive process evolves where looking at the self operates on the products of its own operations. This is an active process, where the researching therapist constructs an experiential reality. Given the accountability that accrues from constructing such a reality, a focus on pragmatic, aesthetic and ethical criteria is incorporated. This research, like the practice of therapy, is a departure from attempts to demonstrate what is already known to modes of research that are recursive and improvisational. / Psychology / D. Litt. et Phil.

The theoretical and methodological assumptions of this research imply a move away from a positivist empiricist approach with its reliance on the real, the measurable and the predictable towards an interactive and collaborative methodology situated in a constructivist and social epistemology. This thesis comprises a recursive intervention in the researching therapist's life. The author sets out on a voyage of self-research to investigate her "choreography of coexistence" (Maturana & Varela, 1987, p.248), due to curiosity about personal and professional impact. The purpose is to create a map of relational modes that in itself creates shifts: in the therapist. This invention-orientated research creates the context of the researcher and moves through processes: the writing of autobiography, detailed contextual description, the interpreting of feedback, and deconstruction. The contents that pour forth are many different narratives tracking the evolvement of the self in the original family, through further definition in new relationships and the expansion of roles in many professional systems both with clients and colleagues. " Extracts from conversations provide new perspectives and feedback about impact. Thus a continual 'provoking of voices' becomes a thesis theme that highlights the researcher's structure, organisation and interpersonal processes. An emancipatory and developmental process is documented through the researching therapist's positioning as actor, observer and then critic in relationship to the data. Patterns and themes emerge that facilitate both self- differentiation and connectedness and many new head-heart connections. This new knowledge could enable the professional's skilful and intuitive use of self. The self comes into being as it reflects itself, ·so a recursive process evolves where looking at the self operates on the products of its own operations. This is an active process, where the researching therapist constructs an experiential reality. Given the accountability that accrues from constructing such a reality, a focus on pragmatic, aesthetic and ethical criteria is incorporated. This research, like the practice of therapy, is a departure from attempts to demonstrate what is already known to modes of research that are recursive and improvisational. / Psychology / D. Litt. et Phil.