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Item Modelling of mereotopological relationships in multidimensional space : a thesis submitted for the degree of Doctor of Philosophy, School of Natural and Computational Sciences, Massey University(Massey University, 2021) Izadi, AzadehInferences based on spatial knowledge play an important role in human lives. Humans are easily able to deal with spatial knowledge without any need to refer to numerical computation. The field of Qualitative Spatial Representation and Reasoning (QSRR) aims to model human common sense of space. Among the various types of qualitative relationship between spatial objects, connectivity (or topology) and parthood (or mereology) serve as the most basic underlying aspects. Most current mereotopological theories are restricted to objects with the same dimension. However, sometimes spatial entities of different dimensions must be considered for many practical applications (e.g. map reading, spatial analysis). The inability of current theories to interact with entities of different dimensions has motivated the foundation of multidimensional spatial theories. However, these theories are less efficient in terms of reasoning power. Moreover, their set of introduced mereotopological relations has not been cognitively validated. This research presents a multidimensional mereotopological theory using part of and boundary part as primitive concepts. We introduce a set of nine spatial relations with the jointly exhaustive and pairwise disjoint property based on these primitives. This property allows us to develop an efficient reasoning strategy (i.e. constraint-based reasoning) which makes our approach more practical than previous works. We used automated theorem provers and finite model finders to aid the formal verification of the theory, proving its properties and generating the composition table for reasoning purposes. This work is the first multidimensional mereotopological theory that not only has properties that are verified by traditional logical deduction techniques (like the other multidimensional mereotopological theories), but that also it supports an efficient reasoning strategy that was not being available before. Furthermore, we verified the cognitive adequacy of our proposed set of relations using human subjects experiments, applying clustering and thematic analyses to empirical data. Our study is the first to pro- vide evidence for the cognitive plausibility of a multidimensional mereotopological theory (going beyond previous studies that have only shown cognitive adequacy for equidimensional mereotopological theories) supporting its closeness to human cognition. In addition, we demonstrate our multidimensional theory by applying it to a real-world scenario (i.e. a flood event).Item A class of absolute retracts : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University(Massey University, 1973) Tyree, Alan LA restricted version of the Tietze Theorem is that a continuous mapping of a closed subspace of a metric space ranging in a closed interval may be extended to a continuous function defined upon the whole metric space. This may be viewed as a property of the closed interval and is expressed by saying that the interval is an absolute extensor. Thus, absolute extensors may be viewed as a generalisation of real intervals, and many of the desirable properties of intervals have been generalised to the class of absolute extensors. In 1951, Dugundji showed that every convex subset of a locally convex linear topological space is an absolute extensor, thus dramatically extending the Tietze theorem. In this thesis, a class of subsets of a normed linear space is defined. This new class of sets includes the convex sets and it is shown that these new sets are also absolute extensors.