<p> This thesis deals with the lattice of all topologies which may be put on an arbitrary set E. The structure of the lattice is investigated together with its lattice properties. A chapter is included on the co-atoms of the lattice, the ultra-topologies, wherein various topological properties which do and do not hold are investigated. Various topological properties are considered as to which topologies are minimal and maximal such and also which topological properties are preserved under lattice operations and relations.</p> / Thesis / Master of Science (MSc)

<p> Commutative semigroup equations are described, and rules of inference for them are given. Then a skeleton sublattice of the lattice of equational classes of commutative semigroups is described, and a partial description is given of the way in which the rest of the lattice hangs on the skeleton.</p> / Thesis / Doctor of Philosophy (PhD)

<p> Fo. determinative curves for lattice planes (301), (311), and (401) were calculated from four natural olivines. 2θ values for each olivine were computed from the lattice parameters using the reciprocal lattice d* for the orthorhombic crystal system d*hkl = [h^2/a^2 + k^2/b^2 + l^2/c^2]^1/2 combined with the Bragg equation λCuKα = 2 sin θhk / d*hkl </p> <p> XRD patterns for 10 natural olivines were used to determine their respective Fo. composition. Errors were calculated for the Fo. composition of the natural olivines yielding an accuracy from the determinative curves of± 6 wt. % Fo. An olivine of 90 wt. % Fo. was X-rayed and plotted along the determinative curves giving values of Fo. composition for (301), (311) and (401) of 89, 87 and 85 wt. % respectively.</p> <p> The effect of Ni, Mn and Ca substituting into the (Mg, Fe) olivine structure causing Δ2θ shifts was also considered and found to be negligible for natural (Mg, Fe) olivines.</p> / Thesis / Bachelor of Science (BSc)

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The last 15 years have seen a significant progress in the development of general purpose algorithms and software for polyhedral computation. Many polytopes of practical interest have enormous output complexity and are often highly degenerate, posing severe difficulties for known general purpose algorithms. They are, however, highly structured and attention has turned to exploiting this structure, particularly symmetry. We focus on polytopes arising from combinatorial optimization problems. In particular, we study the face lattice of the metric polytope associated with the well-known maxcut and multicommodity flow problems, as well as with finite metric spaces. Exploiting the high degree of symmetry, we provide the first complete orbitwise description of the higher layers of the face lattice of the metric polytope for any dimension. Further computational and combinatorial issues are presented. / Thesis / Master of Applied Science (MASc)

The thesis is focused on the theory of special lattices that are important in cryptography, namely ideal, cyclic and NTRU lattices. Specifically, we expand and generalise the work of Ding and Lindner on Identifying Ideal Lattices. The algorithm for identifying ideal lattices is included, along with illustrative examples and more detailed proofs of propositions on which the algorithm is based. In the section about Lattice Isomorphism there is also included a generalised theorem from the paper. We extend the result of identifying the NTRU lattices and supplement it with several examples. The thesis also contains Chapter Applications in Cryptography where we describe a cryptographic hash function based on ideal lattices. And finally, we provide a brief overview of the cryptographic algorithms using NTRU lattices.