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Subject: search.filters.subject.490407 Mathematical logic, set theory, lattices and universal algebra

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    The (5, p)-arithmetic hyperbolic lattices in three dimensions : a dissertation in Mathematics, presented to the Massey University in partial fulfillment of the requirements for the degree of Doctor of Philosophy
    (Massey University, 2024-02-10) Salehi, Keyvan
    The group $Isom^+(\mathbb{H}^3)\cong PSL(2,\mathbb{C})$ contains an unlimited number of lattices of orientation-preserving isometries of hyperbolic 3-space (equivalently Kleinian groups of finite co-volume) that may be produced by using two elements of finite orders $p$ and $q$ as generators. For example, all but a finite number of $(p, 0)$-$(q, 0)$ orbifold Dehn surgery on any of the infinite number of hyperbolic two-bridge links (or knots if $p = q$) would have (orbifold) fundamental groups that are such uniform (co-compact) lattices. However, it was demonstrated in \cite{MM} that, up to conjugacy, two elements of finite order could generate only a finite number of arithmetic lattices. In fact, it is proved in \cite{MM} that there are only a finite number of {\em nearly arithmetic} groups, that is groups generated by two elements of finite order that are discrete subgroups of arithmetic groups and are not free on the two generators. The main result of this thesis is the determination of all the finitely many arithmetic lattices $\Gamma$ in the orientation preserving isometry group of hyperbolic $3$-space $\mathbb{H}^3$ generated by an element of order $5$ and an element of order $p\geq 2$, along with the determination of all the associated nearly arithmetic groups. These groups $\Gamma$ will have a presentation of the form \[ \Gamma\cong\langle f,g: f^5=g^p=w(f,g)=\cdots=1 \rangle \] In this Thesis, we find that necessarily \begin{itemize} \item $p\in \{2,3,4,5\}$ \item The total degree of the invariant trace field \[ k\Gamma=\mathbb{Q}(\{tr^2(h):h\in\Gamma\})\] is at most $6$ and at most $4$ for lattices. \item Each orbifold is either a two bridge link of slope $r/s$ surgered with $(5,0)$, $(p,0)$ orbifold Dehn surgery or a Heckoid group with rational slope $r/s\in [0,1]$ and $w(f,g)=(w_{r/s})^r$ with $r\in \{2,3,4,5\}$, and $w_{r/s}$ is a Farey word - described later. \end{itemize} For each such group, we find a discrete and faithful representation in $PSL(2,\mathbb{C})$, identify the rational slope $r/s$ and identify the associated number theoretic data.
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    Towards fast converging lattice sums : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
    (Massey University, 2023) Burrows, Antony
    In the field of solid state physics there are many open questions surrounding the best configuration of packing spheres to calculating binding energies to J/mol accuracy. Many of these problems have attracted attention from individuals in many faculties from mathematics, physics and chemistry over the course of the last four centuries. A significant amount of work has been done modernizing interaction potentials from the early twentieth century by the use of modern computers and quantum chemical software programs extending versions of the most common two-body potential. The historical survey of the methods leading up until the late nineteen eighties serves as the basis for where we step off for much of the analytic techniques for evaluating lattice sums and their use in answering these open questions. Investigations in to the stability of certain packing configurations compared to others in the solid state can be made with the use of fast techniques to evaluate the properties of such systems, many of which are developed here and used throughout the work in the various projects seen below. The aim of this work is to show that the evaluation of lattice constants and the formulae to calculate them can be given in a concise and efficient form with the use of mathematical and numerical methods. Analytical expressions can be found that are given in terms of real exponents and these expressions can be evaluated to arbitrary precision within a satisfactory amount of computer time. In contrast to the infinite structure that forms the lattice in the physical world, the techniques to calculate its sum have evolved from an infinite direct summation to methods that treat the sum associated with the quadratic form of the lattice re-expressing it as a sum of simple functions using number theoretic techniques and treating sums in terms of fast converging series or sums of hyperbolic functions. The results of this investigation are multiple new formulae for the cubic lattice systems, including expressions for the simple cubic lattice and famous Madelung constant in N--dimensions. A new expression was found for the hexagonal close packed structure that is computationally elegant and allowed the examination of the behaviour of the two-body Lennard--Jones potential in terms of the lattice parameters. A single parameter sum was found for the simple cubic system that was used to investigate the effect of pressure on body centered cubic system compared to the face centered cubic system.
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    Efficient Markov bases for Z-polytope sampling : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Manawatū, New Zealand
    (Massey University, 2021) McVeagh, Michael
    In this thesis we study the use of lattice bases for fibre sampling, with particular attention paid to applications in volume network tomography. We use a geometric interpretation of the fibre as a Z-polytope to provide insight into the connectivity properties of lattice bases. Fibre sampling is used when we are interested in fitting a statistical model to a random process that may only be observed indirectly via the underdetermined linear system y = Ax. We consider the observed data y and random variable of interest x to contain count data. The likelihood function for such models requires a summation over the fibre Fy, the set of all non-negative integer vectors x satisfying this equation for some particular y. This can be computationally infeasible when Fy is large. One approach to addressing this problem involves sampling from Fy using a Markov Chain Monte Carlo algorithm, which amounts to taking a random walk through Fy . This is facilitated by a Markov basis: a set of moves that can be used construct such a walk, which is therefore a subset of the kernel of the configuration matrix A. Algebraic algorithms for finding Markov bases based on the theory of Gröbner bases are available, but these can fail when the configuration matrix is large and the calculations become computationally infeasible. Instead, we propose constructing a sampler based on a type of lattice basis we call a column partition lattice basis, defined by a matrix U. Constructing such a basis is computationally much cheaper than constructing a Gröbner basis. It is known that lattice bases are not necessarily Markov bases. We give a condition on the matrix U that guarantees that it is a Markov basis, and show for a certain class of configuration matrices how a U matrix that is a Markov basis can be constructed. Construction of lattice bases that are Markov bases is facilitated when the configuration matrix is unimodular, or has unimodular partitions. We consider configuration matrices from volume network tomography, and give classes of traffic network that have configuration matrices with these desirable properties. If a Markov basis cannot be found, one alternative is to sample from some larger set that includes Fy . We give some larger sets that can be used, subject to certain conditions.