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Free-surface modelling (FSM) is a highly relevant and computationally intensive area of study in modern computational fluid dynamics. The Elemental software suite currently under development offers FSMcapability, and employs a preconditioned GMRES solver in an attempt to effect fast solution times. In terms of potential solver performance however, multigrid methods can be considered state-of-the-art. This work details the investigation into the use of AlgebraicMultigrid (AMG) as a high performance solver tool for use as black box plug-in for Elemental FSM. Special attention was given to the development of novel and robust methods of addressing AMG setup costs in addition to transcribing the solver to efficient C++ object-oriented code. This led to the development of the so-called Freeze extension of the basic algebraic multigrid method in an object-oriented C++ programming environment. The newly developed Freeze method reduces setup costs by periodically performing the setup procedure in an automatic and robust manner. The developed technology was evaluated in terms of robustness, stability and speed by applying it to benchmark FSM problems on structured and unstructured meshes of various sizes. This evaluation yielded a number of conclusive findings. First, the developed Freeze method reduced setup times by an order of magnitude. Second, the developed AMG solver offered substantial performance increases over the preconditioned GMRES method. In this way, it is proposed that this work has furthered the state-of-the-art of algebraic multigrid methods applied in the context of free-surface modelling. / Dissertation (MEng)--University of Pretoria, 2011. / Mechanical and Aeronautical Engineering / unrestricted

In this thesis, we define a certain group of order 160, which we call a hyperpentagonal group, and we prove that every totally real D5-extension of the rationals ramified at only one prime is contained in a hyperpentagonal extension of the rationals. This generalizes a result of Doud and Childers (originally conjectured by Wong) that every totally real S3 extension of the rationals ramified at only one prime is contained in an S4 extension.

<div> We introduce the quantum toroidal superalgebra E_m|nassociated with the Lie superalgebra gl_m|n and initiate its study. For each choice of parity "s" of gl_m|n, a corresponding quantum toroidal superalgebra E_s is defined. </div><div> </div><div><br></div><div>To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed. </div><div><br></div><div>The superalgebra E_s contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra U_q sl̂_m|n with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E_s, which exchanges the vertical and horizontal subalgebras.</div><div><br></div><div>If <i>m</i> and <i>n</i> are different and "s" is standard, we give a construction of level 1 E_m|n-modules through vertex operators. We also construct an evaluation map from E_m|n(q₁,q₂,q₃) to the quantum affine algebra U_q gl̂_m|n at level c=q₃^(m-n)/2.</div>

This thesis deals with the problem of estimating statistical distributions from data. In the first part, the method of moments is used in combination with computational algebraic techniques in order to estimate parameters coming from local Dirac mixtures and their convolutions. The second part focuses on the nonparametric setting, in particular on combinatorial and algebraic aspects of the estimation of log-concave distributions.

<div>Let <i>K₀</i>(Var<i>_k</i>) be the Grothendieck group of varieties over a field <i>k</i>. We construct an exact category, denoted Add(Var_<i>k</i>)_<i>S</i>, such that there is a surjection <i>K₀</i>(Var<i>k</i>)→<i>K₀</i>(Add(Var<i>_k</i>)_<i>S</i>).If we consider only zero dimensional varieties, then this surjection is an isomorphism. Like <i>K₀</i>(Var<i>_k</i>), the group K_<i>0</i>(Add(Var_<i>k</i>)<i>_S</i>) is also generated by isomorphism classes of varieties,and we construct motivic measures on <i>K₀</i>(Add(Var<i>_k</i>)<i>_S</i>) including the Euler characteristic if <i>k</i>=<i>C</i>, and point counting measures and the zeta function if <i>k</i> is finite.<br></div>

In this thesis we explore the geometric representation theory of shifted quantum affine algebras 𝒜^𝜇, using the critical K-theory of certain moduli spaces of infinite flags of quiver representations resembling the moduli of quasimaps to Nakajima quiver varieties. These critical K-theories become 𝒜^𝜇-modules via the so-called critical R-matrix 𝑅, which generalizes the geometric R-matrix of Maulik, Okounkov, and Smirnov. In the asymptotic limit corresponding to taking infinite instead of finite flags, singularities appear in 𝑅 and are responsible for the shift in 𝒜^𝜇. The result is a geometric construction of interesting infinite-dimensional modules in the category 𝒪 of 𝒜^𝜇, including e.g. the pre-fundamental modules previously introduced and studied algebraically by Hernandez and Jimbo. Following Nekrasov, we provide a very natural geometric definition of qq-characters for our asymptotic modules compatible with the pre-existing definition of q-characters. When 𝒜^𝜇 is the shifted quantum toroidal gl₁ algebra, we construct asymptotic modules DT_𝜇 and PT_𝜇 whose combinatorics match those of (1-legged) vertices in Donaldson--Thomas and Pandharipande--Thomas theories. Such vertices control enumerative invariants of curves in toric 3-folds, and finding relations between (equivariant, K-theoretic) DT and PT vertices with descendent insertions is a typical example of a wall-crossing problem. We prove a certain duality between our DT_𝜇 and PT_𝜇 modules which, upon taking q-/qq-characters, provides one such wall-crossing relation.

In this dissertation, we study morphisms of smooth complex projective varieties to the projective line with at most two singular fibres. We show that if such a morphism has at most one singular fibre, then the domain of the morphism is uniruled and the morphism admits algebraic sections. We reach the same conclusions, but with algebraic genus zero multisections instead of algebraic sections, if the morphism has at most two singular fibres and the first Chern class of the domain of the morphism is supported in a single fibre of the morphism. To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon's virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.

Knot theory has rapidly expanded in recent years. New representations of braid groups led to an extremely powerful polynomial invariant, the Jones polynomial. Combinatorics applied to knot and link diagrams led to generalizations. Knot theory also has connections with other fields such as statistical mechanics and quantum field theory, and has applications in determining how certain enzymes act on DNA molecules, for example. The principal objective of this dissertation is to study the relations between knots and algebraic structures called quandles. A quandle is a set with a binary operation satisfying some properties related to the three Reidemeister moves. The study of quandles in relation to knot theory was intitiated by Joyce and Matveev. Later, racks and their (co)homology theory were defined by Fenn and Rourke. The rack (co)homology was also studied by Grana from the viewpoint of Hopf algebras. Furthermore, a modified definition of homology theory for quandles was introduced by Carter, Jelsovsky, Kamada, Langford, and Saito to define state-sum invariants for knots and knotted surfaces, called quandle cocycle invariants. This dissertation studies the quandle cocycle invariants using extensions of quandles and knot colorings. We obtain a coloring of a knot by assigning elements of a quandle to the arcs of the knot diagram. Such colorings are used to define knot invariants by state-sum. For a given coloring, a 2-cocycle is assigned at each crossing as the Boltzmann weight. The product of the weights over all crossings is the contribution to the state-sum, which is the formal summation of the contributions over all possible colorings of the given knot diagram by a given quandle. Generalizing the cocycle invariant for knots to links, we define two kinds of invariants for links: a component-wise invariant, and an invariant defined as families of vectors. Abelian extensions of quandles are also defined and studied. We give a formula for creating infinite families of abelian extensions of Alexander quandles. These extensions give rise to explicit formulas for computing 2-cocycles. The theory of quandle extensions parallels that of groups. Moreover, we investigate the notion of extending colorings of knots using quandle extensions. In particular, we show how the obstruction to extending the coloring contributes to the non-trivial terms of the cocycle invariants for knots and links. Moreover, we demonstrate the relation between these new cocycle invariants and Alexander matrices.

Indiana University-Purdue University Indianapolis (IUPUI) / We study the Gaudin model associated to Lie algebras of classical types. First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple. Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian.