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We generalise the study of cyclotomic matrices - those with all eigenvalues in the interval [-2; 2] - from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined. We introduce the notion of `4-cyclotomic' matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. Six rings OQ( p d) for d = -1;-2;-3;-7;-11;-15 give rise to examples not found in the rational-integer case; in four (d = -1;-2;-3;-7) we recover infinite families as well as sporadic cases. For d = -15;-11;-7;-2, we demonstrate that a maximal cyclotomic graph is necessarily 4- cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.

Numerical methods based on rational Krylov spaces have become an indispensable tool of scientific computing. In this thesis we study rational Krylov spaces by considering rational Krylov decompositions; matrix relations which, under certain conditions, are associated with these spaces. We investigate the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. We derive standard and harmonic Ritz extraction strategies for approximating the eigenpairs of a matrix and for approximating the action of a matrix function onto a vector. While these topics have been considered previously, our approach does not require the last pole to be infinite, which makes the extraction procedure computationally more efficient. Typically, the computationally most expensive component of the rational Arnoldi algorithm for computing a rational Krylov basis is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become poorly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this non orthogonal basis. As a consequence we obtain a more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using our high performance C++ implementation. We develop an iterative algorithm for solving nonlinear rational least squares problems. The difficulty is in finding the poles of a rational function. For this purpose, at each iteration a rational Krylov decomposition is constructed and a modified linear problem is solved in order to relocate the poles to new ones. Our numerical results indicate that the algorithm, called RKFIT, is well suited for model order reduction of linear time invariant dynamical systems and for optimisation problems related to exponential integration. Furthermore, we derive a strategy for the degree reduction of the approximant obtained by RKFIT. The rational function obtained by RKFIT is represented with the aid of a scalar rational Krylov decomposition and an additional coefficient vector. A function represented in this form is called an RKFUN. We develop efficient methods for the evaluation, pole and root finding, and for performing basic arithmetic operations with RKFUNs. Lastly, we discuss RKToolbox, a rational Krylov toolbox for MATLAB, which implements all our algorithms and is freely available from http://rktoolbox.org. RKToolbox also features an extensive guide and a growing number of examples. In particular, most of our numerical experiments are easily reproducible by downloading the toolbox and running the corresponding example files in MATLAB.

Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.

Rational tetrahedra are tetrahedra with rational edges. Heron tetrahedra are tetrahedra with integer edges, integer faces areas and integer volume --- the three-dimensional analogue of Heron triangles. Of course, if a rational tetrahedron has rational face areas and volume then it is easy to scale it up to get a Heron tetrahedron. So we also use `Heron tetrahedra' when we mean tetrahedra with rational edges, areas and volume. The work in this thesis is motivated by Buchholz's paper {\it Perfect Pyramids} [4]. Buchholz examined certain configurations of rational tetrahedra, looking first for tetrahedra with rational volume, and then for Heron tetrahedra. Buchholz left some of the cases he examined unsolved and Chapter 2 is largely devoted to the resolution of these. In Chapters 3 and 4 we expand upon some of Buchholz's results to find infinite families of Heron tetrahedra corresponding to rational points on certain elliptic curves. In Chapters 5 and 6 we blend the ideas of Buchholz in [4] and of Buchholz and MacDougall in [7], and consider rational tetrahedra with edges in arithmetic (AP) or geometric (GP) progression. It turns out that there are no Heron AP or GP tetrahedra, but AP tetrahedra can have rational volume. They can also have one rational face area, although only one AP tetrahedron has been found with a rational face area and rational volume. For GP tetrahedra there are still unsolved cases, but we show that if GP tetrahedra with rational volume exist, then there are only finitely many. The faces of a rational GP tetrahedron are never rational. Much of the work in these two chapters also appeared in the author's Honours thesis, but has been refined and extended here, and is included to give a more complete picture of the work on Heron tetrahedra which has been done to date. In the final chapter we use a different approach and concentrate on the face areas first, instead of the volume. To make it easier (hopefully) to find tetrahedra with all faces having rational area, we place restrictions on the types of faces and number of different faces the tetrahedra have. / Masters Thesis

Providing a logical characterisation of rational agent reasoning has been a long standing challenge in artificial intelligence (AI) research. It is a challenge that is not only of interest for the construction of AI agents, but is of equal importance in the modelling of agent behaviour. The goal of this thesis is to contribute to the formalisation of agent reasoning by showing that the computational limitations of agents is a vital component of modelling rational behaviour. To achieve this aim, both motivational and formal aspects of resource-bounded agents are examined. It is a central argument of this thesis that accounting for computational limitations is critical to the success of agent reasoning, yet has received only limited attention from the broader research community. Consequently, an important contribution of this thesis is in its advancing of motivational arguments in support of the need to account for computational limitations in agent reasoning research. As a natural progression from the motivational arguments, the majority of this thesis is devoted to an examination of propositional approximate logics. These logics represent a step towards the development of resource-bounded agents, but are also applicable to other areas of automated reasoning. This thesis makes a number of contributions in mapping the space of approximate logics. In particular, it draws a connection between approximate logics and knowledge compilation, by developing an approximate knowledge compilation method based on Cadoli and Schaerf??s S-3 family of approximate logics. This method allows for the incremental compilation of a knowledge base, thus reducing the need for a costly recompilation process. Furthermore, each approximate compilation has well-defined logical properties due to its correspondence to a particular S-3 logic. Important contributions are also made in the examination of approximate logics for clausal reasoning. Clausal reasoning is of particular interest due to the efficiency of modern clausal satisfiability solvers and the related research into problem hardness. In particular, Finger's Logics of Limited Bivalence are shown to be applicable to clausal reasoning. This is subsequently shown to logically characterise the behaviour of the well-known DPLL algorithm for determining boolean satisfiability, when subjected to restricted branching.

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No description available.

No description available.