In this study the evolution of the genetic composition of certain species will be replaced by the evolution of the traits that represent these genetic compositions. Depending on the nature of the trait of interest, a scalar valued parameter called the strategy parameter will be assigned to this trait making the simulation of strategy evolution possible. The trait of interest, and therefore the strategy associated, will be the ability of a population to keep its densities within the carrying capacity of the environment they find themselves in. The Shepherd function, on account of its wide use in population simulations as well as composing of exactly such a density parameter, will be the density curbing mechanism of choice in the age-structured population model designed here. An algorithm will be designed to simulate strategy evolution towards an evolutionary stable strategy or ESS that will ensure not only an optimal fit for this environment but also render the population immune against future invasion by other members of the population practising slight variations of this strategy. There are two ways to come by such an optimal strategy without directly involving genetics. The first is game theory, allowing strategists to compete for this position, and the second is with the use of adaptive dynamics, converting winning and loosing instead into tangible mathematics. Combining these two classics will show that the quest is an excersize in strategy optimization, not only from the point of view of an already established population but also from the point of view of an initially small one. It will be interesting!

SUMMARY Mathematical Literacy, Mathematics and Mathematical Sciences is a learning area in the intermediate phase, which forms part of the General Education and Training band. This learning band is level one of the National Qualifications Framework and is overseen by the South African Qualifications Authority. The curriculum of this learning area consists of various learning strands, namely numbers and operations, fractions, patterns, shapes and space, measurement, and data. When learners fail to meet the expectations of the curriculum, mathematics becomes a major assessment concern. If this problem is not identified it could hinder the acquisition of more advanced mathematical skills. The first task in helping a learner who is struggling with mathematics is to identify the problem. For learners to succeed at mathematics they need to go through various developmental phases. Various cognitive processes form part of these phases. Often in a young child�s functioning, cognitive problems arise such as the inability to perform various mathematical tasks. For this reason a cognitive model for mathematics was used to reflect upon six key concepts that influence learning and teaching in the Mathematical Literacy, Mathematics and Mathematical Sciences learning area. These concepts include the categories of representing experience; motivation; individual differences; cognitive categories and cognitive processes; instructional procedures; and conceptual learning. Learners need to make sense out of what is going on during a mathematics lesson. To help learners develop meaning, a teacher provides experiences that foster mental manipulations. Psychologists refer to these mental manipulations as cognitive processes. When a learner is unable to carry out the cognitive processes necessary for task completion, mathematics becomes a major assessment concern. If a learner fails to meet the expectations of the curriculum or fails to carry out the cognitive processes necessary for successful task completion, then, in accordance with the aim of this study, the Intermediate Phase Mathematics Proficiency Test can be used to identify and address this problem. During the construction of this test, care was taken to ensure that the test was cross-culturally adapted. Differential Item Functioning was used to limit the possibility of cultural bias. The Item Response Theory and the Classical Test Theory were also used for item analysis and selection. The test was standardised for English-, Afrikaans- and Sesotho-speaking grade four, five and six learners. During standardisation, separate norms for each term were calculated. These norms are available in both stanines and percentile ranks. The test can also be used qualitatively to determine not only the learning strand in which the learner may be experiencing problems, but also the specific cognitive process, such as receiving, interpreting, organising, applying, remembering and problem solving, which might be preventing the learner from reaching his or her full mathematical potential. The Intermediate Phase Mathematics Proficiency Test is also a reliable and valid measuring instrument since the bias of the assessment measure has been decreased. This was done by eliminating any item that was biased towards a specific cultural group. The test can therefore be used in practice with confidence. In a multicultural society like South Africa, the adaptation of assessment measures and the elimination of bias from psychometric tools forms a vital part of the transformation process. The Intermediate Phase Mathematics Proficiency Test is a multicultural test with South African norms.

The main purpose of this thesis is to give an exposition of and expand the theory of near vector spaces, as first introduced by Andr´e [1]. The notion of a vector space is well known. For this reason the material in this thesis is presented in such a way that the parallels between near vector spaces and vector spaces are apparent. In Chapter 1 several elementary definitions and properties are given. In addition, some important examples that will be referred to throughout this paper are cited. In Chapter 2 the theory of near vector spaces is presented. We start off with some preliminary results in 2.1 and build up to the definition of a regular near vector space in 2.5. In addition, we show how a near vector space can be decomposed into maximal regular subspaces. We conclude this chapter by showing when a near vector space will in fact be a vector space. We follow the format of De Bruynâs thesis; however, both De Bruyn and Andr´e make use of left nearfields to define the near vector spaces. In light of the material we want to present in Chapter 4, it is more standard to use the notation as in the papers by van der Walt, [12], [13]. Thus we develop the material using right nearfields with scalar multiplication on the right of vectors. The third chapter contains some examples of near vector spaces and serves as an illustration of much of the work of Chapter 2. Examples 1, 2 and 3 were used in De Bruynâs thesis. However, on closer inspection, it was revealed that in Example 2, the element (a, 0, 0, d) is omitted as an element of Q(V ). This error is corrected. And in keeping with our use of right nearfields, the necessary changes are made to Example 1 and 3. In particular, the definition of ⦠in Example 3 is adapted and the necessary adjustments are made. We conclude this chapter by developing a theory that allows us to characterise all finite dimensional near vector spaces over Zp, for p a prime. In Chapter 4 we turn our attention to the work done by van der Walt in [12] and [13]. In Section 4.1 we consider the effects that âperturbationsâ in the action of a (right) nearfield F has on the well known structures, the ring of linear transformations of V and the nearring of homogeneous functions of V into itself. This first section sets the scene for the more generalised situation described in 4.2 and leads to the introduction of the nearring of matrices determined by n multiplicatively isomorphic nearfields and a matrix of isomorphisms. We conclude this chapter by summarising some properties of this nearring in 4.3 and 4.4. Note that throughout this paper, â will be used to convey a proper subset, whereas â will convey the possibility of equality.

Thesis (Ed.D.)--Marshall University, 2008. / Title from document title page. Includes abstract. Document formatted into pages: contains x, 222 p. Includes bibliographical references (p. 150-172).

In this thesis, we explore the use of on-shell scattering amplitudes as a way to understand various gravitational phenomena. We show that amplitudes are a viable way of studying certain aspects of gravity and showcase three such novel results here. First is the computation of the deflection angle of both light and gravitational waves due to a massive static body. We compute this from a purely on-shell amplitude perspective and find that the result is in complete agreement with the corresponding calculation in General Relativity. The second is the ability to derive classical results from the amplitudes. In this section we use on-shell scattering amplitudes to derive the perturbative metric of a rotating black hole in a generic form of Einstein gravity that has additional terms cubic in the Riemann tensor. We show that the metric we derive reduces to correct static metric in the zero angular momentum limit. We show that at first order in the coupling, the classical potential can be written to all orders in spin as a differential operator acting on the non-rotating potential. Further we compute the classical impulse and scattering angle of such a black hole. The third is the resolution of a classical discontinuity in N = 1 super gravity. Here we use on-shell methods for massive particles and use them to compute the supersymmetric version of the van Damme-Veltman-Zakharov (vDVZ) discontinuity. We construct the amplitudes of massive gravitinos (the superpartner of massive gravitons) and show that in the massless limit of the gravitinos there is the same discontinuity as found in massive gravity. This method sheds light on intricacies of the discontinuity that is obscured when handled classically.

Includes abstract. / Includes bibliographical references (leaves 42). / Grid diagrams are essential in the new combinatorial version [MOST07] of the Heegaard Floer knot homology, and proving that these homologies are actually knot and link invariants depends on knowing that two grid diagrams representing isotopic links are related by grid moves. The purpose of this paper is to prove this fact. This result has already been proved by Cromwell [CrogS] and Dynnikov [Dyn06]. We present a new proof which is built upon Markov's theorem involving moves on braid words and link isotopy.

Includes bibliographical references. / The Baryon Acoustic Oscillations offer a powerful method of measuring cosmological distances and the expansion history of the Universe. Understanding of the BAO comes from linear physics and allows for accurate predictions of the BAO scale. This will result in accurate measurements of the parameters of the Universe. Currently, most BAO measurements assume a flat cosmology; this work seeks to investigate if the assumption of flatness provides inaccuracies in the measurement process.

Bibliography: leaves 69-71. / Empirical evidence suggesting that world financial markets are incomplete leads to the question of how best to price and hedge contingent claims and derivative securities in incomplete markets. The focus of this dissertation is on a model proposed by Carr, Geman and Madam [7], which combines elements of arbitrage pricing theory with expected utility maximisation to decide whether a risky investment opportunity is worth undertaking or not. An account of the state of the art of pricing and hedging in incomplete markets is followed by a detailed exposition of the new model. A chapter which details the issues which arise when the model is extended treats multiple time periods, continuous time, and an infinite state space. It is not entirely obvious in each case how the model may be extended, and current work is considered along with some new suggestions to address these issues. A small battery of computer simulations based on the proposed multiple period model is performed using a trinomial tree structure. A justification for using the new model rather than finite difference or classical multinomial tree methods is provided in the form of an argument which establishes the validity of a new approach in cases when the Black-Scholes formulation cannot be applied, chiefly when the market is incomplete.

Bibliography: pages 90-94. / A convergence of a mixed finite element method for three-dimensional curved beams with arbitary geometry is investigated. First, the governing equations are derived for linear elastic curved beams with uniformly loaded based on the Timoshenko-Reissner-Mindlin hypotheses. Then, standard and mixed variational problems are formulated. A new norm, equivalent to H¹- type norm, is introduced. By making use of this norm, sufficient conditions for existence and uniqueness of the solutions of the above problems are established for both continuous and discrete cases. The estimates of the optimal order and minimal regularity are then derived for errors in the generalised displacement vector and the internal force vector. These analytical findings are compared with numerical results, verifying the role of reduced integration and the accuracy of the methods.

Includes abstract. / Includes bibliographical references. / In this thesis we use the 1+1+2 covariant approach to General Relativity to study exact solutions and perturbations of rotationally symmetric spacetimes in f(R) gravity, one of the most widely studied classes of fourth order gravity. We begin by introducing f(R) theories of gravity and present the general equations for these theories. We investigate the problem of matching different regions of spacetime, shedding light on the problem of constructing realistic inhomogeneous cosmologies in the context of f(R) gravity. We also study strong lensing in these fourth order theories of gravity derive the lens mass and magnification for the gravitational lens system. We provide an extensive review of both the 1+3 and 1+1+2 covariant approaches to f(R) theories of gravity and give the full system of evolution, propagation and constraint equations of LRS spacetimes. We then determine the conditions for the existence of spherically symmetric vacuum solutions of these fourth order field equations and prove a Jebsen-Birkhoff like theorem for f(R) theories of gravity and the necessary conditions required for the existence of Schwarzschild solution in these theories.