Pure Mathematics

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Centre manifold theory with an application in population modelling.
(2009) Phongi, Eddy Kimba.; Banasiak, Jacek.
There are basically two types of variables in population modelling, global and local variables. The former describes the behavior of the entire population while the latter describes the behavior of individuals within this population. The description of the population using local variables is more detailed, but it is also computationally costly. In many cases to study the dynamics of this population, it is sufficient to focus only on global variables. In applied sciences, to achieve this, the method of aggregation of variables is used. One of methods used to mathematically justify variables aggregation is the centre manifold theory. In this dissertation we provide detailed proofs of basic results of the centre manifold theory and discuss some examples of applications in population modelling.
Complete symmetry groups : a connection between some ordinary differential equations and partial differential equations.
(2008) Myeni, Senzosenkosi Mandlakayise.; Leach, Peter Gavin Lawrence.; O’Hara, J. G.
The concept of complete symmetry groups has been known for some time in applications to ordinary differential equations. In this Thesis we apply this concept to partial differential equations. For any 1+1 linear evolution equation of Lie’s type (Lie S (1881) Uber die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung Archiv fur Mathematik og Naturvidenskab 6 328-368 (translation into English by Ibragimov NH in CRC Handbook of Lie Group Analysis of Differential Equations 2 473-508) containing three and five exceptional point symmetries and a nonlinear equation admitting a finite number of Lie point symmetries, the representation of the complete symmetry group has been found to be a six-dimensional algebra isomorphic to sl(2,R) s A3,1, where the second subalgebra is commonly known as the Heisenberg-Weyl algebra. More generally the number of symmetries required to specify any partial differential equations has been found to equal the number of independent variables of a general function on which symmetries are to be acted. In the absence of a sufficient number of point symmetries which are not solution symmetries one must look to generalized or nonlocal symmetries to remove the deficiency. This is true whether the evolution equation be linear or not. We report Ans¨ atze which provide a route to the determination of the required nonlocal symmetry or symmetries necessary to supplement the point symmetries for the complete specification of the equations. Furthermore we examine the connection of ordinary differential equations to partial differential equations through a common realisation of complete symmetry group. Lastly we revisit the notion of complete symmetry groups and further extend it so that it refers to those groups that uniquely specify classes of equations or systems. This is based on some recent developments pertaining to the properties and the behaviour of such groups in differential equations under the current definition, particularly their representations and realisations for Lie remarkable equations. The results seem to be quite astonishing.
Consequences of architecture and resource allocation for growth dynamics of bunchgrass clones.
(2005) Tomlinson, Kyle Warwick.; O'Connor, Timothy Gordon.; Hearne, John W.; Swart, Johan.
In order to understand how bunchgrasses achieve dominance over other plant growth forms and how they achieve dominance over one another in different environments, it is first necessary to develop a detailed understanding of how their growth strategy interacts with the resource limits of their environment. Two properties which have been studied separately in limited detail are architecture and disproportionate resource allocation. Architecture is the structural layout of organs and objects at different hierarchical levels. Disproportionate resource allocation is the manner in which resources are allocated across objects at each level of hierarchy. Clonal architecture and disproportionate resource allocation may interact significantly to determine the growth ability of clonal plants. These interactions have not been researched in bunchgrasses. This thesis employs a novel simulation technique, functional-structural plant modelling, to investigate how bunchgrasses interact with the resource constraints imposed in humid grasslands. An appropriate functional-structural plant model, the TILLERTREE model, is developed that integrates the architectural growth of bunchgrasses with environmental resource capture and disproportionate resource allocation. Simulations are conducted using a chosen model species Themeda triandra, and the environment is parameterised using characteristics of the Southern Tall Grassveld, a humid grassland type found in South Africa. Behaviour is considered at two levels, namely growth of single ramets and growth of multiple ramets on single bunchgrass clones. In environments with distinct growing and non-growing seasons, bunchgrasses are subjected to severe light depletion during regrowth at the start of each growing season because of the accumulation of dead material in canopy caused by the upright, densely packed manner in which they grow. Simulations conducted here indicate that bunchgrass tillers overcome this resource bottleneck through structural adaptations (etiolation, nonlinear blade mass accretion, residual live photosynthetic surface) and disproportionate resource allocation between roots and shoots of individual ramets that together increase the temporal resource efficiency of ramets by directing more resources to shoot growth and promoting extension of new leaves through the overlying dead canopy. The architectural arrangement of bunchgrasses as collections of tillers and ramets directly leads to consideration of a critical property of clonal bunchgrasses: tiller recruitment. Tiller recruitment is a fundamental discrete process limiting the vegetative growth of bunchgrass clones. Tiller recruitment occurs when lateral buds on parent tillers are activated to grow. The mechanism that controls bud outgrowth has not been elucidated. Based on a literature review, it is here proposed that lateral bud outgrowth requires suitable signals for both carbohydrate and nitrogen sufficiency. Subsequent simulations with the model provide corroborative evidence, in that greatest clonal productivity is achieved when both signals are present. Resource allocation between live structures on clones may be distributed proportionately in response to sink demand or disproportionately in response to relative photosynthetic productivity. Model simulations indicate that there is a trade-off between total clonal growth and individual tiller growth as the level of disproportionate allocation between ramets on ramet groups and between tillers on ramets increases, because disproportionate allocation reduces tiller population size and clonal biomass, but increases individual tiller performance. Consequently it is proposed that different life strategies employed by bunchgrasses, especially annual versus perennial life strategies, may follow more proportionate and less proportionate allocation strategies respectively, because the former favours maximal resource capture and seed production while the latter favours individual competitive ability. Structural disintegration of clones into smaller physiologically integrated units (here termed ramet groups) that compete with one another for resources is a documented property of bunchgrasses. Model simulations in which complete clonal integration is enforced are unable to survive for long periods because resource bottlenecks compromise all structures equally, preventing them from effectively overcoming resource deficits during periods when light is restrictive to growth. Productivity during the period of survival is also reduced on bunchgrass clones with full integration relative to clones that disintegrate because of the inefficient allocation of resources that arises from clonal integration. This evidence indicates that clonal disintegration allows bunchgrass clones both to increase growth efficiency and pre-empt potential death, by promoting the survival of larger ramet groups and removing smaller ramet groups from the system. The discrete nature of growth in bunchgrasses and the complex population dynamics that arise from the architectural growth and the temporal resource dynamics of the environment, may explain why different bunchgrass species dominate under different environments. In the final section this idea is explored by manipulating two species tiller traits that have been shown to be associated with species distributions across non-selective in defoliation regimes, namely leaf organ growth rate and tiller size (mass or height). Simulations with these properties indicate that organ growth rate affects daily nutrient demands and therefore the rate at which tillers are terminated, but had only a small effect on seasonal resource capture. Tiller mass size affects the size of the live tiller population where smaller tiller clones maintain greater numbers of live tillers, which allows them to them to sustain greater biomass over winter and therefore to store more reserves for spring regrowth, suggesting that size may affect seasonal nitrogen capture. The greatest differences in clonal behaviour are caused by tiller height, where clones with shorter tillers accumulate substantially more resources than clones with taller tillers. This provides strong evidence there is trade-off for bunchgrasses between the ability to compete for light and the ability to compete for nitrogen, which arises from their growth architecture. Using this evidence it is proposed that bunchgrass species will be distributed across environments in response to the nitrogen productivity. Shorter species will dominate at low nitrogen productivity, while taller species dominate at high nitrogen productivity. Empirical evidence is provided in support of this proposal.
Analysis of mixed convection in an air filled square cavity.
(2010) Ducasse, Deborah S.; Sibanda, Precious.
A steady state two-dimensional mixed convection problem in an air filled square unit cavity has been numerically investigated. Two different cases of heating are investigated and compared. In the first case, the bottom wall was uniformly heated, the side walls were linearly heated and the top moving wall was heated sinusoidally. The second case differed from the first in that the side walls were instead uniformly cooled. This investigation is an extension of the work by Basak et al. [6, 7] who investigated mixed convection in a square cavity with similar boundary conditions to the cases listed above with the exception of the top wall which was well insulated. In this dissertation, their work is extended to include a sinusoidally heated top wall. The nonlinear coupled equations are solved using the Penalty Galerkin Finite Element Method. Stream function and isotherm results are found for various values of the Reynolds number and the Grashof number. The strength of the circulation is seen to increase with increasing Grashof number and to decrease with increasing Reynolds number for both cases of heating. A comparison is made between the stream function and isotherm results for the two cases. The results for the rate of heat transfer in terms of the Nusselt number are discussed. Both local and average Nusselt number results are presented and discussed. The average Nusselt number is found using Simpson's 1/3rd rule. The rate of heat transfer is found to be higher at all four walls for the case of cooled side walls than that of linearly heated side walls.
Fischer-Clifford theory and character tables of group extensions.
(1998) Mpono, Zwelethemba Eugene.; Moori, Jamshid.
The smallest Fischer sporadic simple group Fi22 is generated by a conjugacy class D of 3510 involutions called 3-transpositions such that the product of any noncommuting pair is an element of order 3. In Fi22 there are exactly three conjugacy classes of involutions denoted by D, T and N and represented in the ATLAS [26] by 2A, 2B and 2C, containing 3510, 1216215 and 36486450 elements with corresponding centralizers 2·U(6,2), (2 x 2~+8:U(4,2)):2 and 25+8:(83 X 32:4) respectively. In Fi22 , we have Npi22(26) = 26:8P(6,2), where 26 is a 2B-pure group, and thus the maximal subgroup 26:8P(6, 2) of Fi22 is a 2-local subgroup. The full automorphism group of Fi22 is denoted by Fi22 . In Fi22 , there are three involutory outer automorphisms of Fi22 which are denoted bye, f and 0 and represented in the ATLAS [26] by 2D, 2F and 2E respectively. We obtain that Fi22 = Fi22 :(e) and it can be easily shown that Fi22 = Fi22 :(e) = Fi22 :(f) = Fi22 :(0). As e, f and 0 act on Fi22 , then we obtain the subgroups CPi22 (e) rv 0+(8,2):83, CPi22 (f) rv 8P(6,2) x 2 and CPi22 (()) rv 26:0-(6,2) of Fi22 which are generated by CD(e), Cn(f) and CD(0) respectively. In this thesis we are concerned with the construction of the character tables of certain groups which are associated with Fi22 and its automorphism group Fi22 . We use the technique of the Fischer-Clifford matrices to construct the character tables of these groups, which are split extensions. These groups are 26:8P(6, 2), 26:0-(6,2) and 27:8P(6, 2). The study of the group 26:8P(6, 2) is essential, as the other groups studied in this thesis are related to it. The groups 8P(6,2) and 0- (6,2) of 6 x 6 matrices over GF(2), played crucial roles in our construction of the group 8P(6, 2) as a group of 7 x 7 matrices over GF(2) which would act on 27 . Also the character table of 25:86 , the affine subgroup of 8P(6, 2) fixing a nonzero vector in 26 , is constructed by using the technique of the Fischer-Clifford matrices. This character table is used in the construction of the character table 26:SP(6, 2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest version of GAP.
Coherent structures and symmetry properties in nonlinear models used in theoretical physics.
(1994) Harin, Alexander O.; Leach, Peter Gavin Lawrence.; Barashenkov, I. V.
This thesis is devoted to two aspects of nonlinear PDEs which are fundamental for the understanding of the order and coherence observed in the underlying physical systems. These are symmetry properties and soliton solutions. We analyse these fundamental aspects for a number of models arising in various branches of theoretical physics and appli ed mathematics. We start with a fluid model of a plasma in the case of a general polytropic process. We propose a method of the analysis of unmagnetized travelling structures, alternative to the conventional formalism of Sagdeev 's pseudopotential. This method is then utilized to obtain the existence domain for compressive solitons and to establish the absence of rarefactive solitons and monotonic double layers in a two-component plasma. The second class of models under consideration arises in (2+1)-dimensional condensed matter physics. These are the Abelian gauge theories with Chern-Simons term, which are currently considered as candidates for the description of high-Te superconductivity and fra ctional quantum Hall effect. The emphasis here is on nonrelativistic theories. The standard model of a self-gravitating gas of nonrelativistic bosons coupled to the Chern-Simons gauge field is capable of describing asymptotically vanishing field configurations , such as lump-like solitons. We formulate an alternative model, which describes systems of repulsive particles with a background electric charge and allows to incorporate asymptotically nonvanishing configurations, such as condensate and its topological excitations. We demonstrate the absence of the condensate state in the standard nonrelativistic gauge theory and relate this fact to the inadequate Lagrangian formulation of its nongauged precursor. Using an appropriate modification of this Lagrangian as a basis for the gauge theory naturally leads to the new model. Reformulating it as a constrained Hamiltonian system allows us to find two self-duality limit s and construct a large variety of self-dual solutions. We demonstrate the equivalence of the model with the background charge and the standard model in the external magnetic field. Finally we discuss nontopological bubble solutions in Chem-Simons-Maxwell theories and demonstrate their absence in nonrelativistic theories. Finally, we consider a model of a nonhomogeneous nonlinear string. We continue the group theoretical classification of the string equations initiated by Ibragimov et al. and present their preliminary group classification with respect to a countable dimensional subalgebra of their equivalence algebra. This subalgebra is an extension of the 10-dimensional subalgebra considered by Ibragimov et al. Our main result here is a table of non-equivalent equations possessing an additional symmetry.
Extensions and generalisations of Lie analysis.
(1995) Govinder, Keshlan Sathasiva.; Leach, Peter Gavin Lawrence.
The Lie theory of extended groups applied to differential equations is arguably one of the most successful methods in the solution of differential equations. In fact, the theory unifies a number of previously unrelated methods into a single algorithm. However, as with all theories, there are instances in which it provides no useful information. Thus extensions and generalisations of the method (which classically employs only point and contact transformations) are necessary to broaden the class of equations solvable by this method. The most obvious extension is to generalised (or Lie-Backlund) symmetries. While a subset of these, called contact symmetries, were considered by Lie and Backlund they have been thought to be curiosities. We show that contact transformations have an important role to play in the solution of differential equations. In particular we linearise the Kummer-Schwarz equation (which is not linearisable via a point transformation) via a contact transformation. We also determine the full contact symmetry Lie algebra of the third order equation with maximal symmetry (y'''= 0), viz sp(4). We also undertake an investigation of nonlocal symmetries which have been shown to be the origin of so-called hidden symmetries. A new procedure for the determination of these symmetries is presented and applied to some examples. The impact of nonlocal symmetries is further demonstrated in the solution of equations devoid of point symmetries. As a result we present new classes of second order equations solvable by group theoretic means. A brief foray into Painleve analysis is undertaken and then applied to some physical examples (together with a Lie analysis thereof). The close relationship between these two areas of analysis is investigated. We conclude by noting that our view of the world of symmetry has been clouded. A more broad-minded approach to the concept of symmetry is imperative to successfully realise Sophus Lie's dream of a single unified theory to solve differential equations.
Parameters related to fractional domination in graphs.
(1995) Erwin, D. J.; Swart, Hendrika Cornelia Scott.; Dankelmann, Peter A.
The use of characteristic functions to represent well-known sets in graph theory such as dominating, irredundant, independent, covering and packing sets - leads naturally to fractional versions of these sets and corresponding fractional parameters. Let S be a dominating set of a graph G and f : V(G)~{0,1} the characteristic function of that set. By first translating the restrictions which define a dominating set from a set-based to a function-based form, and then allowing the function f to map the vertex set to the unit closed interval, we obtain the fractional generalisation of the dominating set S. In chapter 1, known domination-related parameters and their fractional generalisations are introduced, relations between them are investigated, and Gallai type results are derived. Particular attention is given to graphs with symmetry and to products of graphs. If instead of replacing the function f : V(G)~{0,1} with a function which maps the vertex set to the unit closed interval we introduce a function f' which maps the vertex set to {0, 1, ... ,k} (where k is some fixed, non-negative integer) and a corresponding change in the restrictions on the dominating set, we obtain a k-dominating function. In chapter 2 corresponding k-parameters are considered and are related to the classical and fractional parameters. The calculations of some well known fractional parameters are expressed as optimization problems involving the k- parameters. An e = 1 function is a function f : V(G)~[0,1] which obeys the restrictions that (i) every non-isolated vertex u is adjacent to some vertex v such that f(u)+f(v) = 1, and every isolated vertex w has f(w) = 1. In chapter 3 a theory of e = 1 functions and parameters is developed. Relationships are traced between e = 1 parameters and those previously introduced, some Gallai type results are derived for the e = 1 parameters, and e = 1 parameters are determined for several classes of graphs. The e = 1 theory is applied to derive new results about classical and fractional domination parameters.
Algebraic properties of ordinary differential equations.
(1995) Leach, Peter Gavin Lawrence.
In Chapter One the theoretical basis for infinitesimal transformations is presented with particular emphasis on the central theme of this thesis which is the invariance of ordinary differential equations, and their first integrals, under infinitesimal transformations. The differential operators associated with these infinitesimal transformations constitute an algebra under the operation of taking the Lie Bracket. Some of the major results of Lie's work are recalled. The way to use the generators of symmetries to reduce the order of a differential equation and/or to find its first integrals is explained. The chapter concludes with a summary of the state of the art in the mid-seventies just before the work described here was initiated. Chapter Two describes the growing awareness of the algebraic properties of the paradigms of differential equations. This essentially ad hoc period demonstrated that there was value in studying the Lie method of extended groups for finding first integrals and so solutions of equations and systems of equations. This value was emphasised by the application of the method to a class of nonautonomous anharmonic equations which did not belong to the then pantheon of paradigms. The generalised Emden-Fowler equation provided a route to major development in the area of the theory of the conditions for the linearisation of second order equations. This was in addition to its own interest. The stage was now set to establish broad theoretical results and retreat from the particularism of the seventies. Chapters Three and Four deal with the linearisation theorems for second order equations and the classification of intrinsically nonlinear equations according to their algebras. The rather meagre results for systems of second order equations are recorded. In the fifth chapter the investigation is extended to higher order equations for which there are some major departures away from the pattern established at the second order level and reinforced by the central role played by these equations in a world still dominated by Newton. The classification of third order equations by their algebras is presented, but it must be admitted that the story of higher order equations is still very much incomplete. In the sixth chapter the relationships between first integrals and their algebras is explored for both first order integrals and those of higher orders. Again the peculiar position of second order equations is revealed. In the seventh chapter the generalised Emden-Fowler equation is given a more modern and complete treatment. The final chapter looks at one of the fundamental algebras associated with ordinary differential equations, the three element 8£(2, R), which is found in all higher order equations of maximal symmetry, is a fundamental feature of the Pinney equation which has played so prominent a role in the study of nonautonomous Hamiltonian systems in Physics and is the signature of Ermakov systems and their generalisations.
The paradigms of mechanics : a symmetry based approach.
(1996) Lemmer, Ryan Lee.; Leach, Peter Gavin Lawrence.
An overview of the historical developments of the paradigms of classical mechanics, the free particle, oscillator and the Kepler problem, is given ito (in terms of) their conserved quantities. Next, the orbits of the three paradigms are found from quadratic forms. The quadratic forms are constructed using first integrals found by the application of Poisson's theorem. The orbits are presented ito expanding surfaces defined by the quadratic forms. The Lie and Noether symmetries of the paradigms are investigated. The free particle is discussed in detail and an overview of the work done on the oscillator and Kepler problem is given. The Lie and Noether theories are compared from various aspects. A technical description of Lie groups and algebras is given. This provides a basis for a discussion of the historical development of the paradigms of mechanics ito their group properties. Lastly the paradigms are discussed ito of Quantum Mechanics.
Noether's theorem and first integrals of ordinary differential equations.
(1997) Moyo, Sibusiso.; Leach, Peter Gavin Lawrence.
The Lie theory of extended groups is a practical tool in the analysis of differential equations, particularly in the construction of solutions. A formalism of the Lie theory is given and contrasted with Noether's theorem which plays a prominent role in the analysis of differential equations derivable from a Lagrangian. The relationship between the Lie and Noether approach to differential equations is investigated. The standard separation of Lie point symmetries into Noetherian and nonNoetherian symmetries is shown to be irrelevant within the context of nonlocality. This also emphasises the role played by nonlocal symmetries in such an approach.
First integrals for the Bianchi universes : supplementation of the Noetherian integrals with first integrals obtained by using Lie symmetries.
(1997) Pantazi, Hara.; Leach, Peter Gavin Lawrence.
No abstract available.
Ermakov systems : a group theoretic approach.
(1993) Govinder, Keshlan Sathasiva.; Leach, Peter Gavin Lawrence.
The physical world is, for the most part, modelled using second order ordinary differential equations. The time-dependent simple harmonic oscillator and the Ermakov-Pinney equation (which together form an Ermakov system) are two examples that jointly and separately describe many physical situations. We study Ermakov systems from the point of view of the algebraic properties of differential equations. The idea of generalised Ermakov systems is introduced and their relationship to the Lie algebra sl(2, R) is explained. We show that the 'compact' form of generalized Ermakov systems has an infinite dimensional Lie algebra. Such algebras are usually associated only with first order equations in the context of ordinary differential equations. Apart from the Ermakov invariant which shares the infinite-dimensional algebra of the 'compact' equation, the other three integrals force the dimension of the algebra to be reduced to the three of sl(2, R). Subsequently we establish a new class of Ermakov systems by considering equations invariant under sl(2, R) (in two dimensions) and sl(2, R) EB so(3) (in three dimensions). The former class contains the generalized Ermakov system as a special case in which the force is velocity-independent. The latter case is a generalization of the classical equation of motion of the magnetic monopole which is well known to possess the conserved Poincare vector. We demonstrate that in fact there are three such vectors for all equations of this type.
On the theory and examples of group extensions.
(1999) Rodrigues, Bernardo Gabriel.; Moori, Jamshid.
The work described in this dissertation was largely motivated by the aim of producing a survey on the theory of group extensions. From the broad scope of the theory of group extensions we single out two aspects to discuss, namely the study of the split and the non-split cases and give examples of both. A great part of this dissertation is dedicated to the study of split extensions. After setting the background theory for the study of the split extensions we proceed in exploring the ramifications of this concept within the development of the group structure and consequently investigate well known products which are its derived namely the holomorph, and the wreath product. The theory of group presentations provides in principle the necessary tools that permit the description of a group by means of its generators and relators. Through this knowledge we give presentations for the groups of order pq,p2q and p3. Subsequently using a classical result of Gaschutz we investigate the split extensions of non-abelian groups in which the normal subgroup is either a non-abelian normal nilpotent group or a non-abelian normal solvable group. We also study other cases of split extensions such as the affine subgroups of the general linear and the symplectic groups. It is expected that some of the results obtained will provide a theoretical algorithm to describe these affine subgroups. A particular case of the non-split extensions is discussed as the Frattini extensions. In fact a simplest example of a Frattini extension is a non-split extension in which the kernel of an epimorphism e is an irreducible G-module.
Clifford-Fischer theory applied to certain groups associated with symplectic, unitary and Thompson groups.
(2012) Basheer, Ayoub Basheer Mohammed.; Moori, Jamshid.
The character table of a finite group is a very powerful tool to study the groups and to prove many results. Any finite group is either simple or has a normal subgroup and hence will be of extension type. The classification of finite simple groups, more recent work in group theory, has been completed in 1985. Researchers turned to look at the maximal subgroups and automorphism groups of simple groups. The character tables of all the maximal subgroups of the sporadic simple groups are known, except for some maximal subgroups of the Monster M and the Baby Monster B. There are several well-developed methods for calculating the character tables of group extensions and in particular when the kernel of the extension is an elementary abelian group. Character tables of finite groups can be constructed using various theoretical and computational techniques. In this thesis we study the method developed by Bernd Fischer and known nowadays as the theory of Clifford-Fischer matrices. This method derives its fundamentals from the Clifford theory. Let G = N·G, where N C G and G/N = G, be a group extension. For each conjugacy class [gi]G, we construct a non-singular square matrix Fi, called a Fischer matrix. Once we have all the Fischer matrices together with the character tables (ordinary or projective) and fusions of the inertia factor groups into G, the character table of G is then can be constructed easily. In this thesis we apply the coset analysis technique (this is a method to find the conjugacy classes of group extensions) together with theory of Clifford-Fischer matrices to calculate the ordinary character tables of seven groups of extensions type, in which four are non-split and three are split extensions. These groups are of the forms: 21+8 + ·A9, 37:Sp(6, 2), 26·Sp(6, 2), 25·GL(5, 2), 210:(U5(2):2), 21+6 − :((31+2:8):2) and 22n·Sp(2n, 2) and 28·Sp(8, 2). In addition we give some general results on the non-split group 22n·Sp(2n, 2).
Singularity and symmetry analysis of differential sequences.
(2009) Maharaj, Adhir.; Leach, Peter Gavin Lawrence.; Euler, Marianna.
We introduce the notion of differential sequences generated by generators of sequences. We discuss the Riccati sequence in terms of symmetry analysis, singularity analysis and identification of the complete symmetry group for each member of the sequence. We provide their invariants and first integrals. We propose a generalisation of the Riccati sequence and investigate its properties in terms of singularity analysis. We find that the coefficients of the leading-order terms and the resonances obey certain structural rules. We also demonstrate the uniqueness of the Riccati sequence up to an equivalence class. We discuss the properties of the differential sequence based upon the equation ww''−2w12 = 0 in terms of symmetry and singularity analyses. The alternate sequence is also discussed. When we analyse the generalised equation ww'' − (1 − c)w12 = 0, we find that the symmetry properties of the generalised sequence are the same as for the original sequence and that the singularity properties are similar. Finally we discuss the Emden-Fowler sequence in terms of its singularity and symmetry properties.
Evolutionary dynamics of coexisting species.
(2000) Muir, Peter William.; Apaloo, Joseph.; Hearne, John W.
Ever since Maynard-Smith and Price first introduced the concept of an evolutionary stable strategy (ESS) in 1973, there has been a growing amount of work in and around this field. Many new concepts have been introduced, quite often several times over, with different acronyms by different authors. This led to other authors trying to collect and collate the various terms (for example Lessard, 1990 & Eshel, 1996) in order to promote better understanding ofthe topic. It has been noticed that dynamic selection did not always lead to the establishment of an ESS. This led to the development ofthe concept ofa continuously stable strategy (CSS), and the claim that dynamic selection leads to the establishment of an ESSif it is a CSS. It has since been proved that this is not always the case, as a CSS may not be able to displace its near neighbours in pairwise ecological competitions. The concept of a neighbourhood invader strategy (NIS) was introduced, and when used in conjunction with the concept of an ESS, produced the evolutionary stable neighbourhood invader strategy (ESNIS) which is an unbeatable strategy. This work has tried to extend what has already been done in this field by investigating the dynamics of coexisting species, concentrating on systems whose dynamics are governed by Lotka-Volterra competition models. It is proved that an ESNIS coalition is an optimal strategy which will displace any size and composition of incumbent populations, and which will be immune to invasions by any other mutant populations, because the ESNIS coalition, when it exists, is unique. It has also been shown that an ESNIS coalition cannot exist in an ecologically stable state with any finite number of strategies in its neighbourhood. The equilibrium population when the ESNIS coalition is the only population present is globally stable in a n-dimensional system (for finite n), where the ESNIS coalition interacts with n - 2 other strategies in its neighbourhood. The dynamical behaviour of coexisting species was examined when the incumbent species interacted with various invading species. The different behaviour ofthe incumbent population when invaded by a coalition using either an ESNIS or an NIS phenotype underlines the difference in the various strategies. Similar simulations were intended for invaders who were using an ESS phenotype, but unfortunately the ESS coalition could not be found. If the invading coalition use NIS phenotypes then the outcome is not certain. Some, but not all of the incumbents might become extinct, and the degree to which the invaders flourish is very dependent on the nature ofthe incumbents. However, if the invading species form an ESNIS coalition, one is certain of the outcome. The invaders will eliminate the incumbents, and stabilise at their equilibrium populations. This will occur regardless of the composition and number of incumbent species, as the ESNIS coalition forms a globally stable equilibrium point when it is at its equilibrium populations, with no other species present. The only unknown fact about the outcome in this case is the number ofgenerations that will pass before the system reaches the globally stable equilibrium consisting ofjust the ESNIS. For systems whose dynamics are not given by Lotka-Volterra equations, the existence ofa unique, globally stable ESNIS coalition has not been proved. Moreover, simulations of a non Lotka-Volterra system designed to determine the applicability ofthe proof were inconclusive, due to the ESS coalition not having unique population sizes. Whether or not the proof presented in this work can be extended to non Lotka-Volterra systems remains to be determined.
Measurement and modelling of households' demand and access to basic water in relation to the rapidly increasing household numbers in South Africa.
(2010) Chidozie, Nnadozie Remigius.; Leach, Peter Gavin Lawrence.
Service delivery in post-apartheid South Africa has become a topical issue both in the academia and the political arena . The rise of social movements, the xenophobic tensions of May 2008 and protest actions could be noted as the major traits of post-apartheid South Africa. Though there are divergent views on the underlying causes of these protests, lack of service delivery has most significantly been at the centre stage. In this thesis we investigate the relationship between household/population changes and the demand for piped-water connection in South Africa. There is an ample, albeit at times of questionable accuracy, supply of statistics from official and other sources. These statistics are both the source of inspiration of particular societal measures to be investigated and a gauge of the accuracy of the mathematical/statistical modelling which is the central feature of this project. We construct mathematical/statistical models which take into account demographic constituents of the problem using differential equations for modelling household dynamics and we also investigate the interaction of demographic parameters and the demand for piped-water connection using multivariate statistical techniques. The results show that with a boost in delivery the rich provinces seem to be in better standing of meeting targets and that the increasing demand in household-based services could be most significantly attributed to the fragmentation of households against other demographic processes like natural increase in population and net migration. The results imply that in as much as service delivery policies and programmes should focus on formerly disadvantaged poor communities, adequate provisions for increasing service demands in urban centres should also be a priority in view of the increasing in-migration from rural areas as households fragment. Most of the findings/results are in tabular and graphical forms for easy understanding of the reader.
On free convection and heat transfer in a micropolar fluid flow past a moving semi-infinite plate.
(2012) Tessema, Kassahun Mengist.; Sibanda, Precious.
In this dissertation we investigate free convective heat and mass transfer in micropolar fluid flow past a moving semi-infinite vertical porous plate in the presence of a magnetic field. The aim of this study was to use recent semi-numerical methods such as the successive linearisation method and the spectral-homotopy analysis method to study the effects of viscous heating and the effects of different fluid parameters. The governing boundary layer equations for linear momentum, angular momentum (microrotation), temperature and concentration profiles are transformed to a system of ordinary differential equations and solved using the successive linearisation method and the spectral-homotopy analysis method. The accuracy of the solutions was determined by comparison with numerical approximations obtained using the Matlab bvp4c solver. The influences of the micropolar parameter, Darcy number, Prandtl number, Schmidt number, magnetic parameter, heat absorption parameter, Soret and Dufour numbers, local Reynolds number and Grashof number on velocity, microrotation, temperature and concentration profiles were determined. The results obtained are presented graphically and in tabular form.
On the theory of the frobenius groups.
(2012) Perumal, Pragladan.; Moori, Jamshid.
The Frobenius group is an example of a split extension. In this dissertation we study and describe the properties and structure of the group. We also describe the properties and structure of the kernel and complement, two non-trivial subgroups of every Frobenius group. Examples of Frobenius groups are included and we also describe the characters of the group. Finally we construct the Frobenius group 292 : SL(2, 5) and then compute it's Fischer matrices and character table.

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