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31 
Stratification and domination in graphs.January 2006 (has links)
In a recent manuscript (Stratification and domination in graphs. Discrete Math. 272 (2003), 171185) a new mathematical framework for studying domination is presented. It is shown that the domination number and many domination related parameters can be interpreted as restricted 2stratifications or 2colorings. This framework places the domination number in a new perspective and suggests many other parameters of a graph which are related in some way to the domination number. In this thesis, we continue this study of domination and stratification in graphs. Let F be a 2stratified graph with one fixed blue vertex v specified. We say that F is rooted at the blue vertex v. An Fcoloring of a graph G is a redblue coloring of the vertices of G such that every blue vertex v of G belongs to a copy of F (not necessarily induced in G) rooted at v. The Fdomination number yF(GQ of G is the minimum number of red vertices of G in an Fcoloring of G. Chapter 1 is an introduction to the chapters that follow. In Chapter 2, we investigate the Xdomination number of prisms when X is a 2stratified 4cycle rooted at a blue vertex where a prism is the cartesian product Cn x K2, n > 3, of a cycle Cn and a K2. In Chapter 3 we investigate the Fdomination number when (i) F is a 2stratified path P3 on three vertices rooted at a blue vertex which is an endvertex of the F3 and is adjacent to a blue vertex and with the remaining vertex colored red. In particular, we show that for a tree of diameter at least three this parameter is at most twothirds its order and we characterize the trees attaining this bound. (ii) We also investigate the Fdomination number when F is a 2stratified K3 rooted at a blue vertex and with exactly one red vertex. We show that if G is a connected graph of order n in which every edge is in a triangle, then for n sufficiently large this parameter is at most (n — /n)/2 and this bound is sharp. In Chapter 4, we further investigate the Fdomination number when F is a 2 stratified path P3 on three vertices rooted at a blue vertex which is an endvertex of the P3 and is adjacent to a blue vertex with the remaining vertex colored red. We show that for a connected graph of order n with minimum degree at least two this parameter is bounded above by (n —1)/2 with the exception of five graphs (one each of orders four, five and six and two of order eight). For n > 9, we characterize those graphs that achieve the upper bound of (n — l)/2. In Chapter 5, we define an fcoloring of a graph to be a redblue coloring of the vertices such that every blue vertex is adjacent to a blue vertex and to a red vertex, with the red vertex itself adjacent to some other red vertex. The fdomination number yz{G) of a graph G is the minimum number of red vertices of G in an fcoloring of G. Let G be a connected graph of order n > 4 with minimum degree at least 2. We prove that (i) if G has maximum degree A where A 4 with maximum degree A where A 5 with maximum degree A where 3 / Thesis (Ph.D.)University of KwaZuluNatal, Pietermaritzburg, 2006.

32 
Realistic charged stellar modelsKomathiraj, Kalikkuddy. January 2007 (has links)
In this thesis we seek exact solutions to the isotropic EinstienMaxwell system that model the interior of relativistic stars. The field equations are transformed to a simpler form using the transformation of Durgapal and Bannerji (1983); the integration of the system is reduced to solving the condition of pressure isotropy. This condition is a recurrence relation with variable rational coe±cients which can be solved in general. New classes of solutions of linearly independent functions are obtained in terms of special functions and elementary functions for different spatial geometries. Our results contain models found previously including the superdense Tikekar (1990) neutron star model, the uncharged isotropic Maharaj and Leach (1996) solutions, the Finch and Skea (1989) model and the Durgapal and Bannerji (1983) superdense neutron star. Our general class of solutions also contain charged relativistic spheres found previously, including the model of Hansraj and Maharaj (2006) and the model of Thirukkanesh and Maharaj (2006). In addition, two exact analytical solutions describing the interior of a charged strange quark star are obtained by applying the MIT bag equation of state. We regain the Mak and Harko (2004) solution for a charged quark star as a special case. / Thesis (Ph. D.) University of KwaZuluNatal, Westville, 2007.

33 
Coagulationfragmentation dynamics in size and position structured population models.Noutchie, Suares Cloves Oukouomi January 2008 (has links)
One of the most interesting features of fragmentation models is a possibility to breach / Thesis (Ph.D.)University of KwaZuluNatal, Westville, 2008.

34 
On the logics of algebra.Barbour, Graham. January 2008 (has links)
We present and consider a number of logics that arise naturally from universal algebraic considerations, but which are ‘inherently unalgebraizable’ in the sense of [BP89a], essentially because they have no theo rems. Of particular interest is the membership logic of a quasivariety, which is determined by its theorems, which are the relative congruence classes of the term algebra together with the emptyset in the case that the quasivariety is nontrivial. The membership logic arises by a more general technique developed in this text, for inducing deductive systems from closed systems on the free algebras of quasivarieties. In order to formalize this technique, we develop a theory of logics over constructs, where constructs are concrete categories. With this theory in place, we are able to view a closed system over an algebra as a logic, and in particular a structural logic, structural with respect to a suitable construct, typically the construct con sisting of all algebras in a quasivariety and all algebra homomorphisms between these algebras. Of course, in such a case, none of these logics are generally sentential (i.e., structural and finitary deductive systems in the sense of [BP89a]), since the formulae of sentential logics arise from the terms of the absolutely free term algebra, which is generally not a member of the quasivariety under interest. In such cases, where the term algebra is not a member of a quasivariety, the free algebra of the quasivariety on denumerably countable free generators takes on the role played by the term algebra in sentential logics. Many of the logics that we encounter in this text arise most naturally as finitary logics on this free algebra of the quasivariety and generally are structural with respect to the quasivariety. We call such logics canons, and show how such structural canons induce sentential calculi, which we call the induced ideal ; the filters of the ideal on the free algebra are precisely the theories of the canon. The membership logic is the ideal of the cannon whose theories are the relative congruence classes on the free algebra. The primary aim of this thesis is to provide a unifying framework for logics of this type which extends the BlokPigozzi theory of elementarily algebraizable (and protoalgebraic) deductive systems. In this extension there are two parameters: a set of formulae and a variable. When the former is empty or consists of theorems, the BlokPigozzi theory is recovered, and the variable is redundant. For the membership logic, the appropriate variant of equivalent algebraic semantics encompasses the relatively congruence regular quasivarieties. These results have appeared in [BR03]. The secondary aim of this thesis is to analyse our theory of parameterized algebraization from a non parameterized perspective. To this end, we develop a theory of protoalgebraic logics over constructs and equivalence between logics from different constructs, which we then use to explain the results we obtained in our parameterized theories of protoalgebraicity, algebraic semantics and equivalent algebraic semantics. We relate this theory to the theory of deductively equivalent institutions [Vou03], and as a consequence obtain a number of improved and new results in the field of categorical abstract algebraic logic. We also use our theory of protoalgebraic logics over constructs to obtain a new and simpler characterization of structural finitary ndeductive systems, which we then use to close the program begun in [BR99], by extending those results for 1deductive systems to ndeductive systems, and in particular characterizing the protoalgebraicity of the sentential ndeductive system Sn(K,N), which is the natural extension of the 1deductive system S(K, ) introduce in [BR99], in terms of the quasivariety K having hK,Nicoherent Nclasses (we cannot see how to obtain this result from the standard characterization of protoalgebraic n deductive systems of [Pal03], which is very complex). With respect to this program of completing [BR99], we also show that a quasivariety K is an equivalent algebraic semantics for a ndeductive system with defining equations N iff K is hK,Niregular; a notion of regularity that we introduce and characterize by a quasiMal’cev condition. The third aim of this text is to unify as many disparate arguments and notions in algebraic logic under the banner of continuous translations between closed systems, where our use of the term continuous is in the topological sense rather than in the ordertheoretic sense, and, where possible, to give elementary, i.e. first order, definitions and proofs. To this end, we show that closed systems, closure operators and conse quence relations can all be characterized elementarily over orders, and put into onetoone correspondence that reflects exactly, the standard correspondences between the wellknown concrete notions with the same name. We show that when the order is the complete power order over a set, then these elementary structures coincide with their wellknown counterparts with the same name. We also introduce two other elementary structures over orders, namely the closed equivalence relation and something we term the protoLeibniz relation; these elementary structures are also in onetoone correspondence with the earlier mentioned structures; we have not seen concrete versions of these structures. We then characterize the structure homomorphisms between these structures, as well as considering galois relations between them; galois relations are pairs of orderpreserving function in opposite directions; we call these translations, and they are elementary notions. We demonstrate how notions as disparate as structurality, semantics, algebraic semantics, the filter correspondence property, filters, models, semantic consequence, protoalge braicity and even the logic S(K, ) of [BR99] and our logic Sn(K,N), all fall within this framework, as does much of our parameterized theory and much of the theory of institutions. A brief summary of the standard theory of deductive systems and their algebraization is provided for the reader unfamiliar with algebraic logics, as well as the necessary background material, including construct and category theory, the theory of structures and algebras, and the model theory of structures with and without equality. / Thesis (Ph.D.)University of KwaZuluNatal, Westville, 2008.

35 
On purity relative to an hereditary torsion theory.Gray, Derek Johanathan. January 1992 (has links)
The thesis is mainly concerned with properties of the concept
"σpurity" introduced by J. Lambek in "Torsion Theories, Additive
Semantics and Rings of Quotients", (SpringerVerlag, 1971).
In particular we are interested in modul es M for which every exact
sequence of the form O→M→K→L→O (or O→K→M→L→O or O→K→L→M→O) is σpure
exact. Modules of the first type turn out to be precisely the
σ injective modules of O. Goldman (J. Algebra 13, (1969), 1047).
This characterization allows us to study σ injectivity from the perspective of purity.
Similarly the demand that every short exact sequence of modules of the form O→K→M→L→O or O→K→L→M→O be σpure exact leads to concepts which generalize regularity and flatness respectively. The questions of which properties of regularity and flatness extend to these more general concepts of σ regularity and σflatness are investigated.
For various classes of rings R and torsion radicals σ on Rmod, certain conditions equivalent to the σregularity and the σinjectivity of R are found.
We also introduce some new dimensions and study semiσflat and
semiσinjective modules (defined by suitably restricting conditions
on σflat and σinjective modules). We further characterize those rings R for which every Rmodule is semi σflat.
The related concepts of a projective cover and a perfect ring
(introduced by H. Bass in Trans. Amer. Math. Soc. 95, (1960), 466488)
are extended in a 'natural way and, inter alia , we obtain a generalization of a famous theorem of Bass.
Lastly, we develop a relativized version of the Jacobson Radical which is shown to have properties analogous to both the classical Jacobson Radical and a radical due to J.S. Golan. / Thesis (Ph.D.)University of Natal, 1992.

36 
Complete symmetry groups : a connection between some ordinary differential equations and partial differential equations.Myeni, Senzosenkosi Mandlakayise. January 2008 (has links)
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 328368 (translation into English by Ibragimov NH in CRC Handbook of Lie Group Analysis of Differential Equations 2 473508) 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 sixdimensional algebra isomorphic to sl(2,R) s A3,1, where the second subalgebra is commonly known as the HeisenbergWeyl 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. / Thesis (Ph.D.)University of KwaZuluNatal, Westville, 2008.

37 
Extensions and generalisations of Lie analysis.Govinder, Kesh S. January 1995 (has links)
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 LieBacklund) 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 KummerSchwarz 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 socalled 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 broadminded approach to the concept of symmetry is imperative to successfully realise Sophus Lie's dream of a single unified theory to
solve differential equations. / Thesis (Ph.D.)University of Natal, 1995

38 
CliffordFischer theory applied to certain groups associated with symplectic, unitary and Thompson groups.Basheer, Ayoub Basheer Mohammed. January 2012 (has links)
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 welldeveloped 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 CliffordFischer 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 nonsingular 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 CliffordFischer matrices to calculate the ordinary character tables of seven
groups of extensions type, in which four are nonsplit 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 nonsplit group 22n·Sp(2n, 2). / Thesis (Ph.D.)University of KwaZuluNatal, Pietermaritzburg, 2012.

39 
Singularity and symmetry analysis of differential sequences.Maharaj, Adhir. January 2009 (has links)
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 leadingorder 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 EmdenFowler sequence in terms of its singularity and symmetry properties. / Thesis (Ph.D.)University of KwaZuluNatal, Westville, 2009.

40 
On convection and flow in porous media with crossdiffusion.Khidir, Ahmed A. January 2012 (has links)
In this thesis we studied convection and crossdiffusion effects in porous media.
Fluid flow in different flow geometries was investigated and the equations for momentum, heat and mass transfer transformed into a system of ordinary differential
equations using suitable dimensionless variables. The equations were solved using a
recent successive linearization method. The accuracy, validity and convergence of the
solutions obtained using this method were tested by comparing the calculated results
with those in the published literature, and results obtained using other numerical
methods such as the RungeKutta and shooting methods, the inbuilt Matlab bvp4c
numerical routine and a local nonsimilarity method.
We investigated the effects of different fluid and physical parameters. These
include the Soret, Dufour, magnetic field, viscous dissipation and thermal radiation
parameters on the fluid properties and heat and mass transfer characteristics.
The study sought to (i) investigate crossdiffusion effects on momentum, heat and
mass transport from a vertical flat plate immersed in a nonDarcy porous medium
saturated with a nonNewtonian powerlaw fluid with viscous dissipation and thermal
radiation effects, (ii) study crossdiffusion effects on vertical an exponentially stretching surface in porous medium and (iii) apply a recent hybrid linearizationspectral
technique to solve the highly nonlinear and coupled governing equations. We further
sought to show that this method is accurate, efficient and robust by comparing it with established methods in the literature.
In this study the nonNewtonian behaviour of the fluid is characterized using the
Ostwaldde Waele powerlaw model. Crossdiffusion effects arise in a broad range
of fluid flow situations in many areas of science and engineering. We showed that
crossdiffusion has a significant effect on heat and masstransfer processes and cannot
be neglected. / Thesis (Ph.D.)University of KwaZuluNatal, Pietermaritzburg, 2012.

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