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71 
Codes of designs and graphs from finite simple groups.Rodrigues, Bernardo Gabriel. 10 February 2014 (has links)
No abstract available. / Thesis (Ph.D.)University of Natal, Pietermaritzburg, 2002.

72 
Codes of designs and graphs from finite simple groups.Rodrigues, Bernardo Gabriel. January 2002 (has links)
Discrete mathematics has had many applications in recent years and this is
only one reason for its increasing dynamism. The study of finite structures is
a broad area which has a unity not merely of description but also in practice,
since many of the structures studied give results which can be applied to other, apparently dissimilar structures. Apart from the applications, which themselves generate problems, internally there are still many difficult and interesting problems in finite geometry and combinatorics. There are still many puzzling features about substructures of finite projective spaces, the minimum weight of the dual codes of polynomial codes, as well as about finite projective planes. Finite groups are an ever strong theme for several reasons. There is still much work to be done to give a clear geometric identification of the finite simple groups. There are also many problems in characterizing structures which either have a particular group acting on them or which have some degree of symmetry from a group action.
Codes obtained from permutation representations of finite groups have been given particular attention in recent years. Given a representation of group elements of a group G by permutations we can work modulo 2 and obtain a representation of G on a vector space V over lF2 . The invariant subspaces (the subspaces of V taken into themselves by every group element) are then all the binary codes C for which G is a subgroup of Aut(C). Similar methods produce codes over arbitrary fields. Through a moduletheoretic approach, and based on a study of monomial actions and projective representations, codes with given transitive permutation group were determined by various authors. Starting with well known simple groups and defining designs and codes through the primitive actions of the groups will give structures that have this group in their automorphism groups. For each of the primitive representations, we construct the permutation group and form the orbits of the stabilizer of a point.
Taking these ideas further we have investigated the codes from the primitive permutation representations of the simple alternating and symplectic groups of odd characteristic in their natural rank3 primitive actions. We have also investigated alternative ways of constructing these codes, and these have come about by noticing that the codes constructed from the primitive permutations of the groups could also be obtained from graphs. We achieved this by constructing codes from the span of adjacency matrices of graphs. In particular we have constructed codes from the triangular graphs and from the graphs on triples.
The simple symplectic group PSp2m(q), where m is at least 2 and q is any prime power, acts as a primitive rank3 group of degree q2m1/q1 on the points of the projective (2m1)space PG2m1(IFq ). The codes obtained from the primitive rank3 action of the simple projective symplectic groups PSp2m(Q), where Q= 2t with t an integer such that t ≥ 1, are the well known binary subcodes of the
projective generalized ReedMuller codes. However, by looking at the simple symplectic groups PSp2m(q), where q is a power of an odd prime and m ≥ 2, we observe that in their rank3 action as primitive groups of degree q2m1/q1 these groups have 2modular representations that
give rise to selforthogonal binary codes whose properties can be linked to those of
the underlying geometry. We establish some properties of these codes, including
bounds for the minimum weight and the nature of some classes of codewords.
The knowledge of the structures of the automorphism groups has played a key
role in the determination of explicit permutation decoding sets (PDsets) for the
binary codes obtained from the adjacency matrix of the triangular graph T(n) for n ≥ 5 and similarly from the adjacency matrices of the graphs on triples.
The successful decoding came about by ordering the points in such a way that the
nature of the information symbols was known and the action of the automorphism
group apparent.
Although the binary codes of the triangular graph T(n) were known, we have
examined the codes and their duals further by looking at the question of minimum weight generators for the codes and for their duals. In this way we find bases
of minimum weight codewords for such codes. We have also obtained explicit
permutationdecoding sets for these codes.
For a set Ω of size n and Ω{3} the set of subsets of Ω of size 3, we investigate the binary codes obtained from the adjacency matrix of each of the three graphs with
vertex set Ω{3}1 with adjacency defined by two vertices as 3sets being adjacent if they have zero, one or two elements in common, respectively. We show that
permutation decoding can be used, by finding PDsets, for some of the binary codes obtained from the adjacency matrix of the graphs on (n3) vertices, for n ≥ 7. / Thesis (Ph.D.)University of Natal, Pietermaritzburg, 2002.

73 
Developing an integrated decision support system for an oil refinery.Azizi, Abbas. January 1998 (has links)
This thesis considers the problem of residue upgrading operations in an oil
refinery. Visbreaking is a residueupgrading process that improves profitability
of a refinery. The economics of converting the heavy residue into the lighter and
more valuable streams, coupled with the installation of a modem visbreaker unit
at the Engen Refinery in Durban, provides sufficient motives to develop a
mathematical model to simulate the unit's capability and estimate the economics
of the visbreaking process and fuel oil operations. Furthermore, the proposed
model should provide a crudedependent visbreaking yield that can be used in
the refinery's global linear programme (LP), employed to evaluate and select the
crude and to optimise refinery's operations.
Traditionally, kinetically based models have been used to simulate and
study the refining reaction processes. In this case, due to the complexity of the
process and some unknown reactions, the performances of existing visbreaking
simulators are not fully satisfactory. Consequently, a neural network model of
the visbreaking process and fuel oil blending operation is developed.
The proposed model is called the adaptive visbreaker paradigm, since it is
formed using neuroengineering, a technique that fabricates empiricallybased
neural network models. The network operates in supervised mode to predict the
visbreaking yields and the residue quality.
It was observed that due to the fluctuation in the quality of feedstock, and
plant operating conditions, the prediction accuracy of the model needs to be
improved. To improve the system's predictability, a network reciprocation
procedure has been devised. Network reciprocation is a mechanism that controls
and selects the input data used in the training of a neural network system.
Implementation of the proposed procedure results in a considerable
improvement in the performance ofthe network.
3
To facilitate the interaction between the simulation and optimisation
routines, an integrated system to incorporate the fuel oil blending with the
neurallybased module is constructed. Under an integrated system, the
economics of altering the models' decision variables can be monitored.
To account for the visbreakability of the various petroleum crudes, the yield
predicted by the adaptive visbreaker paradigm should enter into the visbreaker,s
submodel of the global refinery LP. To achieve this, a mechanism to calculate
and update the visbreaking yields of various crude oils is also developed.
The computational results produced by the adaptive visbreaker paradigm
prove that the economics of the visbreaking process is a multidimensional
variable, greatly influenced by the feed quality and the unit's operating
condition. The results presented show the feasibility of applying the proposed
model to predict the cracking reaction yields. Furthermore, the model allows a
dynamic monitoring of the residue properties as applicable to fuel oil blending
optimisation.
In summary, the combination of the proposed models forms an integrated
decision support system suitable for studying the visbreaking and associated
operations, and to provide a visbreaking yield pattern that can be incorporated
into the global refinery LP model. Using an integrated decision support system,
refinery planners are able to see through the complex interactions between
business and the manufacturing process by performing predictive studies using
these models. / Thesis (Ph.D.)University of Natal, Pietermaritzburg, 1998.

74 
Applications of symmetry analysis of partial differential and stochastic differential equations arising from mathematics of finance.Nwobi, Felix Noyanim. January 2011 (has links)
In the standard modeling of the pricing of options and derivatives as generally understood these days the underlying process is taken to be a Wiener Process or a Levy Process. The stochastic process is modeled as a stochastic differential equation. From this equation a partial
differential equation is obtained by application of the FeynmanKac Theorem. The resulting partial differential equation is of HamiltonJacobiBellman type. Analysis of the partial differential equations arising from Mathematics of Finance using the methods of the Lie Theory of Continuous Groups has been performed over the last twenty years, but it is only in recent years that there has been a concerted effort to make full use of the Lie theory. We propose an extension of Mahomed and Leach's (1990) formula for the
nthprolongation of an nthorder ordinary differential equation to the nthprolongation of the generator of an hyperbolic partial differential equation with p dependent and k independent variables. The symmetry analysis of this partial differential equation shows that the associated
Lie algebra is {sl(2,R)⊕W₃}⊕s ∞A₁ with 12 optimal systems.
A modeling approach based upon stochastic volatility for modeling prices in the deregulated Pennsylvania State Electricity market is adopted for application. We propose a dynamic linear model (DLM) in which switching structure for the measurement matrix is incorporated into a twostate Gaussian mixture/firstorder autoregressive (AR (1)) configuration in a nonstationary independent process defined by timevarying probabilities. The estimates of maximum likelihood of the parameters from the "modified" Kalman filter showed a significant meanreversion rate of 0.9363 which translates to a halflife price of electricity of nine months. Associated with this meanreversion is the high measure of price volatility at 35%. Within the last decade there has been some work done upon the symmetries of stochastic differential equations. Here empirical results contradict earliest normality hypotheses on logreturn series in favour of asymmetry of the probability distribution describing the process. Using the
Akaike Information Criterion (AIC) and the Loglikelihood estimation (LLH) methods as selection criteria, the normal inverse Gaussian (NIG) outperformed four other candidate probability distributions among the class of Generalized Hyperbolic (GH) distributions in describing the heavy tails present in the process. Similarly, the Skewed Student's t (SSt) is the best fit for Bonny Crude Oil and Natural Gas logreturns. The observed volatility measures of these three
commodity prices were examined. The Weibull distribution gives the best fit both electricity and crude oil data while the Gamma distribution is selected for natural gas data in the volatility profiles among the five candidate probability density functions (Normal, Lognormal, Gamma, Inverse Gamma and the Inverse Gaussian) considered. / Thesis (Ph.D.)University of KwaZuluNatal, Westville, 2011.

75 
FischerClifford theory for split and nonsplit group extensions.January 2001 (has links)
The character table of a finite group provides considerable amount of information about the group, and hence is of great importance in Mathematics as well as in Physical Sciences. Most of the maximal subgroups of the finite simple groups and their automorphisms are of extensions of elementary abelian groups, so methods have been developed for calculating the character tables of extensions of elementary abelian groups. Character tables of finite groups can be constructed using various techniques. However Bernd Fischer presented a powerful and interesting technique for calculating the character tables of group extensions. This technique, which is known as the technique of the FischerClifford matrices, derives its fundamentals from the Clifford theory. If G=N.G is an appropriate extension of N by G, the method involves the construction of a nonsingular matrix for each conjugacy class of G/N~G. The character table of G can then be determined from these FischerClifford matrices and the character table of certain subgroups of G, called inertia factor groups. In this dissertation, we described the FischerClifford theory and apply it to both split and nonsplit group extensions. First we apply the technique to the split extensions 2,7:Sp6(2) and 2,8:SP6(2) which are maximal subgroups of Sp8(2) and 2,8:08+(2) respectively. This technique has also been discussed and used by many other researchers, but applied only to split extensions or to the case when every irreducible character of N can be extended to an irreducible character of its inertia group in G. However the same method can not be used to construct character tables of certain nonsplit group extensions. In particular, it can not be applied to the nonsplit extensions of the forms 3,7.07(3) and 3,7.(0,7(3):2) which are maximal subgroups of Fischer's largest sporadic simple group Fi~24 and its automorphism group Fi24 respectively. In an attempt to generalize these methods to such type of nonsplit group extensions, we need to consider the projective representations and characters. We have shown that how the technique of FischerClifford matrices can be applied to any such type of nonsplit extensions. However in order to apply this technique, the projective characters of the inertia factors must be known and these can be difficult to determine for some groups. We successfully applied the technique of FischerClifford matrices and determined the FischerClifford matrices and hence the character tables of the nonsplit extensions 3,7.0,7(3) and 3,7.(0,7(3):2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest versions. / Thesis (Ph.D)University of Natal, Pietermaritzburg, 2001.

76 
Optimal designs for linear mixed models.Debusho, Legesse Kassa. January 2004 (has links)
The research of this thesis deals with the derivation of optimum designs for linear mixed models. The problem of constructing optimal designs for linear mixed models is very broad. Thus the thesis is mainly focused on the design theory for random coefficient regression models which are a special case of the linear mixed model. Specifically, the major objective of the thesis is to construct optimal designs for the simple linear and the quadratic regression
models with a random intercept algebraically. A second objective is to investigate the nature of optimal designs for the simple linear random coefficient regression model numerically. In all models time is considered as an explanatory variable and its values are assumed to belong the set {a, 1, ... , k}. Two sets of individual designs, designs with nonrepeated time points comprising up to k + 1 distinct time points and designs with repeated time points comprising up to k + 1 time points not necessarily distinct, are used in the thesis. In the first case there are 2k+  1 individual designs while in the second case there are ( 2 2k k+ 1 )  1 such designs. The problems of constructing population designs, which allocate weights to the individual designs in such a way that the information associated with
the model parameters is in some sense maximized and the variances associated with the mean responses at a given vector of time points are in some sense minimized, are addressed. In particular D and Voptimal designs are discussed. A geometric approach is introduced to confirm the global optimality of D and Voptimal designs for the simple linear regression
model with a random intercept. It is shown that for the simple linear regression model with a random intercept these optimal designs are robust to the choice of the variance ratio. A comparison of these optimal designs over the sets of individual designs with repeated and nonrepeated points for that model is also made and indicates that the D and Voptimal
iii population designs based on the individual designs with repeated points are more efficient than the corresponding optimal population designs with nonrepeated points. Except for the onepoint case, D and Voptimal population designs change with the values of the variance ratio for the quadratic regression model with a random intercept. Further numerical results show that the Doptimal designs for the random coefficient models are dependent on the choice of variance components. / Thesis (Ph.D.)  University of KwaZuluNatal, Pietermaritzburg, 2004.

77 
A new approach to illposed evolution equations : Cregularized and B bounded semigroups.Singh, Virath Sewnath. January 2001 (has links)
The theory of semigroups of linear operators forms an integral part of Functional
Analysis with substantial applications to many fields of the natural sciences. In
this study we are concerned with the application to equations of mathematical
physics. The theory of semigroups of bounded linear operators is closely related to
the solvability of evolution equations in Banach spaces that model time dependent
processes in nature.
Wellposed evolution problems give rise to a semigroup of bounded linear operators.
However, in many important and interesting cases the problem is illposed
making it inaccessible to the classical semigroup theory. One way of dealing
with this problem is to generalize the theory of semigroups.
In this thesis we give an outline of the theory of two such generalizations, namely,
Cregularized semigroups and Bbounded semigroups, with the interrelations
between them and show a number of applications to illposed problems. / Thesis (Ph.D.)University of Natal, Durban, 2001.

78 
An algebraic study of residuated ordered monoids and logics without exchange and contraction.Van Alten, Clint Johann. January 1998 (has links)
Please refer to the thesis for the abstract. / Thesis (Ph.D.)University of Natal, Durban, 1998.

79 
On the status of the geodesic law in general relativity.Nevin, Jennifer Margaret. January 1998 (has links)
The geodesic law for test particles is one of the fundamental principles of general
relativity and is extensively used. It is thought to be a consequence of the field laws
but no rigorous proof exists. This thesis is concerned with a precise formulation of
the geodesic law for test particles and with the extent of its validity. It will be shown
to be true in certain cases but not in others.
A rigorous version of the Infeld/Schild theorem is presented. Several explicit
examples of both geodesic and nongeodesic motion of singularities are given. In the
case of a test particle derived from a test body with a regular internal stressenergy
tensor, a proof of the geodesic law for an ideal fluid test particle under plausible,
explicitly stated conditions is given. It is also shown that the geodesic law is not
generally true, even for weak fields and slow motion, unless the stressenergy tensor
satisfies certain conditions. An explicit example using postNewtonian theory is given
showing how the geodesic law can be violated if these conditions are not satisfied. / Thesis (Ph.D.)University of Natal, Durban, 1998.

80 
Anisotropic stars in general relativity.Chaisi, Mosa. January 2004 (has links)
In this thesis we seek new solutions to the anisotropic Einstein field equations which are important in the study of highly dense stellar structures. We first adopt the approach used by Maharaj & Maartens (1989) to obtain an exact anisotropic solution in terms of elementary functions for a particular choice of the energy density. This class of solution contains the Maharaj & Maartens (1989) and Gokhroo & Mehra (1994) models as special cases. In addition, we obtain six other new solutions following the same approach for different choices of the energy density. All the solutions in this section reduce to one with the energy density profile fL ex r2 . Two new algorithms are generated, Algorithm A and Algorithm B, which produce a new anisotropic solution to the Einstein field equations from a given isotropic solution. For any new anisotropic solution generated with the help of these algorithms, the original isotropic seed solution is regained as a special case. Two examples of known isotropic solutions are used to demonstrate how Algorithm A and Algorithm B work, and to obtain new anisotropic solutions for the Einstein and de Sitter models. Anisotropic isot~ermal sphere models are generated given the corresponding isotropic (fL ex r2 ) solution of the Einstein field equations. Also, anisotropic interior Schwarzschild sphere models are found given the corresponding isotropic (fL ex constant) solution of the field equations. The exact solutions and line elements are given in each case for both Algorithm A and Algorithm B. Note that the solutions have a simple form and are all expressible in terms of elementary functions. Plots for the anisotropic factor S = J3(Pr  pJJ/2 (where Pr and Pl. are radial and tangential pressure respectively) are generated and these point to physically viable models. / Thesis (Ph.D.)University of Natal, Durban, 2004.

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