Thesis (PhD)--Stellenbosch University, 2012. / Please refer to full text for abstract.

Thesis (PhD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: Research on transfer processes in various types of porous media has become important for the optimization of high technology engineering devices and processes. In this study the micro-structural parameters of different types of porous media, namely granular media, foamlike media and fibre beds, are characterized and quantified. Existing analytical modelling procedures for the three different types of porous media have been unified and refined to improve their predictive capabilities. Deterministic equations are proposed for predicting the streamwise pressure gradient, permeability and inertial coefficient of each type of porous medium. The equations are applicable over the entire porosity range and steady laminar flow regime and well suited as drag models in numerical computations. It is shown that the improved granular model can be regarded as qualitative and quantitative proof of the extensively used semi-empirical Ergun equation. The proposed model is used to provide physical meaning to the empirical coefficients. An Ergun-type equation is also proposed for foamlike media by remodelling the interstitial geometric configuration and accompanying flow conditions. The range of applicability of the existing foam model has been extended by incorporating the effect of developing flow in the pressure drop prediction. An equation is proposed in which the variation in the cross-sectional shape of the fibres can be incorporated into the interstitial form drag coefficient used in the foam model. This serves as an improvement on the constant value previously used. The existing foam model is also adapted to account for anisotropy resulting from compression. Two case studies are considered, namely compression of a non-woven glass fibre filter and compression of a soft polyester fibre material. The significant effect of compression on permeability is illustrated. In each case study the permeability values range over more than an order of magnitude for the narrow porosity ranges involved. The pressure drop prediction of the foam model is furthermore adapted to account for the combined effects of compression and developing flow. The newly proposed model diminishes the significant over-prediction of the existing foam model. An equation is furthermore proposed for predicting the permeability of Fontainebleau sandstones in which the effect of blocked throats is accounted for. Lastly, equations are proposed for predicting diffusivity ratios of unconsolidated arrays of squares and cubes. The prediction of the diffusivity ratio proposed in the present study, as opposed to model predictions from the literature, takes into account diffusion that may take place in stagnant fluid volumes. It is shown that a specific weighted average model proposed in the literature is not adequate to predict the diffusivity ratio of fully staggered arrays of squares, since it is shown not to be applicable to rectangular unit cells. Instead a new weighted average model is proposed which is applicable over the entire porosity range and for both staggered and non-staggered arrays of solid squares and cubes. The proposed weighted average model provides satisfactory agreement with experimental data from the literature and numerical data generated in the present study. / AFRIKAANSE OPSOMMING: Navorsing op oordragsprosesse in verskeie tipes poreuse media het belangrik geword vir die optimisering van ho¨e-tegnologie ingenieurstoestelle- en prosesse. In hierdie studie word die mikro-struktuur parameters van verskillende tipes poreuse media, naamklik korrelagtige media, sponsatige media en veselbeddens gekarakteriseer en gekwantifiseer. Bestaande analitiese modelleringsprosedures vir die drie verskillende tipes poreuse media is verenig en verfyn om die voorspelbare bekwaamheid daarvan te verbeter. Deterministiese vergelykings is voorgestel vir die voorspelling van die stroomsgewyse gradi¨ent, permeabiliteit en inersi¨ele ko¨effisi¨ent van elke tipe poreuse medium. Die vergelykings is geldig oor die hele porositeitsgrens en gestadigde laminˆere vloeigrens en goed geskik as weerstandsmodelle in numeriese berekeninge. Dit is aangetoon dat die verbeterde korrelmodel beskou kan word as kwalitatiewe en kwantitatiewe bewys van die ekstensiewe gebruikte semi-empiriese Ergun vergelyking. Die voorgestelde model is gebruik om fisiese betekenis aan die empiriese ko¨effisi¨ente te gee. ’n Ergun-tipe vergelyking is ook voorgestel vir sponsagtige media deur hermodellering van die tussenruimtelike geometriese konfigurasie en gepaardgaande vloeikondisies. Die grense van toepaslikheid van die bestaande sponsmodel is uitgebrei deur die inkorporering van die effek van ontwikkelende vloei in die voorspelling van die drukval. ’n Vergelyking is voorgestel waarin die variasie in die deursnit vorm van die vesels ingesluit is in die sponsmodel. Dit dien as verbetering op die konstante waarde wat voorheen gebruik is. Die bestaande sponsmodel is ook aangepas om voorsiening te maak vir anisotropie as gevolg van kompressie. Twee gevallestudies is oorweeg, naamlik kompressie van ’n nie-geweefde glasvesel filter en kompressie van ’n sagte polyester veselmateriaal. Die beduidende effek van kompressie op permeabiliteit is aangetoon. In elke gevallestudie strek die permeabiliteit waardes oor meer as ’n grootte orde vir die skrale porositeitgrense betrokke. Die drukvalvoorspelling van die sponsmodel is verder aangepas om voorsiening te maak vir die gekombineerde effekte van kompressie en ontwikkelende vloei. Die nuwe voorgestelde model verminder die beduidende oor-voorspelling van die bestaande sponsmodel. ’n Vergelyking is verder voorgestel vir die voorspelling van die permeabiliteit van Fontainebleau sandsteen waarin die effek van geblokte porie¨e in ag geneem is. Laastens is vergelykings voorgestel vir die voorspelling van die diffusiwiteitsverhoudings van nie-konsoliderende rangskikkings van vierkante en kubusse. Die diffusiwiteitsverhouding voorspel in die huidige studie, teenoor modelvoorspellings vanaf die literatuur, neem diffusie in ag wat plaasvind in die stagnante vloeistofvolumes. Dit is aangetoon dat ’n geweegde gemiddelde model, voorgestel in die literatuur, nie in staat is om die diffusiwiteitsverhouding van ten volle verspringende rangskikkings van vierkante te voorspel nie, aangesien dit nie toepaslik is vir reghoekige eenheidselle nie. ’n Nuwe geweegde model is in plaas daarvan voorgestel wat toepaslik is oor die hele porositeitsgrens en vir beide verspringende en nieverspringende rangskikkings van soliede vierkante en kubusse. Die voorgestelde geweegde gemiddelde model bied bevredigende ooreenstemming met eksperimentele data uit die literatuur en numeriese data gegenereer in die huidige studie.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: This thesis deals with a function eld analog of the André-Oort conjecture. The (classical) André-Oort conjecture concerns the distribution of special points on Shimura varieties. In our case we consider the André-Oort conjecture for special points in the product of Drinfeld modular varieties. We in particular manage to prove the André- Oort conjecture for subvarieties in a product of two Drinfeld modular surfaces under a characteristic assumption. / AFRIKAANSE OPSOMMING: Hierdie tesis handel van 'n funksieliggaam analoog van die André-Oort Vermoeding. Die (Klassieke) André-Oort Vermoeding het betrekking tot die verspreiding van spesiale punte op Shimura varietiete. Ons geval beskou ons die André-Oort Vermoeding vir spesiale punte op die produk Drinfeldse modulvarietiete. In die besonders, bewys ons die André-Oort Vermoeding vir ondervarieteite van 'n produk van twee Drinfeldse modulvarietiete, onderhewig aan 'n karakteristiek-aanname.

Thesis (DSc (Mathematical Sciences))--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: (Math symbols have changed) Wavelet and subdivision techniques have developed, over the last two decades, into powerful mathematical tools, for example in signal analysis and geometric modelling. Both wavelet and subdivision analysis are based on the concept of a matrix–refinable function, i.e. a finitely supported matrix function which is self-replicating in the sense that it can be expressed as a linear combination of the integer shifts of its own dilation with factor 2: F = TAF = å k∈Z F(2 ・ −k)Ak. The coefficients Ak, k ∈ Z of d × d matrices, of this linear combination constitute the so-called matrix- mask sequence. Wavelets are in fact constructed as a specific linear combination of the integer shifts of the 2-dilation of a matrix- refinable function cf. [2; 9], whereas the convergence of the associated matrix- subdivision scheme c0 = c, cr+1 = SAcr, r ∈ Z+, SA : c = (ck : k ∈ Z) 7→ SAc = å ℓ∈Z Ak−2ℓ cℓ : k ∈ Z ! , subject to the necessary condition that rank := dim   \ ǫ∈{0,1} n y ∈ Rd : Qǫy = y o   > 0, Qǫ := å j∈Z Aǫ+2j, ǫ ∈ {0, 1}, ( cf. [26]) , implies the existence of a finitely supported matrix- function which is refinable with respect to the mask coefficients defining the refinement equation and the subdivision scheme. Throughout this thesis, we investigate in time–domain for a given matrix mask sequence, the related issues of the existence of a matrix–refinable function and the convergence of the corresponding matrix– cascade algorithm, and finally we apply some results to the particular research area of Hermite interpolatory subdivision schemes. The dissertation is organized as follows: In order to provide a certain flexibility or freedom over the project, we established in Chapter 1 the equivalence relation between the matrix cascade algorithm and the matrix subdivision scheme, subject to a well defined class of initial iterates. Despite the general noncommutativity of matrices, we make use in the full rank case Qǫ = I, ǫ ∈ {0, 1}, of a symbol factorization, to develop in Chapter 2 some useful tools, yielding a convergence result which comes as close to the scalar case as possible: we obtained a concrete sufficient condition on the mask sequence based on the matrix version of the generating function introduced in [3, page 22] for existence and convergence. Whilst the conjecture on nonnegative masks was confirmed in 2005 by Zhou [29], our result on scalar case provided a progress for general mask sequences. We then applied to obtain a new one-parameter family of refinable functions which includes the cardinal splines as a special case, as well as corresponding convergent subdivision schemes. With the view to broaden the class of convergent matrix-masks, we replaced in chapter 3 the full rank condition by the rank one condition Qǫu = u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, then improved the paper by Dubuc and Merrien [13] by using the theory of rank subdivision schemes by Micchelli and Sauer [25; 26], and end up this improvement with a generalization of [13, Theorem 13, p.8] in to the context of rank subdivision schemes. In Chapter 4, we translated the concrete convergence criteria of the general theory from Theorem 3.2, based on the r-norming factor introduced in [13, Definition 6, p.6], into the context of rank, factorization and spectral radius (cf. [26]), and presented a careful analysis of the relationship between the two concepts. We then proceed with generalizations and improvements: we classified the matrix cascade algorithms in term of rank = 1, 2, . . . , d, and provided a complete characterization of each class with the use of a more general r−norming factor namely τ(r)-norming factor. On the other hand, we presented numerical methods to determine, if possible, the convergence of each class of matrix cascade algorithms. In both the scalar and matrix cases above, we also obtained explicitly the geometric constant appearing in the estimate for the geometric convergence of thematrix-cascade algorithm iterates to the matrix- refinable function. This same geometric convergence rate therefore also holds true for the corresponding matrix–cascade algorithm. Finally, in Chapter 5, we apply the theory and algorithms developed in Chapter 4 to the particular research area of Hermite interpolatory subdivision schemes: we provided a new convergence criterium, and end up with new convergence ranges of the parameters’ values of the famous Hermite interpolatory subdivision scheme with two parameters, due to Merrien [23]. / AFRIKAANSE OPSOMMING :(Wiskundige simbole het verander) Golfie en subdivisietegnieke het oor die afgelope twee dekades ontwikkel in kragtige wiskundige gereedskap, byvoorbeeld in seinanalise en geometriesemodellering. Beide golfie en subdivisie analise is gebaseer op die konsep van ’n matriks-verfynbare funksie; oftewel ’n eindig-ondersteunde matriksfunksie F wat selfreproduserend is in die sin dat dit uitgedruk kan word as ’n lineêre kombinasie van die heelgetalskuiwe van F se eie dilasie met faktor 2: F = Σ F(2 · −α)A(α), met A(α), α ∈ Z, wat aandui die sogenaamde matriks-masker ry. Golfies kan dan gekonstrueer word as ’n spesifieke lineêre kombinasie van die funksie ry {F(2 · −α) : α ∈ Z} (sien [2; 9]), terwyl die konvergensie van die ooreenstemmende matriks-subdivisie skema cº = c, cr+1 =(Σ β∈Z A(α − 2β) cr(β) : α ∈ Z ! , r ∈ Z+, onderhewig aan die nodige voorwaarde dat rank := dim   \ ǫ∈{0,1} n y ∈ Rd : Qǫy = y o   > 0, Qǫ := å α∈Z A(ǫ + 2α), ǫ ∈ {0, 1}, (sien [27]) die bestaan impliseer van ’n eindig-ondersteunde matriksfunksie F wat verfynbaar ismet betrekking tot diemaskerko¨effisi¨entewat die subdivisieskema definieer, en in terme waarvan die limietfunksie F van die subdivisieskema uitgedruk kan word as F = å α∈Z F(· − α)c(α). Ons hoofdoel hier is om , in die tydgebied, en vir ’n gegewematriks-masker ry, die verwante kwessies van die bestaan van ’nmatriks-verfynbare funksie en die konvergensie van die ooreenstemmende matriks-kaskade algoritme, en matriks-subdivisieskema, te ondersoek, en om uiteindelik sommige van ons resultate toe te pas op die spesifieke kwessie van die konvergensie van Hermite interpolerende subdivisieskemas. Summary v Eerstens, in Hoofstuk 1, ondersoek ons die verwantskap tussen matriks-kaskade algoritmes en matriks-subdivisie skemas, met verwysing na ’n goedgedefinieerde klas van begin-iterate. Vervolgens beskou ons die volle rang geval Qǫ = I, ǫ ∈ {0, 1}, om, in Hoofstuk 2, nuttige gereedskap te ontwikkel, en wat daarby ’n konvergensie resultaat met ’n sterk konneksie ten opsigte van die skalaar-geval oplewer. Met die doelstelling om ons klas van konvergente matriks-maskers te verbreed, vervang ons, in Hoofstuk 3, die volle rang voorwaarde met die rang een voorwaarde Qǫu = u, ǫ ∈ {0, 1}, u := (1, . . . , 1)T, en verkry ons dan ’n verbetering op ’n konvergensieresultaat in die artikel [14] deur Dubuc en Merrien, deur gebruik te maak van die teorie van rang subdivisieskemas van Micchelli en Sauer [26; 27], waarna ons die resultaat [14, Stelling 13, page 8] na die konteks van rang subdivisieskemas veralgemeen. InHoofstuk 4 herlei ons die konkrete konvergensie kriteria van Stelling 3.2, soos gebaseer op die r-normerende faktor gedefinieer in [14, Definisie 6, page 6] , na die konteks van rang, faktorisering en spektraalradius (sien [27]), en gee ons ’n streng analise van die verwantskap tussen die twee konsepte. Verder stel ons dan bekend ’n nuwe klassifikasie van matriks-kaskade algoritmes ten opsigte van rang, en verskaf ons ’n volledige karakterisering van elke klasmet behulp van ’nmeer algemene r-normerende faktor, nl. die τ(r)-normerende faktor. Daarby gee ons doeltreffende numeriesemetodes vir die implementering van ons teoretiese resultate. Ons verkry ook eksplisiet die geometriese konstante wat voorkom in die afskatting van die geometriese konvergensie van die matriks-kaskade algoritme iterate na die matriks-verfynbare funksie. Ten slotte, in Hoofstuk 5, pas ons die teorie en algoritmes ontwikkel in Hoofstuk 4 toe om die konvergensie van Hermite-interpolerende subdivisieskemas te analiseer. Spesifiek lei ons ’n nuwe konvergensie kriterium af, wat ons dan toepas om nuwe konvergensie gebiede vir die parameter waardes te verkry vir die beroemde Hermite interpolerende subdivisieskema met twee parameters, soos toegeskryf aan Merrien [24].

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: Rank 2 Drinfeld modular forms have been studied for more than 30 years, and while it is known that a higher rank theory could be possible, higher rank Drinfeld modular forms have only recently been de ned. In 1988 Gekeler published [Ge2] in which he studies the coe cients of rank 2 Drinfeld modular forms. The goal of this thesis is to perform a similar study of the coe cients of higher rank Drinfeld modular forms. The main results are that the coe cients themselves are (weak) Drinfeld modular forms, a product formula for the discriminant function, the rationality of certain naturally de ned modular forms, and the computation of some Hecke eigenforms and their eigenvalues. / AFRIKAANSE OPSOMMING: Drinfeld modulêre vorme van rang 2 word al vir meer as 30 jaar bestudeer en alhoewel dit lankal bekend is dat daar Drinfeld modulêre vorme van hoër rang moet bestaan, is die de nisie eers onlangs vasgepen. In 1988 het Gekeler die artikel [Ge2] gepubliseer waarin hy die koeffisiënte van Fourier reekse van rang 2 Drinfeld modulêre vorme bestudeer. Die doel van hierdie proefskrif is om dieselfde studie vir Drinfeld modulêre vorme van hoër rang uit te voer. Die hoofresultate is dat die koeffi siënte self (swak) Drinfeld modulêre vorme is, `n produk formule vir die diskriminant funksie, die feit dat sekere natuurlik gede finiëerde modulêre vorme rasionaal is, en die vasstelling van Hecke eievorme en hul eiewaardes.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The interest rates market has changed dramatically since the 2007 credit crunch with the explosion of basis spreads between rates of different tenors and currencies. Consequently, the classical replication of FRA rates with spot LIBOR rates is no longer valid. Moreover, the 2007 credit crunch yields a separation between the curve used for discounting and the forward or projection curves that estimate all future cash-fl ows. Another impact of the credit crunch in risk management is that market participants have started to give more importance to the difference between collateralized and uncollateralized trades. Nowadays, the wide spread use of collateral, especially in swap contracts, has made the overnight index swap (OIS) rate the appropriate benchmark for discounting collateralized trades. Inspired by the seminal works of Mercurio (2010a,b), Kijima et al. (2008), Fujii et al. (2011), Bianchetti (2010b), with the contributions of other authors, and motivated by the evolution of the interest rates market and models, this thesis examines a new framework that uses multiple-curves to value interest rate derivatives which is compatible with the current market practice. Firstly, we discuss the roots of the 2007 credit crunch and its serious consequences for pricing interest rate derivatives. We underscore the necessity of a multiple-curve pricing framework for interest rate derivatives. This is followed by a discussion on the importance of collateralization and OIS discounting in pricing Over-The-Counter (OTC) derivatives. The central part of the thesis discusses the modern theoretical framework and the practical implementation of the multiple curve pricing method. We present a bootstrapping algorithm used to construct and fit the multiple-yield curves to market prices of plainvanilla contracts. Secondly, starting with the single-currency economy, the extended version of the LIBOR Market Model, developed by Mercurio (2010a,b), which proposes a joint model of FRA rates, implied forward rates and their corresponding spread is investigated. Analogously, the extended version of short-rate model in a multiple-curve setup and in the presence of basis spread, proposed by Kijima et al. (2008), is presented and discussed. This work provides a detailed analysis of these extensions and the corresponding closed formulae for liquid products such as caps and swaptions. Finally, in the multiple-currencies case, the HJM model with stochastic basis spreads, introduced by Fujii et al. (2011), consistent with the foreign exchange and cross-currency swaps markets that includes the effect of collateralization is examined thoroughly. / AFRIKAANSE OPSOMMING: Die rentekoers mark het dramaties verander sedert die 2007 krediet krisis met 'n ontplo ng van basisverspreidings tussen koerse van verskillende looptye ("tenor") en geldeenhede. As gevolg, is die klassieke replikasie van FRA koerse met LIBOR sigkoerse nie langer geldig nie. Verder het die 2007 kredietkrisis 'n skeiding veroorsaak tussen die kromme wat gebruik word vir diskontering en die voorwaardse of vooruitskattings krommes wat toekomstige kontantvloei voorspel. 'n Verdere impak van die kredietkrisis in risikobestuur is dat mark deelnemers begin het om meer klem te lê op verskille tussen aangevulde en onaangevulde handel. Deesdae, met die algemene gebruik van kollaterale sekuriteit, veral in ruiltransaksiekontrakte, is die oornagse indeks ruiltransaksie (overnight index swap, OIS) koers die geskikte maatstaf om aangevulde handel te diskonteer. Geïnspireer deur die gedagteryke werk van Mercurio (2010a,b), Kijima et al. (2008), Fujii et al. (2011), Bianchetti (2010b), met bydrae van menige outeurs, en gemotiveer deur die evolusie van die rentekoers markte en modelle, ondersoek hierdie tesis 'n nuwe raamwerk wat multikrommes gebruik om rentekoers afgeleide effekte te waardeer wat versoenbaar is met die lopende mark praktyk. Eerstens, bespreek ons die oorsake van die 2007 kredietkrisis en die ernstige nagevolge vir die waardering van rentekoers afgeleide effekte. Ons beklemtoon die noodsaaklikheid van 'n multikromme waarderings raamwerk vir rentekoers afgeleide effekte. Dit word gevolg deur 'n bespreking oor die belangrikheid van aanvulling en OIS diskontering in die waardering van oor-die-toonbank (over-the-counter, OTC) effekte. Die teoretiese raamwerk en die praktiese implimentering van die multikromme waarderings metode word bespreek. Ons stel ook ten toon 'n skoenlus ("bootstrapping") algoritme wat gebruik kan word om meervoudige opbrengs krommes saam te stel en die dan te pas op mark pryse van vanielje kontrakte. Tweedens, met 'n enkel geldeenheid ekonomie as beginpunt, word die uitgebreide weergawe van die LIBOR Mark Model (ontwikkel deur Mercurio (2010a,b), wat 'n gesamentlike model van FRA koerse voorstel), geïmpliseerde termyn koerse en hul ooreenstemmende verspreiding bestudeer. Ooreenkomstig word die uitgebreide weergawe van die kort koers model in 'n multikromme opset en in die aanwesigheid van basisspreiding (voorgestel deur Kijima et al. (2008)) uiteengesit en bespreek. Hierdie werk verskaf 'n uitvoerige analise van hierdie uitbreidings en die ooreenstemmende geslote formules vir vloeibare produkte soos perke en ruiltransaksie opsies. Ten slotte, in die multi-geldeenheid geval, word die HJM model met stogastiese basisverspreiding (voorgestel deur Fujii et al. (2011)), nie-strydig met buitelandse valuta en kruisvaluta ruiltransaksie markte wat die effekte van aanvulling insluit word deuglik bestudeer.

Exotic options are derivatives which have features that makes them more complex than vanilla traded products. Thus, finding their fair value is not always an easy task. We look at a particular example of the exotic options - the power option - whose payoffs are nonlinear functions of the underlying asset price. Previous analyses of the power option have only obtained solutions using probability methods for the case of the constant stock volatility and interest rate. Using Lie symmetry analysis we obtain an optimal system of the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. In addition, we find a new analytical solution to the asymmetric type of the power option. We also focus on the more practical and realistic case of time dependent parameters: volatility and interest rate. Utilizing Lie symmetries, we are able to provide a new exact solution for the terminal pay off case. We also consider the power parameter of the option as a time dependent factor. A new solution is once again obtained for this scenario. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2013.

No abstract available. / Thesis (M.Sc.)-University of Natal, 1997.

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. / Thesis (M.Sc.)-University of Natal, 1993.

We consider the study of symmetry analysis of difference equations. The original work done by Lie about a century ago is known to be one of the best methods of solving differential equations. Lie's theory of difference equations on the contrary, was only first explored about twenty years ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some work has been done in the field of symmetries of difference equations for the past years. Given an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to analyze the problem. It is commonly known that the number of independent variables can be reduced after the symmetries of the equation are obtained. One can determine the optimal system of the equation in order to get a reduction of the independent variables. In addition, using the method, one can obtain new solutions from known ones. This feature is interesting because some differential equations have apparently useless trivial solutions, but applying Lie symmetries to them, more interesting solutions are obtained. The question arises when it happens that our equation contains a discrete quantity. In other words, we aim at investigating steps to be performed when we have a difference equation. Doing so, we find symmetries of difference equations and use them to linearize and reduce the order of difference equations. In this work, we analyze the work done by some researchers in the field and apply their results to some examples. This work will focus on the topical review of symmetries of difference equations and going through that will enable us to make some contribution to the field in the near future. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.