# School Mathematics, Statistics and Computer Science

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Item Degree theory in nonlinear functional analysis.(1989) Pillay, Paranjothi.; Hill, C. K.The objective of this dissertation is to expand on the proofs and concepts of Degree Theory, dealt with in chapters 1 and 2 of Deimling [28], to make it more readable and accessible to anyone who is interested in the field. Chapter 1 is an introduction and contains the basic requirements for the subsequent chapters. The remaining chapters aim at defining a ll-valued map D (the degree) on the set M = {(F, Ω, y) / Ω C X open, F : Ὠ → X, y ɇ F(∂Ω)} (each time, the elements of M satisfying extra conditions) that satisfies : (D1) D(I, Ω, y) = 1 if y Є Ω. (D2) D(F, Ω, y) = D(F, Ω1 , y) + D(F, Ω2, y) if Ω1 and Ω2 are disjoint open subsets of Ω o such that y ɇ F(Ὠ \ Ω1 U Ω2 ). (D3) D(I - H(t, .), Ω, y(t)) is independent of t if H : J x Ὠ →X and y : J → X. An important property that follows from these three properties is (D4) F-1(y) ≠ Ø if D(F, Ω, y) ≠ 0. This property ensures that equations of the form Fx = y have solutions if D(F, Ω, y) ≠ 0. Another property that features in these chapters is the Borsuk property which gives us conditions under which the degree is odd and hence nonzero.Item Planarity testing and embedding algorithms.(1990) Carson, D. I.; Oellermann, Ortrud Ruth.This thesis deals with several aspects of planar graphs, and some of the problems associated with non-planar graphs. Chapter 1 is devoted to introducing some of the fundamental notation and tools used in the remainder of the thesis. Graphs serve as useful models of electronic circuits. It is often of interest to know if a given electronic circuit has a layout on the plane so that no two wires cross. In Chapter 2, three efficient algorithms are described for determining whether a given 2-connected graph (which may model such a circuit) is planar. The first planarity testing algorithm uses a path addition approach. Although this algorithm is efficient, it does not have linear complexity. However, the second planarity testing algorithm has linear complexity, and uses a recursive fragment addition technique. The last planarity testing algorithm also has linear complexity, and relies on a relatively new data structure called PQ-trees which have several important applications to planar graphs. This algorithm uses a vertex addition technique. Chapter 3 further develops the idea of modelling an electronic circuit using a graph. Knowing that a given electronic circuit may be placed in the plane with no wires crossing is often insufficient. For example, some electronic circuits often have in excess of 100 000 nodes. Thus, obtaining a description of such a layout is important. In Chapter 3 we study two algorithms for obtaining such a description, both of which rely on the PQ-tree data structure. The first algorithm determines a rotational embedding of a 2-connected graph. Given a rotational embedding of a 2-connected graph, the second algorithm determines if a convex drawing of a graph is possible. If a convex drawing is possible, then we output the convex drawing. In Chapter 4, we concern ourselves with graphs that have failed a planarity test of Chapter 2. This is of particular importance, since complex electronic circuits often do not allow a layout on the plane. We study three different ways of approaching the problem of an electronic circuit modelled on a non-planar graph, all of which use the PQ-tree data structure. We study an algorithm for finding an upper bound on the thickness of a graph, an algorithm for determining the subgraphs of a non-planar graph which are subdivisions of the Kuratowski graphs K5 and K3,3, and lastly we present a new algorithm for finding an upper bound on the genus of a non-planar graph.Item A practical investigation of meteor-burst communications.(1991) Melville, Stuart William.; Sartori-Angus, Alan G.; Oellermann, Ortrud Ruth.This study considers the meteor-burst communication (MBC) environment at three levels. At the lowest level, the trails themselves are studied and analysed. Then individual links are studied in order to determine the data throughput and wait time that might be expected at various data rates. Finally, at the top level, MBC networks are studied in order to provide information on the effects of routing strategies, topologies, and connectivity in such networks. A significant amount of theoretical work has been done in the classification of meteor trails, and the analysis of the throughput potential of the channel. At the same time the issues of wait time on MBC links, and MBC network strategies, have been largely ignored. The work presented here is based on data captured on actual monitoring links, and is intended to provide both an observational comparison to theoretical predictions in the well-researched areas, and a source of base information for the others. Chapter 1 of this thesis gives an overview of the field of meteor-burst communications. Prior work in the field is discussed, as are the advantages and disadvantages of the channel, and current application areas. Chapter 2 describes work done on the classification of observed meteor trails into distinctive 'families'. The rule-based system designed for this task is discussed as well as the eventual classification schema produced, which is far more comprehensive and consistent than previously proposed schemas. Chapter 3 deals with the throughput potential of the channel, based on the observed trails. A comparison to predicted results, both as regards fixed and adaptive data-rates, is made with some notable differences between predicted v results and observed results highlighted. The trail families with the largest contribution to the throughput capacity of the channel are identified. Chapter 4 deals with wait time in meteor-burst communications. The data rates at which wait time is minimised in the links used are found, and compared to the rates at which throughput was optimised. These are found to be very different, as indeed are the contributions of the various trail families at these rates. Chapter 5 describes a software system designed to analyse the effect of routing strategies in MBC networks, and presents initial results derived from this system. Certain features of the channel, in particular its sporadic nature, are shown to have significant effects on network performance. Chapter 6 continues the presentation of network results, specifically concentrating on the effect of topologies and connectivity within MBC networks. Chapter 7 concludes the thesis, highlighting suggested areas for further research as well as summarising the more important results presented.Item Packing problems on a PC.(1991) Deighton, Andrew George.; Meyerowitz, Jane Julia.Bin packing is a problem with many applications in various industries. This thesis addresses a specific instance of the this problem, known as the Container Packing problem. Special attention is paid to the Pallet Loading problem which is a restricted sub-problem of the general Container Packing problem. Since the Bin Packing problem is NP-complete, it is customary to apply a heuristic measure in order to approximate solutions in a reasonable amount of computation time rather than to attempt to produce optimal results by applying some exact algorithm. Several heuristics are examined for the problems under consideration, and the results produced by each are shown and compared where relevant.Item Filter characterisations of the extendibility of continuous functions.(1991) Maltby, Gavin Richard.; Swart, Johan.Abstract available in PDF.Item Some Mal'cev conditions for varieties of algebras.(1991) Moses, Mogambery.; Sturm, Teo.This dissertation deals with the classification of varieties according to their Mal'cev properties. In general the so called Mal'cev-type theorems illustrate an interplay between first order properties of a given class of algebras and the lattice properties of the congruence lattices of algebras of the considered class. CHAPTER 1. A survey of some notational conventions, relevant definitions and auxiliary results is presented. Several examples of less frequently used algebras are given together with the important properties of some of them. The term algebra T(X) and useful results concerning 'term' operations are established. A K-reflection is defined and a connection between a K-reflection of an algebra and whether a class K satisfies an identity of the algebra is established. CHAPTER 2. The Mal'cev-type theorems are presented in complete detail for varieties which are congruence permutable, congruence distributive, arithmetical, congruence modular and congruence regular. Several examples of varieties which exhibit these properties are presented together with the necessary verifications. CHAPTER 3. A general scheme of algorithmic character for some Mal'cev conditions is presented. R. Wille (1970) and A. F. Pixley (1972) provided algorithms for the classification of varieties which exhibit strong Mal'cev properties. This chapter is largely devoted to a modification of the Wille-Pixley schemes. It must be noted that this modification is quite different from all such published schemes. The results are the same as in Wille's scheme but slightly less general than in Pixley's. The text presented here, however is much simpler. As an example, the scheme is used to confirm Mal'cev's original theorem on congruence permutable varieties. Finally, the so-called Chinese var£ety is defined and Mal'cev conditions are established for such a variety of algebras . CHAPTER 4. A comprehensive survey of literature concerning Mal'cev conditions is given in this chapter.Item Built-in tests for a real-time embedded system.(1991) Olander, Peter Andrew.Beneath the facade of the applications code of a well-designed real-time embedded system lies intrinsic firmware that facilitates a fast and effective means of detecting and diagnosing inevitable hardware failures. These failures can encumber the availability of a system, and, consequently, an identification of the source of the malfunction is needed. It is shown that the number of possible origins of all manner of failures is immense. As a result, fault models are contrived to encompass prevalent hardware faults. Furthermore, the complexity is reduced by determining syndromes for particular circuitry and applying test vectors at a functional block level. Testing phases and philosophies together with standardisation policies are defined to ensure the compliance of system designers to the underlying principles of evaluating system integrity. The three testing phases of power-on self tests at system start up, on-line health monitoring and off-line diagnostics are designed to ensure that the inherent test firmware remains inconspicuous during normal applications. The prominence of the code is, however, apparent on the detection or diagnosis of a hardware failure. The authenticity of the theoretical models, standardisation policies and built-in test philosophies are illustrated by means of their application to an intricate real-time system. The architecture and the software design implementing the idealogies are described extensively. Standardisation policies, enhanced by the proposition of generic tests for common core components, are advocated at all hierarchical levels. The presentation of the integration of the hardware and software are aimed at portraying the moderately complex nature of the task of generating a set of built-in tests for a real-time embedded system. In spite of generic policies, the intricacies of the architecture are found to have a direct influence on software design decisions. It is thus concluded that the diagnostic objectives of the user requirements specification be lucidly expressed by both operational and maintenance personnel for all testing phases. Disparity may exist between the system designer and the end user in the understanding of the requirements specification defining the objectives of the diagnosis. It is thus essential for complete collaboration between the two parties throughout the development life cycle, but especially during the preliminary design phase. Thereafter, the designer would be able to decide on the sophistication of the system testing capabilities.Item Partial exchangeability and related topics.(1991) North, Delia Elizabeth.; Dale, Andrew Ian.Partial exchangeability is the fundamental building block in the subjective approach to the probability of multi-type sequences which replaces the independence concept of the objective theory. The aim of this thesis is to present some theory for partially exchangeable sequences of random variables based on well-known results for exchangeable sequences. The reader is introduced to the concepts of partially exchangeable events, partially exchangeable sequences of random variables and partially exchangeable o-fields, followed by some properties of partially exchangeable sequences of random variables. Extending de Finetti's representation theorem for exchangeable random variables to hold for multi-type sequences, we obtain the following result to be used throughout the thesis: There exists a o-field, conditional upon which, an infinite partially exchangeable sequence of random variables behaves like an independent sequence of random variables, identically distributed within types. Posing (i) a stronger requirement (spherical symmetry) and (ii) a weaker requirement (the selection property) than partial exchangeability on the infinite multi-type sequence of random variables, we obtain results related to de Finetti's representation theorem for partially exchangeable sequences of random variables. Regarding partially exchangeable sequences as mixtures of independent and identically distributed (within types) sequences, we (i) give three possible expressions for the directed random measures of the partially exchangeable sequence and (ii) look at three possible expressions for the o-field mentioned in de Finetti's representation theorem. By manipulating random measures and using de Finetti's representation theorem, we point out some concrete ways of constructing partially exchangeable sequences. The main result of this thesis follows by extending de Finetti's represen. tation theorem in conjunction with the Chatterji principle to obtain the following result: Given any a.s. limit theorem for multi-type sequences of independent random variables, identically distributed within types, there exists an analogous theorem satisfied by all partially exchangeable sequences and by all sub-subsequences of some subsequence of an arbitrary dependent infinite multi-type sequence of random variables, tightly distributed within types. We finally give some limit theorems for partially exchangeable sequences of random variables, some of which follow from the above mentioned result.Item Conformal symmetries : solutions in two classes of cosmological models.(1991) Moodley, Manikam.; Maharaj, Sunil Dutt.In this thesis we study the conformal symmetries in two locally rotationally symmetric spacetimes and the homothetic symmetries of a Bianchi I spacetime. The conformal Killing equation in a class AIa spacetime (MacCallum 1980), with a G4 of motions, is integrated to obtain the general solution subject to integrability conditions. These conditions are comprehensively analysed to determine the restrictions on the metric functions. The Killing vectors are contained in the general conformal solution. The homothetic vector is obtained and the explicit functional dependence of the metric functions determined. The class AIa spacetime does not admit a nontrivial special conformal factor. We also integrate the conformal Killing equation in the anisotropic locally rotationally symmetric spacetime of class A3 (MacCallum 1980), with a G4 of motions, to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. The Killing vectors are obtained as a special case from the general conformal solution. The homothetic vector is found for a nonzero constant conformal factor. The explicit functional form of the metric functions is determined for the existence of this homothetic vector. The spatially homogeneous and anisotropic A3 spacetime also does not admit a nontrivial special conformal vector. In the Bianchi I spacetime, with a G3 of motions, the conformal Killing equation is integrated for a constant conformal factor to generate the homothetic symmetries. The integrability conditions are solved to determine the functional dependence of the three time-dependent metric functions.Item Application of backpropagation-like generative algorithms to various problems.(1992) Powell, Alan Roy.; Sartori-Angus, Alan G.Artificial neural networks (ANNs) were originally inspired by networks of biological neurons and the interactions present in networks of these neurons. The recent revival of interest in ANNs has again focused attention on the apparent ability of ANNs to solve difficult problems, such as machine vision, in novel ways. There are many types of ANNs which differ in architecture and learning algorithms, and the list grows annually. This study was restricted to feed-forward architectures and Backpropagation- like (BP-like) learning algorithms. However, it is well known that the learning problem for such networks is NP-complete. Thus generative and incremental learning algorithms, which have various advantages and to which the NP-completeness analysis used for BP-like networks may not apply, were also studied. Various algorithms were investigated and the performance compared. Finally, the better algorithms were applied to a number of problems including music composition, image binarization and navigation and goal satisfaction in an artificial environment. These tasks were chosen to investigate different aspects of ANN behaviour. The results, where appropriate, were compared to those resulting from non-ANN methods, and varied from poor to very encouraging.Item On Stephani universes.(1992) Moopanar, Selvandren.; Maharaj, Sunil Dutt.In this dissertation we study conformal symmetries in the Stephani universe which is a generalisation of the Robertson-Walker models. The kinematics and dynamics of the Stephani universe are discussed. The conformal Killing vector equation for the Stephani metric is integrated to obtain the general solution subject to integrability conditions that restrict the metric functions. Explicit forms are obtained for the conformal Killing vector as well as the conformal factor . There are three categories of solution. The solution may be categorized in terms of the metric functions k and R. As the case kR - kR = 0 is the most complicated, we provide all the details of the integration procedure. We write the solution in compact vector notation. As the case k = 0 is simple, we only state the solution without any details. In this case we exhibit a conformal Killing vector normal to hypersurfaces t = constant which is an analogue of a vector in the k = 0 Robertson-Walker spacetimes. The above two cases contain the conformal Killing vectors of Robertson-Walker spacetimes. For the last case in - kR = 0, k =I 0 we provide an outline of the integration process. This case gives conformal Killing vectors which do not reduce to those of RobertsonWalker spacetimes. A number of the calculations performed in finding the solution of the conformal Killing vector equation are extremely difficult to analyse by hand. We therefore utilise the symbolic manipulation capabilities of Mathematica (Ver 2.0) (Wolfram 1991) to assist with calculations.Item Speech recognition and blackboard expert systems.(1992) Loureiro, Guy Marchand.; Sartori-Angus, Alan G.Spoken language is used by people to communicate naturally with one another. A simplistic view of the communication process is as follows. Person A wishes to communicate an idea to person B. The idea, initiated in the mind/brain of person A is encoded into speech signals by means of the person A's speech production mechanism, the vocal apparata in the vocal tract. Various kinds of noise may interfere with the speech signals as they travel to person B. The resulting signal is captured by person B's speech receiving mechanism, the ear. It is then analysed and decoded into a meaningful message by the brain of person B. This thesis concerns itself with the investigation of and attempt to automate the receiving and decoding of English sentences using a machine - that is to perform the task of person B in the above scenario using a computer. The aim is not only to produce a sequence of phonetic sounds, but to look at the problems of building in the 'mind of the machine', a picture of the meanings, intentions, absurdities and realities of the spoken message. The various models, algorithms and techniques of speech recognition and speech understanding systems are examined. Speech signals are captured and digitised by hardware. The digital samples are analysed and the important distinguishing features of all speech sounds are identified. These are then used to classify speech sounds in subsequent spoken words. The way speech sounds are joined together to form syllables and words introduces difficult problems to the automatic recognition process. Speech sounds are blurred, overlapped or left out due to the effects of coarticulation. Finally, natural language processing issues, such as the importance of syntax (the structure) and semantics (the meaning) of sentences, are studied. A system to control and unite all the above processing is considered. The blackboard expert system model of the widely reported HEARSAY-II speech recognition system is reviewed as the system with the best potential for the above tasks.Item An application of some inventory control techniques.(1992) Samuels, Carol Anne.; Moolman, W. H.; Ryan, K. C.No abstract available.Item The theory of option valuation.(1992) Sewambar, Soraya.; Murray, Michael.Although options have been traded for many centuries, it has remained a relatively thinly traded financial instrument. Paradoxically, the theory of option pricing has been studied extensively. This is due to the fact that many of the financial instruments that are traded in the market place have an option-like structure, and thus the development of a methodology for option-pricing may lead to a general methodology for the pricing of these derivative-assets. This thesis will focus on the development of the theory of option pricing. Initially, a fundamental principle that underlies the theory of option valuation will be given. This will be followed by a discussion of the different types of option pricing models that are prevalent in the literature. Special attention will then be given to a detailed derivation of both the Black-Scholes and the Binomial Option pricing models, which will be followed by a proof of the convergence of the Binomial pricing model to the Black-Scholes model. The Black-Scholes model will be adapted to take into account the payment of dividends, the possibility of a changing inter est rate and the possibility of a stochastic variance for the rate of return on the underlying as set. Several applications of the Black-Scholes model will finally be presented.Item Conformal motions in Bianchi I spacetime.(1992) Lortan, Darren Brendan.; Maharaj, Sunil Dutt.In this thesis we study the physical properties of the manifold in general relativity that admits a conformal motion. The results obtained are general as the metric tensor field is not specified. We obtain the Lie derivative along a conformal Killing vector of the kinematical and dynamical quantities for the general energy-momentum tensor of neutral matter. Equations obtained previously are regained as special cases from our results. We also find the Lie derivative of the energy-momentum tensor for the electromagnetic field. In particular we comprehensively study conformal symmetries in the Bianchi I spacetime. The conformal Killing vector equation is integrated to obtain the general conformal Killing vector and the conformal factor subject to integrability conditions. These conditions place restrictions on the metric functions. A particular solution is exhibited which demonstrates that these conditions have a nonempty solution set. The solution obtained is a generalisation of the results of Moodley (1991) who considered locally rotationally symmetric spacetimes. The Killing vectors are regained as special cases of the conformal solution. There do not exist any proper special conformal Killing vectors in the Bianchi I spacetime. The homothetic vector is found for a nonvanishing constant conformal factor. We establish that the vacuum Kasner solution is the only Bianchi I spacetime that admits a homothetic vector. Furthermore we isolate a class of vectors from the solution which causes the Bianchi I model to degenerate into a spacetime of higher symmetry.Item 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.Item On the integrity of domination in graphs.(1993) Smithdorf, Vivienne.; Swart, Hendrika Cornelia Scott.This thesis deals with an investigation of the integrity of domination in a.graph, i.e., the extent to which domination properties of a graph are preserved if the graph is altered by the deletion of vertices or edges or by the insertion of new edges. A brief historical introduction and motivation are provided in Chapter 1. Chapter 2 deals with kedge-( domination-)critical graphs, i.e., graphsG such that )'(G) = k and )'(G+e) < k for all e E E(G). We explore fundamental properties of such graphs and their characterization for small values of k. Particular attention is devoted to 3-edge-critical graphs. In Chapter 3, the changes in domination number brought aboutby vertex removal are investigated. \ Parameters )'+'(G) (and "((G)), denoting the smallest number of vertices of G in a set 5 such that )'(G-5) > )'(G) ()'(G -5) < )'(G), respectively), are investigated, as are'k-vertex-critical graphs G (with )'(G) = k and )'(G-v) < k for all v E V(O)). The existence of smallest'domination-forcing sets of vertices of graphs is considered. The bondage number 'Y+'(G), i.e., the smallest number of edges of a graph G in a set F such that )'(G- F) > )'(0), is investigated in Chapter 4, as are associated extremal graphs. Graphs with dominating sets or domination numbers that are insensitive to the removal of an arbitrary edge are considered, with particular reference to such graphs of minimum size. Finally, in Chapter 5, we-discuss n-dominating setsD of a graph G (such that each vertex in G-D is adjacent to at least n vertices in D) and associated parameters. All chapters but the first and fourth contain a listing of unsolved problems and conjectures.Item Relativistic spherical stars.(1993) Mkhwanazi, Wiseman Thokozani.; Maharaj, Sunil Dutt.In this thesis we study spherically symmetric spacetimes which are static with a perfect fluid source. The Einstein field equations, in a number of equivalent forms, are derived in detail. The physical properties of a relativistic star are briefly reviewed. We specify two particular choices for one of the gravitational potentials. The behaviour of the remaining gravitational potential is governed by a second order differential equation. This equation has solutions in terms of elementary functions for some cases. The differential equation, in other cases, may be expressed as Bessel, confluent hypergeometric and hypergeometric equations. In such instances the solution is given in terms of special functions. A number of solutions to the Einstein field equations are generated. We believe that these solutions may be used to model realistic stars. Many of the solutions found are new and have not been published previously. In some cases our solutions are generalisations of cases considered previously. For some choices of the gravitational potential our solutions are equivalent to well-known results documented in the literature; in these cases we explicitly relate our solutions to those published previously. We have utilised the computer package MATHEMATICA Version 2.0 (Wolfram 1991) to assist with calculations, and to produce figures to describe the gravitational field. In addition, we briefly investigate the approach of specifying an equation of state relating the energy density and the pressure. The solution of the Einstein field equations, for a linear equation of state, is reduced to integrating Abel's equation of the second kind.Item Possible Models Diagrams - a new approach to teaching propositional logic.(1994) Clarke, Matthew Charles.; Dempster, Robert.; Grayson, Diane J.Abstract available in PDF.Item 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.