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 sub-structures 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 module-theoretic 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 rank-3 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 rank-3 group of degree q2m-1/q-1 on the points of the projective (2m-1)-space PG2m-1(IFq ). The codes obtained from the primitive rank-3 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 Reed-Muller 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 rank-3 action as primitive groups of degree q2m-1/q-1 these groups have 2-modular representations that give rise to self-orthogonal 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 (PD-sets) 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 permutation-decoding 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 3-sets being adjacent if they have zero, one or two elements in common, respectively. We show that permutation decoding can be used, by finding PD-sets, 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.

Bibliography: leaves 185-195 / 195 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1992

We give several characterizations of partition lattices and projective geometries. Most of these characterizations use characteristic polynomials. A geometry is non—splitting if it cannot be expressed as the union of two of its proper flats. A geometry G is upper homogeneous if for all k, k = 1, 2, ... , r(G), and for every pair x, y of flats of rank k, the contraction G/x is isomorphic to the contraction G/y. Given a signed graph, we define a corresponding signed—graphic geometry. We give a characterization of supersolvable signed graphs. Finally, we give the following characterization of non—splitting supersolvable signed-graphic geometries : If a non-splitting supersolvable ternary geometry does not contain the Reid geometry as a subgeometry, then it is signed—graphic.

Yes / In CAD, when considering the question of new designs of complex mechanical parts, such as engine pistons, a parametric representation of the design is usually defined. However, in general there is a lack of efficient tools to create and manipulate such parametrically defined shapes. In this paper, we show how the geometry of complex mechanical parts can be parameterised efficiently enabling a designer to create and manipulate such geometries within an interactive environment. For surface generation we use the PDE method which allows surfaces to be defined in terms of a relatively small number of design parameters. The PDE method effectively creates surfaces by using the information contained at the boundaries (edges) of the surface patch. An interactively defined parameterisation can then be introduced on the boundaries (which are defined by means of space curves) of the surface. Thus, we show how complex geometries of mechanical parts, such as engine pistons, can be efficiently parameterised for geometry manipulation allowing a designer to create alternative designs.

No description available.

A numerical procedure has been developed for the computation of inviscid flows over arbitrary, complex two-dimensional geometries. The Euler equations are solved using a finite-volume method with a non-body-fitted Cartesian grid. A new numerical formulation for complicated body geometries is developed in conjunction with implicit flux-splitting schemes. A variety of numerical computations have been performed to validate the numerical methodologies developed. Computations for supersonic flow over a flat plate with an impinging shock wave are used to verify the numerical algorithm, without geometric considerations. The supersonic flow over a blunt body is utilized to show the accuracy of the non-body-fitted Cartesian grid, along with the shock resolution of flux-vector splitting scheme. Geometric complexities are illustrated with the flow through a two-dimensional supersonic inlet with and without an open bleed door. The ability of the method to deal with subsonic and transonic flows is illustrated by computations over a non-lifting NACA 0012 airfoil. The method is shown to be accurate, efficient and robust and should prove to be particularly useful in a preliminary design mode, where flows past a wide variety of complex geometries can be computed without complicated grid generation procedures. / Ph. D.

Doctor of Philosophy / Department of Mathematics / Louis Crane / We exhibit a static, cylindrically symmetric, exact solution to the Euler-Heisenberg field equations (EHFE) and prove that its effective geometry contains (optical) black holes. It is conjectured that there are also soliton solutions to the EHFE which contain black hole geometries.

Thesis (MEng.)-- University of Stellenbosch, 1992. 140 leaves on single pages, preliminary pages i-xi and numbered pages 1-113. Includes bibliography. Digitized at 600 dpi grayscale to pdf format (OCR),using an Bizhub 250 Konica Minolta Scanner and at 300 dpi grayscale to pdf format (OCR), using a Hp Scanjet 8250 Scanner. / Thesis (MEng (Mechanical and Mechatronic Engineering))--University of Stellenbosch, 1992 / ENGLISH ABSTRACT: The large variety of turbo-machinery blade root geometries in use in industry prompted the question if a optimum geometry could be found. An optimum blade root was defined as a root with a practical geometry which, when loaded, returns the minimum fillet stress concentration factor. A literature survey on the subject provided guidelines but very little real data to work from. An initial optimization was carried out using a formula developed by Heywood to determine loaded projection fillet stresses. The method was found to produce unsatisfactory results, prompting a photoelastic investigation. This experimental optimization was conducted in two stages. A single tang defined load stage and a single tang in-rotor stage which modeled the practical situation. The defined load stage was undertaken in three phases. The first phase was a preliminary investigation, the second phase was a parameter optimization and the third phase was a geometric optimization based on a material utilization optimization. This material optimization approach produced good results. From these experiments a practical optimum geometry was defined. A mathematical model which predicts the fillet stress concentration factor for a given root geometry is presented. The effect of expanding the single tang optimum to a three tang root was examined. / AFRIKAANSE OPSOMMING: Die groot verskeidenheid lemwortelgeometrieë wat in turbomasjiene gebruik word het die vraag na 'n optimum geometrie laat ontstaan. Vir hierdie ondersoek is 'n optimum geometrie gedefineer as 'n praktiese geometrie wat, as dit belas word, die mimimum vloeistukspanningskonsentrasiefaktor laat ontstaan. 'n Literatuur studie het riglyne aan die navorsing gegee maar het wynig spesifieke en bruikbare data opgelewer. Die eerste optimering is met die Heywood formule, wat vloeistukspannings in belaste projeksies bepaal, aangepak. Die metode het nie bevredigende resultate opgelewer nie. 'n Fotoelastiese ondersoek het die basis vir verdere optimeering gevorm. Hierdie eksperimentele optimering is in twee stappe onderneem. 'n Enkelhaak gedefineerde lasgedeelte en 'n enkelhaak in-rotor gedeelte het die praktiese situasie gemodeleer. Die gedefineerde lasgedeelte is in drie fases opgedeel. Die eerste fase was n voorlopige ondersoek. Die tweede fase was 'n parameter optimering. 'n Geometrie optimering gebasseer op 'n materiaal benuttings minimering het die derde fase uitgemaak. Die materiaal optimerings benadering het goeie resultate opgelewer. Vanuit hierdie eksperimente is 'n optimum praktiese geometrie bepaal. 'n Wiskundige model is ontwikkel, wat die vloeistukspanningskonsentrasiefaktor vir 'n gegewe wortelgeometrie voorspel. Die resultaat van 'n geometriese uitbreiding van die enkelhaaklemwortel na 'n driehaaklemwortel op die spanningsverdeling is ondersoek.

[pt] Iniciamos com o estudo das orbifolds, que são espaços topológicos localmente homeomorfos a quocientes de Rn por grupos finitos. Estudamos em seguida os fibrados de Seifert de dimensão três, que consistem-se de folheações por círculos que podem ser vistas como fibrados sobre orbifolds. Esse material é usado em seguida no estudo das geometrias modelo. Uma geometria modelo (ou geometria de Thurston) é um par (G;X), onde X é uma variedade conexa e simplesmente conexa e G é um grupo de difeomorfismos de X com certas propriedades que nos permite encontrar uma métrica riemanniana em X tal que G é o grupo de todas as isometrias. A classificação das geometrias modelo é muito útil na classificação topológica das variedades que admitem uma métrica localmente homogênea e foi feita por Thurston em Three-Dimensional Geometry and Topology, vol.1, Princeton University Press, 1997. Na seqüência, apresentamos uma breve descrição de cada geometria modelo bem como parte da prova do teorema de classificação das geometrias modelo. / [en] We begin by studying orbifolds, i.e., topological spaces locally homeomorphic to quotients of Rn by finite groups. Then we study Seifert fiber spaces of dimension three which are certain type of foliations by circles that can be seen as fiber bundles over orbifolds. This material is useful in the subsequent study of Thurston model geometries. A Thurston model geometry is a pair (G;X), where X is a connected and simply connected manifold and G is a group of diffeomorfisms of X with certain properties that allow us to find a riemannian metric on X such that G is the group of all isometries. The classification of the model geometries is very useful in the topological classification of manifolds that admit a locally-homogeneous metric and was done by Thurston in Three-Dimensional Geometry and Topology, vol.1, Princeton University Press, 1997. Then we give a brief description of each one of these eight geometries and present part of Thurston s classification theorem.

In this dissertation, we study some open problems concerning the existence or non-existence of some combinatorial designs. We give the construction or proof of non-existence of some Steiner systems, large sets of designs, and graph designs, with prescribed automorphism groups. / by Emre Kolotoæglu. / Thesis (Ph.D.)--Florida Atlantic University, 2013. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.