An n-Venn diagram is defined as a collection of n finitely intersecting closed curves dividing the plane into 2^n distinct regions, where each region is in the interior of a unique subset of the curves. A Venn diagram is simple if at most two curves intersect at any point, and it is monotone if it has some embedding on the plane in which all curves are convex. An n-Venn diagram has n-fold rotational symmetry if a rotation of 180 degrees about a centre point in the plane leaves the diagram unchanged, up to a relabeling of the curves. It has been known that rotationally symmetric Venn diagrams could exist only if the number of curves is prime. Moreover, non-simple Venn diagrams with rotational symmetry have been proven to exist for any prime number of curves. However, the largest prime for which a simple rotationally symmetric Venn diagram was known prior to this, was 7. In this thesis, we are concerned with generating simple monotone Venn diagrams, especially those that have some type(s) of symmetry. Several representations of these diagrams are introduced and different backtracking search algorithms are provided based on these representations. Using these algorithms we show that there are 39,020 non-isomorphic simple monotone 6-Venn diagrams in total. In the case of drawing Venn diagrams on a sphere, we prove that there exists a simple symmetric n-Venn diagram, for any n >= 6, with the following set(s) of isometries : (a) a 4-fold rotational symmetry about the polar axis, together with an additional involutional symmetry about an axis through the equator, or (b) an involutional symmetry about the polar axis together with two reflectional symmetries about orthogonal planes that intersect at the polar axis. Finally, we introduce a new type of symmetry of Venn diagrams which leads us to the discovery of the first simple rotationally symmetric Venn diagrams of 11 and 13 curves. / Graduate / 0984 / Khalegh@GMail.com

Sistemas biológicos podem ser representados por redes que armazenam não apenas informações de conectividade, mas também informações de características de seus nós. No contexto biomolecular, esses nós podem representar proteínas, metabólitos, entre outros tipos de moléculas. Cada molécula possui características anotadas e armazenadas em bases de dados como o Gene Ontology. A comparação visual dessas redes depende de ferramentas que permitam o usuário identificar diferenças e semelhanças entre as anotações feitas sobre as moléculas (atributos) e também sobre as interações conhecidas (conexões). Neste trabalho de mestrado, buscou-se desenvolver técnicas que facilitem a comparação desses atributos sobre as moléculas, tentando manter no processo a visualização das redes em que essas moléculas estão inseridas. Como resultado, obteve-se a ferramenta VisPipeline-MultiNetwork, que permite comparar até seis redes, utilizando operações de conjuntos sobre as redes e sobre seus atributos. Dessa forma, diferentemente da maioria das ferramentas conhecidas para a visualização de redes biológicas, o VisPipeline-MultiNetwork permite a criação de redes cujos atributos são derivados das redes originais por meio de operações de união, intersecção e valores exclusivos. A comparação visual das redes é feita pela visualização do resultado dessas operações de conjuntos sobre as redes, por meio de um método de comparação lado-a-lado. Já a comparação dos atributos armazenados nos nós das redes é feita por meio de diagramas de Venn. Para auxiliar este tipo de comparação, a técnica InteractiVenn foi desenvolvida, em que o usuário pode interagir com um diagrama de Venn efetuando operações de união entre conjuntos. Essas operações de união aplicadas sobre os conjuntos são também aplicadas sobre as respectivas formas no diagrama. Esta característica da técnica a diferencia das outras ferramentas de criação de diagramas de Venn. Integrando essas funcionalidades, o usuário é capaz de comparar redes sob diversas perspectivas. Para exemplificar a utilização do VisPipeline-MultiNetwork, dois casos no contexto biomolecular foram estudados. Adicionalmente, uma ferramenta web para a comparação de listas de cadeias de caracteres por meio de diagramas de Venn foi desenvolvida. Ela também implementa a técnica InteractiVenn e foi denominada InteractiVenn website. / Biological systems can be represented by networks that store not only connectivity information, but also node feature information. In the context of molecular biology, these nodes may represent proteins, metabolites, and other types of molecules. Each molecule has features annotated and stored in databases such as Gene Ontology. A visual comparison of networks requires tools that allow the user to identify differences and similarities between nodes attributes as well as known interactions between nodes (connections). In this dissertation, we sought to develop a technique that would facilitate the comparison of these biological networks, striving to maintain in the process the visualization of the network connectivities. As a result, we have developed the VisPipeline-MultiNetwork tool, which allows comparison of up to six networks, using sets of operations on networks and on their attributes. Unlike most known tools for visualizing biological networks, VisPipeline-MultiNetwork allows the creation of networks whose attributes are derived from the original networks through operations of union, intersection and unique values. A visual comparison of the networks is achieved by visualizing the outcome of such joint operations through a all-in-one comparison method. The comparison of nodes attributes is performed using Venn diagrams. To assist this type of comparison, the InteractiVenn technique was developed, in which the user can interact with a Venn diagram, performing union operations between sets and their corresponding diagrams. This diagram union feature differs from other tools available for creating Venn diagrams. With these tools, users manage to compare networks from different perspectives. To exemplify the use of VisPipeline-MultiNetwork, two case studies were carried out in the biomolecular context. Additionally, a web tool for comparing lists of strings by means of Venn diagrams was made available. It also implements the InteractiVenn technique and its site has been named InteractiVenn.

Sistemas biológicos podem ser representados por redes que armazenam não apenas informações de conectividade, mas também informações de características de seus nós. No contexto biomolecular, esses nós podem representar proteínas, metabólitos, entre outros tipos de moléculas. Cada molécula possui características anotadas e armazenadas em bases de dados como o Gene Ontology. A comparação visual dessas redes depende de ferramentas que permitam o usuário identificar diferenças e semelhanças entre as anotações feitas sobre as moléculas (atributos) e também sobre as interações conhecidas (conexões). Neste trabalho de mestrado, buscou-se desenvolver técnicas que facilitem a comparação desses atributos sobre as moléculas, tentando manter no processo a visualização das redes em que essas moléculas estão inseridas. Como resultado, obteve-se a ferramenta VisPipeline-MultiNetwork, que permite comparar até seis redes, utilizando operações de conjuntos sobre as redes e sobre seus atributos. Dessa forma, diferentemente da maioria das ferramentas conhecidas para a visualização de redes biológicas, o VisPipeline-MultiNetwork permite a criação de redes cujos atributos são derivados das redes originais por meio de operações de união, intersecção e valores exclusivos. A comparação visual das redes é feita pela visualização do resultado dessas operações de conjuntos sobre as redes, por meio de um método de comparação lado-a-lado. Já a comparação dos atributos armazenados nos nós das redes é feita por meio de diagramas de Venn. Para auxiliar este tipo de comparação, a técnica InteractiVenn foi desenvolvida, em que o usuário pode interagir com um diagrama de Venn efetuando operações de união entre conjuntos. Essas operações de união aplicadas sobre os conjuntos são também aplicadas sobre as respectivas formas no diagrama. Esta característica da técnica a diferencia das outras ferramentas de criação de diagramas de Venn. Integrando essas funcionalidades, o usuário é capaz de comparar redes sob diversas perspectivas. Para exemplificar a utilização do VisPipeline-MultiNetwork, dois casos no contexto biomolecular foram estudados. Adicionalmente, uma ferramenta web para a comparação de listas de cadeias de caracteres por meio de diagramas de Venn foi desenvolvida. Ela também implementa a técnica InteractiVenn e foi denominada InteractiVenn website. / Biological systems can be represented by networks that store not only connectivity information, but also node feature information. In the context of molecular biology, these nodes may represent proteins, metabolites, and other types of molecules. Each molecule has features annotated and stored in databases such as Gene Ontology. A visual comparison of networks requires tools that allow the user to identify differences and similarities between nodes attributes as well as known interactions between nodes (connections). In this dissertation, we sought to develop a technique that would facilitate the comparison of these biological networks, striving to maintain in the process the visualization of the network connectivities. As a result, we have developed the VisPipeline-MultiNetwork tool, which allows comparison of up to six networks, using sets of operations on networks and on their attributes. Unlike most known tools for visualizing biological networks, VisPipeline-MultiNetwork allows the creation of networks whose attributes are derived from the original networks through operations of union, intersection and unique values. A visual comparison of the networks is achieved by visualizing the outcome of such joint operations through a all-in-one comparison method. The comparison of nodes attributes is performed using Venn diagrams. To assist this type of comparison, the InteractiVenn technique was developed, in which the user can interact with a Venn diagram, performing union operations between sets and their corresponding diagrams. This diagram union feature differs from other tools available for creating Venn diagrams. With these tools, users manage to compare networks from different perspectives. To exemplify the use of VisPipeline-MultiNetwork, two case studies were carried out in the biomolecular context. Additionally, a web tool for comparing lists of strings by means of Venn diagrams was made available. It also implements the InteractiVenn technique and its site has been named InteractiVenn.

Anecdotal observation of the way in which data visualisation techniques are utilised to present relationships in data to audiences informed the author's view that data visualisation had not evolved to utilise the capabilities of ubiquitous business computer equipment. In an information rich but attention poor business environment, a search for a new tool was undertaken to supplement those techniques available to help audiences understand statistical relationships in presentation data. This search resulted in the development of a practical software tool based on animated Venn diagrams (Dvenn) that attempted to exploit the inherent human ability to perceive quantities visually, a faculty described herein as visual numeracy. The exploitation of this faculty is considered here to be a valuable aid for group understanding of business presentation data. The development of the tool was an essential part of the research that was undertaken and the resulting software forms a significant portion of this practise based research. The aim of the software development was to develop a readily accessible tool that could be utilised in a non-specialist business environment to better facilitate an honest shared meaning of numerical data between a presenter and their audience. The development of the tool progressed through a number of iterations and the software that accompanies this work is an important component that needs to be viewed in conjunction with the text. The test of the final version was undertaken with undergraduate University students in an attempt to validate the efficacy of the data visualisation technique. The test of the Dvenn software was made against the mature yardstick of scatter-plots. Interestingly, the correlations presented by scatter-plot were not as readily identified as would have been assumed, however, the results for the Dvenn tests were not supportive of the technique for widespread adoption. Nevertheless, further research into the best method of harnessing visual numeracy would seem to be justified.

A previous study on students approach to problems of Set Theory in Iran brought me somehypotheses. These encouraged me to test the hypotheses in this supplementary study. Mypurpose of this study was to consider some students' cognitive difficulties in Set Theory. Iinvestigated students’ conceptual understanding of two major concepts of Set Theory – theconcepts of inclusion and belonging. I also studied the ways students use Venn diagram tofigure out problems in Set Theory. I wanted to examine how students figure out the differentmeanings of words in the natural and formal language. To do so, I analyzed six experiments. Icompared three experiments with the experiments of my previous study in Iran. My researchquestions suggested using a qualitative research method. My theoretical framework built aroundtheories of semiotic activities, which were used in the analysis.The results indicated that students can make unfortunate use of Euler–Venn diagrams. A set ofsets was a difficult concept for students. Some words in natural language, even the word "set" inboth natural and formal language, caused confusion for students when introduced in Settheoretic contexts. Students failed sometimes to distinguish between sets and elements. Theexperiments showed that students’ cultural context had affect on students' cognition regardingmathematical objects.

A diagram on a surface is a collection of coloured simple closed curves which generally intersect only at points, and a Venn diagram of n curves has the additional property that there are exactly 2^n faces in the diagram, each corresponding to a unique intersection of the interiors of a subset of the curves. A diagram has rotational symmetry if it can be constructed by rotating a single closed curve in the plane n times, each time by 2 \pi /n, and changing the colour of the curve for each rotation; equivalently, the diagram can be constructed from a region forming a "pie-slice" of the diagram and containing a section of each curve, and then copying and rotating this region n times, recolouring the sections of curves in the region appropriately. This and reflective symmetries are the only non-trivial ways a finite plane diagram can have some kind of symmetry. In this thesis, we extend the notion of planar symmetries for diagrams onto the sphere by constructing and projecting diagrams onto the sphere and examining the much richer symmetry groups that result. Restricting our attention to Venn diagrams gives a rich combinatorial structure to the diagrams that we examine and exploit. We derive several constructions of Venn diagrams with interesting symmetries on the sphere by modifying the landmark work of Griggs, Killian and Savage from 2004 which provided some important answers to questions about planar symmetric diagrams. We examine a class of diagrams that exhibit a rotary reflection symmetry (a rotation of the sphere followed by a reflection), in which we make some initial steps towards a general construction for n-Venn diagrams realizing a very rich symmetry group of order 2n, for n prime or a power of two. We also provide a many-dimensional construction of very simple Venn diagrams which realize any subgroup of an important type of symmetry group that use only reflection symmetries. In summary, we exhibit and examine at least one Venn diagram realizing each of the 14 possible different classes of finite symmetry groups on the sphere, many of these diagrams with different types of colour symmetry. All of these investigations are coupled with a theoretical and practical framework for further investigation of symmetries of diagrams and discrete combinatorial objects on spheres and higher-dimensional surfaces.

Specification plays a vital role in software engineering to facilitate the development of highly dependable software. The importance of specification in software development is to serve, amongst others, as a communication tool for stakeholders in the software project. The specification also adds to the understanding of operations, and describes the properties of a system. Various techniques may be used for specification work. Z is a formal specification language that is based on a strongly-typed fragment of Zermelo-Fraenkel set theory and first-order logic to provide for precise and unambiguous specifications. Z uses mathematical notation to build abstract data, which is necessary for a specification. The role of abstraction is to describe what the system does without prescribing how it should be done. Diagrams, on the other hand, have also been used in various areas, and in software engineering they could be used to add a visual component to software specifications. It is plausible that diagrams may also be used to reason in a semi-formal way about the properties of a specification. Many diagrammatic languages are based on contours and set theory. Examples of these languages are Euler-, Spider-, Venn- and Pierce diagrams. Euler diagrams form the foundation of most diagrams that are based on closed curves. Diagrams, on the other hand, have also been used in various areas, and in software engineering they could be used to add a visual component to software specifications. It is plausible that diagrams may also be used to reason in a semi-formal way about the properties of a specification. Many diagrammatic languages are based on contours and set theory. Examples of these languages are Euler-, Spider-, Venn- and Pierce diagrams. Euler diagrams form the foundation of most diagrams that are based on closed curves. The purpose of this research is to demonstrate the extent to which diagrams can be used to represent a Z specification. A case study is used to transform the specification modelled with Z language into a diagrammatic specification. Euler, spider, Venn and Pierce diagrams are combined for this purpose, to form one diagrammatic notation that is used to transform a Z specification / School of Computing / M. Sc. (Information Systems)

This dissertation focuses on how requirements for a new software development system are elicited and what pitfalls could cause a software development project to fail if the said requirements are not captured correctly. A number of existing requirements elicitation methods, namely: JAD (Joint Application Design), RAD (Rapid Application Development), a Formal Specifications Language (Z), Natural Language, UML (Unified Modelling Language) and Prototyping are covered. The aforementioned techniques are then integrated into existing software development life cycle models, such as the Waterfall model, Rapid Prototyping model, Build and Fix model, Spiral model, Incremental model and the V-Process model. Differences in the domains (knowledge and experience of an environment) of a client and that of the software development team are highlighted and this is done diagrammatically using the language of Venn diagrams. The dissertation also refers to a case study highlighting a number of problems during the requirements elicitation process, amongst other the problem of tacit knowledge not surfacing during elicitation. Two new requirements elicitation methodologies are proposed namely: the SRE (Solitary Requirements Elicitation) and the DDI (Developer Domain Interaction) methodology. These two methods could potentially be more time consuming than other existing requirements elicitation methods, but the benefits could outweigh the cost of their implementation, since the new proposed methods have the potential to further facilitate the successful completion of a software development project. Following the introduction of the new requirements elicitation methods, they are then applied to the aforementioned case study and highlight just how the hidden domain of the client may become more visible, because the software development team has gained a deeper understanding of the client’s working environment. They have therefore increased their understanding of how the final product needs to function in order to fulfil the set out requirements correctly. Towards the end of the dissertation a summary and a conclusion as well as future work that could be undertaken in this area are provided. / Computer Science / M. Sc. (Computer Science)

This dissertation focuses on how requirements for a new software development system are elicited and what pitfalls could cause a software development project to fail if the said requirements are not captured correctly. A number of existing requirements elicitation methods, namely: JAD (Joint Application Design), RAD (Rapid Application Development), a Formal Specifications Language (Z), Natural Language, UML (Unified Modelling Language) and Prototyping are covered. The aforementioned techniques are then integrated into existing software development life cycle models, such as the Waterfall model, Rapid Prototyping model, Build and Fix model, Spiral model, Incremental model and the V-Process model. Differences in the domains (knowledge and experience of an environment) of a client and that of the software development team are highlighted and this is done diagrammatically using the language of Venn diagrams. The dissertation also refers to a case study highlighting a number of problems during the requirements elicitation process, amongst other the problem of tacit knowledge not surfacing during elicitation. Two new requirements elicitation methodologies are proposed namely: the SRE (Solitary Requirements Elicitation) and the DDI (Developer Domain Interaction) methodology. These two methods could potentially be more time consuming than other existing requirements elicitation methods, but the benefits could outweigh the cost of their implementation, since the new proposed methods have the potential to further facilitate the successful completion of a software development project. Following the introduction of the new requirements elicitation methods, they are then applied to the aforementioned case study and highlight just how the hidden domain of the client may become more visible, because the software development team has gained a deeper understanding of the client’s working environment. They have therefore increased their understanding of how the final product needs to function in order to fulfil the set out requirements correctly. Towards the end of the dissertation a summary and a conclusion as well as future work that could be undertaken in this area are provided. / Computer Science / M. Sc. (Computer Science)

An Euler diagram C = {c_1, c_2,..., c_n} is a collection of n simple closed curves (i.e., Jordan curves) that partition the plane into connected subsets, called regions, each of which is enclosed by a unique combination of curves. Typically, Euler diagrams are used to visualize the distribution of discrete characteristics across a sample population; in this case, each curve represents a characteristic and each region represents the sub-population possessing exactly the combination of containing curves' properties. Venn diagrams are a subclass of Euler diagrams in which there are 2^n regions representing all possible combinations of curves (e.g., two partially overlapping circles). In this dissertation, we study the Euler Diagram Generation Problem (EDGP), which involves constructing an Euler diagram with a prescribed set of regions. We describe a graph-theoretic model of an Euler diagram's structure and use this model to develop necessary-and-sufficient existence conditions. We also use the graph-theoretic model to prove that the EDGP is NP-complete. In addition, we study the related Area-Proportional Euler Diagram Generation Problem (w-EDGP), which involves constructing an Euler diagram with a prescribed set of regions, each of which has a prescribed area. We develop algorithms for constructing area-proportional Euler diagrams composed of up to three circles and rectangles, as well as diagrams with an unbounded number of curves and a region of common intersection. Finally, we present implementations of our algorithms that allow the dynamic manipulation and real-time construction of area-proportional Euler diagrams.