Spinor Bose-Einstein Condensates and Topological Defects

This thesis concentrates on the topological defects of spin-1 and spin-2 Bose-Einstein condensates, the ground states of spin-3 condensates, and the inert states of spinor condensates with arbitrary spin. Our work is based on the description of a spinor condensate of spin-S atoms in terms of a state vector of a spin-S particle. The results of the homotopy theory are used to study the existence and structure of the topological defects in spinor condensates. We construct examples of defects, study their energetics, and examine how their stability is affected by the presence of an external magnetic field. The ground states of spin-3 condensates are calculated using analytical and numerical means. Special emphasis is put on the ground states of a chromium condensate, whose dependence on the magnetic dipole-dipole interaction is studied. A simple geometrical method for the calculation of inert states of spinor condensates is presented. This method is used to find candidates for the ground states of spin-S condensates.

A Topological Approach to Fuzzy Sets

This thesis presents a topological approach to studying fuzzy setsby means of modifier operators. Modifier operators are mathematical models, e.g., for hedges, and we present briefly different approaches to studying modifier operators. We are interested in compositional modifier operators, modifiers for short, and these modifiers depend on binary relations. We show that if a modifier depends on a reflexive and transitive binary relation on U, then there exists a unique topology on U such that this modifier is the closure operator in that topology. Also, if U is finite then there exists a lattice isomorphism between the class of all reflexive and transitive relations and the class of all topologies on U. We define topological similarity relation "≈" between L-fuzzy sets in an universe U, and show that the class LU/ ≈ is isomorphic with the class of all topologies on U, if U is finite and L is suitable. We consider finite bitopological spaces as approximation spaces, and we show that lower and upper approximations can be computed by means of α-level sets also in the case of equivalence relations. This means that approximations in the sense of Rough Set Theory can be computed by means of α-level sets. Finally, we present and application to data analysis: we study an approach to detecting dependencies of attributes in data base-like systems, called information systems.

Fuzzy subgroups, algebraic and topological points of view and complex analysis

Fuzzy subsets and fuzzy subgroups are basic concepts in fuzzy mathematics. We shall concentrate on fuzzy subgroups dealing with some of their algebraic, topological and complex analytical properties. Explorations are theoretical belonging to pure mathematics. One of our ideas is to show how widely fuzzy subgroups can be used in mathematics, which brings out the wealth of this concept. In complex analysis we focus on Möbius transformations, combining them with fuzzy subgroups in the algebraic and topological sense. We also survey MV spaces with or without a link to fuzzy subgroups. Spectral space is known in MV algebra. We are interested in its topological properties in MV-semilinear space. Later on, we shall study MV algebras in connection with Riemann surfaces. In fact, the Riemann surface as a concept belongs to complex analysis. On the other hand, Möbius transformations form a part of the theory of Riemann surfaces. In general, this work gives a good understanding how it is possible to fit together different fields of mathematics.

On Topological Solitons in The Faddeev-Skyrme Model and its Extensions

The topological solitons of two classical field theories, the Faddeev-Skyrme model and the Ginzburg-Landau model are studied numerically and analytically in this work. The aim is to gain information on the existence and properties of these topological solitons, their structure and behaviour under relaxation. First, the conditions and mechanisms leading to the possibility of topological solitons are explored from the field theoretical point of view. This leads one to consider continuous deformations of the solutions of the equations of motion. The results of algebraic topology necessary for the systematic treatment of such deformations are reviewed and methods of determining the homotopy classes of topological solitons are presented. The Faddeev-Skyrme and Ginzburg-Landau models are presented, some earlier results reviewed and the numerical methods used in this work are described. The topological solitons of the Faddeev-Skyrme model, Hopfions, are found to follow the same mechanisms of relaxation in three different domains with three different topological classifications. For two of the domains, the necessary but unusual topological classification is presented. Finite size topological solitons are not found in the Ginzburg-Landau model and a scaling argument is used to suggest that there are indeed none unless a certain modification to the model, due to R. S. Ward, is made. In that case, the Hopfions of the Faddeev-Skyrme model are seen to be present for some parameter values. A boundary in the parameter space separating the region where the Hopfions exist and the area where they do not exist is found and the behaviour of the Hopfion energy on this boundary is studied.

Space–Time Block Codes and the Complexity of Sphere Decoding

In wireless communications the transmitted signals may be affected by noise. The receiver must decode the received message, which can be mathematically modelled as a search for the closest lattice point to a given vector. This problem is known to be NP-hard in general, but for communications applications there exist algorithms that, for a certain range of system parameters, offer polynomial expected complexity. The purpose of the thesis is to study the sphere decoding algorithm introduced in the article On Maximum-Likelihood Detection and the Search for the Closest Lattice Point, which was published by M.O. Damen, H. El Gamal and G. Caire in 2003. We concentrate especially on its computational complexity when used in space–time coding. Computer simulations are used to study how different system parameters affect the computational complexity of the algorithm. The aim is to find ways to improve the algorithm from the complexity point of view. The main contribution of the thesis is the construction of two new modifications to the sphere decoding algorithm, which are shown to perform faster than the original algorithm within a range of system parameters.

Complexity Optimization in Product Development with Modularization

A company’s competence to manage its product portfolio complexity is becoming critically important in the rapidly changing business environment. The continuous evolvement of customer needs, the competitive market environment and internal product development lead to increasing complexity in product portfolios. The companies that manage the complexity in product development are more profitable in the long run. The complexity derives from product development and management processes where the new product variant development is not managed efficiently. Complexity is managed with modularization which is a method that divides the product structure into modules. In modularization, it is essential to take into account the trade-off between the perceived customer value and the module or component commonality across the products. Another goal is to enable the product configuration to be more flexible. The benefits are achieved through optimizing complexity in module offering and deriving the new product variants more flexibly and accurately. The developed modularization process includes the process steps for preparation, mapping the current situation, the creation of a modular strategy and implementing the strategy. Also the organization and support systems have to be adapted to follow-up targets and to execute modularization in practice.

Overcoming the Complexity of Software Product Management

This thesis describes an approach to overcoming the complexity of software product management (SPM) and consists of several studies that investigate the activities and roles in product management, as well as issues related to the adoption of software product management. The thesis focuses on organizations that have started the adoption of SPM but faced difficulties due to its complexity and fuzziness and suggests the frameworks for overcoming these challenges using the principles of decomposition and iterative improvements. The research process consisted of three phases, each of which provided complementary results and empirical observation to the problem of overcoming the complexity of SPM. Overall, product management processes and practices in 13 companies were studied and analysed. Moreover, additional data was collected with a survey conducted worldwide. The collected data were analysed using the grounded theory (GT) to identify the possible ways to overcome the complexity of SPM. Complementary research methods, like elements of the Theory of Constraints were used for deeper data analysis. The results of the thesis indicate that the decomposition of SPM activities depending on the specific characteristics of companies and roles is a useful approach for simplifying the existing SPM frameworks. Companies would benefit from the results by adopting SPM activities more efficiently and effectively and spending fewer resources on its adoption by concentrating on the most important SPM activities.

The European Union and Assembling Biofuel Development – topological investigations concerning the associations between law, policy and space

Biofuels for transport are a renewable source of energy that were once heralded as a solution to multiple problems associated with poor urban air quality, the overproduction of agricultural commodities, the energy security of the European Union (EU) and climate change. It was only after the Union had implemented an incentivizing framework of legal and political instruments for the production, trade and consumption of biofuels that the problems of weakening food security, environmental degradation and increasing greenhouse gases through land-use changes began to unfold. In other words, the difference between political aims for why biofuels are promoted and their consequences has grown – which is also recognized by the EU policy-makers. Therefore, the global networks of producing, trading and consuming biofuels may face a complete restructure if the European Commission accomplishes its pursuit to sideline crop-based biofuels after 2020. My aim with this dissertation is not only to trace the manifold evolutions of the instruments used by the Union to govern biofuels but also to reveal how this evolution has influenced the dynamics of biofuel development. Therefore, I study the ways the EU’s legal and political instruments of steering biofuels are coconstitutive with the globalized spaces of biofuel development. My analytical strategy can be outlined through three concepts. I use the term ‘assemblage’ to approach the operations of the loose entity of actors and non-human elements that are the constituents of multi-scalar and -sectorial biofuel development. ‘Topology’ refers to the spatiality of this European biofuel assemblage and its parts whose evolving relations are treated as the active constituents of space, instead of simply being located in space. I apply the concept of ‘nomosphere’ to characterize the framework of policies, laws and other instruments that the EU applies and construes while attempting to govern biofuels. Even though both the materials and methods vary in the independent articles, these three concepts characterize my analytical strategy that allows me to study law, policy and space associated with each other. The results of my examinations underscore the importance of the instruments of governance of the EU constituting and stabilizing the spaces of producing and, on the other hand, how topological ruptures in biofuel development have enforced the need to reform policies. This analysis maps the vast scope of actors that are influenced by the mechanism of EU biofuel governance and, what is more, shows how they are actively engaging in the Union’s institutional policy formulation. By examining the consequences of fast biofuel development that are spatially dislocated from the established spaces of producing, trading and consuming biofuels such as indirect land use changes, I unfold the processes not tackled by the instruments of the EU. Indeed, it is these spatially dislocated processes that have pushed the Commission construing a new type of governing biofuels: transferring the instruments of climate change mitigation to land-use policies. Although efficient in mitigating these dislocated consequences, these instruments have also created peculiar ontological scaffolding for governing biofuels. According to this mode of governance, the spatiality of biofuel development appears to be already determined and the agency that could dampen the negative consequences originating from land-use practices is treated as irrelevant.