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.

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.

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.

Equivalence and translation strategies in multilingual thesaurus construction

In the present dissertation, multilingual thesauri were approached as cultural products and the focus was twofold: On the empirical level the focus was placed on the translatability of certain British-English social science indexing terms into the Finnish language and culture at a concept, a term and an indexing term level. On the theoretical level the focus was placed on the aim of translation and on the concept of equivalence. In accordance with modern communicative and dynamic translation theories the interest was on the human dimension. The study is qualitative. In this study, equivalence was understood in a similar way to how dynamic, functional equivalence is commonly understood in translation studies. Translating was seen as a decision-making process, where a translator often has different kinds of possibilities to choose in order to fulfil the function of the translation. Accordingly, and as a starting point for the construction of the empirical part, the function of the source text was considered to be the same or similar to the function of the target text, that is, a functional thesaurus both in source and target context. Further, the study approached the challenges of multilingual thesaurus construction from the perspectives of semantics and pragmatics. In semantic analysis the focus was on what the words conventionally mean and in pragmatics on the ‘invisible’ meaning - or how we recognise what is meant even when it is not actually said (or written). Languages and ideas expressed by languages are created mainly in accordance with expressional needs of the surrounding culture and thesauri were considered to reflect several subcultures and consequently the discourses which represent them. The research material consisted of different kinds of potential discourses: dictionaries, database records, and thesauri, Finnish versus British social science researches, Finnish versus British indexers, simulated indexing tasks with five articles and Finnish versus British thesaurus constructors. In practice, the professional background of the two last mentioned groups was rather similar. It became even more clear that all the material types had their own characteristics, although naturally not entirely separate from each other. It is further noteworthy that the different types and origins of research material were not used to represent true comparison pairs, and that the aim of triangulation of methods and material was to gain a holistic view. The general research questions were: 1. Can differences be found between Finnish and British discourses regarding family roles as thesaurus terms, and if so, what kinds of differences and which are the implications for multilingual thesaurus construction? 2. What is the pragmatic indexing term equivalence? The first question studied how the same topic (family roles) was represented in different contexts and by different users, and further focused on how the possible differences were handled in multilingual thesaurus construction. The second question was based on findings of the previous one, and answered to the final question as to what kinds of factors should be considered when defining translation equivalence in multilingual thesaurus construction. The study used multiple cases and several data collection and analysis methods aiming at theoretical replication and complementarity. The empirical material and analysis consisted of focused interviews (with Finnish and British social scientists, thesaurus constructors and indexers), simulated indexing tasks with Finnish and British indexers, semantic component analysis of dictionary definitions and translations, coword analysis and datasets retrieved in databases, and discourse analysis of thesauri. As a terminological starting point a topic and case family roles was selected. The results were clear: 1) It was possible to identify different discourses. There also existed subdiscourses. For example within the group of social scientists the orientation to qualitative versus quantitative research had an impact on the way they reacted to the studied words and discourses, and indexers placed more emphasis on the information seekers whereas thesaurus constructors approached the construction problems from a more material based solution. The differences between the different specialist groups i.e. the social scientists, the indexers and the thesaurus constructors were often greater than between the different geo-cultural groups i.e. Finnish versus British. The differences occurred as a result of different translation aims, diverging expectations for multilingual thesauri and variety of practices. For multilingual thesaurus construction this means severe challenges. The clearly ambiguous concept of multilingual thesaurus as well as different construction and translation strategies should be considered more precisely in order to shed light on focus and equivalence types, which are clearly not self-evident. The research also revealed the close connection between the aims of multilingual thesauri and the pragmatic indexing term equivalence. 2) The pragmatic indexing term equivalence is very much context-depended. Although thesaurus term equivalence is defined and standardised in the field of library and information science (LIS), it is not understood in one established way and the current LIS tools are inadequate to provide enough analytical tools for both constructing and studying different kinds of multilingual thesauri as well as their indexing term equivalence. The tools provided in translation science were more practical and theoretical, and especially the division of different meanings of a word provided a useful tool in analysing the pragmatic equivalence, which often differs from the ideal model represented in thesaurus construction literature. The study thus showed that the variety of different discourses should be acknowledged, there is a need for operationalisation of new types of multilingual thesauri, and the factors influencing pragmatic indexing term equivalence should be discussed more precisely than is traditionally done.

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.

Fuzzy tolerance/equivalence relations from intersection, union and complement operators and their application to a clustering method

This master thesis work introduces the fuzzy tolerance/equivalence relation and its application in cluster analysis. The work presents about the construction of fuzzy equivalence relations using increasing generators. Here, we investigate and research on the role of increasing generators for the creation of intersection, union and complement operators. The objective is to develop different varieties of fuzzy tolerance/equivalence relations using different varieties of increasing generators. At last, we perform a comparative study with these developed varieties of fuzzy tolerance/equivalence relations in their application to a clustering method.

On the Properties and Cosmology of Q-balls

The properties and cosmological importance of a class of non-topological solitons, Q-balls, are studied. Aspects of Q-ball solutions and Q-ball cosmology discussed in the literature are reviewed. Q-balls are particularly considered in the Minimal Supersymmetric Standard Model with supersymmetry broken by a hidden sector mechanism mediated by either gravity or gauge interactions. Q-ball profiles, charge-energy relations and evaporation rates for realistic Q-ball profiles are calculated for general polynomial potentials and for the gravity mediated scenario. In all of the cases, the evaporation rates are found to increase with decreasing charge. Q-ball collisions are studied by numerical means in the two supersymmetry breaking scenarios. It is noted that the collision processes can be divided into three types: fusion, charge transfer and elastic scattering. Cross-sections are calculated for the different types of processes in the different scenarios. The formation of Q-balls from the fragmentation of the Aflieck-Dine -condensate is studied by numerical and analytical means. The charge distribution is found to depend strongly on the initial energy-charge ratio of the condensate. The final state is typically noted to consist of Q- and anti-Q-balls in a state of maximum entropy. By studying the relaxation of excited Q-balls the rate at which excess energy can be emitted is calculated in the gravity mediated scenario. The Q-ball is also found to withstand excess energy well without significant charge loss. The possible cosmological consequences of these Q-ball properties are discussed.

Development of data sheets for statistical evaluation of fatigue data

To enable a mathematically and physically sound execution of the fatigue test and a correct interpretation of its results, statistical evaluation methods are used to assist in the analysis of fatigue testing data. The main objective of this work is to develop step-by-stepinstructions for statistical analysis of the laboratory fatigue data. The scopeof this project is to provide practical cases about answering the several questions raised in the treatment of test data with application of the methods and formulae in the document IIW-XIII-2138-06 (Best Practice Guide on the Statistical Analysis of Fatigue Data). Generally, the questions in the data sheets involve some aspects: estimation of necessary sample size, verification of the statistical equivalence of the collated sets of data, and determination of characteristic curves in different cases. The series of comprehensive examples which are given in this thesis serve as a demonstration of the various statistical methods to develop a sound procedure to create reliable calculation rules for the fatigue analysis.

Similarity measure based classification

Luokittelujärjestelmää suunniteltaessa tarkoituksena on rakentaa systeemi, joka pystyy ratkaisemaan mahdollisimman tarkasti tutkittavan ongelma-alueen. Hahmontunnistuksessa tunnistusjärjestelmän ydin on luokitin. Luokittelun sovellusaluekenttä on varsin laaja. Luokitinta tarvitaan mm. hahmontunnistusjärjestelmissä, joista kuvankäsittely toimii hyvänä esimerkkinä. Myös lääketieteen parissa tarkkaa luokittelua tarvitaan paljon. Esimerkiksi potilaan oireiden diagnosointiin tarvitaan luokitin, joka pystyy mittaustuloksista päättelemään mahdollisimman tarkasti, onko potilaalla kyseinen oire vai ei. Väitöskirjassa on tehty similaarisuusmittoihin perustuva luokitin ja sen toimintaa on tarkasteltu mm. lääketieteen paristatulevilla data-aineistoilla, joissa luokittelutehtävänä on tunnistaa potilaan oireen laatu. Väitöskirjassa esitetyn luokittimen etuna on sen yksinkertainen rakenne, josta johtuen se on helppo tehdä sekä ymmärtää. Toinen etu on luokittimentarkkuus. Luokitin saadaan luokittelemaan useita eri ongelmia hyvin tarkasti. Tämä on tärkeää varsinkin lääketieteen parissa, missä jo pieni tarkkuuden parannus luokittelutuloksessa on erittäin tärkeää. Väitöskirjassa ontutkittu useita eri mittoja, joilla voidaan mitata samankaltaisuutta. Mitoille löytyy myös useita parametreja, joille voidaan etsiä juuri kyseiseen luokitteluongelmaan sopivat arvot. Tämä parametrien optimointi ongelma-alueeseen sopivaksi voidaan suorittaa mm. evoluutionääri- algoritmeja käyttäen. Kyseisessä työssä tähän on käytetty geneettistä algoritmia ja differentiaali-evoluutioalgoritmia. Luokittimen etuna on sen joustavuus. Ongelma-alueelle on helppo vaihtaa similaarisuusmitta, jos kyseinen mitta ei ole sopiva tutkittavaan ongelma-alueeseen. Myös eri mittojen parametrien optimointi voi parantaa tuloksia huomattavasti. Kun käytetään eri esikäsittelymenetelmiä ennen luokittelua, tuloksia pystytään parantamaan.

Puumassapesurin suotautumisolojen kartoitus ja simulointi

Työn tavoitteena oli tutkia sellumassan suotautumiseen liittyviä tekijöitä Ahlstrom Machineryn Drum Displacer™ -puumassapesurissa (DD-pesuri). Teoriaosassa tarkasteltiin aluksi suotautumisen teoriaa kuitu-vesi-suspensiossa, minkä jälkeen esiteltiin suotautumisnopeuteen vaikuttavia fysikaalisia ja kemiallisia vaikutusmekanismeja. Seuraavaksi kuvattiin massan pesun yleisiä perusteita sekä teoriaa puumassapesureissa. Lopuksi tarkasteltiin pesurien kytkeytymistä muuhun kuitulinjaan sekä prosessista johtuvia pesun toiminnan ulkoisia häiriötekijöitä. Kokeellisen osan aluksi tarkasteltiin paine- ja lämpötilamittauksien avulla massapesurissa vallitsevia prosessioloja. Mittaustulosten perusteella pumppausolot pesurin suodoslinjoissa ovat vaikeahkot ja häiriötilanteita voi esiintyä, mutta käytäntö on osoittanut tästä olevan vain harvoin haittaa prosessin toiminnalle. Pesureissa toteutuneet syrjäytysnopeudet laskettiin ja niitä verrattiin syrjäytystestien antamiin tuloksiin. Kuitulinjasta riippuen testin vastaavuus tehdasprosessiin vaihteli suuresti. Syrjäytystesteillä kokeiltiin myös tehdasprosesseissa usein esiintyvien muuttujien vaikutusta sellukakun syrjäytettävyyteen. Kakun paksuus ja syrjäytyslämpötila vaikuttivat syrjäytysnopeuteen Darcyn lain mukaisesti. Alipaineen massakakun alapuolella havaittiin huonontavan syrjäytysnopeutta verrattuna tilanteeseen, jossa kakun alla vallitsi ilmanpaine. Tämä havainto on selvästi ristiriidassa suotautumisen teorian kanssa. Massakakun muodostumis-pH osoittautui ratkaisevaksi lopulliselle syrjäytysnopeudelle, sillä alkalisissa oloissa muodostetun kuitukakun syrjäytysnopeus ei enää parantunut happamalla syrjäytysnesteellä. Happamissa oloissa muodostetun kakun syrjäytysnopeus oli alkalisista parempi, mutta se alkoi hitaasti alentua, kun syrjäytysneste vaihtui alkaliseen. Massan laimentaminen ennen syrjäytystä alkalisella tehdassuodoksella puhtaan veden sijasta alensi ligniinipitoisella massalla lopullista syrjäytysnopeutta. Shirato-Tillerin mallilla ja Jönssonin staattisella mallilla simuloitiin numeerisesti syrjäytystestiä kahdessa eri pH:ssa, ja simulointituloksia verrattiin vastaavissa oloissa tehtyihin syrjäytystesteihin. Shirato-Tillerin mallin antamien syrjäytysnopeuksien havaittiin olevan lähellä syrjäytystestien nopeuksia, kun Jönssonin mallin antamat tulokset jäivät huomattavasti testituloksia alemmiksi. Herkkyystarkastelussa havaittiin mallien olevan varsin herkkiä parametrien virheille. Hajonta vaadittavien kuituparametrien määrityksissä ja menetelmien työläys rajoittavat numeerisen simuloinnin käytettävyyttä, sillä kuituparametrien määrityksen vaatima työmäärä on ainakin toistaiseksi syrjäytystestiä suurempi. Lopuksi todettiin, että oikeiden syrjäytysolosuhteiden käyttö on ensiarvoisen tärkeää oikeiden tulosten saamiseksi sekä kokeellisessa että numeerisessa simuloinnissa. Nykyinen syrjäytystestilaitteisto on pienin muutoksin käyttökelpoinen, kun massan testaus prosessioloissa tulee rutiininomaiseksi.

Ekvivalenssin ongelma vertailututkimuksessa

Abstract: The problem of equivalence in comparative research

Many valued algebraic structures as measures of comparison

This thesis studies the properties and usability of operators called t-norms, t-conorms, uninorms, as well as many valued implications and equivalences. Into these operators, weights and a generalized mean are embedded for aggregation, and they are used for comparison tasks and for this reason they are referred to as comparison measures. The thesis illustrates how these operators can be weighted with a differential evolution and aggregated with a generalized mean, and the kinds of measures of comparison that can be achieved from this procedure. New operators suitable for comparison measures are suggested. These operators are combination measures based on the use of t-norms and t-conorms, the generalized 3_-uninorm and pseudo equivalence measures based on S-type implications. The empirical part of this thesis demonstrates how these new comparison measures work in the field of classification, for example, in the classification of medical data. The second application area is from the field of sports medicine and it represents an expert system for defining an athlete's aerobic and anaerobic thresholds. The core of this thesis offers definitions for comparison measures and illustrates that there is no actual difference in the results achieved in comparison tasks, by the use of comparison measures based on distance, versus comparison measures based on many valued logical structures. The approach has been highly practical in this thesis and all usage of the measures has been validated mainly by practical testing. In general, many different types of operators suitable for comparison tasks have been presented in fuzzy logic literature and there has been little or no experimental work with these operators.

On the Application of Beam on Elastic Foundation Theory to the Analysis of Stiffened Plate Strips

The main objective of this thesis is to show that plate strips subjected to transverse line loads can be analysed by using the beam on elastic foundation (BEF) approach. It is shown that the elastic behaviour of both the centre line section of a semi infinite plate supported along two edges, and the free edge of a cantilever plate strip can be accurately predicted by calculations based on the two parameter BEF theory. The transverse bending stiffness of the plate strip forms the foundation. The foundation modulus is shown, mathematically and physically, to be the zero order term of the fourth order differential equation governing the behaviour of BEF, whereas the torsion rigidity of the plate acts like pre tension in the second order term. Direct equivalence is obtained for harmonic line loading by comparing the differential equations of Levy's method (a simply supported plate) with the BEF method. By equating the second and zero order terms of the semi infinite BEF model for each harmonic component, two parameters are obtained for a simply supported plate of width B: the characteristic length, 1/ λ, and the normalized sum, n, being the effect of axial loading and stiffening resulting from the torsion stiffness, nlin. This procedure gives the following result for the first mode when a uniaxial stress field was assumed (ν = 0): 1/λ = √2B/π and nlin = 1. For constant line loading, which is the superimposition of harmonic components, slightly differing foundation parameters are obtained when the maximum deflection and bending moment values of the theoretical plate, with v = 0, and BEF analysis solutions are equated: 1 /λ= 1.47B/π and nlin. = 0.59 for a simply supported plate; and 1/λ = 0.99B/π and nlin = 0.25 for a fixed plate. The BEF parameters of the plate strip with a free edge are determined based solely on finite element analysis (FEA) results: 1/λ = 1.29B/π and nlin. = 0.65, where B is the double width of the cantilever plate strip. The stress biaxial, v > 0, is shown not to affect the values of the BEF parameters significantly the result of the geometric nonlinearity caused by in plane, axial and biaxial loading is studied theoretically by comparing the differential equations of Levy's method with the BEF approach. The BEF model is generalised to take into account the elastic rotation stiffness of the longitudinal edges. Finally, formulae are presented that take into account the effect of Poisson's ratio, and geometric non linearity, on bending behaviour resulting from axial and transverse inplane loading. It is also shown that the BEF parameters of the semi infinite model are valid for linear elastic analysis of a plate strip of finite length. The BEF model was verified by applying it to the analysis of bending stresses caused by misalignments in a laboratory test panel. In summary, it can be concluded that the advantages of the BEF theory are that it is a simple tool, and that it is accurate enough for specific stress analysis of semi infinite and finite plate bending problems.