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.

Catalytic transformations of fatty acid derivatives for food, oleochemicals and fuels over carbon supported platinum group metals

Växtoljor som utgör en förnybar naturresurs används som sådana eller i modifierade former i många industriella processer, som är av stor betydelse för vårt vardagliga liv. Växtoljor används i livsmedel, i kemiska och farmaceutiska produkter, i textilindustrin, för framställning av färgämnen och beläggningsmaterial samt som miljövänliga bränslekomponenter. Fetter och oljor hör till de äldsta kemiska komponenterna som utnyttjas av människan. De består huvudsakligen av glycerolestrar och fettsyror. Fetter och oljor har typiskt en kolkedja med kol-koldubbelbindningar samt karboxyl- och estergrupper, som kan genom hydrering eller dekarboxylering konverteras till nyttiga och miljövänliga produkter med hjälp av ädelmetallkatalysatorer. Aktivt kol (C) används som bärare på katalysatorerna. Väteaddition, d.v.s. hydrering av växtoljor har varit föremål för omfattande forskning i över hundra års tid. Hydreringen är en viktig process, för den tillämpas på produktion av fetter och margarin. Omättade fettsyror hydreras traditionellt på nickelbaserade heterogena katalysatorer. Samtidigt med en partiell hydrering av fettsyrorna och fettsyraestrarna som har två dubbelbindningar pågår också isomeringsreaktioner, vilka ger cis- och transisomerer av reaktantmolekylerna. Den största nackdelen med nickelkatalysatorerna är deras giftighet samt bildning av ohälsosamma transisomerer i reaktionsprodukterna. Dessutom deaktiveras nickelkatalysatorn snabbt p.g.a. att nickeltvålar bildas i reaktionsblandningen. Platinabaserade katalysatorer lider däremot inte av dessa begränsningar. Metaller i platinagruppen i det periodiska systemet studerades i detalj för att avslöja kinetiska effekter i hydreringen av cis-metyloleat. Palladium, rutenium, rhodium, platina och iridium användes som katalytiska metaller. Metallhalten på aktivkolbärare var 1 vikt-%. De olika platinametallerna undersöktes för att kartlägga konkurrerande hydrerings- och isomeringsrutter på metallerna. Det visade sig att metallerna i andra raden av det periodiska systemet (Ru, Rh, Pd) är aktivare i isomeringsprocesserna, medan metallerna i tredje raden (Ir, Pt) har en lägre aktivitet. Pd/C valdes bland platinametallerna, för att den är attraktiv ur ekonomisk synvinkel och den är mycket aktiv och selektiv, speciellt jämfört med nickel. Tyngdpunkten i arbetet var utvecklingen av en alternativ, palladiumbaserad hydreringsteknologi som skulle ersätta den traditionella teknologin som är baserad på användningen av nickelkatalysatorer. Palladiumbaserade katalysatorer kan återcirkuleras, de är aktivare och mera resistenta mot syror och de bildar mindre mängder av skadliga transisomerer. För att denna teknologi skall bli ekonomiskt hållbar och konkurrenskraftig, måste den basera sig på de bästa möjliga katalysatorerna, vilket innebär att en optimal kombination av hög aktivitet och selektivitet samt en lång livstid för katalysatorn krävs. Därför inkluderades teknologiska aspekter kraftigt i forskningen. Mycket arbete satsades på design av palladium på en mesoporös kolbärare och undersökning av korrelationerna mellan katalysatorns egenskaper och dess aktivitet i isomeriseringsreaktionerna och i hydreringen av kol-koldubbelbindningarna i reaktantmolekylen. Katalysatorerna karakteriserades med många fysikaliska och kemiska metoder (transmissionselektronmikroskopi (TEM), röntgendiffraktion (XRD), röntgenfotoelektronspektroskopi (XPS), temperaturprogrammerad reduktion (TPR), temperaturprogrammerad desorption (TPD) av kolmonoxid, kemisorption av kolmonoxid, fysisorption av kväve). Temperaturens, vätetryckets och katalysatorkoncentrationens inverkan på fettsyra- och isomersammansättningen hos de hydrerade oljorna bestämdes under kinetiska betingelser, i frånvaro av massöverföringseffekter. Syreavspjälkning genom fullständig dekarboxylering av karboxylgruppen i fettsyramolekylen är det hittills bästa sättet att framställa miljövänlig dieselolja, eftersom linjära paraffiner fås som reaktionsprodukter och en tillsats av dyr vätgas undviks. Deoxygeneringen undersöktes systematiskt på en Pd/C-katalysator (Sibunit) genom att använda mättade fettsyror C16-C20 och C22 som råvara. Produktmolekylen blev en dieselliknande kolvätemolekyl, med en kolatom färre än i utgångsmolekylen. Lika stora dekarboxyleringshastigheter observerades för rena, mättade fettsyror. En jämförelse av deoxygenereringshastigheterna för stearin-, olein- och linolsyra som råvara vid 300oC i närvaro av 1-volymprocent väte på mesoporös Pd/C (Sibunit) avslöjade att katalysatorns aktivitet och selektivitet ökade med en ökande mättningsgrad av reaktantmolekylen. Då stearinsyra användes som utgångsmolekyl, bestod huvudprodukterna av önskade C17-kolväten, medan mängden av aromatiska C17-komponenter ökade, då olein- och linolsyra användes som utgångsmolekyler. Katalysatordeaktiveringen var relativt påfallande vid deoxygeneringen av linolsyra så att endast 3% av fettsyrorna omsattes till produkter i 330 min. Deaktiveringen orsakades av aromatiska C17-komponenter samt av fettsyradimerer, som bildades via en Diels-Alderreaktion. Hydreringen av omättade fettsyror kan därför rekommenderas som ett primärt kemiskt steg i framställningen av miljövänliga dieselprodukter. Målet var också att öka förståelsen av palladiummetallernas roll i nanoskala, speciellt effekten av metallpartiklarna i katalytisk hydrering och deoxygenering. Pd/C-katalysatorer med lika stora halter av Pd syntetiserades och metallens dispersion på bärarmaterialet varierades systematiskt genom en kontrollerad uppväxt av palladiumnanopartiklar på aktiv kolbärare. Metalldispersionens effekt på hydrerings-hastigheten och cis-transförhållandet undersöktes i detalj. En optimal metalldispersion som gav den högsta dekarboxyleringshastigheten hittades. Massöverföringens inverkan på reaktionens hastighet studerades experimentellt och temperaturprogrammerad desorption av kolmonoxid från katalysatorytan undersöktes ingående. Hydrering av växtoljor genomfördes under satsvisa och kontinuerliga betingelser. Både finfördelat Pd/C och katalysatorgranulat användes i experimenten. Ett av målen med arbetet var uppskalningen av hydreringsprocesserna. Med tanke på stora produktionsvolymer var det logiskt att undersöka kontinuerliga hydrerings- och dekarboxyleringsteknologier. En kontinuerlig packad bäddreaktor studerades i laboratorieskala, vilket gav viktig information om katalysatorns långtidsstabilitet och deaktivering. Effekten av rena fettsyror och triglycerider som råvara samt metallpartikelstorleken och palladiumhalten studerades med hjälp av den kontinuerliga reaktorn. Produktionskapaciteten som erhölls med satsvis och kontinuerlig drift jämfördes. Dekarboxyleringen av stearinsyra undersöktes också i en kontinuerlig packad bädd. Omsättningsgraden blev 15% för en stabil katalysator.

Teaching mathematics and programming : new approaches with empirical evaluation

Programming and mathematics are core areas of computer science (CS) and consequently also important parts of CS education. Introductory instruction in these two topics is, however, not without problems. Studies show that CS students find programming difficult to learn and that teaching mathematical topics to CS novices is challenging. One reason for the latter is the disconnection between mathematics and programming found in many CS curricula, which results in students not seeing the relevance of the subject for their studies. In addition, reports indicate that students' mathematical capability and maturity levels are dropping. The challenges faced when teaching mathematics and programming at CS departments can also be traced back to gaps in students' prior education. In Finland the high school curriculum does not include CS as a subject; instead, focus is on learning to use the computer and its applications as tools. Similarly, many of the mathematics courses emphasize application of formulas, while logic, formalisms and proofs, which are important in CS, are avoided. Consequently, high school graduates are not well prepared for studies in CS. Motivated by these challenges, the goal of the present work is to describe new approaches to teaching mathematics and programming aimed at addressing these issues: Structured derivations is a logic-based approach to teaching mathematics, where formalisms and justifications are made explicit. The aim is to help students become better at communicating their reasoning using mathematical language and logical notation at the same time as they become more confident with formalisms. The Python programming language was originally designed with education in mind, and has a simple syntax compared to many other popular languages. The aim of using it in instruction is to address algorithms and their implementation in a way that allows focus to be put on learning algorithmic thinking and programming instead of on learning a complex syntax. Invariant based programming is a diagrammatic approach to developing programs that are correct by construction. The approach is based on elementary propositional and predicate logic, and makes explicit the underlying mathematical foundations of programming. The aim is also to show how mathematics in general, and logic in particular, can be used to create better programs.

Catalysis by gold for biomass transformations

The development of new technologies to supplement fossil resources has led to a growing interest in the utilization of alternative routes. Biomass is a rich renewable feedstock for producing fine chemicals, polymers, and a variety of commodities replacing petroleumderived chemicals. Transformation of biomass into diverse valuable chemicals is the key concept of a biorefinery. Catalytic conversion of biomass, which reduces the use of toxic chemicals is one of the important approaches to improve the profitability of biorefineries. Utilization of gold catalysts allows conducting reactions under environmentally-friendly conditions, with a high catalytic activity and selectivity. Gold-catalyzed valorization of several biomass-derived compounds as an alternative approach to the existing technologies was studied in this work. Isomerization of linoleic acid via double bond migration towards biologically active conjugated linoleic acid isomers (CLA) was investigated. The activity and selectivity of various gold catalysts towards cis-9,trans-11-CLA and trans-10,cis-12-CLA were investigated in a semi-batch reactor, showing that the yield of the desired products varied, depending on the catalyst support. The structure sensitivity in the selective oxidation of arabinose was demonstrated using a series of gold catalysts with different Au cluster sizes in a shaker reactor operating in a semibatch mode. The gas-phase selective oxidation of ethanol was studied and the influence of the catalyst support on the catalytic performance was investigated. The selective oxidation of the lignan hydroxymatairesinol (HMR), extracted from the Norway spruce (Picea abies) knots, to the lignan oxomatairesinol (oxoMAT) was extensively investigated. The influence of the reaction conditions and catalyst properties on the yield of oxoMAT was evaluated. In particular, the structure sensitivity of the reaction was demonstrated. The catalyst deactivation and regeneration procedures were studied. The reaction kinetics and mechanism were advanced.

Conceptions of mathematics teacher education : thoughts among teacher educators in Tanzania

This study addresses the question of teacher educators’ conceptions of mathematics teacher education (MTE) in teacher colleges in Tanzania, and their thoughts on how to further develop it. The tension between exponents of content as opposed to pedagogy has continued to cause challenging conceptual differences, which also influences what teacher educators conceive as desirable in the development of this domain. This tension is connected to the dissatisfaction of parents and teachers with the failure of school mathematics. From this point of view, the overall aim was to identify and describe teacher educators’ various conceptions of MTE. Inspired by the debate among teacher educators about what the balance should be between subject matter and pedagogical knowledge, it was important to look at the theoretical faces of MTE. The theoretical background involved the review of what is visible in MTE, what is yet to be known and the challenges within the practice. This task revealed meanings, perspectives in MTE, professional development and assessment. To do this, two questions were asked, to which no clear solutions satisfactorily existed. The questions to guide the investigation were, firstly, what are teacher educators’ conceptions of MTE, and secondly, what are teacher educators’ thoughts on the development of MTE? The two questions led to the choice of phenomenography as the methodological approach. Against the guiding questions, 27 mathematics teacher educators were interviewed in relation to the first question, while 32 responded to an open-ended questionnaire regarding question two. The interview statements as well as the questionnaire responses were coded and analysed (classified). The process of classification generated patterns of qualitatively different ways of seeing MTE. The results indicate that MTE is conceived as a process of learning through investigation, fostering inspiration, an approach to learning with an emphasis on problem solving, and a focus on pedagogical knowledge and skills in the process of teaching and learning. In addition, the teaching and learning of mathematics is seen as subject didactics with a focus on subject matter and as an organized integration of subject matter, pedagogical knowledge and some school practice; and also as academic content knowledge in which assessment is inherent. The respondents also saw the need to build learner-educator relationships. Finally, they emphasized taking advantage of teacher educators’ neighbourhood learning groups, networking and collaboration as sustainable knowledge and skills sharing strategies in professional development. Regarding desirable development, teacher educators’ thoughts emphasised enhancing pedagogical knowledge and subject matter, and to be determined by them as opposed to conventional top-down seminars and workshops. This study has revealed various conceptions and thoughts about MTE based on teacher educators´ diverse history of professional development in mathematics. It has been reasonably substantiated that some teacher educators teach school mathematics in the name of MTE, hardly distinguishing between the role and purpose of the two in developing a mathematics teacher. What teacher educators conceive as MTE and what they do regarding the education of teachers of mathematics revealed variations in terms of seeing the phenomenon of interest. Within limits, desirable thoughts shed light on solutions to phobias, and in the same way low self-esteem and stigmatization call for the building of teacher educator-student teacher relationships.

Mathematics inspired by Darwin. Adaptive dynamics of dispersal and cooperation

In 1859, Charles Darwin published his theory of evolution by natural selection, the process occurring based on fitness benefits and fitness costs at the individual level. Traditionally, evolution has been investigated by biologists, but it has induced mathematical approaches, too. For example, adaptive dynamics has proven to be a very applicable framework to the purpose. Its core concept is the invasion fitness, the sign of which tells whether a mutant phenotype can invade the prevalent phenotype. In this thesis, four real-world applications on evolutionary questions are provided. Inspiration for the first two studies arose from a cold-adapted species, American pika. First, it is studied how the global climate change may affect the evolution of dispersal and viability of pika metapopulations. Based on the results gained here, it is shown that the evolution of dispersal can result in extinction and indeed, evolution of dispersalshould be incorporated into the viability analysis of species living in fragmented habitats. The second study is focused on the evolution of densitydependent dispersal in metapopulations with small habitat patches. It resulted a very surprising unintuitive evolutionary phenomenon, how a non-monotone density-dependent dispersal may evolve. Cooperation is surprisingly common in many levels of life, despite of its obvious vulnerability to selfish cheating. This motivated two applications. First, it is shown that density-dependent cooperative investment can evolve to have a qualitatively different, monotone or non-monotone, form depending on modelling details. The last study investigates the evolution of investing into two public-goods resources. The results suggest one general path by which labour division can arise via evolutionary branching. In addition to applications, two novel methodological derivations of fitness measures in structured metapopulations are given.

Terpenoid transformations over gold catalysts

Nowadays biomass transformation has a great potential for the synthesis of value-added compounds with a wide range of applications. Terpenoids, extracted from biomass, are inexpensive and renewable raw materials which often have a biological activity and are widely used as important organic platform molecules in the development of new medicines as well as in the synthesis of fine chemicals and intermediates. At the same time, special attention is devoted to the application of gold catalysts to fine chemical synthesis due to their outstanding activity and/or selectivity for transformations of complex organic compounds. Conversion of renewable terpenoids in the presence of gold nanoparticles is one of the new and promising directions in the transformation of biomass to valuable chemicals. In the doctoral thesis, different kinds of natural terpenoids, such as α-pinene, myrtenol and carvone were selected as starting materials. Gold catalysts were utilized for the promising routes of these compounds transformation. Investigation of selective α-pinene isomerization to camphene, which is an important step in an industrial process towards the synthesis of camphor as well as other valuable substrates for the pharmaceutical industry, was performed. A high activity of heterogeneous gold catalysts in the Wagner-Meerwein rearrangement was demonstrated for the first time. Gold on alumina carrier was found to reach the α-pinene isomerization conversion up to 99.9% and the selectivity of 60-80%, thus making this catalyst very promising from an industrial viewpoint. A detailed investigation of kinetic regularities including catalyst deactivation during the reaction was performed. The one-pot terpene alcohol amination, which is a promising approach to the synthesis of valuable complex amines having specific physiological properties, was investigated. The general regularities of the one-pot natural myrtenol amination in the presence of gold catalysts as well as a correlation between catalytic activity, catalyst redox treatment and the support nature were obtained. Catalytic activity and product distribution were shown to be strongly dependent on the support properties, namely acidity and basicity. The gold-zirconia (Au/ZrO2) catalyst pretreated under oxidizing atmosphere was observed to be rather active, resulting in the total conversion of myrtenol and the selectivity to the corresponding amine of about 53%. The reaction kinetics was modelled based on the mechanistic considerations with the catalyst deactivation step incorporated in the mechanism. Carvone hydrogenation over a gold catalyst was studied with the general idea of investigating both the activity of gold catalysts in competitive hydrogenation of different functional groups and developing an approach to the synthesis of valuable carvone derivatives. Gold was found to promote stereo- and chemoselective carvone hydrogenation to dihydrocarvone with a predominant formation of the trans-isomer, which generally is a novel synthetic method for an industrially valuable dihydrocarvone. The solvent effect on the catalytic activity as well as on the ratio between trans- and cis-dihydrocarvone was evaluated.

Utilising student profiles in mathematics course arrangements

This thesis develops a method for identifying students struggling in their mathematical studies at an early stage. It helps in directing support to students needing and benefiting from it the most. Thus, frustration felt by weaker students may decrease and therefore, hopefully, also drop outs of potential engineering students. The research concentrates on a combination of personality and intelligence aspects. Personality aspects gave information on conation and motivation for learning. This part was studied from the perspective of motivation and self-regulation. Intelligence aspects gave information on declarative and procedural knowledge: what had been taught and what was actually mastered. Students answered surveys on motivation and self-regulation in 2010 and 2011. Based on their answers, background information, results in the proficiency test, and grades in the first mathematics course, profiles describing the students were formed. In the following years, the profiles were updated with new information obtained each year. The profiles used to identify struggling students combine personality (motivation, selfregulation, and self-efficacy) and intelligence (declarative and procedural knowledge) aspects at the beginning of their studies. Identifying students in need of extra support is a good start, but methods for providing support must be found. This thesis also studies how this support could be taken into account in course arrangements. The methods used include, for example, languaging and scaffolding, and continuous feedback. The analysis revealed that allocating resources based on the predicted progress does not increase costs or lower the results of better students. Instead, it will help weaker students obtain passing grades.