The drinking water production in Sweden is facing challenges because of climatic changes, eutrophication and brownification, which contributes to an increase of algal blooms. This can lead to a production of different types of toxins, and the most common group of algal toxins is the hepatotoxic microcystins (MC). Other algal toxins that occur in Sweden are anatoxin-A (ATX-A) and the paralytic shellfish toxins (PST), which are neurotoxic. Flocculation and filtration has been shown to reduce intracellular MC, but to reduce extracellular MC it seems like other techniques are needed, such as membrane filtration, active carbon or ozone. It has also been shown that dissolved organic carbon (DOC) might affect the reduction of MC, either through occupying adsorption spots on the membrane, or through bindings between MC and DOC. In this project, the reduction of four types of added MC (microcystin-LR, microcystin-RR, microcystin-LY and (D-Asp3)microcystin-LR) with two nanofilters (NF) (with molecular weight cutoff (MWCO) 300-500 Da and 1000 Da) and one ultrafilter (UF) (MWCO 10000 Da) was studied. Further, the sorption between toxins and membrane or between toxins and DOC was studied. Water samples from three Swedish lakes were used to achieve a varying DOC-character. The reduction was also studied in synthetic water (MilliQ-water with adjusted pH and conductivity). This was done in order to study if the reduction was affected of the presence of DOC. The reduction of ATX-A and PST was studied in one experiment, with an NF-membrane (MWCO 1000 Da). The studied PST’s were saxitoxin (STX), N-sulfocarbamoyl-gonyautoxin-2 (C1) and Gonyuatoxin‑2 (GTX2). MC was reduced to under the limit of detection for all NF-experiments, while a low or no reduction was observed with UF. MC-RR was the toxin that, to the biggest extent, bound to the membrane or to DOC. MC seemed to bind to the membrane or to DOC to a higher extent when the DOC was hydrophilic and low molecular autochthonous, as compared to when the DOC was allochthonous. The reduction of ATX-A and PST was 20 – 40 %, with a reduction that declined during the experiment. C1, the PST with highest molecular weight and lowest net charge, was reduced to the lowest extent, while STX, the PST with the lowest molecular weight and the greatest net charge, was reduced to the highest extent. This implies that size exclusion was not a contributing reduction mechanism for ATX-A or PST, while electrostatic mechanisms probably were more important.

This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures. We then expanded our study to more thoroughly developed theory. This comprehensive theory was more abstract and required the use of a different, more universal, notation. We continued examining least upper and greatest lower bounds but extended our knowledge to subalgebras and families of subsets. The notions of cardinality, cellularity, and pairwise disjoint families were investigated, defined, and then used to understand the Erdös-Tarski Theorem. Lastly, this study concluded with the investigation of denseness and incomparability as well as normal forms and the completion of Boolean algebras.

A Boolean partition algebra is a pair $(B,F)$ where $B$ is a Boolean algebra and $F$ is a filter on the semilattice of partitions of $B$ where $\bigcup F=B\setminus\{0\}$. In this dissertation, we shall investigate the algebraic theory of Boolean partition algebras and their connection with uniform spaces. In particular, we shall show that the category of complete non-Archimedean uniform spaces is equivalent to a subcategory of the category of Boolean partition algebras, and notions such as supercompleteness of non-Archimedean uniform spaces can be formulated in terms of Boolean partition algebras.

É bem conhecido o fato de que todo grupo compacto tem sequências não triviais convergentes. A existência de grupos enumeravelmente compactos sem sequências não triviais convergentes, foi provada usando axiomas adicionais à axiomática usual ZFC: A. Hajnal e I. Juhász sob CH, E. K. van Douwen sob MA, A. H. Tomita sob MA(sigma-centrada) e R.E. Madariaga-Garcia e A. H. Tomita usando ultrafiltros seletivos. Neste trabalho, estudaremos algumas construções recentes relacionadas com as citadas acima, usando o Axioma de Martin, ultrafiltros seletivos e forcing. Essas construções estão relacionadas com algumas questões indicadas por A.D. Wallace, E. van Douwen, M. Tkachenko, D. Dikranjan e D. Shakhmatov / It is well known that every compact group has non-trivial convergent sequences. The existence of countably compact groups without non-trivial convergent sequences was proved using extra set-theoretical assumptions: A. Hajnal and I. Juhasz under CH, E. K. van Douwen under MA, A.H.Tomita under MA(centered) and R.E.Madariaga-Garcia and A.H. Tomita using a selective ultrafilter. I n this work, we study some recent constructions related to the ones given above using Martin Axiom, selective ultrafilters and forcing, related to questions raised by A.D. Wallace, E. van Douwen, M. Tkacenko, D. Dikranjan and D. Shakhmatov.

É bem conhecido o fato de que todo grupo compacto tem sequências não triviais convergentes. A existência de grupos enumeravelmente compactos sem sequências não triviais convergentes, foi provada usando axiomas adicionais à axiomática usual ZFC: A. Hajnal e I. Juhász sob CH, E. K. van Douwen sob MA, A. H. Tomita sob MA(sigma-centrada) e R.E. Madariaga-Garcia e A. H. Tomita usando ultrafiltros seletivos. Neste trabalho, estudaremos algumas construções recentes relacionadas com as citadas acima, usando o Axioma de Martin, ultrafiltros seletivos e forcing. Essas construções estão relacionadas com algumas questões indicadas por A.D. Wallace, E. van Douwen, M. Tkachenko, D. Dikranjan e D. Shakhmatov / It is well known that every compact group has non-trivial convergent sequences. The existence of countably compact groups without non-trivial convergent sequences was proved using extra set-theoretical assumptions: A. Hajnal and I. Juhasz under CH, E. K. van Douwen under MA, A.H.Tomita under MA(centered) and R.E.Madariaga-Garcia and A.H. Tomita using a selective ultrafilter. I n this work, we study some recent constructions related to the ones given above using Martin Axiom, selective ultrafilters and forcing, related to questions raised by A.D. Wallace, E. van Douwen, M. Tkacenko, D. Dikranjan and D. Shakhmatov.

This thesis studies ultrafilters and their various applications in topology, social choice theory and construction of a nonstandard universe. First of all, we introduce basic properties of ultrafilters and show how to use them to construct nonstandard framework. Next, we prove Arrow's impossibility theorem which states that every electoral system with a finite set of voters satisfying certain natural conditions necessarily admits at least one dictator who determines the society's preferences. However, if the set of voters is infinite, this is not true anymore and ultrafilters play a key role in the proof. We present two counterexamples in the infinite case using nonstandard framework. A similar theorem holds in the case where the preferences are real functions. Again, we show two examples of electoral systems that are not dictatorial - one using Banach limits and the other using hyperfinite sums. Finally, we use the ultrafilters to construct the Čech-Stone compactification of natural numbers. We show that the nonstandard enlargement of natural numbers equipped with suitable topology is the Čech-Stone compactification of the set of natural numbers. 1

No description available.

We study the existence of special points in extremally disconnected compact topological spaces that witness their nonhomogeneity. Via Stone duality, we are looking for ultrafilters on complete Boolean algebras with special combinatorial properties. We introduce the notion of a coherent ultrafilter (coherent P-point, coherently selective). We show that generic existence of such ultrafilters on every complete ccc Boolean algebra of weight not exceeding the continuum is consistent with set theory, and that they witness the nonhomogeneity of the corresponding Stone spaces. We study the properties of the order-sequential property on σ-complete Boolean algebras and its relation to measure-theoretic properties. We ask whether the order-sequential topology can be compact in a nontrivial case, and partially answer the question in a special case of the Suslin algebra associated with a Suslin tree.

This dissertation will seek to explore if the properties of some of the key results on semigroups and their compacti cations under the operation of addition also apply under the operation of multiplication. Consider- able emphasis will be placed on the semigroup N of the set of natural numbers and its compacti cation N. Furthermore, the dissertation will discuss the smallest ideal of a semi- group and highlight some of its fundamental properties. / Mathematics / M. Sc. (Mathematics)

Vattentillgången på Öland har under en lång tid varit mycket ansträngd och har kommit till en punkt då det inte längre är hållbart. För att nå en mer långsiktig lösning planerar Mörbylånga kommun med hjälp av Norconsult att bygga ett vattenreningsverk. Detta kommer att rena bräckt vatten ifrån Kalmarsund samt processvatten ifrån Guldfågeln AB, som tillsammans kommer att fungera som ett komplement till det redan befintliga sötvattnet. Två pilotanläggningar som renar processvatten ifrån en industri, genom ultrafilter kombinerat med omvänd osmos, har utvärderats. En metod för att sammanställa de kostsamma driftparametrarna elförbrukning, förbrukning av kemikalier samt underhållsarbete vid drift och membranens inköpskostnad genom en LCC har tagits fram. Denna metod ger en ungefärlig totalkostnad per kubikmeter av det renade vattnet. Även vattenkvaliteten på ingående råvatten, vattnet mellan membranen samt det utgående vattnet har analyserats. Detta för att undersöka att processerna fungerar som den ska, samt ge en indikation på vilka mineraler som kommer att behöva tillsättas i efterhand. Pilot B blir ca fem kronor dyrare per kubikmeter jämfört med Pilot A. De totala kostnaderna, 9.70 – 10.79 kr/m3 för Pilot A och 15.19 – 15.28 kr/m3 för Pilot B, kan jämföras med den nuvarande kostnaden för en kubikmeter vatten i Mörbylånga, 15 kr/m3, samt den kostnad kommunen betalade sommaren 2016 för att frakta vatten i tankbilar, 70 kr/m3.