We study relative homological algebra and relative Hochschild cohomology. We dualise the construction in [Cib+21b] for a ring extension B ⊆ A to construct a long nearly exact sequence for the relative Hochschild cohomology HH∗(A|B), the Hochschild cohomology HH∗(A) and the Hochschild cohomology HH∗(B,A). Parallel to this we also study corings and the associated Cartier cohomology and Hochschild cohomology. Given an A-coring C and its right algebra R we have induced maps ExtiA(M, N) → ExtiR(R⊗A M, R⊗A N) by the induction functor. We characterise the vanishing of the Hochschild cohomology of the coring in terms of these induced maps being isomorphisms for degrees greater than or equal to one.

Quasi-hom-Lie algebras (qhl-algebras) were introduced by Larsson and Silvestrov (2004) as a generalisation of hom-Lie algebras, which are a deformation of Lie algebras. Lie algebras are defined by an operation called bracket, [·,·], and a three-term Jacobi identity. By the theorem from Hartwig, Larsson, and Silvestrov (2003), this bracket and the three-term Jacobi identity are deformed into a new bracket operation, <·,·>, and a six-term Jacobi identity, making it a quasi-hom-Lie algebra. Throughout this thesis we deform the Lie algebra sl2(F), where F is a field of characteristic 0. We examine the quasi-deformed relations and six-term Jacobi identities of the following polynomial algebras: F[t], F[t]/(t2), F[t]/(t3), F[t]/(t4), F[t]/(t5), F[t]/(tn), where n is a positive integer ≥2, and F[t]/((t-t0)3). Larsson and Silvestrov (2005) and Larsson, Sigurdsson, and Silvestrov (2008) have already examined some of these cases, which we repeat for the reader's convenience. We further investigate the following σ-twisted derivations, and how they act in the different cases of mentioned polynomial algebras: the ordinary differential operator, the shifted difference operator, the Jackson q-derivation operator, the continuous q-difference operator, the Eulerian operator, the divided difference operator, and the nilpotent imaginary derivative operator. We also introduce a new, general, σ-twisted derivation operator, which is σ(t) as a polynomial of degree k.

A definition of abstract logic is presented. This is used to explore and compare some abstract logics, such as logics with generalised quantifiers and infinitary logics, and their properties. Special focus is given to the properties of completeness, compactness, and the Löwenheim-Skolem property. A method of comparing different logics is presented and the concept of equivalent logics introduced. Lastly a proof is given for Lindström's theorem, which provides a characterization of elementary logic, also known as first-order logic, as the strongest logic for which both the compactness property and the Löwenheim-Skolem property, holds.

Quasi-hereditary algebras are an important class of algebras with many appli-cations in representation theory, most notably the representation theory of semi-simple complex Lie-algebras. Such algebras sometimes admit an exact Borel sub-algebra, that is a subalgebra satisfying similar formal properties to the Borel sub-algebras from Lie theory. This thesis is divided into two parts. In the first part we classify quasi-hereditary algebras with two simple modules over perfect fields up to Morita equivalence, generalizing a similar result by Membrillo-Hernandez for thealgebraically closed case. In the second part, we take a poset X, a certain set M of constants, and a finite set ρ of paths in the Hasse-diagram of X and construct analgebra A(X, M, ρ) that generalizes the twisted double incidence algebras originally introduced by Deng and Xi. We provide necessary and sufficient conditions for this algebra to be quasi-hereditary when X is a tree, and we show that A(X, M, ρ) admits an exact Borel subalgebra when these conditions are satisfied. Following this, we compute the Ext-algebra of the standard modules of A(X, M, ρ).

Dagens samhälle blir mer digitaliserat för varje år som går och skolan följer den utvecklingen; därmed ökar digitaliseringen och användandet av digitala läromedel även i skolan. Därför är undersökningen relevant för den nutida skolans utveckling. Digitala läromedel är något som används frekvent i matematikundervisning, vilket är något vi har observe- rat under våra VFU-perioder. Dessutom uppmärksammade vi att många yrkesverksamma lärare är mindre insatta i innehållet i digitala läromedel, till skillnad från innehållet i de analoga läromedlen som används. Vi såg att det fanns studier kring digitala läromedel, men att det verkade vara brist på studier med fokus på hur algebra behandlas i elevuppgifter i digitala läromedel. Matematikområdet algebra är i sig ett område där tidigare forskning lyfter behovet av vidare forskning. Därav valde vi att specificera vår undersökning om hur algebra behandlas i elevuppgifter i digitala läromedel. Analysen i undersökningen grundar sig i ett analysschema med kategorier och kriterier utformade efter teorier från Sjödén (2014), Choppin m.fl. (2014), Blanton m.fl. (2015) samt Madej (2021). Teorierna har hjälpt oss att analysera det digitala och algebraiska innehållet i elevuppgifterna och Blantons m.fl. (2015) Big ideas framhäver väsentliga delar av algebraundervisning. Syftet med undersökningen är att belysa vilka potentiella mervärden som skapas genom olika digitala läro- medel inom området algebra för årskurs 6. Det görs genom insamling av data för en kvantitativ innehålls- analys av elevuppgifter innehållande algebra i tre digitala läromedel som används i grundskolan. Valet av läromedel gjordes utifrån vilka digitala läromedel vi har sett används av yrkesverksamma lärare i mellan- stadiet. Resultatet som kan utläsas av analysen är att digitala läromedel erbjuder olika möjligheter som inte erbjuds i analoga läromedel, samt att det algebraiska innehållet varierar i de olika digitala läromedlen. / Today's society is becoming more digitized every year and school follows that evolution; therefore the digitization and the use of digital curriculum materials increases in school as well. This study is relevant for today’s school development. Digital curriculum materials are frequently used in mathematics teaching, which we have noticed during our work-based education (VFU). We also noticed that teachers have less insight about the digital curricu- lum materials than in the books that are used. We found studies about digital curriculum materials, but there seemed to be a lack of research with focus on how algebra is processed in pupil tasks in digital curriculum materials. The mathematical field of algebra is an area where previous research requires further research. Hence, we chose to specify our investigation about how algebra is processed in pupil tasks in digital cur- riculum materials. The analysis in our inquiry is based on an analysis schedule with categories and criteria designed from theories from Sjödén (2014), Choppin et.al. (2014), Blanton et.al. (2015) and Madej (2021). The theories were helpful when we analyzed the digital and algebraic content in the pupil tasks and Blantons et.al. (2015) Big ideas highlights essential parts concerning algebra teaching. The study aims to highlight potential values created through various digital curriculum materials in the area of algebra for grade 6. It is done by collecting data for a content analysis of pupil tasks containing algebra in three digital curriculum materials used in primary school. The choice of resources was made based on which digital curriculum materials we have seen been used by professional teachers in middle school. The result that can be deduced from the analysis is that digital curriculum materials offer different potential values that are not offered in analog curriculum materials, and that the algebraic content where digital values occur varies depending on the digital curriculum materials.

This thesis was conducted with the objective of reducing water consumption byoptimizing the cooling systems of steam sterilizers. As water is a precious resourcewith great environmental effects, it is important not to waste it. Consequently, thereis a need for a more resource-efficient cooling water system. The project focuses onthe development of a system that more efficiently regulates the cooling water utilization by optimizing temperatures. The goal of the project is to achieve a 20% reduction in water consumption of the GSS-91413 model steam sterilizer manufacturedby Getinge. In order to achieve the goal, changes were made to the cooling systemand control logic of the cooling system. By integrating a proportional valve at theoutlet of the cooling system, the system was pressurized with the coolant resultingin greater energy transfer between the condensate and the coolant. The developedcontrol logic incorporates process data combined with an equation-based approachthat utilizes temperature data to adjust the proportional valve leading to increasedcontrol of the flow of the coolant. As a result, the overall water consumption of thesystem was reduced by more than 50% while the maximal temperature of the systemdid not rise more than 1.5%. / Detta examensarbete genomfördes med målet att minska vattenförbrukningen genom att optimera kylsystemen i autoklaver. Eftersom vatten är en värdefull resursmed stora miljöeffekter är det viktigt att den inte slösas. Följaktligen finns ett behovav ett mer resurseffektivt kylvattensystem. Projektet fokuserar på utveckling av ettsystem som mer effektivt reglerar kylvattenutnyttjandet genom att optimera temperaturer. Målet med projektet är att uppnå en 20-procentig minskning av vattenförbrukningen för autoklavmodellen GSS-91413 tillverkad av Getinge. För att nå måletgjordes ändringar i kylsystemet och i kylsystemets styrlogik. Genom att integrera enproportionell ventil vid utloppet av kylsystemet trycksattes systemet med kylvätskanvilket resulterade i större energiöverföring mellan kondensatet och kylvätskan. Denutvecklade styrlogiken inkorporerar processdata kombinerat med ett ekvationsbaserat tillvägagångssätt som använder temperaturdata för att justera proportionalventilen vilket leder till ökad kontroll av kylvätskans flöde. Som ett resultat minskade dentotala vattenförbrukningen i systemet med mer än 50% medan systemets maximalatemperatur inte steg mer än 1.5%.