Browsing by Subject "congruence"

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  • Forsman, David (Helsingin yliopisto, 2020)
    We develop the theory of categories from foundations up. The thesis culminates in a theorem in which we assert that any concrete functor between categories of models of algebraic theories, where the codomain categories' alphabet does not contain relational information, has a left adjoint functor. This theorem is based on The General Adjoint Functor Theorem by Peter Freyd. The first chapter is about the set theoretic foundations of category theory. We present the needed ideas about recursion so that we may define what is meant by first order predicate logic. The first chapter ends in the exposition of the connection between the Grothendieck universes and the inaccessible cardinals. The second chapter starts our conversation about categories and functors between categories. We define properties of morphisms, subobjects, quotient objects and Cartesian closed categories. Furthermore, we talk about embedding and identification morphisms of concrete categories. Much of the third chapter is to show that the category of small categories is a Cartesian closed category. This leads us to talk about natural transformation and canonical constructions relating to functors. To define equivalences and their generalizations, adjoint functors, natural transformations are needed. The fourth chapter enlarges our knowledge about hom-functors and their adjacent functors, representable functors. The study of representable functors yields a profound lemma called Yoneda lemma. Yoneda lemma implies the fully faithfulness of Yoneda embedding. The fifth chapter concentrates to limit operations in a category, which leads us to talk about completeness. We find out how limit procedures are preserved in constructions and how they behave when functors pass them forward. The last chapter is about adjoint functors. The general and the special adjoint functor theorems, due to Peter Freyd, are proven. Using The General Adjoint Functor Theorem, we prove the existence of a left adjoint functor for all suitable forgetful functors among algebraic categories.
  • Tolonen, Kimmo T.; Karjalainen, Juha; Hämäläinen, Heikki; Nyholm, Kristiina; Rahkola-Sorsa, Minna; Cai, Yongjiu; Heino, Jani (Springer Link, 2020)
    Aquatic Ecology 54 3 (2020)
    Lake littoral environments are heterogeneous, and different organisms typically show specific responses to this environmental variation. We examined local environmental and spatial factors affecting lake littoral biodiversity and the structuring of assemblages of phytoplankton, zooplankton and macroinvertebrates within and among three basins of a large lake system. We explored congruence of species composition and species richness among the studied organism groups to evaluate their general indicator potential to represent spatial variation in other groups. We expected that effects of water chemistry on plankton assemblages were stronger than effects of habitat characteristics. In contrast, we anticipated stronger effects of habitat on macroinvertebrates due to their mainly benthic mode of life. We also expected that within-basin spatial effects would be strongest on macroinvertebrates and weakest on phytoplankton. We predicted weak congruence in assemblage composition and species richness among the organism groups. Phytoplankton assemblages were mainly structured by the shared effects of water chemistry and large-scale spatial factors. In contrast to our expectations, habitat effects were stronger than water chemistry effects on zooplankton assemblages. However, as expected, macroinvertebrate species composition and richness were mainly affected by habitat conditions. Among-group congruence was weak for assemblage composition and insignificant for richness. Albeit weak, congruence was strongest between phytoplankton and zooplankton assemblages, as we expected. In summary, our analyses do not support the idea of using a single organism group as a wholesale biodiversity indicator.
  • Bäckman, Hanna (Helsingfors universitet, 2007)
    The Purpose of the Finnish Hygiene Act (The Act on Food Hygiene of Foodstuff of Animal Origin 1195/1996), which was in force until the beginning of March 2006, was to ensure the hygienic quality of foodstuffs of animal origin and prevent the spread of infections from animals to humans via foodstuffs. The Hygiene Act applied to handling, hygienic quality requirements and inspections prior to retail of foodstuff of animal origin. The aim of official food control is to ensure that establishments fulfil the requirements imposed on them by the legislation. According to The Hygiene Act, abattoirs and establishments connected to them were controlled by an official veterinarian working for The National Food Agency, and other establishments controlled by the municipal authorities. Each municipality or federation of municipality was responsible for the official control in its own region. The inspection frequencies depended on the type of establishment, but all establishments had to be inspected regularly. The purpose of this study was to investigate how official control was perceived in establishments covered by the Hygiene Act in terms of effects and congruence of the official control and guidance received from the authorities. The aim was also to investigate the influence of control frequency on perceptions on effects, congruence and guidance. The research was performed in spring 2006 using a questionnaire, which was issued to all establishments in meat branch, establishments handling fishery products, dairy plants, egg packing centres and warehouses of foodstuffs of animal origin registered as approved establishments by The National Food Agency in 2005. 459 answers were received, which was 36 % of the questionnaires sent. The results show that the food control had improved the hygiene of the establishments according to the perceptions of the establishments. Product safety was considered to be improved by the official control more in small and medium-sized establishments than in large establishments. EU-establishments in meat branch have made more changes to their production processes and line of production due to food control than other types of establishments. Approximately one half of the respondents were unable to provide a view on congruence of official control. Incongruence was experienced most frequently in low-capacity slaughterhouses. High inspection frequency was found to be connected to the experience of incongruence. The higher the inspection frequency, the higher was the perceived incongruence of official control. Most establishments were satisfied with the amount of guidance concerning the legislation they had been given by the official inspector. However, more guidance was needed on the planning of the own-check, construction of production facilities, expansion and repair of production facilities and correction of the shortcomings found during inspections. The frequency of inspections was also found to have an effect on perceived benefit of official control. The more frequent the control inspections performed by the municipal control authorities were, the higher was the perceived positive impact on hygiene.