In this licentiate thesis, we study the action of Hecke operators on Drinfeld cusp forms via the theory of crystals over function fields. The thesis contains one preliminary chapter, in which we recall some basic theory of Drinfeld modules and Drinfeld modular forms, as well as the Eichler-Shimura theory developed by Böckle. The core of the thesis consists of Chapter II, in which we prove a Lefschetz trace formula for crystals over stacks and deduce a Ramanujan bound for Drinfeld modular forms, and Chapter III, in which we compute traces and slopes of Hecke operators. We formulate several questions and conjectures based on our data. We also include an appendix in which we discuss the relationship between traces of an operator in positive characteristic and its eigenvalues.

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This paper investigates how physical prototyping modules can be designed to facilitate for interactiondesigners to visualize their ideas early on in a design process. As technology is getting cheaper andeasier to use, it has opened up the possibility for others than just engineers to build with technology.We see technology being more and more used by designers for building prototypes and testing ideas.A setback with using technology is that it is time consuming and error occurs easily. By testing a set ofthree prototypes I will investigate the problems at hand for interaction designers and come up with adesign solution to facilitate their design process. I will come to conclusion about making the modulesflexible, functional and user friendly to meet the user’s demands.

Despite being the world’s second most important low-carbon source of electricity, the development of nuclear capacities is limited and does not comply with the International Energy Agency’s Sustainable Development Scenario. One of the main reasons for this lack of development is high and increasing costs of new nuclear capacities. In fact, capital costs can account for more than 80% of the Levelized Cost of Energy of a new nuclear power plant. Design accounts for about 10% and construction accounts for about 20% of the total capital cost.The importance of design and construction in nuclear power plants’ costs is partly due to limitations of traditional construction methods regarding both technical possibilities and realization schedules. Modular construction methods are proposed to reduce new nuclear power plants’ construction costs, and in particular steel-plate composite structures modular construction. Steel-plate composite structures consist of a structural interaction between thin steel plates and precast concrete. The aim of this thesis was to evaluate the capacity of modular construction, using steel-plate composite panels, to fulfil nuclear safety and functional requirements; as well as to identify potential related gains and opportunities. Advantages and constraints of steel-plate composite modules were identified in scientific literature and intergovernmental opportunity studies, and were related to the specificities of nuclear structures’ design and construction.Steel-plate composite structures modular construction is based on the manufacturing of steel concrete composite modules outside of the civil works site. The higher productivity of workshops and the possible task parallelization may lead to significant construction schedules shortening and capital costs reduction. In addition, steel-plate composite modules offer new technical possibilities which may help solving some constructability issues. However, steel-plate composite structures modular construction requires a reconsideration of the traditional design and construction methods, as they imply new challenges and constraints. In particular, the modularization issue should be addressed as soon as possible in the design, and the module manufacturing capacities should be quickly identified or created in order to deliver properly manufactured modules on-time.Considering the advantages and constraints of steel-plate composite structures modular construction, it appears that steel-plate composite modules may fit advantageously most of the nuclear design requirements. However, due to the lack of feedback, it is proposed that the usage of steel-plate composite modules be limited to critical concrete structures of the containment building. In particular, it appears that the containment dome and the reactor pit construction may benefit from steel-plate composite structures construction methods.

Let <i>R</i> be a commutative ring, <i>I</i> be an ideal in <i>R</i> and let <i>M</i> be a <i>R/ I</i> -module. In this thesis we construct a <i>R/ I</i> -projective resolution of <i>M</i> using given <i>R</i>-projective resolutions of <i>M</i> and <i>I</i>. As immediate consequences of our construction we give descriptions of the canonical maps Ext_R/I<i>(M,N)</i> -> Ext_R<i>(M,N)</i> and Tor^R_N<i>(M, N)</i> -> Tor^R/I_n<i>(M, N)</i> for a <i>R/I</i> module <i>N</i> and we give a new proof of a theorem of Gulliksen [6] which states that if <i>I</i> is generated by a regular sequence of length r then ∐∞_n=o Tor^R/I_n <i>(M, N)</i> is a graded module over the polynomial ring </i>R/ I</i> [X₁. .. X_r] with deg X_i = -2, 1 ≤ i ≤ r. If <i>I</i> is generated by a regular element and if the <i>R</i>-projective dimension of <i>M</i> is finite, we show that <i>M</i> has a <i>R/ I</i>-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when <i>R</i> = (<i>R</i>, m) is a Noetherian local ring and <i>M</i> is a finitely generated <i>R/ I</i> -module, we discuss the minimality of the constructed resolution. If it is minimal we call (<i>M, I</i>) a Golod pair over <i>R</i>. We give a direct proof of a theorem of Levin [10] which states thdt if (<i>M,I</i>) is a Golod pair over <i>R</i> then (Ωⁿ_R/IR/I(M),I) is a Golod pair over <i>R</i> where Ωⁿ_R/IR/I(M) is the nth syzygy of the constructed <i>R/ I</i> -projective resolution of <i>M</i>. We show that the converse of the last theorem is not true and if (Ω¹_R/IR/I(M),I) is a Golod pair over <i>R</i> then we give a necessary and sufficient condition for (<i>M, I</i>) to be a Golod pair over <i>R</i>. Finally we prove that if (<i>M, I</i>) is a Golod pair over <i>R</i> and if a ∈ <i>I</i> - m<i>I</i> is a regular element in </i>R</i> then (<i>M</i>, (a)) and (1/(a), (a)) are Golod pairs over <i>R</i> and (<i>M,I</i>/(a)) is a Golod pair over <i>R</i>/(a). As a corrolary of this result we show that if the natural map π : <i>R</i> → <i>R/1</i> is a Golod homomorphism ( this means (<i>R</i>/m, <i>I</i>) is a Golod pair over <i>R</i> ,Levin [8]), then the natural maps π₁ : <i>R</i> → <i>R</i>/(a) and π₂ : <i>R</i>/(a) → <i>R/1</i> are Golod homomorphisms. / Ph. D.

The response of plants to environmental stress spans several orders of magnitude in time and space, causing system-wide changes. These changes comprise of both protective responses and adverse reactions in the plant. Stresses like water deficit or drought cause a drastic effect in crop yield, while concomitantly agriculture consumes 1/3rd of the fresh water available to us and there is widespread water scarcity around the world. It is, hence, a fundamental goal of modern biology and applied biotechnology to unravel this complex stress response in laboratory model plants like Arabidopsis and crop models like rice. Such an understanding, especially at the cellular level, will aid in informed engineering of stress tolerance in plants. We have developed and used integrative functional genomics approaches to characterize environmental stress response at various levels of organization including genes, modules and networks in Arabidopsis and rice. We have also applied these methods in problems concerning bioenergy. Since the poor knowledge of the cellular roles of a large portion of plant genes remains a fundamental barrier to using such approaches, we have further explored the problem of 'gene function prediction'. And, finally, as a contribution to the community, we have curated a large mutant resource for the crop model, rice, and established a web resource for exploratory analysis of abiotic stress in this model. All together, this work presents insights into several facets of stress response, offers numerous novel predictions for experimental validation, and provides principled analysis frameworks for systems level analysis of environmental stress response in plants. / Ph. D.