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Let G be a reductive algebraic group defined over an algebraically closed field of characteristic p. A subgroup H of G is called G-complete reducible whenever H is contained in a parabolic subgroup P of G, it is contained in some Levi subgroup of P. In this thesis, we present a pair of reductive subgroups H and M of G of type E_7 such that H<M and H is G-completely reducible but not M-completely reducible.

This Thesis is motivated by two problems, each concerning representations (homomorphisms) of groups into a connected reductive algebraic group G over an algebraically closed field k. The first problem is due to B. Külshammer and is to do with representations of finite groups in G: Let Γ be a finite group and suppose k has characteristic p. Let Γp be a Sylow p-subgroup of Γ and let ρ : Γp → G be a representation. Are there only finitely many conjugacy classes of representations ρ' : Γ → G whose restriction to Γp is conjugate to ρ? The second problem follows the work of M. Liebeck and G. Seitz: describe the representations of connected reductive algebraic H in G. These two problems have been settled as long as the characteristic p is large enough but not much is known in the case where the characteristic p is a so called bad prime for G, which will be the setting for our work. At the intersection of these two problems lies another problem which we call the algebraic version of Külshammer's question where we no longer suppose Γ is finite. This new variation of Külshammer's question is interesting in its own right, and a counterexample may provide insight into Külshammer's original question. Our approach is to convert these problems into problems in the nonabelian 1-cohomology. Let K be a reductive algebraic group, P a parabolic subgroup of G with Levi subgroup L < P, V the unipotent radical of P. Let ρ₀ : K → L be a representation. Then the representations ρ : K → P that equal ρ₀ under the canonical projection P → L are in bijective correspondence with elements of the space of 1-cocycles Z¹(K,V ) where K acts on V by xv = ρ₀(x)vρ₀(x)⁻¹. We can then interpret P- and G-conjugacy classes of representations in terms of the 1-cohomology H¹(K,V ). We state and prove the conditions under which a collection of representations from K to P is a finite union of conjugacy classes in terms of the 1-cohomology in Theorem 4.22. Unlike other approaches, we work directly with the nonabelian 1-cohomology. Even so, we find that the 1-cocycles in Z¹(K,V ) often take values in an abelian subgroup of V (Lemmas 5.10 and 5.11). This is interesting, for the question "is the restriction map of 1-cohomologies H¹(H,V) → H¹(U,V) induced by the inclusion of U in K injective?" is closely linked to the question of Külshammer, and has positive answer if V is abelian and H = SL₂k) (Example 3.2). We show that for G = B4 there is a family of pairwise non-conjugate embeddings of SL₂in G, a direction provided by Stewart who proved the result for G = F4. This is important as an example like this is first needed if one hopes to find a counterexample to the algebraic version of Külshammer's question.

The effects of noble metal (NM) promotion and deposition order (co-deposition of NM with the final Co deposition [co-dep] or sequential deposition of NM after Co deposition [seq-dep]) on surface area, pore size, metal retention, crystallite size, noble metal distribution and bonding in Co Fischer Tropsch (FT) catalysts were studied as were the resulting Co reducibility and Fischer Tropsch activity/selectivity properties. Catalysts containing nominally 25wt% Co with either 0.3 wt% Ru, 0.58 wt% Pt, 0.55wt% Re, or no NM on a La-stabilized-Al2O3 support were prepared by wet deposition. The Co, Pt, and Re were uniformly dispersed, but Ru distribution and retention were problematic and deposition-order dependent—85% was lost with co-dep, but it was uniformly distributed while 54% was lost with seq-dep and it was concentrated at the pellet edge. The co-dep catalysts all have smaller reduced Co crystallite size than their corresponding seq-dep catalysts. The average crystallite diameters for all 3 co-dep catalysts are between 4.1 and 4.3nm and ~90% of the crystallites are < 6nm. XAFS measurements showed that after reduction at 360°C, Pt is bonded with Co even with mild calcination between the final Co and the Pt deposition. On the other hand, neither Ru nor Re formed direct bonds with Co. Ru remained in a separate metal phase after reduction even at low loadings. Re remained as Re2O7 and still promoted Co reduction well (e.g. 42% reduced to Co metal compared to none for the unpromoted catalyst). By all measures of reducibility (TPR, EOR, H2 uptake), all NM promoted catalysts were more reducible than the unpromoted catalyst. The co-dep catalysts have lower TPR peak temperatures, but lower extents of reduction than their corresponding seq-dep catalysts. The NM type effect on overall extent of reduction trend was Co/Pt-seq>Co/Re-seq>Co/Ru-seq=Co/Pt-co>Co/Re-co>Co/Ru-co>Co. The Co/Pt-co catalyst was the most active of all the catalysts both on rate per mass and per site basis. The co-dep catalysts were all more active than the corresponding sequentially deposited catalysts. The co-dep Pt and Re catalyst activity is greater due to higher activity per site, while co-dep Ru activity is greater due to a higher abundance of active sites.

The ordering of concept classes under PAC reducibility is nonlinear, even when restricted to particular concrete examples. We construct two effective concept classes of incomparable PAC degrees to show that there exist incomparable PAC degrees, analogous to incomparable Turing degrees. The non-learnability of concept classes in the PAC learning model is explained by the existence of PAC incomparable degrees. It was necessary to deal with the size of an effective concept class thus we propose a method to compute the size of the effective concept class using Kolmogorov complexity. To define the jump operator for PACi degrees the join of effective concept classes is constructed and explores the possibility of embedding known degrees to PACi or PAC degrees. If an embedding exists it will enable proving properties of known degrees for PACi and PAC degrees without explicitly proving them.

This thesis presents a critical analysis of normalisation by evaluation as a technique for speeding up compilation of typed functional programming languages. Our investigation focuses on the SML.NET compiler and its typed intermediate language MIL. We implement and measure the performance of normalisation by evaluation for MIL across a range of benchmarks. Taking a different approach, we also implement and measure the performance of a graph-based shrinking reductions algorithm for SML.NET. MIL is based on Moggi’s computational metalanguage. As a stepping stone to normalisation by evaluation, we investigate strong normalisation of the computational metalanguage by introducing an extension of Girard-Tait reducibility. Inspired by previous work on local state and parametric polymorphism, we define reducibility for continuations and more generally reducibility for frame stacks. First we prove strong normalistion for the computational metalanguage. Then we extend that proof to include features of MIL such as sums and exceptions. Taking an incremental approach, we construct a collection of increasingly sophisticated normalisation by evaluation algorithms, culminating in a range of normalisation algorithms for MIL. Congruence rules and alpha-rules are captured by a compositional parameterised semantics. Defunctionalisation is used to eliminate eta-rules. Normalisation by evaluation for the computational metalanguage is introduced using a monadic semantics. Variants in which the monadic effects are made explicit, using either state or control operators, are also considered. Previous implementations of normalisation by evaluation with sums have relied on continuation-passing-syle or control operators. We present a new algorithm which instead uses a single reference cell and a zipper structure. This suggests a possible alternative way of implementing Filinski’s monadic reflection operations. In order to obtain benchmark results without having to take into account all of the features of MIL, we implement two different techniques for eliding language constructs. The first is not semantics-preserving, but is effective for assessing the efficiency of normalisation by evaluation algorithms. The second is semantics-preserving, but less flexible. In common with many intermediate languages, but unlike the computational metalanguage, MIL requires all non-atomic values to be named. We use either control operators or state to ensure each non-atomic value is named. We assess our normalisation by evaluation algorithms by comparing them with a spectrum of progressively more optimised, rewriting-based normalisation algorithms. The SML.NET front-end is used to generate MIL code from ML programs, including the SML.NET compiler itself. Each algorithm is then applied to the generated MIL code. Normalisation by evaluation always performs faster than the most naıve algorithms— often by orders of magnitude. Some of the algorithms are slightly faster than normalisation by evaluation. Closer inspection reveals that these algorithms are in fact defunctionalised versions of normalisation by evaluation algorithms. Our normalisation by evaluation algorithms perform unrestricted inlining of functions. Unrestricted inlining can lead to a super-exponential blow-up in the size of target code with respect to the source. Furthermore, the worst-case complexity of compilation with unrestricted inlining is non-elementary in the size of the source code. SML.NET alleviates both problems by using a restricted form of normalisation based on Appel and Jim’s shrinking reductions. The original algorithm is quadratic in the worst case. Using a graph-based representation for terms we implement a compositional linear algorithm. This speeds up the time taken to perform shrinking reductions by up to a factor of fourteen, which leads to an improvement of up to forty percent in total compile time.

We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of reducibility. We define variables which in some sense count the number of nowhere-zero flows of certain type in a graph and then deduce equalities and inequalities that must hold for all graphs. We then show how to use these algebraic expressions to prove reducibility. In our case, these inequalities and equalities are linear. We can thus use the well developed theory of linear programming to obtain certificates of these proof. We make publicly available computer programs we wrote to generate the algebraic expressions and obtain the certificates.

We study reducibility for nowhere-zero flows. A reducibility proof typically consists of showing that some induced subgraphs cannot appear in a minimum counter-example to some conjecture. We derive algebraic proofs of reducibility. We define variables which in some sense count the number of nowhere-zero flows of certain type in a graph and then deduce equalities and inequalities that must hold for all graphs. We then show how to use these algebraic expressions to prove reducibility. In our case, these inequalities and equalities are linear. We can thus use the well developed theory of linear programming to obtain certificates of these proof. We make publicly available computer programs we wrote to generate the algebraic expressions and obtain the certificates.

The effect of physical, chemical and mineralogical properties on reducibility of iron containing raw materials were studied with the use of two pellets, one sinter and one lump iron ore sample provided by Erdemir integrated iron and steel works. Although Erdemir lump iron ore contained hematite, it was found to be less reducible than Erdemir sinter since porous structures are easier to reduce and in general sinters have a higher porosity as compared to lump ores. Experimental findings indicated that Erdemir pellet with a code B had the highest reducibility. On the other hand, the results of Erdemir samples were compared with those results obtained from the projects carried out in the Metallurgical and Materials Engineering Department of METU in 1980&rsquo / s. In mentioned projects, samples of various lump iron ores and a concentrate, pellet and sinter from Turkish sources and imported lump iron ores of CVRD from Brazil and ISCOR from the Republic of South Africa were tested. Within the context of this thesis, a mathematical model that would fit to the reduction kinetics was studied and the porous solid model was found to be the best for Erdemir samples.

On utilise la théorie de Floquet-Lin pour des systèmes différentiels linéaires quasi- périodiques pour établir des résultats d'existence et d'unicité et de dépendance continue des systèmes différentiels non linéaires quasi-périodiques. Et dans un second temps on établit un résultat de réductibilité d'un système différentiel linéaire presque-périodique en un système différentiel linéaire triangulaire supérieur avec conservation du nombre des solutions presque-périodiques indépendantes. Ensuite, un résultat d’existence et d’unicité et de dépendance continue des systèmes différentiels non linéaires presque-périodiques par rapport au terme du contrôle. / We use a Floquet theory for quasi-periodic linear ordinary differential equations due to Zhensheng Lin to obtain results, of existence, unicity, continuous and differentiable dependence, on the quasi-periodic solutions of quasi-periodic nonlinear ordinary differential equations. in a second time we establish the reducibility of linear systems of almost periodic differential equations into upper triangular systems of a. p. differential equations. This is done while the number of independent a. p. solutions is conserved. We prove existence and uniqueness of a. p. solutions of a nonlinear system with an a. p. linear part. Also we prove the continuous dependence of a. p. solutions of a nonlinear system with respect to an a. p. control term.