Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

Inaug.--diss.--Halle-Wittenberg. / Cover title. Lebenslauf.

Inaug.--diss.--Halle-Wittenberg. / Cover title. Lebenslauf.

A variety of techniques used to model metabolic networks are examined, both kinetic (ODE) models and flux balance (FB) models. These models are applied to a case study network describing CO and CO2 metabolism in Clostridium autoethanogenum, bacteria which can produce both ethanol and butanediol from a source of carbon monoxide. ODE and FB methods are also used to model a variety of simpler networks. By comparing the results from these simpler networks, the strengths and weaknesses of each examined method are highlighted, and ultimately, insight is gained into the conclusions that can be drawn from each model. ODE models have commonly been used to model metabolism in both in vivo and in vitro contexts, allowing the dynamic behaviour of wildtype bacteria to be examined, as well as that of mutants. An ODE model is formed for the C. autoethanogenum network. By exploring a range of parameter schemes, the possible long timescale behaviours of the model are fully determined. The model is able to exhibit both steady states, and also states in which metabolite concentrations grow indefinitely in time. By considering the scalings of these concentrations in the long timescale, six different non-steady behaviours are categorised and one steady. For a small range of parameter schemes, the model is able to exhibit both steady and unsteady behaviours in the long timescale, depending on initial conditions. FB methods are also applied to the same network. First flux balance analysis (FBA) is used to model the network in steady state. By imposing a range of constraints on the model, limits on levels of flux in the network that are required for a steady-state are found. In particular, boundaries on the ratio of inputs into the network are calculated, outside of which steady states cannot exist. Comparing the steady state regions predicted by FBA and our ODE model, it is found that the FBA model predicts a wider range of conditions leading to steady state. FBA is only able to observe a network in steady state, so an extension of FBA, known as dynamic flux balance analysis (dFBA), is used to examine non-steady-state behaviours. dFBA predicts similar long term non-steady behaviour to the ODE models, with states in which concentrations of some metabolites are able to grow indefinitely in time. These dFBA states do not precisely match those found by the ODE model, and states that cannot be observed in the ODE model are also found, suggesting that other ODE models for the same network could exhibit different long timescale behaviours. The examples considered clarify the strengths and weaknesses of each approach and the nature of insight into metabolic behaviour each provides.

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There are now engulfing theorems for topological, piecewise linear, and differentiable manifolds. Differentiable engulfing so far was reduced to piecewise linear engulfing using the J. H. C. Whitehead triangulation of a differentiable manifold and J. R. Munkres' theory of obstructions to the smoothing of piecewise-differentiable homeomorphisms. In the first part of the thesis we observe that the method of proof of M. H. A. Newman's topological engulfing theorem applies, up to a local lemma, simultaneously to all three categories of manifolds. We prove this local lemma in the differentiable case and thus obtain a differentiable engulfing theorem which has a direct proof. Then we solve the problem of the existence of a stretching diffeomorphism between complementary subcomplexes of a simplicial complex in Euclidean space which is crucial for all applications of engulfing. Next we prove a theorem concerning the uniqueness of open differentiable cylinders which is the differentiable analogue of the uniqueness theorem for open cones. A consequence of this theorem is that if M₁ and M₂ are compact differentiable manifolds with diffeomorphic interiors then M₁x R and M₂xR are diffeomorphic, where (R denotes the real line. Another consequence is that if a differentiable manifold is the monotone union of open differentiable cells it is diffeomorphic to Euclidean space. We present several applications of differentiable engulfing which actually hold in all three categories of manifolds. Our methods are such that they apply also to noncompact manifolds. Theorem: [formulae omitted] This theorem has several corollaries. For instance, if M is a k-connected differentiable manifold of dimension n without boundary, k ≤ n - 3 if k>0, and if [formula omitted] then M may be covered by m open differentiable n-cells. Using this result, we give a new and direct proof of the uniqueness of the differentiable structure of Euclidean n-space for n ≥ 5. Finally, we prove a general h-cobordism theorem. Theorem: Let M be a connected differentiable manifold of dimension n, n ≥ 5, with two connected boundary components N₁, and N₂ such that the inclusion of N₁ into M is a homotopy equivalence, i = 1,2. Then there is a diffeomorphism of N₁x(0,oo) onto M - N₂. / Science, Faculty of / Mathematics, Department of / Graduate

Theorem : Let X be simply connected . H[sub q](X) be finitely generated for each q. π[sub q](X) be finite for each q < n. n>> 1. Then , H[sub q] , π[sub q](X) --> H[sub q](X) has finite kernel for q < 2n has finite cokernel for q < 2n+l ker h[sub 2n+1] Q = ker u where , u is the cup product or the square free cup product on R[sup n+1](x) depending on whether n+1 is even or odd , respectively . ( R[sup N+1](X) is a quotient group of H[sub Q][sup n+1](x) to be defined in this thesis ) / Science, Faculty of / Mathematics, Department of / Graduate

The syntopogenous structures were introduced by Á. Császár. These are generalisations of classical continuity structures such as topologies, proximities and uniformities. In his book, Foundations of General Topology (1963) (Preceded by a French (1960) and a German (1963) edition), Császár treated many properties of syntopgenous structures. Among these properties were completeness and compactness, but not realcompactness. Our purpose was to extend the definition of realcompactness from uniformisable topologies to arbitrary syntopogenous structures and to produce a real compact reflection for arbitrary syntopogenous structures. We did not fully accomplish this purpose. We have, in fact, first defined a notion of quasirealcompactness for arbitrary syntopogenous structures. For uniformisable Hausdorff topologies, realcompactness implies quasirealcompactness; we could not prove or disprove the converse implication. Nevertheless, we were able to give a characterisation of realcompactness for a uniformisable Hausdorff topology in terms of quasirealcompactness of a certain induced proximity; moreover, we produced a double quasirealcompact reflection in the category of separated syntopogenous structures, and from this retrieved the classical Hewitt realcompact reflection.

In this thesis connections between categories of interior algebras and categories of topological spaces, and generalizations of topological concepts to interior algebras, are investigated. The following are some of the most significant results we obtain: The establishment of a duality between topological spaces and complete atomic interior algebras formalized in terms of a category-theoretic co-equivalence between the category of topological spaces and continuous maps and the category of complete atomic interior algebras and maps known as complete topomorphisms (Theorem 2.1.7). Under this co-equivalence, continuous open maps correspond to complete homomorphisms (Theorem 2.1.8). We also establish a duality between arbitrary interior algebras and structures known as Stone fields in terms of a co-equivalence between the category of interior algebras and topomorphisms (see Definition 1.1.8) and the category of Stone fields and their morphisms the field maps (Theorem 2.2.14). Under this co-equivalence weakly open field maps (see Definition 2.2.17) correspond to homomorphisms (Theorem 2.2.18). The well-known connection between pre-ordered sets and interior algebras is shown to be a special case of topological duality (see section 4 of chapter 2). The topological concepts of neighbourhoods, convergence and accumulation are generalized to interior algebras (Chapter 3), and are used to generalize the topological separation and compactness properties to interior algebras (Chapter 4 and Chapter 5). What is particularly interesting with regard to the separation properties is that most of them are first order properties of interior algebras (see Theorem 4.5.11). This should be contrasted with the situation for frames/locales [12] and topological model theory [10]. By generalizing the concept of α-separation to interior algebras we obtain an ω chain of strictly elementary classes of interior algebras all of which have hereditarily undecidable first order theories (Theorem 4.3.14). Characterizations of irreducibility properties for interior algebras are also found. These properties (subdirect irreducibility, finite subdirect irreducibility, direct indecomposability, simplicity and semi-simplicity) can be characterized in many different ways. Characterizations in terms of open elements (fixed points of the interior operator) are found (Theorem 1.3.18 and Theorem 1.3.21) and these are used to obtain further characterizations. In particular a characterization in terms of topological properties of Stone spaces of interior algebras is obtained (Theorem 2.3.9). We also find characterizations of the irreducibility properties in the power set interior algebras of topological spaces (Theorem 2.1.15) and in interior algebras obtained from pre-ordered sets (Theorem 2.4.16). What is particularly striking is that the irreducibility properties correspond to very natural topological properties. (Other results characterizing or related to the irreducibility properties are 2.4.11, 2.4.17, 5.1.13, and 5.1.15). Bibliography: pages 134-135.

This thesis consists of two chapters, of which the first presents a categorial study of the concept of initiality (also known as projective generation) and the second gives applications in the theory of uniform and quasi-uniform spaces.