acase@tulane.edu

acase@tulane.edu

In this thesis we undertake the study of the categories CCP(,o)-G(, )and CP(,o)-G, of chain-complete posets and continuous posets,(, )respectively, with Scott-continuous Galois-connections as their morphisms in both cases, inspired by the work of Dana S. Scott (see {Sc-72}) In Chapter 0 we establish definitions, theorems and terminology needed for the rest of the thesis. We initiate, in Section 1 of Chapter I, a systematic study of chain-complete posets and continuous posets having Scott-continuous Galois-connections as their morphisms. A main structural result is Theorem 1.17. In Section 2 of Chapter I, we study the function spaces of these objects, obtaining partial results in the case of continuous posets. In Section 3, we establish the existence of inverse limits of chain-complete posets, and we lay the mathematical foundation of denotational semantics of programming languages using chain-complete posets as ground objects. In Section 4 we characterize profinite posets, which arise in the semantics of parallel programming In Chapter II we develop several dualities for the categories introduced in Chapter I, using the Lawson duality between continuous posets and completely distributive lattices In Chapter III, we establish the existence of fixed point functors^in the category CCP(,o)-G and give two examples of these functors.(, )^We use these examples to initiate the study of topological combinatory algebras. We define the category TCA, of topological combinatory algebras, and proceed to study this category establishing several typical closedness properties. The category of continuous lattices provided the first examples topological combinatory algebras, using the existence in this category of topological spaces which are isomorphic to their own space of self-functions. Topological combinatory algebra D with D (TURNEQ) {D (--->) D}, are such that every self-function f can be obtained by application, i.e., there is x(,f) (ELEM) D such that f(y) = x(,f)(y) for every y (ELEM) D. For any poset X and topological combinatory algebra D, we form the topological combinatory algebra {X (--->) D} of Scott-continuous functions between X and D and we show that there are self-functions in these algebras which cannot be obtained by application / acase@tulane.edu

This dissertation presents a numerical method to study the two dimensional incompressible thermal boundary layer flows. The governing equations are the Prandtl's thermal boundary layer equations. Analytically, these equations are difficult to solve. Numerically, standard methods such as finite difference method may suffer from problems associated with the use of grids and the discretization of gradients, especially within an extremely thin boundary layer where large velocity and temperature gradient exist The proposed numerical method is a particle method in which the computational elements carry concentrations of vorticity and heat flux. These elements are convected at the fluid velocity and undergo a random walk to simulate the diffusion. New elements are created to simulate the interaction between the vorticity and the heat flux and to satisfy the boundary conditions. At any instant of time, the velocity and temperature field can be recovered from the location and the concentration of the elements by integration The method is grid-free and it does not suffer from problems associated with the use of a very fine grid in the boundary layer where sharp gradients exist. It introduces no numerical diffusion and automatically concentrates computational effort near steep gradients The method is applied to a forced parallel flow past a heated horizontal flat plate and a natural flow around a heated vertical plate. The numerical results are compared with the steady-state similarity solutions and good agreement is observed / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

This dissertation is an attempt to understand Aristotle's De Anima as a unified whole--a unity, I argue, which is only as problematic as is the unity of the soul of which it speaks. By focusing on Aristotle's account of the faculties of sight and touch, and the tension between these two powers by which the activity of knowledge has been metaphorically understood throughout much of the history of philosophy, I believe I have come close to capturing the essence of what Aristotle means by entelecheia. The problem of soul understood as an entelecheia amounts to the problem, as it has been expressed by Nietzsche, of 'How One Becomes What One Is.' For Aristotle, I try to show, becoming what one is already, as a potentiality, is an activity of becoming not a knower but rather a student, or an activity that involves learning how to learn / acase@tulane.edu

acase@tulane.edu

After transferring the concept of a basic subgroup from abelian group theory to torsion-free modules over valuation domains, we investigate three classes of indecomposable torsion-free modules over almost maximal valuation domains. The most attractive class is the one of purely indecomposable modules, i.e., modules all of whose pure submodules are indecomposable. We obtain a complete and independent set of invariants for this class. If n is the basic rank of some torsion-free module, the reduced part of its n-th exterior power is purely indecomposable. We give a necessary and sufficient criterion involving this exterior power for the module in question to be totally indecomposable. A module is totally indecomposable if every pure submodule is either completely decomposable or indecomposable. Co-purely indecomposable modules are also discussed. The basic rank of these modules is one less than their finite rank. We investigate the endomorphism rings of the members of all three classes. In particular, a counter-example to a theorem of D. M. Arnold {2} is given Finally we turn our attention to the question when the module R('R) is separable. Under additional assumptions we can show that this is the case if and only if the valuation domain R is either discrete of maximal. In particular, this answers a question raised by B. Zimmermann in {19} / acase@tulane.edu