The steady-state positive solution of a predator-prey interacting system has been widely investigated in recent years. Since it had already been shown that the only possible positive solution must be constant under the Neumann homogeneous boundary condition, researchers naturally studied the physically important Dirichlet boundary condition problem. For years researchers have worked to obtain sufficient conditions for the existence of positive solutions of systems with quadratic nonlinearities. In this dissertation, we give necessary and sufficient conditions for more general predator-prey systems, under Dirichlet boundary conditions. Some uniqueness results are obtained in Sections 7 and 8. We also discuss the behavior of positive solutions on large regions / acase@tulane.edu

This dissertation argues that Collingwood, although a relativist, is neither sceptical nor irrational in his theory of knowledge, and that his metaphysical commitment is to a reality that is intelligible and interconnected in all of its parts. I present the problem of relations in order to show that Collingwood rejects identification with traditional positions. The concept 'relation' has been divided into two classes: internal and external. Some philosophers have denied that an internal relation can have any degree of externality, and others that an external relation can have any degree of internality. I show that for Collingwood the term 'relation' resists such classification. Relations, as all other philosophical concepts denote overlapping classes. This overlap is not random and haphazard, but exhibits a definite structure, which Collingwood describes as a scale of forms. This has important epistemological consequences I illustrate Collingwood's theory of relation by means of an analysis of historical knowledge, and examine his case against realism. I review Collingwood's theory of Question-and-answer and his theory of absolute presupposition and show that they are not only a reaction to the realists' doctrine of external relations but that they represent a position also removed from traditional idealism I present Toulmin's criticism of the theory of absolute presuppositions and show that Collingwood's theory does not lead to an irrational relativism. Collingwood's theory of the overlap of classes provides him with the possibility of rationally comparing and evaluating different constellations of absolute presuppositions. This interpretation is supported with reference to his own examples of metaphysical analysis. According to Collingwood, knowledge is not simply a matter of assertion, but also involves absolute presuppositions as essential elements of the rational process. Knowing is creative as well as cognitive, and knowledge is acquired through a process of faith seeking understanding / acase@tulane.edu

In this essay I construct a theory of epistemic justification. Specifially, I utilize and modify techniques adopted from cognitive decision theory to develop an account of the conditions under which an agent is justified in believing some proposition. Thus, my principal interest is in an internalist analysis of justified belief. That is, the conditions of justified belief are a function of the agent's subjective conception of his cognitive situation and his assessments of the relationships between the various parameters of justification. The proposed theory focuses on two aspects of justification, the relationship between evidence and belief, and the relationship between what an agent takes to be his evidence for some belief and his other beliefs. These two aspects I refer to as the methodological and cognitive components, respectively Chapter I considers and rejects some alternative internalist analyses of justification. I suggest these analyses are inadequate principally because they do not provide adequate specification of the methodological components Chapter II considers Levi's account of inductive acceptance. While I take Levi's analysis to be a significant advance, I reject it since the methodological component is too idealized and the cognitive component is insufficiently specified In Chapter III I present my analysis of the methodological component. I develop a notion of epistemic probability and show how the methodological component may be analyzed utilizing only comparative probabilities. A significant feature of this approach is that it reflects the limitations of ordinary epistemic agents Chapter IV presents my analysis of the cognitive component. Utilizing a notion of a reliable characterization of the evidence, which specifies the relationship between what an agent takes to be his evidence for some belief and his other beliefs, and the results of Chapter III, I then present an analysis of justified belief In Chapter V I first reject externalism as an adequate approach to justification. After arguing that an adequate analysis of knowledge must be externalist, I claim that the conditions of epistemic rationality are to be sought in a theory of epistemic justification and not in an analysis of knowledge / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

We study quasi-projective modules over integral domains. A module M is called quasi-projective if it has the projective property relative to all exact sequences of the form 0 → N → M → M/N → 0, where N is a submodule of M. Quasi-projective modules have been introduced by Miyashita as a generalization of projective modules The main goals of our research are to generalize the results on quasi-projective modules over valuation domains to arbitrary integral domains and to study special types of quasi-projective modules, e.g. uniserial modules The dissertation consists of two parts The first part is concerned with quasi-projective modules over general domains. The main results of the first part are the following. (1) The so-called 1½-generated ideals are quasi-projective, moreover, projective. (2) The quotient field Q of an integral domain R is a quasi-projective R-module if and only if every proper submodule of Q is complete in its R -topology. (3) Integral domains all of whose ideals are quasi-projective are exactly the almost maximal Prufer domains The second part of the dissertation is primarily devoted to quasi-projective uniserial modules over valuation domains. The main results of the second part are the following. (1) Uniserial module U is quasi-projective if and only if it is weakly quasi-projective and an additional technical requirement is satisfied. (2) For torsion free modules of rank 1, quasi-projectivity is equivalent to the weak quasi-projectivity, and the latter is determined by completeness of certain endomorphism rings in their ring topologies. (3) The archimedean ideals of a valuation domain R with non-principal maximal ideal P are quasi-projective if and only if R/K is complete in the R/ K-topology for each archimedean ideal K, not isomorphic to P In conclusion we investigate the influence of quasi-projectivity on the decomposability of modules over valuation domains as well as on the properties of direct sums of such modules. We show that a torsion-free quasi-projective module M over a valuation domain which has a dense basic submodule is completely decomposable and that direct sums of ℵ -generated uniserial modules of cardinality less than ℵ are quasi-projective / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

This dissertation presents a computational model for the solution of the equations for the turbulent combustion of a pre-mixed gas. The method presented is a hybrid random choice method With the assumption of incompressibility, the Navier-Stokes equations governing the fluid motion may be uncoupled from the equations governing the chemistry. The solution is modelled using a modification of the random vortex method The reaction-diffusion equations governing time and spatial dependence of temperature and species mass-fractions are solved numerically using the random choice method with allowance for the reaction-diffusion terms. This technique is tested on two problems with analytic solutions. The combustion model is applied first to a mixing layer flow and then to a flow past a back-facing step. The method demonstrates an ability to resolve turbulent effects on flame front propagation / acase@tulane.edu