In this thesis we investigate a generalization of purity for modules which has applications to the study of direct sums of cyclic modules over valuation domains. Kaplansky {Trans. Amer. Math. Soc. 72 (1952), page 332, footnote 4} suggested the following as a possible generalization for the purity of a submodule A of an R-module B for a general ring R : for any b + A (ELEM) B/A there exists an a (ELEM) A such that Ann(b + A) = Ann(b - a). We call a submodule with this property cyclically-pure in B. The notion is stronger than RD-purity (relative-divisibility), but coincides with RD-purity and purity in the sense of Cohn for Dedekind domains We investigate in the thesis some general properties of cyclic-purity as well as some of the homological aspects associated with it. In particular, we look at the relationship between cyclic-purity and purity, showing that Prufer and noetherian domains can be characterized in terms of this relationship. We also consider the cyclically-pure-projective modules, which are just the summands of direct sums of cyclics (direct sums of cyclics over local domains), and a cyclically-pure-projective dimension of a module, which measures, in a sense, how far a module is from being a direct sum of cyclics. We show that some basic results from the theory of projective dimensions carry over, with appropriate modifications, to cyclically-pure-projective dimensions, and by restricting to valuation domains, we obtain results connecting the cyclically-pure-projective dimension of a module with the cardinality of a set of generators of minimal cardinality. The results are then applied in the investigation of submodules of direct sums of cyclics over valuation domains / acase@tulane.edu

acase@tulane.edu

The primary task of my dissertation is to develop non-circular necessary and sufficient conditions for personal identity. Personal identity refers to cross-temporal numerical identity. Before constructing criteria of the identity of a person, I first defend my method of using thought experiments. Then I examine various such attempts to determine the necessary and sufficient conditions for personal identity by reductionist philosophers from John Locke to Derek Parfit and Sydney Shoemaker. An analysis of each of their theories yields important flaws, but consequently leads me to develop my own. I maintain that A and B are cross-temporally identical persons if and only if the following conditions are met: (1) There is either psychological continuity between A and B, or else, if there is a break in continuity, B, the later person, must have over fifty percent psychological connectedness to A; (2) There must be no person other than B who also has either psychological continuity or over fifty percent connectedness to A; (3) The cause, as long as it is not copying, of either the continuity or the connectedness is irrelevant; and (4) Continuity and connectedness include not only memories, but personality characteristics, etc Before any theory of personal identity is complete, it is essential that it explain how we can identify a person as a single person. We cannot understand how the persons at two temporal stages of a life are identical unless we can understand how the mental components of a person at a single time hold together in one person. Thus, I address what should be called 'the unity of a person's mental life'. I argue that this unity exists because of a functional integration of stimuli within the person. This integration comes about due to a single interpreter within the person, rather like a central processing unit in a computer Finally I argue that Parfit's belief that persons are not what truly matters is not well founded, and his subsequent contention that his theory renders utilitarianism more plausible is consequently unsubstantiated / acase@tulane.edu

Since it is one of Descartes' basic beliefs that any attempt to explain natural phenomena, as well as to discover natural laws, must be grounded in metaphysics, scholars have interpreted Descartes as if he intended logically to deduce physics, an empirical science, from metaphysics. Consequently, they maintain that Descartes' study of nature results in an a priori system, which is independent of experience and observation. It is the main thesis of this dissertation to reject that interpretation of Descartes' philosophy of nature. It is true that Descartes uses the term 'deduction' to describe the relation between physics and metaphysics. However, it will be argued here that Descartes does not interpret this term in the strict sense of logical deduction. He uses it, rather, in a looser sense to describe the foundational role of metaphysics. This role is to provide a general conceptual context within which principles of physics first arise as hypotheses. But these hypotheses can be confirmed only in the light of observations and well-constructed experiments. Thus, Descartes' conception of the foundational role of metaphysics does not commit him to treating physics as an a priori system / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

Global existence and uniqueness results are established for mixed initial-boundary value problems for degenerate nonlinear parabolic equations of the type ${\partial {\rm u}}\over{\partial {\rm t}}$ = $\phi$(x,$\nabla$u)$\Delta$u + f(x,u,$\nabla$u) where u = u(x,t) is real-valued, x is in a smooth bounded domain $\Omega$ in $\IR\sp{\rm n}$, and t $\geq$ 0. By 'degenerate' we mean that $\phi$(x,$\xi)$ $>$ 0 for x $\epsilon\Omega$ and $\xi\ \epsilon\ \IR\sp{\rm n}$ but possibly $\phi$(x,$\xi)$ = 0 for x $\epsilon\ \partial\ \Omega$. We also consider a variety of boundary conditions, which can be either linear (e.g. Dirichlet, Neumann, Robin or periodic) or nonlinear, in which case it takes the form $-{\partial{\rm u}\over\partial{\rm n}}$ $\epsilon$ $\beta$(u(x,t)) for x $\epsilon\ \partial\Omega$, t $\geq$ 0. Here $\nu$ is the unit outer normal to $\partial\Omega$ at x and $\beta$ is a maximal monotone graph in $\IR$ x $\IR$ containing (0,0). In particular, both the equation and the boundary condition can be nonlinear / acase@tulane.edu

acase@tulane.edu

Central to the study of modules is the extent to which decompositions into indecomposable summands are unique. Our investigations into these matters are motivated by the Krull-Schmidt theorem and its connection with local endomorphism rings. Results have been found in the following three settings (1) Finitely generated modules. The main result states that all indecomposable finitely generated modules over break Henselian local rings have local endomorphism rings. This generalizes a result of Swan-Evans from Noetherian to arbitrary Henselian local rings (2) Finite rank torsion free modules. Extending a result of Lady from the very special discrete case, it is shown that finite rank torsion free indecomposable modules over Henselian valuation domains have local endomorphism rings. The discrete valuation domains are precisely the Noetherian valuation domains, thus an extension similar to that found in the finitely generated case is obtained. The proof utilizes a recent result of Dubrovin. A converse is also proved, giving a characterization of Henselian valuation domains. A more satisfying though conditional characterization is realized via a partial extension of Corner's theorem on endomorphism rings of torsion free abelian groups to modules over rank 1 valuation domains of characteristic $\ne$2 (3) Quasi-decompositions and quasi-endomorphism rings. Failing to show that a given indecomposable module has a local endomorphism ring, we study the weaker notion of quasi-decompositions. Let E denote the quasi-endomorphism ring of a finite rank torsion free module M over a commutative domain R. We prove that E is local if and only if M is strongly indecomposable. This gives a result similar to Azumaya's theorem for quasi-decompositions of finite rank torsion free modules, generalizing the result of Reid on abelian groups / acase@tulane.edu