In this dissertation, we construct homogeneous continua by using a fibered product of a homogeneous continuum X with itself. The space X must have a continuous decomposition into continua, and it must possess a certain type of homogeneity property with respect to this decomposition It is known that the points of any one-dimensional, homogeneous continuum can be 'blown up' into pseudo-arcs to form a new continuum with a continuous decomposition into pseudo-arcs. We will show that these continua can be used in the above construction Finally, we will show that the continuum constructed by using the pseudo-arcs, the circle of pseudo-arcs, or the solenoid of pseudo-arcs is not homeomorphic to any known homogeneous continuum / acase@tulane.edu

We derive a priori and a posteriori estimates for the error of the bi-linear interpolation polynomial for finite difference approximations of the solutions of parabolic systems on grids with irregular nodes. The estimates are developed for the $L\sp2$ norm, the $H\sp1$ semi-norm, and the $H\sp1$ norm of the error. We use the a posteriori error estimates of the interpolation polynomial to determine the 'high error' regions which require a finer mesh for computation. We derive and implement consistent computational stencils for the spatial derivatives at the nodes on the interface of regions of different levels of refinement. We use local error estimation and global computation / acase@tulane.edu

Recent work on classifying compact four-manifolds with positive or nonnegative sectional curvature has included the study of such manifolds that admit circle actions or $\doubz\sb{p}$-actions. This dissertation continues the study of compact four-manifolds that admit finite group actions. Specifically, we consider positively pinched manifolds that admit a $\doubz\sb{p}$-action and nonnegatively curved manifolds that admit a $\doubz\sb{p}\times\doubz\sb{p}$ action / acase@tulane.edu

The primary task of my dissertation is to analyze Heidegger's thought of 'community' from his early concept of 'being-in-the-world-with-one-another' to his later concept of Ereignis (occurrence of appropriation), with an aim to show how Heidegger's thought of community constitutes one of the most important aspects of his inquiry into being First of all, I use Heidegger's criticism of both Kant's attempt of proving the existence of the external world and Husserl's transcendental intentionality to clarify how Heidegger's concept of Dasein as 'being-in-the-world-with-one-another' distinguishes itself from those philosophies of subjectivity in the modern history and thus starts a new, post-subjective way of thinking. In defense of my interpretation I argue with some American pragmatists who misinterpreted Heidegger's Dasein as a 'practical agent' or 'subject of action,' and with Jurgen Habermas who claimed that there is an essential inconsistency between Heidegger's anti-subjective 'world-analysis' and his subjective 'who-analysis.' Secondly, I try to show that Heidegger' s thought of 'community' culminates in his later concept of Ereignis, which is based on his understanding of the essence of truth as untruth, i.e., Geheimnis (mysterious home), and is interpreted as both 'belonging-together' (Zusammengehoren) and 'setting-apart-from-one-another' (Aus-einander-setzung). Compared with the concept of 'being-in-the-world-with-one-another,' which is still Dasein oriented, the concept of Ereignis is being oriented. Because of this, I argue that Heidegger's Ereignis as community is not only a 'human community' but also a 'cosmo-logos community,' which is a humanistic but anti-anthropocentric fourfold (Geviert) Finally, I present some critical discussions of Heidegger's ontological holism, of the 'holy' character of Heidegger' s cosmo-logos community, and a brief comparison between Heidegger's and the traditional Confucian idea of the relation between individual human beings, nature and community. By doing these, I attempt to show both the limitations of and some future possibilities for enrichment of Heidegger's way of thinking of community / acase@tulane.edu

acase@tulane.edu

acase@tulane.edu

It is generally known that the set of finite subsets F(X) of a set X forms a lattice under the operations of union and intersection, and that F(X) with union as the principal operation is the free semilattice on the set X. Michael, in {43} topologized (GAMMA)(X), the space of closed subsets of a topological space X, with an intrinsic topology arising from the topology on X itself and obtained results of a topological nature about (GAMMA)(X) and F(X). We denote this topology, called the finite topology by Michael and the exponential topology by Kuratowski {35}, by v in honor of L. Vietoris {57} Lawson in {37} studied topological semilattices which have a basis of subsemilattice neighborhoods at each point. We call these Lawson semilattices. We show in Chapter I that (F(X), v) is the free Lawson semilattice on the space X. We give F(X) a second, finer topology, denoted by t, which makes F(X) into the free topological semilattice on X. From the relationship between these topologies we develop the study of the topology t. These results parallel some of those of B. V. S. Thomas in her study {55} of free topological groups and exemplify the statement by O. Wyler {60} that any free algebra functor lifts to a free topological algebra functor We introduce a third topology k on F(X) which refines t and make F(X) into the free k-semilattice on the k-space X. The relation v (L-HOOK) t (L-HOOK) k leads to results about the free k-semilattice which are similar to the findings of W. F. LaMartin in his study {36} of free k-groups Continuous lattices were defined by Scott in {47} in order to find a model for the Church-Curry (lamda)-calculus in logic and subsequently to lay the foundation for a theory of computation in {48}. His concept was shown to be equivalent to the already existing idea of a compact Lawson semilattice in topological algebra after the work of Hofmann and Stralka {27} In Chapter II we study the closure operators on a continuous lattice and show that the construction of the closure operator space is functorial. Moreover, the closure operators determine the subalgebras of a continuous lattice. Scott's work {47} showed that a function space functors on his category of continuous lattices preserved projective limits. His construction was amplified and elucidated in {19}. We show that the closure operator functor is a subfunctor of Scott's functor and also preserves projective limits. Using this fact, we show that each continuous lattice is the quotient of a continuous lattice which is isomorphic to its space of closure operators Complete Heyting algebras were shown to be important in intuitionistic logic by Skolem in {52}, in sheaf theory by Fourman and Scott in {15}, in topos theory by Freyd in {16}, and in an investigation of the algebraic nature of topology (under the name frames) by Dowker and Strauss in {7}, {8}, and {9}. D. S. Macnab studied the algebraic theory of modal operators on a complete Heyting algebra in {41} and {42} We examine the continuous Heyting algebras in Chapter III and combine the study of continuous lattices with that of complete Heyting algebras. We show that the Scott continuous modal operators on continuous Heyting algebras determine the subalgebras just as the closure operators do for continuous lattices. Furthermore, the construction of the modal operator space is functorial. These results parallel those of Chapter II. One open question which remains is whether the Scott continuous modal operator space on a continuous Heyting algebra is again a continuous Heyting algebra. We offer an affirmative answer in the case that the <<-relation on the algebra is multiplicative / acase@tulane.edu

acase@tulane.edu

Let X be a compact space and Y a closed subset of X. For M(,k), the complex k x k-matrices, consider the C*-algebra of continuous functions f : X (--->) M(,k) with the property that f(x) is a diagonal matrix for all x (ELEM) Y. We shall study the K-theory of this C*-algebra and some closely related C*-algebras for various spaces X and Y. The tools used in this study are a Mayer-Vietoris Sequence and a Puppe Sequence for K-theory of C*-algebras, both of which reduce to the respective sequence in K-theory of locally compact spaces if the involved C*-algebras are commutative First we set up K-theory of unital C*-algebras, following the approach of Karoubi. We define relative K-groups K(,(alpha))((phi)) for unital C*-morphisms (phi) and prove two excision theorems, which will allow us to define K-theory of non-unital C*-algebras. Moreover, we show that the K-functors do not distinguish between homotopic C*-morphisms. This will enable us to define K(,n) of a C*-algebra for all n (ELEM) and to establish a long exact sequence in K-theory associated to a short exact sequence of C*-algebras. We also define a cup product in K-theory of C*-algebras, which will be a (,2)-graded bilinear map K(,*)(A) x K(,*)(B) (--->) K(,*)(A(' )(CRTIMES)(' )B), give some of its basic properties, and use it to define module structures on the K-groups Finally we prove a non-commutative splitting principle which generalizes the well known splitting principle for vector bundles over compact spaces / acase@tulane.edu

Sierra Las Navajas (20&deg;16'N, 98&deg;3 'W), an extinct Plio-Pleistocene volcano located on the northern edge of the eastern Mexican Volcanic Belt (MVB), consists of peralkaline comendite lavas interbedded with pyroclastic flow and fall deposits. The volcanic history of Sierra Las Navajas can be divided into four major flow complexes, each of which is composed of one or more lava flows that tapped the same sector of a stratified magma chamber. One of these units bears green obsidian that has been a major source of obsidian to Mesoamerican societies for more than 3,000 years, and has been the subject of focused archaeological study the implications of which are addressed herein. Peralkaline trachytes of Cerro Gordo to the west and mildly alkaline basalts that emanated from numerous cinder cones along the flanks of Las Navajas were erupted in association with the comendites of Sierra Las Navajas A series of pyroclastic eruptions late in the first stage of volcanism caused slope destabilization, which resulted in sector collapse of the northern flank. This collapse generated a catastrophic debris avalanche estimated at 32 +/- 9 km3 volume. The debris avalanche deposit that resulted was later dissected by downcutting of the Rio Grande Tulancingo to the north and is now well exposed in the walls of the canyon. 40Ar/ 39Ar dates of basalts immediately underlying (2.41 +/- 0.08 Ma) and overlying (1.87 +/- 0.03 Ma) the debris avalanche deposit constrict its age to Late Pliocene. Slightly less peralkaline rhyolite flows that subsequently filled the collapse amphitheater define the second major stage of volcanism Several models for magma petrogenesis were tested, including magma mixing, partial melting, and fractional crystallization. Based on major and trace element geochemical signatures, which partially or completely negate the other possibilities, fractional crystallization with volatile transfer appears to be the most viable model for rhyolite genesis at Sierra Las Navajas. Multiple liquid lines of descent are present, indicating a constantly changing fractionating assemblage. The peralkaline trachytes however are better explained by partial melting of a mildly alkalic basalt source. Peralkaline volcanism is limited to this area in the eastern MVB, where it appears to be related to back-arc extensional tectonic forces / acase@tulane.edu