• A fully two-dimensional flux-corrected transport algorithm for hyperbolic partial differential equations

      Huang, Sen-Wei; Gislason, Gary (1989)
      Numerical solutions of the hyperbolic partial differential equation, $\partial p\over\partial t$ + $\vec\nabla \cdot (p\vec u)$ = 0, will generally encounter the difficulties of large diffusion and oscillations near steep gradients or discontinuities. The method of Flux-Corrected Transport (FCT) developed by Boris and Book has conquered these difficulties for the one-dimensional case. Motivated by this one-dimensional FCT algorithm, a fully two-dimensional FCT algorithm is developed in this present work. This fully two-dimensional FCT algorithm is a two-step procedure: (1) the transport scheme, and (2) the antidiffusion scheme. The second step of the procedure could also be replaced by an application of the one-dimensional antidiffusion algorithm in the x direction and the y direction separately. The stability, phase shift errors and positivity for the fully two-dimensional transport scheme are analyzed. Test results are presented. The possibility of the extension of the FCT method to three dimensions are discussed.
    • Homology and cohomology of diagrams of topological spaces

      Sarnowski, Krzysztof; Piacenza, Robert J.; Cooper, Charles L.; Coutant, Basil W.; Morack, John L.; Santoni, Larry; Tape, Walter R. (1994)
      Homology and cohomology of objects other than ordinary topological spaces have been investigated by several authors. Let X be a G-space and ${\cal F}$ be a family of all closed subgroups of G. The equivariant cellular structures are obtained by attaching n-cells of the form G/H $\times$ B$\sp{\rm n}$, where B$\sp{\rm n}$ is the unit n-ball and H $\in\ {\cal F}$. A construction of the equivariant singular homology and cohomology for X and ${\cal F}$ was given by Soren Illman. In this thesis ${\cal F}$ is replaced by a small topological category and the G-space X is replaced by a functor taking values in k-spaces and called a diagram of topological spaces. The cellular structures for diagrams are obtained by attaching cells D$\sb{j}\times\rm B\sp{n}$ where D$\sb{j}$ is a representable functor. The purpose of this thesis is to investigate homology and cohomology for diagrams. We review the foundation of homotopy theory and cellular theory for diagrams. We propose axioms for homology and cohomology modeled on Eilenberg-Steenrod system of axioms and prove that they, as in the classical case, determine uniquely homology and cohomology for finite cellular diagrams. We give the generalization of Illman's equivariant singular homology and cohomology to diagrams of topological spaces and we prove that this generalization satisfies all introduced axioms. Also we prove the comparison theorem between the sheaf cohomology for diagrams developed by Robert J. Piacenza and the singular cohomology for diagrams developed in this thesis.
    • The algebra of Green and Mackey functors

      Luca, Florian; Piacenza, Robert (1996)
      We investigate various Green and Mackey functor analogs of concepts from the theory of rings and modules. In particular, we consider ideals, chain conditions, Krull dimension, decomposition theorems and completion for these functors. We characterize the Jacobson radical and the prime and maximal ideals of an arbitrary Green functor A. We prove various properties of these ideals. We also investigate the Krull dimension of a commutative Green functor. We analyze the Green and Mackey functors satisfying various chain conditions. For left-modules over Green functors A satisfying a certain noetherian-like condition we study the analog of the tertiary decomposition theorem. For the case when A is commutative we study the analog of the primary decomposition theorem. We also give induction theorems for various special types of Green and Mackey functors such as, prime and simple Green functors A, simple left-A-modules, cotertiary and coprimary left-A-modules. We end with an induction theory for the completion of a Green functor in a left ideal. This work generalizes most of the major topics from classical algebra to the category of Green and Mackey functors.
    • Transfer and Steenrod squares

      Litvanyi, Peter Ivo; Piacenza, Robert (1993)
      Commutators between the transfer and Steenrod squares have been investigated by several authors. Let X be a finite simplicial complex and $\tau$ be a regular involution on X. If $\tau$ has no fixed point, then the commutator is trivial by certain results in generalized cohomology theory. For involutions with possible fixed points, the commutator was first expressed by Bott as $\Delta\sp*$Sq$\sp{\rm i}$ + Sq$\sp{\rm i}\Delta\sp* = \mu$Sq$\sp{\rm i-1}\Delta\sp*.$ Here $\Delta\sp*$ is the transfer map and $\mu$ denotes the connecting morphism of the Smith sequence. Another formula, closely related to the one above, was given by Kubelka and gives the commutator in terms of the cohomology class restricted to the fixed point set and certain characteristic classes arising from the double cover of the complement to the fixed point set. In this thesis, I prove the generalization of the formulas above for sheaf cohomology. As one of the consequences, due to the powerful nature of sheaf theory we gain the results without serious restrictions on the space: X is required to be paracompact, Hausdorff. In Chapter 1, I review the standard sheaf-theoretical constructions for both the transfer and the Steenrod powers based on Bredon's results. I state and prove a few technical lemmas on Smith sequences that are necessary in my setting. In Chapter 2, I state and prove the analogue of Bott's formula for paracompact Hausdorff spaces. In Chapter 3, we derive a generalization of Kubelka's formula for spaces as above.