Applied General Topology - Vol 08, No 1 (2007)
Permanent URI for this collection
Tabla de contenidos
- Quasicontinuous functions, domains, and extended calculus
- Computational differential topology
- Boundaries in digital spaces
Browse
Recent Submissions
Now showing 1 - 3 of 3
- PublicationBoundaries in digital spaces(Universitat Politècnica de València, 2007-04-01) Herman, Gabor T.; National Institutes of Health, EEUU; National Science Foundation, EEUU[EN] Intuitively, a boundary in an N-dimensional digital space is a connected component of the (N − 1)-dimensional surface of a connected object. In this paper we make these concepts precise, and show that the boundaries so specified have properties that are intuitively desirable. We provide some efficient algorithms for tracking such boundaries. We illustrate that the algorithms can be used, in particular, for computer graphic display of internal structures (such as the skull and the spine) in the human body based on the output of medical imaging devices (such as CT scanners). In the process some interesting mathematical results are proven regarding “digital Jordan boundaries,” such as a specification of a local condition that guarantees the global condition of “Jordanness.”
- PublicationComputational differential topology(Universitat Politècnica de València, 2007-04-01) Blackmore, Denis; Mileyko, Yuriy; National Science Foundation, EEUU[EN] Some of the more differential aspects of the nascent field of computational topology are introduced and treated in considerable depth. Relevant categories based upon stratified geometric objects are proposed, and fundamental problems are identified and discussed in the context of both differential topology and computer science. New results on the triangulation of objects in the computational differential categories are proven, and evaluated from the perspective of effective computability (algorithmic solvability). In addition, the elements of innovative, effectively computable approaches for analyzing and obtaining computer generated representations of geometric objects based upon singularity/stratification theory and obstruction theory are formulated. New methods for characterizing complicated intersection sets are proven using differential analysis and homology theory. Also included are brief descriptions of several implementation aspects of some of the approaches described, as well as applications of the results in such areas as virtual sculpting, virtual surgery, modeling of heterogeneous biomaterials, and high speed visualizations.
- PublicationQuasicontinuous functions, domains, and extended calculus(Universitat Politècnica de València, 2007-04-01) Cazacu, Rodica; Lawson, Jimmie D.[EN] One of the aims of domain theory is the construction of an embedding of a given structure or data type as the maximal or “ideal” elements of an enveloping domain of “approximations,” sometimes called a domain environment. Typically the goal is to provide a computational model or framework for recursive and algorithmic reasoning about the original structure. In this paper we consider the function space of (natural equivalence classes of) quasicontinuous functions from a locally compact space X into L, an n-fold product of the extended reals [−1,1] (more generally, into a bicontinuous lattice). We show that the domain of all “approximate maps” that assign to each point of X an order interval of L is a domain environment for the quasicontinuous function space. We rely upon the theory of domain environments to introduce an interesting and useful function space topology on the quasicontinuous function space. We then apply this machinery to define an extended differential calculus in the quasicontinuous function space, and draw connections with viscosity solutions of Hamiltonian equations. The theory depends heavily on topological properties of quasicontinuous functions that have been recently uncovered that involve dense sets of points of continuity and sections of closed relations and USCO maps. These and other basic results about quasicontinuous functions are surveyed and presented in the early sections.