Computing the bounded subcomplex of an unbounded polyhedron

We study efficient combinatorial algorithms to produce the Hasse diagram of the poset of bounded faces of an unbounded polyhedron, given vertex-facet incidences. We also discuss the special case of simple polyhedra and present computational results.Comment: 16 page

Real time simulation using position sensing

An interactive exercise system including exercise equipment having a resistance system, a speed sensor, a controller that varies the resistance setting of the exercise equipment, and a playback device for playing pre-recorded video and audio. The controller, operating in conjunction with speed information from the speed sensor and terrain information from media table files, dynamically varies the resistance setting of the exercise equipment in order to simulate varying degrees of difficulty while the playback device concurrently plays back the video and audio to create the simulation that the user is exercising in a natural setting such as a real-world exercise course

Comparing Cartesian-closed Categories of (Core) Compactly Generated Spaces

It is well known that, although the category of topological spaces is not cartesian closed, it possesses many cartesian closed full subcategories, e.g.: (i) compactly generated Hausdorff spaces; (ii) quotients of locally compact Hausdorff spaces, which form a larger category; (iii) quotients of locally compact spaces without separation axiom, which form an even larger one; (iv) quotients of core compact spaces, which is at least as large as the previous; (v) sequential spaces, which are strictly included in (ii); and (vi) quotients of countably based spaces, which are strictly included in the category (v). We give a simple and uniform proof of cartesian closedness for many categories of topological spaces, including (ii)–(v), and implicitly (i), and we also give a self-contained proof that (vi) is cartesian closed. Our main aim, however, is to compare the categories (i)–(vi), and others like them. When restricted to Hausdorff spaces, (ii)–(iv) collapse to (i), and most non-Hausdorff spaces of interest, such as those which occur in domain theory, are already in (ii). Regarding the cartesian closed structure, finite products coincide in (i)–(vi). Function spaces are characterized as coreflections of both the Isbell and natural topologies. In general, the function spaces differ between the categories, but those of (vi) coincide with those in any of the larger categories (ii)–(v). Finally, the topologies of the spaces in the categories (i)–(iv) are analysed in terms of Lawson duality