Davide P. Cervone

Vertex-minimal simplicial immersions of the Klein bottle in three space

Although the Klein bottle cannot be embedded inR3, it can be immersed there, and in more than one way. Smooth examples of these immersions have been studied extensively, but little is known about their simplicial versions. The vertices of a triangulation play a crucial role in understanding immersions, so it is reasonable to ask: How few vertices are required to immerse the Klein bottle inR3? Several examples that use only nine vertices are given in Section 3, and since any triangulation of the Klein bottle must have at least eight vertices, the question becomes: Can the Klein bottle be immersed inR3 using only eight vertices? In this paper, we show that, in fact, eight isnot enough, nine are required. The proof consists of three parts: first exhibiting examples of 9-vertex immersions; second determining all possible 8-vertex triangulations ofK2; and third showing that none of these can be immersed inR3.

Voting with rubber bands, weights, and strings

We introduce some new voting rules based on a spatial version of the median known as the mediancentre, or Fermat-Weber point. Voting rules based on the mean include many that are familiar: the Borda Count, Kemeny rule, approval voting, etc. (see Zwicker (2008a,b)). These mean rules can be implemented by “voting machines” (interactive simulations of physical mechanisms) that use ideal rubber bands to achieve an equilibrium among the competing preferences of the voters. One consequence is that in any such rule, a voter who is further from consensus exerts a stronger tug on the election outcome, because her rubber band is more stretched.

A Virtual Reconstruction of a Virtual Exhibit

The traveling exhibit “Para Alem da Terceira Dimensao”, now touring a number of Portuguese cities, was the result of an international cooperation between two mathematicians in the United States and a team of mathematicians and Computer Professionals in Portugal. Inspired by the virtual art gallery “Surfaces Beyond the Third Dimension”, itself based on a physical exhibit originally staged by the Providence Art Club in 1996, this updated version includes new images, new movies, new languages, and a new layout. We describe here the process, and some of the difficulties, of putting together such a project.

A tight polyhedral immersion of the twisted surface of Euler characteristic −3

Abstract Early studies of tight surfaces showed that almost every surface can be immersed tightly in three-space. For these surfaces we can ask: How many significantly different tight immersions are there? If we consider two such immersions to be the same when they are image homotopic, then we can answer the question by determining the number of homotopy classes where tight immersions are possible. A complete description of all classes of immersions under image homotopy already exists, and tight examples are known in all but three of the classes where tight immersions are possible. In this paper we produce examples in two of those three missing classes, and conjecture that no tight immersion exists in the third.

Employing semantic analysis for enhanced accessibility features in MathJax

Recent changes in the landscape for assistive technology solutions for Mathematics on the web have prompted the development of MathJax into a single rendering and accessibility solution. We present our current efforts that depend on a novel semantic interpretation of Presentation MathML expressions. This allows us to introduce a new notion of responsive equations, on which we build advanced accessibility features with improved reflow of content, selective highlighting, and dynamic speech text generation, as well as an innovative interaction technique based on abstracting and intelligently summarising sub-expressions.

Towards universal rendering in MathJax

Mathematics support on the web is currently very limited mainly due to the lack of browser support for the dedicated markup language MathML. While for visual display JavaScript libraries like MathJax can ensure flawless rendering of formulas across all platforms and browsers, there currently exists no system independent assistive technology solutions for Mathematics. We report on the attempt to transform MathJax into a universal rendering solution, providing accessibility also for readers with visual, print, and motor impairments. As our goal is to integrate seamlessly with other assistive technology solutions, we are faced with a number of technical challenges to provide a consistent user experience, regardless of the particular combination of platform, browser, and screen reader employed by readers. This paper presents MathJaxs accessibility features and some of the technical challenges to motivate a wish-list for future standards to enhance web accessibility for STEM content.

Math Awareness Month 2000: An Interactive Experience

The centerpiece for each MAM campaign is a poster, designed to be displayed in classrooms and on bulletin boards at colleges and universities, to draw attention to the theme and to lead students and teachers to delve further into the topic by reading ancillary materials, usually in the form of short essays on sheets distributed along with the poster. For MAM2000, we decided to design and implement an electronic poster that would put the additional materials directly at the hands of the viewer, and appeal to audiences at different levels.

The StageTools Package for Creating Geometry for the Web

A number of commercial software packages for generating three-dimensional graphics are available, but some useful features are missing from most of these products. The StageTools modules for Geomview try to fill this gap by including such features as interactive controls (sliders, type-in areas, check-boxes, etc.), an object-oriented approach to geometry allowing one object to be based on another (e.g., a tube around a curve), hierarchical grouped objects, objects in arbitrary dimensions with easily specified projections and transformations, and sophisticated scripting capabilities for creating electronic movie clips.

The changing landscape of retirement rules of thumb

Purpose - – The purpose of this paper is to evaluate the continued adherence to “standing” rules of thumb for the percentage of pre-retirement income which should be available to retirees. Design/methodology/approach - – An analysis of census data to determine both the cause and magnitude of the debt load which retirees are carrying into their post-working years. Findings - – The “standing” rules of thumb appear to provide less than adequate levels of income for retirees to service their continuing debt load which they have chosen to carry into their retirement years. Research limitations/implications - – Census data are subject to the accuracy of “captured information” provided by the surveyed individuals. In this case, the information captured is consistent with generally reported data on the sufficiency of retirement income. Practical implications - – Financial planners need to “get the word out early” that individuals need to consider earlier/greater funding of their anticipated retirement income. Social implications - – Rising retirees may be “precluded” from retiring as anticipated because of the insufficiency of the replacement income they will have during their retirement years. Originality/value - – Detailed census data have not been reviewed in detail with a focus on “individual debt load” as we have performed in this research study.