Billiards Mathematically Treated

Mathematically Natural Concrete Incompleteness

Every order invariant R ⊆ Q has a maximal nonnegative root, where S1...n|>n = S0...n-1|>n. Every order invariant R ⊆ Q has a # basis containing its upper shift. These are proved here by going well beyond the usual ZFC axioms for mathematics, using the consistency of the system SRP (Con(SRP)). Here SRP is the extension of ZFC by certain well studied large cardinal hypotheses (existence of cardin...

Track Billiards

We study a class of planar billiards having the remarkable property that their phase space consists up to a set of zero measure of two invariant sets formed by orbits moving in opposite directions. The tables of these billiards are tubular neighborhoods of differentiable Jordan curves that are unions of finitely many segments and arcs of circles. We prove that under proper conditions on the seg...

Quantum Billiards

Statistics for the Mathematically Challenged

In this paper I argue how we can impart deep understanding of inferential statistics by the early introduction of two approaches that students normally encounter only late in their studies, if at all. These are non-parametric methods and ANOVA. As illustration, we imagine a marathon race where firstly there are just two teams with two runners each, and secondly where there is a third team with ...

SPREADSPACES: Mathematically-Intelligent Graphical Spreadsheets

Starting from existing spreadsheet software, like Lotus 1-23 ©, Excel ©, or Spreadsheet 2000 ©, we propose a sequence of enhancements to fully integrate constraint-based reasoning, culminating in a system for reactive, graphical, mathematical constructions. This is driven by our view of constraints as the essence of (spreadsheet) computation, rather than as an add-on tool for expert users. We c...