Conference Agenda

In the board game Go, connected stones form groups whose survival depends on their liberties, the adjacent empty points (outer boundary). We determine the maximum number of liberties a group can achieve on an $N \times N$ board by establishing a useful duality between outer boundaries of groups and minimum connected dominating sets in grid graphs. Using previous results on the connected domination number of grids, we derive an explicit formula for all board sizes and prove that the maximum on the standard $19 \times 19$ board is exactly 229 liberties.
We provide explicit constructions for all $N$ and enumerate all optimal groups up to symmetry for $N \leq 7$. We also consider groups with special properties: non-branching, non-separating (doubly connected), and null-homotopic. In particular, we prove that all optimal groups are null-homotopic, that minimum connected dominating sets are path-shaped only for $N \in \{1,2,3,5,6\}$, and characterize when optimal groups are doubly connected.

A mathematical coordinate system for music

_{by Rolphe F. FEHLMANN}

Abstract

Mathematical music theory is most often based on the two historical milestones: on the one hand it is the system of the ‘sectio canonis’ of the Greek EUCLID [300 B.C.], i.e. the division of a string on the monochord leading to the mathematical proportions of musical tones. And on the other hand it is the 12-tone tempering system introduced by the Dutch mathematician Simon STEVIN [NL, 1548–1620] in his treatise ‘Van de spiegeling der singconst’ (published posthum in 1884).

In this presentation the starting point is the sequence of the natural resonance modes [NRMs], aka the harmonic sequence. The reflection of this sequence is usually called the subharmonics and has evoked lots of discussions around its existence. In this alternative approach the numbering of the NRMs as it is done within the ethnic music in Switzerland, (alphorn, buechel, tiba and natural yodel) serves as the basis for a one-to-one mapping to the mathematical number line of the common coordinate system of DESCARTES [F, 1596-1650]

Generally speaking it will not only facilitate the correspondence between mathematics and music, but also more specifically, basic mathematical operations can easily be applied to music. It turns out that chords can be interpreted as vectors and progressions as matrices.