1,645 research outputs found
On the multiple Borsuk numbers of sets
The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the
smallest value of m such that S can be partitioned into m sets of diameters
less than d. Our aim is to generalize this notion in the following way: The
k-fold Borsuk number of such a set S is the smallest value of m such that there
is a k-fold cover of S with m sets of diameters less than d. In this paper we
characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give
bounds for those of centrally symmetric sets, smooth bodies and convex bodies
of constant width, and examine them for finite point sets in the Euclidean
3-space.Comment: 16 pages, 3 figure
Colouring quadrangulations of projective spaces
A graph embedded in a surface with all faces of size 4 is known as a
quadrangulation. We extend the definition of quadrangulation to higher
dimensions, and prove that any graph G which embeds as a quadrangulation in the
real projective space P^n has chromatic number n+2 or higher, unless G is
bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996),
219-227]. The family of quadrangulations of projective spaces includes all
complete graphs, all Mycielski graphs, and certain graphs homomorphic to
Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser
theorem
Generalized Tonnetze and Zeitnetze, and the topology of music concepts
The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies – whether orientable, bounded, manifold, etc. – reveal some of the topological character of musical concepts.Accepted manuscrip
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