Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
On the computation of elementary divisors of integer matrices
Journal of Symbolic Computation
Δ additive and Δ ultra-additive maps, Gromov's trees, and the Farris transform
Discrete Applied Mathematics
Geometry of Cuts and Metrics
Computing Phylogenetic Diversity for Split Systems
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
Discrete Applied Mathematics
Constructing and Drawing Regular Planar Split Networks
IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB)
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An important procedure in the mathematics of phylogenetic analysis is to associate, to any collection of weighted splits, the metric given by the corresponding linear combination of split metrics. In this note, we study necessary and sufficient conditions for a collection of splits of a given finite set X to give rise to a linearly independent collection of split metrics. In addition, we study collections of splits called affine split systems induced by a configurations of lines and points in the plane. These systems not only satisfy the linear-independence condition, but also provide a Z-basis of the Z-lattice D"e"v"e"n(X|Z) consisting of all integer-valued symmetric maps D:XxX-Z defined on X that vanish on the diagonal and for which, in addition, D(x,y)+D(y,z)+D(z,x)=0mod2 holds for all x,y,z@?X. This Z-lattice is generated by all split metrics considered as vectors in the real vectorspace D(X|R) consisting of all real-valued symmetric maps defined on X that vanish on the diagonal - and, hence, is also an R-basis of that vectorspace.