Abstract A metric relation by definition is symmetric. Since many data sets are non-symmetric, in this paper we develop a systematic theory of non-symmetric cost functions. Betweenness relations play an important role. We also introduce the notion of a Dress group in the non-symmetric setting and indicate a notion of curvature.

Abstract Step systems of connected graphs were introduced by Ladislav Nebeský as a means of describing shortest paths solely in terms of local information. A ternary relation T encodes, for each point u , the first step x towards a target v . Eight first-order axioms characterize the ternary relation T that are step systems. Here we show that one of Nebeský’s eight axioms is in fact redundant. Moreover, we characterize the step systems of bipartite graphs, partial cubes, and weakly modular graphs in terms of first-order axioms. In each case we show that the corresponding sets of axioms are non-redundant.

Abstract We develop novel tools for computing the likelihood correspondence of an arrangement of hypersurfaces in a projective space. This uses the module of logarithmic derivations. This object is well-studied in the linear case, when the hypersurfaces are hyperplanes. We here focus on nonlinear scenarios and their applications in statistics and physics.

Abstract Under some technical assumptions of a global nature, we establish the weight part of Serre’s conjecture for mod p Galois representations for CM fields that are tamely ramified and sufficiently generic at p .