Discrete polymatroids.
Let E be a Banach space and let and denote the space of all Baire-one and affine Baire-one functions on the dual unit ball , respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between and , where f is an affine function on . If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.
Let two lattices have the same number of points on each hyperbolic surface . We investigate the case when Λ’, Λ” are sublattices of of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.
Dürer's engraving Melencolia I famously includes a perspective view of a solid polyhedral block of which the visible portion is an 8-circuit bounding a pentagon-triple+triangle patch. The polyhedron is usually taken to be a cube truncated on antipodal corners, but an infinity of others are compatible with the visible patch. Construction of all cubic polyhedra compatible with the visible portion (i.e., Dürer Polyhedra) is discussed, explicit graphs and symmetries are listed for small cases ( ≤ 18...