Boolean representation of manifolds and functions

It is shown that a smooth n dimensional manifold with a boundary in R^n admits a Boolean representation in terms of closed half spaces defined by the tangent hyperplanes at the points on its boundary. A similar result is established for smooth functions on closed convex domains in R^n. The latter result is considered as a form of the Legendre transformation.Comment: 6 pages, 1 figur

Discrete piecewise linear functions

The concept of permutograph is introduced and properties of integral functions on permutographs are established. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is used to establish lattice polynomial representations of piecewise linear functions on convex domains and continuous selectors on linear orders.Comment: 17 pages, 4 figures, Fig.2.1 label correcte

Partial cubes: structures, characterizations, and constructions

AbstractPartial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković’s and Winkler’s relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given.The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated