11,998 research outputs found
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Weighted model spaces and Schmidt subspaces of Hankel operators
For a bounded Hankel matrix , we describe the structure of the
Schmidt subspaces of , namely the eigenspaces of
corresponding to non zero eigenvalues. We prove that these subspaces are in
correspondence with weighted model spaces in the Hardy space on the unit
circle. Here we use the term "weighted model space" to describe the range of an
isometric multiplier acting on a model space. Further, we obtain similar
results for Hankel operators acting in the Hardy space on the real line.
Finally, we give a streamlined proof of the Adamyan-Arov-Krein theorem using
the language of weighted model spaces.Comment: Final version, to appear in Journal of the London Mathematical
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