5,374 research outputs found
Non-homeomorphic topological rank and expansiveness
Downarowicz and Maass (2008) have shown that every Cantor minimal
homeomorphism with finite topological rank is expansive. Bezuglyi,
Kwiatkowski and Medynets (2009) extended the result to non-minimal cases. On
the other hand, Gambaudo and Martens (2006) had expressed all Cantor minimal
continuou surjections as the inverse limit of graph coverings. In this paper,
we define a topological rank for every Cantor minimal continuous surjection,
and show that every Cantor minimal continuous surjection of finite topological
rank has the natural extension that is expansive
Constructing Simplicial Branched Covers
Branched covers are applied frequently in topology - most prominently in the
construction of closed oriented PL d-manifolds. In particular, strong bounds
for the number of sheets and the topology of the branching set are known for
dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain
a simplicial covering space (the partial unfolding) of a given simplicial
complex, thus obtaining a simplicial branched cover [Adv. Geom. 3(2):191-255,
2003]. We present a large class of branched covers which can be constructed via
the partial unfolding. In particular, for d<=4 every closed oriented PL
d-manifold is the partial unfolding of some polytopal d-sphere.Comment: 15 pages, 8 figures, typos corrected and conjecture adde
Hierarchical Models as Marginals of Hierarchical Models
We investigate the representation of hierarchical models in terms of
marginals of other hierarchical models with smaller interactions. We focus on
binary variables and marginals of pairwise interaction models whose hidden
variables are conditionally independent given the visible variables. In this
case the problem is equivalent to the representation of linear subspaces of
polynomials by feedforward neural networks with soft-plus computational units.
We show that every hidden variable can freely model multiple interactions among
the visible variables, which allows us to generalize and improve previous
results. In particular, we show that a restricted Boltzmann machine with less
than hidden binary variables can approximate
every distribution of visible binary variables arbitrarily well, compared
to from the best previously known result.Comment: 18 pages, 4 figures, 2 tables, WUPES'1
Path coverings with prescribed ends in faulty hypercubes
We discuss the existence of vertex disjoint path coverings with prescribed
ends for the -dimensional hypercube with or without deleted vertices.
Depending on the type of the set of deleted vertices and desired properties of
the path coverings we establish the minimal integer such that for every such path coverings exist. Using some of these results, for ,
we prove Locke's conjecture that a hypercube with deleted vertices of each
parity is Hamiltonian if Some of our lemmas substantially
generalize known results of I. Havel and T. Dvo\v{r}\'{a}k. At the end of the
paper we formulate some conjectures supported by our results.Comment: 26 page
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