42,086 research outputs found
Bihomogeneity and Menger manifolds
For every triple of integers a, b, and c, such that a>O, b>0, and c>1, there
is a homogeneous, non-bihomogeneous continuum whose every point has a
neighborhood homeomorphic the Cartesian product of three Menger compacta m^a,
m^b, and m^c. In particular, there is a homogeneous, non-bihomogeneous, Peano
continuum of covering dimension four.Comment: 9 page
Transitive simple subgroups of wreath products in product action
A transitive simple subgroup of a finite symmetric group is very rarely
contained in a full wreath product in product action. All such simple
permutation groups are determined in this paper. This remarkable conclusion is
reached after a definition and detailed examination of `Cartesian
decompositions' of the permuted set, relating them to certain `Cartesian
systemsof subgroups'. These concepts, and the bijective connections between
them, are explored in greater generality, with specific future applications in
mind.Comment: Submitte
Disjointness properties for Cartesian products of weakly mixing systems
For we consider the class JP() of dynamical systems whose every
ergodic joining with a Cartesian product of weakly mixing automorphisms
() can be represented as the independent extension of a joining of the
system with only coordinate factors. For we show that, whenever
the maximal spectral type of a weakly mixing automorphism is singular with
respect to the convolution of any continuous measures, i.e. has the
so-called convolution singularity property of order , then belongs to
JP(). To provide examples of such automorphisms, we exploit spectral
simplicity on symmetric Fock spaces. This also allows us to show that for any
the class JP() is essentially larger than JP(). Moreover, we
show that all members of JP() are disjoint from ergodic automorphisms
generated by infinitely divisible stationary processes.Comment: 24 pages, corrected versio
On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy
In this article we examine the adjacency and Laplacian matrices and their
eigenvalues and energies of the general product (non-complete extended -sum,
or NEPS) of signed graphs. We express the adjacency matrix of the product in
terms of the Kronecker matrix product and the eigenvalues and energy of the
product in terms of those of the factor signed graphs. For the Cartesian
product we characterize balance and compute expressions for the Laplacian
eigenvalues and Laplacian energy. We give exact results for those signed
planar, cylindrical and toroidal grids which are Cartesian products of signed
paths and cycles.
We also treat the eigenvalues and energy of the line graphs of signed graphs,
and the Laplacian eigenvalues and Laplacian energy in the regular case, with
application to the line graphs of signed grids that are Cartesian products and
to the line graphs of all-positive and all-negative complete graphs.Comment: 30 page
Bounds for the annealed return probability on large finite percolation clusters
Bounds for the expected return probability of the delayed random walk on
finite clusters of an invariant percolation on transitive unimodular graphs are
derived. They are particularly suited for the case of critical Bernoulli
percolation and the associated heavy-tailed cluster size distributions. The
upper bound relies on the fact that cartesian products of finite graphs with
cycles of a certain minimal size are Hamiltonian. For critical Bernoulli bond
percolation on the homogeneous tree this bound is sharp. The asymptotic type of
the expected return probability for large times t in this case is of order of
the 3/4'th power of 1/t.Comment: New result for the particular case of homogeneous trees illustrates
sharpness of the boun
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