20,010 research outputs found
Between Treewidth and Clique-width
Many hard graph problems can be solved efficiently when restricted to graphs
of bounded treewidth, and more generally to graphs of bounded clique-width. But
there is a price to be paid for this generality, exemplified by the four
problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that
are all FPT parameterized by treewidth but none of which can be FPT
parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7,
8]. We therefore seek a structural graph parameter that shares some of the
generality of clique-width without paying this price. Based on splits, branch
decompositions and the work of Vatshelle [18] on Maximum Matching-width, we
consider the graph parameter sm-width which lies between treewidth and
clique-width. Some graph classes of unbounded treewidth, like
distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph
Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized
by sm-width
Reduced spectral synthesis and compact operator synthesis
We introduce and study the notion of reduced spectral synthesis, which
unifies the concepts of spectral synthesis and uniqueness in locally compact
groups. We exhibit a number of examples and prove that every non-discrete
locally compact group with an open abelian subgroup has a subset that fails
reduced spectral synthesis. We introduce compact operator synthesis as an
operator algebraic counterpart of this notion and link it with other
exceptional sets in operator algebra theory, studied previously. We show that a
closed subset of a second countable locally compact group satisfies
reduced local spectral synthesis if and only if the subset of satisfies compact operator synthesis. We apply
our results to questions about the equivalence of linear operator equations
with normal commuting coefficients on Schatten -classes.Comment: 43 page
Frequency spanning homoclinic families
A family of maps or flows depending on a parameter which varies in an
interval, spans a certain property if along the interval this property depends
continuously on the parameter and achieves some asymptotic values along it. We
consider families of periodically forced Hamiltonian systems for which the
appropriately scaled frequency is spanned, namely it covers
the semi-infinite line Under some natural assumptions on the
family of flows and its adiabatic limit, we construct a convenient labelling
scheme for the primary homoclinic orbits which may undergo a countable number
of bifurcations along this interval. Using this scheme we prove that a properly
defined flux function is in Combining this proof with previous
results of RK and Poje, immediately establishes that the flux function and the
size of the chaotic zone depend on the frequency in a non-monotone fashion for
a large class of Hamiltonian flows
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