10,561 research outputs found
Going higher in the First-order Quantifier Alternation Hierarchy on Words
We investigate the quantifier alternation hierarchy in first-order logic on
finite words. Levels in this hierarchy are defined by counting the number of
quantifier alternations in formulas. We prove that one can decide membership of
a regular language to the levels (boolean combination of
formulas having only 1 alternation) and (formulas having only 2
alternations beginning with an existential block). Our proof works by
considering a deeper problem, called separation, which, once solved for lower
levels, allows us to solve membership for higher levels
On the number of types in sparse graphs
We prove that for every class of graphs which is nowhere dense,
as defined by Nesetril and Ossona de Mendez, and for every first order formula
, whenever one draws a graph and a
subset of its nodes , the number of subsets of which are of
the form
for some valuation of in is bounded by
, for every . This provides
optimal bounds on the VC-density of first-order definable set systems in
nowhere dense graph classes.
We also give two new proofs of upper bounds on quantities in nowhere dense
classes which are relevant for their logical treatment. Firstly, we provide a
new proof of the fact that nowhere dense classes are uniformly quasi-wide,
implying explicit, polynomial upper bounds on the functions relating the two
notions. Secondly, we give a new combinatorial proof of the result of Adler and
Adler stating that every nowhere dense class of graphs is stable. In contrast
to the previous proofs of the above results, our proofs are completely
finitistic and constructive, and yield explicit and computable upper bounds on
quantities related to uniform quasi-wideness (margins) and stability (ladder
indices)
Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics
We consider a magnetic Schroedinger operator in a planar infinite strip with
frequently and non-periodically alternating Dirichlet and Robin boundary
conditions. Assuming that the homogenized boundary condition is the Dirichlet
or the Robin one, we establish the uniform resolvent convergence in various
operator norms and we prove the estimates for the rates of convergence. It is
shown that these estimates can be improved by using special boundary
correctors. In the case of periodic alternation, pure Laplacian, and the
homogenized Robin boundary condition, we construct two-terms asymptotics for
the first band functions, as well as the complete asymptotics expansion (up to
an exponentially small term) for the bottom of the band spectrum
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