74,794 research outputs found
On Second-Order Monadic Monoidal and Groupoidal Quantifiers
We study logics defined in terms of second-order monadic monoidal and
groupoidal quantifiers. These are generalized quantifiers defined by monoid and
groupoid word-problems, equivalently, by regular and context-free languages. We
give a computational classification of the expressive power of these logics
over strings with varying built-in predicates. In particular, we show that
ATIME(n) can be logically characterized in terms of second-order monadic
monoidal quantifiers
Categorification of Hopf algebras of rooted trees
We exhibit a monoidal structure on the category of finite sets indexed by
P-trees for a finitary polynomial endofunctor P. This structure categorifies
the monoid scheme (over Spec N) whose semiring of functions is (a P-version of)
the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base
change to Z and collapsing H_0). The monoidal structure is itself given by a
polynomial functor, represented by three easily described set maps; we show
that these maps are the same as those occurring in the polynomial
representation of the free monad on P.Comment: 29 pages. Does not compile with pdflatex due to dependency on the
texdraw package. v2: expository improvements, following suggestions from the
referees; final version to appear in Centr. Eur. J. Mat
Assembling homology classes in automorphism groups of free groups
The observation that a graph of rank can be assembled from graphs of
smaller rank with leaves by pairing the leaves together leads to a
process for assembling homology classes for and from
classes for groups , where the generalize
and . The symmetric group
acts on by permuting leaves, and for trivial
rational coefficients we compute the -module structure on
completely for . Assembling these classes then
produces all the known nontrivial rational homology classes for and
with the possible exception of classes for recently discovered
by L. Bartholdi. It also produces an enormous number of candidates for other
nontrivial classes, some old and some new, but we limit the number of these
which can be nontrivial using the representation theory of symmetric groups. We
gain new insight into some of the most promising candidates by finding small
subgroups of and which support them and by finding
geometric representations for the candidate classes as maps of closed manifolds
into the moduli space of graphs. Finally, our results have implications for the
homology of the Lie algebra of symplectic derivations.Comment: Final version for Commentarii Math. Hel
Tightness and efficiency of irreducible automorphisms of handlebodies
Among (isotopy classes of) automorphisms of handlebodies those called
irreducible (or generic) are the most interesting, analogues of pseudo-Anosov
automorphisms of surfaces. We consider the problem of isotoping an irreducible
automorphism so that it is most efficient (has minimal growth rate) in its
isotopy class. We describe a property, called tightness, of certain invariant
laminations, which we conjecture characterizes this efficiency. We obtain
partial results towards proving the conjecture. For example, we prove it for
genus two handlebodies. We also show that tightness always implies efficiency.
In addition, partly in order to provide counterexamples in our study of
properties of invariant laminations, we develop a method for generating a class
of irreducible automorphisms of handlebodies.Comment: This is the version published by Geometry & Topology on 4 March 200
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