42 research outputs found
The Isomorphism Relation Between Tree-Automatic Structures
An -tree-automatic structure is a relational structure whose domain
and relations are accepted by Muller or Rabin tree automata. We investigate in
this paper the isomorphism problem for -tree-automatic structures. We
prove first that the isomorphism relation for -tree-automatic boolean
algebras (respectively, partial orders, rings, commutative rings, non
commutative rings, non commutative groups, nilpotent groups of class n >1) is
not determined by the axiomatic system ZFC. Then we prove that the isomorphism
problem for -tree-automatic boolean algebras (respectively, partial
orders, rings, commutative rings, non commutative rings, non commutative
groups, nilpotent groups of class n >1) is neither a -set nor a
-set
How large are the level sets of the Takagi function?
Let T be Takagi's continuous but nowhere-differentiable function. This paper
considers the size of the level sets of T both from a probabilistic point of
view and from the perspective of Baire category. We first give more elementary
proofs of three recently published results. The first, due to Z. Buczolich,
states that almost all level sets (with respect to Lebesgue measure on the
range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states
that the average number of points in a level set is infinite. The third result,
also due to Lagarias and Maddock, states that the average number of local level
sets contained in a level set is 3/2. In the second part of the paper it is
shown that, in contrast to the above results, the set of ordinates y with
uncountably infinite level sets is residual, and a fairly explicit description
of this set is given. The paper also gives a negative answer to a question of
Lagarias and Maddock by showing that most level sets (in the sense of Baire
category) contain infinitely many local level sets, and that a continuum of
level sets even contain uncountably many local level sets. Finally, several of
the main results are extended to a version of T with arbitrary signs in the
summands.Comment: Added a new Section 5 with generalization of the main results; some
new and corrected proofs of the old material; 29 pages, 3 figure
Measurable versions of the LS category on laminations
We give two new versions of the LS category for the set-up of measurable
laminations defined by Berm\'udez. Both of these versions must be considered as
"tangential categories". The first one, simply called (LS) category, is the
direct analogue for measurable laminations of the tangential category of
(topological) laminations introduced by Colman Vale and Mac\'ias Virg\'os. For
the measurable lamination that underlies any lamination, our measurable
tangential category is a lower bound of the tangential category. The second
version, called the measured category, depends on the choice of a transverse
invariant measure. We show that both of these "tangential categories" satisfy
appropriate versions of some well known properties of the classical category:
the homotopy invariance, a dimensional upper bound, a cohomological lower bound
(cup length), and an upper bound given by the critical points of a smooth
function.Comment: 22 page
Countable structures with a fixed group of automorphisms
We prove that, given a countable group G, the set of countable structures (for a suitable language L)U_G whose automorphism group is isomorphic to G is a complete coanalytic set and if G ≄ H then U_G is Borel inseparable from U_H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures for L whose automorphism group is isomorphic to ℤ_p^ℕ ,p a prime number, is Π^1_1&Σ^1_1-complete