1,309 research outputs found
Extremal sequences of polynomial complexity
The joint spectral radius of a bounded set of real matrices is
defined to be the maximum possible exponential growth rate of products of
matrices drawn from that set. For a fixed set of matrices, a sequence of
matrices drawn from that set is called \emph{extremal} if the associated
sequence of partial products achieves this maximal rate of growth. An
influential conjecture of J. Lagarias and Y. Wang asked whether every finite
set of matrices admits an extremal sequence which is periodic. This is
equivalent to the assertion that every finite set of matrices admits an
extremal sequence with bounded subword complexity. Counterexamples were
subsequently constructed which have the property that every extremal sequence
has at least linear subword complexity. In this paper we extend this result to
show that for each integer , there exists a pair of square matrices
of dimension for which every extremal sequence has subword
complexity at least .Comment: 15 page
Subproduct systems and Cartesian systems; new results on factorial languages and their relations with other areas
We point out that a sequence of natural numbers is the dimension sequence of
a subproduct system if and only if it is the cardinality sequence of a word
system (or factorial language). Determining such sequences is, therefore,
reduced to a purely combinatorial problem in the combinatorics of words. A
corresponding (and equivalent) result for graded algebras has been known in
abstract algebra, but this connection with pure combinatorics has not yet been
noticed by the product systems community. We also introduce Cartesian systems,
which can be seen either as a set theoretic version of subproduct systems or an
abstract version of word systems. Applying this, we provide several new results
on the cardinality sequences of word systems and the dimension sequences of
subproduct systems.Comment: New title; added references; to appear in Journal of Stochastic
Analysi
Definable maximal cofinitary groups of intermediate size
Using almost disjoint coding, we show that for each
consistently ,
where is witnessed by a maximal cofinitary
group.Comment: 22 page
Decidability in the logic of subsequences and supersequences
We consider first-order logics of sequences ordered by the subsequence
ordering, aka sequence embedding. We show that the \Sigma_2 theory is
undecidable, answering a question left open by Kuske. Regarding fragments with
a bounded number of variables, we show that the FO2 theory is decidable while
the FO3 theory is undecidable
Laver's results and low-dimensional topology
In connection with his interest in selfdistributive algebra, Richard Laver
established two deep results with potential applications in low-dimensional
topology, namely the existence of what is now known as the Laver tables and the
well-foundedness of the standard ordering of positive braids. Here we present
these results and discuss the way they could be used in topological
applications
Almost Every Simply Typed Lambda-Term Has a Long Beta-Reduction Sequence
It is well known that the length of a beta-reduction sequence of a simply
typed lambda-term of order k can be huge; it is as large as k-fold exponential
in the size of the lambda-term in the worst case. We consider the following
relevant question about quantitative properties, instead of the worst case: how
many simply typed lambda-terms have very long reduction sequences? We provide a
partial answer to this question, by showing that asymptotically almost every
simply typed lambda-term of order k has a reduction sequence as long as
(k-1)-fold exponential in the term size, under the assumption that the arity of
functions and the number of variables that may occur in every subterm are
bounded above by a constant. To prove it, we have extended the infinite monkey
theorem for strings to a parametrized one for regular tree languages, which may
be of independent interest. The work has been motivated by quantitative
analysis of the complexity of higher-order model checking
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