3,124 research outputs found
More Kolakoski Sequences
Our goal in this article is to review the known properties of the mysterious
Kolakoski sequence and at the same time look at generalizations of it over
arbitrary two letter alphabets. Our primary focus will here be the case where
one of the letters is odd while the other is even, since in the other cases the
sequences in question can be rewritten as (well-known) primitive substitution
sequences. We will look at word and letter frequencies, squares, palindromes
and complexity.Comment: 17 pages, 3 tables, 1 figur
Putting an Edge to the Poisson Bracket
We consider a general formalism for treating a Hamiltonian (canonical) field
theory with a spatial boundary. In this formalism essentially all functionals
are differentiable from the very beginning and hence no improvement terms are
needed. We introduce a new Poisson bracket which differs from the usual
``bulk'' Poisson bracket with a boundary term and show that the Jacobi identity
is satisfied. The result is geometrized on an abstract world volume manifold.
The method is suitable for studying systems with a spatial edge like the ones
often considered in Chern-Simons theory and General Relativity. Finally, we
discuss how the boundary terms may be related to the time ordering when
quantizing.Comment: 36 pages, LaTeX. v2: A manifest formulation of the Poisson bracket
and some examples are added, corrected a claim in Appendix C, added an
Appendix F and a reference. v3: Some comments and references adde
Self-Similar Algebras with connections to Run-length Encoding and Rational Languages
A self-similar algebra is an associative
algebra with a morphism of algebras , where is the set of matrices with coefficients from
. We study the connection between self-similar algebras with
run-length encoding and rational languages. In particular, we provide a curious
relationship between the eigenvalues of a sequence of matrices related to a
specific self-similar algebra and the smooth words over a 2-letter alphabet. We
also consider the language of words in
where such that is a unit in . We
prove that is rational and provide an asymptotic formula for the number
of words of a given length in
Empirical processes, typical sequences and coordinated actions in standard Borel spaces
This paper proposes a new notion of typical sequences on a wide class of
abstract alphabets (so-called standard Borel spaces), which is based on
approximations of memoryless sources by empirical distributions uniformly over
a class of measurable "test functions." In the finite-alphabet case, we can
take all uniformly bounded functions and recover the usual notion of strong
typicality (or typicality under the total variation distance). For a general
alphabet, however, this function class turns out to be too large, and must be
restricted. With this in mind, we define typicality with respect to any
Glivenko-Cantelli function class (i.e., a function class that admits a Uniform
Law of Large Numbers) and demonstrate its power by giving simple derivations of
the fundamental limits on the achievable rates in several source coding
scenarios, in which the relevant operational criteria pertain to reproducing
empirical averages of a general-alphabet stationary memoryless source with
respect to a suitable function class.Comment: 14 pages, 3 pdf figures; accepted to IEEE Transactions on Information
Theor
Vertical representation of -words
We present a new framework for dealing with -words, based on
their left and right frontiers. This allows us to give a compact representation
of them, and to describe the set of -words through an infinite
directed acyclic graph . This graph is defined by a map acting on the
frontiers of -words. We show that this map can be defined
recursively and with no explicit references to -words. We then show
that some important conjectures on -words follow from analogous
statements on the structure of the graph .Comment: Published in Theoretical Computer Scienc
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