631 research outputs found
A Probabilistic Approach to Generalized Zeckendorf Decompositions
Generalized Zeckendorf decompositions are expansions of integers as sums of
elements of solutions to recurrence relations. The simplest cases are base-
expansions, and the standard Zeckendorf decomposition uses the Fibonacci
sequence. The expansions are finite sequences of nonnegative integer
coefficients (satisfying certain technical conditions to guarantee uniqueness
of the decomposition) and which can be viewed as analogs of sequences of
variable-length words made from some fixed alphabet. In this paper we present a
new approach and construction for uniform measures on expansions, identifying
them as the distribution of a Markov chain conditioned not to hit a set. This
gives a unified approach that allows us to easily recover results on the
expansions from analogous results for Markov chains, and in this paper we focus
on laws of large numbers, central limit theorems for sums of digits, and
statements on gaps (zeros) in expansions. We expect the approach to prove
useful in other similar contexts.Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive
linear recurrence relations, distribution of gaps, longest gap, Markov
processe
Dynamical Directions in Numeration
International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper
Mean asymptotic behaviour of radix-rational sequences and dilation equations (Extended version)
The generating series of a radix-rational sequence is a rational formal power
series from formal language theory viewed through a fixed radix numeration
system. For each radix-rational sequence with complex values we provide an
asymptotic expansion for the sequence of its Ces\`aro means. The precision of
the asymptotic expansion depends on the joint spectral radius of the linear
representation of the sequence; the coefficients are obtained through some
dilation equations. The proofs are based on elementary linear algebra
On p/q-recognisable sets
Let p/q be a rational number. Numeration in base p/q is defined by a function
that evaluates each finite word over A_p={0,1,...,p-1} to some rational number.
We let N_p/q denote the image of this evaluation function. In particular, N_p/q
contains all nonnegative integers and the literature on base p/q usually
focuses on the set of words that are evaluated to nonnegative integers; it is a
rather chaotic language which is not context-free. On the contrary, we study
here the subsets of (N_p/q)^d that are p/q-recognisable, i.e. realised by
finite automata over (A_p)^d. First, we give a characterisation of these sets
as those definable in a first-order logic, similar to the one given by the
B\"uchi-Bruy\`ere Theorem for integer bases numeration systems. Second, we show
that the natural order relation and the modulo-q operator are not
p/q-recognisable
Pattern statistics and Vandermonde matrices
In this paper we determine some limit distributions of pattern statistics in rational stochastic models. We present a general approach to analyze these statistics in rational models having an arbitrary number of strongly connected components. We explicitly establish the limit distributions in most significant cases; they are characterized by a family of unimodal density functions defined by means of confluent Vandermonde matrices
North East Indian linguistics 7 (NEIL 7)
This volume includes papers presented at the seventh and eighth meetings of the North East
Indian Linguistics Society (NEILS), held in Guwahati, India, in 2012 and 2014. As with
previous conferences, these meetings were held at the Don Bosco Institute in Guwahati,
Assam, and hosted in collaboration with Gauhati University. This volume continues the
NEILS tradition of papers by both local and international scholars, with half of them by
linguists from universities in the North East, several of whom are native speakers of the
languages they are writing about. In addition we have papers written by scholars from France,
Japan, Russia, Switzerland and USA. The selection of papers presented in this volume
encompass languages from the Sino-Tibetan, Austroasiatic, Indo-European, and Tai-Kadai
language families, and describe aspects of the languages� phonology, morphosyntax, and
history
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