Search results for "Alphabet"

Quantum, stochastic, and pseudo stochastic languages with few states

2014

Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all th…

On the least number of palindromes contained in an infinite word

2013

We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then investigate the same problem when the alphabet has size two.

Mahonian STAT on rearrangement class of words

2017

In 2000, Babson and Steingr\'{i}msson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on $R(w)$, the rearrangement class of a given word $w$. We construct such an involution in this paper, which yields a comparable jo…

Pattern statistics in faro words and permutations

2021

We study the distribution and the popularity of some patterns in $k$-ary faro words, i.e. words over the alphabet $\{1, 2, \ldots, k\}$ obtained by interlacing the letters of two nondecreasing words of lengths differing by at most one. We present a bijection between these words and dispersed Dyck paths (i.e. Motzkin paths with all level steps on the $x$-axis) with a given number of peaks. We show how the bijection maps statistics of consecutive patterns of faro words into linear combinations of other pattern statistics on paths. Then, we deduce enumerative results by providing multivariate generating functions for the distribution and the popularity of patterns of length at most three. Fina…

Constructing Antidictionaries in Output-Sensitive Space

2021

A word $x$ that is absent from a word $y$ is called minimal if all its proper factors occur in $y$. Given a collection of $k$ words $y_1,y_2,\ldots,y_k$ over an alphabet $\Sigma$, we are asked to compute the set $\mathrm{M}^{\ell}_{y_{1}\#\ldots\#y_{k}}$ of minimal absent words of length at most $\ell$ of word $y=y_1\#y_2\#\ldots\#y_k$, $\#\notin\Sigma$. In data compression, this corresponds to computing the antidictionary of $k$ documents. In bioinformatics, it corresponds to computing words that are absent from a genome of $k$ chromosomes. This computation generally requires $\Omega(n)$ space for $n=|y|$ using any of the plenty available $\mathcal{O}(n)$-time algorithms. This is because a…

Fractal surfaces from simple arithmetic operations

2015

Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is shown that these models give rise to a roughness exponent $H$ that shapes the resulting spatial patterns, larger values of the exponent leading to coarser surfaces.

Rigaer Handels-Archiv: alphabetisches Register zum Rigaer Handels-Archiv und zu den demselben vorausgegangenen Druckschriften des Rigaer Börsen-Comité

1883

Rigaer Handels-Archiv: alphabetisches Register zum Rigaer Handels-Archiv und zu den demselben vorausgegangenen Druckschriften des Rigaer Börsen-Comit…

1898

Deux plaidoyers mystiques en faveur du « no pensar nada » : le Troisième Abécédaire spirituel (1527) et la Loi d'amour (1530) de Francisco de Osuna

2017

Two mystical pleas for the «no pensar nada»: the Third Spiritual alphabet (1527) and the Law of love (1530) by Francisco de Osuna A mystic principle rooted in negative theology (from Pseudo-Dionysius the Areopagite to Saint John of the Cross), the « no pensar nada » ("thinking-nothing") was strongly defended by Franciscan preacher and spiritual guide of the « Recollection », Francisco de Osuna, in a context of spiritual polemics marked by the condemnation of Toledo's « Alumbrados » ("the Illuminated") in 1525. Despite the Holy Office's strong pressure, and the threat of censorship, Osuna openly advocated the « thinking-nothing » principle in a chapter of the Third Spiritual Alphabet (1527),…

Finite Alphabet Control of Logistic Networks with Discrete Uncertainty

2014

We consider logistic networks in which the control and disturbance inputs take values in finite sets. We derive a necessary and sufficient condition for the existence of robustly control invariant (hyperbox) sets. We show that a stronger version of this condition is sufficient to guarantee robust global attractivity, and we construct a counterexample demonstrating that it is not necessary. Being constructive, our proofs of sufficiency allow us to extract the corresponding robust control laws and to establish the invariance of certain sets. Finally, we highlight parallels between our results and existing results in the literature, and we conclude our study with two simple illustrative exampl…