Avoiding abelian squares in partial words

AbstractErdős raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes. Several of our constructions are based on iterating morphisms. In the case of one hole, we prove that the minimal alphabet size is four, while in the case of more than one hole, we prove that it is five. We also investigate the number of partial words of length n with a fixed number of holes over a five-letter alphabet that avoid abelian squares and show that this number grows exponentially with n

Counting distinct squares in partial words

A well known result of Fraenkel and Simpson states that the number of distinct squares in a word of length n is bounded by 2n since at each position there are at most two distinct squares whose last occurrence start. In this paper, we investigate the problem of counting distinct squares in partial words, or sequences over a finite alphabet that may have some "do not know" symbols or "holes" (a (full) word is just a partial word without holes). A square in a partial word over a given alphabet has the form uu' where u is compatible with u, and consequently, such square is compatible with a number of full words over the alphabet that are squares. We consider the number of distinct full squares compatible with factors in a partial word with h holes of length n over a k-letter alphabet, and show that this number increases polynomially with respect to k in contrast with full words, and give bounds in a number of cases. For partial words with one hole, it turns out that there may be more than two squares that have their last occurrence starting at the same position. We prove that if such is the case, then the hole is in the shortest square. We also construct a partial word with one hole over a k-letter alphabet that has more than k squares whose last occurrence start at position zero

A NOTE ON THE NUMBER OF SQUARES IN A PARTIAL WORD WITH ONE HOLE

A well known result of Fraenkel and Simpson states that the number of distinct squares in a word of length n is bounded by 2n since at each position there are at most two distinct squares whose last occurrence starts. In this paper, we investigate squares in partial words with one hole, or sequences over a finite alphabet that have a “do not know” symbol or “hole”. A square in a partial word over a given alphabet has the form uv where u is compatible with v, and consequently, such square is compatible with a number of words over the alphabet that are squares. Recently, it was shown that for partial words with one hole, there may be more than two squares that have their last occurrence starting at the same position. Here, we prove that if such is the case, then the length of the shortest square is at most half the length of the third shortest square. As a result, we show that the number of distinct squares compatible with factors of a partial word with one hole of length n is bounded by (symbol cannot be rendered)

Determination of flavonoids and phenolic acids in the extract of bamboo leaves using near-infrared spectroscopy and multivariate calibration

The content of flavonoids and phenolic acids in the extract of bamboo leaves (EBL) was determined using near-infrared spectroscopy. Optimal partial least squares models and least squares-support vector machine models were developed to enable rapid and non-destructive validation based on EBL contents as determined by photocolorimetric methods and high performance liquid chromatography. The least squares-support vector machine models performed slightly better than the partial least squares models. For least squares-support vector machine models, the correlation coefficients (R) and root mean square errors (RMSE) of calibration and validation were R(cal)= 0.9998, RMSE(cal)= 0.05, R(val)= 0.9418 and RMSE(val)= 3.91 for total flavonoids (TF) and R(cal)= 0.9778, RMSE(cal)= 2.49, R(val)= 0.9535, and RMSE(val)= 3.61 for total phenolic acids (TP). For optimal partial least squares models, the corresponding values were 0.9962, 1.12, 0.9124 and 4.97 for TF and 0.9966, 0.98, 0.9325 and 4.33 for TP. The results indicated that near-infrared spectroscopy with least squares-support vector machine could be utilized as a precise method for the determination of flavonoids and phenolic acids in EBL.Key words: Near-infrared spectroscopy, extract of bamboo leaves (EBL), flavonoids, phenolic acids, partial least squares models, least squares-support vector machine models

Efficient Enumeration of Non-Equivalent Squares in Partial Words with Few Holes

International audienceA partial word is a word with holes (also called don't cares: special symbols which match any symbol). A p-square is a partial word matching at least one standard square without holes (called a full square). Two p-squares are called equivalent if they match the same sets of full squares. Denote by psquares(T) the number of non-equivalent p-squares which are subwords of a partial word T. Let PSQUARES k (n) be the maximum value of psquares(T) over all partial words of length n with k holes. We show asympthotically tight bounds: c1 · min(nk 2 , n 2) ≤ PSQUARES k (n) ≤ c2 · min(nk 2 , n 2) for some constants c1, c2 > 0. We also present an algorithm that computes psquares(T) in O(nk 3) time for a partial word T of length n with k holes. In particular, our algorithm runs in linear time for k = O(1) and its time complexity near-matches the maximum number of non-equivalent p-squares

Abelian-Square-Rich Words

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $\Theta(n^2)$ distinct factors, and there exist words of length $n$ containing $\Theta(n^2)$ distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length $n$ grows quadratically with $n$. More precisely, we say that an infinite word $w$ is {\it abelian-square-rich} if, for every $n$, every factor of $w$ of length $n$ contains, on average, a number of distinct abelian-square factors that is quadratic in $n$; and {\it uniformly abelian-square-rich} if every factor of $w$ contains a number of distinct abelian-square factors that is proportional to the square of its length. Of course, if a word is uniformly abelian-square-rich, then it is abelian-square-rich, but we show that the converse is not true in general. We prove that the Thue-Morse word is uniformly abelian-square-rich and that the function counting the number of distinct abelian-square factors of length $2n$ of the Thue-Morse word is $2$-regular. As for Sturmian words, we prove that a Sturmian word $s_{\alpha}$ of angle $\alpha$ is uniformly abelian-square-rich if and only if the irrational $\alpha$ has bounded partial quotients, that is, if and only if $s_{\alpha}$ has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the proof of Proposition

Combinatorics on Words. New Aspects on Avoidability, Defect Effect, Equations and Palindromes

In this thesis we examine four well-known and traditional concepts of combinatorics on words. However the contexts in which these topics are treated are not the traditional ones. More precisely, the question of avoidability is asked, for example, in terms of k-abelian squares. Two words are said to be k-abelian equivalent if they have the same number of occurrences of each factor up to length k. Consequently, k-abelian equivalence can be seen as a sharpening of abelian equivalence. This fairly new concept is discussed broader than the other topics of this thesis. The second main subject concerns the defect property. The defect theorem is a well-known result for words. We will analyze the property, for example, among the sets of 2-dimensional words, i.e., polyominoes composed of labelled unit squares. From the defect effect we move to equations. We will use a special way to define a product operation for words and then solve a few basic equations over constructed partial semigroup. We will also consider the satisfiability question and the compactness property with respect to this kind of equations. The final topic of the thesis deals with palindromes. Some finite words, including all binary words, are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. The famous Thue-Morse word has the property that for each positive integer n, there exists a factor which cannot be generated by fewer than n palindromes. We prove that in general, every non ultimately periodic word contains a factor which cannot be generated by fewer than 3 palindromes, and we obtain a classification of those binary words each of whose factors are generated by at most 3 palindromes. Surprisingly these words are related to another much studied set of words, Sturmian words.Siirretty Doriast

Development of the cat-owner relationship scale (CORS)

Characteristics of the human-animal bond can be influenced by both owner-related and pet-related factors, which likely differ between species. Three studies adapted the Monash Dog-Owner Relationship Scale (MDORS) to permit assessment of human-cat interactions as perceived by the cat's owner. In Study 1293 female cat owners completed a modified version of the MDORS, where 'dog' was replaced with 'cat' for all items. Responses were compared with a matched sample of female dog owners. A partial least squares discriminant analysis revealed systematic differences between cat and dog owners in the Dog (Cat)-Owner Interaction subscale (MDORS subscale 1), but not for Perceived Emotional Closeness or Perceived Costs (Subscales 2 and 3). Study 2 involved analysis of free-text descriptions of cat-owner interactions provided by 61 female cat owners. Text mining identified key words which were used to create additional questions for a new Cat-Owner Interaction subscale. In Study 3, the resulting cat-owner relationship scale (CORS) was tested in a group of 570 cat owners. The main psychometric properties of the scale, including internal consistency and factor structure, were evaluated. We propose that this scale can be used to accurately assess owner perceptions of their relationship with their cat. A modified scale, combining items from the CORS and MDORS (a C/DORS), is also provided for when researchers would find it desirable to compare human-cat and human-dog interactions. (C) 2017 Elsevier B.V. All rights reserved

Words with the Maximum Number of Abelian Squares

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain $\Theta(n^2)$ distinct factors that are abelian squares. We study infinite words such that the number of abelian square factors of length $n$ grows quadratically with $n$.Comment: To appear in the proceedings of WORDS 201