Palindromic complexity of trees

We consider finite trees with edges labeled by letters on a finite alphabet $\varSigma$. Each pair of nodes defines a unique labeled path whose trace is a word of the free monoid $\varSigma^*$. The set of all such words defines the language of the tree. In this paper, we investigate the palindromic complexity of trees and provide hints for an upper bound on the number of distinct palindromes in the language of a tree.Comment: Submitted to the conference DLT201

On Words with the Zero Palindromic Defect

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word

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

Repetitions in infinite palindrome-rich words

Rich words are characterized by containing the maximum possible number of distinct palindromes. Several characteristic properties of rich words have been studied; yet the analysis of repetitions in rich words still involves some interesting open problems. We address lower bounds on the repetition threshold of infinite rich words over 2 and 3-letter alphabets, and construct a candidate infinite rich word over the alphabet $\Sigma_2=\{0,1\}$ with a small critical exponent of $2+\sqrt{2}/2$. This represents the first progress on an open problem of Vesti from 2017.Comment: 12 page

On the Number of Closed Factors in a Word

A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special. We investigate the structure of closed factors of words. We show that a word of length $n$ contains at least $n+1$ distinct closed factors, and characterize those words having exactly $n+1$ closed factors. Furthermore, we show that a word of length $n$ can contain $\Theta(n^{2})$ many distinct closed factors.Comment: Accepted to LATA 201

Controlled non uniform random generation of decomposable structures

Consider a class of decomposable combinatorial structures, using different types of atoms \Atoms = \{\At_1,\ldots ,\At_{|{\Atoms}|}\}. We address the random generation of such structures with respect to a size $n$ and a targeted distribution in $k$ of its \emph{distinguished} atoms. We consider two variations on this problem. In the first alternative, the targeted distribution is given by $k$ real numbers \TargFreq_1, \ldots, \TargFreq_k such that 0 < \TargFreq_i < 1 for all $i$ and \TargFreq_1+\cdots+\TargFreq_k \leq 1. We aim to generate random structures among the whole set of structures of a given size $n$, in such a way that the {\em expected} frequency of any distinguished atom \At_i equals \TargFreq_i. We address this problem by weighting the atoms with a $k$-tuple \Weights of real-valued weights, inducing a weighted distribution over the set of structures of size $n$. We first adapt the classical recursive random generation scheme into an algorithm taking \bigO{n^{1+o(1)}+mn\log{n}} arithmetic operations to draw $m$ structures from the \Weights-weighted distribution. Secondly, we address the analytical computation of weights such that the targeted frequencies are achieved asymptotically, i. e. for large values of $n$. We derive systems of functional equations whose resolution gives an explicit relationship between \Weights and \TargFreq_1, \ldots, \TargFreq_k. Lastly, we give an algorithm in \bigO{k n^4} for the inverse problem, {\it i.e.} computing the frequencies associated with a given $k$-tuple \Weights of weights, and an optimized version in \bigO{k n^2} in the case of context-free languages. This allows for a heuristic resolution of the weights/frequencies relationship suitable for complex specifications. In the second alternative, the targeted distribution is given by a $k$ natural numbers $n_1, \ldots, n_k$ such that $n_1+\cdots+n_k+r=n$ where $r \geq 0$ is the number of undistinguished atoms. The structures must be generated uniformly among the set of structures of size $n$ that contain {\em exactly} $n_i$ atoms \At_i ($1 \leq i \leq k$). We give a \bigO{r^2\prod_{i=1}^k n_i^2 +m n k \log n} algorithm for generating $m$ structures, which simplifies into a \bigO{r\prod_{i=1}^k n_i +m n} for regular specifications

Computing the kk-binomial complexity of the Thue--Morse word

Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $\mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue--Morse word is aperiodic, its $k$-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form $\Psi^\ell(w)$ for an arbitrary morphism $\Psi$. We also thoroughly describe the factors of the Thue--Morse word by introducing a relevant new equivalence relation

The impact of managerial characteristics on business strategies under the environmental change: an investigation of the Israeli diamond industry

The changing business environment has required firms to adopt new strategies to facilitate efficient organizational adaptation. Previous upper echelons studies suggested that the demographic characteristics of top managers influence their choice of strategies, and ultimately the firms’ performance. These studies tended to examine firms in a homogenous way. The characteristics of family-owned firms, together with the unique external environment within which they operate, have been largely ignored. This study aims to fill this gap and examine the relationships between the top managers’ characteristics and their choice of exchange strategy within the diamond industry in light of the environmental changes. This paper illustrates the evolutionary stages of the diamond industry and how players adjust their strategies in-line with the environment. We interviewed 100 diamond firm managers to gather the empirical data. The results have shown that certain managerial characteristics, such as family background and marketing experience, have positively influenced the choice of using the arm’s length market exchange strategies. Our findings have also reflected that the relationship between managers’ characteristics and strategic choice is social and normative. Many managers follow the family or community traditions and imitate other key industry players in order to achieve legitimacy from the stakeholders and improve competitiveness

Antibacterial Fractions from Erodium cicutarium Exposed—Clinical Strains of Staphylococcus aureus in Focus

Followed by a buildup of its phytochemical profile, Erodium cicutarium is being subjected to antimicrobial investigation guided with its ethnobotanical use. The results of performed in vitro screening on Staphylococcus aureus, Pseudomonas aeruginosa, and Candida albicans strains, show that E. cicutarium has antimicrobial activity, with a particular emphasis on clinical S. aureus strains— both the methicillin sensitive (MSSA) and the methicillin resistant (MRSA) S. aureus. Experimental design consisted of general methods (the serial microdilution broth assay and the agar well diffusion assay), as well as observing bactericidal/bacteriostatic activity through time (the “time‐kill” assay), investigating the effect on cell wall integrity and biofilm formation, and modulation of bacterial hemolysis. Observed antibacterial activity from above‐described methods led to further activity‐ guided fractionation of water and methanol extracts using bioautography coupled with UHPLCLTQ OrbiTrap MS4. It was determined that active fractions are predominantly formed by gallic acid derivatives and flavonol glycosides. Among the most active phytochemicals, galloyl‐shikimic acid was identified as the most abundant compound. These results point to a direct connection between galloyl‐shikimic acid and the observed E. cicutarium antibacterial activity, and open several new research approaches for future investigatio