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Lyndon words have been extensively studied in different contexts of free Lie algebra and combinatorics. All Lyndon and Nyldon words are primitive and any primitive class of words contains an unique Lyndon and an unique Nyldon word. This property motivated the study of Lyndon partial word which is primitive but all primitive classes of partial words may not contain a Lyndon partial word. In this paper we introduce two variants of Lyndon partial words namely Nyldon and inverse Lyndon partial words which are constructed from the decreasing alphabetical order. We compare the properties of the variants with those of Lyndon partial words. [ABSTRACT FROM AUTHOR]

As a natural extension of the Parikh matrix of a picture array induced by an alphabet, this paper investigates the concept of extending the Parikh matrix of a picture array induced by a word with some elementary properties. We also examine the relationship between an extending Parikh matrix and a Parikh matrix. Iii addition. we analyze u-ambiguity of picture arrays and discuss its properties. Furthermore, we extend Parikh friendly permutation to picture arrays in the context of Parikh matrices. With the aid of extending Parikh matrix, we analyze the properties of power of the picture arrays. [ABSTRACT FROM AUTHOR]

Following the Covid-19 pandemic, the rapid spread of online education and tests demanded the implementation of cheating detection tools to ensure academic integrity. While advances in technology such as face recognition, face expression recognition, head posture analysis, eye gaze tracking, network data traffic analysis, and IP spoofing detection have shown promising results in detecting fraudulent behavior, their integration raises ethical concerns that must be carefully considered. This work presents a cognitive computing strategy for investigating the ethical implications of using cheating detection systems in online tests. This study attempts to examine the potential impact on students' privacy, fairness, and trust in the examination process by employing cognitive computing, which models human cognitive capacities. A thorough literature review is used in the process to uncover existing ethical norms and regulatory frameworks linked to online assessments and cheating detection. Soft computing approaches are also used to evaluate the effectiveness and dependability of the aforementioned cheating detection strategies. The study looks into how far facial recognition and expression analysis can go in terms of privacy, as well as the possibility of bias in head posture analysis and eye gaze tracking algorithms. Furthermore, it investigates the ethical implications of monitoring network data traffic and detecting IP spoofing, with a focus on data security and user permission. The cognitive computing model, based on the analysis, presents a comprehensive framework for ethical decision-making when installing cheating detection technologies. The findings of this study contribute to the continuing discussion about the ethical concerns of using modern technologies to identify cheating in online exams. It provides educational institutions and policymakers with practical ideas for striking a balance between academic integrity and protecting students' rights and dignity. By emphasizing ethical issues, this study aims to ensure that the implementation of cheating detection systems adheres to values of fairness, transparency, and privacy protection, promoting a trusting and supportive online learning environment for all parties involved. [ABSTRACT FROM AUTHOR]

We take into account the problem of identifying the repeated structure in a given pword v◊ of length l. We show that a pword v◊ is a cover of a pword u◊ if every letter of u◊ lies within an occurrence of v◊ in u◊ and v◊ is a border of u◊. Here, we examine string issues that are concerned with identifying recurring patterns in a given total word x. The total word's period p, a common regularity, captures x's repetition, since x is a prefix of a string created by concatenating p. We think about a challenge developed by expanding the scope of this repetitiveness idea by permitting overlaps between the segments that are repeated. We focus on a key issue in string processing: the compact representation of a word by its most frequent factors. The frequency cover, or the longest repeating factor, is a useful and simple form of quasi-periodicity in words that is proposed in this paper. [ABSTRACT FROM AUTHOR]

Many classes of finite words have noticeable properties with reference to their palindromic factors and one among them are the words having zero palindromic defect i.e., words rich in palindromes. In this paper we introduce rich partial word and discuss its combinatorial properties. We show that the palindromic richness of a partial word can be studied by including the positions of the missing symbols in that word. The significant difference between rich and rich partial word is that a rich word of length n contains exactly n + 1 distinct palindromic factors whereas a rich partial word of length n contains at least n + 1 distinct palindromic factors. These factors differ from the classical palindromes due to the presence of holes. [ABSTRACT FROM AUTHOR]

Combinatorial and algorithmic properties of Lyndon words have been investigated. Lyndon words have wide usage in algebra and pattern matching. In this paper, we introduce Lyndon partial words and show that the language of all Lyndon partial words over the binary alphabet is not context free. [ABSTRACT FROM AUTHOR]