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The combinatorial properties of Fibonacci words and arrays are now a popularly studied concept in formal language theory. One such property is the palindromic property of words and arrays. In this paper, we construct algorithms to generate the involutive Fibonacci words and involutive Fibonacci arrays and to check, if a given word is involutive Fibonacci or not. We verify the algorithms using python programming. And we investigate the number of palindromes present in the sequences of involutive Fibonacci words and involutive Fibonacci arrays which has various applications. [ABSTRACT FROM AUTHOR]

It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large moduli, offering upper bounds for moments and averages of certain products closely related to exponential sums over palindromes. [ABSTRACT FROM AUTHOR]

We answer a question of Bardakov (Kourovka Notebook, Problem 19.8) which asks for the existence of a pair of natural numbers (c , m) with the property that every element in the free group on the two-element set { a , b } can be represented as a concatenation of c , or fewer, m -almost-palindromes in letters a ± 1 , b ± 1. Here, an m -almost-palindrome is a word which can be obtained from a palindrome by changing at most m letters. We show that no such pair (c , m) exists. In fact, we show that the analogous result holds for all non-abelian free groups. [ABSTRACT FROM AUTHOR]

A palindromic substring T[i..j] of a string T is said to be a shortest unique palindromic substring (SUPS) in T for an interval [p, q] if T[i..j] is a shortest palindromic substring such that T[i..j] occurs only once in T, and [i, j] contains [p, q]. The SUPS problem is, given a string T of length n, to construct a data structure that can compute all the SUPSs for any given query interval. It is known that any SUPS query can be answered in O (α) time after O(n)-time preprocessing, where α is the number of SUPSs to output (Inoue in J Discrete Algorithms 52-53:122–132, 2018). In this paper, we first show that α is at most 4, and the upper bound is tight. We also show that the total sum of lengths of minimal unique palindromic substrings of string T, which is strongly related to SUPSs, is O(n). Then, we present the first O(n)-bits data structures that can answer any SUPS query in constant time. Also, we present an algorithm to solve the SUPS problem for a sliding window that can answer any query in O (log log W) time and update data structures in amortized O (log σ + log log W) time, where W is the size of the window, and σ is the alphabet size. Furthermore, we consider the SUPS problem in the after-edit model and present an efficient algorithm. Namely, we present an algorithm that uses O(n) time for preprocessing and answers any k SUPS queries in O (log n log log n + k log log n) time after single character substitution. Finally, as a by-product, we propose a fully-dynamic data structure for range minimum queries (RmQs) with a constraint where the width of each query range is limited to poly-logarithmic. The constrained RmQ data structure can answer such a query in constant time and support a single-element edit operation in amortized constant time. [ABSTRACT FROM AUTHOR]

Let C 2 be the free center-by-metabelian Lie algebra of rank 2 freely generated by the set { x 1 , x 2 }. Let P C 2 be the Lie subalgebra of symmetric polynomials in C 2 , that is, P C 2 consists of all the elements f (x 1 , x 2) in C 2 such that f (x 1 , x 2) = f (x 2 , x 1). We give a linear basis and a minimal infinite generating set for P C 2 , thus extending results of Ş. Findik and N. Ögüşlü [Palindromes in the free metabelian Lie algebras, Internat. J. Algebra Comput. 29(5) (2019) 885–891]. [ABSTRACT FROM AUTHOR]

Verses of impossibility are those that refer to events that can never or rarely happen, tasks that are difficult or impossible to perform, people or things that are of no use, things that are impossible to find or futile efforts. Impossibilities can be classified into logical, physical, causal, nomological, temporary, permanent, partial, total, objective or subjective impossibility. This study sought to analyze verses of impossibility in the Holy Quran, their grammatical structure, rhetorical features, types of questions and conditional, emphatic, and/or negative particles used. Results of the analysis showed two kinds of impossibility (i) those in which Allah challenges the disbelievers and those in which the disbelievers ask the Prophets for what they consider impossible demands. In addition, verses of impossibility in the current study use a combination of grammatical structures and rhetorical devices that mainly contain conditional sentences beginning with conditional particles .../?an/ (if), ..., /Hatta:/ (until), ... laula:/ (if), negative, imperative, interrogative, and emphatic structures, few modals and rhetorical questions, some metaphors, hyperbole, antitheses and counterfactuals to describe challenges, futile efforts, stubbornness of the disbelievers and their incongruous behavior. They also focus on logical, temporary and permanent impossibilities. Other rhetorical features of the Holy Quran, such as antiphrasis, asyndeton, cadence, chiasmus, epizeuxis, equivoque, isocolon, palindrome, polyptoton, and synecdoche are still open for further investigation by future researchers. [ABSTRACT FROM AUTHOR]

The palindromization map has been defined initially by Aldo de Luca in the context of Sturmian words. It was extended to the free group of rank 2 by Kassel and the second author. We extend their construction to arbitrary alphabets. We also investigate the suffix automaton and compact suffix automaton of the words obtained by palindromization. [ABSTRACT FROM AUTHOR]

A panel of 106 insertion/deletion (InDel) polymorphisms and a method of their genotyping on biochips were proposed as a new approach to genetic personal identification. Short lengths and low mutation rates are basic properties of InDel markers, which thus have significant advantages over short tandem repeats (STRs) widely used in forensics. The allele frequency distributions of all known InDel polymorphisms were studied in the five largest world populations (European, East Asian, South Asian, African, and American). Markers were selected to meet the following criteria: the minor allele frequency (MAF) is higher than 0.30; the physical distance between markers is greater than 3 Mb; there are no polymorphisms, tandem repeats, and palindromes in the flanking sequences; the AT/GC ratio is close to 1. A panel of 106 polymorphisms was thus formed; the average MAF was estimated at 0.396 in the five populations. The method developed for panel genotyping included one-step multiplex PCR and subsequent hybridization on a biological microarray. The average amplicon length was 72 bp. A sample of 201 residents of Moscow and St. Petersburg was tested to determine the main characteristics of the panel: the random matching probability (MP) was 1.89× 10–43 and the combined probability of paternity exclusion (CPE) was 0.99999999063. The method provides an alternative to molecular genetic personal identification based on the STR length variations. [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]

Stabilization by a parametric distributed control function plays a very important role in aeronautics, aerospace and physics. Choosing the right parameters is necessary for handling the distributed control. In the current paper, we introduce stabilization criteria for an n-order functional-differential equation with a parametric distributed control function in n-term integrals and 2 n parameter sets. In our article, we use properties of unimodal and log-concave polynomials. [ABSTRACT FROM AUTHOR]

Due to the specific base-pairing recognition, clear nanostructure, programmable sequence and responsiveness of the DNA molecule, DNA materials have attracted extensive attention and been widely used in controlled release, drug delivery and tissue engineering. Generally, the strategies for preparing DNA materials are based on the assembly of multiple DNA strands. The construction of DNA materials using only one DNA strand can not only save time and cost, but also avoid defects in final assemblies generated by the inaccuracy of DNA ratios, which potentially promote the large-scale production and practical application of DNA materials. In order to use one DNA strand to form assemblies, the sequences have to be palindromes with lengths that need to be controlled carefully. In this review, we introduced the development of DNA assembly and mainly summarized current reported materials formed by one DNA strand. We also discussed the principle for the construction of DNA materials using one DNA strand. [ABSTRACT FROM AUTHOR]

Longest common substring (LCS), longest palindrome substring (LPS), and Ulam distance (UL) are three fundamental string problems that can be classically solved in near linear time. In this work, we present sublinear time quantum algorithms for these problems along with quantum lower bounds. Our results shed light on a very surprising fact: Although the classic solutions for LCS and LPS are almost identical (via suffix trees), their quantum computational complexities are different. While we give an exact O ~ (n) time algorithm for LPS, we prove that LCS needs at least time Ω ~ (n 2 / 3) even for 0/1 strings. [ABSTRACT FROM AUTHOR]

The mammalian male-specific Y chromosome plays a critical role in sex determination and male fertility. However, because of its repetitive and haploid nature, it is frequently absent from genome assemblies and remains enigmatic. The Y chromosomes of great apes represent a particular puzzle: their gene content is more similar between human and gorilla than between human and chimpanzee, even though human and chimpanzee share a more recent common ancestor. To solve this puzzle, here we constructed a dataset including Ys from all extant great ape genera. We generated assemblies of bonobo and orangutan Ys from short and long sequencing reads and aligned them with the publicly available human, chimpanzee, and gorilla Y assemblies. Analyzing this dataset, we found that the genus Pan, which includes chimpanzee and bonobo, experienced accelerated substitution rates. Pan also exhibited elevated gene death rates. These observations are consistent with high levels of sperm competition in Pan Furthermore, we inferred that the great ape common ancestor already possessed multicopy sequences homologous to most human and chimpanzee palindromes. Nonetheless, each species also acquired distinct ampliconic sequences. We also detected increased chromatin contacts between and within palindromes (from Hi-C data), likely facilitating gene conversion and structural rearrangements. Our results highlight the dynamic mode of Y chromosome evolution and open avenues for studies of male-specific dispersal in endangered great ape species.

Let b and b1 be distinct positive integers larger than 1, and let Ab(n) and Ab1(n) be the number of palindromes in bases b and b1 that are less than or equal to n, respectively. In this article, we finish the comparative study of the functions Ab(n) and Ab1(n). As a result, we present the full picture of the asymptotic behavior of their difference. [ABSTRACT FROM AUTHOR]

In the tortoise-and-hare algorithm, when the fast pointer reaches the end of a finite list, the slow pointer points to the middle of this list. In the early 2000's, this property was found to make it possible to program a palindrome detector for immutable lists that operates in one recursive traversal of the given list and performs the smallest possible number of comparisons, using the "There And Back Again" (TABA) recursion pattern. In this article, this palindrome detector is reconstructed in OCaml, formalized with the Coq Proof Assistant, and proved to be correct. More broadly, this article presents a compositional account of the tortoise-andhare algorithm for finite lists. Concretely, compositionality means that programs that use a fast and a slow pointer can be expressed with an ordinary fold function for lists and reasoned about using ordinary structural induction on the given list. This article also contains a dozen new applications of the TABA recursion pattern and of its tail-recursive variant, "There and Forth Again". [ABSTRACT FROM AUTHOR]

The analysis of deletions may reveal evolutionary trends and provide new insight into the surprising variability and rapidly spreading capability that SARS-CoV-2 has shown since its emergence. To understand the factors governing genomic stability, it is important to define the molecular mechanisms of deletions in the viral genome. In this work, we performed a statistical analysis of deletions. Specifically, we analyzed correlations between deletions in the SARS-CoV-2 genome and repetitive elements and documented a significant association of deletions with runs of identical (poly-) nucleotides and direct repeats. Our analyses of deletions in the accessory genes of SARS-CoV-2 suggested that there may be a hypervariability in ORF7A and ORF8 that is not associated with repetitive elements. Such recurrent search in a "sequence space" of accessory genes (that might be driven by natural selection) did not yet cause increased viability of the SARS-CoV-2 variants. However, deletions in the accessory genes may ultimately produce new variants that are more successful compared to the viral strains with the conventional architecture of the SARS-CoV-2 accessory genes. [ABSTRACT FROM AUTHOR]

We consider some families of one relator groups arising as fundamental groups of 3-dimensional manifolds, and calculate their character varieties in SL(2, C). Then we give simple geometrical descriptions of such varieties, and determine the number of their irreducible components. Our paper relates to the work of Baker-Petersen, Qazaqzeh and Morales-Marc'en on the character variety of certain classes of one relator groups, but we use different methods based on the concept of palindrome presentations of given groups. [ABSTRACT FROM AUTHOR]

The poem "Palindrome" by Nicole Stockburger is presented. First Line: What is this arrangement- Last Line: what is this arrangement?

The staphylococcal plasmid pC194 is replicated through the rolling-circle mechanism. Its replication protein RepA has been proposed to initiate replication by making a bond between Y214 and DNA phosphate via transesterification and to terminate it by hydrolyzing DNA with E210 and carrying out strand transfer. We tested this model by examining the catalytic functions of the protein with purified RepA proteins and single-stranded DNA oligomers. The wild-type RepA formed a covalent bond with the DNA phosphate at the predicted initiation site. It hydrolyzed the phosphodiester bond at the site, which activity was found to depend on the presence of a large pseudopalindrome contained in the replication origin. The protein carried out a strand-transfer reaction which mimicked the termination step of replication. A Y214F and an E210A mutant respectively lacked the transesterification and the hydrolytic activity. These results are consistent with the previously proposed model, which was based solely on molecular genetics results. In addition, an E142A mutant was found to lack both activities, suggesting that the residue may coordinate the divalent cation necessary for them. A possible role of the pseudopalindrome in controlling the two activities of RepA during a replication cycle is also discussed. [Display omitted] • DNA hydrolysis by plasmid replication protein RepA is demonstrated. • The hydrolysis by RepA is activated by the palindrome in the replication origin. • An auxiliary glutamate may be coordinating Mg(II) for catalytic residues. [ABSTRACT FROM AUTHOR]

The development of a sensitive and isothermal technique with a greatly enhanced miRNA detection signal is still technically problematic due to the low abundance of miRNA and high sequence similarities with homologous miRNAs. Herein, we propose a novel fluorescence approach for sensitive and reliable miRNA detection by integrating the palindrome sequence mediated target recycling with self-priming assisted signal reaction. In this method, a dual toehold DNA nano-probe (HT) with two functional arms is designed to mediate specific target recognition and signal amplification. In the presence of target miRNA, it binds to the recognition module of HT probe, releasing the "2" sequence to initiate strand displacement amplification (SDA) and a self-priming-induced signal reaction. Based on the elegant design, the proposed method exhibits a wide linear response range exceeding five orders of magnitude and a low limit of detection of 0.96 fM according to the 3δ rule. The non-specific signal is below 5 % for non-target miRNA detection. Taking the merits of excellent sensitivity, desirable specificity, and superior anti-interference ability, the proposed approach shows a promising prospect for detecting miRNAs in complicated biological environments and early diagnosis of diseases. [Display omitted] • Sensitive and reliable miRNA detection is crucial for the early-diagnosis of various disease. • We constructed a dual toehold DNA nano-probe (HT) with two functional arms for high-sensitive detection of miRNA-21. • With a linear response range exceeding five orders of magnitude, miRNA-21 can be detected at concentrations as low as 0.96 fM. [ABSTRACT FROM AUTHOR]

Research pertaining to SARS-CoV-2 is in full swing to understand the origin and evolution of this deadly virus that can lead to its rapid detection. To achieve this, atypical genomic sequences which may be unique to SARS-CoV-2 or Coronaviridae family in general may be investigated. Such sequences in virus genomes may be responsible for target prediction, replication, defence mechanisms and viral packaging. This fact has motivated us to explore the different types of repeats such as palindromes, mirror repeats and inverted repeats in SARS-CoV-2, MERS-CoV and SARS-CoV-1. For this purpose, the respective reference sequence of SARS-CoV-2, MERS-CoV and SARS-CoV-1 is divided into descriptors of sequences of length ${k}$ using ${k}$ -mer technique. Thereafter, these descriptors are represented as a collection of tokens which are subsequently used for the identification of palindrome, mirror repeat and inverted repeat in the respective reference sequence. The highest number of palindromes, mirror repeats and inverted repeats are identified for descriptor length 10. As a result, for palindromes such values are 38, 42 and 33 and for mirror repeats they are 52, 38 and 33 for SARS-CoV-2, MERS-CoV and SARS-CoV-1 respectively. For inverted repeats, with a descriptor length 10 and intervening length 5, the values are 59, 56 and 70 respectively. Moreover, the identified repeats are then searched for in 108246, 291 and 340 SARS-CoV-2, MERS-CoV and SARS-CoV-1 virus sequences respectively to find the population coverage of such repeats. It surpasses 99% in most cases and even 100% for some. Furthermore, GC contents which mostly lie between 20%-50% are evaluated for these repeats as well in order to understand their binding efficacy.

Divisibility is an old topic that to this day intrigues and fascinates researchers and scholars. Several rules are well-known in particular the divisibility by eleven, since, for example, a palindrome with an even number of digits is divisible by eleven. In current times, divisibility has its applications, for example, in cryptography. Here, in this paper, we will show that applying two somewhat intuitive procedures to the lines of Pascal's triangle shall always yield numbers divisible by eleven. Illustrative examples are presented.

The palindromic length of a finite word v is the minimum number of palindromes whose concatenation is equal to v. It was conjectured in 2013 that for every aperiodic infinite word x , the palindromic length of its factors is not bounded. Let x be an infinite aperiodic word with a bounded palindromic length and containing infinitely many occurrences of the fifth power u 5 of some word u. In this paper, we prove that we can erase from x extra powers of u and of its reversal u R to get another infinite aperiodic word of bounded palindromic length which does not contain u 5 nor (u R) 5. This result is not sufficient to prove that x does not exist but may give a hint for this proof. [ABSTRACT FROM AUTHOR]

A two-dimensional (2D) word is a 2D palindrome if it is equal to its reverse and it is an HV-palindrome if all its columns and rows are 1D palindromes. We characterize such words and study some of their combinatorial and structural properties. We also find the number of possible palindromic conjugates of a 2D word. We investigate an upper bound on the number of distinct non-empty HV-palindromic subwords in any finite 2D word, thus, proving the conjecture given by Anisiu et al. We also identify the minimum number of HV-palindromic subwords in an infinite 2D word over a finite alphabet. [ABSTRACT FROM AUTHOR]

Stabilization by a parametric distributed control function plays a very important role in aeronautics, aerospace and physics. Choosing the right parameters is necessary for handling the distributed control. In the current paper, we introduce stabilization criteria for an n-order functional-differential equation with a parametric distributed control function in n-term integrals and 2n parameter sets. In our article, we use properties of unimodal and log-concave polynomials.

A finite word w is called rich if it contains | w | + 1 distinct palindromic factors including the empty word. For every finite rich word w there are distinct nonempty palindromes w 1 , w 2 , ... , w p such that w = w p w p − 1 ⋯ w 1 and w i is the longest palindromic suffix of w p w p − 1 ⋯ w i , where 1 ≤ i ≤ p. This palindromic factorization is called UPS-factorization. Let luf (w) = p be the length of UPS-factorization of w. In 2017, it was proved that there is a constant c such that if w is a finite rich word and n = | w | then luf (w) ≤ c n ln n . We improve this result as follows: There are positive constants μ , κ such that if w is a finite rich word and n = | w | then luf (w) ≤ μ n e κ ln n . The constants c , μ , κ depend on the size of the alphabet. [ABSTRACT FROM AUTHOR]

The longest common palindromic subsequence (LCPS) problem is a variant of the longest common subsequence (LCS) problem. Given two input sequences A and B , the LCPS problem is to find the common subsequence of A and B such that the answer is a palindrome with the maximal length. The LCPS problem was first proposed by Chowdhury et al. (2014) [9] , who proposed a dynamic programming algorithm with O (m 2 n 2) time, where m and n denote the lengths of A and B , respectively. This paper proposes a diagonal-based algorithm for solving the LCPS problem with O (L (m − L) R log ⁡ n) time and O (R L) space, where R denotes the number of match pairs between A and B , and L denotes the LCPS length. In our algorithm, the 3-dimensional minima finding algorithm is invoked to overcome the domination problem. As experimental results show, our algorithm is efficient practically compared with some previously published algorithms. [ABSTRACT FROM AUTHOR]

We study the properties of the uncountable set of Stewart words. These are Toeplitz words specified by infinite sequences of Toeplitz patterns of the form αβγ , where α , β , γ is any permutation of the symbols 0,1,?. We determine the critical exponent of the Stewart words, prove that they avoid the pattern x x y y x x , find all factors that are palindromes, and determine their subword complexity. An interesting aspect of our work is that we use automata-theoretic methods and a decision procedure for automata to carry out the proofs. [ABSTRACT FROM AUTHOR]

Ventricle pathology changes and their severity overlap among the dementia classes to understand the pathogenesis of this disorder. In the present work, left and right ventricle variations in the severity classes of dementia were observed using optimizations, along with the dual deep learning techniques (DDLT). Segmentation of ventricle region was carried out using a multilevel threshold-based grey wolf optimization and palindrome detection method was executed to identify its symmetry. AlexNet and ResNet were used to extract the DDLT features which are then used for classification. The obtained results showed that the ventricle region was accurately delineated with a higher degree of correspondence which was >0.9. Furthermore, it was observed that the DDLT with multi-class SVM provided improved accuracy in the left ventricle with 84.8% and the right ventricle with 81.2%. Thus, the left ventricle variation was claimed to be a distinct indicator in demarcating different classes of dementia. [ABSTRACT FROM AUTHOR]

Everybody has certainly heard about palindromes: words that stay the same when read backwards. For instance, kayak, radar, or rotor. Mathematicians are interested in palindromic numbers: positive integers whose expansion in a certain integer base is a palindrome. The following problems are studied: palindromic primes, palindromic squares and higher powers, multi-base palindromic numbers, etc. In this paper, we define and study antipalindromic numbers: positive integers whose expansion in a certain integer base is an antipalindrome. We present new results concerning divisibility and antipalindromic primes, antipalindromic squares and higher powers, and multi-base antipalindromic numbers. We provide a user-friendly application for all studied questions.

The presence of 5-hydroxymethyl cytosine in DNA has been previously associated with ageing. Using in silico analysis of normal liver samples we presently observed that in 5-hydroxymethyl cytosine sequences, DNA methylation is dependent on the co-presence of G-quadruplexes and palindromes. This association exhibits discrete patterns depending on G-quadruplex and palindrome densities. DNase-Seq data show that 5-hydroxymethyl cytosine sequences are common among liver nucleosomes (p

This article, titled "Counting Parity Palindrome Compositions," discusses the concept of parity palindrome compositions (ppcs), which are compositions of positive integers that read the same forward and backward when reduced modulo 2. The article presents a recursive proof for the number of ppcs of 2n and 2n+1, which is given by the formula 2 * 3^(n-1). The proof involves rules for producing new ppcs based on the first and last parts of existing ppcs. The article provides examples and demonstrates that every ppc is produced precisely once. The theorem is proven for all n greater than or equal to 3. [Extracted from the article]

In human languages, a palindrome reads the same forward as backward (e.g., 'madam'). In regulatory DNA, a palindrome is an inverted sequence repeat that allows a transcription factor to bind as a homodimer or as a heterodimer with another type of transcription factor. Regulatory palindromes are typically imperfect, that is, the repeated sequences differ in at least one base pair, but the functional significance of this asymmetry remains poorly understood. Here, we review the use of imperfect palindromes in Drosophila photoreceptor differentiation and mammalian steroid receptor signaling. Moreover, we discuss mechanistic explanations for the predominance of imperfect palindromes over perfect palindromes in these two gene regulatory contexts. Lastly, we propose to elucidate whether specific imperfectly palindromic variants have specific regulatory functions in steroid receptor signaling and whether such variants can help predict transcriptional outcomes as well as the response of individual patients to drug treatments. [ABSTRACT FROM AUTHOR]

Genome editing technologies hold tremendous potential in biomedical research and drug development. Therefore, it is imperative to discover gene editing tools with superior cutting efficiency, good fidelity, and fewer genomic restrictions. Here, we report a CRISPR/Cas9 from Faecalibaculum rodentium, which is characterized by a simple PAM (5′-NNTA-3′) and a guide RNA length of 21–22 bp. We find that FrCas9 could achieve comparable efficiency and specificity to SpCas9. Interestingly, the PAM of FrCas9 presents a palindromic sequence, which greatly expands its targeting scope. Due to the PAM sequence, FrCas9 possesses double editing-windows for base editor and could directly target the TATA-box in eukaryotic promoters for TATA-box related diseases. Together, our results broaden the understanding of CRISPR/Cas-mediated genome engineering and establish FrCas9 as a safe and efficient platform for wide applications in research, biotechnology and therapeutics. Gene editing tools have tremendous potential for biomedical and basic research. Here the authors report a Cas9 from Faecalibaculum rodentium (FrCas9) that achieves efficient and specific gene editing in human cells with a NNTA palindrome PAMs for targeting optimal sites at TATA-boxes to enhance CRISPRa/i screening. [ABSTRACT FROM AUTHOR]

In this paper we consider a number of natural decision problems involving k -regular sequences. Specifically, they arise from considering • lower and upper bounds on growth rate; in particular boundedness, • images, • regularity (recognizability by a deterministic finite automaton) of preimages, and • factors, such as squares and palindromes, of such sequences. We show that these decision problems are undecidable. [ABSTRACT FROM AUTHOR]

This article discusses four newly discovered technopaegnia in Musaeus's Hero and Leander. The author argues that they are interconnected and constitute a complex literary game that spreads almost over the whole poem. The characteristic elements of the technopaegnia discussed in the article are the apostrophes, their palindromic character, and the numbers 9 and 10. Based on the latter feature, the author tries to show that lines 1-10 hide references to the Pythagoreans' teachings about the first ten numbers. [ABSTRACT FROM AUTHOR]

It is known that each word of length n contains at most n + 1 distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric d-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow us to construct new rich words from already known rich words. We focus on morphisms in Class Pret. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism ϕ in the class there exists a palindrome w such that ϕ(a)w is a first complete return word to w for each letter a. We characterize Pret morphisms which preserve richness over a binary alphabet. We also study marked Pret morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class Pret and that it preserves richness. [ABSTRACT FROM AUTHOR]

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A ν-palindrome, recently introduced by the author, is a natural number n, neither a multiple of 10 nor a decimal palindrome, such that the sum of the prime factors of n and their corresponding exponents larger than 1 is equal to that of the number formed by reversing the decimal digits of n. If n(k) denotes the k-copy repeated concatenation of the decimal digits of a natural number n, then whether n(k) is a ν-palindrome depends only on k modulo a natural number ω. The indicator function of n is defined to be the periodic function In: Z → {0, 1} such that for k ≥ 1, In(k) = 1 if n(k) is a ν-palindrome and In(k) = 0 if not. We give a procedure to express In as a kind of linear combination with some "subscripts", which can be implemented on a computer. We then show that the smallest possible ω, the fundamental period, is simply the least common multiple of those "subscripts". [ABSTRACT FROM AUTHOR]

We introduce two new classes of integers. The first class consists of numbers N for which there exists at least one integer A, such that the sum of A and the sum of digits of N, added to the reversal of the sum, gives N. The second class consists of numbers N for which there exists at least one integer A, such that the sum of A and the sum of the digits of N, multiplied by the reversal of the sum, gives N. All palindromes that either have an even number of digits or an odd number of digits and the middle digit even belong to the first class, and all squares of palindromes with at least two digits belong to the second class. These classes contain and are strictly larger than the classes of b-ARHardy numbers, respectively b-MRH numbers introduced in Nițică. [ABSTRACT FROM AUTHOR]