Odd Length Research Articles

On reversible DNA codes over the ring $${\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ based on the deletion distance

AbstractLet $${\mathfrak {R}}= {\mathbb {Z}}_4[u,v]/\langle u^2-2,uv-2,v^2,2u,2v\rangle$$ R = Z 4 [ u , v ] / ⟨ u 2 - 2 , u v - 2 , v 2 , 2 u , 2 v ⟩ be a ring, where $${\mathbb {Z}}_{4}$$ Z 4 is a ring of integers modulo 4. This ring $${\mathfrak {R}}$$ R is a local non-chain ring of characteristic 4. The main objective of this article is to construct reversible cyclic codes of odd length n over the ring $${\mathfrak {R}}.$$ R . Employing these reversible cyclic codes, we obtain reversible cyclic DNA codes of length n, based on the deletion distance over the ring $${\mathfrak {R}}.$$ R . We also construct a bijection $$\Gamma$$ Γ between the elements of the ring $${\mathfrak {R}}$$ R and $$S_{D_{16}}.$$ S D 16 . As an application of $$\Gamma ,$$ Γ , the reversibility problem which occurs in DNA k-bases has been solved. Moreover, we introduce a Gray map $$\Psi _{\hom }:{\mathfrak {R}}^{n}\rightarrow {\mathbb {F}}_{2}^{8n}$$ Ψ hom : R n → F 2 8 n with respect to homogeneous weight $$w_{\hom }$$ w hom over the ring $${\mathfrak {R}}$$ R . Further, we discuss the GC-content of DNA cyclic codes and their deletion distance. Moreover, we provide some examples of reversible DNA cyclic codes.

Shortest odd paths in undirected graphs with conservative weight functions

We consider the Shortest Odd Path problem, where given an undirected graph G, a weight function on its edges, and two vertices s and t in G, the aim is to find an (s,t)-path with odd length and, among all such paths, of minimum weight. For the case when the weight function is conservative, i.e., when every cycle has non-negative total weight, the complexity of the Shortest Odd Path problem had been open for 20 years, and was recently shown to be NP-hard.We give a polynomial-time algorithm for the special case when the weight function is conservative and the set E− of negative-weight edges forms a single tree. Our algorithm exploits the strong connection between Shortest Odd Path and the problem of finding two internally vertex-disjoint paths between two terminals in an undirected edge-weighted graph. It also relies on solving an intermediary problem variant called Shortest Parity-Constrained Odd Path where for certain edges we have parity constraints on their position along the path.Also, we exhibit two FPT algorithms for solving Shortest Odd Path. The first FPT algorithm is parameterized by |E−|, the number of negative edges, or more generally, by the maximum size of a matching in the subgraph of G spanned by E−, when the weight function is conservative. Our second FPT algorithm is parameterized by the treewidth of G, and the algorithm does not rely on conservativeness.

On reversibility problem in DNA bases over a class of rings

Let R k = Z 4 [ u 1 , u 2 , … , u k ] / ⟨ u i 2 − u i , u i u j − u j u i ⟩ be a non-chain ring of characteristic 4, where 1 ≤ i , j ≤ k and k ≥ 1 . In this article, we discuss reversible cyclic codes of odd lengths over the ring R k . We construct bijections between the elements of the ring R k and DNA- 2 k bases for k = 1, 2 in such a manner that the reversibility problem is solved. Employing these bijections, reversible complement cyclic codes of odd lengths are generated. Furthermore, we construct a Gray map Φ : R k n → Z 4 2 k n and as an application of the Gray map Φ, we obtain the GC-content of cyclic codes of arbitrary odd length over the ring R k . Meanwhile, we provide some examples of reversible cyclic codes of odd lengths over the ring R k for different values of k, and also obtain the Lee distances of these codes.

The signless Laplacian spectral radius of graphs without intersecting odd cycles

Let $F_{a_1,\dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,\dots, 2a_k+1,$ respectively, which intersect in exactly one common vertex, where $k\geq1$ and $a_1\ge a_2\ge \cdots\ge a_k\ge 1$. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all $F_{a_1,\dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.

Nearest-Neighbour Correlation Functions for the Supersymmetric XYZ Spin Chain and Painlevé VI

We study nearest-neighbour correlation functions for the ground state of the supersymmetric XYZ spin chain with odd length and periodic boundary conditions. Under a technical assumption related to the Q-operator of the corresponding eight-vertex model, we show that they can be expressed exactly in terms of the Painlevé VI tau functions sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s_n$$\\end{document} and s¯n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\bar{s}}}_n$$\\end{document} introduced by Bazhanov and Mangazeev. Furthermore, we give an interpretation of the correlation functions in terms of the Painlevé VI Hamiltonian.

The chromatic number of heptagraphs

AbstractA pentagraph is a graph without cycles of length 3 or 4 and without induced cycles of odd length at least 7, and a heptagraph is one without cycles of length less than 7 and without induced cycles of odd length at least 9. Chudnovsky and Seymour proved that every pentagraph is 3‐colorable. In this paper, we show that every heptagraph is 3‐colorable.

Kondo screening cloud scaling: impurity entanglement and magnetization

The screening of an impurity spin in the Kondo model occurs over a characteristic length scale ξ K , that defines the size of the Kondo screening cloud or ‘mist’. The presence of such a length scale in experimental and numerical results is rather subtle. A consistent way to show the presence of the screening cloud is to demonstrate scaling in the spatial correlations in terms of the single variable, r/ξK instead of the correlations depending on r and ξ K separately. Here we study the paradigmatic one channel Kondo model using a spin chain representation, with an impurity spin at one end of the chain coupled with a strength JK′ . Using Fermi liquid theory combined with numerical results, we obtain new high precision estimates of the non-universal terms in the entanglement entropy which leads to a verification of the expected non-integer ground-state degeneracy, g. This then allows us to study the impurity contribution to the entanglement in detail. If the impurity coupling JK′ is varied, a precise determination of ξ K can then be obtained. The length scale, ξ K , is then shown to characterize the scaling of both the uniform and alternating part of a measure of the magnetization of part of an odd length chain.

$ (\theta_i, \lambda) $-constacyclic codes and DNA codes over $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $

<p>In this paper, three new automorphisms were identified over the ring $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $ where $ u^3 = u^2 $. With the help of these automorphisms, the characteristic structures of the generator polynomials for the $ \theta_i $-cyclic codes and $ (\theta_i, \lambda) $-constacyclic codes of odd length on this ring were investigated. Also, for all the units over the ring, $ \mathbb{Z}_{4} $-images of $ \theta_i $-cyclic and $ (\theta_i, \lambda) $-constacyclic codes were reviewed with the associated codes based on determined transformations. Using these observations, new and optimal codes were obtained and presented in the table. In addition, a new transformation was identified that involved DNA base pairs with the elements of $ \mathbb{Z}_{4} $. Moreover, a unit reverse polynomial was created, and in this way a new generation method has been built to construct reversible DNA codes over this ring. Finally, this article was further enhanced with supporting examples of the DNA as a part of the study.</p>

Cycle Embedding in Enhanced Hypercubes with Faulty Vertices

The enhanced hypercube is a well-known variant of the hypercube and can be constructed from a hypercube by adding an edge to every pair of vertices with complementary addresses. Let Fv denote the set of faulty vertices in an n-dimensional enhanced hypercube Qn,k(1≤k≤n−1). In this paper, we conclude that if n≥2, then every fault-free edge of Qn,k−Fv lies on a fault-free cycle of every even length from 4 to 2n−2|Fv|, and if n(≥2) and k have the different parity, then every fault-free edge of Qn,k−Fv lies on a fault-free cycle of every possible odd length from n−k+4 to 2n−2|Fv|−1, where |Fv|≤n−2.

Determining the defect type in a graphene lattice

We consider four common defects that appear in an infinite single-layer graphene lattice, i.e. single vacancy V 1 ( 5 − 9 ) , Stone-Wales SW ( 55 − 77 ) , and double vacancies V 2 ( 5 − 8 − 5 ) and V 2 ( 555 − 777 ) . Investigating the honeycomb pattern of the graphene, we make use of the concept of shortest closed paths (periodic orbits) in the underlying topological structure. Using properties of the shortest closed paths of odd length we present and prove mathematically an algorithm that classifies which one of these defects occurs.

Role and interaction of bacterial sphingolipids in human health.

Sphingolipids, present in both higher animals and prokaryotes, involving in cell differentiation, pathogenesis and apoptosis in human physiological health. With increasing attention on the gut microbiome and its impact on wellbeing, there is a renewed focus on exploring bacterial sphingolipids. This review aims to consolidate the current understanding of bacterial sphingolipids and their impact on host health. Compared to mammalian sphingolipids, bacterial sphingolipids are characterized by odd chain lengths due to the presence of branched alkyl chains. Additionally, intestinal microbial sphingolipids can migrate from the gut to various host organs, affecting the immune system and metabolism. Furthermore, the intricate interplay between dietary sphingolipids and the gut microbiota is explored, shedding light on their complex relationship. Despite limited knowledge in this area, this review aims to raise awareness about the importance of bacterial sphingolipids and further our understanding of more uncharacterized bacterial sphingolipids and their significant role in maintaining host health.

VLSI Implementation of 3-Parallel FIR Filters through Coefficient Symmetry Property and Fast FIR Algorithm

In multirate digital signal processing systems, polyphase filters are frequently used. The challenges in multirate digital signal processing systems primarily relate to efficiently processing signals at different rates while maintaining high-quality output. These challenges include computational complexity, power consumption, and signal delay. The coefficient symmetry property plays a key role in systematically arranging of filter taps in symmetric manner to improve operating efficiency. With a major focus on using the coefficient symmetry characteristic, this paper addresses the VLSI implementation of traditional Type-1 polyphase FIR filters. Furthermore, this paper incorporated 3-parallel FIR filters especially for sequences with odd lengths in order to take full advantage of the coefficient symmetry characteristic. The Fast FIR Algorithm (FFA) method is used to increase overall efficiency. Furthermore, in order to achieve better performance, a modified FFA technique is introduced to enhance overall performance. The paper thoroughly investigates important VLSI factors, such as power consumption, utilization, and delay, and shows how, by utilizing coefficient symmetry and applying this modified FFA technique, filters that are optimized for both power consumption and area utilization are able to be implemented. The results illustrate a significant reduction in both area and power consumption, particularly for longer filters, attributed to the reduced number of multiplication operations inherent in the fast FIR algorithm approach. Additionally, image processing application using 3x3 FIR filter is executed.

On the chromatic number of graphs of odd girth without longer odd holes

An odd hole is an induced odd cycle of length at least five. Let l≥2 be an integer, and let Gl denote the family of graphs which have girth 2l+1 and have no holes of odd length at least 2l+3. Chudnovsky and Seymour proved that every graph in G2 is three-colorable. Following the idea of Chudnovsky and Seymour, Wu, Xu and Xu proved that every graph in G3 is three-colorable. In 2022, Wu, Xu and Xu conjectured that every graph G∈⋃l≥2Gl is three-colorable. In this paper, we prove that every graph G∈Gl with radius at most l+3 is three-colorable.

Completing the solution of the directed Oberwolfach problem with cycles of equal length

AbstractIn this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two‐table case with tables of equal odd length. We prove that the complete symmetric digraph on vertices, denoted , admits a resolvable decomposition into directed cycles of odd length . This completely settles the directed Oberwolfach problem with tables of uniform length.

Super oriented cycles in permutations

The degree of dissimilarity between genome sequences of homologous species is a measure of the evolutionary distance between them. It serves as a metric in the construction of phylogenetic trees, which depict the evolutionary relationships and common ancestry among different species. Given two genome sequences, evolutionary distance is determined by estimating the number of global mutations that transform one sequence to the other. The computation of the evolutionary distance is done by modelling a genome with the corresponding permutation. Global rearrangement operations such as transposition that model a particular genomic mutation are studied by employing a combinatorial structure known as a cycle graph of the corresponding permutation. A cycle in a cycle graph that has odd length is called an odd cycle. In the context of the problem of sorting by transpositions (SBT), a valid 2-move is a transposition that increases the number of odd cycles in the cycle graph by two. A super oriented cycle (SOC) is an odd cycle C where C and one of the resultant cycles admit valid 2-moves. The minimum number of mutations required to transform a species S into a related species T is the distance from S to T under that mutation. Christie opined that characterizing SOCs will improve the lower bound of the transposition distance. We characterize super oriented cycles. Equivalent transformations on permutations like reduction and (g,b)-split preserve the transposition distance of a given permutation and map SBT to the corresponding SBT on a transformed simpler permutation. We introduce merge, a novel equivalent transformation. These results have applications in computing transposition and other distances between related species.

Cycle Existence for All Edges in Folded Hypercubes under Scope Faults

The reliability of large-scale networks can be compromised by various factors such as natural disasters, human-induced incidents such as hacker attacks, bomb attacks, or even meteorite impacts, which can lead to failures in scope of processors or links. Therefore, ensuring fault tolerance in the interconnection network is vital to maintaining system reliability. The n-dimensional folded hypercube network structure, denoted as FQn, is constructed by adding an edge between every pair of vertices with complementary addresses from an n-dimensional hypercube, Qn. Notably, FQn exhibits distinct characteristics based on the dimensionality: it is bipartite for odd integers n≥3 and non-bipartite for even integers n≥2. Recently, in terms of the issue of how FQn performs in communication under regional or widespread destruction, we mentioned that in FQn, even when a pair of adjacent vertices encounter errors, any fault-free edge can still be embedded in cycles of various lengths. Additionally, even when the smallest communication ring experiences errors, it is still possible to embed cycles of any length. The smallest communication ring in FQn is observed to be the four-cycle ring. In order to further investigate the communication capabilities of FQn, we further discuss whether every fault-free edge will still be a part of every communication ring with different lengths when the smallest communication ring is compromised in FQn. In this study, we consider a fault-free edge e=(u,v) and F4={f1, f2,f3, f4} as the set of faulty extreme vertices for any four cycles in FQn. Our research focuses on investigating the cycle-embedding properties in FQn−F4, where the fault-free edges play a significant role. The following properties are demonstrated: (1) For n≥4 in FQn−F4, every even length cycle with a length ranging from 4 to 2n−4 contains a fault-free edge e; (2) For every even n≥4 in FQn−F4, every odd length cycle with a length ranging from n+1 to 2n−5 contains a fault-free edge e. These findings provide insights into the cycle-embedding capabilities of FQn, specifically in the context of fault tolerance when considering certain sets of faulty vertices.

On discrete properties of monotone mappings

Let [Formula: see text] and [Formula: see text] be a partial order on [Formula: see text]. We deal with properties of oriented graphs which correspond to the algebra [Formula: see text] in the case that [Formula: see text] is monotone with respect to [Formula: see text]. We derive that every mono-unary algebra except connected one with a cycle of odd length has the property that there exists a nontrivial partial order such that [Formula: see text] is monotone with respect to it. All mono-unary algebras such that there exists a linear order such that [Formula: see text] is monotone with respect to this order will be described; if the number of components of [Formula: see text] is infinite, then the number of such orders is equal to the cardinality of the power set of [Formula: see text].

Revisiting to N‐alkylation of multi‐dentate amines ligand for ATRP

AbstractThe ligands role in normal atom transfer radical polymerization (ATRP) is not only for controlling the system but also regulate the rate of polymerization depending the solubility in the reaction media. Solubility of the catalyst/ligand complexes in organic media is of particular importance to attain homogeneous polymerization conditions. The linear multi‐dentate amine ligands with a long aliphatic chain on the nitrogen atoms provide solubility of its metal complexes in organic solvents. However, the increasing length of the alkyl substituents induces steric effects and can alter the redox potential of the metal center. Herein, to understand the effect of alkyl length, we report the synthesis of tri‐, tetra‐dentate linear amine ligands, diethylenetriamine, triethylenetetramine having n‐alkyl halides containing odd carbon chain lengths (C3H7‐, C5H9‐), respectively. Well‐defined polymers were obtained in ATRP of S and MMA using synthesized ligands. Apparently, increasing the alkyl chain length was sufficient to provide homogeneous ATRP medium, which was not achieved with methyl‐substituted ligands (e.g., pentamethyldiethylenetriamine, PMDETA and hexamethyltriethylenetetramine, HMTETA). In combining ATRP results for odd (synthesized) and even carbon number alkylated ligands (literature), linear decreasing trend was observed on apparent rate constant (kpapp) by the increasing the alkyl chain length.

Poly(3-hydroxybutyrate) nanocomposites modified with even and odd chain length polyhydroxyalkanoates

Poly(3-hydroxybutyrate) (PHB) was blended with medium-chain-length PHAs (mcl-PHAs) for improving its flexibility while nanocellulose (NC) was added as a reinforcing agent. Even and odd-chain-length PHAs, having as main component poly(3-hydroxyoctanoate) (PHO) or poly(3-hydroxynonanoate) (PHN) were synthesized and served as PHB modifiers. The effects of PHO and PHN on the morphology, thermal, mechanical and biodegradation behaviors of PHB were different, especially in the presence of NC. The addition of mcl-PHAs decreased the storage modulus (E') of PHB blends by about 40 %. The further addition of NC mitigated this decrease bringing the E' of PHB/PHO/NC close to that of PHB and having a minor effect on the E' of PHB/PHN/NC. The biodegradability of PHB/PHN/NC was higher than that of PHB/PHO/NC, the latter's being close to that of neat PHB after soil burial for four months. The results showed a complex effect of NC, which enhanced the interaction between PHB and mcl-PHAs and decreased the size of PHO/PHN inclusions (1.9 ± 0.8/2.6 ± 0.9 μm) while increasing the accessibility of water and microorganisms during soil burial. The blown film extrusion test showed the ability of mcl-PHA and NC modified PHB to stretch forming uniform tube and supports the application of these biomaterials in the packaging sector.

The Complexity of the Matroid Homomorphism Problem

We show that for every binary matroid $N$ there is a graph $H_*$ such that for the graphic matroid $M_G$ of a graph $G$, there is a matroid-homomorphism from $M_G$ to $N$ if and only if there is a graph-homomorphism from $G$ to $H_*$. With this we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial time solvable if $N$ has a loop or has no circuits of odd length, and is otherwise $\rm{NP}$-complete. We also get dichotomies for the list, extension, and retraction versions of the problem.