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The main aim of the paper is to give the crossing number of join product W₅ + Dn for the wheel W₅ on six vertices, and Dn consisting of n isolated vertices. In the proofs, it will be extend the idea of the minimum numbers of crossings between two different subgraphs from the family of subgraphs which do not cross the edges of the graph W₅ onto the family of subgraphs that cross the edges of W₅ at least twice. Further, we give a conjecture that the crossing number of Wm + Dn is equal to Z ( m + 1 ) Z ( n ) + ( Z ( m ) − 1 ) ⌊ 1 2 ⌋ + n for m at least three, and where the Zarankiewicz’s number Z ( n ) = ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ is defined for n ≥ 1. Recently, our conjecture was proved for the graphs Wm + Dn , for any n = 3, 4, 5, by Klešč et al., and also for W₃ + Dn and W₄ + Dn due to the result by Klešč, Schrötter and by Staš, respectively. Clearly, the main result of the paper confirms the validity of this conjecture for the graph W₅ + Dn .

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p-dimensional dynamical systems of the form φ(b1,b2,⋯,bp)=(up(bp),u1(b1),⋯,up−1(bp−1)), where bj∈Hj (j∈{1,2,⋯,p}), p≥2 is an integer, and Hj (j∈{1,2,⋯,p}) are compact subintervals of the real line R=(−∞,+∞). uj:Hj→Hj+1(j=1,2,…,p−1) and up:Hp→H1 are continuous maps. Necessary and sufficient conditions for a class of cyclic permutation maps to have Li–Yorke chaos, distributional chaos in a sequence, distributional chaos, or Li–Yorke sensitivity are given. These results extend the existing ones.

An algorithm of digital logarithm calculation for the Galois field G F ( 257 ) is proposed. It is shown that this field is coupled with one of the most important existing standards that uses a digital representation of the signal through 256 levels. It is shown that for this case it is advisable to use the specifics of quasi-Mersenne prime numbers, representable in the form p = 2 n + 1 , which includes the number 257. For fields G F ( 2 n + 1 ) , an alternating encoding can be used, in which non-zero elements of the field are displayed through binary characters corresponding to the numbers + 1 and − 1. In such an encoding, multiplying a field element by 2 is reduced to a quasi-cyclic permutation of binary symbols (the permuted symbol changes sign). Proposed approach makes it possible to significantly simplify the design of computing devices for calculation of digital logarithm and multiplication of numbers modulo 257. A concrete scheme of a device for digital logarithm calculation in this field is presented. It is also shown that this circuit can be equipped with a universal adder modulo an arbitrary number, which makes it possible to implement any operations in the field under consideration. It is shown that proposed digital algorithm can also be used to reduce 256-valued logic operations to algebraic form. It is shown that the proposed approach is of significant interest for the development of UAV on-board computers operating as part of a group.

In the paper, we extend known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn , where the graph G consists of two leaves incident with two opposite vertices of one 4-cycle, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for a calculating the distances between all (k – 1)! vertices of the graph. Finally, by adding new edges to the graph G, we are able to obtain the crossing number of the join product with the discrete graph Dn for two other graphs. The methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations.

In this paper, we introduce and study a tripled system of three associated fractional differential equations. Prior to proceeding to the main results, the proposed system is converted into an equivalent integral form by the help of fractional calculus. Our approach is based on using the addressed tripled system with cyclic permutation boundary conditions. The existence and uniqueness of solutions are investigated. We employ the Banach and Krasnoselskii fixed point theorems to prove our main results. Illustrative examples are presented to explain the theoretical results.

The main aim of the paper is to establish the crossing numbers of the join products of the paths and the cycles on n vertices with a connected graph on five vertices isomorphic to the graph K1,1,3\\e obtained by removing one edge e incident with some vertex of order two from the complete tripartite graph K1,1,3. The proofs are done with the help of well-known exact values for the crossing numbers of the join products of subgraphs of the considered graph with paths and cycles. Finally, by adding some edges to the graph under consideration, we obtain the crossing numbers of the join products of other graphs with the paths and the cycles on n vertices.

Traditional gene set enrichment analysis falters when applied to large genomic domains, where neighboring genes often share functions. This spatial dependency creates misleading enrichments, mistaking mere physical proximity for genuine biological connections. Here we present Spatial Adjusted Gene Ontology (SAGO), a novel cyclic permutation-based approach, to tackle this challenge. SAGO separates enrichments due to spatial proximity from genuine biological links by incorporating the genes’ spatial arrangement into the analysis. We applied SAGO to various datasets in which the identified genomic intervals are large, including replication timing domains, large H3K9me3 and H3K27me3 domains, HiC compartments and lamina-associated domains (LADs). Intriguingly, applying SAGO to prostate cancer samples with large copy number alteration (CNA) domains eliminated most of the enriched GO terms, thus helping to accurately identify biologically relevant gene sets linked to oncogenic processes, free from spatial bias.

The main aim of the paper is to give the crossing number of the join product G + D n for the disconnected graph G of order five consisting of one isolated vertex and of one vertex incident with some vertex of the three-cycle, and D n consists of n isolated vertices. In the proofs, the idea of the new representation of the minimum numbers of crossings between two different subgraphs that do not cross the edges of the graph G by the graph of configurations G D in the considered drawing D of G + D n will be used. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph D n and with the path P n on n vertices for three other graphs.

In the paper, we extend known results concerning crossing numbers of join products of small graphs of order six with discrete graphs. The crossing number of the join product G ∗ + D n for the graph G ∗ on six vertices consists of one vertex which is adjacent with three non-consecutive vertices of the 5-cycle. The proofs were based on the idea of establishing minimum values of crossings between two different subgraphs that cross the edges of the graph G ∗ exactly once. These minimum symmetrical values are described in the individual symmetric tables.