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Finding a cycle that contains certain prescribed vertices or edges within a graph is a classical problem in graph theory. It has been established that every $3$-connected cubic graph contains a cycle through any nine vertices, which is referred to as the well-known nine-point theorem. In this paper, it is demonstrated that every $3$-connected cubic graph contains a cycle through any four prescribed vertices and a $2$-path. This result is optimal.

Balanced hypercube is a variant of the hypercube structure; it offers desirable properties such as connectivity, diameter, and fault-tolerance. Cycles and tori are popular interconnection topologies and have been widely used in distributed-memory parallel computers. Scholars have developed diverse and numerous parallel algorithms of cycles and tori. This paper proposes an efficient algorithm, called reflected edge label sequence, to generate cycle structures in balanced hypercubes. With this algorithm, we can embed cycles of length $2^{2k+1}$, $1\le k<n$, into an $n$-dimensional balanced hypercube. The corresponding algorithm for embedding 2-dimensional $2^{2k}\times 2^{2k}$ tori can be constructed on the basis of this approach, and the time complexity is linear with respect to the size of a balanced hypercube.

Conventional combinational circuits are generally acyclic (feed-forward) but these circuits can have feedbacks (cycles) which will give more minimized expressions as compared to conventional combinational circuits. Deliberate incorporation of such cycles or feedbacks in conventional combinational circuits eventually results in reduction in number of literals in the expression of the combinational circuits. The reduction in literal counts decreases the number of gates required to implement the expressions of the combinational circuits. Hence, the decrease in number of gates leads to reduction in transistor counts or layout area for the circuits. A cyclic combinational circuit (CCC) is defined as the circuit whose output depends on present inputs only, but at the same time contains one or more feedbacks (cycles). This paper presents a simplified methodology for introduction of cycles (feedbacks) and finding expressions for the CCC. The methodology is applied on LGSynth93 benchmark circuits and a reduction up to 28% in literal counts for expressions of CCC was found which is higher than the reduction achieved by other methodologies. Further the methodology is applied to implement binary comparator which has got three multiple outputs using cyclic combinational technique. The circuits are verified through simulation in cadence virtuoso tools using 45nm technology. Based on simulation results, performance parameters like power consumption, propagation delay and layout area of CCC are compared with the conventional circuits.

We present some sufficient conditions for the existence of isochronous sections of planar differential systems. We consider isochronous sections at a critical point, a cycle, the boundary of a central or attraction region.

We study the stochastically forced Chen system in its parameter zone under the transition to chaos via period-doubling bifurcations. We suggest a stochastic sensitivity function technique for the analysis of stochastic cycles. We show that this approach allows to construct the dispersion ellipses of random trajectories for any Poincaré sections, and these ellipses reflect the essential features of a spatial arrangement of random trajectories near deterministic cycles. For the Chen system, we demonstrate a growth of stochastic sensitivity of the forced cycles under transition to chaos.

In this article we extend the theory of local bifurcation in one-dimensional autonomous maps to one-dimensional nonautonomous periodic maps. We give the necessary conditions for the main types of local bifurcation in one-dimensional periodic maps.

The stretch factor and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and the metric space, respectively. In this paper we show that computing the stretch factor of a rectilinear path in L₁ plane has a lower bound of Ω(n log n) in the algebraic computation tree model and describe a worst-case O(σn log² n) time algorithm for computing the stretch factor or maximum detour of a path embedded in the plane with a weighted fixed orientation metric defined by σ ≥ 2 vectors and a worst-case O(n log^d n) time algorithm to d ≥ 3 dimensions in L₁-metric. We generalize the algorithms to compute the stretch factor or maximum detour of trees and cycles in O(σn log^d+1 n) time. We also obtain an optimal O(n) time algorithm for computing the maximum detour of a monotone rectilinear path in L₁ plane.

Given a cycle of length k on a triangulated 2-manifold, we determine if it is null-homologous (bounds a surface) in O(n+k) optimal time and space where n is the size of the triangulation. Further, with a preprocessing step of O(n) time we answer the same query for any cycle of length k in O(g+k) time, g the genus of the 2-manifold. This is optimal for k < g.

Let G be a finite simple graph, and $J_G$ denote the binomial edge ideal of G. In this paper, we first compute the $\mathrm{\text{v}}$-number of binomial edge ideals corresponding to Cohen–Macaulay closed graphs. As a consequence, we obtain the $\mathrm{\text{v}}$-number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for $\mathrm{\text{v}}(J_G)$ using the number of vertices of the graph. We characterize all connected graphs G with $\mathrm{\text{v}}(J_G)=2$. We show that for a given pair $(k,m),k\le m$, there exists a graph G with an associated monomial edge ideal I having $\mathrm{\text{v}}$-number equal to k and regularity m. We also show that if $2k\le m$, then there exists a binomial edge ideal with $\mathrm{\text{v}}$-number k and regularity m. Finally, we compute $\mathrm{\text{v}}$-number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the $\mathrm{\text{v}}$-number of powers of a graded ideal having linear powers, for the class of binomial edge ideals.

We consider a prey-predator-superpredator model. Assuming that the growth rates of the three populations are highly diversified, we prove that under some conditions on the parameters of the model, the three populations can coexist through a cycle for which we give the period. Besides, we give the laws (entrance-exit functions) that govern the prey and predator regeneration when they tend to extinction.

On the basis of a conceptual analysis of the Informational Macrodynamics' equations, this paper introduces a unified information systemic model of evolution including the information functional mechanisms of self-organization, mutation, adaptation, control, the double spiral's genetics with coding language, the system's generation, decaying, and heredity, considered as the information regularities of developing macrosystems.

Here we have considered an n-standby system with one and more than one repair facility (ies) for interference (or stress–strength) models. The system is working under impacts of stresses. Each impact is called a cycle. When a component fails it goes for repair; the repair policy is first-come-first-serve (FCFS). The failure times of components, system and repair times all are measured in cycles. The reliability of the system at the Nth cycle is evaluated in different cases. When stress–strength distributions both are either exponential or gamma or normal the reliability of the system is numerically evaluated for given n, N and for some particular values of the parameters involved and are tabulated. The numerical values of the system reliability are on the expected line.

An efficient solution to the misleading solutions of decision-making problems is the study of consistency when the decision makers express their opinions by means of fuzzy preference relations. To elucidate the consistency of HFPRs, the S-ordinal consistency of HFPRs was proposed, and S-OCI was also proposed to evaluate the degree of consistency of a hesitant fuzzy preference relation by calculating the unreasonable 3-cycles in the directed graph. Two novel methods were also proposed for calculation of the S-OCI. Moreover, the inconsistent judgment in hesitation fuzzy preference relation was developed. In order to repair the inconsistency of a hesitant fuzzy preference relation, an algorithm for finding and removing 3-cycles in digraph was developed. Finally, some illustrative examples were given to prove the effectiveness of the method.

For a finite commutative ring R and a positive integer k ≥ 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = a^k. In this paper, we investigate the iteration digraphs G(𝔽_p^rC_n, k) of 𝔽_p^rC_n, the group ring of a cyclic group C_n over a finite field 𝔽_p^r. We study the cycle structure of G(𝔽_p^rC_n, k), and explore the symmetric digraphs. Finally, we obtain necessary and sufficient conditions on 𝔽_p^rC_n and k such that G(𝔽_p^rC_n, k) is semiregular.

Let $\mathrm{\Delta }$ be a one-dimensional simplicial complex on $\{1,2,\dots ,s\}$ and $S$ the polynomial ring $K[x_1,\dots ,x_s]$ over a field $K$. The explicit formula for $a_0(S/I_{\mathrm{\Delta }}^n)$ is presented when $\mathrm{\text{girth}}(\mathrm{\Delta })\ge 4$. If $\mathrm{\text{girth}}(\mathrm{\Delta })=3$ we characterize the simplicial complexes $\mathrm{\Delta }$ for which $a_0(S/I_{\mathrm{\Delta }}^n)=3n-1$ or $3n-2$.

In this study, the effect of the dynamic bike fitting, which was defined as lower limb vertical alignment by inserted wedges, was evaluated. Twelve non-cyclists are participated in this experiment. For comparing pedaling performance between before and after fitting, sub-maximal pedaling tests were conducted for 2 min. The results showed that both the range of motion of the knee joint (medio-lateral direction) which represented pedaling trajectory and the coefficient of variance of pedaling power were decreased in non-cyclists. This result indicated that the dynamic fitting had a positive effect such as increased pedaling kinematic stability and kinetic efficiency on pedaling performance. Supplementary pedaling test was performed by one of the best cyclist, who was a member of the national team and a middle and long distance cyclist. The result of the cyclist test was also similar to the patterns of 12 non-cyclists. It is believed that this study has the meaning of the preliminary study to evaluate the effects of the dynamic fitting. Further study is necessary to evaluate the effects of the fitting through enough subjects and repeated tests by measuring muscle activities and forces.

Minimal edge controllability of directed networks is investigated in this paper. A new edge dynamics model is first introduced with two nonzero parameters describing the linear relationship between the node states and the edge states. Three different digraphs as skeleton structures for minimal edge controllability are analyzed. The conditions ensuring both node controllability and edge controllability for these three digraphs are presented, respectively. It is found that cycles in these networks play an important role in edge controllability. The notion of minimal edge controllability is then extended to signed digraphs. It is shown that the minimal edge controllability of a signed cycle depends on the number of edges with negative weights, regardless of the placement of the negative weights on the edges. Some examples are presented for illustration and verification.

Monounary algebras are the most simple type of an algebraic structure. Oriented graphs with one outgoing arrow from every vertex represent them. The aim of this paper is to point out the interdisciplinary relationships concerning this structure. Bernoulli shift is a paradigmatic mapping in dynamical systems. It is also called dyadic, bit shift, doubling or sawtooth. We offer a look at the properties of this mapping via monounary algebras.

PVC disulfide (2SPVC) was synthesized by solution crosslink and its molecular structure was confirmed by infrared spectrum. 2SPVC's specific area is 36.1 m²·g^-1 tested by stand BET method, and granularity experiment gives out the particle size of d_0.5 = 11.3 μm. With SEM (Scanning Electron Microscope) experiment the surface morphology and particle shape of 2SPVC were observed. Cyclic voltammetry (scan rate: 0.5 mV·s^-1) shows that 2SPVC experience an obvious S–S redox reaction in charge-discharge process. When 2SPVC was used as cathode material for secondary lithium battery in a 1 mol·L^-1 solution of lithium bis(trifluoromethylsulfonyl) imide (Li(CF₃SO₂)₂N) in a 5:45:50 volume ratio mixture of o-xylene (oxy), diglyme (DG) and dimethoxymethane (DME) at 30°C, the first discharge capacity of 2SPVC is about 400.3 mAh·g^-1 which is very close to its theoretical value (410.5 mAh·g^-1) at a constant discharge current of 15 mA·g^-1. It can retain at about 346.1 mAh·g^-1 of discharge capacity after 30 charge-discharge cycles. So 2SPVC is a very promising cathode candidate for rechargeable lithium batteries.

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k₁) ≃ G(H, k₂). We also discuss the problem when G(H₁, k) is isomorphic to G(H₂, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd(p, k)=1.