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The problem of seeking Nash equilibrium (NE) in multicluster games is studied. Different from the existing multicluster game studies, the participants considered in this article have Euler–Lagrange dynamics and cannot directly obtain the information of nonneighbor participants. Agents can only communicate through directed communication graphs. In addition, there are coupling constraints between agent decisions in the same cluster. Under the widely used assumptions, two distributed algorithms are proposed to solve the NE of the multicluster game based on gradient descent, state feedback, and consensus protocol. The convergence of the two algorithms is analyzed by singular perturbation analysis and variational analysis. The theoretical results show that the first algorithm achieves exponential convergence when the parameters are certain, while the other algorithm also achieves global asymptotic convergence when the parameters are uncertain. Finally, the effectiveness of the search strategy is verified by numerical examples.

In this article, we investigate the problem of seeking Nash equilibrium (NE) in multiagent systems within cooperation–competition networks. Each agent aims to optimize a total cost function that accounts for its own interests as well as those of its cooperators, considering both cooperative and noncooperative interactions with other agents. It is worth noting that, unlike in existing N-coalition games, the agents in this study only have knowledge of whether they are cooperative or noncooperative with their neighboring agents; they do not have information about non-neighboring agents who might be cooperators. As a result, due to potential disconnections in the communication topology within the cluster, it is not possible to consider the entire cluster as a virtual player to optimize its objective functions. To address this issue, we developed an algorithm using the singular perturbation technique, which divides the system into two distinct timescales. We propose a novel estimation algorithm to estimate the total cost function of disconnected subnetworks within the fast system. In the slow system, the search for NE is based on a gradient algorithm, while the Lyapunov stability theory is utilized to analyze the convergence of the algorithms. Furthermore, we extend the problem to accommodate scenarios where multiple subnetworks exist within the network. Numerical simulations are conducted to demonstrate their capability for resolving the noncooperative game problem in cooperation–competition networks.

Origami tessellations have shown the potential to be utilized for approximating curved target surfaces by folding a set of elemental units. As those units are in partially folded states, the origami approximation captures the outline of the target while demonstrating a corrugated surface. In this paper, we focus on the cylindrical surface and propose a method for constructing its discrete version via square or rectangular units with crease patterns inspired by the waterbomb tessellation. We unify the unit size through optimization to realize cost-efficient constructions. Three-dimensional curved structures are deployed by folding collinear creases between adjacent rows of units and the other creases in each row are remained unfolded. The deployed structure can still be folded to a compact state by folding all creases. To flexibly approximate the outline of the target surface, we analyse relationships between the width and height of the rectangular unit and utilize variations of the waterbomb tessellation to control the region of the origami structure when fully folded. The proposed method provides a novel solution to the so-called inverse-origami-design problem for generating foldable cylindrical surfaces using unfolded waterbomb origami units.Graphical Abstract

In this paper, a zero-sum game Nash equilibrium computation problem with event-triggered communication is investigated under an undirected weight-balanced multi-agent network. A novel distributed event-triggered projection subgradient algorithm is developed to reduce the communication burden within the subnetworks. In the proposed algorithm, when the difference between the current state of the agent and the state of the last trigger time exceeds a given threshold, the agent will be triggered to communicate with its neighbours. Moreover, we prove that all agents converge to Nash equilibrium by the proposed algorithm. Finally, two simulation examples verify that our algorithm not only reduces the communication burden but also ensures that the convergence speed and accuracy are close to that of the time-triggered method under the appropriate threshold.

In this article, we study the event-triggered distributed optimization problem of second-order nonlinear multiagent systems under undirected and connected communication topologies. To reduce the cost of information exchange in an optimization problem, a distributed event-triggered communication-based algorithm is proposed. In an event-triggered mechanism, the information interaction process of the network is controlled by a set of trigger conditions. The trigger condition of each agent only requires its own state information and its neighboring agents’ state information. Each agent updates its own states only at the triggering time. In order to achieve the optimal result of the optimization problem and reduce the cost of communicatio, we propose a distributed event-triggered algorithm based on gradient descent. Moreover, we analyze the convergence of the algorithm by constructing a suitable Lyapunov function. Furthermore, we prove that the Zeno behavior can be avoid. Finally, we provide examples to validate the effectiveness of the obtained result.

In this paper, a zero-sum game Nash equilibrium computation problem with a common constraint set is investigated under two time-varying multi-agent subnetworks, where the two subnetworks have opposite payoff function. A novel distributed projection subgradient algorithm with random sleep scheme is developed to reduce the calculation amount of agents in the process of computing Nash equilibrium. In our algorithm, each agent is determined by an independent identically distributed Bernoulli decision to compute the subgradient and perform the projection operation or to keep the previous consensus estimate, it effectively reduces the amount of computation and calculation time. Moreover, the traditional assumption of stepsize adopted in the existing methods is removed, and the stepsizes in our algorithm are randomized diminishing. Besides, we prove that all agents converge to Nash equilibrium with probability 1 by our algorithm. Finally, a simulation example verifies the validity of our algorithm.

The aim of this study was to explore the association between fat distribution and volumetric bone mineral density (vBMD) in Chinese Adults. Using a cross-sectional investigation of 867 participants including 521 women and 346 men from China, vBMD and subcutaneous adipose tissue (SCAT) or visceral adipose tissue (VAT) were measured accordingly by quantitative computed tomography. The peak vBMD values of the spine were observed at ages 30–39 years in women and at ages 20–29 years in men. In women, the peak values of VAT and SCAT were observed, respectively, at ages ≥70 years and at the age range of 60–69 years. In men, the peak values of VAT and SCAT were observed, respectively, at ages ≥70 years and at ages 30–39 years. Using the correlation tests, there was no correlation between SCAT and vBMD in both genders. Most relationships between VAT and BMD were negative (r= −0.204, P< 0.01, in premenopausal women; r= −0.150, P< 0.05, in postmenopausal women; and r= −0.181, P< 0.05, in middle-aged men). After multiple linear regression analysis, no correlations were observed. There appears to be no correlation between fat distribution and vBMD in Chinese adults, and further studies are needed to explore associations between fat distribution and vBMD.

It is known that tibial diaphyseal fractures are often associated with the posterior malleolar fracture (PMF). There are a few studies on tibial shaft fractures with respect to posterior malleolus fracture. However, we found that the incidence of PMF was higher than the previously reported.