Equilibrium Solution Research Articles

Critical and near-critical relaxation of holographic superfluids

We investigate the relaxation of holographic superfluids after quenches, when the end state is either tuned to be exactly at the critical point, or very close to it. By solving the bulk equations of motion numerically, we demonstrate that in the former case the system exhibits a power law falloff, as well as an emergent discrete scale invariance. The latter case is in the regime dominated by critical slowing down, and we show that there is an intermediate time range before the onset of late-time exponential falloff, where the system behaves similarly to the critical point with its power law falloff. We further postulate a phenomenological Gross-Pitaevskii-like equation (corresponding to model F of Hohenberg and Halperin) that is able to make quantitative predictions for the behavior of the holographic superfluid after near-critical quenches into the superfluid and normal phase. Intriguingly, all parameters of our phenomenological equation, which describes the nonlinear time evolution, may be fixed with information from the static equilibrium solutions and linear response theory. Published by the American Physical Society 2024

Adaptive event‐triggered safety control for multiplayer mixed zero‐sum game with partial inputs delay

AbstractThis paper proposes an adaptive safety control method applicable to a multiplayer mixed zero‐sum (MZS) game of nonlinear systems with partial inputs delay. Firstly, a framework is introduced involving N players, where player 1 and player N form a zero‐sum (ZS) game, and player 1 and players 2 to N‐1 form nonzero‐sum (NZS) games, with some players experiencing time delays. Subsequently, the system's value function is augmented with a control barrier function (CBF) to ensure that the system's state remains within a safe region. Secondly, to approximate Nash equilibrium solutions, the study employs adaptive dynamic programming (ADP) and utilizes a critic‐only neural network (NN) to approximate optimal solutions. Diverging from traditional time‐trigger methods, computational and communication load reduction is achieved by introducing a state‐related event trigger condition. The stability of the system is then meticulously analyzed using the Lyapunov theorem. Finally, to validate the effectiveness of the proposed method, the study provides a simulation example demonstrating its performance. In summary, this research introduces an efficient adaptive safety control method for addressing multiplayer MZS games with partial inputs delay, incorporating CBFs, ADP, and state‐related event triggering.

Intelligent Decision-Making Algorithm for UAV Swarm Confrontation Jamming: An M2AC-Based Approach

Unmanned aerial vehicle (UAV) swarm confrontation jamming offers a cost-effective and long-range countermeasure against hostile swarms. Intelligent decision-making is a key factor in ensuring its effectiveness. In response to the low-timeliness problem caused by linear programming in current algorithms, this paper proposes an intelligent decision-making algorithm for UAV swarm confrontation jamming based on the multi-agent actor–critic (M2AC) model. First, based on Markov games, an intelligent mathematical decision-making model is constructed to transform the confrontation jamming scenario into a symbolized mathematical problem. Second, the indicator function under this learning paradigm is designed by combining the actor–critic algorithm with Markov games. Finally, by employing a reinforcement learning algorithm with multithreaded parallel training–contrastive execution for solving the model, a Markov perfect equilibrium solution is obtained. The experimental results indicate that the algorithm based on M2AC can achieve faster training and decision-making speeds, while effectively obtaining a Markov perfect equilibrium solution. The training time is reduced to less than 50% compared to the baseline algorithm, with decision times maintained below 0.05 s across all simulation conditions. This helps alleviate the low-timeliness problem of UAV swarm confrontation jamming intelligent decision-making algorithms under highly dynamic real-time conditions, leading to more effective and efficient UAV swarm operations in various jamming and electronic warfare scenarios.

Light-Powered Directional Ion Transport via PFN-Br/MoS2 Heterogeneous Membranes: Band Alignment and Activation Energy Barrier Engineering.

Biological photoresponsive ion transport systems consistently attract researchers' attention owing to their remarkable functions of harvesting energy from nature and participating in visual perception systems. Designing and constructing artificial light-driven ion transport devices to mimic biological counterparts remains a challenge owing to fabrication limitations in nanoconfined spaces. Herein, a typical conjugated polyelectrolyte (PFN-Br) was assembled onto a laminated MoS2M using simple solution-processing vacuum filtration, resulting in a heterogeneous three- and two-dimensional nanoporous membrane. The designed band alignment between PFN-Br and MoS2 enables effective directional ion transport under irradiation in an equilibrium solution, even against a 30-fold concentration gradient. The staggered energy structure of PFN-Br and MoS2 enhances charge separation and establishes a photogenerated potential as the driving force for ion transport. Additionally, the activation energy barrier for ion transport across the heterogeneous membrane decreased by 60% after light irradiation, considerably improving ion transport flux. The easy fabrication and high performance of the membrane in light-powered ion transport provide promising approaches for designing nanofluidic devices with possible applications in energy conversion, light-enhanced biosensing, and photoresponsive ionic devices.

Guaranteed cost solution for discrete-time uncertain/nonlinear dynamic games

Motivated by an example of fiscal and monetary policy interaction of a national economy, the problem of uncertain/nonlinear two players discrete-time noncooperative games is investigated. Since the models of the systems are uncertain, the notion of Nash equilibrium solution is not suitable, instead, new Nash guaranteeing strategies and Nash guaranteed costs are defined. The system’s uncertainties and/or nonlinearities are assumed to be of quadratically bounded type. First, conditions of the Nash guaranteeing strategies are derived for general uncertain nonlinear systems. These results are specified for systems that have linear nominal part and quadratic cost functions. Approximate solutions are obtained by tractable quadratic matrix inequalities. To illustrate the application of the proposed method, two numerical examples are given.

A Hidden Convexity of Nonlinear Elasticity

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive ‘Hadamard instability;’ this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response.

Optimal transfer prices and technology in decentralized business groups

Decentralized (parent-affiliates) business groups encompass independent firms sharing production resources under common administrative or financial control. Due to the complexity of integrated decision-making, business groups often delegate pricing and technology decisions to separated units. With the growing prevalence of transfer technology in multi-plant operations, there arises a demand for modeling frameworks capturing the joint impact of transfer prices and technology on affiliates’ production. In this paper, we propose an operational methodology to integrate transfer prices and technology, as incentive mechanisms driving affiliates’ production decisions. This consists of a single-leader-multi-follower game where a parent firm (the leader) operates in a single-product market and delegates production to profit-maximizing affiliate firms (the followers). In addition to providing data-driven support for integrated decision-making, our framework reveals how managerial structures respond to the growing significance of intangible technology. In this line, based on the parent’s control of the value chain we present two specifications and solution approaches: full delegation and partial delegation. For the former, we provide an exact characterization of the equilibrium solution; whereas for business groups operating under full delegation, we provide tight lower and upper bounds, computable by state-of-the-art optimization solvers. Our empirical tests rely on a customized version of the Orbis data set. Figures show that our methodology not only integrates the substitutability between transfer prices and technology, as well as their dependency on the parent’s control of the value chain, but also gives rise to a flexible computational approach that is applicable to a wide range of decentralized business groups.

A bathtub model with nonlinear velocity–density relation

The bathtub model is an applicable and effective tool for simulating the characteristics of traffic dynamics in downtown areas during morning rush hour, especially the characteristics of hypercongested traffic. In order to reduce the complexity of the research, previous studies have adopted a linear velocity–density relationship. This has limitations in analyzing the cost and duration of hypercongestion in traffic dynamics. In this paper, we develop a bathtub model with one kind of nonlinear velocity–density Relation, i.e., Pipes–Munjal Relation, based on continuous scheduling preferences and analyze its solutions in both no-toll user equilibrium and social optimum states. Due to the intractability in solving this bathtub model, there exists local analytical solutions, but for a complete solution of the model, only numerical solutions are available. The results indicate that as the value of n increases, the model can simulate traffic dynamics with higher cost and longer duration of hypercongestion under equilibrium, and can simulate traffic states with lower system cost under social optimum. We also explore the results of implementing optimal time-varying toll to realize social optimum. This study expands the relevant theories of bathtub models and provides a new perspective for traffic congestion.

Location Is Everything: Influence of His-Tag Fusion Site on Properties of Adenylosuccinate Synthetase from Helicobacter pylori.

The requirement for fast and dependable protein purification methods is constant, either for functional studies of natural proteins or for the production of biotechnological protein products. The original procedure has to be formulated for each individual protein, and this demanding task was significantly simplified by the introduction of affinity tags. Helicobacter pylori adenylosuccinate synthetase (AdSS) is present in solution in a dynamic equilibrium of monomers and biologically active homodimers. The addition of the His6-tag on the C-terminus (C-His-AdSS) was proven to have a negligible effect on the characteristics of this enzyme. This paper shows that the same enzyme with the His6-tag fused on its N-terminus (N-His-AdSS) has a high tendency to precipitate. Circular dichroism and X-ray diffraction studies do not detect any structural change that could explain this propensity. However, the dynamic light scattering, differential scanning fluorimetry, and analytical ultracentrifugation measurements indicate that the monomer of this construct is prone to aggregation, which shifts the equilibrium towards the insoluble precipitant. In agreement, enzyme kinetics measurements showed reduced enzyme activity, but preserved affinity for the substrates, in comparison with the wild-type and C-His-AdSS. The presented results reinforce the notion that testing the influence of the tag on protein properties should not be overlooked.

Target Damage Calculation Method of Nash Equilibrium Solution Based on Particle Swarm between Projectile and Target Confrontation Game

In order to scientifically evaluate the target damage effect of uncertain intersection between projectile and target confrontation game, this paper establishes a basic model of target damage in the intersection between projectile and target game, which is through the spatial relationship between the explosive position of the projectile and the target position and gives a calculation method of target damage evaluation. The study applies Nash equilibrium theory to analyze income-loss functions in an adversarial game model. According to the definition of the possibility degree of interval number in uncertain multi-attribute decision-making, we proposed a method of solving the Nash equilibrium value of the income matrix of both sides of the game, which is using the interval possibility degree. We used quantitative calculations to analyze game strategies and changes in target damage under different states, such as fragment numbers, velocities, and angles of hit in the confrontation game between projectile and target. Based on the Nash equilibrium value solution method of the particle swarm optimization algorithm, we quantitatively calculated the fitness change relationship between the projectile and the target under different strategy sets. Based on the actual projectile confrontation game test, we analyze the damage probability effects of two projectiles at different explosion positions and different rendezvous angle angles on the three cabins. The results show that in the same area, the intersection angle is smaller, the probability of target damage is greater, and the intersection distance is farther, the damage probability is smaller.

Dynamics around the Earth–Moon triangular points in the Hill restricted 4-body problem

This paper investigates the motion of a small particle moving near the triangular points of the Earth–Moon system. The dynamics are modeled in the Hill restricted 4-body problem (HR4BP), which includes the effect of the Earth and Moon as in the circular restricted 3-body problem (CR3BP), as well as the direct and indirect effect of the Sun as a periodic time-dependent perturbation of the CR3BP. Due to the periodic perturbation, the triangular points of the CR3BP are no longer equilibrium solutions; rather, the triangular points are replaced by periodic orbits with the same period as the perturbation. Additionally, there is a 2:1 resonant periodic orbit that persists from the CR3BP into the HR4BP. In this work, we investigate the dynamics around these invariant objects by performing a center manifold reduction and computing families of 2-dimensional invariant tori and their linear normal behavior. We identify bifurcations and relationships between families. Mechanisms for transport between the Earth, L4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_4$$\\end{document}, and the Moon are discussed. Comparisons are made between the results presented here and in the bicircular problem (BCP).

A two-stage game-theoretical framework for Pricing & Management problem of shared mobility in a competitive market

ABSTRACT With the global proliferation of shared mobility systems, the vehicle-sharing market has witnessed an influx of companies and investors, thereby intensifying competition and potentially leading to suboptimal decision-making. Consequently, there is a pressing need for a modelling framework that specifically addresses operational and strategic-level policy design for multiple vehicle-sharing companies in a competitive market. In this study, we develop a two-stage game-theoretical framework to simulate the competitive process between multiple vehicle-sharing companies, facilitating their business decisions regarding pricing and management (P&M) problem, where management mainly include service area design, vehicle deployment, redirecting, and rebalancing. In the first stage of the framework, two companies operating under the same shared mobility service form a strategic alliance to compete against another vehicle-sharing service. Subsequently, in the second stage, these two companies engage in competition to maximise their individual profits. Based on the factors such as generalised travel cost, average acceptance level and vehicle availability, the competition process of each stage is formulated as a generalised Nash equilibrium problem. A linearisation method is employed for solving the equilibrium solution. Finally, we apply our proposed models to various vehicle-sharing systems and application scenarios to demonstrate their applicability. The numerical results illustrate that our model assists companies in making rational management decisions amidst intense competition while also revealing market competition landscape of vehicle-sharing systems based on different travel distances in the shared mobility market.

Influence of corn stalk and corn stalk biochar application on the strategy of phosphorus sorption and release in brown earth

AbstractThe characteristics of soil phosphorus (P) fixation and release have a critical impact on the availability of P and the risk of environmental loss. Investigation of the effects of corn stalk and corn stalk biochar on the P adsorption–desorption characteristics of brown earth can provide a theoretical basis for the rational return of crop stalk to the field and improving the availability of soil P. The Langmuir adsorption isotherm equation provided a better fit than the Freundlich equation for the curve of soil P adsorption and equilibrium solution P concentration. Biochar application increased the maximum adsorption capacity (MAC) and maximum buffer capacity (MBC) of the soil for P by 8.5% and 17.2%, respectively. Corn stalk and its biochar reduced the P adsorption saturation (DPS) by 41.8% and 31.9%, respectively, but increased the amount of readily desorbable P (RDP) by 39.7% and 154.4%, respectively. The application of corn stalk and its biochar improved the soil P desorption rate at substrate P concentrations of less than 5 mg L−1, whereas significant reductions in the soil P desorption rate were observed between 5 mg L−1 and 30 mg L−1, and no significant effects were observed at P concentrations over 40 mg L−1. In addition, the total P and Olsen‐P contents of the soil were significantly associated with the P adsorption–desorption process (p < .05). Therefore, corn stalk and corn stalk biochar can enhance the capacity of soil for P adsorption, with a stronger effect observed for biochar than for corn stalk, and the rate of P desorption from treated soil is dependent on P concentration in the soil solution.

Orbital Impulsive Pursuit–Evasion Game Formulation and Stackelberg Equilibrium Solutions

The spacecraft pursuit–evasion game in orbit with impulsive maneuvers is investigated based on dynamic game theory. First, the game models are developed progressively from the single-impulse game to the multi-impulse game. In the models, the two-sided optimal strategies of both players form a Stackelberg equilibrium in single-impulse games and constitute a multistage Stackelberg equilibrium in multi-impulse games. Then, the optimality necessary conditions for the two equilibriums are derived and simplified as two multipoint boundary value problems (MPBVPs). To handle the complicated MPBVPs, a rolling backward induction based on the Lambert algorithm accompanied by a homotopy shooting method is proposed to solve the equilibrium solutions. Simulations showed the effectiveness of the proposed methods and verified that the obtained solutions are the (multistage) Stackelberg equilibrium. Meanwhile, the differences between the Stackelberg equilibrium and the Nash equilibrium are illustrated in detail. Finally, an analysis of the influence of the observation delay on the game results is provided.

Energy-informed pathways: A fundamental approach to designing ballistic lunar transfers

Ballistic Lunar Transfers (BLTs) offer propellant-efficient paths to the Moon. The nature of such transfers relies on the combined gravitational influence of the Earth, Moon, and Sun to reach the lunar vicinity. These transfers are often constructed as point designs to deliver results for a specific application. This investigation focuses on a broader, more foundational approach to BLTs seeking general design characteristics. This investigation introduces dynamics-based approaches for characterizing low-energy cislunar transfers. Principles from dynamical systems theory are applied in the Earth-Moon-Sun Bicircular Restricted Four-body Problem to enable the construction of such paths. Structures such as the instantaneous equilibrium solutions and the evolution of the Hamiltonian offer a dynamical foundation for the mechanisms that govern low-energy transfers. A targeting framework enables the methodical construction of BLT families by applying dynamically informed initial conditions from perigee maps and equilibrium solutions. With solutions generated in the four-body model, several strategies are incorporated to transition the solutions into a higher-fidelity force model governed by the ephemerides of the celestial bodies. Incorporating transfer geometries from the Earth-Moon and Sun-B1 rotating frame offers an effective transition strategy for ballistic lunar transfers to such a higher-fidelity force model. Connections between the underlying dynamics that govern ballistic lunar transfers and the practical results in the mission design process are highlighted.

Dual-channel decision models for the transnational supply chain considering strategic inputs and compensation

Based on a transnational dual-channel supply chain consisting of the domestic manufacturer and retailer, this paper constructs four models, namely, the without tariff model, the tariff model, the retailer’s strategic inputs model, and the manufacturer’s compensation model, to investigate the impact of the tariff imposing on supply chain decisions, the effect of the retailer’s strategic inputs to hedge against tariffs, and the incentive effect of the manufacturer’s compensation. The results show that the tariff imposed by the foreign government leads to higher product prices and lower sales volumes, resulting in welfare losses for foreign consumers. When the domestic retailer makes strategic inputs, the prices of products in the foreign market decrease and the sales volumes increase, which increases the profits of the domestic retailer and manufacturer and improves the welfare of foreign consumers. The equilibrium solutions of the models also show that the manufacturer has an incentive to compensate the retailer for its strategic inputs; when the manufacturer compensates for the retailer’s strategic inputs, the profit of the retailer and the manufacturer will be improved again, and thus, the whole supply chain will achieve Pareto improvement.

Phase field modeling and numerical algorithm for two-phase dielectric fluid flows

We develop a method for modeling and simulating a class of two-phase flows consisting of two immiscible incompressible dielectric fluids and their interactions with imposed external electric fields in two and three dimensions. We first present a thermodynamically-consistent and reduction-consistent phase field model for two-phase dielectric fluids. The model honors the conservation laws and thermodynamic principles, and has the property that, if only one fluid component is present in the system, the two-phase formulation will exactly reduce to that of the corresponding single-phase system. In particular, this model accommodates an equilibrium solution that is compatible with the zero-velocity requirement based on physics. This property leads to a simpler method for simulating the equilibrium state of two-phase dielectric systems. We further present an efficient numerical algorithm, together with a spectral-element (for two dimensions) or a hybrid Fourier-spectral/spectral-element (for three dimensions) discretization in space, for simulating this class of problems. This algorithm computes different dynamic variables successively in an un-coupled fashion, and involves only coefficient matrices that are time-independent in the resultant linear algebraic systems upon discretization, even when the physical properties (e.g. permittivity, density, viscosity) of the two dielectric fluids are different. This property is crucial and enables us to employ fast Fourier transforms for three-dimensional problems. Ample numerical simulations of two-phase dielectric flows under imposed voltage are presented to demonstrate the performance of the method herein and to compare the simulation results with theoretical models and experimental data.

Differentiated pricing for the retail electricity provider optimizing demand response to renewable energy fluctuations

As renewable energy expands, its fluctuations challenge grid security. Demand response (DR) is a crucial solution for ensuring grid stability. This paper designs a two-stage optimization model for the retail electricity provider (REP), aiming to develop differentiated prices to incentivize multiple types of users in DR. In the first stage, the REP utilizes energy storage to adjust electricity purchases. In the second stage, a bi-layer optimization model is constructed. The REP participates in incentive-based DR and sets differentiated prices for lower-level users. Based on these prices, multiple types of users modify their electricity consumption behavior, aiming to maximize the combined satisfaction with the comfort and economy of electricity consumption. Finally, the Karush-Kuhn-Tucker (KKT) method is used to derive the equilibrium solution between user response quantity and price. The results indicate that (1) Combining energy storage and user-side DR boosts responsiveness by 28.5% compared to storage alone and 23.2% compared to user-side DR alone. (2) Energy storage allows the REP to create more compelling prices for residential users. (3) Differentiated pricing boosts industrial users participating in DR, resulting in over 50% higher response rate than uniform pricing. This study provides policy implications for grid managers to develop effective DR programs.

Solid-Liquid Phase Equilibria of the Pb2+, Ca2+, Mg2+//Cl--H2O Quaternary System and Its Subsystems at 303.2 K.

The solid-liquid phase equilibria of the ternary systems Pb2+, Ca2+//Cl--H2O, Pb2+, Mg2+//Cl--H2O, and Ca2+, Mg2+//Cl--H2O were investigated at atmospheric pressure and T = 303.2 K using the isothermal dissolution equilibrium method. Additionally, solid phase equilibria of the quaternary system Pb2+, Mg2+, and Ca2+//Cl--H2O were determined, and the corresponding stable phase diagrams and density-composition diagrams were constructed. The results indicate that the phase diagrams of Pb2+, Ca2+//Cl--H2O mainly consist of a ternary invariant point, two solubility curves, and four crystalline regions, while there are two ternary invariant points, three solubility curves, and six crystalline regions in the Pb2+, Mg2+//Cl--H2O and Ca2+, Mg2+//Cl--H2O systems. The results of the density-versus-w(CaCl2) plots of the various ternary systems confirm that the density of the equilibrium solution tends to go upward with the increase in the mass fraction of CaCl2. The density of various ternary systems reaches the maximum and equilibrium at the corresponding invariant point, and there is no significant change with the further increase in the CaCl2 mass fraction. Furthermore, the phase diagram of the Pb2+, Mg2+, Ca2+//Cl--H2O quaternary system includes two invariant points, five isothermal dissolution curves, and five crystalline regions. The order of the relative areas of the crystalline regions for the five salts is PbCl2 > CaCl2·2MgCl2·12H2O > 2PbCl2·3MgCl2·18H2O > MgCl2·6H2O > CaCl2·4H2O.

Uncoupled analysis of piles stabilized transmission tower slopes

It is a common engineering practice to adopt piles to improve the stability of potentially unstable slopes. In this paper, parametric analysis of a homogeneous transmission tower slope stabilized with a row of small diameter steel piles is conducted using an uncoupled numerical method in which the pile response and slope stability are considered separately. The transmission tower slope stability is analyzed using the limit equilibrium solutions, and the stabilizing pile-soil system is modeled as a simple beam on subgrade reaction foundation. Attempts have been made to study the effect of soil property, slope geometry, pile property, pile spacing and pile position on the safety factor of the transmission tower slope stabilized with small diameter steel piles. The study shows that they have a combined effect on the performance of small diameter steel piles to stabilize transmission tower slopes.