Equilibrium Solution Research Articles

The Navier-Stokes equation and a fully developed turbulence

In fairly general conditions we give explicit (smooth) solutions for the potential flow. We show that, rigorously speaking, the equations of the fluid mechanics have not rotational solutions. However, within the usual approximations of an incompressible fluid and an isentropic flow, the Navier-Stokes equation has approximate vorticial (rotational) solutions, generated by viscosity. However, in general, these vortices are unstable, and a discrete distribution of vorticial solutions is not in me chanical equilibrium; it forms an unstable vorticial liquid. On the other hand, these solutions may exhibit turbulent, fluctuating instabilities for large variations of the velocity over short distances. We represent a fully developed turbulence as a homogeneous, isotropic and highly-fluctuating distribution of singular centres of turbulence. A regular mean flow can be included. In these circumstances the Navier-Stokes equation exhibits three time scales. The equations of the mean flow can be disentangled from the equations of the fluctuating part, which is reduced to a vanishing inertial term. This latter equation is not satisfied after averaging out the temporal fluctuations. For a homogeneous and isotropic distribution of non-singular turbulence centres the equation for the inertial term is satisfied trivially, i.e. both the average fluctuating velocity and the average fluctuating inertial term are zero. If the velocity is singular at the turbulence centres, we are left with a quasi-ideal classical gas of singularities, or a solution of singularities (solute) in quasi thermal equilibrium in the background fluid (solvent). This is an example of an emergent dynamics. We give three examples of vorticial liquids.

Static analysis of masonry polygonal domes with membrane forces

Masonry polygonal domes are parts of several historic monuments and masonry buildings. The analysis of polygonal domes with a static approach is here presented, considering the structure made of vault cells and ribs. The vaults are portions of a cylindrical surface with single curvature in the meridian direction, and translation of the generatrix curve in the direction of the cylinder axis. The system is divided into rigid elements with interfaces. The first part of the work regards the geometry of the rigid elements, in particular the angles that characterize the shape of these. Then the equilibrium equations applied to every element are presented, with a procedure to calculate the membrane response for the self-weight in a non-redundant system, under symmetry conditions. The sum of the solution for the vaults is combined with effects of the weight of ribs and radial walls. The use of the equilibrium solution for pointed domes, domes with a lantern and oculus or additional load to the dome self-weight is shown. A limit analysis static approach is formulated calculating the maximum load relative to the equilibrium of the rigid elements considering the interfaces tensile strength and limited cohesion and friction for the shear resistance.

Signaling the Capability of New Counterterrorism Technologies in the Face of a Strategic Threat

ABSTRACTWhen implementing new security technologies (e.g., cargo scanners at ports), security and defense agencies face important decisions regarding the disclosure of related information to the public. Public disclosure of these technologies can grant adversarial organizations access to critical security‐related information, enabling them to potentially develop or adjust their attack strategies to circumvent defense systems. Alternatively, withholding information about newly deployed technologies can create uncertainties for adversaries regarding the effectiveness of defense systems. In this work, we study defensive information disclosure related to the deployment of new security technology, focusing on the level of technical information released by the defender and how this affects the adversary's (i) belief about the technology's capability (in detecting or otherwise thwarting an attack) and (ii) attack plan. Specifically, we introduce a novel signaling model in which the defender has private information about the (level of) capability of her technology, and decides on the level of related technical detail to be disclosed. The attacker, upon receiving the defender's signal, updates his belief about the capability of the technology and subsequently decides whether to launch an attack. We obtain and prove closed‐form equilibrium solutions to the model and conduct numerical sensitivity analyses to understand how the equilibrium strategies change as the model's data varies. Based on the equilibrium conditions, we also characterize and analyze four unique scenarios that partition the solution space based on relations between the defender's signaling costs.

Equilibrium Control in Uncertain Linear Quadratic Differential Games with V-Jumps and State Delays: A Case Study on Carbon Emission Reduction

Uncertainty, time delays, and jumps often coexist in dynamic game problems due to the complexity of the environment. To address such issues, we can utilize uncertain delay differential equations with jumps to depict the dynamic changes in differential game problems that involve uncertain noise, delays, and jumps. In this paper, we first examine a linear quadratic differential game optimistic value problem within an uncertain environment characterized by jumps and delays. By applying the Z(x,y) transform, we convert the infinite-dimensional problem into a finite-dimensional one. We then demonstrate that the condition for the existence of a Nash equilibrium strategy is equivalent to the existence of solutions to two cross-coupled matrix Riccati equations. Furthermore, we explore the saddle point equilibrium strategy of the linear quadratic differential game optimistic value model and derive the corresponding saddle point equilibrium solution. Finally, we apply our results to solve a carbon emission reduction game problem.

Automatic construction and verification algorithm for smart contracts based on formal verification

As an emerging technology, blockchain demonstrates strong potential for applications in digital finance. As a core component of blockchain, the security and reliability of smart contracts is crucial. To ensure the high reliability of smart contracts, this study employs formal construction and verification techniques based on game theory. Initially, the profit function is defined using distortion techniques, and a game model for supply chain participation is designed. However, the equilibrium solution of the two-party game does not represent the optimal solution for the supply chain system. Therefore, the study introduces third-party participation to optimize the equilibrium solution. Finally, a probability model detection method is used to verify the constructed smart contract model. The results show that the supply chain model, analyzed through formal methods, has attributes consistent with theoretical analysis. Consequently, the research on automatic construction and verification algorithms for smart contracts based on formal verification proves to be effective and feasible in practical applications.

Well-balanced High-order Finite Difference Weighted Essentially Nonoscillatory Schemes for a First-order Z4 Formulation of the Einstein Field Equations

We develop a new class of high-order accurate well-balanced finite difference (FD) weighted essentially nonoscillatory (WENO) methods for numerical general relativity (GR), which can be applied to any first-order reduction of the Einstein field equations, even if nonconservative terms are present. We choose the first-order nonconservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. By introducing auxiliary variables, the vacuum Einstein equations in first-order form constitute a partial differential equation system of 54 equations that is naturally nonconservative. We show how to design FD-WENO schemes that can handle nonconservative products. Different variants of FD WENO are discussed, with an eye to their suitability for higher-order accurate formulations for numerical GR. We successfully solve a set of fundamental tests of numerical GR with up to ninth-order spatial accuracy. Due to their intrinsic robustness, flexibility, and ease of implementation, FD-WENO schemes can effectively replace traditional central finite differencing in any first-order formulation of the Einstein field equations, without any artificial viscosity. When used in combination with well-balancing, the new numerical schemes preserve stationary equilibrium solutions of the Einstein equations exactly. This is particularly relevant in view of the numerical study of the quasi-normal modes of oscillations of relevant astrophysical sources. In conclusion, general relativistic high-energy astrophysics could benefit from this new class of numerical schemes and the ecosystem of desirable capabilities built around them.

Dynamics of delay epidemic model with periodic transmission rate

We introduce novel epidemic models with single and two strains described by systems of delay differential equations with a periodic time-dependent disease transmission rate, and based on the number of newly infected individuals. Transitions between infected, recovered, and returning to susceptible compartments due to waning immunity are determined by the accompanying time delays. Positiveness, existence and uniqueness of solutions are demonstrated with the help of fixed point theory. Reducing delay differential equations to integral equations facilitates determining the analytical estimation of the equilibrium solutions. When there are two strains, they compete together, and the strain with a larger basic reproduction number dominates in the population. However, both strains coexist, and the magnitudes of epidemic outbreaks are governed by the basic reproduction numbers. The outlines of this work are verified through comparison with seasonal influenza data.

Off-policy reinforcement-learning-based fault-tolerant [formula omitted] control for topside separation systems with time-varying uncertainties

The topside separation system plays a pivotal role in the treatment of produced water within offshore oil and gas production operations. Due to high-humidity and salt-infested marine environments, topside separation systems are susceptible to dynamic model variations and valve faults. In this work, fault-tolerant control (FTC) of topside separation systems subject to structural uncertainties and slugging disturbances is studied. The system is configured as a cascade structure, comprising a water level control subsystem and a pressure-drop-ratio (PDR) control subsystem. A fault-tolerant H∞ control framework is developed to cope with actuator faults and slugging disturbances. To enhance control performance in the presence of actuator faults and model uncertainties while reducing sensitivity to slugging disturbances, the fault-tolerant H∞ control problem for the topside separation system is established as the two-player differential game problem. In addition, a Nash equilibrium solution for the fault-tolerant H∞ control problem is achieved through the solution of the game algebraic Riccati equation (GARE). A model-free approach is presented to implement the proposed fault-tolerant H∞ control method using off-policy reinforcement learning (RL). Simulation studies demonstrate the effectiveness of the solution.

Control-Oriented Free-Boundary Equilibrium Solver for Tokamaks

A free-boundary equilibrium solver for an axisymmetric tokamak geometry was developed based on the finite difference method and Picard iteration in a rectangular computational area. The solver can run either in forward mode, where external coil currents are prescribed until the converged magnetic flux function ψ(R,Z) map is achieved, or in inverse mode, where the desired plasma boundary, with or without an X-point, is prescribed to determine the required coil currents. The equilibrium solutions are made consistent with prescribed plasma parameters, such as the total plasma current, poloidal beta, or safety factor at a specified flux surface. To verify the mathematical correctness and accuracy of the solver, the solution obtained using this numerical solver was compared with that from an analytic fixed-boundary equilibrium solver based on the EAST geometry. Additionally, the proposed solver was benchmarked against another numerical solver based on the finite-element and Newton-iteration methods in a triangular-based mesh. Finally, the proposed solver was compared with equilibrium reconstruction results in DIII-D experiments.

A fractional order mathematical model for the omicron: a new variant of COVID-19

For the purpose of analyzing the Omicron pandemic, we build a novel SEIaIsIoHR mathematical model. The fundamental properties of the model are studied, including boundedness, uniqueness, and existence. The boundedness of the model’s solution is examined by solving the fractional Gronwall’s inequality using the Laplace transform method. Using the Picard-Lindelöf theorem, we verify the solution’s existence and uniqueness. The next-generation matrix is used to compute Ro (the basic reproduction number), which is significant in the mathematical modeling of infectious diseases. It is demonstrated that both the endemic and disease-free equilibrium solutions are globally and locally asymptotically stable. We also conducted a sensitivity analysis of the parameters based on Ro . Moreover, the model is extended by employing optimal control theory to examine the effects of certain control strategies. Finally, numerical simulations are performed to validate the model.

Modeling analysis framework for metallic currency system

Small coins, large coins, and ingots circulated in parallel in the metallic currency system, and they were sorted by different mechanisms based on the transaction value of each form. This is the first attempt to model their boundaries. We construct a model of the market mechanism of the mint which derives two models to achieve our goal. The first model derived is a supply-demand model of small coins that determines the demarcation between small coins and large coins, and the second is a game model whose equilibrium solution determines the dividing line between large coins and ingots.

Simulation of the Stackelberg–Hotelling Game

This work studies the Hotelling game with sequential choice of prices, that is, the Stackelberg–Hotelling (SHOT) game. The game is studied through numerical simulation, which provides the subgame perfect equilibrium solution not only in the unrestricted game but also in the game with reservation cost and with elastic demand. The simulation technique is tested first in the unconstrained game, where the analytical subgame perfect equilibrium solution was already known. Then, the numerical procedure is generalized to cope with the SHOT game with reservation cost and with elastic demand. These enriched formulations of the SHOT game have not been studied so far, so this article provides an exploratory study of them.

Studying Dynamical and Hydraulic Characteristics of the Hydraulically Suspended Passive Shutdown Subassembly (HS-PSS) and Validating with a Prototypic Test Sample

The pool-type demonstrative China Fast Reactor 600 (CFR-600) adopted a series of improved safety designs, among which the hydraulic suspended passive shutdown subassembly (HS-PSS) is employed for inherent safety enhancement, especially suitable against unprotected loss of flow accident (ULOF). In this article, functional requirements for HS-PSS design in hydro-dynamic aspects are proposed, with the corresponding performance indicators discussed. To address these functional requirements, qualitative analysis on the equilibrium solution properties of the nonlinear dynamical model equation of the hydraulic moving body (HMB) in the HS-PSS are conducted, which leads to the determination of an applicable design parameter domain of the HMB for its practical design from the broad range of structural and parametric design options available for HS-PSS. Furtherly, hydraulically characterizing and modeling the constituent paths and consequent fluid network, hydraulic characteristics of the HS-PSS, as well as the coupled hydro-dynamic motion behaviors of the HMB for suspension and dropping states, were simulated and test-validated. Considering the HS-PSS hydro-dynamic behaviors, key indicators such as critical flowrate, drop time, hydraulic self-tightening performance, as well as the hydraulic characteristics curve are for fulfilling the functional requirements. Meanwhile, through sensitivity study of some structural parameters‘ impact on hydraulic characteristics, some most sensible structural parameters for adjusting and optimizing detailed design are observed. The work is quite significant in supporting the conceptual design of the HS-PSS as well as its engineering improvement.

Comparison and Contrast of Calcite vs. Dolomite after Heat Treatment to Enhance Toluidine Blue Removal from Water

Significant increase in use of color dyes in modern society exerted a great pressure on environmental and water qualities. As such, studies for the removal of color dyes from water have been conducted extensively. In this study, common Earth materials dolomite and calcite were evaluated and contrasted for their removal of toluidine blue (TB), a cationic dye, before and after heat treatment. An increase by a factor of 3 in TB removal capacity from 3.5 to 10 mmol/kg was achieved after dolomite (Dol) was heated to 800 °C (designated as HDol). In contrast, the TB removal capacity increased by more than 100 times from 2 to 220 mmol/kg after calcite (Cal) was heated to 1000 °C (designated as HCal). For Dol and HDol, the TB removal increased as the solution’s pH increased but decreased with increases in the solution’s ionic strength. For Cal and HCal, the influence of the equilibrium solution’s pH and ionic strength on TB removal was negligible. The free energy of TB sorption on Dol, HDol, and Cal were −7 to −15 kJ/mol. The results suggested different removal mechanism for TB by Dol vs. Cal. X-ray diffraction data for Dol showed a slight increase in calcite content after heat treatment. For Cal, CaO was produced after heating, which converted back to calcite after 24 h of mixing with TB solutions. The significant TB removal by HCal could be attributed to its reaction with CaO. Thus, the best solution is to use freshly treated Cal for the removal of TB from solution.

Carbon emission reduction dynamic decision-making in the electricity supply chain with different carbon emission allowance principles

To address the impacts of typical principles for allocating initial carbon emission allowance, namely, the grandfathering and benchmarking principles on long-term emission reduction strategies, this paper presents four differential game models for an electricity supply chain consisting of a single electricity supplier and a single electricity retailer. The equilibrium solutions for different models are obtained and comparatively analyzed. The results show that the power supplier exhibits greater emission reduction efforts and, therefore, that the retailer has higher profits under the benchmarking principle than under the grandfathering principle. Moreover, under the two principles, the level of the power supplier’s emission reduction efforts is related to the decision-making modes of the electricity supply chain. Under specific conditions, the power supplier’s profit is also higher under the benchmarking principle than under the grandfathering principle. In addition, this study extends the models to investigate the constrained differential game models and obtains the game equilibrium solutions, which can be proved as saddle point equilibria.

Tipping prediction of a class of large-scale radial-ring neural networks

Understanding the emergence and evolution of collective dynamics in large-scale neural networks remains a complex challenge. This paper seeks to address this gap by applying dynamical systems theory, with a particular focus on tipping mechanisms. First, we introduce a novel (n+mn)-scale radial-ring neural network and employ Coates’ flow graph topological approach to derive the characteristic equation of the linearized network. Second, through deriving stability conditions and predicting the tipping point using an algebraic approach based on the integral element concept, we identify critical factors such as the synaptic transmission delay, the self-feedback coefficient, and the network topology. Finally, we validate the methodology’s effectiveness in predicting the tipping point. The findings reveal that increased synaptic transmission delay can induce and amplify periodic oscillations. Additionally, the self-feedback coefficient and the network topology influence the onset of tipping points. Moreover, the selection of activation function impacts both the number of equilibrium solutions and the convergence speed of the neural network. Lastly, we demonstrate that the proposed large-scale radial-ring neural network exhibits stronger robustness compared to lower-scale networks with a single topology. The results provide a comprehensive depiction of the dynamics observed in large-scale neural networks under the influence of various factor combinations.

Mixed-Zero-Sum-Game-Based Memory Event-Triggered Cooperative Control of Heterogeneous MASs Against DoS Attacks.

This article studies the problem of memory event-triggered cooperative adaptive control of heterogeneous nonlinear multiagent systems (MASs) under denial-of-service (DoS) attacks based on the multiplayer mixed zero-sum (ZS) game strategy. First, a neural-network-based reinforcement learning scheme is structured to obtain the Nash equilibrium solution of the proposed multiplayer mixed ZS game scheme. Then, a memory-based event-triggered mechanism considering the historical data is proposed. This effectively avoids incorrect triggering information caused by unknown external factors. Moreover, thanks to the idea of switching topology, the mixed ZS game problem under the influence of node-based DoS attacks is solved efficiently. In accordance with the Lyapunov stability theory, it is proved that all signals of heterogeneous MASs are bounded, all heterogeneous followers can track the trajectory of the leader during the no-attack period, the attacked follower can achieve stabilization control during the attack period, and the remaining nonattacked followers can achieve cooperative control during the attack period. Finally, the effectiveness of the designed memory-event-triggered-based mixed ZS game cooperative control strategy is tested by the given simulation results.

A multi‐dimensional incentive mechanism based on age of update in hierarchical federated learning

AbstractFederated learning represents a decentralized approach to machine learning, enabling numerous devices to collaboratively contribute to model training while ensuring the privacy of individual data. However, the existing incentive mechanism of hierarchical federated learning (HFL) only considers the data contribution of a single round, which needs to be revised. For non‐IID data sets, the continuous selection of any end devices will cause the weights to diverge in a specific direction. Therefore, a new metric is needed to avoid continuously selecting a certain end device to ensure the overall effectiveness. We introduce a metric to describe the importance of updates: age of update (AoU), which can help select end devices not selected in the previous round to promote a faster model convergence. We put forward an incentive mechanism based on AoU, reputation, and data quantity in HFL (ARDHFL). We have derived the optimal equilibrium solution for the three‐stage Stackelberg game. Based on this solution, we can ensure maximum edge‐cloud utility while incentivizing end devices to engage actively in HFL tasks and providing superior data to train the HFL model. Finally, we conducted extensive experiments to prove that ARDHFL can effectively improve the performance. Compared with the fixed scheme, random scheme, FMore and InFEDge, the testing accuracy of ARDHFL in the MNIST dataset has been improved by 29.7%, 9.3%, 6.8% and 6.1%, respectively. In the CIFAR‐10 dataset, it has been improved by 40.2%, 33.1%, 16.4% and 14.2%, respectively, and demands fewer communication iterations to achieve the same testing accuracy.

Three Duopoly Game-Theoretic Models for the Smart Grid Demand Response Management Problem

Demand response management (DRM) significantly influences the prospective advancement of electricity smart grids. This paper introduces three distinct game-theoretic duopoly models for the smart grid demand response management problem. It delineates several rational assumptions regarding the model variables, functions, and parameters. The first model adopts a Cournot duopoly form, offering a unique closed-form equilibrium solution. The second model adopts a Stackelberg duopoly structure, also providing a unique closed-form equilibrium solution. Following a comparison of the economic viability of the two model equilibria and an assessment of their sensitivity to parametric changes, the paper proposes a third model with a Cartel structure and discusses its advantages over the earlier models. Finally, the paper examines how demand forecasting affects the equilibrium quantities and pricing solutions of each model.

Zero-sum games subject to uncertain stochastic noncausal systems considering polynomial control functions

An uncertain stochastic noncausal system (USNS) is an uncertain stochastic singular system that is expected to be regular but not impulse-free. This study investigates zero-sum games (ZSGs) under the constraint of nonlinear uncertain stochastic noncausal systems (USNSs) considering polynomial control functions. Firstly, a method is introduced to convert USNSs into subsystems, including forward uncertain stochastic difference equations as well as backward uncertain stochastic difference equations. Equilibrium equations are then derived to determine the saddle-point equilibrium solution of the zero-sum game (ZSG) for USNSs. Subsequently, a unified framework is developed to find the saddle-point equilibrium and equilibrium value of the ZSG. Additionally, a numerical example and an analysis of the treatment of industrial wastewater are provided to demonstrate the effectiveness of the proposed framework.