Equilibrium Theorem Research Articles

Proximity among protected area networks promotes functional connectivity for wintering waterfowl

The equilibrium theorem provided a fundamental framework for understanding species’ distributions and movement in fragmented ecosystems. Wetland-dependent avian species are model organisms to test insular predictions within protected area networks because their mobility allows surveillance of isolated patches without landscape barriers. We hypothesized size and isolation would influence functional connectivity of sanctuaries by GPS-marked wintering mallards (Anas platyrhynchos) within a mesocosm protected sanctuary area network. We evaluated functional connectivity and sanctuary use, measured by movements between sanctuaries, using a multistate modeling framework. Proximity drove connectivity, underscoring that patch isolation—not size—influenced connectivity, even for an avian species with no ascertainable landscape resistance or barriers. We also found that sanctuary use increased overwintering survival by reducing harvest mortality. Our test of equilibrium theory predictions demonstrated that isolation of protected sanctuary areas supersedes their size in determining functional connectivity for mallards and access to these areas may have direct fitness consequences. Our findings could refine land acquisition, restoration, and management practices with equal or greater emphasis on adjacency in protected area network design, especially for wetland-dependent migratory gamebirds.

Three-Stage Duty Cycle-Based Deadbeat Predictive Torque Control for Three-Phase SPMSMs With CMV Reduction

Model Predictive Torque Control (MPTC) is highly regarded for its simple structure, rapid dynamic response, and optimized control performance when applied to actual motor driving systems. However, conventional MPTC techniques often lead to undesirable effects in three-phase inverter-based surface-mounted permanent magnet synchronous motors (SPMSMs), such as high electromagnetic torque, stator flux ripples, and significant current harmonic distortion. Notably, the generalized MPTC approach generates a substantial common mode voltage (CMV) that can cause damage to the motor bearing and winding insulation in SPMSMs. To address these issues, this paper proposes a three-stage duty cycle-based deadbeat predictive torque control (DCDPTC) technique for SPMSMs. The proposed technique leverages the volt-second equilibrium theorem by aligning the duty cycles with the predicted electromagnetic torque, stator flux, and their respective target values while accounting for extreme cases, accordingly electromagnetic torque and stator flux ripples as well as the total harmonic distortion (THD) of the stator current are minimized. In order to improve the system robustness, the duty cycle is limited to avoid the overmodulation. Furthermore, optimized switching sequences are meticulously arranged to ensure the CMV is lowered and the minimum number of switching transitions is maintained while utilizing a fixed switching frequency. The proposed technique is simplified. Its resultant performance is validated by experimental results.

Generalized Equilibrium Problems

If X is a convex subset of a topological vector space and f is a real bifunction defined on X×X, the problem of finding a point x0∈X such that f(x0,y)≥0 for all y∈X, is called an equilibrium problem. When the bifunction f is defined on the cartesian product of two distinct sets X and Y we will call it a generalized equilibrium problem. In this paper, we study the existence of the solutions, first for generalized equilibrium problems and then for equilibrium problems. In the obtained results, apart from the bifunction f, another bifunction is introduced, the two being linked by a certain compatibility condition. The particularity of the equilibrium theorems established in the last section consists of the fact that the classical equilibrium condition (f(x,x)=0, for all x∈X) is missing. The given applications refer to the Minty variational inequality problem and quasi-equilibrium problems.

Bearing capacity of foundations resting on rock masses subjected to Rayleigh waves

This study investigates the ultimate bearing capacity of shallow foundations resting on rock masses under earthquakes using upper-bound limit analysis. A modified pseudodynamic method (MPD) that considers the influence of Rayleigh waves is initially used to characterize the spatiotemporal variation in earthquake acceleration. The strength of rock masses is governed by the generalized Hoek‒Brown criterion and the plastic flow at failure is assumed to respect the associated flow rule. The generalized tangential technique is introduced to provide a linear approximation of the strength envelope for capturing the equivalent Mohr‒Coulomb strength parameters. The classic nonsymmetrical multi-block translational mechanism, which is capable of assessing a rock-foundation system under asymmetrical load, is reconstructed to portray the velocity discontinuity of the plastic failure region. Based upon the energy equilibrium theorem of the limit analysis, an analytical solution of seismic bearing capacity is derived and then formulated as a multivariate optimization problem. Comparisons show good agreement with the literature and validate the correctness and rationality of this work. Parametric study indicates that the seismic foundation capacity obtained by MPD tends to be conservative compared to the pseudostatic method with the same seismic coefficients, and that the period of a Rayleigh wave has only a slight effect on the bearing capacity. The failure region is more sensitive to the disturbance coefficient than it is to the other rock strength parameters.

Inverse of Berge’s maximum theorem in locally convex topological vector spaces and its applications

Abstract In this paper, the inverse of Berge’s maximum theorem is established in a locally convex topological vector space. Using this result, the generalized Gale–Nikaido–Debreu’s lemma and the generalized coincidence point theorem are derived from the equilibrium theorem of generalized games. By combining the inverse of Berge’s maximum theorem with the generalized coincidence point theorem, the equilibrium theorem of the generalized game is obtained immediately. Finally, we show that Fan–Glicksberg’s fixed point theorem and von Neumann’s lemma can be derived directly from the coincidence point theorem.

Chandrasekhar’s Integral Equilibrium Theorems Modified in the Context of Non-Gaussian Kappa Statistics for a Spherically Symmetric Protostar Cloud

Chandrasekhar’s Integral Equilibrium Theorems Modified in the Context of Non-Gaussian Kappa Statistics for a Spherically Symmetric Protostar Cloud

Non-compact Quantum Spin Chains as Integrable Stochastic Particle Processes

In this paper we discuss a family of models of particle and energy diffusion on a one-dimensional lattice, related to those studied previously in Sasamoto and Wadati (Phys Rev E 58:4181–4190, 1998), Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2017) and Povolotsky (J Phys A 46(46):465205, 2013) in the context of KPZ universality class. We show that they may be mapped onto an integrable $${\mathfrak {sl}}(2)$$ Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature. Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in non-equilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a “dual model” of probability theory is possible and useful. The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of $$\mathcal {N}=4$$ super Yang–Mills theory (SYM), in imaginary-time. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the $${\mathfrak {sl}}(2|1)$$ superstring that has been derived directly from $$\mathcal {N}=4$$ SYM.

Ekeland-type variational principle with applications to nonconvex minimization and equilibrium problems

The aim of the present paper is to establish a variational principle in metric spaces without assumption of completeness when the involved function is not lower semicontinuous. As consequences, we derive many fixed point results, nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem in noncomplete metric spaces. Examples are also given to illustrate and to show that obtained results are proper generalizations.

Equivalence results between Nash equilibrium theorem and some fixed point theorems

AbstractWe show that the Kakutani and Brouwer fixed point theorems can be obtained by directly using the Nash equilibrium theorem. The corresponding set-valued problems, such as the Kakutani fixed point theorem, Walras equilibrium theorem (set-valued excess demand function), and generalized variational inequality, can be derived from the Nash equilibrium theorem, with the aid of an inverse of the Berge maximum theorem. For the single-valued situation, we derive the Brouwer fixed point theorem, Walras equilibrium theorem (single-valued excess demand function), KKM lemma, and variational inequality from the Nash equilibrium theorem directly, without any recourse.

Equivalent forms of the Brouwer fixed point theorem

In this paper we survey a set of Brouwer fixed point theorem equivalents. These equivalents are divided into four loops related to (1) the Borsuk retraction theorem, (2) the Himmelberg fixed point theorem, (3) the Gale lemma and (4) the Nash equilibrium theorem.

Singular perturbation margin and generalised gain margin for nonlinear time-invariant systems

ABSTRACTIn this paper, singular perturbation margin (SPM) and generalised gain margin (GGM) are proposed as the classical phase margin and gain margin like stability metrics for nonlinear systems established from the view of the singular perturbation and the regular perturbation, respectively. The problem of SPM and GGM assessment of a nonlinear nominal system is formulated. The SPM and GGM formulations are provided as the functions of radius of attraction (ROA), which is introduced as a conservative measure of the domain of attraction (DOA). Furthermore, the ROA constrained SPM and GGM analysis are processed through two stages: (1) the SPM and GGM assessment for nonlinear systems at the equilibrium point, based on the SPM and GGM equilibrium theorems, including time-invariant and time-varying cases (Theorem 5.3, Theorem 5.2, Theorem 5.4 and Theorem 5.5); (2) the establishment of the relationship between the SPM or GGM and the ROA for nonlinear time-invariant systems through the construction of the Lyapunov function for the singularly perturbed model (Theorem 6.1 and Section 6.2.3).

An equilibrium theorem for excess demand correspondences

An equilibrium theorem for excess demand correspondences

A Non-cooperative Game Theoretic Approach to Energy-efficient Power Control in Wireless Sensor Networks

Aiming at the limited energy characteristics of wireless sensor networks, we apply game theory to solve the power control problem to reduce energy consumption in wireless sensor networks. In this paper, a distributed power control algorithm based on non-cooperative game theory under incomplete information is proposed, which adopts Signal-to-Interference Noise Ratio (SINR) as utility function. The purpose of power control algorithm for noncooperative game is to achieve the largest utility by optimal power control strategy, thus improve the total network energy efficiency. Moreover, Bayesian Nash equilibrium theorem is introduced to study the existence and uniqueness proof of Nash equilibrium algorithm. Simulation results show that there exist points for each of the cost functions considered, which give the maximum net utility given the strategies taken by all other nodes as fixed. And the proposed algorithm is efficient and can achieve better performance.

New results on equilibria of abstract fuzzy economies

In this paper, we consider the model of an abstract fuzzy economy and we obtain new equilibrium theorems for the situations when the involved correspondences satisfy properties which are weaker than upper-semicontinuity. In our case, the constraint or preference correspondences are almost weakly upper semicontinuous or have the e-upper semicontinuous selection property. The obtained results either generalize or improve corresponding ones present in the literature.

Continuous Time Finite State Mean Field Games

In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study the $N+1$-player problem, which the mean field model attempts to approximate. Our main result is the convergence as $N\to \infty$ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.

FIXED POINTS AND ALTERNATIVE PRINCIPLES

In a recent paper, M. Balaj [B] established an alternative principle. The principle was applied to a matching theorem of Ky Fan type, an analytic alternative, a minimax inequality, and existence of solutions of a vector equilibrium theorem. Based on the first author's fixed point theorems, in the present paper, we obtain generalizations of the main result of Balaj [B] and their applications.

Fixed Point and Equilibrium Theorems in a Generalized Convexity Framework

We study the fixed point property of set-valued maps and the existence of equilibria in the framework of \(\mathbb{B}\)-convexity, recently defined by W. Briec and Ch. Horvath. We introduce some classes of the set-valued maps with generalized convexity and prove continuous selection and fixed point properties for them. Finally, we obtain results concerning the existence of quasi-equilibria for W.K. Kim’s new model.

Intersection theorems and their applications in general almost convex spaces

In this paper, we extend the concept of the almost convex condition and establish an approximation property for convex continuous correspondences. We use the approximation property to generalize von Neumann’s intersection theorems and related topics. We obtain, as applications, some new equilibrium theorems for the existence of a maximal element, generalized games, and qualitative games of U-majorized correspondences.

The Fan minimax inequality implies the Nash equilibrium theorem

We show that in an abstract convex space ( E , D ; Γ ) , the partial KKM principle implies the Ky Fan minimax inequality, from which we deduce a generalization of the Nash equilibrium theorem.

Side Constrained Traffic Assignment Model Based on Network Dual Equilibrium

One side constrained traffic assignment model is presented based on the local nearsighted user equilibrium principle by taking advantage of network dual equilibrium theorem. The traffic flow and travel time are regarded as a pair of dual variables. The analysis is started from the basic elements of network. At first, the dual relationships between the conservation conditions of node's flow and the shortest travel time from origin to the node are considered. Then the dual relationships between link flow and travel time constrains on link with nearsighted users are also analyzed. At last, through combining these dual relationships and adding generalized side constraints, new traffic assignment model is built. How to embody two behavioral assumptions (priority of playing cards and elasticity of route adjusting on route) in the proposed algorithm to the model is analyzed. A searching method of effective path set is given by using the model's solution results of unique link flow with respect to origin and destination pair. Several numeral examples are given to prove the validity and efficiency of the model and related algorithm. The results for the side constrained model are explained by congestion pricing and queue delay on links.