Equilibrium Pair Research Articles

An Investigation of Dynamical Behavior of a Wing Model

AbstractBifurcations of equilibria of a wing model have been investigated in this paper. It is shown that the quintic nonlinear terms in the pitch and the plunge coordinate have affected the bifurcation structure of nontrivial equilibria in different degree. In contrast with the quintic stiffening parameter in plunge, the quintic parameter in pitch has a relatively significant effect, which will affect the number, position and stability of nontrivial equilibria. Therein two pairs of nontrivial equilibria with opposite stability coexist or disappear by two fold bifurcations. If the freestream velocity has been taken as a continuation parameter, it will affect the bifurcation structure of all the equilibria, including the trivial and the nontrivial. Wherein the equilibria vary from a trivial to two nontrivial ones by a pitchfork bifurcation. Then one of nontrivial equilibria experiences a supercritical Hopf bifurcation and the bifurcated limit cycles form an ellipsoidal structure with the limit cycles bifurcated from the trivial equilibrium.

Spatial metapopulation dynamics with local and global colonization

We revisit a spatial metapopulation model on continuous space as a stochastic point pattern dynamics. In the model, local patches as points are distributed with a certain spatial configuration and status of each patch changes stochastically between occupied and empty: an occupied patch becomes empty by local extinction and an empty patch becomes occupied both by local and global colonization. We carry out simulation analysis and derive an analytical model in terms of singlet, pair and triplet probabilities that describe the stochastic dynamics. Using a simple closure that approximates triplet probabilities by singlet and pair probabilities, we show that equilibrium singlet and pair probabilities can be analytically derived. The derived equilibrium properties successfully describe simulation results under a certain condition where the range of local colonization and the proportion of global colonization play key roles. Our model is an extension of the classical non-spatial Levins model to a spatially explicit metapopulation model. We appeal the advantage of point pattern approach to study spatial dynamics in general ecology and call for the need to deepen our understanding of mathematical tools to explore point pattern dynamics.

Nonequilibrium interactions between multi-scale colloids regulate the suspension microstructure and rheology.

Understanding nonequilibrium interactions of multi-component colloidal suspensions is critical for many dynamical settings such as self-assembly and material processing. A key question is how the nonequilibrium distributions of individual components influence the effective interparticle interactions and flow behavior. In this work, we develop a first-principle framework to study a bidisperse suspension of colloids and depletants using a Smoluchowski equation and corroborated by Brownian dynamics (BD) simulations. Using nonlinear microrheology as a case study, we demonstrate that effective depletion interactions between driven colloids are sensitive to particle timescales out of equilibrium and cannot be predicted by equilibrium-based pair potentials like Asakura-Oosawa. Furthermore, we show that the interplay between Brownian relaxation timescales of different species plays a critical role in governing the viscosity of multi-component suspensions. Our model highlights the limitations of using equilibrium pair potentials to approximate interparticle interactions in nonequilibrium processes such as hydrodynamic flows and presents a useful framework for studying the transport of driven, interacting suspensions.

The Choice of Trade Policy in an International Differentiated Duopoly with Process R&D Investment under Uncertainty

Purpose - The purpose of this paper is to investigate equilibrium pairs of modes of government intervention in an international differentiated duopoly with process R&D investment under uncertainty. Governments in each of the two countries choose direct R&D investment and quantity controls, or R&D subsidies to shift profits towards domestic firms.
 Design/Methodology/Approach - For the analysis, we extend the model of Haaland and Kind (2008) to that with uncertainty and consider direct R&D investment and quantity controls besides R&D subsidies. Firms in each of the two countries produce differentiated products and governments in each compete via direct R&D investment and quantity controls, or R&D subsidies in a third-country market. In environments with low, high, and intermediate uncertainty, equilibrium pairs of modes of government intervention are examined.
 Findings - It is shown that (i) in an environment with low uncertainty, direct R&D investment and quantity controls by each government is a unique equilibrium if the goods are substitutes and not too close, and R&D subsidies by each government is a unique equilibrium if the goods are complements and not too close; (ii) in an environment with high uncertainty, R&D subsidies by each government is a unique equilibrium regardless of the nature of the goods, except in the cases of too close substitutes and complements; and (iii) in an environment with intermediate uncertainty, R&D subsidies by each government is a unique equilibrium if the goods are complements and not too close. Moreover, in an environment with intermediate uncertainty, there are two symmetric equilibria if the goods are substitutes and not too close and there are two asymmetric equilibria if the goods are substitutes and very close.
 Research Implications - By analyzing the effects of direct R&D investment and quantity controls, and R&D subsidies in international markets under uncertainty, we can expect which government intervention appears in equilibrium.

Nonlinear State Observer and Control Design for Triangular Tethered Satellite Formation

This article researches the control of triangular tethered satellite formation (TTSF) system and nonlinear state observer (NSO) design when velocity vector is unmeasured. First, the stability around the equilibrium of open-loop system is discussed, which is stable around certain equilibrium pairs. To improve the system performance, linear feedback control is designed using Jacobian matrix of original nonlinear system, which is locally stable around linearized point. Besides, nonlinear control is designed subsequently which owns global stability domain. Then in case of the unmeasured velocity states, NSO is designed which contains a nonlinear function of estimation error to regulate the observation bias. With the designed nonlinear control and NSO, the simulation of the deployment of TTSF is made thereafter. Compared with linear feedback control and general observer with linear regulation, the proposed nonlinear control together with NSO has global stability domain and less chattering, which is promising in practical application.

Equilibrium pairs trading under delayed cointegration

Cointegration analysis is an econometric tool used to identify equilibrium among assets and construct a pairs trading portfolio. The discrete-time vector error correction model (VECM) for identifying cointegration includes lag difference terms as explanatory variables, thus permitting delayed adjustment of the deviations from equilibrium. The continuous-time limit of the VECM becomes a stochastic delay differential equation. We investigate the dynamic open-loop equilibrium mean–variance pairs trading strategy under such a delayed cointegration model. The existence and uniqueness results for the equilibrium are offered. We prove in general that the equilibrium strategy can lead to statistical arbitrage under certain conditions that are related to the roots of the corresponding characteristic equation. We obtain an explicit solution for a case with distributed delay. Our empirical study demonstrates the superiority of our strategy over its Markovian counterpart when the model selection result prefers a high-order VECM.

A variational framework for the inverse Henderson problem of statistical mechanics

The inverse Henderson problem refers to the determination of the pair potential which specifies the interactions in an ensemble of classical particles in continuous space, given the density and the equilibrium pair correlation function of these particles as data. For a canonical ensemble in a bounded domain, it has been observed that this pair potential minimizes a corresponding convex relative entropy functional, and that the Newton iteration for minimizing this functional coincides with the so-called inverse Monte Carlo (IMC) iterative scheme. In this paper, we show that in the thermodynamic limit analogous connections exist between the specific relative entropy introduced by Georgii and Zessin and a proper formulation of the IMC iteration in the full space. This provides a rigorous variational framework for the inverse Henderson problem, valid within a large class of pair potentials, including, for example, Lennard-Jones-type potentials. It is further shown that the pressure is strictly convex as a function of the pair potential and the chemical potential, and that the specific relative entropy at fixed density is a strictly convex function of the pair potential. At a given reference potential and a corresponding density in the gas phase, we determine the gradient and the Hessian of the specific relative entropy, and we prove that the Hessian extends to a symmetric positive semidefinite quadratic functional in the space of square integrable perturbations of this potential.

Adversarially robust sequential hypothesis testing

The problem of sequential hypothesis testing is studied, where samples are taken sequentially, and the goal is to distinguish between the null hypothesis where the samples are generated according to a distribution p and the alternative hypothesis where the samples are generated according to a distribution q. The defender (decision maker) aims to distinguish the two hypotheses using as few samples as possible subject to false alarm constraints. The problem is studied under the adversarial setting, where the data generating distributions under the two hypotheses are manipulated by an adversary, whose goal is to deteriorate the performance of the defender—for example, increasing the probability of error and expected sample sizes—with minimal cost. Specifically, under the null hypothesis, the adversary picks a distribution with cost and under the alternative hypothesis, the adversary picks a distribution with cost This problem is formulated as a non-zero-sum game between the defender and the adversary. A pair of strategies (the adversary’s strategy and the defender’s strategy) is proposed and proved to be a Nash equilibrium pair for the non-zero-sum game between the adversary and the defender asymptotically. The defender’s strategy is a sequential probability ratio test and thus is computationally efficient for practical implementation.

Complex magmatic processes recorded by clinopyroxene phenocrysts in a magmatic plumbing system: A case study of mafic volcanic rocks from the Laiyang Basin, southeastern North China Craton

Understanding the evolution of magmatic plumbing systems requires an understanding of the mechanisms of magma ascent from the magma source to the surface. We undertook textural and in situ geochemical studies of clinopyroxene phenocrysts in Early Cretaceous mafic volcanic rocks from the Laiyang Basin, China. Four distinct clinopyroxene populations were recognized based on their textures and zoning patterns in the mafic volcanic rocks (types 1, 2, 3, and 4). Type 1 is reversely zoned and contains rounded, irregularly resorbed, low-Mg cores (Mg# = 60–75) surrounded by clear, high-Mg mantles (Mg# = 75–90). Type 2 phenocrysts are typically normally zoned and characterized by relatively high-Mg cores (Mg# > 85) and thin lower Mg# rims (Mg# = 70–85). Type 3 comprises strong oscillatory zoned crystals (Mg# = 72–85) that are generally normally zoned and, in some cases, exhibit substantial sector zoning. Type 4 crystals are euhedral and unzoned phenocrysts. The textural and compositional features of the clinopyroxenes record complex magmatic processes including fractional crystallization, recycling of early formed crystals, magma convection, and magma recharge and subsequent mixing. Clinopyroxene–melt equilibrium pairs in the volcanic rocks yielded a relatively wide range of temperatures (984–1264°C) and pressures (0.1–10.1 kbar). The P–T array defines two main magma storage zones: (1) a deep, high-T, magma reservoir (~22–28 km), and (2) a shallow, low-T, magma reservoir (~8–19 km). At pressures of >3 kbar, the amount of water in melts in equilibrium with the clinopyroxene is in the range 1.5–3.0 wt.%, but at pressures of <3 kbar, the melt water contents decrease to 0.1 wt.%, indicative of extensive magma degassing. The clinopyroxene phenocrysts have a wide range of 87Sr/86Sr ratios from 0.70683 to 0.71366. The broad range of 87Sr/86Sr ratios (0.70796–0.71171) for the most primitive clinopyroxene (Mg# > 89) might have been inherited from a heterogeneous mantle source. However, some type 1 crystals with more radiogenic Sr isotopic compositions may record assimilation of crustal material during evolution of the mantle-derived magma. The textural and compositional features of the studied clinopyroxene phenocrysts suggest that Laiyang volcano is fed by a vertically extensive magmatic plumbing system that extends from shallow to deep levels, in which geochemically distinct magmas crystallized and mixed under variable physicochemical conditions.

Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

Abstract We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . In addition, they will be minimal in number and of reduced dimension; more precisely, 𝒖 will be of dimension ( d - 1 ) (d-1) , to include necessarily its 𝑑-th component, and 𝑣 will be of dimension 1. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L 3 ⁢ ( Ω ) \boldsymbol{L}^{3}(\Omega) for d = 3 d=3 ) and a corresponding Besov space for the thermal component, q &gt; d q&gt;d . Unique continuation inverse theorems for suitably over-determined adjoint static problems play a critical role in the constructive solution. Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80s.

Time-Inconsistent Consumption-Investment Problems in Incomplete Markets under General Discount Functions

In this paper, we study a time-inconsistent consumption-investment problem with random endowments in a possibly incomplete market under general discount functions. We provide a necessary condition and a verification theorem for an open-loop equilibrium consumption-investment pair in terms of a coupled forward-backward stochastic differential equation. Moreover, we prove the uniqueness of the open-loop equilibrium pair by showing that the original time-inconsistent problem is equivalent to an associated time-consistent one.

Equilibrium Pairs Trading Under Delayed Cointegration

Equilibrium Pairs Trading Under Delayed Cointegration

Cooperative advertising with two local advertising options in a retailer duopoly

This paper considers the issue of cooperative advertising with local advertising options in a channel with three players, including a manufacturer and two retailers. The current study, expands the cooperative advertising literature to a case where there exist two options for local advertising investment. Moreover, this paper compares two cases of presence and absence of cooperative advertising, which has almost been neglected in cooperative advertising literature. The purpose is to determine equilibrium strategy of retailers’ advertising options, players’ advertising expenditures and the manufacturer’ participation rates on retailers’ investment. The aforementioned problem is analyzed as a three-stage game, using backward induction. In the first and second stages, advertising investments of players are determined analytically. Then, in the third stage, the Nash equilibrium pair of advertising options can be found using numerical study. The problem is solved using illustrative examples in two cases of presence and absence of the cooperative advertising contract. Finally, the conditions for which offering the contract is win-win for all players, are identified. A Sensitivity analysis has been carried out to explain the efficacy of the model.

Integral equation theory of thermodynamics, pair structure, and growing static length scale in metastable hard sphere and Weeks-Chandler-Andersen fluids.

We employ the Ornstein-Zernike integral equation theory with the Percus-Yevick (PY) and modified-Verlet (MV) closures to study the equilibrium structural and thermodynamic properties of metastable monodisperse hard sphere and continuous repulsion Weeks-Chandler-Andersen (WCA) fluids under density and temperature conditions where the system is strongly overcompressed or supercooled, respectively. The theoretical results are compared to crystal-avoiding simulations of these dense monodisperse model one-component fluids. The equation of state (EOS) and dimensionless compressibility are computed using both the virial and compressibility routes. For hard spheres, the MV-based virial route EOS and dimensionless compressibility are in very good agreement with simulation for all packing fractions, much better than the PY analogs. The corresponding MV-based predictions for the static structure factor are also very good. The amplitude of density fluctuations on the local cage scale and in the long wavelength limit, and three technically different measures of the density correlation length, are studied with both closures. All five properties grow in a roughly exponential manner with density in the metastable regime up to packing fractions of 58% with no sign of saturation. The MV-based results are in good agreement with our crystal-avoiding simulations. Interestingly, the density dependences of long and short wavelength quantities are closely related. The MV-based theory is also quite accurate for the thermodynamics and structure of supercooled monodisperse WCA fluids. Overall our findings are also relevant as critical input to microscopic theories that relate the equilibrium pair correlation function or static structure factor to dynamical constraints, barriers, and activated relaxation in glass-forming liquids.

Numerical Stackelberg--Nash Control for the Heat Equation

This paper deals with a strategy to solve numerically control problems of the Stackelberg--Nash kind for heat equations with Dirichlet boundary conditions. We assume that we can act on the system through several controls, respecting an order and a hierarchy: a first control (the leader) is assumed to choose the policy; then, a Nash equilibrium pair, determined by the choice of the leader and corresponding to a noncooperative multiple-objective optimization strategy, is found (these are the followers). Our method relies on a formulation inspired by the work of Fursikov and Imanuvilov. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up at the final time. The use of the weights is crucial to ensure the existence of the controls and the associated state in a reasonable space. We present several mixed formulations of the problems and, then, associated mixed finite element approximations that are easy to handle. In a final step, we exhibit some numerical experiments making use of the Freefem++ package.

Uniform stabilization of Boussinesq systems in critical &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbf{L}^q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-based Sobolev and Besov spaces by finite dimensional interior localized feedback controls

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain, with homogeneous boundary conditions, and subject to external sources, assumed to cause instability. The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, feedback controls, which are localized on an arbitrarily small interior subdomain. In addition, they will be minimal in number, and of reduced dimension: more precisely, they will be of dimension $(d-1)$ for the fluid component and of dimension $1$ for the heat component. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to $\mathbf{L}^3(\Omega$) for $ d = 3 $) and the space $L^q(\Omega$) for the thermal component, $ q > d $. Thus, this paper may be viewed as an extension of \cite{LPT.1}, where the same interior localized uniform stabilization outcome was achieved by use of finite dimensional feedback controls for the Navier-Stokes equations, in the same Besov setting.

Microscopic theory of the influence of strong attractive forces on the activated dynamics of dense glass and gel forming fluids.

We theoretically study the nonmonotonic (re-entrant) activated dynamics associated with a finite time scale kinetically defined repulsive glass to fluid to attractive glass transition in high volume fraction particle suspensions interacting via strong short range attractive forces. The classic theoretical "projection" approximation that replaces all microscopic forces by a single effective force determined solely by equilibrium pair correlations is revisited based on the "projectionless dynamic theory" (PDT). A hybrid-PDT approximation is formulated that explicitly quantifies how attractive forces induce dynamical constraints, while singular hard core interactions are treated based on the projection approach. Both the effects of interference between repulsive and attractive forces, and structural changes due to attraction-induced bond formation that competes with caging, are included. Combined with the microscopic Elastically Collective Nonlinear Langevin Equation (ECNLE) theory of activated relaxation, the resultant approach appears to properly capture both the re-entrant dynamic crossover behavior and the strong nonmonotonic variation of the activated structural relaxation time with attraction strength and range at very high volume fractions as observed experimentally and in simulations. Testable predictions are made. Major differences compared to both ideal mode coupling theory and ECNLE theory based on the full force projection approximation are identified. Calculations are also performed for smaller time and length scale intracage dynamics relevant to the non-Gaussian parameter based on analyzing the dynamic free energy that controls particle trajectories. Implications of the new theory for thermal glass forming liquids with relatively long range attractive forces are briefly analyzed.

A circle criterion for strong integral input-to-state stability

We present sufficient conditions for integral input-to-state stability (iISS) and strong iISS of the zero equilibrium pair of continuous-time forced Lur’e systems, where by strong iISS we mean the conjunction of iISS and small-signal ISS. Our main results are reminiscent of the complex Aizerman conjecture and the well-known circle criterion. We derive a number of corollaries, including a result on stabilisation by static feedback in the presence of input saturation. In particular, we identify classes of forced Lur’e systems with saturating nonlinearities which are strongly iISS, but not ISS.

A note on the uniqueness result for the inverse Henderson problem

The inverse Henderson problem of statistical mechanics is the theoretical foundation for many bottom-up coarse-graining techniques for the numerical simulation of complex soft matter physics. This inverse problem concerns classical particles in continuous space which interact according to a pair potential depending on the distance of the particles. Roughly stated, it asks for the interaction potential given the equilibrium pair correlation function of the system. In 1974, Henderson proved that this potential is uniquely determined in a canonical ensemble and he claimed the same result for the thermodynamical limit of the physical system. Here, we provide a rigorous proof of a slightly more general version of the latter statement using Georgii’s variant of the Gibbs variational principle.

Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls

We present some exact controllability results for parabolic equations in the context of hierarchic control using Stackelberg–Nash strategies. We analyze two cases: in the first one, the main control (the leader) acts in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in the second one, we consider a leader acting on the boundary while the followers are of the distributed kind. In both cases, for each leader, an associated Nash equilibrium pair is found; then, we obtain a leader that leads the system exactly to a prescribed (but arbitrary) trajectory. We consider linear and semilinear problems.