Perturbation Vanishes Research Articles

On Hurwicz Preferences in Psychological Games

The literature on strategic ambiguity in classical games provides generalized notions of equilibrium in which each player best responds to ambiguous or imprecise beliefs about his opponents’ strategic choices. In a recent paper, strategic ambiguity has been extended to psychological games, by taking into account ambiguous hierarchies of beliefs and max–min preferences. Given that this kind of preference seems too restrictive as a general method to evaluate decisions, in this paper we extend the analysis by taking into account α-max–min preferences in which decisions are evaluated by a convex combination of the worst-case (with weight α) and the best-case (with weight 1−α) scenarios. We define the α-max–min psychological Nash equilibrium; an illustrative example shows that the set of equilibria is affected by the parameter α and the larger the ambiguity, the greater the effect. We also provide a result of stability of the equilibria with respect to perturbations that involve the attitudes toward ambiguity, the structure of ambiguity, and the payoff functions: converging sequences of equilibria of perturbed games converge to equilibria of the unperturbed game as the perturbation vanishes. Surprisingly, a final example shows that the existence of equilibria is not guaranteed for every value of α.

(Avoided) crossings in the spectra of matrices with globally degenerate eigenvalues

(Avoided) crossings are ubiquitous in physics and are connected to many physical phenomena such as hidden symmetries, the Berry phase, entanglement, Landau–Zener processes, the onset of chaos, etc. A pedagogical approach to cataloging (avoided) crossings has been proposed in the past, using matrices whose eigenvalues avoid or cross as a function of some parameter. The approach relies on the mathematical tool of the discriminant, which can be calculated from the characteristic polynomial of the matrix, and whose roots as a function of the parameter being varied yield the locations as well as degeneracies of the (avoided) crossings. In this article we consider matrices whose symmetries force two or more eigenvalues to be degenerate across the entire range of variation of the parameter of interest, thus leading to an identically vanishing discriminant. To show how this case can be handled systematically, we introduce a perturbation to the matrix and calculate the roots of the discriminant in the limit as the perturbation vanishes. We show that this approach correctly generates a nonzero ‘reduced’ discriminant that yields the locations and degeneracies of the (avoided) crossings. We illustrate our technique using the matrix Hamiltonian for benzene in Hückel theory, which has recently been discussed in the context of (avoided) crossings in its spectrum.

Vanishing flux perturbation, pressure, and magnetic field limit in a Chaplygin magnetogasdynamics

The Riemann problem for a Chaplygin magnetogasdynamics system of compressible fluid flow with a triple parameter perturbation containing flux, pressure, and magnetic field is first solved. Second, it is shown that as the triple parameter flux perturbation, pressure, and magnetic field vanish, any Riemann solution containing two shock waves tends to a delta-shock solution to the transport equations and any Riemann solution containing two rarefaction waves tends to a vacuum solution to the transport equations. Third, it is also proved that as the double parameter flux perturbation and magnetic field vanish, any Riemann solution containing two shock waves approaches a delta-shock solution to the Chaplygin gas equations and any Riemann solution containing two rarefaction waves tends to a solution containing two contact discontinuities of the Chaplygin gas equations. Finally, some numerical results are presented to be consistent with the theoretical analysis.

Flux-Approximation Limits of Solutions to the Brio System with Two Independent Parameters

By the flux-approximation method, we study limits of Riemann solutions to the Brio system with two independent parameters. The Riemann problem of the perturbed system is solved analytically, and four kinds of solutions are obtained constructively. It is shown that, as the two-parameter flux perturbation vanishes, any two-shock-wave and two-rarefaction-wave solutions of the perturbed Brio system converge to the delta-shock and vacuum solutions of the transport equations, respectively. In addition, we specially pay attention to the Riemann problem of a simplified system of conservation laws derived from the perturbed Brio system by neglecting some quadratic term. As one of the purebred parameters of the Brio system goes to zero, the solution of which consisting of two shock waves tends to a delta-shock solution to this simplified system. By contrast, the solution containing two rarefaction waves converges to a contact discontinuity and a rarefaction wave of the simplified system. What is more, the formation mechanisms of delta shock waves under flux approximation with both two parameters and only one parameter are clarified.

Delta shocks and vacuum states in the Euler equations for nonisentropic magnetogasdynamics with the flux perturbation

In this paper, we study the formation of delta shocks and vacuum states in Riemann solutions to nonisentropic magnetogasdynamics with the flux perturbation as the pressure, magnetic field and flux perturbation all vanish. First, the Riemann problem of this model is solved. Second, it is rigorously shown that, as the pressure, magnetic field and flux perturbation vanish, any Riemann solution consisting of two shocks and a possible one-contact-discontinuity to nonisentropic magnetogasdynamics with the flux perturbation tends to a delta shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted $$\delta $$-measure which forms the delta shock wave; any Riemann solution consisting of two rarefaction waves and a possible one-contact-discontinuity tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results are presented to verify the formation of delta shocks and vacuum states.

On the excursion area of perturbed Gaussian fields

We investigate Lipschitz-Killing curvatures for excursion sets of random fields on ℝ2under a very specific perturbation, namely a small spatial-invariant random perturbation with zero mean. An expansion formula for mean curvatures is derived when the magnitude of the perturbation vanishes, which recovers the Gaussian Kinematic Formula at the limit by contiguity of the model. We develop an asymptotic study of the perturbed excursion area behaviour that leads to a quantitative non-Gaussian limit theorem, in Wasserstein distance, for fixed small perturbations and growing domain. When letting both the perturbation vanish and the domain grow, a standard Central Limit Theorem follows. Taking advantage of these results, we propose an estimator for the perturbation variance which turns out to be asymptotically normal and unbiased, allowing to make inference through sparse information on the field.

Dynamics of 2D Incompressible Non-autonomous Navier–Stokes Equations on Lipschitz-like Domains

This paper concerns the tempered pullback dynamics of 2D incompressible non-autonomous Navier–Stokes equations with a non-homogeneous boundary condition on Lipschitz-like domains. With the presence of a time-dependent external force f(t) which only needs to be pullback translation bounded, we establish the existence of a minimal pullback attractor with respect to a universe of tempered sets for the corresponding non-autonomous dynamical system. We then give estimates on the finite fractal dimension of the attractor based on trace formula. Under the additional assumption that the external force is perturbed from a stationary force by a time-dependent perturbation, we also prove the upper semi-continuity of the attractors as the non-autonomous perturbation vanishes. Lastly, we investigate the regularity of these attractors when smoother initial data are given. Our results are new even for smooth domains.

The existence of inner synchronized stationary distribution for stochastic coupled systems on networks

This paper is concerned with the existence of inner synchronized stationary distribution for stochastic coupled systems on networks (SCSNs), which is the first time to consider this problem. Compared with the existing results on inner synchronization problem, we study the problem based on Lyapunov method and Kirchhoff’s Matrix Tree Theorem in graph theory without utilizing Kronecker product method and Linear matrix inequalities, which simplifies some complex analysis and avoids difficulties. Then some new sufficient conditions are presented to guarantee the existence of inner synchronized stationary distribution for SCSNs. These conditions show that the existence domain of inner synchronized stationary distribution has a close relationship with stochastic perturbation intensity. And when stochastic perturbation vanishes, inner synchronized stationary distribution will become complete synchronization. To illustrate the practicability of theoretical results, an application about stochastic coupled oscillators is given with a numerical example being carried out.

On critical properties of the Berry curvature in the Kitaev honeycomb model

We analyse the Kitaev honeycomb model, by means of the Berry curvature with respect to Hamiltonian parameters. We concentrate on the ground-state vortex-free sector, which allows us to exploit an appropriate Fermionisation technique. The parameter space includes a time-reversal breaking term which provides an analytical headway to study the curvature in phases in which it would otherwise vanish. The curvature is then analysed in the limit in which the time-reversal-symmetry-breaking perturbation vanishes. This provides remarkable information about the topological phase transitions of the model. The Berry curvature in itself exhibits no singularities at criticality, nevertheless it distinguishes different phases by showing different behaviours. In particular, the analysis of the first derivative shows a critical behaviour around the transition point.

Delta Shock Waves as Flux-Approximation Limit of Solutions to the Modified Chaplygin Gas Equations

By introducing a triple parameter flux perturbation including pressure in the modified Chaplygin gas equations, we prove that, as the triple parameter flux perturbation vanishes, any two-shock Riemann solution and any two-rarefaction-wave Riemann solution to the perturbed modified Chaplygin gas equations tend to a delta-shock and a vacuum solution to the transport equations, respectively. Then we show that, as a double parameter flux perturbation vanishes, any two-shock Riemann solution under a certain condition to the perturbed modified Chaplygin gas equations tends to a delta shock wave solution to the generalized Chaplygin gas equations. In addition, we also show that, as a single parameter flux perturbation vanishes, any two-shock solution satisfying certain initial data and any parameterized delta shock wave solution to the perturbed generalized Chaplygin gas equations tend to a delta shock wave solution to the generalized Chaplygin gas equations. Finally, we exhibit some representative numerical simulations to confirm the theoretical analysis.

Pullback dynamics of 3D Navier–Stokes equations with nonlinear viscosity

This paper is concerned with pullback dynamics of 3D Navier–Stokes equations with variable viscosity and subject to time-dependent external forces. Our main result establishes the existence of finite-dimensional pullback attractors in a general setting involving tempered universes. We also present a sufficient condition on the viscosity coefficients that guarantees the attractors are nontrivial. We end the paper by showing the upper semi-continuity of pullback attractors as the non-autonomous perturbation vanishes.

Large deviation principle for dynamical systems coupled with diffusion–transmutation processes

In this paper, we introduce a mathematical apparatus that is relevant for understanding a dynamical system with small random perturbations and coupled with the so-called transmutation process — where the latter jumps from one mode to another, and thus modifying the dynamics of the system. In particular, we study the exit problem, i.e., an asymptotic estimate for the exit probabilities with which the corresponding processes exit from a given bounded open domain, and then formally prove a large deviation principle for the exit position joint with the type occupation times as the random perturbation vanishes. Moreover, under certain conditions, we also determine the exit place and the type of distribution at the exit time and, as a consequence of this, such information also give the limit of the Dirichlet problem for the corresponding partial differential equation systems with a vanishing small parameter.

Delta-shocks and vacuums in the relativistic Euler equations for isothermal fluids with the flux approximation

We demonstrate the flux-approximation problem of the isothermal relativistic Euler equations describing a perfect fluid flow in special relativity. First, the Riemann problem of the isothermal relativistic Euler equations under flux perturbation is discussed, and four kinds of solutions are obtained. Second, we rigorously prove that, as the flux perturbation vanishes, any two-shock Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations and the intermediate density between the two shocks tends to a weighted δ-measure which forms a delta shock wave. Correspondingly, any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution to the pressureless relativistic Euler equations and the nonvacuum intermediate state in between tends to a vacuum state.

Perturbation Theory for Weak Measurements in Quantum Mechanics, Systems with Finite-Dimensional State Space

The quantum theory of indirect measurements in physical systems is studied. The example of an indirect measurement of an observable represented by a self-adjoint operator $${\mathcal {N}}$$ with finite spectrum is analyzed in detail. The Hamiltonian generating the time evolution of the system in the absence of direct measurements is assumed to be given by the sum of a term commuting with $${\mathcal {N}}$$ and a small perturbation not commuting with $${\mathcal {N}}$$ . The system is subject to repeated direct (projective) measurements using a single instrument whose action on the state of the system commutes with $${\mathcal {N}}$$ . If the Hamiltonian commutes with the observable $${\mathcal {N}}$$ (i.e., if the perturbation vanishes), the state of the system approaches an eigenstate of $${\mathcal {N}}$$ , as the number of direct measurements tends to $$\infty $$ . If the perturbation term in the Hamiltonian does not commute with $${\mathcal {N}}$$ , the system exhibits “jumps” between different eigenstates of $${\mathcal {N}}$$ . We determine the rate of these jumps to leading order in the strength of the perturbation and show that if time is rescaled appropriately a maximum likelihood estimate of $${\mathcal {N}}$$ approaches a Markovian jump process on the spectrum of $${\mathcal {N}}$$ , as the strength of the perturbation tends to 0.

Synchronized stationary distribution of hybrid stochastic coupled systems with applications to coupled oscillators and a Chua’s circuits network

In this paper, the existence of synchronized stationary distribution for hybrid stochastic coupled systems (HSCSs) (here, also known as stochastic coupled systems with Markovian switching) is concerned. By the existence theory of stationary distribution as well as Lyapunov method and graph theory, two kinds of sufficient criteria are presented to promise the existence of synchronized stationary distribution for HSCSs. Our results exhibit that the existence region of synchronized stationary distribution has a close relationship with the intensity of stochastic perturbation. And when stochastic perturbation vanishes, synchronized stationary distribution will become complete synchronization. Then the proposed theoretical results are successfully applied to stochastic coupled oscillators and a Chua’s circuits network. Some existence criteria of synchronized stationary distribution are also obtained. The corresponding numerical simulations are carried out to verify the validity of the theoretical results.

Delta-shocks and vacuums as limits of flux approximation for the pressureless type system

In this paper, we investigate the phenomena of concentration and cavitation and the formation of delta-shocks and vacuum states in solutions of the pressureless type system with flux approximation. First, the Riemann problem of the pressureless type system with a flux perturbation is considered. A parameterized delta-shock and generalized constant density solution are obtained. Then we rigorously prove that, as the flux perturbation vanishes, they converge to the delta-shock and vacuum state of the pressureless type system, respectively. Secondly, by adding an artificial pressure term in the pressureless type system, we solve the Riemann problem of the system with a double parameter flux approximation including pressure. It is shown that, as the flux perturbations vanish, any two-shock Riemann solution tends to a delta-shock solution to the pressureless type system; any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution to the pressureless type system and the intermediate nonvacuum state in between tends to a vacuum state.

The mass and angular momentum of reconstructed metric perturbations

We prove a key result regarding the mass and angular momentum content of linear vacuum perturbations of the Kerr metric obtained through the formalism developed by Chrzarnowski, Cohen, and Kegeles (CCK). More precisely, we prove that the Abbott–Deser mass and angular momentum integrals of any such perturbation vanish when that perturbation was obtained from a regular Fourier mode of the Hertz potential. As a corollary we obtain a generalization of previous results on the completion of the ‘no string’ radiation gauge metric perturbation generated by a point particle. We find that for any bound orbit around a Kerr black hole, the mass and angular momentum perturbations completing the CCK metric are simply the energy and angular momentum of the particle ‘outside’ the orbit and vanish ‘inside’ the orbit.

Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation

In this paper, we consider a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation. Under the assumption that the potential may be vanishing at infinity, the existence of both the ground state and the ground state sign-changing solutions is established. Furthermore, the behavior of these solutions is studied when the perturbation vanishes. It is a surprise that we find an interesting phenomenon about the monotonicity for the quotient function as a byproduct.

Flux approximation to the isentropic relativistic Euler equations

The isentropic relativistic Euler equations for polytropic gas under flux perturbations are studied. The Riemann problem of the pressureless relativistic Euler equations with a flux approximation is firstly solved, and a family of delta-shock and U-shaped pseudo-vacuum state solutions are constructed. Then it is shown that, as the flux approximation vanishes, the limits of the family of delta-shock and U-shaped pseudo-vacuum solutions are exactly the delta-shock and vacuum state solutions to the pressureless relativistic Euler equations, respectively. Secondly, we study the Riemann problem of the isentropic relativistic Euler equations with a double parameter flux approximation including pressure term. We further prove that, as the pressure and two-parameter flux perturbation vanish, respectively, any two-shock Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations, and the intermediate density between the two shocks tends to a weighted δ-measure which forms a delta shock wave; any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution to the pressureless relativistic Euler equations, and the nonvacuum intermediate state in between tends to a vacuum state.

Flux-approximation limits of solutions to the relativistic Euler equations for polytropic gas

The flux-approximation problem of the relativistic Euler equations for polytropic gas in special relativity is studied. At first, we solve the Riemann problem of the pressureless relativistic Euler equations with a flux approximation, and obtain two kinds of solutions involving a family of delta shock wave and pseudo-vacuum state. Then, as the flux approximation vanishes, we show that the limits of the family of delta-shock and pseudo-vacuum solutions are exactly the delta-shock and vacuum state solutions of the pressureless relativistic Euler equations, respectively. Next, the Riemann problem of the relativistic Euler equations with a double parameter flux approximation including pressure is solved analytically. Furthermore, it is rigorously proved that, as the double parameter flux perturbation vanishes, any two-shock Riemann solution tends to a delta-shock solution to the pressureless relativistic Euler equations; any two-rarefaction Riemann solution tends to a two-contact-discontinuity solution to the pressureless relativistic Euler equations and the nonvacuum intermediate state in between tends to a vacuum state.