General Perturbation Research Articles

Perturbed obstacle problems in Lipschitz domains: linear stability and nondegeneracy in measure

We consider the classical obstacle problem on bounded, connected Lipschitz domains $D \subset \mathbb R^n$. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right-hand side and the boundary data for obstacle problems. In particular, we show that the Lebesgue measure of the symmetric difference between two contact sets is linearly comparable to the $L^1$-norm of perturbations in the data.

Stability Analysis of a Stochastic Viral Infection Model with General Infection Rate and General Perturbation Terms

In this paper, we propose a stochastic viral infection model with general incidence rate. In the proposed model, the stochastic perturbations are modeled by general functions. Further, the global existence and positivity of solutions are investigated. In addition, the stochastic stability of the model is established by using the direct Lyapunov method. Finally, an application of the hepatitis B virus (HBV) is given to validate our theoretical results.

Temporal variation in the winding number due to dynamical symmetry breaking and associated transport in a driven Su-Schrieffer-Heeger chain

Considering a BDI symmetric one-dimensional SSH model, we explore the fate of the bulk topological invariant, namely, the winding number under a generic time dependent perturbation; the effective Hamiltonian, that generates the temporal evolution of the initial (ground) state of the completely symmetric initial Hamiltonian, may have the same or different symmetries. To exemplify, we consider the following protocols, namely (i) a perfectly periodic protocol (ii) a periodic protocol with noisy perturbations and also (iii) sudden changes in the parameters of the initial Hamiltonian. We establish that the topological invariant may change in some cases when the effective Hamiltonian (or the Floquet Hamiltonian in the periodic situation when observed stroboscopically) does not respect all BDI symmetries; this is manifested in the associated particle (polarisation) or heat current in the bulk. Our results establish a strong connection between the time evolution of the winding number (thus, the associated transport of currents) and the symmetry of the Hamiltonian generating the time evolution which has been illustrated considering an exhaustive set of possibilities.

Temporal stability analysis for multiple similarity solutions of viscous incompressible flows in porous channels with moving walls

• We consider the temporal stability of multiple solutions for flows in a porous channel with moving walls for the first time. • We construct second order finite difference schemes for eigenproblems associated with the linear stability analysis . • We verify the linear stability results for the first time by solving the perturbed nonlinear time dependent equation. In this paper, a viscous, incompressible laminar fluid flow along a uniformly porous channel with expanding or contracting walls is considered. We present multiple symmetric steady-state solutions of this flow problem at several different expanding ratios, and use the linear stability theory to analyse the temporal stability for these solutions under symmetric, antisymmetric and general perturbations. We construct second order finite difference schemes for the eigenvalue problems with boundary conditions associated with those perturbations, and observe that most of these solutions which are stable under symmetric perturbations are unstable under antisymmetric perturbations. Furthermore, we verify the linear stability analysis results by directly solving the original perturbed nonlinear time dependent problem, and find that both stability results are consistent.

Perturbations against a Q-ball. II. Contribution of nonoscillation modes

In the present paper, discussion of perturbations against a Q-ball solution is continued. It is shown that in order to correctly describe perturbations containing nonoscillation modes, it is also necessary to consider nonlinear equations of motion for the perturbations, like in the case of oscillation modes only. It is also shown that the additivity of the charge and the energy of different modes holds for the most general nonlinear perturbation consisting of oscillation and nonoscillation modes.

The Parameterized Post-Friedmannian Framework for Nonminimal Derivative Coupling with General Cosmological Perturbation Metric

We study the parameterization of cosmological model where derivative of an additional scalar field coupled to curvature tensor. We extend the Parameterized Post-Friedmannian framework for interacting dark energy theories to the model. Starting from general cosmological perturbation metric, we derive perturbed energy-momentum tensor for scalar field and parameterized the tensor. Based on the value of the parameters, we compare the model with relevant models in current literature. As the results, we find that parameterization for type 1 theories of dark energy which is explicitly coupled to dark matter, gives only 4 non-zero Parameterized Post-Friedmannian coefficients.

Holographic CFT states for localized perturbations to AdS black holes

In this note, we describe a holographic CFT construction of states dual to scalar perturbations of the maximally extended three-dimensional AdS-Schwarzschild black hole. The states are constructed by adding sources for a scalar operator to the path integral that constructs the thermofield double state. For a scalar field of arbitrary mass, we provide the general map between sources and scalar field perturbations at linear order. With this, we investigate to what extent it is possible using this construction to produce perturbations localized to one side of the black hole horizon. Our results suggest that it is possible to produce arbitrarily localized perturbations (and thus, plausibly, general linear perturbations to the black hole initial data slice), but that the amplitude of the perturbation must be taken small as the perturbation becomes more localized in order that the sources do not diverge, as found for the pure AdS case considered in arXiv:1709.10101.

A potential role for rare species in ecosystem dynamics

The ecological importance of common species for many ecosystem processes and functions is unquestionably due to their high abundance. Yet, the importance of rare species is much less understood. Here we take a theoretical approach, exposing dynamical models of ecological networks to small perturbations, to explore the dynamical importance of rare and common species. We find that both species types contribute to the recovery of communities following generic perturbations (i.e. perturbations affecting all species). Yet, when perturbations are selective (i.e. affects only one species), perturbations to rare species have the most pronounced effect on community stability. We show that this is due to the strong indirect effects induced by perturbations to rare species. Because indirect effects typically set in at longer timescales, our results indicate that the importance of rare species may be easily overlooked and thus underrated. Hence, our study provides a potential ecological motive for the management and protection of rare species.

Vison Crystals in an Extended Kitaev Model on the Honeycomb Lattice.

We introduce an extension of the Kitaev honeycomb model by including four-spin interactions that preserve the local gauge structure and, hence, the integrability of the original model. The extended model has a rich phase diagram containing five distinct vison crystals, as well as a symmetric π-flux spin liquid with a Fermi surface of Majorana fermions and a sequence of Lifshitz transitions. We discuss possible experimental signatures and, in particular, present finite-temperature MonteCarlo calculations of the specific heat and the static vison structure factor. We argue that our extended model emerges naturally from generic perturbations to the Kitaev honeycomb model.

Strategic Asset Allocation for Life Insurers with Stochastic Liability

The problem of strategic asset allocation and product mix choice of a life insurancecompany is considered where account is taken of the stochastic risk associated with both assets and liabilities. Using the methods of stochastic dynamic programming we derive equations for optimal weights of both risky and riskless assets under continuous time. The resulting equations can be solved exactly for some parameter values and utility functions. When this is not possible a general perturbation expansion method is set up for which explicit solutions are derived for the first terms but the method can be generalized to any order in the expansion parameter.

Linear stability of inviscid vortex rings to axisymmetric perturbations

We consider the linear stability to axisymmetric perturbations of the family of inviscid vortex rings discovered by Norbury (J. Fluid Mech., vol. 57, 1973, pp. 417–431). Since these vortex rings are obtained as solutions to a free-boundary problem, their stability analysis is performed using recently developed methods of shape differentiation applied to the contour-dynamics formulation of the problem in the three-dimensional axisymmetric geometry. This approach allows us to systematically account for the effects of boundary deformations on the linearized evolution of the vortex ring. We investigate the instantaneous amplification of perturbations assumed to have the same the circulation as the vortex rings in their equilibrium configuration. These stability properties are then determined by the spectrum of a singular integro-differential operator defined on the vortex boundary in the meridional plane. The resulting generalized eigenvalue problem is solved numerically with a spectrally accurate discretization. Our results reveal that while thin vortex rings remain neutrally stable to axisymmetric perturbations, they become linearly unstable to such perturbations when they are sufficiently ‘fat’. Analysis of the structure of the eigenmodes demonstrates that they approach the corresponding eigenmodes of Rankine’s vortex and Hill’s vortex in the thin-vortex and fat-vortex limit, respectively. This study is a stepping stone on the way towards a complete stability analysis of inviscid vortex rings with respect to general perturbations.

Impact of nonideal cycles on the efficiency of quantum heat engines

Given a quantum heat engine that operates in a cycle that reaches maximal efficiency for a time-dependent Hamiltonian H(t) of the working substance, with overall controllable driving H(t) = g(t) H, we study the deviation of the efficiency from the optimal value due to a generic time-independent perturbation in the Hamiltonian. We show that for a working substance consisting of two two-level systems, by suitably tuning the interaction, the deviation can be suppressed up to the third order in the perturbation parameter-and thus almost retaining the optimality of the engine.

A possible solution to the Hubble constant discrepancy: Cosmology where the local volume expansion is driven by the domain average density

Abstract The Hubble constant problem is the discrepancy between different measurements of the Hubble constant on different scales. We show that this problem can be resolved within the general relativistic framework of the perturbation theory in the inhomogeneous universe, with the help of a spatial averaging procedure over a finite local domain in the $t={\rm{const.}}$ hypersurface. The idea presented in this paper is unique in the sense that it has all of the following properties. a) It is based on the general relativistic perturbation theory, with ordinary dust matter only. No strange matter nor energy components are required. b) The employment of the spatially invariant averaging procedure on the finite domain is essential. c) The key is the first-order effect of the inhomogeneities in the linear perturbation theory. No nonlinear effects are required.

Stationary distribution of an HIV model with general nonlinear incidence rate and stochastic perturbations

We present a stochastic HIV infection model with logistic target cell growth, general nonlinear incidence rate, CTL immune response and parameter perturbations. Through a rigorous analysis of the model, we obtain that the solution of the model is positive and global. A critical condition R0s of the model is derived, which depends not only on the general incidence function but also on the noise intensities. Under certain assumptions, by constructing suitable Lyapunov functions, we find that the system has a unique ergodic stationary distribution when R0s>1. We further explore the effect of the noise intensity on model behavior. Our conclusion improves and generalizes the results of the existing HIV stochastic models.

A comparison principle for convolution measures with applications

AbstractWe establish the general form of a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd which holds for arbitrary n and d. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [3].

General Perturbation Method for Satellite Constellation Reconfiguration Using Low-Thrust Maneuvers

A general perturbation solution to a restricted low-thrust Lambert rendezvous problem, considering circular-to-circular in-plane maneuvers using tangential thrust and including a coast arc, is developed. This provides a fully analytical solution to the satellite reconnaissance problem. The solution requires no iteration. Its speed and simplicity allow problems involving numerous spacecraft and maneuvers to be studied; this is demonstrated through two case studies. In the first, a range of maneuvers providing a rapid flyover of Los Angeles is generated, giving an insight to the trade space and allowing the maneuver that best fulfills the mission to be selected. A reduction in flyover time from 13.8 to 1.6 days is possible using a less than velocity change. A comparison with a numerical propagator including atmospheric friction and an 18th-order tesseral model shows 4 s of difference in the time of flyover. A second study considers a constellation of 24 satellites that can maneuver into repeating ground track orbits to provide persistent coverage of a region. A set of maneuvers for all satellites is generated for four sequential targets, allowing the most suitable maneuver strategy to be selected. Improvements in coverage of greater than 10 times are possible as compared to a static constellation using 35% of the propellant available across the constellation.

Topology and exceptional points of massive Dirac models with generic non-Hermitian perturbations

Recently, there has been a lot of activity in the research field of topological non-Hermitian physics, partly driven by fundamental interests and partly driven by applications in photonics. However, despite these activities, a general classification and characterization of non-Hermitian Dirac models that describe the experimental systems is missing. Here, we present a systematic investigation of massive Dirac models on periodic lattices, perturbed by general non-Hermitian terms. We find that there are three different types of non-Hermitian terms. For each case we determine the bulk exceptional points, the boundary modes, and the band topology. Our findings serve as guiding principles for the design of applications, for example, in photonic lattices. For instance, periodic Dirac systems with non-Hermitian mass terms can be used as topological lasers. Periodic Dirac systems with non-Hermitian anti-commuting terms, on the other hand, exhibit exceptional points at the surface, whose non-trivial topology could be utilized for optical devices.

The ECMWF operational ensemble reanalysis–analysis system for ocean and sea ice: a description of the system and assessment

Abstract. The ECMWF OCEAN5 system is a global ocean and sea-ice ensemble of reanalysis and real-time analysis. This paper gives a full description of the OCEAN5 system, with the focus on upgrades of system components with respect to its predecessors, ORAS4 and ORAP5. An important novelty in OCEAN5 is the ensemble generation strategy that includes perturbation of initial conditions and a generic perturbation scheme for observations and forcing fields. Other upgrades include revisions to the a priori bias correction scheme, observation quality control and assimilation method for sea-level anomalies. The OCEAN5 historical reconstruction of the ocean and sea-ice state is the ORAS5 reanalysis, which includes five ensemble members and covers the period from 1979 onwards. Updated versions of observation data sets are used in ORAS5 production, with special attention devoted to the consistency of sea surface temperature (SST) and sea-ice observations. Assessment of ORAS5 through sensitivity experiments suggests that all system components contribute to an improved fit to observation in reanalyses, with the most prominent contribution from direct assimilation of ocean in situ observations. Results of observing system experiments further suggest that the Argo float is the most influential observation type in our data assimilation system. Assessment of ORAS5 has also been carried out for several key ocean state variables and verified against reference climate data sets from the ESA CCI (European Space Agency Climate Change Initiative) project. With respect to ORAS4, ORAS5 has improved ocean climate state and variability in terms of SST and sea level, mostly due to increased model resolution and updates in assimilated observation data sets. In spite of the improvements, ORAS5 still underestimates the temporal variance of sea level and continues exhibiting large SST biases in the Gulf Stream and its extension regions which are possibly associated with misrepresentation of front positions. Overall, the SST and sea-ice uncertainties estimated using five ORAS5 ensemble members have spatial patterns consistent with those of analysis error. The ensemble spread of sea ice is commensurable with the sea-ice analysis error. On the contrary, the ensemble spread is under-dispersive for SST.

Quasinormal modes of black holes in a toy-model for cumulative quantum gravity

The idea that quantum gravity effects might “leak” outside the horizon of a black hole has recently been intensively considered. In this study, we calculate the quasinormal modes as a function of the location and amplitude of a generic metric perturbation distorting to the Schwarzschild spacetime. We conclude on the possible observability of quantum metric corrections by current and future gravitational wave experiments.

Semi-analytical model for third-body perturbations including the inclination and eccentricity of the perturbing body

A general third-body perturbation problem, considering the perturbing body in an elliptic and inclined orbit, is investigated by using a semi-analytical theory. Previous works have contributed to deriving the averaged third-body-perturbed dynamics, but did not provide a transformation between osculating and mean elements in the general case. In this paper, an analytical transformation between osculating and mean elements is developed explicitly using von Zeipel’s method, in addition to developing the long-term dynamical equations. The resulting dynamical model is improved, because the disturbing function is averaged as a whole, instead of separating the disturbing function into many terms and averaging them independently. The simulation results indicate that the new singly averaged dynamical model behaves much better than the doubly averaged dynamics in propagating the long-term evolution of the orbital elements. Moreover, it is shown that the perturbing body’s inclination and eccentricity have a vital influence on the evolution of the satellite’s inclination and eccentricity.