Perturbation Framework Research Articles

Arctic sea-ice decline weakens the Atlantic Meridional Overturning Circulation

The ongoing decline of Arctic sea ice exposes the ocean to anomalous surface heat and freshwater fluxes, resulting in positive buoyancy anomalies that can affect ocean circulation. In this study, we use an optimal flux perturbation framework and comprehensive climate model simulations to estimate the sensitivity of the Atlantic Meridional Overturning Circulation (AMOC) to such buoyancy forcing over the Arctic and globally, and more generally to sea-ice decline. It is found that on decadal timescales, flux anomalies over the subpolar North Atlantic have the largest impact on the AMOC, while on multi-decadal timescales (longer than 20 years), flux anomalies in the Arctic become more important. These positive buoyancy anomalies spread to the North Atlantic, weakening the AMOC and its poleward heat transport. Therefore, the Arctic sea-ice decline may explain the suggested slow-down of the AMOC and the ‘Warming Hole’ persisting in the subpolar North Atlantic. Sea-ice loss has exposed the Arctic Ocean to anomalous heat and freshwater. Climate model experiments suggest that these changes, particularly on multi-decadal timescales, may explain the observed slow-down in the Atlantic Meridional Overturning Circulation.

The relativistic Boltzmann equation for soft potentials

The paper concerns the Cauchy problem on the relativistic Boltzmann equation for soft potentials in a periodic box. We show that the global-in-time solutions around relativistic Maxwellians exist in the weighted L∞ perturbation framework and also approach equilibrium states in large time in the weighted L2 framework at the rate of exp⁡(−λtβ) for some λ>0 and β∈(0,1). The proof is based on the nonlinear L2 energy method and nonlinear L∞ pointwise estimate with appropriate exponential weights in momentum. The results extend those on the classical Boltzmann equation by Caflisch [2,3] and Strain and Guo [31] to the relativistic version, and also improve the recent result on almost exponential time-decay by Strain [28] to the exponential rate. Moreover, we study the propagation of spatial regularity for the obtained solutions and also the large time behavior in the corresponding regular Sobolev space, provided that the spatial derivatives of initial data are bounded, not necessarily small.

Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics

We are concerned with the global well-posedness of the diffusion approximation model in radiation hydrodynamics, which describe the compressible fluid dynamics taking into account the radiation effect under the non-local thermal equilibrium case. The model consist of the compressible Navier-Stokes equations coupled with the radiative transport equation with non-local terms. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The existence of global solution is proved based on the classical energy estimates, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between photons and matter.

Global well-posedness and large time behavior of classical solutions to the Vlasov–Fokker–Planck and magnetohydrodynamics equations

We are concerned with the global well-posedness of the fluid-particle system which describes the evolutions of disperse two-phase flows. The system consists of the Vlasov–Fokker–Planck equation for the dispersed phase (particles) coupled to the compressible magnetohydrodynamics equations modelling a dense phase (fluid) through the friction forcing. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The existence of global solution and decay rate of the solution are proved based on the classical energy estimates and Fourier multiplier technique, which are considerably complicated and some new ideas and techniques are thus required. Moreover, it is shown that neither shock waves nor vacuum and concentration in the solution are developed in a finite time although there is a complex interaction between particle and fluid.

On the conditions of exponential stability in active disturbance rejection control based on singular perturbation analysis

ABSTRACTStability of active disturbance rejection control (ADRC) is analysed in the presence of unknown, nonlinear, and time-varying dynamics. In the framework of singular perturbations, the closed-loop error dynamics are semi-decoupled into a relatively slow subsystem (the feedback loop) and a relatively fast subsystem (the extended state observer), respectively. It is shown, analytically and geometrically, that there exists a unique exponential stable solution if the size of the initial observer error is sufficiently small, i.e. in the same order of the inverse of the observer bandwidth. The process of developing the uniformly asymptotic solution of the system reveals the condition on the stability of the ADRC and the relationship between the rate of change in the total disturbance and the size of the estimation error. The differentiability of the total disturbance is the only assumption made.

Quantum gravity effects on charged microblack holes thermodynamics

The charged black hole thermodynamics is corrected in terms of the quantum gravity effects. Most of the quantum gravity theories support the idea that near the Planck scale, the standard Heisenberg uncertainty principle should be reformulated by the so-called Generalized Uncertainty Principle (GUP) which provides a perturbation framework to perform required modifications of the black hole quantities. In this paper, we consider the effects of the minimal length and maximal momentum as GUP type I and the minimal length, minimal momentum and maximal momentum as GUP type II on thermo dynamics of the charged TeV-scale black holes. We also generalized our study to the universe with the extra dimensions based on the ADD model. In this framework, the effect of the electrical charge on thermodynamics of the black hole and existence of the charged black hole remnants as a potential candidate for the dark matter particles are discussed.

Large-time behavior for fluid and kinetic plasmas with collisions

The motion of collisional plasmas can be governed either by the Euler-Maxwell system with damping at the fluid level or by the Vlasov-Maxwell-Boltzmann system at the kinetic level. In the note, we present some recent results in [8] and [7] for the study of the non-trivial large-time behavior of solutions to the Cauchy problem on the related models in perturbation framework.

Global Classical Solutions and Large-Time Behavior of the Two-Phase Fluid Model

We study the global existence of a unique strong solution, and its large-time behavior, of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier--Stokes equations through a drag forcing term. The coupled system can be derived as the hydrodynamic limit of the Vlasov--Fokker--Planck/isentropic Navier--Stokes equations with strong local alignment forces. When the initial data are sufficiently small and regular, we establish the unique existence of the global $H^s$-solutions in a perturbation framework. We also provide the large-time behavior of classical solutions showing the alignment between two fluid velocities exponentially fast as time evolves. For this, we construct a Lyapunov function measuring the fluctuations of momentum and mass from its averaged quantities.

Global Well-Posedness in Spatially Critical Besov Space for the Boltzmann Equation

The unique global strong solution in the Chemin–Lerner type space to the Cauchy problem on the Boltzmann equation for hard potentials is constructed in a perturbation framework. Such a solution space is of critical regularity with respect to the spatial variable, and it can capture the intrinsic properties of the Boltzmann equation. For the proof of global well-posedness, we develop some new estimates on the nonlinear collision term through the Littlewood–Paley theory.

Attribute association based privacy preservation for multi trust level environment

Enormous amount of e-data is collected world-wide by organizations for the purpose of their research and decision making. The availability of this heterogeneous, sensitive information in e-databases poses a threat to the privacy of the individual or organization on which the data is collected. Privacy Preserving Data Mining [PPDM] is a field of research which concentrates on preserving data privacy during the process of data mining. This paper proposes a two level partition and perturbation frame work to release multiple copies of privacy preserved datasets in Multi Trust Level [MTL] scenario that can prevent linking and diversity attack. The framework proposes two methods namely, Entropy based Attribute Privacy Preservation [EAPP] and Information Gain based Attribute Privacy Preservation [IGAPP] for privacy preservation in MTL environment. The two methods perform vertical and horizontal partitioning of data for privacy preservation. Simple K-Means clustering algorithm with cluster size 2 using both Euclidean and Manhattan distance functions are used for horizontal partitioning. The vertical partitioning of attributes within the cluster is performed based on their entropy value that indicates its one way association with its class in EAPP method and Information Gain [IG] value of the attributes that indicates the two way associations with class in IGAPP method. The attributes in the clusters are subjected to privacy preservation technique based on their entropy and IG values in EAPP and IGAPP methods, respectively. The effect of distance in clustering the data points on privacy preservation and the ability of the privacy preserved datasets generated using the proposed methods to prevent privacy attacks are studied using variance, rank distortion and utility metrics. Real life medical and bench mark adult data sets have been used here for experimentation. The results show that the generated datasets exhibit good variance and rank distortion values and hence can prevent diversity and linking attacks in MTL environment. Also, the privacy preserved datasets have comparable utility on selected classification and clustering algorithms with original and L-Diversified datasets.

Calmness modulus of fully perturbed linear programs

This paper provides operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping in linear optimization under uniqueness of nominal optimal solutions. Our analysis is developed in two different parametric settings. First, in the framework of canonical perturbations (i.e., perturbations of the objective function and the right-hand-side of the constraints), the paper provides a computationally tractable formula for the calmness modulus, which goes beyond some preliminary results of the literature. Second, in the framework of full perturbations (perturbations of all coefficients), after characterizing the calmness property for the optimal set mapping, the paper provides an operative upper bound for the corresponding calmness modulus, as well as some illustrative examples. We provide two applications related to algorithms traced out from the literature: the first one to a descent method in LP, and the second to a regularization method for linear programs with complementarity constraints.

Adaptive sliding mode controller design for nonlinear teleoperation systems using singular perturbation method

In the presence of communication latency in a bilateral teleoperation system, stability and transparency are highly regarded. The major problem with teleoperation systems is complex dynamic and subsequently designing the controller. This essay presents a novel method of controller design, based on singular perturbation framework for the bilateral teleoperation of nonlinear robots through the Internet. Using this method makes it much easier to design the controller because controller designing is done for reduced-order subsystem not full-order system. In this paper, teleoperation system dynamics was decomposed into slave (slow subsystem) and error (fast subsystem, different from master and slave), which contains position and velocity, and an adaptive sliding mode controller was designed for each of them. For constant and time-varying delay, the positions of master–slave were compared together and controlling signal was applied to the slave; therefore, they could track each other in the shortest possible time. In the simulation, the stability and performance results of teleoperation system were obtained under the proposed controller and compared with those of another approach. In this research, having higher speed and lower control signal amplitude, at the same time, elucidates the absolute superiority of the new approach.

A complete analysis of a classical Poisson–Nernst–Planck model for ionic flow

In this work, we examine the stationary one-dimensional classical Poisson–Nernst–Planck (cPNP) model for ionic flow – a singularly perturbed boundary value problem (BVP). For the case of zero permanent charge, we provide a complete answer concerning the existence and uniqueness of the BVP. The analysis relies on a number of ingredients: a geometric singular perturbation framework for a reduction to a singular BVP, a reduction of the singular BVP to a matrix eigenvalue problem, a relation between the matrix eigenvalues and zeros of a meromorphic function, and an application of the Cauchy Argument Principle for identifying zeros of the meromorphic function. Once the zeros of the meromorphic function in a stripe are determined, an explicit solution of the singular BVP is available. It is expected that this work would be useful for studies of other PNP systems.

How Can the Modified Dispersion Relation Affect Friedmann Equations?

The appearance of the quantum gravitational effects in a very high energy regime necessitates some corrections to the thermodynamics of Friedmann-Robertson-Walker (FRW) universe. The modified dispersion relation (MDR) as a phenomenological approach to investigate the high energy physics provides a perturbation framework upon which the FRW universe thermodynamics can be corrected. In this letter, we obtain the corrected entropy-area relation of the apparent horizon of FRW universe by utilizing the extra dimensional form of MDR, leading to the modification of Friedmann equations. The influence of MDR on the Friedmann equations provides a good insight into the understanding of the FRW universe dynamics in the final quantum gravity theory.

MicroBlack Holes Thermodynamics in the Presence of Quantum Gravity Effects

Black hole thermodynamics is corrected in the presence of quantum gravity effects. Some phenomenological aspects of quantum gravity proposal can be addressed through generalized uncertainty principle (GUP) which provides a perturbation framework to perform required modifications of the black hole quantities. In this paper, we consider the effects of both a minimal measurable length and a maximal momentum on the thermodynamics of TeV-scale black holes. We then extend our study to the case that there are all natural cutoffs as minimal length, minimal momentum, and maximal momentum simultaneously. We also generalize our study to the model universes with large extra dimensions (LED). In this framework existence of black holes remnants as a possible candidate for dark matter is discussed. We study probability of black hole production in the Large Hadronic Collider (LHC) and we show this rate decreasing for sufficiently large values of the GUP parameter.

Metastable behavior in Markov processes with internal states

A perturbation framework is developed to analyze metastable behavior in stochastic processes with random internal and external states. The process is assumed to be under weak noise conditions, and the case where the deterministic limit is bistable is considered. A general analytical approximation is derived for the stationary probability density and the mean switching time between metastable states, which includes the pre exponential factor. The results are illustrated with a model of gene expression that displays bistable switching. In this model, the external state represents the number of protein molecules produced by a hypothetical gene. Once produced, a protein is eventually degraded. The internal state represents the activated or unactivated state of the gene; in the activated state the gene produces protein more rapidly than the unactivated state. The gene is activated by a dimer of the protein it produces so that the activation rate depends on the current protein level. This is a well studied model, and several model reductions and diffusion approximation methods are available to analyze its behavior. However, it is unclear if these methods accurately approximate long-time metastable behavior (i.e., mean switching time between metastable states of the bistable system). Diffusion approximations are generally known to fail in this regard.

ClogPalk: a method for predicting alkane/water partition coefficient

Alkane/water partition coefficients (P(alk)) are less familiar to the molecular design community than their 1-octanol/water equivalents and access to both data and prediction tools is much more limited. A method for predicting alkane/water partition coefficient from molecular structure is introduced. The basis for the ClogP(alk) model is the strong (R² = 0.987) relationship between alkane/water partition coefficient and molecular surface area (MSA) that was observed for saturated hydrocarbons. The model treats a molecule as a perturbation of a saturated hydrocarbon molecule with the same MSA and uses increments defined for functional groups to quantify the extent to which logP(alk) is perturbed by the introduction each functional group. Interactions between functional groups, such as intramolecular hydrogen bonds are also parameterized within a perturbation framework. The functional groups and interactions between them are specified substructurally in a transparent and reproducible manner using SMARTS notation. The ClogP(alk) model was parameterized using data measured for structurally prototypical compounds that dominate the literature on alkane/water partition coefficients and then validated using an external test set of 100 alkane/water logP measurements, the majority of which were for drugs.

Macroscopic descriptions of non-linear electromagnetic interactions in many-electron systems

Reduced-density-matrix descriptions are developed for linear and non-linear (possibly coherent) electromagnetic interactions of many-electron systems. Applications of interest include pump-probe optical phenomena in warm atomic vapors, partially-ionized plasmas, and condensed matter. Collision processes are treated within the framework of environmental perturbations, giving rise to decoherence and relaxation phenomena, and externally applied magnetic fields are taken into account on an equal footing with the electromagnetic fields. Time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations are developed in a unified manner. The standard Born (lowest-order-perturbation) and Markov (short-memory-time) approximations are systematically introduced within the framework of the general non-perturbative and non-Markovian formulations. A preliminary semi-classical perturbation-theory treatment of the electromagnetic interaction is adopted. However, it is emphasized that a quantized-electromagnetic-field approach will be necessary for a fully self-consistent quantum-mechanical formulation. The primary quantities of interest are the linear and the non-linear macroscopic electromagnetic-response tensors. Coherent initial electronic excitations and the full tetradic-matrix form of the Liouville-space self-energy operator representing the environmental interactions in the Markov approximation can incorporated in the expressions for these macroscopic electromagnetic-response tensors. Collisional interactions can be treated in various approximations for the Liouville-space self-energy operator, and the influence of Zeeman coherences on the macroscopic electromagnetic response can be investigated.

DISSOLVING OF CUSP FORMS: HIGHER‐ORDER FERMI'S GOLDEN RULES

For a hyperbolic surface, embedded eigenvalues of the Laplace operator are unstable and tend to dissolve into scattering poles i.e. become resonances. A sufficient dissolving condition was identified by Phillips–Sarnak and is elegantly expressed in Fermi's golden rule. We prove formulas for higher approximations and obtain necessary and sufficient conditions for dissolving a cusp form with eigenfunction into a resonance. In the framework of perturbations in character varieties, we relate the result to the special values of the -series . This is the Rankin–Selberg convolution of with , where is the antiderivative of a weight two cusp form. In an example we show that the above-mentioned conditions force the embedded eigenvalue to become a resonance in a punctured neighborhood of the deformation space.

Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force

We are concerned with a two-phase flow system consisting of the Vlasov-Fokker-Planck equation for particles coupled to the compressible Euler equations for the fluid through the friction force. Global well-posedness of the Cauchy problem is established in perturbation framework, and rates of convergence of solutions toward equilibrium, which are algebraic in the whole space and exponential on torus, are also obtained under some additional conditions on initial data. The proof is based on the classical energy estimates.