Perturbation Level Research Articles

On the Problem of Global Localization of Discontinuity Lines for a Function of Two Variables

We consider the ill-posed problem of localizing (finding the position of) the discontinuity lines of a function of two variables that is smooth outside the discontinuity lines and has a discontinuity of the first kind at each point of such lines. A uniform square grid with step τ is considered, and it is assumed that the mean values of a perturbed function over squares with side τ are known at each node of the grid. The perturbed function approximates the exact function in the space L2(ℝ2). The perturbation level δ is known. To solve the problem under consideration, we design and study global discrete algorithms that are based on averaging procedures and approximate the discontinuity lines by a set of points of a uniform grid. The main result of the paper is the development of an approach to the global study of localization algorithms. We formulate conditions for the exact function, thus defining a class of correctness. Within this class, we perform a theoretical study of the proposed algorithms, introduce the characteristics to be estimated (the concept of approximating a set of discontinuity lines by a set of points of a uniform grid), and develop methods for deriving the estimates. To achieve this goal, we use a simplified statement: the discontinuity lines are straight line segments, and the proposed localization algorithm has the simplest thinning block. It is established that the localization error of the algorithm has order O(δ). Estimates of other important parameters characterizing the localization algorithm are given.

Two-field mimetic gravity revisited and Hamiltonian analysis

We revisit the two-field mimetic gravity model with shift symmetries recently proposed in the literature, especially the problems of degrees of freedom and stabilities. We first study the model at the linear cosmological perturbation level by quadratic Lagrangian and Hamiltonian formulations. We show that there are actually two (instead of one) scalar degrees of freedom in this model in addition to two tensor modes. We then push on the study to the full non-linear level in terms of the Hamiltonian analysis, and confirm our result from the linear perturbation theory. We also consider the case where the kinetic terms of the two mimetic scalar fields have opposite signs in the constraint equation. We point out that in this case the model always suffers from the ghost instability problem.

Characteristics of medial-lateral postural control while exposed to the external perturbation in step initiation

Controllability of posture in the medial-lateral direction is critical for balance maintenance, particularly in step initiation. The objective of the current study was to examine the effects of external perturbation and landing orientation on medial-lateral control stability in step initiation. Eleven young healthy participants stood on the force platform and waited for the instruction of taking a step while experiencing a pendulum perturbation applied at the lateral side of the right shoulder. Eight experimental conditions were conducted by two levels of step side (right or left), two levels of perturbation (with or without), and two levels of landing orientation (forward or diagonal). The center of pressure (COP), pelvic movements, and muscle activities were recorded and analyzed as the onset of COP and pelvic movement, the COP displacement, and cocontraction and reciprocal muscle activation pattern. The temporal events of COP and pelvic movement were not significantly different in all experimental conditions. However, COP and pelvic movement were significantly later in the diagonal condition. Most of the segments showed reciprocal muscle activation patterns in relation to the perturbation released time. Subsequently, all segments showed cocontraction muscle activation patterns, which was significantly affected by step side, perturbation, and orientation. The results suggest that how the CNS initiated a step was identical with the COP then pelvic movement. The outcome highlights the importance of external perturbation and foot landing orientation effects on postural adjustments, which may provide a different approach to help step initiation.

Evaluating climate emulation: fundamental impulse testing of simple climate models

Abstract. Simple climate models (SCMs) are numerical representations of the Earth's gas cycles and climate system. SCMs are easy to use and computationally inexpensive, making them an ideal tool in both scientific and decision-making contexts (e.g., complex climate model emulation, parameter estimation experiments, climate metric calculations, and probabilistic analyses). Despite their prolific use, the fundamental responses of SCMs are often not directly characterized. In this study, we use fundamental impulse tests of three chemical species (CO2, CH4, and black carbon – BC) to understand the fundamental gas cycle and climate system responses of several comprehensive (Hector v2.0, MAGICC 5.3, MAGICC 6.0) and idealized (FAIR v1.0, AR5-IR) SCMs. We find that while idealized SCMs are widely used, they fail to capture the magnitude and timescales of global mean climate responses under emissions perturbations, which can produce biased temperature results. Comprehensive SCMs, which have physically based nonlinear forcing and carbon cycle representations, show improved responses compared to idealized SCMs. Even the comprehensive SCMs, however, fail to capture the response timescales to BC emission perturbations seen recently in two general circulation models. Some comprehensive SCMs also generally respond faster than more complex models to a 4×CO2 concentration perturbation, although this was not evident for lower perturbation levels. These results suggest where improvements should be made to SCMs. Further, we demonstrate here a set of fundamental tests that we recommend as a standard evaluation suite for any SCM. Fundamental impulse tests allow users to understand differences in model responses and the impact of model selection on results.

Reduced-Order Modeling of Coupled Flow and Quasistatic Geomechanics

Summary A reduced-order-modeling (ROM) framework is developed and applied to simulate coupled flow/quasistatic-geomechanics problems. The reduced-order model is constructed using POD-TPWL, in which proper orthogonal decomposition (POD), which enables representation of the solution unknowns in a low-dimensional subspace, is combined with trajectory-piecewise linearization (TPWL), where solutions with new sets of well controls are represented by means of linearization around previously simulated (training) solutions. The overdetermined system of equations is projected into the low-dimensional subspace using a least-squares Petrov-Galerkin (LSPG) procedure, which has been shown to maintain numerical stability in POD-TPWL models. The states and derivative matrices required by POD-TPWL, generated by an extended version of the Stanford University Automatic Differentiation General Purpose Research Simulator (AD-GPRS), are provided in an offline (preprocessing or training) step. Offline computational requirements correspond to the equivalent of five to eight full-order simulations, depending on the number of training runs used. Run-time (online) speedups of O(100) or more are typically achieved for new POD-TPWL test-case simulations. The POD-TPWL model is tested extensively for a 2D coupled problem involving oil/water flow and geomechanics. It is shown that POD-TPWL provides predictions of reasonable accuracy, relative to full-order simulations, for well-rate quantities, global pressure and saturation fields, global maximum- and minimum-principal-stress fields, and the Mohr-Coulomb rock-failure criterion, for the cases considered. A systematic study of POD-TPWL error is conducted using various training procedures for different levels of perturbation between test and training cases. The use of randomness in the well-bottomhole-pressure (BHP) profiles used in training is shown to be beneficial in terms of POD-TPWL solution accuracy. The procedure is also successfully applied to a prototype 3D example case.

Cosmological constrains on minimally and non-minimally coupled scalar field models

Abstract We study the minimally and non-minimally coupled scalar field models as possible alternatives for dark energy, the mysterious energy component that is driving the accelerated expansion of the universe. After discussing the dynamics at both the background and perturbation level, we confront the two models with the latest cosmological data. After obtaining updated constraints on their parameters we perform model selection using the basic information criteria. We found that the ΛCDM model is strongly favored when the local determination of the Hubble constant is not considered and that this statement is weakened once local H0 is included in the analysis. We calculate the parameter combination $S_8=\sigma _8\sqrt{\Omega _{m}/0.3}$ and show the decrement of the tension with respect to the Planck results in the case of minimally and non-minimally coupled scalar field models. Finally, for the coupling constant between DE and gravity, we obtain the constraint $\xi \simeq -0.06^{+0.19}_{-0.19}$, approaching the one from solar system tests |ξ| ≲ 10−2 and comparable to the conformal value ξ = 1/6 at 1σ uncertainty.

Economic impact assessment of natural disaster with multi-criteria decision making for interdependent infrastructures

Natural disasters, such as flood, earthquake or hurricane, can cause environmental damage, infrastructural destruction, economic disruption and result in the loss of human lives. The goal of this paper is to assess vulnerability of the interdependent sectors and to determine the impact of a disaster. As part of our investigation, we explore the inoperability input–output model (IIM) to obtain ballpark estimate of losses. IIM can quantify the effect on production level and measure ripple effect of perturbation on the interdependent system by calculating the economic loss and inoperability level. Furthermore, PROMETHEE (preference ranking organization method for enrichment evaluations) is integrated with IIM, incorporating the criteria of economic system linkages, economic losses and percentage of the inoperability. It will help to analyze the effect of disaster and rank the most vulnerable sectors—based on the defined criterion. A case study is performed in Pakistan (an Asian developing country) which was hit by floods in 2011–2012, perturbing the demand in various sectors. To analyze the perturbation level and system linkages, an interdependent input–output matrix—based on the Pakistan’s economy—is constructed. Further research analysis gives perspicacity in terms of describing criticality and sensitivity of the independent economic system. It is concluded that the agriculture and service sectors have suffered with the highest inoperability level. The results also identify significant sectors of the economy after flood strikes; these sectors can be prioritized for policymaking activities to reduce the sector-specific impact in the aftermath of a catastrophic event.

The BAO+BBN take on the Hubble tension

Many attempts to solve the Hubble tension with extended cosmological models combine an enhanced relic radiation density, acting at the level of background cosmology, with new physical ingredients affecting the evolution of cosmological perturbations. Several authors have pointed out the ability of combined Baryon Acoustic Oscillation (BAO) and Big Bang Nucleosynthesis (BBN) data to probe the background cosmological history independently of both CMB maps and supernovae data. Using state-of-the-art assumptions on BBN, we confirm that combined BAO, deuterium, and helium data are in tension with the SH0ES measurements under the ΛCDM assumption at the 3.2σ level, while being in close agreement with the CMB value. We subsequently show that floating the radiation density parameter Neff only reduces the tension down to the 2.6σ level. This conclusion, totally independent of any CMB data, shows that a high Neff accounting for extra relics (either free-streaming or self-interacting) does not provide an obvious solution to the crisis, not even at the level of background cosmology. To circumvent this strong bound, (i) the extra radiation has to be generated after BBN to avoid helium bounds, and (ii) additional ingredients have to be invoked at the level of perturbations to reconcile this extra radiation with CMB and LSS data.

Imprint of a Steep Equation of State in the growth of structure

We study the cosmological properties of a dynamical of dark energy (DE) component determined by a Steep Equation of State (SEoS) w(z)=w0+wi(z/zT)q1+(z/zT)q. The SEoS has a transition at zT between two pivotal values (wi, w0) which can be taken as an early time and present day values of w and the steepness is given by q. We describe the impact of this dynamical DE at background and perturbative level. The steepness of the transition has a better cosmological fit than a conventional CPL model with w=w0+wa(1−a). Furthermore, we analyze the impact of steepness of the transition in the growth of matter perturbations and structure formation. This is manifest in the linear matter power spectrum, P(k), the logarithmic growth function, fσ8(z), and the differential mass function dn/dlogM(z=0). The differences in these last three quantities is at a percent-level using the same cosmological baseline parameters in our SEoS and a ΛCDM model. However, we find an increase in the power spectrum, producing a bump at k ≈ kT with kT ≡ aTH(aT) the mode associated to the time of the steep transition (aT=1/(1+zT)). Different dynamics of DE lead to a different amount of DM at present time which has an impact in Power Spectrum and accordingly in structure formation.

Non-fragile H∞ consensus tracking of nonlinear multi-agent systems with switching topologies and transmission delay via sampled-data control

In this paper, the sampled-data non-fragile H∞ consensus tracking problem for Lipschitz nonlinear multi-agent systems with switching topologies and exogenous disturbances is investigated. Each possible interaction topology in the switching topologies set is assumed to contain a directed spanning tree. With introducing a sampled-data mechanism, the information is only capable of being aperiodically transmitted in the network at each sampling instant and unavoidably subject to a transmission delay. Then a protocol collecting the delayed sampled-data information from neighboring agents is proposed not only to provide robustness against some level of controller gain perturbations, but also to regulate consensus performance with an H∞ disturbance attenuation level. By using tools from algebraic graph theory and Lyapunov–Krasovskii functional technique, it is proved that the concerned consensus tracking problem is solvable if the resultant consensus error system can be asymptotically stabilized. Simulation results verify the theoretical analysis.

Predicted threshold against forward and backward loss of balance for perturbed walking

The biomechanical mechanisms of loss of balance have been studied before for slip condition but have not been investigated for arbitrary perturbation profiles under non-slip conditions in sagittal plane. This study aimed to determine the thresholds of center of mass (COM) velocity and position relative to the base of support (BOS) that predict forward and backward loss of balance during walking with a range of BOS perturbations. Perturbations were modeled as sinusoidal BOS motions in the vertical or anterior-posterior direction or as sagittal rotation. The human body was modeled using a seven-link model. Forward dynamics alongside with dynamic optimization were used to find the thresholds of initial COM velocity for each initial COM position that would predict forward or backward loss of balance. The effects of perturbation frequency and amplitude on these thresholds were modeled based on the simulation data. Experimental data were collected from 15 able-bodied individuals and three individuals with disability during perturbed walking. The simulation results showed similarity with the stability region reported for slip and non-slip conditions. The feasible stability region shrank when the perturbation frequency and amplitude increased, especially for larger initial COM velocities. 89.5% (70.9%) and 82.4% (68.2%) of the measured COM position and velocity combinations during low (high) perturbations were located inside the simulated limits of the stability region, for able-bodied and disabled individuals, respectively. The simulation results demonstrated the effects of different perturbation levels on the stability region. The obtained stability region can be used for developing rehabilitative programs in interactive environments.

Inflation from fermions with curvature-dependent mass

A model of inflation realization driven by fermions with curvature-dependent mass is studied. Such a term is derived from the Covariant Canonical Gauge Theory of gravity (CCGG) incorporating Dirac fermions. We obtain an initial de Sitter phase followed by a successful exit, and moreover, we acquire the subsequent thermal history, with an effective matter era, followed finally by a dark-energy epoch. This behavior is a result of the effective `weakening' of gravity at early times, due to the decreased curvature-dependent fermion mass. Through the perturbation level, we obtain the scalar spectral index and the tensor-to-scalar ratio, which are in agreement with the Planck observations. The efficiency of inflation from fermions with curvature-dependent mass, at both the background and perturbation level, reveals the capabilities of the scenario and makes it a good candidate for the description of nature.

Nonlinear perturbations of Reissner-Nordström black holes

We develop a nonlinear perturbation theory of Reissner-Nordstr\"om black holes. We show that, at each perturbation level, Einstein-Maxwell equations can be reduced to four inhomogeneous wave equations, two for polar and two for axial sector. Gravitational part of these equations is similar to Regge-Wheeler and Zerilli equations with source and additional coupling to the electromagnetic sector. We construct solutions to the inhomogeneous part of wave equations in terms of sources for Einstein-Maxwell equations. We discuss $\ell=0$ and $ \ell=1$ cases separately.

Dark energy perturbations in N-body simulations

We present N-body simulations which are fully compatible with general relativity, with dark energy consistently included at both the background and perturbation level. We test our approach for dark energy parameterised as both a fluid, and using the parameterised post-Friedmann (PPF) formalism. In most cases, dark energy is very smooth relative to dark matter so that its leading effect on structure formation is the change to the background expansion rate. This can be easily incorporated into Newtonian N-body simulations by changing the Friedmann equation. However, dark energy perturbations and relativistic corrections can lead to differences relative to Newtonian N-body simulations at the tens of percent level for scales k < (10−3–10−2) Mpc−1, and given the accuracy of upcoming large scale structure surveys such effects must be included. In this paper we will study both effects in detail and highlight the conditions under which they are important. We also show that our N-body simulations exactly reproduce the results of the Boltzmann solver CLASS for all scales which remain linear.

Linear growth index of matter perturbations in Rastall gravity

Rastall gravity theory shows notable features consistent with physical observations in comparison to the standard Einstein theory. Recently, there has been a debate about the equivalence of Rastall gravity and general relativity. Motivated by this open issue, in the present work, we attempt to shed some light on this debate by analyzing the evolution of the Rastall based cosmological model at the background as well as perturbation level. Employing the dynamical system techniques, we found that at late times, the dynamics of the model resembles the ΛCDM model at the background level irrespective of the choice of Rastall's parameter. However, at the perturbation level, we found that the evolution of the growth index heavily depends on the Rastall's parameter and displays a significant deviation from the ΛCDM model.

Impact of kinetic and potential self-interactions on scalar dark matter

We consider models of scalar dark matter with a generic interaction potential and non-canonical kinetic terms of the K-essence type that are subleading with respect to the canonical term. We analyze the low-energy regime and derive, in the nonrelativistic limit, the effective equations of motions. In the fluid approximation they reduce to the conservation of matter and to the Euler equation for the velocity field. We focus on the case where the scalar field mass $10^{-21} \ll m \lesssim 10^{-4} \, {\rm eV}$ is much larger than for fuzzy dark matter, so that the quantum pressure is negligible on cosmological and galactic scales, while the self-interaction potential and non-canonical kinetic terms generate a significant repulsive pressure. At the level of cosmological perturbations, this provides a dark-matter density-dependent speed of sound. At the nonlinear level, the hydrostatic equilibrium obtained by balancing the gravitational and scalar interactions imply that virialized structures have a solitonic core of finite size depending on the speed of sound of the dark matter fluid. For the most relevant potential in $\lambda_4 \phi^4/4$ or K-essence with a $(\partial \phi)^4$ interaction, the size of such stable cores cannot exceed 60 kpc. Structures with a density contrast larger than $10^6$ can be accommodated with a speed of sound $c_s\lesssim 10^{-6}$. We also consider the case of a cosine self-interaction, as an example of bounded nonpolynomial self-interaction. This gives similar results in low-mass and low-density halos whereas solitonic cores are shown to be absent in massive halos.

Globally Accurate Full-Dimensional Potential Energy Surface for H2 + HCl Inelastic Scattering.

A globally accurate full-dimensional potential energy surface (PES) for the inelastic scattering between H2 and HCl is developed on the basis of a large number of points calculated at the coupled-cluster singles, doubles, and perturbative triples level of theory. The machine-learned PES is trained with 42 417 ab initio points using the permutation invariant polynomial-neural network method, resulting in a root-mean-square fitting error of 5.6 cm-1. Both full- and reduced-dimensional quantum calculations for rotationally inelastic scattering are performed on this new PES and good agreement is obtained with previous quantum dynamical results on a reduced-dimensional model. Furthermore, strong resonances are identified at collision energies below 100 K, including cold conditions. This new PES provides a reliable platform for future studies of scattering dynamics with vibrationally excited collision partners in a wide range of collision energies extending to cold and ultracold conditions.

Hamilton-Jacobi approach to holographic renormalization of massive gravity

Recently, a practical approach to holographic renormalization has been developed based on the Hamilton-Jacobi formulation. Using a simple Einstein-scalar theory, we clarify that this approach does not conflict with the Hamiltonian constraint as it seems. Then we apply it to the holographic renormalization of massive gravity. We assume that the shift vector is falling off fast enough asymptotically. We derive the counterterms up to the boundary dimension d = 4. Interestingly, we find that the conformal anomaly can even occur in odd dimensions, which is different from the Einstein gravity. We check that the counterterms cancel the divergent part of the on-shell action at the background level. At the perturbation level, they are also applicable in several time-dependent cases.

Cross-sectional reshaping of perturbed/unperturbed rectangular jets

Purpose This study aims to investigate the cross-sectional reshaping in transitioning/starting rectangular jets of aspect ratio 2 under various inlet perturbation conditions at the Reynolds number of Re = UDh/v = 17,750. Design/methodology/approach Large eddy simulation results compared with the phase-locked particle image velocimetry data exhibit the cross-sectional jet deformations from rectangular to rounder shapes. Inflow velocity oscillations are introduced at the fundamental frequency associated with the Kelvin–Helmholtz instability characterized by the spectral analysis of the hotwire data and the linear stability predictions. Findings The initially rectangular cross-section of the jet reshapes into the rounder geometries with increased downstream distance while the edges of the jet become distorted due to the shear layer instability more significantly observed near the high curvature corners. The different expansion rates in the longer and shorter edges of the jet and the consequent cross-sectional reshaping are found to be sensitive to small levels of random inlet perturbations. In addition, introducing controlled sinusoidal oscillations results in the formation of more organized trailing shear layer where the stronger vortex rings go through the curvature-induced deformations. Originality/value Spatio-temporal study of vortex dynamics in transitioning rectangular jets reveals important information about the effect of the controlled jet forcing on local entrainment. Dynamics of the leading vortex dominates the entrainment in transitioning jets which are commonly used in practical applications. Near-field entrainment is also promoted proportional to the amplitude of the controlled inlet oscillations within the trailing vortex rings.

About stability of difference equations with continuous time and fading stochastic perturbations

The long term behavior of solutions of difference equations with continuous time and fading stochastic perturbations is investigated. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square summable sequence, then an asymptotically stable and a square summable solution of a deterministic difference equation remains to be an asymptotically mean square stable and a mean square summable solution by stochastic perturbations.