Smooth Perturbations Research Articles

Scintillation Level of Scattered Radio Waves in the Equatorial Ionosphere

Scintillation effects of propagating and scattered radio waves (RWs) are considered analytically and numerically in the equatorial terrestrial ionosphere applying the statistical characteristics. Analytical calculations of the second-order statistical moments of the phase fluctuations of the ordinary and extraordinary RWs in the conductive collision ionospheric magnetoplasma (COCOIMA) are carried out for the first time using the Wentzel-Kramers-Brilluen method and the modified smooth perturbation method in consideration of the diffraction effects. Polarization coefficients of these waves in the equatorial ionosphere have been derived for the first time. The scintillation level includes the variance and the correlation functions of scattered RWs. Statistical moments contain complex refractive index, anisotropy coefficient characterizing irregular plasmonic structures, tilt angle of prolate irregularities concerning to the external magnetic field, permittivity components of the equatorial ionosphere, Hall's, Pedersen and longitudinal conductivities. A new feature of the "Fountain Effect" has been discovered in the equatorial ionosphere at weak and moderate scintillations caused due to anisotropy parameters of electron density fluctuations. Anisotropy parameters of electron density inhomogeneities influence the scintillation index of both the ordinary and extraordinary radio waves shifting maximums of the curves in the opposite directions at weak and moderate scintillations. Numerical calculations are carried out using the hybrid anisotropic correlation function of electron density fluctuations containing both exponential and power-law spectral functions having arbitrary spectral index, applying the experimental data.

Three Levels of Neural Control Contributing to Performance-stabilizing Synergies in Multi-finger Tasks

We tested a hypothesis on force-stabilizing synergies during four-finger accurate force production at three levels: (1) The level of the reciprocal and coactivation commands, estimated as the referent coordinate and apparent stiffness of all four fingers combined; (2) The level of individual finger forces; and (3) The level of firing of individual motor units (MU). Young, healthy participants performed accurate four-finger force production at a comfortable, non-fatiguing level under visual feedback on the total force magnitude. Mechanical reflections of the reciprocal and coactivation commands were estimated using small, smooth finger perturbations applied by the “inverse piano” device. Firing frequencies of motor units in the flexor digitorum superficialis (FDS) and extensor digitorum communis (EDC) were estimated using surface recording. Principal component analysis was used to identify robust MU groups (MU-modes) with parallel changes in the firing frequency. The framework of the uncontrolled manifold hypothesis was used to compute synergy indices in the spaces of referent coordinate and apparent stiffness, finger forces, and MU-mode magnitudes. Force-stabilizing synergies were seen at all three levels. They were present in the MU-mode spaces defined for MUs in FDS, in EDC, and pooled over both muscles. No effects of hand dominance were seen. The synergy indices defined at different levels of analysis showed no correlations across the participants. The findings are interpreted within the theory of control with spatial referent coordinates for the effectors. We conclude that force stabilization gets contributions from three levels of neural control, likely associated with cortical, subcortical, and spinal circuitry.

Temperature Field Around Nearly Spherical Cavities in Uniform Heat Flow

Abstract The problem of determining temperature distribution near a spherical cavity in an otherwise infinite medium which is under uniform heat flow is a classical problem of linear heat conduction theory. Extensive reviews of relevant work can be found in [1-5]. It is clear from these studies is that what has been achieved in this regard is largely related to cavities with highly canonical shapes, mostly spherical. Solutions of similar problems for cavities with non-canonical shapes are rare. In the present article, we consider the case of a cavity in an infinite solid in a uniform heat flow, whose shape deviates slightly from a perfectly spherical shape (hereafter called nearly spherical cavity). To the first order in the small parameter characterizing the boundary perturbation, we are able to derive closed-form expressions for the temperature field around the nearly spherical cavity for sufficiently smooth boundary perturbations that are arbitrary functions of the azimuthal and polar angles.

Orthonormal Strichartz estimate for dispersive equations with potentials

In this paper we prove the orthonormal Strichartz estimates for the higher order and fractional Schrödinger, wave, Klein-Gordon and Dirac equations with potentials. As in the case of the Schrödinger operator, the proofs are based on the smooth perturbation theory by T. Kato. However, for the Klein-Gordon and Dirac equations, we also use a method of the microlocal analysis in order to prove the estimates for wider range of admissible pairs. As applications we prove the global existence of a solution to the higher order or fractional Hartree equation with potentials which describes the dynamics of infinitely many particles. We also give a local existence result for the semi-relativistic Hartree equation with electromagnetic potentials. As another application, the refined Strichartz estimates are proved for higher order and fractional Schrödinger, wave and Klein-Gordon equations.

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

In this paper, we examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold M exists for all time t and uniformly converges to a solution to the Yamabe problem on M as t→∞. We show that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on M in the infinite time. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

The limit cycle bifurcations of a whirling pendulum with piecewise smooth perturbations

The limit cycle bifurcations of a whirling pendulum with piecewise smooth perturbations

Crustal density structure investigation of the East China Sea and adjacent regions using wavenumber domain 3D density imaging method

The East China Sea, situated at the intersection of the Eurasian, Philippine Sea, and Pacific plates, is characterized by complex geology influenced by tectonic phenomena such as plate movements, volcanism, faults, and uplifts. Crustal density structure inversion provides a thorough understanding of the region's geological history as well as Earth's dynamical evolution, providing critical insights into seismic disaster mitigation, resource exploration, marine environmental protection, and maritime safety. The inversion process, on the other hand, presents challenges in data quality, quantity, model complexity, uncertainty, and computational resources. With the advancement of next-generation satellite gravity measurements and developing inversion techniques, the inversion of marine crustal density structures promises to be more precise and comprehensive. We explored the density distribution in the East China Sea and surrounding areas using an innovative wavenumber domain three-dimensional density imaging method along with high-precision global satellite gravity data. By overcoming data quality and computing resource constraints, wavenumber domain three-dimensional density imaging has transformed the discipline of marine geophysics, successfully delivering accurate density distributions in the study area. We were able to get a more precise and comprehensive characterization of the crustal density structure by combining high-precision satellite gravity data and cutting-edge imaging methods. Our investigation has unveiled previously unknown details about density distribution in the East China Sea and its environs. The East China Sea shelf displays smooth low-density perturbations at 18 km depth, whereas the trench–arc–basin region exhibits increasing density perturbations. Notably, the Okinawa Trough, which is surrounded by the Tokara Volcanic Ridge and the Ryukyu Trench, displays strong positive anomalies with a north–northeastern to northeastern orientation. In contrast, the Ryukyu Ridge and the Philippine Sea Basin exhibit smaller negative values and substantial northwestward positive density trends, respectively. These findings indicate diverse material distribution, which provides important insights into the area’s geological evolution and tectonic processes. This study adds new insights into density distribution in the East China Sea and adjacent regions, offering information on the geological complexity of the region. The research lays the groundwork for future research on crustal dynamics and enhances the field of marine geophysics and related disciplines.Graphical abstract

CardioDefense: Defending against adversarial attack in ECG classification with adversarial distillation training

In clinics, doctors rely on electrocardiograms (ECGs) to assess severe cardiac disorders. Owing to the development of technology and the increase in health awareness, ECG signals are currently obtained by using medical and commercial devices. Deep neural networks (DNNs) can be used to analyze these signals because of their high accuracy rate. However, researchers have found that adversarial attacks can significantly reduce the accuracy of DNNs. Studies have been conducted to defend ECG-based DNNs against traditional adversarial attacks, such as projected gradient descent (PGD), and smooth adversarial perturbation (SAP) which targets ECG classification; however, to the best of our knowledge, no study has completely explored the defense against adversarial attacks targeting ECG classification. Thus, we did different experiments to explore the effects of defense methods against white-box adversarial attack and black-box adversarial attack targeting ECG classification, and we found that some common defense methods performed well against these attacks. Besides, we proposed a new defense method based on adversarial distillation training (named CardioDefense) which comes from defensive distillation and can effectively improve the generalization performance of DNNs. The results show that our method performed more effectively against adversarial attacks targeting on ECG classification than the other baseline methods, namely, adversarial training, defensive distillation, Jacob regularization, and noise-to-signal ratio regularization. Furthermore, we found that our method performed better against PGD attacks with low noise levels, which means that our method has stronger robustness.

Feedback Stabilization of a Two-Fluid Surface Tension System Modeling the Motion of a Soap Bubble at Low Reynolds Number: The Two-Dimensional Case

The aim of this paper is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of mathbb {R}^2 into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as the sum of two feedback operators: one is explicit, the second one is finite-dimensional. They enable us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to translations, and up to non-contact with the outer boundary.

Jet from a very large, axisymmetric, surface-gravity wave

We demonstrate that gravity acting alone at large length scales (compared to the capillary length) can produce a jet from a sufficiently steep, axisymmetric surface deformation imposed on a quiescent, deep pool of liquid. Mechanistically, the jet owes it origin to the focusing of a concentric, surface wave towards the axis of symmetry, quite analogous to such focusing of capillary waves and resultant jet formation observed during bubble collapse at small scales. A weakly nonlinear theory based on the method of multiple scales in the potential flow limit is presented for a modal (single-mode) initial condition representing the solution to the primary Cauchy–Poisson problem. A pair of novel, coupled, amplitude equations are derived governing the modulation of the primary mode. For moderate values of the perturbation parameter $\epsilon$ (a measure of the initial perturbation steepness), our second-order theory captures the overshoot (incipient jet) at the axis of symmetry quite well, demonstrating good agreement with numerical simulation of the incompressible, Euler equation with gravity (Popinet 2014, Basilisk. http://basilisk.fr) and no surface tension. We demonstrate that the underlying wave focusing mechanism may be understood in terms of radially inward motion of nodal points of a linearised, axisymmetric, standing wave. This explanation rationalises the ubiquitous observation of such jets accompanying cavity collapse phenomena, spanning length scales from microns to several metres. Expectedly, our theory becomes inaccurate as $\epsilon$ approaches unity. In this strongly nonlinear regime, slender jets form with surface accelerations exceeding gravity by more than an order of magnitude. In this inertial regime, we compare the jets in our simulations with the inertial, self-similar, analytical solution by Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121) and find qualitative agreement with the same. This analysis demonstrates, from first principles, an example of a jet created purely under gravity from a smooth initial perturbation and provides support to the analytical model of Longuet-Higgins (J. Fluid Mech., 1983, vol. 127, pp. 103–121).

Time series adversarial attacks: an investigation of smooth perturbations and defense approaches

AbstractAdversarial attacks represent a threat to every deep neural network. They are particularly effective if they can perturb a given model while remaining undetectable. They have been initially introduced for image classifiers, and are well studied for this task. For time series, few attacks have yet been proposed. Most that have are adaptations of attacks previously proposed for image classifiers. Although these attacks are effective, they generate perturbations containing clearly discernible patterns such as sawtooth and spikes. Adversarial patterns are not perceptible on images, but the attacks proposed to date are readily perceptible in the case of time series. In order to generate stealthier adversarial attacks for time series, we propose a new attack that produces smoother perturbations. We introduced a function to measure the smoothness for time series. Using it, we find that smooth perturbations are harder to detect both visually, by the naked eye and by deep learning models. We also show two ways of protection against adversarial attacks: the first one by detecting the attacks using a deep model; the second one by using adversarial training to improve the robustness of a model against a specific attack, thus making it less vulnerable.

Evaluating the impact of correlated noise for time‐lapse transient electromagnetic (TEM) monitoring studies

AbstractTypically, transient electromagnetic (TEM) hydrogeological applications focus on spatial characterization. However, there is scope for utilizing time‐lapse TEM to monitor dynamic processes. Changes related to hydrogeological processes are subtle; consequently, measurements must be of exceptional quality, and the presence and influence of noise sources must be well understood. A potentially problematic noise source is radio waves operating in the 3–300 kHz range, as they introduce correlated error into TEM measurements. For instance, it was anticipated that such radio waves could introduce smooth perturbations in the data that could be erroneously interpreted in the TEM inversion results. This work evaluates the presence of correlated noise sources, their temporal dependence and their significance for TEM monitoring. This work combines fully sampled data and normalized covariance matrices of gated data to identify correlated noise sources. Radio signals operating in the low and very low frequency (LF and VLF) bands were found to vary across the investigation period; however, their influence on the TEM inversion results was minimal. Measurement stacking remains an effective way to improve the signal‐to‐noise ratio of TEM data affected by LF and VLF noise sources as they do not introduce bias, for example smooth perturbation, into the TEM data explored in this work.

Robust Training of Neural Networks against Bias Field Perturbations

We introduce the problem of training neural networks such that they are robust against a class of smooth intensity perturbations modelled by bias fields. We first develop an approach towards this goal based on a state-of-the-art robust training method utilising Interval Bound Propagation (IBP). We analyse the resulting algorithm and observe that IBP often produces very loose bounds for bias field perturbations, which may be detrimental to training. We then propose an alternative approach based on Symbolic Interval Propagation (SIP), which usually results in significantly tighter bounds than IBP. We present ROBNET, a tool implementing these approaches for bias field robust training. In experiments networks trained with the SIP-based approach achieved up to 31% higher certified robustness while also maintaining a better accuracy than networks trained with the IBP approach.

Simultaneous linearization of diffeomorphisms of isotropic manifolds

Suppose that M is a closed isotropic Riemannian manifold and that R_1,\dots ,R_m generate the isometry group of M . Let f_1,\dots ,f_m be smooth perturbations of these isometries. We show that the f_i are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian [Duke Math. J. {136}, 475–505 (2007)] from S^n to real, complex, and quaternionic projective spaces. In addition, we identify and remedy an oversight in that earlier work.

Piecewise Smooth Perturbations to a Class of Planar Cubic Centers

This paper studies the limit cycle bifurcations of a class of planar cubic isochronous centers. For different values of two key parameters, we give an estimate of the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems under arbitrarily small piecewise smooth polynomial perturbation. The main method and technique are based on the first order averaging theory for discontinuous systems and the Argument Principle in complex analysis.

Geometric Aspects of Shape Optimization

We present a review of known results in shape optimization from the point of view of Geometric Analysis. This paper is devoted to the mathematical aspects of the shape optimization theory. We focus on the theory of gradient flows of objective functions and their regularizations. Shape optimization is a part of calculus of variations which uses the geometry. Shape optimization is also related to the free boundary problems in the theory of Partial Differential Equations. We consider smooth perturbations of geometrical domains in order to develop the shape calculus for the analysis of shape optimization problems. There are many applications of such a framework, in solid and fluid mechanics as well as in the solution of inverse problems. For the sake of simplicity we consider model problems, in principle in two spatial dimensions. However, the methods presented are used as well in three spatial dimensions. We present a result on the convergence of the shape gradient method for a model problem. To our best knowledge it is the first result of convergence in shape optimization. The complete proofs of some results are presented in report (Plotnikov and Sokolowski, Gradient flow for Kohn–Vogelius functional).

Global Well-Posedness and Large-Time Behavior for the Equilibrium Diffusion Model in Radiation Hydrodynamics

In this paper, we investigate the global well-posedness of the equilibrium diffusion model in radiation hydrodynamics. The model consists of the compressible Navier-Stokes equations coupled with radiation effect terms described by the fourth power of temperature. The global existence of classical solutions to the Cauchy problem in the whole space is established when initial data is a small smooth perturbation of a constant equilibrium state: moreover, an algebraic rate of convergence of solutions toward equilibrium is obtained under additional conditions on initial data. The proof is based on the refined energy method and Fourier’s analysis.

Rigidity properties for some isometric extensions of partially hyperbolic actions on the torus

This paper studies local rigidity for some isometric toral extensions of partially hyperbolic Z k \mathbb {Z}^k ( k ⩾ 2 k\geqslant 2 ) actions on the torus. We prove a C ∞ C^\infty local rigidity result for such actions, provided that the smooth perturbations of the actions satisfy the intersection property. We also give a local rigidity result within a class of volume preserving actions. Our method mainly uses a generalization of the Kolmogorov-Arnold-Moser iterative scheme.

Stability of ion-acoustic shock profiles to one-dimensional compressible Navier-Stokes-Poisson equations with large density oscillation

It is shown in [3] that the one-dimensional compressible Navier-Stokes-Poisson system admits a unique (up to a shift) small-amplitude smooth traveling wave and this traveling wave is time-asymptotically nonlinear stable under suitably small smooth zero-mass perturbations. This paper will show that similar results still hold even for certain large initial data. Our main strategy is to use the smallness of the strength of traveling wave solution to regulate the possible growth of the solutions of the perturbation equation, and then obtain the desired uniform-in-time upper and lower bounds on the specific volume.

Long‐term stability of periodic smooth irrotational flows to the electrons Euler–Poisson system in three dimensions

We are concerned with the one‐fluid Euler–Poisson system for the electrons on the cubes in three dimensions. With the bootstrap estimates, we prove long‐term stability of periodic solutions of this system with small smooth irrotational perturbations of a constant background. Our main result is that the solutions are well‐posed for a time at least , where is the size of the initial data, the sufficiently large is the side length of the cubes, and is the index of the classical Sobolev spaces .