Smooth Perturbations Research Articles

Multilinear Oscillatory Integral Operators and Geometric Stability

In this article we prove a sharp decay estimate for certain multilinear oscillatory integral operators of a form inspired by the general framework of Christ et al. (Duke Math 130(2):321–351, 2005). A key purpose of this work is to determine when such estimates are stable under smooth perturbations of both the phase and corresponding projections, which are typically only assumed to be linear. The proof is accomplished by a novel decomposition which mixes features of Gabor or windowed Fourier bases with features of wavelet or Littlewood–Paley decompositions. This decomposition very nearly diagonalizes the problem and seems likely to have useful applications to other geometrically-inspired objects in Fourier analysis.

The Discrete Spectrum of an Infinite Kirchhoff Plate in the Form of a Locally Perturbed Strip

We study the discrete spectra of boundary value problems for the biharmonic operator describing oscillations of a Kirchhoff plate in the form of a locally perturbed strip with rigidly clamped or simply supported edges. The two methods are applied: variational and asymptotic. The first method shows that for a narrowing plate the discrete spectrum is empty in both cases, whereas for a widening plate at least one eigenvalue appears below the continuous spectrum cutoff for rigidly clamped edges. The presence of the discrete spectrum remains an open question for simply supported edges. The asymptotic method works only for small variations of the boundary. While for a small smooth perturbation the construction of asymptotics is generally the same for both types of boundary conditions, the asymptotic formulas for eigenvalues can differ substantially even in the main correction term for a perturbation with corner points.

Finite-frequency traveltime tomography using the Generalized Rytov approximation

SUMMARY Wave-equation-based traveltime tomography has been extensively applied in both global tomography and seismic exploration. Typically, the traveltime Fréchet derivative is obtained using the first-order Born approximation, which is only satisfied for weak velocity perturbations and small phase shifts (i.e. the weak-scattering assumption). Although the small phase-shift restriction can be handled with the Rytov approximation, the weak velocity-perturbation assumption is still a major limitation. The recently developed generalized Rytov approximation (GRA) method can achieve an improved phase accuracy of the forward-scattered wavefield, in the presence of large-scale and strong velocity perturbations. In this paper, we combine GRA with the classical finite-frequency theory and propose a GRA-based traveltime sensitivity kernel (GRA-TSK), which overcomes the weak-scattering limitation of the conventional finite-frequency methods. Numerical examples demonstrate that the accumulated time delay of forward-scattered waves caused by large-scale smooth perturbations can be correctly handled by the GRA-TSK, regardless of the magnitude of the velocity perturbations. Then, we apply the new sensitivity kernel to solve the traveltime inverse problem, and we propose a matrix-free Gauss–Newton method that has a faster convergence rate compared with the gradient-based method. Numerical tests show that, compared with the conventional adjoint traveltime tomography, the proposed GRA-based traveltime tomography can obtain a more accurate model with a faster convergence rate, making it more suited for recovering the large-intermediate scale of the velocity model, even for strong-perturbation and complex subsurface structures.

The Vlasov-Poisson-Landau system near a local Maxwellian

In this paper, we construct the global solutions near a local Maxwellian to the Vlasov-Poisson-Landau system with slab symmetry for the physical Coulomb interaction. The fluid quantities of this local Maxwellian are the approximate rarefaction wave solutions to the associated one-dimensional compressible Euler equations. We prove for the first time that for the Cauchy problem on this system, such a local Maxwellian is time asymptotically stable under suitably small smooth perturbations and global solutions tend in large time to the rarefaction waves of the compressible Euler system with the corresponding Riemann data. As a byproduct, the nonlinear stability of the same rarefaction waves for the pure Landau equation is also proved. This illustrates in our setting that the electric field does not affect the propagation of rarefaction waves to the Landau equation.

A Framework for Ensemble Classification and Sensitivity Analysis in Privacy Preserving Data Mining

The perturbation mechanism for data streams is a challenging task. In the emerging world, data is erupting from various sources. The core applications need care on the data streams for further analysis and experimentation. As the micro data available with the core applications shall not be revealed to the public without taking any chance of breach, the perturbation challenges the analysis to get through the like results as of on the original datasets. In this paper, we have applied a concept of Perlin noise to distract the original data from the eyes of the analysts, however allowing them to perform their activities well. Applying security dynamically on such data is a challenging task. This paper deals about the concepts of generation of smooth noise and syntactic perturbation mechanism on the selective tuples as selective perturbation.

Influence Characteristics of Laser Transmission Amplitude Fluctuation Based on Turbulent Medium

In order to study the transmission characteristics of laser in atmospheric turbulent medium and understand the influence degree of various factors on amplitude fluctuation, by means of smooth perturbation method, this paper establishes a theory model of amplitude fluctuation of laser propagation in turbulent medium by using the smooth perturbation method and reflects the amplitude fluctuation degree, carries out specific discussion on each influence factor. The results show that the larger the wavelength, the more stable the amplitude fluctuation. With the increase of laser section radius, the amplitude fluctuates sharply and then decreases slowly after reaching the peak. Transmission distance is the main influence factor of amplitude fluctuation. With the increase of transmission distance, the amplitude fluctuation will become more obvious. The amplitude acquisition can be comprehensively modulated in a specific transmission distance by wavelength and section ra-dius, so as to ensure the stability of the received laser and provide a theoretical basis for the interferometry technology.

Rigidity of Some Abelian-by-Cyclic Solvable Group Actions on $${\mathbb {T}}^N$$

In this paper, we study a natural class of groups that act as affine transformations of $${\mathbb {T}}^N$$ . We investigate whether these solvable, “abelian-by-cyclic," groups can act smoothly and nonaffinely on $${\mathbb {T}}^N$$ while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by $$C^r$$ diffeomorphisms can be conjugated to the affine action by $$C^{r-\varepsilon }$$ conjugacy. Next, we show that in any dimension, $$C^1$$ small perturbations can be conjugated to an affine action via $$C^{1+\varepsilon }$$ conjugacy. The method is a generalization of the Herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics.

On wave propagation in a rotating random micropolar generalized thermoelastic medium

This article examines the problem of wave propagation in a rotating random micropolar generalized thermoelastic medium. The entire frame of reference is assumed to be rotating with a uniform angular velocity where is a unit vector representing the direction of the axis of rotation. The smooth perturbation technique amenable to stochastic differential equations has been employed. The generalized dispersion equation has been derived and analyzed. The effects due to random variations of micropolar elastic and generalized thermal parameters have been observed. The effect of rotation on wave propagation is discussed. Change of phase speed occurs on account of random fluctuations of inhomogeneities of the medium. Attenuation coefficients for both random and nonrandom media have been computed both theoretically and numerically. Some illustrative graphs have been drawn to demonstrate the effects of rotation and generalized thermoelastic parameters on wave propagation in the micropolar medium.

Inviscid Damping Near the Couette Flow in a Channel

We prove asymptotic stability of shear flows in a neighborhood of the Couette flow for the 2D Euler equations in the domain $\T\times[0,1]$. More precisely we prove that if we start with a small and smooth perturbation (in a suitable Gevrey space) of the Couette flow, then the velocity field converges strongly to a nearby shear flow. The vorticity, which is initially assumed to be supported in the interior of the channel, will remain supported in the interior of the channel, will be driven to higher frequencies by the linear flow, and will converge weakly to $0$ as $t\to\infty$, modulo the shear flows (zero mode in $x$).

Quantum skyrmionics

Skyrmions are topological solitons that emerge in many physical contexts. In magnetism, they appear as textures of the spin-density field stabilized by different competing interactions and characterized by a topological charge that counts the number of times the order parameter wraps the sphere. They behave as classical objects when the spin texture varies slowly on the scale of the microscopic lattice of the magnet. However, the fast development of experimental tools to create and stabilize skyrmions in thin magnetic films has led to a rich variety of textures, sometimes of atomistic sizes. In this paper, we discuss, in a pedagogical manner, how to introduce quantum interference in the translational dynamics of skyrmion textures, starting from the micromagnetic equations of motion for a classical soliton. We study how the nontrivial topology of the spin texture manifests in the semiclassical regime, when the microscopic lattice potential is treated quantum-mechanically, but the external driving forces are taken as smooth classical perturbations. We highlight close relations to the fields of noncommutative quantum mechanics, Chern–Simons theories, and the quantum Hall effect.

Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory

In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and quasimorphisms by incorporating \emph{bulk deformations}, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher [Us4] in a slightly less general context. Then we explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasimorphisms and new Lagrangian intersection results on toric manifolds. The most novel part of this paper is to use open-closed Gromov-Witten theory (operator $\frak q$ in [FOOO1] and its variant involving closed orbits of periodic Hamiltonian system) to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasimorphism to the Lagrangian Floer theory (with bulk deformation). We use this open-closed Gromov-Witten theory to produce new examples. Especially using the calculation of Lagrangian Floer homology with bulk deformation in [FOOO3,FOOO4], we produce examples of compact toric manifolds $(M,\omega)$ which admits uncountably many independent quasimorphisms $\widetilde{\operatorname{Ham}}(M,\omega) \to \mathbb R$. We also obtain a new intersection result of Lagrangian submanifolds on $S^2 \times S^2$ discovered in [FOOO6]. Many of these applications were announced in [FOOO3,FOOO4,FOOO6].

Polarimetric Parameters of Scattered Radiation in the Magnetized Plasma

Second order statistical moments of scattered electromagnetic waves in the turbulent magnetized plasma slab with electron density fluctuations are calculated applying the modify stochastic smooth perturbation theory and the boundary conditions. The obtained results are valid for arbitrary correlation function of electron density fluctuations. Stokes parameters are analyzed both analytically and numerically. The theory predicts that depolarization effect caused by second Stokes parameter may be important in scintillation effects. Numerical calculations are carried out for new spectral function of electron density fluctuations containing both anisotropic Gaussian and power-law spectral functions using the experimental data. Polarimetric parameters are calculated for different anisotropy factor and inclination angle of elongated small-scale irregularities with respect to the magnetic lines of forces. The relationship between the scintillations and the polarimetric parameters is important.

Stability in shape optimization with second variation

We are interested in the question of stability in the field of shape optimization, with focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the Hessian of the considered shape function, so that critical stable domains (i.e. such that the first order derivative vanishes and the second order one is positive) are local minima for smooth perturbations; as we are in an infinite dimensional framework, and that in most applications there is a norm-discrepancy phenomenon, this type of result require a lot of work. We show that these hypotheses are satisfied by classical functionals, involving the perimeter, the Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how we can easily deal with constraints and/or invariance of the functionals. As an application, we retrieve or improve previous results from the existing literature, and provide new local stability results. We finally test the sharpness of our results by showing that the local minimality is in general not valid for non-smooth perturbations.

Parabolic Perturbations of Unipotent Flows on Compact Quotients of SL $${(3,\mathbb{R})}$$ ( 3 , R )

We consider a family of smooth perturbations of unipotent flows on compact quotients of SL $${(3,\mathbb{R})}$$ which are not time-changes. More precisely, given a unipotent vector field, we perturb it by adding a non-constant component in a commuting direction. We prove that, if the resulting flow preserves a measure equivalent to Haar, then it is parabolic and mixing. The proof is based on a geometric shearing mechanism together with a non-homogeneous version of Mautner Phenomenon for homogeneous flows. Moreover, we characterize smoothly trivial perturbations and we relate the existence of non-trivial perturbations to the failure of cocycle rigidity of parabolic actions in SL $${(3,\mathbb{R})}$$ .

Complete Wave Operators in Nonselfadjoint Kato Model of Smooth Perturbation Theory

In the present paper complete wave operators are constructed in the framework of nonselfadjoint Kato model for a wide class of not necessarily small relatively smooth perturbations. For one-dimensional Schroedinger operator with complex potential possessing finite first momentum a criterion of similarity to a selfadjoint operator is obtained which extends and supplements Kato sufficient condition.

Laser-induced formation of holograms for generation of plasmons

Methods of fabrication of precisely adjusted surface structures are indispensable to development of new technologies. Different surface structuring techniques resulting in solitary nanobumps, random surface structures and laser induced periodic surface structures were in focus in the past. Here we consider a physical model and numerical simulation allowing understanding a high field plasmonics formation process of a smooth periodic perturbation of surface on a metal film with a period equal to the surface plasmon-polariton wavelength at a frequency of laser wave. Such surface structure is a hologram produced by thermomechanical response to interference between an incident laser wave, a reflected laser wave, and an electromagnetic field in the running plasmon-polariton wave. The following laser irradiation of the manufactured hologram generates a new plasmon-polariton wave identical to that used for formation of the hologram.

Inverse elastic surface scattering with far-field data

A rigorous mathematical model and an efficient computational method are proposed to solving the inverse elastic surface scattering problem which arises from the near-field imaging of periodic structures. We demonstrate how an enhanced resolution can be achieved by using more easily measurable far-field data. The surface is assumed to be a small and smooth perturbation of an elastically rigid plane. By placing a rectangular slab of a homogeneous and isotropic elastic medium with larger mass density above the surface, more propagating wave modes can be utilized from the far-field data which contributes to the reconstruction resolution. Requiring only a single illumination, the method begins with the far-to-near (FtN) field data conversion and utilizes the transformed field expansion to derive an analytic solution for the direct problem, which leads to an explicit inversion formula for the inverse problem. Moreover, a nonlinear correction scheme is developed to improve the accuracy of the reconstruction. Results show that the proposed method is capable of stably reconstructing surfaces with resolution controlled by the slab's density.

Sharp-interface approach for simulating solid-state dewetting in two dimensions: A Cahn–Hoffman [formula omitted]-vector formulation

By using a Cahn–Hoffman ξ-vector formulation, we propose a sharp-interface approach for solving solid-state dewetting problems in two dimensions. First, based on the thermodynamic variation and smooth vector-field perturbation method, we rigorously derive a sharp-interface model with weakly anisotropic surface energies, and this model describes the interface evolution which occurs through surface diffusion flow and contact line migration. Second, a parametric finite element method in terms of the ξ-vector formulation is proposed for numerically solving the sharp-interface model. By performing numerical simulations, we examine several specific evolution processes for solid-state dewetting of thin films, e.g., the evolution of small islands, pinch-off of large islands and power-law retraction dynamics of semi-infinite step films, and these simulation results are consistent with experimental observations. Finally, we also include the strong surface energy anisotropy into the sharp-interface model and design its corresponding numerical scheme via the ξ-vector formulation.

On Wick Polynomials of Boson Fields in Locally Covariant Algebraic QFT

This work presents some results about Wick polynomials of a vector field renormalization in locally covariant algebraic quantum field theory in curved spacetime. General vector fields are pictured as sections of natural vector bundles over globally hyperbolic spacetimes and quantized through the known functorial machinery in terms of local $^*$-algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles, in addition to the Lorentzian metric. The mass and the coupling constants are in particular viewed as background fields. Wick powers of the quantized vector field are axiomatically defined imposing in particular local covariance, scaling properties and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is specialized to the case of general tensor fields. In particular, the case of a vector Klein-Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. The notion of local smooth dependence on polynomial scalar invariants is made precise in the text.

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.