Статті в журналах: "Linear perturbation theory"

1

Hwang, Jai-Chan. "COSMOLOGICAL LINEAR PERTURBATION THEORY." Publications of The Korean Astronomical Society 26, no. 2 (July 6, 2011): 55–70. http://dx.doi.org/10.5303/pkas.2011.26.2.055.

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2

Dudkin, M. E., and O. Yu Dyuzhenkova. "Singularly perturbed rank one linear operators." Matematychni Studii 56, no. 2 (December 26, 2021): 162–75. http://dx.doi.org/10.30970/ms.56.2.162-175.

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The basic principles of the theory of singularly perturbed self-adjoint operatorsare generalized to the case of closed linear operators with non-symmetric perturbation of rank one.Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function.Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${\mathcal H}_{-1}$-class and ${\mathcal H}_{-2}$-class.The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.Namely, in some cases, when the perturbed by a vectors from ${\mathcal H}_{-2}$ operator can be conveniently described by methods of class ${\mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $\delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.

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3

NYE, V. A. "Perturbation Theory for Degenerate Linear Systems." IMA Journal of Mathematical Control and Information 2, no. 4 (1985): 261–73. http://dx.doi.org/10.1093/imamci/2.4.261.

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4

Renegar, James. "Some perturbation theory for linear programming." Mathematical Programming 65, no. 1-3 (February 1994): 73–91. http://dx.doi.org/10.1007/bf01581690.

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5

Fedorov, A. K., and A. I. Ovseevich. "Perturbation theory of observable linear systems." Mathematical Notes 98, no. 1-2 (July 2015): 216–21. http://dx.doi.org/10.1134/s0001434615070226.

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6

Pixius, C., S. Celik, and M. Bartelmann. "Kinetic field theory: perturbation theory beyond first order." Journal of Cosmology and Astroparticle Physics 2022, no. 12 (December 1, 2022): 030. http://dx.doi.org/10.1088/1475-7516/2022/12/030.

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Abstract We present recent improvements in the perturbative treatment of particle interactions in Kinetic Field Theory (KFT) for inertial Zel'dovich trajectories. KFT has been developed for the systematic analytical calculation of non-linear cosmic structure formation on the basis of microscopic phase-space dynamics. We improve upon the existing treatment of the interaction operator by deriving a more rigorous treatment of phase-space trajectories of particles in an expanding universe. We then show how these results can be applied to KFT perturbation theory by calculating corrections to the late-time dark matter power spectrum at second order in the interaction operator. We find that the modified treatment of interactions w.r.t. inertial Zel'dovich trajectories improves the agreement of KFT with simulation results on intermediate scales compared to earlier results. Additionally, we illustrate that including particle interactions up to second order leads to a systematic improvement of the non-linear power spectrum compared to the first-order result.

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7

Nájera, Antonio, and Amanda Fajardo. "Cosmological perturbation theory in f(Q,T) gravity." Journal of Cosmology and Astroparticle Physics 2022, no. 03 (March 1, 2022): 020. http://dx.doi.org/10.1088/1475-7516/2022/03/020.

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Abstract We developed the cosmological linear theory of perturbations for f(Q,T) gravity, which is an extension of symmetric teleparallel gravity, with Q the non-metricity and T the trace of the stress-energy tensor. By considering an ansatz of f(Q,T) = f 1(Q)+f 2(T), which has been broadly studied in the literature and the coincident gauge where the connection vanishes, we got equations consistent with f(Q) gravity when fT = 0. In the case of the tensor perturbations, the propagation of gravitational waves was found to be identical to f(Q), as expected. For scalar perturbations, outside the limit fT = 0, we got that the coupling between Q and T in the Lagrangian produces a coupling between the perturbation of the density and the pressure. This coupling is preserved when considering the weak coupling limit between Q and T. On the other hand, in the strong coupling limit with a generic function of the form f 2(T) = αT + β T 2, the perturbative equations are heavily driven by the f 2(T) derivatives when β ≠ 0. However, when β = 0, the perturbative equations are identical to the weak coupling limit even though this case is a non-minimally coupling one. The presence of T in the Lagrangian breaks the equation of the conservation of energy, which in turn breaks the standard ρ' + 3𝓗 (ρ+p) = 0 relation. We also derived a coupled system of differential equations between δ, the density contrast and v in the 𝓗 ≪ k limit and with negligible time derivative of the scalar perturbation potentials, which will be useful in future studies to see whether this class of theories constitute a good alternative to dark matter. These results might also enable to test f(Q,T) gravity with CMB and standard siren data that will help to determine if these models can reduce the Hubble constant tension and if they can constitute an alternative to the ΛCDM model.

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8

Zhang, J., L. Hui, and Z. Haiman. "A linear perturbation theory of inhomogeneous reionization." Monthly Notices of the Royal Astronomical Society 375, no. 1 (February 11, 2007): 324–36. http://dx.doi.org/10.1111/j.1365-2966.2006.11311.x.

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9

KOLB, EDWARD W., SABINO MATARRESE, ALESSIO NOTARI, and ANTONIO RIOTTO. "COSMOLOGICAL INFLUENCE OF SUPER-HUBBLE PERTURBATIONS." Modern Physics Letters A 20, no. 35 (November 20, 2005): 2705–10. http://dx.doi.org/10.1142/s0217732305018682.

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The existence of cosmological perturbations of wavelength larger than the Hubble radius is a generic prediction of the inflationary paradigm. We provide the derivation beyond perturbation theory of a conserved quantity which generalizes the linear comoving curvature perturbation. As a by-product, we show that super-Hubble-radius (super-Hubble) perturbations have no physical influence on local observables (e.g. the local expansion rate) if cosmological perturbations are of the adiabatic type.

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10

Ángyán, János G. "Rayleigh-Schrödinger perturbation theory for nonlinear Schrödinger equations with linear perturbation." International Journal of Quantum Chemistry 47, no. 6 (September 15, 1993): 469–83. http://dx.doi.org/10.1002/qua.560470606.

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11

Liu, Yan. "Invariance of Deficiency Indices of Second-Order Symmetric Linear Difference Equations under Perturbations." Journal of Function Spaces 2020 (February 13, 2020): 1–6. http://dx.doi.org/10.1155/2020/1940481.

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This paper focuses on the invariance of deficiency indices of second-order symmetric linear difference equations under perturbations. By applying the perturbation theory of Hermitian linear relations, the invariance of deficiency indices of the corresponding minimal subspaces under bounded and relatively bounded perturbations is built. As a consequence, the invariance of limit types of second-order symmetric linear difference equations under bounded and relatively bounded perturbations is obtained.

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12

GHEORGHE, DANA. "A KATO PERTURBATION-TYPE RESULT FOR OPEN LINEAR RELATIONS IN NORMED SPACES." Bulletin of the Australian Mathematical Society 79, no. 1 (February 2009): 85–101. http://dx.doi.org/10.1017/s0004972708001056.

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AbstractUsing some techniques of perturbation theory for Banach space complexes, we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation under small (with respect to the gap topology) perturbations with linear relations.

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13

Gandini, Augusto. "The heuristically-based generalized perturbation theory." EPJ Nuclear Sciences & Technologies 7 (2021): 7. http://dx.doi.org/10.1051/epjn/2021003.

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The generalized perturbation method is described relevant to ratios of bi-linear functionals of the real and adjoint neutron fluxes of critical multiplying systems. Simple linear analysis for optimization and sensitivity studies are then feasible relative to spectrum and space-dependent quantities, such as Doppler and coolant void reactivity effects in fast reactors.

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14

Seelmann, Albrecht. "Semidefinite perturbations in the subspace perturbation problem." Journal of Operator Theory 81, no. 2 (March 15, 2019): 321–33. http://dx.doi.org/10.7900/jot.2018feb07.2186.

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The variation of spectral subspaces for linear self-adjoint operators under an additive bounded semidefinite perturbation is considered. A variant of the Davis-Kahan sin2Θ theorem adapted to this situation is proved. Under a certain additional geometric assumption on the separation of the spectrum of the unperturbed operator, this leads to a sharp estimate on the norm of the difference of the spectral projections associated with isolated components of the spectrum of the perturbed and unperturbed operators, respectively. Without this additional geometric assumption on the isolated components of the spectrum of the unperturbed operator, a corresponding estimate is obtained by transferring a known optimization approach for general perturbations to the present situation.

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15

Marié, L. "A study of the phase instability of quasi-geostrophic Rossby waves on the infinite β-plane to zonal flow perturbations". Nonlinear Processes in Geophysics 17, № 1 (2 лютого 2010): 49–63. http://dx.doi.org/10.5194/npg-17-49-2010.

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Abstract. The problem of the linear instability of quasi-geostrophic Rossby waves to zonal flow perturbations is investigated on an infinite β-plane using a phase dynamics formalism. Equations governing the coupled evolutions of a zonal velocity perturbation and phase and amplitude perturbations of a finite-amplitude wave are obtained. The analysis is valid in the limit of infinitesimal, zonally invariant perturbation components, varying slowly in the meridional direction and with respect to time. In the case of a slow sinusoidal meridional variation of the perturbation components, analytical expressions for the perturbation growth rates are obtained, which are checked against numerical codes based on standard Floquet theory.

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16

Kalhous, Miloš, Lubomír Skála, Jaroslav Zamastil, and Jiří Čížek. "New Version of the Rayleigh-Schrödinger Perturbation Theory." Collection of Czechoslovak Chemical Communications 68, no. 2 (2003): 295–306. http://dx.doi.org/10.1135/cccc20030295.

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New version of the Rayleigh-Schrödinger perturbation theory based on the linear dependence of the perturbation wavefunctions on the perturbation energies is summarized. It is shown that this method is suitable also for multidimensional problems and the linear dependence can be used at an arbitrary point inside the integration region. The resulting perturbation theory is simple and can be used at large orders. As an example, the method is applied to the Barbanis hamiltonian.

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17

PANI, PAOLO. "ADVANCED METHODS IN BLACK-HOLE PERTURBATION THEORY." International Journal of Modern Physics A 28, no. 22n23 (September 20, 2013): 1340018. http://dx.doi.org/10.1142/s0217751x13400186.

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Black-hole perturbation theory is a useful tool to investigate issues in astrophysics, high-energy physics, and fundamental problems in gravity. It is often complementary to fully-fledged nonlinear evolutions and instrumental to interpret some results of numerical simulations. Several modern applications require advanced tools to investigate the linear dynamics of generic small perturbations around stationary black holes. Here, we present an overview of these applications and introduce extensions of the standard semianalytical methods to construct and solve the linearized field equations in curved space–time. Current state-of-the-art techniques are pedagogically explained and exciting open problems are presented.

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18

Jungnickel, D. U., and C. Wetterich. "The linear meson model and chiral perturbation theory." European Physical Journal C 2, no. 3 (1998): 557. http://dx.doi.org/10.1007/s100520050161.

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19

Jungnickel, D. U., and C. Wetterich. "The linear meson model and chiral perturbation theory." European Physical Journal C 2, no. 3 (April 1998): 557–67. http://dx.doi.org/10.1007/s100529800704.

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20

Parusiński, Adam, and Guillaume Rond. "Multiparameter perturbation theory of matrices and linear operators." Transactions of the American Mathematical Society 373, no. 4 (January 23, 2020): 2933–48. http://dx.doi.org/10.1090/tran/8061.

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21

Falkovsky, L. A. "Perturbation theory for a hamiltonian linear in quasimomentum." JETP Letters 94, no. 9 (January 2012): 723–27. http://dx.doi.org/10.1134/s0021364011210053.

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22

Casotto, Stefano. "The gravitational perturbation spectrum in linear satellite theory." Celestial Mechanics & Dynamical Astronomy 62, no. 1 (May 1995): 1–22. http://dx.doi.org/10.1007/bf00692066.

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23

Kloeckner, Benoit R. "Effective perturbation theory for simple isolated eigenvalues of linear operators." Journal of Operator Theory 81, no. 1 (December 15, 2018): 175–94. http://dx.doi.org/10.7900/jot.2017dec22.2179.

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We propose a new approach to the spectral theory of perturbed linear operators in the case of a simple isolated eigenvalue. We obtain two kinds of results: ``radius bounds'' which ensure perturbation theory applies for perturbations up to an explicit size, and ``regularity bounds'' which control the variations of eigendata to any order. Our method is based on the implicit function theorem and proceeds by establishing differential inequalities on two natural quantities: the norm of the projection to the eigendirection, and the norm of the reduced resolvent. We obtain completely explicit results without any assumption on the underlying Banach space. In companion articles, on the one hand we apply the regularity bounds to Markov chains, obtaining non-asymptotic concentration and Berry-Esseen inequalities with explicit constants, and on the other hand we apply the radius bounds to transfer operators of intermittent maps, obtaining explicit high-temperature regimes where a spectral gap occurs.

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24

SZPAK, NIKODEM. "LINEAR AND NONLINEAR TAILS I: GENERAL RESULTS AND PERTURBATION THEORY." Journal of Hyperbolic Differential Equations 05, no. 04 (December 2008): 741–65. http://dx.doi.org/10.1142/s0219891608001684.

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For nonlinear wave equations with a potential term, we prove pointwise space-time decay estimates and develop a perturbation theory for small initial data. We show that the perturbation series has a positive convergence radius by a method which reduces the wave equation to an algebraic one. We demonstrate that already first and second perturbation orders, satisfying linear equations, can provide precise information about the decay of the full solution to the nonlinear wave equation.

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25

O'MALLEY, ROBERT E. "NAIVE SINGULAR PERTURBATION THEORY." Mathematical Models and Methods in Applied Sciences 11, no. 01 (February 2001): 119–31. http://dx.doi.org/10.1142/s0218202501000787.

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The paper demonstrates, via extremely simple examples, the shocks, spikes, and initial layers that arise in solving certain singularly perturbed initial value problems for first-order ordinary differential equations. As examples from stability theory, they are basic to many asymptotic techniques. First, we note that limiting solutions of linear homogeneous equations [Formula: see text] on t≥0 are specified by the zeros of [Formula: see text], rather than by the turning points where a(t) becomes zero. Furthermore, solutions to the solvable equations [Formula: see text] for k=1, 2 or 3 can feature canards, where the trivial limit continues to apply after it becomes repulsive. Limiting solutions of the separable equation [Formula: see text] may likewise involve switchings between the zeros of c(x) located immediately above and below x(0), if they exist, at zeros of A(t). Finally, limiting solutions of many other problems follow by using asymptotic expansions for appropriate special functions. For example, solutions of [Formula: see text] can be given in terms of the Bessel functions Kj(t4/4ε) and Ij(t4/4ε) for j=3/8 and -5/8.

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26

Bora, Shreemayee, and Volker Mehrmann. "Linear Perturbation Theory for Structured Matrix Pencils Arising in Control Theory." SIAM Journal on Matrix Analysis and Applications 28, no. 1 (January 2006): 148–69. http://dx.doi.org/10.1137/040609355.

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27

Yano, Masayuki, and Anthony T. Patera. "A space–time variational approach to hydrodynamic stability theory." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2155 (July 8, 2013): 20130036. http://dx.doi.org/10.1098/rspa.2013.0036.

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We present a hydrodynamic stability theory for incompressible viscous fluid flows based on a space–time variational formulation and associated generalized singular value decomposition of the (linearized) Navier–Stokes equations. We first introduce a linear framework applicable to a wide variety of stationary- or time-dependent base flows: we consider arbitrary disturbances in both the initial condition and the dynamics measured in a ‘data’ space–time norm; the theory provides a rigorous, sharp (realizable) and efficiently computed bound for the velocity perturbation measured in a ‘solution’ space–time norm. We next present a generalization of the linear framework in which the disturbances and perturbation are now measured in respective selected space–time semi-norms ; the semi-norm theory permits rigorous and sharp quantification of, for example, the growth of initial disturbances or functional outputs. We then develop a (Brezzi–Rappaz–Raviart) nonlinear theory which provides, for disturbances which satisfy a certain (rather stringent) amplitude condition, rigorous finite-amplitude bounds for the velocity and output perturbations. Finally, we demonstrate the application of our linear and nonlinear hydrodynamic stability theory to unsteady moderate Reynolds number flow in an eddy-promoter channel.

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28

Arnal, Ana, Fernando Casas, and Cristina Chiralt. "Exponential Perturbative Expansions and Coordinate Transformations." Mathematical and Computational Applications 25, no. 3 (August 13, 2020): 50. http://dx.doi.org/10.3390/mca25030050.

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We propose a unified approach for different exponential perturbation techniques used in the treatment of time-dependent quantum mechanical problems, namely the Magnus expansion, the Floquet–Magnus expansion for periodic systems, the quantum averaging technique, and the Lie–Deprit perturbative algorithms. Even the standard perturbation theory fits in this framework. The approach is based on carrying out an appropriate change of coordinates (or picture) in each case, and it can be formulated for any time-dependent linear system of ordinary differential equations. All of the procedures (except the standard perturbation theory) lead to approximate solutions preserving by construction unitarity when applied to the time-dependent Schrödinger equation.

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29

Nishihara, K., J. G. Wouchuk, C. Matsuoka, R. Ishizaki, and V. V. Zhakhovsky. "Richtmyer–Meshkov instability: theory of linear and nonlinear evolution." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368, no. 1916 (April 13, 2010): 1769–807. http://dx.doi.org/10.1098/rsta.2009.0252.

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A theoretical framework to study linear and nonlinear Richtmyer–Meshkov instability (RMI) is presented. This instability typically develops when an incident shock crosses a corrugated material interface separating two fluids with different thermodynamic properties. Because the contact surface is rippled, the transmitted and reflected wavefronts are also corrugated, and some circulation is generated at the material boundary. The velocity circulation is progressively modified by the sound wave field radiated by the wavefronts, and ripple growth at the contact surface reaches a constant asymptotic normal velocity when the shocks/rarefactions are distant enough. The instability growth is driven by two effects: an initial deposition of velocity circulation at the material interface by the corrugated shock fronts and its subsequent variation in time due to the sonic field of pressure perturbations radiated by the deformed shocks. First, an exact analytical model to determine the asymptotic linear growth rate is presented and its dependence on the governing parameters is briefly discussed. Instabilities referred to as RM-like, driven by localized non-uniform vorticity, also exist; they are either initially deposited or supplied by external sources. Ablative RMI and its stabilization mechanisms are discussed as an example. When the ripple amplitude increases and becomes comparable to the perturbation wavelength, the instability enters the nonlinear phase and the perturbation velocity starts to decrease. An analytical model to describe this second stage of instability evolution is presented within the limit of incompressible and irrotational fluids, based on the dynamics of the contact surface circulation. RMI in solids and liquids is also presented via molecular dynamics simulations for planar and cylindrical geometries, where we show the generation of vorticity even in viscid materials.

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30

NAKAMURA, KOUJI. "GAUGE-INVARIANT VARIABLES IN GENERAL-RELATIVISTIC PERTURBATIONS: GLOBALIZATION AND ZERO-MODE PROBLEM." International Journal of Modern Physics D 21, no. 11 (October 2012): 1242004. http://dx.doi.org/10.1142/s0218271812420047.

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An outline of a proof of the local decomposition of linear metric perturbations into gauge-invariant and gauge-variant parts on an arbitrary background spacetime is briefly explained. We explicitly construct the gauge-invariant and gauge-variant parts of the linear metric perturbations based on some assumptions. We also point out the zero-mode problem is an essential problem to globalize of this decomposition of linear metric perturbations. The resolution of this zero-mode problem implies the possibility of the development of the higher-order gauge-invariant perturbation theory on an arbitrary background spacetime in a global sense.

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31

FAKHFAKH, FATMA, and MAHER MNIF. "Perturbation theory of lower semi-Browder multivalued linear operators." Publicationes Mathematicae Debrecen 78, no. 3-4 (April 1, 2011): 595–606. http://dx.doi.org/10.5486/pmd.2011.4799.

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32

Maurer, Simon A., Matthias Beer, Daniel S. Lambrecht, and Christian Ochsenfeld. "Linear-scaling symmetry-adapted perturbation theory with scaled dispersion." Journal of Chemical Physics 139, no. 18 (November 14, 2013): 184104. http://dx.doi.org/10.1063/1.4827297.

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33

Wu, Xuejun, Chongming Xu, and Michael Soffel. "General-relativistic Linear Perturbation Theory on Elastical Astronomical Bodies." Symposium - International Astronomical Union 202 (2004): 247–49. http://dx.doi.org/10.1017/s0074180900218007.

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The general relativistic perturbations of a uniformly rotating and axisymmetric elastic deformable astronomical body in the first post-Newtonian approximation of Einstein's general relativity is discussed in a co-rotating coordinates. The main new results presented here are the post-Newtonian variations of the energy and Euler equation, which at the Newtonian level of accuracy is so called Jeffreys-Vicente equation and it play a fundamental role for the description of global geodynamics in the classical geophysics. Our results will be useful to treat the relativistic nutation and precession of the Earth and other planets.

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34

Frank, L. S., and H. W. Norde. "On a singular perturbation in the linear soliton theory." Asymptotic Analysis 4, no. 1 (1991): 17–59. http://dx.doi.org/10.3233/asy-1991-4102.

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35

Blas, Diego, Mathias Garny, and Thomas Konstandin. "On the non-linear scale of cosmological perturbation theory." Journal of Cosmology and Astroparticle Physics 2013, no. 09 (September 23, 2013): 024. http://dx.doi.org/10.1088/1475-7516/2013/09/024.

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36

Brumberg, Eugene, Victor A. Brumberg, Thomas Konrad, and Michael Soffel. "Analytical linear perturbation theory for highly eccentric satellite orbits." Celestial mechanics and dynamical astronomy 61, no. 4 (1995): 369–87. http://dx.doi.org/10.1007/bf00049516.

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37

Gulliksson, M�rten, and Per-�ke Wedin. "Perturbation theory for generalized and constrained linear least squares." Numerical Linear Algebra with Applications 7, no. 4 (2000): 181–95. http://dx.doi.org/10.1002/1099-1506(200005)7:4<181::aid-nla193>3.0.co;2-d.

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38

Moore, Andrew M., Cristina L. Perez, and Javier Zavala-Garay. "A Non-normal View of the Wind-Driven Ocean Circulation." Journal of Physical Oceanography 32, no. 9 (September 1, 2002): 2681–705. http://dx.doi.org/10.1175/1520-0485-32.9.2681.

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Abstract Generalized linear stability theory is applied to the wind-driven ocean circulation in the form of a double gyre described by the barotropic quasigeostrophic vorticity equation. The development of perturbations on this circulation is considered. The circulation fields are inhomogeneous, and regions of straining flow render non-normal the tangent linear operators that describe the time evolution of perturbation energy and enstrophy. When the double-gyre circulation is asymptotically stable, growth of perturbation energy and enstrophy is still possible due to linear interference of its nonorthogonal eigenmodes. The sources and sinks of perturbation energy and enstrophy associated with the interference process are traditionally associated with the interaction of perturbation stresses with the mean flow. These ideas are used to understand the response of an asymptotically stable double-gyre circulation to stochastic wind stress forcing. Calculation of the optimal forcing patterns (stochastic optimals) reveals that much of the stochastically induced variability can be explained by one pattern. Variability induced by this pattern is maintained by long and short Rossby waves that interact with the western boundary currents, and perturbation growth occurs through barotropic processes. The perturbations that maintain the stochastically induced variance in this way have a large projection on some of the most non-normal, least-damped eigenmodes of the double-gyre circulation. Perturbation growth in nonautonomous and asymptotically unstable systems is also considered in the same framework. The Lyapunov vectors of unstable flows are found to have a large projection on some of the most non-normal, least-damped eigenmodes of the time mean circulation.

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39

Álvarez, Teresa, and Diane Wilcox. "Perturbation theory of multivalued atkinson operators in normed spaces." Bulletin of the Australian Mathematical Society 76, no. 2 (October 2007): 195–204. http://dx.doi.org/10.1017/s0004972700039587.

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We prove several stability results for Atkinson linear relations under additive perturbation by small norm, strictly singular and strictly cosingular multivalued linear operators satisfying some additional conditions.

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40

Ding, Jiu. "Perturbation of systems of linear algebraic equations∗." Linear and Multilinear Algebra 47, no. 2 (April 2000): 119–27. http://dx.doi.org/10.1080/03081080008818637.

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41

Alinea, Allan L., and Takahiro Kubota. "Transformation of primordial cosmological perturbations under the general extended disformal transformation." International Journal of Modern Physics D 30, no. 08 (May 11, 2021): 2150057. http://dx.doi.org/10.1142/s0218271821500577.

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Primordial cosmological perturbations are the seeds that were cultivated by inflation and the succeeding dynamical processes, eventually leading to the current universe. In this work, we investigate the behavior of the gauge-invariant scalar and tensor perturbations under the general extended disformal transformation, namely, [Formula: see text], where [Formula: see text] and [Formula: see text], with [Formula: see text] and [Formula: see text] being a general functional of [Formula: see text]. We find that the tensor perturbation is invariant under this transformation. On the other hand, the scalar curvature perturbation receives a correction due the conformal term only; it is independent of the disformal term at least up to linear order. Within the framework of the full Horndeski theory, the correction terms turn out to depend linearly on the gauge-invariant comoving density perturbation and the first time-derivative thereof. In the superhorizon limit, all these correction terms vanish, leaving only the original scalar curvature perturbation. In other words, it is invariant under the general extended disformal transformation in the superhorizon limit, in the context of full Horndeski theory. Our work encompasses a chain of research studies on the transformation or invariance of the primordial cosmological perturbations, generalizing their results under our general extended disformal transformation.

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42

Jiménez-Mejía, Raúl E., Rodrigo Acuna Herrera, and Pedro Torres. "Analysis of Spatially Doped Fused Silica Fiber Optic by Means of a Hamiltonian Formulation of the Helmholtz Equation." Advances in Materials Science and Engineering 2018 (2018): 1–11. http://dx.doi.org/10.1155/2018/5806947.

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This paper discusses an alternative method for calculating modal parameters in optical fibers such as propagation constants, transverse distributions, and anisotropy, due to linear and nonlinear phenomena acting as perturbations caused by doped silica regions. This method is based on a Hamiltonian formulation of the Helmholtz equation and the stationary perturbation theory, which allows a full-vectorial description of the electric field components when linear anisotropic inhomogeneities and Kerr nonlinearity are included. Linear and nonlinear parameters can be found for each propagating mode, and its accuracy has been successfully tested when compared to numerical calculations from the vector finite element method, and the results are published in the literature. This method facilitates the calculation of the spatial-distributed perturbation effects on individual electric field components for each propagating mode.

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43

Ntahompagaze, Joseph, Amare Abebe, and Manasse Mbonye. "A study of perturbations in scalar–tensor theory using 1 + 3 covariant approach." International Journal of Modern Physics D 27, no. 03 (February 2018): 1850033. http://dx.doi.org/10.1142/s0218271818500335.

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This work discusses scalar–tensor theories of gravity, with a focus on the Brans–Dicke sub-class, and one that also takes note of the latter’s equivalence with [Formula: see text] gravitation theories. A [Formula: see text] covariant formalism is used in this case to discuss covariant perturbations on a background Friedmann–Laimaître–Robertson–Walker (FLRW) spacetime. Linear perturbation equations are developed based on gauge-invariant gradient variables. Both scalar and harmonic decompositions are applied to obtain second-order equations. These equations can then be used for further analysis of the behavior of the perturbation quantities in such a scalar–tensor theory of gravitation. Energy density perturbations are studied for two systems, namely for a scalar fluid-radiation system and for a scalar fluid-dust system, for [Formula: see text] models. For the matter-dominated era, it is shown that the dust energy density perturbations grow exponentially, a result which agrees with those already existing in the literatures. In the radiation-dominated era, it is found that the behavior of the radiation energy–density perturbations is oscillatory, with growing amplitudes for [Formula: see text], and with decaying amplitudes for [Formula: see text]. This is a new result.

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44

Arnoldi, Jean-François, and Bart Haegeman. "Unifying dynamical and structural stability of equilibria." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2193 (September 2016): 20150874. http://dx.doi.org/10.1098/rspa.2015.0874.

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We exhibit a fundamental relationship between measures of dynamical and structural stability of linear dynamical systems—e.g. linearized models in the vicinity of equilibria. We show that dynamical stability, quantified via the response to external perturbations (i.e. perturbation of dynamical variables), coincides with the minimal internal perturbation (i.e. perturbations of interactions between variables) able to render the system unstable. First, by reformulating a result of control theory, we explain that harmonic external perturbations reflect the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect to constant changes of its coefficients. However, for this equivalence to hold, imaginary changes of the Jacobian’s coefficients have to be allowed. The connection with dynamical stability is thus lost for real dynamical systems. We show that this issue can be avoided, thus recovering the fundamental link between dynamical and structural stability, by considering stochastic noise as external and internal perturbations. More precisely, we demonstrate that a linear system’s response to white-noise perturbations directly reflects the intensity of internal white-noise disturbance that it can accommodate before becoming stochastically unstable.

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45

Lissy, Pierre, Yannick Privat, and Yacouba Simporé. "Insensitizing control for linear and semi-linear heat equations with partially unknown domain." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 50. http://dx.doi.org/10.1051/cocv/2018035.

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We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝN (N ∈ ℕ*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.

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46

Almeida, Juan P. Beltrán, Josué Motoa-Manzano, Jorge Noreña, Thiago S. Pereira, and César A. Valenzuela-Toledo. "Structure formation in an anisotropic universe: Eulerian perturbation theory." Journal of Cosmology and Astroparticle Physics 2022, no. 02 (February 1, 2022): 018. http://dx.doi.org/10.1088/1475-7516/2022/02/018.

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Abstract We present an effective Eulerian description, in the non-relativistic regime, of the growth of cosmological perturbations around a homogeneous but anisotropic Bianchi I spacetime background. We assume a small deviation from isotropy, sourced at late times for example by dark energy anisotropic stress. We thus derive an analytic solution for the linear dark matter density contrast, and use it in a formal perturbative approach which allows us to derive a second order (non-linear) solution. As an application of the procedure followed here we derive analytic expressions for the power spectrum and the bispectrum of the dark matter density contrast. The power spectrum receives a quadrupolar correction as expected, and the bispectrum receives several angle-dependent corrections. Quite generally, we find that the contribution of a late-time phase of anisotropic expansion to the growth of structure peaks at a finite redshift between CMB decoupling and today, tough the exact redshift value is model-dependent.

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47

Agullo, Ivan, Javier Olmedo, and Vijayakumar Sreenath. "xAct Implementation of the Theory of Cosmological Perturbation in Bianchi I Spacetimes." Mathematics 8, no. 2 (February 20, 2020): 290. http://dx.doi.org/10.3390/math8020290.

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This paper presents a computational algorithm to derive the theory of linear gauge invariant perturbations on anisotropic cosmological spacetimes of the Bianchi I type. Our code is based on the tensor algebra packages xTensor and xPert, within the computational infrastructure of xAct written in Mathematica. The algorithm is based on a Hamiltonian, or phase space formulation, and it provides an efficient and transparent way of isolating the gauge invariant degrees of freedom in the perturbation fields and to obtain the Hamiltonian generating their dynamics. The restriction to Friedmann–Lemaître–Robertson–Walker spacetimes is straightforward.

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48

Nadkarni-Ghosh, Sharvari, and David F. Chernoff. "Modelling non-linear evolution using Lagrangian perturbation theory re-expansions." Monthly Notices of the Royal Astronomical Society 431, no. 1 (March 8, 2013): 799–823. http://dx.doi.org/10.1093/mnras/stt217.

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49

Heck, B. S., and A. H. Haddad. "Singular perturbation theory for piecewise–linear systems with random inputs." Stochastic Analysis and Applications 7, no. 3 (January 1989): 273–89. http://dx.doi.org/10.1080/07362998908809182.

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50

Baskakov, A. G. "Krylov-Bogolyubov substitution in the perturbation theory of linear operators." Ukrainian Mathematical Journal 36, no. 5 (1985): 451–55. http://dx.doi.org/10.1007/bf01086768.

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