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Journal articles on the topic 'Lower order perturbation'

Author: Grafiati

Published: 22 February 2023

Last updated: 19 July 2025

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1

Hu, Yu-Min, Bi-Chu Li, Yang Yu, Martin Krššák, Emmanuel N. Saridakis, and Yi-Fu Cai. "Lowering the strong coupling mode of modified teleparallel gravity theories." Journal of Cosmology and Astroparticle Physics 2025, no. 05 (2025): 088. https://doi.org/10.1088/1475-7516/2025/05/088.

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Abstract We investigate the strong coupling problem in modified teleparallel gravity theories using the effective field theory (EFT) approach, demonstrating that it is possible to shift the emergence of new degrees of freedom (DoFs) to lower orders in perturbation theory. We first focus on the case of f(T) gravity, and we show that in its conformally equivalent form the scalar perturbations are non-dynamical up to the cubic action. We then propose a simple modification of the theory, which lowers the appearance of new DoFs to cubic order, compared to the quartic order in standard f(T) gravity. Our work opens a new avenue to address the issue of strong coupling in modified teleparallel gravity, and suggests a new classification scheme of these theories based on the perturbative order at which new DoFs appear.

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AZHAR, Muhammad Raees Furquan, Umair ZULFIQAR, and Muwahida LIAQUAT. "Lower order controller design using weighted singular perturbation approximation." TURKISH JOURNAL OF ELECTRICAL ENGINEERING & COMPUTER SCIENCES 27, no. 5 (2019): 3935–50. http://dx.doi.org/10.3906/elk-1807-4.

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3

DASH, JAN W. "MULTIVARIATE INTEGRAL PERTURBATION TECHNIQUES I: THEORY." International Journal of Theoretical and Applied Finance 10, no. 08 (2007): 1287–304. http://dx.doi.org/10.1142/s0219024907004652.

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We present a quasi-analytic perturbation expansion for multivariate N-dimensional Gaussian integrals. The perturbation expansion is an infinite series of lower-dimensional integrals (one-dimensional in the simplest approximation). This perturbative idea can also be applied to multivariate Student-t integrals. We evaluate the perturbation expansion explicitly through 2nd order, and discuss the convergence, including enhancement using Padé approximants. Brief comments on potential applications in finance are given, including options, models for credit risk and derivatives, and correlation sensitivities.

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4

GUERRERO, J., and V. ALDAYA. "A PERTURBATIVE APPROACH TO THE RELATIVISTIC HARMONIC OSCILLATOR AND UNITARITY." Modern Physics Letters A 14, no. 25 (1999): 1689–99. http://dx.doi.org/10.1142/s0217732399001784.

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A quantum realization of the relativistic Harmonic oscillator is achieved in terms of the spatial variable x and [Formula: see text] (the minimal canonical representation). The Hamiltonian operator is found (at lower order) by using a perturbative expansion in the constant c-1. Unlike the Foldy–Wouthuysen version of the relativistic hydrogen atom, conventional perturbation theory cannot be applied and a perturbation of the scalar product itself is required to make the theory unitary.

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5

Grzywny, Tomasz, Tomasz Jakubowski, and Grzegorz Żurek. "Green function for gradient perturbation of unimodal Lévy processes." Probability and Mathematical Statistics 37, no. 1 (2018): 119–43. http://dx.doi.org/10.19195/0208-4147.37.1.5.

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GREEN FUNCTION FOR GRADIENT PERTURBATION OF UNIMODAL LÉVY PROCESSESWe prove that the Green function of a generator of isotropic unimodal Lévy processes with the weak lower scaling order greater thanone and the Green function of its gradient perturbations are comparable for bounded smooth open sets if the drift function is from an appropriate Kato class.

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6

Liu, J. K., and H. C. Chan. "Universal Matrix Perturbation Method for Structural Dynamic Reanalysis of General Damped Gyroscopic Systems." Journal of Vibration and Control 10, no. 4 (2004): 525–41. http://dx.doi.org/10.1177/1077546304036230.

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We investigate an effective matrix perturbation method for structural dynamic reanalysis of general damped gyroscopic systems. By using the complex eigensubspace condensation and the or thogonal decomposition procedures, two greatly reduced generalized eigenvalue equations are obtained. The lower-order perturbations of eigensolutions (i.e. complex eigenvalues and the corresponding left and right eigenvectors) are then determined by solving the two reduced eigenvalue problems. The higher-order perturbations of eigensolutions are obtained by executing a singular value decomposition procedure for a complex matrix. The proposed method is a universal perturbation method, for it is universally applicable to the reanalysis of general damped gyroscopic systems with all three cases of complex eigenvalues: distinct, repeated, and closely spaced eigenvalues. Numerical examples corresponding to the three different cases of eigenvalues are presented. The perturbed eigensolutions are computed using the present method and compared with the exact solutions.

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7

Takeuchi, Akitsugu. "On the Convective Stability of Solar Photospheric Flux Tubes." Publications of the Astronomical Society of Japan 47, no. 3 (1995): 331–36. https://doi.org/10.1093/pasj/47.3.331.

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Abstract One-dimensional magnetohydrodynamic (MHD) numerical simulations were performed in order to investigate the convective stability of an intense photosheric flux tube, adopting the thin flux-tube approximation. The radiative heat exchange between the flux tube and its surroundings was also taken into account using Newton's law of cooling. To study the stability, we examined the temporal evolution of a small perturbation (downflow) added to an initial atmosphere. When a closed-boundary condition is adopted at the lower boundary, the perturbation evolves into a longitudinal overstable oscillation. However, an open (flow-through) condition at the lower boundary should be more suitable for representing the actual solar conditions, because a closed condition excludes the leakage of perturbations through the boundary. We thus repeated the simulation while adopting an open condition for the lower boundary. As a result, it was shown that the perturbation does not evolve into an overstable oscillation, but, rather, leaks out completely through the lower boundary. Our results imply that the actual photospheric flux tubes are in a convectively stable state, consistent with the recent observational results.

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8

Chabrowski, J. "On a critical Neumann problem with a perturbation of lower order." Acta Mathematicae Applicatae Sinica, English Series 24, no. 3 (2008): 441–52. http://dx.doi.org/10.1007/s10255-008-8038-5.

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9

Rubakov, V. A., and O. Yu Shvedov. "Sphalerons and large order behaviour of perturbation theory in lower dimension." Nuclear Physics B 434, no. 1-2 (1995): 245–63. http://dx.doi.org/10.1016/0550-3213(94)00447-m.

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10

Sawicki, Jerzy T., and T. V. V. L. N. Rao. "A Nonlinear Model for Prediction of Dynamic Coefficients in a Hydrodynamic Journal Bearing." International Journal of Rotating Machinery 10, no. 6 (2004): 507–13. http://dx.doi.org/10.1155/s1023621x04000508.

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This paper investigates the variation of nonlinear stiffness and damping coefficients in a journal orbit with respect to equilibrium position. The journal orbit is obtained by the combined solution of equations of motion and Reynolds equation. In the linearized dynamic analysis, dynamic pressure is written as a perturbation of static pressure and pressure gradients at equilibrium position. However, in order to obtain nonlinear dynamic coefficients about equilibrium position, the dynamic pressure gradients in the orbit are also written as the first order perturbation of static pressure gradients and higher order pressure gradients for displacement and velocity perturbations. The dynamic coefficients are functions of bearing displacement and velocity perturbations. The higher order pressure gradients at equilibrium position are evaluated at various eccentricity ratios and L/D ratios of 0.5 and 1.0. The variation of nonlinear dynamic coefficients is analyzed for three Sommerfeld numbers of a two-axial groove journal bearing under the action of an external synchronous load along and perpendicular to the radial journal load. Results indicate that the oil film nonlinearities affect the journal motion at lower eccentricity ratios (higher Sommerfeld numbers) with wide variation in stiffness and damping coefficients.

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11

Ikehata, Ryo, and Marina Soga. "Asymptotic profiles for a strongly damped plate equation with lower order perturbation." Communications on Pure & Applied Analysis 14, no. 5 (2015): 1759–80. http://dx.doi.org/10.3934/cpaa.2015.14.1759.

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12

Yang, Zhanshan. "An analysis of the mixed least squares-total least squares problems." Filomat 36, no. 12 (2022): 4195–209. http://dx.doi.org/10.2298/fil2212195y.

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In this paper, we first get further consideration of the first order perturbation with normwise condition number of the MTLS problem. For easy estimation, we show a lower bound for the normwise condition number which is proved to be optimal. In order to overcome the problems encountered in calculating the normwise condition number, we give an upper bound for computing more effectively and nonstandard and unusual perturbation bounds for the MTLS problem. Both of the two types of the perturbation bounds can enjoy storage and computational advantages. For getting more insight into the sensitivity of the MTLS technique with respect to perturbations in all data, we analyze the corrections applied by MTLS to the data in Ax ? b to make the set compatible and indicate how closely the data A, b fit the so-called general errors-in-variables model. On how to estimate the conditioning of the MTLS problem more effectively, we propose statistical algorithms by taking advantage of the superiority of small sample statistical condition estimation (SCE) techniques.

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13

Peng, Zhenhua, and Zhongping Wan. "Second-Order Composed Contingent Derivative of the Perturbation Map in Multiobjective Optimization." Asia-Pacific Journal of Operational Research 37, no. 02 (2020): 2050002. http://dx.doi.org/10.1142/s0217595920500025.

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In view of the structural advantage of second-order composed derivatives, the purpose of this paper is to analyze quantitatively the behavior of perturbation maps for the first time by using this concept. First, new concepts of the second-order composed adjacent derivative and the second-order composed lower Dini derivative are introduced. Some relationships among the second-order composed contingent derivative, the second-order composed adjacent derivative and the second-order composed lower Dini derivative are discussed. Second, the relationships between second-order composed lower Dini derivable and Aubin property are provided. Third, by virtue of second-order composed contingent derivatives and the above relationships, some results concerning second-order sensitivity analysis are established without the assumption of the locally Lipschitz property or the locally Hölder continuity. Finally, we give some complete characterizations of second-order composed contingent derivatives of the perturbation maps.

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14

Lindner, Peter, and Per-Olov Löwdin. "Upper and lower bounds in second-order perturbation theory and the unsÖld approximation." International Journal of Quantum Chemistry 2, S2 (2009): 161–73. http://dx.doi.org/10.1002/qua.560020717.

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15

Kuić, Domagoj, Alemka Knapp, and Diana Šaponja-Milutinović. "Finite Time Path Field Theory and a New Type of Universal Quantum Spin Chain Quench Behavior." Universe 11, no. 7 (2025): 230. https://doi.org/10.3390/universe11070230.

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We discuss different quench protocols for Ising and XY spin chains in a transverse magnetic field. With a sudden local magnetic field quench as a starting point, we generalize our approach to a large class of local non-sudden quenches. Using finite time path field theory (FTPFT) perturbative methods, we show that the difference between the sudden quench and a class of quenches with non-sudden switching on the perturbation vanishes exponentially with time, apart from non-substantial modifications that are systematically accounted for. As the consequence of causality and analytic properties of functions describing the discussed class of quenches, this is true at any order of perturbation expansion and thus for the resummed perturbation series. The only requirements on functions describing the perturbation strength switched on at a finite time t=0 are as follows: (1) their Fourier transform f(p) is a function that is analytic everywhere in the lower complex semiplane, except at the simple pole at p=0 and possibly others with ℑ(p)<0; and (2) f(p)/p converges to zero at infinity in the lower complex semiplane. A prototypical function of this class is tanh(ηt), to which the perturbation strength is proportional after the switching at time t=0. In the limit of large η, such a perturbation approaches the case of a sudden quench. It is shown that, because of this new type of universal behavior of Loschmidt echo (LE) that emerges in an exponentially short time scale, our previous results for the sudden local magnetic field quench of Ising and XY chains, obtained by the resummation of the perturbative expansion, extend in the long-time limit to all non-sudden quench protocols in this class, with non-substantial modifications systematically taken into account. We also show that analogous universal behavior exists in disorder quenches, and ultimately global ones. LE is directly connected to the work probability distribution, and the described universal behavior is therefore appropriate in potential concepts of quantum technology related to spin chains.

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16

Li, Xiaowang, Haitao Zhao, Jiping Huang, and Ji’an Chen. "Force Reconstruction for Uncertain Structure Based on Interval Model and Second-Order Perturbation Theory." International Journal of Computational Methods 18, no. 01 (2019): 1950040. http://dx.doi.org/10.1142/s0219876219500403.

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In order to reconstruct the upper and lower bounds of dynamic excitations applied on the uncertain structure, an algorithm based on interval model and second-order perturbation theory is presented in this paper. First, interval model is built up by expressing the uncertain parameters of structure in interval form. Next according to second-order perturbation theory, structure characteristic matrices and input force vector are approximated as second-order Taylor polynomial expansion at the midpoint of uncertain parameters. After that the input force’s midpoint, first-order and second-order partial derivatives are respectively calculated by existing step-by-step integration method. Then addition and subtraction of the three components obtained in previous step are operated. Ultimately the upper and lower bounds of dynamic load can be identified. Numerical simulation results demonstrate this method is with the characteristic of high efficiency and precision. In addition, it is able to remain a relatively strong robustness under noise turbulence.

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17

KASHIMA, YOHEI. "A RIGOROUS TREATMENT OF THE PERTURBATION THEORY FOR MANY-ELECTRON SYSTEMS." Reviews in Mathematical Physics 21, no. 08 (2009): 981–1044. http://dx.doi.org/10.1142/s0129055x09003797.

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Four point correlation functions for many electrons at finite temperature in periodic lattice of dimension d (≥1) are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite dimensional Grassmann integrals. A lower bound on the radius of convergence and an upper bound on the perturbation series are obtained by evaluating the Taylor expansion of logarithm of the finite dimensional Grassmann Gaussian integrals. The perturbation series up to second-order is numerically implemented along with the volume-independent upper bounds on the sum of the higher order terms in the 2-dimensional case.

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18

Lee, Anna, Tanvi Bhatt, Xuan Liu, Yiru Wang, Shuaijie Wang, and Yi-Chung (Clive) Pai. "Can Treadmill Slip-Perturbation Training Reduce Longer-Term Fall Risk Upon Overground Slip Exposure?" Journal of Applied Biomechanics 36, no. 5 (2020): 298–306. http://dx.doi.org/10.1123/jab.2019-0211.

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The purpose was to examine and compare the longer-term generalization between 2 different practice dosages for a single-session treadmill slip-perturbation training when reexposed to an overground slip 6 months later. A total of 45 older adults were conveniently assigned to either 24 or 40 slip-like treadmill perturbation trials or a third control group. Overground slips were given immediately after initial training, and at 6 months after initial training in order to examine immediate and longer-term effects. The performance (center of mass stability and vertical limb support) and fall percentage from the laboratory-induced overground slips (at initial posttraining and at 6 mo) were measured and compared between groups. Both treadmill slip-perturbation groups showed immediate generalization at the initial posttraining test and longer-term generalization at the 6-month retest. The higher-practice-dosage group performed significantly better than the control group (P < .05), with no difference between the lower-practice-dosage and the control groups at the 6-month retest (P > .05). A single session of treadmill slip-perturbation training showed a positive effect for reducing older adults’ fall risk for laboratory-induced overground slips. A higher-practice dosage of treadmill slip perturbations could be more beneficial for further reducing fall risk.

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19

Mao, X., S. J. Sherwin, and H. M. Blackburn. "Non-normal dynamics of time-evolving co-rotating vortex pairs." Journal of Fluid Mechanics 701 (May 16, 2012): 430–59. http://dx.doi.org/10.1017/jfm.2012.171.

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AbstractTransient energy growth of disturbances to co-rotating pairs of vortices with axial core flows is investigated in an analysis where vortex core expansion and vortex merging are included by adopting a time-evolving base flow. The dynamics of pairs are compared with those of individual vortices in order to highlight the effect of vortex interaction. Three typical vortex pair cases are studied, with the pairs comprised respectively of individually inviscidly unstable vortices at the streamwise wavenumber that maximizes the individual instabilities, viscously unstable vortices also at the streamwise wavenumber maximizing the individual instabilities and asymptotically stable vortices at streamwise wavenumber zero. For the inviscidly unstable case, the optimal perturbation takes the form of a superposition of two individual helical unstable modes and the optimal energy growth is similar to that predicted for an individual inviscid unstable vortex, while where the individual vortices are viscously unstable, the optimal disturbances within each core have similar spatial distributions to the individually stable case. For both of these cases, time horizons considered are much lower than those required for the merger of the undisturbed vortices. However, for the asymptotically stable case, large linear transient energy growth of optimal perturbations occurs for time horizons corresponding to vortex merging. Linear transient disturbance energy growth exhibited by pairs in this stable case is two to three orders of magnitude larger than that for a corresponding individual vortex. The superposition of the perturbation and the base flow shows that the perturbation has a displacement effect on the vortices in the base flow. Direct numerical simulations of stable pairs seeded by optimal initial perturbations have been carried out and acceleration/delay of vortex merging associated with a dual vortex meandering and vortex breakup related to axially periodic acceleration and delay of vortex merging are observed. For axially invariant cases, the sign of perturbation has an effect, as well as magnitude; the sign dependence relates to whether or not the perturbation adds to or subtracts from the swirl of the base flow. For a two-dimensional perturbation that adds to the swirl of the base flow, seeding with the linear optimal disturbance at a relative energy level $1\ensuremath{\times} 1{0}^{\ensuremath{-} 4} $ induces the pair to move towards each other and approximately halves the time required for merger. Direct numerical simulation shows that the optimal three-dimensional perturbation can induce the vortex system to break up before merging occurs, since the two-dimensional nature of vortex merging is broken by the development of axially periodic perturbations.

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20

Luz, Paulo, and Sante Carloni. "Gauge invariant perturbations of static spatially compact LRS II spacetimes." Classical and Quantum Gravity 41, no. 23 (2024): 235012. http://dx.doi.org/10.1088/1361-6382/ad8a14.

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Abstract We present a framework to describe completely general first-order perturbations of static, spatially compact, and locally rotationally symmetric class II spacetimes within the theory of general relativity. The perturbation variables are by construction covariant and identification gauge invariant and encompass the geometry and the thermodynamics of the fluid sources. The new equations are then applied to the study of isotropic, adiabatic perturbations. We discuss how the choice of frame in which perturbations are described can significantly simplify the mathematical analysis of the problem and show that it is possible to change frames directly from the linear level equations. We find explicitly that the case of isotropic, adiabatic perturbations can be reduced to a singular Sturm–Liouville eigenvalue problem, and lower bounds for the values of the eigenfrequencies can be derived. These results lay the theoretical groundwork to analytically describe linear, isotropic, and adiabatic perturbations of static, spherically symmetric spacetimes.

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21

Braun Rodrigues, N., G. Chinni, P. D. Cordaro, and M. R. Jahnke. "Lower order perturbation and global analytic vectors for a class of globally analytic hypoelliptic operators." Proceedings of the American Mathematical Society 144, no. 12 (2016): 5159–70. http://dx.doi.org/10.1090/proc/13178.

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22

Abdellaoui, M. "Perturbation effects for some nonlinear parabolic equations with lower order term and $$L^{1}$$-data." Journal of Elliptic and Parabolic Equations 6, no. 2 (2020): 599–631. http://dx.doi.org/10.1007/s41808-020-00071-z.

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23

MOUS, SIPKO L. J., and JOHAN GRASMAN. "TWO METHODS FOR ASSESSING THE SIZE OF EXTERNAL PERTURBATIONS IN CHAOTIC PROCESSES." Mathematical Models and Methods in Applied Sciences 03, no. 04 (1993): 577–93. http://dx.doi.org/10.1142/s0218202593000291.

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This paper deals with the assessment of an external perturbation in nonlinear chaotic dynamical processes using either a modified sentinel function or an extended Kalman filter treatment. We consider processes that can be modeled by a system of nonlinear ordinary differential equations. The sentinel function is used to detect an external perturbation, that is not included in the model of the process. In cases where the time dependency of the external perturbation is known but the size of the perturbation is unknown, the sentinel function is also used to estimate the size of this perturbation. We have compared this approach to estimate the size of a perturbation with an extended Kalman filter treatment and as an example to illustrate these two approaches we have analyzed a lower order spectral model of the atmospheric circulation with a perturbed equator-pole temperature gradient.

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24

CORSI, LIVIA, and GUIDO GENTILE. "Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems." Ergodic Theory and Dynamical Systems 35, no. 4 (2014): 1079–140. http://dx.doi.org/10.1017/etds.2013.92.

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AbstractWe consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeros of a suitable function, called the Melnikov function—by analogy with the periodic case. We show that, if the Melnikov function has a zero of odd order and under some further condition on the sign of the perturbation parameter, then there exists at least one resonant solution which continues an unperturbed solution. If the Melnikov function is identically zero then one can push perturbation theory up to the order where a counterpart of Melnikov function appears and does not vanish identically: if such a function has a zero of odd order and a suitable positiveness condition is met, again the same persistence result is obtained. If the system is Hamiltonian, then the procedure can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the result follows that at least one resonant quasi-periodic solution always exists with no assumption on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.

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25

Vulanović, Relja. "An Almost Sixth-Order Finite-Difference Method for Semilinear Singular Perturbation Problems." Computational Methods in Applied Mathematics 4, no. 3 (2004): 368–83. http://dx.doi.org/10.2478/cmam-2004-0020.

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AbstractThe discretization meshes of the Shishkin type are more suitable for high- order finite-difference schemes than Bakhvalov-type meshes. This point is illustrated by the construction of a hybrid scheme for a class of semilinear singularly perturbed reaction-diffusion problems. A sixth-order five-point equidistant scheme is used at most of the mesh points inside the boundary layers, whereas lower-order three-point schemes are used elsewhere. It is proved under certain conditions that this combined scheme is almost sixth-order accurate and that its error does not increase when the perturbation parameter tends to zero.

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26

Ayala, C., S. V. Mikhailov, A. V. Pimikov та N. G. Stefanis. "Form factor π0γ*γ in lightcone sum rules combined with renormalization-group summation vs experimental data." EPJ Web of Conferences 222 (2019): 03017. http://dx.doi.org/10.1051/epjconf/201922203017.

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We consider the lightcone sum-rule (LCSR) description of the pionphoton transition form factor in combination with the renormalization group of QCD. The emerging scheme represents a certain version of Fractional Analytic Perturbation Theory and significantly extends the applicability domain of perturbation theory towards lower momenta Q2 ≲ 1 GeV2. We show that the predictions calculated herewith agree very well with the released preliminary data of the BESIII experiment, which have very small errors just in this region, while the agreement with other data at higher Q2 is compatible with the LCSR predictions obtained recently by one of us using fixed-order perturbation theory.

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Abbas, Salman H. "Fourth-Order Block Methods for the Numerical Solution of First Order Initial Value Problems." Sarajevo Journal of Mathematics 2, no. 2 (2024): 247–58. http://dx.doi.org/10.5644/sjm.02.2.12.

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Block methods of order two and three for the numerical solution of initial value problems are extended to four order. The proposed two fourth order block methods might be efficient for implementation in multiprocessor computers. The matrix coefficients like block methods of order two and three of these methods are chosen so that lower powers of blocksize appear in the principle local truncation errors. The stability polynomial is shown to be a perturbation of the $(p + 1)^{th}$ order explicit Runge-Kutta method, scaled according to block size. In order to show the linear stability properties of the block predictor corrector methods, the maximum absolute errors using Type I and Type II methods with blocksize $k = 10$ and various step sizes are investigated numerically. 2000 Mathematics Subject Classification. 65L05, 65Y05

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Zhang, Xinguang, Jingsong Chen, Lishuang Li, and Yonghong Wu. "A Singular Tempered Sub-Diffusion Fractional Equation with Changing-Sign Perturbation." Axioms 13, no. 4 (2024): 264. http://dx.doi.org/10.3390/axioms13040264.

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In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points.

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29

Bin, Honghua, and Zhenkun Huang. "Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/707686.

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We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

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Vavryčuk, Václav. "Weak anisotropy-attenuation parameters." GEOPHYSICS 74, no. 5 (2009): WB203—WB213. http://dx.doi.org/10.1190/1.3173154.

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Velocity anisotropy and attenuation in weakly anisotropic and weakly attenuating structures can be treated uniformly using weak anisotropy-attenuation (WAA) parameters. The WAA parameters are constructed in a way analogous to weak anisotropy (WA) parameters designed for weak elastic anisotropy. The WAA parameters generalize WA parameters by incorporating attenuation effects. They can be represented alternatively by one set of complex values or by two sets of real values. Assuming high-frequency waves and using the first-order perturbation theory, all basic wave quantities such as the slowness vector, the polarization vector, propagation velocity, attenuation, and the quality factor are linear functions of WAA parameters. Numerical modeling shows that perturbation equations have different accuracy for different wave quantities. The propagation velocity usually is calculated with high accuracy. However, the attenuation and quality factor can be reproduced with appreciably lower accuracy. This happens mostly when the strength of velocity anisotropy is higher than 10% and attenuation is moderate or weak [Formula: see text]. In this case, the errors of the attenuation or [Formula: see text]-factor can attain values comparable to the strength of anisotropy or even higher. A simple modification of the equations by including some higher-order perturbations improves accuracy by three to four times.

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31

Nicasio, José, Adnan Bashir, Ulrich D. Jentschura, and James P. Edwards. "Landau-Khalatnikov-Fradkin Gauge Transformations for the Propagator and Vertex in QED and QED₂." Journal of Physics: Conference Series 2667, no. 1 (2023): 012023. http://dx.doi.org/10.1088/1742-6596/2667/1/012023.

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Abstract The non-perturbative Landau-Khalatnikov-Fradkin (LKF) transformations describe how Green functions in quantum field theory transform under a change in the photon field’s linear covariant gauge parameter (denoted ξ). The transformations are framed most simply in coordinate space where they are multiplicative. They imply that information on gauge-dependent contributions from higher order diagrams in the perturbative series is contained in lower order contributions, which is useful in multi-loop calculations. We study the LKF transformations for the propagator and the vertex in both scalar and spinor QED, in some particular dimensions. A novelty of our work is to derive momentum-space integral representations of these transformations; our expressions are also applicable to the longitudinal and transverse parts of the vertex. Applying these transformations to the tree-level Green functions, we show that the one-loop terms obtained from the LKF transformation agree with the gauge dependent parts obtained from perturbation theory. Our results will be presented in more comprehensive form elsewhere.

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32

Du, Jiayin, Lu Xu, and Yong Li. "An infinite dimensional KAM theorem with normal degeneracy." Nonlinearity 37, no. 6 (2024): 065021. http://dx.doi.org/10.1088/1361-6544/ad45a1.

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Abstract In this paper, we consider a classical Hamiltonian normal form with degeneracy in the normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. Instead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to verify, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.

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33

Fedorov, Vladimir E. "Generators of Analytic Resolving Families for Distributed Order Equations and Perturbations." Mathematics 8, no. 8 (2020): 1306. http://dx.doi.org/10.3390/math8081306.

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Linear differential equations of a distributed order with an unbounded operator in a Banach space are studied in this paper. A theorem on the generation of analytic resolving families of operators for such equations is proved. It makes it possible to study the unique solvability of inhomogeneous equations. A perturbation theorem for the obtained class of generators is proved. The results of the work are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable.

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34

Economou, J. T., A. Tsourdos, and B. A. White. "Fuzzy logic consequent perturbation analysis for electric vehicles." Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 221, no. 7 (2007): 757–64. http://dx.doi.org/10.1243/09544070jauto538.

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This paper presents a methodology for the estimation of upper and lower bounds due to sensor errors for the class of fuzzy Sugeno systems with linear consequent polynomials. The proposed method was also applied to the electrical permanent magnet actuator of an electric vehicle in order to demonstrate its effectiveness. Fuzzy logic Sugeno modelling approaches mainly concentrate on inserting uncertainty in the fuzzy antecedent. In this paper the focus is on perturbations in the consequent polynomial parameters for every possible rule.

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35

Zhang, Xinguang, Jingsong Chen, Peng Chen, Lishuang Li, and Yonghong Wu. "Nonlocal Changing-Sign Perturbation Tempered Fractional Sub-Diffusion Model with Weak Singularity." Fractal and Fractional 8, no. 6 (2024): 337. http://dx.doi.org/10.3390/fractalfract8060337.

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In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables.

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36

Aleroev, Temirkhan, Hedi Aleroeva, and Lyudmila Kirianova. "One method for the boundary value problem eigenvalues calcuating for a second-order differential equation with a fractional derivative." International Journal of Modeling, Simulation, and Scientific Computing 10, no. 01 (2019): 1941004. http://dx.doi.org/10.1142/s1793962319410046.

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In this paper, we give a formula for computing the eigenvalues of the Dirichlet problem for a differential equation of second-order with fractional derivatives in the lower terms. We obtained this formula using the perturbation theory for linear operators. Using this formula we can write out the system of eigenvalues for the problem under consideration.

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37

Xia, Q., K. T. Yang, and D. Mukutmoni. "Effect of Imposed Wall Temperature Oscillations on the Stability of Natural Convection in a Square Enclosure." Journal of Heat Transfer 117, no. 1 (1995): 113–20. http://dx.doi.org/10.1115/1.2822289.

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The present numerical study is directed toward buoyancy-driven laminar flows in a two-dimensional square enclosure with differential heating at the vertical walls. The top and bottom walls are insulated. A time-dependent temperature varying sinusoidal perturbation is imposed on the hot vertical wall. The cold vertical wall is maintained at a constant temperature. The fluid is air with a Prandtl number of 0.72. Computations were carried out at one imposed frequency, which is of the same order as the first natural frequency of the system. It was found that the perturbations destabilized the flow in that higher amplitudes lead to lower critical Rayleigh numbers for the flow transitions. Computations spanned four regimes: periodic, quasi-periodic with two frequencies, quasi-periodic with three-frequencies, and chaotic.

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38

Yin, W. Y. "The Lower- and Higher-Order Perturbation Effects of a Biisotropic Chiral Sphere in Cylindrical and Spherical Cavities." Progress In Electromagnetics Research 20 (1998): 125–51. http://dx.doi.org/10.2528/pier97122300.

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39

Abdellaoui, M. "Correction to: Perturbation effects for some nonlinear parabolic equations with lower order term and $$L^{1}$$-data." Journal of Elliptic and Parabolic Equations 6, no. 2 (2020): 947. http://dx.doi.org/10.1007/s41808-020-00090-w.

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40

Sawyer, W. G., and J. A. Tichy. "Non-Newtonian Lubrication With the Second-Order Fluid." Journal of Tribology 120, no. 3 (1998): 622–28. http://dx.doi.org/10.1115/1.2834596.

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In certain applications where the lubricant is subjected to rapidly changing conditions along its flowing path (such as an elastohydrodynamic contact), the inherently time dependent nature of the lubricant may be significant. The simplest type of model to correctly account for such time dependence is the second-order fluid, which is a systematic small departure from Newtonian behavior, involving higher order rate-of-rate-of strain tensors. As in a companion paper using the Maxwell model, the formalities of applying such a model to thin film flow are emphasized. Using a regular perturbation in the Deborah number, with the conventional lubrication solution as the leading term, a solution can be obtained. Viscoelasticity may raise or lower pressure depending on the nature of edge boundary conditions.

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41

Wu, Y., X. Y. Lin, H. X. Jiang, and A. G. Cheng. "Finite Element Analysis of the Uncertainty of Physical Response of Acoustic Metamaterials with Interval Parameters." International Journal of Computational Methods 17, no. 08 (2019): 1950052. http://dx.doi.org/10.1142/s021987621950052x.

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The physical response of acoustic metamaterials may change due to the variation of the material properties in the manufacture process. Thus, an interval perturbation finite element method is formulated to study the mechanical response of acoustic metamaterials with the interval parameters, which includes the uncertainty effects on the band structure, resonance mode and frequency response of acoustic metamaterials. By virtue of the first-order Taylor series expansion and sensitivity analysis of dynamic properties of acoustic metamaterials with respect to the interval parameters, the interval perturbation finite element method established in this work can predict the upper and lower bounds of the dynamic properties of the acoustic metamaterials. Three numerical examples are studied to validate the effectiveness of the interval perturbation finite element method to analyze the physical response of acoustic metamaterials with the interval parameters, and the results calculated by Monte Carlo method are regarded as the reference results to validate the interval perturbation finite element method. The uncertainty model constructed by interval perturbation finite element method provides a great help in the design of acoustic metamaterials.

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42

Amendola, Luca, Marco Marinucci, Massimo Pietroni, and Miguel Quartin. "Improving precision and accuracy in cosmology with model-independent spectrum and bispectrum." Journal of Cosmology and Astroparticle Physics 2024, no. 01 (2024): 001. http://dx.doi.org/10.1088/1475-7516/2024/01/001.

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Abstract A new and promising avenue was recently developed for analyzing large-scale structure data with a model-independent approach, in which the linear power spectrum shape is parametrized with a large number of freely varying wavebands rather than by assuming specific cosmological models. We call this method FreePower. Here we show, using a Fisher matrix approach, that precision of this method for the case of the one-loop power spectrum is greatly improved with the inclusion of the tree-level bispectrum. We also show that accuracy can be similarly improved by employing perturbation theory kernels whose structure is entirely determined by symmetries instead of evolution equations valid in particular models (like in the usual Einstein-de Sitter approximation). The main result is that with the Euclid survey one can precisely measure the Hubble function, distance and (k-independent) growth rate f(z) in seven redshift bins in the range z ∈ [0.6, 2.0]. The typical errors for the lower zbins are around 1% (for H), 0.7–1% (for D), and 2–3% (for f). The use of general perturbation theory allows us, for the first time, to study constraints on the nonlinear kernels of cosmological perturbations, that is, beyond the linear growth factor, showing that they can be probed at the 10–20% level. We find that the combination of spectrum and bispectrum is particularly effective in constraining the perturbation parameters, both at linear and quadratic order.

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43

Bhattacharyya, Sombuddha, and Tuhin Ghosh. "Inverse Boundary Value Problem of Determining Up to a Second Order Tensor Appear in the Lower Order Perturbation of a Polyharmonic Operator." Journal of Fourier Analysis and Applications 25, no. 3 (2018): 661–83. http://dx.doi.org/10.1007/s00041-018-9625-3.

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44

Kleinert, H. "Five-Loop Critical Temperature Shift in Weakly Interacting Homogeneous Bose–Einstein Condensate." Modern Physics Letters B 17, no. 19 (2003): 1011–20. http://dx.doi.org/10.1142/s0217984903006074.

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Using variational perturbation theory, we calculate the shift in the critical temperature Tcup to five loops to lowest order in the scattering length a and find [Formula: see text], where n is the particle density. Our result is slightly lower than the latest Monte Carlo result (1.32±0.02) an1/3.

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45

Zhang, Liye, Ye Xia, Jose A. Lozano-Galant, and Limin Sun. "Mass-Stiffness Combined Perturbation Method for Mode Shape Monitoring of Bridge Structures." Shock and Vibration 2019 (November 6, 2019): 1–14. http://dx.doi.org/10.1155/2019/7320196.

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Identification of the mode shapes through monitoring is one of the key problems in damage diagnosis based on modal parameters especially for damaged structures. In order to obtain mode shapes of damaged structures easily and accurately, the mass-stiffness combined perturbation (MSCP) method is proposed in this paper. To do so, the relationship between the stiffness perturbation mode shapes of damaged and intact structures is firstly derived and established. Then, the principle of similar frequency is applied to optimize the objective function of the most suitable mass perturbation model. Both numerical analyses and experimental tests on simple and complex structures demonstrate that the proposed MSCP method achieves higher precision than traditional mode shape identification methods. The additional advantages of the MSCP method include (i) lower requirement on the frequency analysis of only damaged structures and (ii) higher effectiveness for minor damage scenarios. In fact, the lower the damage, the higher the precision achieved by the MSCP method. As illustrated in the paper, the proposed technique has excellent applications in mode shapes identification and structural health monitoring.

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46

Rasulov, T. H., and M. Sh Sharipova. "Spectral estimates for the bounds of an operator matrix of order three." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 6 (June 29, 2025): 88–93. https://doi.org/10.26907/0021-3446-2025-6-88-93.

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In this paper we consider $3 \times 3$ operator matrix ${\mathcal A}_\mu$ with spectral parameter $\mu>0$ related with the Hamiltonian of a system with nonconserved and no more than three particles on a one-dimensional lattice. Essential and discrete spectra of the operator matrix ${\mathcal A}_\mu$ are described. It is established that the operator matrix ${\mathcal A}_\mu$ has at most four simple eigenvalues outside of the essential spectrum. Spectral estimates for the lower and upper bounds of the operator matrix ${\mathcal A}_\mu$ are obtained using cubic numerical range, Gershgorin enclosures and classical perturbation theory.

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47

Hastings, Alan. "MULTILOCUS POPULATION GENETICS WITH WEAK EPISTASIS. II. EQUILIBRIUM PROPERTIES OF MULTILOCUS MODELS: WHAT IS THE UNIT OF SELECTION?" Genetics 112, no. 1 (1986): 157–71. http://dx.doi.org/10.1093/genetics/112.1.157.

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ABSTRACT Using perturbation techniques, I study the equilibrium of deterministic discrete time multilocus models with weak epistasis. The most important results are on the relationship between epistasis and disequilibrium. Disequilibrium involving a particular set of loci reflects only epistasis simultaneously involving those loci. Moreover, all the disequilibria of all orders vary approximately as the inverse of the probability of at least one recombination event among the loci involved. Finally, higher order disequilibria among loci will be lower than lower order ones, even if the level of epistasis is the same at all orders. In this sense, the unit of selection is small. However, given the larger number of higher order disequilibria, these higher order disequilibria may play an important role in the computation of gametic frequencies from allelic frequencies in models with a large number of loci. Finally, I show that epistasis between blocks of loci will be averages of epistatic effects, not additions of epistatic effects. Thus, failure to find significant epistasis on a chromosomal basis does not rule out the importance of epistatic effects.

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48

Gou, Hai-De. "Existence of ω-periodic solutions for second order delay differential equation in Banach spaces". Filomat 36, № 16 (2022): 5347–58. http://dx.doi.org/10.2298/fil2216347g.

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The propose of the paper is devoted to study the existence of ?-periodic solutions for second order delay differential equation in abstract Banach space. Firstly, we build a new maximum principle for the ?-periodic solutions of the corresponding linear equation. Secondly, with the help of this maximum principle, we study the existence of the minimal and maximal periodic solutions for our concerns problem by means of perturbation method and monotone iterative technique of the lower and upper solutions. In addition, an example is presented to show the application of our main results.

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49

Shved, G. M., V. P. Ogibalov, and A. I. Pogoreltsev. "Effect of planetary waves on cooling the upper mesosphere and lower thermosphere by the CO<sub>2</sub> 15-µm emission." Annales Geophysicae 22, no. 10 (2004): 3383–94. http://dx.doi.org/10.5194/angeo-22-3383-2004.

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Abstract. The steady-state 2-D linearized model of global-scale waves, calibrated according to available observations, is used to evaluate planetary-wave perturbations of temperature from the surface up to the height of about 165km. The maximum order of perturbation amplitudes in the upper mesosphere and lower thermosphere is found to be 15K for the ultra-fast Kelvin wave (UFKW) of the 3.5-day period, 8K for the 10-day wave, 5K for the 2- and 16-day waves, and 2K for the 5-day wave. The wave-caused variation in heat influx in the CO2 15-µm band, averaged over the wave period, depends on both the amplitude of temperature and temperature profile in the atmosphere unperturbed by the wave. An additional increase in radiative cooling is the prevailing effect of planetary waves in the upper mesosphere and lower thermosphere. The UFKW results in an increasing cooling rate up to ~0.1 of the cooling rate in the unperturbed atmosphere. The tangible contributions of the 2-, 10-, and 16-day waves are questionable. The contribution of the 5-day wave is negligible.

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50

Fueglistaler, S., and M. B. Baker. "A modelling study of the impact of cirrus clouds on the moisture budget of the upper troposphere." Atmospheric Chemistry and Physics Discussions 5, no. 5 (2005): 9769–99. http://dx.doi.org/10.5194/acpd-5-9769-2005.

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Abstract. We present a modelling study of the effect of cirrus clouds on the moisture budget of the layer wherein the cloud formed. Our framework simplifies many aspects of cloud microphysics and collapses the problem of sedimentation onto a 0-dimensional box model, but retains essential feedbacks between saturation mixing ratio, particle growth, and water removal through particle sedimentation. The water budget is described by two coupled first-order differential equations for dimensionless particle number density and saturation point temperature, where the parameters defining the system (layer depth, reference temperature, amplitude and time scale of temperature perturbation and inital particle number density, which may or may not be a function of reference temperature and cooling rate) are encapsulated in a single coefficient. This allows us to scale the results to a broad range of atmospheric conditions, and to test sensitivities. Results of the moisture budget calculations are presented for a range of atmospheric conditions (T: 238–205 K; p: 325–180 hPa) and a range of time scales τT of the temperature perturbation that induces the cloud formation. The cirrus clouds are found to efficiently remove water for τT longer than a few hours, with longer perturbations (τT≳10 h) required at lower temperatures (T≲210 K). Conversely, temperature perturbations of duration order 1 hour and less (a typical timescale for e.g., gravity waves) are not found to efficiently dehydrate over most of the upper troposphere. A consequence is that (for particle densities typical of current cirrus clouds) the assumption of complete dehydration to the saturation mixing ratio may yield valid predictions for upper tropospheric moisture distributions if it is based on the large scale temperature field, but this assumption is not necessarily valid if it is based on smaller scale temperature fields.

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