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Journal articles on the topic 'Singular Perturbations'

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Author: Grafiati

Published: 4 June 2021

Last updated: 25 May 2023

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1

Malamud, M., and H. Neidhardt. "Perturbation determinants for singular perturbations." Russian Journal of Mathematical Physics 21, no. 1 (March 2014): 55–98. http://dx.doi.org/10.1134/s1061920814010051.

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2

Lasserre, Jean B. "A formula for singular perturbations of Markov chains." Journal of Applied Probability 31, no. 3 (September 1994): 829–33. http://dx.doi.org/10.2307/3215160.

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We give formulas for updating both the steady-state probability distribution and the fundamental matrices of a singularly perturbed Markov chain. This formula generalizes Schweitzer's regular perturbation formulas to the case of singular perturbations.

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3

Lasserre, Jean B. "A formula for singular perturbations of Markov chains." Journal of Applied Probability 31, no. 03 (September 1994): 829–33. http://dx.doi.org/10.1017/s0021900200045381.

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We give formulas for updating both the steady-state probability distribution and the fundamental matrices of a singularly perturbed Markov chain. This formula generalizes Schweitzer's regular perturbation formulas to the case of singular perturbations.

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4

Charafi, A. "Singular perturbation I. (Spaces and singular perturbations on manifolds without boundary)." Engineering Analysis with Boundary Elements 9, no. 2 (January 1992): 191–92. http://dx.doi.org/10.1016/0955-7997(92)90069-j.

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5

Snyder, Chris, and Gregory J. Hakim. "Cyclogenetic Perturbations and Analysis Errors Decomposed into Singular Vectors." Journal of the Atmospheric Sciences 62, no. 7 (July 1, 2005): 2234–47. http://dx.doi.org/10.1175/jas3458.1.

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Abstract Singular vectors (SVs) have been applied to cyclogenesis, to initializing ensemble forecasts, and in predictability studies. Ideally, the calculation of the SVs would employ the analysis error covariance norm at the initial time or, in the case of cyclogenesis, a norm based on the statistics of initial perturbations, but the energy norm is often used as a more practical substitute. To illustrate the roles of the choice of norm and the vertical structure of initial perturbations, an upper-level wave with no potential vorticity perturbation in the troposphere is considered as a typical cyclogenetic perturbation or analysis error, and this perturbation is then decomposed by its projection onto each energy SV. All calculations are made, for simplicity, in the context of the quasigeostrophic Eady model (i.e., for a background flow with constant vertical shear and horizontal temperature gradient). Viewed in terms of the energy SVs, the smooth vertical structure of the typical perturbation, as well as its evolution, results from strong cancellation between the growing and decaying SVs, most of which are highly structured and tilted in the vertical. A simpler picture, involving less cancellation, follows from decomposition of the typical perturbation into SVs using an alternative initial norm, which is based on the relation between initial norms and the statistics of initial perturbations together with the empirical assumption that the initial perturbations are not dominated by interior potential vorticity. Differences between the energy SVs and those based on the alternative initial norm can be understood by noting that the energy norm implicitly assumes initial perturbations with second-order statistics given by the covariance matrix whose inverse defines the energy norm. Unlike the “typical” perturbation, perturbations with those statistics have large variance of potential vorticity in the troposphere and fine vertical structure. Finally, a brief assessment is presented of the extent to which the upper wave, and more generally the alternative initial norm, is representative of cyclogenetic perturbations and analysis errors. There is substantial evidence supporting deep perturbations with little vertical structure as frequent precursors to cyclogenesis, but surrogates for analysis errors are less conclusive: operational midlatitude analysis differences have vertical structure similar to that of the perturbations implied by the energy norm, while short-range forecast errors and analysis errors from assimilation experiments with simulated observations are more consistent with the alternative norm.

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6

Biyarov, Bazarkan, Dimitry Svistunov, and Gulnara Abdrasheva. "CORRECT SINGULAR PERTURBATIONS OF THE LAPLACE OPERATOR." Eurasian Mathematical Journal 11, no. 4 (2020): 25–34. http://dx.doi.org/10.32523/2077-9879-2020-11-4-25-34.

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7

Lax, Christian, and Sebastian Walcher. "Singular perturbations and scaling." Discrete & Continuous Dynamical Systems - B 25, no. 1 (2020): 1–29. http://dx.doi.org/10.3934/dcdsb.2019170.

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8

Astaburuaga, M. A., V. H. Cortés, C. Fernández, and R. Del Río. "Singular rank one perturbations." Journal of Mathematical Physics 63, no. 2 (February 1, 2022): 023502. http://dx.doi.org/10.1063/5.0061250.

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9

Soner, H. Mete. "Singular Perturbations in Manufacturing." SIAM Journal on Control and Optimization 31, no. 1 (January 1993): 132–46. http://dx.doi.org/10.1137/0331010.

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10

Ikegami, Gik? "Singular perturbations in foliations." Inventiones Mathematicae 95, no. 2 (June 1989): 215–46. http://dx.doi.org/10.1007/bf01393896.

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11

Arai, Asao. "Supersymmetry and singular perturbations." Journal of Functional Analysis 60, no. 3 (February 1985): 378–93. http://dx.doi.org/10.1016/0022-1236(85)90046-1.

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12

Grimmer, Ronald, and Hetao Liu. "Singular Perturbations in Viscoelasticity." Rocky Mountain Journal of Mathematics 24, no. 1 (March 1993): 61–75. http://dx.doi.org/10.1216/rmjm/1181072452.

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13

YOSHITOMI, KAZUSHI. "INVERSE SPECTRAL PROBLEMS FOR SINGULAR RANK-ONE PERTURBATIONS OF A HILL OPERATOR." Journal of the Australian Mathematical Society 87, no. 3 (December 2009): 421–28. http://dx.doi.org/10.1017/s1446788709000135.

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AbstractWe investigate an inverse spectral problem for the singular rank-one perturbations of a Hill operator. We give a necessary and sufficient condition for a real sequence to be the spectrum of a singular rank-one perturbation of the Hill operator.

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14

Misiats, Oleksandr, Oleksandr Stanzhytskyi, and Nung Kwan Yip. "Asymptotic analysis and homogenization of invariant measures." Stochastics and Dynamics 19, no. 02 (March 27, 2019): 1950015. http://dx.doi.org/10.1142/s0219493719500151.

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In this paper, we study limiting behavior of the invariant measures for reaction–diffusion equations in the whole space [Formula: see text] with regular and singular perturbations. In the regular case, we show the convergence of the unique stationary solution of [Formula: see text] to a stationary solution of the limiting equation [Formula: see text]. We also consider the asymptotic behavior of the stationary solution under the perturbations of spectrum. Finally, for the singular perturbation of homogenization type, we show the weak convergence of invariant measure to its homogenized limit.

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15

Tolsa, Javier, and Miquel Salichs. "Convergence of singular perturbations in singular linear systems." Linear Algebra and its Applications 251 (January 1997): 105–43. http://dx.doi.org/10.1016/0024-3795(95)00556-0.

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16

Kellogg, R. Bruce. "Corner singularities and singular perturbations." ANNALI DELL UNIVERSITA DI FERRARA 47, no. 1 (December 2001): 177–206. http://dx.doi.org/10.1007/bf02838182.

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17

DEMİREKLER, MÜBECCEL, and M. YILDIRIM ÜÇTUĞ. "Singular perturbations with origin eigenvalues." International Journal of Control 45, no. 5 (May 1987): 1823–34. http://dx.doi.org/10.1080/00207178708933848.

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18

Devaney, Robert L. "Singular perturbations of complex polynomials." Bulletin of the American Mathematical Society 50, no. 3 (April 2, 2013): 391–429. http://dx.doi.org/10.1090/s0273-0979-2013-01410-1.

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19

Dmitriev, M. G., and G. A. Kurina. "Singular perturbations in control problems." Automation and Remote Control 67, no. 1 (January 2006): 1–43. http://dx.doi.org/10.1134/s0005117906010012.

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20

MACGILLIVRAY, A. D. "AN INTRODUCTION TO SINGULAR PERTURBATIONS." Natural Resource Modeling 13, no. 2 (June 28, 2008): 181–217. http://dx.doi.org/10.1111/j.1939-7445.2000.tb00033.x.

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21

Howland, James S. "Random perturbations of singular spectra." Proceedings of the American Mathematical Society 112, no. 4 (April 1, 1991): 1009. http://dx.doi.org/10.1090/s0002-9939-1991-1037208-5.

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22

DEVANEY, ROBERT L., KREŠIMIR JOSIĆ, and YAKOV SHAPIRO. "SINGULAR PERTURBATIONS OF QUADRATIC MAPS." International Journal of Bifurcation and Chaos 14, no. 01 (January 2004): 161–69. http://dx.doi.org/10.1142/s0218127404009259.

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We give a complete description of the dynamics of the mapping fε(z)=z2+(ε/z) for positive real values of ε. We then consider two generalizations: the case of complex ε and the mapping z→zn+(ε/zm), where ε is positive and real. In both cases we provide a full characterization of the map for a certain set of parameters, and give observations based on numerical evidence for all other parameter values. The dynamics of all maps that we consider bears striking resemblance to that of complex quadratic maps.

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23

Bezuidenhout, Carol. "Singular Perturbations of Degenerate Diffusions." Annals of Probability 15, no. 3 (July 1987): 1014–43. http://dx.doi.org/10.1214/aop/1176992078.

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24

Huddell, Walter B., and Rhonda J. Hughes. "Smooth approximation of singular perturbations." Journal of Mathematical Analysis and Applications 282, no. 2 (June 2003): 512–30. http://dx.doi.org/10.1016/s0022-247x(03)00163-x.

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25

Anosov, D. V. "Some examples of singular perturbations." Journal of Dynamical and Control Systems 2, no. 2 (April 1996): 289–98. http://dx.doi.org/10.1007/bf02259529.

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26

Kohn, Robert V., and Peter Sternberg. "Local minimisers and singular perturbations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 111, no. 1-2 (1989): 69–84. http://dx.doi.org/10.1017/s0308210500025026.

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SynopsisWe construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problemIt is shown that in certain nonconvex domains Ω ⊂ ℝn and for ε small, there exist nonconstant local minimisers uε satisfying uε ≈ ± 1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit uε →u0, the hypersurface separating the states u0 = 1 and u0 = −1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing |∇u|2.

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27

Bobrowski, Adam. "Singular perturbations involving fast diffusion." Journal of Mathematical Analysis and Applications 427, no. 2 (July 2015): 1004–26. http://dx.doi.org/10.1016/j.jmaa.2015.02.029.

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28

Najman, B. "Quasilinear parabolic-parabolic singular perturbations." Nonlinear Analysis: Theory, Methods & Applications 26, no. 2 (January 1996): 277–97. http://dx.doi.org/10.1016/0362-546x(94)00280-u.

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29

Fouque, J. P., G. Papanicolaou, R. Sircar, and K. Solna. "Singular Perturbations in Option Pricing." SIAM Journal on Applied Mathematics 63, no. 5 (January 2003): 1648–65. http://dx.doi.org/10.1137/s0036139902401550.

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30

Baía, Margarida, Ana Cristina Barroso, and José Matias. "Singular perturbations for phase transitions." São Paulo Journal of Mathematical Sciences 6, no. 2 (December 30, 2012): 117. http://dx.doi.org/10.11606/issn.2316-9028.v6i2p117-134.

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31

O’Malley, Jr., Robert E. "On Singular Perturbations, Especially Matching." SIAM Review 36, no. 3 (September 1994): 413–14. http://dx.doi.org/10.1137/1036096.

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32

Gostev, V. B., V. S. Mineev, and A. R. Frenkin. "Singular perturbations of discrete spectrum." Soviet Physics Journal 31, no. 3 (March 1988): 223–27. http://dx.doi.org/10.1007/bf00898228.

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33

Mease, K. D. "Geometry of Computational Singular Perturbations." IFAC Proceedings Volumes 28, no. 14 (June 1995): 855–61. http://dx.doi.org/10.1016/s1474-6670(17)46936-9.

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34

O’Malley, Jr, R. E. "Singular Perturbations I. Spaces and Singular Perturbations on Manifolds without Boundaries (L. S. Frank)." SIAM Review 34, no. 1 (March 1992): 142. http://dx.doi.org/10.1137/1034029.

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35

DelSole, Timothy. "Optimal Perturbations in Quasigeostrophic Turbulence." Journal of the Atmospheric Sciences 64, no. 4 (April 1, 2007): 1350–64. http://dx.doi.org/10.1175/jas3875.1.

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Abstract This paper tests the hypothesis that optimal perturbations in quasigeostrophic turbulence are excited sufficiently strongly and frequently to account for the energy-containing eddies. Optimal perturbations are defined here as singular vectors of the propagator, for the energy norm, corresponding to the equations of motion linearized about the time-mean flow. The initial conditions are drawn from a numerical solution of the nonlinear equations associated with the linear propagator. Experiments confirm that energy is concentrated in the leading evolved singular vectors, and that the average energy in the initial singular vectors is within an order of magnitude of that required to explain the average energy in the evolved singular vectors. Furthermore, only a small number of evolved singular vectors (4 out of 4000) are needed to explain the dominant eddy structure when total energy exceeds a predefined threshold. The initial singular vectors explain only 10% of such events, but this discrepancy was similar to that of the full propagator, suggesting that it arises primarily due to errors in the propagator. In the limit of short lead times, energy conservation can be expressed in terms of suitable singular vectors to constrain the energy distribution of the singular vectors in statistically steady equilibrium. This and other connections between linear optimals and nonlinear dynamics suggests that the positive results found here should carry over to other systems, provided the propagator and initial states are chosen consistently with respect to the nonlinear system.

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36

Besova, Margarita, and Vasiliy Kachalov. "Axiomatic Approach in the Analytic Theory of Singular Perturbations." Axioms 9, no. 1 (January 16, 2020): 9. http://dx.doi.org/10.3390/axioms9010009.

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Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

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37

DEMUTH, M., and J. A. VAN CASTEREN. "ON SPECTRAL THEORY OF SELFADJOINT FELLER GENERATORS." Reviews in Mathematical Physics 01, no. 04 (January 1989): 325–414. http://dx.doi.org/10.1142/s0129055x89000158.

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Upon introducing the theory of (symmetric) Feller semigroups, regularly and singularly perturbations of generators of Feller semigroups are studied. For the regular perturbations potentials belonging to the Kato-Feller class are admitted. The singular perturbations arise from potential barriers with increasing height. The Feynman-Kac representation is derived for the singular case. Selfadjointness conditions are given for these perturbed generators of Feller semigroups. Up to some extent we give explicit forms. The second main part consists of spectral theoretical considerations for these perturbed Feller generators. We give some examples which indicate the usefulness of the Feynman-Kac formula in spectral theory. We are interested in compactness properties of semigroup differences, which consist of trace class or Hilbert-Schmidt operators.

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38

Zhang, Lijun, Maoan Han, Mingji Zhang, and Chaudry Masood Khalique. "A New Type of Solitary Wave Solution of the mKdV Equation Under Singular Perturbations." International Journal of Bifurcation and Chaos 30, no. 11 (September 15, 2020): 2050162. http://dx.doi.org/10.1142/s021812742050162x.

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In this work, we examine the solitary wave solutions of the mKdV equation with small singular perturbations. Our analysis is a combination of geometric singular perturbation theory and Melnikov’s method. Our result shows that two families of solitary wave solutions of mKdV equation, having arbitrary positive wave speeds and infinite boundary limits, persist for selected wave speeds after small singular perturbations. More importantly, a new type of solitary wave solution possessing both valley and peak, named as breather in physics, which corresponds to a big homoclinic loop of the associated dynamical system is observed. It reveals an exotic phenomenon and exhibits rich dynamics of the perturbed nonlinear wave equation. Numerical simulations are performed to further detect the wave speeds of the persistent solitary waves and the nontrivial one with both valley and peak.

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39

Cots, Olivier, Joseph Gergaud, and Boris Wembe. "Homotopic approach for turnpike and singularly perturbed optimal control problems." ESAIM: Proceedings and Surveys 71 (August 2021): 43–53. http://dx.doi.org/10.1051/proc/202171105.

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The first aim of this article is to present the link between the turnpike property and the singular perturbations theory: the first one being a particular case of the second one. Then, thanks to this link, we set up a new framework based on continuation methods for the resolution of singularly perturbed optimal control problems. We consider first the turnpike case, then, we generalize the approach to general control problems with singular perturbations (that is with fast but also slow variables). We illustrate each step with an example.

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40

Yudenkov, Aleksey V., Aleksandr M. Volodchenkov, and Liliya P. Rimskaya. "STABILITY OF SYSTEMS OF SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNEL." T-Comm 14, no. 9 (2020): 48–55. http://dx.doi.org/10.36724/2072-8735-2020-14-9-48-55.

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Singular Cauchy integral equations have been widely used for mathematical simulation of the actual physical and technical systems. They are considered universal at every level of simulation beginning with quantum field theory and up to strength analysis of the underground constructions. Therefore investigating system stability of such models under perturbation of their absolute terms and coefficients appears an urgent scientific task. The aim of the study is to show various aspects of stability of singular Cauchy integral sets of equations which are generalizing simulation models of the primal problems of the elasticity theory for homogeneous isotropic bodies. The methods of study are based on the properties of the Cauchy singular integral, on the general theory of Fredholm operators. When in use, systems of the singular integral equations are reduced to a set of Fredholm integral equations of the second kind and a set of the boundary value problems for analytic functions. The key results of the study are the following: development of the general determination method of the system index for singular integral equations, proof of the system stability against perturbations of the absolute terms of the set. Against perturbations of the boundary coefficients, the singular integral system is unstable. Demonstration of the stability of the singular integral Cauchy sets generalizing primal problems of the elasticity theory appears a significantly new result. The research of singular integral equations sets has been performed conducted on the space of functions satisfying the Holder condition. However the main research results prove to be true if we operate random functions converting in mean square. Stability of singular integral equations sets against perturbations of the absolute terms lays a foundation for calculus of approximations in real world tasks of defining the built-in stress of an elastic complex body.

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41

Achatz, Ulrich, and Gerhard Schmitz. "Shear and Static Instability of Inertia–Gravity Wave Packets: Short-Term Modal and Nonmodal Growth." Journal of the Atmospheric Sciences 63, no. 2 (February 1, 2006): 397–413. http://dx.doi.org/10.1175/jas3636.1.

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Abstract The problem of nonmodal instabilities of inertia–gravity waves (IGW) in the middle atmosphere is addressed, within the framework of a Boussinesq model with realistic molecular viscosity and thermal diffusion, by singular-vector analysis of horizontally homogeneous vertical profiles of wind and buoyancy obtained from IGW packets at their statically least stable or most unstable horizontal location. Nonmodal growth is always found to be significantly stronger than that of normal modes, most notably at wave amplitudes below the static instability limit where normal-mode instability is very weak, whereas the energy gain between the optimal perturbation and singular vector after one Brunt–Väisälä period can be as large as two orders of magnitude. Among a multitude of rapidly growing singular vectors for this optimization time, small-scale (wavelengths of a few 100 m) perturbations propagating in the horizontal parallel to the IGW are most prominent. These parallel optimal perturbations are amplified by a roll mechanism, while transverse perturbations (with horizontal scales of a few kilometers) are to a large part subject to an Orr mechanism, both controlled by the transverse wind shear in the IGW at its statically least stable altitude, but further enhanced by reduced static stability. The elliptic polarization of the IGW leaves its traces in an additional impact of the roll mechanism via the parallel wind shear on the leading transverse optimal perturbation.

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42

DOTTI, GUSTAVO, REINALDO J. GLEISER, and IGNACIO F. RANEA-SANDOVAL. "INSTABILITIES IN KERR SPACETIMES." International Journal of Modern Physics E 20, supp01 (December 2011): 27–31. http://dx.doi.org/10.1142/s0218301311040049.

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We present a generalization of previous results regarding the stability under gravitational perturbations of nakedly singular super extreme Kerr spacetime and Kerr black hole interior beyond the Cauchy horizon. To do so we study solutions to the radial and angular Teukolsky's equations with different spin weights, particulary s = ±1 representing electromagnetic perturbations, s = ±1/2 representing a perturbation by a Dirac field and s = 0 representing perturbations by a scalar field. By analizing the properties of radial and angular eigenvalues we prove the existence of an infinite family of unstable modes.

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43

GHOSHAL, DEBASHIS, PORUS LAKDAWALA, and SUNIL MUKHI. "PERTURBATION OF THE GROUND VARIETIES OF c=1 STRING THEORY." Modern Physics Letters A 08, no. 33 (October 30, 1993): 3187–99. http://dx.doi.org/10.1142/s0217732393002129.

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We discuss the effect of perturbations on the ground rings of c=1 string theory at the various compactification radii defining the AN points of the moduli space. We argue that perturbations by plus-type moduli define ground varieties which are equivalent to the unperturbed ones under redefinitions of the coordinates and hence cannot smoothen the singularity. Perturbations by the minus-type moduli, on the other hand, lead to semi-universal deformations of the singular varieties that can smoothen the singularity under certain conditions. To first order, the cosmological perturbation by itself can remove the singularity only at the self-dual (A1) point.

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44

MANSOUR, I., and M. KATARY. "SINGULAR PERTURBATIONS AND AIRCRAFT LONGITUDINAL DYNAMICS." International Conference on Applied Mechanics and Mechanical Engineering 2, no. 2 (May 1, 1986): 193–201. http://dx.doi.org/10.21608/amme.1986.56820.

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45

Mannucci, Paola, Claudio Marchi, and Nicoletta Tchou. "Singular perturbations for a subelliptic operator." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 4 (October 2018): 1429–51. http://dx.doi.org/10.1051/cocv/2017063.

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We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.

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46

Liu, James H. "Singular Perturbations in a Nonlinear Viscoelasticity." Journal of Integral Equations and Applications 9, no. 2 (June 1997): 99–112. http://dx.doi.org/10.1216/jiea/1181075999.

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47

Marotta, Sebastian M. "Singular perturbations in the quadratic family." Journal of Difference Equations and Applications 14, no. 6 (June 2008): 581–95. http://dx.doi.org/10.1080/10236190701702429.

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48

Derkach, Vladimir, Seppo Hassi, and Henk de Snoo. "Singular Perturbations of Self-Adjoint Operators." Mathematical Physics, Analysis and Geometry 6, no. 4 (April 2003): 349–84. http://dx.doi.org/10.1023/b:mpag.0000007189.09453.fc.

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49

Alvarez, J. J., M. Gadella, L. M. Glasser, L. P. Lara, and L. M. Nieto. "One dimensional systems with singular perturbations." Journal of Physics: Conference Series 284 (March 1, 2011): 012009. http://dx.doi.org/10.1088/1742-6596/284/1/012009.

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50

Zhao Weili. "Singular perturbations for nonlinear Robin problems." Journal of Computational and Applied Mathematics 81, no. 1 (June 1997): 59–74. http://dx.doi.org/10.1016/s0377-0427(97)00010-1.

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