Relevant bibliographies by topics / Stationary and stable solution

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Stationary and stable solution'

Author: Grafiati
Published: 4 June 2021
Last updated: 12 February 2022

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Journal articles on the topic "Stationary and stable solution"

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AMBROSO, ANNALISA. "STABILITY FOR SOLUTIONS OF A STATIONARY EULER–POISSON PROBLEM." Mathematical Models and Methods in Applied Sciences 16, no. 11 (November 2006): 1817–37. http://dx.doi.org/10.1142/s0218202506001728.

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We study the stability of the solutions of a stationary Euler–Poisson problem modeling a plasma diode. The model was presented in a previous paper, where we proved the existence of multiple solutions. Since, physically speaking, we expect only one solution to be present, we propose here two strategies for selecting the physical stationary regime. On the one hand, we study the energy functional associated with the stationary problem and order the different solutions in terms of their energy level and as stationary points for the functional. On the other hand, starting from the remark that a stable stationary state is the state which is reached after an evolution, we solve numerically the corresponding time-dependent Euler–Poisson problem and show that only one of the possible solutions is indeed stable in this sense. These two independent approaches select the same solution that we label therefore as stable.
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Dikansky, Arnold. "Asymptotically stable stationary solutions to the reaction-diffusion equations." Bulletin of the Australian Mathematical Society 47, no. 2 (April 1993): 273–86. http://dx.doi.org/10.1017/s0004972700012508.

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We assume that there exists an asymptotically stable stationary solution of a Galerkin approximation for the reaction-diffusion system. It is shown that there exists a nearby stationary solution of the full reaction-diffusion system provided the order of the Galerkin approximation is high enough. The Lyapunov second method is used to prove the asymptotic stability of the stationary solution.
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Bass, L., A. J. Bracken, K. Holmåker, and B. R. F. Jefferies. "Integro-differential equations for the self-organisation of liver zones by competitive exclusion of cell-types." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 29, no. 2 (October 1987): 156–94. http://dx.doi.org/10.1017/s0334270000005701.

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AbstractA model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.
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Lan, Xiangjun, Zhihua Feng, and Fan Lv. "Stochastic Principal Parametric Resonances of Composite Laminated Beams." Shock and Vibration 2014 (2014): 1–17. http://dx.doi.org/10.1155/2014/617828.

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This paper presents a detailed study on the stochastic stability, jump, and bifurcation of the motion of the composite laminated beams subject to axial load. The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is quantificationally resolved and the results show that the increase of the bandwidth facilitates the almost sure stability of the trivial response. The stochastic jump and bifurcation of the response are numerically calculated through the stationary joint probability and the results reveal that (a) the higher the excitation frequency is, the more probable the jump from the stable stationary nontrivial solution to the stable stationary trivial one is; (b) the most probable motion is around the nontrivial solution when the bandwidth is smaller; (c) the outer flabellate peak decreases, while the central volcano peak increases as the value of the excitation load decreases; and (d) the overall tendency of the response is that the probable motion jumps from the stable stationary nontrivial branch to the stable stationary trivial one as the fiber orientation angle of the first lamina with respect to thex-axis of the beam increases from zero to a smaller angle.
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SUZUKI, TAKASHI. "A NOTE ON THE STABILITY OF STATIONARY SOLUTIONS TO A SYSTEM OF CHEMOTAXIS." Communications in Contemporary Mathematics 02, no. 03 (August 2000): 373–83. http://dx.doi.org/10.1142/s0219199700000189.

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In 1970 a system of parabolic equations was proposed by Keller and Segel to describe the chemotactic feature of cellular slime molds. It has L1 preserving property for the first component and in use of this the stationary problem is reduced to a single elliptic problem concerning the second component. This problem has a variational structure and several features of the solutions are derived from it. In this paper we study linearized stable solutions in this sense and show that any of them is stable as a stationary solution to the original system of parabolic equations.
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FELLNER, KLEMENS, and GAËL RAOUL. "STABLE STATIONARY STATES OF NON-LOCAL INTERACTION EQUATIONS." Mathematical Models and Methods in Applied Sciences 20, no. 12 (December 2010): 2267–91. http://dx.doi.org/10.1142/s0218202510004921.

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In this paper, we are interested in the large-time behaviour of a solution to a non-local interaction equation, where a density of particles/individuals evolves subject to an interaction potential and an external potential. It is known that for regular interaction potentials, stable stationary states of these equations are generically finite sums of Dirac masses. For a finite sum of Dirac masses, we give (i) a condition to be a stationary state, (ii) two necessary conditions of linear stability w.r.t. shifts and reallocations of individual Dirac masses, and (iii) show that these linear stability conditions imply local non-linear stability. Finally, we show that for regular repulsive interaction potential Wε converging to a singular repulsive interaction potential W, the Dirac-type stationary states [Formula: see text] approximate weakly a unique stationary state [Formula: see text]. We illustrate our results with numerical examples.
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Finkelshtein, Dmitri, Yuri Kondratiev, Stanislav Molchanov, and Pasha Tkachov. "Global stability in a nonlocal reaction-diffusion equation." Stochastics and Dynamics 18, no. 05 (September 12, 2018): 1850037. http://dx.doi.org/10.1142/s0219493718500375.

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We study stability of stationary solutions for a class of nonlocal semilinear parabolic equations. To this end, we prove the Feynman–Kac formula for a Lévy processes with time-dependent potentials and arbitrary initial condition. We propose sufficient conditions for asymptotic stability of the zero solution, and use them to the study of the spatial logistic equation arising in population ecology. For this equation, we find conditions which imply that its positive stationary solution is asymptotically stable. We consider also the case when the initial condition is given by a random field.
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Kong, Liang. "Existence of Positive Solutions of Fisher-KPP Equations in Locally Spatially Variational Habitat with Hybrid Dispersal." Journal of Mathematics Research 9, no. 1 (January 2, 2017): 1. http://dx.doi.org/10.5539/jmr.v9n1p1.

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The current paper investigate the persistence of positive solutions of KPP type evolution equations with random/nonlocal dispersal in locally spatially inhomogeneous habitat. By the constructions of super/sub solutions and comparison principle, we prove that such an equation has a unique globally stable positive stationary solution.
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Surgailis, Donatas. "A quadratic ARCH(∞) model with long memory and Lévy stable behavior of squares." Advances in Applied Probability 40, no. 04 (December 2008): 1198–222. http://dx.doi.org/10.1017/s0001867800003025.

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We introduce a modification of the linear ARCH (LARCH) model (Giraitis, Robinson, and Surgailis (2000)) - a special case of Sentana's (1995) quadratic ARCH (QARCH) model - for which the conditional variance is a sum of a positive constant and the square of an inhomogeneous linear combination of past observations. Necessary and sufficient conditions for the existence of a stationary solution with finite variance are obtained. We give conditions under which the stationary solution with infinite fourth moment can exhibit long memory, the leverage effect, and a Lévy-stable limit behavior of partial sums of squares.
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Surgailis, Donatas. "A quadratic ARCH(∞) model with long memory and Lévy stable behavior of squares." Advances in Applied Probability 40, no. 4 (December 2008): 1198–222. http://dx.doi.org/10.1239/aap/1231340170.

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We introduce a modification of the linear ARCH (LARCH) model (Giraitis, Robinson, and Surgailis (2000)) - a special case of Sentana's (1995) quadratic ARCH (QARCH) model - for which the conditional variance is a sum of a positive constant and the square of an inhomogeneous linear combination of past observations. Necessary and sufficient conditions for the existence of a stationary solution with finite variance are obtained. We give conditions under which the stationary solution with infinite fourth moment can exhibit long memory, the leverage effect, and a Lévy-stable limit behavior of partial sums of squares.
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Dissertations / Theses on the topic "Stationary and stable solution"

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Biesdorf, João. "Mínimos locais de funcionais com dependência especial via Γ convergência: com e sem vínculo." Universidade Federal de São Carlos, 2011. https://repositorio.ufscar.br/handle/ufscar/5822.

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Made available in DSpace on 2016-06-02T20:27:39Z (GMT). No. of bitstreams: 1 3744.pdf: 1323892 bytes, checksum: 71a7a7180d61db167b8cbec4db2bbe8b (MD5) Previous issue date: 2011-05-30
Universidade Federal de Sao Carlos
We address the question of existence of stationary stable solutions to a class of reaction-diffusion equations with spatial dependence in 2 and 3-dimensional bounded domains. The approach consists of proving the existence of local minimizer of the corres-ponding energy functional. For existence, it was enough to give sufficient conditions on the diffusion coefficient and on the reaction term to ensure the existence of isolated mi¬nima of the Γlimit functional of the energy functional family. In the second part we take the techniques developed in the first part to minimize functional in 2 and 3-dimensional rectangles, with and without constraint, solving in a more general form this problem, which was originaly proposed in 1989 by Robert Kohn and Peter Sternberg.
Na primeira parte deste trabalho, abordamos a existência de soluções estacioná-rias estáveis para uma classe de equações de reação-difusão com dependência espacial em domínios limitados 2 e 3-dimensionais. Esta abordagem foi feita via existência de míni¬mos locais dos funcionais de energia correspondentes. Para tal, foi suficiente encontrar condições no coeficiente de difusão e no termo de reação que garantam existência de míni¬mos isolados do funcional Γlimite da família de funcionais de energia. Na segunda parte, aproveitamos as técnicas desenvolvidas na primeira parte para minimizar funcionais em retângulos e paralelepípedos, com e sem vínculo, resolvendo de forma bem mais geral este problema, originalmente proposto em 1989 por Robert Kohn e Peter Sternberg.
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Flandoli, Franco, and Michael Högele. "A solution selection problem with small stable perturbations." Universität Potsdam, 2014. http://opus.kobv.de/ubp/volltexte/2014/7120/.

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The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift.
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Knappett, Daniel. "Numerical solution of the stationary FPK equation using Shannon wavelets." Thesis, University of Nottingham, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.367109.

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Hewett, Caspar Julian Mnaser. "Unconditionally stable finite difference schemes for the solution of problems in hydraulics." Thesis, University of Newcastle Upon Tyne, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275595.

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Routledge, Jack. "Exploring interactions between anions and kinetically stable lanthanide complexes in aqueous solution." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:69e73701-0689-475a-ac33-ee260fa8baea.

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This work utilises the interactions between lanthanide complexes and anions in solution to explore the nature of coordination chemistry in the f-elements, and to define fundamental behaviour in paramagnetic complexes, as well as investigating the factors responsible for ensuring high lanthanide luminescence quantum yields. Chapter one describes characteristic lanthanide properties with a focus on luminescence and the current understanding of lanthanide complexes. Illustrating the response of luminescent lanthanide complexes to a range of analytes in aqueous solution. Chapter two explores the factors affecting luminescence quantum yield in arylacylDO3A complexes as well studying the pH behaviour of these species, showing that the whole energy transfer cascade involved in sensitisation of lanthanide luminescence must be considered when optimising the properties of lanthanide complexes. Chapter three describes the effect of fluoride binding in lanthanide tetra-amide complexes appended with fluorinated benzyl groups on the lanthanide crystal field, demonstrating that the effect of a strong axial donor on the overall ligand field can determine the nature of the magnetic anisotropy at the lanthanide centre. This in turn determines both the optical spectra and the NMR spectra of the complexes. The results obtained illustrate that the whole structure of the complex (and not just the donor set) need to be considered when defining the behaviour of the lanthanide complex. Chapter four investigates the interaction between fluoride and lanthanide complexes of the Lehn cryptand. In this case, the relatively symmetric donor set associated with the cryptand, combined with exchange between isomers, gives rise to small observed anisotropies in the absence of fluoride that are dramatically enhanced by fluoride binding. A range of fluoride responsive behaviour has been identified. Chapter five describes the synthesis and study of a group of halogenated phenacylDO3A lanthanide complexes. It is shown how such complexes respond to changes in cyanide concentration but are essentially inert towards fluoride as a consequence of the reduced residual charge on the lanthanide centre. The interaction with cyanide is assigned to the formation of a cyanohydrin which is assisted by the proximate lanthanide ion, which acts as a Lewis acid. Chapter six draws together the work described in earlier chapters. Chapter seven provides experimental procedures along with characterisation data for the compounds studied.
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Millis, Kathryn A. (Kathryn Ann). "Distributed measures of solution existence and its optimality in stationary electric power systems : scattering approach." Thesis, Massachusetts Institute of Technology, 2000. http://hdl.handle.net/1721.1/86430.

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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2000.
Includes bibliographical references (p. [149]-151).
by Kathryn A. Millis.
Ph.D.
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Yevik, Andrei. "Numerical approximations to the stationary solutions of stochastic differential equations." Thesis, Loughborough University, 2011. https://dspace.lboro.ac.uk/2134/7777.

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This thesis investigates the possibility of approximating stationary solutions of stochastic differential equations using numerical methods. We consider a particular class of stochastic differential equations, which are known to generate random dynamical systems. The existence of stochastic stationary solution is proved using global attractor approach. Euler's numerical method, applied to the stochastic differential equation, is proved to generate a discrete random dynamical system. The existence of stationary solution is proved again using global attractor approach. At last we prove that the approximate stationary point converges in mean-square sense to the exact one as the time step of the numerical scheme diminishes.
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Ortoleva, Cecilia Maria. "Asymptotic properties of the dynamics near stationary solutions for some nonlinear Schrödinger équations." Phd thesis, Université Paris-Est, 2013. http://tel.archives-ouvertes.fr/tel-00825627.

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The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The first model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a {point} (or contact) interaction with strength $alpha$, which consists of a singular perturbation of the Laplacian described by a self adjoint operator $H_{alpha}$, and letting the strength $alpha$ depend on the wave function: $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$.It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to $|x - x_0|^{-1}$, where $x_0$is the location of the point interaction. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of itssingular part, then, in order to introduce a nonlinearity, we let the strength $alpha$ depend on $u$ according to the law $alpha=-nu|q|^sigma$, with $nu > 0$. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form $u (t)=e^{iomega t}Phi_{omega}$, which are orbitally stable in the range $sigma in (0,1)$, and orbitally unstable for $sigma geq 1.$ Moreover, we show that for $sigma in(0,frac{1}{sqrt 2}) cup left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$ every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted $L^p$ space which allows dispersive stimates, the following resolution holds: $u(t) =e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}}+U_t*psi_{infty} +r_{infty}$, where $U_t$ is the free Schrödinger propagator,$omega_{infty} > 0$ and $psi_{infty}$, $r_{infty} inL^2(R^3)$ with $| r_{infty} |_{L^2} = O(t^{-p}) quadtextrm{as} ;; t right arrow +infty$, $p = frac{5}{4}$,$frac{1}{4}$ depending on $sigma in (0, 1/sqrt{2})$, $sigma in (1/sqrt{2}, 1)$, respectively, and finally $l(t)$ is a logarithmic increasing function that appears when $sigma in (frac{1}{sqrt{2}},sigma^*)$, for a certain $sigma^* in left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right]$. Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation $i frac{du}{dt}=-Delta u-|u|^4 u$. In this case we prove, for any $nu$ and $alpha_0$ sufficiently small, the existence of radial finite energy solutions of the form$u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDeltat}zeta^*+o_{dot H^1} (1)$ as $tright arrow +infty$, where$alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$,$W(x)=(1+frac13|x|^2)^{-1/2}$ is the ground state and $zeta^*$is arbitrarily small in $dot H^1$
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Perella, Andrew James. "A class of Petrov-Galerkin finite element methods for the numerical solution of the stationary convection-diffusion equation." Thesis, Durham University, 1996. http://etheses.dur.ac.uk/5381/.

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A class of Petrov-Galerkin finite element methods is proposed for the numerical solution of the n dimensional stationary convection-diffusion equation. After an initial review of the literature we describe this class of methods and present both asymptotic and nonasymptotic error analyses. Links are made with the classical Galerkin finite element method and the cell vertex finite volume method. We then present numerical results obtained for a selection of these methods applied to some standard test problems. We also describe extensions of these methods which enable us to solve accurately for derivative values of the solution.
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Seif, Wael. "The development of an efficient and stable solution to the advection dispersion equation for saline groundwater flow." Thesis, University of Leeds, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.426829.

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Books on the topic "Stationary and stable solution"

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Baumeister, Johann. Stable Solution of Inverse Problems. Wiesbaden: Vieweg+Teubner Verlag, 1987. http://dx.doi.org/10.1007/978-3-322-83967-1.

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Stable solution of inverse problems. Braunschweig: F. Vieweg, 1986.

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Pipiras, Vladas, and Murad S. Taqqu. Stable Non-Gaussian Self-Similar Processes with Stationary Increments. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-62331-3.

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Greenberg, Joseph. Stable standards of behavior: A unifying approach to solution concepts. Stanford, Calif: Institute for Mathematical Studies in the Social Sciences, Stanford University, 1986.

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Greenberg, Joseph. Perfect equilibria paths in repeated games: The unique maximal stationary stable standard of behavior. Stanford, Calif: Institute for Mathematical Studies in the Social Sciences, Stanford University, 1986.

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Hough, Patricia D. Stable and efficient solution of weighted least-squares problems with applications in interior point methods. Ithaca, N.Y: Cornell Theory Center, Cornell University, 1996.

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McCallum, Bennett T. The unique minimum state variable re solution is e-stable in all well formulated linear models. Cambridge, Mass: National Bureau of Economic Research, 2003.

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Inglese, G. Identification of the drift coefficient of a Fokker-Plank equation from the moment discretization of its stationary solution. Ithaca, N.Y: Cornell Theory Center, Cornell University, 1995.

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Stable Solution of Inverse Problems. Wiesbaden: Vieweg+Teubner Verlag, 1987.

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Taqqu, Murad S., and Vladas Pipiras. Stable Non-Gaussian Self-Similar Processes with Stationary Increments. Springer, 2017.

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Book chapters on the topic "Stationary and stable solution"

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Luo, Jiaowan. "Exponentially Stable Stationary Solutions for Delay Stochastic Evolution Equations." In Stochastic Analysis with Financial Applications, 169–78. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0097-6_11.

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Davydova, M. A., N. N. Nefedov, and S. A. Zakharova. "Asymptotically Lyapunov-Stable Solutions with Boundary and Internal Layers in the Stationary Reaction-Diffusion-Advection Problems with a Small Transfer." In Finite Difference Methods. Theory and Applications, 216–24. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-11539-5_23.

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Nolan, John P. "Bounded Stationary Stable Processes and Entropy." In Stable Processes and Related Topics, 101–5. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4684-6778-9_5.

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Kôno, Norio, and Makoto Maejima. "Self-Similar Stable Processes with Stationary Increments." In Stable Processes and Related Topics, 275–95. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4684-6778-9_13.

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Arov, D. Z., and J. Rovnyak. "Stable Dissipative Linear Stationary Dynamical Scattering Systems." In Interpolation Theory, Systems Theory and Related Topics, 99–136. Basel: Birkhäuser Basel, 2002. http://dx.doi.org/10.1007/978-3-0348-8215-6_6.

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Scott, L. Ridgway, and Dexuan Xie. "Parallel Linear Stationary Iterative Methods." In Parallel Solution of Partial Differential Equations, 31–55. New York, NY: Springer New York, 2000. http://dx.doi.org/10.1007/978-1-4612-1176-1_2.

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Nečasová, Šárka, and Stanislav Kračmar. "Fundamental Solution of the Stationary Problem." In Atlantis Briefs in Differential Equations, 25–38. Paris: Atlantis Press, 2016. http://dx.doi.org/10.2991/978-94-6239-231-1_4.

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Janicki, Aleksander, and Aleksander Weron. "Spectral Representations of Stationary Processes." In Simulation and Chaotic Behavior of α-Stable Stochastic Processes, 111–40. Boca Raton: CRC Press, 2021. http://dx.doi.org/10.1201/9781003208877-5.

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Janicki, Aleksander, and Aleksander Weron. "Chaotic Behavior of Stationary Processes." In Simulation and Chaotic Behavior of α-Stable Stochastic Processes, 231–62. Boca Raton: CRC Press, 2021. http://dx.doi.org/10.1201/9781003208877-9.

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Podgórski, Krzysztof, and Aleksander Weron. "Characterizations of ergodic stationary stable processes via the dynamical functional." In Stable Processes and Related Topics, 317–28. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4684-6778-9_16.

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Conference papers on the topic "Stationary and stable solution"

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Pan, Ruigui, and Huw G. Davies. "Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0245.

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Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.
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Aidanpa¨a¨, Jan-Olov. "Multiple Solutions in an Amplitude Limited Jeffcott Rotor Including Rubbing and Stick-Slip Effect." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84616.

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The non-linear behaviour of rub-impact rotors have been studied in several papers. In such systems rich dynamics have been found together with the coexistence of solutions within some specific parameter ranges. In this paper an attempt is made to find all stable solutions for an amplitude limited Jeffcott rotor including rubbing and stick-slip effect. The recently suggested “multi bifurcation diagram method” is used to find and extract stable sets of bifurcation diagrams. A system is chosen where the linear stationary amplitude only exceeds the clearance in a narrow region near the natural frequency. Therefore large regions in frequency are expected to have only the linear stationary response. The results show that it is only for very low frequencies that one single solution exists. Even though periodic motions are dominant, there exist large ranges in frequency with quasi-periodic or chaotic motions. For the studied cases, three coexisting stable solutions are most common. In one case as many as four stable solutions was found to coexist. For rotors with large clearances (no impacts necessary) it is still possible to find several coexisting motions. For all cases the stick motion is the most severe one with large amplitudes and high backward whirl frequencies. In real situations the consequence of this stick motion is machine failure. These high amplitude motions were found to be stable over large frequency ranges. From the stability analysis it was found that this rolling motion can be avoided by low spin speed, low contact stiffness, low coefficient of friction, small ratio of disc radius/clearance or high damping ratio. In a design situation the parameters are seldom known with high accuracy. Therefore, it is of interest to know all solutions for parameter intervals. The multi-bifurcation diagram can be used in such situations to design a robust machine or at least be prepared for unwanted dynamics.
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Ganesan, R., and T. S. Sankar. "Resonant Oscillations and Stability of Asymmetric Rotors." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0099.

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Abstract Non-stationary oscillations of an asymmetric rotor while passing through primary resonance and the associated stability behaviour are analyzed. Solutions are developed based on a Jeffcott rotor model and the equations of motion are rewritten in a form suitable for applying the method of multiple scales. The many-variable version using “slow” and “fast” lime scales is applied to obtain the uniform expansions of amplitudes of motion. Similar general expressions for amplitude and frequency modulation functions are explicitly obtained and are specialized to yield steady-state solutions. Frequency-amplitude relationships resulting from combined parametric and mass unbalance excitations, for the nonlinear vibration are derived. Stability regions in the parameter space are obtained for a stable solution in terms of the perturbed steady-state solutions of the governing equations of motion. Also, trivial solutions are examined for stability. The sensitivity of vibration amplitudes to various rotor-dynamic system parameters is illustrated through a numerical study.
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Cohen, Nadav, and Izhak Bucher. "The Dynamics of a Bi-Stable Energy Harvester: Exploration via Slow-Fast Decomposition and Analytical Modeling." In ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/esda2012-83013.

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The paper discusses the advantages of the bi-stable energy harvester over linear oscillators in the low frequency excitation regime. When excited by low-frequency base motions, a bistable vibration-based energy harvester’s response is characterized by a combination of a slow, and a non-stationary fast component. By decomposing the response of the bi-stable system into fast and slow components, some new physical insights into the dynamical properties of the system are obtained. Properties such as mechanical frequency up-conversion, asymmetry in the bi-stable potential of the system and extraction of the backbone curve are explored. The proposed decomposition is demonstrated and explained via numerical and experimental results. A simple, approximate analytical model, for the bi-stable oscillator is proposed and its ability to detect migration towards different vibration regimes is illustrated. An expression for the power output of the harvester is derived from the analytical solution allowing us to tune the bi-stable potential towards optimum performance. The analytical model sheds light on the occurrences of bifurcations in the response of such nonlinear systems and on the optimal values of potential barrier vs. excitation levels.
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Childs, Dara W. "The Multiple Contributions of Jorgen Lund’s Ph.D. Dissertation, “Self-Excited, Stationary Whirl Orbits of a Journal in Sleeve Bearings,” RPI, 1966, Engineering Mechanics." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21370.

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Abstract Lund set out to define the circumstances under which stable limit-cycle orbits could exist for the linearly unstable motion of a rigid rotor. He also undertook to examine the nature of these stable limit cycles when they are demonstrated to exist. He obviously succeeded in meeting both these objectives; however, Lund’s really remarkable and most useful contributions are covered “incidentally” in the course of developing his nonlinear analytical/computational solutions.
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Banik, A. K., and T. K. Datta. "Stochastic Response and Stability Analysis of Single Leg Articulated Tower." In ASME 2003 22nd International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2003. http://dx.doi.org/10.1115/omae2003-37032.

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The stationary response and asymptotic stability in probability of an articulated tower under random wave excitation are investigated. The articulated tower is modelled as a SDOF system having stiffness nonlinearity, damping nonlinearity and parametric excitation. Using a stochastic averaging procedure and Fokker-Plank-Kolomogorov equation (FPK), the probability density function of the stationary solution is obtained for random sea state represented by a P-M sea spectrum. The method involves a Van-Der-Pol transformation of the nonlinear equation of motion to convert it to the Ito’s stochastic differential equation with averaged drift and diffusion coefficients. The asymptotic stability in probability of the system is investigated by obtaining the averaged Ito’s equation for the Hamiltonian of the system. The asymptotic stability is examined approximately by investigating the asymptotic behaviour of the diffusion process Y(t) at its two boundaries Y = 0 and ∞. As an illustrative example, an articulated tower in a sea depth of 150 m is considered. The tower consists of hollow cylinder of varying diameter along the height, providing the required buoyancy of the system. Wave forces on the structure are calculated using Morrison’s equation. The stochastic response and the stability conditions are obtained for a sea state represented by P-M spectrum with 16m significant wave height. The results of the study indicate that the probability density of the stationary response obtained by the stochastic averaging procedure is in very good agreement with that obtained from digital simulation. Further, the articulated tower is found to be asymptotically stable under the parametric excitation arising due to hydrodynamic damping.
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Ishida, Yukio, Kimihiko Yasuda, and Shin Murakami. "Nonstationary Vibration of a Rotating Shaft With Nonlinear Spring Characteristics During Acceleration Through a Major Critical Speed: A Discussion by the Asymptotic Method and the FFT Method." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0120.

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Abstract Nonstationary vibrations at the major critical speed of a rotating shaft with nonlinear spring characteristics are discussced. Firstly, the first order approximate solutions of steady-state and nonstationary oscillations are obtained by the asymptotic method. The relations between these approximate solutions and the nonlinear components in the polar coordinate expression are investigated. It is clarified that, similar to the case of the stationary oscillations, only the isotropic nonlinear component has influence on nonstationary oscillations in the first order approximation. Secondly, the complex-FFT method where non-stationary time histories obtained by numerical integrations of the equations of motion are treated as complex numbers in the complex plane which coincides with the whirling plane are proposed. By this method, the amplitude variation curves of each vibration component are obtained. From the comparison of the amplitude variation curves of the first approximation of the asymptotic method, the solution of the complex-FFT method, and direct numerical integration, it is clarified that, although all these solutions coincide well in the case of stationary solutions, the first approximation of the asymptotic method has comparatively large quantitative error in the case of nonstationary solutions. In addition, the influences of the anisotropic nonlinear components which do not appear in the first approximation of the asymptotic method are investigated.
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Hill, D. Lee. "Sectional Modeling of a Centrifugal Compressor." In 2002 4th International Pipeline Conference. ASMEDC, 2002. http://dx.doi.org/10.1115/ipc2002-27172.

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The prediction of compressor performance using steady-state methods has had limited success for multistage configurations. Accepted limitations of the computational methods such as streamline curvature effects have been ignored because of a more dominant issue of the interface between the rotating and stationary components and the change in pitch between the vaned-stationary components. The most common model reported in the literature for centrifugal compressor stage analysis is known as the frozen-rotor or implicit model. Its selection, however, is normally driven for numerical stability reasons not for accuracy. The literature has shown that this method does not provide physically correct solutions for off-design predictions. The current work attempts to improve the steady-state modeling approach by employing an interface modeling that assumes that the tip speed is much greater than the through flow velocity. This model is referred to as circumferential averaging and is less stable than the frozen rotor model. The next proposed improvement is to model all of the vaned-stationary passages in order to preserve geometric periodicity. The last novel improvement is in the area of the diffuser inlet region where a portion of the secondary flow path is included to resolve the entrance loss into the diffuser. This approach was used to model two sections of a high pressure centrifugal compressor. The results are presented for design and off-design flows. The calculations are compared with test data taken from full scale testing.
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Zhu, W. D., and K. Wu. "Dynamic Stability of Translating and Stationary Strings With Sinusoidally Varying Velocities." In ASME 2012 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/imece2012-86182.

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Parametric instability in a system is caused by periodically varying coefficients in its governing differential equations. While parametric excitation of lumped-parameter systems has been extensively studied, that of distributed-parameter systems has been traditionally analyzed by applying Floquet theory to their spatially discretized equations. In this work, parametric instability regions of a second-order non-dispersive distributed structural system, which consists of a translating string with a constant tension and a sinusoidally varying velocity, and two boundaries that axially move with a sinusoidal velocity relative to the string, are obtained using the wave solution and the fixed point theory without spatially discretizing the governing partial differential equation. There are five cases that involve non-trivial combinations of string and boundary motions: I) a translating string with a sinusoidally varying velocity and two stationary boundaries; II) a translating string with a sinusoidally varying velocity, a sinusoidally moving boundary, and a stationary boundary; III) a translating string with a sinusoidally varying velocity and two sinusoidally moving boundaries; IV) a stationary string with a sinusoidally moving boundary and a stationary boundary; and V) a stationary string with two sinusoidally moving boundaries. Unlike parametric instability regions of lumped-parameter systems that are classified as principal, secondary, and combination instability regions, the parametric instability regions of the class of distributed structural systems considered here are classified as period-1 and period-i (i>1) instability regions. Period-1 parametric instability regions are analytically obtained; an equivalent total velocity vector is introduced to express them for all the cases considered. While period-i (i>1) parametric instability regions can be numerically calculated using bifurcation diagrams, it is shown that only period-1 parametric instability regions exist in case IV, and no period-i (i>1) parametric instability regions can be numerically found in case V. Unlike parametric instability in a lumped-parameter system that is characterized by an unbounded displacement, the parametric instability phenomenon discovered here is characterized by a bounded displacement and an unbounded vibratory energy, due to formation of infinitely compressed shock-like waves. There are seven independent parameters in the governing equation and boundary conditions, and the parametric instability regions in the seven-dimensional parameter space can be projected to a two-dimensional parameter plane if five parameters are specified. Period-1 parametric instability occurs in certain excitation frequency bands centered at the averaged natural frequencies of the systems in all the cases. If the parameters are chosen to be in the period-i (i≥1) parametric instability region corresponding to an integer k, an initial smooth wave will be infinitely compressed to k shock-like waves as time approaches infinity. The stable and unstable responses of the linear model in case I are compared with those of a corresponding nonlinear model that considers the coupled transverse and longitudinal vibrations of the translating string and an intermediate linear model that includes the effect of the tension change due to axial acceleration of the string on its transverse vibration. The parametric instability in the original linear model can exist in the nonlinear and intermediate linear models.
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Liu, Yoncai, Hamdi Sheibani, Susumu Sakai, Yasunori Okano, and Sadik Dost. "A Three Dimensional Simulation Model for Liquid Phase Electroepitaxy Under Magnetic Field." In ASME 2002 Pressure Vessels and Piping Conference. ASMEDC, 2002. http://dx.doi.org/10.1115/pvp2002-1537.

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A three-dimensional numerical simulation for the Liquid Phase Electroepitaxial (LPEE) growth of GaAs under a vertical stationary magnetic field was carried out. The effect of magnetic field intensity on the flow field in the liquid solution was investigated. Numerical results show that the flow patterns exhibit three distinct stability characteristics: a stable flow field up to a magnetic field level of Ha = 150, a transitional flow between Ha = 150 and Ha = 220, and an unstable flow above Ha = 220. In the stable region, the applied magnetic field suppresses the flow field, and the flow intensity decreases with increasing magnetic field. In the transitional region, the flow intensity increases dramatically with increase in the magnetic field strength. The flow patterns are significantly different from those in the stable region. The flow field is no longer axisymmetric but still stable. In the unstable region, the flow structure and intensity change with time. Under a strong magnetic field, the flow cells are confined to the vicinity of the vertical wall and exhibit significant non-uniformity near the growth interface. Such strong flow fluctuations and non-uniformities near the growth interface may have an adverse effect on the growth process and lead to an unsatisfactory growth.
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Reports on the topic "Stationary and stable solution"

1

Cambanis, Stamatis, and Makoto Maejima. Two Classes of Self-Similar Stable Processes with Stationary Increments. Fort Belvoir, VA: Defense Technical Information Center, January 1988. http://dx.doi.org/10.21236/ada192842.

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2

Young, D. M., and D. R. Kincaid. Linear stationary second-degree methods for the solution of large linear systems. Office of Scientific and Technical Information (OSTI), July 1990. http://dx.doi.org/10.2172/674848.

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McCallum, Bennett. The Unique Minimum State Variable RE Solution is E-Stable in All Well Formulated Linear Models. Cambridge, MA: National Bureau of Economic Research, September 2003. http://dx.doi.org/10.3386/w9960.

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Galuszka-Muga, Barbara, and Luis M. Muga. The Influence of Radiation on Pit Solution Chemistry as it Pertains to the Transition from Metastable to Stable Pitting in Steels. Office of Scientific and Technical Information (OSTI), December 2006. http://dx.doi.org/10.2172/892996.

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Lillard, Scott, and Robert Hanrahan. The Influence of Radiation on Pit Solution Chemistry as it Pertains to the Transition from Metastable to Stable Pitting in Steels. Office of Scientific and Technical Information (OSTI), June 2005. http://dx.doi.org/10.2172/893224.

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Lillard, R. Scott, and Robert J. Hanrahan. The Influence of Radiation on Pit Solution Chemistry as it Pertains to the Transition from Metastable to Stable Pitting in Steels. Office of Scientific and Technical Information (OSTI), June 2002. http://dx.doi.org/10.2172/835035.

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Lillard, R. Scott, and Robert J. Hanrahan. The Influence of Radiation on Pit Solution Chemistry as it Pertains to the Transition from Metastable to Stable Pitting in Steels. Office of Scientific and Technical Information (OSTI), June 2003. http://dx.doi.org/10.2172/835036.

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Galuszka-Muga, Barbara. The Influence of Radiation on Pit Solution Chemistry as it Pertains to the Transition from Metastable to Stable Pitting in Steels. Office of Scientific and Technical Information (OSTI), May 2005. http://dx.doi.org/10.2172/840166.

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Lillard, Scott, and Robert Hanrahan. The Influence of Radiation on Pit Solution Chemistry as it Pertains to the Transition from Metastable to Stable Pitting in Steels. Office of Scientific and Technical Information (OSTI), June 2004. http://dx.doi.org/10.2172/839104.

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NORTHWEST RESEARCH ASSOCIATES INC BELLEVUE WA. Surface Layer Flux Sources and Parameterization Failure in Stable Conditions from CASES-99 Data Analysis: Impacts of Intermittent Turbulence its Sources and a Proposed Solution. Fort Belvoir, VA: Defense Technical Information Center, April 2003. http://dx.doi.org/10.21236/ada415238.

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