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Journal articles on the topic 'Limit equilibrium'

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Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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1

Yu, H. S., R. Salgado, S. W. Sloan, and J. M. Kim. "Limit Analysis versus Limit Equilibrium for Slope Stability." Journal of Geotechnical and Geoenvironmental Engineering 124, no. 1 (January 1998): 1–11. http://dx.doi.org/10.1061/(asce)1090-0241(1998)124:1(1).

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2

Leshchinsky, Dov, H. S. Yu, R. Salgado, S. W. Sloan, and J. M. Kim. "Limit Analysis versus Limit Equilibrium for Slope Stability." Journal of Geotechnical and Geoenvironmental Engineering 125, no. 10 (October 1999): 914–18. http://dx.doi.org/10.1061/(asce)1090-0241(1999)125:10(914).

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3

Krahn, John. "The 2001 R.M. Hardy Lecture: The limits of limit equilibrium analyses." Canadian Geotechnical Journal 40, no. 3 (June 1, 2003): 643–60. http://dx.doi.org/10.1139/t03-024.

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Limit equilibrium types of analysis have been in use in geotechnical engineering for a long time and are now used routinely in geotechnical engineering practice. Modern graphical software tools have made it possible to gain a much better understanding of the inner numerical details of the method. A closer look at the details reveals that the limit equilibrium method of slices has some serious limitations. The fundamental shortcoming of limit equilibrium methods, which only satisfy equations of statics, is that they do not consider strain and displacement compatibility. This limitation can be overcome by using finite element computed stresses inside a conventional limit equilibrium framework. From the finite element stresses both the total shear resistance and the total mobilized shear stress on a slip surface can be computed and used to determine the factor of safety. Software tools that make this feasible and practical are now available, and they hold great promise for advancing the technology of analyzing the stability of earth structures.Key words: limit equilibrium, stability, factor of safety, finite element, ground stresses, slip surface.

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4

Michel, Bernard. "Limit equilibrium of ice jams." Cold Regions Science and Technology 20, no. 2 (February 1992): 107–17. http://dx.doi.org/10.1016/0165-232x(92)90011-i.

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5

Chowdhury, Prabal Roy. "Limit-pricing as Bertrand equilibrium." Economic Theory 19, no. 4 (June 1, 2002): 811–22. http://dx.doi.org/10.1007/s001990100171.

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6

Myerson, Roger B., and Philip J. Reny. "Perfect Conditional ε‐Equilibria of Multi‐Stage Games With Infinite Sets of Signals and Actions." Econometrica 88, no. 2 (2020): 495–531. http://dx.doi.org/10.3982/ecta13426.

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We extend Kreps and Wilson's concept of sequential equilibrium to games with infinite sets of signals and actions. A strategy profile is a conditional ε‐equilibrium if, for any of a player's positive probability signal events, his conditional expected utility is within ε of the best that he can achieve by deviating. With topologies on action sets, a conditional ε‐equilibrium is full if strategies give every open set of actions positive probability. Such full conditional ε‐equilibria need not be subgame perfect, so we consider a non‐topological approach. Perfect conditional ε‐equilibria are defined by testing conditional ε‐rationality along nets of small perturbations of the players' strategies and of nature's probability function that, for any action and for almost any state, make this action and state eventually (in the net) always have positive probability. Every perfect conditional ε‐equilibrium is a subgame perfect ε‐equilibrium, and, in finite games, limits of perfect conditional ε‐equilibria as ε → 0 are sequential equilibrium strategy profiles. But limit strategies need not exist in infinite games so we consider instead the limit distributions over outcomes. We call such outcome distributions perfect conditional equilibrium distributions and establish their existence for a large class of regular projective games. Nature's perturbations can produce equilibria that seem unintuitive and so we augment the game with a net of permissible perturbations.

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7

Enoki, Meiketsu, Norio Yagi, Ryuichi Yatabe, and Elzaburo Ichimoto. "Relation of Limit Equilibrium Method to Limit Analysis Method." Soils and Foundations 31, no. 4 (December 1991): 37–47. http://dx.doi.org/10.3208/sandf1972.31.4_37.

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8

Huang, Jicai, Xiaojing Xia, Xinan Zhang, and Shigui Ruan. "Bifurcation of Codimension 3 in a Predator–Prey System of Leslie Type with Simplified Holling Type IV Functional Response." International Journal of Bifurcation and Chaos 26, no. 02 (February 2016): 1650034. http://dx.doi.org/10.1142/s0218127416500346.

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It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positive equilibrium which is a degenerate Bogdanov–Takens singularity (focus case) of codimension 3 for other values of parameters. We prove that the model exhibits degenerate focus type Bogdanov–Takens bifurcation of codimension 3 around the unique degenerate positive equilibrium. Numerical simulations, including the coexistence of three hyperbolic positive equilibria, two limit cycles, bistability states (one stable equilibrium and one stable limit cycle, or two stable equilibria), tristability states (two stable equilibria and one stable limit cycle), a stable limit cycle enclosing a homoclinic loop, a homoclinic loop enclosing an unstable limit cycle, or a stable limit cycle enclosing three unstable hyperbolic positive equilibria for various parameter values, confirm the theoretical results.

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9

Florig, Michael, and Jorge Rivera. "Walrasian equilibrium as limit of competitive equilibria without divisible goods." Journal of Mathematical Economics 84 (October 2019): 1–8. http://dx.doi.org/10.1016/j.jmateco.2019.05.001.

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10

Renault, Jérôme, and Bruno Ziliotto. "Limit Equilibrium Payoffs in Stochastic Games." Mathematics of Operations Research 45, no. 3 (August 2020): 889–95. http://dx.doi.org/10.1287/moor.2019.1015.

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We study the limit of equilibrium payoffs, as the discount factor goes to one, in non-zero-sum stochastic games. We first show that the set of stationary equilibrium payoffs always converges. We then provide two-player examples in which the whole set of equilibrium payoffs diverges. The construction is robust to perturbations of the payoffs and to the introduction of normal-form correlation.

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11

Alexakis, Haris, and Nicos Makris. "Limit equilibrium analysis of masonry arches." Archive of Applied Mechanics 85, no. 9-10 (November 26, 2014): 1363–81. http://dx.doi.org/10.1007/s00419-014-0963-6.

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12

Morel, Jim E., and Jeffery D. Densmore. "A two-component equilibrium-diffusion limit." Annals of Nuclear Energy 31, no. 17 (November 2004): 2049–57. http://dx.doi.org/10.1016/j.anucene.2004.07.011.

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13

Morel, Jim E., and Jeffery D. Densmore. "A two-component equilibrium-diffusion limit." Annals of Nuclear Energy 32, no. 2 (January 2005): 233–40. http://dx.doi.org/10.1016/j.anucene.2004.08.010.

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14

Beghian, L. E. "Thermostatic equilibrium in the classical limit." Il Nuovo Cimento B Series 11 107, no. 12 (December 1992): 1437–44. http://dx.doi.org/10.1007/bf02722854.

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15

Fudenberg, Drew, Giacomo Lanzani, and Philipp Strack. "Limit Points of Endogenous Misspecified Learning." Econometrica 89, no. 3 (2021): 1065–98. http://dx.doi.org/10.3982/ecta18508.

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We study how an agent learns from endogenous data when their prior belief is misspecified. We show that only uniform Berk–Nash equilibria can be long‐run outcomes, and that all uniformly strict Berk–Nash equilibria have an arbitrarily high probability of being the long‐run outcome for some initial beliefs. When the agent believes the outcome distribution is exogenous, every uniformly strict Berk–Nash equilibrium has positive probability of being the long‐run outcome for any initial belief. We generalize these results to settings where the agent observes a signal before acting.

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16

Dai, Yanfei, and Yulin Zhao. "Hopf Cyclicity and Global Dynamics for a Predator–Prey System of Leslie Type with Simplified Holling Type IV Functional Response." International Journal of Bifurcation and Chaos 28, no. 13 (December 12, 2018): 1850166. http://dx.doi.org/10.1142/s0218127418501663.

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This paper is concerned with a predator–prey model of Leslie type with simplified Holling type IV functional response, provided that it has either a unique nondegenerate positive equilibrium or three distinct positive equilibria. The type and stability of each equilibrium, Hopf cyclicity of each weak focus, and the number and distribution of limit cycles in the first quadrant are studied. It is shown that every equilibrium is not a center. If the system has a unique positive equilibrium which is a weak focus, then its order is at most [Formula: see text] and it has Hopf cyclicity [Formula: see text]. Moreover, some explicit conditions for the global stability of the unique equilibrium are established by applying Dulac’s criterion and constructing the Lyapunov function. If the system has three distinct positive equilibria, then one of them is a saddle and the others are both anti-saddles. For two anti-saddles, we prove that the Hopf cyclicity for the anti-saddle with smaller abscissa (resp., bigger abscissa) is [Formula: see text] (resp., [Formula: see text]). Furthermore, if both anti-saddle positive equilibria are weak foci, then they are unstable weak foci of order one. Moreover, one limit cycle can bifurcate from each of them simultaneously. Numerical simulations show that there is also a big stable limit cycle enclosing these two small limit cycles.

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17

Jiang, Jiao, and Yongli Song. "Stability and Bifurcation Analysis of a Delayed Leslie-Gower Predator-Prey System with Nonmonotonic Functional Response." Abstract and Applied Analysis 2013 (2013): 1–19. http://dx.doi.org/10.1155/2013/152459.

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A delayed Leslie-Gower predator-prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone iterative scheme, a sufficient condition independent of delay for the global stability of the positive equilibrium is obtained. Hopf bifurcations induced by the ratio of the intrinsic growth rates of the predator and prey and by delay, respectively, are found. Employing the normal form theory, the direction and stability of Hopf bifurcations can be explicitly determined by the parameters of the system. Some numerical simulations are given to support and extend our theoretical results. Two limit cycles enclosing an equilibrium, one limit cycle enclosing three equilibria and different types of heteroclinic orbits such as connecting two equilibria and connecting a limit cycle and an equilibrium are also found by using analytic and numerical methods.

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18

Leshchinsky, Ben, and Spencer Ambauen. "Limit Equilibrium and Limit Analysis: Comparison of Benchmark Slope Stability Problems." Journal of Geotechnical and Geoenvironmental Engineering 141, no. 10 (October 2015): 04015043. http://dx.doi.org/10.1061/(asce)gt.1943-5606.0001347.

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19

Powrie, W. "Limit equilibrium analysis of embedded retaining walls." Géotechnique 46, no. 4 (December 1996): 709–23. http://dx.doi.org/10.1680/geot.1996.46.4.709.

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20

Mundell, C., P. McCombie, C. Bailey, A. Heath, and P. Walker. "Limit-equilibrium assessment of drystone retaining structures." Proceedings of the Institution of Civil Engineers - Geotechnical Engineering 162, no. 4 (August 2009): 203–12. http://dx.doi.org/10.1680/geng.2009.162.4.203.

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21

Ferguson, J. M., J. E. Morel, and R. Lowrie. "The equilibrium-diffusion limit for radiation hydrodynamics." Journal of Quantitative Spectroscopy and Radiative Transfer 202 (November 2017): 176–86. http://dx.doi.org/10.1016/j.jqsrt.2017.07.031.

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22

Zhu, Jian-Zhou. "Continuum limit of electrostatic gyrokinetic absolute equilibrium." Physics of Plasmas 19, no. 6 (June 2012): 062304. http://dx.doi.org/10.1063/1.4725725.

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23

Bartoszek, K. "A Central Limit Theorem for punctuated equilibrium." Stochastic Models 36, no. 3 (May 5, 2020): 473–517. http://dx.doi.org/10.1080/15326349.2020.1752242.

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24

GOETTLER, RONALD L., CHRISTINE A. PARLOUR, and UDAY RAJAN. "Equilibrium in a Dynamic Limit Order Market." Journal of Finance 60, no. 5 (September 16, 2005): 2149–92. http://dx.doi.org/10.1111/j.1540-6261.2005.00795.x.

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25

Burakovsky, L., and L. P. Horwitz. "Galilean limit of equilibrium relativistic mass distribution." Journal of Physics A: Mathematical and General 27, no. 8 (April 21, 1994): 2623–31. http://dx.doi.org/10.1088/0305-4470/27/8/003.

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26

Lysova, S. S., T. A. Starikova, and Yu E. Zevatskii. "Limit of concentration constant of protolytic equilibrium." Russian Journal of General Chemistry 84, no. 8 (August 2014): 1634–35. http://dx.doi.org/10.1134/s1070363214080325.

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27

Zhu, D. Y., and C. F. Lee. "Explicit limit equilibrium solution for slope stability." International Journal for Numerical and Analytical Methods in Geomechanics 26, no. 15 (2002): 1573–90. http://dx.doi.org/10.1002/nag.260.

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28

Hungr, O., and F. Amann. "Limit equilibrium of asymmetric laterally constrained rockslides." International Journal of Rock Mechanics and Mining Sciences 48, no. 5 (July 2011): 748–58. http://dx.doi.org/10.1016/j.ijrmms.2011.04.008.

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29

Renault, Jérôme, and Bruno Ziliotto. "Hidden stochastic games and limit equilibrium payoffs." Games and Economic Behavior 124 (November 2020): 122–39. http://dx.doi.org/10.1016/j.geb.2020.08.001.

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30

Lee, Joonkyum, and Bumsoo Kim. "Airline Booking Limit Competition Game Under Differentiated Fare Structure." Journal of Applied Business Research (JABR) 33, no. 3 (April 28, 2017): 615–22. http://dx.doi.org/10.19030/jabr.v33i3.9950.

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We address a two-firm booking limit competition game in the airline industry. We assume aggregate common demand, and differentiated ticket fare and capacity, to make this study more realistic. A game theoretic approach is used to analyze the competition game. The optimal booking limits and the best response functions are derived. We show the existence of a pure Nash equilibrium and provide the closed-form equilibrium solution. The location of the Nash equilibrium depends on the relative magnitude of the ratios of the full and discount fares. We also show that the sum of the booking limits of the two firms remains the same regardless of the initial allocation proportion of the demand.

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31

Fang, Lidong, Apala Majumdar, and Lei Zhang. "Surface, size and topological effects for some nematic equilibria on rectangular domains." Mathematics and Mechanics of Solids 25, no. 5 (February 26, 2020): 1101–23. http://dx.doi.org/10.1177/1081286520902507.

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We study nematic equilibria on rectangular domains, in a reduced two-dimensional Landau–de Gennes framework. These reduced equilibria carry over to the three-dimensional framework at a special temperature. There is one essential model variable, [Formula: see text], which is a geometry-dependent and material-dependent variable. We compute the limiting profiles exactly in two distinguished limits: the [Formula: see text] 0 limit relevant for macroscopic domains and the [Formula: see text] limit relevant for nanoscale domains. The limiting profile has line defects near the shorter edges in the [Formula: see text] limit, whereas we observe fractional point defects in the [Formula: see text] 0 limit. The analytical studies are complemented by some bifurcation diagrams for these reduced equilibria as a function of [Formula: see text] and the rectangular aspect ratio. We also introduce the concept of ‘non-trivial’ topologies and study the relaxation of non-trivial topologies to trivial topologies mediated via point and line defects, with potential consequences for non-equilibrium phenomena and switching dynamics.

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32

Makarenkov, Oleg, and Lakmi Niwanthi Wadippuli Achchige. "Bifurcations of Finite-Time Stable Limit Cycles from Focus Boundary Equilibria in Impacting Systems, Filippov Systems, and Sweeping Processes." International Journal of Bifurcation and Chaos 28, no. 10 (September 2018): 1850126. http://dx.doi.org/10.1142/s0218127418501262.

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We establish a theorem on bifurcation of limit cycles from a focus boundary equilibrium of an impacting system, which is universally applicable to prove the bifurcation of limit cycles from focus boundary equilibria in other types of piecewise-smooth systems, such as Filippov systems and sweeping processes. Specifically, we assume that one of the subsystems of the piecewise-smooth system under consideration admits a focus equilibrium that lie on the switching manifold at the bifurcation value of the parameter. In each of the three cases, we derive a linearized system which is capable of concluding the occurrence of a finite-time stable limit cycle from the above-mentioned focus equilibrium when the parameter crosses the bifurcation value. Examples illustrate how conditions of our theorems lead to closed-form formulas for the coefficients of the linearized system.

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33

Liu, Weiping, Lina Hu, Yongxuan Yang, and Mingfu Fu. "Limit Support Pressure of Tunnel Face in Multi-Layer Soils Below River Considering Water Pressure." Open Geosciences 10, no. 1 (December 31, 2018): 932–39. http://dx.doi.org/10.1515/geo-2018-0074.

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AbstractThis paper presents a method to determine the limit support pressure of tunnel face in multi-layer soils below river considering the water pressure. The proposed method is based on the 3D Terzaghi earth pressure theory and the wedge theory considering the water pressure. The limit support pressures are investigated using the limit equilibrium method and compared to those calculated using a numerical method, such as FLAC3D. Four cases focusing different combinations of three layers are analyzed. The results obtained by the numerical method agree well with the predictions of the proposed limit equilibrium method. The limit support pressure obtained using the limit equilibrium method is greater than that obtained by the numerical method. The limit equilibrium method is safe and conservative in obtaining the limit support pressure. The proposed limit equilibrium method is expected to be easily adaptable and to enhance the reliability of tunnel design and construction in multi-layer soils below river.

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34

Kuang, Lin, Ai Zhong Lv, and Yu Zhou. "Slope Stability Analysis of Elastic Limit Equilibrium Method." Applied Mechanics and Materials 275-277 (January 2013): 1423–26. http://dx.doi.org/10.4028/www.scientific.net/amm.275-277.1423.

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Based on finite element analysis software ANSYS, slope stability analysis is carried out by Elastic limiting equilibrium method proposed in this paper. A series of sliding surface of the slope can be assumed firstly, and then stress field along the sliding surface is analyzed as the slope is in elastic state. The normal and tangential stresses along each sliding surface can be obtained, respectively. Then the safety factor for each slip surface can be calculated, the slip surface which the safety factor is smallest is the most dangerous sliding surface. This method is different from the previous limit equilibrium method. For the previous limit equilibrium method, the normal and tangential stresses along the sliding surface are calculated based on many assumptions. While, the limit equilibrium method proposed in this paper has fewer assumptions and clear physical meaning.

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35

Albouy, Alain, and Yanning Fu. "Relative equilibria of four identical satellites." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2109 (June 10, 2009): 2633–45. http://dx.doi.org/10.1098/rspa.2009.0115.

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We consider the Newtonian 5-body problem in the plane, where four bodies have the same mass m , which is small compared with the mass M of the remaining body. We consider the (normalized) relative equilibria in this system and follow them to the limit when m / M →0. In some cases, two small bodies will coalesce at the limit. We call the other equilibria the relative equilibria of four separate identical satellites. We prove rigorously that there are only three such equilibria, all already known after the numerical researches by H. Salo and C. F. Yoder. Our main contribution is to prove that any equilibrium configuration possesses a symmetry, a statement indicated by J. Llibre as the missing key to proving that there is no other equilibrium.

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36

Huang, Chuan Zhi, Yong Hua Cao, and Wan He Sun. "Generalized Limit Equilibrium Method for Slope Stability Analysis." Applied Mechanics and Materials 170-173 (May 2012): 557–68. http://dx.doi.org/10.4028/www.scientific.net/amm.170-173.557.

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On the basis of the limit equilibrium method and the physical significance of Coulomb’s yield criteria, extremum conditions of yield functions is established, which will be the fundamental equations for the limit analysis of soil mass. Once the stress equation along a sliding surface is available, the normal stress on the sliding surface can be obtained, a new limit analysis method, generalized limit equilibrium method (GLEM), can be established. With the generalized limit equilibrium method, an analysis method to solve the problem of slope stability can be obtained without introducing any other assumptions or simplified conditions but the sliding surface. With this algorithm, any discretionally possible sliding surface can be trial calculated and the least value of the calculated results of different sliding surfaces is taken as the safety factor. As long as a selected sliding surface is close to the true sliding surface, the derived safety factor will be approximate to the genuine solution to a problem of slope stability.

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37

Chen, R. H., and J. L. Chameau. "Discussion: Three-dimensional limit equilibrium analysis of slopes." Géotechnique 35, no. 2 (June 1985): 215–16. http://dx.doi.org/10.1680/geot.1985.35.2.215.

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38

Gunderson, L. M., and A. Bhattacharjee. "A Model of Solar Equilibrium: The Hydrodynamic Limit." Astrophysical Journal 870, no. 1 (January 4, 2019): 47. http://dx.doi.org/10.3847/1538-4357/aad55f.

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39

Castagnino, Mario. "The equilibrium limit of the Casati–Prosen model." Physics Letters A 357, no. 2 (September 2006): 97–100. http://dx.doi.org/10.1016/j.physleta.2006.04.024.

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40

TAKI, Masakazu, and Takuo YAMAGAMI. "Limit Equilibrium Slope Stability Analysis Considering Progressive Failure." Landslides 34, no. 3 (1997): 34–40. http://dx.doi.org/10.3313/jls1964.34.3_34.

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41

YAMABE, Satoru, and Takuo YAMAGAMI. "Limit Equilibrium Slope Stability Assessment Considering Progressive Failure." Landslides 38, no. 3 (2001): 180–86. http://dx.doi.org/10.3313/jls1964.38.3_180.

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42

Prakash, K., and A. Sridharan. "Determination of liquid limit from equilibrium sediment volume." Géotechnique 52, no. 9 (November 2002): 693–96. http://dx.doi.org/10.1680/geot.2002.52.9.693.

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43

Klier, J., F. Schletterer, P. Leiderer, and V. Shikin. "Equilibrium helium film in the thick-film limit." Low Temperature Physics 29, no. 9 (September 2003): 716–19. http://dx.doi.org/10.1063/1.1614175.

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44

Haan, Marco, and Hans Maks. "Stackelberg and Cournot competition under equilibrium limit pricing." Journal of Economic Studies 23, no. 5/6 (December 1996): 110–27. http://dx.doi.org/10.1108/01443589610154090.

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45

Chowdhury, R. N., and S. Zhang. "Convergence aspect of limit equilibrium methods for slopes." Canadian Geotechnical Journal 27, no. 1 (February 1, 1990): 145–51. http://dx.doi.org/10.1139/t90-013.

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This note is concerned with the multiplicity of solutions for the factor of safety that may be obtained on the basis of the method of slices. Discontinuities in the function for the factor of safety are discussed and the reasons for false convergence in any iterative solution process are explored, with particular reference to the well-known Bishop simplified method (circular slip surfaces) and Janbu simplified or generalized method (slip surfaces of arbitrary shape). The note emphasizes that both the solution method and the method of searching for the critical slip surface must be considered in assessing the potential for numerical difficulties and false convergence. Direct search methods for optimization (e.g., the simplex reflection method) appear to be superior to the grid search or repeated trial methods in this respect. To avoid false convergence, the initially assumed value of factor of safety F0 should be greater than β1(=−tan α1 tan [Formula: see text]) where α1 and [Formula: see text] are respectively the base inclination and internal friction angle of the first slice near the toe of a slope, the slice with the largest negative reverse inclination. A value of F0 = 1 + β1, is recommended on the basis of experience. If there is no slice with a negative slope for any of the slip surfaces generated in the automatic, search process, then any positive value of F0 will lead to true convergence for F. It is necessary to emphasize that no slip surface needs to be rejected for computational reasons except for Sarma's methods and similarly no artificial changes need to be made to the value of [Formula: see text] except for Sarma's methods. Key words: slope stability, convergence, limit equilibrium, analysis, optimization, slip surfaces, geological discontinuity, simplex reflection technique.

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46

Morrison Jr., Ernest E., and Robert M. Ebeling. "Limit equilibrium computation of dynamic passive earth pressure." Canadian Geotechnical Journal 32, no. 3 (June 1, 1995): 481–87. http://dx.doi.org/10.1139/t95-050.

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Few solution techniques exist for the determination of pseudostatic dynamic passive earth pressures for cohesionless soils. The widely accepted Mononobe–Okabe equation can result in the computing of unconservative values if the wall interface friction angle is greater than half the soil internal friction angle. As an alternate solution, equilibrium equations were formulated assuming a log spiral failure surface, and a research computer program was written to calculate the dynamic passive earth pressure coefficient. The primary purpose of this paper is to present a comparison of results obtained using the Mononobe–Okabe equation with those obtained using the log spiral formulation. Key words : pseudostatic, dynamic, passive earth pressure.

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47

Mehta, Prateek, Patrick M. Barboun, Yannick Engelmann, David B. Go, Annemie Bogaerts, William F. Schneider, and Jason C. Hicks. "Plasma-Catalytic Ammonia Synthesis beyond the Equilibrium Limit." ACS Catalysis 10, no. 12 (May 18, 2020): 6726–34. http://dx.doi.org/10.1021/acscatal.0c00684.

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48

GREIDER, C. W. "Telomerase RNA Levels Limit the Telomere Length Equilibrium." Cold Spring Harbor Symposia on Quantitative Biology 71 (January 1, 2006): 225–29. http://dx.doi.org/10.1101/sqb.2006.71.063.

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49

LACHOWICZ, MIROSŁAW, and DARIUSZ WRZOSEK. "NONLOCAL BILINEAR EQUATIONS: EQUILIBRIUM SOLUTIONS AND DIFFUSIVE LIMIT." Mathematical Models and Methods in Applied Sciences 11, no. 08 (November 2001): 1393–409. http://dx.doi.org/10.1142/s0218202501001380.

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This paper deals with the qualitative analysis of a class of bilinear systems of equations describing the dynamics of individuals undergoing kinetic (stochastic) interactions. A corresponding evolution problem is formulated in terms of integro-differential (nonlocal) system of equations. A general existence theory is provided. Under the assumption of periodic boundary conditions and the interaction rates expressed in terms of convolution operators two classes of equilibrium solutions are distinguished. The first class contains only constant functions and the second one contains some nonconstant functions. In the scalar case (one equation) under suitable scaling, related to the shrinking of interaction range of each individual, the limit to the corresponding "macroscopic" equation is studied. The limiting equation turns out to be the (nonlinear) porous medium equation.

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Morozov, D. Kh, S. N. Vafin, and A. A. Mavrin. "Thermal Equilibrium and Density Limit in Tokamak-Reactor." Universal Journal of Physics and Application 9, no. 4 (August 2015): 201–5. http://dx.doi.org/10.13189/ujpa.2015.090405.

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