American Institute of Mathematical Sciences

Advanced
March  2012, 32(3): 753-793. doi: 10.3934/dcds.2012.32.753

Instability for a priori unstable Hamiltonian systems: A dynamical approach

Abed Bounemoura 1, and  Edouard Pennamen 2,

1. 

IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460-320, Brazil
2. 

IMJ, 4 place Jussieu, 75252 Paris Cedex, France

Received  October 2010 Revised  May 2011 Published  October 2011

In this article, we consider an a priori unstable Hamiltonian system with three degrees of freedom, for which we construct a drifting solution with an optimal time of instability. Such a result has been already proved by Berti, Bolle and Biasco using variational arguments, and by Treschev using his separatrix map theory. Our approach is new: it is based on a special type of symbolic dynamics corresponding to the random iteration of a family of twist maps of the annulus, and it gives the first concrete application of this idea introduced by Moeckel in an abstract setting and further studied by Marco. Our method should also be useful in obtaining the optimal time of instability in the more difficult context of a priori stable Hamiltonian systems.
Keywords: Arnold diffusion., Hamiltonian systems, Perturbation theory.
Mathematics Subject Classification: Primary: 37J40; Secondary: 37D0.
Citation: Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753
References:
[1]

V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations, Russ. Math. Surv., 18 (1963), 9-36. doi: 10.1070/RM1963v018n05ABEH004130.  Google Scholar

[2]

V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.  Google Scholar

[3]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," [Dynamical Systems III], Transl. from the Russian original by E. Khukhro, Third edition, Encyclopedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006.  Google Scholar

[4]

M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time, J. Math. Pures Appl., 82 (2003), 613-664. doi: 10.1016/S0021-7824(03)00032-1.  Google Scholar

[5]

P. Bernard, Perturbation of a partially hyperbolic Hamiltonian system, [Perturbation d'un hamiltonien partiellement hyperbolique], C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 189-195.  Google Scholar

[6]

P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615–-669. doi: 10.1090/S0894-0347-08-00591-2.  Google Scholar

[7]

U. Bessi, An approach to Arnol'd's diffusion through the calculus of variations, Nonlinear Anal., 26 (1996), 1115-1135. doi: 10.1016/0362-546X(94)00270-R.  Google Scholar

[8]

A. Bounemoura and J.-P. Marco, Improved exponential stability for near-integrable quasi-convex Hamiltonians, Nonlinearity, 24 (2011), 97-112. doi: 10.1088/0951-7715/24/1/005.  Google Scholar

[9]

A. Bounemoura, Effective stability for Gevrey and finitely differentiable prevalent Hamiltonians, to appear in Communication in Mathematical Physics, 2011. Google Scholar

[10]

A. Bounemoura, Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians, Journal of Differential Equations, 249 (2010), 2905-2920. doi: 10.1016/j.jde.2010.06.004.  Google Scholar

[11]

L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. Henri Poincaré, Phys. Théor., 60 (1994), 144 pp.  Google Scholar

[12]

J. Cresson and C. Guillet, Periodic orbits and Arnold diffusion, Discrete Contin. Dyn. Syst., 9 (2003), 451-470.  Google Scholar

[13]

M. Chaperon, Stable manifolds and the Perron-Irwin method, Ergodic Theory Dynam. Systems, 24 (2004), 1359-1394. doi: 10.1017/S0143385703000701.  Google Scholar

[14]

C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, Journal of Differential Geometry, 67 (2004), 457-517.  Google Scholar

[15]

C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, Journal of Differential Geometry, 82 (2009), 229-277.  Google Scholar

[16]

A. Delshams, R. de la Llave and T. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Heuristics and rigorous verification on a model, Mem. Amer. Math. Soc., 179 (2006), viii+141 pp.  Google Scholar

[17]

M. R. Herman, Some open problems in dynamical systems, in "Proceedings of the International Congress of Mathematicians," Vol. II (Berlin, 1998), Doc. Math., 1998, Extra Vol. II, 797-808.  Google Scholar

[18]

M. Chaperon, The Lipschitzian core of some invariant manifold theorems, Ergodic Theory Dynam. Systems, 28 (2008), 1419-1441. doi: 10.1017/S0143385707000910.  Google Scholar

[19]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, (1977), ii+149 pp.  Google Scholar

[20]

V. Kaloshin, M. Levi and M. Saprykina, An example of nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension, preprint, 2010. Google Scholar

[21]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.  Google Scholar

[22]

V. Kaloshin, K. Zhang and Y. Zheng, "Almost Dense Orbit on Energy Surface," Proceedings of the XVI-th ICMP, Prague, (2009), 314-322. Google Scholar

[23]

P. Lochak and J.-P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems, Central European Journal of Mathematics, 3 (2005), 342-397. doi: 10.2478/BF02475913.  Google Scholar

[24]

P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems, Mem. Amer. Math. Soc., 163 (2003), viii+145 pp.  Google Scholar

[25]

P. Lochak, Arnold diffusion; a compendium of remarks and questions, in "Hamiltonian Systems with Three or More Degrees of Freedom" (S'Agaró, 1995), 168-183, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999.  Google Scholar

[26]

J.-P. Marco, Uniform lower bounds of the splitting for analytic symplectic systems, preprint, 2005. Google Scholar

[27]

J.-P. Marco, Models for skew-products and polysystems, C. R. Acad. Sci. Paris, 346, (2008), 203-208.  Google Scholar

[28]

J.-P. Marco, Arnold diffusion in a priori stable systems on $\A^3$, in preparation, 2010. Google Scholar

[29]

J.-P. Marco, Generic properties of classical systems on the torus $\T^2$, in preparation, 2010. Google Scholar

[30]

J.-P. Marco, Nets of hyperbolic annuli in generic nearly integrable systems on $\A^3$, in preparation, 2010. Google Scholar

[31]

J.-P. Marco, Skew-products and polysystems in the neighborhood of hyperbolic annuli, in preparation, 2010. Google Scholar

[32]

N. D. Mèzer, Arnol'd diffusion. I. Announcement of results, J. Math. Sci. (N. Y.), 124 (2004), 5275-5289. doi: 10.1023/B:JOTH.0000047353.78307.09.  Google Scholar

[33]

R. Moeckel, Generic drift on Cantor sets of annuli, in "Celestial Mechanics" (Evanston, IL, 1999), 163-171, Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002.  Google Scholar

[34]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., II (1962), 1-20.  Google Scholar

[35]

J. Moser, "Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics," Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, (1973), viii+198 pp.  Google Scholar

[36]

J.-P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems,, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 199-275.  Google Scholar

[37]

J.-P. Marco and D. Sauzin, Wandering domains and random walks in Gevrey near-integrable systems, Ergodic Theory Dynam. Systems, 24 (2004), 1619-1666. doi: 10.1017/S0143385703000786.  Google Scholar

[38]

N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5–-66.  Google Scholar

[39]

N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk., 5 (1979), 5-50.  Google Scholar

[40]

D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity, 17 (2004), 1803-1841. doi: 10.1088/0951-7715/17/5/014.  Google Scholar

[41]

J.-C. Yoccoz, Introduction to hyperbolic dynamics, in "Real and Complex Dynamical Systems" (Hillerød, 1993), 265-291, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995.  Google Scholar

[42]

K. Zhang, Speed of Arnold diffusion for analytic Hamiltonian systems, to appear in Inventiones Mathematicae, 2011. Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations, Russ. Math. Surv., 18 (1963), 9-36. doi: 10.1070/RM1963v018n05ABEH004130.  Google Scholar

[2]

V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12.  Google Scholar

[3]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," [Dynamical Systems III], Transl. from the Russian original by E. Khukhro, Third edition, Encyclopedia of Mathematical Sciences, 3, Springer-Verlag, Berlin, 2006.  Google Scholar

[4]

M. Berti, L. Biasco and P. Bolle, Drift in phase space: A new variational mechanism with optimal diffusion time, J. Math. Pures Appl., 82 (2003), 613-664. doi: 10.1016/S0021-7824(03)00032-1.  Google Scholar

[5]

P. Bernard, Perturbation of a partially hyperbolic Hamiltonian system, [Perturbation d'un hamiltonien partiellement hyperbolique], C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 189-195.  Google Scholar

[6]

P. Bernard, The dynamics of pseudographs in convex Hamiltonian systems, J. Amer. Math. Soc., 21 (2008), 615–-669. doi: 10.1090/S0894-0347-08-00591-2.  Google Scholar

[7]

U. Bessi, An approach to Arnol'd's diffusion through the calculus of variations, Nonlinear Anal., 26 (1996), 1115-1135. doi: 10.1016/0362-546X(94)00270-R.  Google Scholar

[8]

A. Bounemoura and J.-P. Marco, Improved exponential stability for near-integrable quasi-convex Hamiltonians, Nonlinearity, 24 (2011), 97-112. doi: 10.1088/0951-7715/24/1/005.  Google Scholar

[9]

A. Bounemoura, Effective stability for Gevrey and finitely differentiable prevalent Hamiltonians, to appear in Communication in Mathematical Physics, 2011. Google Scholar

[10]

A. Bounemoura, Nekhoroshev estimates for finitely differentiable quasi-convex Hamiltonians, Journal of Differential Equations, 249 (2010), 2905-2920. doi: 10.1016/j.jde.2010.06.004.  Google Scholar

[11]

L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. Henri Poincaré, Phys. Théor., 60 (1994), 144 pp.  Google Scholar

[12]

J. Cresson and C. Guillet, Periodic orbits and Arnold diffusion, Discrete Contin. Dyn. Syst., 9 (2003), 451-470.  Google Scholar

[13]

M. Chaperon, Stable manifolds and the Perron-Irwin method, Ergodic Theory Dynam. Systems, 24 (2004), 1359-1394. doi: 10.1017/S0143385703000701.  Google Scholar

[14]

C.-Q. Cheng and J. Yan, Existence of diffusion orbits in a priori unstable Hamiltonian systems, Journal of Differential Geometry, 67 (2004), 457-517.  Google Scholar

[15]

C.-Q. Cheng and J. Yan, Arnold diffusion in Hamiltonian systems: A priori unstable case, Journal of Differential Geometry, 82 (2009), 229-277.  Google Scholar

[16]

A. Delshams, R. de la Llave and T. Seara, A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Heuristics and rigorous verification on a model, Mem. Amer. Math. Soc., 179 (2006), viii+141 pp.  Google Scholar

[17]

M. R. Herman, Some open problems in dynamical systems, in "Proceedings of the International Congress of Mathematicians," Vol. II (Berlin, 1998), Doc. Math., 1998, Extra Vol. II, 797-808.  Google Scholar

[18]

M. Chaperon, The Lipschitzian core of some invariant manifold theorems, Ergodic Theory Dynam. Systems, 28 (2008), 1419-1441. doi: 10.1017/S0143385707000910.  Google Scholar

[19]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, (1977), ii+149 pp.  Google Scholar

[20]

V. Kaloshin, M. Levi and M. Saprykina, An example of nearly integrable Hamiltonian system with a trajectory dense in a set of maximal Hausdorff dimension, preprint, 2010. Google Scholar

[21]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.  Google Scholar

[22]

V. Kaloshin, K. Zhang and Y. Zheng, "Almost Dense Orbit on Energy Surface," Proceedings of the XVI-th ICMP, Prague, (2009), 314-322. Google Scholar

[23]

P. Lochak and J.-P. Marco, Diffusion times and stability exponents for nearly integrable analytic systems, Central European Journal of Mathematics, 3 (2005), 342-397. doi: 10.2478/BF02475913.  Google Scholar

[24]

P. Lochak, J.-P. Marco and D. Sauzin, On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems, Mem. Amer. Math. Soc., 163 (2003), viii+145 pp.  Google Scholar

[25]

P. Lochak, Arnold diffusion; a compendium of remarks and questions, in "Hamiltonian Systems with Three or More Degrees of Freedom" (S'Agaró, 1995), 168-183, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999.  Google Scholar

[26]

J.-P. Marco, Uniform lower bounds of the splitting for analytic symplectic systems, preprint, 2005. Google Scholar

[27]

J.-P. Marco, Models for skew-products and polysystems, C. R. Acad. Sci. Paris, 346, (2008), 203-208.  Google Scholar

[28]

J.-P. Marco, Arnold diffusion in a priori stable systems on $\A^3$, in preparation, 2010. Google Scholar

[29]

J.-P. Marco, Generic properties of classical systems on the torus $\T^2$, in preparation, 2010. Google Scholar

[30]

J.-P. Marco, Nets of hyperbolic annuli in generic nearly integrable systems on $\A^3$, in preparation, 2010. Google Scholar

[31]

J.-P. Marco, Skew-products and polysystems in the neighborhood of hyperbolic annuli, in preparation, 2010. Google Scholar

[32]

N. D. Mèzer, Arnol'd diffusion. I. Announcement of results, J. Math. Sci. (N. Y.), 124 (2004), 5275-5289. doi: 10.1023/B:JOTH.0000047353.78307.09.  Google Scholar

[33]

R. Moeckel, Generic drift on Cantor sets of annuli, in "Celestial Mechanics" (Evanston, IL, 1999), 163-171, Contemp. Math., 292, Amer. Math. Soc., Providence, RI, 2002.  Google Scholar

[34]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., II (1962), 1-20.  Google Scholar

[35]

J. Moser, "Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics," Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, (1973), viii+198 pp.  Google Scholar

[36]

J.-P. Marco and D. Sauzin, Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems,, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 199-275.  Google Scholar

[37]

J.-P. Marco and D. Sauzin, Wandering domains and random walks in Gevrey near-integrable systems, Ergodic Theory Dynam. Systems, 24 (2004), 1619-1666. doi: 10.1017/S0143385703000786.  Google Scholar

[38]

N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5–-66.  Google Scholar

[39]

N. N. Nehorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk., 5 (1979), 5-50.  Google Scholar

[40]

D. Treschev, Evolution of slow variables in a priori unstable Hamiltonian systems, Nonlinearity, 17 (2004), 1803-1841. doi: 10.1088/0951-7715/17/5/014.  Google Scholar

[41]

J.-C. Yoccoz, Introduction to hyperbolic dynamics, in "Real and Complex Dynamical Systems" (Hillerød, 1993), 265-291, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 464, Kluwer Acad. Publ., Dordrecht, 1995.  Google Scholar

[42]

K. Zhang, Speed of Arnold diffusion for analytic Hamiltonian systems, to appear in Inventiones Mathematicae, 2011. Google Scholar

[1]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61
[2]

Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 795-811. doi: 10.3934/dcds.2002.8.795
[3]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623
[4]

Claude Froeschlé, Massimiliano Guzzo, Elena Lega. First numerical evidence of global Arnold diffusion in quasi-integrable systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 687-698. doi: 10.3934/dcdsb.2005.5.687
[5]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[6]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684
[7]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[8]

Jacky Cresson. The transfer lemma for Graff tori and Arnold diffusion time. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 787-800. doi: 10.3934/dcds.2001.7.787
[9]

Massimiliano Berti. Some remarks on a variational approach to Arnold's diffusion. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 307-314. doi: 10.3934/dcds.1996.2.307
[10]

Amadeu Delshams, Rodrigo G. Schaefer. Arnold diffusion for a complete family of perturbations with two independent harmonics. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6047-6072. doi: 10.3934/dcds.2018261
[11]

John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations & Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001
[12]

Amadeu Delshams, Rafael de la Llave and Tere M. Seara. A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: Announcement of results. Electronic Research Announcements, 2003, 9: 125-134.

[13]

Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545
[14]

Kenneth R. Meyer, Jesús F. Palacián, Patricia Yanguas. Normally stable hamiltonian systems. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1201-1214. doi: 10.3934/dcds.2013.33.1201
[15]

Antonio Giorgilli. Unstable equilibria of Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 855-871. doi: 10.3934/dcds.2001.7.855
[16]

Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375
[17]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
[18]

Florian Dorsch, Hermann Schulz-Baldes. Random Möbius dynamics on the unit disc and perturbation theory for Lyapunov exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021076
[19]

Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2019-2034. doi: 10.3934/dcdss.2019130
[20]

Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences