American Institute of Mathematical Sciences

Advanced
September  2021, 20(9): 2941-2963. doi: 10.3934/cpaa.2021090

The curved symmetric $ 2 $– and $ 3 $–center problem on constant negative surfaces

Sawsan Alhowaity 1, , Ernesto Pérez-Chavela 2, and  Juan Manuel Sánchez-Cerritos 3,4,,

1. 

Department of Mathematics, Shaqra University, Saudi Arabia
2. 

Department of Mathematics, Instituto Tecnológico Autónomo de México, Mexico City, Mexico
3. 

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, China
4. 

Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing, China

* Corresponding author

Received  May 2020 Revised  April 2021 Published  September 2021 Early access  June 2021

Fund Project: The second author has been partially supported by Conacyt-México project A1-S-10112 and Asociación Mexicana de Cultura A.C.

We study the motion of the negative curved symmetric two and three center problem on the Poincaré upper semi plane model for a surface of constant negative curvature $ \kappa $, which without loss of generality we assume $ \kappa = -1 $. Using this model, we first derive the equations of motion for the $ 2 $-and $ 3 $-center problems. We prove that for $ 2 $–center problem, there exists a unique equilibrium point and we study the dynamics around it. For the motion restricted to the invariant $ y $–axis, we prove that it is a center, but for the general two center problem it is unstable. For the $ 3 $–center problem, we show the non-existence of equilibrium points. We study two particular integrable cases, first when the motion of the free particle is restricted to the $ y $–axis, and second when all particles are along the same geodesic. We classify the singularities of the problem and introduce a local and a global regularization of all them. We show some numerical simulations for each situation.

Keywords: Negative curved $ N- $body problem, $ 2 $ and $ 3 $-center problem, local and global regularization.
Mathematics Subject Classification: Primary: 37C83, 34C14; Secondary: 70F10, 70F15.
Citation: Sawsan Alhowaity, Ernesto Pérez-Chavela, Juan Manuel Sánchez-Cerritos. The curved symmetric $ 2 $– and $ 3 $–center problem on constant negative surfaces. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2941-2963. doi: 10.3934/cpaa.2021090
References:
[1]

S. Alhowaity, F. Diacu and E. Pérez-Chavela, Relative equilibria in curved restricted 4-body problems, Can. Math. Bull., 61 (2018), 673-687.  doi: 10.4153/CMB-2018-019-9.

[2]

J. Andrade, N. Dávila, E. Pérez-Chavela and C. Vidal, Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature, Can. J. Math., 69 (2017), 961-991.  doi: 10.4153/CJM-2016-014-5.

[3]

S. V. Bolotin and P. Negrini, Chaotic behavior in the 3-center problem, J. Differ. Equ., 190 (2003), 539-558. doi: 10.1016/S0022-0396(03)00024-X.

[4]

W. Bolyai and J. Bolyai, Geometrische Untersuchungen, Teubner, Leipzig-Berlin, 1913.

[5]

F. Diacu, On the singularities of the curved $N$-body problem, Trans. Amer. Math. Soc., 363 (2011), 2249-2264.  doi: 10.1090/S0002-9947-2010-05251-1.

[6] F. Diacu, Relative equilibria of the curved $N$-body problem, in Atlantis Studies in Dynamical Systems, vol. 1, Atlantis Press, Amsterdam, 2012.  doi: 10.2991/978-94-91216-68-8.
[7]

F. Diacu, Relative equilibria of the 3-dimensional curved $n$-body problem, Memoirs Amer. Math. Soc., 228 (2013), 1071.

[8]

F. Diacu, The curved $N$-body problem: risks and rewards, Math. Intell., 35 (2013), 24-33.  doi: 10.1007/s00283-013-9397-1.

[9]

F. Diacu and S. Kordlou, Rotopulsators of the curved $N$-body problem, J. Differ. Equ., 255 (2013), 2709-2750.  doi: 10.1016/j.jde.2013.07.009.

[10]

F. Diacu, R. Martínez, E. Pérez-Chavela and C. Simó, On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, Physica D, 256 (2013), 21-35.  doi: 10.1016/j.physd.2013.04.007.

[11]

F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved $3$-body problem, J. Differ. Equ., 250 (2011), 340-366.  doi: 10.1016/j.jde.2010.08.011.

[12]

F. Diacu, E. Pérez-Chavela and M. Santoprete, Saari's conjecture for the collinear $N$-body problem, Trans. Amer. Math. Soc., 357 (2005), 4215-4223.  doi: 10.1090/S0002-9947-04-03606-2.

[13]

F. Diacu, E. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part I: Relative equilibria, J. Nonlinear Sci., 22 (2012), 247-266.  doi: 10.1007/s00332-011-9116-z.

[14]

F. Diacu, E. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part II: Singularities, J. Nonlinear Sci., 22 (2012), 267-275.  doi: 10.1007/s00332-011-9117-y.

[15]

F. Diacu, E. Pérez-Chavela and G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The negative curvature case, J. Differ. Equ., 252 (2012), 4529-4562.  doi: 10.1016/j.jde.2012.01.002.

[16]

F. Diacu and §. Popa, All Lagrangian relative equilibria have equal masses, J. Math. Phys., 55 (2014), 112701. doi: 10.1063/1.4900833.

[17]

F. Diacu and B. Thorn, Rectangular orbits of the curved 4-body problem, Proc. Amer. Math. Soc., 143 (2015), 1583-1593.  doi: 10.1090/S0002-9939-2014-12326-4.

[18]

L. C. García-Naranjo, J. C. Marrero, E. Pérez-Chavela and M. Rodríguez-Olmos, Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2, J. Differ. Equ., 260 (2016), 6375-6404.  doi: 10.1016/j.jde.2015.12.044.

[19]

W. Killing, Die Rechnung in den nichteuklidischen Raumformen, J. Reine Angew. Math., 89 (1880), 265-287.  doi: 10.1515/crll.1880.89.265.

[20]

W. Killing, Die Mechanik in den nichteuklidischen Raumformen, J. Reine Angew. Math., 98 (1885), 1-48.  doi: 10.1515/crll.1885.98.1.

[21]

W. Killing, Die Nicht-Euklidischen Raumformen in Analytischer Behandlung, Teubner, Leipzig, 1885.

[22]

V.V. Kozlov and A. O. Harin, Kepler's problem in constant curvature spaces, Celestial Mech. Dynam. Astronom, 54 (1992), 393-399.  doi: 10.1007/BF00049149.

[23]

H. Kragh, Is space Flat? Nineteenth century astronomy and non-Euclidean geometry, J. Astr. Hist. Heritage, 15 (2012), 149-158. 

[24]

R. Lipschitz, Extension of the planet-problem to a space of $n$ dimensions and constant integral curvature, Quart. J. Pure Appl. Math., 12 (1873), 349-370. 

[25]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels [in Russian], 1835-1838, in Collected Works, vol. 2, GITTL, Moscow, 1949.

[26]

R. Martínez and C. Simó, On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb S^2$, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013) 1157–1175. doi: 10.3934/dcds. 2013.33.1157.

[27]

R. Martínez and C. Simó, Relative equilibria of the restricted 3-body problem in curved spaces, Celes.t Mech. Dyn. Ast.r, 128 (2017), 221-259.  doi: 10.1007/s10569-016-9750-8.

[28]

E. Pérez-Chavela and J. G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The positive curvature case, Trans. Amer. Math. Soc., 364 (2012), 3805-3827.  doi: 10.1090/S0002-9947-2012-05563-2.

[29]

E. Schering, Die Schwerkraft im Gaussischen Räume, Nachr. Königl. Ges. Wiss. Gött., 15, (1870), 311–321.

[30]

E. Schering, Die Schwerkraft in mehrfach ausgedehnten Gaussischen und Riemmanschen Räumen, Nachr. Königl. Ges. Wiss. Gött., 6 (1873), 149–159.

[31]

A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A: Math. Gen., 39, (2006), 5787-5806; corrected version at math. DS/0601382. doi: 10.1088/0305-4470/39/20/011.

[32]

P. Tibboel, Polygonal homographic orbits in spaces of constant curvature, Proc. Amer. Math. Soc., 141 (2013), 1465-1471.  doi: 10.1090/S0002-9939-2012-11410-8.

[33]

P. Tibboel, Existence of a class of rotopulsators, J. Math. Anal. Appl., 404 (2013), 185-191.  doi: 10.1016/j.jmaa.2013.02.066.

[34]

P. Tibboel, Existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature, J. Math. Anal. Appl., 416 (2014), 205-211.  doi: 10.1016/j.jmaa.2014.02.036.

[35]

T. Vozmischeva, Integrable problems of celestial mechanics in spaces of constant curvature, in Astrophysics and Space Science Library, Volume 295, Springer, 2003. doi: 10.1007/978-94-017-0303-1.

[36]

S. Zhu, Eulerian relative equilibria of the curved 3-body problems in $\mathbb S^2$, Proc. Amer. Math. Soc., 142 (2014), 2837-2848.  doi: 10.1090/S0002-9939-2014-11995-2.

show all references

References:
[1]

S. Alhowaity, F. Diacu and E. Pérez-Chavela, Relative equilibria in curved restricted 4-body problems, Can. Math. Bull., 61 (2018), 673-687.  doi: 10.4153/CMB-2018-019-9.

[2]

J. Andrade, N. Dávila, E. Pérez-Chavela and C. Vidal, Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature, Can. J. Math., 69 (2017), 961-991.  doi: 10.4153/CJM-2016-014-5.

[3]

S. V. Bolotin and P. Negrini, Chaotic behavior in the 3-center problem, J. Differ. Equ., 190 (2003), 539-558. doi: 10.1016/S0022-0396(03)00024-X.

[4]

W. Bolyai and J. Bolyai, Geometrische Untersuchungen, Teubner, Leipzig-Berlin, 1913.

[5]

F. Diacu, On the singularities of the curved $N$-body problem, Trans. Amer. Math. Soc., 363 (2011), 2249-2264.  doi: 10.1090/S0002-9947-2010-05251-1.

[6] F. Diacu, Relative equilibria of the curved $N$-body problem, in Atlantis Studies in Dynamical Systems, vol. 1, Atlantis Press, Amsterdam, 2012.  doi: 10.2991/978-94-91216-68-8.
[7]

F. Diacu, Relative equilibria of the 3-dimensional curved $n$-body problem, Memoirs Amer. Math. Soc., 228 (2013), 1071.

[8]

F. Diacu, The curved $N$-body problem: risks and rewards, Math. Intell., 35 (2013), 24-33.  doi: 10.1007/s00283-013-9397-1.

[9]

F. Diacu and S. Kordlou, Rotopulsators of the curved $N$-body problem, J. Differ. Equ., 255 (2013), 2709-2750.  doi: 10.1016/j.jde.2013.07.009.

[10]

F. Diacu, R. Martínez, E. Pérez-Chavela and C. Simó, On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, Physica D, 256 (2013), 21-35.  doi: 10.1016/j.physd.2013.04.007.

[11]

F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved $3$-body problem, J. Differ. Equ., 250 (2011), 340-366.  doi: 10.1016/j.jde.2010.08.011.

[12]

F. Diacu, E. Pérez-Chavela and M. Santoprete, Saari's conjecture for the collinear $N$-body problem, Trans. Amer. Math. Soc., 357 (2005), 4215-4223.  doi: 10.1090/S0002-9947-04-03606-2.

[13]

F. Diacu, E. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part I: Relative equilibria, J. Nonlinear Sci., 22 (2012), 247-266.  doi: 10.1007/s00332-011-9116-z.

[14]

F. Diacu, E. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part II: Singularities, J. Nonlinear Sci., 22 (2012), 267-275.  doi: 10.1007/s00332-011-9117-y.

[15]

F. Diacu, E. Pérez-Chavela and G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The negative curvature case, J. Differ. Equ., 252 (2012), 4529-4562.  doi: 10.1016/j.jde.2012.01.002.

[16]

F. Diacu and §. Popa, All Lagrangian relative equilibria have equal masses, J. Math. Phys., 55 (2014), 112701. doi: 10.1063/1.4900833.

[17]

F. Diacu and B. Thorn, Rectangular orbits of the curved 4-body problem, Proc. Amer. Math. Soc., 143 (2015), 1583-1593.  doi: 10.1090/S0002-9939-2014-12326-4.

[18]

L. C. García-Naranjo, J. C. Marrero, E. Pérez-Chavela and M. Rodríguez-Olmos, Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2, J. Differ. Equ., 260 (2016), 6375-6404.  doi: 10.1016/j.jde.2015.12.044.

[19]

W. Killing, Die Rechnung in den nichteuklidischen Raumformen, J. Reine Angew. Math., 89 (1880), 265-287.  doi: 10.1515/crll.1880.89.265.

[20]

W. Killing, Die Mechanik in den nichteuklidischen Raumformen, J. Reine Angew. Math., 98 (1885), 1-48.  doi: 10.1515/crll.1885.98.1.

[21]

W. Killing, Die Nicht-Euklidischen Raumformen in Analytischer Behandlung, Teubner, Leipzig, 1885.

[22]

V.V. Kozlov and A. O. Harin, Kepler's problem in constant curvature spaces, Celestial Mech. Dynam. Astronom, 54 (1992), 393-399.  doi: 10.1007/BF00049149.

[23]

H. Kragh, Is space Flat? Nineteenth century astronomy and non-Euclidean geometry, J. Astr. Hist. Heritage, 15 (2012), 149-158. 

[24]

R. Lipschitz, Extension of the planet-problem to a space of $n$ dimensions and constant integral curvature, Quart. J. Pure Appl. Math., 12 (1873), 349-370. 

[25]

N. I. Lobachevsky, The new foundations of geometry with full theory of parallels [in Russian], 1835-1838, in Collected Works, vol. 2, GITTL, Moscow, 1949.

[26]

R. Martínez and C. Simó, On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb S^2$, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013) 1157–1175. doi: 10.3934/dcds. 2013.33.1157.

[27]

R. Martínez and C. Simó, Relative equilibria of the restricted 3-body problem in curved spaces, Celes.t Mech. Dyn. Ast.r, 128 (2017), 221-259.  doi: 10.1007/s10569-016-9750-8.

[28]

E. Pérez-Chavela and J. G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The positive curvature case, Trans. Amer. Math. Soc., 364 (2012), 3805-3827.  doi: 10.1090/S0002-9947-2012-05563-2.

[29]

E. Schering, Die Schwerkraft im Gaussischen Räume, Nachr. Königl. Ges. Wiss. Gött., 15, (1870), 311–321.

[30]

E. Schering, Die Schwerkraft in mehrfach ausgedehnten Gaussischen und Riemmanschen Räumen, Nachr. Königl. Ges. Wiss. Gött., 6 (1873), 149–159.

[31]

A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A: Math. Gen., 39, (2006), 5787-5806; corrected version at math. DS/0601382. doi: 10.1088/0305-4470/39/20/011.

[32]

P. Tibboel, Polygonal homographic orbits in spaces of constant curvature, Proc. Amer. Math. Soc., 141 (2013), 1465-1471.  doi: 10.1090/S0002-9939-2012-11410-8.

[33]

P. Tibboel, Existence of a class of rotopulsators, J. Math. Anal. Appl., 404 (2013), 185-191.  doi: 10.1016/j.jmaa.2013.02.066.

[34]

P. Tibboel, Existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature, J. Math. Anal. Appl., 416 (2014), 205-211.  doi: 10.1016/j.jmaa.2014.02.036.

[35]

T. Vozmischeva, Integrable problems of celestial mechanics in spaces of constant curvature, in Astrophysics and Space Science Library, Volume 295, Springer, 2003. doi: 10.1007/978-94-017-0303-1.

[36]

S. Zhu, Eulerian relative equilibria of the curved 3-body problems in $\mathbb S^2$, Proc. Amer. Math. Soc., 142 (2014), 2837-2848.  doi: 10.1090/S0002-9939-2014-11995-2.

Figure 1.  Phase space for $ y(t),\dot{y}(t) $ ($ x(t) = 0 $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The vertical case
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Phase portrait in non-regularized and regularized coordinates
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The geodesic case
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The geodesic case, with $ M $ between $ \mu $ and $ m_1 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Phase portrait in n non-regularized and regularized coordinates
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Shaped area corresponds to the points $ ( \xi, \eta) $ with $ \cos \xi \cosh \eta<\frac{B}{A} $ with $ \bar{y}_1 = 2 $. Region 1 is for $ x>0 $, $ y>0 $; and region 2 for $ x<0 $, $ y>0 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  An orbit in the $ xy $-plane and its corresponding in the regularized plane
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  An orbit in the $ xy $-plane and its corresponding in the regularized plane
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 10.  We plot the same constrained graphic as Figure 9b where we can see the positions of the centers plotted
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1441-1460. doi: 10.3934/dcds.2018059
[2]

Holger R. Dullin, Jürgen Scheurle. Symmetry reduction of the 3-body problem in $ \mathbb{R}^4 $. Journal of Geometric Mechanics, 2020, 12 (3) : 377-394. doi: 10.3934/jgm.2020011
[3]

Peili Li, Xiliang Lu, Yunhai Xiao. Smoothing Newton method for $ \ell^0 $-$ \ell^2 $ regularized linear inverse problem. Inverse Problems and Imaging, 2022, 16 (1) : 153-177. doi: 10.3934/ipi.2021044
[4]

Genni Fragnelli, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. Operators of order 2$ n $ with interior degeneracy. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3417-3426. doi: 10.3934/dcdss.2020128
[5]

Pak Tung Ho. Prescribing $ Q $-curvature on $ S^n $ in the presence of symmetry. Communications on Pure and Applied Analysis, 2020, 19 (2) : 715-722. doi: 10.3934/cpaa.2020033
[6]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067
[7]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306
[8]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129
[9]

Kailu Yang, Xiaomiao Wang, Menglong Zhang, Lidong Wang. Some progress on optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021012
[10]

Zijun Chen, Shengkun Wu. Local well-posedness for the Zakharov system in dimension $ d = 2, 3 $. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4307-4319. doi: 10.3934/cpaa.2021161
[11]

Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079
[12]

Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3603-3631. doi: 10.3934/jimo.2020135
[13]

Jin-Yun Guo, Cong Xiao, Xiaojian Lu. On $ n $-slice algebras and related algebras. Electronic Research Archive, 2021, 29 (4) : 2687-2718. doi: 10.3934/era.2021009
[14]

Carlos García-Azpeitia. Relative periodic solutions of the $ n $-vortex problem on the sphere. Journal of Geometric Mechanics, 2019, 11 (3) : 427-438. doi: 10.3934/jgm.2019021
[15]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[16]

Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062
[17]

Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $ k $-facility location problem with linear penalties. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056
[18]

Sascha Kurz, Ivan Landjev, Assia Rousseva. Classification of $ \mathbf{(3 \!\mod 5)} $ arcs in $ \mathbf{ \operatorname{PG}(3,5)} $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021066
[19]

Tian-Xiao He, Peter J.-S. Shiue. Identities for linear recursive sequences of order $ 2 $. Electronic Research Archive, 2021, 29 (5) : 3489-3507. doi: 10.3934/era.2021049
[20]

Liu Liu, Justyna Jarczyk, Witold Jarczyk, Weinian Zhang. Iterative roots of type $ \mathcal {T}_2 $. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022082

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (203)
  • HTML views (178)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy