doi: 10.3934/cpaa.2021090

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

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  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.

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 & Applied Analysis, doi: 10.3934/cpaa.2021090
References:
[1]

S. AlhowaityF. 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.  Google Scholar

[2]

J. AndradeN. DávilaE. 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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar
[7]

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

[8]

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

[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.  Google Scholar

[10]

F. DiacuR. MartínezE. 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.  Google Scholar

[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.  Google Scholar

[12]

F. DiacuE. 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.  Google Scholar

[13]

F. DiacuE. 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.  Google Scholar

[14]

F. DiacuE. 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.  Google Scholar

[15]

F. DiacuE. 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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

L. C. García-NaranjoJ. C. MarreroE. 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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. AlhowaityF. 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.  Google Scholar

[2]

J. AndradeN. DávilaE. 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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar
[7]

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

[8]

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

[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.  Google Scholar

[10]

F. DiacuR. MartínezE. 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.  Google Scholar

[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.  Google Scholar

[12]

F. DiacuE. 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.  Google Scholar

[13]

F. DiacuE. 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.  Google Scholar

[14]

F. DiacuE. 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.  Google Scholar

[15]

F. DiacuE. 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.  Google Scholar

[16]

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

[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.  Google Scholar

[18]

L. C. García-NaranjoJ. C. MarreroE. 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.  Google Scholar

[19]

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

[20]

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

[21]

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

[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.  Google Scholar

[23]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Phase space for $ y(t),\dot{y}(t) $ ($ x(t) = 0 $)
Figure 2.  The vertical case
Figure 3.  Phase portrait in non-regularized and regularized coordinates
Figure 4.  The geodesic case
Figure 5.  The geodesic case, with $ M $ between $ \mu $ and $ m_1 $
Figure 6.  Phase portrait in n non-regularized and regularized coordinates
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 8.  An orbit in the $ xy $-plane and its corresponding in the regularized plane
Figure 9.  An orbit in the $ xy $-plane and its corresponding in the regularized plane
Figure 10.  We plot the same constrained graphic as Figure 9b where we can see the positions of the centers plotted
[1]

Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete & 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]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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 & Management Optimization, 2020  doi: 10.3934/jimo.2020135

[11]

Jin-Yun Guo, Cong Xiao, Xiaojian Lu. On $ n $-slice algebras and related algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021009

[12]

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

[13]

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 & Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062

[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]

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 & Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056

[16]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228

[17]

Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094

[18]

Jianqin Zhou, Wanquan Liu, Xifeng Wang, Guanglu Zhou. On the $ k $-error linear complexity for $ p^n $-periodic binary sequences via hypercube theory. Mathematical Foundations of Computing, 2019, 2 (4) : 279-297. doi: 10.3934/mfc.2019018

[19]

Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009

[20]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (32)
  • HTML views (33)
  • Cited by (0)

[Back to Top]