American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure
  • NHM Home
  • This Issue
  • Next Article
    Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model
June  2021, 16(2): 257-281. doi: 10.3934/nhm.2021006

Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations

Mayte Pérez-Llanos 1, , Juan Pablo Pinasco 2, and  Nicolas Saintier 2,

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, C/ Tarfia s/n, and Instituto de Matemáticas Antonio de Castro Brzezicki, Edificio Celestino Mutis, Avda. de la Reina Mercedes s/n, Universidad de Sevilla, Campus de Reina Mercedes, (41012) Sevilla, Spain
2. 

IMAS UBA-CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Av Cantilo s/n, Ciudad Universitaria, (1428) Buenos Aires, Argentina

Received  August 2020 Revised  November 2020 Published  June 2021 Early access  February 2021

Fund Project: This work was partially supported by Universidad de Buenos Aires under grants 20020150100154BA and 20020130100283BA, by ANPCyT PICT2012 0153 and PICT2014-1771, CONICET (Argentina) PIP 11220150100032CO and 5478/1438. J.P. Pinasco and N. Saintier are members of CONICET, Argentina. M. Pérez-Llanos thanks to Junta de Andalucía FQM-131, Spain

In this work we study the formation of consensus in homogeneous and heterogeneous populations, and the effect of attractiveness or fitness of the opinions. We derive the corresponding kinetic equations, analyze the long time behavior of their solutions, and characterize the consensus opinion.

Keywords: Opinion formation models, Boltzmann equation, grazing limit, non-local transport equation.
Mathematics Subject Classification: 91C20; 82B21; 60K35.
Citation: Mayte Pérez-Llanos, Juan Pablo Pinasco, Nicolas Saintier. Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations. Networks and Heterogeneous Media, 2021, 16 (2) : 257-281. doi: 10.3934/nhm.2021006
References:
[1]

G. Aletti, A. K. Naimzada and G. Naldi., Mathematics and physics applications in sociodynamics simulation: The case of opinion formation and diffusion, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Birkhäuser Boston, (2010), 203–221. doi: 10.1007/978-0-8176-4946-3_8.

[2]

G. Aletti, G. Naldi and G. Toscani, First-order continuous models of opinion formation, SIAM J. Appl. Math., 67 (2007), 837-853.  doi: 10.1137/060658679.

[3]

S. E. Asch, Opinions and social pressure, Scientific American, 193 (1955), 31-35.  doi: 10.1038/scientificamerican1155-31.

[4]

R. B. Ash, Real Analysis and Probability, Probability and Mathematical Statistics, Academic Press, New York-London, 1972.

[5]

P. Balenzuela, J. P. Pinasco and V. Semeshenko, The undecided have the key: Interaction driven opinion dynamics in a three state model, PLoS ONE, 10 (2016), e0139572, 1–21.

[6]

B. O. Baumgaertner, R. C. Tyson and S. M. Krone, Opinion strength influences the spatial dynamics of opinion formation, The Journal of Mathematical Sociology, 40 (2016), 207-218.  doi: 10.1080/0022250X.2016.1205049.

[7]

B. O. Baumgaertner, P. A. Fetros, R. C. Tyson and S. M. Krone, Spatial Opinion Dynamics and the Efects of Two Types of Mixing, Phys Rev E., 98 (2018), 022310.

[8]

N. Bellomo, Modeling Complex Living Systems A Kinetic Theory and Stochastic Game Approach, Birkhauser, 2008.

[9]

N. Bellomo, G. Ajmone Marsan and A. Tosin, Complex Systems and Society. Modeling and Simulation, SpringerBriefs in Mathematics, 2013. doi: 10.1007/978-1-4614-7242-1.

[10]

K. C. Border and C. D. Aliprantis, Infinite Dimensional Analysis - A Hitchhiker's Guide, 3rd Edition, Springer, 2006.

[11]

E. Burnstein and A. Vinokur, What a person thinks upon learning he has chosen differently from others: Nice evidence for the persuasive-arguments explanation of choice shifts, Journal of Experimental Social Psychology, 11 (1975), 412-426.  doi: 10.1016/0022-1031(75)90045-1.

[12]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[13]

R. B. Cialdini and M. R. Trost, Social influence: Social norms, conformity and compliance, The Handbook of Social Psychology, McGraw-Hill, (1998), 151–192.

[14]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[15]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98.  doi: 10.1142/S0219525900000078.

[16]

P. Embrechts and M. Hofert, A note on generalized inverse, Mathematical Methods of Operations Research, 77 (2013), 423-432.  doi: 10.1007/s00186-013-0436-7.

[17]

N. E. Friedkin and E. C. Johnsen, Social influence and opinions, Journal of Mathematical Sociology, 15 (1990), 193-206.  doi: 10.1080/0022250X.1990.9990069.

[18]

T. Fujimoto, A simple model of consensus formation, Okayama Economic Review, 31 (1999), 95-100. 

[19]

S. Galam, Sociophysics: A Physicist's Modeling of Psycho-Political Phenomena, , Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-2032-3.

[20]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002).

[21]

R. A. Holley and T. M. Liggett, Ergodic theorems for weakly interacting infinite systems and the voter model, The Annals of Probability, 3 (1975), 643-663.  doi: 10.1214/aop/1176996306.

[22]

C. La Rocca, L. A. Braunstein and F. Vázquez, The influence of persuasion in opinion formation and polarization, Europhys. Letters, 106 (2014), 40004. doi: 10.1209/0295-5075/106/40004.

[23]

B. Latané, The psychology of social impact, American Psychologist, 36 (1981), 343-356. 

[24]

J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838.  doi: 10.1142/S0129183107011789.

[25]

M. Mäs and A. Flache, Differentiation without distancing. Explaining bi-polarization of opinions without negative influence, PloS one, 8 (2013), e74516.

[26]

P. Milgrom and I. Segal, Envelope theorems for arbitrary choice sets, Econometrica, 70 (2002), 583-601.  doi: 10.1111/1468-0262.00296.

[27]

R. Ochrombel, Simulation of Sznajd sociophysics model with convincing single opinions, International Journal of Modern Physics C, 12 (2001), 1091-1092.  doi: 10.1142/S0129183101002346.

[28] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014. 
[29]

L. Pedraza, J. P. Pinasco and N. Saintier, Measure-valued opinion dynamics, M3AS: Mathematical Models and Methods in Applied Sciences, 30 (2020), 225-260.  doi: 10.1142/S0218202520500062.

[30]

M. Pérez-Llanos, J. P. Pinasco and N. Saintier, Opinion attractiveness and its effect in opinion formation models, Phys. A, 559 (2020), 125017, 9 pp. doi: 10.1016/j.physa.2020.125017.

[31]

M. Pŕez-Llanos, J. P. Pinasco, N. Saintier and A. Silva, Opinion formation models with heterogeneous persuasion and zealotry, SIAM Journal on Mathematical Analysis, 50 (2018), 4812-4837.  doi: 10.1137/17M1152784.

[32]

F. Vazquez, N. Saintier and J. P. Pinasco, The role of voting intention in public opinion polarization, Phys. Rev. E, 101 (2020), 012101, 13pp.

[33]

N. Saintier, J. P. Pinasco and F. Vazquez, A model for a phase transition between political mono-polarization and bi-polarization, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30 (2020), 063146, 17 pp. doi: 10.1063/5.0004996.

[34]

J. P. Pinasco, V. Semeshenko and P. Balenzuela, Modeling opinion dynamics: Theoretical analysis and continuous approximation, Chaos, Solitons & Fractals, 98 (2017), 210-215.  doi: 10.1016/j.chaos.2017.03.033.

[35]

F. Slanina and H. Lavicka, Analytical results for the Sznajd model of opinion formation, The European Physical Journal B, 35 (2003) 279–288. doi: 10.1140/epjb/e2003-00278-0.

[36] D. W. Stroock, Probability Theory, An Analytic View, Cambridge University Press, 1993. 
[37]

K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics - C, 11 (2000), 1157-1165.  doi: 10.1142/S0129183100000936.

[38]

G. Toscani, Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.

[39]

C. Villani, Topics in optimal transportation, Grad.Studies in Math., American Mathematical Soc., (2003). doi: 10.1090/gsm/058.

show all references

References:
[1]

G. Aletti, A. K. Naimzada and G. Naldi., Mathematics and physics applications in sociodynamics simulation: The case of opinion formation and diffusion, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Birkhäuser Boston, (2010), 203–221. doi: 10.1007/978-0-8176-4946-3_8.

[2]

G. Aletti, G. Naldi and G. Toscani, First-order continuous models of opinion formation, SIAM J. Appl. Math., 67 (2007), 837-853.  doi: 10.1137/060658679.

[3]

S. E. Asch, Opinions and social pressure, Scientific American, 193 (1955), 31-35.  doi: 10.1038/scientificamerican1155-31.

[4]

R. B. Ash, Real Analysis and Probability, Probability and Mathematical Statistics, Academic Press, New York-London, 1972.

[5]

P. Balenzuela, J. P. Pinasco and V. Semeshenko, The undecided have the key: Interaction driven opinion dynamics in a three state model, PLoS ONE, 10 (2016), e0139572, 1–21.

[6]

B. O. Baumgaertner, R. C. Tyson and S. M. Krone, Opinion strength influences the spatial dynamics of opinion formation, The Journal of Mathematical Sociology, 40 (2016), 207-218.  doi: 10.1080/0022250X.2016.1205049.

[7]

B. O. Baumgaertner, P. A. Fetros, R. C. Tyson and S. M. Krone, Spatial Opinion Dynamics and the Efects of Two Types of Mixing, Phys Rev E., 98 (2018), 022310.

[8]

N. Bellomo, Modeling Complex Living Systems A Kinetic Theory and Stochastic Game Approach, Birkhauser, 2008.

[9]

N. Bellomo, G. Ajmone Marsan and A. Tosin, Complex Systems and Society. Modeling and Simulation, SpringerBriefs in Mathematics, 2013. doi: 10.1007/978-1-4614-7242-1.

[10]

K. C. Border and C. D. Aliprantis, Infinite Dimensional Analysis - A Hitchhiker's Guide, 3rd Edition, Springer, 2006.

[11]

E. Burnstein and A. Vinokur, What a person thinks upon learning he has chosen differently from others: Nice evidence for the persuasive-arguments explanation of choice shifts, Journal of Experimental Social Psychology, 11 (1975), 412-426.  doi: 10.1016/0022-1031(75)90045-1.

[12]

J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.  doi: 10.1142/S0218202511005131.

[13]

R. B. Cialdini and M. R. Trost, Social influence: Social norms, conformity and compliance, The Handbook of Social Psychology, McGraw-Hill, (1998), 151–192.

[14]

F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842.

[15]

G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98.  doi: 10.1142/S0219525900000078.

[16]

P. Embrechts and M. Hofert, A note on generalized inverse, Mathematical Methods of Operations Research, 77 (2013), 423-432.  doi: 10.1007/s00186-013-0436-7.

[17]

N. E. Friedkin and E. C. Johnsen, Social influence and opinions, Journal of Mathematical Sociology, 15 (1990), 193-206.  doi: 10.1080/0022250X.1990.9990069.

[18]

T. Fujimoto, A simple model of consensus formation, Okayama Economic Review, 31 (1999), 95-100. 

[19]

S. Galam, Sociophysics: A Physicist's Modeling of Psycho-Political Phenomena, , Springer Science & Business Media, 2012. doi: 10.1007/978-1-4614-2032-3.

[20]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002).

[21]

R. A. Holley and T. M. Liggett, Ergodic theorems for weakly interacting infinite systems and the voter model, The Annals of Probability, 3 (1975), 643-663.  doi: 10.1214/aop/1176996306.

[22]

C. La Rocca, L. A. Braunstein and F. Vázquez, The influence of persuasion in opinion formation and polarization, Europhys. Letters, 106 (2014), 40004. doi: 10.1209/0295-5075/106/40004.

[23]

B. Latané, The psychology of social impact, American Psychologist, 36 (1981), 343-356. 

[24]

J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838.  doi: 10.1142/S0129183107011789.

[25]

M. Mäs and A. Flache, Differentiation without distancing. Explaining bi-polarization of opinions without negative influence, PloS one, 8 (2013), e74516.

[26]

P. Milgrom and I. Segal, Envelope theorems for arbitrary choice sets, Econometrica, 70 (2002), 583-601.  doi: 10.1111/1468-0262.00296.

[27]

R. Ochrombel, Simulation of Sznajd sociophysics model with convincing single opinions, International Journal of Modern Physics C, 12 (2001), 1091-1092.  doi: 10.1142/S0129183101002346.

[28] L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, Oxford, 2014. 
[29]

L. Pedraza, J. P. Pinasco and N. Saintier, Measure-valued opinion dynamics, M3AS: Mathematical Models and Methods in Applied Sciences, 30 (2020), 225-260.  doi: 10.1142/S0218202520500062.

[30]

M. Pérez-Llanos, J. P. Pinasco and N. Saintier, Opinion attractiveness and its effect in opinion formation models, Phys. A, 559 (2020), 125017, 9 pp. doi: 10.1016/j.physa.2020.125017.

[31]

M. Pŕez-Llanos, J. P. Pinasco, N. Saintier and A. Silva, Opinion formation models with heterogeneous persuasion and zealotry, SIAM Journal on Mathematical Analysis, 50 (2018), 4812-4837.  doi: 10.1137/17M1152784.

[32]

F. Vazquez, N. Saintier and J. P. Pinasco, The role of voting intention in public opinion polarization, Phys. Rev. E, 101 (2020), 012101, 13pp.

[33]

N. Saintier, J. P. Pinasco and F. Vazquez, A model for a phase transition between political mono-polarization and bi-polarization, Chaos: An Interdisciplinary Journal of Nonlinear Science, 30 (2020), 063146, 17 pp. doi: 10.1063/5.0004996.

[34]

J. P. Pinasco, V. Semeshenko and P. Balenzuela, Modeling opinion dynamics: Theoretical analysis and continuous approximation, Chaos, Solitons & Fractals, 98 (2017), 210-215.  doi: 10.1016/j.chaos.2017.03.033.

[35]

F. Slanina and H. Lavicka, Analytical results for the Sznajd model of opinion formation, The European Physical Journal B, 35 (2003) 279–288. doi: 10.1140/epjb/e2003-00278-0.

[36] D. W. Stroock, Probability Theory, An Analytic View, Cambridge University Press, 1993. 
[37]

K. Sznajd-Weron and J. Sznajd, Opinion evolution in closed community, International Journal of Modern Physics - C, 11 (2000), 1157-1165.  doi: 10.1142/S0129183100000936.

[38]

G. Toscani, Kinetic models of opinion formation, Communications in Mathematical Sciences, 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1.

[39]

C. Villani, Topics in optimal transportation, Grad.Studies in Math., American Mathematical Soc., (2003). doi: 10.1090/gsm/058.

Figure 1.  Evolution of the opinion of 10 agents (blue) from a homogeneous population of $ N = 1000 $ agents interacting according the interaction rule (16) studied in this paper (left) or the interaction rule (22) considered in [2,38] (right). The red dashed line indicates the theoretical limit opinion in both cases ($ m_\infty\approx -0.35 $ for interaction rule (16), (left), and $ \tilde m_\infty\approx 0.41 $ for interaction rule (22), (right)) The early evolution is shown in inset
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Evolution of $ \ln(\max\,w-\min\,w) $, where the $ \max $ and $ \min $ are taken on the support of the distribution of opinions, for interaction rule (16) studied in this paper (left), and interaction rule (22) considered in [2,38] (right)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Evolution of the opinion of 10 agents (blue) from a population of $ N = 1000 $ agents with $ \alpha_0 = 2\% $ (left) and $ \alpha_0 = 60\% $ (right) stubborn agents. The red dashed line indicates the theoretical limit opinion $ m_\infty = 1/2 $, and the blue dotted lines the opinion of the stubborn agents (half with opinion $ 1/4 $ and the other half with opinion $ 3/4 $). The early evolution is shown in inset
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Evolution of the distribution of $ (w,q) $ among the non-stubborn population during one simulation ($ w $ in the horizontal axis and $ q $ in the vertical axis) with $ \lambda(w) = (w-0.5)^2 + \varepsilon $, $ \varepsilon = 0.01 $. From left to right and top to bottom, figures show the distribution of $ (w,q) $ at time $ 1,500,1000,1500,3000,5000,10000,15000,30000 $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35
[2]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203
[3]

Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115
[4]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037
[5]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159
[6]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic and Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395
[7]

Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029
[8]

Tao Wang. Global dynamics of a non-local delayed differential equation in the half plane. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2475-2492. doi: 10.3934/cpaa.2014.13.2475
[9]

Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701
[10]

Zhaoquan Xu, Chufen Wu. Spreading speeds for a class of non-local convolution differential equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4479-4492. doi: 10.3934/dcdsb.2020108
[11]

Valeriano Comincioli, Lucia Della Croce, Giuseppe Toscani. A Boltzmann-like equation for choice formation. Kinetic and Related Models, 2009, 2 (1) : 135-149. doi: 10.3934/krm.2009.2.135
[12]

Yanan Li, Alexandre N. Carvalho, Tito L. M. Luna, Estefani M. Moreira. A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5181-5196. doi: 10.3934/cpaa.2020232
[13]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic and Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205
[14]

Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789
[15]

Hirotada Honda. Global-in-time solution and stability of Kuramoto-Sakaguchi equation under non-local Coupling. Networks and Heterogeneous Media, 2017, 12 (1) : 25-57. doi: 10.3934/nhm.2017002
[16]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[17]

Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935
[18]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032
[19]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations and Control Theory, 2022, 11 (1) : 301-324. doi: 10.3934/eect.2021014
[20]

Felisia Angela Chiarello, Paola Goatin. Non-local multi-class traffic flow models. Networks and Heterogeneous Media, 2019, 14 (2) : 371-387. doi: 10.3934/nhm.2019015

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (195)
  • HTML views (243)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy