American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms
  • DCDS-B Home
  • This Issue
  • Next Article
    Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico
doi: 10.3934/dcdsb.2019201

Takens–Bogdanov singularity for age structured models

Zhihua Liu and  Rong Yuan

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received  January 2019 Revised  March 2019 Published  September 2019

Fund Project: Research was partially supported by NSFC, the Fundamental Research Funds for the Central Universities, and Laboratory of Mathematics and Complex Systems, Ministry of Education

The main purpose of this article is to derive a easily feasible method for the determination of Takens–Bogdanov singularity in age structured models. We present a SIR epidemic model with age structure as an example to illustrate the theoretical results.

Keywords: Takens–Bogdanov singularity, age structured model, epidemic model.
Mathematics Subject Classification: 34K18, 35K90, 37L10, 37G10.
Citation: Zhihua Liu, Rong Yuan. Takens–Bogdanov singularity for age structured models. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019201
References:
[1]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Grundlehren der Mathematischen Wissenschaften, 250, Springer-Verlag, New York-Berlin, 1983. Google Scholar

[2]

R. I. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta Math. Soviet, 1 (1981), 373-388. Google Scholar

[3]

R. I. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Funct. Anal. i Priloežn, 9 (1975), 63. Google Scholar

[4]

J. Z. Cao and R. Yuan, Bogdanov-Takens bifurcation for neutral functional differential equations, Electronic Journal of Differential Equations, 2013 (2013), 12 pp. Google Scholar

[5]

S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982. Google Scholar

[6] S.-N. Chow, C. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639. Google Scholar
[7]

J. X. Chu, A. Ducrot, P. Magal and S. G. Ruan, Hopf bifurcation in a size structured population dynamic model with random growth, J. Differ. Equ., 247 (2009), 956-1000. doi: 10.1016/j.jde.2009.04.003. Google Scholar

[8]

J. X. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase, Discrete Contin. Dyn. Syst., 33 (2013), 4891-4921. doi: 10.3934/dcds.2013.33.4891. Google Scholar

[9]

J. X. Chu, P. Magal and R. Yuan, Hopf bifurcation for a maturity structured population dynamic model, J. Nonlinear Sci., 21 (2011), 521-562. doi: 10.1007/s00332-010-9091-9. Google Scholar

[10]

J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. Google Scholar

[11]

A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518. doi: 10.1016/j.jmaa.2007.09.074. Google Scholar

[12]

F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals, Lecture Notes in Math, 1480, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0098353. Google Scholar

[13]

T. Faria, Bifurcation aspects for some delayed population models with diffusion, in Differential Equations with Applications to Biology, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 21 (1999), 143–158. Google Scholar

[14]

T. Faria, Normal form and Hopf bifurcation for partial differential equations with delays, Transactions of the American Mathematical Society, 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7. Google Scholar

[15]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182. Google Scholar

[16]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2. Google Scholar

[17]

J. K. Hale, L. T. Magalh$\widetilde{a}$es and W. M. Oliva, Dynamics in Infinite Dimensions, Applied Math. Sciences, 47, Springer-Verlag, New York, 2002. doi: 10.1007/b100032. Google Scholar

[18]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. Math. Monographs C. N. R., Vol. 7, Giadini Editori e Stampatori, Pisa, 1994.Google Scholar

[19]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998. Google Scholar

[20]

W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162. Google Scholar

[21]

W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956. Google Scholar

[22]

Z. H. Liu and N. W. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonlinear Sci., 25 (2015), 937-957. doi: 10.1007/s00332-015-9245-x. Google Scholar

[23]

Z. H. Liu, P. Magal and S. G. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Zeitschrift fur Angewandte Mathematik und Physik, 62 (2011), 191-222. doi: 10.1007/s00033-010-0088-x. Google Scholar

[24]

Z. H. Liu, P. Magal and S. G. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations, 257 (2014), 921-1011. doi: 10.1016/j.jde.2014.04.018. Google Scholar

[25]

Z. H. Liu, P. Magal and S. G. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 537-555. doi: 10.3934/dcdsb.2016.21.537. Google Scholar

[26]

Z. H. Liu, P. Magal and D. M. Xiao, Bogdanov-Takens bifurcation in a predator-prey model, Zeitschrift fur Angewandte Mathematik und Physik, 67 (2016), Art. 137, 29 pp. doi: 10.1007/s00033-016-0724-1. Google Scholar

[27]

Z. H. Liu and R. Yuan, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Science China Mathematics, 60 (2017), 1371-1398. doi: 10.1007/s11425-016-0371-8. Google Scholar

[28]

Z. H. Liu and R. Yuan, The effect of diffusion for a predator-prey system with nonmonotonic functional response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 4309-4316. doi: 10.1142/S0218127404011867. Google Scholar

[29]

P. Magal and S. G. Ruan, Center manifolds for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7. Google Scholar

[30]

F. Takens, Forced oscillations and bifurcations, Comm. Math. Inst. Rijksuniv. Utrecht, Math. Inst. Rijksuniv. Utrecht, Utrecht, (1974), 1-59. Google Scholar

[31]

F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., (1974), 47-100. Google Scholar

[32]

H. Tang and Z. H. Liu, Hopf bifurcation for a predator-prey model with age structure, Applied Mathematical Modelling, 40 (2016), 726-737. doi: 10.1016/j.apm.2015.09.015. Google Scholar

[33]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. Google Scholar

[34]

H. R. Thieme, Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem, J. Evol. Equ., 8 (2008), 283-305. doi: 10.1007/s00028-007-0355-2. Google Scholar

[35]

H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447. doi: 10.1016/0022-247X(90)90074-P. Google Scholar

[36]

Z. Wang and Z. H. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150. doi: 10.1016/j.jmaa.2011.07.038. Google Scholar

[37]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985. Google Scholar

[38]

Y. X. Xu and M. Y. Huang, Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity, J. Differential Equations, 244 (2008), 582-598. doi: 10.1016/j.jde.2007.09.003. Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Grundlehren der Mathematischen Wissenschaften, 250, Springer-Verlag, New York-Berlin, 1983. Google Scholar

[2]

R. I. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta Math. Soviet, 1 (1981), 373-388. Google Scholar

[3]

R. I. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Funct. Anal. i Priloežn, 9 (1975), 63. Google Scholar

[4]

J. Z. Cao and R. Yuan, Bogdanov-Takens bifurcation for neutral functional differential equations, Electronic Journal of Differential Equations, 2013 (2013), 12 pp. Google Scholar

[5]

S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982. Google Scholar

[6] S.-N. Chow, C. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639. Google Scholar
[7]

J. X. Chu, A. Ducrot, P. Magal and S. G. Ruan, Hopf bifurcation in a size structured population dynamic model with random growth, J. Differ. Equ., 247 (2009), 956-1000. doi: 10.1016/j.jde.2009.04.003. Google Scholar

[8]

J. X. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase, Discrete Contin. Dyn. Syst., 33 (2013), 4891-4921. doi: 10.3934/dcds.2013.33.4891. Google Scholar

[9]

J. X. Chu, P. Magal and R. Yuan, Hopf bifurcation for a maturity structured population dynamic model, J. Nonlinear Sci., 21 (2011), 521-562. doi: 10.1007/s00332-010-9091-9. Google Scholar

[10]

J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. Google Scholar

[11]

A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518. doi: 10.1016/j.jmaa.2007.09.074. Google Scholar

[12]

F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals, Lecture Notes in Math, 1480, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0098353. Google Scholar

[13]

T. Faria, Bifurcation aspects for some delayed population models with diffusion, in Differential Equations with Applications to Biology, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 21 (1999), 143–158. Google Scholar

[14]

T. Faria, Normal form and Hopf bifurcation for partial differential equations with delays, Transactions of the American Mathematical Society, 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7. Google Scholar

[15]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463. doi: 10.1006/jmaa.2000.7182. Google Scholar

[16]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2. Google Scholar

[17]

J. K. Hale, L. T. Magalh$\widetilde{a}$es and W. M. Oliva, Dynamics in Infinite Dimensions, Applied Math. Sciences, 47, Springer-Verlag, New York, 2002. doi: 10.1007/b100032. Google Scholar

[18]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. Math. Monographs C. N. R., Vol. 7, Giadini Editori e Stampatori, Pisa, 1994.Google Scholar

[19]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998. Google Scholar

[20]

W. M. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162. Google Scholar

[21]

W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956. Google Scholar

[22]

Z. H. Liu and N. W. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonlinear Sci., 25 (2015), 937-957. doi: 10.1007/s00332-015-9245-x. Google Scholar

[23]

Z. H. Liu, P. Magal and S. G. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Zeitschrift fur Angewandte Mathematik und Physik, 62 (2011), 191-222. doi: 10.1007/s00033-010-0088-x. Google Scholar

[24]

Z. H. Liu, P. Magal and S. G. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations, 257 (2014), 921-1011. doi: 10.1016/j.jde.2014.04.018. Google Scholar

[25]

Z. H. Liu, P. Magal and S. G. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 537-555. doi: 10.3934/dcdsb.2016.21.537. Google Scholar

[26]

Z. H. Liu, P. Magal and D. M. Xiao, Bogdanov-Takens bifurcation in a predator-prey model, Zeitschrift fur Angewandte Mathematik und Physik, 67 (2016), Art. 137, 29 pp. doi: 10.1007/s00033-016-0724-1. Google Scholar

[27]

Z. H. Liu and R. Yuan, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Science China Mathematics, 60 (2017), 1371-1398. doi: 10.1007/s11425-016-0371-8. Google Scholar

[28]

Z. H. Liu and R. Yuan, The effect of diffusion for a predator-prey system with nonmonotonic functional response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 4309-4316. doi: 10.1142/S0218127404011867. Google Scholar

[29]

P. Magal and S. G. Ruan, Center manifolds for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7. Google Scholar

[30]

F. Takens, Forced oscillations and bifurcations, Comm. Math. Inst. Rijksuniv. Utrecht, Math. Inst. Rijksuniv. Utrecht, Utrecht, (1974), 1-59. Google Scholar

[31]

F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., (1974), 47-100. Google Scholar

[32]

H. Tang and Z. H. Liu, Hopf bifurcation for a predator-prey model with age structure, Applied Mathematical Modelling, 40 (2016), 726-737. doi: 10.1016/j.apm.2015.09.015. Google Scholar

[33]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. Google Scholar

[34]

H. R. Thieme, Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem, J. Evol. Equ., 8 (2008), 283-305. doi: 10.1007/s00028-007-0355-2. Google Scholar

[35]

H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447. doi: 10.1016/0022-247X(90)90074-P. Google Scholar

[36]

Z. Wang and Z. H. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150. doi: 10.1016/j.jmaa.2011.07.038. Google Scholar

[37]

G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985. Google Scholar

[38]

Y. X. Xu and M. Y. Huang, Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity, J. Differential Equations, 244 (2008), 582-598. doi: 10.1016/j.jde.2007.09.003. Google Scholar

[1]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643
[2]

Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69
[3]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
[4]

Fadia Bekkal-Brikci, Khalid Boushaba, Ovide Arino. Nonlinear age structured model with cannibalism. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 201-218. doi: 10.3934/dcdsb.2007.7.201
[5]

Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva. An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences & Engineering, 2010, 7 (1) : 123-147. doi: 10.3934/mbe.2010.7.123
[6]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
[7]

Yanxia Dang, Zhipeng Qiu, Xuezhi Li. Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (4) : 901-931. doi: 10.3934/mbe.2017048
[8]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
[9]

Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[10]

Fred Brauer. A model for an SI disease in an age - structured population. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 257-264. doi: 10.3934/dcdsb.2002.2.257
[11]

Ryszard Rudnicki, Radosław Wieczorek. On a nonlinear age-structured model of semelparous species. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2641-2656. doi: 10.3934/dcdsb.2014.19.2641
[12]

Janet Dyson, Eva Sánchez, Rosanna Villella-Bressan, Glenn F. Webb. An age and spatially structured model of tumor invasion with haptotaxis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 45-60. doi: 10.3934/dcdsb.2007.8.45
[13]

Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019226
[14]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[15]

Cameron Browne. Immune response in virus model structured by cell infection-age. Mathematical Biosciences & Engineering, 2016, 13 (5) : 887-909. doi: 10.3934/mbe.2016022
[16]

Georgi Kapitanov, Christina Alvey, Katia Vogt-Geisse, Zhilan Feng. An age-structured model for the coupled dynamics of HIV and HSV-2. Mathematical Biosciences & Engineering, 2015, 12 (4) : 803-840. doi: 10.3934/mbe.2015.12.803
[17]

Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999
[18]

Xichao Duan, Sanling Yuan, Kaifa Wang. Dynamics of a diffusive age-structured HBV model with saturating incidence. Mathematical Biosciences & Engineering, 2016, 13 (5) : 935-968. doi: 10.3934/mbe.2016024
[19]

Diène Ngom, A. Iggidir, Aboudramane Guiro, Abderrahim Ouahbi. An observer for a nonlinear age-structured model of a harvested fish population. Mathematical Biosciences & Engineering, 2008, 5 (2) : 337-354. doi: 10.3934/mbe.2008.5.337
[20]

Georgi Kapitanov. A double age-structured model of the co-infection of tuberculosis and HIV. Mathematical Biosciences & Engineering, 2015, 12 (1) : 23-40. doi: 10.3934/mbe.2015.12.23

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (21)
  • HTML views (88)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences