-
Previous Article
Modelling contact with isotropic and anisotropic friction by the bipotential approach
- DCDS-S Home
- This Issue
-
Next Article
Parametric nonlinear PDEs with multiple solutions: A PGD approach
Comparison between Borel-Padé summation and factorial series, as time integration methods
1. | Laboratoire des Sciences de l'Ingénieur pour l'Environnement - UMR 7356, Université de La Rochelle, 17042 La Rochelle Cedex 1, France, France, France |
References:
[1] |
V. Adukov and O. Ibryaeva, A new algorithm for computing padé approximants,, \arXiv{1112.5694}., ().
|
[2] |
G. Baker, J. Gammel and J. Wills, An investigation of the applicability of the Padé approximant method, Journal of Mathematical Analysis and Applications, 2 (1961), 405-418.
doi: 10.1016/0022-247X(61)90019-1. |
[3] |
B. Beckermann and A. Ana Matos, Algebraic properties of robust Padé approximants, Journal of Approximation Theory, 190 (2015), 91-115, arXiv:1310.2438.
doi: 10.1016/j.jat.2014.05.018. |
[4] |
J. Boyd, Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals, Journal of Scientific Computing, 2 (1987), 99-109.
doi: 10.1007/BF01061480. |
[5] |
C. Brezinski, Rationnal approximation to formal power serie, Journal of Approximation Theory, 25 (1979), 295-317.
doi: 10.1016/0021-9045(79)90019-4. |
[6] |
C. Brezinski and J. Van Iseghem, Padé approximations, in Handbook of Numerical Analysis (eds. P. G. Ciarlet and J. L. Lions), Elsevier, 3 (1994), 47-222,
doi: 10.1016/S1570-8659(05)80016-X. |
[7] |
A. Bultheel, Recursive algorithms for nonnormal Pade tables, SIAM Journal on Applied Mathematics, 39 (1980), 106-118.
doi: 10.1137/0139009. |
[8] |
O. Costin, G. Luo and S. Tanveer, Divergent expansion, Borel summability and three-dimensional Navier-Stokes equation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366 (2008), 2775-2788.
doi: 10.1098/rsta.2008.0052. |
[9] |
P. J. Davis and P. Rabinowitz, Ignoring the singularity in approximate integration, Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis, 2 (1965), 367-383.
doi: 10.1137/0702029. |
[10] |
A. Deeb, A. Hamdouni, E. Liberge and D. Razafindralandy, Borel-Laplace summation method used as time integration scheme, ESAIM: Procedings and Surveys, 45 (2014), 318-327.
doi: 10.1051/proc/201445033. |
[11] |
E. Delabaere and J.-M. Rasoamanana, Sommation effective d'une somme de Borel par séries de factorielles, Annales de l'institut Fourier, 57 (2007), 421-456.
doi: 10.5802/aif.2263. |
[12] |
W. Gautschi, Gauss-type quadrature rules for rational functions, Numerical Integration IV, the series ISNM International Series of Numerical Mathematics, 112 (1993), 111-130, arXiv:math/9307223.
doi: 10.1007/978-3-0348-6338-4_9. |
[13] |
W. Gautschi, The use of rational functions in numerical quadrature, Journal of Computational and Applied Mathematics, 133 (2001), 111-126.
doi: 10.1016/S0377-0427(00)00637-3. |
[14] |
W. Gautschi, Quadrature formulae on half-infinite intervals, BIT Numerical Mathematics, 31 (1991), 438-446.
doi: 10.1007/BF01933261. |
[15] |
J. Gilewicz, Approximants de Padé, vol. 667 of Lecture Notes in Mathematics, Springer-Verlag, 1978. |
[16] |
J. Gilewicz and Y. Kryakin, Froissart doublets in Padé approximation in the case of polynomial noise, Journal of Computational and Applied Mathematics, 153 (2003), 235-242.
doi: 10.1016/S0377-0427(02)00674-X. |
[17] |
J. Gilewicz and M. Pindor, Padé approximants and noise: A case of geometric series, Journal of Computational and Applied Mathematics, 87 (1997), 199-214.
doi: 10.1016/S0377-0427(97)00185-4. |
[18] |
P. Gonnet, S. Güttel and L. Trefethen, Robust Padé approximation via SVD, SIAM Review, 55 (2013), 101-117.
doi: 10.1137/110853236. |
[19] |
N. Hall, Interview of sir michael berry by nina hall: Caustics, catastrophes and quantum chaos,, Nexus News, (): 4.
|
[20] |
M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and An Introduction to Chaos, Elsevier, 2013.
doi: 10.1016/B978-0-12-382010-5.00001-4. |
[21] |
H. Kleinert and V. Schulte-Frohlinde, Critical Properties of $\Phi^4$-Theories, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
doi: 10.1142/9789812799944. |
[22] |
V. Kowalenko, The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation, Bentham, 2009.
doi: 10.2174/97816080501091090101. |
[23] |
R. Kumar and M. K. Jain, Quadrature formulas for semi-infinite integrals, Mathematics of Computation, 28 (1974), 499-503.
doi: 10.1090/S0025-5718-1974-0343549-5. |
[24] |
D. Lubinsky, Reflections on the Baker-Gammel-Wills (Padé), in Analytic Number Theory, Approximation Theory, and Special Functions (eds. G. V. Milovanović and M. T. Rassias), Springer New York, 2014, 561-571. |
[25] |
D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands, Mathematics of Computation, 43 (1984), 219-242.
doi: 10.1090/S0025-5718-1984-0744932-2. |
[26] |
D. Lutz, M. Miyake and R. Schäfke, On the Borel summability of divergent solutions of the heat equation, Nagoya Mathematical Journal, 154 (1999), 1-29. |
[27] |
G. Lysik, Borel summable solutions of the Burgers equation, Annales Polonici Mathematici, 95 (2009), 187-197.
doi: 10.4064/ap95-2-9. |
[28] |
G. Lysik and S. Michalik, Formal solutions of semilinear heat equations, Journal of Mathematical Analysis and Applications, 341 (2008), 372-385.
doi: 10.1016/j.jmaa.2007.10.005. |
[29] |
W. Mascarenhas, Robust Padé approximants can diverge,, , ().
|
[30] |
N. Nielsen, Recherches sur les séries de factorielles, Annales Scientifiques de l'E.N.S. 3è série, 19 (1902), 409-453. |
[31] |
N. Nielsen, Les séries de factorielles et les opérations fondamentales, Mathematische Annalen, 59 (1904), 355-376.
doi: 10.1007/BF01445147. |
[32] |
N. Nielsen, Sur les séries de factorielles et la fonction gamma (extrait d'une lettre adressée à M.-N. de Sonin à Saint-Pétersbourg), Annales Scientifiques de l'E.N.S. 3è série, 23 (1906), 145-168. |
[33] |
N. Nörlund, Vorlesungen Über Differenzenrechnung, Srpinger Verlag, 1924. |
[34] |
N. Nörlund, Leçons Sur Les Séries D'interpolation, Gauthier-Villard et Cie, 1926. |
[35] |
S. Pincherle, Sulle serie di fattoriali. nota II, Atti della Reale Accademia dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali. Series 5, 11 (1902), 417-426. |
[36] |
J.-P. Ramis, Séries divergentes et théories asymptotiques, in Journées X-UPS 1991, 1991, 7-67. |
[37] |
J.-P. Ramis, Les développements asymptotiques après poincaré: Continuité et... divergences,, Gazettes des Mathématiciens., ().
|
[38] |
D. Razafindralandy and A. Hamdouni, Time integration algorithm based on divergent series resummation, for ordinary and partial differential equations, Journal of Computational Physics, 236 (2013), 56-73.
doi: 10.1016/j.jcp.2012.10.022. |
[39] |
H. Stahl, Conjectures around the Baker-Gammel-Wills conjecture, Constructive Approximation, 13 (1997), 287-292,
doi: 10.1007/s003659900044. |
[40] |
H. Stahl, Spurious poles in Padé approximation, Journal of Computational and Applied Mathematics, 99 (1998), 511-527.
doi: 10.1016/S0377-0427(98)00180-0. |
[41] |
J. Thomann, Resommation des séries formelles. Solutions d'équations différentielles linéaires ordinaires du second ordre dans le champ complexe au voisinage de singularités irrégulières, Numerische Mathematik, 58 (1990), 503-535.
doi: 10.1007/BF01385638. |
[42] |
J. Thomann, Procédés formels et numériques de sommation de séries solutions d'équations différentielles, in Journées X-UPS 1991, Séries divergentes et procédés de resommation (ed. C. de mathématiques), 1991, 101-114. |
[43] |
J. Thomann, Formal and Numerical Summation of Formal Power Series Solutions of ODE's, Technical report, CIRM Luminy, 2000. |
[44] |
F. Thomlinson, Generalized factorial series,, Transactions of the American Mathematical Society, 31 ().
|
[45] |
M. Thomson, The Calculus Of Finite Differences, Macmillan and Company, 1933. |
[46] |
J. van Deun, A. Bultheel and P. González Vera, On computing rational Gauss-Chebyshev quadrature formulas, Mathematics of Computation, 75 (2006), 307-326.
doi: 10.1090/S0025-5718-05-01774-6. |
[47] |
G. N. Watson, The transformation of an asymptotic series into a convergent series of inverse factorials, Rendiconti del Circolo Matematico di Palermo, 34 (1912), 41-88. |
[48] |
E. Weniger, Summation of divergent power series by means of factorial series, Applied Numerical Mathematics, 60 (2010), 1429-1441.
doi: 10.1016/j.apnum.2010.04.003. |
show all references
References:
[1] |
V. Adukov and O. Ibryaeva, A new algorithm for computing padé approximants,, \arXiv{1112.5694}., ().
|
[2] |
G. Baker, J. Gammel and J. Wills, An investigation of the applicability of the Padé approximant method, Journal of Mathematical Analysis and Applications, 2 (1961), 405-418.
doi: 10.1016/0022-247X(61)90019-1. |
[3] |
B. Beckermann and A. Ana Matos, Algebraic properties of robust Padé approximants, Journal of Approximation Theory, 190 (2015), 91-115, arXiv:1310.2438.
doi: 10.1016/j.jat.2014.05.018. |
[4] |
J. Boyd, Exponentially convergent Fourier-Chebshev quadrature schemes on bounded and infinite intervals, Journal of Scientific Computing, 2 (1987), 99-109.
doi: 10.1007/BF01061480. |
[5] |
C. Brezinski, Rationnal approximation to formal power serie, Journal of Approximation Theory, 25 (1979), 295-317.
doi: 10.1016/0021-9045(79)90019-4. |
[6] |
C. Brezinski and J. Van Iseghem, Padé approximations, in Handbook of Numerical Analysis (eds. P. G. Ciarlet and J. L. Lions), Elsevier, 3 (1994), 47-222,
doi: 10.1016/S1570-8659(05)80016-X. |
[7] |
A. Bultheel, Recursive algorithms for nonnormal Pade tables, SIAM Journal on Applied Mathematics, 39 (1980), 106-118.
doi: 10.1137/0139009. |
[8] |
O. Costin, G. Luo and S. Tanveer, Divergent expansion, Borel summability and three-dimensional Navier-Stokes equation, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 366 (2008), 2775-2788.
doi: 10.1098/rsta.2008.0052. |
[9] |
P. J. Davis and P. Rabinowitz, Ignoring the singularity in approximate integration, Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis, 2 (1965), 367-383.
doi: 10.1137/0702029. |
[10] |
A. Deeb, A. Hamdouni, E. Liberge and D. Razafindralandy, Borel-Laplace summation method used as time integration scheme, ESAIM: Procedings and Surveys, 45 (2014), 318-327.
doi: 10.1051/proc/201445033. |
[11] |
E. Delabaere and J.-M. Rasoamanana, Sommation effective d'une somme de Borel par séries de factorielles, Annales de l'institut Fourier, 57 (2007), 421-456.
doi: 10.5802/aif.2263. |
[12] |
W. Gautschi, Gauss-type quadrature rules for rational functions, Numerical Integration IV, the series ISNM International Series of Numerical Mathematics, 112 (1993), 111-130, arXiv:math/9307223.
doi: 10.1007/978-3-0348-6338-4_9. |
[13] |
W. Gautschi, The use of rational functions in numerical quadrature, Journal of Computational and Applied Mathematics, 133 (2001), 111-126.
doi: 10.1016/S0377-0427(00)00637-3. |
[14] |
W. Gautschi, Quadrature formulae on half-infinite intervals, BIT Numerical Mathematics, 31 (1991), 438-446.
doi: 10.1007/BF01933261. |
[15] |
J. Gilewicz, Approximants de Padé, vol. 667 of Lecture Notes in Mathematics, Springer-Verlag, 1978. |
[16] |
J. Gilewicz and Y. Kryakin, Froissart doublets in Padé approximation in the case of polynomial noise, Journal of Computational and Applied Mathematics, 153 (2003), 235-242.
doi: 10.1016/S0377-0427(02)00674-X. |
[17] |
J. Gilewicz and M. Pindor, Padé approximants and noise: A case of geometric series, Journal of Computational and Applied Mathematics, 87 (1997), 199-214.
doi: 10.1016/S0377-0427(97)00185-4. |
[18] |
P. Gonnet, S. Güttel and L. Trefethen, Robust Padé approximation via SVD, SIAM Review, 55 (2013), 101-117.
doi: 10.1137/110853236. |
[19] |
N. Hall, Interview of sir michael berry by nina hall: Caustics, catastrophes and quantum chaos,, Nexus News, (): 4.
|
[20] |
M. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and An Introduction to Chaos, Elsevier, 2013.
doi: 10.1016/B978-0-12-382010-5.00001-4. |
[21] |
H. Kleinert and V. Schulte-Frohlinde, Critical Properties of $\Phi^4$-Theories, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
doi: 10.1142/9789812799944. |
[22] |
V. Kowalenko, The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation, Bentham, 2009.
doi: 10.2174/97816080501091090101. |
[23] |
R. Kumar and M. K. Jain, Quadrature formulas for semi-infinite integrals, Mathematics of Computation, 28 (1974), 499-503.
doi: 10.1090/S0025-5718-1974-0343549-5. |
[24] |
D. Lubinsky, Reflections on the Baker-Gammel-Wills (Padé), in Analytic Number Theory, Approximation Theory, and Special Functions (eds. G. V. Milovanović and M. T. Rassias), Springer New York, 2014, 561-571. |
[25] |
D. S. Lubinsky and P. Rabinowitz, Rates of convergence of Gaussian quadrature for singular integrands, Mathematics of Computation, 43 (1984), 219-242.
doi: 10.1090/S0025-5718-1984-0744932-2. |
[26] |
D. Lutz, M. Miyake and R. Schäfke, On the Borel summability of divergent solutions of the heat equation, Nagoya Mathematical Journal, 154 (1999), 1-29. |
[27] |
G. Lysik, Borel summable solutions of the Burgers equation, Annales Polonici Mathematici, 95 (2009), 187-197.
doi: 10.4064/ap95-2-9. |
[28] |
G. Lysik and S. Michalik, Formal solutions of semilinear heat equations, Journal of Mathematical Analysis and Applications, 341 (2008), 372-385.
doi: 10.1016/j.jmaa.2007.10.005. |
[29] |
W. Mascarenhas, Robust Padé approximants can diverge,, , ().
|
[30] |
N. Nielsen, Recherches sur les séries de factorielles, Annales Scientifiques de l'E.N.S. 3è série, 19 (1902), 409-453. |
[31] |
N. Nielsen, Les séries de factorielles et les opérations fondamentales, Mathematische Annalen, 59 (1904), 355-376.
doi: 10.1007/BF01445147. |
[32] |
N. Nielsen, Sur les séries de factorielles et la fonction gamma (extrait d'une lettre adressée à M.-N. de Sonin à Saint-Pétersbourg), Annales Scientifiques de l'E.N.S. 3è série, 23 (1906), 145-168. |
[33] |
N. Nörlund, Vorlesungen Über Differenzenrechnung, Srpinger Verlag, 1924. |
[34] |
N. Nörlund, Leçons Sur Les Séries D'interpolation, Gauthier-Villard et Cie, 1926. |
[35] |
S. Pincherle, Sulle serie di fattoriali. nota II, Atti della Reale Accademia dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali. Series 5, 11 (1902), 417-426. |
[36] |
J.-P. Ramis, Séries divergentes et théories asymptotiques, in Journées X-UPS 1991, 1991, 7-67. |
[37] |
J.-P. Ramis, Les développements asymptotiques après poincaré: Continuité et... divergences,, Gazettes des Mathématiciens., ().
|
[38] |
D. Razafindralandy and A. Hamdouni, Time integration algorithm based on divergent series resummation, for ordinary and partial differential equations, Journal of Computational Physics, 236 (2013), 56-73.
doi: 10.1016/j.jcp.2012.10.022. |
[39] |
H. Stahl, Conjectures around the Baker-Gammel-Wills conjecture, Constructive Approximation, 13 (1997), 287-292,
doi: 10.1007/s003659900044. |
[40] |
H. Stahl, Spurious poles in Padé approximation, Journal of Computational and Applied Mathematics, 99 (1998), 511-527.
doi: 10.1016/S0377-0427(98)00180-0. |
[41] |
J. Thomann, Resommation des séries formelles. Solutions d'équations différentielles linéaires ordinaires du second ordre dans le champ complexe au voisinage de singularités irrégulières, Numerische Mathematik, 58 (1990), 503-535.
doi: 10.1007/BF01385638. |
[42] |
J. Thomann, Procédés formels et numériques de sommation de séries solutions d'équations différentielles, in Journées X-UPS 1991, Séries divergentes et procédés de resommation (ed. C. de mathématiques), 1991, 101-114. |
[43] |
J. Thomann, Formal and Numerical Summation of Formal Power Series Solutions of ODE's, Technical report, CIRM Luminy, 2000. |
[44] |
F. Thomlinson, Generalized factorial series,, Transactions of the American Mathematical Society, 31 ().
|
[45] |
M. Thomson, The Calculus Of Finite Differences, Macmillan and Company, 1933. |
[46] |
J. van Deun, A. Bultheel and P. González Vera, On computing rational Gauss-Chebyshev quadrature formulas, Mathematics of Computation, 75 (2006), 307-326.
doi: 10.1090/S0025-5718-05-01774-6. |
[47] |
G. N. Watson, The transformation of an asymptotic series into a convergent series of inverse factorials, Rendiconti del Circolo Matematico di Palermo, 34 (1912), 41-88. |
[48] |
E. Weniger, Summation of divergent power series by means of factorial series, Applied Numerical Mathematics, 60 (2010), 1429-1441.
doi: 10.1016/j.apnum.2010.04.003. |
[1] |
Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511 |
[2] |
Chuang Peng. Minimum degrees of polynomial models on time series. Conference Publications, 2005, 2005 (Special) : 720-729. doi: 10.3934/proc.2005.2005.720 |
[3] |
Ruiqi Li, Yifan Chen, Xiang Zhao, Yanli Hu, Weidong Xiao. Time series based urban air quality predication. Big Data & Information Analytics, 2016, 1 (2&3) : 171-183. doi: 10.3934/bdia.2016003 |
[4] |
Yu-Ting Lin, John Malik, Hau-Tieng Wu. Wave-shape oscillatory model for nonstationary periodic time series analysis. Foundations of Data Science, 2021, 3 (2) : 99-131. doi: 10.3934/fods.2021009 |
[5] |
Antonella Falini, Francesca Mazzia, Cristiano Tamborrino. Spline based Hermite quasi-interpolation for univariate time series. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022039 |
[6] |
Armengol Gasull, Francesc Mañosas. Subseries and signed series. Communications on Pure and Applied Analysis, 2019, 18 (1) : 479-492. doi: 10.3934/cpaa.2019024 |
[7] |
Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial and Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107 |
[8] |
Zhi Liu, Tie Zhang. An improved ARMA(1, 1) type fuzzy time series applied in predicting disordering. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 355-366. doi: 10.3934/naco.2020007 |
[9] |
Hassan Khodaiemehr, Dariush Kiani. High-rate space-time block codes from twisted Laurent series rings. Advances in Mathematics of Communications, 2015, 9 (3) : 255-275. doi: 10.3934/amc.2015.9.255 |
[10] |
Annalisa Pascarella, Alberto Sorrentino, Cristina Campi, Michele Piana. Particle filtering, beamforming and multiple signal classification for the analysis of magnetoencephalography time series: a comparison of algorithms. Inverse Problems and Imaging, 2010, 4 (1) : 169-190. doi: 10.3934/ipi.2010.4.169 |
[11] |
Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, Stock price fluctuation prediction method based on time series analysis. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-915. doi: 10.3934/dcdss.2019061 |
[12] |
Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903 |
[13] |
Nikita Kalinin, Mikhail Shkolnikov. Introduction to tropical series and wave dynamic on them. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2827-2849. doi: 10.3934/dcds.2018120 |
[14] |
Geir Bogfjellmo. Algebraic structure of aromatic B-series. Journal of Computational Dynamics, 2019, 6 (2) : 199-222. doi: 10.3934/jcd.2019010 |
[15] |
Ricardo García López. A note on L-series and Hodge spectrum of polynomials. Electronic Research Announcements, 2009, 16: 56-62. doi: 10.3934/era.2009.16.56 |
[16] |
G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure and Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509 |
[17] |
Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure and Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229 |
[18] |
Ghobad Barmalzan, Ali Akbar Hosseinzadeh, Narayanaswamy Balakrishnan. Stochastic comparisons of series-parallel and parallel-series systems with dependence between components and also of subsystems. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021101 |
[19] |
Mario Pulvirenti, Sergio Simonella, Anton Trushechkin. Microscopic solutions of the Boltzmann-Enskog equation in the series representation. Kinetic and Related Models, 2018, 11 (4) : 911-931. doi: 10.3934/krm.2018036 |
[20] |
Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]