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Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system
First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions
Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany |
Mixed-moment models, introduced in [
References:
[1] |
G. W. Alldredge, C. D. Hauck, D. P. O'Leary and A. L. Tits,
Adaptive change of basis in entropy-based moment closures for linear kinetic equations, Journal of Computational Physics, 258 (2014), 489-508.
doi: 10.1016/j.jcp.2013.10.049. |
[2] |
G. W. Alldredge, C. D. Hauck and A. L. Tits,
High-Order Entropy-Based Closures for Linear Transport in Slab Geometry Ⅱ: A Computational Study of the Optimization Problem, SIAM Journal on Scientific Computing, 34 (2012), B361-B391.
doi: 10.1137/11084772X. |
[3] |
G. W. Alldredge and F. Schneider,
A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, Journal of Computational Physics, 295 (2015), 665-684.
doi: 10.1016/j.jcp.2015.04.034. |
[4] |
U. Ascher, S. Ruuth and R. Spiteri,
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, 25 (1997), 151-167.
doi: 10.1016/S0168-9274(97)00056-1. |
[5] |
G. I. Bell and S. Glasstone,
Nuclear Reactor Theory Technical report, Division of Technical Information, US Atomic Energy Commission, 1970. |
[6] |
M. A. Blanco, M. Flórez and M. Bermejo,
Evaluation of the rotation matrices in the basis of real spherical harmonics, Journal of Molecular Structure, 419 (1997), 19-27.
doi: 10.1016/S0166-1280(97)00185-1. |
[7] |
T. A. Brunner and J. P. Holloway,
One-dimensional Riemann solvers and the maximum entropy closure, Journal of Quantitative Spectroscopy and Radiative Transfer, 69 (2001), 543-566.
doi: 10.1016/S0022-4073(00)00099-6. |
[8] |
T. A. Brunner and J. P. Holloway,
Two-dimensional time dependent Riemann solvers for neutron transport, Journal of Computational Physics, 210 (2005), 386-399.
doi: 10.1016/j.jcp.2005.04.011. |
[9] |
J. A. Carrillo, A. Klar and A. Roth,
Single to double mill small noise transition via semi-lagrangian finite volume methods, Commun. Math. Sci., 14 (2016), 1111-1136.
doi: 10.4310/CMS.2016.v14.n4.a12. |
[10] |
R. E. R. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math, 17 (1991), 603–635, URL https://www.math.uh.edu/~hjm/v017n4/
0603CURTO.pdf |
[11] |
B. Dubroca and J.-L. Feugeas,
Entropic moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915-920.
doi: 10.1016/S0764-4442(00)87499-6. |
[12] |
B. Dubroca and A. Klar,
Half-moment closure for radiative transfer equations, Journal of Computational Physics, 180 (2002), 584-596.
doi: 10.1006/jcph.2002.7106. |
[13] |
A. S. Eddington,
The Internal Constitution of the Stars Dover, 1926. |
[14] |
A. Ern and J. L. Guermond,
Theory and Practice of Finite Elements Applied Mathematical Sciences, Springer New York, 2004, https://books.google.de/books?id=CCjm79FbJbcC.
doi: 10.1007/978-1-4757-4355-5. |
[15] |
G. D. Fies and M. Vianello, Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012), 102–112, URL http://www.emis. ams.org/journals/ETNA/vol.39.2012/pp102-112.dir/pp102-112.pdf. |
[16] |
G. D. Fies and M. Vianello,
Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012), 102-112.
|
[17] |
M. Frank,
Partial moment entropy approximation to radiative heat transfer, Pamm, 5 (2005), 659-660.
doi: 10.1002/pamm.200510306. |
[18] |
M. Frank, B. Dubroca and A. Klar,
Partial moment entropy approximation to radiative heat transfer, Journal of Computational Physics, 218 (2006), 1-18.
doi: 10.1016/j.jcp.2006.01.038. |
[19] |
M. Frank, H. Hensel and A. Klar,
A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy, SIAM Journal on Applied Mathematics, 67 (2007), 582-603.
doi: 10.1137/06065547X. |
[20] |
W. Fulton,
Eigenvalues of sums of Hermitian matrices, Séminaire Bourbaki, 40 (1998), 255-269.
|
[21] |
C. K. Garrett and C. D. Hauck,
A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport Theory and Statistical Physics, 42 (2013), 203-235.
doi: 10.1080/00411450.2014.910226. |
[22] |
E. M. Gelbard,
Simplified Spherical Harmonics Equations and Their Use in Shielding Problems Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. |
[23] |
C. D. Hauck,
High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.
doi: 10.4310/CMS.2011.v9.n1.a9. |
[24] |
C. D. Hauck, M. Frank and E. Olbrant,
Perturbed, entropy-based closure for radiative transfer, SIAM Journal on Applied Mathematics, 6 (2013), 557-587.
doi: 10.3934/krm.2013.6.557. |
[25] |
H. Hensel, R. Iza-Teran and N. Siedow,
Deterministic model for dose calculation in photon radiotherapy, to appear in Phys. Med. Biol., 51 (2006), 675-693.
doi: 10.1088/0031-9155/51/3/013. |
[26] |
J. H. Jeans,
The equations of radiative transfer of energy, Monthly Notices Royal Astronomical Society, 78 (1917), 28-36.
doi: 10.1093/mnras/78.1.28. |
[27] |
M. Junk,
Maximum entropy for reduced moment problems, Math. Meth. Mod. Appl. Sci., 10 (2000), 1001-1025.
doi: 10.1142/S0218202500000513. |
[28] |
C. Kelley,
Solving Nonlinear Equations with Newton's Method Society for Industrial and Applied Mathematics, 2003.
doi: 10.1137/1. 9780898718898. |
[29] |
D. S. Kershaw, Flux limiting nature's own way: A new method for numerical solution of the transport equation,
http://www.osti.gov/bridge/product.biblio.jsp?osti_id=104974. |
[30] |
C. J. Knight and A. C. R. Newbery,
Trigonometric and Gaussian quadrature, Mathematics of Computation, 24 (1970), 575-581.
doi: 10.1090/S0025-5718-1970-0275672-4. |
[31] |
V. I. Lebedev and D. N. Laikov, A quadrature formula for the sphere of the 131st algebraic
order of accuracy, in Doklady. Mathematics, vol. 59, MAIK Nauka/Interperiodica, 1999,477–
481. |
[32] |
C. D. Levermore,
Relating eddington factors to flux limiters, Journal of Quantitative Spectroscopy and Radiative Transfer, 31 (1984), 149-160.
doi: 10.1016/0022-4073(84)90112-2. |
[33] |
C. D. Levermore,
Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[34] |
W. R. Martin, The application of the finite element method to the neutron transport equation,
1–232. |
[35] |
G. N. Minerbo,
Maximum entropy Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978), 541-545.
doi: 10.1016/0022-4073(78)90024-9. |
[36] |
P. Monreal,
Moment Realizability and Kershaw Closures in Radiative Transfer PhD thesis, TU Aachen, 2012. |
[37] |
P. Monreal and M. Frank, Higher order minimum entropy approximations in radiative transfer,
arXiv preprint, arXiv: 0812.3063, 1–18, URL http://arxiv.org/abs/0812.3063. |
[38] |
G. C. Pomraning,
The Fokker-Planck operator as an asymptotic limit, Math. Mod. Meth. Appl. Sci., 2 (1992), 21-36.
doi: 10.1142/S021820259200003X. |
[39] |
A. Roth,
Numerical Schemes for Kinetic Equations with Applications to Fibre Lay-Down and Interacting Particles Verlag Dr. Hut, 2014. |
[40] |
A. Roth, A. Klar, B. Simeon and E. Zharovsky,
A semi-lagrangian method for 3-d fokker planck equations for stochastic dynamical systems on the sphere, Journal of Scientific Computing, 61 (2014), 513-532.
doi: 10.1007/s10915-014-9835-z. |
[41] |
F. Schneider, First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions: Code 2016.
doi: 10.5281/zenodo.48753. |
[42] |
F. Schneider, Implicit-explicit, realizability-preserving first-order scheme for moment models with Lipschitz-continuous source terms, arXiv preprint, http://arxiv.org/abs/1611.01314. |
[43] |
F. Schneider,
Kershaw closures for linear transport equations in slab geometry Ⅰ: Model derivation, Journal of Computational Physics, 322 (2016), 905-919.
doi: 10.1016/j.jcp.2016.02.080. |
[44] |
F. Schneider,
Moment Models in Radiation Transport Equations Dr. Hut Verlag, 2016. |
[45] |
F. Schneider, G. W. Alldredge, M. Frank and A. Klar,
Higher order mixed-moment approximations for the Fokker--Planck equation in one space dimension, SIAM Journal on Applied Mathematics, 74 (2014), 1087-1114.
doi: 10.1137/130934210. |
[46] |
F. Schneider, G. W. Alldredge and J. Kall,
A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kinetic and Related Models, 9 (2016), 193-215.
doi: 10.3934/krm.2016.9.193. |
[47] |
B. Seibold and M. Frank,
StaRMAP—A Second Order Staggered Grid Method for Spherical Harmonics Moment Equations of Radiative Transfer, ACM Transactions on Mathematical Software, 41 (2014), 1-28.
doi: 10.1145/2590808. |
[48] |
H. Weyl,
Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Mathematische Annalen, 71 (1912), 441-479.
doi: 10.1007/BF01456804. |
show all references
References:
[1] |
G. W. Alldredge, C. D. Hauck, D. P. O'Leary and A. L. Tits,
Adaptive change of basis in entropy-based moment closures for linear kinetic equations, Journal of Computational Physics, 258 (2014), 489-508.
doi: 10.1016/j.jcp.2013.10.049. |
[2] |
G. W. Alldredge, C. D. Hauck and A. L. Tits,
High-Order Entropy-Based Closures for Linear Transport in Slab Geometry Ⅱ: A Computational Study of the Optimization Problem, SIAM Journal on Scientific Computing, 34 (2012), B361-B391.
doi: 10.1137/11084772X. |
[3] |
G. W. Alldredge and F. Schneider,
A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, Journal of Computational Physics, 295 (2015), 665-684.
doi: 10.1016/j.jcp.2015.04.034. |
[4] |
U. Ascher, S. Ruuth and R. Spiteri,
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, 25 (1997), 151-167.
doi: 10.1016/S0168-9274(97)00056-1. |
[5] |
G. I. Bell and S. Glasstone,
Nuclear Reactor Theory Technical report, Division of Technical Information, US Atomic Energy Commission, 1970. |
[6] |
M. A. Blanco, M. Flórez and M. Bermejo,
Evaluation of the rotation matrices in the basis of real spherical harmonics, Journal of Molecular Structure, 419 (1997), 19-27.
doi: 10.1016/S0166-1280(97)00185-1. |
[7] |
T. A. Brunner and J. P. Holloway,
One-dimensional Riemann solvers and the maximum entropy closure, Journal of Quantitative Spectroscopy and Radiative Transfer, 69 (2001), 543-566.
doi: 10.1016/S0022-4073(00)00099-6. |
[8] |
T. A. Brunner and J. P. Holloway,
Two-dimensional time dependent Riemann solvers for neutron transport, Journal of Computational Physics, 210 (2005), 386-399.
doi: 10.1016/j.jcp.2005.04.011. |
[9] |
J. A. Carrillo, A. Klar and A. Roth,
Single to double mill small noise transition via semi-lagrangian finite volume methods, Commun. Math. Sci., 14 (2016), 1111-1136.
doi: 10.4310/CMS.2016.v14.n4.a12. |
[10] |
R. E. R. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math, 17 (1991), 603–635, URL https://www.math.uh.edu/~hjm/v017n4/
0603CURTO.pdf |
[11] |
B. Dubroca and J.-L. Feugeas,
Entropic moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915-920.
doi: 10.1016/S0764-4442(00)87499-6. |
[12] |
B. Dubroca and A. Klar,
Half-moment closure for radiative transfer equations, Journal of Computational Physics, 180 (2002), 584-596.
doi: 10.1006/jcph.2002.7106. |
[13] |
A. S. Eddington,
The Internal Constitution of the Stars Dover, 1926. |
[14] |
A. Ern and J. L. Guermond,
Theory and Practice of Finite Elements Applied Mathematical Sciences, Springer New York, 2004, https://books.google.de/books?id=CCjm79FbJbcC.
doi: 10.1007/978-1-4757-4355-5. |
[15] |
G. D. Fies and M. Vianello, Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012), 102–112, URL http://www.emis. ams.org/journals/ETNA/vol.39.2012/pp102-112.dir/pp102-112.pdf. |
[16] |
G. D. Fies and M. Vianello,
Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012), 102-112.
|
[17] |
M. Frank,
Partial moment entropy approximation to radiative heat transfer, Pamm, 5 (2005), 659-660.
doi: 10.1002/pamm.200510306. |
[18] |
M. Frank, B. Dubroca and A. Klar,
Partial moment entropy approximation to radiative heat transfer, Journal of Computational Physics, 218 (2006), 1-18.
doi: 10.1016/j.jcp.2006.01.038. |
[19] |
M. Frank, H. Hensel and A. Klar,
A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy, SIAM Journal on Applied Mathematics, 67 (2007), 582-603.
doi: 10.1137/06065547X. |
[20] |
W. Fulton,
Eigenvalues of sums of Hermitian matrices, Séminaire Bourbaki, 40 (1998), 255-269.
|
[21] |
C. K. Garrett and C. D. Hauck,
A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport Theory and Statistical Physics, 42 (2013), 203-235.
doi: 10.1080/00411450.2014.910226. |
[22] |
E. M. Gelbard,
Simplified Spherical Harmonics Equations and Their Use in Shielding Problems Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. |
[23] |
C. D. Hauck,
High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.
doi: 10.4310/CMS.2011.v9.n1.a9. |
[24] |
C. D. Hauck, M. Frank and E. Olbrant,
Perturbed, entropy-based closure for radiative transfer, SIAM Journal on Applied Mathematics, 6 (2013), 557-587.
doi: 10.3934/krm.2013.6.557. |
[25] |
H. Hensel, R. Iza-Teran and N. Siedow,
Deterministic model for dose calculation in photon radiotherapy, to appear in Phys. Med. Biol., 51 (2006), 675-693.
doi: 10.1088/0031-9155/51/3/013. |
[26] |
J. H. Jeans,
The equations of radiative transfer of energy, Monthly Notices Royal Astronomical Society, 78 (1917), 28-36.
doi: 10.1093/mnras/78.1.28. |
[27] |
M. Junk,
Maximum entropy for reduced moment problems, Math. Meth. Mod. Appl. Sci., 10 (2000), 1001-1025.
doi: 10.1142/S0218202500000513. |
[28] |
C. Kelley,
Solving Nonlinear Equations with Newton's Method Society for Industrial and Applied Mathematics, 2003.
doi: 10.1137/1. 9780898718898. |
[29] |
D. S. Kershaw, Flux limiting nature's own way: A new method for numerical solution of the transport equation,
http://www.osti.gov/bridge/product.biblio.jsp?osti_id=104974. |
[30] |
C. J. Knight and A. C. R. Newbery,
Trigonometric and Gaussian quadrature, Mathematics of Computation, 24 (1970), 575-581.
doi: 10.1090/S0025-5718-1970-0275672-4. |
[31] |
V. I. Lebedev and D. N. Laikov, A quadrature formula for the sphere of the 131st algebraic
order of accuracy, in Doklady. Mathematics, vol. 59, MAIK Nauka/Interperiodica, 1999,477–
481. |
[32] |
C. D. Levermore,
Relating eddington factors to flux limiters, Journal of Quantitative Spectroscopy and Radiative Transfer, 31 (1984), 149-160.
doi: 10.1016/0022-4073(84)90112-2. |
[33] |
C. D. Levermore,
Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[34] |
W. R. Martin, The application of the finite element method to the neutron transport equation,
1–232. |
[35] |
G. N. Minerbo,
Maximum entropy Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978), 541-545.
doi: 10.1016/0022-4073(78)90024-9. |
[36] |
P. Monreal,
Moment Realizability and Kershaw Closures in Radiative Transfer PhD thesis, TU Aachen, 2012. |
[37] |
P. Monreal and M. Frank, Higher order minimum entropy approximations in radiative transfer,
arXiv preprint, arXiv: 0812.3063, 1–18, URL http://arxiv.org/abs/0812.3063. |
[38] |
G. C. Pomraning,
The Fokker-Planck operator as an asymptotic limit, Math. Mod. Meth. Appl. Sci., 2 (1992), 21-36.
doi: 10.1142/S021820259200003X. |
[39] |
A. Roth,
Numerical Schemes for Kinetic Equations with Applications to Fibre Lay-Down and Interacting Particles Verlag Dr. Hut, 2014. |
[40] |
A. Roth, A. Klar, B. Simeon and E. Zharovsky,
A semi-lagrangian method for 3-d fokker planck equations for stochastic dynamical systems on the sphere, Journal of Scientific Computing, 61 (2014), 513-532.
doi: 10.1007/s10915-014-9835-z. |
[41] |
F. Schneider, First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions: Code 2016.
doi: 10.5281/zenodo.48753. |
[42] |
F. Schneider, Implicit-explicit, realizability-preserving first-order scheme for moment models with Lipschitz-continuous source terms, arXiv preprint, http://arxiv.org/abs/1611.01314. |
[43] |
F. Schneider,
Kershaw closures for linear transport equations in slab geometry Ⅰ: Model derivation, Journal of Computational Physics, 322 (2016), 905-919.
doi: 10.1016/j.jcp.2016.02.080. |
[44] |
F. Schneider,
Moment Models in Radiation Transport Equations Dr. Hut Verlag, 2016. |
[45] |
F. Schneider, G. W. Alldredge, M. Frank and A. Klar,
Higher order mixed-moment approximations for the Fokker--Planck equation in one space dimension, SIAM Journal on Applied Mathematics, 74 (2014), 1087-1114.
doi: 10.1137/130934210. |
[46] |
F. Schneider, G. W. Alldredge and J. Kall,
A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kinetic and Related Models, 9 (2016), 193-215.
doi: 10.3934/krm.2016.9.193. |
[47] |
B. Seibold and M. Frank,
StaRMAP—A Second Order Staggered Grid Method for Spherical Harmonics Moment Equations of Radiative Transfer, ACM Transactions on Mathematical Software, 41 (2014), 1-28.
doi: 10.1145/2590808. |
[48] |
H. Weyl,
Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Mathematische Annalen, 71 (1912), 441-479.
doi: 10.1007/BF01456804. |














Model | Abs. runtime [s] | Rel. runtime | ||||
5 | 1.305e+05 | 3.885e-01 | ||||
13 | 3.359e+05 | 1.0 | ||||
5 | 3.408e+01 | 1.015e-04 | ||||
3 | 9.418e+04 | 2.804e-01 | ||||
6 | 2.269e+05 | 6.757e-01 | ||||
12 | 5.309e+01 | 1.581e-04 |
Model | Abs. runtime [s] | Rel. runtime | ||||
5 | 1.305e+05 | 3.885e-01 | ||||
13 | 3.359e+05 | 1.0 | ||||
5 | 3.408e+01 | 1.015e-04 | ||||
3 | 9.418e+04 | 2.804e-01 | ||||
6 | 2.269e+05 | 6.757e-01 | ||||
12 | 5.309e+01 | 1.581e-04 |
Model | Abs. runtime [s] | Rel. runtime | ||||
5 | 2.489e+05 | 1.0 | ||||
13 | 9.526e+04 | 3.828e-01 | ||||
5 | 1.535e+02 | 6.169e-04 | ||||
3 | 6.639e+04 | 2.668e-01 | ||||
6 | 1.189e+05 | 4.777e-01 | ||||
12 | 2.232e+05 | 8.967e-01 |
Model | Abs. runtime [s] | Rel. runtime | ||||
5 | 2.489e+05 | 1.0 | ||||
13 | 9.526e+04 | 3.828e-01 | ||||
5 | 1.535e+02 | 6.169e-04 | ||||
3 | 6.639e+04 | 2.668e-01 | ||||
6 | 1.189e+05 | 4.777e-01 | ||||
12 | 2.232e+05 | 8.967e-01 |
Model | $n$ | Abs. runtime [s] | Rel. runtime | |||
${\rm{M}}{{\rm{M}}_1}$ | 5 | 1.638e+05 | 2.091e-01 | |||
${\rm{M}}{{\rm{M}}_2}$ | 13 | 3.491e+05 | 4.456e-01 | |||
${\rm{M}}{{\rm{M}}_3}$ | 25 | 7.834e+05 | 1.0 | |||
${\rm{M}}{{\rm{K}}_1}$ poly | 5 | 1.888e+02 | 2.409e-04 | |||
${\rm{M}}{{\rm{K}}_1}$ | 5 | 2.898e+04 | 3.700e-02 | |||
${{\rm{M}}_1}$ | 3 | 5.458e+04 | 6.968e-02 | |||
${{\rm{M}}_2}$ | 6 | 9.696e+04 | 1.238e-01 | |||
${{\rm{M}}_3}$ | 10 | 1.823e+05 | 2.327e-01 |
Model | $n$ | Abs. runtime [s] | Rel. runtime | |||
${\rm{M}}{{\rm{M}}_1}$ | 5 | 1.638e+05 | 2.091e-01 | |||
${\rm{M}}{{\rm{M}}_2}$ | 13 | 3.491e+05 | 4.456e-01 | |||
${\rm{M}}{{\rm{M}}_3}$ | 25 | 7.834e+05 | 1.0 | |||
${\rm{M}}{{\rm{K}}_1}$ poly | 5 | 1.888e+02 | 2.409e-04 | |||
${\rm{M}}{{\rm{K}}_1}$ | 5 | 2.898e+04 | 3.700e-02 | |||
${{\rm{M}}_1}$ | 3 | 5.458e+04 | 6.968e-02 | |||
${{\rm{M}}_2}$ | 6 | 9.696e+04 | 1.238e-01 | |||
${{\rm{M}}_3}$ | 10 | 1.823e+05 | 2.327e-01 |
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