• Previous Article
    Pseudospectral discretization of delay differential equations in sun-star formulation: Results and conjectures
  • DCDS-S Home
  • This Issue
  • Next Article
    Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations
doi: 10.3934/dcdss.2020221

Space-time kernel based numerical method for generalized Black-Scholes equation

Department of Basic Sciences, University of Engineering and Technology Peshawar, Pakistan

* Corresponding author: Marjan Uddin

Received  February 2019 Revised  June 2019 Published  December 2019

Fund Project: This work is supported by HEC Pakistan.

In approximating time-dependent partial differential equations, major error always occurs in the time derivatives as compared to the spatial derivatives. In the present work the time and the spatial derivatives are both approximated using time-space radial kernels. The proposed numerical scheme avoids the time stepping procedures and produced sparse differentiation matrices. The stability and accuracy of the proposed numerical scheme is tested for the generalized Black-Scholes equation.

Citation: Marjan Uddin, Hazrat Ali. Space-time kernel based numerical method for generalized Black-Scholes equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020221
References:
[1]

V. R. Ambati and O. Bokhove, Space-time discontinuous Galerkin finite element method for shallow water flows, J. Comput. Appl. Math., 204 (2007), 452-462.  doi: 10.1016/j.cam.2006.01.047.  Google Scholar

[2]

C. Canuto, M. Y. Hussaini, A. Z. Quarteroni and T. Zang, Option Pricing: Mathematical Models and Computation, Springer, 1993. Google Scholar

[3]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation, Springer, Berlin, 2007.  Google Scholar

[4]

C. Chen, A. Karageorghis and Y. Smyrlis, The Method of Fundamental Solutions: A Meshless Method, Dynamic Publishers Atlanta, 2008. Google Scholar

[5]

A. Cohen, Numerical Analysis of Wavelet Methods, Studies in Mathematics and its Applications, 32. North-Holland Publishing Co., Amsterdam, 2003.  Google Scholar

[6]

J. C. CoxS. A. Ross and M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229-263.  doi: 10.1016/0304-405X(79)90015-1.  Google Scholar

[7]

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, With 1 CD-ROM (Windows, Macintosh and UNIX), Interdisciplinary Mathematical Sciences, 6. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6437.  Google Scholar

[8]

G. E. Fasshauer and M. McCourt, Kernel-Based Approximation Methods Using Matlab, World Scientific pub. Co, 2015. Google Scholar

[9]

G. E. Fasshauer and J. G. Zhang, On choosing optimal shape parameters for RBF approximation, Numerical Algorithms, 45 (2007), 345-368.  doi: 10.1007/s11075-007-9072-8.  Google Scholar

[10]

R. Geske and K. Shastri, Valuation by approximation: A comparison of alternative option valuation techniques, Journal of Financial and Quantitative Analysis, 20 (1985), 45-71.  doi: 10.2307/2330677.  Google Scholar

[11]

M. HamaidiA. Naji and A. Charafi, Space-time localized radial basis function collocation method for solving parabolic and hyperbolic equations, Eng. Anal. Bound. Elem., 67 (2016), 152-163.  doi: 10.1016/j.enganabound.2016.03.009.  Google Scholar

[12]

J. Hull and A. White, The use of the control variate technique in option pricing, Journal of Financial and Quantitative analysis, 23 (1988), 237-251.  doi: 10.2307/2331065.  Google Scholar

[13]

M. K. KadalbajooL. P. Tripathi and A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation, Math. Comput. Modelling, 55 (2012), 1483-1505.  doi: 10.1016/j.mcm.2011.10.040.  Google Scholar

[14]

C. M. KlaijaJ. J. W. van der Vegta and H. van der Venb, Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations, J. Comput. Phys., 217 (2006), 589-611.  doi: 10.1016/j.jcp.2006.01.018.  Google Scholar

[15]

M. LiW. Chen and C. S. Chen, The localized RBFs collocation methods for solving high dimensional PDEs, Eng. Anal. Bound. Elem., 37 (2013), 1300-1304.  doi: 10.1016/j.enganabound.2013.06.001.  Google Scholar

[16]

Z. Li and X. Z. Mao, Global multiquadric collocation method for groundwater contaminant source identification, Environmental modelling & software, 26 (2011), 1611-1621.  doi: 10.1016/j.envsoft.2011.07.010.  Google Scholar

[17]

Z. Li and X. Z. Mao, Global space-time multiquadric method for inverse heat conduction problem, Internat. J. Numer. Methods Engrg., 85 (2011), 355-379.  doi: 10.1002/nme.2975.  Google Scholar

[18]

F. Moukalled, L. Mangani and M. Darwish, The Finite Volume Method in Computational Fluid Dynamics, Fluid Mechanics and its Applications, 113. Springer, Cham, 2016. doi: 10.1007/978-3-319-16874-6.  Google Scholar

[19]

H. Netuzhylov, A Space-Time Meshfree Collocation Method for Coupled Problems on Irregularly-Shaped Domains, Ph.D thesis, Zugl., Braunschweig, Techn. Univ., Diss., 2008. Google Scholar

[20]

R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing, Comput. Math. Appl., 69 (2015), 777-797.  doi: 10.1016/j.camwa.2015.02.018.  Google Scholar

[21]

S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, 39. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-540-68093-2.  Google Scholar

[22]

J. J.SudirhamJ. J. W. van der Vegt and R. M. J. van Damme, Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math., 56 (2006), 1491-1518.  doi: 10.1016/j.apnum.2005.11.003.  Google Scholar

[23]

T. E. TezduyarS. SatheR. Keedy and K. Stein, Space-time finite element techniques for computation of fluid-structure interactions, Comput. Methods Appl. Mech. Engrg., 195 (2006), 2002-2027.  doi: 10.1016/j.cma.2004.09.014.  Google Scholar

[24]

C. TurchettiM. ContiP. Crippa and S. Orcioni, On the approximation of stochastic processes by approximate identity neural networks, IEEE Transactions on Neural Networks, 9 (1998), 1069-1085.  doi: 10.1109/72.728353.  Google Scholar

[25]

C. TurchettiP. CrippaM. Pirani and G. Biagetti, Representation of nonlinear random transformations by non-Gaussian stochastic neural networks, IEEE transactions on neural networks, 19 (2008), 1033-1060.  doi: 10.1109/TNN.2007.2000055.  Google Scholar

[26]

M. UddinH. Ali and A. Ali, Kernel-based local meshless method for solving multi-dimensional wave equations in irregular domain, CMES-Computer Modeling In Engineering & Sciences, 107 (2015), 463-479.   Google Scholar

[27]

M. Uddin and H. Ali, The space time kernel based numerical method for Burgers equations, Mathematics, 6 (2018), 212-222.  doi: 10.3390/math6100212.  Google Scholar

[28]

M. UddinK. KamranM. Usman and A. Ali, On the Laplace-transformed-based local meshless method for fractional-order diffusion equation, Int. J. Comput. Methods Eng. Sci. Mech., 19 (2018), 221-225.  doi: 10.1080/15502287.2018.1472150.  Google Scholar

[29]

B. M. Vaganan and E. E. Priya, Generalized Cole-Hopf transformations for generalized Burgers equations, Pramana, 85 (2015), 861-867.  doi: 10.1007/s12043-015-1107-4.  Google Scholar

[30]

D. L. YoungC. C. TsaiK. MurugesanaC. M. Fan and C. W. Chen, Time-dependent fundamental solutions for homogeneous diffusion problems, Engineering Analysis with Boundary Elements, 28 (2004), 1463-1473.  doi: 10.1016/j.enganabound.2004.07.003.  Google Scholar

[31]

H. ZhangF. LiuI. Turner and Q. Yang, Numerical solution of the time fractional Black-Scholes model governing European options, Comput. Math. Appl., 71 (2016), 1772-1783.  doi: 10.1016/j.camwa.2016.02.007.  Google Scholar

show all references

References:
[1]

V. R. Ambati and O. Bokhove, Space-time discontinuous Galerkin finite element method for shallow water flows, J. Comput. Appl. Math., 204 (2007), 452-462.  doi: 10.1016/j.cam.2006.01.047.  Google Scholar

[2]

C. Canuto, M. Y. Hussaini, A. Z. Quarteroni and T. Zang, Option Pricing: Mathematical Models and Computation, Springer, 1993. Google Scholar

[3]

C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods, Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation, Springer, Berlin, 2007.  Google Scholar

[4]

C. Chen, A. Karageorghis and Y. Smyrlis, The Method of Fundamental Solutions: A Meshless Method, Dynamic Publishers Atlanta, 2008. Google Scholar

[5]

A. Cohen, Numerical Analysis of Wavelet Methods, Studies in Mathematics and its Applications, 32. North-Holland Publishing Co., Amsterdam, 2003.  Google Scholar

[6]

J. C. CoxS. A. Ross and M. Rubinstein, Option pricing: A simplified approach, J. Financ. Econ., 7 (1979), 229-263.  doi: 10.1016/0304-405X(79)90015-1.  Google Scholar

[7]

G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, With 1 CD-ROM (Windows, Macintosh and UNIX), Interdisciplinary Mathematical Sciences, 6. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/6437.  Google Scholar

[8]

G. E. Fasshauer and M. McCourt, Kernel-Based Approximation Methods Using Matlab, World Scientific pub. Co, 2015. Google Scholar

[9]

G. E. Fasshauer and J. G. Zhang, On choosing optimal shape parameters for RBF approximation, Numerical Algorithms, 45 (2007), 345-368.  doi: 10.1007/s11075-007-9072-8.  Google Scholar

[10]

R. Geske and K. Shastri, Valuation by approximation: A comparison of alternative option valuation techniques, Journal of Financial and Quantitative Analysis, 20 (1985), 45-71.  doi: 10.2307/2330677.  Google Scholar

[11]

M. HamaidiA. Naji and A. Charafi, Space-time localized radial basis function collocation method for solving parabolic and hyperbolic equations, Eng. Anal. Bound. Elem., 67 (2016), 152-163.  doi: 10.1016/j.enganabound.2016.03.009.  Google Scholar

[12]

J. Hull and A. White, The use of the control variate technique in option pricing, Journal of Financial and Quantitative analysis, 23 (1988), 237-251.  doi: 10.2307/2331065.  Google Scholar

[13]

M. K. KadalbajooL. P. Tripathi and A. Kumar, A cubic B-spline collocation method for a numerical solution of the generalized Black-Scholes equation, Math. Comput. Modelling, 55 (2012), 1483-1505.  doi: 10.1016/j.mcm.2011.10.040.  Google Scholar

[14]

C. M. KlaijaJ. J. W. van der Vegta and H. van der Venb, Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations, J. Comput. Phys., 217 (2006), 589-611.  doi: 10.1016/j.jcp.2006.01.018.  Google Scholar

[15]

M. LiW. Chen and C. S. Chen, The localized RBFs collocation methods for solving high dimensional PDEs, Eng. Anal. Bound. Elem., 37 (2013), 1300-1304.  doi: 10.1016/j.enganabound.2013.06.001.  Google Scholar

[16]

Z. Li and X. Z. Mao, Global multiquadric collocation method for groundwater contaminant source identification, Environmental modelling & software, 26 (2011), 1611-1621.  doi: 10.1016/j.envsoft.2011.07.010.  Google Scholar

[17]

Z. Li and X. Z. Mao, Global space-time multiquadric method for inverse heat conduction problem, Internat. J. Numer. Methods Engrg., 85 (2011), 355-379.  doi: 10.1002/nme.2975.  Google Scholar

[18]

F. Moukalled, L. Mangani and M. Darwish, The Finite Volume Method in Computational Fluid Dynamics, Fluid Mechanics and its Applications, 113. Springer, Cham, 2016. doi: 10.1007/978-3-319-16874-6.  Google Scholar

[19]

H. Netuzhylov, A Space-Time Meshfree Collocation Method for Coupled Problems on Irregularly-Shaped Domains, Ph.D thesis, Zugl., Braunschweig, Techn. Univ., Diss., 2008. Google Scholar

[20]

R. Mohammadi, Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing, Comput. Math. Appl., 69 (2015), 777-797.  doi: 10.1016/j.camwa.2015.02.018.  Google Scholar

[21]

S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, 39. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-540-68093-2.  Google Scholar

[22]

J. J.SudirhamJ. J. W. van der Vegt and R. M. J. van Damme, Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains, Appl. Numer. Math., 56 (2006), 1491-1518.  doi: 10.1016/j.apnum.2005.11.003.  Google Scholar

[23]

T. E. TezduyarS. SatheR. Keedy and K. Stein, Space-time finite element techniques for computation of fluid-structure interactions, Comput. Methods Appl. Mech. Engrg., 195 (2006), 2002-2027.  doi: 10.1016/j.cma.2004.09.014.  Google Scholar

[24]

C. TurchettiM. ContiP. Crippa and S. Orcioni, On the approximation of stochastic processes by approximate identity neural networks, IEEE Transactions on Neural Networks, 9 (1998), 1069-1085.  doi: 10.1109/72.728353.  Google Scholar

[25]

C. TurchettiP. CrippaM. Pirani and G. Biagetti, Representation of nonlinear random transformations by non-Gaussian stochastic neural networks, IEEE transactions on neural networks, 19 (2008), 1033-1060.  doi: 10.1109/TNN.2007.2000055.  Google Scholar

[26]

M. UddinH. Ali and A. Ali, Kernel-based local meshless method for solving multi-dimensional wave equations in irregular domain, CMES-Computer Modeling In Engineering & Sciences, 107 (2015), 463-479.   Google Scholar

[27]

M. Uddin and H. Ali, The space time kernel based numerical method for Burgers equations, Mathematics, 6 (2018), 212-222.  doi: 10.3390/math6100212.  Google Scholar

[28]

M. UddinK. KamranM. Usman and A. Ali, On the Laplace-transformed-based local meshless method for fractional-order diffusion equation, Int. J. Comput. Methods Eng. Sci. Mech., 19 (2018), 221-225.  doi: 10.1080/15502287.2018.1472150.  Google Scholar

[29]

B. M. Vaganan and E. E. Priya, Generalized Cole-Hopf transformations for generalized Burgers equations, Pramana, 85 (2015), 861-867.  doi: 10.1007/s12043-015-1107-4.  Google Scholar

[30]

D. L. YoungC. C. TsaiK. MurugesanaC. M. Fan and C. W. Chen, Time-dependent fundamental solutions for homogeneous diffusion problems, Engineering Analysis with Boundary Elements, 28 (2004), 1463-1473.  doi: 10.1016/j.enganabound.2004.07.003.  Google Scholar

[31]

H. ZhangF. LiuI. Turner and Q. Yang, Numerical solution of the time fractional Black-Scholes model governing European options, Comput. Math. Appl., 71 (2016), 1772-1783.  doi: 10.1016/j.camwa.2016.02.007.  Google Scholar

Figure 1.  A typical centers arrangements in global space-time domain as well as in a local sub-domain, and sparsity of descretized operator of problem 1, where $ m = 100 $, $ n = 10 $
Figure 2.  The exact solution versus the approximate solution in space-time domain corresponding to problem 1, when $ m = 1600 $ and $ n = 10 $ in domain $ (x,t)\in (-2,2)\times(0,1) $
Figure 3.  The numerical solution and error in space-time domain, corresponding to problem 2 when $ m = 400 $ and $ n = 10 $ in the domain $ (x,t)\in (0,1)\times(0,1) $
Figure 4.  Double barrier option prices obtained by space-time local kernel method
Figure 5.  Call option prices obtained by space-time local kernel method
Figure 6.  Put option prices obtained by space-time local kernel method
Table 1.  Sapce-time (ST) solution of problem 1 for different total collocation points $ m $ and stencil size $ n $, and time integration (TI) solution in domain $ (x,t)\in (-2,2)\times(0,1) $
$ m $ $ n $ $ L_{\infty} $ ST method (C.time) TI method (C.time)
100 10 1.23E-02 0.6783 3.2891
400 9.49E-02 0.7123 6.2821
1600 1.08E-04 0.9129 10.2370
2500 1.26E-04 1.1234 15.2950
100 15 2.37E-02 0.8234 4.3491
400 1.45E-03 10.2356 9.2371
1600 2.45E-03 11.2916 13.9820
2500 3.45E-04 13.1087 20.3582
100 20 2.22E-02 9.1835 10.10491
400 2.89E-03 11.2349 12.7146
1600 9.88E-03 14.8679 21.3812
2500 7.23E-04 16.1955 25.0492
$ m $ $ n $ $ L_{\infty} $ ST method (C.time) TI method (C.time)
100 10 1.23E-02 0.6783 3.2891
400 9.49E-02 0.7123 6.2821
1600 1.08E-04 0.9129 10.2370
2500 1.26E-04 1.1234 15.2950
100 15 2.37E-02 0.8234 4.3491
400 1.45E-03 10.2356 9.2371
1600 2.45E-03 11.2916 13.9820
2500 3.45E-04 13.1087 20.3582
100 20 2.22E-02 9.1835 10.10491
400 2.89E-03 11.2349 12.7146
1600 9.88E-03 14.8679 21.3812
2500 7.23E-04 16.1955 25.0492
Table 2.  Observed maximum absolute error for example 1 in Reza [20] and Kadabajoo [13] for different $ \theta $ in domain $ (x,t)\in (-2,2)\times(0,1) $
$ M = N $ 10 20 40 80 160
[20] for $ \theta=1 $ 7.24E-02 3.12E-02 1.39E-02 6.08E-02 2.71E-04
[20] for $ \theta=\frac{1}{2} $ 1.12E-03 2.08E-02 3.91E-05 7.19E-06 1.31E-06
[13] for $ \theta=1 $ 7.44E-02 3.94E-02 2.02E-02 1.02E-02 5.16E-03
[13] for $ \theta=\frac{1}{2} $ 5.89E-03 1.46E-03 3.64E-04 9.10E-05 2.27E-05
$ M = N $ 10 20 40 80 160
[20] for $ \theta=1 $ 7.24E-02 3.12E-02 1.39E-02 6.08E-02 2.71E-04
[20] for $ \theta=\frac{1}{2} $ 1.12E-03 2.08E-02 3.91E-05 7.19E-06 1.31E-06
[13] for $ \theta=1 $ 7.44E-02 3.94E-02 2.02E-02 1.02E-02 5.16E-03
[13] for $ \theta=\frac{1}{2} $ 5.89E-03 1.46E-03 3.64E-04 9.10E-05 2.27E-05
[1]

Rodrigue Gnitchogna Batogna, Abdon Atangana. Generalised class of Time Fractional Black Scholes equation and numerical analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 435-445. doi: 10.3934/dcdss.2019028

[2]

Rehana Naz, Imran Naeem. Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020122

[3]

Erik Ekström, Johan Tysk. A boundary point lemma for Black-Scholes type operators. Communications on Pure & Applied Analysis, 2006, 5 (3) : 505-514. doi: 10.3934/cpaa.2006.5.505

[4]

Kais Hamza, Fima C. Klebaner. On nonexistence of non-constant volatility in the Black-Scholes formula. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 829-834. doi: 10.3934/dcdsb.2006.6.829

[5]

Junkee Jeon, Jehan Oh. (1+2)-dimensional Black-Scholes equations with mixed boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (2) : 699-714. doi: 10.3934/cpaa.2020032

[6]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216

[7]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

[8]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[9]

Yanzhao Cao, Li Yin. Spectral Galerkin method for stochastic wave equations driven by space-time white noise. Communications on Pure & Applied Analysis, 2007, 6 (3) : 607-617. doi: 10.3934/cpaa.2007.6.607

[10]

Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212

[11]

Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020223

[12]

Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845

[13]

Susanne Pumplün, Thomas Unger. Space-time block codes from nonassociative division algebras. Advances in Mathematics of Communications, 2011, 5 (3) : 449-471. doi: 10.3934/amc.2011.5.449

[14]

Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599

[15]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713

[16]

Frédérique Oggier, B. A. Sethuraman. Quotients of orders in cyclic algebras and space-time codes. Advances in Mathematics of Communications, 2013, 7 (4) : 441-461. doi: 10.3934/amc.2013.7.441

[17]

Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167

[18]

David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35

[19]

Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257

[20]

Vincent Astier, Thomas Unger. Galois extensions, positive involutions and an application to unitary space-time coding. Advances in Mathematics of Communications, 2019, 13 (3) : 513-516. doi: 10.3934/amc.2019032

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (54)
  • HTML views (158)
  • Cited by (0)

Other articles
by authors

[Back to Top]