• Previous Article
    Adaptive attitude determination of bionic polarization integrated navigation system based on reinforcement learning strategy
  • MFC Home
  • This Issue
  • Next Article
    Concept and attribute reduction based on rectangle theory of formal concept
doi: 10.3934/mfc.2021044
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

On a special class of modified integral operators preserving some exponential functions

Department of Computer Technologies, Division of Technology of Security of Informatics, Karabuk University, Karabuk, Turkey

Received  August 2021 Revised  December 2021 Early access January 2022

In the present paper, we consider a general class of operators enriched with some properties in order to act on $ C^{\ast }( \mathbb{R} _{0}^{+}) $. We establish uniform convergence of the operators for every function in $ C^{\ast }( \mathbb{R} _{0}^{+}) $ on $ \mathbb{R} _{0}^{+} $. Then, a quantitative result is proved. A quantitative Voronovskaya-type estimate is obtained. Finally, some applications are provided concerning particular kernel functions.

Citation: Gümrah Uysal. On a special class of modified integral operators preserving some exponential functions. Mathematical Foundations of Computing, doi: 10.3934/mfc.2021044
References:
[1]

T. Acar, A. Aral and H. Gonska, On Szász-Mirakyan operators preserving $e^2ax$, $a>0$, Mediterr. J. Math., 14 (2017), Paper No. 6, 14 pp. doi: 10.1007/s00009-016-0804-7.

[2]

T. Acar, M. Mursaleen and S. N. Deveci, Gamma operators reproducing exponential functions, Adv. Difference Equ., (2020), Paper No. 423, 13 pp. doi: 10.1186/s13662-020-02880-x.

[3]

O. AgratiniA. Aral and E. Deniz, On two classes of approximation processes of integral type, Positivity, 21 (2017), 1189-1199.  doi: 10.1007/s11117-016-0460-y.

[4]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics 17., Walter De Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.

[5]

G. A. Anastassiou and S. G. Gal, Approximation Theory. Moduli of Continuity and Global Smoothness Preservation, Springer, Birkhäuser, Boston, 2000. doi: 10.1007/978-1-4612-1360-4.

[6]

L. Angeloni and G. Vinti, A review on approximation results for integral operators in the space of functions of bounded variation, J. Funct. Spaces, 2016 (2016), Art. ID 3843921, 11 pp. doi: 10.1155/2016/3843921.

[7]

P. M. Anselone and I. H. Sloan, Integral equations on the half line, J. of Integral Equations, 9 (1985), 3-23. 

[8]

A. Aral, On generalized Picard integral operators, Advances in Summability and Approximation Theory, (2018), 157–168. doi: 10.1007/978-981-13-3077-3_9.

[9]

A. AralD. Cárdenas-Morales and P. Garrancho, Bernstein-type operators that reproduce exponential functions, J. Math. Inequal., 12 (2018), 861-872.  doi: 10.7153/jmi-2018-12-64.

[10]

A. AralD. Inoan and I. Raşa, Approximation properties of Szász–Mirakyan operators preserving exponential functions, Positivity, 23 (2019), 233-246.  doi: 10.1007/s11117-018-0604-3.

[11]

A. Aral, B. Yılmaz and E. Deniz, A new construction of Picard operators on the semi-real axis, (2018), to appear.

[12]

F. Barbieri, Approximation by moment kernels, (Italian), Atti Sem. Mat. Fis. Univ. Modena, 32 (1983), 308-328. 

[13]

C. Bardaro and I. Mantellini, Voronovskaja-type estimates for Mellin convolution operators, Results Math., 50 (2007), 1-16.  doi: 10.1007/s00025-006-0231-3.

[14]

C. Bardaro and I. Mantellini, A quantitative Voronovskaya formula for Mellin convolution operators, Mediterr. J. Math., 7 (2010), 483-501.  doi: 10.1007/s00009-010-0062-z.

[15]

C. Bardaro and I. Mantellini, Multivariate moment type operators: Approximation properties in Orlicz spaces, J. Math. Inequal., 2 (2008), 247-259.  doi: 10.7153/jmi-02-22.

[16]

C. Bardaro, I. Mantellini, G. Uysal and B. Yılmaz, A class of integral operators that fix exponential functions, Mediterr. J. Math., 18 (2021), Paper No. 179, 21 pp. doi: 10.1007/s00009-021-01819-0.

[17]

C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications 9., Walter De Gruyter & Co., Berlin, 2003. doi: 10.1515/9783110199277.

[18]

H. Bohman, On approximation of continuous and of analytic functions, Ark. Mat., 2 (1952), 43-56.  doi: 10.1007/BF02591381.

[19]

B. D. Boyanov and V. M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 14 (1970), 9-13. 

[20] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation Vol. 1: One-Dimensional Theory, Pure and Applied Mathematics, Vol. 40. Academic Press, New York-London, 1971.  doi: 10.1007/978-3-0348-7448-9.
[21]

P. L. Butzer and S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3 (1997), 325-376.  doi: 10.1007/BF02649101.

[22]

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, Springer Texts Electrical Eng., Springer, New York, (1993), 157–183. doi: 10.1007/978-1-4613-9757-1_5.

[23]

D. Costarelli and G. Vinti, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim., 34 (2013), 819-844.  doi: 10.1080/01630563.2013.767833.

[24]

D. Costarelli and G. Vinti, Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators, Mathematical Foundations of Computing, 3 (2020), 41-50.  doi: 10.3934/mfc.2020004.

[25]

A. D. Gadžiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin's theorem, (Russian), Dokl. Akad. Nauk SSSR, 218 (1974), 1001-1004. 

[26]

V. Gupta and V. K. Singh, Modified Post-Widder operators preserving exponential functions, Advances in Mathematical Methods and High Performance Computing, 41 (2019), 181-192.  doi: 10.1007/978-3-030-02487-1_10.

[27]

V. Gupta and G. Tachev, On approximation properties of Phillips operators preserving exponential functions, Mediterr. J. Math., 14 (2017), Paper No. 177, 12 pp. doi: 10.1007/s00009-017-0981-z.

[28]

A. Holhoş, The rate of approximation of functions in an infinite interval by positive linear operators, Stud. Univ. Babeş–Bolyai Math., 55 (2010), 133–142.

[29]

A. Holhoş, Quantitative estimates of Voronovskaya type in weighted spaces, Results Math., 73 (2018), Paper No. 53, 11 pp. doi: 10.1007/s00025-018-0814-9.

[30]

H. Karslı, Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters, Appl. Anal., 85 (2006), 781-791.  doi: 10.1080/00036810600712665.

[31]

J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar., 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.

[32]

P. P. Korovkin, On convergence of linear positive operators in the spaces of continuous functions, (Russian), Doklady Akad. Nauk. SSSR (N.S.), 90 (1953), 961-964. 

[33]

P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corp., Delhi, 1960.

[34]

A. Lupaş and M. Müller, Approximationseigenschaften der Gammaoperatoren, (German), Math. Z., 98 (1967), 208-226.  doi: 10.1007/BF01112415.

[35]

R. G. Mamedov, The Mellin Transform and Approximation Theory, (Russian) "Elm", Baku, 1991.

[36]

C. P. May, Saturation and inverse theorems for combinations of a class of exponential-type operators, Canadian J. Math., 28 (1976), 1224-1250.  doi: 10.4153/CJM-1976-123-8.

[37]

I. P. Natanson, Theory of Functions of a Real Variable Vol. Ⅱ., Frederick Ungar Pub. Co., New York, 1961.

[38]

R. S. Phillips, An inversion formula for Laplace transforms and semi-groups of linear operators, Ann. of Math., 59 (1954), 325-356.  doi: 10.2307/1969697.

[39]

L. Rempulska and K. Tomczak, On some properties of the Picard operators, Arch. Math. (Brno), 45 (2009), 125-135. 

[40] L. L. Schumaker, Spline Functions: Basic Theory, 3$^rd$ edition, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618994.
[41]

T. Świderski and E. Wachnicki, Nonlinear singular integrals depending on two parameters, Comment. Math. (Prace Mat.), 40 (2000), 181-189. 

[42]

E. V. Voronovskaya, Determination of the asymptotic form of approximation of functions by the polynomials of S. N. Bernstein, Dokl. Akad. Nauk SSSR, A, (1932), 79–85.

[43]

E. Wachnicki and G. Krech, Approximation of functions by nonlinear singular integral operators depending on two parameters, Publ. Math. Debrecen, 92 (2018), 481-494.  doi: 10.5486/PMD.2018.8080.

[44] D. V. Widder, The Laplace Transform, Princeton Mathematical Series, Vol. 6. Princeton Univ. Press, Princeton, 1941. 
[45]

B. YılmazG. Uysal and A. Aral, Reconstruction of two approximation processes in order to reproduce $e^ax$ and $e^2ax$, $a>0$, J. Math. Inequal., 15 (2021), 1101-1118.  doi: 10.7153/jmi-2021-15-75.

show all references

References:
[1]

T. Acar, A. Aral and H. Gonska, On Szász-Mirakyan operators preserving $e^2ax$, $a>0$, Mediterr. J. Math., 14 (2017), Paper No. 6, 14 pp. doi: 10.1007/s00009-016-0804-7.

[2]

T. Acar, M. Mursaleen and S. N. Deveci, Gamma operators reproducing exponential functions, Adv. Difference Equ., (2020), Paper No. 423, 13 pp. doi: 10.1186/s13662-020-02880-x.

[3]

O. AgratiniA. Aral and E. Deniz, On two classes of approximation processes of integral type, Positivity, 21 (2017), 1189-1199.  doi: 10.1007/s11117-016-0460-y.

[4]

F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics 17., Walter De Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110884586.

[5]

G. A. Anastassiou and S. G. Gal, Approximation Theory. Moduli of Continuity and Global Smoothness Preservation, Springer, Birkhäuser, Boston, 2000. doi: 10.1007/978-1-4612-1360-4.

[6]

L. Angeloni and G. Vinti, A review on approximation results for integral operators in the space of functions of bounded variation, J. Funct. Spaces, 2016 (2016), Art. ID 3843921, 11 pp. doi: 10.1155/2016/3843921.

[7]

P. M. Anselone and I. H. Sloan, Integral equations on the half line, J. of Integral Equations, 9 (1985), 3-23. 

[8]

A. Aral, On generalized Picard integral operators, Advances in Summability and Approximation Theory, (2018), 157–168. doi: 10.1007/978-981-13-3077-3_9.

[9]

A. AralD. Cárdenas-Morales and P. Garrancho, Bernstein-type operators that reproduce exponential functions, J. Math. Inequal., 12 (2018), 861-872.  doi: 10.7153/jmi-2018-12-64.

[10]

A. AralD. Inoan and I. Raşa, Approximation properties of Szász–Mirakyan operators preserving exponential functions, Positivity, 23 (2019), 233-246.  doi: 10.1007/s11117-018-0604-3.

[11]

A. Aral, B. Yılmaz and E. Deniz, A new construction of Picard operators on the semi-real axis, (2018), to appear.

[12]

F. Barbieri, Approximation by moment kernels, (Italian), Atti Sem. Mat. Fis. Univ. Modena, 32 (1983), 308-328. 

[13]

C. Bardaro and I. Mantellini, Voronovskaja-type estimates for Mellin convolution operators, Results Math., 50 (2007), 1-16.  doi: 10.1007/s00025-006-0231-3.

[14]

C. Bardaro and I. Mantellini, A quantitative Voronovskaya formula for Mellin convolution operators, Mediterr. J. Math., 7 (2010), 483-501.  doi: 10.1007/s00009-010-0062-z.

[15]

C. Bardaro and I. Mantellini, Multivariate moment type operators: Approximation properties in Orlicz spaces, J. Math. Inequal., 2 (2008), 247-259.  doi: 10.7153/jmi-02-22.

[16]

C. Bardaro, I. Mantellini, G. Uysal and B. Yılmaz, A class of integral operators that fix exponential functions, Mediterr. J. Math., 18 (2021), Paper No. 179, 21 pp. doi: 10.1007/s00009-021-01819-0.

[17]

C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications 9., Walter De Gruyter & Co., Berlin, 2003. doi: 10.1515/9783110199277.

[18]

H. Bohman, On approximation of continuous and of analytic functions, Ark. Mat., 2 (1952), 43-56.  doi: 10.1007/BF02591381.

[19]

B. D. Boyanov and V. M. Veselinov, A note on the approximation of functions in an infinite interval by linear positive operators, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.), 14 (1970), 9-13. 

[20] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation Vol. 1: One-Dimensional Theory, Pure and Applied Mathematics, Vol. 40. Academic Press, New York-London, 1971.  doi: 10.1007/978-3-0348-7448-9.
[21]

P. L. Butzer and S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl., 3 (1997), 325-376.  doi: 10.1007/BF02649101.

[22]

P. L. Butzer and R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, Springer Texts Electrical Eng., Springer, New York, (1993), 157–183. doi: 10.1007/978-1-4613-9757-1_5.

[23]

D. Costarelli and G. Vinti, Approximation by nonlinear multivariate sampling Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim., 34 (2013), 819-844.  doi: 10.1080/01630563.2013.767833.

[24]

D. Costarelli and G. Vinti, Asymptotic expansions and Voronovskaja type theorems for the multivariate neural network operators, Mathematical Foundations of Computing, 3 (2020), 41-50.  doi: 10.3934/mfc.2020004.

[25]

A. D. Gadžiev, A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P. P. Korovkin's theorem, (Russian), Dokl. Akad. Nauk SSSR, 218 (1974), 1001-1004. 

[26]

V. Gupta and V. K. Singh, Modified Post-Widder operators preserving exponential functions, Advances in Mathematical Methods and High Performance Computing, 41 (2019), 181-192.  doi: 10.1007/978-3-030-02487-1_10.

[27]

V. Gupta and G. Tachev, On approximation properties of Phillips operators preserving exponential functions, Mediterr. J. Math., 14 (2017), Paper No. 177, 12 pp. doi: 10.1007/s00009-017-0981-z.

[28]

A. Holhoş, The rate of approximation of functions in an infinite interval by positive linear operators, Stud. Univ. Babeş–Bolyai Math., 55 (2010), 133–142.

[29]

A. Holhoş, Quantitative estimates of Voronovskaya type in weighted spaces, Results Math., 73 (2018), Paper No. 53, 11 pp. doi: 10.1007/s00025-018-0814-9.

[30]

H. Karslı, Convergence and rate of convergence by nonlinear singular integral operators depending on two parameters, Appl. Anal., 85 (2006), 781-791.  doi: 10.1080/00036810600712665.

[31]

J. P. King, Positive linear operators which preserve x2, Acta Math. Hungar., 99 (2003), 203-208.  doi: 10.1023/A:1024571126455.

[32]

P. P. Korovkin, On convergence of linear positive operators in the spaces of continuous functions, (Russian), Doklady Akad. Nauk. SSSR (N.S.), 90 (1953), 961-964. 

[33]

P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan Publishing Corp., Delhi, 1960.

[34]

A. Lupaş and M. Müller, Approximationseigenschaften der Gammaoperatoren, (German), Math. Z., 98 (1967), 208-226.  doi: 10.1007/BF01112415.

[35]

R. G. Mamedov, The Mellin Transform and Approximation Theory, (Russian) "Elm", Baku, 1991.

[36]

C. P. May, Saturation and inverse theorems for combinations of a class of exponential-type operators, Canadian J. Math., 28 (1976), 1224-1250.  doi: 10.4153/CJM-1976-123-8.

[37]

I. P. Natanson, Theory of Functions of a Real Variable Vol. Ⅱ., Frederick Ungar Pub. Co., New York, 1961.

[38]

R. S. Phillips, An inversion formula for Laplace transforms and semi-groups of linear operators, Ann. of Math., 59 (1954), 325-356.  doi: 10.2307/1969697.

[39]

L. Rempulska and K. Tomczak, On some properties of the Picard operators, Arch. Math. (Brno), 45 (2009), 125-135. 

[40] L. L. Schumaker, Spline Functions: Basic Theory, 3$^rd$ edition, Cambridge University Press, Cambridge, 2007.  doi: 10.1017/CBO9780511618994.
[41]

T. Świderski and E. Wachnicki, Nonlinear singular integrals depending on two parameters, Comment. Math. (Prace Mat.), 40 (2000), 181-189. 

[42]

E. V. Voronovskaya, Determination of the asymptotic form of approximation of functions by the polynomials of S. N. Bernstein, Dokl. Akad. Nauk SSSR, A, (1932), 79–85.

[43]

E. Wachnicki and G. Krech, Approximation of functions by nonlinear singular integral operators depending on two parameters, Publ. Math. Debrecen, 92 (2018), 481-494.  doi: 10.5486/PMD.2018.8080.

[44] D. V. Widder, The Laplace Transform, Princeton Mathematical Series, Vol. 6. Princeton Univ. Press, Princeton, 1941. 
[45]

B. YılmazG. Uysal and A. Aral, Reconstruction of two approximation processes in order to reproduce $e^ax$ and $e^2ax$, $a>0$, J. Math. Inequal., 15 (2021), 1101-1118.  doi: 10.7153/jmi-2021-15-75.

Figure 1.  $ a = \frac{1}{2} $ and $ n = 5 $
Figure 2.  $ a = \frac{3}{4} $ and $ n = 5 $
Figure 3.  $ a = \frac{3}{10} $ and $ n = 5 $
Figure 4.  $ a = \frac{1}{2} $ and $ n = 5 $
[1]

Michele Campiti. Korovkin-type approximation of set-valued and vector-valued functions. Mathematical Foundations of Computing, 2022, 5 (3) : 231-239. doi: 10.3934/mfc.2021032

[2]

Parveen Bawa, Neha Bhardwaj, P. N. Agrawal. Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. Mathematical Foundations of Computing, 2022, 5 (4) : 269-293. doi: 10.3934/mfc.2022003

[3]

Márcio Cavalcante, Chulkwang Kwak. Local well-posedness of the fifth-order KdV-type equations on the half-line. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2607-2661. doi: 10.3934/cpaa.2019117

[4]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[5]

Purshottam Narain Agrawal, Şule Yüksel Güngör, Abhishek Kumar. Better degree of approximation by modified Bernstein-Durrmeyer type operators. Mathematical Foundations of Computing, 2022, 5 (2) : 75-92. doi: 10.3934/mfc.2021024

[6]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028

[7]

António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163

[8]

Virginie Bonnaillie-Noël. Harmonic oscillators with Neumann condition on the half-line. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2221-2237. doi: 10.3934/cpaa.2012.11.2221

[9]

İsmail Aslan, Türkan Yeliz Gökçer. Approximation by pseudo-linear discrete operators. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021037

[10]

Lucian Coroianu, Danilo Costarelli, Sorin G. Gal, Gianluca Vinti. Approximation by multivariate max-product Kantorovich-type operators and learning rates of least-squares regularized regression. Communications on Pure and Applied Analysis, 2020, 19 (8) : 4213-4225. doi: 10.3934/cpaa.2020189

[11]

Peter Howard, Alim Sukhtayev. The Maslov and Morse indices for Sturm-Liouville systems on the half-line. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 983-1012. doi: 10.3934/dcds.2020068

[12]

Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841

[13]

Bogdan Sasu, Adina Luminiţa Sasu. On the dichotomic behavior of discrete dynamical systems on the half-line. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3057-3084. doi: 10.3934/dcds.2013.33.3057

[14]

Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems and Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475

[15]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511

[16]

Xiaomei Chen, Xiaohui Yu. Liouville type theorem for Hartree-Fock Equation on half space. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2079-2100. doi: 10.3934/cpaa.2022050

[17]

Cristina Stoica. An approximation theorem in classical mechanics. Journal of Geometric Mechanics, 2016, 8 (3) : 359-374. doi: 10.3934/jgm.2016011

[18]

Danilo Costarelli. Preface: Special issue on approximation by linear and nonlinear operators with applications. Part Ⅱ. Mathematical Foundations of Computing, 2022, 5 (3) : ⅰ-ⅱ. doi: 10.3934/mfc.2022010

[19]

Angkana Rüland, Mikko Salo. Quantitative approximation properties for the fractional heat equation. Mathematical Control and Related Fields, 2020, 10 (1) : 1-26. doi: 10.3934/mcrf.2019027

[20]

Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control and Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007

 Impact Factor: 

Article outline

Figures and Tables

[Back to Top]