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Abstract
In finite-dimensional commutative associative algebra, the concept of σ-monogenic function is introduced. Necessary and sufficient conditions for σ-monogeneity have been established. In some low-dimensional algebras, with a special choice of σ, the representation of σ-monogenic functions is obtained using holomorphic functions of a complex variable. We proposed the application of σ-monogenic functions with values in two-dimensional biharmonic algebra to representation of solutions of two-dimensional biharmonic equation.
Keywords:
commutative associative algebra, monogenic function, $\sigma$-monogenic function, constructive description of $\sigma$-monogenic function, generalization of pseudoanalytic functions, generalization of $(p,q)$-analytic functions
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How to Cite
Shpakivskyi, V. (2023). σ-monogenic functions in commutative algebras. Proceedings of the International Geometry Center, 16(1), 17-41. https://doi.org/10.15673/tmgc.v16i1.2421
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References
1. A. K. Bakhtin. Analytic functions of vector argument and partially-conformal mappings in multidimensional complex spaces. In Progress in Analysis, Proceedings of the 8th Congress of the International Society for Analysis, its Applications, and Computation, pages 1-8, 2011.
2. A. K. Bakhtin. A generalization of some results in the theory of schlicht functions to multidimensional complex spaces. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (3):7-11, 2011.
3. L. Bers. Theory of pseudo-analytic functions. New York University, Institute for Mathematics and Mechanics, New York, 1953.
4. L. Bers and A. Gelbart. On a class of differential equations in mechanics of continua. Quart. Appl. Math., 1:168-188, 1943. https://doi.org/10.1090/qam/8556 pathdoi:10.1090/qam/8556.
5. E. Cartan. Les groupes bilineaires et les syst`emes de nombres complexes. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 12(1):B1-B64, 1898. URL: http://www.numdam.org/item?id=AFST_1898_1_12_1_B1_0.
6. S. V. Grishchuk and S. A. Plaksa. Monogenic functions in a biharmonic algebra. Ukrainian Math. J., 61(12):1587-1596, 2009. https://doi.org/10.1007/s11253-010-0319-5 pathdoi:10.1007/s11253-010-0319-5.
7. V. V. Kravchenko. Applied pseudoanalytic function theory. Frontiers in Mathematics. Birkhauser Verlag, Basel, 2009. With a foreword by Wolfgang Sproessig. https://doi.org/10.1007/978-3-0346-0004-0 pathdoi:10.1007/978-3-0346-0004-0.
8. E. R. Lorch. The theory of analytic function in normed abelian vector rings. Trans. Amer. Math. Soc., 54(17):414-425, 1943. https://doi.org/10.2307/1990255 pathdoi:10.2307/1990255.
9. M. E. Luna-Elizarraras, M. Shapiro, D. C. Struppa, and A. Vajiac. Bicomplex holomorphic functions: The Algebra, Geometry and Analysis of Bicomplex Numbers. 2015. https://doi.org/10.1007/978-3-319-24868-4 pathdoi:10.1007/978-3-319-24868-4.
10. H. Malonek. Generalizing the (f,g)-derivative in the sense of bers. In: Clifford Algebras and Their Application in Mathematical Physics, Dordrecht: Kluwer Academic Publishers, pages 247-257, 1996. https://doi.org/10.1007/978-94-011-5036-1_20 pathdoi:10.1007/978-94-011-5036-1_20.
11. I. P. Mel'nichenko. The representation of harmonic mappings by monogenic functions. Ukrainian Math. J., 27(5):499-505, 1975. https://doi.org/10.1007/BF01089142 pathdoi:10.1007/BF01089142.
12. E. Picard. Sur une generalization des equations de la theorie des functions d'une variable complex. C. R. Math. Acad. Sci. Paris, 112:139-140, 1891.
13. E. Picard. Sur une syst`eme des equations aux derivees partielles. C. R. Math. Acad. Sci. Paris, 112:685-688, 1891.
14. S. A. Plaksa and R. P. Pukhtaievych. Monogenic functions in a finite-dimensional semi-simple commutative algebra. An. cStiint. Univ. \"Ovidius\" Constanta Ser. Mat., 22(1):221-235, 2014. https://doi.org/10.2478/auom-2014-0018 pathdoi:10.2478/auom-2014-0018.
15. S. A. Plaksa and V. S. Shpakivskyi. Constructive description of monogenic functions in a harmonic algebra of the third rank. Ukrainian Math. J., 62(8):1078-1091, 2010. https://doi.org/10.1007/s11253-011-0427-x pathdoi:10.1007/s11253-011-0427-x.
16. S. A. Plaksa and V. S. Shpakivskyi. Cauchy theorem for a surface integral in commutative algebras. Complex Var. Elliptic Equ., 59(1):110-119, 2014. https://doi.org/10.1080/17476933.2013.845178 pathdoi:10.1080/17476933.2013.845178.
17. G. N. Polozhii. On p-analytic functions of a complex variable. Dokl. Akad. Nauk SSSR, 58:1275-1278, 1947.
18. G. N. Polozhii. On the question of (p,q)-analytic functions of a complex variable and their applications. Revue de Math. pures et appl. Acad. de la Rep. pop. Roumaine, 2:331-361, 1957.
19. G. N. Polozhii. The theory and application of p-analytic and and (p,q)-analytic functions. generalizations of the theory of analytic functions of a complex variable. 2nd ed. Naukova Dumka, Kyiv, 1973.
20. D. Rochon. On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrodinger equation. Complex Var. Elliptic Equ., 53(6):501-521, 2008. https://doi.org/10.1080/17476930701769058 pathdoi:10.1080/17476930701769058.
21. M. N. Rosculet. Funcctii monogene pe algebre comutative. Editura Academiei Republicii Socialiste Rom^ania, Bucharest, 1975. With a French summary and table of contents.
22. V. S. Shpakivskyi. Integral theorems for monogenic functions in commutative algebras. Zb. Pr. Inst. Mat. NAN Ukr., 12(4):313-328, 2015.
23. V. S. Shpakivskyi. Monogenic functions in finite-dimensional commutative associative algebras. Zb. Pr. Inst. Mat. NAN Ukr., 12(3):251-268, 2015.
24. V. S. Shpakivskyi. Constructive description of monogenic functions in a finite-dimensional commutative associative algebra. Adv. Pure Appl. Math., 7(1):63-75, 2016. https://doi.org/10.1515/apam-2015-0022 pathdoi:10.1515/apam-2015-0022.
25. V. S. Shpakivskyi. Curvilinear integral theorems for monogenic functions in commutative associative algebras. Adv. Appl. Clifford Algebr., 26(1):417-434, 2016. https://doi.org/10.1007/s00006-015-0561-x pathdoi:10.1007/s00006-015-0561-x.
26. V. S. Shpakivskyi. Hypercomplex method for solving linear partial differential equations. Proc. of the Inst. Appl. Math. Mech. NAS Ukraine, 32:160-181, 2018. https://doi.org/10.37069/1683-4720-2018-32-16 pathdoi:10.37069/1683-4720-2018-32-16.
27. V. S. Shpakivskyi. On monogene functions on extensions of commutative algebra. Proc. Int. Geom. Cent., 11(3):1-18, 2018. https://doi.org/10.15673/tmgc.v11i3.1200 pathdoi:10.15673/tmgc.v11i3.1200.
28. V. S. Shpakivskyi. On monogenic functions defined in different commutative algebras. J. Math. Sci. (N. Y.), 239(1):92-109, 2019. https://doi.org/10.1007/s10958-019-04291-0 pathdoi:10.1007/s10958-019-04291-0.
29. I. N. Vekua. Generalized Analytic Functions. ADIWES international series in mathematics. Pergamon Press, 1962. URL: https://books.google.com.ua/books?id=PfQyAAAAMAAJ.
30. N. Virchenko and O. Ovcharenko. The main properties of q-functions. Research Bulletin of the National Technical University of Ukraine Kyiv Politechnic Institute, (4):32-38, 2017
2. A. K. Bakhtin. A generalization of some results in the theory of schlicht functions to multidimensional complex spaces. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (3):7-11, 2011.
3. L. Bers. Theory of pseudo-analytic functions. New York University, Institute for Mathematics and Mechanics, New York, 1953.
4. L. Bers and A. Gelbart. On a class of differential equations in mechanics of continua. Quart. Appl. Math., 1:168-188, 1943. https://doi.org/10.1090/qam/8556 pathdoi:10.1090/qam/8556.
5. E. Cartan. Les groupes bilineaires et les syst`emes de nombres complexes. Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 12(1):B1-B64, 1898. URL: http://www.numdam.org/item?id=AFST_1898_1_12_1_B1_0.
6. S. V. Grishchuk and S. A. Plaksa. Monogenic functions in a biharmonic algebra. Ukrainian Math. J., 61(12):1587-1596, 2009. https://doi.org/10.1007/s11253-010-0319-5 pathdoi:10.1007/s11253-010-0319-5.
7. V. V. Kravchenko. Applied pseudoanalytic function theory. Frontiers in Mathematics. Birkhauser Verlag, Basel, 2009. With a foreword by Wolfgang Sproessig. https://doi.org/10.1007/978-3-0346-0004-0 pathdoi:10.1007/978-3-0346-0004-0.
8. E. R. Lorch. The theory of analytic function in normed abelian vector rings. Trans. Amer. Math. Soc., 54(17):414-425, 1943. https://doi.org/10.2307/1990255 pathdoi:10.2307/1990255.
9. M. E. Luna-Elizarraras, M. Shapiro, D. C. Struppa, and A. Vajiac. Bicomplex holomorphic functions: The Algebra, Geometry and Analysis of Bicomplex Numbers. 2015. https://doi.org/10.1007/978-3-319-24868-4 pathdoi:10.1007/978-3-319-24868-4.
10. H. Malonek. Generalizing the (f,g)-derivative in the sense of bers. In: Clifford Algebras and Their Application in Mathematical Physics, Dordrecht: Kluwer Academic Publishers, pages 247-257, 1996. https://doi.org/10.1007/978-94-011-5036-1_20 pathdoi:10.1007/978-94-011-5036-1_20.
11. I. P. Mel'nichenko. The representation of harmonic mappings by monogenic functions. Ukrainian Math. J., 27(5):499-505, 1975. https://doi.org/10.1007/BF01089142 pathdoi:10.1007/BF01089142.
12. E. Picard. Sur une generalization des equations de la theorie des functions d'une variable complex. C. R. Math. Acad. Sci. Paris, 112:139-140, 1891.
13. E. Picard. Sur une syst`eme des equations aux derivees partielles. C. R. Math. Acad. Sci. Paris, 112:685-688, 1891.
14. S. A. Plaksa and R. P. Pukhtaievych. Monogenic functions in a finite-dimensional semi-simple commutative algebra. An. cStiint. Univ. \"Ovidius\" Constanta Ser. Mat., 22(1):221-235, 2014. https://doi.org/10.2478/auom-2014-0018 pathdoi:10.2478/auom-2014-0018.
15. S. A. Plaksa and V. S. Shpakivskyi. Constructive description of monogenic functions in a harmonic algebra of the third rank. Ukrainian Math. J., 62(8):1078-1091, 2010. https://doi.org/10.1007/s11253-011-0427-x pathdoi:10.1007/s11253-011-0427-x.
16. S. A. Plaksa and V. S. Shpakivskyi. Cauchy theorem for a surface integral in commutative algebras. Complex Var. Elliptic Equ., 59(1):110-119, 2014. https://doi.org/10.1080/17476933.2013.845178 pathdoi:10.1080/17476933.2013.845178.
17. G. N. Polozhii. On p-analytic functions of a complex variable. Dokl. Akad. Nauk SSSR, 58:1275-1278, 1947.
18. G. N. Polozhii. On the question of (p,q)-analytic functions of a complex variable and their applications. Revue de Math. pures et appl. Acad. de la Rep. pop. Roumaine, 2:331-361, 1957.
19. G. N. Polozhii. The theory and application of p-analytic and and (p,q)-analytic functions. generalizations of the theory of analytic functions of a complex variable. 2nd ed. Naukova Dumka, Kyiv, 1973.
20. D. Rochon. On a relation of bicomplex pseudoanalytic function theory to the complexified stationary Schrodinger equation. Complex Var. Elliptic Equ., 53(6):501-521, 2008. https://doi.org/10.1080/17476930701769058 pathdoi:10.1080/17476930701769058.
21. M. N. Rosculet. Funcctii monogene pe algebre comutative. Editura Academiei Republicii Socialiste Rom^ania, Bucharest, 1975. With a French summary and table of contents.
22. V. S. Shpakivskyi. Integral theorems for monogenic functions in commutative algebras. Zb. Pr. Inst. Mat. NAN Ukr., 12(4):313-328, 2015.
23. V. S. Shpakivskyi. Monogenic functions in finite-dimensional commutative associative algebras. Zb. Pr. Inst. Mat. NAN Ukr., 12(3):251-268, 2015.
24. V. S. Shpakivskyi. Constructive description of monogenic functions in a finite-dimensional commutative associative algebra. Adv. Pure Appl. Math., 7(1):63-75, 2016. https://doi.org/10.1515/apam-2015-0022 pathdoi:10.1515/apam-2015-0022.
25. V. S. Shpakivskyi. Curvilinear integral theorems for monogenic functions in commutative associative algebras. Adv. Appl. Clifford Algebr., 26(1):417-434, 2016. https://doi.org/10.1007/s00006-015-0561-x pathdoi:10.1007/s00006-015-0561-x.
26. V. S. Shpakivskyi. Hypercomplex method for solving linear partial differential equations. Proc. of the Inst. Appl. Math. Mech. NAS Ukraine, 32:160-181, 2018. https://doi.org/10.37069/1683-4720-2018-32-16 pathdoi:10.37069/1683-4720-2018-32-16.
27. V. S. Shpakivskyi. On monogene functions on extensions of commutative algebra. Proc. Int. Geom. Cent., 11(3):1-18, 2018. https://doi.org/10.15673/tmgc.v11i3.1200 pathdoi:10.15673/tmgc.v11i3.1200.
28. V. S. Shpakivskyi. On monogenic functions defined in different commutative algebras. J. Math. Sci. (N. Y.), 239(1):92-109, 2019. https://doi.org/10.1007/s10958-019-04291-0 pathdoi:10.1007/s10958-019-04291-0.
29. I. N. Vekua. Generalized Analytic Functions. ADIWES international series in mathematics. Pergamon Press, 1962. URL: https://books.google.com.ua/books?id=PfQyAAAAMAAJ.
30. N. Virchenko and O. Ovcharenko. The main properties of q-functions. Research Bulletin of the National Technical University of Ukraine Kyiv Politechnic Institute, (4):32-38, 2017