2020 - Doctor of Sciences (in Physics and Mathematics, Mathematical Analysis).
Dissertation Title: Monogenic functions in commutative algebras, Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine
2018 - 2020 - Doctoral studies, Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine
2011 - Ph.D. (in Physics and Mathematics, Mathematical Analysis).
Theses: Integral theorems for monogenic functions in three-dimensional harmonic algebra,
Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine
2008 - 2011 - Ph.D. studies under the guidance of Professor S. A. Plaksa,
Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv, Ukraine
2003 - 2008 - Master's degree, diploma with honors, Faculty of Physics and Mathematics,
Zhytomyr Ivan Franko State University, Zhytomyr, Ukraine
1. We have developed an algebraic-analytic approach to
equations in mathematical physics. The idea of this approach is to find commutative Banach
algebras such that the monogenic functions given in them form an algebra and have components
that satisfy the given partial differential equations. We have obtained a constructive description
of monogenic functions defined in the domains of special subspaces of an arbitrary finite-dimensional
commutative associative algebra over the field C, with values in this algebra using holomorphic functions of a complex variable.
2. Analogues of integral theorems (Cauchy integral theorem
for curvilinear and surface integrals, Cauchy integral formula, Morera theorem) are proved
for monogenic functions with values in an arbitrary finite-dimensional commutative associative
algebra, as well as in an infinite-dimensional commutative algebra and a topological vector space
with commutative multiplication, associated with the three-dimensional Laplace equation.
3. The relations between monogenic functions
with values in algebras that form a sequence of extensions of commutative algebras of
a certain class is established. The indicated monogenic functions are used to construct
an infinite-dimensional family of solutions of linear homogeneous PDEs with constant coefficients.
4. Various types of differentiability in the algebra of
complex quaternions are studied. A class of quaternion G-monogenic mappings is introduced and
their constructive description is obtained using holomorphic functions of a complex variable;
analogues of integral theorems for mappings of this class are proved. A relation is established
between the known classes of quaternion differential functions and functions analytic according
to Hausdorff, and a connection is established between the known definitions of derivatives and the Hausdorff derivative.
5. An algorithm for constructing functions belonging to
the kernel of the Dirac operator in generalized Cayley-Dickson algebras has been developed.
A constructive description of various classes of hyperholomorphic functions in the algebra of complex quaternions has been obtained.
2023 - Plaksa S.A. and Shpakivskyi V.S.Monogenic functions in spaces with commutative multiplication and applications.
Frontiers in Mathematics, Birkhäuser Cham. DOI: 10.1007/978-3-031-32254-9
Chapters in books or edited volumes:
2009 - Shpakivskyi V. S., Plaksa S. A. Integral theorems in a commutative three-dimensional harmonic.
In: Progress in Analysis and its Applications: Proc. of the 7th Intern. ISAAC Congress, London: World Scientific, 2009, 232 – 239.
DOI: 10.1142/7791
2010 - Shpakivskyi V. S. Power series and Laurent series in three-dimensional harmonic algebra.
In: Proc. of the Institute of Mathematics of the National Academy of Sciences of Ukraine,
9,No. 2, 384 – 391.
2012 - Shpakivskyi V. S.On the isomorphism of functional algebras in a harmonic algebra with a two-dimensional radical.
In: Proc. of the Institute of Mathematics of the National Academy of Sciences of Ukraine,
7,No. 2, 314 – 321.
2013 - Shpakivskyi V. S., Kuzmenko T. S. Monogenic functions of double variable,
In: Proc. of the Institute of Mathematics of the National Academy of Sciences of Ukraine,
10,No. 4-5, 372 – 378.
2015 - Shpakivskyi V. S. Monogenic functions in finite-dimensional commutative associative algebras ,
In: Proc. of the Institute of Mathematics of the National Academy of Sciences of Ukraine, 12,No. 3, 251 – 268.
2015 - Shpakivskyi V. S. Integral theorems for monogenic functions in commutative algebras ,
In: Proc. of the Institute of Mathematics of the National Academy of Sciences of Ukraine, 12,No. 4, 313 – 328.
2019 - Kuzmenko T. S., Shpakivskyi V. S. A theory of quaternionic G-monogenic mappings in E_3 ,
In: Models and Theories in Social Systems (Eds. C. Flaut etc.). – Springer, Vol. 179. – P. 451 – 508. DOI: 10.1007/978-3-030-00084-4
2019 - Shpakivskyi V. Monogenic functions in commutative algebras ,
In:Analysis, Probability, Applications, and Computation, Trends in Mathematics, (Eds. K.-O. Lindahl et al.). – Springer, 2019. – P. 171 – 178. DOI: 10.1007/978-3-030-04459-6
2022 - Shpakivskyi V. S. Sigma-monogenic functions in commutative algebras (Ukrainian),
In: Proc. of the Institute of Mathematics of the National Academy of Sciences of Ukraine,
19,No. 1, 231 – 256.
2023 - Plaksa S. A., Shpakivskyi V. S. Integral theorems in finite-dimensional commutative algebra (Ukrainian),
In: Proc. of the Institute of Mathematics of the National Academy of Sciences of Ukraine,
20,No. 1, 911 – 946.
Selected publications in scientific journals:
2008 - Mierzejewski D. A., Szpakowski V. S. On solutions of some types of quaternionic quadratic equations,
Bulletin de la Société des Sciences et des Lettres de Łódź 58, Ser. Recherches sur les déformations, Vol.55, 49 – 58.
2009 - Szpakowski V. S. Solution of general quadratic quaternionic equations ,
Bulletin de la Société des Sciences et des Lettres de Łódź 59, Ser. Recherches sur les déformations, Vol. 58, 45 – 58.
2010 - Plaksa S.A. and Shpakivskyi V.S. Integral theorems for differentiable functions in three-dimensional harmonic algebra,
Reports of the National Academy of Sciences of Ukraine, 5, 23 - 30.
2010 - Shpakivskyi V.S. and Plaksa S.A. Integral theorems and a Cauchy formula in a commutative three-dimensional harmonic algebra,
Bulletin de la Societe des Sciences et des Lettres de Lodz, 60(2), 47-54
2011 - Plaksa S.A. and Shpakivskyi V.S. Constructive description of monogenic functions in a harmonic algebra of the third rank,
Ukr. Math. J., 62(8), 1078-1091. DOI: 10.1007/s11253-011-0427-x
2011 - Shpakivskyi V. S. Linear quaternionic equations and their systems,
Advances in Applied Clifford Algebras, Vol. 21, No. 3, 637 – 645. DOI: 10.1007/s00006-010-0264-2
2011 - Plaksa S.A., Gryshchuk S.V. and Shpakivskyi V.S. Commutative algebras of monogenic functions associated with classic equations of mathematical physics,
Contemporary Mathematics, 553, Amer. Math. Soc., Providence, 245-258. DOI: 10.1090/conm/553
2012 - Plaksa S.A. and Shpakivskyi V.S. Limiting values of Cauchy type integral in a three-dimensional harmonic algebra,
Eurasian Math. J., 3(2), 120-128
2012 - Plaksa S.A. and Shpakivskyi V.S. A description of spatial potential fields by means of monogenic functions in infinite-dimensional spaces with a commutative multiplication,
Bulletin de la Societe des Sciences et des Lettres de Lodz, 62(2), 55-65
2013 - Plaksa S.A. and Shpakivskyi V.S. On the logarithmic residues of monogenic functions in a three-dimensional harmonic algebra with two-dimensional radical,
Ukr. Math. J., 65(7), 1079-1086. DOI: 10.1007/s11253-013-0843-1
2013 - Plaksa S.A. and Shpakivskyi V.S. On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical,
Annales Universitatis Mariae Curie-Sklodowska. Sectio A, 67(1), 57-64
2013 - Flaut C., Shpakivskyi V. Some identities in algebras obtained by the Cayle-Dickson process,
Advances in Applied Clifford Algebras, Vol. 23, No. 1, 63 – 76. DOI: 10.1007/s00006-012-0344-6
2013 - Flaut C., Shpakivskyi V. Real matrix representations for the complex quaternions ,
Advances in Applied Clifford Algebras, Vol. 23, No. 3, 657 – 671. DOI: 10.1007/s00006-013-0387-3
2013 - Flaut C., Shpakivskyi V. On Generalized Fibonacci Quaternions and Fibonacci-Narayana Quaternions ,
Advances in Applied Clifford Algebras, Vol. 23, No. 3, 673 – 688. DOI: 10.1007/s00006-013-0388-2
2014 - Plaksa S.A. and Shpakivskyi V.S. Cauchy theorem for a surface integral in commutative algebras,
Complex Variables and Elliptic Equations, 59(1), 110-119. DOI: 10.1080/17476933.2013.845178
2014 - Plaksa S.A. and Shpakivskyi V.S. Monogenic functions in a finite-dimensional algebra with unit and radical of maximal dimensionality,
Journal of Algerian Mathematical Society, 1, 1-13.
2015 - Flaut C., Shpakivskyi V. Holomorphic functions in generalized Cayley-Dickson algebras,
Advances in Applied Clifford Algebras, Vol. 25, No. 1, 95 – 112. DOI: 10.1007/s00006-014-0479-8
2015 - Flaut C., Shpakivskyi V. An efficient method for solving equations in generalized quaternion and octonion algebras,
Advances in Applied Clifford Algebras, Vol. 25, No. 2, 337 – 350. DOI: 10.1007/s00006-014-0493-x
2015 - Shpakivskyi V. Constructive description of monogenic functions in finite-dimensional commutative algebras (Ukrainian),
Reports of the National Academy of Sciences of Ukraine, 4, 23 - 28.
2015 - Flaut C., Shpakivskyi V. Some remarks about Fibonacci elements in an arbitrary algebra,
Bulletin de la Société des Sciences et des Lettres de Łódź, Ser. Recherches sur les déformations, 65, No. 3, 63 – 73. DOI: http://www.ltn.lodz.pl/streszczenia/Bulletin-65-3.pdf
2016 - Shpakivskyi V. S., Kuzmenko T. S. On one class of quaternion mappings,
Ukr. Math. J. 68, 127 - 143. DOI: 10.1007/s11253-016-1213-6
2016 - Shpakivskyi V. S. Constructive description of monogenic functions in a finite-dimensional commutative associative algebra,
Adv. Pure Appl. Math., Vol. 7, No. 1, 63 – 75. DOI: 10.1515/apam-2015-0022
2016 - Shpakivskyi V. S., Kuzmenko T. S. Integral theorems for the quaternionic G-monogenic mappings,
An. Şt. Univ. Ovidius Constanţa, 24, No. 2, 271 – 281. DOI: 10.1515/auom-2016-0042
2016 - Shpakivskyi V. S. Curvilinear integral theorems for monogenic functions in commutative associative algebras,
Advances in Applied Clifford Algebras, Vol. 26, No. 1, P. 417 – 434. DOI: 10.1007/s00006-015-0561-x"
2016 - Shpakivskyi V. S., Kuzmenko T. S. On monogenic mappings of a quaternion variable,
J. Math. Sci., 221, 712 – 726. DOI: 10.1007/s10958-017-3260-4
2016 - Kuzmenko T. S., Shpakivskyi V. S. Generalized integral theorems for the quaternionic G-monogenic mappings,
J. Math. Sci., 224, 530 – 540. DOI: 10.1007/s10958-017-3433-1
2016 - Shpakivskyi V. S. Hypercomplex representation of analytical solutions of one hydrodynamic equation (Russian),
Proceedings of the IPMM NAS of Ukraine, 30, 155 – 164. DOI: http://dspace.nbuv.gov.ua/handle/123456789/140867
2017 - Plaksa S.A. and Shpakivskyi V.S. An extension of monogenic functions and spatial potentials,
Lobachevskii Journal of Mathematics, 38(2), 330-337. DOI: 10.1134/S1995080217020160
2017 - Flaut C., Shpakivskyi V., Vlad E. Some remarks regarding h(x)–Fibonacci polynomials in an arbitrary algebra,
Chaos, Solitons and Fractals, Vol. 99, 32 – 35. DOI: 10.1016/j.chaos.2017.03.040
2018 - Shpakivskyi V. S., Kuzmenko T. S. Quaternionic G-monogenic mappings in E_m,
Int. J. Adv. Res. Math., Vol. 12, 1 – 34.
2018 - Plaksa S.A. and Shpakivskyi V.S. Integral theorems for monogenic functions in an infinite-dimensional space with a commutative multiplication,
Bulletin de la Societe des Sciences et des Lettres de Lodz, 68(2), 25-36
2018 - Shpakivskyi V. S. On monogenic functions defined in different commutative algebras,
J. Math. Sci. 239, 92 – 109. DOI: 10.1007/s10958-019-04291-0
2018 - Shpakivskyi V. S. On monogene functions on extensions of commutative algebra (Ukrainian),
Proceedings of the International Geometry Center, 11 (3), 1 - 18. DOI: 10.15673/tmgc.v11i3.1200
2019 - Luna-Elizarrarás M. E., Shapiro M., Shpakivskyi V. On the Hausdorff Analyticity for Quaternion-Valued Functions ,
Complex Analysis and Operator Theory, Vol. 13, No. 6, 2863 – 2880. DOI: 10.1007/s11785-018-0856-8
2022 - Kuzmenko T. S., Shpakivskyi V. S. G-monogenic mappings in a three-dimensional noncommutative algebra,
Complex Variables and Elliptic Equations, Vol. 67, No. 11, 2759 – 2769. DOI: 10.1080/17476933.2021.1947257
2022 - Kuzmenko T. S., Shpakivskyi V. S. Differentiable functions in a three-dimensional associative noncommutative algebra ,
Advances in the Theory of Nonlinear Analysis and its Applications. Vol. 6, No. 1, 66 – 73. DOI: https://atnaea.org/index.php/journal/article/view/101
2022 - Shpakivskyi V. S., Kuzmenko T. S. Hausdorff analytic functions in the three-dimensional associative noncommutative algebra,
J. Math. Sci. 262, 207 – 221. DOI: 10.1007/s10958-022-05811-1
2023 - Shpakivskyi V. S. Sigma-monogenic functions in commutative algebras,
Proceedings of the International Geometry Center, Vol. 16, No. 1, 17 – 41. DOI: 10.15673/tmgc.v16i1.2421
2023 - Shpakivskyi V. S. Conformable fractional derivative in commutative algebras,
Journal of Mathematical Sciences, Vol. 274, No. 3, 392 – 402. DOI: 10.1007/s10958-023-06608-6
2024 - Kuzmenko T. S., Shpakivskyi V. S. Representations of Some Classes of Quaternionic Hyperholomorphic Functions,
Complex Analysis and Operator Theory, 18(5):116. DOI: 10.1007/s11785-024-01561-x
2024 - Shpakivskyi V. S. Construction of solutions of PDEs using holomorphic functions of several variables,
Electronic Journal of Differential Equations,Vol. 2024, No. 71, 1 - 12. DOI: 10.58997/ejde.2024.71
2024 - Shpakivskyi V. S. Construction of an infinite-dimensional family of exact solutions of a three-dimensional biharmonic equation by the hypercomplex method,
Advances in Applied Clifford Algebras, 35:5, 12 pp. DOI: 10.1007/s00006-024-01365-4
2006 - The XI Kravchuk International Scientific Conference, (Ukraine, Kyiv);
2007 - Conference “Modern problems of mathematics and its application in natural sciences and information technologies”,
(Ukraine, Kharkiv);
2007 - Bogolyubov Readings 2007. International Conference of Complex Analysis and Wave Processes in Mechanics, (Ukraine, Zhytomyr);
2008 - The student scientific conference of the applied mathematics and informatics,
(Ukraine, Lviv);
2009 - International Scientific Conference dedicate to Prof. M.I. Nagnibida,
(Ukraine, Chernivtsi);
2009 - International Scientific Conference „Analytic methods of mechanics and complex analysis” dedicated to
N. A. Kilchevskii and V.A.Zmorovich on the occasion of their birthday centenary”,
(Ukraine, Kyiv) ;
2009 - Hypercomplex Seminar 2009: From Schauder Basis to Hypercomplex,
Randers-Ingarden and Fractal Structures, and Nanostructures, (Poland, Będlewo);
2009 - 7th ISAAC Congress , (UK, London);
2010 - Hypercomplex Seminar 2010: (Hyper)Complex and Randers-Ingarden Structures in Mathematics and Physics, (Poland, Będlewo)
;
2010 - 6-th International Conference ”Finsler Extensions of Relativity Theory”, (Moscow, Russia);
2011 - Conference on Analytic Functions and Related Topics, (Chelm, Poland);
2013 - International workshop “A new approach in theoretical and applied methods in algebra and analysis”, (Constanta, Romania);
2013 - International Conference "Complex Analysis, Potential Theory and Applications", (Kyiv, Ukraine);
2013 - Joint events of Colloquium on Differential Geometry and its Applications and IX-th International
Conference on Finsler Extensions of Relativity Theory,
(Debrecen, Hungary);
2014 - International Conference “Differential equations, computational mathematics, theory of functions and mathematical methods of mechanics”
dedicate to 100th anniversary of Prof. G. M. Polozhii, (Kyiv, Ukraine);
2015 - 10th ISAAC Congress, (Macau, China);
2015 - X Summer School "Algebra, Topology, Analysis",
(Odessa, Ukraine);
2017 - 11th ISAAC Congress,
(Växjö, Sweden);
2018 - Hypercomplex Seminar 2018: (Hyper)Complex Analysis in Differential Equations, Geometry and Physical Applications,
(Będlewo, Poland);
2019 - 12th ISAAC Congress,
(Aveiro, Portugal);
2021 - (Hyper)Complex Seminar,
(Bedlewo, Poland);
2023 - International Conference of Young Mathematicians, (Kyiv, Ukraine);
2024 - (Hyper)Complex Seminar, (Bedlewo, Poland);
2024 - International conference dedicated to the 145th anniversary of the birth of Hans Hahn, Yu.
Fedkovych Chernivtsi National University, (Chernivtsi, Ukraine);
