Shavarovskii Bogdan Zenoviyovych

Education: Drohobych Ivan Franko State Pedagogikal University (specialty – mathematics and physics, 1973), Postgraduate study at Pidstryhach IAPMM) (01.01.06 – algebra and number theory, 1979 ) 

Scientific degree: Doctor of sciences in physics and mathematics (doctoral thesis "Reduction and decomposition of matrices over polynomial rings", 2013 ð., 01.01.06 – algebra and number theory.)

Scientific title: Senior Research Fellow (1995)

Position: Leading Research Fellow

In Institute: from 1974

Research interests: linear algebra, matrix theory, invariant theory, matrix equations, computational methods of linear algebra

Direction of scientific researches: Reduction of matrix over rings and reduction of sets of matrices over fields: search invariants, including their complete systems; build to simple form; special types of transformation. Decomposition of matrices over ring: factorization of matrices into a product of regular factors of lower degrees. Solving polynomial equations of higher degrees and systems of linear equations bilateral matrix.

Main scientific results:

 

  1. Developed and applied a method of semiscalar equivalence of polynomial matrices to the similarity problem of finite sets of numerical matrices.
  2. The problem of reduction of rather wide classes of polynomial matrices to direct sum of summands has been solved. This enabled to reduce complex problems of semiscalar equivalence and similarity of matrices to analogous problems for matrices of lower dimension.
  3. For some classes of finite sets of numerical matrices constructed canonical and related to them forms and found systems of invariants with respect to similarity transformations.
  4. Investigated the equivalence and similarity matrices by different transformations using the matrices of special block structure.
  5. The decomposabiliti into factors (in particular linear) of some classes of matrix polynomials with elementary divisors of degrees more than two has been proved, the classes including the majority of known now classes of factorable matrix polynomials.
  6. The results obtained have been applied to the problems of construction the canonical forms matrix polynomials relating to similarity transformation and also to solution of various types of matrix algebraic equations (polynomial higher degrees, linear two-sided, Riccati equations).

 

Major publications:

 

  1. Shavarovskii B.Z. On the triangular form of a polynomial matrix of simple structure and its invariants with respect to semi-scalar equivalence // Mat. Metody Fiz.-Mekh. Polya.  – 2023. – 66, ¹ 1–2. – P. 16–22. (in Ukrainian).
  2. Shavarovskii B.Z. On the Triangular Form of 3 X 3-Matrix of Simple Structure Relative to Semiscalar Equivalence // Mat. Metody Fiz.-Mekh. Polya. – 2022. – 65, ¹ 3-4. – P. 5–28.
  3. Shavarovskii B.Z. Conditions of semiscalar equivalence of one class 3x3 matrices of simple structure // Journal of Mathematics. − 2022. − vol. 2022 Article ID 8395922, 13 pages. − https://doi.org/10.1155/2022/8395922. (Scopus, 1.4, Q1)
  4. Shavarovskii B.Z. Oriented by characteristic roots reduced matrices in the class of semiscalarly equivalent // Journal of Mathematics. − 2021. − vol. 2021 Article ID 5592756, 6 pages. − https://doi.org/10.1155/2021/5592756. (Scopus, 1.555, Q1)
  5. Shavarovskii B.Z. Reduction to a Canonical Form of a Third-order Polynomial Matrix with One Characteristic Root by means of Semiscalarly Equivalent Transformations // Journal of Mathematics, vol. 2020, Article ID 7960987, 10pages, 2020. https://doi.org/10.1155/2020/7960987 (Scopus, 0.971, Q2)
  6. Shavarovskii B.Z. Non-singular solutions of one class of matrix equations over a polynomial ring // Applied problems of mechanics and mathematics. – 2019. – Issue 17. – P. 17–26.
  7. Shavarovskii B.Z. Canonical Form of Reduced 3-by-3 Matrix with One Characteristic Root and with Some Zero Subdiagonal Elements // Journal of Mathematics. Volume 2019, Article ID 7646132, 10 pages, 2019. (https://doi.org/10.1155/2019/7646132). (Scopus, 0.913, Q3)
  8. Shavarovskii B.Z. About some invariants of polynomial matrices with respect to semiscalar equivalence // Applied problems of mechanics and mathematics. – 2018. – Issue 16. – P. 14–18. https://doi.org/10.15407/apmm2018.16.14-18 (in Ukrainian)
  9. Shavarovskii B.Z. Reduced Triangular Form of Polynomial 3-by-3 Matrices with One Characteristic Root and Its Invariants // Journal of Mathematics. Volume 2018, (2018), Article ID 3127984, 6 pages. http://doi.org/10.1155/2018/3127984 (Scopus, 0.627, Q4)
  10. Shavarovskii B. Z. Semiscalar equivalence and quasidiagonal similarity of the matrices // Journal of Mathematical Science 2018. – 228, ¹ 1. – P.11–25. https://doi.org/10.1007/s10958-017-3602-2 (Scopus, 0.569, Q3)
  11. Shavarovskii B.Z. Toeplitz Matrices in the Problem of Semiscalar Equivalence of Second-Order Polynomial Matrices // International Journal of Analysis. Volume 2017, (2017), Article ID 6701078, 14 pages. http://doi.org/10.1155/2017/6701078
  12. Shavarovskii B.Z. Conditions of Semiscalar Equivalence of Polynomial 2-by-2 Matrices // Mathematical Bulletin of Taras Shevchenko Scientific Society. – 2017, v. 14. – P. 7–28.
  13. Shavarovskii B.Z. Block-Diagonal Similarity and Semisñalar Equivalence of Matrices // Journal of Mathematical Sciences. – 2017. – 222, No. 1 – P. 35-49. (Scopus, Web of Science, 0.517, Q3)
  14. Shavarovskii B.Z. On invariants and canonical form of matrices of second order with respect to semiscalar equivalence // BASM. – 2016. – n.3 (82). – P. 12–23.
  15. Shavarovskii B. Z. The block-diagonal similarity and semisñalar equivalence of polynomial matrices // Mathematical methods and physico-mechanical fields. – 2015. – 58, No 1. – P. 29–41. (in Ukrainian)
  16. Shavarovskii B. Z. The semiscalar equivalence and quasi-diagonal similarity of matrices  // Mathematical methods and physico-mechanical fields. – 2015. – 58, No 4. – P. 15–26. (in Ukrainian)
  17. Shavarovskii B. Z. On nonsingular solutions of one type of systems of matrix equations // Applied problems of mechanics and mathematics. 2015. – Issue. 13. – P. 11–17,
  18. Shavarovskii B. Z. Semisñalar equivalence of polynomial matrices and triangular similarity of numerical matrices // Applied problems of mechanics and mathematics. 2013. – Issue. 11. – P. 63–69.
  19. Shavarovskii B.Z. On the Triangular Form of a Polynomial Matrix and Its Invariants with Respect to the Semiscalar Equivalence // J. Math. Sci. (N.Y.) – 2013. – 194, No 2. – P. 133–148.
  20. Shavarovskii B.Z. Canonical form of polynomial matrices with all identical elementary divisors // Ukr. Math. J. – 2012. – 64, No. 2. – P. 282–297.
  21. Shavarovskii B.Z. Factorability of polynomial matrices depending on the powers of their elementary divisors // Zh Vychisl. Mat. Mat Fiz– 2012. – 52, ¹ 7. – P. 1171–1184.
  22. Shavarovskii B. Z. On a triangular form of polynomial matrix and its invariants with respect to semiscalar equivalence // Mathematical methods and physico-mechanical fields. – 2012. – 55, No 3. – P. 7–20.
  23. Shavarovskii B. Z. Canonical form for pair of matrices, in which one matrix has pairwise relatively prime elementary divisors of degree at most two // Applied problems of mechanics and mathematics. 2012. – Issue. 10. – P. 62–70.
  24. Shavarovskii B. Z On the similarity of matrix polynomials of simple structure // Applied problems of mechanics and mathematics. 2011. – Issue. 9. – P. 74–81.
  25. Shavarovskii B.Z. Decomposability of matrix polynomials with commuting coefficients into a product of linear factors // Ukr. Math. J. – 2011. – 62, No. 8. – P. 1295–1306.
  26. Shavarovskii B.Z. Similarity Transformations of Decomposable Matrix Polynomials and Related Questions // Computational Mathematics and Mathematical Physics. Vol. 49, No. 9, 2009. P. 1469–1482.
  27. Shavarovskii B.Z. Classes of Factorable and Unfactorable Polynomial Matrices and Their Relative Position // Doklady Mathematics . – 2007 – 75, ¹.2. – Ð. 318–321.
  28. Shavarovskii B.Z. On the Similarity of Matrices of Even Order. // Mathematical Notes. – 81, no. 3. – 2007. – P. 392–407.
  29. Shavarovskii B.Z. Finding a Complete Set of Solutions or Proving Unsolvability for Certain Classes of Matrix Polynomial Equations with Commuting Coefficients // Computational Mathematics and Mathematical Physics. – 2007. – 47, ¹. 12. – Ð. 1902–1911.
  30. Shavarovskii B.Z. Multiplicative Structure of a Polynomial Matrix Depending on Its Elementary Divisors// Doklady Mathematics , 2007. – 76, ¹3. – Ð. 843–847.
  31. Shavarovskii B.Z. Solvability of Matrix Equations in Rings of Quasi-Diagonal Matrices and Similarity of Matrix Polynomials // Computational Mathematics and Mathematical Physics. – 46. ¹. 8. – 2006. – P. 1283–1292.
  32. Shavarovskii B.Z. On Some ‘Tame” and “Wild” Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices // Mathematical Notes. – 76, no. 1. – 2004. – P. 111–123.
  33. Shavarovskii B.Z. Solving a System of Linear  Homogeneous Two-Sided Matrix Equations in Two Unknowns. // Computational Mathematics and Mathematical Physics. – 44. ¹ 11. – 2004. – P. 1853–1866.
  34. Shavarovskii B.Z. A Complete System of Invariants with Respect to Similarity for Pairs of Matrices of Even Order with Characteristic Matrix of Special Smith form // Mathematical Notes. – 73, no. 6. – 2003. – P. 871–888.
  35. Shavarovskii B.Z. On a Method for Reducing a Pair of Matrices to Block Diagonal Form by a Similarity Transformation // Computational Mathematics and Mathematical Physics . – 43. ¹ 9. – 2003. – P. 1231–1240.
  36. Shavarovskii B.Z. Factorable polynomial matrices // Mathematical Notes. – 2000. – 68, ¹4. – Ñ. 507-518
  37. Shavarovskii B.Z. On Semiscalar and Quasidiagonal Equivalences of Matrices // Ukr. Math. J. – 2000. – 52, No. 10. – P. 1638–1643.
  38. Shavarovskii B.Z. Decomposition of polynomial matrices into a direct sum of triangular summands // Ukr. Math. J. – 1999. – 51, No. 8. – P. 1291–1295.
  39. Shavarovskii B.Z. On decomposition of polynomial matrices into direct sum of matrices of smaller order // Reports of the Academy of Sciences of Ukraine. – 1999. – ¹4. – P. 49–53.
  40. Shavarovskii B.Z., Reduction of matrices of equivalent transformations and similarity transformations // Mathematical Notes. – 1998. – 64, ¹5. – P. 663–673.

List of Conference Proceedings:

1.     Shavarovskii Bohdan. A complete system of invariants with respect to semi-scalar equivalence for a 3õ3-matrix of a simple structure // Current problems of Mechanics and Mathematics - 2023: Collection of Scientific Papers / Edited by Roman M. Kushnir (Academician of NAS of Ukraine) and Volodymyr O. Pelykh (Corresponding Member of NAS of Ukraine) [Published Online] // Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine. - 2023. - 454 p. - Access by the link: http://iapmm.lviv.ua/mpmm2023/materials/proceedings.mpmm2023.pdf P. 397–398. – http://iapmm.lviv.ua/mpmm2023/materials/ma10_08.pdf

2.     Shavarovskii Bohdan.The triangular form of the polynomial 3 × 3-matrix of a simple structure and its invariants with respect to semiscalar equivalence // 14th Ukraine Algebra Conference, July 3-7, 2023 Sumy, Ukraine. Book of Abstracts: Sumy State Pedagogical University named after A.S. Makarenko, Sumy, Ukraine. – 150 p. – P. 123.

  1. Shavarovskii Bohdan Semiscalar equivalence of polynomial 3-by-3 matrices of simple structure // The 13th International Algebraic Conference in Ukraine. July 6–9, 2021, Taras Shevchenko National University of Kyiv. Book of Abstracts: Taras Shevchenko National University of Kyiv, Kyiv, Ukraine. – 94 p. – Ðåæèì äîñòóïó äî ðåñóðñó: https://bit.ly/3hcTbKe – P. 75.
  2. Bohdan Shavarovskii. Canonical form of a reduced matrix with all non-zero sub-diagonal elements // XI International Skorobohatko Mathematical Conference (October 26–30, 2020, Lviv, Ukraine). Abstracts. Lviv – 2020. P. 106.
  3. Shavarovskii B.Z. Canonical form of a reduced matrix with one zero sub-diagonal element // Book of abstracts of the International mathematical conference dedicated to the 60th anniversary of the department of algebra and mathematical logic of Taras Shevchenko National University of Kyiv, 14-17 July 2020, Kyiv, Ukraine. – 93 p. – [Electronic resource]. – Access mode: https://bit.ly/2ZIyqMs– P. 74.
  4. Shavarovskii B. Improved triangular form of the 3-by-3-matrices with one characteristic root in the class of semiscalarly equivalent // Modern problems of Mechanics and Mathematics: collection of scientific papers in 3 vol. / Edited by A.Ì. Samoilenko, R.M. Kushnir [Electronic resource] // Pidstryhach Institute for Applied Problems of Mechanics and Mathematics NAS of Ukraine. – 2018. – Vol. 3. – Access mode: www.iapmm.lviv.ua/mpmm2018. P. 241–242.
  5. Shavarovskii B.Z. Improved triangular form of the matrix and its invariants // Book of abstracts of the XI International Algebraic Conference in Ukraine dedicated to the 75th anniversary of V.V.Kirichenko (July, 2017, Kyiv, Ukraine). P.123.
  6. B.Z. Shavarovskii. When polynomial matrices of second order semiscalarly are equivalent? // X International Algebraic Conference in Ukraine dedicated to the 70th anniversary of Yu. A. Drozd. August 20 – 27, 2015. Odessa, Ukraine. Abstracts. P. 100.
  7. B.Z. Shavarovskii. Quasi-diagonal similarity and semiscalar equivalence of matrices //  International Algebraic Conference dedicated to 100th anniversary of L.A. Kaluzhnin. Book of Abstracts. Jule 7–12, 2014. Kyiv. P. 76 – 77.
  8. B.Z. Shavarovskii. Reduction of polynomial matrices by semiscalar equivalent transformation // 9th International Algebraic Conference in Ukraine. July 8–13, 2013, L’viv, Ukraine. Book of abstracts. P. 172.
  9. Bogdan Shavarovskii On the canonical form for a certain class of matrices with respect to similarity // International Conference dedicated to the 120th anniversary of Stefan Banach, Lviv, Ukraine, 17–21 September 2012. Abstracts of Reports. – P. 264.
  10. B. Shavarovskii. On similarity of pairs of matrices, in which one matrix has only one elementary divisor // International mathematical conference: abstracts of talks. – Mykolayiv: Published by Mykolayiv V.O. Suchomlinsky National University, 2012. – P. 134.
  11. B. Shavarovskii Canonical forms for matrices under semiscalar equivalence // Book of abstracts of the 8th International Algebraic Conference in Ukraine (July 5 – 12, 2011, Lugansk Taras Shevchenko National University, Ukraine), P. 181.
  12. B. Shavarovskii. On the invariants of polynomial matrices with respect to semiscalar equivalence // 7 International Algebraic Conference in Ukraine: abstracts of talks ( 18 – 23 August, 2009, Kharkov) – Kiev, 2009. – 168 p. P. 127 – 128.
  13. Bogdan Shavarovskii. Solving a System of Linear Two-Sided Matrix Equations // Óêðà¿íñüêèé ìàòåìàòè÷íèé êîíãðåñ – 2009, Êè¿â. – www.imath.kiev.ua/~congress2009/Abstracts/Shavarovski.pdf
  14. Shavarovs’kii B. On the class of factorable polynomial matrices // 6th International Algebraic Conference in Ukraine. Kamyanets-Podilsky, July 1-7, 2007. Abstracts. –Kamyanets-Podilsky, 2007. – P. 187-188.
  15. Shavarovs’kii B.Z. On similarity of some collection of matrices // 5th International Algebraic Conference in Ukraine. Odessa, July 20-27, 2005. Abstracts. – Odessa, 2005. – P. 186-187.

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E-mail: bshavarovskii@gmail.com