Publications

  1. Zatorsky R.A. Counting m-submultisets through their secondary specifications. Combinatorial analysis. Issue 7. Mosc. Univer. Publ., 1987. Pp. 136-145. (Russian).
    Abstract: General formulas for counting the cardinality of the submultiset and permutation set of a multiset are established, along with some classes of multisets for which these formulas are simplified.
  2. R. A. Zatorskii, “On the number of standard Young tableaux of skew diagrams,” Izv. Kievsk. Un-ta. Ser. Fiz.-Matem. Nauki, No. 4, pp. 24–33 (2000). (Ukrainian).
    Abstract: The new way of counting the number of maximum chains between two divisions is considered here by using the slanting diagram that looks like a triangular matrix. Using this approach an important for the theory of the symmetrical groups introduction class of the so called staged slanting diagrams is selected and the formula for counting the quantity of the corresponding maximum chains is given. The formula that expresses the number of standard Young tables of the Ferrier diagram through the Vandermonds determination is also presented.
  3. Zatorsky R.А. Of the number of combinations on multisets. Izv. Kievsk. Un-ta. Ser. Fiz.-Matem. Nauki, No. 3, pp. 42–47 (2000). (Ukrainian).
    Abstract: A new formula for the number of combinations on multisets is obtained using the corresponding generating function/ An efficient algorithm for counting of such number and the corresponding Q-Basic program is described. There found a class of multisets for which these numbers are relatively easy to calculate.
  4. R. A. Zatorsky. Continued Fractions, k-Polynomials, and Parapermanents. Mathematical Methods and Physico-Mechanical Fields. 2001- 44, No.4. pp. 12-21. (Ukrainian).
    Abstract: A new methodology for investigating the properties of continued fractions is proposed, with the main tool being parapermanents. New algorithms for computing convergents are formulated, allowing the establishment of new convergence conditions and error estimates for periodic and mixed periodic continued fractions.
  5. Zatorsky R.A. On paradeterminants and parapermanents of triangular matrices. Mat. Stud. 17 (2002), No. 1, pp. 3-17.  (Ukrainian).
    Abstract: Properties of some functions on the set of triangular matrices are studied.
  6. Zatorsky R.A. On an algorithm for calculation of the number of m-permutations on multisets. Matem. Stud., 17, No. 2, (2002) pp. 215-219. (Ukrainian).
    Abstract: Using a generating function/formula for the number of m-permutations on a multiset, a recursive relationship is derived, upon which an algorithm for calculating the number of m-permutations on the multiset is constructed.
  7. Zatorsky R.A.Paradeterminants and parapermanents of triangular matrices. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 2002, No. 8, pp.21-28. (Ukrainian).
    Abstract: Properties of some functions on the set of triangular matrices are studied.
  8. R. Zatorsky Determinants of triangular matrices and trajectories on Ferre diagrams. Journal “Mathematical Notes”, 2002, Volume 72, Issue 5-6, pp. 768-783.
    Abstract: Some functions defined on the set of integer triangular matrices and their modifications axe considered. These functions axe analogs of the classical determinant and permanent functions and provide a powerful apparatus for studying sequences generated by linear recursive equations. Modifications of these functions are used to solve the problem about trajectories on Ferre diagrams and to establish new combinatorial identities.
  9.  R. Zatorsky  Paradeterminants and formal operations on series // 4th International Algebraic Conference in Ukraine, Lviv, 2003.,  242.
  10. Atamaniuk O.B., Zatorsky R.A. Application of triangular matrices functions for solving certain systems of equations. In: Materials of the International Conference dedicated to the 125th anniversary of Hans Hahn’s birth. Chernivtsi, 2004, pp. 7-8.
    Abstract: Some infinite systems of equations whose solutions are functions of Hessenberg matrices are being considered.
  11. Zatorsky R.A., Fedoruk P.I., Diakiw N.M. Random number generator in a remote knowledge control system. // Mathematical Machines and Systems. – 2004. – No. 4. – pp.98 – 107. Abstract: Article is dedicated to the problem of random numbers generator (RNG) that may be used in systems of distance education and knowledge control. Weak points of existing RNG applied for typical tasks in this area have been analyzed and a new type of generators based on described n f -function has been offered. Conducted researches convincingly show that offered RNG can be effectively used not only in systems of distance education and knowledge control where its approbation has been carried out but also in other calculating and mathematical systems йrequiring RNG with the most uniform frequency distribution on given set.
  12. Ганюшкін О., Заторський Р., Безущак О. Видатний український математик Ростислав Іванович Григорчук. /Журнал «У світі математики», Т.11, вип. 2, 2005 р.
  13. N. М. Diakiv, R. A. Zatorsky.  On -paradeterminants and  -parapermanents of triangular matrices.  Mat. Metodi Fiz.-Mekh. Polya 48 (2005), No. 1, pp. 21-26.
    Abstract: It is proved that-paradeterminants and -parapermanents are particular cases of paradeterminants and parapermanents. The relation between the upper and lover -paradeterminants and upper and lover -parapermanents is established. Certain important statements on -parapermanents, beind counterparts of the known facts about -parapermanents, are proved..
  14. Oleksandr G. Ganyushkin, Roman A. Zatorsky, Ivan I. Lishchynsky. On paradeterminants and parapermanents of triangular matrices. Izv. Kievsk. Un-ta. Ser. Fiz.-Matem. Nauki, No. 1, 35–41 (2005).
    Abstract: Some new properties of paradeterminants and parapermanents are proved and the algorithm with O(n2) actions for calculating nthorder paradeterminants and parapermanents  is provided.
  15. R.A. Zatorsky, I.I. Lishchynskyi. On connection between determinants and paradeterminants, Matematychni Studii, V.25 (2006), pp. 97-102.
    Abstract: Some connections between determinants and paradeterminants are investigated.
  16. Zatorsky R.A. Some Methods and Problems of Combinatorial Analysis (Special Mathematics Course). – Ivano-Frankivsk: Lik, 2006. – 136 p. (Ukrainian).
    Abstract: The book presents, for the first time, the general theorems of combinatorics on multisets. The central focus of the book is on fundamental concepts and notions, developed by the author, of the theory of paradeterminants and parapermanents of triangular matrices and their application in number theory, the theory of continued fractions, and solving certain classes of combinatorial problems. The book will be useful for specialists in the field of combinatorial analysis and algebra, as well as for students and postgraduates majoring in Mathematics.
  17. R.A.Zatorsky. Theory of paradeterminants and its applications. Algebra and Discrete Mathematics Number 1. (2007). Pp 108-137.
    Abstract: We consider elements of linear algebra based on triangular tables with entries in some number field and their functions, analogical to the classical notions of a matrix, determinant and permanent. Some properties are investigated and applications in various areas of mathematics are given.
  18. Zatorsky R.A. Parafunctions and Combinatorial Identities. Scientific Bulletin of Chernivtsi University. Collection of Scientific Works. Issue 336-337. Mathematics. – Chernivtsi: Chernivtsi National University, 2007. – P. 79-84.
    Abstract: The applications of the parafunctions of the triangle matrix to generation and classification of combinatorial identities are considered.
  19. Zatorsky R.A., Malarchuk A.R. The infinite linear recurrent equations and paradeterminants,
    Precarpathian Journal of Mathematics, 1, 1 (2008) 134-145.
    Abstract: The infinite linear recurrent equations by means of parapermanents waves of triangular matrices are investigated.
  20. R.А. Zatorsky, І.М. Lytvynenko. Application of Parapermanents to Linear Recurrent Equations. Scientific News of NTUU ‘KPI’, Theoretical and Applied Problems of Physical and Mathematical Sciences. 2008 / 5. pp. 1-7. (Ukrainian).
    Abstract: The applications of parapermanents of triangular matrices in studying the properties of numerical sequences generated by linear recurrent relations of the k-th order with constant coefficients are considered. In particular, some general statements regarding the properties of theoretical-numerical sequences generated by linear recurrent equations of the second order are proven.
  21. R. Zatorsky. Scalar product of a vector with the paradiagonal determinant of a triangular matrix and its application.  Carpathian Herald of the Shevchenko Scientific Society. Number. № 1(1) (2008)
    Abstract: The definition of the scalar product of a vector on a paradiagonal tri-diagonal matrix is presented; the application of this operation to investigate formal operations with formal power series is considered. The author derives formulas for n-fold differentiation of a composite function and generalizes the polynomial partitions of Bell numbers.
  22. R.A. Zatorsky. Operations with formal power series with have nontrivial free numbers. Bulletin of University of Kyiv Series: Physics & Mathematics – 2008. –  V. 1.  –  pp. 36–39. (Ukrainian).
    Abstract: Operations with formal power series with have nontrivial free coefficient are considered.
  23. Zatorsky R.A. Paradeterminants and partitions polynomials. Ukrainian Mathematical Journal.  –  2008. –  V. 60, No. 11.  –  pp. 1457–1469.
    Abstract: We study Bell polynomials by using functions of triangular matrices (parapermanents and paradeterminants). Some combinatorial sdentiies and the relationships between these functions fnd the Stirling numbers of the first and second kind are established.
  24. R. A. Zatorskii  and A. R. Malyarchuk. Triangular Matrices and Combinatorial Inversion Formulas. Mathematical Notes, 2009, Vol. 85, No. 1, pp. 11–19.
    Abstract: The paper is devoted to the construction of the matrix inverse of an infinite triangular matrix and to finding the connection coefficients between polynomial sequences and general combinatorial inversion formulas.
  25. Zatorsky R.A., Malyarchuk O.R. Infinite linear recurrent equations and parapermanents. Carpathian Mathematical Publications. – 2009,  V.1, No.1, pp. 35-46.
    Abstract: Using parapermanents of triangular matrices, infinite linear recurrent equations are investigated.
  26. R. Zatorsky. Recursions and positional notations.  Carpathian Herald of the Shevchenko Scientific Society. Number. № 1(5) (2009).
    Abstract: Generalizing classical positional numeral systems.
  27. Tarakanov, V.E.; Zatorskii, R.A. A relationship between determinants and permanents. (English)
    Math. Notes 85, No. 2, 267-273 (2009); translation from Mat. Zametki 85, No. 2, 292-299 (2009).
  28. Zatorsky R.A., Malyarchuk O.R. Factorial Powers and Triangular Matrices. Carpathian Mathematical Publications. – 2009,  Т.1, No2, pp. 161-171.
    Abstract: Increasing and decreasing factorial degrees as well as identities of Nerlund and Vandermonde are generalized in the article. By means  of factorial degrees with a step it is selected so called class of factorial numerical triangles, elements of which satisfy some recurrent relacion.
  29. Заторський Р., Семенчук А. Зображення деяких класів кубічних ірраціональностей періодичними рекурентними дробами.// Матеріали Міжнародної наукової конференції, присвяченої 50-річчю кафедри алгебри і математичної логіки КНУ ім. Тараса Шевченка. Київ 2009, с. 45.
  30. Zatorsky R. Calculus of Triangular Matrices (Tables).  Ivano-Frankivsk: Publishing House « Simyk », 2010, p. 508. ISBN 978-966-8067-63-1. (Ukrainian).
    Abstract: The monograph represents the first systematic exposition of « Calculation of Triangular Matrices, » a new branch of linear algebra actively developed by the author and his students. The book also explores the applications of this calculation to various problems in combinatorial analysis and number theory. No knowledge beyond the introductory courses in linear algebra and analysis is required from the reader. The book will undoubtedly interest specialists in linear algebra, discrete mathematics, number theory, and other branches of mathematics. It will also be beneficial for students and postgraduates in mathematical disciplines.
  31.  R.A. Zatorsky, A.R. Malarchuk. Reduction of Paradeterminants of a triangular matrix to  k-diagonal kind. Mat. Stud. V.34 (2010), pp. 20-29.
    Abstract: Reduction of paradeterminants of triangular matrices to paradeterminants of k-diagonal matrices is considered.
  32. Zatorsky R.A., Semenchuk A.V. Periodic recurrent fractions of third degree. , Carpathian Mathematical Publications, 2, 1 (2010), 72–81.
    Abstract: Periodic recurrent fractions of third degree are investigated by means of arapermanents of triangular matrices.
  33. Заторський Р.А., Федак І.В. Про деякі явні представлення чисел Ферма квадратами натуральних чисел. Тринадцята міжнародна наукова конференція імені академіка Михайла Кравчука. 13-15 травня 2010 року, м. Київ.
    Анотація: Показано, що кожне число Ферма може бути представлене у вигляді суми трьох різних, або чотирьох квадратів натуральних чисел. При цьому подано явні формули таких представлень.
  34. R.A. Zatorsky. Operations with formal power series with zero constant term. Bulletin of University of Kyiv Series: Physics & Mathematics – 2010. –  V. 1.  –  pp. 7–10. (Ukrainian).
    Abstract: Formal Operations with formal power series with zero constant term are considered.
  35. Zatorsky R.A., Fedoruk P.I., Diakiw N.M. Random number generator in a remote knowledge control system.
    Abstract: Article is dedicated to the problem of random numbers generator (RNG) that may be used in systems of distance education and knowledge control. Weak points of existing RNG applied for typical tasks in this area have been analyzed and a new type of generators based on described n f -function has been offered. Conducted researches convincingly show that offered RNG can be effectively used not only in systems of distance education and knowledge control where its approbation has been carried out but also in other calculating and mathematical systems йrequiring RNG with the most uniform frequency distribution on given set.
  36. Заторський Р. А. Парадетермiнантний добуток трикутних матриць та його властивостi. Математичний вiсник НТШ.  –  2011  – Т. 8.  –  C.  87–98.
    Анотація: Введено комутативні операції парадетермінантного та параперманентного добутків трикутних матриць, для яких парадетермінант та параперманент є мультиплікативними функціями трикутних матриць.
  37. R. A. Zatorskii, A. R. Malyarchuk, Generalization of the Ballot Numbers, Mat. Zametki, 2011, Volume 90, Issue 4, 527–529.
    Abstract: The paper is devoted to a generalization of the ballot numbers related to a random walk.
  38. Zatorsky R.A. Rational Approximations of Higher Order Algebraic Numbers and Some Parameterization of Generalized Pell Equations. arXiv:1103.5772v1 [math.NT] 29 Mar 2011. (Russian)
    Abstract: A new algebraic object is introduced – recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some parametrization for generalized Pell’s equations is constructed.
  39. Zatorsky R.A. Researching of Hessenberg’s matrix functions. Carpathian Math. Publ. 2011, 3 (1), 49–55. (Ukrainian)
    Abstract: In this work we research the connections of Hessenberg’s matrix functions with paradeterminants and parapermanents.
  40. Б.В. Василишин, П.Б. Василишин, Т.П. Гой, Б.М. Дрінь, Р.А. Заторський, А.І. Казмерчук та ін. Збірник тестових завдань з математики /Під ред. О.Р. Никифорчина та ін. – 3-тє вид. випр. і доп. – Івано-Франківськ, ПНУ, «Сімик». – 2011. 268 с.
    Анотація: Збірник тестових завдань закритої форми для підготовки до зовнішнього оцінювання знань з математики. Розроблено на основі програми з математики для загальноосвітніх шкіл Міністерства освіти і науки, молоді та спорту України.
  41. R.A. Zatorsky. Parafunctions of matrices with skew structure and polynomial factorizations. Scientific Bulletin of Chernivtsi National University. Mathematics.V.1 No.4, 2011, pp. 59-66.
    Abstract: Parafunctions of three-cornered matrices of sloping structure are considered, some combinatorics equalities and recurrent correlations for the polynomials of partitions are proved.
  42. Bodnar D. I., Zatorsky R.A.  Generalization continued fractions. I.  Mat. Metodi Fiz.-Mekh. Polya 54 (2011), No. 1, pp. 57-64.
    Abstract: The study of continued fractions expressed through parapermanents of triangular matrices, initiated in the monograph ‘Computations of triangular matrices and its applications’, has been extended.
  43. Заторський Р.А. Обернення многочленів розбиттів. Прикладні задачі математики. Всеукраїнська наукова конференція. М. Яремче, 13-15 жовтня 2011 р. с. 47-48
    Анотація: У доповіді йдеться про застосування методів лінійної алгебри до обернення многочленів розбиттів.
  44. Bodnar D. I., Zatorsky R.A.  Generalization continued fractions. II.  Mat. Metodi Fiz.-Mekh. Polya 54 (2011), No. 2, pp. 1-8.
    Abstract: The work continues the study of continued fractions initiated in [2]. Algorithms have been developed for computing rational approximations of a certain continued fraction. Based on the alues of this expression, a series of conclusions are made regarding the nature and convergence speed of he rational reductions of the continued fraction to its value.
  45. Dovbniak T.S.. Zatorsky R.A. Parafunctions of matrices of vertical and horizontal structure. Carpathian Mathematical Publications, 4. 1 (2012), 36-42.
    Abstract: The properties of parafunctions of matrices of vertical and horizontal structure are investigated. Some general formulas of inversion of polynomial sequences are built.
  46. Заторський Р.А., Стефлюк С.Д. Узагальнення арифметичної функції Мебіуса. // Матеріали XІІІ Міжнар. наук. конф. ім. академіка М. Кравчука.– К: НТУУ (КПІ). – 2012. – с. 123.
  47. Гой Т.П., Заторський Р.А. Про нові функції, породжені зростаючими факторіальними степенями, та їх властивості / Матеріали Міжнар. наук.-практ. інтернет-конф. «Мате­матичне моделювання прикладних задач математики, фізики, механіки». – Харків: Екограф, 2013. – С. 103-106
  48. R. Zatorsky, S. Stefluk, On one class of partition polynomials. Algebra Discrete Math., 2013, Volume 16, Issue 1, 127–133.
    Abstract: We consider relations between one class of partition polynomials, parafunctions of triangular matrices (tables), and linear recurrence relations.
  49. Гой Т.П., Заторський Р.А. Диференціальні рівняння функцій, породжених зростаючими факторіальними степенями. // Матеріали Міжнародної математичної конференції «Боголюбовські читання DIF-2013. Диференціальні рівняння, теорія функцій та їх застосування.- м. Севастополь, 23-30 червня 2013 р. – 83-84 с.
  50. R.A. Zatorsky. On one Function defined on the Cartesian Product and Guinness Number. arXiv:1309.1436v1 [math.GM] 3 Sep 2013.
    Abstract: New numbers, called Guinness numbers, are introduced using certain function of natural argument. Few problems related to these numbers are formulated.
  51. Goy T.P., Zatorsky R.A. New integral functions generated by rising factorial powers. Carpathian Mathematical Publications 2013, 5 (2), 217–224.
    Abstract: We consider new nonelementary functions such as the Fresnel integrals, generated by rising factorial powers. Graphs of such functions are plotted and some of their properties are proved. It is shown, that new integral functions are solutions of second order ordinary differential equations with variable coefficients.
  52. Zatorsky R. Guinness number . part I , Ivano-Frankivsk: Publishing House « Simyk »,  2013,   p. 1830.
  53. Zatorsky R. Guinness number . part IІ , Ivano-Frankivsk: Publishing House « Simyk »,  2013,   p. 1829.
  54. Б.К. Остафійчук, І.М. Будзуляк, М.М. Кузишин, Б.І.Рачій, Р.А. Заторський, Р.П. Лісовський, В.І. Мандзюк Азотовмісне нанопористе вугілля для електродів суперконденсаторів.  Журнал нано та електронної фізики Т.5 №3, 03049(6сс) (2013) (Ukrainian).
  55. Колектив авторів. Боднарчук Юрій Вікторович (некролог) Карпатські математичні публікації. Т.5, №2 (2013) стор. 345-347.
  56. R. A. Zatorsky, T. P. Goy . Generalized factorial powers and some of its application.  Carpathian Herald of the Shevchenko Scientific Society. Number. № 1(21) (2013).
    Abstract: We   consider   the   identities   of   generalized   factorial   powers.   In   particular, we generalized polynomial formula for arbitrary finite number of terms.  The  connection  of  some  non-elementary  real  functions  generated  by  rising factorials with the generalized hypergeometric function is established.
  57. І.М. Будзуляк, Б.К. Остафійчук, Р.В. Ільницький, Л.М. Гуменюк, В.М. Пилипів, Р.А. Заторський Строение и свойства наноразмерных и мезоскопических материалов. Металлофиз. Новейшие технол. 2013, т. 35, № 12, сс 1609-1616. (Russian)
  58. Roman Zatorsky, Svitlana Stefluk. On one class of partition polynomials. Algebra and Discrete Mathematics. Volume 16 (2013). Number 1. pp. 127 – 133.
    Abstract: We consider relations between one class of partition polynomials, parafunctions of triangular matrices (tables), and linear recurrence relations.
  59. Zatorsky R., Pylypiv V., Hulka S.  Together, some interconnected listed problems. Carpathian Math. Publ. 2013, 5 (2), 242–248 (Ukrainian)
    Abstract: The work is dedicated to enumerating the partitions of certain classes of multisets, n-vertex regular graphs of the second degree, and bitransversals of square matrices of order n.
  60. Zatorsky R.A. , Semenchuk A.V. Calculation Algorithm of  rational estimations of recurrence periodical fourth order Fraction. Carpathian  Math. Publ. 2014, 6 (1), 32–43.
    Abstract:  Recurrence fourth order fractions are studied. Connection with algebraic fourth order equations is established. Calculation algorithms of rational contractions of such fractions are built.
  61. T.P. Goy, R.A. Zatorsky. New functions generated by growing factorials and their properties. Bukovinian Mathematical Journal. №1-2, T.1 (2014), pp. 30-35.
    Abstract: We consider new functions defined by increasing factorial powers. We sketch graphs of such functions and find some of their properties and related formulas. It is shown that constructed functions are solutions of ordinary differential equations derived in the paper.
  62. R. A. Zatorsky, I. I. Lishchynskyj, А. V. Sеmenchuk. Relationship algebraic equations with (n,m)-forms, their degrees and recurrence fractions. .  Carpathian Herald of the Shevchenko Scientific Society. Number. № 1(25) (2014).
    Abstract:  The paper explores the connections of (n, m)-forms with generalized Pell equations, algebraic equations of nth degree, and continued fractions.
  63. R.A. Zatorsky, V.M. Pylypiv, S.D. Stefluk. Polynomial partitions. Bukovinian Mathematical Journal. №1,T.2 (2014), pp. 43-47.
    Abstract: In this paper, the authors propose a uni_ed approach to the study of partition polynomials.
  64. Goy T.P., Zatorsky R.A. On One Nonelementary Function of the Dawson Integral Type.// Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics & Mathematics.- No 1 (2014) p. 15-19.
  65. Пилипів В.М., Заторський Р.А., Ліщинський І.І. Кільце поліномів: навчально-методичний посібник. – Івано-Франківськ: Плай. – 2014. – 100 с. (Ukrainian)
  66. В.М.Пилипів, Р.А.Заторський І.І.Ліщинський. Класичні основи теорії чисел. Навчальний посібник Ів.Фр.:Плай, 2014, – 68 с. (Ukrainian).
    Анотація: В посібнику розглянуто основні аспекти вступу до теорії чисел, зокрема, поняття про системи числення, найважливіші числові функції, елементи теорії конгруенцій. Посібник є поєднанням курсу лекцій, розв’язника характерних завдань та задачника для самостійної роботи. Посібник розроблено для користування студентами класичних і технічних університетів при вивченні теорії чисел та її частин.
  67. Загороднюк А.В., Заторський Р.А., Копач М.І. та ін. Практикум з математичного аналізу. Частина 2. Івано-Франківськ: Сімик, 2014. – 90 с. (Ukrainian)
    Анотація: У посібнику наведені короткі відомості з теоретичного курсу математичного аналізу, а також вправи та приклади розв’язування деяких з них. Друга частина посібника розкриває наступні теми: границя та неперервність функції багатьох змінних, диференціювання функцій багатьох змінних, похідна функції за напрямом, градієнт функції, дослідження функції багатьох змінних на екстремум, умовний екстремум функції багатьох змінних, властивості неявних функцій та знаходження їх похідних.
    Для студентів математичних та технічних напрямів підготовки, які вивчають курси «Математичний аналіз І» та «Математичний аналіз ІІ».
  68. Заторський Р.А., Василишин П.Б., Гаврилків В.М. та ін. Збірник тестових завдань для вступників на освітньо-кваліфікаційний рівень «Спеціаліст» спеціальності «Прикладна математика»/ За ред. д.ф.-м.н. Р.А. Заторського, к.ф.-м.н. О.В.Махнея. – Івано-Франківськ: Сімик, 2014. 204 с. (Ukrainian)
    Анотація: Збірник тестових завдань закритої форми для фахового вступного випробування на освітньо-кваліфікаційний рівень спеціаліста за спеціальністю 7.04030101 «Прикладна математика».  Розроблено на основі освітньо-професійної програми підготовки бакалавра за напрямом 6.040301 «Прикладна математика». Збірник також може використовуватись для підготовки до вступних випробувань на освітньо-кваліфікаційний рівень магістра за цією ж спеціальністю.
  69. T. P. Goy, R. A. Zatorsky. Solving of quadratic Diophantine equations using systems of linear recurrence equations.  Mathematics and Natural Sciences. Theory and Practice. Issue 10. Yaroslavl, 2015.P. 32-34. (Russian).
    Abstract: We have shown the possibility of using systems of linear second order recurrence equations with constant coefficients for solving quadratic Diophantine equations.
  70. T. Goy, R. Zatorsky. Using triangular matrices for construction of ordinary differential equations for the known fundamen system of solutions. NaUKMA (National University of Kyiv-Mohyla Academy) Scientific Notes. Volume 165.,2015.  Physical and Mathematical Sciences (Ukrainian).
    Abstract: Using triangular matrices, we propose an algorithm for construction of linear homogeneous ordinary differential equations for a given fundamental system of solutions.
  71. R. Zatorsky. Introduction to the Theory of Triangular Matrices (Tables). In: Advances in Linear Algebra Research, 2015, Nova Science Publishers, New York, pp. 185-237.
    Abstract: We consider elements of linear algebra based on triangular tables with entries in some number field and their functions, analogical to the classical notions of a matrix, determinant and permanent. Some properties are investigated and applications in various areas of mathematics are given.
  72. R. A. Zatorsky, S. D. Stefliuk. General classes of mutually inverse polynomials of partition. Carpathian Herald of the Shevchenko Scientific Society. Number. № 1(29) (2015).
    Abstract: The paper is devoted to the study of general class of mutually inverse polynomials of partitions.
  73. Roman Zatorsky and Taras Goy. Parapermanents of Triangular Matrices and Some General Theorems on Number Sequences. Journal of Integer Sequences, Vol. 19 (2016).
    Abstract: Applying the apparatus of triangular matrices, this paper determines general relations between the terms of the sequences generated by linear homogeneous recurrence equations. We single out, in particular, the class of normal linear recurrence equations, for which the corresponding number sequences have some interesting number-theoretic properties.
  74. T.P. Goy, R.A. Zatorsky. About a Class of Mutually Inverse Polynomial Partitions. Carpathian Herald of the Shevchenko Scientific Society. Number. № 1(33) (2016).
    Abstract: We study a class of mutually inverse partition polynomials, which are represented as a functions of some triangular matrices.
  75. Колектив авторів. Сущанський Віталій Іванович (некролог) Карпатські математичні публікації. Т.8, №2 (2016) стор. 323-325.
  76. Svitlana Stefluk and Roman Zatorsky. Parafunctions of triangular matrices and m-ary partitions of numbers.  Algebra and Discrete Mathematics Volume 21 (2016). Number 1, pp. 144–152.
    Abstract: Using the machinavy of paradeterminants and parapermanents developed in [2] we get new relations for some number-theoretical functions natural argument that were studied in [3].
  77. Zatorskyi R., Dudka O., Vlasii O. The Role Of Information And Communication Technologies In Vizualization Of Learning Mathematics. Physical and Mathematical Education : scientific journal. – 2017. – Issue 3(13). – Р. 39-44.
    Abstract: The article analyzes the issue of information visualization in the study of mathematics. The necessity to expand and improve existing methods of visualization throughout the entire process of studying mathematics is established. The potential of utilizing modern information and communication technologies to enhance the toolkit of mathematical visualization is examined. The use of online educational resources is proposed as a means to activate students’ cognitive activity. A systematic review of educational resources and applications is conducted, which can be used for both learning new material and reinforcing acquired skills and knowledge through various forms of educational organization. The expediency of using some of the discussed resources in the study of the school mathematics course is demonstrated through illustrative examples. The need to study computerized tools for information visualization in the training of future mathematics teachers as a necessary component in the development of professional competencies is emphasized.
  78. Goy T., Zatorsky R. Infinite linear recurrence relation and superposition of linear recurrence equations, Journal of Integer Sequences (2017), 20(5), Article 17.5.3.
    Abstract: We consider the applications of triangular matrices and functions of them to the study of number sequences generated by infinite-order linear recurrence equations with constant coefficients. We also introduce the concept of superposition of linear recurrence equations and establish some of its properties.
  79. Goy T., Zatorsky R. On Oresme numbers and their connection with Fibonacci and Pell numbers, Fibonacci Quarterly (2019), 57 (3), P. 238–245.
    Abstract: In this paper, we consider determinants for some families of Toeplitz-Hessenberg matrices the entries of which are the Oresme numbers. These determinant formulas may also be rewritten as identities involving sums of products of Oresme numbers and multinomial coefficients. In particular, we establish a connection between the Oresme and the Fibonacci and Pell sequences via Toeplitz-Hessenberg determinants.
  80. Колектив авторів. Кириченко Володимир Васильович (некролог) Карпатські математичні публікації. Т115, №1 (2019) стор. 206-207.
  81. Goy T.P., Zatorsky R.A.,  Lishchynsky I.I. Symmetric polynomials and functions of triangular matrices (tables). Carpathian Herald of the Shevchenko Scientific Society. – 2019. – № 1(53). (Ukrainian).
    Abstract:   Using the apparatus of triangular matrices, certain connections have been established between specific classes of symmetric polynomials in many variables.
  82. Goy T., Zatorsky R. Hessenberg matrices: Properties and some applications, in: I. Kyrchei (ed.), Hot Topics in Linear Algebra, New York, Nova Science Publishers, 2020, P. 163–204.
    Abstract: We consider a new approach to studying the properties of Hessenberg matrices and propose an effective algorithm for calculating the determinants and permanents of such matrices. Also, we present the applications of Hessenberg matrices to the problems associated with linear recurrence relations and partition polynomials.
  83. Makhnei, O.V.; Pylypiv, V.M.; Zatorskii, R.A. m-submultisets and m-permutations of multisets elements. Journal Carpathian Math. Publ. 2021, 13 (1), 240–258.
    Abstract:  The article is devoted to two classical combinatorial problems on multisets, which in the existing literature are given unjustifiably little space. Namely, the calculation of the number of all submultisets of power m for an arbitrary multiset and the number of m-permutations of such multisets. The first problem is closely related to the width of a partially ordered set of all submultisets of a multiset with the inclusion ⊆. The article contains some important classes of multisets. Combinatorial proofs of problems on the number of m-submultisets and m-permutations of multiset elements are considered. In the article, on the basis of the generatrix method, thrifty algorithms for calculating m-submultisets and m-permutations of multiset elements are constructed. The paper also provides a brief overview of the results that are related to this area of research.
  84. Makhnei, O.V.; Pylypiv, V.M.; Zatorskii, R.A. Algebroids and combinatorial interpretations. Ivano-Frankivsk Herald of the Shevchenko Scientific Society..  2022. – No. 17(64).
    Abstract: The construction of a combinatorial-algebraic structure is considered – a set of algebraic expressions, whose operands belong to a certain multiset M, and commutative operations with a specified priority – to a certain set O.
  85. Пилипів В.М., Заторський Р.А., Ліщинський І.І.  Кільце поліномів: навчально-методичний посібник (видання друге, доповнене). – Івано-Франківськ: ЛІК. – 2023. – 120 с.
    Анотація: В посібнику розглянуто алгебраїчні та функціональні аспекти теорії поліномів, описано структуру кільця поліномів над полем та його властивості. Матеріал насичений багатьма прикладами та вправами для самостійної роботи.
    Посібник розроблено для користування студентами класичних, технічних та економічних університетів при вивченні алгебри та алгебраїчної частини вищої математики.
  86. Роман Заторський.  ЯЄ (Наука всіх наук)/З Божою допомогою, Заторський Р.А., – Івано-Франківськ: Голіней О.В., 2023. – 202 с. ISBN 978-617-95377-2-1.
    Анотація: Ця невеличка книга може розглядатися як наука про Християнство для освічених істиною людей. Вона є результатом тривалих роздумів математика над трьома одвічними запитаннями: «Звідки ЯЄ?», «Для чого ЯЄ?» і «Куди ЯЄ?». Відповіді на ці запитання природно привели до виділення такої системи понять, аксіом та законів, які дозволяють адекватно інтерпретувати Слово Святого Письма Старого і нового Завіту та, в цілому, погоджуються із словом сучасної науки.