Vitalii S. Shpakivskyi

• Dr. Sci.
• Senior Researcher at Institute of Mathematics of the National Academy of Sciences of Ukraine

Institute of Mathematics of the National Academy of Sciences of Ukraine Municipal Institution "Zhytomyr Regional Institute of Postgraduate Pedagogical Education" of Zhytomyr Regional Council Hlybochytsia Village Council of Zhytomyr District of Zhytomyr Region invite you to take part in the International Conference «Complex and hypercomplex analys...

The infinite-dimensional family of exact solutions of the Klein--Gordon equation is constructed by the hypercomplex method.

An infinite-dimensional family of exact solutions of a three-dimensional biharmonic equation was constructed by the hypercomplex method.

The work develops Humbert's method of solving PDEs. It applies method for constructing solutions of the three-dimensional Laplace, Helmholtz, and Poisson equations in the form of components of holomorphic functions of several variables (complex and hypercomplex). For more information see https://ejde.math.txstate.edu/Volumes/2024/71/abstr.html

An infinite-dimensional family of exact solutions of a three-dimensional biharmonic equation was constructed by the hypercomplex method.

In the algebra of complex quaternions H(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H(C)}$$\end{document} we consider the left– and right–ψ\documentclass...

In this paper, an analog of the conformable fractional derivative is defined in an arbitrary finite-dimensional commutative associative algebra. Functions taking values in the indicated algebras and having derivatives in the sense of a conformable fractional derivative are called $\varphi$% -monogenic. A relation between the concepts of $\varphi$-m...

A class of H-analytic (differentiable by Hausdorff) functions in a three-dimensional noncommutative algebra e A~2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\over...

We consider a three-dimensional associative noncommutative algebra à over the field C, which contains the algebra of bicomplex numbers B(C) as a subalgebra. In this paper we consider functions of the form Φ(ζ)=f(ξ,ξ,ξ)I+f(ξ,ξ,ξ)I+f(ξ,ξ,ξ)ρ of the variable ζ=ξI+ξI+ξρ, where ξ, ξ, ξ are independent complex variables and f, f, f are holomorphic functi...

A class of H-analytic (differentiable by Hausdorff) functions in a three-dimensional noncommutative algebra $\mathbb{\widetilde{A}}_{2}$ over the field $\mathbb{C}$ is considered. All $H$-analytic functions are described in the explicit form. The obtained description is applied to the integral representation of these functions, and the mentioned fu...

We consider a three-dimensional associative noncommutative algebra Ae2 over the field C with the basis {I1, I2, ρ}, where I1, I2 are idempotents and ρ is nilpotent. The algebra Ae2 contains the algebra of bicomplex numbers B(C) as a subalgebra. In this paper we consider functions of the form Φ(ζ) = f1(ξ1, ξ2, ξ3)I1 + f2(ξ1, ξ2, ξ3)I2 + f3(ξ1, ξ2, ξ...

In this paper, we consider some three-dimensional noncommutative algebra A~2 over the field C, which contains the algebra of bicomplex numbers B(C) as a subalgebra. Locally bounded and Gâteaux-differentiable mappings defined in the domains of the three-dimensional subspace of the algebra B(C) and taking values in the algebra A~2 are considered. Suc...

We consider the concept of the Hausdorff analyticity for functions ranged in real algebras and the corresponding notion of the Hausdorff derivative. Both apply to the real algebra H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{...

A correspondence between a monogenic function in an arbitrary finite-dimensional commutative associative algebra and a finite set of monogenic functions in a special commutative associative algebra is established.

We consider a class of so-called quaternionic G-monogenic (differentiable in the sense of Gâteaux) mappings and propose a description of all mappings in this class by using four analytic functions of complex variable. For G-monogenic mappings we generalize some analogues of classical integral theorems of the holomorphic function theory of one compl...

Let A n be an arbitrary n-dimensional commutative associative algebra over the field of complex numbers. Let e 1 = 1, e 2 , e 3 be elements of A n which are linearly independent over the field of real numbers. We consider monogenic (i.e., continuous and differentiable in the sense of Gateaux) functions of the variable xe 1 + ye 2 + ze 3 , where x,...

Algebraic-analytic approach to constructing solutions for given partial differential equations were investigated in many papers. In particular, in papers [1-14]. It involves solving two problems. Problem (P 1) is to describe all the sets of vectors \( e_1, e_2, \ldots, e_d \), which satisfy the characteristic equation (or specify the procedure by w...

In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on some sequences of commutative associative algebras over the field of complex numbers. To achieve this goal, we f...

We consider a class of so-called quaternionic G-monogenic mappings associatedwith m-dimensional (m 2 f2; 3; 4g) partial differential equations and propose a description of allmappings from this class by using four analytic functions of complex variable. For G-monogenicmappings we generalize some analogues of classical integral theorems of the holom...

A correspondence between a monogenic function in an arbitrary finite-dimensional commutative associative algebra and a finite set of monogenic functions in a special commutative associative algebra is established.

For G-monogenic mappings taking values in the algebra of complex quaternions we generalize some analogues of classical integral theorems of the holomorphic function theory of a complex variable (the surface and the curvilinear Cauchy integral theorems).

In this paper, we introduce h(x) – Fibonacci polynomials in an arbitrary finite-dimensional unitary algebra over a field These polynomials generalize h(x) – Fibonacci quaternion polynomials andh(x) – Fibonacci octonion polynomials. For h(x) – Fibonacci polynomials in an arbitrary algebra, we provide generating function, Binet-style formula, Catalan...

In the paper [1] we consider a new class, so-called, $G$-monogenic (differentiable in the sense of Gateaux) quaternionic mappings. In the present paper we introduce quaternionic $H$-monogenic (differentiable in the sense of Hausdorff) mappings and establish the relation between $G$-monogenic and $H$-monogenic mappings. The equivalence of different...

We obtain explicitly principal extensions of analytic functions of the complex variable into an infinite-dimensional commutative Banach algebra associated with the three-dimensional Laplace equation. We consider an extension of differentiable in the sense of Gâteaux functions with values in a topological vector space being an expansion of the menti...

Известно, что в некоторых коммутативных ассоциативных алгебрах экспоненциальная функция, при определенном выборе базисных векторов алгебры, является решением линейного дифференциального уравнения в частных производных с постоянными коэффициентами. В данной работе предложена процедура, позволяющая для любого натурального числа $N$ выписать все компо...

In this paper, we introduce h(x)-Fibonacci polynomials in an arbitrary finite-dimensional unitary algebra over a field K (K = R,C), which generalize both h(x)-Fibonacci quaternion polynomials and h(x)-Fibonacci octonion polynomials. For h(x)-Fibonacci polynomials in such an arbitrary algebra, we prove summation formula, generating function, Binet-s...

Построены аналитические решения одного уравнения гидродинамики и получены их представления в виде действительнозначных компонент от некоторых функций со значениями в двумерной коммутативной алгебре. ================================================================================ We construct analytic solutions of one equation of hydrodynamics. We...

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.~e. continuous and differentiable in the...

In this paper, we prove some relations between Fibonacci elements in an arbitrary algebra. Moreover, we define imaginary Fibonacci quaternions and imaginary Fibonacci octonions and we prove that always three arbitrary imaginary Fibonacci quaternions are linear independents and the mixed product of three arbitrary imaginary Fibonacci octonions is ze...

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.e. continuous and differentiable in the...

We consider an arbitrary finite-dimensional commutative associative algebra, $\mathbb{A}_n^m$, with unit over the field of complex number with $m$ idempotents. Let $e_1=1,e_2,e_3$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.e. continuous and differentiable in the sense of G...

In the paper [1] considered a new class of quaternionic mappings, so-called $G$-monogenic mappings. In this paper we prove analogues of classical integral theorems of the holomorphic function theory: the Cauchy integral theorems for surface and curvilinear integrals, and the Cauchy integral formula for $G$-monogenic mappings.

We consider a new class of quaternionic mappings, associated with the spatial partial differential equations. We describe all mappings from this class using four analytic functions of the complex variable.

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,e_3$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We consider monogenic (i.e. continuous and differentiable in the sense of Gateaux) functions o...

Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the g...

Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the a...

In this paper, we generalized De Moivre's formula and Euler's formula to octonions and find the roots of generalized octonions using these formulae.

In this paper, we generalized De Moivre's formula and Euler's formula to octonions and find the roots of generalized octonions using these formulae.

Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the a...

We prove an analogue of the Cauchy integral theorem for hyperholomorphic functions given in three-dimensional domains with non piece-smooth boundaries and taking values in an arbitrary finite-dimensional commutative associative Banach algebra.

We obtain a constructive description of monogenic functions taking values in a �nite-dimensional commutative algebra with unit and radical of maximal dimensionality by means of holomorphic functions of the complex variable. We prove that the mentioned monogenic functions have the Gateaux derivatives of all orders, and analogues of classical theorem...

For monogenic (continuous and Gâteaux-differentiable) functions taking values in a three-dimensional harmonic algebra with two-dimensional radical, we compute the logarithmic residue. It is shown that the logarithmic residue depends not only on the roots and singular points of a function but also on the points at which the function takes values in...

In this paper we investigated holomorphic functions (belonging to the kernel of the Dirac operator) in Cayley-Dickson algebras. For this purpose, we study the structure of Cayley-Dickson algebras. We also provide an algorithm for the construction of such functions.

We consider a certain analog of the Cauchy integral taking values in a three-dimensional harmonic algebra with two-dimensional radical. We establish sufficient conditions for the existence of limiting values of this integral on the curve of integration.

We consider a certain analog of Cauchy type integral taking values in a three-dimensional harmonic algebra with two-dimensional radical. We establish sufficient conditions for an existence of limiting values of this integral on the curve of integration.

We establish a constructive description of twice-monogenic functions of two variables by means of twice-differentiable functions of one real variable.

In this paper, we investigate some properties of generalized Fibonacci quaternions and Fibonacci-Narayana quaternions.

Starting from known results, due to Y. Tian in [Ti; 00], referring to the real matrix representations of the real quaternions, in this paper we will investigate the left and right real matrix representations for the complex quaternions and we give some examples in the special case of the complex Fibonacci quaternions.

Polynomial identities in algebras are the central objects of Polynomial Identities Theory. They play an important role in learning of algebras properties. In particular, the Hall identity is fulfilled in the quaternion algebra and does not hold in other non-commutative associative algebras. For this reason, the Hall identity is important for the qu...

We establish sufficient conditions for the existence of the limiting values of a certain analog of the Cauchy type integral taking values in a three-dimensional harmonic algebra with two-dimensional radical.

We establish sufficient conditions for the existence of limiting values of a certain analog of the Cauchy type integral taking values in a three-dimensional harmonic algebra with two-dimensional radical.

The general linear quaternionic equation with one unknown and systems of linear quaternionic equations with two unknown are solved. Examples of equations and their systems are considered. Mathematics Subject Classification (2010)15A06–11R52

By using analytic functions of a complex variable, we give a constructive description of monogenic functions that take values in a commutative harmonic algebra of the third rank over the field of complex numbers. We establish an isomorphism between algebras of monogenic functions in the case of transition from one harmonic basis to another.

The idea of an algebraic-analytic approach to equations of mathematical physics means finding commutative Banach algebras such that mono-genic functions defined on them form an algebra and have components satisfying previously given equations with partial derivatives. We obtain constructive descriptions of monogenic functions taking values in commu...

The idea of an algebraic-analytic approach to equations of mathematical physics means to find commutative Banach algebras such that monogenic functions defined on them form an algebra and have components satisfying previously given equations with partial derivatives. We obtain constructive descriptions of monogenic functions taking values in commut...

For monogenic functions taking values in a three-dimensional commutative harmonic algebra with unit element and a two-dimensional radical, we prove analogs of classical integral theorems of the theory of analytic functions of one complex variable: the Cauchy integral theorems for a surface integral and a curvilinear integral, the Morera theorem and...

For monogenic functions taking values in a three-dimensional commutative harmonic algebra with the unit and two-dimensional radical, we prove analogs of classical integral theorems of the theory of analytic functions of the complex variable: the Cauchy integral theorems for surface integral and curvilinear integral, the Morera theorem and the Cauch...

We investigate a certain wide class of quadratic quaternioinic equations. In the article [A. Pogorui and M. Shapiro, Complex Variables, Theory Appl. 49, No. 6, 379–389 (2004; Zbl 1160.30353)], equations of the form ∑ l=0 n a l x l are investigated and it is proved that the roots can consist of isolated points and spheres. In our article, we show th...

The authors investigate the set of all solutions of any quaternionic quadratic equation of the form ax 2 +xa+bx+c=0, by reducing it to one of the form x 2 +px+q=0. This set can be one point, or two points, or a two-dimensional sphere perpendicular to the real axis (but with any centre, unlike the case x 2 +ax+b=0 investigated by A. Pogorui and M. S...