S. A. Plaksa

• Professor
• Head of Department at 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 aim of this work is to prove an analog of Menchov–Trokhimchuk theorem on weakening conditions of monogeneity for functions given in a concrete three-dimensional commutative algebra over the field of complex numbers. The property of monogeneity of a function is understood as a combination of its continuity with the existence of its Gâteaux deriv...

For monogenic (continuous and differentiable in the sense of G\^ateaux) functions given in special real subspaces of an arbitrary finite-dimensional commutative associative algebra over the complex field and taking values in this algebra, we establish basic properties analogous to properties of holomorphic functions of a complex variable. Methods f...

We develop a functionally analytic method for effective solving boundary problems in a meridian plane of a spatial axial-symmetric potential field. This method is based on integral expressions for the axial-symmetric potential and Stokes’ flow function in an arbitrary simply connected domain symmetric with respect to an axis that are presented in C...

We consider special topological vector spaces with a commutative multiplication for some of elements of the spaces and monogenic functions taking values in these spaces.Monogenic functions are understood as continuous and differentiable in the sense of G\^ateaux functions.We describe relations between the mentioned monogenic functions and harmonic...

The aim of the present work is to weaken the conditions of monogeneity for functions taking values in a given three-dimensional commutative algebra over the field of complex numbers. The monogeneity of a function is understood as a combination of its continuity with the existence of Gâteaux derivative.

A commutative algebra \( \mathbbm{B} \) over the complex field with a basis {e1, e2} satisfying the conditions \( {\left({e}_1^2+{e}_2^2\right)}^2=0,{e}_1^2+{e}_2^2\ne 0 \) is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type Φ (xe1+...

A commutative algebra $\mathbb{B}$ over the complex field with a basis $% \{e_{1},e_{2}\}$ satisfying the conditions $(e_{1}^{2}+e_{2}^{2})^{2}=0$, $% e_{1}^{2}+e_{2}^{2}\neq 0$ is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type $\...

УДК 517.54 Послаблено умови моногенності функцій зі значеннями в певній тривимірній комутативній алгебрі над полем комплексних чисел.Під моногенністю мається на увазі неперервність та існування похідної Гато.

We develop a hypercomplex method of solving of boundary value problems for biharmonic functions. This method is based on a relation between biharmonic functions and monogenic functions taking values in a commutative algebra associated with the biharmonic equation. We consider Schwarz-type boundary value problems for monogenic functions that have re...

The aim of this work is to weaken the conditions of monogenity for functions that take values in one concrete three-dimensional commutative algebras over the field of complex numbers. The monogenity of the function understood as a combination of its continuity with the existence of a Gato derivative.

We consider a generalized Cauchy–Riemann system with a rectilinear singular interval of the real axis. Schwarz boundary value problems for generalized analytic functions which satisfy the mentioned system are reduced to the Fredholm integral equations of the second kind under natural assumptions relating to the boundary of a domain and the given bo...

The methods involving the functions analytic in a complex plane for plane potential fields inspire the search for the analogous efficient methods for solving the spatial and multidimensional problems of mathematical physics. Many such methods are based on the mappings of hypercomplex algebras. The essence of the algebraic-analytic approach to ellip...

In the paper we consider a certain analog of the Cauchy type integral taking values in a three-dimensional commutative algebra over the field of complex numbers with one-dimensional radical. We have established sufficient conditions for the existence of limiting values for such an integral. It is also shown that analogues of Sokhotskii–Plemelj form...

We consider a generalized Cauchy-Riemann system with a rectilinear singular interval of the real axis. Schwarz boundary value problems for generalized analytic functions which satisfy the mentioned system are reduced to the Fredholm integral equations of the second kind under natural assumptions relating to the boundary of a domain and the given bo...

We consider axial-symmetric stationary flows of the ideal incompressible fluid as an important case of potential solenoid fields. We use an integral expression of the Stokes flow function via the corresponding complex analytic function for solving a boundary value problem with respect to a steady streamline of the ideal incompressible fluid along a...

We consider axial-symmetric stationary flows of the ideal incompressible fluid as an important case of potential solenoid vector fields. We establish relations between axial-symmetric potential solenoid fields and principal extensions of complex analytic functions into a special topological vector space containing an infinite-dimensional commutativ...

We consider Schwartz-type boundary value problems for monogenic functions in a commutative algebra \(\mathbb {B}\) over the field of complex numbers with the bases {e1, e2} satisfying the conditions \((e_1^2+e_2^2)^2=0\), \(e_1^2+e_2^2\ne 0\). The algebra \(\mathbb {B}\) is associated with the biharmonic equation, and considered problems have relat...

We consider a commutative algebra B over the field of complex numbers with a basis {e 1 , e 2 } satisfying the conditions (e12+e22)2=0, e12+e22≠0. We consider a Schwarz-type boundary value problem for “analytic” B-valued functions in a simply connected domain. This problem is associated with BVPs for biharmonic functions. Using a hypercomplex analo...

We consider monogenic functions taking values in a three-dimensional commutative algebra A 2 over the field of complex numbers with one- dimensional radical. We calculate the logarithmic residues of monogenic functions acting from a three-dimensional real subspace of A 2 into A 2 . It is shown that the logarithmic residue depends not only on zeros...

We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {e1, e2} satisfying the conditions (e12+e22)2=0,e12+e22≠0. $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i...

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...

A commutative algebra $\mathbb{B}$ over the field of complex numbers with the bases $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$, is considered. The algebra $\mathbb{B}$ is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem on finding a monogenic function of the type $\Phi(xe_...

We consider a commutative algebra B over the field of complex numbers with a basis {e_1, e_2} satisfying the conditions (e_1^{2}+e_2^{2})^2=0, e_1^2+e_2^2 ><0. Let D be a bounded domain in the Cartesian plane xOy and Dζ = {xe_1 +ye_2 : (x, y) ∈ D}. Components of every monogenic function Φ(xe_1 + ye_2) =U_{1}(x, y) e_1+U_{2}(x, y) ie_1+U_{3}(x, y) e...

We consider a commutative algebra $\mathbb{B}$ over the field of complex numbers with a basis $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$. Let $D$ be a bounded domain in the Cartesian plane $xOy$ and $D_{\zeta}=\{xe_1+ye_2 : (x,y)\in D\}$. Components of every monogenic function $\Phi(xe_1+ye_2)=U_{1}(x,y)\,e_1+U_...

Одержано конструктивний опис моногенних функцiй, що набувають значень в скiнченновимiрнiй напiвпростiй комутативнiй алгебрi, за допомогою аналiтичних функцiй комплексної змiнної. Доведено, що такi моногеннi функцiї мають похiднi Гато усiх порядкiв.

We obtained a constructive description of monogenic functions taking values in a finite-dimensional semi-simple commutative algebra by means of analytic functions of the complex variable. We proved that the mentioned monogenic functions have the Gateaux derivatives of all orders. We have proved analogs of classical integral theorems of the theory o...

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 obtained a constructive description of monogenic functions taking values in a three-dimensional commutative harmonic semi-simple algebra and of monogenic functions taking values in a three-dimensional harmonic algebra with the one-dimensional radical by means of holomorphic functions of the complex variable. We proved that the mentioned monogeni...

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...

We present a constructive description of monogenic functions that take values in a three-dimensional commutative harmonic algebra with one-dimensional radical by using analytic functions of complex variable. It is shown that monogenic functions have the Gâteaux derivatives of all orders.

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 establish integral theorems for monogenic functions taking values in an infinite-dimensional commutative Banach algebra associated with spatial potential solenoid fields symmetric with respect to an axis. We also establish integral theorems for monogenic functions taking values in a topological vector space being an expansion of the mentioned al...

We consider monogenic functions given in a biharmonic plane and taking values in a commutative algebra associated with the biharmonic equation. For the mentioned functions, we establish basic properties analogous to properties of holomorphic functions of the complex variable: the Cauchy integral theorem and integral formula, the Morera theorem, the...

Тамразов Промарз Мелiкович — вiдомий вчений-математик, що працював, насамперед, в комплексному аналiзi i теорiї потенцiалу, професор, член-кореспондент НАН України з 2006 року.

The idea of an algebraic-analytic approach to equations of mathematical physics means to find a commutative Banach algebra such that monogenic functions with values in this algebra have components satisfying to given equations with partial derivatives. We obtain here a constructive description of monogenic functions taking values in a commutative a...

80-річчя від дня народження) Юрій Іванович Самойленко — член-кореспондент НАН України, відомий учений, наукова ді-яльність якого пов'язана з математичним моделюванням фізичних процесів, побудовою основ теорії просторово розподілених систем керування швидкоплинними фізичними процесами. Він є одним із фундаторів фізичної кібернетики. Народився Ю. І....

We consider a two-dimensional commutative algebra B over the field of complex numbers. The algebra B is associated with the biharmonic equation. For monogenic functions with values in B, we consider a Schwartz-type boundary value problem (associated with the main biharmonic problem) for a half-plane and for a disk of the biharmonic plane. We obtain...

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.

Член-кореспондент НАН України, головний науковий спiвробiтник вiддiлу комплексного аналiзу i теорiї потенцiалу Iнституту математики НАН України Ю.I. Самойленко (08.IV.1932 — 11.XII.2008) був i залишається вiдомим вченим, наукова дiяльнiсть якого пов’язана з математичним моделюванням фiзичних процесiв, побудовою основ теорiї просторово розподiлених...

The Riemann boundary-value problem is solved for the classes of open rectifiable Jordan curves extended as compared with previous results and functions defined on these curves.

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 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 present a constructive description of monogenic functions that take values in a commutative biharmonic algebra by using analytic functions of complex variables. We establish an isomorphism between algebras of monogenic functions defined in different biharmonic planes. It is proved that every biharmonic function in a bounded simply connected doma...

We obtain integral representations of generalized axially symmetric potentials via analytic functions of a complex variable that are defined in an arbitrary simply connected bounded domain symmetric with respect to the real axis. We prove that these integral representations establish a one-to-one correspondence between analytic functions of a compl...

In an infinite dimensional commutative Banach algebra, we construct explicitly monogenic functions, which have components satisfying the three dimensional Laplace equation. We also describe a relation between these monogenic functions and harmonic vectors.

We extend classes of closed rectifiable Jordan curves and given functions in the theory of the piecewise-continuous Riemann boundary-value problem and the characteristic singular integral equation with Cauchy kernel related to this problem.

We establish sufficient conditions for the differentiability of a singular Cauchy integral with piecewise-continuous density. Formulas for the nth-order derivatives of a singular Cauchy integral and for the boundary values of the nth-order derivatives of a Cauchy-type integral are obtained.

For investigation of equations with partial derivatives we develop a method analogous to the analytic function method in the complex plane. We have obtained expressions of solutions of elliptic equations degenerating on an axis via components of analytic functions taking values in a commutative associative Banach algebra.

For investigation of elliptic type equations degenerating on an axis we develop a method analogous to the method of analytic function in the complex plane. We have obtained expressions of generalized axial-symmetric potentials via components of analytic functions taking values in a commutative associative Banach algebra.

Sufficient conditions for Noetherian properties of singular integral equation with the Cauchy kernel with piecewise-continuous coefficients on a closed Jordan rectifiable curve are established.

We prove generalized Noether theorems for a singular integral equation with Cauchy kernel on a closed rectifiable Jordan curve in classes of piecewise-continuous functions with oscillation-type discontinuities. We obtain results concerning the normal solvability of operators associated with the equation and acting into a Banach space and incomplete...

We develop a method for the reduction of the Dirichlet problem for the Stokes flow function in a simply-connected domain of the meridian plane to the Cauchy singular integral equation. For the case where the boundary of the domain is smooth and satisfies certain additional conditions, the regularization of the indicated singular integral equation i...

We develop new methods for solving boundary problems for spatial axial-symmetric potential solenoid fields depending on the nature and specific features of axial-symmetric problems. The Dirichlet problems for the axial-symmetric potential and the Stokes flow function in a simply connected domain of the meridian plane are reduced to the Cauchy singu...

For an unbounded domain of the meridian plane with bounded smooth boundary that satisfies certain additional conditions, we develop a method for the reduction of the Dirichlet problem for an axisymmetric potential to Fredholm integral equations. In the case where the boundary of the domain is a unit circle, we obtain a solution of the exterior Diri...

We develop a method for the reduction of the Dirichlet problem for an axisymmetric potential in a simply connected domain of the meridian plane to a Cauchy singular integral equation. In the case where the boundary of the domain is smooth and satisfies certain additional conditions, we regularize the indicated singular integral equation.

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.

We obtain new integral representations for an axisymmetric potential and the Stokes flow function in an arbitrary simply-connected domain of the meridian plane. The boundary properties of these integral representations are studied for domains with closed rectifiable Jordan boundary.

We develop new methods for the solution of boundary-value problems in the meridian plane of an antisymmetric potential solenoidal field with regard for the nature and specific features of axisymmetric problems. We determine the solutions of the Dirichlet problems for an axisymmetric potential and the Stokes flow function in a disk in an explicit fo...

System of equations $$y\frac{{\partial\varphi(x,y)}}{{{\partial_x}}}=\frac{{\partial\psi(x,y)}}{{{\partial_y}}},y\frac{{\partial\varphi (x,y)}}{{{\partial_y}}}=-\frac{{\partial\psi(x,y)}}{{{\partial_x}}}$$ (1) describes the spatial potential solenoid field, which is symmetrycal with respect to the axis Ox,in its meridianalplane xOy.Here φ,o is the...

We obtain new representations of the potential and flow function of three-dimensional potential solenoidal fields with axial symmetry, study principal algebraic analytic properties of monogenic functions of vector variables with values in an infinite-dimensional Banach algebra of even Fourier series, and establish the relationship between these fun...

We obtain a new representation of potential and flow functions for space potential solenoidal fields with axial symmetry. We study principal algebraic-analytical properties of monogenic functions of a vector variable with values in an infinite-dimensional Banach algebra of even Fourier series and describe the relationship between these functions an...

The principal biharmonic problem for a quadrant with piecewise-continuous boundary conditions is reduced to a system of nonsingular integral equations.

The paper deals with the theory of a complete singular integral equation with a Cauchy kernel. The classes of curves and given functions are extended and generalizations of the classical Noether theorems are proved. As a consequence of these theorems, the Noether property is established for the operators associated with this equation, which act int...

Under the minimal assumptions imposed on given spaces, the sufficient conditions are established, under which the compositionBA of operatorsA andB has the Noether property and is normally solvable. Similar conditions, guaranteeing that the operatorA is normally solvable or possesses the Noether property, are obtained for the operatorsB andBA.

The problem of perturbation of a semi-Noether operator is studied under minimal restrictions imposed on given spaces.

The perturbation problem is considered for a semi-Noether operator under minimal assumptions imposed on given spaces.

We study the inhomogeneous Riemann boundary problem with infinite index of logarithmic order on an open rectifiable spiral-form Jordan contour where the influence of the contour on the solvability of the problem is comparable with the influence of the argument of its coefficient. A solution of the problem is constructed explicitly in the class of f...

We study the Riemann boundary problem with infinite index of logarithmic order on an open rectifiable Jordan spiral-form contour, here the influence of the contour on the solvability of the problem is comparable with the influence of the argument of its coefficient. An explicit solution of the problem is constructured in the class of functions admi...

The author investigates the piecewise-continuous Riemann boundary-value problem with index minus infinity on a closed rectifiable Jordan curve; the index of the problem is a measure of the integral effect exerted on the solvability of the problem by the argument and modulus of the coefficient, and also by the properties of the junction curve. Disco...

A piecewise-continuous Riemann boundary problem with index plus-infinity on a closed rectifiable Jordan curve is studied; here the index of the problem takes into account the integral influence on its solvability of the argument and modulus of the coefficient of the problem and also the properties of the line of conjugation. One permits discontinui...