Vyacheslav Boyko - Academia.edu (original) (raw)
Papers by Vyacheslav Boyko
We construct nonlinear representations of the Poincare, Galilei, and conformal algebras on a set ... more We construct nonlinear representations of the Poincare, Galilei, and conformal algebras on a set of the vector-functions Psi=(vecE,vecH)\Psi =(\vec E, \vec H)Psi=(vecE,vecH). A nonlinear complex equation of Euler type for the electromagnetic field is proposed. The invariance algebra of this equation is found.
We generalize the classical Lie results on a basis of differential invariants for a one-parameter... more We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal invariant of a one-parameter group is known then a complete set of functionally independent differential invariants can be constructed via one quadrature and differentiations.
Journal of Physics: Conference Series, 2015
Admissible point transformations of classes of rth order linear ordinary differential equations (... more Admissible point transformations of classes of rth order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational and Laguerre-Forsyth canonical forms) are exhaustively described. Using these results the group classification of such equations is carried out within the algebraic approach in three different ways.
Journal of Mathematical Analysis and Applications
Lie symmetries of systems of second-order linear ordinary differential equations with constant co... more Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
ABSTRACT We study reduction operators (called also nonclassical or conditional symmetries) of the... more ABSTRACT We study reduction operators (called also nonclassical or conditional symmetries) of the (1+1)-dimensional linear rod equation. In particular, we prove and illustrate a new theorem on linear reduction operators of linear partial differential equations.
Linear Algebra and its Applications, 2008
The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly up... more The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartan's method of moving frames.
Journal of Physics A: Mathematical and Theoretical, 2007
An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators... more An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], which deals with low-dimensional Lie algebras, here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable Lie algebras of general dimension n < ∞ restricted only by a required structure of the nilradical.
Journal of Physics A: Mathematical and Theoretical, 2007
The invariants of solvable triangular Lie algebras with one nilindependent diagonal element are s... more The invariants of solvable triangular Lie algebras with one nilindependent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartan's method of moving frames and the special technique developed for for triangular and closed algebras in [J. Phys. A: Math. Theor., 2007, V.40, 7557]. The conjecture of Tremblay and Winternitz [J. Phys. A: Math. Gen., 2001, V.34, 9085] on the number and form of elements in the bases is completed and proved.
Journal of Physics A: Mathematical and Theoretical, 2007
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular ma... more Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.
Journal of Physics A: Mathematical and General, 2006
A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir ... more A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. The algorithm is applied, in particular, to computation of invariants of real low-dimensional Lie algebras. A number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature. Bases of invariants of the real solvable Lie algebras up to dimension five, the real six-dimensional nilpotent Lie algebras and the real six-dimensional solvable Lie algebras with four-dimensional nilradicals are newly calculated and listed in tables.
Journal of Physics A: Mathematical and General, 2003
Using a new powerful technique based on the notion of megaideal, we construct a complete set of i... more Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject.
Journal of Nonlinear Mathematical Physics, 1994
We construct nonlinear representations of the Poincaré, Galilei, and conformal algebras on a set ... more We construct nonlinear representations of the Poincaré, Galilei, and conformal algebras on a set of the vector-functions Ψ = ( E, H). A nonlinear complex equation of Euler type for the electromagnetic field is proposed. The invariance algebra of this equation is found.
Journal of Mathematical Analysis and Applications, 2013
Lie symmetries of systems of second-order linear ordinary differential equations with constant co... more Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
... Maryna O. NESTERENKO and Vyacheslav M. BOYKO Kyiv Taras Shevchenko National University,... more ... Maryna O. NESTERENKO and Vyacheslav M. BOYKO Kyiv Taras Shevchenko National University, 60 Volodymyrs&amp;amp;amp;#x27;ka Str., Kyiv, Ukraine E-mail: appmath@imath.kiev.ua Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kyiv 4, Ukraine E-mail: boyko ...
We construct nonlinear representations of the Poincare, Galilei, and conformal algebras on a set ... more We construct nonlinear representations of the Poincare, Galilei, and conformal algebras on a set of the vector-functions Psi=(vecE,vecH)\Psi =(\vec E, \vec H)Psi=(vecE,vecH). A nonlinear complex equation of Euler type for the electromagnetic field is proposed. The invariance algebra of this equation is found.
We generalize the classical Lie results on a basis of differential invariants for a one-parameter... more We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal invariant of a one-parameter group is known then a complete set of functionally independent differential invariants can be constructed via one quadrature and differentiations.
Journal of Physics: Conference Series, 2015
Admissible point transformations of classes of rth order linear ordinary differential equations (... more Admissible point transformations of classes of rth order linear ordinary differential equations (in particular, the whole class of such equations and its subclasses of equations in the rational and Laguerre-Forsyth canonical forms) are exhaustively described. Using these results the group classification of such equations is carried out within the algebraic approach in three different ways.
Journal of Mathematical Analysis and Applications
Lie symmetries of systems of second-order linear ordinary differential equations with constant co... more Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
ABSTRACT We study reduction operators (called also nonclassical or conditional symmetries) of the... more ABSTRACT We study reduction operators (called also nonclassical or conditional symmetries) of the (1+1)-dimensional linear rod equation. In particular, we prove and illustrate a new theorem on linear reduction operators of linear partial differential equations.
Linear Algebra and its Applications, 2008
The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly up... more The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartan's method of moving frames.
Journal of Physics A: Mathematical and Theoretical, 2007
An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators... more An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], which deals with low-dimensional Lie algebras, here the effectiveness of the algorithm is demonstrated by its application to computation of invariants of solvable Lie algebras of general dimension n < ∞ restricted only by a required structure of the nilradical.
Journal of Physics A: Mathematical and Theoretical, 2007
The invariants of solvable triangular Lie algebras with one nilindependent diagonal element are s... more The invariants of solvable triangular Lie algebras with one nilindependent diagonal element are studied exhaustively. Bases of the invariant sets of all such algebras are constructed using an original algebraic algorithm based on Cartan's method of moving frames and the special technique developed for for triangular and closed algebras in [J. Phys. A: Math. Theor., 2007, V.40, 7557]. The conjecture of Tremblay and Winternitz [J. Phys. A: Math. Gen., 2001, V.34, 9085] on the number and form of elements in the bases is completed and proved.
Journal of Physics A: Mathematical and Theoretical, 2007
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular ma... more Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.
Journal of Physics A: Mathematical and General, 2006
A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir ... more A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan's method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. The algorithm is applied, in particular, to computation of invariants of real low-dimensional Lie algebras. A number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature. Bases of invariants of the real solvable Lie algebras up to dimension five, the real six-dimensional nilpotent Lie algebras and the real six-dimensional solvable Lie algebras with four-dimensional nilradicals are newly calculated and listed in tables.
Journal of Physics A: Mathematical and General, 2003
Using a new powerful technique based on the notion of megaideal, we construct a complete set of i... more Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject.
Journal of Nonlinear Mathematical Physics, 1994
We construct nonlinear representations of the Poincaré, Galilei, and conformal algebras on a set ... more We construct nonlinear representations of the Poincaré, Galilei, and conformal algebras on a set of the vector-functions Ψ = ( E, H). A nonlinear complex equation of Euler type for the electromagnetic field is proposed. The invariance algebra of this equation is found.
Journal of Mathematical Analysis and Applications, 2013
Lie symmetries of systems of second-order linear ordinary differential equations with constant co... more Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
... Maryna O. NESTERENKO and Vyacheslav M. BOYKO Kyiv Taras Shevchenko National University,... more ... Maryna O. NESTERENKO and Vyacheslav M. BOYKO Kyiv Taras Shevchenko National University, 60 Volodymyrs&amp;amp;amp;#x27;ka Str., Kyiv, Ukraine E-mail: appmath@imath.kiev.ua Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kyiv 4, Ukraine E-mail: boyko ...