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Darboux developed an ingenious algebraic mechanism to construct infinite chains of ''integrable'' second-order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and…
What is a vector field on a C*-algebra is defined. Its relation to semigroups of endomorphisms was researched. Some results given about those vector fields and semigroups. There are also various constructions of semigroups including one…
We study completeness in partial differential varieties. We generalize many results from ordinary differential fields to the partial differential setting. In particular, we establish a valuative criterion for differential completeness and…
Let $k$ be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over $k$ via Tschirnhausen transformation. Based on the general result in the former part, we give an…
A new family of solutions of the Jacobi partial differential equations for finite-dimensional Poisson systems is investigated. This family is mathematically remarkable, as the functional dependences of the solutions appear to be associated…
The main aim of this paper is to investigate Darboux rectifying curves on a smooth surface immersed in the Euclidean space. First, we discuss the component of the position vector of a Darboux rectifying curve on a smooth immersed surface…
In this paper we study vector fields on the big phase space of Gromov-Witten theory which are idempotents of the quantum product. Such vector fields can be used to simplify universal equations for higher genus Gromov-Witten invariants.
We use the notion of isomorphism between two invariant vector fields to shed new light on the issue of linearization of an invariant vector field near a relative equilibrium. We argue that the notion is useful in understanding the passage…
In this paper we study polynomial maps of vector spaces and their eigenvectors and eigenvalues. The new quantity called complanart is defined. Complanarts determine complanarity of solution vectors of systems of polynomial equations.…
General covariance is a crucial notion in the study of field theories in curved spacetime. A field theory defined with respect to a semi-Riemannian metric is generally covariant if two metrics which are related by a diffeomorphism produce…
We develop the Darboux procedure for the case of the two-level system. In particular, it is demonstrated that one can construct the Darboux intertwining operator that does not violate the specific structure of the equations of the two-level…
In this paper, we classify all polynomial vector fields in $\mathbb{R}^3$ of degree up to three such that their flow makes the torus $$\mathbb{T}^2=\{(x,y,z)\in \mathbb{R}^3:(x^2+y^2-a^2)^2+z^2-1=0\}~\mbox{with}~a\in (1,\infty)$$ invariant.…
We show that for any finite configuration of closed curves $\Gamma\subset \mathbb{R}^2$, one can construct an explicit planar polynomial vector field that realizes $\Gamma$, up to homeomorphism, as the set of its limit cycles with…
We develop method that allows to derive reductions and solutions to hyperbolic systems of partial differential equations. The method is based on using functions that are constant in the direction of characteristics of the system. These…
Working within the path-integral framework we first establish a duality between the partion functions of two $U(1)$ gauge theories with a theta term in $d=4$ space-time dimensions. Then, after a dimensional reduction to $d=3$ dimensions we…
N-order Darboux transformation operator is defined on the basis of a general notion of transformation operators. Factorisation properties of this operator are studied. The Darboux transformation operator technique is applied to construct…
We prove that certain classical groups $G\subseteq {\rm GL}(d,\mathbb{R}^d)$ serve to characterize ordinary polynomials in $d$ real variables as elements of finite-dimensional subspaces of $C(\mathbb{R}^d)$ that are invariant by changes of…
Darboux transformations are non-group type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation $ML=L_1M$ or the intertwining relation $ML=L_1N$ in the cases when the…
This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe regular and singular mechanical systems), and certain cases of multisymplectic manifolds, and extends it…
We apply the Darboux theory of integrability to polynomial ODE's of dimension 3. Using this theory and computer algebra, we study the existence of first integrals for the 3-dimensional Lotka-Volterra systems with polynomial invariant…