Related papers: Integrable ODEs on Associative Algebras
The symmetries play important roles in physical systems. We study the symmetries of a Hamiltonian system by investigating the asymmetry of the Hamiltonian with respect to certain algebras. We define the asymmetry of an operator with respect…
This work develops a framework to discover relations between the components of the solution to a given initial-value problem for a first-order system of ordinary differential equations. This is done by using sparse identification techniques…
For a system of ordinary differential equations (ODEs) or, more generally, an involutive distribution of vector fields, the problem of its integration is considered. Among the many approaches to this problem, solvable structures provide a…
In this paper, we study Hamiltonian operators which are sum of a first order operator and of a Poisson tensor, in two spatial independent variables. In particular, a complete classification of these operators is presented in two and three…
A theorem that is of aid in computing the domain of the adjoint operator is provided. It may serve e.g. as a criterion for selfadjointness of a symmetric operator, for normality of a formally normal operator or for $H$--selfadjointness of…
Let $K$ be a field of arbitrary characteristic, $\CA$ be a commutative $K$-algebra which is a domain of essentially finite type (eg, the algebra of functions on an irreducible affine algebraic variety), $\ga_r$ be its {\em Jacobian ideal},…
This expository article presents a self-contained introduction to simplicial homology for finite simplicial complexes, emphasizing concrete computation and geometric intuition. Beginning with orientations of simplices and the construction…
The group gradings on the symmetric composition algebras over arbitrary fields are classified. Applications of this result to gradings on the exceptional simple Lie algebras are considered too.
We consider systems of ODEs with the right hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M.…
The notion of differentiation index for DAE systems of arbitrary order with generic second members is discussed by means of the study of the behavior of the ranks of certain Jacobian associated sub-matrices. As a by-product, we obtain upper…
The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators…
All exactly integrable systems connected with the semisimple algebras of the second rank with an arbitrary choice of the grading in them are presented in explicit form. General solution of such systems are expressed in terms of the matrix…
In the context of the integration over algebras introduced in a previous paper, we obtain several results for a particular class of associative algebras with identity. The algebras of this class are called self-conjugated, and they include,…
As an abstraction and generalization of the integral operator in analysis, integral operators (known as Rota-Baxter operators of weight zero) on associative algebras and Lie algebras have played an important role in mathematics and physics.…
First order integro-differential operators on a finite interval are studied. Properties of spectral characteristic are established, and the uniqueness theorem is proved for the inverse problem of recovering operators from their spectral…
We introduce a suitable notion of integral operators (comprising the fractional Laplacian as a particular case) acting on functions with minimal requirements at infinity. For these functions, the classical definition would lead to divergent…
Most ideas about what an algorithm is are very similar. Basic operations are used for transforming objects. The evaluation of internal and external states by relations has impact on the further process. A more precise definition can lead to…
We present a distributed algorithm to compute the first homology of a simplicial complex. Such algorithms are very useful in topological analysis of sensor networks, such as its coverage properties. We employ spanning trees to compute a…
A Lagrangian formalism for variational second-order delay ordinary differential equations (DODEs) is developed. The Noether operator identity for a DODE is established, which relates the invariance of a Lagrangian function with the…
All non-equivalent integrable evolution equations of the fifth order of the form $u_t=D_x\frac{\delta H}{\delta u}$ are found.