Related papers: A Note on Algebraic Linear Partial Differential Eq…
For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie…
We consider a matrix nonlinear partial differential equation that generalizes Heisenberg ferromagnet equation. This generalized Heisenberg ferromagnet equation is completely integrable with a linear bundle Lax pair related to the…
We present a short proof, relaying on the divergence theorem, verifying that minimal sets in the plane are trivial.
In this paper we provide new examples of geometrically trivial strongly minimal differential algebraic varieties living on nonisotrivial curves over differentially closed fields of characteristic zero. Our technique involves developing a…
In this paper, we discuss the solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop…
In this paper by exploiting critical point theory, the existence of two distinct nontrivial solutions for a nonlinear algebraic system with a parameter is established. Our goal is achieved by requiring an appropriate behavior of the…
We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary…
A notion of an algebroid - a generalization of a Lie algebroid structure is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on T^M can be obtained in…
A simple algorithm to compute all the zeros of a generic polynomial is proposed.
We give upper bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over any field of constants $K$ of characteristic $0$. Let $\vec{x}$ be a set of $n$ differential variables,…
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
This short communication develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the…
A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely…
The algebraic degeneracy of holomorphic curves in a semi-Abelian variety omitting a divisor is proved (Lang's conjecture generalized to semi-Abelian varieties) by making use of the {\it jet-projection method} and the logarithmic Wronskian…
We describe a notion of categorical model for unitless fragments of (multiplicative) linear logic. The basic definition uses promonoidal categories, and we also give an equivalent elementary axiomatisation.
The paper contains a review of results on linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general inhomogeneous boundary conditions in Sobolev spaces. The character of the…
We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A…
A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…
Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if the solution to the partial differential equation is smooth…
We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a re duction of the associated dynamical…