Nonstandard methods for bounds in differential polynomial rings

Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt ["Bounds in the theory of polynomial rings over fields. A nonstandard approach.", Inventionnes Mathematicae, 76:77--91, 1984] are here extended to differential polynomial rings over differential fields. Among the standard consequences of this work are: a partial answer to the primality problem, the equivalence of this problem with several others related to the Ritt problem, and the existence of bounds for characteristic sets of minimal prime differential ideals and for the differential Nullstellensatz.Comment: 18 page

A nonpolynomial collocation method for fractional terminal value problems

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational and Applied Mathematics, 275, February 2015, doi: 10.1016/j.cam.2014.06.013In this paper we propose a non-polynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α, 0 < α < 1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a non-polynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.The work was supported by an International Research Excellence Award funded through the Santander Universities scheme

On Projective Equivalence of Univariate Polynomial Subspaces

We pose and solve the equivalence problem for subspaces of ${\mathcal P}_n$, the $(n+1)$ dimensional vector space of univariate polynomials of degree $\leq n$. The group of interest is ${\rm SL}_2$ acting by projective transformations on the Grassmannian variety ${\mathcal G}_k{\mathcal P}_n$ of $k$-dimensional subspaces. We establish the equivariance of the Wronski map and use this map to reduce the subspace equivalence problem to the equivalence problem for binary forms

Quasi-exact solvability in a general polynomial setting

Our goal in this paper is to extend the theory of quasi-exactly solvable Schrodinger operators beyond the Lie-algebraic class. Let \cP_n be the space of n-th degree polynomials in one variable. We first analyze "exceptional polynomial subspaces" which are those proper subspaces of \cP_n invariant under second order differential operators which do not preserve \cP_n. We characterize the only possible exceptional subspaces of codimension one and we describe the space of second order differential operators that leave these subspaces invariant. We then use equivalence under changes of variable and gauge transformations to achieve a complete classification of these new, non-Lie algebraic Schrodinger operators. As an example, we discuss a finite gap elliptic potential which does not belong to the Treibich-Verdier class.Comment: 29 pages, 10 figures, typed in AMS-Te

Solution of polynomial Lyapunov and Sylvester equations

A two-variable polynomial approach to solve the one-variable polynomial Lyapunov and Sylvester equations is proposed. Lifting the problem from the one-variable to the two-variable context gives rise to associated lifted equations which live on finite-dimensional vector spaces. This allows for the design of an iterative solution method which is inspired by the method of Faddeev for the computation of matrix resolvents. The resulting algorithms are especially suitable for applications requiring symbolic or exact computation

Language-based Abstractions for Dynamical Systems

Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of effectively performing analyses. This has motivated a large body of research, across many disciplines, into abstraction techniques that provide smaller ODE systems while preserving the original dynamics in some appropriate sense. In this paper we give an overview of a recently proposed computer-science perspective to this problem, where ODE reduction is recast to finding an appropriate equivalence relation over ODE variables, akin to classical models of computation based on labelled transition systems.Comment: In Proceedings QAPL 2017, arXiv:1707.0366

The derived moduli space of stable sheaves

We construct the derived scheme of stable sheaves on a smooth projective variety via derived moduli of finite graded modules over a graded ring. We do this by dividing the derived scheme of actions of Ciocan-Fontanine and Kapranov by a suitable algebraic gauge group. We show that the natural notion of GIT-stability for graded modules reproduces stability for sheaves