Ring Of Linear Differential Operators Research Articles

Some remarks on Kähler differentials and ordinary differentials in nonlinear control theory

In the algebraic approach to nonlinear control systems two similar notions, namely Kähler differentials and the formal vector space of differential one-forms having the properties of ordinary differentials, are frequently used to study the systems. This technical note explains that the formal vector space of differential one-forms is isomorphic to a quotient space (module) of Kähler differentials. These two modules coincide when they are modules over a ring of linear differential operators over the field of algebraic functions. Some remarks and examples demonstrating when the use of Kähler differentials might not be appropriate are included.

Ideals of differential operators and transformations of linear partial differential equations

A ring of linear differential operators with smooth coefficients generated by two differentiations is considered. Concepts of operators closed with respect to commutation, a resultant of two operators, and a two-dimensional analogue of Wronskian are introduced. Sufficient conditions that two differential operators are generators of a left ideal annihilating a finite-dimensional space of functions are found. Differential operators annihilating given functions are constructed. The operators obtained transform solutions of one secondorder differential equation into solutions of another equation of the same order.

Commuting differential operators

We consider a model problem, related to Bessel functions, in the theory of commutative rings of linear differential operators with one independent variable. We construct new series of such commutative rings and give simple examples of rings with three generators.

Minimal resolutions of geometric [formula omitted]-modules

In this paper, we study minimal free resolutions for modules over rings of linear differential operators. The resolutions we are interested in are adapted to a given filtration, in particular to the so-called V -filtrations (see Oaku and Takayama (2001) [18] and Granger and Oaku (2004) [9]). We are interested in the module D x , t f s associated with germs of functions f 1 , … , f p , which we call a geometric module, and it is endowed with V -filtration along t 1 = ⋯ = t p = 0 . The Betti numbers of the minimal resolution associated with this data lead to analytical invariants for the germ of space defined by f 1 , … , f p . For p = 1 , we show that, under some natural conditions on f , the computation of the Betti numbers is reduced to a commutative algebra problem. This includes the case of an isolated quasi-homogeneous singularity, for which we give the Betti numbers explicitly. Moreover, for an isolated singularity, we characterize the quasi-homogeneity in terms of the minimal resolution.

Janet?s approach to presentations and resolutions for polynomials and linear pdes

Janet’s algorithm to create normal forms for systems of linear pdes is outlined and used as a tool to construct resolutions for finitely generated modules over polynomial rings over fields as well as over rings of linear differential operators with coefficients in a differential field. The main result is that a Janet basis for a module allows to read off a Janet basis for the syzygy module. Two concepts are introduced: The generalized Hilbert series allowing to read off a basis (over the ground field) of the modules, once the Janet basis is constructed, and the Janet graph, containing all the relevant information connected to the Janet basis. In the context of pdes, the generalized Hilbert series enumerates the free Taylor coefficients for power series solutions. Rather than presenting Janet’s algorithm as a powerful computational tool competing successfully with more commonly known Grobner basis techniques, it is used here to prove theoretical results.

Une interprétation algébrique de la transformation de laplace et des matrices de transfert

The tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where, even in the time-varying case, a (formal) Laplace transform and the transfer matrix are most naturally defined. Several classic problems are examined in this algebraic setting: the relationship between left (right) coprime matrix decomposition and controllability (observability), the state-variable canonical realization, the transfer algebra with respect to parallel and series connections, the input-output inversion, and model matching.

An Algebraic Definition of Time-Varying Transfer Matrices

Abstract Tensoring the module of a linear system with the quotient field of the ring of linear differential operators yields an algebraic framework where transfer matrices most naturally appear and which easily applies to the time-varying case. Several basic structural properties are examined : transfer algebra, matrix fraction decompositions and their connections with controllability and observability, realization, input-output inversion, model-matching.

Nonlinear (2+1)-dimensional systems solvable through asymptotic modules

Some nonlinear differential systems in (2+1) dimensions are characterized by means of asymptotic modules involving two poles and a ring of linear differential operators with scalar coefficients.Rational and soliton-like are exhibited. If these coefficients are rational functions, the formalism leads to nonlinear evolution equations with constraints.

Global Dimension of Differential Operator Rings. II

The aim of this paper is to find the global homological dimension of the ring of linear differential operators $R[{\theta _1}, \ldots ,{\theta _u}]$ over a differential ring $R$ with $u$ commuting derivations. When $R$ is a commutative noetherian ring with finite global dimension, the main theorem of this paper (Theorem 21) shows that the global dimension of $R[{\theta _1}, \ldots ,{\theta _u}]$ is the maximum of $k$ and $q + u$, where $q$ is the supremum of the ranks of all maximal ideals $M$ of $R$ for which $R/M$ has positive characteristic, and $k$ is the supremum of the sums $rank(P) + diff\;dim(P)$ for all prime ideals $P$ of $R$ such that $R/P$ has characteristic zero. [The value $diff\;dim(P)$ is an invariant measuring the differentiability of $P$ in a manner defined in §3.] In case we are considering only a single derivation on $R$, this theorem leads to the result that the global dimension of $R[\theta ]$ is the supremum of gl $dim(R)$ together with one plus the projective dimensions of the modules $R/J$, where $J$ is any primary differential ideal of $R$. One application of these results derives the global dimension of the Weyl algebra in any degree over any commutative noetherian ring with finite global dimension.