Recent progress in an algebraic analysis approach to linear systems

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized

Lagrangian constraints and differential Thomas decomposition

publisher: Elsevier articletitle: Lagrangian constraints and differential Thomas decomposition journaltitle: Advances in Applied Mathematics articlelink: http://dx.doi.org/10.1016/j.aam.2015.09.005 content_type: article copyright: Crown copyright © 2015 Published by Elsevier Inc. All rights reserved

Singularities of Algebraic Differential Equations

We combine algebraic and geometric approaches to general systems of algebraic ordinary or partial differential equations to provide a unified framework for the definition and detection of singularities of a given system at a fixed order. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notion that is ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main algorithmic tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.Comment: 45 pages, 5 figure

The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs

We present the Maple package TDDS (Thomas Decomposition of Differential Systems) for decomposition of polynomially nonlinear differential systems, which in addition to equations may contain inequations, into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given

On the General Analytical Solution of the Kinematic Cosserat Equations

Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.Comment: 14 pages, 3 figure

Thomas Decomposition and Nonlinear Control Systems

This paper applies the Thomas decomposition technique to nonlinear control systems, in particular to the study of the dependence of the system behavior on parameters. Thomas' algorithm is a symbolic method which splits a given system of nonlinear partial differential equations into a finite family of so-called simple systems which are formally integrable and define a partition of the solution set of the original differential system. Different simple systems of a Thomas decomposition describe different structural behavior of the control system in general. The paper gives an introduction to the Thomas decomposition method and shows how notions such as invertibility, observability and flat outputs can be studied. A Maple implementation of Thomas' algorithm is used to illustrate the techniques on explicit examples

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

The Development of Practice Recommendations for Drug-Disease Interactions by Literature Review and Expert Opinion

Background Drug-disease interactions negatively affect the benefit/risk ratio of drugs for specific populations. In these conditions drugs should be avoided, adjusted, or accompanied by extra monitoring. The motivation for many drug-disease interactions in the Summary of Product Characteristics (SmPC) is sometimes insufficiently supported by (accessible) evidence. As a consequence the translation of SmPC to clinical practice may lead to non-specific recommendations. For the translation of this information to the real world, it is necessary to evaluate the available knowledge about drug-disease interactions, and to formulate specific recommendations for prescribers and pharmacists. The aim of this paper is to describe a standardized method how to develop practice recommendations for drug-disease interactions by literature review and expert opinion. Methods The development of recommendations for drug-disease interactions will follow a six-step plan involving a multidisciplinary expert panel (1). The scope of the drug-disease interaction will be specified by defining the disease and by describing relevant effects of this drug-disease interaction. Drugs possibly involved in this drug-disease interaction are selected by checking the official product information, literature, and expert opinion (2). Evidence will be collected from the official product information, guidelines, handbooks, and primary literature (3). Study characteristics and outcomes will be evaluated and presented in standardized reports, including preliminary conclusions on the clinical relevance and practice recommendations (4). The multidisciplinary expert panel will discuss the reports and will either adopt or adjust the conclusions (5). Practice recommendations will be integrated in clinical decision support systems and published (6). The results of the evaluated drug-disease interactions will remain up-to-date by screening new risk information, periodic literature review, and (re)assessments initiated by health care providers. Actionable Recommendations The practice recommendations will result in advices for specific DDSI. The content and considerations of these DDSIs will be published and implemented in all Clinical Decision Support Systems in the Netherlands. Discussion The recommendations result in professional guidance in the context of individual patient care. The professional will be supported in the decision making in concerning pharmacotherapy for the treatment of a medical problem, and the clinical risks of the proposed medication in combination with specific diseases