Qualitative properties and stabilizability of a model for blood thrombin formation

We perform a qualitative analysis of a differential equation that was originally introduced by Stortelder, Hemker and Hemker to model the formation of thrombin, and discuss issues of controllability and stabilizability. Results include a general proof of convergence to equilibrium, and of local exponential stabilizability.     

The inverse syzygy problem in algebraic systems theory

We present a constructive solution of the inverse syzygy problem over arbitrary coherent rings and show how it can be used to compute certain extension groups. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear exact modeling from multivariate data

We construct a linear system of PDE having certain prescribed solutions and as little additional solutions as possible. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The canonical Cauchy problem for linear systems of partial difference equations with constant coefficients over the completer-dimensional integral lattice ℕ2r

We canonically define and algorithmically solve the problem of the title. Such algorithms are of great significance for the method of finite differences for the solution of partial differential equations and for many technical applications such as image processing. In contrast to the wide (system theoretic) literature for ordinary difference equations and in spite of the great theoretical and practical significance of this problem, until now, there was no systematic theory of these systems and in particular of the corresponding Cauchy problem, let alone an algorithm. In this paper, we give both. The method consists in a transformation of this problem into a naturally associated problem which is defined over the 2r-dimensional natural number lattice ^2r (the upper quadrant in ^2r ) and for which the canonical initial value or Cauchy problem was defined and constructively solved by the second author.     

On the Inverse Syzygy Problem

We present a constructive solution of the inverse syzygy problem over arbitrary coherent rings. By relating the existence of a kernel representation to torsionlessness instead of the more common torsionfreeness, we do not need to assume the existence of a quotient field. As a by-product, we obtain an algorithm to compute the extension groups of finitely presented modules.     

A note on the kinetics of suicide substrates

We determine the limit sets of a system modelling suicide substrate kinetics, and show that a result by Tatsunami et al. (Biochim Biophys Acta 662:226–235, 1981), derived under additional quasi-steady state assumptions, holds generally.     

Computing quasi-steady state reductions

There is a systematic approach to the computation of quasi-steady state reductions, employing the classical theory of Tikhonov and Fenichel, rather than the commonly used ad-hoc method. In the present paper we discuss the relevant case that the local slow manifold (in the asymptotic limit) is a vector subspace, give closed-form expressions for the reduction and compare these to the ones obtained by the customary method. As it turns out, investment of more theory pays off in the form of simpler reduced systems. Applications include a number of standard models for reactions in biochemistry, for which the reductions are extended to the fully reversible setting. In a short final section we illustrate by example that a QSS assumption may be erroneous if the hypotheses for Tikhonov’s theorem are not satisfied.     

An algebraic analysis approach to linear time-varying systems

^** Email: zerz{at}mathematik.uni-kl.de This paper introduces an algebraic analysis approach to lineartime-varying systems. The analysis is carried out in an ‘almosteverywhere’ setting, i.e. the considered signals are smoothexcept for a set of measure zero, and the coefficients of thelinear ordinary differential equations are supposed to be rationalor meromorphic functions. The methodology is based on a normalform for matrices over the resulting ring of differential operators,which is a non-commutative analogue of the Smith form. Thisis used to establish a duality between linear time-varying systemson the one hand and modules over the ring of differential operatorson the other. This correspondence is based on the fact thatthe signal space is an injective cogenerator when consideredas a module over this ring of differential operators.     

Computer algebraic methods for the structural analysis of linear controlsystems

We give an overview of the mathematical background of the Singular control library. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Autonomy,Controllability, and Extension Modules

The so‐called behavioral approach to systems theory, developed by Willems, provides a unified framework for the mathematical treatment of linear systems. In the behavioral context, a linear system is nothing but the solution space of a linear system of (partial) difference or differential equations. For simplicity, the coefficients are supposed to be constant. Oberst proved a duality theorem that builds upon an earlier result of Palamodov. It says that for certain signal spaces of interest, e.g., the smooth functions or the distributions, the solutions spaces of linear systems of partial differential equations, are dual to certain polynomial modules associated to them. Then the solution space and the module contain the same information, and algebraic properties of the module translate to analytic properties of the solution space. Powerful tools from commutative algebra may be used to derive them. As a prominent example, we study two properties that lie at the very heart of systems and control theory: autonomy and controllability. We summarize the characterizations given by several authors, and unify them in the language of extension modules, an algebraic concept which yields a full classification of these systems theoretic notions.