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This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gr\"obner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of $G$-algebras (OLGAs). We split the computation of a normal form of a matrix into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a matrix $M$ over an OLGA we provide a diagonalization algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds and $D$ is diagonal. The fraction-free approach gives us more information on the system of linear functional equations and its solutions, than the classical setup of an operator algebra with rational functions coefficients. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over $K[x]$ and use them in the unimodularity analysis of transformation matrices $U,V$. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system {\sc Singular:Plural} follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. Since we experience moderate swell of coefficients and obtain simple transformation matrices, the method we propose is well suited for solving nontrivial practical problems., Comment: 25 pages, to appear in Journal of Symbolic Computation

Proper orthogonal decomposition and subsequent Galerkin-projection is a common technique to obtain low-dimensional models for distributed parameter systems. In this paper a transformation of boundary conditions into equivalent source terms for linear partial differential equations is used to set up such reduced-order models. This approach provides the means to consider boundary actuation in Galerkin-models, which is essential for the design of feedback controllers. Using the example of a one-dimensional convection-diffusion equation, the transient dynamics between the non-actuated and actuated steady state are described with such a reduced-order-model. Dynamic range and model accuracy are investigated in connection with the employed number of POD modes and the dynamics of actuation.

We focus on Grobner bases for modules of univariate polynomial vectors over a ring. We identify a useful property, the “predictable leading monomial (PLM) property” that is shared by minimal Grobner bases of modules in F [ x ] q , no matter what positional term order is used. The PLM property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property (= row reducedness), a terminology introduced by Forney in the 70’s. Because of the presence of zero divisors, minimal Grobner bases over a finite ring of the type Z p r (where p is a prime integer and r is an integer > 1 ) do not necessarily have the PLM property. In this paper we show how to derive, from an ordered minimal Grobner basis, a so-called “minimal Grobner p-basis” that does have a PLM property. We demonstrate that minimal Grobner p-bases lend themselves particularly well to derive minimal realization parametrizations over Z p r . Applications are in coding and sequences over Z p r .

Given a finite set of polynomial, multivariate, and vector-valued functions, we show that their span can be written as the solution set of a linear system of partial differential equations (PDE) with polynomial coefficients. We present two different but equivalent ways to construct a PDE system whose solution set is precisely the span of the given trajectories. One is based on commutative algebra and the other one works directly in the Weyl algebra, thus requiring the use of tools from non-commutative computer algebra. In behavioral systems theory, the resulting model for the data is known as the most powerful unfalsified model (MPUM) within the class of linear systems with kernel representations over the Weyl algebra, i.e., the ring of differential operators with polynomial coefficients.

S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices dened in (6). If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only nitely many s-extremal strongly modular lattices of even minimum.

The “predictable degree property”, a terminology introduced by Forney in 1970, is a property of polynomial matrices over a field F that has proven itself to be fundamentally useful for a range of applications. In this paper we strengthen this property into the “predictable leading monomial” property, and show that this PLM property is shared by minimal Grobner bases for any positional term order (here: TOP and POT) in F[x]q. The property is useful particularly for minimal interpolationtype problems. Because of the presence of zero divisors, minimal Grobner bases over a finite ring of the type ℤ p r (where p is a prime integer and r is an integer > 1) do not have the PLM property. We show how to construct, from an ordered minimal Grobner basis, a so-called minimal Grobner p-basis that does have a PLM property. The parametrization of all shortest linear recurrence relations of a finite sequence over ℤ p r is a type of problem for which this is useful and we include an illustrative example.

Linear exact modeling is a problem coming from system identification: given a set of observed trajectories, the goal is to find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas operators with varying coefficients were addressed only recently. This question can be tackled either using Gröbner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods for computing “most powerful unfalsified models” (MPUM) and their counterparts with variable coefficients (V MPUM) for polynomial and polynomial–exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind the algorithms and provide several examples.