A note on equivalent systems of linear diophantine equations

Summary Another constructive proof is presented for the fact that a system of linear equations with integer coefficients in bounded integer variables is equivalent to a single equation, which is a linear combination of the original ones. The equation is obtained in a number of steps; in each step two equations are replaced by a single one. This replacement is performed subject to the condition that the remaining equations hold and a final equation with relatively small coefficients is obtained. It may be inefficient however to calculate small coefficients, as the original coefficients can be used to represent the final ones in a suitably chosen number system.

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Zusammenfassung Ein System linearer Gleichungen mit ganzzahligen Koeffizienten in beschränkten ganzzahligen Variablen ist äquivalent zu einer einzigen Gleichung, die sich als Linearkombination der ursprünglichen Gleichungen schreiben läßt. In einem neuen konstruktiven Beweis von diesem Satz wird gezeigt, wie die endgültige Gleichung in einigen Schritten gefunden werden kann. In jedem Schritt werden zwei Gleichungen von einer einzigen ersetzt unter der Voraussetzung, daß die übrigen Gleichungen gültig bleiben.Obwohl die Koeffizienten der Endgleichung verhältnismäßig klein sind, ist es nicht immer zweckmäßig, sie in der angegebenen Weise zu berechnen, da man ein Zahlensystem anwenden kann, in dem die Koeffizienten der ursprünglichen Gleichungen zur Darstellung der neuen Koeffizienten gebraucht werden.

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This note is a slightly revised version of report BW 12/71 (July 1971).     

Normal form for linear systems with respect to its vector relative degree

For multi-input multi-output (MIMO) linear systems with existing vector relative degree a normal form is constructed. This normal form is not only structural simple but allows to characterize the system’s zero dynamics for the design of feedback controllers. A characterization of the zero dynamics in terms of the normal form is given.     

Reducing systems of linear differential equations to a passive form

In this paper, an application of the Riquer-Thomas-Janet theory is described for the problem of transforming a system of partial differential equations into a passive form, i.e., to a special form which contains explicitly both the equations of the initial system and all their nontrivial differential consequences. This special representation of a system markedly facilitates the subsequent integration of the corresponding differential equations. In this paper, the modern approach to the indicated problem is presented. This is the approach adopted in the Knuth-Bendix procedure [13] for critical-pair/completion and then Buchberger's algorithm for completion of polynomial ideal bases [13] (or, alternatively, for the construction of Groebner bases for ideals in a differential operator ring [14]). The algorithm of reduction to the passive form for linear system of partial differential equations and its implementation in the algorithmic language REFAL, as well as the capabilities of the corresponding program, are outlined. Examples illustrating the power and efficiency of the system are presented.     

Reduction of linear systems to a form with relative degree using minimum-phase output transformation

A normal form is one of the canonical forms frequently used in control theory for linear time-invariant systems. Only systems with a relative degree can be reduced to such a form. Although a control system does not necessarily have a relative degree, in a sufficiently general case there exists a stable dynamic output transformation reducing the system to a system with a relative degree. We prove that this dynamic transformation can be chosen in such a way that the inverse transformation is stable as well.     

A weak‐derivative form for linear hyperbolic systems

Difference schemes for linear hyperbolic systems are considered. As a main result, a weak derivative form (WDF) of the governing equations is derived, which is also valid near flow discontinuities. The occurrence of one‐sided derivatives in the WDF structure indicated how to difference near discontinuities. When first‐order differencing is applied to the WDF result, the (linearly identical) schemes by Godunov, Roe, and Steger‐Warming are reproduced. The extension to nonlinear systems is via a local linearization. Choosing Roe's averaging reduces the WDF algorithm to Roe's scheme, whereas other nonlinear WDF schemes are possible. The suitability of various kinds of averaging is numerically investigated. For weak shocks a surprising lack of sensitivity of the method to a particular averaging is exhibited. However, for strong shocks and where the ordinary arithmetic average is used, a slightly more pronounced difference in performance exists between Roe's scheme and WDF. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Reduction of systems of linear differential equations to jordan normal form

Summary Various methods are discussed of finding a non-singular matrix P such that PAP⁻¹=J, where J is theJordan normal form of A, with special reference to the problem of reducing the system of equations x=Ax to the form y=Jy, where y=Px. To Giovanni Sansone on his 70^th birth day.     

Continuous approximation of linear impulsive systems and a new form of robust stability

The time-scale tolerance for linear ordinary impulsive differential equations is introduced. How large the time-scale tolerance is directly reflects the degree to which the qualitative dynamics of the linear impulsive system can be affected by replacing the impulse effect with a continuous (as opposed to discontinuous, impulsive) perturbation, producing what is known as an impulse extension equation. Theoretical properties related to the existence of the time-scale tolerance are given for periodic systems, as are algorithms to compute them. Some methods are presented for general, aperiodic systems. Additionally, sufficient conditions for the convergence of solutions of impulse extension equations to the solutions of their associated impulsive differential equation are proven. Counterexamples are provided.     

Multivariable Euler transform of systems of linear ordinary differential equations of Okubo normal form

The multivariable Euler transform of a solution of the system of linear ordinary differential equations of Okubo normal form is considered. The Pfaffian system satisfied by the transform is derived. Applications to the Appell hypergeometric functions \(F_{1}\), \(F_{3}\), and the Lauricella hypergeometric function \(F_{D}\) are given.     

Maximal minors of a matrix with linear form entries

Let P be a matrix whose entries are homogeneous polynomials in n variables of degree one over an algebraically closed field. We show that the maximal minors, say m-minors, of P generate the linear space of homogeneous polynomials of degree m if P has the maximal rank m at every point of the affine n-space except the origin.