Jiří Gregor

On the design of positive real functions

Construction of all rational positive real functions with a given denominator is described. Examples showing how to respect various requirements on the degrees of the resulting polynomials are given.

Convolutional solutions of partial difference equations

A significant part of the theory of one-dimensional linear shift-invariant systems is based on the concept of weighting function (or impulse response): the output is the convolution of the weighting function with the input. This paper introduces the concept of linear translation-invariant systems and uses this notion in studying impulse response, z-transforms, and transfer functions for multidimensional systems.

Singular systems of partial difference equations

Sufficient conditions for the existence and uniqueness of solutions of singular systems of 2-D difference equations with constant coefficients are formulated. Linear transformation of the matrix coefficients leads to recursive forms of such systems of equations. The results are applied to standard 2-D state-space models of discrete systems and sufficient conditions of BIBO stability of these singular systems are obtained.

The Cauchy Problem for Partial Difference Equations

We want to discuss partial difference equations, first of all with respect to the existence and uniqueness of their solution. These equations are considered with solutions on arbitrary subsets of the n-dimensional grid Zn. The basic theorem enables one to formulate the Cauchy problem for such equations. The solution is proven to be recursively computable for partial difference equations under very mild restrictions. (Variable coefficients for linear equations, systems of equations as well as nonlinear equations are not excluded.) The construction of solutions presented here also allows for some qualitative conclusions, such as boundedness of solutions.

On the singular Fornasini-Marchesini model of 2-D discrete systems

The singular Fornasini-Marchesini model of 2-D discrete systems cannot be handled by commonly used z-transform techniques. A recursive form of the basic equation is found, and consequently existence and uniqueness theorems can be formulated. The recursive form of the singular Fornasini-Marchesini model also yields a sufficient stability condition for this model. A detailed example illustrates the results. >

On the Derivative of Drawdown for Single Well Pumping Tests

The communication explains the physical meaning of the derivative of drawdown for a pumping test.

Collocation and Least Squares Methods

Various methods to adjust given or chosen mathematical models to actual requirements or actual data are presented. Such adjustment, if it exists, means choice of proper values of some parameters, which usually leads to solution of a set of linear or nonlinear equations. When such adjustment is not possible then least squares method gives an alternative.

Maximal and Minimal Values

The well known fact that differentiable functions may have local extremes only at points where their first derivatives vanish is illustrated. In some cases the corresponding equations are not directly accessible or solvable and other considerations have to be applied. Constrained extremal values for real functions of several variables and extremal problems of non-differentiable functions are included.

Center of Mass and Moments

The dynamics of an isolated system of N free particles, each with the mass m i , i=1,2,…,N is completely determined by the collection of their moments (of order 1). Generalizations of this concept opened new ways in applied sciences as well as in mathematics. Interconnections between the moment problem and least squares techniques form the basic idea of orthogonal series, which became an important tool in approximation theory.

Mappings, Composite and Inverse Functions

The chapter deals with with significant special mappings, their inverse mappings,composition, iteration, invariants and fixed points of mappings. It includes examples of functional equations, methods of investigating special functions, examples of mappings f:R 3→R 2.