SMOOTH AND DISCRETE SYSTEMS—ALGEBRAIC, ANALYTIC, AND GEOMETRICAL REPRESENTATIONS ˇ FRANTISEK NEUMAN Received 12 January 2004 What is a differential equation? Certain objects may have different, sometimes equivalent representations. By using algebraic and geometrical methods as well as discrete relations, different representations of objects mainly given as analytic relations, differential equations can be considered. Some representations may be suitable when given data are not sufficiently smooth, or their derivatives are difficult to obtain in a sufficient accuracy; other ones might be better for expressing conditions on qualitative behaviour of their solution spaces. Here, an overview of old and recent results and mainly new. | SMOOTH AND DISCRETE SYSTEMS ALGEBRAIC ANALYTIC AND GEOMETRICAL REPRESENTATIONS FRANTISEK NEUMAN Received 12 January 2004 What is a differential equation Certain objects may have different sometimes equivalent representations. By using algebraic and geometrical methods as well as discrete relations different representations of objects mainly given as analytic relations differential equations can be considered. Some representations may be suitable when given data are not sufficiently smooth or their derivatives are difficult to obtain in a sufficient accuracy other ones might be better for expressing conditions on qualitative behaviour of their solution spaces. Here an overview of old and recent results and mainly new approaches to problems concerning smooth and discrete representations based on analytic algebraic and geometrical tools is presented. 1. Motivation When considering certain objects we may represent them in different often equivalent ways. For example graphs can be viewed as collections of vertices points and edges arcs or as matrices of incidence expressing in their entries aij the number of oriented edges going from one vertex i to the other one j . Another example of different representations are matrices we may look at them as centroaffine mappings of m-dimensional vector space to n-dimensional one or as n X m entries or coefficients of the above mappings in particular coordinate systems of the vector spaces placed at lattice points of rectangles. Still there is another example. Some differential equations can be considered in the form y f x y with the initial condition y x0 y0. For continuous f satisfying Lipschitz condition we get the unique solution of . The solution space of is a set of differentiable functions satisfying and depending on one constant the initial value y0. Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 2 2004 111-120 2000 Mathematics Subject Classification 34A05 39A12 35A05 .
