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- A note on fractal interpolation vs fractal regressionPublication . Serpa, CristinaFractals fascinates both academics and art lovers. They are a form of chaos. A key feature that distinguishes a fractal from other chaotic phenomena is the self-similarity. This is a property that consists of replicating a shape to smaller pieces of the whole. In other words, making zoom in or zoom out gives similar perspectives of the same fractal thing. We may find these shapes everywhere and nature presents many examples of fractal creations. An amazing case is the romanesque cabbage. Mandelbrot is the father of the term fractal and studied various examples (see [3]). Constructing a fractal is a simple task to do, just consider an initial configuration and a replication rule for smaller scales. This is how one gets, for example, the Sierpinski triangle, the dragon curve, or the Koch Snowflake. A simple rule creates complicated shapes with non-classical geometries. Analytically, it is also possible to define fractals as solutions of a system of iterative func tional equations. Barnsley defined such a system in [1]. This non-classical geometric concept has attracted many researchers when they are faced with the need to analyse real data with irregular characteristics.
- Compatibility conditions for systems of iterative functional equations with non-trivial contact setsPublication . Buescu, Jorge; Serpa, CristinaSystems of iterative functional equations with a non-trivial set of contact points are not necessarily solvable, as the resulting intersections may lead to an overdetermination of the system. To obtain existence and uniqueness results additional conditions must be imposed on the system. These are the compatibility conditions, which we define and study in a general setting. An application to the affine and doubly affine cases allows us to solve an open problem in the theory of functional equations. In the last section we consider a special problem in a different perspective, showing that a complex compatibility condition may result in an elegant and simple property of the solution.