Binary Operations in Metric Spaces Satisfying Side Inequalities

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

On the group of isometries of planar IFS fractals*

For any iterated function system (IFS) on R 2 , let K be the attractor. Consider the group of all isometries on K. If K is a self-similar or self-affine set, it is proven that the group must be finite. If K is a bi-Lipschitz IFS fractal, the necessary and sufficient conditions for the infiniteness (or finiteness) of the group are given. For the finite case, the computation of the size of the group is also discussed.

Morphogenesis Analysis for Digital Image Production with L-System

The process of forming an image requires a correct color composition, location and distance between the lines to produce a good image. Human abilities in both creativity and high imagination are very limited, especially in forming new images by utilizing existing image patterns or images that resemble old images. Here we showed the implementation of L-System to generate new image generations with additional flame as a fire effect/glow on images for image transformation. This research used the L-System algorithm, Iterated Function System, and Voronoi Diagram to improve the result of image transformation. The results of this study indicated that mathematical calculations can be applied in the formation of images and the resulting images can be abstract and symmetrical. The next generation of images produced in this research can be in unlimited numbers as the generation of morphogenesis processes. The process of generating images is carried out randomly by merging the two existing images with morphogenesis analogy. The resulting images can be exported into jpg, png, and svg formats. Furthermore, this research showed that the implementation of the calculation for the variation reach the value of 99.48% while the image variation composition has a value of 99.29%.

Piecewise linear iterated function systems on the line of overlapping construction

In this paper we consider iterated function systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition. Moreover, we do not require that the functions of the IFS are injective, but we assume that their derivatives are separated from zero. We prove that if we fix all the slopes but perturb all other parameters, then for all parameters outside of an exceptional set of less than full packing dimension, the Hausdorff dimension of the attractor is equal to the exponent which comes from the most natural system of covers of the attractor.

Some properties of the fractal convolution of functions

We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.