Related papers: On some fractal sets only containing irrationals
The aim of this article is to show the existence, and also give an explicit construction, of infinite sets of orthogonal exponentials for certain families of convex polytopes which include simple-rational polytopes and also non simple…
Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be…
Fractal structures emerge from statistical and hierarchical processes in urban development or network evolution. In a class of efficient and robust geographical networks, we derive the size distribution of layered areas, and estimate the…
Dynamical Systems theory generally deals with fixed point iterations of continuous functions. Computation by Turing machine although is a fixed point iteration but is not continuous. This specific category of fixed point iterations can only…
We forge connections between the theory of fractal sets obtained as attractors of iterated function systems and process calculi. To this end, we reinterpret Milner's expressions for processes as contraction operators on a complete metric…
Nonhamiltonian interaction of hamiltonian systems is considered. Dynamical equations are constructed by use of symmetric designs on Lie algebras. The results of analysis of these equations show that some class of symmetric designs on Lie…
In this paper the author presents the results of the preliminary investigation of fractional dynamical systems based on the results of numerical simulations of fractional maps. Fractional maps are equivalent to fractional differential…
Recent evidence suggests that physiological signals under healthy conditions may have a fractal temporal structure. We investigate the possibility that time series generated by certain physiological control systems may be members of a…
We show that if any four distinct solutions of a rational difference equation are algebraically independent, then any number of distinct solutions to the equation are independent. A nontrivial variant of this result is given for autonomous…
By slight modification of the data of the Sierpinski gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nature and in random models. This is caused by a complicated…
We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these systems have no irrational spectrum and…
In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to…
We introduce a design strategy for neural network macro-architecture based on self-similarity. Repeated application of a simple expansion rule generates deep networks whose structural layouts are precisely truncated fractals. These networks…
We combine several folklore observations to provide a working framework for iterating constructions which contradict the axiom of choice. We use this to define a model in which any kind of structural failure must fail with a proper class of…
Multi-system interaction is an important and difficult problem in physics. Motivated by the experimental result of an electronic circuit element "Fractor", we introduce the concept of dynamic-order fractional dynamic system, in which the…
The main result of this paper is an analogue for a continuous family of tori of Kronecker-Weyl's unique ergodicity of irrational rotations. We show that the notion corresponding in this setup to irrationality, namely asynchronicity, is…
This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with…
A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive…
Fractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to…
Any infinite sequence of substitutions with the same matrix of the Pisot type defines a symbolic dynamical system which is minimal. We prove that, to any such sequence, we can associate a compact set (Rauzy fractal) by projection of the…