Related papers: Fractal tiles associated with shift radix systems
Chaotic transients occur in many experiments including those in fluids, in simulations of the plane Couette flow, and in coupled map lattices and they are a common phenomena in dynamical systems. Superlong chaotic transients are caused by…
Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity mathematically described by fractal geometry. In recent years, the engineering of self-similar structures in…
In this work we completely describe the dynamics of triangle tiling billiards. In the first part of this work, we propose a geometric approach of dynamics by introducing natural foliations associated to it. In the second part, we exploit…
Urbanization is a phenomenon of concern for planning and public health: projections are difficult because of policy changes and natural events, and indicators are multiple. There are previous studies of development that used fractals, but…
Motivated by a host of recent applications requiring some amount of redundancy, frames are becoming a standard tool in the signal processing toolbox. In this paper, we study a specific class of frames, known as discrete Fourier transform…
The basins of convergence, associated with the roots (attractors) of a complex equation, are revealed in the Hill problem with oblateness and radiation, using a large variety of numerical methods. Three cases are investigated, regarding the…
The existence of normal deterministic diffusion in dynamical systems with a two-dimensional phase space tiled by regular triangles (or their unions into regular hexagons) is proven.
An effort has been made to show mathematicians some new ideas applied to image analysis. Gray images are presented as tilings. Based on topological properties of the tiling, a number of gray convex hulls: maximal, minimal, and oriented ones…
Fractal surfaces ('patchwork quilts') are shown to arise under most general circumstances involving simple bitwise operations between real numbers. A theory is presented for all deterministic bitwise operations on a finite alphabet. It is…
We present an overview of a theory of complex dimensions of self-similar fractal strings, and compare this theory to the theory of varieties over a finite field from the geometric and the dynamical point of view. Then we combine the several…
In this paper we study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to…
This paper proposes a new method to analyze the spatial structure of urban systems using ideas from fractals. Regarding a system of cities as a set of "particles" distributed randomly on a triangular lattice, we construct a spatial…
If our aesthetic preferences are affected by fractal geometry of nature, scaling regularities would be expected to appear in all art forms, including music. While a variety of statistical tools have been proposed to analyze time series in…
We count tilings of a rectangle of integer sides m-1 and n-1 by a special set of tiles. The result is obtained fron the study of the kernel of the adjacency matrix of an n x n rectangular graph of Z x Z.
In previous papers by A. Kameyama and by J. Kigami distances on fractals have been discussed having two different but similar properties. One property is that the maps defining the fractal are Lipschitz of prescribed constants less than 1,…
The number of complete tilings of m X n floors for tiles of shape 1 X 2, 1 X 3, 1 X 4 and 2 X 3 is computed numerically for floors up to width m=9 and variable floor lengths n. Counts are obtained for two classes, for fixed tile stack…
Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 80-s of the last century with the purpose to solve some famous problems in mathematics, including the question raising to von…
A standard scientific study comprises two processes: one is to describe a thing, and the other is to understand how the thing works. In order to understand the principle of urban growth, a number of shape indexes are proposed to describe…
The translation symmetry of a lattice is greatly modified when subjected to a perpendicular magnetic field [Zak, Phys. Rev. \textbf{134}, A1602 (1964)]. This change in symmetry can lead to magnetic unit cells that are substantially larger…
In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for…