Coastlines are irregular in nature having (random) fractal geometry and are formed by various natural activities. Fractal dimension is a measure of degree of geometric irregularity present in the coastline. A novel multicore parallel processing algorithm is presented to calculate the fractal dimension of coastline of Australia. The reliability of the coastline length of Australia is addressed by recovering the power law from our computational results. For simulations, the algorithm is implemented on a parallel computer for multi-core processing using the QGIS software, R-programming language and Python codes.

This is the first of a series of papers devoted to the study of h-p spectral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the vertex-edge singularities. In addition, the solution is anisotropic in the neighbourhoods of the edges and vertex-edges. To overcome the singularities which arise in the neighbourhoods of vertices, vertex-edges and edges, we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. The geometrical mesh becomes a quasi-uniform mesh in the new system of coordinates. We then derive differentiability estimates in these new set of variables and state our main stability estimate theorem using a non-conforming h-p spectral element method whose proof is given in a separate paper.

Fractals are geometric shapes and patterns that can describe the roughness (or irregularity) present in almost every object in nature. Many fractals may repeat their geometry at smaller or larger scales. This paper is the second (and last) part of a series of two papers dedicated to an eclectic survey of fractals describing the infinite complexity and amazing beauty of fractals from historical, theoretical, mathematical, aesthetical and technological aspects, including their diverse applications in various fields. In this article, our focus is on engineering, industrial, commercial and futuristic applications of fractals, whereas in the first part, we discussed the basics of fractals, mathematical description, fractal dimension and artistic applications. Among many different applications of fractals, fractal landscape generation (fractal landscapes that can simulate and describe natural terrains and landscapes more precisely by mathematical models of fractal geometry), fractal antennas (fractal-shaped antennas that are designed and used in devices which operate on multiple and wider frequency bands) and fractal image compression (a fractal-based lossy compression method for digital and natural images which uses inherent self-similarity present in an image) are the most creative, engineering-driven, industry-oriented, commercial and emerging applications. We consider each of these applications in detail along with some innovative and future ready applications.

Fractals are geometric shapes and patterns that may repeat their geometry at smaller or larger scales. It is well established that fractals can describe shapes and surfaces that cannot be represented by the classical Euclidean geometry. An eclectic survey of fractals is presented in two parts encompassing applications of fractals in a variety of diverse and innovative fields. The goal of the first part is to focus on the glossary of fractals, their mathematical description, aesthetic, artistic, and architectural applications, while the second part is focused on engineering, industry, commercial, and futuristic applications of fractals.

This is the second of a series of papers devoted to the study of h-p spectral element methods for three dimensional elliptic problems on non-smooth domains. The present paper addresses the proof of the main stability theorem. We assume that the differential operator is a strongly elliptic operator which satisfies Lax-Milgram conditions. The spectral element functions are non-conforming. The stability estimate theorem of this paper will be used to design a numerical scheme which give exponentially accurate solutions to three dimensional elliptic problems on non-smooth domains and can be easily implemented on parallel computers.