While the numerical methods which utilizes partitions of equal-size, including the box-counting method, remain the most popular choice for computing the generalized dimension of multifractal sets, two massoriented methods are investigated by applying them to the one-dimensional generalized Cantor set. We show that both mass-oriented methods generate relatively good results for generalized dimensions for important cases where the box-counting method is known to fail. Both the strengths and limitations of the methods are also discussed.Fractal sets are characterized by self-similarity, and power laws can be associated with them. Examples of fractals in nature are ubiquitous. Their discovery led to the extension of the notion of dimension. For monofractals, the scaling pattern is homogeneous while it varies over the set for multifractals. By introducing the generalized dimension D q , not only a non-integer dimension can be assigned to a set, but also a spectrum of dimensions can be attributed to a single set if the set is a multifractal. In finding the generalized dimensions, the box-counting method has been by far the most popular choice among researchers across various fields. However, it is known that the class of methods which deal with partitions of equal size, including the box-counting method, is illsuited for computing the generalized dimensions on some domain of q. In this paper, two promising methods which utilize mass-oriented partitions, rather than partitions of equal-size, are investigated.

Numerical simulations of self-gravitating systems are generally based on N-body codes, which solve the equations of motion of a large number of interacting particles. This approach suffers from poor statistical sampling in regions of low density. In contrast, Vlasov codes, by meshing the entire phase space, can reach higher accuracy irrespective of the density. Here, we perform one-dimensional Vlasov simulations of a long-standing cosmological problem, namely, the fractal properties of an expanding Einstein-de Sitter universe in Newtonian gravity. The N-body results are confirmed for high-density regions and extended to regions of low matter density, where the N-body approach usually fails.

We investigate structure formation in a one dimensional model of a matterdominated universe using a quasi-newtonian formulation. In addition to dark matter, luminous matter is introduced to examine the potential bias in the distributions. We use multifractal analysis techniques to identify structures, including clusters and voids. Both dark matter and luminous matter exhibit fractal geometry as the universe evolves over a finite range. We present the results for the generalized dimensions computed on various scales for each matter distribution. We compare and contrast the fractal dimensions of two types of matter for the first time and show how dynamical considerations cause them to differ.