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Orthogonal arrays are arguably one of the most fascinating and important statistical tools for efficient data collection. They have a simple, natural definition, desirable properties when used as fractional factorials, and a rich and…
Building upon [1], this study aims to introduce fractal geometry into graph theory, and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore…
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…
The fractal dimension of a surface allows its degree of roughness to be characterized quantitatively. However, limited effort is attempted to calculate the fractal dimension of surfaces computed from precisely known atomic coordinates from…
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…
In this paper, we discuss a connection between geometric measure theory and number theory. This method brings a new point of view for some number-theoretic problems concerning digit expansions. Among other results, we showed that for each…
Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of study area. This phenomenon has been puzzling many researchers. This…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
An algorithm for calculating generalized fractal dimension of a time series using the general information function is presented. The algorithm is based on a strings sort technique and requires $O(N \log_2 N)$ computations. A rough estimate…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with…
Fractal functions that produce smooth and non-smooth approximants constitute an advancement to classical nonrecursive methods of approximation. In both classical and fractal approximation methods emphasis is given for investigation of…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
Current practice in parameter space exploration in euclidean space is dominated by randomized sampling or design of experiment methods. The biggest issue with these methods is not keeping track of what part of parameter space has been…
The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a measure of scale dependence, which indicates the scale-free distribution of urban…
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…
This article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's. We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum…
Topological Data Analysis (TDA) uses insights from topology to create representations of data able to capture global and local geometric and topological properties. Its methods have successfully been used to develop estimations of fractal…
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…
We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves…