Related papers: Counting-Based Effective Dimension and Discrete Re…
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…
We show that the fractal curvature measures of invariant sets of one-dimensional conformal iterated function systems satisfying the open set condition exist, if and only if the associated geometric potential function is nonlattice.…
We propose quantum-mechanical systems in which the number of spatial dimensions is promoted to a dynamical quantum variable, making the effective dimension state-dependent. Interestingly, systems of this form can exhibit enhanced symmetries…
We consider the fractal characteristic of the quantum mechanical paths and we obtain for any universal class of fractons labeled by the Hausdorff dimension defined within the interval 1$ $$ < $$ $$h$$ $$ <$$ $$ 2$, a fractal distribution…
To any spectral triple (A,D,H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that |D|^-d has non trivial logarithmic Dixmier trace. Moreover, when d is…
In this paper, we present a formal quantification of epistemic uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due…
Advances in architectural design, data availability, and compute have driven remarkable progress in semantic segmentation. Yet, these models often rely on relaxed Bayesian assumptions, omitting critical uncertainty information needed for…
It is well known that string theory generates the idea of higher dimensional spacetime instead of the (3+1) dimensions, in which we seem to live. It indicates that the extra space dimensions may remain curled up into very small space. In…
The ability to design the scattering properties of electromagnetic structures is of fundamental interest in optical science and engineering. While there has been great practical success applying local optimization methods to electromagnetic…
Most entropy measures depend on the spread of the probability distribution over the sample space $\mathcal{X}$, and the maximum entropy achievable scales proportionately with the sample space cardinality $|\mathcal{X}|$. For a finite…
Image-based simulation, the use of 3D images to calculate physical quantities, fundamentally relies on image segmentation to create the computational geometry. However, this process introduces image segmentation uncertainty because there is…
Statistical mechanical systems at and near their points of phase transition are expected to exhibit rich, fractal-like behaviour that is independent of the small-scale details of the system but depends strongly on the dimension in which the…
Anderson localization is fundamentally controlled by dimensionality, yet the nature of the Anderson transition in continuously tunable noninteger dimensions remains largely unexplored. Here, we introduce a family of three-dimensional…
The concept of derivative coordinate functions proved useful in the formulation of analytic fractal functions to represent smooth symmetric binary fractal trees [1]. In this paper we introduce a new geometry that defines the fractal space…
In machine learning or statistics, it is often desirable to reduce the dimensionality of a sample of data points in a high dimensional space $\mathbb{R}^d$. This paper introduces a dimensionality reduction method where the embedding…
In the parameter spaces of nonlinear dynamical systems, we investigate the boundaries between periodicity and chaos and unveil the existence of fractal sets characterized by a singular fractal dimension. This dimension stands out from the…
The fractal dimension of large-scale galaxy clustering has been demonstrated to be roughly $D_F \sim 2$ from a wide range of redshift surveys. If correct, this statistic is of interest for two main reasons: fractal scaling is an implicit…
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…
We propose and analyze a novel theoretical and algorithmic framework for structured prediction. While so far the term has referred to discrete output spaces, here we consider more general settings, such as manifolds or spaces of probability…
Fractal nests are sets defined as unions of unit $n$-spheres scaled by a sequence of $k^{-\alpha}$ for some $\alpha>0$. In this article we generalise the concept to subsets of such spheres and find the formulas for their box counting…