Related papers: On concentric fractal spheres and spiral shells
We study the kinetics of random sequential adsorption of a mixture of particles with continuous distribution of sizes for different deposition rules. It appears in the long time limit the resulting system can be described using the fractal…
Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic…
We characterize the existence of certain geometric configurations in the fractal percolation limit set $A$ in terms of the almost sure dimension of $A$. Some examples of the configurations we study are: homothetic copies of finite sets,…
We find the complete rational homology for the finite subset spaces of a $d$-dimensional sphere. We also determine the integral homology in top $d$ degrees and obtain a partial description of it in codimension $d$.
We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra $U(1)^c \times U(1)^c$, or equivalently the linear programming bound for sphere packing in $2c$ dimensions. We give a more…
A fractal oscillatority of solutions of second-order differential equations near infinity is measured by oscillatory and phase dimensions. The phase dimension is defined as a box dimension of the trajectory $(x,\dot{x})$ in $\mathbb{R}^2$…
Spherical $t$-designs are finite point sets on the unit sphere that enable exact integration of polynomials of degree at most $t$ via equal-weight quadrature. This concept has recently been extended to spherical $t$-design curves by the use…
The three dimensional structure of large packings of monosized spheres with volume fractions ranging between 0.58 and 0.64 has been studied with X-ray Computed Tomography. We search for signatures of organization, we classify local…
Thickenings of a metric space capture local geometric properties of the space. Here we exhibit applications of lower bounding the topology of thickenings of the circle and more generally the sphere. We explain interconnections with the…
Homogeneity and isotropy of the universe at sufficiently large scales is a fundamental premise on which modern cosmology is based. Fractal dimensions of matter distribution is a parameter that can be used to test the hypothesis of…
We study the fractal scaling of iso-levels sets of a passive scalar mixed by three-dimensional homogeneous and isotropic turbulence at high Reynolds numbers. The Schmidt number is unity. A fractal box-counting dimension $D_F$ can be…
Following the pattern of the Frobenius structure usually assigned to the 1-dimensional sphere, we investigate the Frobenius structures of spheres in all other dimensions. Starting from dimension $d=1$, all the spheres are commutative…
In this note, we provide equivalent definitions for fractal geometric dimensions through dyadic cube constructions. Given a metric space $X$ with finite Assouad dimension, i.e., satisfying the doubling property, we show that the…
Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a…
Density-based clustering methodology has been widely considered in the statistical literature for classifying Euclidean observations. However, this approach has not been contemplated for directional data yet. In this work, directional…
The Clifford spectrum is a form of joint spectrum for noncommuting matrices. This theory has been applied in photonics, condensed matter and string theory. In applications, the Clifford spectrum can be efficiently approximated using…
We introduce fractal liquids by generalizing classical liquids of integer dimensions $d = 1, 2, 3$ to a fractal dimension $d_f$. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the…
Random packing of spheres inside fractal collectors of dimension 2 < d < 3 is studied numerically using Random Sequential Adsorption (RSA) algorithm. The paper focuses mainly on the measurement of random packing saturation limit.…
A \emph{fractal} is an object exhibiting complexity at arbitrarily small scales. In order to study and characterise fractals, one is often interested in quantifying how they fill up space on small scales. This gives rise to various notions…
We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $\Psi^j$, $j\in…