Related papers: On the cluster structures in Collatz level sets
The Collatz conjecture, which posits that any positive integer will eventually reach 1 through a specific iterative process, is a classic unsolved problem in mathematics. This research focuses on designing an efficient algorithm to compute…
We continue the investigation of problems concerning correlation clustering or clustering with qualitative information, which is a clustering formulation that has been studied recently. The basic setup here is that we are given as input a…
In a separate paper we have discussed the possibility that six quark clusters can affect the rate of double-beta decay. In this article we develop the formalism needed in the evaluation of the energy of all six-quark cluster configurations,…
Stars form predominantly in groups usually denoted as clusters or associations. The observed stellar groups display a broad spectrum of masses, sizes and other properties, so it is often assumed that there is no underlying structure in this…
We prove cyclic sieving phenomena satisfied by corner-rooted plane trees (alias ordered trees). The sets of rooted plane trees that we consider are: (1) all trees with $n$ nodes; (2) all trees with $n$ nodes and $k$ leaves; (3) all trees…
We investigate the internal structure of clusters of galaxies in high-resolution N-body simulations of 4 different cosmologies. There is a higher proportion of disordered clusters in critical-density than in low-density universes, although…
We compute the leading clustering (abelian non-global) logarithms, which arise in the distribution of non-global QCD observables when final-state partons are clustered using the $k_t$ jet algorithm, up to six loops in perturbation theory.…
We introduce multi-soliton sets in the two-dimensional medium with the second-harmonic-generating nonlinearity subject to spatial modulation in the form of a triangle of singular peaks. Various families of symmetric and asymmetric sets are…
A numerical evidence for the dependence of the substructure abundance of cluster halos on the orientation coherence of the surrounding tidal fields is presented. Applying the adapted minimal spanning tree (MST) algorithm to the cluster…
Constructing small-sized coresets for various clustering problems in different metric spaces has attracted significant attention for the past decade. A central problem in the coreset literature is to understand what is the best possible…
The Collatz process is defined on natural numbers by iterating the map $T(x) = T_0(x) = x/2$ when $x\in\mathbb{N}$ is even and $T(x)=T_1(x) =(3x+1)/2$ when $x$ is odd. In an effort to understand its dynamics, and since Generalised Collatz…
Spectral clustering is a well-known technique which identifies $k$ clusters in an undirected graph with weight matrix $W\in\mathbb{R}^{n\times n}$ by exploiting its graph Laplacian $L(W)$, whose eigenvalues $0=\lambda_1\leq \lambda_2 \leq…
We find a duality between two well-known trees, the Calkin-Wilf tree and the Stern-Brocot tree, derived from cluster algebra theory. The vertex sets of these trees are the set of rational numbers, and they have cluster structures induced by…
Motivated by the existence of hierarchies of structure in the Universe, we present four new families of exact initial data for inhomogeneous cosmological models at their maximum of expansion. These data generalise existing black hole…
Define the \emph{Collatz map} $\mathrm{Col} : \mathbb{N}+1 \to \mathbb{N}+1$ on the positive integers $\mathbb{N}+1 = \{1,2,3,\dots\}$ by setting $\mathrm{Col}(N)$ equal to $3N+1$ when $N$ is odd and $N/2$ when $N$ is even, and let…
We obtain a complete classification of components of strata of holomorphic and meromorphic k-differentials. We show that, when genus is at least two and outside of explicit exceptions when k < 4, there is one primitive nonhyperelliptic…
The stability for all generic equilibria of the Lie-Poisson dynamics of the $\mathfrak{so}(4)$ rigid body dynamics is completely determined. It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate…
We study cluster synchronization in networks and show that the stability of all possible cluster synchronization patterns depends on a small set of Lyapunov exponents. Our approach can be applied to clusters corresponding to both orbital…
A comprehensive study on the relative structural stability of various nanostructures of carbon clusters (including fullerenes, cages, onions, icosahedral clusters, bucky-diamond clusters, spherically bulk terminated clusters, and clusters…
This paper considers metric spaces where distances between a pair of nodes are represented by distance intervals. The goal is to study methods for the determination of hierarchical clusters, i.e., a family of nested partitions indexed by a…