Related papers: Finite- and Infinite-Time Cluster Formation for Al…
This paper introduces {\em fusion subspace clustering}, a novel method to learn low-dimensional structures that approximate large scale yet highly incomplete data. The main idea is to assign each datum to a subspace of its own, and minimize…
We study hierarchical clusterings of metric spaces that change over time. This is a natural geometric primitive for the analysis of dynamic data sets. Specifically, we introduce and study the problem of finding a temporally coherent…
We study equilibrium configurations of infinitely many identical particles on the real line or finitely many particles on the circle, such that the (repelling) force they exert on each other depends only on their distance. The main question…
Clusters appear in nature in a diversity of contexts, involving distances as long as the cosmological ones, and down to atoms and molecules and the very small nuclear size. They also appear in several other scenarios, in particular in…
We consider a compressible Euler system with singular velocity alignment, known as the Euler-alignment system, describing the flocking behaviors of large animal groups. We establish a local well-posedness theory for the system, as well as a…
In this paper, we propose several models, which can realize synchronization of complex networks in finite time effectively. The results apply to heterogeneous dynamic networks, too. The mechanism of finite time convergence is revealed.…
We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist…
Temporal data, obtained in the setting where it is only possible to observe one time point per experiment, is widely used in different research fields, yet remains insufficiently addressed from the statistical point of view. Such data often…
In this letter, by regarding finite-time stability as an inverse problem, we reveal the essence of finite-time stability and fixed-time stability. Some necessary and sufficient conditions are given. As application, we give a new approach…
We construct smooth axisymmetric-with-swirl initial data in a periodic cylinder for which the three-dimensional incompressible Euler evolution develops a finite-time boundary singularity. The construction is carried out in the dynamically…
We study two $(1+1)$-dimensional systems, denoted $(R0)$ and $(Z0)$, that are rigorously derived from the three-dimensional axisymmetric Euler equations in a signed polar formulation on the meridian plane. The main point of view in this…
In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping…
The paper outlines an exact combinatorial approach to finite coagulating systems. In this approach, cluster sizes and time are discrete, and the binary aggregation alone governs the time evolution of the systems. By considering the growth…
Single-time and two-time correlators are computed exactly in the $1D$ Glauber-Ising model after a quench to zero temperature and on a periodic chain of finite length $N$, using a simple analytical continuation technique. Besides the general…
We present a survey of some aspects and new results on configurations, i.e. disjoint unions of constellations of infinitely near points, local and global theory, with some applications and results on generalized Enriques diagrams, singular…
We establish finite-time singularity formation for $C^{1,\alpha}$ solutions to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and infinitely smooth except in the radial direction at the origin. The solutions are smooth…
We give an extremely short proof that the free-surface incompressible, irrotational Euler equations with regular initial condition can form a finite time singularity in 2D or 3D. Thus, we provide a simple view of the problem studied by…
Periodic orbits in chaotic systems form clusters, whose elements traverse approximately the same points of the phase space. The distribution of cluster sizes depends on the length n of orbits and the parameter p which controls closeness of…
Galaxy surveys provide finite catalogs of objects observed within bounded volumes, yet clustering statistics are often interpreted using theoretical frameworks developed for infinite point processes. In this work, we formulate key…
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…