Related papers: Shape Coherence and Finite-Time Curvature Evolutio…
Mixing, and coherence are fundamental issues at the heart of understanding transport in fluid dynamics and other non-autonomous dynamical systems. Recently, the notion of coherence has come to a more rigorous footing, and particularly…
We study the effects of inhomogeneities on the evolution of the Universe, by considering a range of cosmological models with discretized matter content. This is done using exact and fully relativistic methods that exploit the symmetries in…
We study the curvature flow of planar nonconvex lens-shaped domains, considered as special symmetric networks with two triple junctions. We show that the evolving domain becomes convex in finite time; then it shrinks homothetically to a…
We study the evolution of coherent structures in arbitrary turbulence phenomena, developing some tools, from non-archimedean analysis and algebraic geometry, in order to model its display. We match the scale-dependent, topological structure…
Modeling the spontaneous evolution of morphology in natural systems and its preservation by proportionate growth remains a major scientific challenge. Yet, it is conceivable that if the basic mechanisms of growth and the coupled kinetic…
The evolution of timelike geodesic congruences in a spherically symmetric, nonstatic, inhomogeneous spacetime representing gravitational collapse of a massless scalar field is studied. We delineate how initial values of the expansion,…
Fourth order curvature driven interface evolution equations frequently appear in the natural sciences. Often axisymmetric geometries are of interest, and in this situation numerical computations are much more efficient. We will introduce…
We show the consistency of a threshold dynamics type algorithm for the anisotropic motion by fractional mean curvature, in the presence of a time dependent forcing term. Beside the consistency result, we show that convex sets remain convex…
Curve evolution is often used to solve computer vision problems. If the curve evolution fails to converge, we would not be able to solve the targeted problem in a lifetime. This paper studies the theoretical aspect of the convergence of a…
The time-dependence of correlation functions under the influence of classical equations of motion is described by an exact evolution equation. For conservative systems thermodynamic equilibrium is a fixed point of these equations. We show…
Geometric evolution represents a fundamental aspect of many physical phenomena. In this paper we consider the geometric evolution of structures that undergo topological changes. Topological changes occur when the shape of an object evolves…
We examine the properties of open quantum systems with respect to their time evolution in different regimes, Markovian and non-Markovian. We analyze their behaviour with respect to their coherent or decoherent time evolution by means of…
The protein folding problem must ultimately be solved on all length scales from the atomic up through a hierarchy of complicated structures. By analyzing the stability of the folding process using physics and mathematics, this paper shows…
Collective cell motions underlie structure formation during embryonic development. Tissues exhibit emergent multicellular characteristics such as jamming, rigidity transitions, and glassy dynamics, but there remain questions about how those…
Inspired by biological systems, we introduce a general framework for quasi-static shape control of human-scale structures under slowly varying external actions or requirements. In this setting, shape control aims to traverse the stable…
In this letter, we study the late-time evolution of a torsion cosmology only with the spin-$0^+$ mode. We find three kinds of analytical solutions with a constant affine scalar curvature. In the first case, it is not physical because the…
In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions…
In this proceedings article we shall survey a series of results on the stability of self-similar solutions of the vortex filament equation. This equation is a geometric flow for curves in $\mathbb R^3$ and it is used as a model for the…
In a given geometry, the kinematics of a congruence of curves is described by a set of three quantities called expansion, rotation, and shear. The equations governing the evolution of these quantities are referred to as kinematic equations.…
We consider the volume preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long time asymptotics…