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Distribution of Cracks in a Chain of Atoms at Low Temperature
We consider a one-dimensional classical many-body system with interaction potential of Lennard–Jones type in the thermodynamic limit at low temperature 1/β∈(0,∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of Nexp(−βesurf/2) with esurf>0 a surface energy. For the proof, the system is mapped to an effective model, which is a low-density lattice gas of defects. The results require conditions on the interactions between defects. We succeed in verifying these conditions for next-nearest neighbor interactions, applying recently derived uniform estimates of correlations.
Local Well-Posedness of Strong Solutions to the Three-Dimensional Compressible Primitive Equations
This work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity. © 2021, The Author(s).
A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions
A boundary control problem for the pure Cahn–Hilliard equations with possibly singular potentialsand dynamic boundary conditions is studied and rst-order necessary conditions for optimality are proved.
Distributed optimal control of a nonstandard nonlocal phase field system
We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.
Time-Warping Invariants of Multidimensional Time Series
In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants. We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties. © 2020, The Author(s).
Assessment of Stability in Partitional Clustering Using Resampling Techniques
The assessment of stability in cluster analysis is strongly related to the main difficult problem of determining the number of clusters present in the data. The latter is subject of many investigations and papers considering different resampling techniques as practical tools. In this paper, we consider non-parametric resampling from the empirical distribution of a given dataset in order to investigate the stability of results of partitional clustering. In detail, we investigate here only the very popular K-means method. The estimation of the sampling distribution of the adjusted Rand index (ARI) and the averaged Jaccard index seems to be the most general way to do this. In addition, we compare bootstrapping with different subsampling schemes (i.e., with different cardinality of the drawn samples) with respect to their performance in finding the true number of clusters for both synthetic and real data.
Novel fixed-time stabilization of quaternion-valued BAMNNs with disturbances and time-varying coefficients
In this paper, with the quaternion number and time-varying coefficients introduced into traditional BAMNNs, the model of quaternion-valued BAMNNs are formulated. For the first time, fixed-time stabilization of time-varying quaternion-valued BAMNNs is investigated. A novel fixed-time control method is adopted, in which the choice of the Lyapunov function is more general than in most previous results. To cope with the noncommutativity of the quaternion multiplication, two different fixed-time control methods are provided, a decomposition method and a non-decomposition method. Furthermore, to reduce the control strength and improve control efficiency, an adaptive fixed-time control strategy is proposed. Lastly, numerical examples are presented to demonstrate the effectiveness of the theoretical results. © 2020 the Author(s), licensee AIMS Press.
Differentiability Properties for Boundary Control of Fluid-Structure Interactions of Linear Elasticity with Navier-Stokes Equations with Mixed-Boundary Conditions in a Channel
In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from (Lasiecka et al. in Nonlinear Anal 44:54–85, 2018). An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of (Wollner and Wick in J Math Fluid Mech 21:34, 2019) to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.
Revealing the co-action of viscous and multistability hysteresis in an adhesive, nominally flat punch: A combined numerical and experimental study
Viscoelasticity is well known to cause a significant hysteresis of crack closure and opening when an elastomer is brought in and out of contact with a flat, rigid counterface. In contrast, the idea that adhesive hysteresis can also result under quasi-static driving due to small-scale, elastic multistability is relatively new. Here, we study a system in which both mechanisms act concurrently. Specifically, we compare the simulated and experimentally measured time evolution of the interfacial force and the real contact area between a soft elastomer and a rigid, flat punch, to which small-scale, single-sinusoidal roughness is added. To this end, we further the Green's function molecular dynamics method and extend recently developed imaging techniques to elucidate the rate- and preload-dependence of the pull-off process. Our results reveal that hysteresis is much enhanced when the saddle points of the topography come into contact, which, however, is impeded by viscoelastic forces and may require sufficiently large preloads. A similar coaction of viscous- and multistability effects is expected to occur in macroscopic polymer contacts and be relevant, e.g., for pressure-sensitive adhesives and modern adhesive gripping devices.
Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and a symmetric deviatoric stress tensor. This stress tensor is transported via the Zaremba-Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray-Hopf-type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion.