We establish a ln-ln-type estimate for the electric potential associated to the zeroth order perturbation of the polyharmonic operator when the Dirichlet-to-Neumann map is taken only on a subset.
We prove the boundary layer separation of the two-dimensional steady Prandtl equation with magnetic damping under an appropriate adverse pressure gradient. We also obtain the separation rate of the solution and the local behavior of the solution near the separation point. Our results show that magnetic damping exerts a suppressive effect on boundary layer separation.
To obtain the null point of the sum for a maximally monotone operator A and a cocoercive operator B, this article is concerned with the asymptotic properties of the trajectories to a second-order continuous damped autonomous dynamical systems with dry friction, which plays a crucial role for achieving stability within a finite time. Additionally, Hessian-driven damping is incorporated, which helps mitigating oscillations for inertial effects. The combination of dry friction and Hessian-driven damping in the context of split monotone inclusions represents a novel contribution, as it simultaneously ensures finite-time convergence and suppresses oscillatory behavior. The existence and uniqueness of global solutions, as well as the convergence of system trajectories are analyzed, the performance of the iterative algorithm generated by the second-order dynamic system discretely are also verified through several numerical experiments.
We investigate regularity criteria for weak solutions to the three-dimensional non-Newtonian MHD-type system for 115⩽ q<52 under a sufficient condition involving only the magnetic field. In addition, weak-strong uniqueness in ℝ3 is established for q⩾52 .
We consider the bounded penalty method to solve the modified Signorini contact problem with nonlocal Coulomb’s friction in electro-elasticity studied in I. El Ouardy, Y. Mandyly, H. Benkhira, and R. Fakhar (2024). We formulate a regularized variational problem and prove the existence and uniqueness of its solution using elliptic quasi-variational inequalities, strongly monotone operators, and Schauder’s fixed point theorem. We also derive error estimates that depend on the penalty parameter ϵ, establishing a convergence rate of O(√(ϵ)) . Finally, we propose an iterative method to numerically solve the regularized problem and prove its convergence.
We study the long-time behavior of solutions to a thermoelastic Timoshenko beam system with dual-phase-lag heat conduction. We prove the existence of solutions and their uniqueness using semigroup methods. Exponential energy decay is shown under the condition 2τ2 > τ1. The analysis preserves full thermal-mechanical coupling for accurate modeling. Numerical simulations using mixed finite elements in space and a multistep time scheme (FEniCSx) confirm the theoretical energy decay and show how the lag parameters and coupling strength influence the decay rate.
We consider a dynamic problem describing frictional contact between a thermo-elasto-viscoplastic body and a foundation. The contact is frictional and bilateral with a moving rigid foundation that results in the wear of the contacting surface. The constitutive law includes a temperature effect described by the first-order evolution equation and a damage effect described by the parabolic inclusion with the homogeneous Neumann boundary condition. We present a variational formulation of the problem and establish the existence and uniqueness of the weak solution. The proof is based on parabolic variational inequalities, first order evolutionary variational equations, and fixed point arguments.
We consider a static frictional contact problem between a magneto-electro-elastic body and a foundation. We assume that the friction law is non-monotone, slip dependent, and associated with a non-monotone normal compliance, which is restricted by a unilateral constraint. We derive a variational formulation for the model, which is a system that couples a variational-hemivariational inequality with a variational equation. Then, we apply abstract results to prove its weak solvability. Finally, under additional assumptions on the contact functions, we prove the uniqueness of the solution.
Convex clustering formulates the clustering problem as a convex optimization task that encourages similar data points to merge into clusters by minimizing a combination of fitting error and a penalty on differences between cluster centroids. This method has attracted considerable attention due to its capacity to address the challenges associated with local optimal solutions and numerical instability prevalent in traditional nonconvex clustering methods. However, it typically relies on the standard Euclidean metric for measuring distance, which can lead to decreased performance, especially in the presence of outlier features. Additionally, it is not particularly robust against outlier samples. To address these issues, we separate the data into cluster and outlier components, effectively eliminating the outlier features. By incorporating self-paced learning, we develop a model that adaptively selects relevant examples while minimizing interference from outlier instances. This approach enhances robustness by removing outlier features and samples, forming a biconvex clustering framework with strong statistical properties. We propose an efficient, convergent algorithm and establish a finite sample bound for prediction error. Experiments on artificial and benchmarking datasets show improved clustering effectiveness compared to classical convex clustering methods.
Based on the technique of T. Ogita and K. Aishima (2020) and J. A. Ezquerro and M. A. Hernández (2012), we designed an improved two-step method for solving the inverse singular value problems. Compared with other existing two-step methods, the proposed method has comparable computational cost. However, computing the product of matrices is simpler than solving linear equations and has no instability problem caused by ill-conditioning in solving linear equations, and thus it seems more stable and greatly reducing computational costs. Under appropriate assumptions, the proposed method is proved to be convergent with the cubic root-convergence rate. The proposed method is applied to the noise reduction for modal parameters estimation and indicates that it can significantly remove noise from measured signals and accurately estimate the modal frequencies and damping ratios. The numerical results demonstrate the effectiveness of the improved method.
After a brief biography of Professor Miloš Zlámal we recall his most important mathematical achievements. First, he dealt with various properties of analytical solutions of ordinary and partial differential equations. He then laid the theoretical foundation for the finite element method as the minimum angle condition, curved finite elements, the numerical solution of heat conduction problem and semiconductor equations, and superconvergence. He developed the so-called mortar finite elements used to cover a strip between two different types of finite elements. Complete chronological list of Zlámal’s publications closes the contribution.
A simplified model is derived for pressure-driven flow between adjacent surfaces of materials modeled as seemingly viscoplastic or truly viscoplastic. The material response to external forces is traditionally described by constitutive relations in which the extra stress tensor S is expressed as a function of the symmetric part of the velocity gradient D. However, for viscoplastic materials, S cannot, in general, be written as a function of D, whereas D can be expressed in terms of S. Motivated by this observation, a model based on constitutive relations of the form D = f(S) is proposed, leading to a system of first-order partial differential equations. A local Poiseuille law is also formulated, and a reduced-dimensional equation for the pressure is derived. Explicit velocity profiles are obtained for selected cases.
The paper examines the evolution of a weak discontinuity, specifically, an acceleration wave in a one-dimensional unsteady plasma flow influenced by an axial magnetic field in the presence of dust particles. We obtained self-similar solutions through the application of Lie group transformations. This study also explores the idea of interaction between the acceleration wave and a strong shock wave, with particular emphasis on the roles played by dust particles and the magnetic field. The effects of various parameters involved in the flow are examined. Additionally, the reflected and transmitted waves following the interactions are analyzed and the results are depicted.
Recent research has shown growing interest in quantifying uncertainty in system lifetimes. This paper investigates the failure extropy (FEx) of an n-component mixed system, conditioned on the failure of all components by a given time t. Using the concept of system signature, explicit expressions for the FEx of the system lifetime are derived, along with key properties and informative bounds. To extend this framework, a divergence measure based on FEx is proposed to assess the complexity of system structures. A new discrimination measure is also proposed, serving as a valuable tool to assess how closely a system resembles a parallel system. An application to redundancy allocation has been carried out to demonstrate the practical relevance of the proposed results and provide insights into optimal system design under uncertainty.
We study the dynamics of similarity classes of tetrahedra generated by the longest-edge bisection (LEB) algorithm. Building on the normalization strategy introduced by F. Perdomo, Á. Plaza (2014) we construct a canonical representation of tetrahedra in a normalized space embedded in the product of the hyperbolic half-plane and the hyperbolic half-space model. This representation allows us to define the left and right refinement maps, ΦL and ΦR, acting on the space of normalized tetrahedral shapes, and to study their iterative orbits as discrete dynamical systems. Using these maps, we show that the orbit of the space-filling Sommerville tetrahedron contains only 4 similarity classes, 3 of which form an attractive cycle corresponding to the orbit of the path tetrahedron. We also show that small perturbations of elements in those orbits still lead to finite orbits. In addition, we study small perturbations of the regular tetrahedron and show that their orbits are also finite. Extensive numerical exploration of orbits for the other types of tetrahedra suggests that the LEB algorithm does not produce degenerating tetrahedra. Our framework provides a geometric and dynamical foundation for analyzing the shape evolution of tetrahedral meshes and offers a possible route towards an analytic proof of the nondegeneracy property for the tetrahedral partitions generated by the LEB refinements. This property is highly desired in e.g., the finite element methods (FEMs).
Perfectly Matched Layers (PML) has become a very common method for the numerical approximation of wave and wave-like equations on unbounded domains. This technique allows one to obtain accurate solutions while working on a finite computational domain, and the technique is relatively simple to implement. Results concerning the accuracy of the PML method have been obtained, but mostly with regard to problems at a fixed frequency. In this paper we provide very explicit time-domain bounds on the accuracy of PML for the inhomogeneous two-dimensional wave equation with a particular type of forcing term, and illustrate our conclusions with some numerical examples.
