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Monotone lattice recurrence relations such as the Frenkel–Kontorova lattice, arise in Hamiltonian lattice mechanics, as models for ferromagnetism and as discretization of elliptic PDEs. Mathematically, they are a multi-dimensional counterpart of monotone twist maps. Such recurrence relations often admit a variational structure, so that the solutions x : Z d → R are the stationary points of a formal action function W ( x ) . Given any rotation vector ω ∈ R d , classical Aubry–Mather theory establishes the existence of a large collection of solutions of ∇ W ( x ) = 0 of rotation vector ω. For irrational ω, this is the well-known Aubry–Mather set. It consists of global minimizers and it may have gaps. In this paper, we study the parabolic gradient flow d x d t = − ∇ W ( x ) and we will prove that every Aubry–Mather set can be interpolated by a continuous gradient-flow invariant family, the so-called ‘ghost circle’. The existence of these ghost circles is known in dimension d = 1 , for rational rotation vectors and Morse action functions. The main technical result of this paper is therefore a compactness theorem for lattice ghost circles, based on a parabolic Harnack inequality for the gradient flow. This implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions. As a consequence, we can give a simple proof of the fact that when an Aubry–Mather set has a gap, then this gap must be filled with minimizers, or contain a non-minimizing solution.

In this paper we consider a class of singularly perturbed domains, obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero. We use an Almgren-type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues, via Courant-Fischer Min-Max characterization and blow-up analysis for scaled eigenfunctions., Comment: 36 pages

This paper involves the dynamic (stability) analysis of distributed drill-string systems. A minimal set of parameters characterizing the linearized, axial-torsional dynamics of a distributed drill string coupled through the bit-rock interaction is derived. This is found to correspond to five parameters for a simple drill string and eight parameters for a two-sectioned drill-string (e.g., corresponding to the pipe and collar sections of a drilling system). These dynamic characterizations are used to plot the inverse gain margin of the system, parametrized in the non-dimensional parameters, effectively creating a stability map covering the full range of realistic physical parameters. This analysis reveals a complex spectrum of dynamics not evident in stability analysis with lumped models, thus indicating the importance of analysis using distributed models. Moreover, it reveals trends concerning stability properties depending on key system parameters useful in the context of system and control design aiming at the mitigation of vibrations.

In this paper, we apply a special finite-volume scheme, limited to smooth temperature distributions and Cartesian grids, to test the importance of connectivity of the finite volumes. The area of application is nuclear fusion plasma with field line aligned temperature gradients and extreme anisotropy. We apply the scheme to the anisotropic heat-conduction equation, and compare its results with those of existing finite-volume schemes for anisotropic diffusion. Also, we introduce a general model adaptation of the steady diffusion equation for extremely anisotropic diffusion problems with closed field lines.

The Berry heuristic has been a long standing \emph{ansatz} about the high energy (i.e. large eigenvalues) behaviour of eigenfunctions (see Berry 1977). Roughly speaking, it states that under some generic boundary conditions, these eigenfunctions exhibit Gaussian behaviour when the eigenvalues grow to infinity. Our aim in this paper is to make this statement quantitative and to establish some rigorous bounds on the distance to Gaussianity, focussing on the hyperspherical case (i.e., for eigenfunctions of the Laplace-Beltrami operator on the normalized $d$-dimensional sphere - also known as spherical harmonics). Some applications to non-Gaussian models are also discussed.

In this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver.

We shall give conditions on the illuminations $\varphi_{i}$ such that the solutions to Maxwell's equations \[ \left\{ \begin{array}{l} {\rm curl} E^{i}=i\omega\mu H^{i}\qquad\text{in }\Omega,\\ {\rm curl} H^{i}=-i(\omega\varepsilon+i\sigma)E^{i}\qquad\text{in }\Omega,\\ E^{i}\times\nu=\varphi_{i}\times\nu\qquad\text{on }\partial\Omega, \end{array}\right. \] satisfy certain non-zero qualitative properties inside the domain $\Omega$, provided that a finite number of frequencies $\omega$ are chosen in a fixed range. The illuminations are explicitly constructed. This theory finds applications in several hybrid imaging problems, where unknown parameters have to be imaged from internal measurements. Some of these examples are discussed. This paper naturally extends a previous work of the author [Inverse Problems 29 (2013) 115007], where the Helmholtz equation was studied., Comment: 24 pages

In this paper, we build a Gibbs measure for the cubic defocusing Schrodinger equation on the real line with a decreasing interaction potential, in the sense that the non-linearity | u | 2 u is multiplied by a function χ which we assume integrable and smooth enough. We prove that this equation is globally well-posed in the support of this measure and that the measure is invariant under the flow of the equation. What is more, the support of the measure (the set of initial data) is disjoint from L 2 .

The original explicit Runge-Kutta-Chebyshev (RKC) method is a stabilized second-order integration method for pure diffusion problems. Recently, it has been extended in an implicit explicit manner to also incorporate highly stiff reaction terms. This implicit-explicit RKC method thus treats diffusion terms explicitly and the highly stiff reaction terms implicitly. The current paper deals with the incorporation of advection terms for the explicit method, thus aiming at the implicit-explicit RKC integration of advection-diffusion-reaction equations in a manner that advection and diffusion terms are treated simultaneously and explicitly and the highly stiff reaction terms implicitly.

This paper builds upon the Lp-stability results for discrete orthogonal projections on the spaces Sh of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L2-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves.

Most reported work on transient fluid/structure interaction (FSI) in liquid-filled pipes has been carried out in the time domain. When needed, information in the frequency domain (e.g., frequency responses) has been deduced by discrete Fourier transforms. In this paper, the analysis is undertaken directly in the frequency domain and has the advantage of enabling (linear) dispersive terms to be included in a fully coupled manner. In principle, time-domain results (e.g., pressure histories) can be obtained by numerical inverse Laplace transforms. In both domains, the mathematical model has one pair of equations for each mode of wave propagation—e.g., pressure waves in the liquid, flexural waves in the pipe. Axial FSI coupling exists in the equations and also in boundary conditions. The development used herein highlights common features between analysis in the frequency domain and analysis by the method of characteristics (MOC) in the time domain. A general formulation, not restricted to pipe systems, is presented. The method is validated by comparison with an alternative exact analytical solution, with results obtained by discrete Fourier transform from an MOC analysis and by comparison with measured data from a laboratory apparatus.

Let Jµ denote the Bessel function of order µ. The system jαn ={jαn(s)}s≥1 = {2√α + 2n + 1 Jα + 2n + 1(ps)/aps|Jα + 1(aps|}s≥1 with n = 0, 1, ..., α > - 1, and where ps denotes the sth positive zero of Jα(ax), is orthonormal in l2(N). In this paper, we study the mean convergence of the Fourier series with respect to this system. Also, we describe the space in which the span of the system is dense.

This paper describes a two-dimensional (2D) upwind residual distribution or fluctuation splitting (FS) scheme (MHD-A) for the numerical solutions of planar magnetohydrodynamics (MHD) equations on structured or unstructured triangular meshes. The scheme is second order in space and time, and utilizes a consistent 2D wave model originating from the eigensystem of a 2D jacobian matrix of the MHD flux vector. The possible waves existing in this wave model are entropy, magnetoacoustic, and (numerical) magnetic monopole waves; however, Alfven waves do not exist since the problem is planar. One of the important features of the method is that the mesh structure has no influence on propagation directions of the waves. These directions are dependent only on flow properties and field gradients (for example, it is shown that the magnetoacoustic waves propagate in the directions of maximum and minimum magnetic strain rates). The other feature is that no flux evaluations and no information from the neighboring cells are needed to obtain a second order, positive, and linearity preserving scheme. A variety of numerical tests carried out by the model on structured and unstructured triangular meshes show that MHD-A produces rather encouraging numerical results even though it is the first FS wave model ever developed for multidimensional MHD. © 1999 Academic Press.