where ν is the unit normal to ∂Ω and the superscript t indicates transposition (in this case, of the matrix ∇~u = (∂ju)j,α). Relying on the method of layer potentials and suitable Rellich-NečasPayne-Weinberger formulas, the boundary value problems (1.2)-(1.3) with ~ f = 0 and ~g ∈ L(∂Ω), 2− ε < p < 2 + ε, have been treated (in all spacedimensions) by B. Dahlberg, C.Kenig, and G.Verchota (cf. [7...

The long-time behaviour of many dynamical systems may be effectively predicted by a low-dimensional model that describes the evolution of a reduced set of variables. We consider the question of how to equip such a low-dimensional model with appropriate initial conditions, so that it faithfully reproduces the long-term behaviour of the original high-dimensional dynamical system. Our method invol...

In this study, multimode chaos observed from two coupled chaotic oscillators with hard nonlinearities is investigated. At first, a simple chaotic oscillator with hard nonlinearities is realized. It is confirmed that in this chaotic oscillator the origin is always asymptotically stable and that the solution, which is excited by giving relatively large initial conditions, undergoes period-doublin...

In this paper we consider the analogue of the Courant nodal domain theorem for the nonlinear eigenvalue problem for the p-Laplacian. In particular we prove that if uln is an eigenfunction associated with the nth variational eigenvalue, ln, then uln has at most 2n−2 nodal domains. Also, if uln has n+k nodal domains, then there is another eigenfunction with at most n−k nodal domains. © 2002 Elsev...

Abstract. Consider the random normal matrix ensemble associated with a potential on the plane which is sufficiently strong near infinity. It is known that, to a first approximation, the eigenvalues obey a certain equilibrium distribution, given by Frostman’s solution to the minimum energy problem of weighted logarithmic potential theory. On a finer scale, one can consider fluctuations of eigenv...

When a system jams it undergoes a transition from a flowing to a rigid state. Despite this important change in the dynamics, the internal structure of the system remains disordered in the solid as well as the fluid phase. In this way jamming is very different from crystallization, the other common way in which a fluid solidifies. Jamming is a paradigm for thinking about how many different types...

BACKGROUND The study examined the effects of dietary fasting on physical balance among young healthy women. METHODS This study undertaken involving 22 young healthy women (age = 22 +/- 1.5) using a within subject counterbalanced 2-week crossover study design. Participants were asked to refrain from consuming any food or beverage for 12 hours prior to the fasting trial and to maintain their re...

The envelope of straight lines normal to a plane curve C is its evolute; the envelope of lines tangent to C is the original curve, together with the entire tangent line at each inflexion of C. We introduce some standard techniques of singularity theory and use them to explain how the first of these envelopes turns into the second, as the (constant) angle between the set of lines forming the env...

Cauchy problem in a product space (that may be particularly involved). For example, consider the heat equation  ∂u ∂t (t, x) = ∆u(t, x), t ≥ 0, x ∈ Ω, ∂w ∂t (t, z) = − ∂w ∂n (t, z), t ≥ 0, z ∈ ∂Ω, w(t, z) = u(t, z), t ≥ 0, z ∈ ∂Ω, where − ∂ ∂n denotes the outer normal derivative. This can be written as U ′(t) = ( ∆ 0 − ∂ ∂n 0 ) U(t), where U = ( u u|∂Ω ) . Observe that a relationship between ...

We apply dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD) methods to flows in highly heterogeneous porous media to extract the dominant coherent structures and derive reduced-order models via Galerkin projection. Permeability fields with high contrast are considered to investigate the capability of these techniques to capture the main flow features and forecast the flo...