We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α-stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument [9]. Irreducibility and uniform strong Feller property play an essential role in both ...

We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schrödinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We...

We investigate existence and asymptotic completeness of the wave operators for nonlinear Klein-Gordon and Schrödinger equations with a defocusing exponential nonlinearity in two space dimensions. A certain threshold is defined based on the value of the conserved Hamiltonian, below which the exponential potential energy is dominated by the kinetic energy via a Trudinger-Moser type inequality. We...

Let Ω be a bounded domain in R . In this paper, we consider the following nonlinear elliptic equation of N -Laplacian type: (0.1) { −∆Nu = f (x, u) u ∈ W 1,2 0 (Ω) \ {0} when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosett...

Let Ω ⊂ R, n > 2, be a bounded connected domain of the class C for some θ ∈ (0, 1]. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem u ∈ W L(Ω), − div ( Φ(|∇u|) ∇u |∇u| ) + V (x)Φ(|u|) u |u| = f(x, u) + μh(x) in Ω, ∂u...

In this paper we provide a general framework to study general class of linear and nonlinear kinetic equations with random uncertainties from the initial data or collision kernels, and their stochastic Galerkin approximations, in both incompressible Navier-Stokes and Euler (acoustic) regimes. First, we show that the general framework put forth in [C. Mouhot and L. Neumann, Nonlinearity, 19, 969-...

Abstract. Hebbian learning is a local learning algorithm and allows an on-line adaptation of the weights. Therefore an artificial neural network with built-in hebbian learning is capable of learning on operation. This paper presents the implementation of this algorithm in a digital Field Programmable Gate Array (FPGA). Nonlinearity is introduced by applying nonlinear low-pass-filtering to all i...

Our paper studies the optical solitons for Kundu–Eckhaus (KE) equation with quintic nonlinearity and Raman effect. By applying improved modified extended tanh-function method, many soliton solutions can be obtained such as bright solutions, dark singular solution. In addition, we obtain various types of namely, periodic exponential rational Jacobi elliptic Weierstrass doubly solutions. Moreover...

Measurement of integral non-linearity (INL) and differential non-linearity (DNL) of an A/D converter using the histogram method incurs large test time. This test time can be a significant percentage of the total test time, especially for high resolution and low sampling-speed A/D converters. This paper describes a test methodology for measuring the INL and DNL specifications of A/D converters b...