In this paper, we study a Fokker-Planck equation of the form ut = I[u] + div(xu) where the operator I, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the only steady state of the associated stationnary equation.

In this paper, we study the nonlinear heat equation ∂ ∂t △u(x, t) − c♦u(x, t) = f(x, t, u(x, t)), where△k is the Laplacian operator iterated k− times and is defined by (1.4)and ♦k is the Diamond operator iterated k− times and is defined by (1.2). We obtain an interesting kernel related to the nonlinear heat equation.

In this paper, we first define the Teodorescu operator [Formula: see text] related to the Helmholtz equation and discuss its properties in quaternion analysis. Then we propose the Riemann boundary value problem related to the Helmholtz equation. Finally we give the integral representation of the boundary value problem by using the previously defined operator.

We can write the polynomial solution of the second order linear differential equation of hypergeometric-type φ(x)y + ψ(x)y + λy = 0, where φ and ψ are polynomials, deg φ ≤ 2, degψ = 1 and λ is a constant, among others, by using the Rodrigues operator Rk(φ,u) (see [3]) where u is certain linear operator which satisfies the distributional equation

We derive a modification of the semiclassical Fermi Golden Rule collision operator based on quantum thermodynamic principles. The resulting operator is nonlocal in space and acknowledges the presence of steep potential gradients and potential barriers. The resulting quantum mechanical transport equation the Wigner quantum Boltzmann equation increases the corresponding quantum mechanical entropy...

The existence of positive semidefinite solutions of the operator equation n ∑ j=1 AXA = Y is investigated by applying grand Furuta inequality. If there exists positive semidefinite solutions of the operator equation, one of the special types of Y is obtained, which extends the related result before. Finally, an example is given based on our result.

In this paper, we study a Fokker-Planck equation of the form ut = I[u] + div(xu) where the operator I, which is usually the Laplacian, is replaced here with a general Lévy operator. We prove by the entropy production method the exponential decay in time of the solution to the only steady state of the associated stationnary equation.

Interactive online help systems are considered to be a fruitful supplement to traditional IT helpdesks, which are often overloaded. They often comprise user-generated FAQ collections playing the role of technology-based conceptual artifacts. Two main questions arise: how the conceptual artifacts should be used, and which factors influence their acceptance in a community of practice (CoP). First...

We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illust...

1 The Boltzmann equation 6 1.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Fluid limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 Other collis...