The purpose of this paper is to study the influence of large or unbounded domains on a stochastic PDE near a change of stability, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider the stochastic Swift-Hohenberg equation and derive rigorously the Ginzburg-Landau equation as a modulation equation for the amplitudes of...

We discuss the use of the formal theory of differential equations in the numerical analysis of general systems of partial differential equations. This theory provides us with a very powerful and natural framework for generalising many ideas from differential algebraic equations to partial differential equations. We study in particular the existence and uniqueness of (formal) solutions, the meth...

In this work, we construct the general solution to the Heat Equation (HE) and to many tensor structures associated to the Heat Equation, such as Symmetries, Lagrangians, Poisson Brackets (PB) and Lagrange Brackets (LB), using newly devised techniques that may be applied to any linear equation (e.g., Schrödinger Equation in field theory, or the small-oscillations problem in mechanics). In partic...

A theoretical framework model that researches the interrelationship of organizational innovation, organizational performance and dynamic core competence of enterprise (DCCE) is proposed on the foundation of the analysis of measure dimensions which include four oriented organizational innovations and the four sub-core capabilities of DCCE and the three levels of variables of performance. By usin...

The concept of livable city was put forward naturally and began to gain more attention when people care more about human needs during the economic development. In this paper, we define a livable city as an urban area takes the residents’ demand as first priority. It has a pleasant ecological environment, a mature community with rich public resources such as culture, transportation and medical s...