Domain of influence

This shows that the usual ideas associated with propagating waves in electromagnetism or fluid dynamics do not describe the behaviors found here. These differences could be expected because of the mathematical structure of reaction-diffusion equations, which owing to their parabolic character propagate information with infinite velocity. On the contrary, in the case of classical wave equations or hyperbolic equations there is a well-defined domain of influence and a characteristic velocity of propagation of information. ... 

Equation (1.27), which has been expanded to different types of liquid chromatography (Knox equation), shows that there is an optimum flow rate for each separation and that this does indeed correspond to the minimum on the curve represented by equation (1.27). The loss in efficiency that occurs when the velocity is increased represents what occurs when trying to rush the chromatographic separation by increasing the flow rate of the mobile phase. However, intuition can hardly predict the loss in efficiency that occurs when the flow is too slow. To explain this phenomenon, the origins of the terms A, B and C have to be considered. Each of these parameters has a domain of influence that can be seen in Fig. 1.9. Essentially, this curve does not depend on the nature of the solute. 

The transient heat equation (Eq. 3.285) often serves as the model for parabolic equations. Here the solution depends on initial conditions, meaning a complete description of T(0, x) for the entire spatial domain at t =0. Furthermore the solution T(t,x) at any spatial position x and time t depends on boundary conditions up to the time t. The shading in Fig. 3.14 indicates the domain of influence for the solution at a point (indicated by the dot). 

Fig. 3.14 Domains of influence for model partial-differential equations of different classifications.

The solution at a given point depends only partly on the initial and boundary conditions, depending on the wave speed a. As illustrated in the lower right-hand panel of Fig. 3.14, the domain of influence slopes away from the point. The higher the sound speed, the shallower becomes the slope. For a truly incompressible fluid, where the sound speed in infinite, the slope approaches zero and characteristics become essentially parabolic. 

An approximate characterization of an electron-pair s domain of influence. 

First order hyperbolic differential equations transmit discontinuities without dispersion or dissipation. Unfortunately, as Carver (10) and Carver and Hinds (11) point out, the use of spatial finite difference formulas introduces unwanted dispersion and spurious oscillation problems into the numerical solution of the differential equations. They suggest the use of upwind difference formulas as a way to diminish the oscillation problem. This follows directly from the concept of domain of influence. For hyperbolic systems, the domain of influence of a given variable is downstream from the point of reference, and therefore, a natural consequence is to use upstream difference formulas to estimate downstream conditions. When necessary, the unwanted dispersion problem can be reduced by using low order upwind difference formulas. 

Therefore, the plane is divided into two domains of influence. The boundary of these domains is the circle. Everything becomes more interesting if complex representations are used instead of the former iteration functions. 

However, intuition can hardly predict the loss in efficiency that occurs when the flow rate is too slow. To explain this phenomenon, the origins of the terms A, B and C must be recalled. Each of these parameters represents a domain of influence which can be perceived on the graph (Figure 1.11). 

A point to be noted is that the selection of the number of cells and hence the cell length. Ax, cannot be totally free in any finite difference scheme. The Courant condition suggests that the time integration should not attempt to calculate beyond the spatial domain of influence by using a temperature at a distance beyond the range of influence determined by the characteristic velocity of temperature propagation. Hence... 

The implementation of either of the above approaches is simple and straightforward. First, at any node J, one should determine the nodes which contain node J within their domain of influence. Then, the construction of Eq. 21 simply involves calculating the second derivative of the shape functions of all relevant nodes at X = Xj. 

Note that this seemingly straightforward use of Equation 7-3 is actually subtle. For our constant density fluid, the pressure at (ij) must depend on its neighbors at i-l,j, i+l,j, i,j-l and i,j+l. That is, the flow at any point is influenced by every other point, and each point affects all other points. The situation is different for hyperbolic problems for example, disturbances created by a supersonic aircraft cannot propagate ahead of the plane, so that a difference approximation that violates domains of influence and dependence cannot be used. Similarly, in unsteady wave propagation, computations cannot depend on future time. Hence, there are areas in physics where use of central differencing throughout is inappropriate, and one-sided models must be used. However, for Laplace s equation, the approximation in Equation 7-15 is perfectly valid. 

My point here is that people need to distinguish between factors they can control on a personal level and factors beyond their domain of influence. Similarly, Covey (1989) recommends we distinguish between our "Circle of Concern" and "Circle of Influence," and focus our efforts in the Circle of Influence. Thus, it is healthy to admit there are things we are concerned about but have little influence over. Then, when negative consequences occur outside our domain of personal influence, we will not attribute personal blame and reduce our sense of self-efficacy, personal control, or optimism. 

Obviously, we cannot have complete control over all factors contributing to an injury. That is why I think it is wrong to say "all injuries are preventable." However, there is much we can do within our own domain of influence, and we can prepare for factors outside our personal control. Thus, we take an umbrella to the golf course in case it rains, and we wear personal protective equipment in case we are exposed to risks beyond our domain of personal control. Likewise, we protect our children from events beyond their control, as illustrated in Figure 16.9. 

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