Patients at 6 US hospitals were randomly assigned to 1 of 3 groups: 604 received intercessory prayer after being informed that they may or may not receive prayer; 597 did not receive intercessory prayer also after being informed that they may or may not receive prayer; and 601 received intercessory prayer after being informed they would receive prayer. Intercessory prayer was provided for 14 days, starting the night before CABG. The primary outcome was presence of any complication within 30 days of CABG. Secondary outcomes were any major event and mortality.
The usual economic analysis of externalities can be illustrated using a standard supply and demand diagram if the externality can be valued in terms of money . An extra supply or demand curve is added, as in the diagrams below. One of the curves is the private cost that consumers pay as individuals for additional quantities of the good, which in competitive markets, is the marginal private cost. The other curve is the true cost that society as a whole pays for production and consumption of increased production the good, or the marginal social cost . Similarly there might be two curves for the demand or benefit of the good. The social demand curve would reflect the benefit to society as a whole, while the normal demand curve reflects the benefit to consumers as individuals and is reflected as effective demand in the market.
Attraction between the plates can be easily understood by focusing on the one-dimensional situation. Suppose that a moveable conductive plate is positioned at a short distance a from one of two widely separated plates (distance L apart). With a << L , the states within the slot of width a are highly constrained so that the energy E of any one mode is widely separated from that of the next. This is not the case in the large region L , where there is a large number (numbering about L / a ) of states with energy evenly spaced between E and the next mode in the narrow slot – in other words, all slightly larger than E . Now on shortening a by d a (< 0), the mode in the narrow slot shrinks in wavelength and therefore increases in energy proportional to −d a / a , whereas all the L / a states that lie in the large region lengthen and correspondingly decrease their energy by an amount proportional to d a / L (note the denominator). The two effects nearly cancel, but the net change is slightly negative, because the energy of all the L / a modes in the large region are slightly larger than the single mode in the slot. Thus the force is attractive: it tends to make a slightly smaller, the plates attracting each other across the thin slot.