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There are ve handles you can grab one on the top, one on the bottom, one on the right, and one on both the upper-right and lower-left corners (Figure 8.3). You can click on any of these and drag to resize the chart to your liking. It is more common to create x-axis (time-based) space as opposed to y-axis (price-based) space, but the extra blank space you form is totally up to you, and of course it can be changed at any time. Once you are satis ed with the new size of your chart, you can click anywhere within the chart to set that size. Let s look at a simple example of how a resized chart might be useful. Figure 8.4 shows a graph of the S&P 500 index over the span of a couple of years, and it has a channel (with a median line) drawn on top of it. The graph shows that the market fell quickly from about 1,330 to about 1,255. Suppose you were interested in shorting this market, but you wanted to wait until the price had climbed back to the median line in order to reduce the risk of the trade. If you had a normal graph, it would not be possible to judge this accurately. However, by being able to see six weeks into the future (albeit without price data), you can easily picture how a retracement might take place, and, using your cursor and the data line, you could point at the likely touch point on the median and get an accurate estimate as to the price the S&P needs to hit before you place your trade.

We used the estimate lV'ml2 ex I/L 2 on the grounds that L is the only relevant length in the problem. By the same token, the thickness of the wall is also proportional to L. Thus, the volume of the wall is proportional to L d, and we conclude that (16.65) Reasoning as in the previous case, we conclude that entropy favors domain creation for d :$ 2. Therefore the lower critical dimension is d = 2 for continuous systems, in agreement with an earlier result. None of the arguments presented above are rigorous. But they give us physical insight into the importance of dimensionality. Rigorous arguments in special models support these conclusions. Hohenberg has shown that the superfluid order parameter must vanish in d::s; 2. Using a similar argument, Mermin

*F. Bloch, Z. Phis. 61,206 (1930); C. H. Herring and C. Kittel, Phys. Rev. 81, 869 (1950).

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and Wagner proved that in d:s: 2 the Heisenberg model can exhibit neither ferromagnetic nor antiferromagnetic order. *

Experimentally it is possible to create metals containing randomly distributed magnetic impurities that are out of thermal equilibrium with the host system, with a very long relaxation time. This is called "quenched" randomness (as opposed to "annealed" randomness, which obtains when the impurities are in thermal equilibrium with the system). The host system sees the impurities as sources of external fields, which vary from point to point at random. Experimentally the nonequilibrium status can be shown by checking the concentration of the impurities against known equilibrium values. The randomness of the impurities can be demonstrated by X-ray diffraction. Suppose a system in the presence of a quenched random external field magnetizes spontaneously. Since the random field varies from point to point, there will be regions in which it would pay for the magnetization to reorient itself, so as to lower the interaction energy in the local external field. Thus a domain will be formed, provided the domain wall does not cost too much energy. To find the energy balance, assume that the random field has Gaussian fluctuations. The value of the external field averaged over a domain will then be of order {ii, where N is the number of impurities in the domain. For a domain of linear size L, we have N a: L d. Hence the volume energy is proportional to - L d / 2 Adding this to the energy of the domain wall, which is proportional to L d - 2 as before, we obtain the energy cost of one domain:

(16.66)

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