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trendline. In this case, two are used, and you could just as easily have three, four, or more. You want to avoid what some call the pick-up sticks effect of having lines scattered all over the chart, because that just creates noise, and what you seek is clarity. The old saw that less is more certainly applies to good charting. The reasons for having multiple trendlines fall into two camps: to show contrary trends (as was the case with Figure 7.18, which showed a general uptrend and a shorter-lived downtrend) or to show varying degrees of momentum for a single trend, such as with Figure 7.19. In this instance, which is a chart of the crude oil commodity market from 2001 through 2006, the lower trendline shows a decisive uptrend that spans virtually the entire graph, and the higher trendline shows how, starting late in 2003, the market accelerated even faster, resulting in a steeper trendline. Both trendlines are valid, and both are useful. If the prices violated the medium-term trendline, that would negate the faster uptrend, but it would not negate the broader, larger uptrend until such time as that trendline was itself crossed.

"It will be noted that the proofs of these lemmas make use only of the general property (15.411 and the special representation (15.42) of {fp.}.

A straightforward calculation shows that S".(O)fA S,,-;,l(O) = fA S". (0) f"S,,-;' 1( 0) S".(O)f.S,,-;,l(O)

*' IL, A *' v)

The eigenvalues of w(ILvIO) are 1 (2n - 2-fold degenerate), and e i8 (nondegenerate). The eigenvalues of S".(O) are e i8/2 (each 2 n - 1-fold degenerate).

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Proof The first part is trivial. The second part can be proved by choosing a special representation for f"f., since the eigenvalues of S".(O) are independent of the representation. As a representation for f" and f. we use (15.43) with X and Z interchanged. Since the number of the f" in (15.43) is not unique we are free to choose any two to be f" and f . What we choose for the remaining 2n - 2 matrices is irrelevant for the proof. We choose

(e- i8 /

e i8 / 2

2 3 4

Thus S".(O) is diagonal. The diagonal elements are either ei8 / 2 or e- i8 / 2, each appearing the same number of times, i.e., 2 n -1 times each.

The time when pro ts are made is when prices push through support or resistance and break out. The longer a price has been trying to push through a certain price level, the more forceful it will be when it nally does make it through (think back to our Red Rover game, and picture a particularly eager youngster who has tried ten times to get through the line and is more determined than ever to do so). Figure 1.3 is an example of how potent this is. The rst half of the stock chart for ALVR shows prices bouncing between about $2.00 and $2.50. For month after month the stock was completely stagnant, and buyers were accumulating at these levels. There were several attempts to push through resistance (represented by the horizontal line), but they failed . . . until the midpoint of the graph, in April. At that point, three important things happened: (1) buyers overcame sellers, pushing prices above resistance; (2) volume increased as excitement began to build around the stock; and (3) when some pro t-taking took place, prices eased back, but they did not go beneath the former line

be a product of n commuting plane rotations:

w(a/1101)w(ySI02)'"

(15.57)

where {a, /1, ... , IL, v} is a permutation of the set of integers {I, 2, ... , 2n - 1, 2n}, and 01 , . , On are complex numbers. Then

(15.58)

(15.591

(15.60 1 with the signs chosen independently. This lemma is an immediate consequence of lemmas 1 and 2 and the fact that [f/Lfv ' f"fp] = O.

By this lemma, the eigenvalues of S( w) can be immediately obtained from those of w, if the eigenvalues of ware of the form (15.59). The usefulness of these lemmas rests on the fact that V2V1 can be expressed in terms of S( w).

FIGURE 1.3 After breaking out of a saucer pattern, ALVR blasted ahead on much bigger volume to a 500% gain in about a year.

2 3 4 1

15.3 THE SOLUTION*

In the absence of an external magnetic field the formulas (15.14) and (15.40) lead to the following:

1 lim - log QI(O, T)

(15.611

where A and (15.631 where Vi is given by (15.37) and V2 by (15.38). These formulas are valid if all eigenvalues of V are positive and if lim n -1 log A exists. Our main task is to diagonalize the matrix V. Throughout the present section, all matrices are understood to be matrices in a definite representation.

(a=I, ... ,n)

(15.64)

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