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where K is an integer called the Kaluza-Klein excitation number. This is an important result without the compacti ed dimension, the center of mass momentum of the string is continuous. Compactifying a dimension quantizes the center of mass momentum along that dimension. Looking at Eq. (8.7) then, the rst term involving the momenta is the total center of mass momentum of the string. We call this the momentum mode. The second term, however, also involves momentum. In fact this term is the winding mode of the string, which satis es

Figure 1-24 Quadrature amplitude modulation (QAM)

The next line of code defines a macro that will be used when APP_STORE_FINAL is defined. You might be asking, Hey, why not just define it to be nothing at all Good question! The problem with that approach is that you might have a variable used in your assert that is used only in the assert. If you re compiling with Treat Warnings as Errors, you could get a whole bunch of errors when you turn off your custom asserts. The following line of code generates a very small amount of assembly while still avoiding those build warnings:

(8.10)

0101 0011 0110 0100 0010 0000 1110 0001

#define MY_ASSERT(STATEMENT) do { (void)sizeof(STATEMENT); } while(0)

1000 1010 1100

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Looking at Eq. (8.3), we see that the winding w can be de ned in terms of the momentum of the left- and right-moving modes as w= 1 nR 25 = p 25 pR 2 L

1011 1101

(8.11)

The do while loop that exists around the macro body is there to avoid nasty scoping errors that can come up if you were to use your custom assert inside an if statement without braces, for example.

quad-bit encoding. This scheme, sometimes called quadrature amplitude modulation (QAM) (pronounced Kw am ), permits a single signal to represent four bits, which means that there is a 4:1 ratio between the bit rate and the signaling rate. Thus, it is possible to achieve higher bit rates in the bandwidth-limited telephony local loop by using multibit encoding techniques such as QAM. The first high bit rate modems (9,600 bitsper-second) used this technique or a variation of it to overcome the

Compactifying a dimension will lead to a modi ed mass spectrum. To obtain the mass spectrum for the state with a compacti ed dimension, let us begin with the Virasoro operators. Recall that L0 =

Finally, you can clearly see that when you re building your debug build, your custom assert just uses the standard assert function (with the do while wrapped around it for safekeeping):

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(8.12)

design limitations of the network. In fact, these multibit schemes are also used by the cable industry to achieve the high bit rates they need to operate their multimedia broadband networks. There is one other limitation that must be mentioned: noise. Look at Figure 1-25. Here we have a typical QAM graph, but now we have added noise in the form of additional points on the graph that have no implied value. When a receiver sees them, however, how does it know which points are noise and which are data Similarly, the oscilloscope trace shown in Figure 1-26 of a high-speed transmission would be difficult to interpret if there were noise spikes intermingled with the data. There is, therefore, a well-known relationship between the noise level in a circuit and the maximum bit rate that is achievable over that circuit, a relationship that was first described by Bell Labs researcher Claude Shannon, who is widely known as the father of information theory. In 1948, Shannon published A Mathematical Theory of Communication, which is now universally accepted as the framework for modern communications. We won t delve into the complex (but fascinating) mathematics that underlie Shannon s Theorem, but simply know that

#define MY_ASSERT(STATEMENT) do { assert(STATEMENT); } while(0)

Note the repeated index which is an upper and lower index on the rst term in the right so we have an implied sum. Here the index ranges over the entire

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space-time, that is, = 0, ..., 2. Now let s write L0 in such a way that we peel of the = 25 term. Then L0 =

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