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where l(x) is the length of the codeword associated with x. Without loss of generality, we can assume that the D-ary alphabet is D = {0, 1, . . . , D 1}. Some examples of codes follow. Example 5.1.1 Let X be a random variable with the following distribution and codeword assignment: Pr(X = 1) = 1 , codeword C(1) = 0 2 Pr(X = 2) = 1 , codeword C(2) = 10 4 Pr(X = 3) = 1 , codeword C(3) = 110 8 Pr(X = 4) = 1 , codeword C(4) = 111. 8 The entropy H (X) of X is 1.75 bits, and the expected length L(C) = El(X) of this code is also 1.75 bits. Here we have a code that has the same average length as the entropy. We note that any sequence of bits can be uniquely decoded into a sequence of symbols of X. For example, the bit string 0110111100110 is decoded as 134213. Example 5.1.2 variable: Consider another simple example of a code for a random Pr(X = 1) = 1 , codeword C(1) = 0 3 Pr(X = 2) = 1 , codeword C(2) = 10 3 Pr(X = 3) =

0 0 0

scaffolds and culture in bioreactors, the structural and functional properties of tissue engineered cartilage were signi cantly enhanced when a human IGF-I cDNA was transfected (Table 5.1). Transgene expression was maintained over 28 days of in vitro cultivation followed by an additional 10 days either in vitro or in vivo [Madry et al., 2001]. In these experiments, the human IGF-I gene served as a model to demonstrate the potential bene ts of gene transfer for tissue engineering. Genetically modi ed cartilaginous constructs overexpressing a growth or transcription factor gene may be used as a tissue substitute that simultaneously provides a stimulus for repair.

Analysis of Variance Source Regression Residual Error Total DF 3 73 76 SS 12246.2 2750.5 14996.8 MS 4082.1 37.7 F 108.34 P 0.000

where (2.85) follows from Jensen s inequality. Since log t is a strictly concave function of t, we have equality in (2.85) if and only if q(x)/p(x) is constant everywhere [i.e., q(x) = cp(x) for all x]. Thus, x A q(x) = c x A p(x) = c. We have equality in (2.87) only if x A q(x) = x X q(x) = 1, which implies that c = 1. Hence, we have D(p||q) = 0 if and only if p(x) = q(x) for all x. Corollary (Nonnegativity of mutual information) variables, X, Y , I (X; Y ) 0, with equality if and only if X and Y are independent. Proof: I (X; Y ) = D(p(x, y)||p(x)p(y)) 0, with equality if and only if p(x, y) = p(x)p(y) (i.e., X and Y are independent). For any two random (2.90)

If mass transfer controls aerosol growth, this immediately gives the equilibration timescale between a continuum-sized aerosol and its surrounding mixture, te = (4 Dv M1 ) 1 , Cns 1 (3.85)

Available in external and internal versions, these devices allow you to copy data onto write-once discs or rewriteable discs There are many types of burners and discs available, which often can be confusing Here is a quick guideline: CD-R: The original recordable CD format Stores 650 to 700 megabytes of data on a single, write-once disc (which means the data is permanently burned to the disc) CD-RW: Has the same capacity as CD-R discs, but can be erased and reused hundreds or even thousands of times DVD-R: Stores up to 47 gigabytes of data on DVD discs that are identical in size to the DVD movies you watch at home Most of these devices are backward compatible with CD-R and CD-RW, meaning they can read and write data from the aforementioned CD discs.

Even for an experienced Linux integrator, this is a cripplingly limited command set, and you can do nothing useful with these commands alone. Should you find yourself in this situation, there are some quick workarounds. If you already have a Parallel ATA (PATA) CD/DVD optical drive, use that instead during the build phase, and then reintroduce the SATA drive once the system is established. Another option using the installer is described later in this chapter, in a section that deals with manual installation.

DVD-RW: Has same storage capacity as DVD-R, but can be erased and reused hundreds or even thousands of times These devices usually have backward compatibility with CD-R and CD-RW DVD+R: Here is where it gets tricky Notice that this type of DVD has a plus sign (+) rather than a minus sign (-) The plus (+) discs cannot be used in minus (-) drives, and vice versa Other than that, both formats have the same 47 gigabyte storage capacity DVD+RW: Same restriction about not mixing plus discs and minus discs, and has the same storage capacity as the other DVD formats Can be erased and reused hundreds or even thousands of times Multi-format DVD burners: These devices bypass the war between the plus discs and the minus discs by allowing you to use any of the four kinds: -R, -RW, +R, or +RW.