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This notation gives a finite presentation of the group 5'3, which is a wellknown symmetric group. It is important to note that a group G rnay have many different finite presentations. For example, Problem 6 asks the reader to show that s:j= ( x , y I Z''I , y2, (n:y) 2 ). Sometimes, thc relations in a finite presentation can be used t o rewrite cvery possible word into a canonical form. For example, using the finite prcsmtatiori S;( = (z,y I 2': y 2 , ( z ~ ) ~ ) ,see that the clernent zyzy = Is:, can we be rcwrittcm as z-'(xy;my) = :cP1 . lLy:,, which simplifics to yzy = zP1(:c3), whicli, in turn, simplifies to yxg = x2 and, finally, y x = :c2y. Using the fact that yx = x2y, along with the fact that z 3 = y2 = 1sZJ, can write cvery we word in 5 3 in the form zi:yj, where i E (0, 1 , 2 } arid j E (0, l}. Consequently, tlic group ~ : cim he viewed as t,Iie set, of elements { l s , , ~ , 2 2 , y , z y . x ' y } , j

Computer software also produces an NP chart. This is just a control chart of nP D, the number of defectives in a sample. The points, center line, and control limits for this chart are just multiples (times n) of the corresponding elements of a P chart. The use of an NP chart avoids the fractions in a P chart.

0.6 0.5 0.4 0.3 0.2 0.1 p

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can record a QuickTime format .mov le approximately 1 hour and 4 minutes long at standard-de nition resolution and approximately 4 hours and 16 minutes in low de nition. Canon recommends memory cards with at least an SD Speed Class Rating of 4 or higher for smoother transfer of video images. Because it s always a good idea to replace your memory cards every few years, now might be a great time to upgrade to faster and higher-capacity cards. You can edit and create any length movie you want by using thirdparty software such as Adobe Premiere Pro or Apple Final Cut Pro and editing sequences together, but it s only the individual continuous clips that are limited to these times.

together with the binary operation of concatenation, and subject t o the relations z3= y2 = zyzy = lss. We are now ready t o give a description of Arithmetica. Suppose that Alice and Bob want to establish a symmetric key. A finitely presented, infinite nonabelian group G is made public. Alice and Bob create their public keys to be subgroups of G, say,

16-8.2

U Chart (Control Chart for Defects per Unit)

It is sometimes necessary to monitor the number of defects in a unit of product rather than the fraction defective. Suppose that in the production of cloth it is necessary to control the number of defects per yard or that in assembling an aircraft wing the number of missing rivets must be controlled. In these situations we may use the control chart for defects per unit, or the U chart. Many defects-per-unit situations can be modeled by the Poisson distribution. If each sample consists of n units and there are C total defects in the sample, U C n

Sg = ( t o , t l , . . . t m - l ) ,

is the average number of defects per unit. A U chart may be constructed for such data. If the number of defects in a unit is a Poisson random variable with parameter , the mean and variance of this distribution are both . Each point on the chart is U, the average number of defects per unit from a sample of n units. Therefore, the mean of U is and the variance of U is n. UCL LCL 3 3 Bn Bn

(16-25)

respectively, and these are made public knowledge. The subgroups S A and Sg can be thought of as being the sets of all formal words in the alphabets si and t j respectively, with the binary operation of concatenation, subject t o the relations of G . Alice and Bob select their private keys

If there are m preliminary samples, and the number of defects per unit in these samples are U1, U2, . . . , Um, the estimator of the average number of defects per unit is U 1 m m ia Ui 1 (16-26)

(16-27)

an-1

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