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The vector [vn , vd ] is the Earth relative velocity of point P represented in the instantaneous tangent plane. The third term in eqn. (3.95) is due to the rotation rate of the geographic frame with respect to the inertial frame and can be derived using the law of Coriolis. The rotation matrix Rg from p platform to geographic frame is de ned as Rg = [ ]2 = p cos( ) sin( ) sin( ) cos( )

But N does not divide either of these individual terms, since x 1 mod N Therefore N must have a nontrivial factor in common with each of (x + 1) and (x 1) In particular, gcd(N, x + 1) is a nontrivial factor of N Example Let N = 15 Then 42 1 mod 15, but 4 1 mod 15 Both gcd(4 1, 15) = 3 and gcd(4 + 1, 15) = 5 are nontrivial factors of 15 To complete the connection with periodicity, we need one further concept De ne the order of x modulo N to be the smallest positive integer r such that x r 1 mod N For instance, the order of 2 mod 15 is 4 Computing the order of a random number x mod N is closely related to the problem of nding nontrivial square roots, and thereby to factoring Here s the link.

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Figure 3-7

Lemma Let N be an odd composite, with at least two distinct prime factors, and let x be chosen uniformly at random between 0 and N 1 If gcd(x, N ) = 1, then with probability at least 1/2, the order r of x mod N is even, and moreover x r/2 is a nontrivial square root of 1 mod N The proof of this lemma is left as an exercise What it implies is that if we could compute the order r of a randomly chosen element x mod N , then there s a good chance that this order is even and that xr/2 is a nontrivial square root of 1 modulo N In which case gcd(x r/2 + 1, N ) is a factor of N Example.

Stepping into the Edit Parameter screen that supports the Commands, we can start to review the configuration of each of the Command buttons. The following illustration shows the initial view of the configuration for the Save View command button. These configuration settings are standard for most command options. You start by confirming the Caption, confirming whether or not you would like to have Quick Info, enabling the command, and selecting what the format would be in this case we will accept the Standard format.

Coaching at all levels in information technology is occurring more and more in organizations to improve the productivity of employees and reduce turnover Coaches help employees to identify the skills they need, to increase their job satisfaction, and to network with other employees Training is the most important ingredient in the success formula for computer professionals It is the lack of good training and development opportunities that causes individuals to become dead-ended early in their careers The first question that a job applicant should ask is, What kind of training and development will the company provide me if I accept this position The recent emphasis on training is due in part to the failure of college curricula in computer and information sciences to educate graduates in the high-demand skill areas Dramatic changes in technology and organizational structure require constant educational program evaluation and modification Curriculum change occurs too slowly.

If x = 2 and N = 15, then the order of 2 is 4 since 2 4 1 mod 15 Raising 2 to half this power, we get a nontrivial root of 1: 2 2 4 1 mod 15 So we get a divisor of 15 by computing gcd(4 + 1, 15) = 5 Hence we have reduced FACTORING to the problem of ORDER FINDING The advantage of this latter problem is that it has a natural periodic function associated with it: x N and x, and consider the function f (a) = xa mod N If r is the order of x, then f (0) = f (r) = f (2r) = = 1, and similarly, f (1) = f (r + 1) = f (2r + 1) = = x Thus f is periodic, with period r.

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