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Windows 2000 enhances the SAM/LSA mechanism by integrating Kerberos 5.0 for network authentication. This third-party authentication system described in the previous chapter on UNIX, is a component of Windows 2000 Active Directory. Kerberos provides additional interoperability between Windows and a number of other vendor operating systems and applications that support the MIT Kerberos specification. Note that Kerberos is an authentication system and does not normally indicate authorization information. Microsoft has enhanced Kerberos by incorporating Windows access control information into an unused field within the Kerberos ticket. The field is ignored by MIT Kerberos systems and does not affect overall interoperability. Windows 2000 clients can be configured to use an MIT Kerberos security server for authentication. Under this configuration, authorization information is collected from Windows 2000 servers via Kerberos transitive trust relationships.

1.62. 1.63.

2 y2 The equation of the ellipse is x 2 + 2 = 1. We first want to define x as a function of y. a b x 2 + y 2 1, x Solve 2 b2 a 2 2 a y a2 y 2 2 2 x a b2 , x a b2

Next we obtain an integral representing the cross-sectional area of the tank. We take the positive solution and double the area, taking advantage of symmetry. a = 10; b = 5; x[y_]= a2 a2 y 2 ; b2

area[h_]= 2

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LIMIT POINTS, BOUNDS, BOLZANO WEIERSTRASS THEOREM 1.64. Given the set of numbers 1; 1:1; :9; 1:01; :99; 1:001; :999; . . . . (a) Is the set bounded (b) Does the set have a l.u.b. and g.l.b. If so, determine them. (c) Does the set have any limit points If so, determine them. (d) Is the set a closed set Ans. (a) Yes (b) l:u:b: 1:1; g:l:b: :9 (c) 1 (d) Yes Give the set :9; :9; :99; :99; :999; :999 answer the questions of Problem 64. Ans. (a) Yes (b) l:u:b: 1; g:l:b: 1 (c) 1; 1 (d) No Give an example of a set which has (a) 3 limit points, (b) no limit points.

x[y] y;

Summary

As a check, we can compute area[0], area[b], and area[2b]. The area enclosed by the ellipse x 2 + y 2 = 1 is ab. a2 b2 area[0] 0 area[b] 25 area[2b] 50

Since the tank has a uniform cross-section, its volume = length cross-sectional area. length = 20; volume[h_] = length * area[h]; To approximate the location on the stick that corresponds to 500 cubic feet, we draw the graph of volume[h]. Then we use FindRoot to obtain a more accurate value. (We cannot use NSolve, as in the previous problem, because volume[h] is a non-algebraic function.) Plot[volume[h], {h, 0, 2b}, AxesLabel {"h", "volume"}]

1.66. 1.67.

From the graph we observe that volume = 500 when h is near 2. FindRoot[volume[h] 500, {h, 2}] {h 2.1623}

9.21 An underground fuel tank is in the shape of an ellipsoid with semi-axes 6, 10, and 6 ft. (This problem, although more difficult than the previous problem, is somewhat more realistic.) To measure the amount of fuel in the tank, we insert a stick vertically through the center of the ellipsoid until it touches the bottom of the tank and measure how high the fuel level is on the stick. How far from the end of the stick should a mark be placed to indicate that only 500 cubic feet of fuel remain

The Windows NT operating system represents a layered architecture with partitioned user and kernel modes similar to those of UNIX. The Windows NT microkernel was based on the Mach microkernel.

2 2 y2 The equation of this ellipsoid is x 2 + 2 + z 2 = 1 with a = 6, b = 10, and c = 6. The intersection of the a b c 2 2 z2 ellipsoid with the plane z = z0 is the ellipse x 2 + y 2 = 1 02 whose area can be computed as a function a b c of z0. We then integrate with respect to z to obtain the volume.

(a) Prove that every point of the interval 0 < x < 1 is a limit point. (b) Are there are limit points which do not belong to the set in (a) Justify your answer. Let S be the set of all rational numbers in 0; 1 having denominator 2n , n 1; 2; 3; . . . . (a) Does S have any limit points (b) Is S closed (a) Give an example of a set which has limit points but which is not bounded. (b) Does this contradict the Bolzano Weierstrass theorem Explain.

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