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In the previous section on domains we talked about how domain user and group credentials and rights are stored in the domain controller's Security Account Manager (SAM) database. Rights are rules that govern access to all the resources in the domain. Rights can be defined per individual or to groups of individuals that share similar needs (Figure 3.12). Group rights have both local and global scope. Membership in a global group permits access to resources that reside in multiple domains. Global group rights are governed by the interdomain trust relationships. Local groups are restricted to resources within a single domain. Access may also be limited by local SAM information that exists on nondomain controllers and NT workstations.

Plot[s[x], {x, 0, 3}, AxesLabel {"x", "s(x)"}]

CHAP. 1]

s (x)

8 6 4 2 x

The graph shows s(2) 5.

COUNTABILITY 1.58. (a) Prove that there is a one to one correspondence between the points of the interval 0 @ x @ 1 and 5 @ x @ 3. (b) What is the cardinal number of the sets in (a) Ans. (b) C, the cardinal number of the continuum. (a) Prove that the set of all rational numbers is countable. (b) What is the cardinal number of the set in (a) Ans. (b) Fo Prove that the set of (a) all real numbers, (b) all irrational numbers is non-countable.

- 52 -

solution = FindRoot[s[x] 5, {x, 2}] {x 2.08401} x = x /. solution;

{x, f[x]}

{2.08401, 4.34308}

1.60. 1.61.

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9.19 A mixing bowl is a hemisphere of radius 5 in. Determine the height of 100 cubic inches of liquid.

The Local Security Authority (LSA) and Net Login service are responsible for verifying access rights to resources and authenticating login requests respectively. The LSA validates each request by checking the user's Resource Identifier (RID) and the domain's Security Identifier (SID) against the access permissions of the requested resource. Resource permissions are defined by Access Control Lists (ACLs).

The equation of the hemisphere in three dimensions is x 2 + y 2 + z 2 = 25, z 0. Its intersection with the plane z = z0 is the circle x 2 + y 2 = 25 z02, whose radius r = 25 z02 and whose area r 2 = (25 z02 ) . Integrating with respect to z, the volume of the shaded region is V (h) = been replaced by z for convenience.) v[h_] = o

As a check, v[5] should give the volume of the hemisphere. T he volume of the hemisphere is 2 r3 = 2 (53 ) = 250 . 3 3 3 v[5] 250 3 Plot v as a function of h. Plot[v[h], {h, 0, 5}, AxesLabel {"h", "v[h]"}]

The intersection of two sets A and B, denoted by A \ B or AB, is the set consisting of all elements belonging to both A and B. Prove that if A and B are countable, so is their intersection. Prove that a countable set of countable sets is countable. Prove that the cardinal number of the set of points inside a square is equal to the cardinal number of the sets of points on (a) one side, (b) all four sides. (c) What is the cardinal number in this case (d) Does a corresponding result hold for a cube Ans. (c) C

Since v[h] is a polynomial function, we can use NSolve to determine the approximate solution to the problem. (From the graph, it looks like h is near 3.) NSolve[v[h] 100] {{h 2.34629}, {h 2.79744}, {h 14.5489}} Obviously, the only realistic solution is h = 2.79744 in.

9.20 An underground fuel tank is in the shape of an elliptical cylinder. The tank has length l = 20 ft, semi-major axis a = 10 ft and semi-minor axis b = 5 ft. b To measure the amount of fuel in the tank, we insert a stick vertically through the center of the cylinder until it touches the bottom of the tank and measure how high the a a 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 h feet of fuel remain

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