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Windows domains are similar to workgroups in that they represent a group of users, services and computers that collectively share a common namespace. Domains differ from workgroups, in that a domain enforces a common security policy and management structure across its membership. Account directory information is stored in a secure central repository called the Security Accounts Manager (SAM). The SAM database is maintained on the primary domain controller and replicated on each of the backup domain controllers (see 3). What this means for the system administrator is that there is a central location for managing the accounts, passwords, access controls and policies for the entire domain. End-users benefit from the domain model in by gaining access to all domain resources via a single logon. Domains improve access security over workgroups because client authentication is performed only once using a secure communication channel with the PDC during domain logon. Once authenticated, client access to resources is governed by applying access controls based on the client's group memberships. Domain groups are a means of categorizing users with similar needs and work profiles. In contrast, each workgroup client maintains a persistent local cache of accounts and passwords required to access network resource. A separate credential cache is created for each active user and stored in the \windows directory as <username>.PWL. Windows opens the cache file using the associated user's logon password as a key. During connection to a network resource, each entry in the user's credential cache is tried until authentication is either accepted or denied. Credentials stored in these cache files are extremely vulnerable to security threats due to the weak encryption employed and their persistent nature across logons by different workstation users.

THE GAMMA FUNCTION 15.28. Evaluate Ans. (a) 7 ; 2 4 3 b 16=105; b c 3 3=2 ; 9=2

n = 5; a = TableRandomInteger[9], {i, 1, n}, {j, 1, n}]; a // MatrixForm 4 5 1 7 0 6 3 6 9 9 5 0 9 7 4 3 5 7 1 5 3 6 7 2 6

matrixofminors = Minors[a]; MatrixForm[matrixofminors, TableAlignments Right] 159 174 171 1350 1584 270 669 549 342 4 96 561 339 8 3231 3333 438 283 216 48 293 78 143 168 497 246

140

3 3=2 8

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signs = Table[( 1)^(i + j), {i, 1, n}, {j, 1, n}]; cofactors = matrixofminors * signs; MatrixForm[cofactors, TableAlignments Right] 171 159 174 283 216 1350 1584 270 140 48 669 549 342 293 78 561 339 96 143 168 7 3231 3333 438 497 246 i=3 (* we expand using the third row *)

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determinant = a[[n i + 1,n j+ 1]]* cofactors[[i,j]] ]

2082 Det[a] 2082

c 1=2 3=2 5=2 .

The (i, j)th element of Minors[a] gives the determinant of the matrix obtained by deleting row n i + 1 and column n j + 1.

1 1 1 2 2 2 12.19 Let x = 3 . Compute x T x = [1 2 3 4 5] 3 and xx T = 3 [1 2 3 4 5] . 4 4 4 5 5 5

Domains also offer a means for authorizing access to resources residing in separate domains through trusts. A trust relationship is a link that logically combines two domains into an administrative unit. The trust relationship may be applied one way (domain A trusts domain B, but domain B does not trust domain A) or two ways (domain A and domain B trust each other).

a 30;

xTx is the dot product of the vector x with itself. xxT, however, is a 5 5 matrix. x.x 55 Outer[Times, x, x] // MatrixForm 1 2 3 4 2 4 6 8 3 6 9 12 4 8 12 16 5 10 15 20 5 10 15 20 25

Mathematica offers a variety of matrix manipulation commands that are quite useful when working problems in linear algebra. Since matrices are actually lists, many of the commands are the same as described in 3.

Join[list1, list2] combines the two lists list1 and list2 into one list consisting of the elements from list1 and from list2. For matrices, this has the effect of placing the rows of list2 under the rows of list1. Join[list1, list2, n] joins the objects at level n in each list. If n = 2, this has the effect of placing the columns of list2 to the right of the columns of list1. ArrayFlatten[{{m11, m12,...}, {m21, m22,...},...}]creates a single flattened matrix from a matrix of matrices mi j. All the matrices in the same row must have the same first dimension, and all the matrices in the same column must have the same second dimension.

1 15.29. Evaluate Ans. (a)

list = {{a, b, c, d}, {e, f, g, h}, {i, j, k, l}}; list[[2]][[3]] g list[[2, 3]] g Part[list, 2, 3] g Part[Part[list, 2], 3] g

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