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"" -1 262 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 263 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 264 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 265 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 266 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "" -1 267 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 206 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 207 "" 0 "" {TEXT -1 19 "Graphs and Matrices" }}{PARA 208 "" 0 "" {TEXT -1 189 "A (DIRECTED) GRAPH is a collection of points, called VERTICES, to gether with a set of directed connections, called EDGES, between the v ertices. For example, consider the following graph G:" }}{PARA 209 "" 0 "" {TEXT -1 67 "We can see that the vertex set for this graph could be labelled as\n" }{TEXT -1 8 " \n" }{TEXT -1 41 " \+ V = \{v1, v2, v3, v4\}.\n" }{TEXT -1 1 "\n" }{TEXT -1 279 "The e dges of the graph are designated by ordered pairs, where the first ent ry of the pair is the source of the edge and the next entry is the des tination of the edge. For example, in our picture above we see that t here are 6 edges, which we could designate in order as the set \n" } {TEXT -1 4 " \n" }{TEXT -1 85 " E = \{ (v1, v2), (v2, v3 ), (v3, v4), (v2, v4), (v1, v3), (v1, v4) \}.\n" }{TEXT -1 1 "\n" }{TEXT -1 589 "Notice that the vertex set V and the edge set E com pletely describe the graph G. Here is a matrix associated with the gr aph, called the INCIDENCE MATRIX of the graph. It, too, completely de scribes the graph. What we do is label the columns of the matrix with the vertices, and the rows by the edges. We then insert in each row -1 under the source vertex and +1 under the destination vertex, and \+ 0's everywhere else. For example, with the above graph G, we have the following incidence matrix (I have put the row and column labels in, \+ but this is not really part of the matrix):\n" }{TEXT -1 1 "\n" }{TEXT -1 35 " v1 v2 v3 v4\n" }{TEXT -1 37 " e1 \+ -1 1 0 0\n" }{TEXT -1 37 " e2 0 -1 \+ 1 0\n" }{TEXT -1 37 " e3 0 0 -1 1\n" } {TEXT -1 37 " e4 0 -1 0 1\n" }{TEXT -1 37 " \+ e5 -1 0 1 0\n" }{TEXT -1 37 " e6 -1 \+ 0 0 1\n" }{TEXT -1 1 "\n" }{TEXT -1 124 "This is a 6 X 4 m atrix, and we could draw a picture of the graph using the information \+ in it. Designate this matrix as A.\n" }{TEXT -1 1 "\n" }{TEXT -1 156 "These ideas have many applications. For example electrical circu its are just graphs with extra features. They are also very useful in computer science. \n" }{TEXT -1 1 "\n" }{TEXT -1 437 "Following up o n the idea of electrical curcuits, we could assign to each vertex a po tential. Suppose that x = (x1, x2, x3, x4) is our (column) vector of potentials. Then the matrix-vector product A &* x has a very inter esting interpretation: each entry represents the jump in potential as you move across the corresponding edge. Let's try it out. We'll st art by defining the incidence matrix A and a given potential vector \+ x." }}{PARA 210 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 211 "> " 0 "" {MPLTEXT 1 0 58 "A := matrix([[-1,1,0,0],[0,-1,1,0],[0,0,-1 ,1],[0,-1,0,1],\n" }{MPLTEXT 1 0 24 "[-1,0,1,0],[-1,0,0,1]]);" }} {PARA 212 "> " 0 "" {MPLTEXT 1 0 24 "x := vector([2,4,-5,6]);" }} {PARA 213 "> " 0 "" {MPLTEXT 1 0 19 "b := evalm(A &* x);" }}{PARA 214 "" 0 "" {TEXT -1 230 "Now check it out: the potential jump across edg e 6, for example, should be the potential at v4 minus potential at v1, that is, 6 - 2. This is exactly what the 6th entry of the vector A \+ &* x is. Likewise for the other entries.\n" }{TEXT -1 1 "\n" }{TEXT -1 977 "Now here is a fascinating idea: what if we have a situation w here vertices represent competing groups, which we shall call teams, a nd \"potential\" represents the strength of a team. (Call the potenti al of a team its \"power rating\" if you prefer!) In particular, if \+ two teams have a match, one would expect the jump in potential from on e team to another to be analogous to the jump in score between the two teams. For example, if team 2 had a potential rating of 100 and team 4 had a potential of 91, one might expect that the outcome of a match between these teams would be that team 2 should win by 9 points. Of \+ course, this doesn't say what the final score would be; simply that th e difference between the teams will be 9 points in favor of team 2. N ow let's expand on this interpretation by thinking of an edge connecti ng two vertices as representing the fact that the teams represented by the vertices have, in fact, played each other, and that the outcome i s known.\n" }{TEXT -1 1 "\n" }{TEXT -1 921 "Let's go back to the examp le graph G above. Suppose that the vertices represent teams 1, 2, 3 , and 4, and each edge represents a match. Suppose a correct potenti al vector for these matches is the vector x given above. Notice that \+ x is not unique. The vector [5, 7, -2, 9] would work just as well, \+ since we get it just by shifting every potential value up 3 units. Thu s differences between the potentials remain unchanged. Now we can see what the difference in scores must have been in each match. Simply f orm the vector b = A &* x. The first coordinate of b represents \+ the difference of team 2's score minus team 1's scores, since the firs t edge goes from vertex 1 to 2. Likewise, the second coordinate of b represents the difference of team 3's score minus that of team 2. A nd so forth. Here is the vector b using a shifted potential vector. \+ Notice that there is no change from what we had above:" }}{PARA 215 " > " 0 "" {MPLTEXT 1 0 24 "x := vector([5,7,-2,9]);" }}{PARA 216 "> " 0 "" {MPLTEXT 1 0 20 "b := evalm( A &* x);" }}{PARA 217 "" 0 "" {TEXT -1 103 "Thus the third entry says that team 4's score was 11 points mo re than that of team 3 in their match. \n" }{TEXT -1 1 "\n" }{TEXT -1 647 "Now consider the possibility that we observe outcomes and want to use these to determine appropriate potential values. These are ju st the entries of the right hand side vector b. What is very likely \+ is that they are inconsistent, in the sense that no assignment of pote ntials will give the correct differences. In other words, the system A &* x = b is inconsistent. This inconsistency is a reflection of the fact that teams play better at some times than others, and upsets do occur. For example, we may have calculated the differences in sc ores in all the matches represented by the 6 edges of our example and \+ come up with this vector:" }}{PARA 218 "> " 0 "" {MPLTEXT 1 0 32 "b := matrix(6,1,[2,7,-5,6,0,3]);" }}{PARA 219 "" 0 "" {TEXT -1 173 "Now he re's the question: is there an assignment of potentials that gives th is outcome? In other words, is the system A &* x = b consistent? Let 's ask Maple for solutions:" }}{PARA 220 "> " 0 "" {MPLTEXT 1 0 15 "li nsolve(A, b);" }}{PARA 221 "" 0 "" {TEXT -1 217 "Notice the absence of any output? This is what happens when Maple can't come up with any s olution. So the system is inconsistent. Here's another way to check \+ it out. Form the augmented matrix and look at its RREF:" }}{PARA 222 "> " 0 "" {MPLTEXT 1 0 21 "augA := augment(A,b);" }}{PARA 223 "> " 0 " " {MPLTEXT 1 0 11 "rref(augA);" }}{PARA 224 "" 0 "" {TEXT -1 116 "As w e can see, the system must be inconsistent since there is a pivot in t he last column of the augmented matrix. \n" }{TEXT -1 1 "\n" }{TEXT -1 339 "So how should we assign potentials to each team. There is a w ay. We can try to find a vector of potentials x that is in some sen se as close as possible to an honest solution, even though no real sol ution exists. We have seen this idea in class. It is the \"least squ ares\" idea. Remember that what we do is form the NORMAL EQUATIONS:\n " }{TEXT -1 1 "\n" }{TEXT -1 54 " transpose(A) &* A &* x = tr anspose(A) &* b.\n" }{TEXT -1 1 "\n" }{TEXT -1 585 "This system will a lways be consistent. If A has full column rank, then it will be inver tible and there will be a unique solution. In our case we won't have \+ uniqueness unless we add an equation that specifies a value for some n ode. A solution to this system is called a \"least squares solution\" to the problem. Here is how to use it on the example above. First, \+ let's add a row to both A and b that says that the value for the last node will be 0. In case we want floating point numbers instead of fr actions, we should enclose the linsolve that gives xsoln inside a \"ev alf( )\"." }}{PARA 225 "> " 0 "" {MPLTEXT 1 0 35 "A := stack(A, matr ix([[0,0,0,1]]));" }}{PARA 226 "> " 0 "" {MPLTEXT 1 0 27 "b := stack(b , vector([0]));" }}{PARA 227 "> " 0 "" {MPLTEXT 1 0 32 "AtA := evalm(t ranspose(A) &* A);" }}{PARA 228 "> " 0 "" {MPLTEXT 1 0 32 "Atb := eval m(transpose(A) &* b);" }}{PARA 229 "> " 0 "" {MPLTEXT 1 0 28 "xsoln := linsolve(AtA, Atb);" }}{PARA 230 "" 0 "" {TEXT -1 39 "DATA FOR THE BI G EIGHT FOOTBALL GRAPH:\n" }{TEXT -1 1 "\n" }{TEXT -1 75 "Here is how \+ we will identify the Big Eight teams as vertices of our graph:\n" } {TEXT -1 1 "\n" }{TEXT -1 29 "1: Colorado University (CU)\n" }{TEXT -1 32 "2: Iowa State University (ISU)\n" }{TEXT -1 34 "3: Kansas Sta te University (KSU)\n" }{TEXT -1 27 "4: Kansas University (KU)\n" } {TEXT -1 30 "5: Missouri University (MU)\n" }{TEXT -1 30 "6: Nebras ka University (NU) \n" }{TEXT -1 36 "7: Oklahoma State University (OS U)\n" }{TEXT -1 29 "8: Oklahoma University (OU)\n" }{TEXT -1 1 "\n" } {TEXT -1 67 "We will always list the home team first in the scores tha t follow.\n" }{TEXT -1 1 "\n" }{TEXT -1 29 "Games played as of 10/10/9 4:\n" }{TEXT -1 1 "\n" }{TEXT -1 14 "ISU 6, OU 34\n" }{TEXT -1 15 "KU 13, KSU 21\n" }{TEXT -1 14 "NU 32, OSU 3\n" }{TEXT -1 14 "MU 23, C U 38\n" }{TEXT -1 1 "\n" }{TEXT -1 29 "Games played as of 10/15/94:\n" }{TEXT -1 18 "KSU 6 , NU 17 \n" }{TEXT -1 17 "ISU 23, KU 41\n" }{TEXT -1 17 "OSU 15, MU 24\n" }{TEXT -1 16 "CU 45, OU 7\n" } {TEXT -1 1 "\n" }{TEXT -1 29 "Games played as of 10/22/94:\n" }{TEXT -1 14 "NU 42, MU 7\n" }{TEXT -1 16 "ISU 31, OSU 31\n" }{TEXT -1 16 "KSU 21, CU 35\n" }{TEXT -1 16 "OU 20, KU 17\n" }{TEXT -1 1 "\n" }{TEXT -1 29 "Games played as of 10/29/94:\n" }{TEXT -1 12 "NU 24, CU \+ 7\n" }{TEXT -1 14 "OU 20, KSU 37\n" }{TEXT -1 14 "ISU 20, MU 34\n" } {TEXT -1 14 "KU 24, OSU 14\n" }{TEXT -1 1 "\n" }{TEXT -1 28 "Games pla yed as of 11/5/94:\n" }{TEXT -1 13 "NU 45, KU 17\n" }{TEXT -1 15 "KSU \+ 38, ISU 20\n" }{TEXT -1 13 "OU 30, MU 13\n" }{TEXT -1 13 "CU 17, OSU 3 \n" }{TEXT -1 1 "\n" }{TEXT -1 29 "Games played as of 11/12/94:\n" } {TEXT -1 14 "ISU 12, NU 28\n" }{TEXT -1 13 "KU 26, CU 51\n" }{TEXT -1 14 "MU 18, KSU 21\n" }{TEXT -1 14 "OSU 14, OU 33\n" }{TEXT -1 1 "\n" } {TEXT -1 75 "Here is how we will identify the Big Eight teams as verti ces of our graph:\n" }{TEXT -1 1 "\n" }{TEXT -1 29 "1: Colorado Unive rsity (CU)\n" }{TEXT -1 32 "2: Iowa State University (ISU)\n" }{TEXT -1 34 "3: Kansas State University (KSU)\n" }{TEXT -1 27 "4: Kansas U niversity (KU)\n" }{TEXT -1 30 "5: Missouri University (MU)\n" } {TEXT -1 30 "6: Nebraska University (NU) \n" }{TEXT -1 36 "7: Oklaho ma State University (OSU)\n" }{TEXT -1 29 "8: Oklahoma University (OU )\n" }{TEXT -1 1 "\n" }{TEXT -1 67 "We will always list the home team \+ first in the scores that follow.\n" }}{PARA 231 "" 0 "" {TEXT -1 384 " As above, I am going to make the system a \"little more unique\". Af ter all, given any assignment of potentials, any uniform translation o f the potentials does exactly the same job as far as the system of equ ations is concerned. Therefore, we will fix a reference point. It re ally doesn't make any difference how, but we will guess at the lowest \+ team and set their potential to 0. " }}{PARA 232 "" 0 "" {TEXT -1 13 "After week 1:" }}{PARA 233 "> " 0 "" {MPLTEXT 1 0 36 "A := matrix(5,8 ,[[0,1,0,0,0,0,0,0],\n" }{MPLTEXT 1 0 37 " [0,-1,0,0,0 ,0,0,1],\n" }{MPLTEXT 1 0 37 " [0,0,1,-1,0,0,0,0],\n" }{MPLTEXT 1 0 37 " [0,0,0,0,0,-1,1,0],\n" }{MPLTEXT 1 0 38 " [1,0,0,0,-1,0,0,0]]);" }}{PARA 234 "> " 0 "" {MPLTEXT 1 0 43 "b := matrix(5,1,[0,34-6,21-13,3-32,38-23]);" }}{PARA 235 "" 0 "" {TEXT -1 13 "After week 2:" }}{PARA 236 "> " 0 "" {MPLTEXT 1 0 40 "newA := matrix(4,8,[[0,0,-1,0,0,1,0,0],\n" }{MPLTEXT 1 0 37 " [0,-1,0,1,0,0,0,0],\n" }{MPLTEXT 1 0 37 " \+ [0,0,0,0,1,0,-1,0],\n" }{MPLTEXT 1 0 38 " [-1,0,0 ,0,0,0,0,1]]);" }}{PARA 237 "> " 0 "" {MPLTEXT 1 0 44 "newb := matrix( 4,1,[17-6,41-23,24-15,7-45]);" }}{PARA 238 "> " 0 "" {MPLTEXT 1 0 19 " A := stack(A,newA);" }}{PARA 239 "> " 0 "" {MPLTEXT 1 0 20 "b := stack (b, newb);" }}{PARA 240 "" 0 "" {TEXT -1 13 "After week 3:" }}{PARA 241 "> " 0 "" {MPLTEXT 1 0 40 "newA := matrix(4,8,[[0,0,0,0,-1,1,0,0], \n" }{MPLTEXT 1 0 37 " [0,1,0,0,0,0,-1,0],\n" } {MPLTEXT 1 0 37 " [-1,0,1,0,0,0,0,0],\n" }{MPLTEXT 1 0 38 " [0,0,0,-1,0,0,0,1]]);" }}{PARA 242 "> " 0 "" {MPLTEXT 1 0 45 "newb := matrix(4,1,[42-7,31-31,21-35,20-17]);" }} {PARA 243 "> " 0 "" {MPLTEXT 1 0 19 "A := stack(A,newA);" }}{PARA 244 "> " 0 "" {MPLTEXT 1 0 20 "b := stack(b, newb);" }}{PARA 245 "" 0 "" {TEXT -1 13 "After week 4:" }}{PARA 246 "> " 0 "" {MPLTEXT 1 0 40 "new A := matrix(4,8,[[-1,0,0,0,0,1,0,0],\n" }{MPLTEXT 1 0 37 " \+ [0,0,-1,0,0,0,0,1],\n" }{MPLTEXT 1 0 37 " [0,1,0, 0,-1,0,0,0],\n" }{MPLTEXT 1 0 38 " [0,0,0,1,0,0,-1,0]] );" }}{PARA 247 "> " 0 "" {MPLTEXT 1 0 45 "newb := matrix(4,1,[24-7,20 -37,20-34,24-14]);" }}{PARA 248 "> " 0 "" {MPLTEXT 1 0 19 "A := stack( A,newA);" }}{PARA 249 "> " 0 "" {MPLTEXT 1 0 20 "b := stack(b, newb);" }}{PARA 250 "" 0 "" {TEXT -1 13 "After week 5:" }}{PARA 251 "> " 0 "" {MPLTEXT 1 0 40 "newA := matrix(4,8,[[0,0,0,-1,0,1,0,0],\n" }{MPLTEXT 1 0 37 " [0,-1,1,0,0,0,0,0],\n" }{MPLTEXT 1 0 37 " \+ [0,0,0,0,-1,0,0,1],\n" }{MPLTEXT 1 0 38 " \+ [1,0,0,0,0,0,-1,0]]);" }}{PARA 252 "> " 0 "" {MPLTEXT 1 0 45 "newb : = matrix(4,1,[45-17,38-20,30-13,17-3]);" }}{PARA 253 "> " 0 "" {MPLTEXT 1 0 19 "A := stack(A,newA);" }}{PARA 254 "> " 0 "" {MPLTEXT 1 0 20 "b := stack(b, newb);" }}{PARA 255 "" 0 "" {TEXT -1 13 "After w eek 6:" }}{PARA 256 "> " 0 "" {MPLTEXT 1 0 39 "newA := matrix(4,8,[[0, 0,0,0,0,0,0,0],\n" }{MPLTEXT 1 0 36 " [0,0,0,0,0,0,0,0 ],\n" }{MPLTEXT 1 0 36 " [0,0,0,0,0,0,0,0],\n" } {MPLTEXT 1 0 37 " [0,0,0,0,0,0,0,0]]);" }}{PARA 257 "> " 0 "" {MPLTEXT 1 0 30 "newb := matrix(4,1,[0,0,0,0]);" }}{PARA 258 " > " 0 "" {MPLTEXT 1 0 19 "A := stack(A,newA);" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 20 "b := stack(b, newb);" }}{PARA 260 "" 0 "" {TEXT -1 13 "After week 7:" }}{PARA 261 "> " 0 "" {MPLTEXT 1 0 39 "newA := matr ix(4,8,[[0,0,0,0,0,0,0,0],\n" }{MPLTEXT 1 0 36 " [0,0, 0,0,0,0,0,0],\n" }{MPLTEXT 1 0 36 " [0,0,0,0,0,0,0,0], \n" }{MPLTEXT 1 0 37 " [0,0,0,0,0,0,0,0]]);" }}{PARA 262 "> " 0 "" {MPLTEXT 1 0 30 "newb := matrix(4,1,[0,0,0,0]);" }} {PARA 263 "> " 0 "" {MPLTEXT 1 0 19 "A := stack(A,newA);" }}{PARA 264 "> " 0 "" {MPLTEXT 1 0 20 "b := stack(b, newb);" }}{PARA 265 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 266 "" 0 "" {TEXT -1 1 "\n" }{TEXT -1 10 "E XERCISE:\n" }}{PARA 267 "" 0 "" {TEXT -1 1 "\n" }{TEXT -1 858 "Read th is notebook. Then enter the correct data for \"After week 6\" in the \+ appropriate cells. It is the last bit of data given above under the h eading of BIG 8 DATA. How you order the edges is up to you. Just ma ke sure that the equation that results correctly reflects the differen ce in score between the teams that played each other. Now imitate th e procedure of the first part of this notebook to set up the normal eq uations for this problem and obtain a power rating for each Big 8 team . Based on this, make a prediction as to the scores of the remaining \+ games. At the end of Thanksgiving week I will add the scores of the r emaining games into our Big8Data file. Use this to come up with a fin al power rating of the Big 8 teams. Remember to click your way throug h the preceding cells, including the \"with(linalg)\" at the start of \+ the notebook. \n" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"';F &\"\"%E\\[l96$\"\"&F&!\"\"6$F&F&F-6$F'\"\"$\"\"!6$\"\"#F&F16$F'F3F16$F 0F)F&6$F&F3F&6$F'F)F&6$F&F)F16$F,F3F16$F)F&F16$F,F0F&6$F,F)F16$F)F3F-6 $F0F&F16$F3F)F16$F)F0F16$F&F0F16$F0F3F16$F3F0F&6$F'F&F-6$F)F)F&6$F0F0F -6$F3F3F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"%E\\[l%F& \"\"#F)F'\"\"$!\"&F'\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\" \"\"\"'E\\[l'F&\"\"#F)!\"*\"\"$\"#6\"\"%F)\"\"&!\"(F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"%E\\[l%F&\"\"&\"\"#\"\"(\"\"$!\"#F '\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6#;\"\"\"\"\"'E\\[l'F&\"\" #F)!\"*\"\"$\"#6\"\"%F)\"\"&!\"(F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 " =6\"6$;\"\"\"\"\"';F&F&E\\[l'6$\"\"&F&\"\"!6$F&F&\"\"#6$F.F&\"\"(6$\" \"%F&F'6$\"\"$F&!\"&6$F'F&F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$; \"\"\"\"\"';F&\"\"&E\\[l?6$F)F&!\"\"6$F&F&F,6$F'\"\"$\"\"!6$\"\"#F&F06 $F'F2F06$F/\"\"%F&6$F2F)\"\"(6$F5F)F'6$F&F2F&6$F'F5F&6$F&F5F06$F)F2F06 $F5F&F06$F)F/F&6$F)F5F06$F5F2F,6$F/F&F06$F2F5F06$F/F)!\"&6$F5F/F06$F'F )F/6$F&F/F06$F/F2F06$F2F/F&6$F&F)F26$F'F&F,6$F5F5F&6$F)F)F06$F/F/F,6$F 2F2F," }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"';F&\"\"&E\\[l ?6$F)F&\"\"!6$F&F&F&6$F'\"\"$F,6$\"\"#F&F,6$F'F1F,6$F/\"\"%!\"\"6$F1F) F,6$F4F)F&6$F&F1F,6$F'F4F,6$F&F4F56$F)F1F,6$F4F&F,6$F)F/F,6$F)F4F,6$F4 F1F,6$F/F&F,6$F1F4F56$F/F)F,6$F4F/F,6$F'F)F,6$F&F/F,6$F/F1F,6$F1F/F,6$ F&F)F,6$F'F&F,6$F4F4F,6$F)F)F,6$F/F/F&6$F1F1F&" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 11 "" 1 "" {XPPMATH 20 "= 6\"6$;\"\"\"\"\"%F%E\\[l16$F&F&\"\"$6$\"\"#F&!\"\"6$F*F'F-6$F&F,F-6$F& F'F-6$F'F&F-6$F'F,F-6$F*F&F-6$F,F'F-6$F'F*F-6$F&F*F-6$F*F,F-6$F,F*F-6$ F'F'F*6$F*F*F*6$F,F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\" \"\"%;F&F&E\\[l%6$F&F&!\"&6$\"\"#F&!#66$F'F&F'6$\"\"$F&\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&F&E\\[l%6$F&F&,&#!\"*F'F& &&I#_tGF#6#F&F1F&6$\"\"#F&,&#!#:F'F&F.F&6$F'F&F.6$\"\"$F&,&F3F&F.F&" } }{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"&;F&\"\")E\\[lI6$F'F&F &6$F&F)\"\"!6$F'\"\"(F-6$F&F&F-6$\"\"#\"\"'F-6$\"\"$F/F-6$F2F&F-6$F5\" \"%!\"\"6$F2F'F-6$F8F'F-6$F&F2F&6$F&F8F-6$F5F)F-6$F'F2F-6$F8F&F-6$F5F3 F-6$F'F)F-6$F'F5F-6$F&F/F-6$F'F8F-6$F&F3F-6$F8F/F&6$F8F2F-6$F5F&F-6$F2 F8F-6$F5F'F-6$F8F5F-6$F2F)F&6$F&F5F-6$F8F)F-6$F2F/F-6$F5F2F-6$F2F5F-6$ F8F3F96$F'F3F-6$F&F'F-6$F8F8F-6$F'F'F96$F5F5F&6$F2F2F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"&;F&F&E\\[l&6$F'F&\"#:6$F&F&\"\"!6$ \"\"#F&\"#G6$\"\"%F&!#H6$\"\"$F&\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&\"\")E\\[lA6$F&F)\"\"!6$F&F&F,6$\"\"#\"\"'F,6$\" \"$\"\"(!\"\"6$F/F&F,6$F2F'F,6$F/\"\"&F,6$F'F8F,6$F&F/F,6$F&F'F,6$F2F) F,6$F'F&F46$F2F0F,6$F&F3F,6$F&F0F&6$F'F3F,6$F'F/F,6$F2F&F,6$F/F'F&6$F2 F8F&6$F'F2F,6$F/F)F,6$F&F2F46$F'F)F&6$F/F3F,6$F2F/F,6$F/F2F,6$F'F0F,6$ F&F8F,6$F'F'F,6$F2F2F,6$F/F/F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$ ;\"\"\"\"\"%;F&F&E\\[l%6$F&F&\"#66$\"\"#F&\"#=6$F'F&!#Q6$\"\"$F&\"\"*" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 11 "" 1 " " {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&\"\")E\\[lA6$F&F)\"\"!6$F&F&F,6$\" \"#\"\"'F,6$\"\"$\"\"(F,6$F/F&F,6$F2F'F,6$F/\"\"&F,6$F'F7F,6$F&F/F,6$F &F'F,6$F2F)F,6$F'F&F,6$F2F0F,6$F&F3F,6$F&F0F&6$F'F3F,6$F'F/F,6$F2F&!\" \"6$F/F'F,6$F2F7F,6$F'F2F,6$F/F)F,6$F&F2F,6$F'F)F&6$F/F3FC6$F2F/F,6$F/ F2F,6$F'F0F,6$F&F7FC6$F'F'FC6$F2F2F&6$F/F/F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&F&E\\[l%6$F&F&\"#N6$\"\"#F&\"\"!6$F' F&\"\"$6$F0F&!#9" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive ass ignment" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&\"\")E\\[lA6$F&F) \"\"!6$F&F&!\"\"6$\"\"#\"\"'F,6$\"\"$\"\"(F,6$F0F&F,6$F3F'F,6$F0\"\"&F ,6$F'F8F,6$F&F0F,6$F&F'F,6$F3F)F,6$F'F&F,6$F3F1F,6$F&F4F,6$F&F1F&6$F'F 4F.6$F'F0F,6$F3F&F,6$F0F'F,6$F3F8F.6$F'F3F,6$F0F)F&6$F&F3F,6$F'F)F,6$F 0F4F,6$F3F0F&6$F0F3F.6$F'F1F,6$F&F8F,6$F'F'F&6$F3F3F,6$F0F0F," }} {PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&F&E\\[l%6$F&F&\"#<6 $\"\"#F&!#<6$F'F&\"#56$\"\"$F&!#9" }}{PARA 8 "" 1 "" {TEXT 205 27 "Err or, recursive assignment" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recur sive assignment" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F& \"\")E\\[lA6$F&F)\"\"!6$F&F&F,6$\"\"#\"\"'F,6$\"\"$\"\"(F,6$F/F&F,6$F2 F'F,6$F/\"\"&F,6$F'F7F,6$F&F/F,6$F&F'!\"\"6$F2F)F&6$F'F&F&6$F2F0F,6$F& F3F,6$F&F0F&6$F'F3F;6$F'F/F,6$F2F&F,6$F/F'F,6$F2F7F;6$F'F2F,6$F/F)F,6$ F&F2F,6$F'F)F,6$F/F3F,6$F2F/F,6$F/F2F&6$F'F0F,6$F&F7F,6$F'F'F,6$F2F2F, 6$F/F/F;" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&F&E\\[l %6$F&F&\"#G6$\"\"#F&\"#=6$F'F&\"#96$\"\"$F&\"#<" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 11 "" 1 "" {XPPMATH 20 "= 6\"6$;\"\"\"\"\"%;F&\"\")E\\[lA6$F&F)\"\"!6$F&F&F,6$\"\"#\"\"'F,6$\"\" $\"\"(F,6$F/F&F,6$F2F'F,6$F/\"\"&F,6$F'F7F,6$F&F/F,6$F&F'F,6$F2F)F,6$F 'F&F,6$F2F0F,6$F&F3F,6$F&F0F,6$F'F3F,6$F'F/F,6$F2F&F,6$F/F'F,6$F2F7F,6 $F'F2F,6$F/F)F,6$F&F2F,6$F'F)F,6$F/F3F,6$F2F/F,6$F/F2F,6$F'F0F,6$F&F7F ,6$F'F'F,6$F2F2F,6$F/F/F," }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\" \"\"\"%;F&F&E\\[l%6$F&F&\"\"!6$\"\"#F&F+6$F'F&F+6$\"\"$F&F+" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&\"\")E\\[lA6$F&F)\"\"!6$F&F&F,6$\"\" #\"\"'F,6$\"\"$\"\"(F,6$F/F&F,6$F2F'F,6$F/\"\"&F,6$F'F7F,6$F&F/F,6$F&F 'F,6$F2F)F,6$F'F&F,6$F2F0F,6$F&F3F,6$F&F0F,6$F'F3F,6$F'F/F,6$F2F&F,6$F /F'F,6$F2F7F,6$F'F2F,6$F/F)F,6$F&F2F,6$F'F)F,6$F/F3F,6$F2F/F,6$F/F2F,6 $F'F0F,6$F&F7F,6$F'F'F,6$F2F2F,6$F/F/F," }}{PARA 11 "" 1 "" {XPPMATH 20 "=6\"6$;\"\"\"\"\"%;F&F&E\\[l%6$F&F&\"\"!6$\"\"#F&F+6$F'F&F+6$\"\"$ F&F+" }}{PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }} {PARA 8 "" 1 "" {TEXT 205 27 "Error, recursive assignment" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }