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-1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 269 1 {CSTYLE "" -1 -1 "Helvetica " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 }1 1 0 0 6 6 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT 258 33 "Matrices and Systems of Equations" }}{PARA 258 "" 0 "" {TEXT 256 4 "HW 5" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 260 "Click on a [+] sign to expand a section. Click on a [-] sign to collapse a section. To do these exercises you will have to insert execution groups. That \+ can be done by clicking on the toolbar icon that looks like \"[>\". It can also be done via the Insert menu." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 279 22 "Student Name and ID: " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 21 "Student Name and ID: \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 314 105 "(If you are working as a group of two students, hand in only one assignme nt. Both names should be given.)" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{PARA 0 "" 0 "" {TEXT -1 9 "In this " }{TEXT 257 5 "MAPLE" }{TEXT -1 3 " " }{HYPERLNK 17 "worksheet" 2 "worksheet" " " }{TEXT -1 28 ", you will be asked to use " }{TEXT 266 5 "MAPLE" } {TEXT -1 25 " to work with matrices. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 7 "Reports" }}{PARA 0 "" 0 "" {TEXT -1 30 "Reports that you p repare with " }{TEXT 267 5 "MAPLE" }{TEXT -1 112 " should be prepare d with the same care that you would devote to laboratory reports in bi ology and chemistry. " }{TEXT 269 46 "A report should not be a diary \+ or history of a" }{TEXT -1 1 " " }{TEXT 268 6 " MAPLE" }{TEXT -1 2 " \+ " }{TEXT 270 1 " " }{TEXT 271 51 "session. Delete what is not needed \+ for the report." }{TEXT -1 25 " All lines of the form " }{TEXT 259 6 "?topic" }{TEXT -1 82 " (that arise from help queries) should be er ased. All errors should be erased. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "When you are printing a " }{TEXT 272 5 "MAPLE" }{TEXT -1 165 " report, think about the toner and paper res ources that you are using. All commands must be terminated - either wi th the standard terminator, the semicolon, or the " }{TEXT 276 17 "sil ent terminator" }{TEXT -1 57 ", the colon. When you assign a variable, for example \n \"" }{TEXT 278 7 "x := 5;" }{TEXT -1 30 " \", there \+ is no need to have " }{TEXT 273 5 "MAPLE" }{TEXT -1 12 " echo back " }{TEXT 260 6 "x := 5" }{TEXT -1 89 ". When this is printed, it simply wastes paper and ink. Choose the silent terminator \"" }{TEXT 261 7 "x := 5:" }{TEXT -1 70 " \" instead. When you load a package (with out the silent terminator)," }{TEXT 262 1 " " }{TEXT -1 2 " " }{TEXT 274 5 "MAPLE" }{TEXT -1 123 " will list the commands that become avail able with the package. This is fine - it will help you become familiar with what " }{TEXT 275 5 "MAPLE" }{TEXT -1 125 " makes available. Ho wever, these commands should not be part of a lab report. Reload the p ackage with the silent terminator. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 79 "Much of the text in this worksheet shoul d be deleted. For example, delete the " }{TEXT 263 12 "Introduction" }{TEXT -1 5 " and " }{TEXT 264 8 "Keywords" }{TEXT -1 34 " sections. D elete this section on " }{TEXT 265 7 "Reports" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 235 "Remember that your worksheet should execute in the order that it has been writ ten. In particular, remember that the ditto refers to the result of t he last executed command - not the result of the command that physical ly precedes it. " }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Background \+ Worksheet" }}{PARA 0 "" 0 "" {TEXT -1 419 "The following worksheet, av ailable for download from the syllabus web page, has examples and dis cussions that will help you do this homework. If they are in the same directory as this worksheet, and if you have retained the filename un der which they were posted, then clicking on the hyperlink below will \+ automatically open them. Use the Window menu to control the view when \+ multiple files are opened simultaneously. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{HYPERLNK 17 "5.1-5.8epR4.m ws" 1 "5.1-5.8epR4.mws" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 "Exercise 1 Eigenvalues" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 310 18 "Backg roud Reading:" }{TEXT -1 5 " " }{HYPERLNK 17 "5.1-5.8epR4.mws" 1 " 5.1-5.8epR4.mws" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Consider the matrix" }{TEXT 283 5 " " }{XPPEDIT 284 1 "A=MATRIX([[2, -1, -1, 2], [0, 6, 5, -6], [0, -1, 2, 2], [0, 1, 2, 1 ]])" "/%\"AG-%'MATRIXG6#7&7&\"\"#,$\"\"\"!\"\",$F+F,F)7&\"\"!\"\"'\"\" &,$F0F,7&F/,$F+F,F)F)7&F/F+F)F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Use the command " }{TEXT 286 5 " det " }{TEXT -1 39 " to calc ulate the two eigenvalues of " }{TEXT 285 1 "A" }{TEXT -1 2 ".\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "with(linalg): # Execute me! " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "A := matrix([[2, -1, -1 , 2], [0, 6, 5, -6], [0, -1, 2, 2], [0, 1, 2, 1]]): # Execute me!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Use the command " }{TEXT 287 13 " eigenvals " }{TEXT -1 25 " to v erify your answer. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 259 "" 0 "" {TEXT -1 25 "Exer cise 2 Eigenvectors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 311 18 "Backgroud Reading:" }{TEXT -1 5 " " }{HYPERLNK 17 "5.1-5.8epR4.mws" 1 "5.1-5.8epR4.mws" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Let us find an eigenvector that corresponds to the smaller of the two eigenvalues:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Id := arra y(identity,1..4,1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IdG-%&arr ayG6&%)identityG;\"\"\"\"\"%F)7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalm(A - ?*Id);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %'MATRIXG6#7&7&\"\"!!\"\"F)\"\"#7&F(\"\"%\"\"&!\"'7&F(F)F(F*7&F(\"\"\" F*F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 " Do row operations using " }{TEXT 290 8 " addrow" }{TEXT -1 2 ", " } {TEXT 291 7 " mulrow" }{TEXT -1 7 ", and " }{TEXT 292 8 " swaprow" } {TEXT -1 42 " until you can determine an eigenvector." }}{PARA 0 "" 0 "" {TEXT -1 72 "(You might need more or less execution groups than t hose entered below.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Name the eigenvector that you have found:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "? := vector([?, ?, ?, ?]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "In a similar way, find an eigenvector for the other eigenvalue.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalm(A - ?*Id);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "(You need n't name this eigenvector just yet.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Confirm your eigenvectors with the " }{TEXT 293 10 "eigenvects" }{TEXT -1 11 " command.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Exercise 3 Generalized Eigenvec tors" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 312 18 " Backgroud Reading:" }{TEXT -1 5 " " }{HYPERLNK 17 "5.1-5.8epR4.mws " 1 "5.1-5.8epR4.mws" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "The larger of the two eigenvalues has a defect. We \+ must find generalized eigenvectors in the amount of that defect." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "B := (A - ?*Id); # Enter the larger eigenvalue and execute!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Compute t he third power of this matrix:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalm(B^3);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The next vector will be one generalized eigenvector. Choose the last entry " }{TEXT 295 13 "appropriately" }{TEXT -1 64 " (based on the result of the precedin g calculation) and name it." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "? := vector([a,b,c,?]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Here is the nex t generalized eigenvector. Name it:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "? := evalm(B &* ? );" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "If all ha s gone well, the next vector should be a nonzero eigenvector (for some specification of " }{TEXT 296 1 "a" }{TEXT -1 2 ", " }{TEXT 297 2 " \+ b" }{TEXT -1 6 ", and " }{TEXT 299 2 " c" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "? := \+ evalm(B &* ? );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "(The last result should be a scalar multiple of the eigen vector that " }{TEXT 304 12 " eigenvects " }{TEXT -1 11 " reported.) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now s ubstitute specific values for " }{TEXT 301 1 "a" }{TEXT -1 2 ", " } {TEXT 302 2 " b" }{TEXT -1 6 ", and " }{TEXT 303 2 " c" }{TEXT -1 1 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "? := subs( \{a = , b = , c = \}, ?); # First generalized eigenvector" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "? := subs( \{a = , b = , c = \}, ?); # Second general ized eigenvector" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "? := su bs( \{a = , b = , c = \}, ?); # Eigenvector" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 294 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 "Exercise 4 The Exponential of a Matrix" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 18 "Backgroud Reading:" }{TEXT -1 5 " " }{HYPERLNK 17 "5.1-5.8epR4.mws" 1 "5.1-5.8epR4.mws" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "Use your eigenv ectors and generalized eigenvectors to calculate a fundamental matrix: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "? := t -> augment(?, ?, ?, ?);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "Use this fundamental matrix to find \+ a fundamental matrix that is the identity matrix when" }{TEXT 306 6 " \+ t = 0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "? := t -> ? ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "Compare this last matrix with t he exponential of " }{TEXT 307 4 " A " }{TEXT -1 25 "to ensure that i t agrees:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalm( ? - ? );" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Solve the initial value problem " }{TEXT 308 2 " " }{XPPEDIT 309 1 "x*`'`(t)= A*x(t),x[1](0)=2,x[2](0)=-1,x[3](0)=0,x[4](0)=-4" "6'/*&%\"xG\"\"\"-% \"'G6#%\"tGF&*&%\"AGF&-F%6#F*F&/-&F%6#F&6#\"\"!\"\"#/-&F%6#F56#F4,$F&! \"\"/-&F%6#\"\"$6#F4F4/-&F%6#\"\"%6#F4,$FGF<" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "? := vector([2,-1,0,-4]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 "Copyright and Author I nformation" }}{EXCHG {PARA 260 "" 0 "" {TEXT -1 45 "05F00R4.mws A \+ MapleV Release 4 worksheet." }}{PARA 261 "" 0 "" {TEXT -1 0 "" }} {PARA 262 "" 0 "" {TEXT -1 43 "Author: Brian E. Blank (23 November 2 000)" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by any medium," }}{PARA 0 "" 0 " " {TEXT -1 55 "including print, disk, and electronic transfer, without " }}{PARA 0 "" 0 "" {TEXT -1 39 "prior written permission of the autho r." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 49 "F or more information, please contact the author:" }}{PARA 265 "" 0 "" {TEXT -1 4 " " }}{PARA 265 "" 0 "" {TEXT -1 32 " Department of \+ Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington Universi ty in St. Louis" }}{PARA 0 "" 0 "" {TEXT -1 26 " St. Louis, MO 6 3130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-6763" }}{PARA 266 "" 0 "" {TEXT -1 44 " e-mail: brian@math.wustl.edu" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 56 "Copyright: \251 2000 B rian E. Blank, All Rights Reserved." }}}}}{MARK "1 4 0" 22 } {VIEWOPTS 1 1 0 3 4 1802 }