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1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 257 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 66 "Maple Programs for Binary Tensor Invariants and Outerplanar Graphs" }}{PARA 19 "" 0 "" {TEXT -1 14 "Alec Mihailovs" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 80 "Invaria nts and decompositions of tensor powers of the standard representation of" }{TEXT 256 3 " SL" }{TEXT -1 4 "(2)." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The dimension of the space of " }{TEXT 257 6 "SL(2)-" } {TEXT -1 14 "invariants in " }{TEXT 258 1 "n" }{TEXT -1 69 "-fold tens or product of the standard 2-dimensional representation of " }{TEXT 259 2 "SL" }{TEXT -1 46 "(2) can be determined using following command :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "DimInvSL2 := n->`if`(n mod 2 = 0, \n binomial(n,n/2)/(n/2+1), \n 0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "For example," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "DimInvSL2(100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" =cWxBOlghvd;Ey>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Here is the li st of the dimensions for " }{TEXT 260 1 "n" }{TEXT -1 14 " from 0 to 2 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "for n from 0 to 20 \n do A[n] := DimInvSL2(n) end do: \n(op(op((A))));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#77/\"\"!\"\"\"/F&F%/\"\"#F&/\"\"$F%/\"\"%F)/\"\"&F%/ \"\"'F//\"\"(F%/\"\")\"#9/\"\"*F%/\"#5\"#U/\"#6F%/\"#7\"$K\"/\"#8F%/F6 \"$H%/\"#:F%/\"#;\"%I9/\"#F%/\"#?\"&'z;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "We can also put the formula in a spreadsh eet, so that it can be used without typing the command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(Spread):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "i2 := CreateSpreadsheet():\nfor n to 10 \n \+ do SetCellFormula(i2,n,2,evaln(DimInvSL2(cat(~A,n)))) od; \nSetSelect ion(i2,1$4):" }}{PARA 0 "" 1 "" {SPREADSHEET {NAME "SpreadSheet009" } {ROWHEIGHTS } {COLWIDTHS 2 124 } {SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }487 291 291 {CELL 1 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "22" 20 "6#\"#A" }0 }{CELL 1 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "DimInvSL2(~A1)" 20 "6#\"&'ye" }0 }{CELL 2 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "24" 20 "6#\"#C" }0 }{CELL 2 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "DimInvSL2(~A2)" 20 "6#\"'7!3#" }0 } {CELL 3 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "26" 20 "6#\"#E" }0 }{CELL 3 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "DimInvSL2(~A3)" 20 "6#\"'+Hu" }0 }{CELL 4 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "2 8" 20 "6#\"#G" }0 }{CELL 4 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ " DimInvSL2(~A4)" 20 "6#\"(SWn#" }0 }{CELL 5 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "30" 20 "6#\"#I" }0 }{CELL 5 2 {CELLOPTS 0 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will produce the dimensions of the invariant spaces in column " }{TEXT 262 1 "B" }{TEXT -1 206 " after the evaluating of \+ the worksheet. For example, try to enter some numbers. Reentering othe r numbers and reevaluating the worksheet will correct the dimensions. \+ Another way of doing it is using Maplets." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "with(Maplets[Elements]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 474 "m_i2 := Maplet( \nWindow( 'title'=\"Dimensions of Invariants\", [\n [\"Enter the tensor power: \", \n TextFiel d['TF1'](26, onchange=Evaluate('TB1' = evaln(DimInvSL2(TF1))))],\n \+ [\"Here is the dimension: \", \n TextBox['TB1']('editable' = ' false',3..40 )],\n [Button(\"Evaluate the dimension\", Evaluate('TB 1' = evaln(DimInvSL2(TF1)))),\n Button(\"Close\", Shutdown(['TF1 ', 'TB1'])),\n Button(\"Clear\", SetOption('TF1' = \"\"))]\n] ) \+ ):\nMaplets[Display]( m_i2 ):" }}{PARA 6 "" 1 "" {TEXT -1 38 "Initiali zing Java runtime environment." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The maplet looks as follows:" }}}{EXCHG {PARA 256 "" 0 "" {BITMAP 442 183 183 1 "?TMGVgA:;JUJNJWA:i?=JFJ:::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::yF^YvU;uQYER:::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::<:;jIsmYHjyF^xSByB::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::jq:;jIsmZY<::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::jq:;jIsmZYLJySB=tI=C:: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::jI=sAyCY=>:y^I=xC;_Z<:::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::vMYvavGJ:vMxFra> N:y^I=xC;_Z<:::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::vMYvavGJ:vMx Fra>N:y^I=xC;_Z<:::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::vMYvavGJ :vMxFra>N:y^I=xC;_Z<:::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::vMYv avGJ:vMxFra>N:y^I=xC;_Z<:::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::jIwkYlq:;jIsmZYLJCb:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: vMYvavGJ:vMxFra>NNNTJr::::::::::::::::::::::: ::::::::;J:NZZLJ;:::::::::::::::BnZZ<::::::::::::::::::::::::::: ::JrZ:>NN::::::::::::::::::::::::::::::ZZLJ;:: :::::::::::::ynAyCY=>:y^I=xC;_Z<::::::::::::::Jb;;?::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::Zj;;Bb> N::::::::::::::::HB:;_Z<::::::::::: ::::Jb;;?:::::::::::::::::::::::::::::Zj;;Bb>N:::::::::::::::jIwkYlq:; 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::::::::::::::::::::Y@vM:::::::jq::y>:::::::vGJ::Y@vM:::::::jq:;:vMxCY =>:y^I=xC;Z:jyCZ;FZ;Gb;?Y_ZDb[yyJrkysqxDjy?Zy>dyyM;_Zlyyqyre>::::::::::::::::::::::ZJJB:;J::::::::::::::::::::::::<;>:::: :::::::::::::::::::ZJJB:;J:::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::ja;y>:::::::vG:jI;:::::: :Y=>:ja;y>:::::::vGJ::y^Ylq:;jIsmZYL:B:vyy?dixi:>j:@jZDbB:yrE[LbN::: ::::vG:jI;:::::::Y=>:ja;y>:::::::vGJ::y^Ylq:;jIsmZYL:B:vYyueymQCZZ:vSxF::::::::B:<:jasm::::::::Z:B::y^YlqB:> :y^ImIsa>:<:vOxD:::::::::;B::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::vMxCY=>:y^I=xC;Z::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::jIsavGJ:vMxFra:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::vG J:vMxFra:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::vGJ:vMxFY@:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::vgZ;;jIsa::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::Z:J:3:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "Similarly, we can produce the procedure, the spreadsheet, and the maplet for the decomposition \+ of the tensor powers of the standard representation of " }{TEXT 263 2 "SL" }{TEXT -1 4 "(2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "M ultSL2 := (n,k)->`if`(n-k mod 2 = 0, \n binomial(n,(n-k)/2)*(k+1)/(n -(n-k)/2+1),\n 0):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For examp le, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "MultSL2(6,2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Here is the corresponding spreadsheet:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "m2 := CreateSpreadsheet():\nfor n to 10 \+ \n do SetCellFormula(m2,n,3,evaln(MultSL2(cat(~A,n),cat(~B,n)))) od; \nSetSelection(m2,1$4):" }}{PARA 0 "" 1 "" {SPREADSHEET {NAME "Spread Sheet002" } {ROWHEIGHTS } {COLWIDTHS 3 150 } {SSOPTS {CELLOPTS 2 10 4 2 1 255 255 255 }1 }487 291 291 {CELL 1 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "2" 20 "6#\"\"#" }0 }{CELL 1 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "0" 20 "6#\"\"!" }0 }{CELL 1 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A1,~B1)" 20 "6#\"\"\"" }0 }{CELL 2 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "4" 20 "6#\"\"%" }0 }{CELL 2 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "2" 20 "6#\"\"#" }0 }{CELL 2 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A2,~B2)" 20 "6# \"\"$" }0 }{CELL 3 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "6" 20 "6 #\"\"'" }0 }{CELL 3 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "4" 20 " 6#\"\"%" }0 }{CELL 3 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL 2(~A3,~B3)" 20 "6#\"\"&" }0 }{CELL 4 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "9" 20 "6#\"\"*" }0 }{CELL 4 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "1" 20 "6#\"\"\"" }0 }{CELL 4 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A4,~B4)" 20 "6#\"#U" }0 }{CELL 5 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "12" 20 "6#\"#7" }0 }{CELL 5 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "12" 20 "6#\"#7" }0 }{CELL 5 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A5,~B5)" 20 "6#\" \"\"" }0 }{CELL 6 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "15" 20 "6 #\"#:" }0 }{CELL 6 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "15" 20 " 6#\"#:" }0 }{CELL 6 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2 (~A6,~B6)" 20 "6#\"\"\"" }0 }{CELL 7 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "20" 20 "6#\"#?" }0 }{CELL 7 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "10" 20 "6#\"#5" }0 }{CELL 7 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A7,~B7)" 20 "6#\"&f1\"" }0 }{CELL 8 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "25" 20 "6#\"#D" }0 }{CELL 8 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "19" 20 "6#\"#>" }0 }{CELL 8 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A8,~B8)" 20 "6# \"%+?" }0 }{CELL 9 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "30" 20 " 6#\"#I" }0 }{CELL 9 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "26" 20 "6#\"#E" }0 }{CELL 9 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL 2(~A9,~B9)" 20 "6#\"$0%" }0 }{CELL 10 1 {CELLOPTS 0 -1 -1 0 0 0 0 0 } {R5MATHOBJ "37" 20 "6#\"#P" }0 }{CELL 10 2 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "17" 20 "6#\"#<" }0 }{CELL 10 3 {CELLOPTS 0 -1 -1 0 0 0 0 0 }{R5MATHOBJ "MultSL2(~A10,~B10)" 20 "6#\"*;l#RA" }0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "Entering the tensor power in the A colum n and the index of the irreducible representation in the B column, we \+ obtain the multiplicity of the representation in the tensor power of t he standard representation of " }{TEXT 264 2 "SL" }{TEXT -1 25 "(2). H ere is the maplet: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 730 "m_m 2 := Maplet( \nWindow( 'title'=\"Multiplicities of Representations\", \+ [\n [\"Enter the tensor power: \", \n TextField['TF1'](26,onc hange=Evaluate('TB1' = evaln(MultSL2(TF1,TF2)))), \n Button(\"Cl ear\", SetOption('TF1' = \"\"))], \n [[\"Enter the index\", \"of th e representation:\"], \n TextField['TF2'](26,onchange=Evaluate(' TB1' = evaln(MultSL2(TF1,TF2)))), \n Button(\"Clear\", SetOption ('TF2' = \"\"))],\n [\"Here is the multiplicity: \", \n Tex tBox['TB1']('editable' = 'false',3..50 )],\n [Button(\"Evaluate the multiplicity\", Evaluate('TB1' = evaln(MultSL2(TF1,TF2)))),\n B utton(\"Close\", Shutdown(['TF1', 'TF2', 'TB1'])),\n Button(\"Cl ear\", SetOption('TB1' = \"\"))]\n] ) ):\nMaplets[Display]( m_m2 ):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The maplet looks as follows:" }} }{EXCHG {PARA 256 "" 0 "" {BITMAP 507 236 236 1 "?TMGSYE:;JUJNjwA:SA=J FJ:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::jI=sAyPjxoq<@::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::>:<:y^Iys:ymJta[v]:::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::rUZ:jIsmZY<::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::x@B:vMxFra>nY@lZxmJ<::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::ZY@lIxMx@B :vMxFra>NNNNN:::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::ZZxM;?:::::::::::::::::::::::::Z:B::::::::::::::::::::::: ::::::::::;_Z<:::::::::::::::::::::::::J:>:::::::::::::::::::::::::::: ::::ZZLJ;::::::::::::::::::::raxrIsUZ:jIsmZYLJCb::::::::::::::::::::>t JysjwgZ?hi?F:::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::>:;Bb>N:::::::::::::::::::::::::::;Bb>N:::::::::::::::::::::xcIxMx@B:vMxFra>NZZLJ;:::::::::::::::::::: :::::BnZZ<:::::::::::::::::::::::::::::JrZ:>NN::::::::::::::::::::::::::::::ZZLJ;::::::::::::::::::::raxrI sUZ:jIsmZYLJCb::::::::::::::::::::>TJJ;::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::BVJ:TJJ;:::::::::::::::::::::::::::::BVJ:NB:;?NB:;?NB:;?NB:;?NB:;?NB:;?NB:;?N B:;?NB:;?NB:;?NB:;?NB:;?NB:;?NB:;? 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