diff --git a/01001 - Matematik 1a/Hjemmeaflevering nr. 3/Hjemmeaflevering3.pdf b/01001 - Matematik 1a/Hjemmeaflevering nr. 3/Hjemmeaflevering3.pdf index 1fbfabc..e34176e 100644 --- a/01001 - Matematik 1a/Hjemmeaflevering nr. 3/Hjemmeaflevering3.pdf +++ b/01001 - Matematik 1a/Hjemmeaflevering nr. 3/Hjemmeaflevering3.pdf @@ -2737,7 +2737,7 @@ endobj << /Type /Font /Subtype /Type0 - /BaseFont /MPFYHC+NewCMMath-Book-Identity-H + /BaseFont /PWBUQK+LiberationSans-Bold /Encoding /Identity-H /DescendantFonts [248 0 R] /ToUnicode 251 0 R @@ -2747,8 +2747,8 @@ endobj 248 0 obj << /Type /Font - /Subtype /CIDFontType0 - /BaseFont /MPFYHC+NewCMMath-Book + /Subtype /CIDFontType2 + /BaseFont /PWBUQK+LiberationSans-Bold /CIDSystemInfo << /Registry (Adobe) /Ordering (Identity) @@ -2756,7 +2756,8 @@ endobj >> /FontDescriptor 250 0 R /DW 0 - /W [0 0 500 1 1 944 2 2 722 3 3 569 4 4 567 5 5 569 6 6 778 7 9 667 10 10 778 11 12 500 13 15 667 16 16 278 17 17 569 18 22 500 23 24 389 25 25 584 26 26 485 27 29 500 30 30 1000 31 31 861 32 32 681 33 33 1000 34 34 778 35 35 681 36 37 500 38 38 681 39 39 778 40 40 278 41 41 681 42 42 569 43 43 715 44 44 1139 45 45 681 46 46 278 47 47 490 48 48 407 49 49 778 50 50 477 51 51 667 52 52 433 53 53 278 54 54 778 55 55 643 56 56 722 57 57 736 58 59 528 60 60 736 61 61 566 62 62 518 63 63 306 64 64 392 65 65 278 66 66 725 67 67 278 68 68 648 69 69 600 70 70 472 71 71 597 72 72 472 73 73 597 74 74 569 75 75 667 76 76 569 77 77 667 78 78 499 79 80 681 81 81 499 82 82 778 83 83 970 84 84 869 85 85 1089 86 86 642 87 87 976 88 88 683 89 89 556 90 90 444 91 91 389 92 92 440 93 98 875 99 100 422 101 101 778 102 102 583 103 103 738 104 104 678 105 105 528 106 106 706 107 107 750 108 108 1000 109 109 361 110 110 332 111 111 394 112 112 556 113 113 500 114 114 556 115 122 902 123 123 1000 124 124 572 125 126 444 127 128 875 129 129 589 130 130 429 131 132 458 133 133 583 134 134 529 135 135 520 136 136 466 137 138 278 139 139 833] + /CIDToGIDMap /Identity + /W [0 0 750 1 1 722.16797 2 2 277.83203 3 3 556.15234 4 4 889.16016 5 7 610.83984 8 9 556.15234 10 10 277.83203 11 13 556.15234 14 14 666.9922 15 15 333.0078 16 18 610.83984 19 19 333.0078 20 20 722.16797 21 21 666.9922 22 22 389.16016 23 23 610.83984 24 24 277.83203 25 25 722.16797 26 26 277.83203 27 27 610.83984 28 29 722.16797 30 30 333.0078 31 32 556.15234 33 33 722.16797 34 34 666.9922 35 35 610.83984 36 36 777.83203] >> endobj @@ -2766,11 +2767,187 @@ endobj /Filter /FlateDecode >> stream -x |X +x endstream endobj 250 0 obj +<< + /Type /FontDescriptor + /FontName /PWBUQK+LiberationSans-Bold + /Flags 131076 + /FontBBox [-15.625 -211.91406 827.14844 724.6094] + /ItalicAngle 0 + /Ascent 728.02734 + /Descent -210.44922 + /CapHeight 687.9883 + /StemV 168.6 + /CIDSet 249 0 R + /FontFile2 252 0 R +>> +endobj + +251 0 obj +<< + /Length 1110 + /Type /CMap + /WMode 0 +>> +stream +%!PS-Adobe-3.0 Resource-CMap +%%DocumentNeededResources: procset CIDInit +%%IncludeResource: procset CIDInit +%%BeginResource: CMap Custom +%%Title: (Custom Adobe Identity 0) +%%Version: 1 +%%EndComments +/CIDInit /ProcSet findresource begin +12 dict begin +begincmap +/CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def +end def +/CMapName /Custom def +/CMapVersion 1 def +/CMapType 0 def +/WMode 0 def +1 begincodespacerange +<0000> +endcodespacerange +36 beginbfchar +<0001> <0048> +<0002> <006A> +<0003> <0065> +<0004> <006D> +<0005> <006F> +<0006> <0070> +<0007> <0067> +<0008> <0061> +<0009> <0076> +<000A> <0020> +<000B> <0033> +<000C> <0030> +<000D> <0031> +<000E> <0053> +<000F> <0074> +<0010> <0075> +<0011> <0064> +<0012> <006E> +<0013> <003A> +<0014> <0044> +<0015> <0050> +<0016> <0072> +<0017> <0062> +<0018> <006C> +<0019> <0041> +<001A> <0069> +<001B> <0054> +<001C> <0055> +<001D> <0042> +<001E> <0029> +<001F> <0032> +<0020> <006B> +<0021> <0043> +<0022> <0045> +<0023> <0046> +<0024> <0047> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +252 0 obj +<< + /Length 8486 + /Filter /FlateDecode +>> +stream +xz XSWZ IxD 1$bUN"V«J4R-tNՙvЗM>jӧvmog3wT;|{k;D +oWVl]n-/l9u+7.} g{ĻfV֥I׺'HJl[yիopb@YݼjYpwFڢy ]2@k\NFMl:Iz ?4Y([1|40 83.6!x^'t <AA ++#؈^x:`܍p#a7Z(var?"k1~8c9l| ?g;Ĕ9B(QQpE(9X%xZ`3tAi`Ѐ>}c)V+f>{0wW^6'6yv xVY/g,90 +T%8h:&tr&14 2)5T^sfW +TϜqZ6}+N4dBa,w'Js)1hc &2z+[`uCE֒<3HMTە-)Wj4/lP~^[յEsOsZB%RԌ3␨^G: 6ytRHs"x-g45f5 nqM \4GX,O.Vԣ9TcUAS˕ +Mm{Sܤarl#%jU,m,q66S%q.+DQT7 c}Ŋy?C;f6hejrhJ2Ԑ!O ;E9R03z~"1y+.iphW3x JdChW h3cy5kDEbP3* aj-瘖T,-ӑ%ByN3+ aJ;D'8XbaU4kL2Y yZM 6P+ǍAmU%Q=nCrGlHh[yz0Z2xvី@l He'dA{Ţb OUbnBxftyjR95 70c`FVv~ek]s 23fꆗB%@QHHix1Pz*@r@K(,F +KGR%\ SsuP1ըJq!=ǫW&tu03k(;3FFTcn8t%0>C ϫHm!JBPkWS,6phWcg:hgz2%x=3bpE uh̠$4=MIOgOS.7Dp&Y5xXZ*edNnGf6%Sii|'$*5 ,gH%4D~5bk^qSDE#d  Uv0b6a 6666YvZ(+*466-_9c { 60CyX>uȚL IU2s`s"ԆY׊V ZQŊc& +9eM"+NQ+~dݔM+f+YlcTnuQLLs ڵq0ˆr!'ӗ~_40m̬%[ =W`ڽG}=l |a` 8tҠM`,+'tGbMȕf&BrMH@Ruh-İXnjFlԂF꫔Q3A*6h` #s&oW[@%׳ǰچna/:S/پoa'W>O]p\ѐadml0dIIgm2D6~gc]6^Gl|d[+!elԲqW6vdcō%L seCJ$:-L.b_;/JO(Ҟx%6+~d%¤7|L~Or g_c+%[V޸_B"ءPM HqU!UXG$Rp͚x0Zh7[,y+}~ dш)0#d?<\緞}u آ㺹jH7LUT5!ѓiO +Y$sUBxp FeeJb<,5Uq̀EVV%!u~og@GF6lX5Y?1c&}W\fL:}.77B:%53!űYV+bf*bsBbuZnc3"9;pY % L(ӗ@i8#x2b%Q4\~CjlSNr\Ve㲲DўdcY-)kBIL~$Wc䒬ՒVUqnpYU%Hf\DËÍԂK}lJ'a zO=N$:u:G#nl~.PHX4AXY\;۞zq&nVM_} +a pK?e 穜z͜ͳH0p+)9r +Ƅa%pVb$!a> r[ +!HGzAb:i%FS,R^wFƒ1yAzGbvKxt4IX.I"MO=n]&)"1q|$<'$$c/8UqRĀ!iҀtZdT=(a/m%,$f-)$>(YrIe+e~${HJ @14jk +®1WS? )15}n0})*z?Cs  }B6<6s=SZBOEDC[e 8Ut2J̺ΰ1N˕ D )b5q@ EU{ă"/l IJ$[K 'D*qqY^hbI/Kb Ǒ =p_zolnwA΁Oً[лi\45^pvcdNr.AH'd6/c-ѐQdS!2یG4'GOfm$LGu>f'Ϟo[_]~7:8/yKDa>Du_, +[-:ɽ! 8$J<`@B̆eEbdB-%^wb,6c<ľWߜ+N j Ǝt\^ɦ)رiYDE,YDF?,2fVx>)0;y 崡a'P&KR`8GP`0fB_ tHSxnpwnYv2Vm-oVIz6C'p@Z$p)IV+ Y]TIөbIS8\zzvT}=NDē%X ̆)Џnc/w̺)~D7~?8 U%Z+HU!d\U!Vp=V \}BXrqt^`DQj.[0E0D֮YUۍ`фަsҲ5mj-\ WĎjE!5=b#e$)! `ߝ=x:t#x*@]f~k٦ʯ OwjĖ7\9 ;V9/0cVav#/:2+6bFtl!#2A#ۈa#VЈ`֏{c#FX6s̈F +P)G)nJ_@#N>M{y1ʲPQFm9?%]x`(ȅԂ\r&}2n]TH˜tA %r]wP5 -YH +7CÆ6TlHʡ6XUʯyJPSC8||wԅY zNJ5,"m%y`0CVv1b&;e؜ˆbD$$:"/A]4.af 9#,v-|_N1FLWbC7}W*nNŃ؛T,H~BR`E)vJQfLt/_N moݛ1GS1t畩2V0M)`O-*$,Ih7gXI?cM3;bIJJ#2`DžЏ\vúG~FeFaS}vPJ[so@nw59yTyxfxtxǃ<Px;rd-wc\1̌$ro "kE L^zr2X+Le萑2ߕ%%}2.:X'c']FNֳ2~3𼌰[ƟRvqE#6iOޑ9#528[(g~q=AXF6Mf?2 cT}-'}2ZeDߑQq73KpX +zJJPr {!GYIS]Imms/Ĭ$ Pʥ +ݬuۈbn6[iMa  cvsUAX"\NdE8`͜=vUKbY5jsh4|yWHֿ4r_u妜i,WY8 xы8Έ#1vΈW ӴD YSt9qb_F\0̜hF#;F|r9)|*n7MtC9s#F|Px"PtzvTՅ/Nw HiBBrn]ⶳ~X/^={ n:_w /Wd[wB PdcəFu+isCfxCmqFGnaL0 "q& E&$eeGrHup7э$8^?+}  9 +5jh13̲iRr8dH0Fe: ' # ɅD5c(yd^< +R-BI1ae؊XyLZa8L~?N }[Vl__h-خ^($"p7Cf~3g|7?ijfwgyCȅC[Vuw,Ȝb"%jmwo.|̜''cyd˕O&$͈, (B8$0D$*\㶤y)1 Ü{TyD܏ИJIO%sbR2/rc>DYN l6Kp(10 :K,'z&N*q'*}~q8,?~c#ޢ{m/~f&Xp1~12?7rӹ{|?ßΐnXdxp"!)an"xu?H-glb+t^:X0`X^GI,; i#d6MB16 xto x[3:6@686+Nc &XNFX +-@ 4C;4K "RVha EPmJX + +̄6X $rC@1ZZ(Pעra5,p5,Ű +B3(̀6X -p%Da ,Bik2"ZJ(0CɏJq1) +a-\ +iR(S1Tv=ZJ.D3BU##V2X˩/RZ1mq uԫQh|sǏGH-n= +'(őXiL( xhL + +a#D$ZW=?\+/cRq5ǘmH ͆Mz JB UP!``0`TX?CT8&dv eCz,R T.Up ?Nz ~=X +g,i8q @=4OafA=f>^GoLJ4 {EN A|s-vqO +N:̈́OaSN={Bq}sbDz\_::Mc¯/23`fic~Mu}1s594p%/c  p(0C5o\9WW3V}/MsA?cEثU6ɫgi.xEf_dԤ1}TQ; 0 0 X4@P1M\ez9s/I<y3azdzTQKh~͏k0\hWo Nu|$#Y?xw]~3@|~\nWw[n||w%mm̝w];p⭁[Ak]j\ѠϕAggo#LkC:n$m+]EtLO֋~^`\A)\ +}E+] YopLp%Yyd"E3[VmfעZPk32jmrjjװUc\1*&TGf\kVCwpLQd\o)23C~s]hQF\f79\en3wbymp WW;_nռVׄ_ЇxO莻chE ZASIAsP=ھ.7h;yyEsssQB!h0͍B.yF17 b"+ d8: Rh;!ȓF#SɱW#l +endstream +endobj + +253 0 obj +<< + /Type /Font + /Subtype /Type0 + /BaseFont /MPFYHC+NewCMMath-Book-Identity-H + /Encoding /Identity-H + /DescendantFonts [254 0 R] + /ToUnicode 257 0 R +>> +endobj + +254 0 obj +<< + /Type /Font + /Subtype /CIDFontType0 + /BaseFont /MPFYHC+NewCMMath-Book + /CIDSystemInfo << + /Registry (Adobe) + /Ordering (Identity) + /Supplement 0 + >> + /FontDescriptor 256 0 R + /DW 0 + /W [0 0 500 1 1 944 2 2 722 3 3 569 4 4 567 5 5 569 6 6 778 7 9 667 10 10 778 11 12 500 13 15 667 16 16 278 17 17 569 18 22 500 23 24 389 25 25 584 26 26 485 27 29 500 30 30 1000 31 31 861 32 32 681 33 33 1000 34 34 778 35 35 681 36 37 500 38 38 681 39 39 778 40 40 278 41 41 681 42 42 569 43 43 715 44 44 1139 45 45 681 46 46 278 47 47 490 48 48 407 49 49 778 50 50 477 51 51 667 52 52 433 53 53 278 54 54 778 55 55 643 56 56 722 57 57 736 58 59 528 60 60 736 61 61 566 62 62 518 63 63 306 64 64 392 65 65 278 66 66 725 67 67 278 68 68 648 69 69 600 70 70 472 71 71 597 72 72 472 73 73 597 74 74 569 75 75 667 76 76 569 77 77 667 78 78 499 79 80 681 81 81 499 82 82 778 83 83 970 84 84 869 85 85 1089 86 86 642 87 87 976 88 88 683 89 89 556 90 90 444 91 91 389 92 92 440 93 98 875 99 100 422 101 101 778 102 102 583 103 103 738 104 104 678 105 105 528 106 106 706 107 107 750 108 108 1000 109 109 361 110 110 332 111 111 394 112 112 556 113 113 500 114 114 556 115 122 902 123 123 1000 124 124 572 125 126 444 127 128 875 129 129 589 130 130 429 131 132 458 133 133 583 134 134 529 135 135 520 136 136 466 137 138 278 139 139 833] +>> +endobj + +255 0 obj +<< + /Length 13 + /Filter /FlateDecode +>> +stream +x |X +endstream +endobj + +256 0 obj << /Type /FontDescriptor /FontName /MPFYHC+NewCMMath-Book @@ -2781,12 +2958,12 @@ endobj /Descent -194 /CapHeight 683 /StemV 95.4 - /CIDSet 249 0 R - /FontFile3 252 0 R + /CIDSet 255 0 R + /FontFile3 258 0 R >> endobj -251 0 obj +257 0 obj << /Length 2530 /Type /CMap @@ -2953,7 +3130,7 @@ end endstream endobj -252 0 obj +258 0 obj << /Length 15242 /Filter /FlateDecode @@ -3035,18 +3212,18 @@ PGKn endstream endobj -253 0 obj +259 0 obj << /Type /Font /Subtype /Type0 /BaseFont /SFNMRI+LiberationSans /Encoding /Identity-H - /DescendantFonts [254 0 R] - /ToUnicode 257 0 R + /DescendantFonts [260 0 R] + /ToUnicode 263 0 R >> endobj -254 0 obj +260 0 obj << /Type /Font /Subtype /CIDFontType2 @@ -3056,14 +3233,14 @@ endobj /Ordering (Identity) /Supplement 0 >> - /FontDescriptor 256 0 R + /FontDescriptor 262 0 R /DW 0 /CIDToGIDMap /Identity /W [0 0 750 1 3 556.15234 4 4 277.83203 5 5 610.83984 6 6 556.15234 7 7 222.16797 8 8 666.9922 9 9 556.15234 10 10 833.0078 11 11 500 12 12 277.83203 13 13 333.0078 14 14 833.0078 15 15 556.15234 16 16 222.16797 17 17 500 18 18 556.15234 19 19 333.0078 20 20 666.9922 21 21 556.15234 22 22 500 23 25 556.15234 26 26 333.0078 27 27 722.16797 28 28 333.0078 29 29 666.9922 30 31 556.15234 32 32 722.16797 33 33 500 34 34 556.15234 35 35 889.16016 36 36 556.15234 37 37 277.83203 38 38 610.83984 39 39 556.15234 40 40 500 41 41 277.83203 42 42 666.9922 43 43 556.15234 44 44 277.83203 45 45 500 46 46 556.15234 47 47 722.16797 48 48 222.16797 49 49 277.83203 50 50 666.9922 51 51 556.15234 52 52 722.16797 53 53 556.15234 54 54 666.9922 55 55 610.83984 56 56 722.16797 57 57 777.83203 58 58 556.15234 59 59 350.09766 60 62 556.15234 63 63 777.83203 64 64 277.83203 65 65 500 66 66 666.9922] >> endobj -255 0 obj +261 0 obj << /Length 13 /Filter /FlateDecode @@ -3073,7 +3250,7 @@ x endstream endobj -256 0 obj +262 0 obj << /Type /FontDescriptor /FontName /SFNMRI+LiberationSans @@ -3084,12 +3261,12 @@ endobj /Descent -210.44922 /CapHeight 687.9883 /StemV 95.4 - /CIDSet 255 0 R - /FontFile2 258 0 R + /CIDSet 261 0 R + /FontFile2 264 0 R >> endobj -257 0 obj +263 0 obj << /Length 1530 /Type /CMap @@ -3195,7 +3372,7 @@ end endstream endobj -258 0 obj +264 0 obj << /Length 11884 /Filter /FlateDecode @@ -3233,183 +3410,6 @@ L endstream endobj -259 0 obj -<< - /Type /Font - /Subtype /Type0 - /BaseFont /PWBUQK+LiberationSans-Bold - /Encoding /Identity-H - /DescendantFonts [260 0 R] - /ToUnicode 263 0 R ->> -endobj - -260 0 obj -<< - /Type /Font - /Subtype /CIDFontType2 - /BaseFont /PWBUQK+LiberationSans-Bold - /CIDSystemInfo << - /Registry (Adobe) - /Ordering (Identity) - /Supplement 0 - >> - /FontDescriptor 262 0 R - /DW 0 - /CIDToGIDMap /Identity - /W [0 0 750 1 1 722.16797 2 2 277.83203 3 3 556.15234 4 4 889.16016 5 7 610.83984 8 9 556.15234 10 10 277.83203 11 13 556.15234 14 14 666.9922 15 15 333.0078 16 18 610.83984 19 19 333.0078 20 20 722.16797 21 21 666.9922 22 22 389.16016 23 23 610.83984 24 24 277.83203 25 25 722.16797 26 26 277.83203 27 27 610.83984 28 29 722.16797 30 30 333.0078 31 32 556.15234 33 33 722.16797 34 34 666.9922 35 35 610.83984 36 36 777.83203] ->> -endobj - -261 0 obj -<< - /Length 13 - /Filter /FlateDecode ->> -stream -x -endstream -endobj - -262 0 obj -<< - /Type /FontDescriptor - /FontName /PWBUQK+LiberationSans-Bold - /Flags 131076 - /FontBBox [-15.625 -211.91406 827.14844 724.6094] - /ItalicAngle 0 - /Ascent 728.02734 - /Descent -210.44922 - /CapHeight 687.9883 - /StemV 168.6 - /CIDSet 261 0 R - /FontFile2 264 0 R ->> -endobj - -263 0 obj -<< - /Length 1110 - /Type /CMap - /WMode 0 ->> -stream -%!PS-Adobe-3.0 Resource-CMap -%%DocumentNeededResources: procset CIDInit -%%IncludeResource: procset CIDInit -%%BeginResource: CMap Custom -%%Title: (Custom Adobe Identity 0) -%%Version: 1 -%%EndComments -/CIDInit /ProcSet findresource begin -12 dict begin -begincmap -/CIDSystemInfo 3 dict dup begin - /Registry (Adobe) def - /Ordering (Identity) def - /Supplement 0 def -end def -/CMapName /Custom def -/CMapVersion 1 def -/CMapType 0 def -/WMode 0 def -1 begincodespacerange -<0000> -endcodespacerange -36 beginbfchar -<0001> <0048> -<0002> <006A> -<0003> <0065> -<0004> <006D> -<0005> <006F> -<0006> <0070> -<0007> <0067> -<0008> <0061> -<0009> <0076> -<000A> <0020> -<000B> <0033> -<000C> <0030> -<000D> <0031> -<000E> <0053> -<000F> <0074> -<0010> <0075> -<0011> <0064> -<0012> <006E> -<0013> <003A> -<0014> <0044> -<0015> <0050> -<0016> <0072> -<0017> <0062> -<0018> <006C> -<0019> <0041> -<001A> <0069> -<001B> <0054> -<001C> <0055> -<001D> <0042> -<001E> <0029> -<001F> <0032> -<0020> <006B> -<0021> <0043> -<0022> <0045> -<0023> <0046> -<0024> <0047> -endbfchar -endcmap -CMapName currentdict /CMap defineresource pop -end -end -%%EndResource -%%EOF -endstream -endobj - -264 0 obj -<< - /Length 8486 - /Filter /FlateDecode ->> -stream -xz XSWZ IxD 1$bUN"V«J4R-tNՙvЗM>jӧvmog3wT;|{k;D -oWVl]n-/l9u+7.} g{ĻfV֥I׺'HJl[yիopb@YݼjYpwFڢy ]2@k\NFMl:Iz ?4Y([1|40 83.6!x^'t <AA -+#؈^x:`܍p#a7Z(var?"k1~8c9l| ?g;Ĕ9B(QQpE(9X%xZ`3tAi`Ѐ>}c)V+f>{0wW^6'6yv xVY/g,90 -T%8h:&tr&14 2)5T^sfW -TϜqZ6}+N4dBa,w'Js)1hc &2z+[`uCE֒<3HMTە-)Wj4/lP~^[յEsOsZB%RԌ3␨^G: 6ytRHs"x-g45f5 nqM \4GX,O.Vԣ9TcUAS˕ -Mm{Sܤarl#%jU,m,q66S%q.+DQT7 c}Ŋy?C;f6hejrhJ2Ԑ!O ;E9R03z~"1y+.iphW3x JdChW h3cy5kDEbP3* aj-瘖T,-ӑ%ByN3+ aJ;D'8XbaU4kL2Y yZM 6P+ǍAmU%Q=nCrGlHh[yz0Z2xvី@l He'dA{Ţb OUbnBxftyjR95 70c`FVv~ek]s 23fꆗB%@QHHix1Pz*@r@K(,F -KGR%\ SsuP1ըJq!=ǫW&tu03k(;3FFTcn8t%0>C ϫHm!JBPkWS,6phWcg:hgz2%x=3bpE uh̠$4=MIOgOS.7Dp&Y5xXZ*edNnGf6%Sii|'$*5 ,gH%4D~5bk^qSDE#d  Uv0b6a 6666YvZ(+*466-_9c { 60CyX>uȚL IU2s`s"ԆY׊V ZQŊc& -9eM"+NQ+~dݔM+f+YlcTnuQLLs ڵq0ˆr!'ӗ~_40m̬%[ =W`ڽG}=l |a` 8tҠM`,+'tGbMȕf&BrMH@Ruh-İXnjFlԂF꫔Q3A*6h` #s&oW[@%׳ǰچna/:S/پoa'W>O]p\ѐadml0dIIgm2D6~gc]6^Gl|d[+!elԲqW6vdcō%L seCJ$:-L.b_;/JO(Ҟx%6+~d%¤7|L~Or g_c+%[V޸_B"ءPM HqU!UXG$Rp͚x0Zh7[,y+}~ dш)0#d?<\緞}u آ㺹jH7LUT5!ѓiO -Y$sUBxp FeeJb<,5Uq̀EVV%!u~og@GF6lX5Y?1c&}W\fL:}.77B:%53!űYV+bf*bsBbuZnc3"9;pY % L(ӗ@i8#x2b%Q4\~CjlSNr\Ve㲲DўdcY-)kBIL~$Wc䒬ՒVUqnpYU%Hf\DËÍԂK}lJ'a zO=N$:u:G#nl~.PHX4AXY\;۞zq&nVM_} -a pK?e 穜z͜ͳH0p+)9r -Ƅa%pVb$!a> r[ +!HGzAb:i%FS,R^wFƒ1yAzGbvKxt4IX.I"MO=n]&)"1q|$<'$$c/8UqRĀ!iҀtZdT=(a/m%,$f-)$>(YrIe+e~${HJ @14jk -®1WS? )15}n0})*z?Cs  }B6<6s=SZBOEDC[e 8Ut2J̺ΰ1N˕ D )b5q@ EU{ă"/l IJ$[K 'D*qqY^hbI/Kb Ǒ =p_zolnwA΁Oً[лi\45^pvcdNr.AH'd6/c-ѐQdS!2یG4'GOfm$LGu>f'Ϟo[_]~7:8/yKDa>Du_, -[-:ɽ! 8$J<`@B̆eEbdB-%^wb,6c<ľWߜ+N j Ǝt\^ɦ)رiYDE,YDF?,2fVx>)0;y 崡a'P&KR`8GP`0fB_ tHSxnpwnYv2Vm-oVIz6C'p@Z$p)IV+ Y]TIөbIS8\zzvT}=NDē%X ̆)Џnc/w̺)~D7~?8 U%Z+HU!d\U!Vp=V \}BXrqt^`DQj.[0E0D֮YUۍ`фަsҲ5mj-\ WĎjE!5=b#e$)! `ߝ=x:t#x*@]f~k٦ʯ OwjĖ7\9 ;V9/0cVav#/:2+6bFtl!#2A#ۈa#VЈ`֏{c#FX6s̈F -P)G)nJ_@#N>M{y1ʲPQFm9?%]x`(ȅԂ\r&}2n]TH˜tA %r]wP5 -YH -7CÆ6TlHʡ6XUʯyJPSC8||wԅY zNJ5,"m%y`0CVv1b&;e؜ˆbD$$:"/A]4.af 9#,v-|_N1FLWbC7}W*nNŃ؛T,H~BR`E)vJQfLt/_N moݛ1GS1t畩2V0M)`O-*$,Ih7gXI?cM3;bIJJ#2`DžЏ\vúG~FeFaS}vPJ[so@nw59yTyxfxtxǃ<Px;rd-wc\1̌$ro "kE L^zr2X+Le萑2ߕ%%}2.:X'c']FNֳ2~3𼌰[ƟRvqE#6iOޑ9#528[(g~q=AXF6Mf?2 cT}-'}2ZeDߑQq73KpX -zJJPr {!GYIS]Imms/Ĭ$ Pʥ -ݬuۈbn6[iMa  cvsUAX"\NdE8`͜=vUKbY5jsh4|yWHֿ4r_u妜i,WY8 xы8Έ#1vΈW ӴD YSt9qb_F\0̜hF#;F|r9)|*n7MtC9s#F|Px"PtzvTՅ/Nw HiBBrn]ⶳ~X/^={ n:_w /Wd[wB PdcəFu+isCfxCmqFGnaL0 "q& E&$eeGrHup7э$8^?+}  9 -5jh13̲iRr8dH0Fe: ' # ɅD5c(yd^< -R-BI1ae؊XyLZa8L~?N }[Vl__h-خ^($"p7Cf~3g|7?ijfwgyCȅC[Vuw,Ȝb"%jmwo.|̜''cyd˕O&$͈, (B8$0D$*\㶤y)1 Ü{TyD܏ИJIO%sbR2/rc>DYN l6Kp(10 :K,'z&N*q'*}~q8,?~c#ޢ{m/~f&Xp1~12?7rӹ{|?ßΐnXdxp"!)an"xu?H-glb+t^:X0`X^GI,; i#d6MB16 xto x[3:6@686+Nc &XNFX --@ 4C;4K "RVha EPmJX - -̄6X $rC@1ZZ(Pעra5,p5,Ű -B3(̀6X -p%Da ,Bik2"ZJ(0CɏJq1) -a-\ -iR(S1Tv=ZJ.D3BU##V2X˩/RZ1mq uԫQh|sǏGH-n= -'(őXiL( xhL - -a#D$ZW=?\+/cRq5ǘmH ͆Mz JB UP!``0`TX?CT8&dv eCz,R T.Up ?Nz ~=X -g,i8q @=4OafA=f>^GoLJ4 {EN A|s-vqO -N:̈́OaSN={Bq}sbDz\_::Mc¯/23`fic~Mu}1s594p%/c  p(0C5o\9WW3V}/MsA?cEثU6ɫgi.xEf_dԤ1}TQ; 0 0 X4@P1M\ez9s/I<y3azdzTQKh~͏k0\hWo Nu|$#Y?xw]~3@|~\nWw[n||w%mm̝w];p⭁[Ak]j\ѠϕAggo#LkC:n$m+]EtLO֋~^`\A)\ -}E+] YopLp%Yyd"E3[VmfעZPk32jmrjjװUc\1*&TGf\kVCwpLQd\o)23C~s]hQF\f79\en3wbymp WW;_nռVׄ_ЇxO莻chE ZASIAsP=ھ.7h;yyEsssQB!h0͍B.yF17 b"+ d8: Rh;!ȓF#SɱW#l -endstream -endobj - 265 0 obj [/ICCBased 267 0 R] endobj @@ -3488,9 +3488,9 @@ endobj /x0 295 0 R >> /Font << - /f0 253 0 R - /f1 259 0 R - /f2 247 0 R + /f0 259 0 R + /f1 247 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -3535,9 +3535,9 @@ endobj /c1 266 0 R >> /Font << - /f0 259 0 R - /f1 253 0 R - /f2 247 0 R + /f0 247 0 R + /f1 259 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -3593,9 +3593,9 @@ endobj /c1 266 0 R >> /Font << - /f0 259 0 R - /f1 253 0 R - /f2 247 0 R + /f0 247 0 R + /f1 259 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -3649,9 +3649,9 @@ endobj /c1 266 0 R >> /Font << - /f0 259 0 R - /f1 253 0 R - /f2 247 0 R + /f0 247 0 R + /f1 259 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -3695,9 +3695,9 @@ endobj /c1 266 0 R >> /Font << - /f0 259 0 R - /f1 253 0 R - /f2 247 0 R + /f0 247 0 R + /f1 259 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -3740,9 +3740,9 @@ endobj /c1 266 0 R >> /Font << - /f0 259 0 R - /f1 253 0 R - /f2 247 0 R + /f0 247 0 R + /f1 259 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -3813,9 +3813,9 @@ endobj /c1 266 0 R >> /Font << - /f0 259 0 R - /f1 253 0 R - /f2 247 0 R + /f0 247 0 R + /f1 259 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -3873,9 +3873,9 @@ endobj /c1 266 0 R >> /Font << - /f0 259 0 R - /f1 253 0 R - /f2 247 0 R + /f0 247 0 R + /f1 259 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -3935,9 +3935,9 @@ endobj /c1 266 0 R >> /Font << - /f0 259 0 R - /f1 253 0 R - /f2 247 0 R + /f0 247 0 R + /f1 259 0 R + /f2 253 0 R >> >> /MediaBox [0 0 595.2756 841.8898] @@ -4193,8 +4193,8 @@ endobj 296 0 obj << /Creator (Typst 0.14.0) - /ModDate (D:20251123224454Z) - /CreationDate (D:20251123224454Z) + /ModDate (D:20251127070017Z) + /CreationDate (D:20251127070017Z) >> endobj @@ -4205,7 +4205,7 @@ endobj /Subtype /XML >> stream -Typst 0.14.0en2025-11-23T22:44:54+00:002025-11-23T22:44:54+00:009application/pdfMDJi4l/9czZ4xUroYkw2Ow==MDJi4l/9czZ4xUroYkw2Ow==proof1.7 +Typst 0.14.0en2025-11-27T07:00:17+00:002025-11-27T07:00:17+00:009application/pdfpCyTvtS3mwQnOhgjbat+WQ==pCyTvtS3mwQnOhgjbat+WQ==proof1.7 endstream endobj @@ -4477,23 +4477,23 @@ xref 0000056716 00000 n 0000056798 00000 n 0000056943 00000 n -0000057119 00000 n -0000058475 00000 n -0000058566 00000 n -0000058815 00000 n -0000061427 00000 n -0000076776 00000 n -0000076941 00000 n -0000078078 00000 n -0000078169 00000 n -0000078458 00000 n -0000080070 00000 n -0000092035 00000 n -0000092205 00000 n -0000092886 00000 n -0000092977 00000 n -0000093268 00000 n -0000094460 00000 n +0000057113 00000 n +0000057794 00000 n +0000057885 00000 n +0000058176 00000 n +0000059368 00000 n +0000067934 00000 n +0000068110 00000 n +0000069466 00000 n +0000069557 00000 n +0000069806 00000 n +0000072418 00000 n +0000087767 00000 n +0000087932 00000 n +0000089069 00000 n +0000089160 00000 n +0000089449 00000 n +0000091061 00000 n 0000103026 00000 n 0000103064 00000 n 0000103102 00000 n @@ -4533,7 +4533,7 @@ trailer /Size 299 /Root 298 0 R /Info 296 0 R - /ID [(MDJi4l/9czZ4xUroYkw2Ow==) (MDJi4l/9czZ4xUroYkw2Ow==)] + /ID [(pCyTvtS3mwQnOhgjbat+WQ==) (pCyTvtS3mwQnOhgjbat+WQ==)] >> startxref 259500 diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.3-7.png b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.3-7.png new file mode 100644 index 0000000..3f6c33c Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.3-7.png differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.4-11.png b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.4-11.png new file mode 100644 index 0000000..4af0b47 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.4-11.png differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-1.png b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-1.png new file mode 100644 index 0000000..81c917b Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-1.png differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-13.png b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-13.png new file mode 100644 index 0000000..ac53b0b Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-13.png differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-3.png b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-3.png new file mode 100644 index 0000000..83f2f31 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Exercise-pictures/10.5-3.png differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.pdf b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.pdf index c3bcc35..9dbee77 100644 Binary files a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.pdf and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.pdf differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.typ b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.typ index e69de29..74cc336 100644 --- a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.typ +++ b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Graphs and the Cantor-Schröder-Bernstein theorem.typ @@ -0,0 +1,184 @@ +#import "@local/dtu-template:0.5.1":* + +#show: dtu-note.with( + course: "01017", + course-name: "Discrete Mathematics", + title: "Lecture - November 27, 2025", + date: datetime(day: 27, month: 11, year: 2025), + author: "Rasmus Rosendahl-Kaa (S255955)", + semester: "2025 Fall", +) + += Graphs + +== Konigsberg Bridge problem +Consider a river with an island, and where it diverges into two streams. There are 7 bridges to the island. + +#image("Lecture-pictures/Konigsberg Bridge Problem.png") + +Can you start somewhere and walk over every bridge without walking over a bridge more than once. + +*Solution:*\ +The layout of the bridges doesn't matter. You can represent the island, and the land areas as points. So 4 points, and lines between them. What you have is a graph. The points are *vertexes* and the lines/curves are *edge* (you can have single-edges, double-edges, and so on). If you have more than 3 vertexes between two vertexes, its called a multiple edge. A loop is an edge looping to the same vertex. + +If there are no loops or multiple edges, the graph is called _simple_. + +#definition(title: "Vertex")[ +A vertex is the points in the graph. + +$V(G)={v_1,v_2,dots,v_n}$ are the vertices in graph $G$ + +An isolated vertex is a singular vertex with no edges. +] + +#proof()[ + Assume $G$ is connected and every vertex has even degree. We shall find a closed Euler walk.\ + You start in a vertex $s$ and go to a random neighbor. Continue this until you can't anymore. You _will_ stop in $s$ because else $s$ will always have degree 1. + + This doesn't prove it necessarily because you don't know if some vertex has another edge. Though you can solve this by starting from the vertex that has more edges, and assuming that is a smaller graph, so you complete that. +] + +#theorem()[ + A graph $G$ has a closed Euler walk if and only if ($<=>$) every vertex has even degree and $G$ is connected. +] + +#definition(title: "Degree")[ + The amount of edges going out of a vertex. +] + +#definition(title: "Euler circuit/walk/tour")[ +*Closed (circuit):*\ +You start in a vertex $s$ and go through every edge precisely once and return in $s$. + +*Open (path):*\ +The same as closed, but you return to $t$, not $s$ +] + +#definition(title: "Connected")[ +A graph is called connected when there's a path between every vertexes ($forall x,y in V(G)$).\ +If not, it is called *disconnected*. + +- *Strongly connected:* A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph. +- *Weekly connected:* A directed graph is weakly connected if there is a path between every two vertices in the +underlying undirected graph. +] +#definition(title: "Connected component (piece)")[ +For a graph, there is not connected, you can have connected components. Is is a subgraph of the graph that is connected. + +You can also call it the maximal connected subgraph +] + +== Degree sequence +#definition(title: "Complete")[ +A complete graph of _n_ vertices, denoted $K_n$, is a simple graph that contains exactly one edge between each pair of distinct vertices. Like a tetrahedron +#image("Lecture-pictures/Tetrahedron.svg",width: 50%) + +A tetrahedron can be realized as a degree sequence ($3,3,3,3$) +] + + + +#definition(title: "Degree Sequence")[ + The *degree sequence* of a graph is a list of the degrees of its vertices, usually written in non-increasing order. + + For example, if a graph has vertices with degrees 3, 2, 2, and 1, its degree sequence is (3, 2, 2, 1). +] + +#lemma()[ +*Hand shaking Lemma* + +For a graph $G$ with vertices $v_1,v_2,dots,v_n$ and degrees\ $deg(v_1),deg(v_2),dots,deg(v_n)$. +$ +sum^(n)_(i=1)deg(v_i)=2|E(G)| +$ +#proof()[ +Each edge is contributing to the total degree twice, so the sum of the degrees are just the twice the amount of edges. +] + +Every graph has an even number of odd-degree vertices, because to get an even degree you can't have a single odd degree. +] + +#definition(title: "Path")[ +The path is a line of vertices. $P_n$ +] +#definition(title: "Cycle")[ +#image("Lecture-pictures/Cycle.png") + +] +#definition(title: "Wheel")[ +#image("Lecture-pictures/Wheel.png") +] +#definition(title: "n-cube")[ +#image("Lecture-pictures/n-Cube.png") +$Q_n$ + +$ V(Q_n)={(x_1,x_2,dots, x_(n))| x_(i) in {0,1}} $ + +Two vertices $(x_1,dots,x_n)$ and $(y_1,dots,y_n)$ are neighbors if and only if $exists ! i : x_i eq.not y_i$ +] + +#definition(title: "Tree and forest")[ +A tree (like a family tree) is also a graph. It is a connected graph without cycle + +A forest is a (_non-connected_) graph without cycle +] + +#definition(title: "Bipartite")[ +A bipartite graph with $"bipartite"(V_1,V_2)$ is a graph with $V(G)=V_1 union V_2$ and all its edges go from $V_1 "to" V_2$ + +] + + +== The marriage problem +You have a collection of men (set $V_1$) and a collection of women (set $V_2$). Some man may know some women. + +The problem is: Can every man marry a women, when a man can only marry _one_ woman.\ +It will not be possible to solve if you have more men than women. Even if you have more women than men, there is a possibility of not all men having married, if, for example, 3 men know some 3 women out of 4, but there is a man who doesn't know any women. + +#theorem(title: "Hall's marriage problem")[ +A bipartite graph $G$ with bipartitian $V_1,V_2$ has a matching with $|V_1|$ edges if and only if: +$ forall S subset.eq V_1, |N(S)| >= |S| $ +which is called Hall's condition. + +So if you take 3 men, they should know at least 3 women. + +Hall's condition solves the marriage problem because you can take a subset containing one man, which then must be at least 1 (so every man will know at least one woman if Hall's condition is true). +] + +#definition(title: "Matching")[ +Every vertex has degree at most 1 +] + +== Isomorphism problem +#definition(title: "Isomorphism problem")[ +#image("Lecture-pictures/Isomorphism.jpg",width:40%) +$ G tilde.eq H <=>^("def") V(G)={v_1,v_2,dots,v_n}, V(H)={u_1,u_2,dots,u_n}\ +v_i v_j "is an edge" <=> u_i u_j "is an edge" +$ +A graph is isomorphism if, for every pair of vertices in one that are neighbors, a likewise pair should be in the other (though they can be arranged in a different way) +] + +== Adjacency matrix +#set math.mat(delim: "[") +#definition(title: "Adjacency matrix")[ +#image("Lecture-pictures/Adjacency-Matrix.jpg", width: 50%) +A matrix consisting of all the neighbors: + +$mat("",v_1,v_2,v_3,v_4;v_1,0,1,0,1;v_2,1,0,1,1;v_3,0,1,0,1;v_4,1,1,1,0)$ +] + +#definition(title: "Directed graph")[ +There is a path through the graph + +*Strongly directed:* A graph $D$ is strongly connected if and only f $forall x,y in V(D):$ D contains a directed path from $x$ to $y$ and from $y$ to $x$ + +*Weekly directed:* If you take all the directed away, you will be left with an undirected graph which is still connected. +] + +Strong component: The directed path is also the maximal connected graph. + +#definition(title: "Complementary")[ + The complementary graph $overline(G)$ of a simple graph $G$ has the same vertices as $G$. Two vertices are adjacent in $G$ if and only if they are not adjacent in $G$. Describe each of these graphs. + + A simple graph G is called self-complementary if $G$ and $overline(G)$ are isomorphic +] diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Cycle.png b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Cycle.png new file mode 100644 index 0000000..5871f7f Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Cycle.png differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Isomorphism.jpg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Isomorphism.jpg new file mode 100644 index 0000000..0f56d81 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Isomorphism.jpg differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_071126036.jpg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_071126036.jpg new file mode 100644 index 0000000..7d8dec6 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_071126036.jpg differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_073437932.jpg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_073437932.jpg new file mode 100644 index 0000000..19643fe Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_073437932.jpg differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075411848.jpg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075411848.jpg new file mode 100644 index 0000000..c9ccde7 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075411848.jpg differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075419885.jpg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075419885.jpg new file mode 100644 index 0000000..e97347d Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075419885.jpg differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075641278.jpg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075641278.jpg new file mode 100644 index 0000000..6815365 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_075641278.jpg differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_083623052.jpg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_083623052.jpg new file mode 100644 index 0000000..a09d7ec Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_083623052.jpg differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_083624803.jpg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_083624803.jpg new file mode 100644 index 0000000..74f1a50 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/PXL_20251127_083624803.jpg differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Tetrahedron.svg b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Tetrahedron.svg new file mode 100644 index 0000000..d04c05c --- /dev/null +++ b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Tetrahedron.svg @@ -0,0 +1,176 @@ + + + + + + + + + + + + + + + + +Strecke g +Strecke g: Strecke E, F + + + + + + + + + +Strecke h +Strecke h: Strecke F, B + + + + + + + + + +Strecke i +Strecke i: Strecke B, E + + + + + + + + + +Strecke g_1 +Strecke g_1: Strecke J, F + + + + + + + + + +Strecke h_1 +Strecke h_1: Strecke J, E + + + + + + + + + +Strecke i_1 +Strecke i_1: Strecke J, B + + + + + + + + + +Punkt B +Punkt B: Punkt auf yAchse + + + + + + + + + +Punkt B +Punkt B: Punkt auf yAchse + + + + + + + + + +Punkt E +Punkt E: Schnittpunkt von c, f + + + + + + + + + +Punkt E +Punkt E: Schnittpunkt von c, f + + + + + + + + + +Punkt F +Punkt F: Schnittpunkt von c, e + + + + + + + + + +Punkt F +Punkt F: Schnittpunkt von c, e + + + + + + + + + +Punkt J +Punkt J: Schnittpunkt von j, k + + + + + + + + + +Punkt J +Punkt J: Schnittpunkt von j, k + + + + + + diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Wheel.png b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Wheel.png new file mode 100644 index 0000000..3b48862 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/Wheel.png differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/n-Cube.png b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/n-Cube.png new file mode 100644 index 0000000..bd68823 Binary files /dev/null and b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Lecture-pictures/n-Cube.png differ diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Opgaver.pdf b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Opgaver.pdf new file mode 100644 index 0000000..7be6545 --- /dev/null +++ b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Opgaver.pdf @@ -0,0 +1,2800 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 4 + /Kids [174 0 R 176 0 R 178 0 R 180 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 15 0 R + /Count 5 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Next 5 0 R + /First 4 0 R + /Last 4 0 R + /Count -1 + /Title (10.1) + /Dest 159 0 R +>> +endobj + +4 0 obj +<< + /Parent 3 0 R + /Title (11) + /Dest 158 0 R +>> +endobj + +5 0 obj +<< + /Parent 2 0 R + /Next 9 0 R + /Prev 3 0 R + /First 6 0 R + /Last 8 0 R + /Count -3 + /Title (10.2) + /Dest 163 0 R +>> +endobj + +6 0 obj +<< + /Parent 5 0 R + /Next 7 0 R + /Title (5) + /Dest 160 0 R +>> +endobj + +7 0 obj +<< + /Parent 5 0 R + /Next 8 0 R + /Prev 6 0 R + /Title (43) + /Dest 161 0 R +>> +endobj + +8 0 obj +<< + /Parent 5 0 R + /Prev 7 0 R + /Title ((53)) + /Dest 162 0 R +>> +endobj + +9 0 obj +<< + /Parent 2 0 R + /Next 12 0 R + /Prev 5 0 R + /First 10 0 R + /Last 11 0 R + /Count -2 + /Title (10.3) + /Dest 166 0 R +>> +endobj + +10 0 obj +<< + /Parent 9 0 R + /Next 11 0 R + /Title ((7)) + /Dest 164 0 R +>> +endobj + +11 0 obj +<< + /Parent 9 0 R + /Prev 10 0 R + /Title (57) + /Dest 165 0 R +>> +endobj + +12 0 obj +<< + /Parent 2 0 R + /Next 15 0 R + /Prev 9 0 R + /First 13 0 R + /Last 14 0 R + /Count -2 + /Title (10.4) + /Dest 169 0 R +>> +endobj + +13 0 obj +<< + /Parent 12 0 R + /Next 14 0 R + /Title (9) + /Dest 167 0 R +>> +endobj + +14 0 obj +<< + /Parent 12 0 R + /Prev 13 0 R + /Title (11) + /Dest 168 0 R +>> +endobj + +15 0 obj +<< + /Parent 2 0 R + /Prev 12 0 R + /First 16 0 R + /Last 18 0 R + /Count -3 + /Title (10.5) + /Dest 173 0 R +>> +endobj + +16 0 obj +<< + /Parent 15 0 R + /Next 17 0 R + /Title ((1)) + /Dest 170 0 R +>> +endobj + +17 0 obj +<< + /Parent 15 0 R + /Next 18 0 R + /Prev 16 0 R + /Title (3) + /Dest 171 0 R +>> +endobj + +18 0 obj +<< + /Parent 15 0 R + /Prev 17 0 R + /Title (13) + /Dest 172 0 R +>> +endobj + +19 0 obj +<< + /Type /StructTreeRoot + /RoleMap << + /Datetime /Span + /Terms /Part + /Title /P + /Strong /Span + /Em /Span + >> + /K [24 0 R] + /ParentTree << + /Nums [0 20 0 R 1 21 0 R 2 22 0 R 3 23 0 R] + >> + /ParentTreeNextKey 4 +>> +endobj + +20 0 obj +[123 0 R 122 0 R 116 0 R 121 0 R 116 0 R 120 0 R 116 0 R 119 0 R 116 0 R 118 0 R 118 0 R 118 0 R 116 0 R 116 0 R 117 0 R 117 0 R 117 0 R 117 0 R 117 0 R 116 0 R 115 0 R 114 0 R 113 0 R 108 0 R 112 0 R 108 0 R 111 0 R 108 0 R 110 0 R 108 0 R 109 0 R 106 0 R 105 0 R 104 0 R 101 0 R 100 0 R 99 0 R 99 0 R 99 0 R 97 0 R 96 0 R 95 0 R 94 0 R 93 0 R 93 0 R 93 0 R 93 0 R 93 0 R 93 0 R 93 0 R 93 0 R 93 0 R 93 0 R 93 0 R 91 0 R 89 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 88 0 R 86 0 R 85 0 R 84 0 R 83 0 R 81 0 R 80 0 R] +endobj + +21 0 obj +[79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 79 0 R 77 0 R 76 0 R 75 0 R 73 0 R 72 0 R 72 0 R 70 0 R 68 0 R 67 0 R 66 0 R 65 0 R 64 0 R 64 0 R 64 0 R 64 0 R 64 0 R 62 0 R 61 0 R 61 0 R 60 0 R 59 0 R] +endobj + +22 0 obj +[57 0 R 56 0 R 55 0 R 53 0 R 51 0 R 49 0 R 47 0 R 45 0 R 42 0 R 41 0 R 41 0 R 41 0 R 39 0 R 38 0 R] +endobj + +23 0 obj +[37 0 R 36 0 R 35 0 R 34 0 R 33 0 R 31 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 32 0 R 30 0 R 29 0 R 29 0 R 27 0 R 26 0 R 25 0 R 25 0 R] +endobj + +24 0 obj +<< + /Type /StructElem + /S /Document + /P 19 0 R + /K [123 0 R 122 0 R 116 0 R 115 0 R 102 0 R 101 0 R 100 0 R 98 0 R 97 0 R 96 0 R 95 0 R 90 0 R 89 0 R 87 0 R 86 0 R 85 0 R 84 0 R 82 0 R 81 0 R 80 0 R 78 0 R 77 0 R 69 0 R 68 0 R 67 0 R 66 0 R 65 0 R 63 0 R 62 0 R 61 0 R 60 0 R 58 0 R 57 0 R 56 0 R 43 0 R 42 0 R 40 0 R 39 0 R 38 0 R 37 0 R 36 0 R 35 0 R 34 0 R 33 0 R 31 0 R 30 0 R 28 0 R 27 0 R 26 0 R 25 0 R] +>> +endobj + +25 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [32 33] + /Pg 180 0 R +>> +endobj + +26 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [31] + /Pg 180 0 R +>> +endobj + +27 0 obj +<< + /Type /StructElem + /S /Figure + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [30] + /Pg 180 0 R +>> +endobj + +28 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [29 0 R] +>> +endobj + +29 0 obj +<< + /Type /StructElem + /S /Em + /P 28 0 R + /K [28 29] + /Pg 180 0 R +>> +endobj + +30 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T (13) + /K [27] + /Pg 180 0 R +>> +endobj + +31 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [5 32 0 R] + /Pg 180 0 R +>> +endobj + +32 0 obj +<< + /Type /StructElem + /S /Formula + /P 31 0 R + /K [6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26] + /Pg 180 0 R +>> +endobj + +33 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [4] + /Pg 180 0 R +>> +endobj + +34 0 obj +<< + /Type /StructElem + /S /Figure + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [3] + /Pg 180 0 R +>> +endobj + +35 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T (3) + /K [2] + /Pg 180 0 R +>> +endobj + +36 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [1] + /Pg 180 0 R +>> +endobj + +37 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [0] + /Pg 180 0 R +>> +endobj + +38 0 obj +<< + /Type /StructElem + /S /Figure + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [13] + /Pg 178 0 R +>> +endobj + +39 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T ((1)) + /K [12] + /Pg 178 0 R +>> +endobj + +40 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [41 0 R] +>> +endobj + +41 0 obj +<< + /Type /StructElem + /S /Em + /P 40 0 R + /K [9 10 11] + /Pg 178 0 R +>> +endobj + +42 0 obj +<< + /Type /StructElem + /S /H1 + /P 24 0 R + /T (10.5) + /K [8] + /Pg 178 0 R +>> +endobj + +43 0 obj +<< + /Type /StructElem + /S /L + /P 24 0 R + /A [<< + /O /List + /ListNumbering /None + >>] + /K [52 0 R 48 0 R 44 0 R] +>> +endobj + +44 0 obj +<< + /Type /StructElem + /S /LI + /P 43 0 R + /K [46 0 R 45 0 R] +>> +endobj + +45 0 obj +<< + /Type /StructElem + /S /LBody + /P 44 0 R + /K [7] + /Pg 178 0 R +>> +endobj + +46 0 obj +<< + /Type /StructElem + /S /Lbl + /P 44 0 R + /K [47 0 R] +>> +endobj + +47 0 obj +<< + /Type /StructElem + /S /Strong + /P 46 0 R + /K [6] + /Pg 178 0 R +>> +endobj + +48 0 obj +<< + /Type /StructElem + /S /LI + /P 43 0 R + /K [50 0 R 49 0 R] +>> +endobj + +49 0 obj +<< + /Type /StructElem + /S /LBody + /P 48 0 R + /K [5] + /Pg 178 0 R +>> +endobj + +50 0 obj +<< + /Type /StructElem + /S /Lbl + /P 48 0 R + /K [51 0 R] +>> +endobj + +51 0 obj +<< + /Type /StructElem + /S /Strong + /P 50 0 R + /K [4] + /Pg 178 0 R +>> +endobj + +52 0 obj +<< + /Type /StructElem + /S /LI + /P 43 0 R + /K [54 0 R 53 0 R] +>> +endobj + +53 0 obj +<< + /Type /StructElem + /S /LBody + /P 52 0 R + /K [3] + /Pg 178 0 R +>> +endobj + +54 0 obj +<< + /Type /StructElem + /S /Lbl + /P 52 0 R + /K [55 0 R] +>> +endobj + +55 0 obj +<< + /Type /StructElem + /S /Strong + /P 54 0 R + /K [2] + /Pg 178 0 R +>> +endobj + +56 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [1] + /Pg 178 0 R +>> +endobj + +57 0 obj +<< + /Type /StructElem + /S /Figure + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [0] + /Pg 178 0 R +>> +endobj + +58 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [59 0 R] +>> +endobj + +59 0 obj +<< + /Type /StructElem + /S /Em + /P 58 0 R + /K [46] + /Pg 176 0 R +>> +endobj + +60 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T (11) + /K [45] + /Pg 176 0 R +>> +endobj + +61 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [43 44] + /Pg 176 0 R +>> +endobj + +62 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [42] + /Pg 176 0 R +>> +endobj + +63 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [64 0 R] +>> +endobj + +64 0 obj +<< + /Type /StructElem + /S /Em + /P 63 0 R + /K [37 38 39 40 41] + /Pg 176 0 R +>> +endobj + +65 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T (9) + /K [36] + /Pg 176 0 R +>> +endobj + +66 0 obj +<< + /Type /StructElem + /S /H1 + /P 24 0 R + /T (10.4) + /K [35] + /Pg 176 0 R +>> +endobj + +67 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [34] + /Pg 176 0 R +>> +endobj + +68 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [33] + /Pg 176 0 R +>> +endobj + +69 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [76 0 R 74 0 R 73 0 R 71 0 R 70 0 R] +>> +endobj + +70 0 obj +<< + /Type /StructElem + /S /Em + /P 69 0 R + /K [32] + /Pg 176 0 R +>> +endobj + +71 0 obj +<< + /Type /StructElem + /S /Formula + /P 69 0 R + /K [72 0 R] +>> +endobj + +72 0 obj +<< + /Type /StructElem + /S /Em + /P 71 0 R + /K [30 31] + /Pg 176 0 R +>> +endobj + +73 0 obj +<< + /Type /StructElem + /S /Em + /P 69 0 R + /K [29] + /Pg 176 0 R +>> +endobj + +74 0 obj +<< + /Type /StructElem + /S /Formula + /P 69 0 R + /K [75 0 R] +>> +endobj + +75 0 obj +<< + /Type /StructElem + /S /Em + /P 74 0 R + /K [28] + /Pg 176 0 R +>> +endobj + +76 0 obj +<< + /Type /StructElem + /S /Em + /P 69 0 R + /K [27] + /Pg 176 0 R +>> +endobj + +77 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T (57) + /K [26] + /Pg 176 0 R +>> +endobj + +78 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [79 0 R] +>> +endobj + +79 0 obj +<< + /Type /StructElem + /S /Formula + /P 78 0 R + /K [0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25] + /Pg 176 0 R +>> +endobj + +80 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [75] + /Pg 174 0 R +>> +endobj + +81 0 obj +<< + /Type /StructElem + /S /Figure + /P 24 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [74] + /Pg 174 0 R +>> +endobj + +82 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [83 0 R] +>> +endobj + +83 0 obj +<< + /Type /StructElem + /S /Em + /P 82 0 R + /K [73] + /Pg 174 0 R +>> +endobj + +84 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T ((7)) + /K [72] + /Pg 174 0 R +>> +endobj + +85 0 obj +<< + /Type /StructElem + /S /H1 + /P 24 0 R + /T (10.3) + /K [71] + /Pg 174 0 R +>> +endobj + +86 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T ((53)) + /K [70] + /Pg 174 0 R +>> +endobj + +87 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [88 0 R] +>> +endobj + +88 0 obj +<< + /Type /StructElem + /S /Formula + /P 87 0 R + /K [56 57 58 59 60 61 62 63 64 65 66 67 68 69] + /Pg 174 0 R +>> +endobj + +89 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [55] + /Pg 174 0 R +>> +endobj + +90 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [94 0 R 92 0 R 91 0 R] +>> +endobj + +91 0 obj +<< + /Type /StructElem + /S /Em + /P 90 0 R + /K [54] + /Pg 174 0 R +>> +endobj + +92 0 obj +<< + /Type /StructElem + /S /Formula + /P 90 0 R + /K [93 0 R] +>> +endobj + +93 0 obj +<< + /Type /StructElem + /S /Em + /P 92 0 R + /K [43 44 45 46 47 48 49 50 51 52 53] + /Pg 174 0 R +>> +endobj + +94 0 obj +<< + /Type /StructElem + /S /Em + /P 90 0 R + /K [42] + /Pg 174 0 R +>> +endobj + +95 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T (43) + /K [41] + /Pg 174 0 R +>> +endobj + +96 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [40] + /Pg 174 0 R +>> +endobj + +97 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [39] + /Pg 174 0 R +>> +endobj + +98 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [99 0 R] +>> +endobj + +99 0 obj +<< + /Type /StructElem + /S /Em + /P 98 0 R + /K [36 37 38] + /Pg 174 0 R +>> +endobj + +100 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T (5) + /K [35] + /Pg 174 0 R +>> +endobj + +101 0 obj +<< + /Type /StructElem + /S /H1 + /P 24 0 R + /T (10.2) + /K [34] + /Pg 174 0 R +>> +endobj + +102 0 obj +<< + /Type /StructElem + /S /L + /P 24 0 R + /A [<< + /O /List + /ListNumbering /Circle + >>] + /K [107 0 R 103 0 R] +>> +endobj + +103 0 obj +<< + /Type /StructElem + /S /LI + /P 102 0 R + /K [106 0 R 104 0 R] +>> +endobj + +104 0 obj +<< + /Type /StructElem + /S /LBody + /P 103 0 R + /K [105 0 R 33] + /Pg 174 0 R +>> +endobj + +105 0 obj +<< + /Type /StructElem + /S /Strong + /P 104 0 R + /K [32] + /Pg 174 0 R +>> +endobj + +106 0 obj +<< + /Type /StructElem + /S /Lbl + /P 103 0 R + /K [31] + /Pg 174 0 R +>> +endobj + +107 0 obj +<< + /Type /StructElem + /S /LI + /P 102 0 R + /K [114 0 R 108 0 R] +>> +endobj + +108 0 obj +<< + /Type /StructElem + /S /LBody + /P 107 0 R + /K [113 0 R 23 112 0 R 25 111 0 R 27 110 0 R 29 109 0 R] + /Pg 174 0 R +>> +endobj + +109 0 obj +<< + /Type /StructElem + /S /Formula + /P 108 0 R + /K [30] + /Pg 174 0 R +>> +endobj + +110 0 obj +<< + /Type /StructElem + /S /Formula + /P 108 0 R + /K [28] + /Pg 174 0 R +>> +endobj + +111 0 obj +<< + /Type /StructElem + /S /Formula + /P 108 0 R + /K [26] + /Pg 174 0 R +>> +endobj + +112 0 obj +<< + /Type /StructElem + /S /Formula + /P 108 0 R + /K [24] + /Pg 174 0 R +>> +endobj + +113 0 obj +<< + /Type /StructElem + /S /Strong + /P 108 0 R + /K [22] + /Pg 174 0 R +>> +endobj + +114 0 obj +<< + /Type /StructElem + /S /Lbl + /P 107 0 R + /K [21] + /Pg 174 0 R +>> +endobj + +115 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [20] + /Pg 174 0 R +>> +endobj + +116 0 obj +<< + /Type /StructElem + /S /P + /P 24 0 R + /K [2 121 0 R 4 120 0 R 6 119 0 R 8 118 0 R 12 13 117 0 R 19] + /Pg 174 0 R +>> +endobj + +117 0 obj +<< + /Type /StructElem + /S /Formula + /P 116 0 R + /K [14 15 16 17 18] + /Pg 174 0 R +>> +endobj + +118 0 obj +<< + /Type /StructElem + /S /Formula + /P 116 0 R + /K [9 10 11] + /Pg 174 0 R +>> +endobj + +119 0 obj +<< + /Type /StructElem + /S /Formula + /P 116 0 R + /K [7] + /Pg 174 0 R +>> +endobj + +120 0 obj +<< + /Type /StructElem + /S /Formula + /P 116 0 R + /K [5] + /Pg 174 0 R +>> +endobj + +121 0 obj +<< + /Type /StructElem + /S /Formula + /P 116 0 R + /K [3] + /Pg 174 0 R +>> +endobj + +122 0 obj +<< + /Type /StructElem + /S /H2 + /P 24 0 R + /T (11) + /K [1] + /Pg 174 0 R +>> +endobj + +123 0 obj +<< + /Type /StructElem + /S /H1 + /P 24 0 R + /T (10.1) + /K [0] + /Pg 174 0 R +>> +endobj + +124 0 obj +<< + /Type /Font + /Subtype /Type0 + /BaseFont /BHXUKT+LibertinusSerif-Bold-Identity-H + /Encoding /Identity-H + /DescendantFonts [125 0 R] + /ToUnicode 128 0 R +>> +endobj + +125 0 obj +<< + /Type /Font + /Subtype /CIDFontType0 + /BaseFont /BHXUKT+LibertinusSerif-Bold + /CIDSystemInfo << + /Registry (Adobe) + /Ordering (Identity) + /Supplement 0 + >> + /FontDescriptor 127 0 R + /DW 0 + /W [0 0 500 1 2 514 3 3 244 4 4 504 5 5 551 6 6 325 7 7 598 8 8 358 9 9 322 10 10 616 11 11 256 12 12 558 13 13 905 14 14 489 15 15 428 16 16 456 17 17 367 18 18 680 19 19 561 20 20 529 21 24 514 25 26 315 27 28 514 29 29 505.99997 30 30 542] +>> +endobj + +126 0 obj +<< + /Length 12 + /Filter /FlateDecode +>> +stream +x?  +endstream +endobj + +127 0 obj +<< + /Type /FontDescriptor + /FontName /BHXUKT+LibertinusSerif-Bold + /Flags 131078 + /FontBBox [0 -232 893 720] + /ItalicAngle 0 + /Ascent 894 + /Descent -246 + /CapHeight 645 + /StemV 168.6 + /CIDSet 126 0 R + /FontFile3 129 0 R +>> +endobj + +128 0 obj +<< + /Length 1030 + /Type /CMap + /WMode 0 +>> +stream +%!PS-Adobe-3.0 Resource-CMap +%%DocumentNeededResources: procset CIDInit +%%IncludeResource: procset CIDInit +%%BeginResource: CMap Custom +%%Title: (Custom Adobe Identity 0) +%%Version: 1 +%%EndComments +/CIDInit /ProcSet findresource begin +12 dict begin +begincmap +/CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def +end def +/CMapName /Custom def +/CMapVersion 1 def +/CMapType 0 def +/WMode 0 def +1 begincodespacerange +<0000> +endcodespacerange +30 beginbfchar +<0001> <0031> +<0002> <0030> +<0003> <002E> +<0004> <0053> +<0005> <006F> +<0006> <006C> +<0007> <0075> +<0008> <0074> +<0009> <0069> +<000A> <006E> +<000B> <003A> +<000C> <0079> +<000D> <006D> +<000E> <0065> +<000F> <0072> +<0010> <0063> +<0011> <0049> +<0012> <0066006C> +<0013> <0078> +<0014> <0076> +<0015> <0032> +<0016> <0035> +<0017> <0034> +<0018> <0033> +<0019> <0028> +<001A> <0029> +<001B> <0037> +<001C> <0039> +<001D> <0061> +<001E> <0062> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +129 0 obj +<< + /Length 3423 + /Filter /FlateDecode + /Subtype /CIDFontType0C +>> +stream +xWyTSwO $r꽴hԥv+Z ,eIkYoBVQY@PZZjqi}y}'Uu^{q|r8ifvRZNvif쵷"YiQ, +&ӣQ$M!53&p99uÝ,E/\x8xiZvRrmz23)!1;l.|mႅE'JÞ53=Y*[5ÙrfٷUiqQ14(lMAqc}8p8o4dL09ɝ}{jhROȫ +t { +:)2WA)t~%=kc&Xí?_a%8a%rg ? !@0#NBI jby oQy.l**y*lwY "FmZSK+b~/ +wMjX'qzlx^S|%BSE\bQ(6Sop#싖,e`mПbh\Tۗ&$SXŮ`cC҇Spgn>ka!Zo/!u4RÛ'S)CރIQ"UAa|}^0 CvQy(HmO?8ӕAnoJ>s8yvT\8\F@0^Hp4@\{:=T9/ElNOImJ@?^f!QlJz#b['mR?`xD~IjM9vpM,d&!iYz&HEg[neE0+:pQ2* BRȒ_U57nګhlY^(V^Mŏi3[PL6N?sK#2U;Xy_a˖HRޥ $[ɄҔm46bh34 6E^{D$v] 8~(nJJb߿3'am5UiKH4nC`<;뺏vjr\e뭪69 \k*2i1zN`?>рwWT)QMWtlmi8ږ(M]mQΔ: ❅5FmY#KIԊ_U ˾+u*J&IM#Wo= "O +i:>}EA^дOF zi1?6;& l>[0GL oOmINDuB=ޕ0kdY" +L9fhGbl>yB BP p(&UM;cͥw;0Jkc1ep5d6lX"_/$ffyLS^\vr΂tA#HyDF,Nc' v 3`󱻰:Vb8&EmHx̯|ibw N"mPNg; wz&z3G8T!i"̢7鵚A̔TOavgS݄]](* ;v:΃?szG~hcD086Adf1^mչYPM4^Д)t4Z`ѺIW}YC9 Je61ed'VMEa!L>{̦va({YA4u :ݵi:{7QP^!c5% WV; E%l򝜴$ +[<m )0|np + ͎|m3-W nL܄ъT}QfL(ɫ>ZD+."aуÛ4,B kz*yKBƥحh :. <~#OvyrB<c s~_qyQLG)Y$NCSiz=4u~)t*e&[0s;3!>А**7r! 7TUHdGkjU^<|/O^QШݣE#$A=?NcO4^"Nd]ECi)TBQDڲ;NT6yԥK')F%(Z~MyS!&Ew(0F?.9"@X! RS= @!P D%)C_`5l rmIyEv 6XLGiop{q跮@*: G8O2J 4zt[b*˄_4v::Hq{Xk؜TËO߼"y HFAsтwQ𢽃(^p7Rk#c[c"K90 `F8q r+bP] !|aҷP9}mSFUV@a:y RDdo׭<:(3Q@$Xگ5bz ONZq!aX p,-XRؠPA+\UԾKax0&9RzNDy 7v4Zȑ"ኴʦ\Z_P +endstream +endobj + +130 0 obj +<< + /Type /Font + /Subtype /Type0 + /BaseFont /TUKHPS+LibertinusSerif-Italic-Identity-H + /Encoding /Identity-H + /DescendantFonts [131 0 R] + /ToUnicode 134 0 R +>> +endobj + +131 0 obj +<< + /Type /Font + /Subtype /CIDFontType0 + /BaseFont /TUKHPS+LibertinusSerif-Italic + /CIDSystemInfo << + /Registry (Adobe) + /Ordering (Identity) + /Supplement 0 + >> + /FontDescriptor 133 0 R + /DW 0 + /W [0 0 500 1 1 616 2 2 486 3 3 518 4 4 250 5 5 353 6 6 276 7 7 783 8 8 489 9 9 266 10 10 401 11 11 477 12 12 357 13 13 519 14 14 475 15 15 307 16 16 688 17 18 444 19 19 472 20 20 389 21 21 447 22 22 314 23 23 489 24 24 405 25 25 673 26 26 503.00003 27 27 491 28 28 521 29 29 667 30 30 219 31 31 555 32 32 526 33 33 444 34 34 259 35 35 457 36 36 333 37 37 478 38 38 306 39 39 454 40 40 444 41 41 313 42 42 499 43 43 447 44 44 219 45 45 486 46 46 280 47 47 534 48 48 444 49 49 611] +>> +endobj + +132 0 obj +<< + /Length 13 + /Filter /FlateDecode +>> +stream +x +endstream +endobj + +133 0 obj +<< + /Type /FontDescriptor + /FontName /TUKHPS+LibertinusSerif-Italic + /Flags 131142 + /FontBBox [-78 -238 786 708] + /ItalicAngle -12 + /Ascent 894 + /Descent -246 + /CapHeight 645 + /StemV 95.4 + /CIDSet 132 0 R + /FontFile3 135 0 R +>> +endobj + +134 0 obj +<< + /Length 1296 + /Type /CMap + /WMode 0 +>> +stream +%!PS-Adobe-3.0 Resource-CMap +%%DocumentNeededResources: procset CIDInit +%%IncludeResource: procset CIDInit +%%BeginResource: CMap Custom +%%Title: (Custom Adobe Identity 0) +%%Version: 1 +%%EndComments +/CIDInit /ProcSet findresource begin +12 dict begin +begincmap +/CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def +end def +/CMapName /Custom def +/CMapVersion 1 def +/CMapType 0 def +/WMode 0 def +1 begincodespacerange +<0000> +endcodespacerange +49 beginbfchar +<0001> <0043> +<0002> <0061> +<0003> <006E> +<0004> <0020> +<0005> <0073> +<0006> <0069> +<0007> <006D> +<0008> <0070> +<0009> <006C> +<000A> <0065> +<000B> <0067> +<000C> <0072> +<000D> <0068> +<000E> <0078> +<000F> <0074> +<0010> <0077> +<0011> <0031> +<0012> <0035> +<0013> <0076> +<0014> <0063> +<0015> <006F> +<0016> <0066> +<0017> <0064> +<0018> <003F> +<0019> <0048> +<001A> <0079> +<001B> <0071> +<001C> <0075> +<001D> <0044> +<001E> <002E> +<001F> <0052> +<0020> <0045> +<0021> <0033> +<0022> <006A> +<0023> <0046> +<0024> <002D> +<0025> <0062> +<0026> <0028> +<0027> <0053> +<0028> <0030> +<0029> <0029> +<002A> <0050> +<002B> <0151> +<002C> <002C> +<002D> <006B> +<002E> <0049> +<002F> <2013> +<0030> <0038> +<0031> <00660074> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +135 0 obj +<< + /Length 6100 + /Filter /FlateDecode + /Subtype /CIDFontType0C +>> +stream +xYy\`TNZSasZjuUQ % >}e /a<(ZZAͮz|9'%3{y}™82xAvnjf^;Ԥݕ: p1CL>'=z413cNȌiq8^sBCBxo((M͏eg&h?= fnL1Eiܙ妈sp&pB8*K奾[gJ=aP] +>So}k;(q`m7 DE6CA(뻙{>_Ofr1u N+r0h?/\FGg֬"k>>ytOhͥJL9\Ozg%pZm)q;m$wD:GGB7m]OU},Z$MYo6g3☤̽a_sjY;%GhtMi{+8:m,nj,CCGj u"hARM +S-].7^DФk'tUetN2LVhh;@nZRL +]W; <ɃVm q)Pa9A־?#3*_[WXkot;6^5 ;Oő]ilclXaZќgjr}I $vs5{n&9 ?Z p.鮸Z],KM롑{Q3 +VNk;D+~Ի$3/ΫϫdRt0<3Q4 ?VMt1],DrPk wh3m: =7-xGA35$IU&^I}EeGh-hE Gfh^G5@%v"6ՍkV'f_)v>BjӲuct4OiNDbEq? &m#y}kh,d|B +Lq Q.&9)C>XE+fDJ>JF#zިЗd, ƚ;2&bA`&dBjo#roSA{=zɃ^nN?%I&"B8M`=K!YM4C|]Yϫ[I(2Un0'mUh|*(s"i3c# RQL>D~jljSe̞7\@C<3yJ\cp\Vos`  cupE4\ +E=+9>r(J|\6|GzgvSEu`Ӗ=xG:=խ Gf`jpshl:B7:~[WVJjNu9+-s0pbܲ%q$:Fe3`*Sn}!)kA*xoOmk}XR2jVۈSG%ѿQ>B!qm^ +&HMxN|s@!gxo+A?uʵk. [;%{O '_Gt]jtw+0S$#91V&_`Syohn^ giQkj*O"#yqyW$~s8uqu +FI9i@LC`dcy''aȘv!ФC|kdp}'L҈vAQ@; +Ӑ@ +UAk(MC(ZeQyry]YV=KK! y[9 Q5ep]&.PZ[Z! GD]MYne>M]77&խdLcdg㹯ċSHt:elJ(i-K=N()wsyy#f5+M̨0(#M+5.tSN>F֡7oWqZe)q[6(G +)0Obt7`0ٲ#IEꖰ54ڝYsvՕGj0G}ѳT񒲒:?WVEۏ7zK::Ƣ̂Ee5h7!c;K[B\Ph-ꡢ3E]6mW^`] +NM͑ UHu&imJ!sK[l64TU{Aiqn'Cb(86G,6MF#kjub.eUDq`=\h"2zId``6uԠtp޴?|f4K.ɪ0 :†^ eCAh  +q\IV}R\"͗*]5)kzp`"~3҂FBK}olwUgͻk}^jC_,._+Ft9? ~ܛ@a:>>6D ||mF".Ծw%]uSr gn"yY"ɝg=Rh6xgue:hglmg™pVS +\OIhauIg]뫋<h6m$:2?Y%KZ뤕b&2⌇̦8{98!w_a= -m +Xm6KØlL0DžϦ Ԫo#"]1UNT7ӑ؃g0ww{g[mV?mOF&ZRG`"mLT:}q~YUx{p=.o=,:Vb$(}@T)l<}/O}0m~>;Zu"<u@QT2!o2~WWJ:]J]\-MOUWԷ'Ik:"@+\׊J^FG\T Q 0 +bv̥"ZNHK*p8i5"@Η^C}nqXxӾƓQph0Wh'k?8X*"N@`+[_󕞆rK G\-+J]Yƶ Д~j]VLΓaZr/?VJDBMPo9[x6r[fSS>>oa>s6|6xsI{ +`2\H1m`;g V2DovAZuF-1wLʨT&*At7`0Aڝm~,9 :CAKU6l'yKv ӾT&P5dEQθ{>*Yټ1- d2c6M#M3Ex:*A47Zj.-ZUM:KZg_Rz/E%SHI8-e3A`Agq^!tv/=Y 6i;4Ʒ4 t\’Kf 0Hh28FifJ5`1NP +`mt6h s'0&jDP=2:m8|$,_if9l5hsIHFގW٭{e;{h[$~@f0 FR +Eya= yj]$"z *@~1$z |Ď[E}q̘d)/_[1'6-LIBO EVhkTkT'f`}=$? bq+7ӈ "])JoCEK@o-Pـ01>aO+ ؂VMO]5q\Q R{ErD%'ݝ>wLLFRWkrkr,F3C):_l8zL&H;Ӌx:Jp}gx]lX!.9|懩IϷ#@y =?-hfmGV+tֶL;ӳoy;>`X‡ipƲd]r F 5o<4䣋HUUUDsE>յX}~pR7"XnqM/ Oiffo ꐌs8i8 Wb?7e:Cˋ6b0d'\XEo*@iށ[m#(nqQ<%xz5c-j\:.V5~:֦gd҃u nXaסrlt`bu9MNԪOXſZokj!]Z3˴> u+]Yk^ڇ68iAцTY5ɄT@"ɟxV_`۽ay`'T4Og`G KWoE0 p#{1Q)0=ߴ;OIߗJ qOny77?SH7 m5>.z Ə_<2r^2U%'b c\LlۚL-G6V5aC'A_w= h"^SW+0?~˞ؕ'VKe}e Q^r+)-xuYk[ZKIx@_7j-&+q#,-碃-7py_ +endstream +endobj + +136 0 obj +<< + /Type /Font + /Subtype /Type0 + /BaseFont /RLTCRZ+C059-Italic-Identity-H + /Encoding /Identity-H + /DescendantFonts [137 0 R] + /ToUnicode 140 0 R +>> +endobj + +137 0 obj +<< + /Type /Font + /Subtype /CIDFontType0 + /BaseFont /RLTCRZ+C059-Italic + /CIDSystemInfo << + /Registry (Adobe) + /Ordering (Identity) + /Supplement 0 + >> + /FontDescriptor 139 0 R + /DW 0 + /W [0 0 278 1 1 611] +>> +endobj + +138 0 obj +<< + /Length 9 + /Filter /FlateDecode +>> +stream +x; +endstream +endobj + +139 0 obj +<< + /Type /FontDescriptor + /FontName /RLTCRZ+C059-Italic + /Flags 131140 + /FontBBox [-60 -202 564 736] + /ItalicAngle -15 + /Ascent 737 + /Descent -263 + /CapHeight 722 + /StemV 95.4 + /CIDSet 138 0 R + /FontFile3 141 0 R +>> +endobj + +140 0 obj +<< + /Length 619 + /Type /CMap + /WMode 0 +>> +stream +%!PS-Adobe-3.0 Resource-CMap +%%DocumentNeededResources: procset CIDInit +%%IncludeResource: procset CIDInit +%%BeginResource: CMap Custom +%%Title: (Custom Adobe Identity 0) +%%Version: 1 +%%EndComments +/CIDInit /ProcSet findresource begin +12 dict begin +begincmap +/CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def +end def +/CMapName /Custom def +/CMapVersion 1 def +/CMapType 0 def +/WMode 0 def +1 begincodespacerange +<0000> +endcodespacerange +1 beginbfchar +<0001> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +141 0 obj +<< + /Length 422 + /Filter /FlateDecode + /Subtype /CIDFontType0C +>> +stream +xcd`aa`ddq60,IL=~H!CQAB}Weïk oeeQqyT@0* ##_cJ~RgJj^IfIs~AeQfzFFhPBD!&ZZ_WB.&FY?g}0m7c5}}'LWXQf*bFFAUY6qScO>{hŏ^تˏH/Z R$UbŋXd%ZÄ. [o!M~H~g K]m֪>e%R+ 6:s1.ҍ-SO>e?X ;ܤY( +endstream +endobj + +142 0 obj +<< + /Type /Font + /Subtype /Type0 + /BaseFont /EXBIOM+LibertinusSerif-Regular-Identity-H + /Encoding /Identity-H + /DescendantFonts [143 0 R] + /ToUnicode 146 0 R +>> +endobj + +143 0 obj +<< + /Type /Font + /Subtype /CIDFontType0 + /BaseFont /EXBIOM+LibertinusSerif-Regular + /CIDSystemInfo << + /Registry (Adobe) + /Ordering (Identity) + /Supplement 0 + >> + /FontDescriptor 145 0 R + /DW 0 + /W [0 0 500 1 1 528 2 2 447 3 3 316 4 4 250 5 5 493 6 6 457 7 7 390 8 8 271 9 9 790 10 10 519 11 11 264 12 12 500 13 13 372 14 14 538 15 15 220 16 16 485 17 17 504 18 18 747 19 19 542 20 20 310 21 21 497 22 22 428 23 23 531 24 24 505.99997 25 25 515 26 26 220 27 27 540 28 28 490 29 29 685 30 30 351 31 31 297 32 32 588 33 33 512 34 34 699 35 35 550 36 36 465 37 37 560 38 38 557 39 39 338 40 40 465 41 41 575 42 42 596 43 43 695] +>> +endobj + +144 0 obj +<< + /Length 14 + /Filter /FlateDecode +>> +stream +x +endstream +endobj + +145 0 obj +<< + /Type /FontDescriptor + /FontName /EXBIOM+LibertinusSerif-Regular + /Flags 131078 + /FontBBox [-1 -238 782 698] + /ItalicAngle 0 + /Ascent 894 + /Descent -246 + /CapHeight 658 + /StemV 95.4 + /CIDSet 144 0 R + /FontFile3 147 0 R +>> +endobj + +146 0 obj +<< + /Length 1220 + /Type /CMap + /WMode 0 +>> +stream +%!PS-Adobe-3.0 Resource-CMap +%%DocumentNeededResources: procset CIDInit +%%IncludeResource: procset CIDInit +%%BeginResource: CMap Custom +%%Title: (Custom Adobe Identity 0) +%%Version: 1 +%%EndComments +/CIDInit /ProcSet findresource begin +12 dict begin +begincmap +/CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def +end def +/CMapName /Custom def +/CMapVersion 1 def +/CMapType 0 def +/WMode 0 def +1 begincodespacerange +<0000> +endcodespacerange +43 beginbfchar +<0001> <004C> +<0002> <0065> +<0003> <0074> +<0004> <0020> +<0005> <0062> +<0006> <0061> +<0007> <0073> +<0008> <0069> +<0009> <006D> +<000A> <0070> +<000B> <006C> +<000C> <0067> +<000D> <0072> +<000E> <0068> +<000F> <002E> +<0010> <0053> +<0011> <006F> +<0012> <0077> +<0013> <006E> +<0014> <0066> +<0015> <0076> +<0016> <0063> +<0017> <0075> +<0018> <0064> +<0019> <0079> +<001A> <002C> +<001B> <0066006C> +<001C> <0078> +<001D> <0047> +<001E> <2022> +<001F> <0049> +<0020> <0042> +<0021> <006B> +<0022> <004E> +<0023> <003D> +<0024> <0035> +<0025> <00660069> +<0026> <0045> +<0027> <002D> +<0028> <0033> +<0029> <0059> +<002A> <00660074> +<002B> <0041> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +147 0 obj +<< + /Length 5236 + /Filter /FlateDecode + /Subtype /CIDFontType0C +>> +stream +xX tN,M"(s 30#\ b](rEd*toSMIl2iI&-V( Xr??)_yywa2&Mb0L:a\bN0#O%1G̛yi9w+) Ca t2~g|SB[fnd0d0d0G? 6dk&d%IH,*&>6-|sτ/ᱭ)Mئǖd30'K:;xtYYF'.loȔsC?d0'z=1.1+ά~|_C{X\&N6O~|wSDSn- !7K>H#?N:5}٠O]i ;R7H}; RsCt( w4g,f6aQ7: ;&6ͧ2\gqu~W ֗!`_VYɷ&TY.I4 YJ(¨pZJ:w;]asJTrhjUB + z)?Jxc ]_!_26#ɱΡ4L.=p P}5@FOot'b~'o0Ui*%8t5g,MHwxBa\Z! H>j)D]ZI*7ЇYwd80ذߏjJRNp$eHu/lkт9F^a~ic@JPL)WI5R\S^j0!0> hvZuާ#@xj)TcCk +r䯕ke"R1tdƦ;|hJPdRZ*,V;)8. LοX-.K;'i֮N74K0``;J}R7@Mr|^/~ +4Җ5tjN JQʥH?WZ-|Sd/mlsVqֈnO!ZQkpEq(h[EGx]&Ti[,/Slan&=/XՌA㫇_XK(t$ɁlpsVSZt6dibxG8 :*mr,Q_́Rd>Tr^L1$`: :h6trxJU24SX*Ez%gXeU 2܄qw؜{:6eK0Գg7e75:mY(eD ޷#`KA]Z1i2$S=aM9"jzlMk|(9 "E^aA.G&Z؋TjTԉMJN3En9l9U$gQaϳtU=4WGVcErնm[چy>j}mFTi-EB6d6X :R)q;B C0Hx}]FjR0 -*+:*;B0.FQwG0ؕFܪF.e'rd߃h=qnzwǣЈRVt>rՎ^N.ʘKO*H1`:5lbBzAM1u\xt?46dK0ˆ>cjR̍]`ibu: øϗ;>.8M4~}]Ù_](ƠΊT&$|g#ڲ0'&ST#-R: roϳ.<~0:3΄D}SՈ0X}iJW/ipaV15 +.8ˌr[@3h+; U`/^Ԏ%Y%/lonY'zw'-T|tLogWʥߨ߯?a~~k( ,ƌRqfxGyp jn7[ێX,ZYZ cwIEJLUaEK_}wy=GNiJY04{\M5F63J78e%<` m(L ;6JWmp\FRaBK10Ŭ FaxAv,x W䷘Goh7522D RgG@Ga+6!5JWBxy˻c߸y(/5ʵrq1۾l˽5xc>߻[u|d;u_@mJbV8r/s.XP0F8.˛!5Y1IhrLu0Bϡyo.\q^Hf\ Cj6^4<91uU6sF"(!+VRA֘3M?&10 m2LcΫ%) qsQa@wb&EOs 9B\17y ZIr֚a2աAbxo2!wt ё|mYI)7)(I^uo逺JLoetʍ +~i2A gv}z#=ek:6Y'/1޴>`P+$9ء [S? +T;^5KsXU< &A7Wi6`CA  D6譪'%x.6tSө,J~.)4ep (US$<Of1RѲ5z<448#(rJQ.;ZS>K_|&?-O"`Ze.,N޵Mrx?喌qTn};<=ݠ_FN +],GVTHAYFQy~x!>^n1I?F?Oz._Qj%JJre +zvB}r.¿σwN7Kt_uǾW x4,_|5 4 B7o\3ݳ £i+lBCSNVqҌ }et znptޭ/AEuOܞO$O._9Я o~ӳ!jW hJ6G +f]ܲ +-bWP:r6}K#,n V67H+G`>BCnq\qv=^[52^tv=Or4րacFBc1C Ic4FhDVc(3$"w,GԢEinF0;?(4_?I/$\0&nSwܼRB!§?cd⺮ IͦӸRlΝc!GMމ5GW]Ady(~>VA,z*e_%ϘǐE]`=ʟ30W :8*WV5Z+9vgT_\FY0j:˼ Y_fWlL6;Zia/})< +nzRLsšN=trc(t]f-$)YoHtOPчJ Py",AtCW <e\`GWn:;y⥁Da;^w~wQ`BۃK׏wMtȣt~Iޯ_B#f;YkW6`vMo5q +'4(tCSX Dqϴ:j!9Yj8%)^Eh+_rw0Rԑ$o(dRqgOWi!܄gpƚ`~^j~P-_ +endstream +endobj + +148 0 obj +<< + /Type /Font + /Subtype /Type0 + /BaseFont /TPCGAJ+NewCMMath-Book-Identity-H + /Encoding /Identity-H + /DescendantFonts [149 0 R] + /ToUnicode 152 0 R +>> +endobj + +149 0 obj +<< + /Type /Font + /Subtype /CIDFontType0 + /BaseFont /TPCGAJ+NewCMMath-Book + /CIDSystemInfo << + /Registry (Adobe) + /Ordering (Identity) + /Supplement 0 + >> + /FontDescriptor 151 0 R + /DW 0 + /W [0 0 500 1 1 786 2 2 759 3 3 572 4 4 485 5 5 500 6 6 278 7 10 500 11 11 569 12 12 778 13 14 569 15 15 778 16 16 500 17 19 667 20 20 620 21 21 501.99997 22 22 510.99997 23 23 595 24 24 569 25 27 667 28 28 600 29 29 715 30 30 706 31 31 529 32 32 429 33 33 466 34 34 520 35 35 433] +>> +endobj + +150 0 obj +<< + /Length 13 + /Filter /FlateDecode +>> +stream +x +endstream +endobj + +151 0 obj +<< + /Type /FontDescriptor + /FontName /TPCGAJ+NewCMMath-Book + /Flags 131076 + /FontBBox [20 -250 760 1500] + /ItalicAngle 0 + /Ascent 806 + /Descent -194 + /CapHeight 683 + /StemV 95.4 + /CIDSet 150 0 R + /FontFile3 153 0 R +>> +endobj + +152 0 obj +<< + /Length 1104 + /Type /CMap + /WMode 0 +>> +stream +%!PS-Adobe-3.0 Resource-CMap +%%DocumentNeededResources: procset CIDInit +%%IncludeResource: procset CIDInit +%%BeginResource: CMap Custom +%%Title: (Custom Adobe Identity 0) +%%Version: 1 +%%EndComments +/CIDInit /ProcSet findresource begin +12 dict begin +begincmap +/CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def +end def +/CMapName /Custom def +/CMapVersion 1 def +/CMapType 0 def +/WMode 0 def +1 begincodespacerange +<0000> +endcodespacerange +31 beginbfchar +<0001> +<0002> +<0003> +<0004> +<0005> <007B> +<0006> <002C> +<0007> <007D> +<0008> <0035> +<0009> <0032> +<000A> <0031> +<000B> <0035> +<000C> <002B> +<000D> <0032> +<000E> <0031> +<000F> <003D> +<0010> <0037> +<0011> <005B> +<0014> +<0015> +<0016> +<0017> +<0018> <0030> +<0019> <005D> +<001C> +<001D> +<001E> +<001F> +<0020> +<0021> +<0022> +<0023> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +153 0 obj +<< + /Length 4425 + /Filter /FlateDecode + /Subtype /CIDFontType0C +>> +stream +xWytSם"d]Q+tNAr:mHڤMr`b0 8,7Y^dm~ɲd˲ɶ, 2ml1 q1[ܤiC MS&,mȣ90m9ONjZ(=3Sqǎ4eΪ EEx2H<(p#w1a-P<*'~$rW><*;fEGr̷<^}FQzfBFf2WySOxq EqɥE\en%9q"E~\ni"SV*T)s26IJ/*Tm=YXjU\iff\RYܚ5JU"EBe5k"ewL^=a㦝IV+ʸ"E\F2-W^:*(^Ԋ۾$.vc-x4nJ,J.YYGtuuMNWLã:YʏEEEEGH[BoFx//-\ lSM"!'-._vol̹bT#Q#GԱkcW`MMOiJĝw%l UMuӷTXΜszRQ[W[W$umgsvu562NLN8CPn2`tYl}5GfsnfëWz9G?Eh͇Fq;dniZPm1 ɽ?A2s?-*I%a7gh4'πjUiёXOovRok̔O$ph`\F[͉>[[9`Pc;PYkPh +zd'Q +Zm nFS}TtK{mm3DpnʘMē4X7KG A'RՌUW*u +saݎ"V;:-`h$|WCרX\[H]ggcw;e ׌Q}ZSq ŎҕwZhYK?V8i=pWPP[ow} +*TVU+zfMU%G mfpJOMD<3~I%m0:u5FOh *nB֞N} 8ٜާY@7^^oxgWi>+`)ɭkմj(Aaxq ^ƈoacY׆&L_Ộ4D-0-BJ[ m}W[Z0AΔw% +h)G?k:‰$.B <څ_8ODCH3di*F8pz띌ܡR.D?otH;f_s2w|xOeK >Y3nb7 ]'d: 7'N<=6{kKBS5|拲E"' #_ܑ$|SE~з%NKE+T|N7ѥ ib?Oߋ9&qU;u,j=Ůt$cpE"=W}d >vaׁ\@*kN6;7~vMMt^-Ϛ:Os^W@h0h1Q jp@#4 Y}P #tSk[@Q+N 0KE+..H4Ӥ[bnȱlmRHn@(KAwfb"O,nܼp\@WaBğ-yR̅F)lo]zowَpcaBD]#54uy~,x}NFwy<~_eV&T]A| ёeenr+,xC},̻p]zO S욚hMT`o*:1t``Yk1Le"hahA<@(z_0 +8[I5I!U\oR*¹،;SBױHԮ +aCեJ*X_T(qݮτgt/GWWAD+ l Mh^y2oП dg˴U`jݤIŷXf2;zj3[;d+*V֔Af@~)>rxޖ_xQ&-ugeU0ߧeB+Fڔ_,xmS?BVN +-䤸Yw|7tq[Ot'&%&lsci4M#4NAT*S {'@'vGד_hE1t0x ]W4w:]8|F <6gwohuhU2pc +(t7@`7D +cTDR^~7'P+oݜrսXE_֙}3.\yo^>C`,5Fּ}Yd];` "Kiu;Sc[GI*sUoF qv-ý >wE zDbk%lPznܭ&7MS=@ltD{ m|wsC s z=) wOOzhחf1͆ؔks32N47 m0M\X/V$N"3G»AD9nd2S fkң(u09OU6bI*R=TО؝R¡mBYtcIO :ӴLV;+ad?,UƪCd@Ӈ9tNwPGDظ>z o*-Ƨ'%.% tXP㶺 FdF1^)/4& +<;\髧qIŏ@j +$#DRU?,|1G)ӚH&Kr4N +-NadfgD4>˛FFQ*9xO +/^JB2 G<!?b{2p3Yv҃c6N/:XE& 8IauT-IؖI9Ѿ+';*[&2oH9@=T˸C;[?Y.*8AI^]p?Jbh4 $I,ފMx?$6y̢00s 7b m4 +endstream +endobj + +154 0 obj +[/ICCBased 156 0 R] +endobj + +155 0 obj +[/ICCBased 157 0 R] +endobj + +156 0 obj +<< + /Length 258 + /N 1 + /Range [0 1] + /Filter /FlateDecode +>> +stream +xuJPFOUvDD@`]\ +FMkIRB|&إnऋ(HrTų9` ըa&ʺ:l +3Ŭ*ުnh)&C|>b纝黓AvCƫ+ y') +̵8+/> +stream +x}Kq?UX 94%MQKSN#ԦQvBˡhFk ! +ՠZP~xx^ޗQx"^P c!H0 + 0l+ߣy7;׫;Q?V._tF3LEK)y z80eIPkT/%[p:8+%d"G `d_{ٹ֖gzm\8rSif1\CU` S] @o _ +endstream +endobj + +158 0 obj +[174 0 R /XYZ 70.86614 755.2506 0] +endobj + +159 0 obj +[174 0 R /XYZ 70.86614 781.0236 0] +endobj + +160 0 obj +[174 0 R /XYZ 70.86614 587.8416 0] +endobj + +161 0 obj +[174 0 R /XYZ 70.86614 479.06262 0] +endobj + +162 0 obj +[174 0 R /XYZ 70.86614 370.43762 0] +endobj + +163 0 obj +[174 0 R /XYZ 70.86614 613.6146 0] +endobj + +164 0 obj +[174 0 R /XYZ 70.86614 316.35065 0] +endobj + +165 0 obj +[176 0 R /XYZ 70.86614 701.333 0] +endobj + +166 0 obj +[174 0 R /XYZ 70.86614 342.12363 0] +endobj + +167 0 obj +[176 0 R /XYZ 70.86614 563.25 0] +endobj + +168 0 obj +[176 0 R /XYZ 70.86614 411.59302 0] +endobj + +169 0 obj +[176 0 R /XYZ 70.86614 589.023 0] +endobj + +170 0 obj +[178 0 R /XYZ 70.86614 244.8471 0] +endobj + +171 0 obj +[180 0 R /XYZ 70.86614 710.15063 0] +endobj + +172 0 obj +[180 0 R /XYZ 70.86614 455.77228 0] +endobj + +173 0 obj +[178 0 R /XYZ 70.86614 314.45508 0] +endobj + +174 0 obj +<< + /Type /Page + /Resources << + /ProcSet [/PDF /Text /ImageC /ImageB] + /ColorSpace << + /c0 154 0 R + /c1 155 0 R + >> + /XObject << + /x0 183 0 R + >> + /Font << + /f0 124 0 R + /f1 142 0 R + /f2 148 0 R + /f3 130 0 R + /f4 136 0 R + >> + >> + /MediaBox [0 0 595.2756 841.8898] + /StructParents 0 + /Parent 1 0 R + /Contents 175 0 R +>> +endobj + +175 0 obj +<< + /Length 2113 + /Filter /FlateDecode +>> +stream +x[KoW:T~by `Q'ɬݣ\a꫞3?s v?h޾ՑAL@ޓ5`Bog4y׽}<컻'4o>"GD<%_y]CsGCe7U0!C*;4QB#4>mBvc*e+n-! " +:0Y8|>Ox>oϦq2@. T[>gcbi鍖|ywɸٹ2=D $xWk˸{[i)Kx- ܫVjԦ5(F JhZkA#jȂAs?wU֬w#M*7P[ 8Fl 6!4:ףH"B$zB2ejz1=jL|PJ MN9yӍRJ>MY¸O,nmܣkG-)b鉓zOdm"P2VHn!3!x,cH9vE@j4Vbo"8/Z%yYfP{- -O!-^9%2^.n|j(co{ټ}ӁO4 scV,?kg"͗>O|p2Dg{L LN_ Q@ BcԼ +LhUw 0UZ QżF,҃"Zdi/E5\u{j9K Z he\rm.o2b 6 l3:ֳLfKs Vٍ֓G~jKWDJQɋEcGq,25sXd709hj& fUcYUWg8Qnzs/߬:s~͢P/ zE=XD 04U٤G2Uw7pZZ5*. Hߨɮέ siB '@ 'yGN^8WY=LpU.9,4L#˫yՏ"E1+o45ݓanj.$ l!{a[Ib3':I 9 BtzCjRr%g8oRr$e ڤ*S !{~ub}A5z$o6O S*sx&-R9Dfvk>N.|U255YY +;Ǿmtʰjut@f-xQ]r3b"7W9:f)W{#())Z#/Be]y/hZe*\JcmךVی_NZ%p9)6m3~4ES +F4uR\ r3N}u 8$D`m P#a]mx[-bЈ->G)[P҉|vfEwmT \SZ~\]7kέ3$`XsSFԿ3D!2K??h"rȹJ>>ΡN)N`jovyT՚/p)*t7 +endstream +endobj + +176 0 obj +<< + /Type /Page + /Resources << + /ProcSet [/PDF /Text /ImageC /ImageB] + /ColorSpace << + /c0 155 0 R + /c1 154 0 R + >> + /Font << + /f0 148 0 R + /f1 124 0 R + /f2 130 0 R + /f3 142 0 R + /f4 136 0 R + >> + >> + /MediaBox [0 0 595.2756 841.8898] + /StructParents 1 + /Parent 1 0 R + /Contents 177 0 R +>> +endobj + +177 0 obj +<< + /Length 1636 + /Filter /FlateDecode +>> +stream +xZK6Whm; @^͡oIl5ڍ $Z)R*zYKZ8Ǚ/_>lيW^77(@@aA:c(A7 Ho:&] o6M;}Úf"ncH: F50m~i޾{\ mݕ2-*-*ֻo.7w7-֯]m^F@x&XBIqB!Ax&zyn߭x~5Lb\ނ@y>•XYx4hRAJK EZ4EQޮ+z e+!+mNB += +@<k*+ TkLV 6~Q& EȈ5?2HP`c + a:V9}GTR k9uDD51qB*c8#AI28YǐH$׼HVAdURt\e g!+cXu TY$e8YTV( c6gRe pIe gdV(HƝT}<:S3::3Sև:8%z+[K3p@^|M+Ple>1SJǓH>}o8R M}jj + A3TfBre1)2="a8?$溏>ZE3VVh%2|qgXîwy.ڇ"]\3i:gKVf> + /XObject << + /x0 185 0 R + /x1 187 0 R + >> + /Font << + /f0 124 0 R + /f1 142 0 R + /f2 130 0 R + >> + >> + /MediaBox [0 0 595.2756 841.8898] + /StructParents 2 + /Parent 1 0 R + /Contents 179 0 R +>> +endobj + +179 0 obj +<< + /Length 858 + /Filter /FlateDecode +>> +stream +xXKo0W \h$3J(vE`?$zt*.Lf<Ň%0 @vz +g'pkh@@q`Ɛ% Vkpg/[qJi 9GHԲ+C \If~z*Yf͔\+)_a?ʊ!Q$I[!L +Y,[n~+s*>c P;l(`Yʫaf +QnGӎo}plV$O|^/w&ٻ }.QC(Պj;hl " T&9Ǎ( 0AB .D>(EBACB+;25PtBd{ԡ g1m(72w%+qj>+FɃXigN`#DNhb(z덝bb ܟ6][Iyf\w2&:OG,E1!GD|cOulû= 6AD|'8uu)j-3M%f:N.TACEmAu}`@EhgaHj$t'=OJٛ_ۊr Nn"^DDA7*M*l{w:B"޶Tt.vߜ1vΝJFhwR;|Ҳ̠+q]%s^ʣ_sno^>Ru&T/8a2Si~j&dR7! +endstream +endobj + +180 0 obj +<< + /Type /Page + /Resources << + /ProcSet [/PDF /Text /ImageC /ImageB] + /ColorSpace << + /c0 155 0 R + /c1 154 0 R + >> + /XObject << + /x0 189 0 R + /x1 191 0 R + >> + /Font << + /f0 124 0 R + /f1 142 0 R + /f2 148 0 R + /f3 130 0 R + >> + >> + /MediaBox [0 0 595.2756 841.8898] + /StructParents 3 + /Parent 1 0 R + /Contents 181 0 R +>> +endobj + +181 0 obj +<< + /Length 1314 + /Filter /FlateDecode +>> +stream +xYKo6Wp"n< 4hE}kzH$na~aYD1$73맿O" +9)%#@=BڬXi#`B](*Ve5"ZH3LvhkL=V!on{~v޿__~RX}.|^nޚC^,*Tu5_"RefXlLfSr=S^%72;9! " +2L|h؈1pD-j6/%k|QWx L~H=AAHd эGWD?#| 8ھ=XdOb*e*t# x3&L`)A6 H" ) ^= JXvnvL~i/QX8`4A'A1xP ^YO$á;Q+p1 cP1xڝQGwKPc ؐ5MpdJPSoDi¡f ha[Xjq_YS6fuj6  4T$&~EWI-V05UgRSNŚj +Ye`@LW[{/}E&5&i*V63ՕXZsX- kL+S+t +_Wu3߈݆c;8,u狣 Y/`]pXϕҞz^^L*;IiD{}32Ŏ8t۶! \R4Sm4p&5pqu'5peـguvp{m[7rӛbs)ob"5%~<κ0Y4 躶#:%_JRϸG۷V"Fcű.8GIIhR*}r%Juh 2H+o@ZզElc GBů@5g|[0TvbRjSI-7^̋1$A8bqrLi^<DM +fUD.l +endstream +endobj + +182 0 obj +<< + /Length 99 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 319 + /Height 247 + /ColorSpace /DeviceGray + /BitsPerComponent 8 +>> +stream +x1 g /aq& +endstream +endobj + +183 0 obj +<< + /Length 7365 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 319 + /Height 247 + /ColorSpace 155 0 R + /BitsPerComponent 8 + /SMask 182 0 R +>> +stream +xսaI!Z{ň$V5^ЗT*5 Phc5 +,A(R5ȥ$$m{7B lArxPvf/}ydw̜5?l TUMd`E:hDjkjriM]xFefۢ\4D"tUWUe+XCoo -]`޼?dt=¬jmiN$ +|:ǛAYN,^<|Ȑ{=y7c0˶$6݉_"9X|iwqcwGuz:]W<%%{{т0fvҩԴz2,]oF +.b556f?]/X_4Lib@]8_dݗ.]ʺy/]n:Jr]hOW݈YetHp X̽ؼio[4Ɵhmi̎6mw"?x7|#{G\5vؐA\Z!б_??|t G#gάyQÆeẅ Cr)lD`0 +con:r~y*qW^){tAܹs?.@>t7M8}^UN[JTgr]]Ü9Ta߯0kN# '}}}Uc Xcǎ})-}3Fal!+_xPntMt;!=dР]]6+}7אW`޼~si*$}9}Y=+%% -`[֜z淿5벥Kƞ|q~籆/晒ex P*K]5vM]ǎU:eaVTŋ^M;C -`|'Yfܳ{w'6xK+W0$ $U57<\-7omX{>0s*hnκ_7qX쭷$+u+9k,ˬ>yuW_M?s#hۻWal~"#}2N2l^}Cm] u}޽onݺyOÞ;wnm;laHl6O0;3bgYw vv*w=曗.][F ]Z&wk@V3a<#~TI 'KzJaw˗o.Z +Ey7.TmOc ]fƚy# !tk^pE7n= ī~_Wˇ~w/Z%qaOoqMeO&Mڲys˗s]˟~ܹsMݔG*kvPTߟgϜ{d۷ϟ;>>yr޽_twc3oMyKiJ&꫗_v,7o5 35J-%_}oC2cPFQ #a9{ 'Jb--IU彳dY 656vvt +K*Mk11|OuU=աltΎK=H/sV_Ƭ-k-1+&U%8hua ShۢD"9obb8"PlOn:bf!br Mm =ȽIv{%82kVM9eR4+Egy|j7L *R7R$X{ALS+RE!#+3:jq/ߊz\=2ozZn4WWWUkBz (C^*zy=IU-~6kh.7ZcbЯ@xuSvl-P>66]#3h~uV Tu^ +E!O"N$pgyBBm_X +5ݵJ2a@yЯK6 X MGf%f8c3ː _{ʵ/f&`9EEzЯ Z<%.r.7.~߀qiWrKQQ&8|Vju"?j{-lBTrkA[d=9%C8ENAy\2J.D؆dv/|z~s(~nEۀC_ ~KUm 2U\6DU}\Me \'/NB/9QC =^[j߳/W7K$ۦiUDVS&u\tB78y/^KR% @PQ= ~*WHms,ҩÇ܇SS8|SBMv9Bff{%@b)_L&G 06u'o¬b\ %*'^.7ϸ~=^Grn;n)DspÎ̱> + Cܰa- +xvұR\ndv- +nT5~=VBqC7,|=n$w֯ucD?p%%*~(# R%zMbUF~\ + hMDJ]%fL[7dRTDVHS~*=甾ti6xʍ8ׯt-^i9H_ѯKm @i3Gx mqTP +$UϢ3gPTSb +cN YBT5ϑ'Μ1C`%| cT޽뇝Y]8o踅~AĆUiM[7__)0vP[E wkTn*ʩ}9Y3=l?~>CGTCm#k*+֖%:2nр" \orcu'Lέu2p<`O=CMo_=0 Pi؎ )7K~oB֣:>.>l[>oS9(*G3,6t&^+*W!OD׮UV (T*{ "KhVŶJQu>]ȫ/)D;kL{l,% ,A@|RȫYHPgf\Z̫62x{ס+}%\@p(qГ^PJl.0h[[S#g12er˕Y.) oq'{I ^Ivע!rSC".)W~]rN'J9ʕ_{H"b(/*+KHKqChyU݉7%2^դi7!4YG'7QWUQ&/n,gM\Th &ZrBCrskng'> ׍mT PxUrk/oEWmM.6aBNAij˖¥TI Gfk9 2!aƕ+^zˢvR„XS-5Mt*eSHF`aB'|\NRU 5;tjȣD[$Ga5*ӝH~+d>G96LH\Gع*ʘA9I ڼ@׏oAs^n*ܣ>AKU*Nx"xEs])HO +y(cvz/[ln9Y4/aRzN!&U!ƹpn/=8ׯswU IItԉê\!u.=A*84z<`V:3Y6A.匓ݢ[XIyN"s2(5i/T.$U.NrR6w4?r׶]rh{Io+׶~mgⰙ ^ /)(;U(i#o9`cz \{%o9`ւwnV./))G, '~ݤJ(W~KQ\ndh r%D}ܒ4B׀vNcobf(R!ITyIATP~$'EZ΅~ uUEZ%ؠ, 4~YRa33Zy +PYK1¤83lLb!fNRi1b}CJ,]*{܀=Zhr~ (WV=3~3nQG.OIIUmjlD9GhU LUX]CRK_An焒]G$0A|+xS)(W~i`n؊ u$U5paTA 򭲼'xnu2IS+7 Dft+(^:<0N~1f]|jMj:dieF.$[@M]Ϭﾰ*$\ +cƏ㪱csݜxbSE1O9_QnF׻֖nr=,5JEY}5D"؎&פu/^07з١CXC$RKWtڶh<(K{Ѹ4P%FzRUX-mD +ŬȚz{,WmdYӴΎmHW1+&!X!1]BQ (%+݉JZaeqȚJR%-X%PhhDm2q{/^*/̣]dQ8ξzD3LS-7bT_ +yͳH$Sm+W_~eЯ@ׯ[W5f .\#Gh:Qgv۶ <5kȬY +c/Z%u\ᡯWػgI;Y,|CF{%xht}{656~7*s'ñ#G*}_ܮ!CoxtZİo?޼*>\ϷRHz Xo+d$ -Lܚj.7s LtRq1BxA/=B"r .x,Iql/Z]Ǎe$,^Pa쑟|gȠA +cǏ ({- - +c˟y|ٳˮ h畞!d-?_u'Nϗ_y!K/O8˟~z /(֭^ۀ~ˇ\9jԌ:uܹgϜ=p ~#֭]kƋ+V(](`(*8IgW;w-xg:<ݹt>I?SAŨm pPWbJ@@GK Jߐ@]Rᕞh %VZ=neIyɊmƲ C7j4ST~۱ߌrqL~Yy6EPnpQi).nTU~'5_"ǔ2N-;\Zf2NWd>H\ tt'(Ґ4=3.=ՓX+O*^Tծx-ujDŬb-l݉D{,5D" HmMMmM 7G0g\, +endstream +endobj + +184 0 obj +<< + /Length 483 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 555 + /Height 856 + /ColorSpace /DeviceGray + /BitsPerComponent 8 +>> +stream +x1 g /` +endstream +endobj + +185 0 obj +<< + /Length 20923 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 555 + /Height 856 + /ColorSpace 155 0 R + /BitsPerComponent 8 + /SMask 184 0 R +>> +stream +xU?$Y%(  J0(K$,, A1Of3 +J(bGY`:,I5u}뮫tuTaLȁTwUs> on{iڴEvC@'O.*,;O~a`/Ph…Av[LO<&.!h+ 48MCn! 4665P($-l3{qРn cgiBz) `k^}Uid7αP-[d7믚}̙g]+ kJKg>+W,n#4!|NUYXP0suoyUMC´VڪUEw K/n\iÇ?pｳz-[(ukf&Ĥ׮xvlv…g,߻wmY$&ĘQB`ٹ&̟?dM)'n`m-^|…kצIَyO?^nNt3(}~g R0o^>}^qr%/._nAT[7oքXdɢ;:_cu_:mSR6[gɛ9SbMivA?;%E"w$䥋Avt]0w5\?%+WjBlݼ|rޜ9{wﶠHӧkBϞ|oZ&ė;vk&WGo?vAuo&Ŀ8wVqߏE_d">6[Gg2P(4_?M9f_v-gOM#kuڵ? ñcY Fмjq~ti1lX9顇4!>x=z$ԯco>M22P?DZ9CnA"sПo>h%K"~+V_F߷O +{nM(juII{?/] QXP ]HiSjBbvMMu^>ʕ+U_~n4V5ׯ_O^ ^hziB߷/y?Пԯ_BBo^b7}C ZV\|ѝЫ/Aߴ\MUjimm]TXx}4E"SR2RS~ZΐhBT|vN13'.]Wp#Oj s&N /租jBHbŲeڅ+'ׯY#Qط#ws5|,XǏOj#4fԨ#NZp￿{.M>Ζ-[~; @RiB̜>8|`߾6l.$̇￯4k:O=cFGK<Yp^˂uϟ7^ >\_R&Dӑl1c<n./_SZɓ̚Uv!aZ[[_^"=%epӦLə8qܹ.^./~bN4H_Ŭ _'׮];tǎ'ygϜI5k8zHSS )qooUU#Pwq6 `qQQmMMđGńaWd&[ꪪ⢢!d>VF>?q><ӥ^T l@a F|n0@?*/ƴ> +d=3<Ic@ "R:UUg 07̂+.*t:$T X`8 ,Q >ěOt`м3oc|v69]tT^V  +;ʠL'FIܜHP0Ȅ 7nK_'J%"*!@ UUɨB;i[6YY7wkc(7'P˗uYC q铓cHհtrN=nMN:2}bDJ (M/˴78 }:)9@P9X>Q*fuZ 6`ORu{!ܓ>(2=ښDA]]}祏|8r!juu8ҧHGlO.lJطJ\HS*RJ3S IEAQRmg#>vFAE`0b[WOmΆpcsTTFs|uT )O'b/':WN.LqƸq -ʉ>E.L@\ج2 (Agצ2 (}5~ L( "(>ؚl>:2Ȟ %lSă0}td @Pﯬ0^ Ŷ#글m#l글@ёAAAu\)>:2& X.GGqKh  XGGEAu\R8 }tdDTl=Ǥ 글-ёAR@P>:2zTl q[#DAu\pI @Pp*}td5 8\>:2TQA GAu\Bt J,N'FdP@PpH.!글DtTҧȠ@Pp>q"DAu\C$'Ƞn#:."} 'IȠn :.ؑ>IEuqLj%T 2dP SȠ@PptdP, 8 +G"2STG:2(: 2( +n2#T!2("6#}l j글 ͑Aa XG( 2#:.`G-d글IA:sL( "nIAT qgչ&}DAu.Iqm@P.`ǑܙAT q0fչ&}mDAu.IpU@P.`UܓAT q!d9&}\ DAuIs|@PS/` sL DAuλI$ a0d9&} ":\ܖAT 3p>pg@P0f)}>ps@P0g)z>H3TI$DAu]N ":.`Ib@PB0郤R. N  S&}`2>*DAu6IHD@P/`? ζ0lATg M9 Ά0[m@P.`6d ":[]lˆDAuI؜2lr>P2p>P}2_c ":0E! NL@i3d]@n@P cH &}02,I8 ":+/`f}@Pe0dz8 Κ KXATgLU, KL KL, KL咝AT AT , KLd%&}2x>@R3H::j@"}dgPuUUZ]RV  ~?8`AE XAOjd@~Ԅ XA7?rsr^@rAK8GyYY2 ʊs5iRI`e5Pכr>w$2"~$©GvȠ2jL`ey?::z'3( 6|V.0ASaUD~隧^wtdge2†,l#`QEYfy`q VWUYF }AqQu?Yd0~k$=:zň' 7gcyPĊ2Ș@`vS +'}d<^2v  6L}GsAu^oyEE|Za >_w 1bذ˗/n).,Qv +;-vqJ6E?yyt{ؑ;ifӺXLa ĨA[ +DLiwbfgen)еyl.ƍQؚJC!`C7BeQ}Qg`kH(==Efz<,3ssrd7@7w(X=l. vAyy>v#.`([]R_YYEE|TQ]U? 啗qXon-Ο?_nxd7. + 5!>ܥ`|MyGvC2b0MS"!9OnEvC֦M{NvC\ߖ,\i֭kmm"vG~X0 })S4!vBoƍU|q0\^ ?!]bwϞ[33_ ߧOz^/_4 +WL5!ٲEvtKsuIq5!~:4!| 6771f ϟ4ػ{wjߙ?5m&VImЉ{\zUiq=X[+i:))?coMC5 G 4 +}cǘQ̚Ӊs^TbϚO666 0  +kЉ;>taAzٍK.NIфk5y3gkй/ qq*>XbȀџP(tkf&'u@'kB̙5KvCtW^iufӧ5 +/ qϝwOnڰ_.[l&Gg=t@bQmmmضMj}whz^tk߾iSn3MЉm{׮ygLPss<"@)){6e +S%B'{LϠ^ӟ]V~Y\"@'MM9oMw'Gټ^&c,;իKjMڕ+Wn !֭ӄHMʗ_ք6e`?{׮߮Џ$ҥg0OKb}pwkBdge4!^R7_mySMiiIjc^}8ںhy/ׄx>Gg˗/k}?m +nIjaĉaSMm[ee,Is睚+Ls 56 pu}c9;z/+.'3jԑÇ}N-Y߽k&G}g˖-o gtbj ̯>}>w g-*,Կ yAϟ?s֗;v߯ ;%l$\8^kkKOz֬}w…_^VR7g?iÆߞ7s|uyڵCpy"\{̙PDKKK}}7_jG5'cmM6<8~&AGЍ1riiM[>q#~o( ~Zk[ P^nNŋg N3Gys;X[ 1lȐ*Б{L/xΞS{7G5pߩS7tS̙OXmʊښo叆ښʊ:7:qțo..*jii6yy慱z 8kE:&q2|쬬*?/lٳwϭ~? %`WYQa^YQþ:2=`vlp`=j8_]U1r'2Bee; +MܜTWUEy")l3GyYf{G aܬښ%%>7@ +c*:Gٗ?i]ny1:w+8Wt~dz<[ +@歅9}0rX.'ML25|vEGC:Gʉh !3b\++*g\9G *7O99}\+ vAyy'ןkQ4|x >%K+M"wҏ٘;4 +dz<(1|%%쬬"[s x^-4|aASs_kkj%%"<`puIqCm"N9EEOCфغys/kĈzlp<¸fjkj"Nϝ4)NA;>ݸYa t4>蒨1%YYL\)~{Dx"1Г^b +&_rP0XYQr(yXi^J%"^buIIiKL'ʨMXem DjCB ;L(_ka[~b{Rhj|o,'daoc!0P* YY9֍]ZCB-qUiXvna]Q1B*l&̘Đlݼ$bM\1Q>px4{0$F[{1e%p8<{ j8lR\,w`8oN(8R3nS5V9)B,u^oGΐ2 D!ƨMĥ7~?({F;N2g?JBE_ږ #~vH)PSxڢN;P2nXƃj/`%T +{]O]/[1~/ q"wP>PZ(*ȒɊ55a齄ܱrǰ^>=I/!k3p~=qا`?;'m(؁z +(5l@z8cȬ> z kfU^@:ٰ0gGecvsgQ>ȶDMs%vS(ڗDK>b +uV(X潄}Z*Pؼ[*B< 2?Pj_>I@K<$*R:(XC^†-@$m\re%̋_nD%Q I\/a( +hpQgSH$m %93 +;/($ҽ3(ܠHG8 QT%i;,L@LQKV׉x|%7Z-ws阙IrS<%$m %śrZq@78Ǘr[˕33S{ ;t*ZPKCB<^dPqz<)7OHgMܤ \@aX3|>~RvKd?;m/j?Ic"Pt@.%$Nӥ ڗkH*rO/!e)y@m./( 7sU/a(F,FB~Pe4 n%,f9xV" Q^GvVK^aK${%W!Y +~n) k{"pOӗoaBn%^MJtFB`uUqUD?6h彄y?8Q+W;0́@yYYR &f%#vԑouIIWsܽ=%OcM(P0X[ScL4C Z)\^1ޅČQs P]Uj.zIG/@F`U~lA?]ߴ4}PKJVTWUu軣GׯYbٲE^KKKr@KDG啚/L#cD{.hQN>7Olkk;zaWFrÎrq"/_7taC<9mȣWR-s%8m;'麡 /]83=˗/'|-QK@'%bYXWMU#q\"rQe7Zjk%bط{v ti={u%Ċ^"ZǎX@~=_V-o;(7nUP<8@/nyہ.};Shw +.lM‚tD6?Q8 +(`pC3 a+hB<2aB 7bB/};u@@v{3ommm'OѴ=٣ill\TXg. *@'%y?;#}r4fQҧ?ݶ3p`L%Ș_( % 82|ؓz BfJή a`sj4SzȘ],%.U%UXKsG9w$+((IJ^Bѝ=@J[@dLkl9ḣ&DؙrͯD[̲73Ff7e};X@vOvVV^T~jM(AOK6Y&uHȂ=p#!/gN&'5(~,} $8Iђ@Fo +i:Jܜa8:-{c+I + vTgLX ;5}x+(%a9xMxQ p0PWyohR\65.Lw +a[VhhR0")6OրB`sfЈ1WH&4&$(8:8x1B-7Ҝ8n(ebϓ';h p0Pc%$s@vܠ{C4\25r7n{b_f Ks;w5 +hgЄYƲBʐʱ?2@v\Hߍ. cЇܣ~%@v\KGaG97k̯j&DiA +nriQw#ob\=(@vͧqʠYjW uQȘDkB`mM͚R=_1 +/Ys ))g4UUFNԘ)Xin<ÄX\eeu^opջn]lFz]y"*;>h1øCo*+*bxm*[&?p>L\9ez P4ꪪ(#5gǰ!Cx YĹLaǐd7v ⢢ښ~?gN>+:]|sX|dge.)1&41~({C`U@#7'_y V܌Wp (lf$jB6K(XVƾNæ(t@Nt2vʥ\"ܦ<n]͋׬3%v'_\T}i,:*äJ틑oH?iqQQ~^^yYYX1Rsʕ.n-*+*fN~ze\ٳn E4!n& ｧ `\ ˜Y4!>C HKKg@Msgn {-Yi6[*EPO'Nbłsז9| BoƍU|q0\^ ?!]PXiQa᭙{wniiyw֜5!V,[&iʕ+<=%|a@]p޻~Ν\xQg.٭`{N7osscܐqyMZ[[0!gO~R?; 7-իR[.ٓڣGǣw ́3{VvӠ-o ̂*+5!>vׯ>\bMii(rǎ1F͙5'd7 :} i={>}|~Ųe6l46RsZ~O)-@ +MnN+W qqyM`#+^xAqd71[5!6n4|wVMÆk{ 1g, C=5!? QXP ul_Ԅ;'[[[7m믿kT/u?ݶM? +.Y--~컣G{҄x}ժk߾iSn3MƎ ɓ'ػ{wwhBlVz^m`  M{O2)pǞ]?喽{=#7 @6hj:tZ/-pw`P?s…ݻv555n5|om"!H3&B% NﾆmA՟<;엞>e7"X+C : x1M7^]vC O?ݿOo$iB:xPvC gAΟ?/-H^ +d:W5!6MvC0?LbSOnD믿jBhBmAb,*,Ԅxg F---<';#r'Mz/]oԄC7IqۈSڪw,,(9}7߼zpS LpgjBxڵ=5!>,uѣ2677zuʕb-ֶ vAWy9 qCFƕ+Wc]pHMW]ׯ_Or3DKJ"Nhҥ~3P@-}ҒF$>ط'~~>}Z^8ql\>o4^>mD}c&k>GiBXL^BwԿO3TnфxqR\T 1st}nڰAjR[=s̛9So"cx\^'ϙ5I +ׯ_;j9qQc;t 1t 9@ccćG`m퍃3h`߾&6bDʊB224!֯]+8S qwG\Vbؐ!2 p_~|VYQ=vlnmVs@ZϞ/X!!:uM7͜iR N?>曋ZZZdΞsذa$[ss3e@~qQQvV>>?/" n*+*|GvVV'ug>4pߟD uf-h_x@55[ +pN߿G}g]+ kJKg>$˔bCY໣GׯYbٲEChmm]vmǣx3 ޻45.)%mO˛ƲGфX]R"]R0o^>}".Y R:xX='6dȓӦ<5!>Kʕ[7o67g&ݻK1EE(VPŋo1|q{~Ѣ4q9O}4m-Gg2P08:J}nkBm(!lCiB|{zH^ +#΅x`޹Svs鴞={tHiB ?gܹZ%iE4|UUu^^~gڵ))T*e7@eRM3fOק >yMS'<O{%T1\MUjimm]TXx}4իR[O{`x#.ɟ=[@vC$XĉaSMm[eyn6:,G Ȅ a7[hj( Xb̨QG:d] +wl۲ۨښN_cJvOi=zݳG?ظpϮ] .ϟ}hK9|;v߯ ;%1?^h,ljcm{L8e)'߀3\v?;fW=sFj{vuYYLHիW?޽SdTz1Oc:@JuX!IqnЧĸ͜.u8obqeTiNU;ک*=ܬ9hc\GGnN > 8XeEEX%%%nڥ'™⢢6鄂Mf 8Ls {!&"jftVWUyyݸވ[ω< ed7ݘAQw)/).*ns Qx +pӍhmwnNcjt1;+kuIImMMGz &f:D8}ױ:#H02dp; +F Fٽ7z=?!J> ʊbzQyM@u1~yen܍'S]Ueޭ&\7ݎGyYӅՂ3PnNOӇU O VS` t|ҥ<_c %ǹﶔ"f8` 66CfW^V&-sm7Zu6c[ܜm\88l<Pj*,~"  "(l[0˦=U*a )em.. @6Y갍꺧}AVFdM(A k<g#㼍]!ඍ-::l@6ljkj*+*x1 \@ tnިN ٭tݻm\Q;?Jݨ\…tns:p ~ҕxQbD@KUy\!p'].|"p']*76T+ZƆS\=tElgD@K瓮6ڨ\%t7zS]B.O ~p 7|ҝ1VW@Kgu>YG㒍 %I+ 9r\tRSw@K8Bj +Up g㒍 %Izㆍ %Iwp/Wp ~?Xȷ:p ~?R p G~a %IwjMoTG.OK8j{b꬚@K8= Q>zqFuN;q}N"EEm.>p |ҝ:%̅ |=Ps|0w.}"o~(@}q^pEփյ?+G. Q^V֦_ms1lTG,Pwp]hwnD'I@ǝYLN +;JP&,7''b8~팗N D&Wk(71Fua qLڔR\T# @:(T+#w g_; Aݍ ;dPrlɠ8>jPɺ$j,nTGOZˢFux~jB̛ʂWXOR`3g5-@~rpbPzy,Jԋzͳb긼h$'] xażʊx~)'ԛͯprz3G +Q\TT]UեgaKYY Pm>'A"ώܣ}M'euIImMMТ:5] (wlsȍ: +[%%u^ocx#7''`X>6>PX77m6xaÃa @u@Wb|af',) {c+oXQtcʫ W,*R _iegeE`nR3DYgF +UqFuHX&gr8IGEEvyFuFezx=Msnhb~^Uܨ.Q̚ ngeOL\RKKg@Msgn ˟ӟ +'ֻ(pgNxiŊs--=r@|09t`[[=n6ʲL-!(Դ̽wukĉ+-4 .M~ki`M_|q'Q**gz)7iq& +4YqGy5`UUrs>2aBZϞ_ǎiBMKzHyF՗^2|wR>wFu]5!Y|r[e&G$I4!6[gyO҄Xs&JoTӧOߐֳӧW,[ i y9:R{ɹvM0@bMNW\:hGС.7/o?NﷶJiI)pLboaާ۶gB%K@ׯ_I"G]y˗e5Ɯ>vFucG҄MMMqOݽ;;4!K+V|weT% nyW_zi7С?ɝwnTg׮A[nٻgO[[ۣ< q+Z N],w6'>ٻg1rݨC/^{׮& 7nYqq@Sm#F z)=fJlT@-))Z?==k֤5⠿uM(Q医 b>y̡C,mӇ&?e?]ݨ. 20 !^ç?={|NyaӦNK ~zϗׄ˟J<́@BiiƎEE&o-[֯]@BIYYKJn!55Maa +\䔗y1QۣSB:tۚe*+*=kVzJ +3Ӽ9s4!6_/!ԩ1FݚyX5J쬬=g~:q_znDؘڣGFjkdqY꫚3xB >_pرϟ4`Ϯ] .1=%EC6n:/](iWkB,_d7qYl&DFj%K>얷&zSӦiBl3 A\7Ne7b9t&tUw˰a[7ooOkBf A&w&_.5 45zՖyȶ&DaA ^e7jBKOrǎlۖ3qbɓb&Ďe7 +BE}?\ccv@d~8#55$!H?ccnvvվiiOϚ%!Wi|U]ֳej-55mmmӦLYTX(EWXf&DuU՚ٳC\`m >*9:w…7xc[ee9/[7o.ZEnL9_)vO? mp ?ԞS\t}kӦGqswÿ +endstream +endobj + +186 0 obj +<< + /Length 104 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 348 + /Height 241 + /ColorSpace /DeviceGray + /BitsPerComponent 8 +>> +stream +x1 o Rg +endstream +endobj + +187 0 obj +<< + /Length 5044 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 348 + /Height 241 + /ColorSpace 155 0 R + /BitsPerComponent 8 + /SMask 186 0 R +>> +stream +x[TU?s}G ےM$,ͭLErr5ևL/YVj +jA$3gs̝{νz/^G<39|}s^&Nw+8yw۶#Gx7D't#[y7@SV뫯ۘJȖ͛y7Gn+q$JHsښgϧPltrn @UΘ6-bA0㵵_xwCC=vݳ{7Q[G ]Yɻ!ZFȌ6 ԍv :;;)!SR^ݻw~anNNnNNü&4See_}gxܿRXP57==';{ْ%є; |KKƜ9EE==^w1QQt}¨t)%C/\bvJHnUWZ,'';R};vA'IbWXq]ŋҧ488ȵiin~;K ._D X|n:>uǯi"r:AJ7Wn }wm:'{橧(!oO5R'% RB*-P)X^_G+WN?.q=-[;wVTk}iivx‚ӧy7MPwNiOw-[D#2ۼoY榧Q}t |챃UZ:/#tÆ~y4bI_Y/n蛫W}¦_g/g„--^ͦkXGMpK3A1tl: &ޱ|ߍT>?`qR1ʇ0P>~W%~@g97̯b~u:\`uL;} t-VwC$P>;Q쮬r[K'2C+~ rsrKh#[P>^| +|"?`"h# +vy1U1+MC +s8a~vF>FZnK~+#cLc Ϸ|$~Lk\>47 ?`Ĥ]A; &ڔ|1;[!.P>ԍ%G.̯19>})Ba~>} L+)1~c`r8 U,?` ?pU|P | f`;`C1A?dAHC\>,~l? 0|NZ>|0j0|ҞP 9Q7PD,Wtx6So<&tk0ͦ)j#`<$&0Bfr5U0>g#`5a f0BfX+';v(O4V0s6ޡ|-Enp@DZR}UQ h ;3HS|*f 0"7~!((@קVU^`g| +1ƾv(@cQe(@N| +K4N?Z^T\a~v(@- +DO ?`0X T;~"/v&p%~e{L+R (0#AA +y^_W/| +{F'~CPP>ǯ^F(;3B`VA1#u +AX,r\` !;ˇnT񧁮r\Q>p)~ued${/>|pN ~aՔhQD&|U>Zr`>5zdj?`݇iɔ^vv{6QTUyE!`wwwrC I:Uց(;{c4zAzǕii!E )z4"]fkׯO֯]vY#}6NCVX+k|W{[e˺^|3PB*_azlgbװ1V w c_~9=%eݚ5==+'NM@._2?H{z{z(!)ve0x\ۨX?:p &*Jygd:Jȼ mn7ySܹsGy?0Hqjypԩ{&LX`|rzYNYBr^%ʝG)![Pʇ;[GԩqˍvGf̠\pAhjln\2gFy!~ѯ_O ),(PX'MrݚEVf&%dʝmmѿq:F'|Pwg_dYQA > h}є׮)ͥdef0$(TP'IWnJmW/RBb\s)!e?1} ʇedV lL ɘ3GCgNov>y˗-,hhؾ=ءBJR{RB>t +tJ̇\paUGRS^o[[۱jo`'R*_wMM/>9w.%c^1XQ K PO8c>|XS'O طoْ%;F ν+###1QQ{&LX/%͞?>k֭~b +KU$WH/v+{;?/Z$&6:`J|gҥo ?㙦.y_}rHᐗYm^V"e\o`yڵ_[=[ZFɠKytlr}*f.ZCϴ5pjAe;T^.IHD 0||Y<0-L2Oo V +:A l 앮f5{GLlAw?t V>7akA @ŒQ>v 3v5A>[> ~:-K/ZB=`66X6ԤqR_c 90;--d6&T99i`|@ @).tH> `PBv]8T|p\ʁiI`ҏjNycq RIehV>47#~ rA [> ~C<`\N (?>w\!: M'#~ `"Hah>0nE/BQ VkC8w=o!>=b9[> AA]a5X> ~`O`!]\Y%c& /-u+0彃[łysϞqiɨvm!L|Q,n>Ǩ7, +``Oȉe\`i"C +[y7xNV !ٰjEEt:uhqQQK/k}}}[LKN~t``)!EE|l{J{+Һ+,|Y#YKqMuww7y2%䛫W^YQuL+w2iz/_駟x'Nnpֲ23 +6otލӠb-AmO INL©')!1QQ> ' p MMNxnx> +stream +x1 g 2t/ +endstream +endobj + +189 0 obj +<< + /Length 6889 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 348 + /Height 294 + /ColorSpace 155 0 R + /BitsPerComponent 8 + /SMask 188 0 R +>> +stream +xwUBBxX:]h *D"`!k.jy1ÈB hg1RtxT.9sϪI::Ug?~*{׷{}R)FWWO-Ξ=yӦ/Y+/߷wj=qb^8a]Ps쁨O2p>~KOvnp,ٽ[0i:::5lX"=ysꪪuVMxW%_hN쁐`}W\IͿeՄ77n4:~e^81ͲBo[[ 8f<g}yٲ+WN:ŋ=[~{ةSTjiS]ҥKFlPW[#5k룏RO?̜9ÊN'3zt4+f({ٳykĉGGǎ]?rXxDWWWYF5Ɩ.^R_1`ywqj?vժab0bȐ:Tc[7g͜1Om5ӦL1cC d.yCSS'O:48_‰kezkIa! J"n9st˗/#?~Mco-Mhv=;Xo^j>v=f_G?".յ~ڴk5lX$aSOi+<(,^X@c,\Y)i\iDP9{-USx V {Ga#ZTԎ8lR S{̖|Lc앗^wp7F 4_~01•(a׼pؖ =`&,]|1vwR۷[2%CmcK.7.$IPaau5?9rpԩsn}O{ڼyM<(@uUش)SĝC~q)NGGGx?ͨQc]{mq͛Ǎ{ϸq>ܹsz+יo'#jnؐ(/1uuWĉ3Oo=qBTɓc%ݻU_޷lIۡƯ˗/jnho??zp6X^x_|Qclܹ֛+*dɢEh\\ӦNݿo_ϫ`Պd{@Gضu+X%%>t 7pkw~;sig۶n}757gnڅrӴDUUXi0X]U7-j0M^Y +_<Ey" &3 %hF3zyM1hD2!RN)7v`'xEyy}BǏs,M4744vwӦN--Gg< ZQ14 6{TgIX(I1dJ!K5ś;@V,hT伩G ʐ&B|6(%v&P +C|g zNxk3%h2d!2 vuJk;Vh2dXbi0hWGe>IHkM@ YY|&PV,x=!6I&!+C<\wzaM BnV,HD_16I&!g+C'ah-jzTh2dM:qNt'h2dBi0\31I-b M@V,8X/8}l+oᛡ*4X Of-S2M]V,؎a<'urɼ@ YHe wc@3? @{ Yw;_-م @ۭ Y.i1V pʐH&@kǩb6! h2d!Ap8,5B(! 9\>_.J,Ø +8jeB%wR W ﳆ&Pi+C)bPW[?&P+C<(]5p pʐO}'98 +feBF6U^X5p pʐN4E%P嬨MV,:)j@(ྕ˂DAr;O@)V&+ |/C@3(p~j|> hdY,˥iA)G +H2)Y +zZ1ϧC( DdAv1͑C1 h[YyY=YAT,R[ xj˂H4Hy:F+oN+* "~İYx pXY=YD>mo)+$ "7h7)o5+! Y¹iy pZYYa)r}- )at@>x\bjIz be?_g8ApN-% 59W==X/sCW}oD}Yq_x beV~/Xٳ |lD\B|-) |ՐsrhNM|-@y+[eA'Bhon8ܩ'_6LIɂmes$A.Ī+n9LӬᇙFD-@y+;1=qp)=VTP $búucomޜsȂ/mekrA{NclՊ~zgnjܹs}~Y,/t݉^< & +0 .]r4'F Ԕ*/ ^t1jsʕoIclʕD]ƌ=sǎcՖoz Ppbc&aO=drʕ}²Ao[)/TY(\x:IHRK-]wʂ׼(oeɡ/LmWgWTh͜>ݮG() z!keKj;7=H=Y,ޝp +OnmѢ5^{W9xGh|,x[V]/qw~{xIXQAɓ&7頼,HE;?օ)~¡&{+4,* ={mA Y&P@ED19P1& +4[*n[F,@(ಕ ێB!7y4t!,Shܴ0 jpb#)Y&P5+K|6|*  +ce +8]UM VXڪMVRUe@G,8VFӾ夆,@(R&<.  +le)E9,  +feM \Y&P +}FţXbY(4|[Ua4<4ʼk@, ը,@()r,h4rSxfDd@C>@bi0]d0-huUV\1>Ԁ,X%尾JX]=V$YY7-'fh7 X^S#r +Pj1dР m(*p@}2A;7+uhDl%T#jq.\6^XWx\RBI̗aOoא=L -[z[fhnbm$.tF=oCg<nohhF1i3gd _|Q9k]w1b 'Lv-coo"{,Opo1ݷ@>N]e k=3g<)S4>رC@|yի,\8/;tuu(a]jSO-]3#-0GT&8J9=.97#Fl޴ŋǏ+{!{ƍ=4`Ǿ뮛:yi=òEWW3s W +|:4`n?,nb9oޗGJ//[1zJ͑CjرC1 &N4qbK4*{Psnomh~ˈ!C6^M^c5-bK.EcN ./_>rpsSӅ[W^re޽?>-{hr +endstream +endobj + +190 0 obj +<< + /Length 96 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 317 + /Height 240 + /ColorSpace /DeviceGray + /BitsPerComponent 8 +>> +stream +x  otom) +endstream +endobj + +191 0 obj +<< + /Length 3364 + /Type /XObject + /Subtype /Image + /Filter /FlateDecode + /Width 317 + /Height 240 + /ColorSpace 155 0 R + /BitsPerComponent 8 + /SMask 190 0 R +>> +stream +xT?wYDHliRЂiMIPJՆ + XD +RҥbbV].!%Y`fCv)7Nfwfs+Cw{=r]ǎmk׮0uڹ}{8`Tjo_|qԩ m6R.~~rA쑡o>l?sNܿoC#GJA#r6ueK.^v GªTF#tbaCѧgInΟyt?_~p[[)ZLaU3wSЧ{‡AUGa{ƍݻv͚>=Ȝ36u=|m|~}R= _ܼĿƹ(I7oڵEF'h #}i{{wmZݶuk_>ta TpXa[W֯/0ݗ%l<> eccc2O"XSW( 7qw-C>Xu+\dɏ-yNrӵau˖7|}Ҽ9sJA07q}Pſ8S9}ȷU|n` ˿dѵ'{{ko{ }^_lT +߽r׮YS?ll[*pWÞߦ&O?}?ϞKVg> :eWgXG~q#sLnO!ٯ}vEu-.SOEo>־fϘҺuj?~upp~裷jy1RR5 QV+?>pO{;gϜiaxxc'{{o߾m0>M&ct +R\(1]H% o4Pi5@ @ȅj03(5@Fےqn!еP|0 'gXtpT?[kX02 +Bl7n- P7(XLA%O' tеfLFN^b3ɗp+Kk;zB~-+inx>Q$Eݷ@M4c^OVjxKщԥv0 %2k€@X*Z; ɯߎ+T}&|.jWxZQ$<8\>#vh?@萰р[C\TF^ma13i+?r6'e*x̟LQ PtӒq)@(^*2R.} */BzϺ<2˂2W2-.,HClq+SW-+JV2¤=،)S8(ʦ_2dfѡe(_7?kuRgf&d_XR[XɴrdƇ2::R[F v#mh=K ՁC8B*4O|TJ o~&nJ {CJu!|QTr&=%%ZC|JʔJ e_7?@qJ*5zJ%*5eM,ҝ@?٥ŠRY^G@t"$IR,d_XRYl8P*5,L@J72Rjr=%n%eJhtJ||ʔJ 7~z&Tj AJ6@RSbSTj o0\ᡏ1J Hn2ixa;^@eK >j},'\ɜ9u* L$3A"Jybk>B)יORͨR#D_^+ڄt$@N)la&&Ԡ2%#"[&tC.ymܤu?0id4σ{S FbN"_1O쒆 9Cm^m4$9C$hIXR٥xFt!:9f.Bԑu&g:,JP\N& ȸe*edEe=e2Ⲥr`[ouҦagy?xDqY|9Hz(ߤeq-&eő{!)SʙOmJ'+$"e꼢AQ\f~*Km/R\f%BafR2ug4Y$ˬuY 2u0eV~La2[ RnsqŴ0 D29LkFTMBZշ2!91RJɩ Lxr"\ڧTj$‚^_|" u]..s eؐebSLU)tLx_TZ,.MAˍ S6 z)j#s(Hc1*-PI˟ +Bz uWS+Xx͝5?]$zdE{W xJr`=ZI[q]td2#-0CUL׬0@=LZEI4>_|..SW&LcseJ%2-=gtQsNxsT0;u$Ca9w +P%Of|PrP|}x*m4dr 9-@;3`iqh@p9+zТ5*7$+b*+i$Z G[43y"L9Ad6 ^^FyY9ؒuv7]46-V6!8@F䈳D@]6tua@FyE6jTFdkhP׵3|aAO}\b*h"F"1}EFUDT'?( +endstream +endobj + +192 0 obj +<< + /Creator (Typst 0.14.0) + /ModDate (D:20251127101821Z) + /CreationDate (D:20251127101821Z) +>> +endobj + +193 0 obj +<< + /Length 996 + /Type /Metadata + /Subtype /XML +>> +stream +Typst 0.14.0en2025-11-27T10:18:21+00:002025-11-27T10:18:21+00:004application/pdfeakdjdGr87FiY+ZUFN5oDA==eakdjdGr87FiY+ZUFN5oDA==proof1.7 +endstream +endobj + +194 0 obj +<< + /Type /Catalog + /Pages 1 0 R + /Metadata 193 0 R + /Lang (en) + /StructTreeRoot 19 0 R + /MarkInfo << + /Marked true + /Suspects false + >> + /ViewerPreferences << + /Direction /L2R + >> + /Outlines 2 0 R +>> +endobj + +xref +0 195 +0000000000 65535 f +0000000016 00000 n +0000000106 00000 n +0000000187 00000 n +0000000312 00000 n +0000000380 00000 n +0000000519 00000 n +0000000600 00000 n +0000000696 00000 n +0000000780 00000 n +0000000922 00000 n +0000001007 00000 n +0000001091 00000 n +0000001234 00000 n +0000001318 00000 n +0000001403 00000 n +0000001532 00000 n +0000001618 00000 n +0000001717 00000 n +0000001802 00000 n +0000002058 00000 n +0000002645 00000 n +0000002993 00000 n +0000003110 00000 n +0000003367 00000 n +0000003801 00000 n +0000003891 00000 n +0000003978 00000 n +0000004122 00000 n +0000004199 00000 n +0000004290 00000 n +0000004388 00000 n +0000004481 00000 n +0000004630 00000 n +0000004716 00000 n +0000004859 00000 n +0000004955 00000 n +0000005041 00000 n +0000005127 00000 n +0000005271 00000 n +0000005370 00000 n +0000005447 00000 n +0000005540 00000 n +0000005639 00000 n +0000005783 00000 n +0000005868 00000 n +0000005958 00000 n +0000006037 00000 n +0000006128 00000 n +0000006213 00000 n +0000006303 00000 n +0000006382 00000 n +0000006473 00000 n +0000006558 00000 n +0000006648 00000 n +0000006727 00000 n +0000006818 00000 n +0000006904 00000 n +0000007047 00000 n +0000007124 00000 n +0000007212 00000 n +0000007310 00000 n +0000007400 00000 n +0000007487 00000 n +0000007564 00000 n +0000007664 00000 n +0000007761 00000 n +0000007861 00000 n +0000007948 00000 n +0000008035 00000 n +0000008140 00000 n +0000008228 00000 n +0000008311 00000 n +0000008402 00000 n +0000008490 00000 n +0000008573 00000 n +0000008661 00000 n +0000008749 00000 n +0000008847 00000 n +0000008924 00000 n +0000009082 00000 n +0000009169 00000 n +0000009313 00000 n +0000009390 00000 n +0000009478 00000 n +0000009577 00000 n +0000009677 00000 n +0000009777 00000 n +0000009854 00000 n +0000009986 00000 n +0000010073 00000 n +0000010164 00000 n +0000010252 00000 n +0000010335 00000 n +0000010453 00000 n +0000010541 00000 n +0000010639 00000 n +0000010726 00000 n +0000010813 00000 n +0000010890 00000 n +0000010984 00000 n +0000011082 00000 n +0000011183 00000 n +0000011325 00000 n +0000011414 00000 n +0000011515 00000 n +0000011609 00000 n +0000011700 00000 n +0000011789 00000 n +0000011931 00000 n +0000012026 00000 n +0000012121 00000 n +0000012216 00000 n +0000012311 00000 n +0000012405 00000 n +0000012496 00000 n +0000012584 00000 n +0000012726 00000 n +0000012833 00000 n +0000012933 00000 n +0000013027 00000 n +0000013121 00000 n +0000013215 00000 n +0000013313 00000 n +0000013413 00000 n +0000013595 00000 n +0000014068 00000 n +0000014158 00000 n +0000014410 00000 n +0000015522 00000 n +0000019051 00000 n +0000019235 00000 n +0000019948 00000 n +0000020039 00000 n +0000020296 00000 n +0000021674 00000 n +0000027880 00000 n +0000028053 00000 n +0000028295 00000 n +0000028381 00000 n +0000028627 00000 n +0000029327 00000 n +0000029854 00000 n +0000030039 00000 n +0000030703 00000 n +0000030795 00000 n +0000031050 00000 n +0000032352 00000 n +0000037694 00000 n +0000037870 00000 n +0000038376 00000 n +0000038467 00000 n +0000038714 00000 n +0000039900 00000 n +0000044431 00000 n +0000044469 00000 n +0000044507 00000 n +0000044866 00000 n +0000045289 00000 n +0000045342 00000 n +0000045395 00000 n +0000045448 00000 n +0000045502 00000 n +0000045556 00000 n +0000045609 00000 n +0000045663 00000 n +0000045715 00000 n +0000045769 00000 n +0000045820 00000 n +0000045874 00000 n +0000045926 00000 n +0000045979 00000 n +0000046033 00000 n +0000046087 00000 n +0000046141 00000 n +0000046546 00000 n +0000048739 00000 n +0000049103 00000 n +0000050819 00000 n +0000051206 00000 n +0000052143 00000 n +0000052548 00000 n +0000053942 00000 n +0000054229 00000 n +0000061797 00000 n +0000062469 00000 n +0000083596 00000 n +0000083889 00000 n +0000089136 00000 n +0000089447 00000 n +0000096539 00000 n +0000096823 00000 n +0000100390 00000 n +0000100507 00000 n +0000101593 00000 n +trailer +<< + /Size 195 + /Root 194 0 R + /Info 192 0 R + /ID [(eakdjdGr87FiY+ZUFN5oDA==) (eakdjdGr87FiY+ZUFN5oDA==)] +>> +startxref +101831 +%%EOF \ No newline at end of file diff --git a/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Opgaver.typ b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Opgaver.typ new file mode 100644 index 0000000..1459b0d --- /dev/null +++ b/Diskret Mat/Graphs and the Cantor-Schröder-Bernstein theorem/Opgaver.typ @@ -0,0 +1,68 @@ +#import "@local/dtu-template:0.5.1":* + += 10.1 +== 11 +Let $G$ be a simple graph. Show that the relation $R$ on the set of vertices of $G$ such that $u R v$ if and only if there is an edge associated to ${u, v}$ is a symmetric, irreflexive relation on G. +#solution[ +- *Symmetric:* If $u$ is connected to $v$, then $v$ is connected to $u$ +- *Irreflexive:* Because you are looking at a simple graph, then there will not be a loop. +] += 10.2 +== 5 +_Can a simple graph exist with 15 vertices each of degree five?_ +#solution[ +No because the total degree would be uneven +] +== 43 +_How many edges does a graph have if its degree sequence is $5, 2, 2, 2, 2, 1$? Draw such a graph._ +#solution[ +$(5+2+2+2+2+1)/2=7$ +] +== (53) += 10.3 +== (7) +_Represent the graph in Exercise 3 with an adjacency matrix._ +#image("Exercise-pictures/10.3-7.png",width: 40%) +#set math.mat(delim: "[") +#solution[ +$mat("",a,b,c,d;a,1,1,1,1;b,0,0,0,1;c,1,1,0,0;d,0,1,1,1)$ +] +== 57 +_For which integers $n$ is $C_n$ self-complementary?_ +#solution[ +n=5 only +] += 10.4 +== 9 +_Explain why in the collaboration graph of mathematicians (see Example 3 in Section 10.1) a vertex representing a mathematician is in the same connected component as the vertex representing Paul Erdős if and only if that mathematician has a finite Erdős number._ + +#solution[ +Because if a mathematician has a finite Erdos number, then there must be a path of mathematicians who have collaborated connecting them to Erdo. +] +== 11 +_Determine whether each of these graphs is strongly connected and if not, whether it is weakly connected._ +#image("Exercise-pictures/10.4-11.png", width: 50%) +#solution[ +/ a): No. Because of a. But it is weekly connected +/ b): No. Because of a-e-b-a. But it is weekly connected. +/ c): Neither, because of the two subgraphs are not connected +] += 10.5 +_In Exercises 1–8 determine whether the given graph has an Euler circuit. Construct such a circuit when one exists. If no Euler circuit exists, determine whether the graph has an Euler path and construct such a path if one exists._ +== (1) +#image("Exercise-pictures/10.5-1.png", width: 40%) +#solution[ +Neither +] +== 3 +#image("Exercise-pictures/10.5-3.png", width: 40%) +#solution[ +No Euler circuit because a has degree 3. It has a path $a,b,e,b,d,e,a,c,e,c,d$ +] +== 13 +_In Exercises 13–15 determine whether the picture shown can be drawn with a pencil in a continuous motion without lifting the pencil or retracing part of the picture._ +#image("Exercise-pictures/10.5-13.png",width: 40%) +#solution[ +Yes, draw the triangles first, and then after draw the pentagon in the middle. Also, imagine if every time the lines cross, its a vertex, all the vertices have even edges. +] +