diff --git a/01001 - Matematik 1a/Mat1a gennemgang/Mat1a gennemgang.pdf b/01001 - Matematik 1a/Mat1a gennemgang/Mat1a gennemgang.pdf index d1ef8e7..f063458 100644 --- a/01001 - Matematik 1a/Mat1a gennemgang/Mat1a gennemgang.pdf +++ b/01001 - Matematik 1a/Mat1a gennemgang/Mat1a gennemgang.pdf @@ -4,8 +4,8 @@ 1 0 obj << /Type /Pages - /Count 7 - /Kids [392 0 R 394 0 R 396 0 R 398 0 R 400 0 R 402 0 R 404 0 R] + /Count 23 + /Kids [1079 0 R 1081 0 R 1083 0 R 1085 0 R 1087 0 R 1089 0 R 1091 0 R 1093 0 R 1095 0 R 1097 0 R 1099 0 R 1101 0 R 1103 0 R 1105 0 R 1107 0 R 1109 0 R 1111 0 R 1113 0 R 1115 0 R 1117 0 R 1119 0 R 1121 0 R 1123 0 R] >> endobj @@ -13,8 +13,8 @@ endobj << /Type /Outlines /First 3 0 R - /Last 15 0 R - /Count 6 + /Last 29 0 R + /Count 9 >> endobj @@ -23,7 +23,7 @@ endobj /Parent 2 0 R /Next 4 0 R /Title (Udsagnslogik) - /Dest 379 0 R + /Dest 1050 0 R >> endobj @@ -33,7 +33,7 @@ endobj /Next 5 0 R /Prev 3 0 R /Title - /Dest 380 0 R + /Dest 1051 0 R >> endobj @@ -46,7 +46,7 @@ endobj /Last 12 0 R /Count -3 /Title (Funktioner) - /Dest 388 0 R + /Dest 1059 0 R >> endobj @@ -58,7 +58,7 @@ endobj /Last 10 0 R /Count -4 /Title (Inkektive, surjektive og bijektive funktioner) - /Dest 385 0 R + /Dest 1056 0 R >> endobj @@ -67,7 +67,7 @@ endobj /Parent 6 0 R /Next 8 0 R /Title (Injektiv) - /Dest 381 0 R + /Dest 1052 0 R >> endobj @@ -77,7 +77,7 @@ endobj /Next 9 0 R /Prev 7 0 R /Title (Surjektiv) - /Dest 382 0 R + /Dest 1053 0 R >> endobj @@ -87,7 +87,7 @@ endobj /Next 10 0 R /Prev 8 0 R /Title (Bijektiv) - /Dest 383 0 R + /Dest 1054 0 R >> endobj @@ -96,7 +96,7 @@ endobj /Parent 6 0 R /Prev 9 0 R /Title (Inverse funktion) - /Dest 384 0 R + /Dest 1055 0 R >> endobj @@ -106,7 +106,7 @@ endobj /Next 12 0 R /Prev 6 0 R /Title (Identitetsfunktionen) - /Dest 386 0 R + /Dest 1057 0 R >> endobj @@ -115,40 +115,186 @@ endobj /Parent 5 0 R /Prev 11 0 R /Title (Sammensatte funktioner) - /Dest 387 0 R + /Dest 1058 0 R >> endobj 13 0 obj << /Parent 2 0 R - /Next 14 0 R + /Next 16 0 R /Prev 5 0 R - /Title (Komplekse tal) - /Dest 389 0 R + /First 14 0 R + /Last 14 0 R + /Count -1 + /Title + /Dest 1062 0 R >> endobj 14 0 obj << - /Parent 2 0 R - /Next 15 0 R - /Prev 13 0 R - /Title (Polynomier) - /Dest 390 0 R + /Parent 13 0 R + /First 15 0 R + /Last 15 0 R + /Count -1 + /Title (Mulige former for komplekse tal til eksamen) + /Dest 1061 0 R >> endobj 15 0 obj << - /Parent 2 0 R - /Prev 14 0 R - /Title (Matricer) - /Dest 391 0 R + /Parent 14 0 R + /Title (Binome ligninger (binomier)) + /Dest 1060 0 R >> endobj 16 0 obj +<< + /Parent 2 0 R + /Next 20 0 R + /Prev 13 0 R + /First 17 0 R + /Last 17 0 R + /Count -1 + /Title (Polynomier) + /Dest 1066 0 R +>> +endobj + +17 0 obj +<< + /Parent 16 0 R + /First 18 0 R + /Last 19 0 R + /Count -2 + /Title + /Dest 1065 0 R +>> +endobj + +18 0 obj +<< + /Parent 17 0 R + /Next 19 0 R + /Title + /Dest 1063 0 R +>> +endobj + +19 0 obj +<< + /Parent 17 0 R + /Prev 18 0 R + /Title (Divisionsalgoritmen) + /Dest 1064 0 R +>> +endobj 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obj +<< + /Parent 2 0 R + /Prev 24 0 R + /Title + /Dest 1076 0 R +>> +endobj + +30 0 obj << /Type /StructTreeRoot /RoleMap << @@ -158,377 +304,112 @@ endobj /Strong /Span /Em /Span >> - /K [24 0 R] + /K [54 0 R] /ParentTree << - /Nums [0 17 0 R 1 18 0 R 2 19 0 R 3 20 0 R 4 21 0 R 5 22 0 R 6 23 0 R] + /Nums [0 31 0 R 1 32 0 R 2 33 0 R 3 34 0 R 4 35 0 R 5 36 0 R 6 37 0 R 7 38 0 R 8 39 0 R 9 40 0 R 10 41 0 R 11 42 0 R 12 43 0 R 13 44 0 R 14 45 0 R 15 46 0 R 16 47 0 R 17 48 0 R 18 49 0 R 19 50 0 R 20 51 0 R 21 52 0 R 22 53 0 R] >> - /ParentTreeNextKey 7 ->> -endobj - -17 0 obj -[356 0 R 353 0 R 350 0 R 349 0 R 349 0 R 348 0 R 347 0 R 345 0 R 344 0 R 341 0 R 340 0 R 339 0 R 338 0 R 336 0 R 334 0 R 334 0 R 334 0 R 332 0 R 332 0 R 332 0 R 330 0 R 330 0 R 328 0 R 328 0 R 328 0 R 326 0 R 326 0 R 326 0 R 323 0 R 321 0 R 319 0 R 317 0 R 315 0 R 313 0 R 311 0 R 308 0 R 306 0 R 304 0 R 302 0 R 300 0 R 298 0 R 296 0 R 293 0 R 291 0 R 289 0 R 287 0 R 285 0 R 283 0 R 281 0 R 278 0 R 276 0 R 274 0 R 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/Type /StructElem - /S /H1 - /P 24 0 R - /T (Matricer) - /K [2] - /Pg 404 0 R ->> -endobj - -26 0 obj -<< - /Type /StructElem - /S /H1 - /P 24 0 R - /T (Polynomier) - /K [1] - /Pg 404 0 R ->> -endobj - -27 0 obj -<< - /Type /StructElem - /S /H1 - /P 24 0 R - /T (Komplekse tal) - /K [0] - /Pg 404 0 R ->> -endobj - -28 0 obj -<< - /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136] - /Pg 402 0 R ->> -endobj - -29 0 obj -<< - /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [106 107 108 109 110 111 112 113 114 115 116 117] - /Pg 402 0 R ->> -endobj - -30 0 obj -<< - /Type /StructElem - /S /H2 - /P 24 0 R - /T (Sammensatte funktioner) - /K [105] - /Pg 402 0 R + /ParentTreeNextKey 23 >> endobj 31 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [104] - /Pg 402 0 R ->> +[1021 0 R 1020 0 R 1019 0 R 1018 0 R 1016 0 R 1015 0 R 1012 0 R] endobj 32 0 obj -<< - /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [95 96 97 98 99 100 101 102 103] - /Pg 402 0 R ->> +[1011 0 R 1010 0 R 1009 0 R 1008 0 R 1006 0 R 1004 0 R 1004 0 R 1004 0 R 1002 0 R 1002 0 R 1002 0 R 1000 0 R 1000 0 R 998 0 R 998 0 R 998 0 R 996 0 R 996 0 R 996 0 R 993 0 R 991 0 R 989 0 R 987 0 R 985 0 R 983 0 R 981 0 R 978 0 R 976 0 R 974 0 R 972 0 R 970 0 R 968 0 R 966 0 R 963 0 R 961 0 R 959 0 R 957 0 R 955 0 R 953 0 R 951 0 R 948 0 R 946 0 R 944 0 R 942 0 R 940 0 R 938 0 R 936 0 R 927 0 R 932 0 R 932 0 R 932 0 R 927 0 R 931 0 R 927 0 R 930 0 R 930 0 R 930 0 R 927 0 R 929 0 R 927 0 R 928 0 R 927 0 R 927 0 R 926 0 R 925 0 R 925 0 R 925 0 R 925 0 R 925 0 R 925 0 R 925 0 R 925 0 R 925 0 R 924 0 R 923 0 R 921 0 R 919 0 R 917 0 R 917 0 R 917 0 R 917 0 R 917 0 R 915 0 R 915 0 R 915 0 R 915 0 R 915 0 R 915 0 R 915 0 R 915 0 R 915 0 R 912 0 R 910 0 R 908 0 R 906 0 R 904 0 R 901 0 R 899 0 R 897 0 R 895 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0 R 784 0 R 784 0 R 784 0 R 783 0 R 782 0 R 782 0 R 781 0 R] endobj 34 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [76 37 0 R 82 36 0 R 85 35 0 R] - /Pg 402 0 R ->> +[780 0 R 780 0 R 780 0 R 780 0 R 780 0 R 780 0 R 780 0 R 780 0 R 780 0 R 780 0 R 780 0 R 780 0 R 779 0 R 778 0 R 778 0 R 778 0 R 778 0 R 778 0 R 778 0 R 778 0 R 778 0 R 778 0 R 778 0 R 775 0 R 777 0 R 777 0 R 777 0 R 777 0 R 775 0 R 775 0 R 775 0 R 776 0 R 776 0 R 776 0 R 776 0 R 776 0 R 776 0 R 776 0 R 774 0 R 773 0 R 773 0 R 773 0 R 773 0 R 773 0 R 773 0 R 773 0 R 773 0 R 773 0 R 773 0 R 772 0 R 771 0 R 770 0 R 769 0 R 768 0 R 768 0 R 767 0 R 765 0 R 766 0 R 765 0 R 765 0 R 764 0 R 763 0 R 763 0 R 762 0 R 762 0 R] endobj 35 0 obj -<< - /Type /StructElem - /S /Formula - /P 34 0 R - /K [86 87 88 89 90 91 92 93] - /Pg 402 0 R ->> +[761 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 760 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 759 0 R 758 0 R 757 0 R 754 0 R 756 0 R 756 0 R 756 0 R 756 0 R 754 0 R 755 0 R 755 0 R 755 0 R 755 0 R 754 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 753 0 R 752 0 R 751 0 R 751 0 R 751 0 R 751 0 R 751 0 R 751 0 R 751 0 R 751 0 R 751 0 R 751 0 R 751 0 R 751 0 R 749 0 R 750 0 R 750 0 R 750 0 R 748 0 R 746 0 R 747 0 R 747 0 R 747 0 R 747 0 R 747 0 R 747 0 R 747 0 R 747 0 R 747 0 R 742 0 R 745 0 R 742 0 R 744 0 R 742 0 R 743 0 R 743 0 R 743 0 R 743 0 R 742 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 741 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 740 0 R 739 0 R 738 0 R 738 0 R 738 0 R 738 0 R 738 0 R 738 0 R 738 0 R 738 0 R 738 0 R 738 0 R] endobj 36 0 obj -<< - /Type /StructElem - /S /Formula - /P 34 0 R - /K [83 84] - /Pg 402 0 R ->> +[737 0 R 734 0 R 736 0 R 736 0 R 736 0 R 736 0 R 736 0 R 734 0 R 735 0 R 735 0 R 735 0 R 735 0 R 734 0 R 734 0 R 733 0 R 732 0 R 732 0 R 732 0 R 732 0 R 732 0 R 732 0 R 732 0 R 732 0 R 732 0 R 732 0 R 732 0 R 732 0 R 731 0 R 729 0 R 730 0 R 730 0 R 729 0 R 729 0 R 728 0 R 727 0 R 727 0 R 727 0 R 727 0 R 727 0 R 727 0 R 727 0 R 727 0 R 727 0 R 727 0 R 727 0 R 727 0 R 726 0 R 724 0 R 724 0 R 725 0 R 725 0 R 725 0 R 725 0 R 725 0 R 725 0 R 725 0 R 725 0 R 725 0 R 725 0 R 725 0 R 725 0 R 723 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 722 0 R 721 0 R 720 0 R 718 0 R 719 0 R 719 0 R 719 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 717 0 R 716 0 R 714 0 R 715 0 R 715 0 R 714 0 R 711 0 R 713 0 R 713 0 R 713 0 R 713 0 R 711 0 R 712 0 R 712 0 R 711 0 R 710 0 R] endobj 37 0 obj -<< - /Type /StructElem - /S /Formula - /P 34 0 R - /K [77 78 79 80 81] - /Pg 402 0 R ->> +[709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 709 0 R 708 0 R 704 0 R 707 0 R 707 0 R 707 0 R 707 0 R 707 0 R 704 0 R 706 0 R 706 0 R 704 0 R 705 0 R 705 0 R 705 0 R 705 0 R 705 0 R 705 0 R 705 0 R 705 0 R 703 0 R 702 0 R 702 0 R 702 0 R 702 0 R 702 0 R 702 0 R 702 0 R 702 0 R 702 0 R 701 0 R 700 0 R 699 0 R 699 0 R 699 0 R 699 0 R 699 0 R 699 0 R 699 0 R 699 0 R 699 0 R 699 0 R 699 0 R 699 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 698 0 R 696 0 R 697 0 R 693 0 R 695 0 R 693 0 R 694 0 R 694 0 R 694 0 R 694 0 R 694 0 R 691 0 R 692 0 R 692 0 R 692 0 R 692 0 R 692 0 R 689 0 R 690 0 R 689 0 R 688 0 R 687 0 R 687 0 R 687 0 R 687 0 R 687 0 R 687 0 R 683 0 R 685 0 R 685 0 R 685 0 R 685 0 R 685 0 R 685 0 R 685 0 R 683 0 R 684 0 R 684 0 R 684 0 R 684 0 R 684 0 R 684 0 R 684 0 R 682 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 681 0 R 679 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 678 0 R 677 0 R] endobj 38 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [75] - /Pg 402 0 R ->> +[676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 676 0 R 675 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 674 0 R 673 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 672 0 R 671 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 670 0 R 669 0 R 668 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 667 0 R 666 0 R 665 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 664 0 R 660 0 R 663 0 R 663 0 R 663 0 R 663 0 R 663 0 R 663 0 R 660 0 R 662 0 R 662 0 R 662 0 R 662 0 R 660 0 R 661 0 R 661 0 R 660 0 R] endobj 39 0 obj -<< - /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74] - /Pg 402 0 R ->> +[657 0 R 657 0 R 657 0 R 657 0 R 657 0 R 657 0 R 657 0 R 657 0 R 656 0 R 656 0 R 656 0 R 656 0 R 656 0 R 656 0 R 656 0 R 656 0 R 655 0 R 654 0 R 653 0 R 652 0 R 651 0 R 650 0 R 649 0 R 648 0 R 647 0 R 646 0 R 645 0 R 644 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 643 0 R 642 0 R 641 0 R 641 0 R 640 0 R 640 0 R 639 0 R 639 0 R 638 0 R 638 0 R 637 0 R 637 0 R 637 0 R 636 0 R 636 0 R 636 0 R 635 0 R 635 0 R 635 0 R 634 0 R 633 0 R 633 0 R 633 0 R 632 0 R 632 0 R 632 0 R 631 0 R 631 0 R 631 0 R 630 0 R 630 0 R 630 0 R 629 0 R 629 0 R 629 0 R 628 0 R 628 0 R 628 0 R 627 0 R 627 0 R 627 0 R 627 0 R 626 0 R 626 0 R 626 0 R 626 0 R 626 0 R 626 0 R 626 0 R 626 0 R 625 0 R 625 0 R 625 0 R 625 0 R 625 0 R 625 0 R 625 0 R 625 0 R 625 0 R 624 0 R 624 0 R 624 0 R 624 0 R 624 0 R 624 0 R 624 0 R 624 0 R 623 0 R 623 0 R 623 0 R 623 0 R 623 0 R 623 0 R 623 0 R 623 0 R 623 0 R 622 0 R 622 0 R 622 0 R 622 0 R 622 0 R 622 0 R 622 0 R 622 0 R 622 0 R 622 0 R 621 0 R 621 0 R 621 0 R 621 0 R 621 0 R 621 0 R 621 0 R 621 0 R 621 0 R 620 0 R 620 0 R 620 0 R 620 0 R 620 0 R 620 0 R 620 0 R 620 0 R 620 0 R 620 0 R 619 0 R 619 0 R 619 0 R 619 0 R 619 0 R 619 0 R 619 0 R 619 0 R 619 0 R 619 0 R 619 0 R 618 0 R 618 0 R 618 0 R 618 0 R 618 0 R 618 0 R 618 0 R 618 0 R 618 0 R 618 0 R 617 0 R 617 0 R 617 0 R 617 0 R 617 0 R 617 0 R 617 0 R 617 0 R 617 0 R 616 0 R 616 0 R 616 0 R 616 0 R 616 0 R 616 0 R 616 0 R 616 0 R 616 0 R 616 0 R 615 0 R 615 0 R 615 0 R 615 0 R 615 0 R 615 0 R 615 0 R 615 0 R 615 0 R 614 0 R 614 0 R 614 0 R 614 0 R 614 0 R 613 0 R 613 0 R 612 0 R 612 0 R 612 0 R 612 0 R 612 0 R 611 0 R 611 0 R 610 0 R 610 0 R 610 0 R 610 0 R 610 0 R 610 0 R 609 0 R 609 0 R 609 0 R 608 0 R 608 0 R 608 0 R 608 0 R 608 0 R 608 0 R 607 0 R 607 0 R 607 0 R 606 0 R 605 0 R 605 0 R 605 0 R 605 0 R 604 0 R 604 0 R 604 0 R 604 0 R 603 0 R 603 0 R 603 0 R 602 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 601 0 R 599 0 R 598 0 R 598 0 R 598 0 R 598 0 R 598 0 R 598 0 R 598 0 R 598 0 R 598 0 R 598 0 R 598 0 R 596 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 595 0 R 592 0 R 590 0 R 588 0 R 585 0 R 583 0 R 581 0 R 578 0 R 578 0 R 576 0 R 576 0 R 576 0 R 574 0 R 574 0 R 571 0 R 571 0 R 569 0 R 569 0 R 569 0 R 567 0 R 567 0 R 567 0 R 564 0 R 564 0 R 562 0 R 562 0 R 560 0 R 560 0 R 560 0 R 557 0 R 557 0 R 555 0 R 553 0 R 659 0 R 547 0 R 546 0 R 544 0 R 543 0 R 542 0 R 539 0 R 540 0 R 539 0 R 536 0 R 535 0 R 532 0 R 534 0 R 534 0 R 534 0 R 534 0 R 534 0 R 532 0 R 533 0 R 533 0 R 533 0 R 533 0 R 533 0 R 533 0 R 533 0 R 533 0 R 533 0 R 533 0 R 533 0 R 529 0 R 528 0 R 526 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 527 0 R 523 0 R 522 0 R 521 0 R 518 0 R 517 0 R 514 0 R 516 0 R 516 0 R 516 0 R 516 0 R 516 0 R 516 0 R 514 0 R 515 0 R 514 0 R 512 0 R 511 0 R 508 0 R 510 0 R 510 0 R 510 0 R 510 0 R 510 0 R 510 0 R 510 0 R 510 0 R 508 0 R 509 0 R 508 0 R 506 0 R 505 0 R 502 0 R 504 0 R 504 0 R 504 0 R 504 0 R 504 0 R 504 0 R 504 0 R 504 0 R 504 0 R 502 0 R 503 0 R 502 0 R 500 0 R 499 0 R 496 0 R 498 0 R 498 0 R 498 0 R 498 0 R 498 0 R 498 0 R 498 0 R 498 0 R 498 0 R 496 0 R 497 0 R 496 0 R 493 0 R 492 0 R 488 0 R 491 0 R 491 0 R 491 0 R 491 0 R 491 0 R 491 0 R 491 0 R 491 0 R 488 0 R 490 0 R 490 0 R 490 0 R 490 0 R 490 0 R 490 0 R 490 0 R 490 0 R 488 0 R 489 0 R 489 0 R 489 0 R 489 0 R 489 0 R 489 0 R 484 0 R 483 0 R 482 0 R 482 0 R 482 0 R 482 0 R 482 0 R] endobj 40 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [35] - /Pg 402 0 R ->> +[481 0 R 481 0 R 481 0 R 479 0 R 480 0 R 480 0 R 480 0 R 479 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 478 0 R 477 0 R 475 0 R 476 0 R 476 0 R 476 0 R 476 0 R 475 0 R 474 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 473 0 R 472 0 R 470 0 R 471 0 R 471 0 R 471 0 R 471 0 R 469 0 R 469 0 R 469 0 R 469 0 R 469 0 R 469 0 R 469 0 R 469 0 R 469 0 R 469 0 R 469 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 467 0 R 466 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R 465 0 R] endobj 41 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [26 43 0 R 31 42 0 R 34] - /Pg 402 0 R ->> +[464 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 463 0 R 461 0 R 462 0 R 462 0 R 462 0 R 462 0 R 461 0 R 460 0 R 457 0 R 459 0 R 457 0 R 458 0 R 458 0 R 458 0 R 458 0 R 458 0 R 458 0 R 456 0 R 455 0 R 452 0 R 452 0 R 454 0 R 454 0 R 454 0 R 454 0 R 452 0 R 453 0 R 453 0 R 453 0 R 453 0 R 453 0 R 453 0 R 453 0 R 452 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 451 0 R 450 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 449 0 R 447 0 R 448 0 R 448 0 R 448 0 R 448 0 R 448 0 R 448 0 R 448 0 R 448 0 R 448 0 R 446 0 R 446 0 R 444 0 R 445 0 R 445 0 R 443 0 R 440 0 R 442 0 R 442 0 R 442 0 R 442 0 R 440 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 441 0 R 437 0 R 439 0 R 439 0 R 439 0 R 439 0 R 437 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 438 0 R 434 0 R 436 0 R 436 0 R 436 0 R 434 0 R 435 0 R 435 0 R 435 0 R 435 0 R 435 0 R 435 0 R 435 0 R 435 0 R 435 0 R 435 0 R 435 0 R 435 0 R 433 0 R 431 0 R 432 0 R 432 0 R 432 0 R 431 0 R 431 0 R 430 0 R 429 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 428 0 R 426 0 R 427 0 R 427 0 R 427 0 R 427 0 R 427 0 R 427 0 R 426 0 R 425 0 R] endobj 42 0 obj -<< - /Type /StructElem - /S /Formula - /P 41 0 R - /K [32 33] - /Pg 402 0 R ->> +[424 0 R 423 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 422 0 R 420 0 R 421 0 R 421 0 R 421 0 R 421 0 R 419 0 R 418 0 R 416 0 R 417 0 R 417 0 R 417 0 R 417 0 R 417 0 R 417 0 R 417 0 R 417 0 R 417 0 R 416 0 R 415 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 414 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 413 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 412 0 R 411 0 R 411 0 R] endobj 43 0 obj -<< - /Type /StructElem - /S /Formula - /P 41 0 R - /K [27 28 29 30] - /Pg 402 0 R ->> +[410 0 R 406 0 R 409 0 R 409 0 R 409 0 R 409 0 R 409 0 R 409 0 R 406 0 R 408 0 R 406 0 R 407 0 R 406 0 R 406 0 R 405 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 404 0 R 403 0 R 402 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 401 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 400 0 R 399 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 398 0 R 397 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 396 0 R 395 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 394 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R 393 0 R] endobj 44 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [22 45 0 R 25] - /Pg 402 0 R ->> +[392 0 R 391 0 R 391 0 R 391 0 R 391 0 R 391 0 R 391 0 R 391 0 R 391 0 R 390 0 R 390 0 R 390 0 R 390 0 R 389 0 R 389 0 R 389 0 R 389 0 R 389 0 R 389 0 R 389 0 R 389 0 R 389 0 R 389 0 R 389 0 R 389 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 387 0 R 386 0 R 385 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 384 0 R 380 0 R 383 0 R 383 0 R 383 0 R 383 0 R 380 0 R 382 0 R 382 0 R 382 0 R 382 0 R 382 0 R 382 0 R 382 0 R 382 0 R 382 0 R 382 0 R 382 0 R 380 0 R 381 0 R 381 0 R 381 0 R 381 0 R 381 0 R 380 0 R 379 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 378 0 R 377 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 376 0 R 375 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 374 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R 373 0 R] endobj 45 0 obj -<< - /Type /StructElem - /S /Formula - /P 44 0 R - /K [23 24] - /Pg 402 0 R ->> +[372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 372 0 R 371 0 R 369 0 R 370 0 R 370 0 R 370 0 R 370 0 R 370 0 R 370 0 R 370 0 R 370 0 R 370 0 R 369 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 368 0 R 366 0 R 367 0 R 367 0 R 367 0 R 367 0 R 367 0 R 367 0 R 367 0 R 365 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 364 0 R 362 0 R 359 0 R 361 0 R 361 0 R 361 0 R 359 0 R 360 0 R 360 0 R 360 0 R 360 0 R 360 0 R 360 0 R 360 0 R 360 0 R 360 0 R 360 0 R 360 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 358 0 R 357 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 356 0 R 355 0 R 353 0 R 354 0 R 354 0 R 354 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 352 0 R 351 0 R 349 0 R 350 0 R 350 0 R 350 0 R 350 0 R 350 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 348 0 R 347 0 R] endobj 46 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [21] - /Pg 402 0 R ->> +[346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 346 0 R 345 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 344 0 R 341 0 R 343 0 R 343 0 R 343 0 R 343 0 R 341 0 R 342 0 R 342 0 R 342 0 R 340 0 R 337 0 R 339 0 R 339 0 R 339 0 R 338 0 R 338 0 R 338 0 R 338 0 R 338 0 R 338 0 R 336 0 R 333 0 R 335 0 R 335 0 R 335 0 R 335 0 R 335 0 R 335 0 R 333 0 R 334 0 R 334 0 R 334 0 R 334 0 R 334 0 R 332 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 331 0 R 330 0 R 327 0 R 329 0 R 329 0 R 329 0 R 327 0 R 328 0 R 328 0 R 328 0 R 328 0 R 328 0 R 326 0 R 323 0 R 325 0 R 325 0 R 325 0 R 325 0 R 325 0 R 323 0 R 324 0 R 324 0 R 324 0 R 324 0 R 324 0 R 324 0 R 324 0 R 322 0 R 319 0 R 321 0 R 321 0 R 321 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 320 0 R 319 0 R 318 0 R 315 0 R 317 0 R 317 0 R 317 0 R 317 0 R 317 0 R 315 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 316 0 R 314 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 313 0 R 309 0 R 312 0 R 312 0 R 312 0 R 312 0 R 312 0 R 309 0 R 311 0 R 311 0 R 311 0 R 311 0 R 311 0 R 311 0 R 311 0 R 311 0 R 309 0 R 310 0 R 310 0 R 310 0 R 310 0 R 310 0 R 310 0 R 308 0 R 307 0 R 307 0 R 307 0 R 307 0 R 307 0 R 307 0 R 307 0 R 307 0 R] endobj 47 0 obj -<< - /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20] - /Pg 402 0 R ->> +[303 0 R 305 0 R 305 0 R 305 0 R 305 0 R 305 0 R 305 0 R 305 0 R 305 0 R 305 0 R 303 0 R 304 0 R 304 0 R 304 0 R 302 0 R 299 0 R 301 0 R 301 0 R 301 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 300 0 R 298 0 R 295 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 297 0 R 295 0 R 296 0 R 296 0 R 296 0 R 296 0 R 296 0 R 294 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 293 0 R 292 0 R 291 0 R 290 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 289 0 R 286 0 R 287 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 285 0 R 284 0 R 284 0 R 284 0 R 284 0 R 284 0 R 284 0 R 284 0 R 282 0 R 283 0 R 282 0 R 281 0 R 280 0 R 279 0 R 279 0 R 279 0 R 279 0 R 279 0 R 279 0 R 279 0 R 279 0 R 279 0 R 279 0 R 279 0 R 279 0 R 277 0 R] endobj 48 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [1 49 0 R] - /Pg 402 0 R ->> +[276 0 R 276 0 R 275 0 R 272 0 R 274 0 R 274 0 R 274 0 R 274 0 R 272 0 R 273 0 R 273 0 R 273 0 R 273 0 R 273 0 R 271 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 270 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 269 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 268 0 R 267 0 R 265 0 R 265 0 R 265 0 R 265 0 R 265 0 R 262 0 R 264 0 R 262 0 R 263 0 R 262 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 261 0 R 259 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 258 0 R 256 0 R 257 0 R 256 0 R 255 0 R 254 0 R 246 0 R 253 0 R 246 0 R 252 0 R 246 0 R 251 0 R 246 0 R 250 0 R 246 0 R 249 0 R 246 0 R 248 0 R 246 0 R 246 0 R 247 0 R 246 0 R] endobj 49 0 obj -<< - /Type /StructElem - /S /Formula - /P 48 0 R - /K [2 3 4] - /Pg 402 0 R ->> +[245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 245 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 244 0 R 242 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 241 0 R 240 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 239 0 R 235 0 R 237 0 R 237 0 R 235 0 R 236 0 R 235 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 234 0 R 231 0 R 233 0 R 233 0 R 233 0 R 231 0 R 232 0 R 232 0 R 232 0 R 230 0 R 229 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 228 0 R 227 0 R 227 0 R 226 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 225 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 223 0 R 221 0 R 221 0 R 221 0 R 221 0 R 221 0 R 221 0 R 221 0 R 221 0 R 221 0 R 221 0 R 221 0 R 221 0 R 219 0 R 218 0 R 218 0 R 217 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 215 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 211 0 R 210 0 R] endobj 50 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [0] - /Pg 402 0 R ->> +[209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 209 0 R 206 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 208 0 R 206 0 R 207 0 R 206 0 R 205 0 R 201 0 R 203 0 R 203 0 R 203 0 R 201 0 R 202 0 R 202 0 R 202 0 R 202 0 R 202 0 R 202 0 R 201 0 R 196 0 R 198 0 R 198 0 R 198 0 R 198 0 R 198 0 R 198 0 R 198 0 R 196 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 197 0 R 195 0 R 195 0 R 195 0 R 194 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 193 0 R 192 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 191 0 R 190 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 189 0 R 188 0 R 187 0 R 185 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 186 0 R 184 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 182 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 178 0 R 177 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R 175 0 R] endobj 51 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [126] - /Pg 400 0 R ->> +[171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 171 0 R 170 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 169 0 R 168 0 R 166 0 R 167 0 R 166 0 R 165 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 163 0 R 157 0 R 156 0 R 156 0 R 156 0 R 156 0 R 156 0 R 156 0 R 156 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 154 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 152 0 R 151 0 R 148 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 149 0 R 146 0 R 141 0 R 144 0 R 144 0 R 144 0 R 144 0 R 144 0 R 144 0 R 144 0 R 144 0 R 141 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 143 0 R 141 0 R 142 0 R 139 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 137 0 R 133 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 132 0 R 131 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 129 0 R 125 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 123 0 R 119 0 R 117 0 R 117 0 R 117 0 R 117 0 R 117 0 R 117 0 R 117 0 R 117 0 R 117 0 R 117 0 R 117 0 R] endobj 52 0 obj -<< - /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [112 113 114 115 116 117 118 119 120 121 122 123 124 125] - /Pg 400 0 R ->> +[112 0 R 111 0 R 111 0 R 111 0 R 111 0 R 111 0 R 111 0 R 111 0 R 111 0 R 111 0 R 111 0 R 111 0 R 108 0 R 109 0 R 109 0 R 109 0 R 109 0 R 108 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 107 0 R 103 0 R 105 0 R 105 0 R 103 0 R 104 0 R 104 0 R 104 0 R 104 0 R 104 0 R 104 0 R 102 0 R 101 0 R 98 0 R 100 0 R 100 0 R 100 0 R 100 0 R 100 0 R 100 0 R 100 0 R 100 0 R 98 0 R 99 0 R 98 0 R 96 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 97 0 R 96 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 95 0 R 92 0 R 94 0 R 92 0 R 93 0 R 93 0 R 93 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 90 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 89 0 R 87 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 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 86 0 R 86 0 R 86 0 R 86 0 R 84 0 R 85 0 R 84 0 R 83 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 82 0 R 79 0 R 81 0 R 79 0 R 80 0 R 80 0 R 80 0 R 80 0 R 80 0 R 80 0 R 80 0 R 77 0 R 78 0 R 78 0 R 78 0 R 78 0 R 78 0 R 78 0 R 78 0 R 78 0 R 74 0 R 76 0 R 76 0 R 76 0 R 76 0 R 76 0 R 74 0 R 75 0 R 75 0 R 75 0 R 75 0 R 75 0 R 75 0 R 75 0 R 75 0 R 75 0 R 73 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 72 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 70 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 66 0 R 65 0 R 64 0 R] endobj 53 0 obj -<< - /Type /StructElem - /S /P - /P 24 0 R - /K [111] - /Pg 400 0 R ->> +[62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 62 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 61 0 R 60 0 R 59 0 R 59 0 R 59 0 R 59 0 R 56 0 R 58 0 R 56 0 R 57 0 R 57 0 R 57 0 R 57 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R 55 0 R] endobj 54 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [97 98 55 0 R] - /Pg 400 0 R + /S /Document + /P 30 0 R + /K [1021 0 R 1020 0 R 1013 0 R 1012 0 R 1011 0 R 1010 0 R 1009 0 R 933 0 R 927 0 R 926 0 R 924 0 R 824 0 R 822 0 R 821 0 R 820 0 R 819 0 R 818 0 R 817 0 R 811 0 R 810 0 R 804 0 R 803 0 R 797 0 R 796 0 R 795 0 R 793 0 R 791 0 R 787 0 R 786 0 R 785 0 R 784 0 R 783 0 R 782 0 R 781 0 R 780 0 R 775 0 R 774 0 R 773 0 R 772 0 R 771 0 R 770 0 R 769 0 R 768 0 R 767 0 R 765 0 R 764 0 R 763 0 R 762 0 R 761 0 R 760 0 R 759 0 R 758 0 R 757 0 R 754 0 R 753 0 R 752 0 R 751 0 R 749 0 R 748 0 R 746 0 R 742 0 R 741 0 R 740 0 R 739 0 R 738 0 R 737 0 R 734 0 R 733 0 R 732 0 R 731 0 R 729 0 R 728 0 R 727 0 R 726 0 R 724 0 R 723 0 R 722 0 R 721 0 R 720 0 R 718 0 R 717 0 R 716 0 R 714 0 R 711 0 R 710 0 R 709 0 R 708 0 R 704 0 R 703 0 R 702 0 R 701 0 R 700 0 R 699 0 R 698 0 R 696 0 R 693 0 R 691 0 R 689 0 R 686 0 R 683 0 R 680 0 R 679 0 R 678 0 R 677 0 R 676 0 R 675 0 R 674 0 R 673 0 R 672 0 R 671 0 R 670 0 R 669 0 R 668 0 R 667 0 R 666 0 R 665 0 R 664 0 R 660 0 R 548 0 R 547 0 R 545 0 R 485 0 R 484 0 R 483 0 R 482 0 R 479 0 R 478 0 R 477 0 R 475 0 R 474 0 R 473 0 R 472 0 R 470 0 R 468 0 R 467 0 R 466 0 R 465 0 R 464 0 R 463 0 R 461 0 R 460 0 R 457 0 R 456 0 R 455 0 R 452 0 R 451 0 R 450 0 R 449 0 R 447 0 R 444 0 R 443 0 R 440 0 R 437 0 R 434 0 R 433 0 R 431 0 R 430 0 R 429 0 R 426 0 R 425 0 R 424 0 R 423 0 R 422 0 R 420 0 R 419 0 R 418 0 R 416 0 R 415 0 R 414 0 R 413 0 R 412 0 R 411 0 R 410 0 R 406 0 R 405 0 R 404 0 R 403 0 R 402 0 R 401 0 R 400 0 R 399 0 R 398 0 R 397 0 R 396 0 R 395 0 R 394 0 R 393 0 R 392 0 R 388 0 R 387 0 R 386 0 R 385 0 R 384 0 R 380 0 R 379 0 R 378 0 R 377 0 R 376 0 R 375 0 R 374 0 R 373 0 R 372 0 R 371 0 R 369 0 R 368 0 R 366 0 R 363 0 R 359 0 R 358 0 R 357 0 R 356 0 R 353 0 R 352 0 R 349 0 R 348 0 R 347 0 R 346 0 R 345 0 R 344 0 R 341 0 R 337 0 R 333 0 R 332 0 R 331 0 R 330 0 R 327 0 R 326 0 R 323 0 R 319 0 R 315 0 R 314 0 R 313 0 R 309 0 R 308 0 R 306 0 R 303 0 R 299 0 R 295 0 R 294 0 R 293 0 R 292 0 R 291 0 R 290 0 R 288 0 R 286 0 R 285 0 R 282 0 R 281 0 R 278 0 R 277 0 R 276 0 R 275 0 R 272 0 R 271 0 R 270 0 R 269 0 R 268 0 R 266 0 R 262 0 R 260 0 R 259 0 R 258 0 R 256 0 R 255 0 R 254 0 R 246 0 R 245 0 R 243 0 R 242 0 R 241 0 R 240 0 R 238 0 R 235 0 R 234 0 R 231 0 R 230 0 R 229 0 R 227 0 R 226 0 R 224 0 R 222 0 R 220 0 R 219 0 R 218 0 R 212 0 R 211 0 R 210 0 R 209 0 R 206 0 R 199 0 R 196 0 R 194 0 R 193 0 R 192 0 R 191 0 R 190 0 R 189 0 R 188 0 R 185 0 R 179 0 R 178 0 R 172 0 R 171 0 R 170 0 R 169 0 R 166 0 R 160 0 R 158 0 R 157 0 R 155 0 R 153 0 R 152 0 R 134 0 R 133 0 R 132 0 R 113 0 R 112 0 R 110 0 R 108 0 R 106 0 R 103 0 R 102 0 R 98 0 R 96 0 R 92 0 R 90 0 R 87 0 R 84 0 R 83 0 R 79 0 R 77 0 R 74 0 R 73 0 R 71 0 R 69 0 R 67 0 R 65 0 R 63 0 R 62 0 R 61 0 R 56 0 R 55 0 R] >> endobj @@ -537,8 +418,12 @@ endobj /Type /StructElem /S /Formula /P 54 0 R - /K [99 100 101 102 103 104 105 106 107 108 109 110] - /Pg 400 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119] + /Pg 1123 0 R >> endobj @@ -546,9 +431,9 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [96] - /Pg 400 0 R + /P 54 0 R + /K [60 0 R 59 0 R 91 58 0 R 93 57 0 R] + /Pg 1123 0 R >> endobj @@ -556,53 +441,53 @@ endobj << /Type /StructElem /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [84 85 86 87 88 89 90 91 92 93 94 95] - /Pg 400 0 R + /P 56 0 R + /K [94 95 96 97] + /Pg 1123 0 R >> endobj 58 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [83] - /Pg 400 0 R + /S /Formula + /P 56 0 R + /K [92] + /Pg 1123 0 R >> endobj 59 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [78 60 0 R 81 82] - /Pg 400 0 R + /S /Formula + /P 56 0 R + /K [87 88 89 90] + /Pg 1123 0 R >> endobj 60 0 obj << /Type /StructElem - /S /Formula - /P 59 0 R - /K [79 80] - /Pg 400 0 R + /S /Strong + /P 56 0 R + /K [86] + /Pg 1123 0 R >> endobj 61 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [77] - /Pg 400 0 R + /S /Formula + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [75 76 77 78 79 80 81 82 83 84 85] + /Pg 1123 0 R >> endobj @@ -610,13 +495,13 @@ endobj << /Type /StructElem /S /Formula - /P 24 0 R + /P 54 0 R /A [<< /O /Layout /Placement /Block >>] - /K [65 66 67 68 69 70 71 72 73 74 75 76] - /Pg 400 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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74] + /Pg 1123 0 R >> endobj @@ -624,29 +509,28 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [64] - /Pg 400 0 R + /P 54 0 R + /K [64 0 R] >> endobj 64 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [51 66 0 R 57 65 0 R 62 63] - /Pg 400 0 R + /S /Strong + /P 63 0 R + /K [300] + /Pg 1121 0 R >> endobj 65 0 obj << /Type /StructElem - /S /Formula - /P 64 0 R - /K [58 59 60 61] - /Pg 400 0 R + /S /P + /P 54 0 R + /K [66 0 R 299] + /Pg 1121 0 R >> endobj @@ -654,9 +538,9 @@ endobj << /Type /StructElem /S /Formula - /P 64 0 R - /K [52 53 54 55 56] - /Pg 400 0 R + /P 65 0 R + /K [286 287 288 289 290 291 292 293 294 295 296 297 298] + /Pg 1121 0 R >> endobj @@ -664,9 +548,8 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [50] - /Pg 400 0 R + /P 54 0 R + /K [68 0 R] >> endobj @@ -674,13 +557,9 @@ endobj << /Type /StructElem /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [40 41 42 43 44 45 46 47 48 49] - /Pg 400 0 R + /P 67 0 R + /K [271 272 273 274 275 276 277 278 279 280 281 282 283 284 285] + /Pg 1121 0 R >> endobj @@ -688,9 +567,8 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [39] - /Pg 400 0 R + /P 54 0 R + /K [70 0 R] >> endobj @@ -698,57 +576,48 @@ endobj << /Type /StructElem /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /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 26 27 28 29 30 31 32 33 34 35 36 37 38] - /Pg 400 0 R + /P 69 0 R + /K [238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270] + /Pg 1121 0 R >> endobj 71 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185] - /Pg 398 0 R + /S /P + /P 54 0 R + /K [72 0 R] >> endobj 72 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [148 75 0 R 150 74 0 R 152 73 0 R 157] - /Pg 398 0 R + /S /Formula + /P 71 0 R + /K [225 226 227 228 229 230 231 232 233 234 235 236 237] + /Pg 1121 0 R >> endobj 73 0 obj << /Type /StructElem - /S /Formula - /P 72 0 R - /K [153 154 155 156] - /Pg 398 0 R + /S /P + /P 54 0 R + /K [224] + /Pg 1121 0 R >> endobj 74 0 obj << /Type /StructElem - /S /Formula - /P 72 0 R - /K [151] - /Pg 398 0 R + /S /P + /P 54 0 R + /K [208 76 0 R 214 75 0 R] + /Pg 1121 0 R >> endobj @@ -756,39 +625,39 @@ endobj << /Type /StructElem /S /Formula - /P 72 0 R - /K [149] - /Pg 398 0 R + /P 74 0 R + /K [215 216 217 218 219 220 221 222 223] + /Pg 1121 0 R >> endobj 76 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [138 77 0 R] - /Pg 398 0 R + /S /Formula + /P 74 0 R + /K [209 210 211 212 213] + /Pg 1121 0 R >> endobj 77 0 obj << /Type /StructElem - /S /Formula - /P 76 0 R - /K [139 140 141 142 143 144 145 146 147] - /Pg 398 0 R + /S /P + /P 54 0 R + /K [199 78 0 R] + /Pg 1121 0 R >> endobj 78 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [137] - /Pg 398 0 R + /S /Formula + /P 77 0 R + /K [200 201 202 203 204 205 206 207] + /Pg 1121 0 R >> endobj @@ -796,9 +665,9 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [133 80 0 R] - /Pg 398 0 R + /P 54 0 R + /K [82 0 R 189 81 0 R 191 80 0 R] + /Pg 1121 0 R >> endobj @@ -807,8 +676,8 @@ endobj /Type /StructElem /S /Formula /P 79 0 R - /K [134 135 136] - /Pg 398 0 R + /K [192 193 194 195 196 197 198] + /Pg 1121 0 R >> endobj @@ -816,37 +685,30 @@ endobj << /Type /StructElem /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [121 122 123 124 125 126 127 128 129 130 131 132] - /Pg 398 0 R + /P 79 0 R + /K [190] + /Pg 1121 0 R >> endobj 82 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [120] - /Pg 398 0 R + /S /Formula + /P 79 0 R + /K [183 184 185 186 187 188] + /Pg 1121 0 R >> endobj 83 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119] - /Pg 398 0 R + /S /H1 + /P 54 0 R + /T + /K [182] + /Pg 1121 0 R >> endobj @@ -854,9 +716,9 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [88 86 0 R 93 85 0 R 98] - /Pg 398 0 R + /P 54 0 R + /K [86 0 R 179 85 0 R 181] + /Pg 1121 0 R >> endobj @@ -865,8 +727,8 @@ endobj /Type /StructElem /S /Formula /P 84 0 R - /K [94 95 96 97] - /Pg 398 0 R + /K [180] + /Pg 1121 0 R >> endobj @@ -875,8 +737,8 @@ endobj /Type /StructElem /S /Formula /P 84 0 R - /K [89 90 91 92] - /Pg 398 0 R + /K [174 175 176 177 178] + /Pg 1121 0 R >> endobj @@ -884,19 +746,19 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [87] - /Pg 398 0 R + /P 54 0 R + /K [89 0 R 149 88 0 R] + /Pg 1121 0 R >> endobj 88 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [86] - /Pg 398 0 R + /S /Formula + /P 87 0 R + /K [150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173] + /Pg 1121 0 R >> endobj @@ -904,38 +766,29 @@ endobj << /Type /StructElem /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85] - /Pg 398 0 R + /P 87 0 R + /K [130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148] + /Pg 1121 0 R >> endobj 90 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32] - /Pg 398 0 R + /S /P + /P 54 0 R + /K [91 0 R 129] + /Pg 1121 0 R >> endobj 91 0 obj << /Type /StructElem - /S /H3 - /P 24 0 R - /T (Inverse funktion) - /K [5] - /Pg 398 0 R + /S /Formula + /P 90 0 R + /K [120 121 122 123 124 125 126 127 128] + /Pg 1121 0 R >> endobj @@ -943,65 +796,59 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [3 4] - /Pg 398 0 R + /P 54 0 R + /K [95 0 R 114 94 0 R 116 93 0 R] + /Pg 1121 0 R >> endobj 93 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [1 2] - /Pg 398 0 R + /S /Formula + /P 92 0 R + /K [117 118 119] + /Pg 1121 0 R >> endobj 94 0 obj << /Type /StructElem - /S /H3 - /P 24 0 R - /T (Bijektiv) - /K [0] - /Pg 398 0 R + /S /Strong + /P 92 0 R + /K [115] + /Pg 1121 0 R >> endobj 95 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [57 96 0 R 59 60] - /Pg 396 0 R + /S /Formula + /P 92 0 R + /K [95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113] + /Pg 1121 0 R >> endobj 96 0 obj << /Type /StructElem - /S /Span - /P 95 0 R - /A [<< - /O /Layout - /TextDecorationType /Underline - >>] - /K [58] - /Pg 396 0 R + /S /P + /P 54 0 R + /K [75 97 0 R 94] + /Pg 1121 0 R >> endobj 97 0 obj << /Type /StructElem - /S /H3 - /P 24 0 R - /T (Surjektiv) - /K [56] - /Pg 396 0 R + /S /Formula + /P 96 0 R + /K [76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93] + /Pg 1121 0 R >> endobj @@ -1009,103 +856,88 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [54 55] - /Pg 396 0 R + /P 54 0 R + /K [101 0 R 63 100 0 R 72 99 0 R 74] + /Pg 1121 0 R >> endobj 99 0 obj << /Type /StructElem - /S /H3 - /P 24 0 R - /T (Injektiv) - /K [53] - /Pg 396 0 R + /S /Formula + /P 98 0 R + /K [73] + /Pg 1121 0 R >> endobj 100 0 obj << /Type /StructElem - /S /Figure - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [52] - /Pg 396 0 R + /S /Formula + /P 98 0 R + /K [64 65 66 67 68 69 70 71] + /Pg 1121 0 R >> endobj 101 0 obj << /Type /StructElem - /S /H2 - /P 24 0 R - /T (Inkektive, surjektive og bijektive funktioner) - /K [51] - /Pg 396 0 R + /S /Formula + /P 98 0 R + /K [62] + /Pg 1121 0 R >> endobj 102 0 obj << /Type /StructElem - /S /Figure - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [50] - /Pg 396 0 R + /S /P + /P 54 0 R + /K [61] + /Pg 1121 0 R >> endobj 103 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [40 41 42 43 44 45 46 47 48 49] - /Pg 396 0 R + /S /P + /P 54 0 R + /K [51 105 0 R 54 104 0 R] + /Pg 1121 0 R >> endobj 104 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [39] - /Pg 396 0 R + /S /Formula + /P 103 0 R + /K [55 56 57 58 59 60] + /Pg 1121 0 R >> endobj 105 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [109 0 R 108 0 R 24 107 0 R 29 30 31 106 0 R] - /Pg 396 0 R + /S /Formula + /P 103 0 R + /K [52 53] + /Pg 1121 0 R >> endobj 106 0 obj << /Type /StructElem - /S /Formula - /P 105 0 R - /K [32 33 34 35 36 37 38] - /Pg 396 0 R + /S /P + /P 54 0 R + /K [107 0 R] >> endobj @@ -1113,57 +945,48 @@ endobj << /Type /StructElem /S /Formula - /P 105 0 R - /K [25 26 27 28] - /Pg 396 0 R + /P 106 0 R + /K [18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50] + /Pg 1121 0 R >> endobj 108 0 obj << /Type /StructElem - /S /Formula - /P 105 0 R - /K [14 15 16 17 18 19 20 21 22 23] - /Pg 396 0 R + /S /P + /P 54 0 R + /K [12 109 0 R 17] + /Pg 1121 0 R >> endobj 109 0 obj << /Type /StructElem - /S /Strong - /P 105 0 R - /K [13] - /Pg 396 0 R + /S /Formula + /P 108 0 R + /K [13 14 15 16] + /Pg 1121 0 R >> endobj 110 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [1 2 3 4 5 6 7 8 9 10 11 12] - /Pg 396 0 R + /S /P + /P 54 0 R + /K [111 0 R] >> endobj 111 0 obj << /Type /StructElem - /S /Figure - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [0] - /Pg 396 0 R + /S /Formula + /P 110 0 R + /K [1 2 3 4 5 6 7 8 9 10 11] + /Pg 1121 0 R >> endobj @@ -1171,48 +994,40 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [193 194] - /Pg 394 0 R + /P 54 0 R + /K [0] + /Pg 1121 0 R >> endobj 113 0 obj << /Type /StructElem - /S /H1 - /P 24 0 R - /T (Funktioner) - /K [192] - /Pg 394 0 R + /S /L + /P 54 0 R + /A [<< + /O /List + /ListNumbering /None + >>] + /K [126 0 R 120 0 R 114 0 R] >> endobj 114 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191] - /Pg 394 0 R + /S /LI + /P 113 0 R + /K [118 0 R 115 0 R] >> endobj 115 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154] - /Pg 394 0 R + /S /LBody + /P 114 0 R + /K [116 0 R] >> endobj @@ -1220,225 +1035,204 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [126] - /Pg 394 0 R + /P 115 0 R + /K [117 0 R] >> endobj 117 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [119 120 0 R 121 119 0 R 123 118 0 R 125] - /Pg 394 0 R + /S /Formula + /P 116 0 R + /K [422 423 424 425 426 427 428 429 430 431 432] + /Pg 1119 0 R >> endobj 118 0 obj << /Type /StructElem - /S /Formula - /P 117 0 R - /K [124] - /Pg 394 0 R + /S /Lbl + /P 114 0 R + /K [119 0 R] >> endobj 119 0 obj << /Type /StructElem - /S /Formula - /P 117 0 R - /K [122] - /Pg 394 0 R + /S /Strong + /P 118 0 R + /K [421] + /Pg 1119 0 R >> endobj 120 0 obj << /Type /StructElem - /S /Formula - /P 117 0 R - /K [120] - /Pg 394 0 R + /S /LI + /P 113 0 R + /K [124 0 R 121 0 R] >> endobj 121 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [111 122 0 R] - /Pg 394 0 R + /S /LBody + /P 120 0 R + /K [122 0 R] >> endobj 122 0 obj << /Type /StructElem - /S /Formula + /S /P /P 121 0 R - /K [112 113 114 115 116 117 118] - /Pg 394 0 R + /K [123 0 R] >> endobj 123 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [104 124 0 R 110] - /Pg 394 0 R + /S /Formula + /P 122 0 R + /K [366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420] + /Pg 1119 0 R >> endobj 124 0 obj << /Type /StructElem - /S /Formula - /P 123 0 R - /K [105 106 107 108 109] - /Pg 394 0 R + /S /Lbl + /P 120 0 R + /K [125 0 R] >> endobj 125 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [103] - /Pg 394 0 R + /S /Strong + /P 124 0 R + /K [365] + /Pg 1119 0 R >> endobj 126 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102] - /Pg 394 0 R + /S /LI + /P 113 0 R + /K [130 0 R 127 0 R] >> endobj 127 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [132 0 R 131 0 R 81 130 0 R 83 129 0 R 85 128 0 R] - /Pg 394 0 R + /S /LBody + /P 126 0 R + /K [128 0 R] >> endobj 128 0 obj << /Type /StructElem - /S /Formula + /S /P /P 127 0 R - /K [86] - /Pg 394 0 R + /K [129 0 R] >> endobj 129 0 obj << /Type /StructElem - /S /Span - /P 127 0 R - /A [<< - /O /Layout - /TextDecorationType /Underline - >>] - /K [84] - /Pg 394 0 R + /S /Formula + /P 128 0 R + /K [345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364] + /Pg 1119 0 R >> endobj 130 0 obj << /Type /StructElem - /S /Formula - /P 127 0 R - /K [82] - /Pg 394 0 R + /S /Lbl + /P 126 0 R + /K [131 0 R] >> endobj 131 0 obj << /Type /StructElem - /S /Formula - /P 127 0 R - /K [78 79 80] - /Pg 394 0 R + /S /Strong + /P 130 0 R + /K [344] + /Pg 1119 0 R >> endobj 132 0 obj << /Type /StructElem - /S /Strong - /P 127 0 R - /K [77] - /Pg 394 0 R + /S /Formula + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343] + /Pg 1119 0 R >> endobj 133 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76] - /Pg 394 0 R + /S /P + /P 54 0 R + /K [296] + /Pg 1119 0 R >> endobj 134 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [139 0 R 138 0 R 55 137 0 R 57 136 0 R 59 135 0 R] - /Pg 394 0 R + /S /L + /P 54 0 R + /A [<< + /O /List + /ListNumbering /None + >>] + /K [147 0 R 140 0 R 135 0 R] >> endobj 135 0 obj << /Type /StructElem - /S /Formula + /S /LI /P 134 0 R - /K [60] - /Pg 394 0 R + /K [138 0 R 136 0 R] >> endobj 136 0 obj << /Type /StructElem - /S /Span - /P 134 0 R - /A [<< - /O /Layout - /TextDecorationType /Underline - >>] - /K [58] - /Pg 394 0 R + /S /LBody + /P 135 0 R + /K [137 0 R] >> endobj @@ -1446,19 +1240,18 @@ endobj << /Type /StructElem /S /Formula - /P 134 0 R - /K [56] - /Pg 394 0 R + /P 136 0 R + /K [237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295] + /Pg 1119 0 R >> endobj 138 0 obj << /Type /StructElem - /S /Formula - /P 134 0 R - /K [52 53 54] - /Pg 394 0 R + /S /Lbl + /P 135 0 R + /K [139 0 R] >> endobj @@ -1466,33 +1259,28 @@ endobj << /Type /StructElem /S /Strong - /P 134 0 R - /K [51] - /Pg 394 0 R + /P 138 0 R + /K [236] + /Pg 1119 0 R >> endobj 140 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50] - /Pg 394 0 R + /S /LI + /P 134 0 R + /K [145 0 R 141 0 R] >> endobj 141 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [146 0 R 145 0 R 29 144 0 R 31 143 0 R 33 142 0 R] - /Pg 394 0 R + /S /LBody + /P 140 0 R + /K [201 144 0 R 210 143 0 R 234 142 0 R] + /Pg 1119 0 R >> endobj @@ -1501,22 +1289,18 @@ endobj /Type /StructElem /S /Formula /P 141 0 R - /K [34] - /Pg 394 0 R + /K [235] + /Pg 1119 0 R >> endobj 143 0 obj << /Type /StructElem - /S /Span + /S /Formula /P 141 0 R - /A [<< - /O /Layout - /TextDecorationType /Underline - >>] - /K [32] - /Pg 394 0 R + /K [211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233] + /Pg 1119 0 R >> endobj @@ -1525,18 +1309,17 @@ endobj /Type /StructElem /S /Formula /P 141 0 R - /K [30] - /Pg 394 0 R + /K [202 203 204 205 206 207 208 209] + /Pg 1119 0 R >> endobj 145 0 obj << /Type /StructElem - /S /Formula - /P 141 0 R - /K [26 27 28] - /Pg 394 0 R + /S /Lbl + /P 140 0 R + /K [146 0 R] >> endobj @@ -1544,141 +1327,127 @@ endobj << /Type /StructElem /S /Strong - /P 141 0 R - /K [25] - /Pg 394 0 R + /P 145 0 R + /K [200] + /Pg 1119 0 R >> endobj 147 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [24] - /Pg 394 0 R + /S /LI + /P 134 0 R + /K [150 0 R 148 0 R] >> endobj 148 0 obj << /Type /StructElem - /S /Formula - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23] - /Pg 394 0 R + /S /LBody + /P 147 0 R + /K [180 149 0 R] + /Pg 1119 0 R >> endobj 149 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [2] - /Pg 394 0 R + /S /Formula + /P 148 0 R + /K [181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199] + /Pg 1119 0 R >> endobj 150 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [1] - /Pg 394 0 R + /S /Lbl + /P 147 0 R + /K [151 0 R] >> endobj 151 0 obj << /Type /StructElem - /S /H1 - /P 24 0 R - /T - /K [0] - /Pg 394 0 R + /S /Strong + /P 150 0 R + /K [179] + /Pg 1119 0 R >> endobj 152 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [153 0 R] + /S /Formula + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178] + /Pg 1119 0 R >> endobj 153 0 obj << /Type /StructElem - /S /Formula - /P 152 0 R - /K [140 141 142 143 144 145 146 147 148 149 150 151 152 153 154] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [154 0 R] >> endobj 154 0 obj << /Type /StructElem - /S /Table - /P 24 0 R - /A [<< - /O /Layout - /BorderColor [0 0 0] - /BorderThickness 1 - >>] - /K [243 0 R 232 0 R 221 0 R 210 0 R 199 0 R 188 0 R 177 0 R 166 0 R 155 0 R] + /S /Formula + /P 153 0 R + /K [117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144] + /Pg 1119 0 R >> endobj 155 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [164 0 R 162 0 R 160 0 R 158 0 R 156 0 R] + /S /P + /P 54 0 R + /K [156 0 R] >> endobj 156 0 obj << /Type /StructElem - /S /TD + /S /Formula /P 155 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [157 0 R] + /K [110 111 112 113 114 115 116] + /Pg 1119 0 R >> endobj 157 0 obj << /Type /StructElem - /S /Formula - /P 156 0 R - /K [139] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [109] + /Pg 1119 0 R >> endobj 158 0 obj << /Type /StructElem - /S /TD - /P 155 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] + /S /P + /P 54 0 R /K [159 0 R] >> endobj @@ -1688,19 +1457,18 @@ endobj /Type /StructElem /S /Formula /P 158 0 R - /K [138] - /Pg 392 0 R + /K [] >> endobj 160 0 obj << /Type /StructElem - /S /TD - /P 155 0 R + /S /L + /P 54 0 R /A [<< - /O /Layout - /BorderStyle /Solid + /O /List + /ListNumbering /None >>] /K [161 0 R] >> @@ -1709,22 +1477,17 @@ endobj 161 0 obj << /Type /StructElem - /S /Formula + /S /LI /P 160 0 R - /K [137] - /Pg 392 0 R + /K [164 0 R 162 0 R] >> endobj 162 0 obj << /Type /StructElem - /S /TD - /P 155 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] + /S /LBody + /P 161 0 R /K [163 0 R] >> endobj @@ -1734,20 +1497,16 @@ endobj /Type /StructElem /S /Formula /P 162 0 R - /K [136] - /Pg 392 0 R + /K [82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108] + /Pg 1119 0 R >> endobj 164 0 obj << /Type /StructElem - /S /TD - /P 155 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] + /S /Lbl + /P 161 0 R /K [165 0 R] >> endobj @@ -1755,32 +1514,30 @@ endobj 165 0 obj << /Type /StructElem - /S /Formula + /S /Strong /P 164 0 R - /K [135] - /Pg 392 0 R + /K [81] + /Pg 1119 0 R >> endobj 166 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [175 0 R 173 0 R 171 0 R 169 0 R 167 0 R] + /S /P + /P 54 0 R + /K [168 0 R 78 167 0 R 80] + /Pg 1119 0 R >> endobj 167 0 obj << /Type /StructElem - /S /TD + /S /Strong /P 166 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [168 0 R] + /K [79] + /Pg 1119 0 R >> endobj @@ -1788,215 +1545,201 @@ endobj << /Type /StructElem /S /Formula - /P 167 0 R - /K [134] - /Pg 392 0 R + /P 166 0 R + /K [77] + /Pg 1119 0 R >> endobj 169 0 obj << /Type /StructElem - /S /TD - /P 166 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [170 0 R] + /K [54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76] + /Pg 1119 0 R >> endobj 170 0 obj << /Type /StructElem - /S /Formula - /P 169 0 R - /K [133] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [53] + /Pg 1119 0 R >> endobj 171 0 obj << /Type /StructElem - /S /TD - /P 166 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [172 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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52] + /Pg 1119 0 R >> endobj 172 0 obj << /Type /StructElem - /S /Formula - /P 171 0 R - /K [132] - /Pg 392 0 R + /S /L + /P 54 0 R + /A [<< + /O /List + /ListNumbering /None + >>] + /K [173 0 R] >> endobj 173 0 obj << /Type /StructElem - /S /TD - /P 166 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [174 0 R] + /S /LI + /P 172 0 R + /K [176 0 R 174 0 R] >> endobj 174 0 obj << /Type /StructElem - /S /Formula + /S /LBody /P 173 0 R - /K [131] - /Pg 392 0 R + /K [175 0 R] >> endobj 175 0 obj << /Type /StructElem - /S /TD - /P 166 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [176 0 R] + /S /Formula + /P 174 0 R + /K [356 357 358 359 360 361 362 363 364 365 366 367 368 369 370] + /Pg 1117 0 R >> endobj 176 0 obj << /Type /StructElem - /S /Formula - /P 175 0 R - /K [130] - /Pg 392 0 R + /S /Lbl + /P 173 0 R + /K [177 0 R] >> endobj 177 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [186 0 R 184 0 R 182 0 R 180 0 R 178 0 R] + /S /Strong + /P 176 0 R + /K [355] + /Pg 1117 0 R >> endobj 178 0 obj << /Type /StructElem - /S /TD - /P 177 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [179 0 R] + /K [267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354] + /Pg 1117 0 R >> endobj 179 0 obj << /Type /StructElem - /S /Formula - /P 178 0 R - /K [129] - /Pg 392 0 R + /S /L + /P 54 0 R + /A [<< + /O /List + /ListNumbering /None + >>] + /K [180 0 R] >> endobj 180 0 obj << /Type /StructElem - /S /TD - /P 177 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [181 0 R] + /S /LI + /P 179 0 R + /K [183 0 R 181 0 R] >> endobj 181 0 obj << /Type /StructElem - /S /Formula + /S /LBody /P 180 0 R - /K [128] - /Pg 392 0 R + /K [182 0 R] >> endobj 182 0 obj << /Type /StructElem - /S /TD - /P 177 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [183 0 R] + /S /Formula + /P 181 0 R + /K [243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266] + /Pg 1117 0 R >> endobj 183 0 obj << /Type /StructElem - /S /Formula - /P 182 0 R - /K [127] - /Pg 392 0 R + /S /Lbl + /P 180 0 R + /K [184 0 R] >> endobj 184 0 obj << /Type /StructElem - /S /TD - /P 177 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [185 0 R] + /S /Strong + /P 183 0 R + /K [242] + /Pg 1117 0 R >> endobj 185 0 obj << /Type /StructElem - /S /Formula - /P 184 0 R - /K [126] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [187 0 R 211 186 0 R] + /Pg 1117 0 R >> endobj 186 0 obj << /Type /StructElem - /S /TD - /P 177 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [187 0 R] + /S /Formula + /P 185 0 R + /K [212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241] + /Pg 1117 0 R >> endobj @@ -2004,123 +1747,122 @@ endobj << /Type /StructElem /S /Formula - /P 186 0 R - /K [125] - /Pg 392 0 R + /P 185 0 R + /K [210] + /Pg 1117 0 R >> endobj 188 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [197 0 R 195 0 R 193 0 R 191 0 R 189 0 R] + /S /P + /P 54 0 R + /K [209] + /Pg 1117 0 R >> endobj 189 0 obj << /Type /StructElem - /S /TD - /P 188 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [190 0 R] + /K [186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208] + /Pg 1117 0 R >> endobj 190 0 obj << /Type /StructElem - /S /Formula - /P 189 0 R - /K [124] - /Pg 392 0 R + /S /H2 + /P 54 0 R + /T + /K [185] + /Pg 1117 0 R >> endobj 191 0 obj << /Type /StructElem - /S /TD - /P 188 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [192 0 R] + /K [132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184] + /Pg 1117 0 R >> endobj 192 0 obj << /Type /StructElem - /S /Formula - /P 191 0 R - /K [123] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [131] + /Pg 1117 0 R >> endobj 193 0 obj << /Type /StructElem - /S /TD - /P 188 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [194 0 R] + /K [89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130] + /Pg 1117 0 R >> endobj 194 0 obj << /Type /StructElem - /S /Formula - /P 193 0 R - /K [122] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [195 0 R 88] + /Pg 1117 0 R >> endobj 195 0 obj << /Type /StructElem - /S /TD - /P 188 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [196 0 R] + /S /Formula + /P 194 0 R + /K [85 86 87] + /Pg 1117 0 R >> endobj 196 0 obj << /Type /StructElem - /S /Formula - /P 195 0 R - /K [121] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [55 198 0 R 63 197 0 R] + /Pg 1117 0 R >> endobj 197 0 obj << /Type /StructElem - /S /TD - /P 188 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [198 0 R] + /S /Formula + /P 196 0 R + /K [64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84] + /Pg 1117 0 R >> endobj @@ -2128,54 +1870,51 @@ endobj << /Type /StructElem /S /Formula - /P 197 0 R - /K [120] - /Pg 392 0 R + /P 196 0 R + /K [56 57 58 59 60 61 62] + /Pg 1117 0 R >> endobj 199 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [208 0 R 206 0 R 204 0 R 202 0 R 200 0 R] + /S /L + /P 54 0 R + /A [<< + /O /List + /ListNumbering /None + >>] + /K [200 0 R] >> endobj 200 0 obj << /Type /StructElem - /S /TD + /S /LI /P 199 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [201 0 R] + /K [204 0 R 201 0 R] >> endobj 201 0 obj << /Type /StructElem - /S /Formula + /S /LBody /P 200 0 R - /K [119] - /Pg 392 0 R + /K [43 203 0 R 47 202 0 R 54] + /Pg 1117 0 R >> endobj 202 0 obj << /Type /StructElem - /S /TD - /P 199 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [203 0 R] + /S /Formula + /P 201 0 R + /K [48 49 50 51 52 53] + /Pg 1117 0 R >> endobj @@ -2183,21 +1922,17 @@ endobj << /Type /StructElem /S /Formula - /P 202 0 R - /K [118] - /Pg 392 0 R + /P 201 0 R + /K [44 45 46] + /Pg 1117 0 R >> endobj 204 0 obj << /Type /StructElem - /S /TD - /P 199 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] + /S /Lbl + /P 200 0 R /K [205 0 R] >> endobj @@ -2205,23 +1940,20 @@ endobj 205 0 obj << /Type /StructElem - /S /Formula + /S /Strong /P 204 0 R - /K [117] - /Pg 392 0 R + /K [42] + /Pg 1117 0 R >> endobj 206 0 obj << /Type /StructElem - /S /TD - /P 199 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [207 0 R] + /S /P + /P 54 0 R + /K [29 208 0 R 39 207 0 R 41] + /Pg 1117 0 R >> endobj @@ -2230,21 +1962,18 @@ endobj /Type /StructElem /S /Formula /P 206 0 R - /K [116] - /Pg 392 0 R + /K [40] + /Pg 1117 0 R >> endobj 208 0 obj << /Type /StructElem - /S /TD - /P 199 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [209 0 R] + /S /Formula + /P 206 0 R + /K [30 31 32 33 34 35 36 37 38] + /Pg 1117 0 R >> endobj @@ -2252,154 +1981,144 @@ endobj << /Type /StructElem /S /Formula - /P 208 0 R - /K [115] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /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 26 27 28] + /Pg 1117 0 R >> endobj 210 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [219 0 R 217 0 R 215 0 R 213 0 R 211 0 R] + /S /P + /P 54 0 R + /K [335] + /Pg 1115 0 R >> endobj 211 0 obj << /Type /StructElem - /S /TD - /P 210 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [212 0 R] + /K [275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334] + /Pg 1115 0 R >> endobj 212 0 obj << /Type /StructElem - /S /Formula - /P 211 0 R - /K [114] - /Pg 392 0 R + /S /L + /P 54 0 R + /A [<< + /O /List + /ListNumbering /None + >>] + /K [213 0 R] >> endobj 213 0 obj << /Type /StructElem - /S /TD - /P 210 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [214 0 R] + /S /LI + /P 212 0 R + /K [216 0 R 214 0 R] >> endobj 214 0 obj << /Type /StructElem - /S /Formula + /S /LBody /P 213 0 R - /K [113] - /Pg 392 0 R + /K [215 0 R] >> endobj 215 0 obj << /Type /StructElem - /S /TD - /P 210 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [216 0 R] + /S /Formula + /P 214 0 R + /K [247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274] + /Pg 1115 0 R >> endobj 216 0 obj << /Type /StructElem - /S /Formula - /P 215 0 R - /K [112] - /Pg 392 0 R + /S /Lbl + /P 213 0 R + /K [217 0 R] >> endobj 217 0 obj << /Type /StructElem - /S /TD - /P 210 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [218 0 R] + /S /Strong + /P 216 0 R + /K [246] + /Pg 1115 0 R >> endobj 218 0 obj << /Type /StructElem - /S /Formula - /P 217 0 R - /K [111] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [244 245] + /Pg 1115 0 R >> endobj 219 0 obj << /Type /StructElem - /S /TD - /P 210 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [220 0 R] + /S /P + /P 54 0 R + /K [243] + /Pg 1115 0 R >> endobj 220 0 obj << /Type /StructElem - /S /Formula - /P 219 0 R - /K [110] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [221 0 R] >> endobj 221 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [230 0 R 228 0 R 226 0 R 224 0 R 222 0 R] + /S /Formula + /P 220 0 R + /K [231 232 233 234 235 236 237 238 239 240 241 242] + /Pg 1115 0 R >> endobj 222 0 obj << /Type /StructElem - /S /TD - /P 221 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] + /S /P + /P 54 0 R /K [223 0 R] >> endobj @@ -2409,20 +2128,16 @@ endobj /Type /StructElem /S /Formula /P 222 0 R - /K [109] - /Pg 392 0 R + /K [208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230] + /Pg 1115 0 R >> endobj 224 0 obj << /Type /StructElem - /S /TD - /P 221 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] + /S /P + /P 54 0 R /K [225 0 R] >> endobj @@ -2432,99 +2147,89 @@ endobj /Type /StructElem /S /Formula /P 224 0 R - /K [108] - /Pg 392 0 R + /K [186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207] + /Pg 1115 0 R >> endobj 226 0 obj << /Type /StructElem - /S /TD - /P 221 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [227 0 R] + /S /P + /P 54 0 R + /K [185] + /Pg 1115 0 R >> endobj 227 0 obj << /Type /StructElem - /S /Formula - /P 226 0 R - /K [107] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [228 0 R 183 184] + /Pg 1115 0 R >> endobj 228 0 obj << /Type /StructElem - /S /TD - /P 221 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [229 0 R] + /S /Formula + /P 227 0 R + /K [165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182] + /Pg 1115 0 R >> endobj 229 0 obj << /Type /StructElem - /S /Formula - /P 228 0 R - /K [106] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [164] + /Pg 1115 0 R >> endobj 230 0 obj << /Type /StructElem - /S /TD - /P 221 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [231 0 R] + /S /H2 + /P 54 0 R + /T (Basis) + /K [163] + /Pg 1115 0 R >> endobj 231 0 obj << /Type /StructElem - /S /Formula - /P 230 0 R - /K [105] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [155 233 0 R 159 232 0 R] + /Pg 1115 0 R >> endobj 232 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [241 0 R 239 0 R 237 0 R 235 0 R 233 0 R] + /S /Formula + /P 231 0 R + /K [160 161 162] + /Pg 1115 0 R >> endobj 233 0 obj << /Type /StructElem - /S /TD - /P 232 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [234 0 R] + /S /Formula + /P 231 0 R + /K [156 157 158] + /Pg 1115 0 R >> endobj @@ -2532,123 +2237,115 @@ endobj << /Type /StructElem /S /Formula - /P 233 0 R - /K [104] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154] + /Pg 1115 0 R >> endobj 235 0 obj << /Type /StructElem - /S /TD - /P 232 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [236 0 R] + /S /P + /P 54 0 R + /K [121 237 0 R 124 236 0 R 126] + /Pg 1115 0 R >> endobj 236 0 obj << /Type /StructElem - /S /Formula + /S /Strong /P 235 0 R - /K [103] - /Pg 392 0 R + /K [125] + /Pg 1115 0 R >> endobj 237 0 obj << /Type /StructElem - /S /TD - /P 232 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [238 0 R] + /S /Formula + /P 235 0 R + /K [122 123] + /Pg 1115 0 R >> endobj 238 0 obj << /Type /StructElem - /S /Formula - /P 237 0 R - /K [102] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [239 0 R] >> endobj 239 0 obj << /Type /StructElem - /S /TD - /P 232 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [240 0 R] + /S /Formula + /P 238 0 R + /K [104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120] + /Pg 1115 0 R >> endobj 240 0 obj << /Type /StructElem - /S /Formula - /P 239 0 R - /K [101] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [103] + /Pg 1115 0 R >> endobj 241 0 obj << /Type /StructElem - /S /TD - /P 232 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [242 0 R] + /K [36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102] + /Pg 1115 0 R >> endobj 242 0 obj << /Type /StructElem - /S /Formula - /P 241 0 R - /K [100] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [35] + /Pg 1115 0 R >> endobj 243 0 obj << /Type /StructElem - /S /TR - /P 154 0 R - /K [252 0 R 250 0 R 248 0 R 246 0 R 244 0 R] + /S /P + /P 54 0 R + /K [244 0 R] >> endobj 244 0 obj << /Type /StructElem - /S /TD + /S /Formula /P 243 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [245 0 R] + /K [19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34] + /Pg 1115 0 R >> endobj @@ -2656,22 +2353,23 @@ endobj << /Type /StructElem /S /Formula - /P 244 0 R - /K [91 92 93 94 95 96 97 98 99] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18] + /Pg 1115 0 R >> endobj 246 0 obj << /Type /StructElem - /S /TD - /P 243 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [247 0 R] + /S /P + /P 54 0 R + /K [192 253 0 R 194 252 0 R 196 251 0 R 198 250 0 R 200 249 0 R 202 248 0 R 204 205 247 0 R 207] + /Pg 1113 0 R >> endobj @@ -2680,44 +2378,38 @@ endobj /Type /StructElem /S /Formula /P 246 0 R - /K [86 87 88 89 90] - /Pg 392 0 R + /K [206] + /Pg 1113 0 R >> endobj 248 0 obj << /Type /StructElem - /S /TD - /P 243 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [249 0 R] + /S /Formula + /P 246 0 R + /K [203] + /Pg 1113 0 R >> endobj 249 0 obj << /Type /StructElem - /S /Formula - /P 248 0 R - /K [85] - /Pg 392 0 R + /S /Strong + /P 246 0 R + /K [201] + /Pg 1113 0 R >> endobj 250 0 obj << /Type /StructElem - /S /TD - /P 243 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [251 0 R] + /S /Formula + /P 246 0 R + /K [199] + /Pg 1113 0 R >> endobj @@ -2725,22 +2417,19 @@ endobj << /Type /StructElem /S /Formula - /P 250 0 R - /K [84] - /Pg 392 0 R + /P 246 0 R + /K [197] + /Pg 1113 0 R >> endobj 252 0 obj << /Type /StructElem - /S /TD - /P 243 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [253 0 R] + /S /Formula + /P 246 0 R + /K [195] + /Pg 1113 0 R >> endobj @@ -2748,9 +2437,9 @@ endobj << /Type /StructElem /S /Formula - /P 252 0 R - /K [83] - /Pg 392 0 R + /P 246 0 R + /K [193] + /Pg 1113 0 R >> endobj @@ -2758,19 +2447,19 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [255 0 R 82] - /Pg 392 0 R + /P 54 0 R + /K [191] + /Pg 1113 0 R >> endobj 255 0 obj << /Type /StructElem - /S /Formula - /P 254 0 R - /K [73 74 75 76 77 78 79 80 81] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [190] + /Pg 1113 0 R >> endobj @@ -2778,19 +2467,19 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [72] - /Pg 392 0 R + /P 54 0 R + /K [187 257 0 R 189] + /Pg 1113 0 R >> endobj 257 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [56 262 0 R 60 261 0 R 62 260 0 R 66 259 0 R 68 258 0 R 70 71] - /Pg 392 0 R + /S /Formula + /P 256 0 R + /K [188] + /Pg 1113 0 R >> endobj @@ -2798,29 +2487,33 @@ endobj << /Type /StructElem /S /Formula - /P 257 0 R - /K [69] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186] + /Pg 1113 0 R >> endobj 259 0 obj << /Type /StructElem - /S /Formula - /P 257 0 R - /K [67] - /Pg 392 0 R + /S /H2 + /P 54 0 R + /T (Polynomiumrum) + /K [166] + /Pg 1113 0 R >> endobj 260 0 obj << /Type /StructElem - /S /Formula - /P 257 0 R - /K [63 64 65] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [261 0 R] >> endobj @@ -2828,78 +2521,68 @@ endobj << /Type /StructElem /S /Formula - /P 257 0 R - /K [61] - /Pg 392 0 R + /P 260 0 R + /K [145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165] + /Pg 1113 0 R >> endobj 262 0 obj << /Type /StructElem - /S /Formula - /P 257 0 R - /K [57 58 59] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [265 0 R 140 264 0 R 142 263 0 R 144] + /Pg 1113 0 R >> endobj 263 0 obj << /Type /StructElem - /S /Table - /P 24 0 R - /A [<< - /O /Layout - /BorderColor [0 0 0] - /BorderThickness 1 - >>] - /K [324 0 R 309 0 R 294 0 R 279 0 R 264 0 R] + /S /Formula + /P 262 0 R + /K [143] + /Pg 1113 0 R >> endobj 264 0 obj << /Type /StructElem - /S /TR - /P 263 0 R - /K [277 0 R 275 0 R 273 0 R 271 0 R 269 0 R 267 0 R 265 0 R] + /S /Formula + /P 262 0 R + /K [141] + /Pg 1113 0 R >> endobj 265 0 obj << /Type /StructElem - /S /TD - /P 264 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [266 0 R] + /S /Formula + /P 262 0 R + /K [135 136 137 138 139] + /Pg 1113 0 R >> endobj 266 0 obj << /Type /StructElem - /S /Formula - /P 265 0 R - /K [55] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [267 0 R] >> endobj 267 0 obj << /Type /StructElem - /S /TD - /P 264 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [268 0 R] + /S /Strong + /P 266 0 R + /K [134] + /Pg 1113 0 R >> endobj @@ -2907,22 +2590,27 @@ endobj << /Type /StructElem /S /Formula - /P 267 0 R - /K [54] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133] + /Pg 1113 0 R >> endobj 269 0 obj << /Type /StructElem - /S /TD - /P 264 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [270 0 R] + /K [91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117] + /Pg 1113 0 R >> endobj @@ -2930,45 +2618,43 @@ endobj << /Type /StructElem /S /Formula - /P 269 0 R - /K [53] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90] + /Pg 1113 0 R >> endobj 271 0 obj << /Type /StructElem - /S /TD - /P 264 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [272 0 R] + /S /P + /P 54 0 R + /K [14] + /Pg 1113 0 R >> endobj 272 0 obj << /Type /StructElem - /S /Formula - /P 271 0 R - /K [52] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [3 274 0 R 8 273 0 R] + /Pg 1113 0 R >> endobj 273 0 obj << /Type /StructElem - /S /TD - /P 264 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [274 0 R] + /S /Formula + /P 272 0 R + /K [9 10 11 12 13] + /Pg 1113 0 R >> endobj @@ -2976,100 +2662,88 @@ endobj << /Type /StructElem /S /Formula - /P 273 0 R - /K [51] - /Pg 392 0 R + /P 272 0 R + /K [4 5 6 7] + /Pg 1113 0 R >> endobj 275 0 obj << /Type /StructElem - /S /TD - /P 264 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [276 0 R] + /S /P + /P 54 0 R + /K [2] + /Pg 1113 0 R >> endobj 276 0 obj << /Type /StructElem - /S /Formula - /P 275 0 R - /K [50] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [0 1] + /Pg 1113 0 R >> endobj 277 0 obj << /Type /StructElem - /S /TD - /P 264 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [278 0 R] + /S /P + /P 54 0 R + /K [208] + /Pg 1111 0 R >> endobj 278 0 obj << /Type /StructElem - /S /Formula - /P 277 0 R - /K [49] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [280 0 R 279 0 R] >> endobj 279 0 obj << /Type /StructElem - /S /TR - /P 263 0 R - /K [292 0 R 290 0 R 288 0 R 286 0 R 284 0 R 282 0 R 280 0 R] + /S /Formula + /P 278 0 R + /K [196 197 198 199 200 201 202 203 204 205 206 207] + /Pg 1111 0 R >> endobj 280 0 obj << /Type /StructElem - /S /TD - /P 279 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [281 0 R] + /S /Strong + /P 278 0 R + /K [195] + /Pg 1111 0 R >> endobj 281 0 obj << /Type /StructElem - /S /Formula - /P 280 0 R - /K [48] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [194] + /Pg 1111 0 R >> endobj 282 0 obj << /Type /StructElem - /S /TD - /P 279 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [283 0 R] + /S /P + /P 54 0 R + /K [284 0 R 191 283 0 R 193] + /Pg 1111 0 R >> endobj @@ -3078,21 +2752,18 @@ endobj /Type /StructElem /S /Formula /P 282 0 R - /K [47] - /Pg 392 0 R + /K [192] + /Pg 1111 0 R >> endobj 284 0 obj << /Type /StructElem - /S /TD - /P 279 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [285 0 R] + /S /Formula + /P 282 0 R + /K [184 185 186 187 188 189 190] + /Pg 1111 0 R >> endobj @@ -3100,22 +2771,23 @@ endobj << /Type /StructElem /S /Formula - /P 284 0 R - /K [46] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183] + /Pg 1111 0 R >> endobj 286 0 obj << /Type /StructElem - /S /TD - /P 279 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [287 0 R] + /S /P + /P 54 0 R + /K [142 287 0 R] + /Pg 1111 0 R >> endobj @@ -3124,20 +2796,16 @@ endobj /Type /StructElem /S /Formula /P 286 0 R - /K [45] - /Pg 392 0 R + /K [143] + /Pg 1111 0 R >> endobj 288 0 obj << /Type /StructElem - /S /TD - /P 279 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] + /S /P + /P 54 0 R /K [289 0 R] >> endobj @@ -3147,44 +2815,40 @@ endobj /Type /StructElem /S /Formula /P 288 0 R - /K [44] - /Pg 392 0 R + /K [113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141] + /Pg 1111 0 R >> endobj 290 0 obj << /Type /StructElem - /S /TD - /P 279 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [291 0 R] + /S /P + /P 54 0 R + /K [112] + /Pg 1111 0 R >> endobj 291 0 obj << /Type /StructElem - /S /Formula - /P 290 0 R - /K [43] - /Pg 392 0 R + /S /H2 + /P 54 0 R + /T (Vektorrum) + /K [111] + /Pg 1111 0 R >> endobj 292 0 obj << /Type /StructElem - /S /TD - /P 279 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [293 0 R] + /S /H1 + /P 54 0 R + /T + /K [110] + /Pg 1111 0 R >> endobj @@ -3192,31 +2856,33 @@ endobj << /Type /StructElem /S /Formula - /P 292 0 R - /K [42] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109] + /Pg 1111 0 R >> endobj 294 0 obj << /Type /StructElem - /S /TR - /P 263 0 R - /K [307 0 R 305 0 R 303 0 R 301 0 R 299 0 R 297 0 R 295 0 R] + /S /P + /P 54 0 R + /K [58] + /Pg 1111 0 R >> endobj 295 0 obj << /Type /StructElem - /S /TD - /P 294 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [296 0 R] + /S /P + /P 54 0 R + /K [298 0 R 35 297 0 R 52 296 0 R] + /Pg 1111 0 R >> endobj @@ -3225,44 +2891,38 @@ endobj /Type /StructElem /S /Formula /P 295 0 R - /K [41] - /Pg 392 0 R + /K [53 54 55 56 57] + /Pg 1111 0 R >> endobj 297 0 obj << /Type /StructElem - /S /TD - /P 294 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [298 0 R] + /S /Formula + /P 295 0 R + /K [36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51] + /Pg 1111 0 R >> endobj 298 0 obj << /Type /StructElem - /S /Formula - /P 297 0 R - /K [40] - /Pg 392 0 R + /S /Strong + /P 295 0 R + /K [34] + /Pg 1111 0 R >> endobj 299 0 obj << /Type /StructElem - /S /TD - /P 294 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [300 0 R] + /S /P + /P 54 0 R + /K [302 0 R 15 301 0 R 300 0 R] + /Pg 1111 0 R >> endobj @@ -3271,44 +2931,38 @@ endobj /Type /StructElem /S /Formula /P 299 0 R - /K [39] - /Pg 392 0 R + /K [19 20 21 22 23 24 25 26 27 28 29 30 31 32 33] + /Pg 1111 0 R >> endobj 301 0 obj << /Type /StructElem - /S /TD - /P 294 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [302 0 R] + /S /Formula + /P 299 0 R + /K [16 17 18] + /Pg 1111 0 R >> endobj 302 0 obj << /Type /StructElem - /S /Formula - /P 301 0 R - /K [38] - /Pg 392 0 R + /S /Strong + /P 299 0 R + /K [14] + /Pg 1111 0 R >> endobj 303 0 obj << /Type /StructElem - /S /TD - /P 294 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [304 0 R] + /S /P + /P 54 0 R + /K [0 305 0 R 10 304 0 R] + /Pg 1111 0 R >> endobj @@ -3317,76 +2971,67 @@ endobj /Type /StructElem /S /Formula /P 303 0 R - /K [37] - /Pg 392 0 R + /K [11 12 13] + /Pg 1111 0 R >> endobj 305 0 obj << /Type /StructElem - /S /TD - /P 294 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [306 0 R] + /S /Formula + /P 303 0 R + /K [1 2 3 4 5 6 7 8 9] + /Pg 1111 0 R >> endobj 306 0 obj << /Type /StructElem - /S /Formula - /P 305 0 R - /K [36] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [307 0 R] >> endobj 307 0 obj << /Type /StructElem - /S /TD - /P 294 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [308 0 R] + /S /Formula + /P 306 0 R + /K [275 276 277 278 279 280 281 282] + /Pg 1109 0 R >> endobj 308 0 obj << /Type /StructElem - /S /Formula - /P 307 0 R - /K [35] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [274] + /Pg 1109 0 R >> endobj 309 0 obj << /Type /StructElem - /S /TR - /P 263 0 R - /K [322 0 R 320 0 R 318 0 R 316 0 R 314 0 R 312 0 R 310 0 R] + /S /P + /P 54 0 R + /K [252 312 0 R 258 311 0 R 267 310 0 R] + /Pg 1109 0 R >> endobj 310 0 obj << /Type /StructElem - /S /TD + /S /Formula /P 309 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [311 0 R] + /K [268 269 270 271 272 273] + /Pg 1109 0 R >> endobj @@ -3394,22 +3039,19 @@ endobj << /Type /StructElem /S /Formula - /P 310 0 R - /K [34] - /Pg 392 0 R + /P 309 0 R + /K [259 260 261 262 263 264 265 266] + /Pg 1109 0 R >> endobj 312 0 obj << /Type /StructElem - /S /TD + /S /Formula /P 309 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [313 0 R] + /K [253 254 255 256 257] + /Pg 1109 0 R >> endobj @@ -3417,45 +3059,43 @@ endobj << /Type /StructElem /S /Formula - /P 312 0 R - /K [33] - /Pg 392 0 R + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251] + /Pg 1109 0 R >> endobj 314 0 obj << /Type /StructElem - /S /TD - /P 309 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [315 0 R] + /S /P + /P 54 0 R + /K [217] + /Pg 1109 0 R >> endobj 315 0 obj << /Type /StructElem - /S /Formula - /P 314 0 R - /K [32] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [318 0 R 192 317 0 R 198 316 0 R] + /Pg 1109 0 R >> endobj 316 0 obj << /Type /StructElem - /S /TD - /P 309 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [317 0 R] + /S /Formula + /P 315 0 R + /K [199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216] + /Pg 1109 0 R >> endobj @@ -3463,45 +3103,39 @@ endobj << /Type /StructElem /S /Formula - /P 316 0 R - /K [31] - /Pg 392 0 R + /P 315 0 R + /K [193 194 195 196 197] + /Pg 1109 0 R >> endobj 318 0 obj << /Type /StructElem - /S /TD - /P 309 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [319 0 R] + /S /Strong + /P 315 0 R + /K [191] + /Pg 1109 0 R >> endobj 319 0 obj << /Type /StructElem - /S /Formula - /P 318 0 R - /K [30] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [322 0 R 164 321 0 R 320 0 R 190] + /Pg 1109 0 R >> endobj 320 0 obj << /Type /StructElem - /S /TD - /P 309 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [321 0 R] + /S /Formula + /P 319 0 R + /K [168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189] + /Pg 1109 0 R >> endobj @@ -3509,77 +3143,69 @@ endobj << /Type /StructElem /S /Formula - /P 320 0 R - /K [29] - /Pg 392 0 R + /P 319 0 R + /K [165 166 167] + /Pg 1109 0 R >> endobj 322 0 obj << /Type /StructElem - /S /TD - /P 309 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [323 0 R] + /S /Strong + /P 319 0 R + /K [163] + /Pg 1109 0 R >> endobj 323 0 obj << /Type /StructElem - /S /Formula - /P 322 0 R - /K [28] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [149 325 0 R 155 324 0 R] + /Pg 1109 0 R >> endobj 324 0 obj << /Type /StructElem - /S /TR - /P 263 0 R - /K [337 0 R 335 0 R 333 0 R 331 0 R 329 0 R 327 0 R 325 0 R] + /S /Formula + /P 323 0 R + /K [156 157 158 159 160 161 162] + /Pg 1109 0 R >> endobj 325 0 obj << /Type /StructElem - /S /TD - /P 324 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [326 0 R] + /S /Formula + /P 323 0 R + /K [150 151 152 153 154] + /Pg 1109 0 R >> endobj 326 0 obj << /Type /StructElem - /S /Formula - /P 325 0 R - /K [25 26 27] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [148] + /Pg 1109 0 R >> endobj 327 0 obj << /Type /StructElem - /S /TD - /P 324 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [328 0 R] + /S /P + /P 54 0 R + /K [138 329 0 R 142 328 0 R] + /Pg 1109 0 R >> endobj @@ -3588,67 +3214,62 @@ endobj /Type /StructElem /S /Formula /P 327 0 R - /K [22 23 24] - /Pg 392 0 R + /K [143 144 145 146 147] + /Pg 1109 0 R >> endobj 329 0 obj << /Type /StructElem - /S /TD - /P 324 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [330 0 R] + /S /Formula + /P 327 0 R + /K [139 140 141] + /Pg 1109 0 R >> endobj 330 0 obj << /Type /StructElem - /S /Formula - /P 329 0 R - /K [20 21] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [137] + /Pg 1109 0 R >> endobj 331 0 obj << /Type /StructElem - /S /TD - /P 324 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout - /BorderStyle /Solid + /Placement /Block >>] - /K [332 0 R] + /K [102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136] + /Pg 1109 0 R >> endobj 332 0 obj << /Type /StructElem - /S /Formula - /P 331 0 R - /K [17 18 19] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [101] + /Pg 1109 0 R >> endobj 333 0 obj << /Type /StructElem - /S /TD - /P 324 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [334 0 R] + /S /P + /P 54 0 R + /K [336 0 R 88 335 0 R 95 334 0 R] + /Pg 1109 0 R >> endobj @@ -3657,44 +3278,38 @@ endobj /Type /StructElem /S /Formula /P 333 0 R - /K [14 15 16] - /Pg 392 0 R + /K [96 97 98 99 100] + /Pg 1109 0 R >> endobj 335 0 obj << /Type /StructElem - /S /TD - /P 324 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [336 0 R] + /S /Formula + /P 333 0 R + /K [89 90 91 92 93 94] + /Pg 1109 0 R >> endobj 336 0 obj << /Type /StructElem - /S /Formula - /P 335 0 R - /K [13] - /Pg 392 0 R + /S /Strong + /P 333 0 R + /K [87] + /Pg 1109 0 R >> endobj 337 0 obj << /Type /StructElem - /S /TD - /P 324 0 R - /A [<< - /O /Layout - /BorderStyle /Solid - >>] - /K [338 0 R] + /S /P + /P 54 0 R + /K [340 0 R 77 339 0 R 338 0 R] + /Pg 1109 0 R >> endobj @@ -3703,122 +3318,120 @@ endobj /Type /StructElem /S /Formula /P 337 0 R - /K [12] - /Pg 392 0 R + /K [81 82 83 84 85 86] + /Pg 1109 0 R >> endobj 339 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [11] - /Pg 392 0 R + /S /Formula + /P 337 0 R + /K [78 79 80] + /Pg 1109 0 R >> endobj 340 0 obj << /Type /StructElem - /S /P - /P 24 0 R - /K [10] - /Pg 392 0 R + /S /Strong + /P 337 0 R + /K [76] + /Pg 1109 0 R >> endobj 341 0 obj << /Type /StructElem - /S /H1 - /P 24 0 R - /T (Udsagnslogik) - /K [9] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [67 343 0 R 72 342 0 R] + /Pg 1109 0 R >> endobj 342 0 obj << /Type /StructElem - /S /Div - /P 24 0 R - /K [346 0 R 343 0 R] + /S /Formula + /P 341 0 R + /K [73 74 75] + /Pg 1109 0 R >> endobj 343 0 obj << /Type /StructElem - /S /Div - /P 342 0 R - /K [345 0 R 344 0 R] + /S /Formula + /P 341 0 R + /K [68 69 70 71] + /Pg 1109 0 R >> endobj 344 0 obj << /Type /StructElem - /S /Span - /P 343 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout /Placement /Block >>] - /K [8] - /Pg 392 0 R + /K [44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66] + /Pg 1109 0 R >> endobj 345 0 obj << /Type /StructElem - /S /Strong - /P 343 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [7] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [43] + /Pg 1109 0 R >> endobj 346 0 obj << /Type /StructElem - /S /Div - /P 342 0 R - /K [348 0 R 347 0 R] + /S /Formula + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42] + /Pg 1109 0 R >> endobj 347 0 obj << /Type /StructElem - /S /Span - /P 346 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [6] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [264] + /Pg 1107 0 R >> endobj 348 0 obj << /Type /StructElem - /S /Strong - /P 346 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout /Placement /Block >>] - /K [5] - /Pg 392 0 R + /K [238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263] + /Pg 1107 0 R >> endobj @@ -3826,147 +3439,7618 @@ endobj << /Type /StructElem /S /P - /P 24 0 R - /K [3 4] - /Pg 392 0 R + /P 54 0 R + /K [351 0 R 232 350 0 R] + /Pg 1107 0 R >> endobj 350 0 obj << /Type /StructElem - /S /Span - /P 24 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [2] - /Pg 392 0 R + /S /Formula + /P 349 0 R + /K [233 234 235 236 237] + /Pg 1107 0 R >> endobj 351 0 obj << /Type /StructElem - /S /Div - /P 24 0 R - /K [355 0 R 354 0 R 352 0 R] + /S /Strong + /P 349 0 R + /K [231] + /Pg 1107 0 R >> endobj 352 0 obj << /Type /StructElem - /S /Div - /P 351 0 R - /K [353 0 R] + /S /Formula + /P 54 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230] + /Pg 1107 0 R >> endobj 353 0 obj << /Type /StructElem - /S /Span - /P 352 0 R - /A [<< - /O /Layout - /Placement /Block - >>] - /K [1] - /Pg 392 0 R + /S /P + /P 54 0 R + /K [355 0 R 202 354 0 R] + /Pg 1107 0 R >> endobj 354 0 obj << /Type /StructElem - /S /Div - /P 351 0 R - /K [] + /S /Formula + /P 353 0 R + /K [203 204 205] + /Pg 1107 0 R >> endobj 355 0 obj << /Type /StructElem - /S /Div - /P 351 0 R - /K [356 0 R] + /S /Strong + /P 353 0 R + /K [201] + /Pg 1107 0 R >> endobj 356 0 obj << /Type /StructElem - /S /Figure - /P 355 0 R + /S /Formula + /P 54 0 R /A [<< /O /Layout /Placement /Block >>] - /K [0] - /Pg 392 0 R + /K [180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200] + /Pg 1107 0 R >> endobj 357 0 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366 367 368] + /Pg 1095 0 R +>> +endobj + +528 0 obj +<< + /Type /StructElem + /S /Strong + /P 526 0 R + /K [351] + /Pg 1095 0 R +>> +endobj + +529 0 obj +<< + /Type /StructElem + /S /Lbl + /P 524 0 R + /K [350] + /Pg 1095 0 R +>> +endobj + +530 0 obj +<< + /Type /StructElem + /S /LI + /P 485 0 R + /K [536 0 R 531 0 R] +>> +endobj + +531 0 obj +<< + /Type /StructElem + /S /LBody + /P 530 0 R + /K [532 0 R] +>> +endobj + +532 0 obj +<< + /Type /StructElem + /S /P + /P 531 0 R + /K [535 0 R 332 534 0 R 338 533 0 R] + /Pg 1095 0 R +>> +endobj + +533 0 obj +<< + /Type /StructElem + /S /Formula + /P 532 0 R + /K [339 340 341 342 343 344 345 346 347 348 349] + /Pg 1095 0 R +>> +endobj + +534 0 obj +<< + /Type /StructElem + /S /Formula + /P 532 0 R + /K [333 334 335 336 337] + /Pg 1095 0 R +>> +endobj + +535 0 obj +<< + /Type /StructElem + /S /Strong + /P 532 0 R + /K [331] + /Pg 1095 0 R +>> +endobj + +536 0 obj +<< + /Type /StructElem + /S /Lbl + /P 530 0 R + /K [330] + /Pg 1095 0 R +>> 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0 obj +<< + /Type /StructElem + /S /P + /P 54 0 R + /K [322] + /Pg 1095 0 R +>> +endobj + +548 0 obj +<< + /Type /StructElem + /S /Div + /P 54 0 R + /K [658 0 R 549 0 R] +>> +endobj + +549 0 obj +<< + /Type /StructElem + /S /Div + /P 548 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [657 0 R 656 0 R 655 0 R 654 0 R 653 0 R 652 0 R 651 0 R 650 0 R 649 0 R 648 0 R 647 0 R 646 0 R 645 0 R 644 0 R 643 0 R 642 0 R 641 0 R 640 0 R 639 0 R 638 0 R 637 0 R 636 0 R 635 0 R 634 0 R 633 0 R 632 0 R 631 0 R 630 0 R 629 0 R 628 0 R 627 0 R 626 0 R 625 0 R 624 0 R 623 0 R 622 0 R 621 0 R 620 0 R 619 0 R 618 0 R 617 0 R 616 0 R 615 0 R 614 0 R 613 0 R 612 0 R 611 0 R 610 0 R 609 0 R 608 0 R 607 0 R 606 0 R 605 0 R 604 0 R 603 0 R 593 0 R 550 0 R] +>> +endobj + +550 0 obj +<< + /Type /StructElem + /S /Table + /P 549 0 R + /A [<< + /O /Layout + /BorderColor [0.6666667 0.6666667 0.6666667] + /BorderThickness 0.3 + >>] + /K [586 0 R 579 0 R 572 0 R 565 0 R 558 0 R 551 0 R] +>> +endobj + +551 0 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/Solid + >>] + /K [585 0 R] +>> +endobj + +585 0 obj +<< + /Type /StructElem + /S /Formula + /P 584 0 R + /K [292] + /Pg 1095 0 R +>> +endobj + +586 0 obj +<< + /Type /StructElem + /S /TR + /P 550 0 R + /K [591 0 R 589 0 R 587 0 R] +>> +endobj + +587 0 obj +<< + /Type /StructElem + /S /TD + /P 586 0 R + /A [<< + /O /Layout + /BorderStyle /Solid + >>] + /K [588 0 R] +>> +endobj + +588 0 obj +<< + /Type /StructElem + /S /Formula + /P 587 0 R + /K [291] + /Pg 1095 0 R +>> +endobj + +589 0 obj +<< + /Type /StructElem + /S /TD + /P 586 0 R + /A [<< + /O /Layout + /BorderStyle /Solid + >>] + /K [590 0 R] +>> +endobj + +590 0 obj +<< + /Type /StructElem + /S /Formula + /P 589 0 R + /K [290] + /Pg 1095 0 R +>> +endobj + +591 0 obj +<< + /Type /StructElem + /S /TD + /P 586 0 R + /A [<< + /O /Layout + /BorderStyle /Solid + >>] + /K [592 0 R] +>> +endobj + +592 0 obj +<< + /Type /StructElem + /S /Strong + /P 591 0 R + /K [289] + /Pg 1095 0 R +>> +endobj + +593 0 obj +<< + /Type /StructElem + /S /Div + /P 549 0 R + /K [600 0 R 597 0 R 594 0 R] +>> +endobj + +594 0 obj +<< + /Type /StructElem + /S /Div + /P 593 0 R + /K [596 0 R 595 0 R] +>> +endobj + +595 0 obj +<< + /Type /StructElem + /S /Formula + /P 594 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288] + /Pg 1095 0 R +>> +endobj + +596 0 obj +<< + /Type /StructElem + /S /Strong + /P 594 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [272] + /Pg 1095 0 R +>> +endobj + +597 0 obj +<< + /Type /StructElem + /S /Div + /P 593 0 R + /K [599 0 R 598 0 R] +>> +endobj + +598 0 obj +<< + /Type /StructElem + /S /Formula + /P 597 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [261 262 263 264 265 266 267 268 269 270 271] + /Pg 1095 0 R +>> +endobj + +599 0 obj +<< + /Type /StructElem + /S /Strong + /P 597 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [260] + /Pg 1095 0 R +>> +endobj + +600 0 obj +<< + /Type /StructElem + /S /Div + /P 593 0 R + /K [602 0 R 601 0 R] +>> +endobj + +601 0 obj +<< + /Type /StructElem + /S /Formula + /P 600 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [246 247 248 249 250 251 252 253 254 255 256 257 258 259] + /Pg 1095 0 R +>> +endobj + +602 0 obj +<< + /Type /StructElem + /S /Strong + /P 600 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [245] + /Pg 1095 0 R +>> +endobj + +603 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [242 243 244] + /Pg 1095 0 R +>> +endobj + +604 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [238 239 240 241] + /Pg 1095 0 R +>> +endobj + +605 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [234 235 236 237] + /Pg 1095 0 R +>> +endobj + +606 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [233] + /Pg 1095 0 R +>> +endobj + +607 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [230 231 232] + /Pg 1095 0 R +>> +endobj + +608 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [224 225 226 227 228 229] + /Pg 1095 0 R +>> +endobj + +609 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [221 222 223] + /Pg 1095 0 R +>> +endobj + +610 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [215 216 217 218 219 220] + /Pg 1095 0 R +>> +endobj + +611 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [213 214] + /Pg 1095 0 R +>> +endobj + +612 0 obj +<< + /Type /StructElem + /S /Formula + /P 549 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [208 209 210 211 212] + /Pg 1095 0 R +>> +endobj 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+0000866975 00000 n +0001426995 00000 n +0001427496 00000 n +0001623412 00000 n +0001623530 00000 n +0001624618 00000 n trailer << - /Size 417 - /Root 416 0 R - /Info 414 0 R - /ID [(rOOz8pGXHyta5b8Oh7VdFA==) (rOOz8pGXHyta5b8Oh7VdFA==)] + /Size 1138 + /Root 1137 0 R + /Info 1135 0 R + /ID [(Rhl3HTK9GIzarsKRz4SuQw==) (Rhl3HTK9GIzarsKRz4SuQw==)] >> startxref -1180525 +1624858 %%EOF \ No newline at end of file diff --git a/01001 - Matematik 1a/Mat1a gennemgang/Mat1a gennemgang.typ b/01001 - Matematik 1a/Mat1a gennemgang/Mat1a gennemgang.typ index b03d649..5990a1b 100644 --- a/01001 - Matematik 1a/Mat1a gennemgang/Mat1a gennemgang.typ +++ b/01001 - Matematik 1a/Mat1a gennemgang/Mat1a gennemgang.typ @@ -1,12 +1,13 @@ #import "@local/dtu-template:0.5.1":* +#import "@preview/cetz-venn:0.1.4" +#import "@preview/cetz:0.4.2" +#import "@preview/cetz-plot:0.1.3": plot, chart -#show: dtu-math-assignment.with( - course: "01001", - course-name: "Mathematics 1a (Polytechnical Foundation)", - title: "Mat1a gennemgang", - due-date: datetime(day: 30, month: 11, year: 2025), - author: "Rasmus Rosendahl-Kaa (S255955)", - semester: "2025 Fall", +#show: dtu-note.with( + title: "Mat1a gennemgang ", + date: datetime(year: 2025, month: 11, day: 30), + author: "Rasmus Rosendahl-kaa", + semester: "Spring 2025" ) #set math.mat(delim: "[") @@ -224,7 +225,983 @@ $ f compose g: A arrow C\ f compose g = f(g(x)) $ -= Komplekse tal += Komplekse tal $CC$ +Er en mængde af tal, hvor vi tillader et tal $i$, som er løsningen til ligningen $i^2=-1$ + +Det gør, at man kan tage kvadratroden af negative tal. $i=sqrt(-1)$ + +Bruger normalt $Z$ til at beksrive dem. + +*Rektangulær form:*\ +$Z=a+i b$ + +Regulære del: $R e(Z)=a$\ +Imaginære del: $I m(Z)=b$ + +*Polær form:*\ +$Z=|Z| dot e^(i dot arg(Z))$ + +Kan betragte dem som funktioner:\ +$ Z_1 = (a_1 + i b_1), Z_2 = (a_2 + i b_2) $ +hvor man kan lægge dem sammen:\ +$ Z_1+Z_2 = a_1 + a_2 + i (b_1 + b_2) $ + +Gange dem sammen:\ +$ Z_1 dot Z_2 &= (a_1 + i b_1) (a_2 + i b_2)=a_1 a_2 + i a_1 b_2 + i b_1 a_2 + i^2 b_1 b_2\ &= a_1 a_2 + i a_1 b_2 + i b_1 a_2 - b_1 b_2 $ + + +Gange sammen på polære form:\ +$ +Z_3 = |Z_3| e^(i arg(Z_3)), Z_4 = |Z_4| e^(i arg(Z_4))\ +Z_3 dot Z_4 = |Z_3||Z_4| e^(i (arg(Z_3) + arg(Z_4))) +$ + +#definition(title: "Eulers formel")[ +$ +e^(i v) = cos(v) + i sin(v)\ +|Z| e^(i arg(Z)) = |Z| cos(arg(Z)) + i |Z| sin(arg(Z)) +$ +] + +#definition(title: "4.3.1: Fra rektangulær til Polær")[ +Længden/Absolutværdien af et komplekst tal er:\ +$ |Z| = sqrt(a^2 + b^2)=sqrt(R e(Z)^2 + I m(Z)^2) $ +Bruger pythagoras. + +Argumentet: +$ +A r g(Z)=cases( +arctan(b/a) &"for" a>=0, +pi/2 &"for" a=0 and b > 0, +arctan(b/a)+pi &"for" a<0 and b>=0, +- pi/2 &"for" a=0 and b < 0, +arctan(b/a) - pi &"for" a <0 and b < 0 +) +$ +Argument ($a r g(Z)$) er hovedargumentet (vinklen fra $0-2 pi$) plus $2 pi$ gange noget. + +] +#align( + center, +)[ + #figure( + caption: [Den Komplette Enhedscirkel - Trigonometri, Komplekse Tal og Koordinater], + )[ + #cetz.canvas( + { + import cetz.draw: * + + let r = 4.0 // Større radius for bedre læsbarhed + + // ═══════════════════════════════════════════════════════════════ + // BAGGRUND: Kvadrant-farver (CAST-reglen) + // ═══════════════════════════════════════════════════════════════ + // Kvadrant I (0° til 90°): ALLE positive - lysegrøn + arc( + (0, 0), + start: 0deg, + stop: 90deg, + radius: r, + mode: "PIE", + fill: rgb(200, 255, 200, 40), + stroke: none, + anchor: "origin", + ) + // Kvadrant II (90° til 180°): SIN positiv - lyseblå + arc( + (0, 0), + start: 90deg, + stop: 180deg, + radius: r, + mode: "PIE", + fill: rgb(200, 200, 255, 40), + stroke: none, + anchor: "origin", + ) + // Kvadrant III (180° til 270°): TAN positiv - lyserød + arc( + (0, 0), + start: 180deg, + stop: 270deg, + radius: r, + mode: "PIE", + fill: rgb(255, 200, 200, 40), + stroke: none, + anchor: "origin", + ) + // Kvadrant IV (270° til 360°): COS positiv - lysegul + arc( + (0, 0), + start: 270deg, + stop: 360deg, + radius: r, + mode: "PIE", + fill: rgb(255, 255, 200, 40), + stroke: none, + anchor: "origin", + ) + + // ═══════════════════════════════════════════════════════════════ + // HOVEDCIRKEL + // ═══════════════════════════════════════════════════════════════ + circle((0, 0), radius: r, stroke: (paint: black, thickness: 1.5pt)) + + // ═══════════════════════════════════════════════════════════════ + // AKSER med pile + // ═══════════════════════════════════════════════════════════════ + // x-akse (Real akse / cos-akse) + line((-r - 1.0, 0), (r + 1.0, 0), stroke: (paint: gray, thickness: 1pt), mark: (end: "stealth", fill: gray)) + // y-akse (Imaginær akse / sin-akse) + line((0, -r - 1.0), (0, r + 1.0), stroke: (paint: gray, thickness: 1pt), mark: (end: "stealth", fill: gray)) + + // Akseetiketter + content((r + 1.3, 0), text(weight: "bold")[$x = cos(theta) = "Re"$], anchor: "west") + content((0, r + 1.3), text(weight: "bold")[$y = sin(theta) = "Im"$], anchor: "south") + + // ═══════════════════════════════════════════════════════════════ + // CAST-REGEL ETIKETTER (i hver kvadrant) + // ═══════════════════════════════════════════════════════════════ + content((r * 0.5, r * 0.5), text(size: 10pt, fill: rgb(0, 100, 0))[ + #box(fill: rgb(200, 255, 200, 150), inset: 3pt, radius: 2pt)[#strong[A]lle $+$] + ]) + content((-r * 0.5, r * 0.5), text(size: 10pt, fill: rgb(0, 0, 150))[ + #box(fill: rgb(200, 200, 255, 150), inset: 3pt, radius: 2pt)[#strong[S]in $+$] + ]) + content((-r * 0.5, -r * 0.5), text(size: 10pt, fill: rgb(150, 0, 0))[ + #box(fill: rgb(255, 200, 200, 150), inset: 3pt, radius: 2pt)[#strong[T]an $+$] + ]) + content((r * 0.5, -r * 0.5), text(size: 10pt, fill: rgb(150, 100, 0))[ + #box(fill: rgb(255, 255, 200, 150), inset: 3pt, radius: 2pt)[#strong[C]os $+$] + ]) + + // ═══════════════════════════════════════════════════════════════ + // KVADRANT-NUMRE (romertal i hjørnerne) + // ═══════════════════════════════════════════════════════════════ + content((r * 0.85, r * 0.85), text(size: 8pt, fill: gray)[I]) + content((-r * 0.85, r * 0.85), text(size: 8pt, fill: gray)[II]) + content((-r * 0.85, -r * 0.85), text(size: 8pt, fill: gray)[III]) + content((r * 0.85, -r * 0.85), text(size: 8pt, fill: gray)[IV]) + + // ═══════════════════════════════════════════════════════════════ + // ALLE 16 VIGTIGE VINKLER MED KOMPLETTE DATA + // Format: (grader, radianer-label, cos-værdi, sin-værdi, kompleks-form) + // ═══════════════════════════════════════════════════════════════ + let angles = ( + // Grad 0 (Kvadrant I/IV grænse) + (0, $0$, $1$, $0$, $1$), + // Kvadrant I + (30, $pi/6$, $sqrt(3)/2$, $1/2$, $e^(i pi/6)$), + (45, $pi/4$, $sqrt(2)/2$, $sqrt(2)/2$, $e^(i pi/4)$), + (60, $pi/3$, $1/2$, $sqrt(3)/2$, $e^(i pi/3)$), + (90, $pi/2$, $0$, $1$, $i$), + // Kvadrant II + (120, $(2pi)/3$, $-1/2$, $sqrt(3)/2$, $e^(i 2pi/3)$), + (135, $(3pi)/4$, $-sqrt(2)/2$, $sqrt(2)/2$, $e^(i 3pi/4)$), + (150, $(5pi)/6$, $-sqrt(3)/2$, $1/2$, $e^(i 5pi/6)$), + (180, $pi$, $-1$, $0$, $-1$), + // Kvadrant III + (210, $(7pi)/6$, $-sqrt(3)/2$, $-1/2$, $e^(i 7pi/6)$), + (225, $(5pi)/4$, $-sqrt(2)/2$, $-sqrt(2)/2$, $e^(i 5pi/4)$), + (240, $(4pi)/3$, $-1/2$, $-sqrt(3)/2$, $e^(i 4pi/3)$), + (270, $(3pi)/2$, $0$, $-1$, $-i$), + // Kvadrant IV + (300, $(5pi)/3$, $1/2$, $-sqrt(3)/2$, $e^(i 5pi/3)$), + (315, $(7pi)/4$, $sqrt(2)/2$, $-sqrt(2)/2$, $e^(i 7pi/4)$), + (330, $(11pi)/6$, $sqrt(3)/2$, $-1/2$, $e^(i 11pi/6)$), + ) + + // ═══════════════════════════════════════════════════════════════ + // TEGN ALLE VINKLER + // ═══════════════════════════════════════════════════════════════ + for (deg, rad_label, cos_val, sin_val, complex_form) in angles { + let rad = deg * calc.pi / 180 + let x = r * calc.cos(rad) + let y = r * calc.sin(rad) + + // Radial linje fra origin til punkt (stiplet) + line((0, 0), (x, y), stroke: (paint: blue.darken(20%), thickness: 0.6pt, dash: "dashed")) + + // Punkt på cirklen + circle((x, y), radius: 0.08, fill: blue, stroke: (paint: white, thickness: 0.5pt)) + + // Projektionslinjer til akserne (hjælpelinjer for at vise cos og sin) + if deg != 0 and deg != 180 { + line((x, 0), (x, y), stroke: (paint: red.lighten(30%), thickness: 0.4pt, dash: "dotted")) + } + if deg != 90 and deg != 270 { + line((0, y), (x, y), stroke: (paint: green.lighten(30%), thickness: 0.4pt, dash: "dotted")) + } + } + + // ═══════════════════════════════════════════════════════════════ + // GRADTAL PÅ YDERSIDEN AF CIRKLEN + // ═══════════════════════════════════════════════════════════════ + let deg_radius = r + 0.35 + for (deg, rad_label, cos_val, sin_val, complex_form) in angles { + let rad = deg * calc.pi / 180 + let x = deg_radius * calc.cos(rad) + let y = deg_radius * calc.sin(rad) + + // Bestem ankerpunkt baseret på vinkel + let anchor = if deg == 0 { "west" } else if deg < 90 { "south-west" } else if deg == 90 { "south" } else if deg < 180 { "south-east" } else if deg == 180 { "east" } else if deg < 270 { "north-east" } else if deg == 270 { "north" } else { "north-west" } + + content((x, y), text(size: 7pt, fill: gray.darken(20%))[#deg°], anchor: anchor) + } + + // ═══════════════════════════════════════════════════════════════ + // RADIANER PÅ INDERSIDEN AF CIRKLEN + // ═══════════════════════════════════════════════════════════════ + let rad_radius = r - 0.45 + for (deg, rad_label, cos_val, sin_val, complex_form) in angles { + let rad = deg * calc.pi / 180 + let x = rad_radius * calc.cos(rad) + let y = rad_radius * calc.sin(rad) + + // Bestem ankerpunkt (modsat af grader) + let anchor = if deg == 0 { "east" } else if deg < 90 { "north-east" } else if deg == 90 { "north" } else if deg < 180 { "north-west" } else if deg == 180 { "west" } else if deg < 270 { "south-west" } else if deg == 270 { "south" } else { "south-east" } + + content((x, y), text(size: 6pt, fill: purple.darken(20%))[$#rad_label$], anchor: anchor) + } + + // ═══════════════════════════════════════════════════════════════ + // KOORDINATER (cos, sin) VED HVERT PUNKT + // Placeret længere ude for læsbarhed + // ═══════════════════════════════════════════════════════════════ + let coord_radius = r + 1.0 + for (deg, rad_label, cos_val, sin_val, complex_form) in angles { + let rad = deg * calc.pi / 180 + let x = coord_radius * calc.cos(rad) + let y = coord_radius * calc.sin(rad) + + // Juster position for akseværdier + let (adj_x, adj_y) = if deg == 0 { (x + 0.2, y - 0.3) } else if deg == 90 { (x + 0.5, y + 0.2) } else if deg == 180 { (x - 0.2, y - 0.3) } else if deg == 270 { (x + 0.5, y - 0.2) } else { (x, y) } + + // Bestem ankerpunkt + let anchor = if deg == 0 { "north-west" } else if deg < 90 { "south-west" } else if deg == 90 { "south-west" } else if deg < 180 { "south-east" } else if deg == 180 { "north-east" } else if deg < 270 { "north-east" } else if deg == 270 { "north-west" } else { "north-west" } + + // Kun vis koordinater for ikke-aksepunkter (aksepunkter vises separat) + if deg != 0 and deg != 90 and deg != 180 and deg != 270 { + content( + (adj_x, adj_y), + text(size: 5.5pt)[#box(fill: white.darken(5%), inset: 1pt, radius: 1pt)[$(#cos_val, #sin_val)$]], + anchor: anchor, + ) + } + } + + // ═══════════════════════════════════════════════════════════════ + // AKSEPUNKTER (specielle værdier) - større og tydeligere + // ═══════════════════════════════════════════════════════════════ + // (1, 0) ved 0° + content((r + 0.15, -0.35), text(size: 7pt)[ + #box(fill: white, inset: 2pt, radius: 2pt, stroke: 0.3pt + gray)[ + $(1, 0)$ \ $= 1$ + ] + ], anchor: "north-west") + + // (0, 1) ved 90° + content((0.35, r + 0.15), text(size: 7pt)[ + #box(fill: white, inset: 2pt, radius: 2pt, stroke: 0.3pt + gray)[ + $(0, 1)$ \ $= i$ + ] + ], anchor: "south-west") + + // (-1, 0) ved 180° + content((-r - 0.15, -0.35), text(size: 7pt)[ + #box(fill: white, inset: 2pt, radius: 2pt, stroke: 0.3pt + gray)[ + $(-1, 0)$ \ $= -1$ + ] + ], anchor: "north-east") + + // (0, -1) ved 270° + content((0.35, -r - 0.15), text(size: 7pt)[ + #box(fill: white, inset: 2pt, radius: 2pt, stroke: 0.3pt + gray)[ + $(0, -1)$ \ $= -i$ + ] + ], anchor: "north-west") + + // ═══════════════════════════════════════════════════════════════ + // AKSEMARKERINGER (tick marks ved ±1) + // ═══════════════════════════════════════════════════════════════ + // x-akse tick marks + line((r, -0.1), (r, 0.1), stroke: black) + line((-r, -0.1), (-r, 0.1), stroke: black) + // y-akse tick marks + line((-0.1, r), (0.1, r), stroke: black) + line((-0.1, -r), (0.1, -r), stroke: black) + + // ═══════════════════════════════════════════════════════════════ + // EKSEMPEL-VINKEL θ MED VISUALISERING + // Tegner en eksempel vinkel ved 45° med forklaring + // ═══════════════════════════════════════════════════════════════ + let example_deg = 45 + let example_rad = example_deg * calc.pi / 180 + let ex = r * calc.cos(example_rad) + let ey = r * calc.sin(example_rad) + + // Tykkere linje for eksemplet + line((0, 0), (ex, ey), stroke: (paint: orange, thickness: 2pt), name: "example-line") + + // Vinkelmarkering (bue fra x-aksen til linjen) + arc( + (0, 0), + start: 0deg, + stop: 45deg, + radius: 0.8, + stroke: (paint: orange, thickness: 1.5pt), + mark: (end: "stealth", fill: orange, scale: 0.5), + ) + + // θ label ved vinklen + content((1.0, 0.35), text(size: 10pt, fill: orange, weight: "bold")[$theta$]) + + // Fremhæv projektioner for eksemplet + line((ex, 0), (ex, ey), stroke: (paint: red, thickness: 1.5pt), name: "sin-line") + line((0, ey), (ex, ey), stroke: (paint: green.darken(20%), thickness: 1.5pt), name: "cos-line") + line((0, 0), (ex, 0), stroke: (paint: green.darken(20%), thickness: 1.5pt)) + + // Labels for cos og sin projektioner + content((ex / 2, -0.25), text(size: 8pt, fill: green.darken(20%), weight: "bold")[$cos(theta)$]) + content((ex + 0.35, ey / 2), text(size: 8pt, fill: red, weight: "bold")[$sin(theta)$]) + + // Radius = 1 label + content((ex / 2 - 0.15, ey / 2 + 0.25), text(size: 7pt, fill: orange)[$r = 1$], anchor: "south-east") + + // ═══════════════════════════════════════════════════════════════ + // NØGLE-FORMLER I BUNDEN + // ═══════════════════════════════════════════════════════════════ + content((0, -r - 2.2), text(size: 8pt)[ + #box(fill: rgb(245, 245, 255), inset: 6pt, radius: 3pt, stroke: 0.5pt + gray)[ + #std.grid( + columns: 3, + gutter: 15pt, + [#strong[Eulers formel:] \ $e^(i theta) = cos(theta) + i sin(theta)$], + [#strong[Koordinat:] \ $(cos(theta), sin(theta))$], + [#strong[Kompleks tal:] \ $z = e^(i theta) = cos(theta) + i sin(theta)$], + ) + ] + ]) + + // ═══════════════════════════════════════════════════════════════ + // EKSTRA: Vigtige trigonometriske værdier tabel (til højre) + // ═══════════════════════════════════════════════════════════════ + content((r + 3.5, 0), text(size: 6pt)[ + #box(fill: white, inset: 4pt, radius: 2pt, stroke: 0.5pt + gray)[ + #table( + columns: 3, + align: center, + stroke: 0.3pt + gray, + inset: 2pt, + [*θ*], + [$cos$], + [$sin$], + [$0$], + [$1$], + [$0$], + [$pi/6$], + [$sqrt(3)/2$], + [$1/2$], + [$pi/4$], + [$sqrt(2)/2$], + [$sqrt(2)/2$], + [$pi/3$], + [$1/2$], + [$sqrt(3)/2$], + [$pi/2$], + [$0$], + [$1$], + ) + ] + ], anchor: "west") + }, + ) + ] +] + +#note-box[ + *Sådan læser du enhedscirklen:* + + 1. *Vinklen $theta$* måles fra den positive $x$-akse (mod uret er positiv) + + 2. *Koordinaterne* $(x, y)$ på cirklen er $(cos(theta), sin(theta))$ + + 3. *Komplekst tal*: Punktet svarer til $z = e^(i theta) = cos(theta) + i sin(theta)$ + + 4. *CAST-reglen* fortæller hvilke funktioner er positive: + - #strong[Kvadrant I] ($0° - 90°$): #strong[A]lle positive + - #strong[Kvadrant II] ($90° - 180°$): Kun #strong[S]in positiv + - #strong[Kvadrant III] ($180° - 270°$): Kun #strong[T]an positiv + - #strong[Kvadrant IV] ($270° - 360°$): Kun #strong[C]os positiv + + 5. *Huske-værdier*: $sqrt(3)/2 approx 0.87$, $sqrt(2)/2 approx 0.71$, $1/2 = 0.5$ +] + +== Mulige former for komplekse tal til eksamen +#lemma(name: "4.6.1")[ +$ +e^Z = w +$ +$Z in CC$ skal findes\ +$w in CC$ er givet. + +$ +Z = ln(|w|) + i (A r g(w) + P 2 pi), p in ZZ +$ + +#important()[ +Angiv altid $, p in ZZ$ for at få fuld point +] +] +#example()[ +$ +e^Z=sqrt(2)+sqrt(2)i\ +w = sqrt(2)+sqrt(2)i\ +|w|=sqrt(2+2)=2\ +A r g(w) = pi/4\ +Z = ln(2) + i (pi/4 + 2p pi), p in ZZ +$ +] + +=== Binome ligninger (binomier) +Er et polynomium $Z^n=w$ + +$Z,w in CC, n in ZZ_(>=0)$ + +$ + Z= root(n, |w|) dot e^(i ((a r g(w))/n) + p (2 pi)/n), p = {0,dots,n-1} +$ + +#example()[ +$ + Z^3=-i\ + n=3\ + w=-i, |w|=1\ + A r g(w)=-pi/2\ + Z = root(3, 1) dot e^(i dot (-pi/6 + p (2 pi)/3)), p in {0,1,2}\ + Z = 1 e^(i ((-pi/2)/3 + p 2 pi/3)) = e^(i (-pi/6 + p 2 pi/3)), quad p = {0,1,2} \ + Z = e^(- pi/6 i), e^(pi/2 i), e^((-5 pi)/6 i) +$ +] = Polynomier +$ +P(Z)&=a_0 Z^0 + a_1 Z^1 + a_2 Z^2 + dots + a_n Z^n\ +&=a_0 + a_1 Z + a_2 Z^2 + dots + a_n Z^n +$ +Hvis $a_n eq.not 0$ kaldes det et n'te grads polynomie. + +== Rødder +Tal $lambda$ sǻ $P(lambda)=0$ + +#definition(title: "Sætning 5.6.1: Algebraets fundamentalsætning")[ +Et n'te grads polynomie har n komplekse rødder regnet med multiplicitet. + +Man kan omskrive alle polynomier som et produkt af førstegradspolynomier.\ +Hvis $p(z)$ har rødder $lambda_1, dots, lambda_n$, kan det omskrives til: +$ +P(z)=a_n(z-lambda_1) dot dots dot (z-lambda_n) +$ +eller med multiplicitet: +$ +P(z)=a_n (z-lambda_1)^(m_1)dot dots dot (lambda_j)^(m_j)\ +lambda_1 eq.not lambda_2 eq.not dots eq.not lambda_j +$ +og $m_1 + dots + m_j = n$ + +$m_k$ kaldes den algebraiske multiplicitet af $lambda_k$ +] + +=== Hvordan finder man rødder +For $n=1:$ $P(z)=a_0+a_1z=0 => lambda = - a_0/a_1$ + +For $n=2:$ $P(z)=c+b z + a z^2 => lambda = (-b plus.minus sqrt(D))/(2 a), D = b^2 - 4 a c$ + +Hvis $D<0$: $lambda=(-b plus.minus i sqrt(|D|))/(2 a)$ + +#note-box()[ +Man kan muligvis få opgaver, hvor man skal gætte en rod ved $n>2$. Her er den typisk mellem -2 og 2 +] + +=== Divisionsalgoritmen + +#example()[ +$P(z)=2z^3-2z^2-8z-12$ givet\ +$p(3)=0$ også givet. + +Find alle rødder: +#solution()[ +Vi ved at: +$ + P(z)=(z-3)q(z) +$ +Bruger divisionsalgoritmen til at finde $q(z)$ +#image("divisionsalgoritmen.png",width:50%) +] +] +Resten ved divisionsalgoritmen skal være 0, ellers er det ikke en rod. + +Alle $P(z) in RR[Z]$ så kan det skrives som produkt af reelle første- og andengradspolynomier. + = Matricer +$ +bold(A) = mat(a_11,a_12,a_13,dots,a_(1 n);a_21,a_22,dots,,a_(2 n);dots.v;a_(m 1),a_(m 2),dots,a_(m n)) in FF^(n times m) +$ + +$ &mat(a_1, c_1; b_1, d_1) + mat(a_2, c_2; b_2, d_2)\ +&= mat(a_1+a_2, c_1+c_2; b_1+b_2, d_1+d_2) +$ + +$ +&mat(a_1, c_1; b_1, d_1) dot mat(a_2, c_2; b_2, d_2)\ +&=mat(a_1 a_2 + c_1 b_2, a_1 c_2 + c_1 d_2; a_1 b_2 + d_1 b_2, b_1 c_2 + d_1 d_2) +$ +Når man ganger matricer sammen skal den anden matrix have samme mængde rækker som den første søjler. + +#note-box()[ +Når der står $bold(A) in FF^(m times n)$ er $m$ antal søjler (hvor mange tal nedad) og $n$ er antal rækker (hvor mange tal til højre) +] + +== Søjlevektor +$ bold(v) in FF^n +bold(v) = mat(v_1;v_2;dots.v;v_n) +$ + + +== Lineært ligningssystem +Mængde af lineære ligninger +$ + cases( + a_11 x_1 + a_12 x_2 + dots + a_(1 n) x_n = b_1, + dots.v, + a_(m 1) x_1 + a_(m 2) x_2 + dots + a_(m n) x_n = b_n + ) +$ +$ + bold(A) bold(x) = bold(b):\ + mat(a_11,dots,,a_(1 n);dots.v;a_(m 1), dots,, a_(m n)) dot mat(x_1;dots.v;x_n) = mat(b_1; dots.v; b_n) +$ + +Gauss elimination + +$ +bold(T) = mat(bold(A), bold(b); augment: #1) = mat(2,1,1,1;4,5,7,2; augment: #(-1)) +$ + +Reduceret trappeform: + +$ +mat(1,0,0;0,1,0;augment: #(-1))\ +mat(1,0;0,1) mat(x_1;x_2) = mat(0;0)\ +1 x_1 + 0x_2 = 0, x_1 = 0\ +0 x_1 + 1 x_2 = 0, x_2 = 0 +$ + +#example()[ +$ +bold(A) = mat(1,-4,0,1;0,0,1,2;0,0,0,0;augment: #(-1)) +bold(x)=mat(x_1;x_2;x_3) +$ +$ +bold(A) dot bold(x) = mat(1,-4,0;0,0,1;0,0,0) dot mat(x_1;x_2;x_3) = mat(x_1 - 4 x_2;x_3;0) = mat(1;2;0) +$ + +#solution()[ + +$x_1-4x_2 = 1$\ +$x_3 = 2$\ +$x_2 = t, x_1 = 1 + 4t$ +$ +mat(x_1;x_2;x_3)&=mat(1+4t;t;2)\ +&=mat(1;0;2)+t mat(4;1;0) +$ +] +] + += Induktion +Ideen er, at vi viser to ting: +$ +cases( + P(1) "holder", + "For alle" n in NN_(>=2) P(n-1) => P(n) +) +$ +Kan også forskydes (så vis $P(2)$ holder og $P(n-1)=> P(n)$ for alle $n in NN_(>=3)$) + +#example()[ +$ +sum^n_(k=1)k=(n(n+1))/2 +$ +Basis (holder): +$ sum^1_(k=1)k=1, (1 (1+1))/2 = 1 $ + +Induktionshypotese:\ +$ +sum^(n-1)_(k=1)k=((n-1)(n-1+1))/2=((n-1)dot n)/2 +$ + +$ +sum^(n)_(k=1)k=underbrace(1+2+3+dots+(n-2)+(n-1)+, sum^(n-1)_(k=1)k) n +$ +$ +sum^(n)_(k=1)k&=sum^(n-1)_(k=1)k + n\ +&=((n-1)n)/2 + (2n)/2\ +&=((n-1)n+2n)/2\ +&= (n^2-n+2n)/2\ +&= (n^2+n)/2 = (n(n+1))/2 +$ +Så hypotesen holder. +] + +#example(title: $(S_1,S_2,dots)$)[ +$ +S_n=cases(1 &"for" n=1, 2 dot S_(n-1)+1 &"for" n>= 2) "for" n in NN +$ +Vis at $S_n=2^n-1$ + +*Basis:*\ +$n=1, S_1=1, 2^1-1=1$ + +*Induktionshypotese:*\ +Antag, at der for et vilkårligt $n>=2$ gælder $S_(n-1)=2^(n-1)-1$ + +$ +S_n&=2 S_(n-1)+1\ +&= 2 dot (2^(n-1)-1)+1\ +&= 2^n-2+1 +&= 2^n-1 + +$ +] + +#example()[ +$ +sum^n_(k=2)2^k=2^(n+1)-4 "for alle" n in NN_(>=2) +$ +*Basis:* $n=2$ +$ +sum^2_(k=2)2^k=2^2=4\ +2^(2+1)-4=8-4=4 +$ +*Induktion:* for $n in NN_(>=2)$ +$ +sum^(n-1)_(k=2)2^k=2^((n-1)+1)-4=2^n-4 +$ +Betragt: +$ +sum^n_(k=2)2^k&=sum^(n-1)_(k=2)2^k+2^n\ +&=2^n-4+2^n\ +&=2 dot 2^n-4\ +&=2^(n+1)-4 +$ + +] + +#example()[ +$ +a_n=cases(2 &"for" n=1,sqrt(2+3 dot a_(n-1)) &"for" n>=2) +$ +Vi vil vise, at $a_n < 4$ for alle $n in NN$ + +*Basis:* $n=1$\ +$a_1 = 4 < 4$ + +*Induktion:* $a_(n-1) < 4$ for $n in NN_(>=2)$ + +Betrakt: +$ +a_n=sqrt(2+3 a_(n-1)) < sqrt(2+3 dot 4) = sqrt(2+12)=sqrt(14)=4)$, vis at $n! >= 2^(n-1)$ + +*Basis:* $n=4$\ +$4!=4 dot 3 dot 2 dot 1 = 24, 2^(4-1) = 2^(3)=8$ Holder + +*Induktion:* Antag for $n in NN_(>=5)$ at $(n-1)! >= 2^(n-1-1) = 2^(n-2)$ + +Betragt +$ +n! &= n dot (n-1)!\ +&>= n dot 2^(n-2)\ +&=1/2 dot n dot 2^(n-1) >= 2^(n-1) +$ +Fordi vi ved at $1/2 n >= 1$, må $1/2 n dot 2^(N-1)$ være større end $1 dot 2^(n-1)$ +] + +#example()[ +$bold(A) = mat(1,2;0,1)$ + +Vis at $bold(A)=mat(1,2n;0,1)$ for alle $n in NN$ + +*Basis:* $n=1$\ +$bold(A)^1 = bold(A), mat(1, 2 dot 1; 0, 1) = bold(A)$ + +*Induktion:*\ +Antag $bold(A)^(n-1)=mat(1, 2 (n-1);0,1)$ for $n in NN_(>=2)$ + +Betragt +$ +&bold(A)^n = bold(A)^(n-1) dot bold(A)\ +&=mat(1, 2 (n-1);0,1) dot mat(1,2;0,1)\ +&= mat(1, 2+2(n-1);0,1)\ +&= mat(1,2n;0,1) +$ +] += Lineære algebra +== Vektorrum +#example()[ +$bold(v_1)=vec(1,2,1,4), bold(v_2)=vec(3,3,0,1), bold(v_3)=vec(7,8,1,6)$ + +Start med at finde en ordnet basis for $bold(V)$ + +$ +mat(bold(v_1),bold(v_2),bold(v_3)) = mat(1,3,7;2,3,8;1,0,1;4,1,6) arrow.long_("RREF") mat(1,0,1;0,1,2;0,0,0;0,0,0) +$ +$(bold(v_1),bold(v_2))$ er en ordnet basis for $bold(V)$. + +Dimensionen for den ordnet basis er antal vektorer i den ordnet basis (her 2). + +*Opskriv lineær kombination for vektoren der ikke er i ordnet basis:*\ +$bold(v_3)=1 dot bold(v_1) + 2 dot bold(v_2)$ +] + + + +#definition(title: "Sætning 9.2.1")[ +Sæt vektor sammen til en matrice, få den på reduceret trappeform. I de søjler der er pivotelementer, er de vektorer der er i den ordnet base. +] + +#definition(title: "Nulrummet")[ +Alle de $bold(x) in RR^n$ således at $bold(A) dot bold(x) = bold(0)$ +] + +#example()[ +$ +bold(A) = mat(1,3,7;2,3,8;1,0,1;4,1,6)\ +r r e f(mat(bold(A),bold(0)))= mat(1,0,1,0;0,1,2,0;0,0,0,0;0,0,0,0)\ +x_1 + x_3 = 0 <=> x_1 = -x_3\ +x_2 + 2x_3 = 0 <=> x_2 = -2x_3 +$ + +$ +vec(x_1,x_2,x_3) = t vec(-1,-2,1) in ker(bold(A)), t in RR +$ + +$ ker(bold(A))="span"{vec(-1,-2,1)} $ + +*Søjlerummet:* + +$bold(A) dot bold(x) = bold(b)$ hvor $bold(x)$ kan variere frit, hvad kan $bold(b)$ så være? + +$"colsp"(bold(A))="span"{vec(1,2,1,4),vec(3,3,0,1)}$ +] + +== Polynomiumrum +$ +W = {a + b Z | a,b in CC} subset.eq CC[Z] +$ +Vis at $W$ er et underrum. + +#lemma(name: "10.4.2")[ +Altid skriv, at du bruger Lemma 10.4.2 til eksamen. + +Lad $V$ være et vektorrum over $FF$ og $W$ en delmængde af $V$, *der ikke er tom*. Da er $W$ et underrum i $V$, hvis følgende er opfyldt: +$ +"for alle" bold(u), bold(v) in W "og alle" c in FF "gælder der, at" bold(u) + c dot bold(v) in W +$ + +$u = a_1+b_1Z\ v=a_2+b_2Z$ + +Betragt +$ +u+ alpha dot v &= (a_1+b_1Z)+alpha (a_2 + b_2 Z) +&= a_1 + b_1Z + alpha a_2+ alpha b_2Z = underbrace((a_1 + alpha a_2), a in CC) + underbrace((b + alpha b_2), b in CC)Z +$ +] + +#example()[ +$W_0 = {a+Z| a in CC} subset.eq CC[Z]$ + +Vis at $W_0$ *ikke* er et underrum, bruger vi Lemma 10.4.2. + +$ +u=1+Z in W_0\ +u+u = 1 + Z + 1 + Z = 2 + 2Z in.not W_0 +$ +Er ikke et underrum da det kun er et underrum hvis du har $1 dot Z$ men vi har her $2 dot Z$ +] + +== Basis +#definition()[ +$beta = {1,Z,Z^2,Z^3,dots,Z^n}$ er lineært uafhængige, og enhver delmængde er også lineær uafhængig. +] + +#example()[ +$W = {a+b Z| a,b in CC}, beta = {1,Z}$ + +$c_1 dot 1 + c_2 dot Z = 0 + 0 dot Z => c_1 = c_2 = 0$ + +$gamma = (4+Z, 2+3Z)$ + +To krav fra definition 10.2.4: + +Hvis antal uafhængige baser er det samme som dimensionen, er de lineært uafhængige og er en ordnet basis. + +/ 1): $c_1 (4+Z) + c_2 (2+3Z) = 0 [=> c_1 = c_2=0]$ +$ 4c_1 + c_1 Z + 2c_2 + 3c_2Z = 0\ +=(4c_1+2c_2)dot 1 +(c_1+3c_2)Z = 0\ +4c_1+2c_2=0\ +c_1+3c_2=0 +$ +Totalmatrix: +$ +mat(4,2,0;1,3,0; augment: #(-1)) arrow.long_("RREF") mat(1,0,0;0,1,0; augment: #(-1))\ +"så" c_1=0 "og" c_2=0 +$ +Fordi $c_1=0 "og" c_2=0$ må $gamma$ være lineært uafhængige. + +/ 2): For $u in W$ så $u=a+b Z$. + +Vi vil vise, at der findes $c_1,c_2 in CC$ således at $c_1(4+Z)+c_2(2+3Z) = a+b Z$ + +$a,b$ er givet i opgaven, så de kendes. + +$ +(4c_1+2c_2)+(c_1+3c_2)Z=a+b Z\ +4c_1+2c_2=a\ +c_1+3c_2=b\ +$ +Totalmatrix: +$ +mat(4,2,a;1,3,b;augment: #(-1)) arrow.long_("RREF") mat(1,0,-b/5+3a/10;0,1,2b/5-a/10;augment: #(-1))\ +c_1=-b/5+3a/10\ +c_2=2b/5-a/10 +$ +] + +== Lineære afbildninger +$ +f: RR^2 arrow RR, f(vec(x_1,x_2)) = x_1+x_2 +$ +#example()[ +$f$ er lineær hvis $f(bold(x)+bold(y)) = f(bold(x))+f(bold(y)) "og" f(c dot bold(x)) = c dot f(bold(x))$ + +/ 1): $bold(x),bold(y) in RR^2 "så vil" bold(x) = vec(x_1,x_2) "og" bold(y) = vec(y_1,y_2)$ +$ + f(vec(x_1,x_2)+vec(y_1,y_2))&=f(vec(x_1,x_2))+f(vec(y_1,y_2))\ + &=(x_1+y_1)+(x_2+y_2)\ + &=(x_1+x_2)+(y_1+y_2)\ + &=f(vec(x_1,x_2))+f(y_1,y_2) +$ +/ 2): $f(c dot vec(x_1,x_2)), c in RR$ +$ + f(c dot vec(x_1,x_2))&=f(vec(c x_1,c x_2))\ + &= c x_1 + c x_2 = c (x_1+x_2)\ + &= c dot f(vec(x_1,x_2)) +$ +] + +#example()[ +$ +g: RR^2 arrow RR, g(vec(x_1,x_2)) = x_1 dot x_2 +$ +$g$ er *ikke* lineær. + +/ 2): $g(c dot vec(x_1,x_2))=g(vec(c x_1, c x_2)), c in RR$ +$ + +$ + +] + + +#lemma(name: "11.3.3")[ +$L: V_1 arrow V_2$ + +$beta = (v_1, dots, v_n) "for" V_1\ gamma = (w_1,dots,w_n) "for" V_2$ + +$ +amat(L, gamma, beta) = mat([L(v_1)]_gamma, [L(v_2)]_gamma, dots, [L(v_n)]_gamma) +$ + +/ 1): Voldregn $L(v_1), L(v_2), dots, L(v_n)$ +/ 2): Find: $[L(v_1)]_gamma$. Altså løs: $L(v_1)=c_1^((1)) w_1 +dots+c_n^((1)) w_n$ for alle $v$ +/ 3): $amat(L, gamma, beta) = mat(c_1^((1)),c_1^((2)),dots,c_1^((n));c_2^((1)),c_2^((2)),dots,c_1^((n));dots.v,dots.v,,dots.v;c_m^((1)),c_m^((2)),dots,c_m^((n)))$ +] + +#example()[ +$ +f: RR^2 arrow RR^2, f(vec(x_1,x_2)) = x_1+x_2\ +beta = (vec(1,1),vec(-1,1)) "for" RR^2 "og" gamma=3 "for" RR +$ +/ 1): $f(vec(1,1))=2, f(vec(-1,1))=0$ + +/ 2): $[f(vec(1,1))]_gamma=c_1^((1)) 3 = 2 => c_1^((1))=2/3\ [f(vec(-1,1))]_gamma = c_1^((2))=0 => c_1^((2))=0$ + +/ 3): $amat(f, gamma, beta) = mat(2/3,0)$ +] + +#example()[ +$amat(f, gamma, beta) = mat(2/3,0)$ + +Hvad er $"Vm"(f)$? Find søjlerummet. + +$"rref"(amat(f, gamma, beta))=mat(1,0)\ "Vm"(f) = "span"{2/3}={2/3 t| t in RR}$ + +Det er alle reelle tal ganget med $2/3$, men det er også bare reelle tal, så $"Vm"(f)=RR$ + +Den er surjektiv fordi dispositionsmængden er det samme som billedet. + +$f$ er injektiv hvis og kun hvis $ker(f)={0}$ (og $f$ er lineær) + +Derfor, find nulrummet. $"rref"(amat(f, gamma, beta))=mat(1,0), x_1 = 0$ dvs. + +$amat(f, gamma, beta)) underbrace(vec(0,t), = t vec(0,1))=0$ for *alle* $t in RR$ + +$f(vec(0,1))=0$. Tilbage til normale koordinater. + +$0 dot vec(1,1) + 1 vec(-1,1) = vec(-1,1)$ så $ker(f)="span"{vec(-1,1)}, f(vec(-1,1))=0$ + +$vec(-1,1)$ er en ordnet basis for kernen. Dimensionen af kernen er 1, så den er ikke tom, aka. $f$ er ikke injektiv +] + += Egenværdier +$bold(A)in FF^(n times n)$ har egenværdier $lambda$, som opfylder $bold(A) dot bold(v) = lambda dot bold(v)$ + +Finder egenværdien $det(bold(A)-lambda bold(I)_n)$ + +Finder vektorene/egenrum ($E_j, bold(v_j)$) ved: $ker(bold(A)- lambda_j bold(I)_n)$ + +#example()[ +$bold(A)=mat(1,2;2,1) in RR^(2 times 2)$ + +$det(bold(A)-lambda bold(I)_2) = det(mat(1-lambda,2;2,1-lambda))=(1-lambda)^2-4=0$ + +$(1-lambda)^2-4 => lambda = cases(-1,3)$ + + +$P(Z)=(1-Z)^2-4$ er det karakteristiske polynomium + +*Egenrum:* +$ +E_(-1)=ker(bold(A)-(-1) bold(I)_2) = ker(mat(2,2;2,2))\ +mat(2,2;2,2)arrow.long_("RREF") = mat(1,1;0,0), v_1+v_2 = 0 => v_2 = t, v_1 = -t\ +bold(V)= vec(v_1,v_2)=t vec(-1,1) +$ +$ +E_3="span"{vec(1,1)} +$ +*Algebraisk multiplicitet:*\ +$"am"(lambda)$ multiplicitet af $lambda$ i $P(z)$ + +$ "gm"(lambda) = dim(E_(lambda))\ 1 <= "gm"(lambda) <= "am"(lambda) $ + +]