Content-Type: multipart/mixed; boundary="-------------0309011025298" This is a multi-part message in MIME format. ---------------0309011025298 Content-Type: text/plain; name="03-396.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-396.keywords" Almost Mathieu Operator, Cantor Spectrum, Ten Martini Problem, Harper equation. ---------------0309011025298 Content-Type: application/postscript; name="0301puig.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="0301puig.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: preprint_final.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips preprint_final.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.08.28:1721 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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R 1318 0 V -1306 15 R 1306 0 V -1293 14 R 1293 0 V -1281 14 R 1281 0 V -1268 14 R 1268 0 V -1256 15 R 1256 0 V -1244 14 R 1244 0 V -1231 14 R 1231 0 V -1219 14 R 1219 0 V -1206 14 R 1206 0 V -1194 15 R 1194 0 V -1181 14 R 1181 0 V -1168 14 R 1168 0 V -1156 14 R 1156 0 V -1143 15 R 1143 0 V -1131 14 R 1131 0 V -1118 14 R 1118 0 V -1105 14 R 1105 0 V -1093 15 R 1093 0 V -1080 14 R 1080 0 V -1067 14 R 1067 0 V -1054 14 R 1054 0 V -1042 15 R 1042 0 V -1029 14 R 1029 0 V -1016 14 R 1016 0 V -1003 14 R 1003 0 V -990 15 R 990 0 V -978 14 R 978 0 V -965 14 R 965 0 V -952 14 R 952 0 V -939 15 R 939 0 V -926 14 R 926 0 V -913 14 R 913 0 V -913 0 R 913 0 V 3.000 UL LT1 1918 480 M -7 14 R 15 0 V -22 14 R 29 0 V -37 15 R 45 0 V -52 14 R 59 0 V -66 14 R 74 0 V -82 14 R 90 0 V -97 15 R 104 0 V -111 14 R 119 0 V -126 14 R 134 0 V -141 14 R 148 0 V -156 15 R 164 0 V -171 14 R 179 0 V -186 14 R 194 0 V -201 14 R 208 0 V -215 15 R 223 0 V -230 14 R 238 0 V -245 14 R 253 0 V -260 14 R 268 0 V -274 15 R 282 0 V -289 14 R 297 0 V -304 14 R 312 0 V -319 14 R 327 0 V -334 15 R 342 0 V -348 14 R 356 0 V -363 14 R 371 0 V -378 14 R 386 0 V -392 14 R 400 0 V -407 15 R 416 0 V -422 14 R 430 0 V -437 14 R 445 0 V -451 14 R 459 0 V -466 15 R 475 0 V -481 14 R 489 0 V -495 14 R 503 0 V -510 14 R 519 0 V -525 15 R 533 0 V -539 14 R 547 0 V -554 14 R 563 0 V -569 14 R 577 0 V -583 15 R 591 0 V -597 14 R 606 0 V -612 14 R 620 0 V -626 14 R 635 0 V -641 15 R 649 0 V -655 14 R 664 0 V -670 14 R 679 0 V -685 14 R 693 0 V -699 15 R 708 0 V -714 14 R 722 0 V -728 14 R 737 0 V -743 14 R 752 0 V -757 14 R 765 0 V -771 15 R 780 0 V -786 14 R 795 0 V -800 14 R 808 0 V -814 14 R 823 0 V -829 15 R 838 0 V -843 14 R 852 0 V -858 14 R 866 0 V -871 14 R 880 0 V -886 15 R 895 0 V -900 14 R 909 0 V -914 14 R 923 0 V -929 14 R 938 0 V -943 15 R 951 0 V -956 14 R 965 0 V -971 14 R 980 0 V -985 14 R 994 0 V -999 15 R 1008 0 V -1013 14 R 1022 0 V -1027 14 R 1036 0 V -1041 14 R 1050 0 V -1055 15 R 1064 0 V -1069 14 R 1078 0 V -1083 14 R 1092 0 V -1097 14 R 1106 0 V -1111 14 R 1120 0 V -1125 15 R 1134 0 V -1139 14 R 1148 0 V -1153 14 R 1162 0 V -1166 14 R 1175 0 V -1180 15 R 1189 0 V -1194 14 R 1203 0 V -1207 14 R 1216 0 V -1221 14 R 1230 0 V -1235 15 R 1244 0 V -1248 14 R 1257 0 V -1262 14 R 1271 0 V -1275 14 R 1284 0 V -1289 15 R 1299 0 V -1303 14 R 1312 0 V -1317 14 R 1326 0 V -1330 14 R 1339 0 V -1343 15 R 1352 0 V -1356 14 R 1365 0 V -1370 14 R 1379 0 V -1383 14 R 1392 0 V -1396 15 R 1406 0 V -1410 14 R 1419 0 V -1423 14 R 1432 0 V -1437 14 R 1446 0 V -1450 14 R 1459 0 V -1463 15 R 1472 0 V -1476 14 R 1485 0 V -1489 14 R 1499 0 V -1502 14 R 1511 0 V -1515 15 R 1524 0 V -1528 14 R 1537 0 V -1541 14 R 1550 0 V -1554 14 R 1563 0 V -1567 15 R 1576 0 V -1579 14 R 1589 0 V -1593 14 R 1602 0 V -1606 14 R 1615 0 V -1618 15 R 1627 0 V -1631 14 R 1640 0 V -1643 14 R 1652 0 V -1656 14 R 1665 0 V -1668 15 R 1677 0 V -1681 14 R 1691 0 V -1694 14 R 1703 0 V -1707 14 R 1716 0 V -1719 15 R 1728 0 V -1731 14 R 1740 0 V -1744 14 R 1753 0 V -1756 14 R 1765 0 V -1768 14 R 1777 0 V -1780 15 R 1789 0 V -1792 14 R 1801 0 V -1805 14 R 1814 0 V -1817 14 R 1826 0 V -1829 15 R 1838 0 V -1841 14 R 1850 0 V -1853 14 R 1862 0 V -1865 14 R 1874 0 V -1876 15 R 1885 0 V -1888 14 R 1897 0 V -1900 14 R 1909 0 V -1912 14 R 1921 0 V -1924 15 R 1933 0 V -1935 14 R 1944 0 V -1947 14 R 1956 0 V -1959 14 R 1968 0 V -1970 15 R 1979 0 V -1982 14 R 1991 0 V -1993 14 R 2002 0 V -2005 14 R 2013 0 V -2015 15 R 2024 0 V -2027 14 R 2036 0 V -2038 14 R 2047 0 V -2050 14 R 2059 0 V -2061 14 R 2069 0 V -2071 15 R 2080 0 V -2082 14 R 2091 0 V -2093 14 R 2102 0 V -2105 14 R 2113 0 V -2115 15 R 2124 0 V -2126 14 R 2135 0 V -2137 14 R 2145 0 V -2147 14 R 2156 0 V -2158 15 R 2166 0 V -2167 14 R 2176 0 V -2178 14 R 2187 0 V -2189 14 R 2197 0 V -2199 15 R 2208 0 V -2209 14 R 2217 0 V -2219 14 R 2228 0 V -2230 14 R 2238 0 V -2239 15 R 2247 0 V -2249 14 R 2258 0 V -2259 14 R 2267 0 V -2269 14 R 2277 0 V -2278 15 R 2287 0 V -2288 14 R 2296 0 V -2297 14 R 2305 0 V -2307 14 R 2315 0 V -2316 14 R 2325 0 V -2326 15 R 2334 0 V -2335 14 R 2343 0 V -2344 14 R 2352 0 V -2353 14 R 2361 0 V -2361 15 R 2369 0 V -2370 14 R 2378 0 V -2379 14 R 2387 0 V -2387 14 R 2395 0 V -2396 15 R 2404 0 V -2405 14 R 2412 0 V -2412 14 R 2420 0 V -2420 14 R 2428 0 V -2429 15 R 2436 0 V -2436 14 R 2444 0 V -2444 14 R 2452 0 V -2452 14 R 2459 0 V -2459 15 R 2467 0 V -2467 14 R 2474 0 V -2474 14 R 2481 0 V -2480 14 R 2488 0 V -2488 15 R 2495 0 V -2494 14 R 2501 0 V -2501 14 R 2508 0 V currentpoint stroke M -2507 14 R 2514 0 V -2513 14 R 2520 0 V -2519 15 R 2525 0 V -2524 14 R 2531 0 V -2530 14 R 2537 0 V -2536 14 R 2542 0 V -2540 15 R 2547 0 V -2546 14 R 2552 0 V -2550 14 R 2557 0 V -2556 14 R 2562 0 V -2560 15 R 2566 0 V -2564 14 R 2571 0 V -2570 14 R 2576 0 V -2574 14 R 2580 0 V -2578 15 R 2584 0 V -2582 14 R 2588 0 V -2586 14 R 2593 0 V -2591 14 R 2597 0 V -2595 15 R 2601 0 V -2599 14 R 2605 0 V -2603 14 R 2609 0 V -2606 14 R 2612 0 V -2610 15 R 2616 0 V -2614 14 R 2619 0 V -2617 14 R 2623 0 V -2620 14 R 2626 0 V -2624 14 R 2630 0 V -2627 15 R 2633 0 V -2631 14 R 2637 0 V -2634 14 R 2639 0 V -2637 14 R 2643 0 V -2640 15 R 2646 0 V -2643 14 R 2649 0 V -2647 14 R 2652 0 V -2649 14 R 2655 0 V -2652 15 R 2658 0 V -2656 14 R 2661 0 V -2658 14 R 2664 0 V -2661 14 R 2667 0 V -2664 15 R 2669 0 V -2666 14 R 2672 0 V -2669 14 R 2674 0 V -2671 14 R 2677 0 V -2674 15 R 2679 0 V -2676 14 R 2682 0 V -2679 14 R 2684 0 V -2681 14 R 2687 0 V -2684 15 R 2689 0 V -2686 14 R 2692 0 V -2689 14 R 2694 0 V -2691 14 R 2697 0 V -2694 14 R 2699 0 V -2696 15 R 2702 0 V -2698 14 R 2703 0 V -2700 14 R 2706 0 V -2703 14 R 2708 0 V -2705 15 R 2710 0 V -2706 14 R 2712 0 V -2709 14 R 2714 0 V -2711 14 R 2716 0 V -2712 15 R 2718 0 V -2715 14 R 2720 0 V -2717 14 R 2723 0 V -2719 14 R 2724 0 V -2721 15 R 2726 0 V -2723 14 R 2729 0 V -2725 14 R 2730 0 V -2727 14 R 2732 0 V -2728 15 R 2734 0 V -2731 14 R 2736 0 V -2732 14 R 2737 0 V -2734 14 R 2739 0 V -2735 15 R 2741 0 V -2738 14 R 2743 0 V -2739 14 R 2744 0 V -2740 14 R 2745 0 V -2742 14 R 2748 0 V -2744 15 R 2749 0 V -2746 14 R 2751 0 V -2747 14 R 2752 0 V -2748 14 R 2754 0 V -2751 15 R 2756 0 V -2752 14 R 2757 0 V -2753 14 R 2758 0 V -2755 14 R 2761 0 V -2757 15 R 2762 0 V -2758 14 R 2763 0 V -2759 14 R 2764 0 V -2761 14 R 2766 0 V -2762 15 R 2768 0 V -2764 14 R 2769 0 V -2765 14 R 2770 0 V -2766 14 R 2771 0 V -2768 15 R 2773 0 V -2769 14 R 2775 0 V -2771 14 R 2776 0 V -2772 14 R 2777 0 V -2773 15 R 2778 0 V -2774 14 R 2779 0 V -2776 14 R 2782 0 V 3.000 UL LT1 3035 480 M -1 0 V 1 14 R -1 0 V 1 14 R 0 15 R 0 14 R 1 14 R -1 0 V 1 14 R -1 0 V 2 15 R -1 0 V 1 14 R -1 0 V 2 14 R -2 0 V 3 14 R -2 0 V 3 15 R -3 0 V 4 14 R -3 0 V 4 14 R -4 0 V 5 14 R -4 0 V 5 15 R -5 0 V 7 14 R -6 0 V 7 14 R -6 0 V 8 14 R -7 0 V 8 15 R -7 0 V 9 14 R -9 0 V 11 14 R -10 0 V 12 14 R -11 0 V 13 15 R -12 0 V 14 14 R -13 0 V 15 14 R -14 0 V 16 14 R -14 0 V 16 14 R -15 0 V 18 15 R -17 0 V 19 14 R -18 0 V 21 14 R -19 0 V 21 14 R -20 0 V 23 15 R -22 0 V 25 14 R -23 0 V 26 14 R -25 0 V 28 14 R -26 0 V 29 15 R -27 0 V 30 14 R -29 0 V 32 14 R -30 0 V 33 14 R -31 0 V 35 15 R -34 0 V 37 14 R -35 0 V 39 14 R -37 0 V 40 14 R -38 0 V 42 15 R -40 0 V 44 14 R -42 0 V 45 14 R -43 0 V 47 14 R -45 0 V 49 15 R -47 0 V 51 14 R -49 0 V 54 14 R -52 0 V 56 14 R -54 0 V 58 14 R -55 0 V 59 15 R -57 0 V 62 14 R -60 0 V 64 14 R -61 0 V 66 14 R -64 0 V 69 15 R -66 0 V 70 14 R -68 0 V 73 14 R -70 0 V 75 14 R -73 0 V 78 15 R -75 0 V 80 14 R -77 0 V 82 14 R -79 0 V 84 14 R -82 0 V 87 15 R -84 0 V 90 14 R -87 0 V 92 14 R -89 0 V 95 14 R -92 0 V 97 15 R -94 0 V 100 14 R -97 0 V 102 14 R -99 0 V 105 14 R -102 0 V 108 15 R -105 0 V 110 14 R -107 0 V 113 14 R -110 0 V 116 14 R -112 0 V 118 14 R -115 0 V 121 15 R -118 0 V 124 14 R -120 0 V 127 14 R -124 0 V 130 14 R -127 0 V 133 15 R -129 0 V 136 14 R -133 0 V 139 14 R -135 0 V 141 14 R -137 0 V 144 15 R -141 0 V 148 14 R -144 0 V 150 14 R -146 0 V 153 14 R -150 0 V 157 15 R -153 0 V 159 14 R -155 0 V 162 14 R -158 0 V 165 14 R -162 0 V 169 15 R -165 0 V 172 14 R -168 0 V 175 14 R -171 0 V 178 14 R -174 0 V 182 15 R -178 0 V 185 14 R -181 0 V 188 14 R -184 0 V 191 14 R -186 0 V 194 14 R -190 0 V 197 15 R -193 0 V 201 14 R -197 0 V 204 14 R -200 0 V 208 14 R -203 0 V 210 15 R -206 0 V 214 14 R -210 0 V 217 14 R -212 0 V 220 14 R -216 0 V 224 15 R -219 0 V 227 14 R -223 0 V 231 14 R -226 0 V 233 14 R -229 0 V 237 15 R -232 0 V 240 14 R -236 0 V 244 14 R -239 0 V 247 14 R -242 0 V 250 15 R -246 0 V 255 14 R -250 0 V 258 14 R -253 0 V 261 14 R -256 0 V 264 15 R -260 0 V 268 14 R -263 0 V 272 14 R -267 0 V 275 14 R -270 0 V 278 14 R -273 0 V 282 15 R -277 0 V 285 14 R -280 0 V 288 14 R -283 0 V 292 14 R -287 0 V 295 15 R -290 0 V 299 14 R -294 0 V 302 14 R -297 0 V 306 14 R -301 0 V 310 15 R -305 0 V 313 14 R -308 0 V 317 14 R -312 0 V 321 14 R -315 0 V 323 15 R -318 0 V 327 14 R -322 0 V 331 14 R -326 0 V 335 14 R -329 0 V 337 15 R -332 0 V 341 14 R -336 0 V 345 14 R -339 0 V 348 14 R -343 0 V 352 15 R -347 0 V 356 14 R -350 0 V 359 14 R -354 0 V 362 14 R -356 0 V 365 14 R -360 0 V 369 15 R -363 0 V 372 14 R -366 0 V 375 14 R -370 0 V 379 14 R -373 0 V 382 15 R -377 0 V 386 14 R -380 0 V 389 14 R -383 0 V 393 14 R -388 0 V 397 15 R -391 0 V 400 14 R -394 0 V 403 14 R -397 0 V 406 14 R -400 0 V 409 15 R -404 0 V 413 14 R -407 0 V 416 14 R -410 0 V 419 14 R -413 0 V 423 15 R -417 0 V 426 14 R -420 0 V 429 14 R -423 0 V 432 14 R -426 0 V 435 15 R -429 0 V 438 14 R -432 0 V 442 14 R -436 0 V 445 14 R -438 0 V 447 14 R -441 0 V 450 15 R -444 0 V 453 14 R -447 0 V 456 14 R -449 0 V 459 14 R -453 0 V 462 15 R -456 0 V 465 14 R -458 0 V 467 14 R -461 0 V 470 14 R -463 0 V 472 15 R -466 0 V 475 14 R -468 0 V 477 14 R -471 0 V 480 14 R -473 0 V 483 15 R -476 0 V 485 14 R -479 0 V 488 14 R -481 0 V 490 14 R -483 0 V 492 15 R -485 0 V 493 14 R -486 0 V 495 14 R -488 0 V 497 14 R -490 0 V 499 15 R -492 0 V 501 14 R -494 0 V 502 14 R -494 0 V 503 14 R -496 0 V currentpoint stroke M 505 14 R -497 0 V 505 15 R -497 0 V 506 14 R -499 0 V 507 14 R -499 0 V 507 14 R -499 0 V 508 15 R -500 0 V 508 14 R -500 0 V 508 14 R -500 0 V 509 14 R -501 0 V 509 15 R -501 0 V 509 14 R -501 0 V 509 14 R -501 0 V 509 14 R -501 0 V 510 15 R -502 0 V 510 14 R -501 0 V 509 14 R -501 0 V 509 14 R -501 0 V 509 15 R -500 0 V 509 14 R -501 0 V 509 14 R -500 0 V 508 14 R -500 0 V 508 15 R -500 0 V 508 14 R -499 0 V 507 14 R -499 0 V 507 14 R -498 0 V 507 14 R -498 0 V 506 15 R -498 0 V 506 14 R -497 0 V 505 14 R -497 0 V 505 14 R -496 0 V 504 15 R -495 0 V 503 14 R -495 0 V 504 14 R -495 0 V 503 14 R -494 0 V 502 15 R -493 0 V 501 14 R -493 0 V 501 14 R -492 0 V 500 14 R -491 0 V 499 15 R -490 0 V 499 14 R -491 0 V 499 14 R -490 0 V 498 14 R -489 0 V 497 15 R -488 0 V 496 14 R -487 0 V 496 14 R -487 0 V 495 14 R -486 0 V 494 15 R -485 0 V 493 14 R -484 0 V 492 14 R -483 0 V 492 14 R -483 0 V 491 14 R -482 0 V 490 15 R -482 0 V 490 14 R -481 0 V 490 14 R -480 0 V 488 14 R -479 0 V 487 15 R -478 0 V 486 14 R -477 0 V 486 14 R -477 0 V 485 14 R -476 0 V 484 15 R -475 0 V 484 14 R -475 0 V 483 14 R -474 0 V 482 14 R -473 0 V 482 15 R -473 0 V 481 14 R -472 0 V 480 14 R -470 0 V 479 14 R -470 0 V 478 15 R -469 0 V 477 14 R -468 0 V 477 14 R -468 0 V 476 14 R -467 0 V 475 15 R -465 0 V 474 14 R -465 0 V 473 14 R -464 0 V 473 14 R -464 0 V 472 14 R -462 0 V 470 15 R -461 0 V 470 14 R -461 0 V 469 14 R -460 0 V 469 14 R -459 0 V 467 15 R -458 0 V 467 14 R -458 0 V 466 14 R -456 0 V 465 14 R -456 0 V 464 15 R -455 0 V 463 14 R -453 0 V 462 14 R -453 0 V 461 14 R -452 0 V 461 15 R -451 0 V 460 14 R -451 0 V 459 14 R -450 0 V 459 14 R -449 0 V 457 15 R -448 0 V 457 14 R -448 0 V 456 14 R -446 0 V 455 14 R -446 0 V 454 15 R -444 0 V 453 14 R -444 0 V 453 14 R -443 0 V 3.000 UL LT1 3509 480 M 0 14 R 0 14 R 0 15 R 0 14 R 0 14 R 1 14 R 0 15 R 0 14 R 1 14 R 0 14 R 1 15 R 0 14 R 1 14 R 1 14 R 0 15 R 1 0 V 0 14 R 1 14 R 1 14 R 1 15 R 1 14 R 0 14 R 1 0 V 1 14 R 1 15 R 1 14 R 1 0 V 0 14 R 1 0 V 0 14 R 1 0 V 0 14 R 1 0 V 1 15 R 1 0 V 0 14 R 1 0 V 0 14 R 2 0 V 0 14 R 2 0 V -1 15 R 2 0 V 0 14 R 2 0 V -1 14 R 3 0 V -1 14 R 2 0 V 0 15 R 2 0 V -1 14 R 3 0 V -1 14 R 3 0 V -1 14 R 3 0 V -1 15 R 3 0 V -1 14 R 3 0 V -2 14 R 5 0 V -3 14 R 5 0 V -3 15 R 5 0 V -3 14 R 6 0 V -3 14 R 5 0 V -3 14 R 6 0 V -4 15 R 6 0 V -4 14 R 7 0 V -5 14 R 7 0 V -4 14 R 7 0 V -5 14 R 8 0 V -6 15 R 9 0 V -6 14 R 9 0 V -7 14 R 9 0 V -6 14 R 10 0 V -8 15 R 11 0 V -8 14 R 11 0 V -9 14 R 12 0 V -9 14 R 12 0 V -9 15 R 12 0 V -9 14 R 13 0 V -11 14 R 14 0 V -11 14 R 15 0 V -12 15 R 15 0 V -12 14 R 16 0 V -13 14 R 16 0 V -13 14 R 17 0 V -14 15 R 18 0 V -15 14 R 19 0 V -16 14 R 20 0 V -17 14 R 20 0 V -17 15 R 22 0 V -18 14 R 22 0 V -19 14 R 23 0 V -20 14 R 24 0 V -20 14 R 24 0 V -21 15 R 25 0 V -21 14 R 26 0 V -23 14 R 27 0 V -24 14 R 29 0 V -25 15 R 29 0 V -25 14 R 30 0 V -27 14 R 31 0 V -27 14 R 32 0 V -28 15 R 33 0 V -30 14 R 35 0 V -31 14 R 36 0 V -32 14 R 37 0 V -33 15 R 38 0 V -34 14 R 39 0 V -35 14 R 40 0 V -36 14 R 41 0 V -37 15 R 42 0 V -38 14 R 43 0 V -39 14 R 45 0 V -41 14 R 46 0 V -42 15 R 48 0 V -44 14 R 49 0 V -44 14 R 50 0 V -46 14 R 51 0 V -47 14 R 53 0 V -48 15 R 54 0 V -50 14 R 56 0 V -51 14 R 57 0 V -53 14 R 59 0 V -54 15 R 60 0 V -56 14 R 62 0 V -57 14 R 63 0 V -58 14 R 64 0 V -60 15 R 66 0 V -61 14 R 67 0 V -62 14 R 69 0 V -64 14 R 70 0 V -66 15 R 73 0 V -68 14 R 74 0 V -69 14 R 76 0 V -71 14 R 77 0 V -72 15 R 79 0 V -74 14 R 81 0 V -76 14 R 82 0 V -76 14 R 83 0 V -78 15 R 85 0 V -80 14 R 87 0 V -82 14 R 89 0 V -83 14 R 90 0 V -85 14 R 92 0 V -87 15 R 94 0 V -88 14 R 96 0 V -91 14 R 98 0 V -92 14 R 99 0 V -94 15 R 102 0 V -96 14 R 103 0 V -98 14 R 105 0 V -99 14 R 107 0 V -101 15 R 108 0 V -102 14 R 110 0 V -105 14 R 113 0 V -107 14 R 114 0 V -108 15 R 116 0 V -110 14 R 118 0 V -112 14 R 120 0 V -114 14 R 122 0 V -116 15 R 124 0 V -118 14 R 126 0 V -120 14 R 128 0 V -122 14 R 130 0 V -124 15 R 132 0 V -125 14 R 133 0 V -127 14 R 135 0 V -129 14 R 138 0 V -131 14 R 139 0 V -133 15 R 141 0 V -135 14 R 144 0 V -137 14 R 145 0 V -139 14 R 148 0 V -141 15 R 149 0 V -142 14 R 151 0 V -145 14 R 153 0 V -146 14 R 155 0 V -148 15 R 157 0 V -151 14 R 159 0 V -152 14 R 161 0 V -154 14 R 163 0 V -156 15 R 165 0 V -158 14 R 167 0 V -160 14 R 168 0 V -161 14 R 170 0 V -163 15 R 172 0 V -165 14 R 174 0 V -167 14 R 176 0 V -169 14 R 178 0 V -171 15 R 180 0 V -172 14 R 182 0 V -175 14 R 184 0 V -177 14 R 186 0 V -178 14 R 187 0 V -180 15 R 189 0 V -181 14 R 190 0 V -183 14 R 193 0 V -185 14 R 194 0 V -187 15 R 196 0 V -188 14 R 198 0 V -190 14 R 199 0 V -192 14 R 201 0 V -193 15 R 203 0 V -195 14 R 204 0 V -196 14 R 205 0 V -197 14 R 207 0 V -199 15 R 208 0 V -200 14 R 210 0 V -202 14 R 211 0 V -203 14 R 213 0 V -204 15 R 213 0 V -205 14 R 215 0 V -207 14 R 216 0 V -207 14 R 216 0 V -208 15 R 218 0 V -209 14 R 218 0 V -209 14 R 219 0 V -210 14 R 219 0 V -210 14 R 219 0 V -210 15 R 220 0 V -211 14 R 220 0 V -211 14 R 220 0 V -211 14 R 220 0 V -210 15 R 220 0 V -211 14 R 220 0 V -211 14 R 220 0 V -210 14 R 220 0 V currentpoint stroke M -210 15 R 219 0 V -210 14 R 219 0 V -209 14 R 218 0 V -209 14 R 219 0 V -209 15 R 218 0 V -208 14 R 217 0 V -207 14 R 217 0 V -207 14 R 216 0 V -206 15 R 215 0 V -206 14 R 216 0 V -206 14 R 215 0 V -205 14 R 215 0 V -204 15 R 213 0 V -203 14 R 213 0 V -203 14 R 212 0 V -202 14 R 212 0 V -202 14 R 211 0 V -201 15 R 211 0 V -201 14 R 210 0 V -199 14 R 209 0 V -199 14 R 208 0 V -198 15 R 208 0 V -197 14 R 206 0 V -196 14 R 206 0 V -196 14 R 206 0 V -195 15 R 204 0 V -194 14 R 204 0 V -193 14 R 203 0 V -193 14 R 203 0 V -192 15 R 201 0 V -191 14 R 201 0 V -190 14 R 200 0 V -190 14 R 200 0 V -189 15 R 199 0 V -188 14 R 197 0 V -187 14 R 197 0 V -186 14 R 196 0 V -185 15 R 195 0 V -185 14 R 195 0 V -184 14 R 194 0 V -183 14 R 193 0 V -182 14 R 192 0 V -182 15 R 192 0 V -181 14 R 191 0 V -180 14 R 190 0 V -179 14 R 189 0 V -178 15 R 188 0 V -177 14 R 187 0 V -176 14 R 186 0 V -175 14 R 186 0 V -175 15 R 185 0 V -174 14 R 184 0 V -174 14 R 184 0 V -172 14 R 182 0 V -171 15 R 181 0 V -170 14 R 181 0 V -170 14 R 180 0 V -169 14 R 179 0 V -168 15 R 179 0 V -168 14 R 178 0 V -167 14 R 177 0 V -166 14 R 177 0 V -166 15 R 176 0 V -164 14 R 174 0 V -163 14 R 174 0 V -163 14 R 173 0 V -162 14 R 173 0 V -162 15 R 172 0 V -160 14 R 171 0 V -160 14 R 170 0 V -159 14 R 170 0 V -159 15 R 169 0 V -157 14 R 168 0 V -157 14 R 167 0 V -156 14 R 167 0 V -155 15 R 166 0 V -155 14 R 165 0 V -153 14 R 164 0 V -153 14 R 163 0 V -152 15 R 163 0 V -151 14 R 162 0 V -151 14 R 162 0 V -150 14 R 160 0 V -149 15 R 160 0 V -148 14 R 159 0 V -148 14 R 158 0 V -146 14 R 157 0 V -146 15 R 157 0 V -145 14 R 156 0 V -145 14 R 156 0 V 3.000 UL LT1 1100 480 M 0 14 R 0 14 R 0 15 R 0 14 R 0 14 R 0 14 R 0 15 R -1 0 V 0 14 R 0 14 R 0 14 R 0 15 R -1 14 R 0 14 R 0 14 R -1 15 R 0 14 R 0 14 R -1 0 V 0 14 R 0 15 R -1 0 V 0 14 R 0 14 R -1 0 V 0 14 R 0 15 R -1 0 V 0 14 R -1 14 R 0 14 R -1 0 V 0 14 R -1 15 R 0 14 R -1 0 V 0 14 R -1 14 R 0 15 R -1 0 V 0 14 R -1 0 V 0 14 R -1 14 R -1 15 R 0 14 R -1 0 V 0 14 R -1 0 V 0 14 R -1 0 V 0 15 R -1 0 V 0 14 R -1 0 V 0 14 R -1 0 V 0 14 R -1 0 V 1 15 R -2 0 V 1 14 R -2 0 V 1 14 R -2 0 V 1 14 R -2 0 V 1 15 R -2 0 V 1 14 R -2 0 V 1 14 R -2 0 V 1 14 R -2 0 V 1 14 R -2 0 V 1 15 R -3 0 V 2 14 R -3 0 V 2 14 R -3 0 V 2 14 R -3 0 V 2 15 R -3 0 V 2 14 R -4 0 V 3 14 R -4 0 V 3 14 R -4 0 V 3 15 R -5 0 V 4 14 R -5 0 V 4 14 R -5 0 V 4 14 R -6 0 V 5 15 R -6 0 V 5 14 R -6 0 V 5 14 R -7 0 V 6 14 R -7 0 V 6 15 R -7 0 V 6 14 R -8 0 V 7 14 R -8 0 V 7 14 R -9 0 V 8 15 R -9 0 V 8 14 R -9 0 V 8 14 R -10 0 V 9 14 R -10 0 V 9 14 R -11 0 V 10 15 R -11 0 V 10 14 R -12 0 V 11 14 R -12 0 V 11 14 R -13 0 V 12 15 R -13 0 V 12 14 R -14 0 V 13 14 R -14 0 V 13 14 R -15 0 V 14 15 R -15 0 V 14 14 R -16 0 V 15 14 R -17 0 V 16 14 R -17 0 V 17 15 R -19 0 V 18 14 R -19 0 V 18 14 R -20 0 V 19 14 R -21 0 V 20 15 R -21 0 V 20 14 R -22 0 V 22 14 R -23 0 V 22 14 R -24 0 V 23 15 R -25 0 V 24 14 R -25 0 V 24 14 R -26 0 V 26 14 R -28 0 V 27 14 R -28 0 V 27 15 R -29 0 V 29 14 R -31 0 V 30 14 R -31 0 V 30 14 R -32 0 V 32 15 R -34 0 V 33 14 R -34 0 V 33 14 R -35 0 V 35 14 R -37 0 V 36 15 R -37 0 V 37 14 R -39 0 V 38 14 R -40 0 V 39 14 R -40 0 V 40 15 R -42 0 V 41 14 R -43 0 V 43 14 R -44 0 V 44 14 R -46 0 V 45 15 R -47 0 V 47 14 R -48 0 V 47 14 R -49 0 V 49 14 R -51 0 V 51 15 R -53 0 V 52 14 R -53 0 V 53 14 R -55 0 V 55 14 R -57 0 V 56 14 R -57 0 V 57 15 R -59 0 V 59 14 R -61 0 V 61 14 R -62 0 V 61 14 R -63 0 V 63 15 R -65 0 V 65 14 R -67 0 V 67 14 R -68 0 V 68 14 R -70 0 V 70 15 R -72 0 V 72 14 R -73 0 V 73 14 R -75 0 V 75 14 R -77 0 V 77 15 R -78 0 V 78 14 R -80 0 V 80 14 R -82 0 V 82 14 R -83 0 V 83 15 R -85 0 V 85 14 R -87 0 V 87 14 R -88 0 V 88 14 R -90 0 V 90 15 R -91 0 V 92 14 R -94 0 V 94 14 R -96 0 V 96 14 R -97 0 V 97 14 R -99 0 V 100 15 R -101 0 V 101 14 R -103 0 V 103 14 R -104 0 V 105 14 R -107 0 V 107 15 R -108 0 V 108 14 R -110 0 V 111 14 R -112 0 V 112 14 R -114 0 V 115 15 R -116 0 V 116 14 R -118 0 V 119 14 R -120 0 V 120 14 R -122 0 V 123 15 R -124 0 V 124 14 R -125 0 V 126 14 R -128 0 V 129 14 R -130 0 V 130 15 R -131 0 V 132 14 R -134 0 V 135 14 R -136 0 V 136 14 R -137 0 V 138 15 R -140 0 V 141 14 R -142 0 V 143 14 R -144 0 V 144 14 R -145 0 V 146 14 R -147 0 V 148 15 R -149 0 V 150 14 R -152 0 V 152 14 R -153 0 V 154 14 R -155 0 V 156 15 R -157 0 V 158 14 R -159 0 V 160 14 R -161 0 V 162 14 R -163 0 V 163 15 R -164 0 V 165 14 R -165 0 V 166 14 R -167 0 V 168 14 R -169 0 V 170 15 R -171 0 V 172 14 R -173 0 V 174 14 R -174 0 V 174 14 R -175 0 V 176 15 R -176 0 V 177 14 R -178 0 V 179 14 R -179 0 V 179 14 R -180 0 V 181 15 R -181 0 V 182 14 R -182 0 V 182 14 R -182 0 V 183 14 R -183 0 V 183 14 R -183 0 V 183 15 R -182 0 V 183 14 R -183 0 V 183 14 R -182 0 V 182 14 R -182 0 V 182 15 R -181 0 V 182 14 R -181 0 V 181 14 R -181 0 V 181 14 R -180 0 V 180 15 R -179 0 V 179 14 R -178 0 V 179 14 R -178 0 V currentpoint stroke M 178 14 R -178 0 V 178 15 R -177 0 V 177 14 R -176 0 V 176 14 R -175 0 V 175 14 R -174 0 V 175 15 R -174 0 V 174 14 R -172 0 V 172 14 R -171 0 V 171 14 R -170 0 V 170 15 R -169 0 V 170 14 R -169 0 V 169 14 R -168 0 V 168 14 R -167 0 V 167 14 R -165 0 V 165 15 R -164 0 V 165 14 R -164 0 V 164 14 R -163 0 V 163 14 R -161 0 V 162 15 R -161 0 V 161 14 R -160 0 V 160 14 R -158 0 V 158 14 R -157 0 V 158 15 R -157 0 V 157 14 R -155 0 V 156 14 R -155 0 V 155 14 R -154 0 V 154 15 R -152 0 V 153 14 R -152 0 V 152 14 R -151 0 V 151 14 R -149 0 V 150 15 R -149 0 V 149 14 R -147 0 V 148 14 R -147 0 V 147 14 R -145 0 V 146 15 R -145 0 V 145 14 R -144 0 V 145 14 R -143 0 V 143 14 R -142 0 V 143 14 R -141 0 V 141 15 R -140 0 V 141 14 R -139 0 V 139 14 R -138 0 V 139 14 R -137 0 V 137 15 R -136 0 V 137 14 R -135 0 V 136 14 R -135 0 V 135 14 R -133 0 V 134 15 R -132 0 V 132 14 R -131 0 V 132 14 R -130 0 V 131 14 R -130 0 V 130 15 R -128 0 V 129 14 R -128 0 V 129 14 R -127 0 V 127 14 R -126 0 V 127 15 R -125 0 V 126 14 R -124 0 V 124 14 R -123 0 V 124 14 R -122 0 V 123 15 R -122 0 V 123 14 R -121 0 V 121 14 R -120 0 V 121 14 R -119 0 V 120 14 R -118 0 V 119 15 R -118 0 V 118 14 R -116 0 V 117 14 R -116 0 V 117 14 R -115 0 V 116 15 R -114 0 V 115 14 R -114 0 V 114 14 R -112 0 V 113 14 R -112 0 V 113 15 R -111 0 V 112 14 R -110 0 V 111 14 R -110 0 V 111 14 R -109 0 V 110 15 R -108 0 V 108 14 R -107 0 V 108 14 R -106 0 V 107 14 R -106 0 V 107 15 R -105 0 V 106 14 R -104 0 V 105 14 R -104 0 V 105 14 R -103 0 V 104 15 R -102 0 V 103 14 R -102 0 V 103 14 R -101 0 V 3.000 UL LT1 3697 480 M 0 14 R 1 14 R -1 0 V 1 15 R 0 14 R 0 14 R 0 14 R 1 15 R 0 14 R 0 14 R 1 14 R 0 15 R 1 14 R 0 14 R 1 14 R 1 15 R -1 0 V 1 14 R 1 14 R 1 14 R 1 15 R 0 14 R 1 14 R 1 14 R 1 15 R 1 14 R 2 14 R 1 14 R 1 14 R 1 15 R 1 14 R 2 14 R 1 14 R 2 15 R 1 14 R 2 14 R -1 0 V 2 14 R 2 15 R 1 14 R 2 14 R 2 14 R 2 15 R 1 14 R 2 14 R 2 14 R 2 15 R 2 14 R 2 14 R 2 14 R 3 15 R -1 0 V 3 14 R 2 14 R 2 14 R 3 14 R -1 0 V 3 15 R 2 14 R 3 14 R 2 14 R 3 15 R 3 14 R -1 0 V 3 14 R 3 14 R 3 15 R -1 0 V 3 14 R 3 14 R 3 14 R 3 15 R 3 14 R 3 14 R 3 14 R 3 15 R 4 14 R -1 0 V 4 14 R -1 0 V 4 14 R -1 0 V 4 15 R -1 0 V 5 14 R -1 0 V 4 14 R -1 0 V 5 14 R -2 0 V 5 14 R -1 0 V 5 15 R -2 0 V 5 14 R -1 0 V 5 14 R -1 0 V 5 14 R -2 0 V 5 15 R -1 0 V 5 14 R -2 0 V 6 14 R -2 0 V 6 14 R -2 0 V 6 15 R -2 0 V 6 14 R -2 0 V 6 14 R -2 0 V 6 14 R -2 0 V 6 15 R -2 0 V 6 14 R -2 0 V 7 14 R -3 0 V 7 14 R -3 0 V 7 15 R -3 0 V 8 14 R -4 0 V 8 14 R -3 0 V 8 14 R -4 0 V 8 15 R -4 0 V 9 14 R -4 0 V 8 14 R -4 0 V 9 14 R -4 0 V 9 14 R -5 0 V 10 15 R -5 0 V 10 14 R -5 0 V 10 14 R -6 0 V 11 14 R -6 0 V 11 15 R -6 0 V 11 14 R -6 0 V 11 14 R -6 0 V 11 14 R -7 0 V 12 15 R -7 0 V 12 14 R -7 0 V 13 14 R -8 0 V 13 14 R -7 0 V 13 15 R -8 0 V 13 14 R -8 0 V 14 14 R -9 0 V 14 14 R -9 0 V 15 15 R -9 0 V 15 14 R -10 0 V 16 14 R -10 0 V 15 14 R -10 0 V 16 15 R -11 0 V 17 14 R -11 0 V 17 14 R -11 0 V 17 14 R -12 0 V 18 14 R -12 0 V 19 15 R -13 0 V 19 14 R -13 0 V 19 14 R -14 0 V 21 14 R -15 0 V 21 15 R -15 0 V 21 14 R -15 0 V 22 14 R -16 0 V 23 14 R -17 0 V 23 15 R -17 0 V 24 14 R -17 0 V 24 14 R -18 0 V 24 14 R -18 0 V 25 15 R -19 0 V 26 14 R -19 0 V 26 14 R -20 0 V 27 14 R -20 0 V 27 15 R -21 0 V 28 14 R -21 0 V 29 14 R -23 0 V 30 14 R -23 0 V 30 15 R -23 0 V 31 14 R -25 0 V 32 14 R -25 0 V 33 14 R -26 0 V 33 14 R -26 0 V 34 15 R -27 0 V 34 14 R -27 0 V 35 14 R -28 0 V 36 14 R -29 0 V 37 15 R -30 0 V 38 14 R -31 0 V 39 14 R -32 0 V 40 14 R -32 0 V 40 15 R -33 0 V 41 14 R -34 0 V 42 14 R -34 0 V 42 14 R -35 0 V 43 15 R -35 0 V 44 14 R -37 0 V 45 14 R -37 0 V 45 14 R -38 0 V 47 15 R -39 0 V 47 14 R -39 0 V 48 14 R -40 0 V 49 14 R -42 0 V 50 15 R -42 0 V 51 14 R -43 0 V 52 14 R -44 0 V 53 14 R -45 0 V 53 14 R -45 0 V 54 15 R -46 0 V 55 14 R -47 0 V 56 14 R -48 0 V 57 14 R -48 0 V 57 15 R -49 0 V 59 14 R -51 0 V 60 14 R -51 0 V 60 14 R -52 0 V 61 15 R -53 0 V 62 14 R -53 0 V 63 14 R -55 0 V 64 14 R -55 0 V 64 15 R -55 0 V 65 14 R -57 0 V 66 14 R -57 0 V 67 14 R -58 0 V 67 15 R -58 0 V 68 14 R -59 0 V 68 14 R -59 0 V 69 14 R -60 0 V 69 15 R -60 0 V 70 14 R -60 0 V 70 14 R -61 0 V 70 14 R -61 0 V 71 14 R -61 0 V 70 15 R -60 0 V 70 14 R -61 0 V 70 14 R -60 0 V 70 14 R -60 0 V 70 15 R -60 0 V 69 14 R -59 0 V 69 14 R -59 0 V 69 14 R -59 0 V 68 15 R -58 0 V 68 14 R -58 0 V 68 14 R -58 0 V 68 14 R -58 0 V 68 15 R -57 0 V 66 14 R -56 0 V 66 14 R -56 0 V 66 14 R -55 0 V 65 15 R -55 0 V 65 14 R -55 0 V 65 14 R -54 0 V 64 14 R -54 0 V 64 15 R -53 0 V 63 14 R -52 0 V 63 14 R -53 0 V 63 14 R -52 0 V 62 14 R -51 0 V 61 15 R -51 0 V 61 14 R -50 0 V 61 14 R -50 0 V 60 14 R -49 0 V currentpoint stroke M 59 15 R -49 0 V 60 14 R -49 0 V 59 14 R -48 0 V 59 14 R -48 0 V 58 15 R -47 0 V 58 14 R -47 0 V 57 14 R -46 0 V 57 14 R -46 0 V 56 15 R -45 0 V 56 14 R -45 0 V 56 14 R -44 0 V 54 14 R -43 0 V 54 15 R -43 0 V 54 14 R -43 0 V 54 14 R -43 0 V 54 14 R -42 0 V 52 15 R -41 0 V 52 14 R -41 0 V 52 14 R -40 0 V 51 14 R -40 0 V 51 14 R -40 0 V 51 15 R -39 0 V 50 14 R -39 0 V 50 14 R -38 0 V 49 14 R -38 0 V 49 15 R -37 0 V 48 14 R -37 0 V 48 14 R -36 0 V 48 14 R -37 0 V 48 15 R -36 0 V 47 14 R -36 0 V 47 14 R -35 0 V 47 14 R -35 0 V 46 15 R -35 0 V 46 14 R -34 0 V 46 14 R -34 0 V 45 14 R -33 0 V 44 15 R -33 0 V 45 14 R -33 0 V 44 14 R -32 0 V 44 14 R -32 0 V 43 15 R -32 0 V 44 14 R -32 0 V 43 14 R -31 0 V 43 14 R -31 0 V 42 14 R -30 0 V 42 15 R -30 0 V 41 14 R -29 0 V 41 14 R -29 0 V 41 14 R -29 0 V 40 15 R -28 0 V 40 14 R -28 0 V 40 14 R -28 0 V 40 14 R -28 0 V 39 15 R -27 0 V 39 14 R -27 0 V 39 14 R -27 0 V 39 14 R -27 0 V 39 15 R -27 0 V 38 14 R -25 0 V 37 14 R -25 0 V 37 14 R -25 0 V 37 15 R -25 0 V 37 14 R -25 0 V 37 14 R -25 0 V 37 14 R -24 0 V 36 15 R -24 0 V 36 14 R -24 0 V 36 14 R -24 0 V 3.000 UL LT1 2651 480 M 0 14 R 0 14 R 0 15 R 0 14 R 1 0 V 0 14 R 0 14 R 1 15 R 1 14 R 0 14 R 1 0 V 0 14 R 1 15 R 1 14 R 1 0 V 1 14 R 1 14 R 1 15 R 2 14 R 1 14 R 2 14 R 2 15 R 1 14 R 1 0 V 1 14 R 2 14 R 2 15 R 1 0 V 2 14 R 2 14 R 2 14 R 3 14 R 2 15 R 3 14 R 2 14 R 1 0 V 2 14 R 3 15 R 3 14 R 3 14 R 3 14 R 3 15 R 4 14 R 3 14 R 3 14 R 4 15 R 3 14 R 4 14 R 4 14 R 4 15 R 3 14 R 1 0 V 3 14 R 4 14 R 4 15 R 1 0 V 4 14 R 4 14 R 4 14 R 4 14 R 1 0 V 4 15 R 4 14 R 1 0 V 4 14 R 4 14 R 1 0 V 4 15 R 5 14 R 5 14 R 5 14 R 4 15 R 1 0 V 4 14 R 1 0 V 4 14 R 1 0 V 5 14 R 5 15 R 5 14 R 5 14 R 1 0 V 4 14 R 1 0 V 5 15 R 5 14 R 1 0 V 5 14 R 5 14 R 1 0 V 5 15 R 5 14 R 1 0 V 5 14 R 1 0 V 4 14 R 1 0 V 5 14 R 1 0 V 5 15 R 1 0 V 5 14 R 1 0 V 4 14 R 2 0 V 4 14 R 2 0 V 4 15 R 2 0 V 4 14 R 2 0 V 4 14 R 2 0 V 4 14 R 2 0 V 4 15 R 2 0 V 4 14 R 2 0 V 4 14 R 3 0 V 3 14 R 3 0 V 4 15 R 2 0 V 4 14 R 2 0 V 4 14 R 3 0 V 3 14 R 3 0 V 4 15 R 3 0 V 3 14 R 3 0 V 3 14 R 4 0 V 3 14 R 3 0 V 3 15 R 4 0 V 2 14 R 4 0 V 3 14 R 4 0 V 2 14 R 5 0 V 2 14 R 4 0 V 2 15 R 5 0 V 2 14 R 5 0 V 1 14 R 5 0 V 2 14 R 5 0 V 1 15 R 6 0 V 1 14 R 6 0 V 0 14 R 7 0 V 0 14 R 7 0 V 0 15 R 7 0 V -1 14 R 7 0 V 0 14 R 7 0 V -1 14 R 8 0 V -1 15 R 8 0 V -1 14 R 8 0 V -2 14 R 9 0 V -2 14 R 9 0 V -2 15 R 9 0 V -3 14 R 11 0 V -4 14 R 11 0 V -4 14 R 11 0 V -5 15 R 12 0 V -5 14 R 12 0 V -5 14 R 12 0 V -5 14 R 12 0 V -6 14 R 13 0 V -6 15 R 14 0 V -7 14 R 14 0 V -8 14 R 15 0 V -8 14 R 15 0 V -8 15 R 16 0 V -9 14 R 16 0 V -10 14 R 17 0 V -10 14 R 17 0 V -10 15 R 18 0 V -12 14 R 19 0 V -12 14 R 19 0 V -12 14 R 20 0 V -13 15 R 20 0 V -14 14 R 21 0 V -14 14 R 22 0 V -15 14 R 22 0 V -16 15 R 23 0 V -16 14 R 24 0 V -17 14 R 24 0 V -18 14 R 26 0 V -19 15 R 26 0 V -19 14 R 26 0 V -20 14 R 28 0 V -21 14 R 28 0 V -21 14 R 29 0 V -23 15 R 30 0 V -23 14 R 31 0 V -24 14 R 31 0 V -25 14 R 32 0 V -25 15 R 33 0 V -26 14 R 33 0 V -27 14 R 35 0 V -28 14 R 35 0 V -28 15 R 36 0 V -30 14 R 37 0 V -30 14 R 38 0 V -32 14 R 39 0 V -32 15 R 40 0 V -33 14 R 40 0 V -34 14 R 42 0 V -35 14 R 43 0 V -36 15 R 43 0 V -37 14 R 45 0 V -38 14 R 45 0 V -39 14 R 47 0 V -40 15 R 47 0 V -40 14 R 48 0 V -42 14 R 50 0 V -43 14 R 50 0 V -44 14 R 52 0 V -45 15 R 52 0 V -45 14 R 53 0 V -47 14 R 55 0 V -48 14 R 55 0 V -48 15 R 56 0 V -50 14 R 58 0 V -51 14 R 58 0 V -51 14 R 59 0 V -53 15 R 60 0 V -53 14 R 61 0 V -54 14 R 62 0 V -56 14 R 63 0 V -56 15 R 64 0 V -57 14 R 65 0 V -58 14 R 65 0 V -59 14 R 67 0 V -60 15 R 67 0 V -60 14 R 68 0 V -61 14 R 69 0 V -62 14 R 69 0 V -62 15 R 70 0 V -63 14 R 70 0 V -63 14 R 70 0 V -63 14 R 71 0 V -63 14 R 70 0 V -63 15 R 70 0 V -63 14 R 70 0 V -62 14 R 69 0 V -62 14 R 70 0 V -62 15 R 69 0 V -62 14 R 69 0 V -61 14 R 68 0 V -61 14 R 68 0 V -60 15 R 67 0 V -59 14 R 66 0 V -59 14 R 66 0 V -58 14 R 65 0 V -58 15 R 65 0 V -57 14 R 64 0 V -56 14 R 63 0 V -56 14 R 63 0 V -55 15 R 62 0 V -54 14 R 61 0 V -54 14 R 62 0 V -54 14 R 61 0 V -53 15 R 60 0 V -53 14 R 60 0 V -52 14 R 59 0 V -51 14 R 58 0 V -50 14 R 57 0 V currentpoint stroke M -50 15 R 57 0 V -49 14 R 56 0 V -48 14 R 55 0 V -48 14 R 55 0 V -47 15 R 54 0 V -46 14 R 54 0 V -47 14 R 54 0 V -46 14 R 53 0 V -45 15 R 52 0 V -44 14 R 51 0 V -44 14 R 51 0 V -43 14 R 50 0 V -42 15 R 49 0 V -42 14 R 50 0 V -42 14 R 49 0 V -41 14 R 48 0 V -40 15 R 47 0 V -40 14 R 47 0 V -39 14 R 46 0 V -38 14 R 45 0 V -38 15 R 46 0 V -38 14 R 45 0 V -37 14 R 44 0 V -37 14 R 44 0 V -36 14 R 43 0 V -35 15 R 43 0 V -36 14 R 43 0 V -35 14 R 42 0 V -34 14 R 41 0 V -34 15 R 41 0 V -33 14 R 40 0 V -32 14 R 40 0 V -32 14 R 39 0 V -32 15 R 39 0 V -31 14 R 38 0 V -30 14 R 38 0 V -31 14 R 38 0 V -30 15 R 37 0 V -30 14 R 37 0 V -29 14 R 36 0 V -28 14 R 36 0 V -29 15 R 36 0 V -28 14 R 35 0 V -27 14 R 34 0 V -27 14 R 35 0 V -27 15 R 34 0 V -26 14 R 33 0 V -26 14 R 33 0 V -25 14 R 33 0 V -25 14 R 32 0 V -25 15 R 32 0 V -24 14 R 31 0 V -23 14 R 31 0 V -24 14 R 31 0 V -23 15 R 30 0 V -23 14 R 31 0 V -23 14 R 30 0 V -22 14 R 29 0 V -22 15 R 29 0 V -21 14 R 29 0 V -21 14 R 28 0 V -21 14 R 28 0 V -20 15 R 28 0 V -20 14 R 27 0 V -20 14 R 27 0 V -19 14 R 27 0 V -20 15 R 27 0 V -19 14 R 26 0 V -18 14 R 26 0 V -19 14 R 26 0 V -18 15 R 25 0 V -17 14 R 25 0 V -18 14 R 25 0 V 3.000 UL LT1 2385 480 M 0 14 R 1 14 R -1 0 V 1 15 R 0 14 R 1 14 R 1 14 R 1 15 R -1 0 V 2 14 R 1 14 R 1 14 R 2 15 R -1 0 V 2 14 R 2 14 R 2 14 R 2 15 R 2 14 R 2 14 R 2 14 R 3 15 R 3 14 R -1 0 V 3 14 R 3 14 R 3 15 R 3 14 R 4 14 R -1 0 V 4 14 R 3 14 R 4 15 R 4 14 R -1 0 V 4 14 R 4 14 R 4 15 R 4 14 R 5 14 R -1 0 V 5 14 R 4 15 R 5 14 R 4 14 R 5 14 R 5 15 R 5 14 R 5 14 R 5 14 R 5 15 R 5 14 R 6 14 R -1 0 V 6 14 R 5 15 R 6 14 R 6 14 R 5 14 R 6 14 R 6 15 R 6 14 R 6 14 R 6 14 R 6 15 R 6 14 R 6 14 R 7 14 R 6 15 R 7 14 R -1 0 V 7 14 R 7 14 R -1 0 V 7 15 R 7 14 R 7 14 R -1 0 V 7 14 R 7 15 R 7 14 R 7 14 R 7 14 R 7 15 R 7 14 R 7 14 R 7 14 R 7 14 R 7 15 R 8 14 R -1 0 V 8 14 R 7 14 R 8 15 R -1 0 V 8 14 R 7 14 R 8 14 R 7 15 R 8 14 R 7 14 R 8 14 R 8 15 R -1 0 V 8 14 R 8 14 R -1 0 V 9 14 R -1 0 V 8 15 R 8 14 R -1 0 V 9 14 R -1 0 V 9 14 R -1 0 V 9 15 R -1 0 V 8 14 R -1 0 V 9 14 R -1 0 V 9 14 R -1 0 V 9 14 R -1 0 V 9 15 R -1 0 V 9 14 R -1 0 V 9 14 R -1 0 V 9 14 R -2 0 V 10 15 R -2 0 V 10 14 R -2 0 V 10 14 R -2 0 V 10 14 R -2 0 V 10 15 R -2 0 V 10 14 R -2 0 V 10 14 R -2 0 V 10 14 R -2 0 V 10 15 R -2 0 V 11 14 R -3 0 V 11 14 R -3 0 V 11 14 R -3 0 V 11 15 R -3 0 V 11 14 R -3 0 V 11 14 R -3 0 V 11 14 R -3 0 V 12 15 R -4 0 V 12 14 R -4 0 V 12 14 R -4 0 V 12 14 R -4 0 V 12 14 R -4 0 V 13 15 R -5 0 V 13 14 R -5 0 V 13 14 R -5 0 V 13 14 R -5 0 V 14 15 R -6 0 V 14 14 R -6 0 V 14 14 R -6 0 V 14 14 R -6 0 V 15 15 R -7 0 V 15 14 R -7 0 V 15 14 R -7 0 V 15 14 R -7 0 V 16 15 R -9 0 V 17 14 R -9 0 V 17 14 R -9 0 V 17 14 R -9 0 V 18 15 R -10 0 V 18 14 R -10 0 V 18 14 R -10 0 V 18 14 R -10 0 V 19 15 R -12 0 V 20 14 R -12 0 V 20 14 R -12 0 V 20 14 R -12 0 V 21 14 R -13 0 V 21 15 R -14 0 V 22 14 R -14 0 V 22 14 R -14 0 V 23 14 R -16 0 V 24 15 R -16 0 V 24 14 R -16 0 V 24 14 R -17 0 V 25 14 R -17 0 V 26 15 R -19 0 V 27 14 R -19 0 V 27 14 R -19 0 V 27 14 R -20 0 V 28 15 R -20 0 V 28 14 R -21 0 V 29 14 R -21 0 V 30 14 R -23 0 V 31 15 R -24 0 V 32 14 R -24 0 V 32 14 R -25 0 V 33 14 R -26 0 V 34 15 R -26 0 V 34 14 R -27 0 V 35 14 R -28 0 V 36 14 R -28 0 V 36 14 R -29 0 V 37 15 R -30 0 V 38 14 R -31 0 V 39 14 R -32 0 V 39 14 R -32 0 V 40 15 R -33 0 V 41 14 R -33 0 V 41 14 R -34 0 V 42 14 R -35 0 V 43 15 R -36 0 V 43 14 R -36 0 V 44 14 R -37 0 V 45 14 R -38 0 V 46 15 R -39 0 V 46 14 R -39 0 V 47 14 R -40 0 V 47 14 R -41 0 V 49 15 R -42 0 V 49 14 R -42 0 V 50 14 R -43 0 V 50 14 R -43 0 V 51 15 R -44 0 V 51 14 R -44 0 V 51 14 R -44 0 V 51 14 R -44 0 V 51 14 R -44 0 V 51 15 R -44 0 V 51 14 R -44 0 V 51 14 R -43 0 V 50 14 R -43 0 V 49 15 R -42 0 V 49 14 R -42 0 V 49 14 R -42 0 V 48 14 R -41 0 V 48 15 R -41 0 V 48 14 R -41 0 V 47 14 R -40 0 V 47 14 R -39 0 V 45 15 R -38 0 V 45 14 R -38 0 V 44 14 R -37 0 V 44 14 R -37 0 V 43 15 R -36 0 V 43 14 R -36 0 V 42 14 R -35 0 V 41 14 R -34 0 V 41 15 R -35 0 V 41 14 R -34 0 V 41 14 R -34 0 V 40 14 R -33 0 V 39 14 R -32 0 V 39 15 R -32 0 V 38 14 R -31 0 V 37 14 R -31 0 V 38 14 R -31 0 V 37 15 R -30 0 V 36 14 R -29 0 V 35 14 R -28 0 V 35 14 R -29 0 V 35 15 R -28 0 V 34 14 R -27 0 V 33 14 R -27 0 V 34 14 R -27 0 V currentpoint stroke M 33 15 R -26 0 V 32 14 R -26 0 V 32 14 R -25 0 V 31 14 R -24 0 V 31 15 R -25 0 V 31 14 R -24 0 V 30 14 R -23 0 V 29 14 R -23 0 V 29 15 R -22 0 V 29 14 R -23 0 V 29 14 R -22 0 V 28 14 R -22 0 V 28 14 R -21 0 V 27 15 R -21 0 V 27 14 R -20 0 V 26 14 R -20 0 V 27 14 R -20 0 V 26 15 R -20 0 V 26 14 R -19 0 V 25 14 R -19 0 V 25 14 R -18 0 V 24 15 R -18 0 V 24 14 R -17 0 V 24 14 R -18 0 V 24 14 R -17 0 V 23 15 R -17 0 V 23 14 R -17 0 V 23 14 R -16 0 V 22 14 R -16 0 V 22 15 R -15 0 V 21 14 R -15 0 V 21 14 R -15 0 V 22 14 R -15 0 V 21 15 R -15 0 V 21 14 R -15 0 V 21 14 R -14 0 V 20 14 R -14 0 V 20 14 R -14 0 V 20 15 R -13 0 V 19 14 R -13 0 V 19 14 R -13 0 V 20 14 R -13 0 V 19 15 R -13 0 V 19 14 R -13 0 V 19 14 R -12 0 V 18 14 R -12 0 V 18 15 R -12 0 V 18 14 R -11 0 V 17 14 R -11 0 V 17 14 R -11 0 V 17 15 R -10 0 V 17 14 R -11 0 V 17 14 R -11 0 V 17 14 R -11 0 V 17 15 R -10 0 V 16 14 R -10 0 V 16 14 R -10 0 V 16 14 R -10 0 V 16 15 R -9 0 V 15 14 R -9 0 V 15 14 R -9 0 V 3.000 UL LT1 3770 480 M 1 0 V 0 14 R 0 14 R 0 15 R 0 14 R 0 14 R 0 14 R 1 15 R 0 14 R 0 14 R 1 14 R 0 15 R 1 14 R 0 14 R 1 14 R 0 15 R 1 14 R 1 14 R 1 14 R 0 15 R 1 14 R 1 14 R 1 14 R 1 15 R 1 14 R 1 14 R 1 14 R 1 14 R 2 15 R 1 14 R 1 14 R 1 14 R 1 0 V 1 15 R 1 14 R 2 14 R 1 14 R 2 15 R 1 14 R 2 14 R 2 14 R 1 15 R 2 14 R 2 14 R 2 14 R 2 15 R 2 14 R 2 14 R 2 14 R 2 15 R 2 14 R 2 14 R 2 14 R 3 14 R 2 15 R 2 14 R 3 14 R 2 14 R 2 15 R 3 14 R 3 14 R 2 14 R 3 15 R 2 14 R 1 0 V 2 14 R 3 14 R 3 15 R 3 14 R 3 14 R 3 14 R 3 15 R 3 14 R 3 14 R 3 14 R 3 15 R 3 14 R 3 14 R 4 14 R 3 14 R 3 15 R 1 0 V 3 14 R 3 14 R 4 14 R 3 15 R 1 0 V 3 14 R 4 14 R 3 14 R 1 0 V 3 15 R 4 14 R 4 14 R 4 14 R 4 15 R 4 14 R 4 14 R 4 14 R 4 15 R 4 14 R 4 14 R 4 14 R 1 0 V 4 15 R 4 14 R 4 14 R 1 0 V 4 14 R 4 14 R 1 0 V 4 15 R 4 14 R 1 0 V 4 14 R 5 14 R 4 15 R 1 0 V 4 14 R 5 14 R 5 14 R 5 15 R 4 14 R 1 0 V 4 14 R 1 0 V 4 14 R 1 0 V 5 15 R 5 14 R 5 14 R 1 0 V 4 14 R 1 0 V 4 15 R 1 0 V 5 14 R 5 14 R 1 0 V 4 14 R 1 0 V 5 15 R 1 0 V 4 14 R 1 0 V 5 14 R 1 0 V 4 14 R 2 0 V 4 14 R 1 0 V 5 15 R 1 0 V 4 14 R 2 0 V 4 14 R 2 0 V 4 14 R 2 0 V 4 15 R 2 0 V 4 14 R 2 0 V 4 14 R 2 0 V 4 14 R 2 0 V 4 15 R 2 0 V 4 14 R 2 0 V 4 14 R 2 0 V 4 14 R 3 0 V 3 15 R 3 0 V 4 14 R 3 0 V 3 14 R 3 0 V 3 14 R 4 0 V 3 15 R 3 0 V 3 14 R 4 0 V 3 14 R 4 0 V 2 14 R 4 0 V 3 15 R 4 0 V 3 14 R 4 0 V 2 14 R 5 0 V 2 14 R 5 0 V 2 14 R 5 0 V 2 15 R 5 0 V 2 14 R 5 0 V 2 14 R 6 0 V 1 14 R 6 0 V 1 15 R 6 0 V 1 14 R 7 0 V 0 14 R 7 0 V 0 14 R 8 0 V 0 15 R 7 0 V 0 14 R 8 0 V -1 14 R 9 0 V -1 14 R 8 0 V -1 15 R 9 0 V -1 14 R 9 0 V -1 14 R 9 0 V -2 14 R 10 0 V -2 15 R 10 0 V -2 14 R 10 0 V -2 14 R 10 0 V -3 14 R 12 0 V -4 15 R 12 0 V -4 14 R 12 0 V -4 14 R 13 0 V -5 14 R 13 0 V -4 14 R 13 0 V -5 15 R 13 0 V -5 14 R 14 0 V -6 14 R 15 0 V -6 14 R 14 0 V -6 15 R 15 0 V -6 14 R 15 0 V -7 14 R 16 0 V -7 14 R 16 0 V -8 15 R 17 0 V -8 14 R 17 0 V -8 14 R 17 0 V -8 14 R 17 0 V -9 15 R 19 0 V -10 14 R 19 0 V -10 14 R 19 0 V -10 14 R 20 0 V -11 15 R 20 0 V -10 14 R 20 0 V -11 14 R 20 0 V -11 14 R 21 0 V -12 15 R 21 0 V -11 14 R 21 0 V -12 14 R 21 0 V -11 14 R 21 0 V -11 14 R 21 0 V -11 15 R 20 0 V -11 14 R 21 0 V -11 14 R 21 0 V -11 14 R 21 0 V -11 15 R 20 0 V -10 14 R 20 0 V -9 14 R 19 0 V -9 14 R 19 0 V -9 15 R 19 0 V -9 14 R 19 0 V -8 14 R 18 0 V -8 14 R 18 0 V -7 15 R 17 0 V -7 14 R 18 0 V -7 14 R 17 0 V -7 14 R 17 0 V -6 15 R 16 0 V -6 14 R 17 0 V -6 14 R 16 0 V -5 14 R 16 0 V -5 15 R 15 0 V -5 14 R 16 0 V -5 14 R 15 0 V -4 14 R 15 0 V -4 14 R 15 0 V -4 15 R 14 0 V -3 14 R 14 0 V -3 14 R 14 0 V -3 14 R 14 0 V -3 15 R 13 0 V -2 14 R 13 0 V -1 14 R 12 0 V -1 14 R 12 0 V -1 15 R 12 0 V -1 14 R 12 0 V 0 14 R 11 0 V 0 14 R 11 0 V 0 15 R 12 0 V 0 14 R 11 0 V 0 14 R 11 0 V 1 14 R 10 0 V 1 15 R 10 0 V 2 14 R 10 0 V 1 14 R 10 0 V 2 14 R 9 0 V 3 15 R 9 0 V 2 14 R 9 0 V 3 14 R 9 0 V 3 14 R 8 0 V currentpoint stroke M 3 14 R 9 0 V 3 15 R 8 0 V 4 14 R 8 0 V 4 14 R 8 0 V 3 14 R 8 0 V 4 15 R 8 0 V 4 14 R 8 0 V 4 14 R 7 0 V 5 14 R 7 0 V 5 15 R 7 0 V 5 14 R 7 0 V 5 14 R 6 0 V 6 14 R 6 0 V 6 15 R 6 0 V 6 14 R 6 0 V 6 14 R 6 0 V 6 14 R 6 0 V 6 15 R 6 0 V 6 14 R 6 0 V 6 14 R 6 0 V 6 14 R 6 0 V 7 15 R 5 0 V 7 14 R 5 0 V 7 14 R 5 0 V 7 14 R 5 0 V 8 14 R 5 0 V 7 15 R 5 0 V 7 14 R 5 0 V 8 14 R 4 0 V 8 14 R 4 0 V 8 15 R 5 0 V 8 14 R 4 0 V 8 14 R 4 0 V 8 14 R 5 0 V 8 15 R 4 0 V 8 14 R 4 0 V 9 14 R 4 0 V 8 14 R 4 0 V 9 15 R 3 0 V 9 14 R 4 0 V 9 14 R 3 0 V 9 14 R 4 0 V 9 15 R 3 0 V 9 14 R 4 0 V 9 14 R 3 0 V 9 14 R 4 0 V 9 15 R 3 0 V 10 14 R 3 0 V 9 14 R 3 0 V 3.000 UL LT1 1419 480 M 0 14 R 0 14 R -1 15 R 1 0 V -1 14 R 0 14 R -1 14 R 0 15 R -1 14 R 0 14 R -1 14 R -1 15 R -1 14 R -1 14 R -1 14 R -1 15 R -1 14 R -1 14 R -2 14 R 1 0 V -2 15 R -1 14 R -2 14 R -1 14 R -2 15 R -2 14 R 1 0 V -2 14 R -2 14 R -2 14 R -2 15 R -2 14 R -2 14 R -2 14 R -2 15 R -2 14 R -2 14 R -2 14 R -3 15 R 1 0 V -3 14 R -2 14 R -3 14 R 1 0 V -3 15 R -2 14 R -3 14 R -2 14 R -3 15 R -3 14 R 1 0 V -3 14 R -3 14 R -2 15 R -3 14 R -3 14 R -2 14 R -3 14 R -3 15 R -3 14 R -2 14 R -3 14 R -3 15 R -3 14 R -3 14 R -3 14 R 1 0 V -3 15 R -3 14 R -3 14 R -3 14 R -3 15 R -3 14 R -3 14 R -3 14 R 1 0 V -3 15 R -3 14 R -3 14 R -3 14 R -3 15 R -3 14 R -3 14 R -3 14 R -3 14 R -3 15 R 1 0 V -3 14 R -3 14 R -3 14 R -3 15 R -3 14 R -3 14 R -3 14 R -3 15 R 1 0 V -3 14 R -3 14 R -3 14 R -3 15 R -3 14 R 1 0 V -3 14 R -3 14 R -3 15 R -3 14 R 1 0 V -3 14 R -3 14 R -3 15 R 1 0 V -4 14 R 1 0 V -3 14 R -3 14 R 1 0 V -4 14 R 1 0 V -3 15 R -3 14 R 1 0 V -3 14 R -3 14 R 1 0 V -4 15 R 1 0 V -3 14 R 1 0 V -4 14 R 1 0 V -3 14 R 1 0 V -4 15 R 1 0 V -3 14 R 1 0 V -4 14 R 1 0 V -3 14 R 1 0 V -4 15 R 2 0 V -4 14 R 1 0 V -3 14 R 1 0 V -4 14 R 2 0 V -4 15 R 1 0 V -3 14 R 1 0 V -4 14 R 2 0 V -4 14 R 2 0 V -4 15 R 2 0 V -4 14 R 2 0 V -5 14 R 2 0 V -4 14 R 2 0 V -4 14 R 2 0 V -4 15 R 2 0 V -4 14 R 2 0 V -4 14 R 2 0 V -5 14 R 3 0 V -5 15 R 4 0 V -6 14 R 4 0 V -6 14 R 4 0 V -6 14 R 4 0 V -6 15 R 4 0 V -6 14 R 5 0 V -6 14 R 4 0 V -6 14 R 4 0 V -6 15 R 5 0 V -7 14 R 5 0 V -7 14 R 6 0 V -8 14 R 6 0 V -7 15 R 6 0 V -8 14 R 6 0 V -8 14 R 7 0 V -8 14 R 6 0 V -8 15 R 7 0 V -9 14 R 8 0 V -9 14 R 8 0 V -10 14 R 8 0 V -9 14 R 8 0 V -10 15 R 9 0 V -10 14 R 9 0 V -10 14 R 9 0 V -11 14 R 10 0 V -11 15 R 10 0 V -12 14 R 11 0 V -12 14 R 12 0 V -13 14 R 12 0 V -13 15 R 12 0 V -13 14 R 12 0 V -14 14 R 14 0 V -15 14 R 14 0 V -15 15 R 15 0 V -16 14 R 15 0 V -16 14 R 16 0 V -17 14 R 16 0 V -17 15 R 17 0 V -18 14 R 18 0 V -18 14 R 17 0 V -18 14 R 18 0 V -19 15 R 19 0 V -20 14 R 20 0 V -20 14 R 20 0 V -21 14 R 21 0 V -22 14 R 22 0 V -22 15 R 22 0 V -23 14 R 23 0 V -23 14 R 23 0 V -24 14 R 25 0 V -25 15 R 25 0 V -26 14 R 26 0 V -26 14 R 27 0 V -27 14 R 27 0 V -28 15 R 28 0 V -28 14 R 29 0 V -29 14 R 29 0 V -29 14 R 30 0 V -30 15 R 30 0 V -31 14 R 32 0 V -32 14 R 33 0 V -33 14 R 33 0 V -33 15 R 34 0 V -33 14 R 34 0 V -34 14 R 34 0 V -34 14 R 35 0 V -35 15 R 36 0 V -35 14 R 35 0 V -35 14 R 36 0 V -35 14 R 36 0 V -36 14 R 36 0 V -35 15 R 36 0 V -35 14 R 35 0 V -34 14 R 35 0 V -34 14 R 35 0 V -34 15 R 34 0 V -33 14 R 34 0 V -33 14 R 33 0 V -32 14 R 33 0 V -32 15 R 33 0 V -31 14 R 31 0 V -30 14 R 31 0 V -30 14 R 31 0 V -30 15 R 31 0 V -29 14 R 29 0 V -28 14 R 29 0 V -27 14 R 28 0 V -27 15 R 28 0 V -27 14 R 28 0 V -26 14 R 27 0 V -26 14 R 27 0 V -25 15 R 26 0 V -25 14 R 26 0 V -24 14 R 25 0 V -23 14 R 24 0 V -23 14 R 24 0 V -22 15 R 24 0 V -22 14 R 23 0 V -22 14 R 23 0 V -21 14 R 22 0 V -20 15 R 22 0 V -20 14 R 21 0 V -19 14 R 20 0 V -19 14 R 21 0 V -19 15 R 20 0 V -18 14 R 20 0 V -18 14 R 19 0 V -17 14 R 19 0 V -17 15 R 18 0 V -16 14 R 18 0 V -16 14 R 17 0 V -15 14 R 17 0 V -15 15 R 17 0 V currentpoint stroke M -15 14 R 16 0 V -14 14 R 16 0 V -14 14 R 16 0 V -14 15 R 16 0 V -14 14 R 15 0 V -13 14 R 15 0 V -12 14 R 14 0 V -12 14 R 14 0 V -12 15 R 14 0 V -12 14 R 14 0 V -12 14 R 14 0 V -11 14 R 13 0 V -11 15 R 13 0 V -11 14 R 13 0 V -11 14 R 13 0 V -10 14 R 12 0 V -10 15 R 12 0 V -10 14 R 12 0 V -9 14 R 11 0 V -9 14 R 11 0 V -9 15 R 11 0 V -8 14 R 10 0 V -8 14 R 11 0 V -8 14 R 10 0 V -8 15 R 10 0 V -7 14 R 9 0 V -7 14 R 9 0 V -7 14 R 10 0 V -7 15 R 9 0 V -7 14 R 9 0 V -6 14 R 9 0 V -7 14 R 9 0 V -6 14 R 8 0 V -6 15 R 9 0 V -6 14 R 8 0 V -5 14 R 7 0 V -5 14 R 8 0 V -5 15 R 7 0 V -5 14 R 8 0 V -5 14 R 7 0 V -5 14 R 7 0 V -4 15 R 7 0 V -4 14 R 6 0 V -4 14 R 7 0 V -4 14 R 6 0 V -3 15 R 6 0 V -4 14 R 6 0 V -3 14 R 6 0 V -3 14 R 6 0 V -4 15 R 6 0 V -3 14 R 6 0 V -3 14 R 5 0 V -3 14 R 6 0 V -3 15 R 5 0 V -2 14 R 5 0 V -2 14 R 5 0 V 3.000 UL LT1 3351 480 M 0 14 R 0 14 R 0 15 R 1 14 R -1 0 V 1 14 R 0 14 R 0 15 R 1 14 R 0 14 R 1 14 R 0 15 R 1 14 R 1 14 R 0 14 R 1 15 R 1 14 R 1 14 R 1 14 R 1 15 R 1 14 R 1 14 R 1 14 R 1 15 R 2 14 R 1 14 R 1 14 R 2 14 R 1 15 R 2 14 R 1 14 R 2 14 R 2 15 R 2 14 R -1 0 V 2 14 R 2 14 R 2 15 R 2 14 R 2 14 R 2 14 R 2 15 R 3 14 R -1 0 V 3 14 R 2 14 R 2 15 R 3 14 R 2 14 R 3 14 R 2 15 R 3 14 R 3 14 R 3 14 R -1 0 V 3 14 R 3 15 R 3 14 R 3 14 R 3 14 R 3 15 R 3 14 R 3 14 R 4 14 R 3 15 R 3 14 R 4 14 R 3 14 R 4 15 R -1 0 V 4 14 R 4 14 R 3 14 R 4 15 R 4 14 R 4 14 R 4 14 R -1 0 V 4 15 R 4 14 R 4 14 R 5 14 R -1 0 V 5 14 R 4 15 R 4 14 R 4 14 R 5 14 R 4 15 R 5 14 R 4 14 R 5 14 R 4 15 R 5 14 R 5 14 R 4 14 R 5 15 R 5 14 R 5 14 R 5 14 R 5 15 R 5 14 R 5 14 R 5 14 R 6 15 R -1 0 V 6 14 R 5 14 R 6 14 R -1 0 V 6 14 R 5 15 R 6 14 R 5 14 R 6 14 R 6 15 R -1 0 V 6 14 R 6 14 R 6 14 R 6 15 R -1 0 V 7 14 R -1 0 V 7 14 R -1 0 V 7 14 R -1 0 V 7 15 R -1 0 V 7 14 R -1 0 V 7 14 R -1 0 V 7 14 R -1 0 V 7 15 R 6 14 R 7 14 R -1 0 V 7 14 R -1 0 V 7 15 R 7 14 R -1 0 V 7 14 R -1 0 V 8 14 R -1 0 V 7 14 R -1 0 V 8 15 R -1 0 V 7 14 R -1 0 V 8 14 R -1 0 V 8 14 R -1 0 V 8 15 R -2 0 V 8 14 R -1 0 V 8 14 R -1 0 V 8 14 R -2 0 V 9 15 R -2 0 V 9 14 R -2 0 V 9 14 R -2 0 V 9 14 R -2 0 V 9 15 R -2 0 V 9 14 R -2 0 V 9 14 R -2 0 V 9 14 R -2 0 V 10 15 R -3 0 V 10 14 R -3 0 V 10 14 R -3 0 V 10 14 R -3 0 V 11 15 R -4 0 V 11 14 R -4 0 V 12 14 R -4 0 V 11 14 R -4 0 V 11 14 R -4 0 V 12 15 R -5 0 V 13 14 R -5 0 V 12 14 R -5 0 V 13 14 R -6 0 V 13 15 R -5 0 V 13 14 R -6 0 V 14 14 R -6 0 V 14 14 R -7 0 V 14 15 R -6 0 V 14 14 R -7 0 V 15 14 R -7 0 V 15 14 R -8 0 V 16 15 R -8 0 V 16 14 R -9 0 V 17 14 R -9 0 V 17 14 R -9 0 V 17 15 R -10 0 V 18 14 R -10 0 V 18 14 R -10 0 V 18 14 R -11 0 V 19 15 R -11 0 V 19 14 R -11 0 V 20 14 R -12 0 V 20 14 R -13 0 V 21 14 R -13 0 V 22 15 R -14 0 V 22 14 R -14 0 V 22 14 R -14 0 V 23 14 R -15 0 V 23 15 R -15 0 V 24 14 R -16 0 V 24 14 R -16 0 V 25 14 R -17 0 V 25 15 R -17 0 V 26 14 R -18 0 V 27 14 R -19 0 V 27 14 R -19 0 V 28 15 R -20 0 V 29 14 R -20 0 V 29 14 R -21 0 V 29 14 R -21 0 V 30 15 R -22 0 V 31 14 R -22 0 V 31 14 R -23 0 V 32 14 R -23 0 V 32 15 R -24 0 V 32 14 R -23 0 V 32 14 R -24 0 V 33 14 R -24 0 V 33 14 R -24 0 V 32 15 R -23 0 V 32 14 R -23 0 V 32 14 R -23 0 V 32 14 R -23 0 V 31 15 R -22 0 V 31 14 R -22 0 V 31 14 R -21 0 V 30 14 R -21 0 V 30 15 R -21 0 V 30 14 R -21 0 V 29 14 R -19 0 V 28 14 R -19 0 V 28 15 R -19 0 V 28 14 R -18 0 V 27 14 R -18 0 V 27 14 R -17 0 V 26 15 R -17 0 V 26 14 R -17 0 V 27 14 R -17 0 V 26 14 R -17 0 V 26 15 R -16 0 V 25 14 R -16 0 V 25 14 R -15 0 V 24 14 R -14 0 V 24 14 R -15 0 V 24 15 R -14 0 V 23 14 R -14 0 V 23 14 R -13 0 V 23 14 R -13 0 V 22 15 R -13 0 V 22 14 R -12 0 V 22 14 R -12 0 V 21 14 R -12 0 V 22 15 R -12 0 V 21 14 R -11 0 V 20 14 R -11 0 V 21 14 R -11 0 V 20 15 R -10 0 V 20 14 R -10 0 V 20 14 R -10 0 V 19 14 R -10 0 V 20 15 R -10 0 V 19 14 R -9 0 V 19 14 R -9 0 V 18 14 R -8 0 V 18 15 R -8 0 V 18 14 R -9 0 V 18 14 R -8 0 V 18 14 R -8 0 V currentpoint stroke M 18 14 R -8 0 V 18 15 R -8 0 V 17 14 R -7 0 V 17 14 R -7 0 V 17 14 R -7 0 V 17 15 R -7 0 V 16 14 R -6 0 V 16 14 R -6 0 V 16 14 R -6 0 V 16 15 R -6 0 V 16 14 R -6 0 V 16 14 R -6 0 V 16 14 R -6 0 V 15 15 R -5 0 V 15 14 R -5 0 V 15 14 R -5 0 V 15 14 R -5 0 V 15 15 R -5 0 V 15 14 R -5 0 V 15 14 R -5 0 V 15 14 R -4 0 V 14 15 R -4 0 V 14 14 R -4 0 V 14 14 R -4 0 V 14 14 R -4 0 V 15 14 R -5 0 V 15 15 R -4 0 V 14 14 R -4 0 V 14 14 R -4 0 V 14 14 R -4 0 V 14 15 R -3 0 V 13 14 R -3 0 V 13 14 R -3 0 V 14 14 R -4 0 V 14 15 R -3 0 V 13 14 R -3 0 V 13 14 R -3 0 V 14 14 R -3 0 V 13 15 R -3 0 V 13 14 R -3 0 V 13 14 R -2 0 V 13 14 R -3 0 V 13 15 R -3 0 V 13 14 R -2 0 V 13 14 R -3 0 V 13 14 R -3 0 V 13 15 R -2 0 V 13 14 R -3 0 V 13 14 R -2 0 V 1.000 UL LT0 6792 480 M -2976 0 V 0 14 V 0 14 V 0 15 V 0 14 V 1 14 V 0 14 V 0 15 V 0 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 0 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 2 14 V 1 15 V 1 14 V 1 14 V 2 14 V 1 15 V 1 14 V 2 14 V 1 14 V 2 15 V 1 14 V 2 14 V 2 14 V 1 15 V 2 14 V 2 14 V 2 14 V 1 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 14 V 2 15 V 2 14 V 3 14 V 2 14 V 3 15 V 2 14 V 3 14 V 2 14 V 3 15 V 2 14 V 3 14 V 3 14 V 2 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 4 14 V 3 14 V 3 15 V 4 14 V 3 14 V 4 14 V 3 15 V 4 14 V 3 14 V 4 14 V 3 15 V 4 14 V 4 14 V 4 14 V 4 15 V 3 14 V 4 14 V 4 14 V 4 15 V 4 14 V 5 14 V 4 14 V 4 15 V 4 14 V 4 14 V 5 14 V 4 14 V 5 15 V 4 14 V 5 14 V 4 14 V 5 15 V 4 14 V 5 14 V 5 14 V 5 15 V 4 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 6 14 V 5 14 V 5 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 14 V 6 15 V 5 14 V 6 14 V 6 14 V 5 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 7 14 V 6 14 V 7 15 V 6 14 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 10 14 V 10 14 V 10 14 V 9 15 V 10 14 V 10 14 V 10 14 V 11 15 V 10 14 V 10 14 V 10 14 V 11 15 V 10 14 V 11 14 V 10 14 V 11 15 V 11 14 V 10 14 V 11 14 V 11 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 12 14 V 11 14 V 11 15 V 12 14 V 11 14 V 12 14 V 11 15 V 12 14 V 11 14 V 12 14 V 11 15 V 12 14 V 12 14 V 11 14 V 12 15 V 12 14 V 12 14 V 12 14 V 11 15 V 12 14 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 13 14 V 12 14 V 12 15 V 12 14 V 12 14 V 13 14 V 12 15 V 12 14 V 13 14 V 12 14 V 12 15 V 13 14 V 12 14 V 13 14 V 12 15 V 12 14 V 13 14 V 12 14 V 13 14 V 12 15 V 13 14 V 13 14 V 12 14 V 13 15 V 12 14 V 13 14 V 13 14 V 12 15 V 13 14 V 13 14 V 13 14 V 12 15 V 13 14 V 13 14 V 13 14 V 13 15 V 12 14 V 13 14 V 13 14 V 13 15 V 13 14 V 13 14 V 913 0 V 1.000 UL LT0 6792 480 M 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 1.000 UL LT0 1918 480 M -7 14 V -7 14 V -8 15 V -7 14 V -7 14 V -8 14 V -7 15 V -7 14 V -7 14 V -7 14 V -8 15 V -7 14 V -7 14 V -7 14 V -7 15 V -7 14 V -7 14 V -7 14 V -6 15 V -7 14 V -7 14 V -7 14 V -7 15 V -6 14 V -7 14 V -7 14 V -6 14 V -7 15 V -6 14 V -7 14 V -6 14 V -7 15 V -6 14 V -6 14 V -7 14 V -6 15 V -6 14 V -7 14 V -6 14 V -6 15 V -6 14 V -6 14 V -6 14 V -6 15 V -6 14 V -6 14 V -6 14 V -6 15 V -6 14 V -6 14 V -6 14 V -5 14 V -6 15 V -6 14 V -5 14 V -6 14 V -6 15 V -5 14 V -6 14 V -5 14 V -6 15 V -5 14 V -5 14 V -6 14 V -5 15 V -5 14 V -6 14 V -5 14 V -5 15 V -5 14 V -5 14 V -5 14 V -5 15 V -5 14 V -5 14 V -5 14 V -5 14 V -5 15 V -5 14 V -5 14 V -4 14 V -5 15 V -5 14 V -4 14 V -5 14 V -5 15 V -4 14 V -5 14 V -4 14 V -5 15 V -4 14 V -5 14 V -4 14 V -4 15 V -4 14 V -5 14 V -4 14 V -4 15 V -4 14 V -4 14 V -5 14 V -4 14 V -4 15 V -4 14 V -4 14 V -3 14 V -4 15 V -4 14 V -4 14 V -4 14 V -4 15 V -3 14 V -4 14 V -4 14 V -3 15 V -4 14 V -3 14 V -4 14 V -3 15 V -4 14 V -3 14 V -4 14 V -3 15 V -3 14 V -4 14 V -3 14 V -3 14 V -3 15 V -3 14 V -4 14 V -3 14 V -3 15 V -3 14 V -3 14 V -3 14 V -2 15 V -3 14 V -3 14 V -3 14 V -3 15 V -2 14 V -3 14 V -3 14 V -2 15 V -3 14 V -2 14 V -3 14 V -2 15 V -3 14 V -2 14 V -3 14 V -2 14 V -2 15 V -2 14 V -2 14 V -3 14 V -2 15 V -2 14 V -2 14 V -2 14 V -2 15 V -1 14 V -2 14 V -2 14 V -2 15 V -1 14 V -2 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -1 14 V -1 14 V -2 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V 0 15 V -1 14 V -1 14 V 0 14 V -1 15 V -1 14 V 0 14 V 0 14 V -1 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 1 14 V 0 15 V 1 14 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 2 14 V 1 14 V 2 15 V 2 14 V 1 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 3 14 V 2 15 V 2 14 V 2 14 V 3 14 V 2 14 V 3 15 V 2 14 V 3 14 V 2 14 V 3 15 V 3 14 V 2 14 V 3 14 V 3 15 V 2 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 3 14 V 4 15 V 3 14 V 3 14 V 4 14 V 3 15 V 3 14 V 4 14 V 3 14 V 4 15 V 3 14 V 4 14 V 3 14 V 4 15 V 3 14 V 4 14 V 4 14 V 3 14 V 4 15 V 3 14 V 4 14 V 4 14 V 3 15 V 4 14 V 4 14 V 3 14 V 4 15 V 4 14 V 4 14 V 3 14 V 4 15 V 4 14 V 4 14 V 4 14 V 3 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 3 14 V 2782 0 V 1.000 UL LT0 1918 480 M 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 9 14 V 8 14 V 9 15 V 8 14 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 8 14 V 9 14 V 9 15 V 9 14 V 8 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 10 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 10 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 9 14 V 8 14 V 9 15 V 8 14 V 9 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 6 14 V 7 14 V 6 14 V 6 15 V 7 14 V 6 14 V 6 14 V 6 15 V 6 14 V 7 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 5 14 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 5 14 V 6 14 V 6 15 V 5 14 V 6 14 V 6 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 14 V 6 15 V 5 14 V 6 14 V 5 14 V 5 15 V 6 14 V 5 14 V 5 14 V 6 15 V 5 14 V 6 14 V 5 14 V 5 15 V 6 14 V 5 14 V 5 14 V 6 15 V 5 14 V 5 14 V 5 14 V 6 15 V 5 14 V 5 14 V 5 14 V 6 14 V 5 15 V 5 14 V 5 14 V 6 14 V 5 15 V 5 14 V 5 14 V 6 14 V 5 15 V 5 14 V 5 14 V 5 14 V 6 15 V 5 14 V 5 14 V 5 14 V 5 15 V 6 14 V 5 14 V 5 14 V 5 15 V 5 14 V 6 14 V 1.000 UL LT0 3035 480 M 0 14 V 0 14 V 0 15 V 0 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 2 14 V 1 14 V 2 14 V 1 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 14 V 3 15 V 2 14 V 3 14 V 2 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 4 15 V 3 14 V 4 14 V 3 14 V 4 15 V 4 14 V 3 14 V 4 14 V 4 15 V 4 14 V 5 14 V 4 14 V 4 14 V 4 15 V 5 14 V 4 14 V 5 14 V 5 15 V 4 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 15 V 6 14 V 5 14 V 6 14 V 6 15 V 5 14 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 7 14 V 6 14 V 6 15 V 7 14 V 6 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 7 14 V 8 14 V 7 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 7 14 V 8 14 V 8 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 9 15 V 8 14 V 9 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 8 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 9 14 V 9 15 V 8 14 V 9 14 V 9 14 V 9 15 V 9 14 V 8 14 V 9 14 V 9 14 V 8 15 V 9 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 15 V 8 14 V 8 14 V 9 14 V 8 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 8 14 V 9 14 V 8 14 V 9 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 9 14 V -443 0 V 1.000 UL LT0 3034 480 M 0 14 V 1 14 V 0 15 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 1 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 1 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 2 14 V 1 14 V 1 15 V 1 14 V 2 14 V 1 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 1 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 3 14 V 2 15 V 2 14 V 3 14 V 2 14 V 3 15 V 2 14 V 3 14 V 2 14 V 3 15 V 3 14 V 3 14 V 2 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 4 14 V 3 14 V 3 15 V 4 14 V 3 14 V 3 14 V 4 15 V 3 14 V 4 14 V 4 14 V 3 15 V 4 14 V 4 14 V 3 14 V 4 15 V 4 14 V 4 14 V 3 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 5 14 V 4 14 V 4 15 V 4 14 V 4 14 V 5 14 V 4 15 V 4 14 V 5 14 V 4 14 V 5 15 V 4 14 V 5 14 V 4 14 V 5 15 V 4 14 V 5 14 V 5 14 V 4 15 V 5 14 V 5 14 V 5 14 V 4 15 V 5 14 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 6 14 V 5 15 V 5 14 V 5 14 V 6 14 V 5 15 V 5 14 V 6 14 V 5 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 14 V 6 15 V 6 14 V 5 14 V 6 14 V 5 15 V 6 14 V 6 14 V 5 14 V 6 15 V 6 14 V 6 14 V 6 14 V 5 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 14 V 6 15 V 6 14 V 7 14 V 6 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 9 15 V 8 14 V 9 14 V 8 14 V 8 15 V 9 14 V 8 14 V 9 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 9 15 V 8 14 V 9 14 V 9 14 V 9 15 V 8 14 V 9 14 V 9 14 V 9 15 V 8 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 14 V 8 15 V 9 14 V 10 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 10 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 10 14 V 9 14 V 10 15 V 9 14 V 10 14 V 1.000 UL LT0 3509 480 M 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 1 14 V 0 15 V 0 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 1 14 V 0 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 0 14 V 2 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 2 14 V 1 14 V 2 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 15 V 2 14 V 2 14 V 3 14 V 2 14 V 2 15 V 3 14 V 2 14 V 3 14 V 2 15 V 3 14 V 2 14 V 3 14 V 3 15 V 3 14 V 2 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 3 14 V 4 14 V 3 15 V 4 14 V 3 14 V 3 14 V 4 15 V 4 14 V 3 14 V 4 14 V 4 15 V 3 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 5 14 V 4 14 V 4 14 V 5 15 V 4 14 V 5 14 V 4 14 V 5 15 V 4 14 V 5 14 V 5 14 V 4 15 V 5 14 V 5 14 V 5 14 V 4 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 6 14 V 5 15 V 5 14 V 5 14 V 6 14 V 5 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 15 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 7 14 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 8 14 V 7 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 10 14 V 10 15 V 9 14 V 10 14 V 9 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 11 15 V 10 14 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 11 14 V 10 14 V 10 15 V 11 14 V 10 14 V 10 14 V 11 15 V 10 14 V 11 14 V 10 14 V 11 15 V 10 14 V 11 14 V 10 14 V 11 15 V 11 14 V 10 14 V 11 14 V 11 15 V 10 14 V 11 14 V 11 14 V 11 14 V 10 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 10 14 V 12 14 V 11 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 11 14 V 11 14 V 11 15 V 12 14 V 11 14 V 11 14 V 11 14 V 11 15 V 12 14 V 11 14 V 11 14 V 11 15 V 12 14 V 11 14 V 11 14 V 12 15 V 11 14 V 12 14 V 11 14 V 11 15 V 12 14 V 11 14 V 12 14 V 11 15 V 12 14 V 11 14 V 12 14 V 11 15 V 12 14 V 11 14 V 156 0 V 1.000 UL LT0 3509 480 M 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 1 14 V 0 15 V 0 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 2 14 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 2 14 V 2 14 V 1 15 V 2 14 V 2 14 V 1 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 15 V 3 14 V 2 14 V 3 14 V 2 15 V 3 14 V 2 14 V 3 14 V 3 14 V 3 15 V 3 14 V 2 14 V 4 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 4 14 V 3 15 V 4 14 V 3 14 V 4 14 V 4 15 V 4 14 V 4 14 V 3 14 V 5 15 V 4 14 V 4 14 V 4 14 V 4 14 V 4 15 V 5 14 V 4 14 V 5 14 V 4 15 V 5 14 V 4 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 6 14 V 5 14 V 6 15 V 5 14 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 7 14 V 6 14 V 7 15 V 6 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 9 14 V 8 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 9 15 V 8 14 V 9 14 V 9 14 V 9 15 V 9 14 V 8 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 10 14 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 10 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 10 14 V 9 15 V 10 14 V 9 14 V 10 14 V 9 15 V 10 14 V 9 14 V 9 14 V 10 15 V 9 14 V 10 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 10 14 V 9 14 V 10 14 V 9 15 V 10 14 V 9 14 V 10 14 V 9 14 V 10 15 V 9 14 V 10 14 V 9 14 V 10 15 V 9 14 V 10 14 V 10 14 V 9 15 V 10 14 V 10 14 V 10 14 V 9 15 V 10 14 V 10 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 11 14 V 10 14 V 10 14 V 11 15 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 11 14 V 10 14 V 11 14 V 10 15 V 11 14 V 10 14 V 11 14 V 10 15 V 11 14 V 10 14 V 11 14 V 11 15 V 10 14 V 11 14 V 10 14 V 11 15 V 11 14 V 11 14 V 10 14 V 11 15 V 11 14 V 10 14 V 11 14 V 11 15 V 11 14 V 11 14 V 1.000 UL LT0 1100 480 M 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V -1 14 V 0 14 V 0 14 V 0 15 V -1 14 V 0 14 V 0 14 V -1 15 V 0 14 V 0 14 V -1 14 V 0 15 V -1 14 V 0 14 V -1 14 V 0 15 V -1 14 V -1 14 V 0 14 V -1 14 V -1 15 V 0 14 V -1 14 V -1 14 V 0 15 V -1 14 V -1 14 V -1 14 V -1 15 V 0 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V 0 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V 0 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V 0 14 V -1 14 V -1 15 V -1 14 V -1 14 V 0 14 V -1 14 V -1 15 V 0 14 V -1 14 V -1 14 V 0 15 V -1 14 V -1 14 V 0 14 V -1 15 V 0 14 V -1 14 V -1 14 V 0 15 V -1 14 V 0 14 V 0 14 V -1 15 V 0 14 V -1 14 V 0 14 V 0 15 V -1 14 V 0 14 V 0 14 V -1 14 V 0 15 V 0 14 V 0 14 V -1 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 1 14 V 0 15 V 0 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 1 14 V 0 15 V 1 14 V 1 14 V 0 14 V 1 15 V 1 14 V 1 14 V 0 14 V 1 14 V 1 15 V 1 14 V 0 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 0 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 0 14 V 1 15 V 1 14 V 1 14 V 0 14 V 1 15 V 1 14 V 0 14 V 1 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 0 15 V 0 14 V 1 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 1 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 1 15 V 0 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 1 14 V 0 15 V 1 14 V 1 14 V 0 14 V 1 15 V 1 14 V 0 14 V 1 14 V 1 15 V 1 14 V 0 14 V 1 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 0 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V -101 0 V 1.000 UL LT0 1100 480 M 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V -1 15 V 0 14 V 0 14 V 0 14 V 0 15 V -1 14 V 0 14 V 0 14 V -1 15 V 0 14 V -1 14 V 0 14 V -1 15 V 0 14 V -1 14 V 0 14 V -1 15 V 0 14 V -1 14 V -1 14 V 0 14 V -1 15 V -1 14 V 0 14 V -1 14 V -1 15 V -1 14 V 0 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 14 V -2 15 V -1 14 V -1 14 V -1 14 V -1 15 V -2 14 V -1 14 V -1 14 V -2 15 V -1 14 V -1 14 V -2 14 V -1 15 V -1 14 V -2 14 V -1 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -1 14 V -2 14 V -1 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -2 14 V -2 14 V -1 14 V -2 15 V -1 14 V -2 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -2 15 V -1 14 V -2 14 V -2 14 V -1 14 V -2 15 V -2 14 V -1 14 V -2 14 V -2 15 V -1 14 V -2 14 V -2 14 V -1 15 V -2 14 V -2 14 V -1 14 V -2 15 V -2 14 V -1 14 V -2 14 V -2 15 V -1 14 V -2 14 V -2 14 V -2 15 V -1 14 V -2 14 V -2 14 V -1 14 V -2 15 V -2 14 V -1 14 V -2 14 V -2 15 V -2 14 V -1 14 V -2 14 V -2 15 V -1 14 V -2 14 V -2 14 V -1 15 V -2 14 V -2 14 V -1 14 V -2 15 V -2 14 V -1 14 V -2 14 V -1 15 V -2 14 V -2 14 V -1 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -1 14 V -2 14 V -1 14 V -1 15 V -2 14 V -1 14 V -1 14 V -2 15 V -1 14 V -1 14 V -1 14 V -1 14 V -1 15 V -2 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V 0 14 V -1 14 V -1 14 V -1 15 V -1 14 V 0 14 V -1 14 V 0 15 V -1 14 V 0 14 V -1 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 1 14 V 0 14 V 1 15 V 1 14 V 0 14 V 1 14 V 1 15 V 1 14 V 1 14 V 0 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 2 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 2 14 V 1 15 V 1 14 V 1 14 V 2 14 V 1 15 V 1 14 V 2 14 V 1 14 V 1 15 V 2 14 V 1 14 V 1 14 V 2 15 V 1 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 1 15 V 1 14 V 2 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 1 14 V 2 15 V 1 14 V 2 14 V 1 14 V 2 15 V 2 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 1 14 V 2 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 1 14 V 2 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 1.000 UL LT0 3697 480 M 0 14 V 1 14 V 0 15 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 1 14 V 0 15 V 1 14 V 0 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 2 14 V 1 14 V 1 14 V 1 15 V 1 14 V 2 14 V 1 14 V 2 15 V 1 14 V 2 14 V 1 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 3 15 V 2 14 V 2 14 V 2 14 V 3 14 V 2 15 V 2 14 V 3 14 V 2 14 V 3 15 V 3 14 V 2 14 V 3 14 V 3 15 V 2 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 4 14 V 3 14 V 4 15 V 3 14 V 4 14 V 4 14 V 3 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 5 14 V 4 14 V 4 15 V 5 14 V 4 14 V 5 14 V 4 15 V 5 14 V 4 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 6 14 V 5 14 V 6 15 V 5 14 V 6 14 V 5 14 V 6 15 V 6 14 V 6 14 V 5 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 8 14 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 9 15 V 8 14 V 9 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 10 14 V 9 14 V 9 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 10 14 V 9 14 V 10 14 V 9 15 V 10 14 V 9 14 V 10 14 V 9 15 V 10 14 V 10 14 V 9 14 V 10 14 V 9 15 V 10 14 V 9 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 9 15 V 10 14 V 10 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 11 14 V 10 14 V 10 14 V 10 15 V 10 14 V 11 14 V 10 14 V 10 15 V 11 14 V 10 14 V 11 14 V 10 15 V 11 14 V 10 14 V 11 14 V 10 15 V 11 14 V 11 14 V 10 14 V 11 15 V 11 14 V 11 14 V 11 14 V 10 15 V 11 14 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 11 14 V 12 14 V 11 15 V 11 14 V 11 14 V 12 14 V 11 15 V 11 14 V 12 14 V 11 14 V 11 15 V 12 14 V 11 14 V 12 14 V 11 15 V 12 14 V 11 14 V 12 14 V 11 14 V 12 15 V 11 14 V 12 14 V 12 14 V 11 15 V 12 14 V 12 14 V 12 14 V 11 15 V 12 14 V 12 14 V 12 14 V 12 15 V 11 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V -24 0 V 1.000 UL LT0 3697 480 M 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 1 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 1 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 2 14 V 1 14 V 1 14 V 1 15 V 1 14 V 2 14 V 1 14 V 2 15 V 1 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 3 14 V 2 14 V 2 14 V 2 14 V 3 15 V 2 14 V 3 14 V 2 14 V 3 15 V 2 14 V 3 14 V 3 14 V 2 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 3 14 V 4 14 V 3 15 V 4 14 V 4 14 V 3 14 V 4 15 V 3 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 5 14 V 4 14 V 4 15 V 5 14 V 4 14 V 5 14 V 4 14 V 5 15 V 5 14 V 4 14 V 5 14 V 5 15 V 5 14 V 5 14 V 4 14 V 5 15 V 5 14 V 5 14 V 6 14 V 5 15 V 5 14 V 5 14 V 5 14 V 6 15 V 5 14 V 6 14 V 5 14 V 5 15 V 6 14 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 7 14 V 6 14 V 6 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 15 V 9 14 V 8 14 V 9 14 V 9 15 V 8 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 10 14 V 9 14 V 9 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 11 15 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 11 14 V 10 14 V 11 15 V 11 14 V 10 14 V 11 14 V 11 14 V 10 15 V 11 14 V 11 14 V 11 14 V 10 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 12 14 V 11 14 V 11 15 V 11 14 V 11 14 V 12 14 V 11 15 V 11 14 V 12 14 V 11 14 V 11 14 V 12 15 V 11 14 V 12 14 V 11 14 V 12 15 V 11 14 V 12 14 V 11 14 V 12 15 V 11 14 V 12 14 V 12 14 V 11 15 V 12 14 V 12 14 V 12 14 V 11 15 V 12 14 V 12 14 V 12 14 V 11 15 V 12 14 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 13 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 13 14 V 12 15 V 12 14 V 12 14 V 1.000 UL LT0 2651 480 M 0 14 V 0 14 V 0 15 V 0 14 V 1 14 V 0 14 V 1 15 V 1 14 V 0 14 V 1 14 V 1 15 V 1 14 V 2 14 V 1 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 3 14 V 2 14 V 2 14 V 3 14 V 2 15 V 3 14 V 2 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 3 14 V 4 15 V 3 14 V 4 14 V 4 14 V 4 15 V 3 14 V 4 14 V 4 14 V 4 15 V 5 14 V 4 14 V 4 14 V 4 14 V 5 15 V 4 14 V 5 14 V 4 14 V 5 15 V 5 14 V 5 14 V 5 14 V 4 15 V 5 14 V 5 14 V 6 14 V 5 15 V 5 14 V 5 14 V 5 14 V 6 15 V 5 14 V 6 14 V 5 14 V 6 15 V 5 14 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 8 15 V 8 14 V 7 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 25 0 V 1.000 UL LT0 2651 480 M 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 1 15 V 1 14 V 1 14 V 0 14 V 1 15 V 2 14 V 1 14 V 1 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 2 14 V 1 14 V 2 14 V 3 15 V 2 14 V 2 14 V 2 14 V 3 14 V 2 15 V 3 14 V 3 14 V 2 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 4 14 V 3 14 V 3 14 V 4 15 V 3 14 V 4 14 V 4 14 V 4 15 V 4 14 V 3 14 V 4 14 V 5 15 V 4 14 V 4 14 V 4 14 V 5 14 V 4 15 V 5 14 V 4 14 V 5 14 V 4 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 6 14 V 5 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 15 V 6 14 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 7 14 V 6 14 V 6 15 V 6 14 V 7 14 V 6 14 V 7 15 V 6 14 V 7 14 V 6 14 V 7 15 V 6 14 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 8 14 V 7 14 V 8 14 V 7 15 V 7 14 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 7 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 7 14 V 7 14 V 8 14 V 7 14 V 7 15 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 7 14 V 8 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 8 14 V 7 14 V 1.000 UL LT0 2385 480 M 0 14 V 1 14 V 0 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 3 15 V 3 14 V 2 14 V 3 14 V 3 15 V 3 14 V 4 14 V 3 14 V 3 14 V 4 15 V 4 14 V 3 14 V 4 14 V 4 15 V 4 14 V 5 14 V 4 14 V 4 15 V 5 14 V 4 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 6 14 V 5 14 V 5 15 V 6 14 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 8 14 V 7 15 V 8 14 V 7 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 6 14 V 7 15 V 6 14 V 7 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 6 14 V 6 15 V 7 14 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 7 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 7 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V -9 0 V 1.000 UL LT0 2385 480 M 0 14 V 0 14 V 1 15 V 0 14 V 1 14 V 1 14 V 0 15 V 2 14 V 1 14 V 1 14 V 1 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 3 15 V 2 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 4 14 V 3 14 V 4 15 V 3 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 5 14 V 4 15 V 5 14 V 4 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 6 14 V 5 15 V 6 14 V 6 14 V 5 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 7 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 15 V 7 14 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 8 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 8 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 6 14 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 6 14 V 7 14 V 6 14 V 7 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 7 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 7 14 V 6 15 V 6 14 V 7 14 V 6 14 V 6 15 V 7 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 1.000 UL LT0 3770 480 M 1 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 1 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 1 14 V 1 14 V 0 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 2 14 V 1 14 V 2 15 V 1 14 V 2 14 V 2 14 V 1 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 3 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 15 V 3 14 V 3 14 V 2 14 V 3 15 V 2 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 4 14 V 3 14 V 3 15 V 4 14 V 3 14 V 4 14 V 3 15 V 4 14 V 4 14 V 3 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 4 14 V 5 15 V 4 14 V 4 14 V 5 14 V 4 14 V 5 15 V 4 14 V 5 14 V 5 14 V 4 15 V 5 14 V 5 14 V 5 14 V 5 15 V 4 14 V 5 14 V 5 14 V 6 15 V 5 14 V 5 14 V 5 14 V 5 15 V 6 14 V 5 14 V 5 14 V 6 15 V 5 14 V 6 14 V 5 14 V 6 14 V 6 15 V 5 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 7 14 V 6 14 V 6 14 V 7 15 V 6 14 V 7 14 V 6 14 V 7 15 V 7 14 V 6 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 8 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 9 15 V 10 14 V 9 14 V 9 14 V 9 15 V 10 14 V 9 14 V 10 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 10 15 V 10 14 V 11 14 V 10 14 V 10 15 V 10 14 V 11 14 V 10 14 V 11 15 V 10 14 V 11 14 V 10 14 V 11 15 V 10 14 V 11 14 V 11 14 V 11 15 V 10 14 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 12 14 V 11 14 V 11 15 V 11 14 V 12 14 V 11 14 V 11 15 V 12 14 V 11 14 V 12 14 V 11 15 V 12 14 V 11 14 V 12 14 V 12 15 V 11 14 V 12 14 V 12 14 V 11 14 V 12 15 V 12 14 V 12 14 V 11 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 13 15 V 12 14 V 12 14 V 12 14 V 13 14 V 12 15 V 12 14 V 13 14 V 12 14 V 12 15 V 13 14 V 12 14 V 12 14 V 13 15 V 12 14 V 13 14 V 12 14 V 13 15 V 12 14 V 13 14 V 12 14 V 13 15 V 12 14 V 13 14 V 12 14 V 13 15 V 13 14 V 12 14 V 3 0 V 1.000 UL LT0 3771 480 M 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 1 15 V 0 14 V 0 14 V 1 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 1 14 V 1 14 V 0 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 1 14 V 2 14 V 1 15 V 1 14 V 2 14 V 1 14 V 2 15 V 1 14 V 2 14 V 2 14 V 1 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 3 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 15 V 3 14 V 3 14 V 2 14 V 3 15 V 3 14 V 2 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 4 14 V 3 14 V 4 15 V 3 14 V 3 14 V 4 14 V 4 15 V 3 14 V 4 14 V 4 14 V 3 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 4 14 V 4 15 V 4 14 V 4 14 V 5 14 V 4 15 V 4 14 V 5 14 V 4 14 V 5 14 V 4 15 V 5 14 V 4 14 V 5 14 V 5 15 V 4 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 6 14 V 5 14 V 5 15 V 5 14 V 6 14 V 5 14 V 6 15 V 5 14 V 6 14 V 6 14 V 5 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 7 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 9 14 V 8 15 V 8 14 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 15 V 9 14 V 9 14 V 10 14 V 9 15 V 10 14 V 9 14 V 10 14 V 9 15 V 10 14 V 9 14 V 10 14 V 10 14 V 9 15 V 10 14 V 10 14 V 10 14 V 9 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 11 14 V 10 14 V 10 14 V 10 15 V 11 14 V 10 14 V 11 14 V 10 15 V 11 14 V 10 14 V 11 14 V 11 14 V 10 15 V 11 14 V 11 14 V 11 14 V 10 15 V 11 14 V 11 14 V 11 14 V 11 15 V 11 14 V 11 14 V 11 14 V 12 15 V 11 14 V 11 14 V 11 14 V 11 15 V 12 14 V 11 14 V 11 14 V 12 15 V 11 14 V 12 14 V 11 14 V 12 14 V 11 15 V 12 14 V 12 14 V 11 14 V 12 15 V 12 14 V 11 14 V 12 14 V 12 15 V 12 14 V 11 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 12 15 V 12 14 V 12 14 V 12 14 V 13 14 V 12 15 V 12 14 V 12 14 V 12 14 V 13 15 V 12 14 V 12 14 V 13 14 V 12 15 V 12 14 V 13 14 V 12 14 V 12 15 V 13 14 V 12 14 V 13 14 V 12 15 V 13 14 V 12 14 V 13 14 V 12 15 V 13 14 V 12 14 V 1.000 UL LT0 1419 480 M 0 14 V 0 14 V -1 15 V 0 14 V 0 14 V -1 14 V 0 15 V -1 14 V 0 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -2 14 V -1 15 V -1 14 V -2 14 V -1 14 V -2 15 V -2 14 V -1 14 V -2 14 V -2 14 V -2 15 V -2 14 V -2 14 V -2 14 V -2 15 V -2 14 V -2 14 V -2 14 V -3 15 V -2 14 V -2 14 V -3 14 V -2 15 V -2 14 V -3 14 V -2 14 V -3 15 V -3 14 V -2 14 V -3 14 V -2 15 V -3 14 V -3 14 V -2 14 V -3 14 V -3 15 V -3 14 V -2 14 V -3 14 V -3 15 V -3 14 V -3 14 V -3 14 V -2 15 V -3 14 V -3 14 V -3 14 V -3 15 V -3 14 V -3 14 V -3 14 V -2 15 V -3 14 V -3 14 V -3 14 V -3 15 V -3 14 V -3 14 V -3 14 V -3 14 V -3 15 V -2 14 V -3 14 V -3 14 V -3 15 V -3 14 V -3 14 V -3 14 V -3 15 V -2 14 V -3 14 V -3 14 V -3 15 V -3 14 V -2 14 V -3 14 V -3 15 V -3 14 V -2 14 V -3 14 V -3 15 V -3 14 V -2 14 V -3 14 V -3 14 V -2 15 V -3 14 V -2 14 V -3 14 V -3 15 V -2 14 V -3 14 V -2 14 V -3 15 V -2 14 V -3 14 V -2 14 V -3 15 V -2 14 V -2 14 V -3 14 V -2 15 V -2 14 V -3 14 V -2 14 V -2 15 V -2 14 V -3 14 V -2 14 V -2 14 V -2 15 V -2 14 V -2 14 V -3 14 V -2 15 V -2 14 V -2 14 V -2 14 V -2 15 V -2 14 V -1 14 V -2 14 V -2 15 V -2 14 V -2 14 V -2 14 V -1 15 V -2 14 V -2 14 V -1 14 V -2 15 V -2 14 V -1 14 V -2 14 V -1 14 V -2 15 V -1 14 V -1 14 V -2 14 V -1 15 V -2 14 V -1 14 V -1 14 V -1 15 V -1 14 V -2 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V 0 14 V -1 14 V -1 15 V -1 14 V 0 14 V -1 14 V -1 14 V 0 15 V -1 14 V 0 14 V -1 14 V 0 15 V -1 14 V 0 14 V 0 14 V -1 15 V 0 14 V 0 14 V 0 14 V 0 15 V -1 14 V 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 2 14 V 1 14 V 1 14 V 1 15 V 2 14 V 1 14 V 2 14 V 1 15 V 1 14 V 2 14 V 1 14 V 2 15 V 1 14 V 2 14 V 2 14 V 1 14 V 2 15 V 2 14 V 1 14 V 2 14 V 2 15 V 2 14 V 2 14 V 1 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 3 14 V 2 14 V 2 15 V 2 14 V 2 14 V 3 14 V 2 15 V 2 14 V 2 14 V 3 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 15 V 3 14 V 2 14 V 3 14 V 2 15 V 3 14 V 2 14 V 2 14 V 3 15 V 2 14 V 3 14 V 2 14 V 3 14 V 2 15 V 3 14 V 3 14 V 2 14 V 3 15 V 2 14 V 3 14 V 2 14 V 3 15 V 3 14 V 2 14 V 3 14 V 3 15 V 2 14 V 3 14 V 3 14 V 2 15 V 3 14 V 3 14 V 2 14 V 3 15 V 3 14 V 3 14 V 5 0 V 1.000 UL LT0 1419 480 M 0 14 V 0 14 V 0 15 V -1 14 V 0 14 V -1 14 V 0 15 V -1 14 V 0 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -2 15 V -1 14 V -2 14 V -1 14 V -2 15 V -1 14 V -2 14 V -2 14 V -2 14 V -2 15 V -2 14 V -2 14 V -2 14 V -2 15 V -2 14 V -2 14 V -2 14 V -2 15 V -3 14 V -2 14 V -2 14 V -3 15 V -2 14 V -3 14 V -2 14 V -3 15 V -2 14 V -3 14 V -3 14 V -2 15 V -3 14 V -3 14 V -2 14 V -3 14 V -3 15 V -3 14 V -2 14 V -3 14 V -3 15 V -3 14 V -3 14 V -2 14 V -3 15 V -3 14 V -3 14 V -3 14 V -3 15 V -3 14 V -3 14 V -2 14 V -3 15 V -3 14 V -3 14 V -3 14 V -3 15 V -3 14 V -3 14 V -3 14 V -3 14 V -2 15 V -3 14 V -3 14 V -3 14 V -3 15 V -3 14 V -3 14 V -3 14 V -2 15 V -3 14 V -3 14 V -3 14 V -3 15 V -2 14 V -3 14 V -3 14 V -3 15 V -2 14 V -3 14 V -3 14 V -2 15 V -3 14 V -3 14 V -2 14 V -3 14 V -3 15 V -2 14 V -3 14 V -2 14 V -3 15 V -2 14 V -3 14 V -2 14 V -3 15 V -2 14 V -3 14 V -2 14 V -2 15 V -3 14 V -2 14 V -2 14 V -3 15 V -2 14 V -2 14 V -2 14 V -2 15 V -2 14 V -3 14 V -2 14 V -2 14 V -2 15 V -2 14 V -2 14 V -2 14 V -1 15 V -2 14 V -2 14 V -2 14 V -2 15 V -1 14 V -2 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -2 14 V -1 14 V -2 14 V -1 15 V -1 14 V -1 14 V -2 14 V -1 14 V -1 15 V -1 14 V -1 14 V -1 14 V -1 15 V -1 14 V 0 14 V -1 14 V -1 15 V -1 14 V 0 14 V -1 14 V 0 15 V -1 14 V 0 14 V -1 14 V 0 15 V 0 14 V -1 14 V 0 14 V 0 15 V 0 14 V 0 14 V 0 14 V 0 14 V 0 15 V 0 14 V 0 14 V 1 14 V 0 15 V 0 14 V 1 14 V 0 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 1 14 V 0 14 V 1 15 V 1 14 V 0 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 0 14 V 1 15 V 0 14 V 1 14 V 1 14 V 0 15 V 1 14 V 0 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 0 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 1 14 V 1 14 V 2 15 V 1 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 15 V 2 14 V 2 14 V 3 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 14 V 3 15 V 2 14 V 2 14 V 3 14 V 2 15 V 3 14 V 2 14 V 2 14 V 3 15 V 2 14 V 3 14 V 2 14 V 3 15 V 2 14 V 3 14 V 3 14 V 2 15 V 3 14 V 2 14 V 3 14 V 2 15 V 3 14 V 3 14 V 1.000 UL LT0 3351 480 M 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 0 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 1 14 V 0 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 2 14 V 1 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 2 14 V 1 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 3 14 V 2 14 V 2 14 V 2 15 V 3 14 V 2 14 V 3 14 V 2 15 V 3 14 V 3 14 V 3 14 V 2 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 4 14 V 3 15 V 3 14 V 4 14 V 3 14 V 4 15 V 3 14 V 4 14 V 3 14 V 4 15 V 4 14 V 4 14 V 4 14 V 3 15 V 4 14 V 4 14 V 5 14 V 4 14 V 4 15 V 4 14 V 4 14 V 5 14 V 4 15 V 5 14 V 4 14 V 5 14 V 4 15 V 5 14 V 5 14 V 4 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 6 15 V 5 14 V 5 14 V 6 14 V 5 14 V 5 15 V 6 14 V 5 14 V 6 14 V 6 15 V 5 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 7 14 V 6 14 V 6 15 V 7 14 V 6 14 V 7 14 V 6 14 V 7 15 V 6 14 V 7 14 V 7 14 V 7 15 V 6 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 7 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 7 14 V 8 15 V 8 14 V 7 14 V 8 14 V 7 15 V 8 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 9 14 V 8 14 V 8 14 V 9 15 V 8 14 V 8 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 9 14 V 8 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 8 14 V 9 14 V 9 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 8 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 8 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 9 14 V 10 14 V 9 15 V 9 14 V 10 14 V 9 14 V 10 15 V 9 14 V 9 14 V 10 14 V 9 15 V 10 14 V 10 14 V 9 14 V 10 15 V 9 14 V 10 14 V 9 14 V 10 15 V 10 14 V 9 14 V 10 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 9 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 11 14 V 10 14 V 10 15 V 11 14 V 10 14 V -2 0 V 1.000 UL LT0 3351 480 M 0 14 V 0 14 V 0 15 V 0 14 V 1 14 V 0 14 V 0 15 V 1 14 V 0 14 V 1 14 V 0 15 V 1 14 V 1 14 V 0 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 1 14 V 1 14 V 1 14 V 1 15 V 2 14 V 1 14 V 1 14 V 2 14 V 1 15 V 2 14 V 1 14 V 2 14 V 2 15 V 1 14 V 2 14 V 2 14 V 2 15 V 2 14 V 2 14 V 2 14 V 2 15 V 2 14 V 3 14 V 2 14 V 2 15 V 3 14 V 2 14 V 3 14 V 2 15 V 3 14 V 3 14 V 2 14 V 3 14 V 3 15 V 3 14 V 3 14 V 3 14 V 3 15 V 3 14 V 3 14 V 4 14 V 3 15 V 3 14 V 4 14 V 3 14 V 3 15 V 4 14 V 4 14 V 3 14 V 4 15 V 4 14 V 4 14 V 3 14 V 4 15 V 4 14 V 4 14 V 4 14 V 5 14 V 4 15 V 4 14 V 4 14 V 5 14 V 4 15 V 5 14 V 4 14 V 5 14 V 4 15 V 5 14 V 5 14 V 4 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 5 14 V 5 14 V 5 14 V 5 15 V 6 14 V 5 14 V 5 14 V 6 14 V 5 15 V 6 14 V 5 14 V 6 14 V 5 15 V 6 14 V 6 14 V 6 14 V 5 15 V 6 14 V 6 14 V 6 14 V 6 15 V 6 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 6 14 V 7 15 V 6 14 V 6 14 V 7 14 V 6 14 V 7 15 V 6 14 V 7 14 V 7 14 V 6 15 V 7 14 V 7 14 V 6 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 7 14 V 7 14 V 7 15 V 7 14 V 8 14 V 7 14 V 7 14 V 7 15 V 8 14 V 7 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 7 14 V 8 15 V 7 14 V 8 14 V 8 14 V 7 15 V 8 14 V 8 14 V 7 14 V 8 15 V 8 14 V 8 14 V 7 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 8 14 V 8 14 V 8 14 V 8 15 V 9 14 V 8 14 V 8 14 V 8 15 V 9 14 V 8 14 V 9 14 V 8 15 V 9 14 V 8 14 V 9 14 V 9 14 V 9 15 V 9 14 V 9 14 V 9 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 9 14 V 10 14 V 9 14 V 9 15 V 10 14 V 9 14 V 10 14 V 9 15 V 9 14 V 10 14 V 9 14 V 10 15 V 9 14 V 10 14 V 10 14 V 9 14 V 10 15 V 9 14 V 10 14 V 10 14 V 9 15 V 10 14 V 10 14 V 9 14 V 10 15 V 10 14 V 9 14 V 10 14 V 10 15 V 10 14 V 10 14 V 9 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 9 14 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 15 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 10 14 V 10 14 V 10 14 V 11 15 V 10 14 V 10 14 V 10 14 V 11 15 V 10 14 V 10 14 V 10 14 V 11 15 V 10 14 V 10 14 V 11 14 V 10 15 V 10 14 V 11 14 V 10 14 V 10 15 V 11 14 V 10 14 V 10 14 V 11 15 V 10 14 V 11 14 V stroke grestore end showpage %%EndDocument @endspecial 515 1855 a currentpoint currentpoint translate 1 0.82678 div 1 0.82678 div scale neg exch neg exch translate 515 1855 a -236 2004 a Fp(Figure)34 b(1:)49 b Fx(Numerical)31 b(computation)i(of)g(the)g(ten)g(biggest)g (sp)s(ectral)f(gaps)h(for)g(the)g(Almost)g(Mathieu)f(op)s(erator)-236 2117 y(with)k(di\013eren)m(t)i(v)-5 b(alues)37 b(of)h Fw(b)g Fx(and)g Fw(!)j Fx(=)c(\()1237 2042 y Fr(p)p 1313 2042 46 4 v 75 x Fx(5)26 b Fr(\000)f Fx(1\))p Fw(=)p Fx(2.)65 b(They)38 b(corresp)s(ond)f(to)h(the)h(\014rst)e Fr(j)p Fw(k)s Fr(j)h Fx(suc)m(h)g(that)h Fr(f)p Fw(k)s(!)s Fr(g)p Fw(=)p Fx(2)-236 2230 y(b)s(elongs)31 b(to)i([1)p Fw(=)p Fx(4)p Fw(;)15 b Fx(1)p Fw(=)p Fx(2].)50 b(The)32 b(coupling)f(parameter)i Fw(b)f Fx(is)f(in)g(the)i(v)m(ertical)f (direction)g(whereas)g(the)g(sp)s(ectral)g(one,)-236 2343 y Fw(a)p Fx(,)e(is)g(in)f(the)h(horizon)m(tal)g(one.)41 b(Note)32 b(that)f(for)f Fw(b)c Fx(=)f(0,)31 b(all)e(gaps)h(except)i (the)e(upp)s(er)f(one)h(are)h(collapsed.)-236 2678 y Fp(See)38 b(Figure)f(1)g(for)g(a)h(n)m(umerical)e(computation)g(of)h (the)h(biggest)f(gaps)h(in)f(the)h(sp)s(ectrum)g(of)f(the)h(Almost)-236 2799 y(Mathieu)32 b(op)s(erator)g(for)g(sev)m(eral)h(v)-5 b(alues)33 b Fo(b)p Fp(.)-90 2919 y(In)28 b(particular)d(if,)i(for)g(a) g(\014xed)h Fo(b)p Fp(,)h(all)c(the)j(sp)s(ectral)f(gaps)h(are)f(op)s (en)g(and)h(the)f(frequency)j Fo(!)g Fp(is)d(irrational,)-236 3039 y(then)37 b(the)g(sp)s(ectrum)f Fo(\033)646 3054 y Fv(b)718 3039 y Fp(is)f(a)i(Can)m(tor)f(set.)56 b(The)37 b(question)g(of)f(the)h(non-collapsing)d(of)i(all)e(sp)s(ectral)i(gaps) -236 3160 y(is)d(sometimes)f(called)g(the)i Fk(Str)-5 b(ong)35 b(\(or)g(Dry\))g(T)-7 b(en)35 b(Martini)g(Pr)-5 b(oblem)p Fp(.)45 b(Ho)m(w)m(ev)m(er,)36 b(if)c(non-collapsed)g(gaps) -236 3280 y(are)37 b(dense)h(in)f(the)g(sp)s(ectrum,)i(then)f(this)e (is)h(still)e(a)h(Can)m(tor)i(set,)h(although)d(some)h(\(p)s(erhaps)g (an)g(in\014nite)-236 3401 y(n)m(um)m(b)s(er\))c(of)f(collapsed)g(gaps) g(ma)m(y)h(also)e(co)s(exist.)-90 3521 y(No)m(w)i(w)m(e)h(can)f(form)m (ulate)e(the)i(second)h(corollary)c(in)i(this)g(pap)s(er:)-236 3733 y Fl(Corollary)k(4)48 b Fk(Assume)42 b(that)f Fo(!)i Fn(2)d Fm(R)52 b Fk(is)41 b(Diophantine.)62 b(Then,)42 b(ther)-5 b(e)41 b(is)g(a)g(c)-5 b(onstant)41 b Fo(C)47 b Fp(=)39 b Fo(C)7 b Fp(\()p Fo(!)t Fp(\))38 b Fo(>)i Fp(0)-236 3853 y Fk(such)c(that)i(if)e Fp(0)31 b Fo(<)h Fn(j)p Fo(b)p Fn(j)f Fo(<)g(C)44 b Fk(or)37 b 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b(Martini)e(Problem")f(b)m(y)k(Simon)c([31])i(\(and)g(remained) g(as)g(Problem)f(4)h(in)-236 4787 y([32)o(]\).)43 b(Sinai)26 b([33],)j(pro)m(v)m(ed)h(that)f(for)f(Diophan)m(tine)f Fo(!)t Fp('s)h(and)g(su\016cien)m(tly)i(large)d(\(or)h(small)e Fn(j)p Fo(b)p Fn(j)p Fp(\),)j(dep)s(ending)-236 4907 y(on)34 b Fo(!)t Fp(,)h(the)g(sp)s(ectrum)g Fo(\033)681 4922 y Fv(b)750 4907 y Fp(is)g(a)f(Can)m(tor)h(set.)50 b(Choi,)35 b(Elliott)d(&)i(Y)-8 b(ui)34 b([8])h(pro)m(v)m(ed)h(that)e (the)i(sp)s(ectrum)f Fo(\033)3645 4922 y Fv(b)3714 4907 y Fp(is)-236 5028 y(a)d(Can)m(tor)h(set)g(for)f(all)f Fo(b)d Fn(6)p Fp(=)f(0)33 b(when)g Fo(!)j Fp(is)c(a)h(Liouville)c(n)m (um)m(b)s(er)k(ob)s(eying)f(the)h(condition)1443 5140 y Fi(\014)1443 5200 y(\014)1443 5260 y(\014)1443 5320 y(\014)1477 5285 y Fo(!)25 b Fn(\000)1673 5217 y Fo(p)p 1673 5262 49 4 v 1674 5353 a(q)1732 5140 y Fi(\014)1732 5200 y(\014)1732 5260 y(\014)1732 5320 y(\014)1793 5285 y Fo(<)i(D)1980 5244 y Ft(\000)p Fv(q)2073 5285 y Fo(;)-236 5547 y Fp(for)33 b(a)g(certain)h(constan)m(t)g Fo(D)f(>)c Fp(1)34 b(and)g(in\014nitely)e(man)m(y)i(rationals)e Fo(p=q)t Fp(.)46 b(In)34 b(particular,)f(this)g(means)h(that)-236 5668 y(for)44 b(a)g Fo(G)95 5683 y Fv(\016)133 5668 y Fp(-dense)i(subset)g(of)e(pairs)g(\()p Fo(b;)17 b(!)t Fp(\))44 b(the)h(sp)s(ectrum)g(is)f(a)g(Can)m(tor)h(set,)j(whic)m(h)d (is)f(the)h(Bellissard-)-236 5788 y(Simon)31 b(result)h([5].)44 b(F)-8 b(or)32 b(generic)g(results)i(on)e(Can)m(tor)h(sp)s(ectrum)g (for)f(almost)f(p)s(erio)s(dic)g(an)i(quasi-p)s(erio)s(dic)1747 6037 y(3)p eop %%Page: 4 4 4 3 bop -236 219 a Fp(Sc)m(hr\177)-49 b(odinger)41 b(op)s(erators)f (see)i(Moser)f([28])f(and)h(Johnson)g([22].)67 b(Nev)m(ertheless,)45 b(collapsed)40 b(gaps)h(app)s(ear)-236 339 y(naturally)34 b(in)h(quasi-p)s(erio)s(dic)e(Sc)m(hr\177)-49 b(odinger)36 b(op)s(erators,)g(as)g(it)e(w)m(as)j(sho)m(wn)g(b)m(y)f(Bro)s(er,)g (Puig)f(&)h(Sim\023)-49 b(o)34 b([6)o(])-236 459 y(and)c(there)g(are)g 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Fp(:)g Fm(T)j Fn(!)c Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))34 b(homotopic)27 b(to)i(the)g(iden)m (tit)m(y)-8 b(.)42 b(F)-8 b(or)28 b(a)g(general)g Fo(A)g Fp(:)g Fm(T)j Fn(!)c Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\),)35 b(this)28 b(last)-236 459 y(prop)s(ert)m(y)j(is)f (not)h(alw)m(a)m(ys)g(true,)h(since)f Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))36 b(is)30 b(not)h(simply)e(connected.)45 b(Indeed,)32 b(its)f(\014rst)g(homotop)m(y)-236 580 y(group)h(is)g (isomorphic)f(to)h Fm(Z)p Fp(,)e(with)i(generator)h(the)g(rotation)e Fo(R)2149 595 y Fu(1)2216 580 y Fp(:)d Fm(T)j Fn(!)c Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))38 b(giv)m(en)33 b(b)m(y)1188 826 y Fo(R)1262 841 y Fu(1)1302 826 y Fp(\()p Fo(\022)s Fp(\))28 b(=)1557 685 y Fi(\022)1672 765 y Fp(cos)17 b Fo(\022)92 b Fn(\000)17 b Fp(sin)f Fo(\022)1678 885 y Fp(sin)g Fo(\022)186 b Fp(cos)17 b Fo(\022)2282 685 y Fi(\023)-236 1072 y Fp(for)32 b(all)e Fo(\022)h Fn(2)d Fm(T)p Fp(.)47 b(In)33 b(our)f(case,)i(the)f 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Fu(\))p Fv(v)p 1867 1740 250 4 v 1867 1797 a Ft(k)p Fv(A)p Fu(\()p Fv(\022)r Fu(\))p Fv(v)r Ft(k)2127 1763 y Fo(;)17 b(\022)25 b Fp(+)d(2)p Fo(\031)t(!)2512 1653 y Fi(\021)3655 1703 y Fp(\(7\))-236 1995 y(is)32 b(also)g(homotopic)g(to)h(the)g(iden)m(tit)m(y)-8 b(.)45 b(Therefore,)34 b(it)e(admits)g(a)g(con)m(tin)m(uous)i(lift)2851 1970 y(~)2829 1995 y Fo(F)42 b Fp(:)29 b Fm(R)k Fn(\002)23 b Fm(T)32 b Fn(!)c Fm(R)34 b Fn(\002)23 b Fm(T)36 b Fp(of)-236 2115 y(the)d(form:)1150 2211 y(~)1128 2236 y Fo(F)13 b Fp(\()p Fo(t;)k(\022)s Fp(\))28 b(=)g(\()p Fo(t)22 b Fp(+)g Fo(f)11 b Fp(\()p Fo(\022)s(;)17 b(t)p Fp(\))p Fo(;)g(\022)25 b Fp(+)d(2)p Fo(\031)t(!)t Fp(\))-236 2396 y(suc)m(h)34 b(that)490 2547 y Fo(f)11 b Fp(\()p Fo(t)22 b Fp(+)g(2)p Fo(\031)t(;)17 b(\022)25 b Fp(+)d(2)p Fo(\031)t(!)t Fp(\))k(=)i Fo(f)11 b Fp(\()p Fo(t;)17 b(\022)s Fp(\))32 b(and)h Fo(p)17 b Fp(\()o Fo(t)23 b Fp(+)f Fo(f)11 b Fp(\()p Fo(t;)17 b(\022)s Fp(\)\))27 b(=)2636 2479 y Fo(A)p Fp(\()p Fo(\022)s Fp(\))p Fo(p)p Fp(\()p Fo(t)p Fp(\))p 2586 2524 457 4 v 2586 2615 a Fn(k)p Fo(A)p Fp(\()p Fo(\022)s Fp(\))p Fo(p)p Fp(\()p Fo(t)p Fp(\))p Fn(k)-236 2780 y Fp(for)41 b(all)f Fo(t)j Fn(2)h Fm(R)52 b Fp(and)42 b Fo(\022)k Fn(2)e Fm(T)p Fp(.)74 b(The)43 b(map)d Fo(f)53 b Fp(is)41 b(indep)s(enden)m(t)i(of)e (the)h(c)m(hoice)g(of)2876 2755 y(~)2853 2780 y Fo(F)56 b Fp(up)42 b(to)f(the)h(addition)-236 2900 y(of)37 b(a)h(constan)m(t)g (2)p Fo(\031)t(k)s Fp(,)h(with)f Fo(k)i Fn(2)d Fm(Z)p Fp(.)56 b(Since)38 b(the)h(map)e Fo(\022)i Fn(7!)e Fo(\022)29 b Fp(+)c(2)p Fo(\031)t(!)41 b Fp(is)c(uniquely)h(ergo)s(dic)f(on)h Fm(T)j Fp(for)d(all)-236 3021 y(\()p Fo(t;)17 b(\022)s Fp(\))27 b Fn(2)i Fm(R)k Fn(\002)22 b Fm(T)p Fp(,)36 b(the)d(limit)1234 3234 y(lim)1199 3296 y Fv(N)7 b Ft(!1)1504 3166 y Fp(1)p 1430 3211 197 4 v 1430 3302 a(2)p Fo(\031)t(N)1653 3109 y Fv(N)g Ft(\000)p Fu(1)1658 3139 y Fi(X)1663 3348 y Fv(n)p Fu(=0)1823 3234 y Fo(f)1898 3123 y Fi(\020)1980 3209 y Fp(~)1958 3234 y Fo(F)2035 3193 y Fv(n)2081 3234 y Fp(\()p Fo(t;)17 b(\022)s Fp(\))2284 3123 y Fi(\021)-236 3482 y Fp(exists,)34 b(it)f(is)g(indep)s(enden)m(t)i(of)e(\()p Fo(t;)17 b(\022)s Fp(\))34 b(and)f(the)h(con)m(v)m(ergence)j(is)c (uniform)e(in)i(\()p Fo(t;)17 b(\022)s Fp(\),)34 b(see)h(Herman)e([18]) g(and)-236 3602 y(Johnson)e(&)g(Moser)h([23].)43 b(This)31 b(ob)5 b(ject)32 b(is)f(called)e(the)j Fk(\014b)-5 b(er)g(e)g(d)33 b(r)-5 b(otation)33 b(numb)-5 b(er)p Fp(,)31 b(whic)m(h)h(will)c(b)s(e) k(denoted)-236 3722 y(as)h Fo(\032)-66 3737 y Fv(f)-21 3722 y Fp(\()p Fo(a;)17 b(b)p Fp(\),)33 b(and)g(it)e(is)h(de\014ned)i (mo)s(dulus)e Fm(Z)p Fp(.)-90 3843 y(F)-8 b(or)27 b(instance,)j(if)d Fo(A)642 3858 y Fu(0)709 3843 y Fn(2)h Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))34 b(is)28 b(a)g(constan)m(t)h(matrix,)e(then) i(the)g(rotation)d(n)m(um)m(b)s(er)j(of)e(the)i(co)s(cycle)-236 3963 y(\()p Fo(A)-125 3978 y Fu(0)-86 3963 y Fo(;)17 b(!)t Fp(\),)37 b(for)f(an)m(y)h(irrational)c Fo(!)t Fp(,)k(is)f(the)g(absolute)h(v)-5 b(alue)36 b(of)g(the)g(argumen)m(t)h (of)f(the)g(eigen)m(v)-5 b(alues)37 b(divided)-236 4083 y(b)m(y)c(2)p Fo(\031)t Fp(.)-90 4204 y(Using)27 b(a)g(susp)s(ension)h (argumen)m(t)f(\(see)i(Johnson)e([24]\))g(it)g(can)g(b)s(e)h(seen)g (that,)g(for)f(the)h(Almost)e(Mathieu)-236 4324 y(co)s(cycle)k(\(lik)m (e)f(for)g(an)m(y)h(quasi-p)s(erio)s(dic)e(Sc)m(hr\177)-49 b(odinger)30 b(co)s(cycle\),)h(the)f(\014b)s(ered)g(rotation)f(n)m(um)m (b)s(er)h(coincides)-236 4445 y(with)37 b(the)i Fk(Sturmian)45 b Fp(de\014nition)36 b(giv)m(en)i(in)f(the)i(in)m(tro)s(duction.)58 b(Note)38 b(that)f(this)h(last)f(rotation)f(n)m(um)m(b)s(er,)-236 4565 y(rot)o(\()p Fo(a;)17 b(b)p Fp(\),)30 b(b)s(elongs)e(to)g(the)h (in)m(terv)-5 b(al,)28 b(whereas)j(the)d(\014b)s(ered)i(rotation)d(n)m (um)m(b)s(er,)j Fo(\045)2805 4580 y Fv(f)2850 4565 y Fp(\()p Fo(a;)17 b(b)p Fp(\),)30 b(is)e(an)h(elemen)m(t)f(of)-236 4685 y Fm(R)5 b Fo(=)p Fm(Z)p Fp(.)45 b(They)31 b(can)e(b)s(e)g(b)s (oth)g(link)m(ed)f(b)m(y)i(means)f(of)g(the)g Fk(inte)-5 b(gr)g(ate)g(d)31 b(density)h(of)f(states)p Fp(,)f(see)g(Avron)f(&)g (Simon)-236 4806 y([3].)43 b(Let)31 b Fo(k)161 4821 y Fv(L)213 4806 y Fp(\()p Fo(a;)17 b(b;)g(\036)p Fp(\))31 b(b)s(e)h(\()p Fo(L)20 b Fn(\000)g Fp(1\))998 4770 y Ft(\000)p Fu(1)1123 4806 y Fp(times)31 b(the)g(n)m(um)m(b)s(er)h(of)f (eigen)m(v)-5 b(alues)31 b(less)h(than)f(or)h(equal)f(to)g Fo(a)g Fp(for)g(the)-236 4926 y(restriction)h(of)g Fo(H)423 4941 y Fv(b;\036)552 4926 y Fp(to)g(the)i(set)f Fn(f)p Fp(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(L)22 b Fn(\000)h Fp(1)p Fn(g)p Fp(,)33 b(for)f(some)h Fo(\036)28 b Fn(2)g Fm(T)p Fp(,)37 b(with)32 b(zero)i(b)s(oundary)f(conditions)f(at)-236 5047 y(b)s(oth)g(ends)h(0)f(and)g Fo(L)p Fp(.)43 b(Then,)34 b(as)e Fo(L)c Fn(!)f(1)p Fp(,)32 b(the)h Fo(k)1622 5062 y Fv(L)1674 5047 y Fp(\()p Fo(a;)17 b(b;)g(\036)p Fp(\))31 b(con)m(v)m(erge)j(to)e(a)g(con)m(tin)m(uous)h(function)e Fo(k)s Fp(\()p Fo(a;)17 b(b)p Fp(\),)-236 5167 y(whic)m(h)33 b(is)f(the)h(in)m(tegrated)f(densit)m(y)i(of)e(states.)44 b(The)34 b(basic)e(relations)f(are)640 5355 y(2rot)o(\()p Fo(a;)17 b(b)p Fp(\))28 b(=)g Fo(k)s Fp(\()p Fo(a;)17 b(b)p Fp(\))195 b(and)h(2)p Fo(\045)2070 5370 y Fv(f)2115 5355 y Fp(\()p Fo(a;)17 b(b)p Fp(\))28 b(=)g Fo(k)s Fp(\()p Fo(a;)17 b(b)p Fp(\))22 b(+)g Fo(l)r(;)-236 5543 y Fp(for)32 b(a)g(suitable)g(in)m(teger)g Fo(l)e Fn(2)e Fm(Z)p Fp(.)41 b(In)33 b(particular,)1139 5770 y(rot)o(\()p Fo(a;)17 b(b)p Fp(\))28 b(=)g Fo(\032)1657 5785 y Fv(f)1702 5770 y Fp(\()p Fo(a;)17 b(b)p Fp(\))33 b(\(mo)s(d)2215 5702 y(1)p 2215 5747 49 4 v 2215 5838 a(2)2273 5770 y Fm(Z)p Fp(\))p Fo(:)1747 6037 y Fp(5)p eop %%Page: 6 6 6 5 bop -90 219 a Fp(In)38 b(what)g(follo)m(ws,)h(the)f(arithmetic)e (nature)i(of)f(the)i(rotation)d(n)m(um)m(b)s(er)i(will)e(b)s(e)i(of)f (imp)s(ortance.)58 b(W)-8 b(e)-236 339 y(will)38 b(sa)m(y)k(that)e(the) h(rotation)e(n)m(um)m(b)s(er)i(is)f Fk(r)-5 b(ational)42 b(or)g(r)-5 b(esonant)49 b Fp(with)41 b(resp)s(ect)h(to)e Fo(!)k Fp(if)39 b(there)i(exists)h(a)-236 459 y(constan)m(t)h Fo(k)k Fn(2)e Fm(Z)40 b Fp(suc)m(h)k(that)e(rot)o(\()p Fo(a;)17 b(b)p Fp(\))45 b(=)f Fn(f)p Fo(k)s(!)t Fn(g)p Fo(=)p Fp(2)d(or)h(equiv)-5 b(alen)m(tly)d(,)44 b Fo(\045)2547 474 y Fv(f)2593 459 y Fp(\()p Fo(a;)17 b(b)p Fp(\))44 b(=)g Fo(k)s(!)t(=)p Fp(2)e(mo)s(dulus)3641 420 y Fu(1)p 3641 436 36 4 v 3641 494 a(2)3686 459 y Fm(Z)p Fp(.)-236 580 y(Also,)32 b(w)m(e)h(sa)m(y)h(that)e(it)g(is)g Fk(Diophantine)38 b Fp(with)33 b(resp)s(ect)g(to)g Fo(!)i Fp(whenev)m(er)h(the)d(b)s (ound)656 674 y Fi(\014)656 734 y(\014)656 793 y(\014)656 853 y(\014)689 818 y Fp(rot\()p Fo(a;)17 b(b)p Fp(\))23 b Fn(\000)1158 751 y(f)p Fo(k)s(!)t Fn(g)p 1158 795 219 4 v 1243 887 a Fp(2)1386 674 y Fi(\014)1386 734 y(\014)1386 793 y(\014)1386 853 y(\014)1447 818 y Fp(=)28 b(min)1575 881 y Fv(l)q Ft(2)p Fj(Z)1730 674 y Fi(\014)1730 734 y(\014)1730 793 y(\014)1730 853 y(\014)1763 818 y Fo(\032)1813 833 y Fv(f)1859 818 y Fp(\()p Fo(a;)17 b(b)p Fp(\))22 b Fn(\000)2203 751 y Fo(k)s(!)p 2203 795 119 4 v 2237 887 a Fp(2)2353 818 y Fn(\000)2471 751 y Fo(l)p 2463 795 49 4 v 2463 887 a Fp(2)2521 674 y Fi(\014)2521 734 y(\014)2521 793 y(\014)2521 853 y(\014)2582 818 y Fn(\025)2729 751 y Fo(K)p 2697 795 153 4 v 2697 887 a Fn(j)p Fo(k)s Fn(j)2807 858 y Fv(\034)2860 818 y Fo(;)-236 1062 y Fp(holds)32 b(for)g(all)f Fo(k)f Fn(2)e Fm(Z)20 b Fn(\000)j(f)p Fp(0)p Fn(g)32 b Fp(and)g(suitable)g(\014xed)i(p)s(ositiv)m(e)e(constan)m(ts)h Fo(K)40 b Fp(and)33 b Fo(\034)11 b Fp(.)-236 1346 y Fh(Reducibilit)l(y) -236 1531 y Fp(A)41 b(main)f(to)s(ol)f(in)i(the)g(study)i(of)e(quasi-p) s(erio)s(dic)e(sk)m(ew-pro)s(duct)k(\015o)m(ws)f(is)f(its)g Fk(r)-5 b(e)g(ducibility)50 b Fp(to)41 b(constan)m(t)-236 1652 y(co)s(e\016cien)m(ts.)87 b(Reducibilit)m(y)45 b(is)h(a)h(concept) g(de\014ned)i(for)d(the)h(con)m(tin)m(uous)g(and)g(discrete)h(case)f (\(for)g(an)-236 1772 y(in)m(tro)s(duction)33 b(see)i(the)g(reviews)g (b)m(y)h(Eliasson)d([14],)i([15)o(])g(and,)g(for)f(more)f(references,)k (the)e(surv)m(ey)i([30)o(])e(b)m(y)-236 1892 y(the)e(author\).)-90 2013 y(A)39 b(quasi-p)s(erio)s(dic)e(sk)m(ew-pro)s(duct)j(\015o)m(w)g (lik)m(e)e(\(3\),)i(or)f(a)f(quasi-p)s(erio)s(dic)f(co)s(cycle)i(lik)m (e)f(\(6\),)i(with)f Fo(A)f Fp(:)-236 2133 y Fm(T)51 b Fn(!)d Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\),)53 b(is)44 b(said)h(to)f(b)s(e)h Fk(r)-5 b(e)g(ducible)45 b(to)h(c)-5 b(onstant)46 b(c)-5 b(o)g(e\016cients)52 b Fp(if)43 b(there)i(is)g(a)f(con)m(tin)m(uous)h(map)-236 2253 y Fo(Z)40 b Fp(:)33 b Fm(T)k Fn(!)32 b Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))41 b(and)36 b(a)g(constan)m(t)h (matrix)d Fo(B)k Fn(2)c Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)t Fp(\),)43 b(called)34 b(the)i(Flo)s(quet)f(matrix,)g(suc)m(h)j (that)-236 2374 y(the)33 b(conjugation)1224 2494 y Fo(A)p Fp(\()p Fo(\022)s Fp(\))p Fo(Z)7 b Fp(\()p Fo(\022)s Fp(\))28 b(=)f Fo(Z)7 b Fp(\()p Fo(\022)25 b Fp(+)d(2)p Fo(\031)t(!)t Fp(\))p Fo(B)1340 b Fp(\(8\))-236 2654 y(is)24 b(satis\014ed)h(for)g(all)d Fo(\022)31 b Fn(2)d Fm(T)p Fo(:)h Fp(When)c Fo(!)k Fp(is)24 b(rational,)g(in)g(whic)m(h)h (case)h(the)f(\015o)m(w)h(is)e(p)s(erio)s(dic,)h(an)m(y)h(sk)m(ew-pro)s (duct)-236 2775 y(\015o)m(w)k(is)g(reducible)g(to)f(constan)m(t)i(co)s (e\016cien)m(ts.)44 b(Ev)m(en)32 b(in)d(this)h(p)s(erio)s(dic)e(case,)k (it)c(is)i(not)g(alw)m(a)m(ys)g(p)s(ossible)g(to)-236 2895 y(reduce)g(with)e(the)h(same)f(frequency)j Fo(!)t Fp(,)e(but)g(with)f Fo(!)t(=)p Fp(2.)41 b(If)28 b(there)i(is)e(a)g (reduction)h(to)f(constan)m(t)h(co)s(e\016cien)m(ts)-236 3016 y(lik)m(e)j(\(8\),)g(then)h(a)f(fundamen)m(tal)g(matrix)f(of)h (solutions)g(of)g(\(3\),)865 3202 y Fo(X)946 3217 y Fv(n)p Fu(+1)1083 3202 y Fp(\()p Fo(\036)p Fp(\))27 b(=)h Fo(A)p Fp(\(2)p Fo(\031)t(n!)d Fp(+)d Fo(\036)p Fp(\))p Fo(X)1986 3217 y Fv(n)2033 3202 y Fp(\()p Fo(\036)p Fp(\))p Fo(;)211 b(n)28 b Fn(2)g Fm(Z)p Fo(;)-236 3389 y Fp(with)k Fo(X)67 3404 y Fu(0)134 3389 y Fp(:)c Fm(T)j Fn(!)c Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))38 b(con)m(tin)m(uous,)c(has)f(the)g (follo)m(wing)c Fk(Flo)-5 b(quet)35 b(r)-5 b(epr)g(esentation)7 b Fp(:)955 3576 y Fo(X)1036 3591 y Fv(n)1083 3576 y Fp(\()p Fo(\036)p Fp(\))28 b(=)f Fo(Z)7 b Fp(\(2)p Fo(\031)t(n!)25 b Fp(+)d Fo(\036)p Fp(\))p Fo(B)1985 3535 y Fv(n)2032 3576 y Fo(Z)7 b Fp(\()p Fo(\036)p Fp(\))2240 3535 y Ft(\000)p Fu(1)2334 3576 y Fo(X)2415 3591 y Fu(0)2454 3576 y Fp(\()p Fo(\036)p Fp(\))1067 b(\(9\))-236 3763 y(for)33 b(all)f Fo(n)e Fn(2)g Fm(Z)h Fp(and)j Fo(\036)29 b Fn(2)i Fm(T)p Fp(.)50 b(This)34 b(giv)m(es)g(a)g(complete)f(description)g(of)h(the)g (qualitativ)m(e)e(b)s(eha)m(viour)i(of)f(the)-236 3884 y(\015o)m(w)g(\(3\).)-90 4004 y(The)f(rotation)d(n)m(um)m(b)s(er)j(of)e (a)h(quasi-p)s(erio)s(dic)e(co)s(cycle)j(is)e(not)h(in)m(v)-5 b(arian)m(t)30 b(through)h(a)f(conjugation)g(lik)m(e)-236 4124 y(\(8\).)43 b(There)34 b(are)e(ho)m(w)m(ev)m(er)j(the)e(follo)m (wing)d(easy)k(relations:)-236 4315 y Fl(Prop)s(osition)h(5)49 b Fk(L)-5 b(et)42 b Fo(!)i Fk(b)-5 b(e)41 b(an)f(irr)-5 b(ational)41 b(numb)-5 b(er)40 b(and)h Fp(\()p Fo(A)2125 4330 y Fu(1)2164 4315 y Fo(;)17 b(!)t Fp(\))40 b Fk(and)h Fp(\()p Fo(A)2658 4330 y Fu(2)2697 4315 y Fo(;)17 b(!)t Fp(\))40 b Fk(b)-5 b(e)41 b(two)g(quasi-p)-5 b(erio)g(dic)-236 4435 y(c)g(o)g(cycles)46 b(on)g Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))p Fn(\002)53 b Fk(homotopic)46 b(to)h(the)g(identity,)j(b)-5 b(eing)46 b Fo(\032)2288 4450 y Fu(1)2375 4435 y Fk(and)g Fo(\032)2626 4450 y Fu(2)2713 4435 y Fk(the)h(c)-5 b(orr)g(esp)g(onding)45 b(\014b)-5 b(er)g(e)g(d)-236 4556 y(r)g(otation)35 b(numb)-5 b(ers.)44 b(Assume)35 b(that)g(ther)-5 b(e)35 b(exists)f(a)h(c)-5 b(ontinuous)35 b(map)f Fo(Z)g Fp(:)28 b Fm(T)j Fn(!)d Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)t Fp(\))41 b Fk(such)35 b(that)1125 4742 y Fo(A)1198 4757 y Fu(1)1238 4742 y Fp(\()p Fo(\022)s Fp(\))p Fo(Z)7 b Fp(\()p Fo(\022)s Fp(\))28 b(=)f Fo(Z)7 b Fp(\()p Fo(\022)25 b Fp(+)d(2)p Fo(\031)t(!)t Fp(\))p Fo(A)2255 4757 y Fu(2)2294 4742 y Fp(\()p Fo(\022)s Fp(\))-236 4929 y Fk(for)34 b(al)5 b(l)35 b Fo(\022)c Fn(2)d Fm(T)p Fk(.)48 b(Then,)34 b(if)h Fo(k)30 b Fn(2)e Fm(Z)33 b Fk(is)h(the)h(de)-5 b(gr)g(e)g(e)34 b(of)h Fo(Z)7 b Fk(,)1244 5116 y Fo(\032)1294 5131 y Fu(1)1362 5116 y Fp(=)27 b Fo(\032)1515 5131 y Fu(2)1577 5116 y Fp(+)22 b Fo(k)s(\013)36 b Fk(mo)-5 b(dulus)34 b Fm(Z)p Fo(:)-90 5307 y Fp(This)29 b(prop)s(osition)e(sho)m(ws)j (that,)f(for)f(an)m(y)i(\014xed)g(irrational)25 b(frequency)31 b Fo(!)t Fp(,)d(the)i(class)e(of)h(quasi-p)s(erio)s(dic)-236 5427 y(co)s(cycles)35 b(with)g(rational)d(rotation)h(n)m(um)m(b)s(er)i (\(resp.)50 b(with)35 b(Diophan)m(tine)e(rotation)g(n)m(um)m(b)s(er\))i (is)f(in)m(v)-5 b(arian)m(t)-236 5547 y(under)43 b(conjugation,)g (although)e(the)h(rotation)f(n)m(um)m(b)s(er)h(itself)f(ma)m(y)h(c)m (hange.)72 b(Also,)44 b(that)e(whenev)m(er)j(a)-236 5668 y(quasi-p)s(erio)s(dic)34 b(sk)m(ew-pro)s(duct)k(\015o)m(w)f(in)f Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))30 b Fn(\002)c Fm(T)39 b Fp(is)d(reducible)g(to)h(a)f(Flo)s(quet)f(matrix)g(with)h (trace)-236 5788 y Fn(\006)p Fp(2,)d(the)g(rotation)e(n)m(um)m(b)s(er)i (m)m(ust)f(b)s(e)h(rational.)1747 6037 y(6)p eop %%Page: 7 7 7 6 bop -236 219 a Fh(Dualit)l(y)46 b(and)f(lac)l(k)g(of)g(co)t (existence)-236 403 y Fp(T)-8 b(o)30 b(end)g(this)g(section,)h(let)e (us)h(presen)m(t)i(a)e(sp)s(eci\014c)g(feature)h(of)e(the)h(Almost)f (Mathieu)h(op)s(erator)f(or,)h(rather,)-236 524 y(of)39 b(the)i(asso)s(ciated)e(eigen)m(v)-5 b(alue)40 b(equation)f(whic)m(h)i (will)d(in)h(the)h(basis)g(of)f(our)h(argumen)m(ts.)66 b(It)40 b(is)f(part)h(of)-236 644 y(what)33 b(is)f(kno)m(wn)i(as)e Fk(A)n(ubry)k(duality)41 b Fp(or)33 b(simply)e Fk(duality)9 b Fp(:)-236 861 y Fl(Theorem)37 b(6)g(\(Avron)g(&)h(Simon)e([3]\))49 b Fk(F)-7 b(or)33 b(every)h(irr)-5 b(ational)34 b Fo(!)t Fk(,)f(the)h(r)-5 b(otation)35 b(numb)-5 b(er)33 b(of)h(\(2\))g(satis-) -236 981 y(\014es)g(the)h(r)-5 b(elation)1251 1102 y Fp(rot)o(\()p Fo(a;)17 b(b)p Fp(\))28 b(=)g(rot\(2)p Fo(a=b;)17 b Fp(4)p Fo(=b)p Fp(\))1313 b(\(10\))-236 1272 y Fk(for)34 b(al)5 b(l)35 b Fo(b)28 b Fn(6)p Fp(=)g(0)34 b Fk(and)h Fo(a)27 b Fn(2)i Fm(R)t Fk(.)-90 1489 y Fp(According)i(to)g (Prop)s(osition)e(3)i(this)g(means)g(that)g(the)h(sp)s(ectrum)f Fo(\033)2435 1505 y Fu(4)p Fv(=b)2541 1489 y Fp(,)g(for)g Fo(b)d Fn(6)p Fp(=)g(0)j(is)f(just)i(a)f(dilatation)-236 1609 y(of)36 b(the)g(sp)s(ectrum)h Fo(\033)535 1624 y Fv(b)570 1609 y Fp(.)55 b(In)36 b(particular,)g Fo(\033)1314 1624 y Fv(b)1385 1609 y Fp(is)g(a)g(Can)m(tor)g(set)h(\(resp.)56 b(none)37 b(of)f(the)g(sp)s(ectral)h(gaps)f(of)g Fo(\033)3643 1624 y Fv(b)3714 1609 y Fp(is)-236 1730 y(collapsed\))24 b(if,)h(and)g(only)g(if)f Fo(\033)848 1745 y Fu(4)p Fv(=b)978 1730 y Fp(is)g(a)h(Can)m(tor)g(set)h(\(resp.)42 b(none)25 b(of)g(the)g(sp)s(ectral)g(gaps)g(of)f Fo(\033)3108 1745 y Fu(4)p Fv(=b)3239 1730 y Fp(is)g(collapsed\).)-90 1850 y(In)31 b(the)g(pro)s(of)f(of)g(our)g(t)m(w)m(o)h(main)e(results)i(w)m (e)g(will)e(use)i(the)g(follo)m(wing)d(argumen)m(t,)j(whic)m(h)g(is)f (analogous)-236 1971 y(to)35 b(Ince's)j(argumen)m(t)e(for)f(the)i (classical)d(Mathieu)i(p)s(erio)s(dic)e(di\013eren)m(tial)g(equation)i (\(see)h([19])f Fn(x)p Fp(7.41\).)53 b(In)-236 2091 y(principle,)39 b(the)g(eigen)m(v)-5 b(alue)38 b(equation)h(of)f(a)h(general)f(quasi-p) s(erio)s(dic)e(Sc)m(hr\177)-49 b(odinger)39 b(op)s(erator)g(ma)m(y)f (ha)m(v)m(e)-236 2211 y(t)m(w)m(o)43 b(linearly)e(indep)s(enden)m(t)j (quasi-p)s(erio)s(dic)d(solutions)h(with)g(frequency)j Fo(!)h Fp(\(or)c Fo(!)t(=)p Fp(2\).)73 b(One)44 b(ma)m(y)e(call)-236 2332 y(this)33 b(phenomenon)i Fk(c)-5 b(o)g(existenc)g(e)40 b Fp(of)33 b(quasi-p)s(erio)s(dic)f(solutions,)h(in)g(analogy)g(with)h (the)g(classical)e(Flo)s(quet)-236 2452 y(theory)j(for)e(second-order)j (p)s(erio)s(dic)c(di\013eren)m(tial)h(equations.)48 b(A)35 b(trivial)c(example)j(of)g(this)g(o)s(ccurs)h(in)e(the)-236 2572 y(Almost)e(Mathieu)h(case)i(for)e Fo(b)c Fp(=)g(0)k(and)h (suitable)e(v)-5 b(alues)33 b(of)f Fo(a)p Fp(.)-90 2693 y(Let)k(us)h(no)m(w)g(sho)m(w)g(that)f(in)f(the)i(Almost)e(Mathieu)h (case)h(this)e(do)s(es)i(not)f(happ)s(en)h(if)e Fo(b)f Fn(6)p Fp(=)f(0,)k(i.e.)54 b(t)m(w)m(o)-236 2813 y(quasi-p)s(erio)s (dic)22 b(solutions)h(with)g(frequency)j Fo(!)i Fp(of)23 b(the)i(eigen)m(v)-5 b(alue)23 b(equation)h(cannot)g(co)s(exist.)41 b(Let)24 b(\()p Fo(x)3556 2828 y Fv(n)3603 2813 y Fp(\))3641 2828 y Fv(n)p Ft(2)p Fj(Z)-236 2934 y Fp(satisfy)32 b(the)h(equation) 686 3144 y Fo(x)741 3159 y Fv(n)p Fu(+1)901 3144 y Fp(+)22 b Fo(x)1054 3159 y Fv(n)p Ft(\000)p Fu(1)1214 3144 y Fp(+)g Fo(b)17 b Fp(cos)q(\(2)p Fo(\031)t(!)t(n)k Fp(+)h Fo(\036)p Fp(\))p Fo(x)2040 3159 y Fv(n)2115 3144 y Fp(=)27 b Fo(ax)2324 3159 y Fv(n)2372 3144 y Fo(;)212 b(n)27 b Fn(2)i Fm(Z)746 b Fp(\(11\))-236 3354 y(for)40 b(some)g Fo(a)p Fp(,)j Fo(b)f Fn(6)p Fp(=)f(0)g(and)f Fo(\036)p Fp(.)68 b(If)40 b(it)g(is)g(quasi-p)s(erio)s(dic)f(with)h(frequency)j Fo(!)t Fp(,)f(there)f(exists)g(a)g(con)m(tin)m(uous)-236 3474 y(function)34 b Fo( )i Fp(:)c Fm(T)j Fn(!)d Fm(R)46 b Fp(suc)m(h)36 b(that)f Fo(x)1134 3489 y Fv(n)1214 3474 y Fp(=)c Fo( )t Fp(\(2)p Fo(\031)t(!)t(n)23 b Fp(+)h Fo(\036)p Fp(\))35 b(for)f(all)f Fo(n)f Fn(2)h Fm(Z)p Fp(.)48 b(The)36 b(F)-8 b(ourier)34 b(co)s(e\016cien)m(ts)i(of)e Fo( )t Fp(,)-236 3595 y(\()p Fo( )-135 3610 y Fv(m)-68 3595 y Fp(\))-30 3610 y Fv(m)p Ft(2)p Fj(Z)160 3595 y Fp(satisfy)f(the)g(follo)m(wing)d(equation:)604 3854 y(2)17 b(cos\(2)p Fo(\031)t(!)t(m)p Fp(\))p Fo( )1197 3869 y Fv(m)1285 3854 y Fp(+)1396 3787 y Fo(b)p 1393 3831 49 4 v 1393 3923 a Fp(2)1452 3854 y(\()p Fo( )1553 3869 y Fv(m)p Fu(+1)1732 3854 y Fp(+)22 b Fo( )1893 3869 y Fv(m)p Ft(\000)p Fu(1)2050 3854 y Fp(\))28 b(=)f Fo(a )2333 3869 y Fv(m)2400 3854 y Fo(;)212 b(m)28 b Fn(2)g Fm(Z)p Fo(;)-236 4090 y Fp(whic)m(h)33 b(is)f(equiv)-5 b(alen)m(t)32 b(to)632 4339 y Fo( )695 4354 y Fv(m)p Fu(+1)874 4339 y Fp(+)22 b Fo( )1035 4354 y Fv(m)p Ft(\000)p Fu(1)1214 4339 y Fp(+)1322 4272 y(4)p 1322 4316 V 1326 4407 a Fo(b)1397 4339 y Fp(cos)q(\(2)p Fo(\031)t(!)t(m)p Fp(\))p Fo( )1925 4354 y Fv(m)2019 4339 y Fp(=)2132 4272 y(2)p Fo(a)p 2132 4316 100 4 v 2162 4407 a(b)2242 4339 y( )2305 4354 y Fv(m)2372 4339 y Fo(;)212 b(m)28 b Fn(2)g Fm(Z)p Fo(:)692 b Fp(\(12\))-90 4584 y(Since)45 b Fo( )j Fp(is)c(at)g(least)g(con)m (tin)m(uous,)k(then)d(\()p Fo( )1634 4599 y Fv(m)1701 4584 y Fp(\))1739 4599 y Fv(m)p Ft(2)p Fj(Z)1941 4584 y Fp(b)s(elongs)f(to)g Fo(l)2465 4548 y Fu(2)2505 4584 y Fp(\()p Fm(Z)p Fp(\).)76 b(No)m(w)45 b(the)g(reason)g(for)f(the)-236 4705 y(absence)37 b(of)f(co)s(existing)f(quasi-p)s(erio)s(dic)f (solutions)g(is)i(clear.)53 b(Indeed,)38 b(if)d(if)g(\()p Fo(y)2742 4720 y Fv(n)2788 4705 y Fp(\))2826 4720 y Fv(n)p Ft(2)p Fj(Z)3000 4705 y Fp(is)g(another)h(linearly)-236 4825 y(indep)s(enden)m(t)h(quasi-p)s(erio)s(dic)e(solution)f(of)i (\(11\))g(with)g(frequency)i Fo(!)t Fp(,)f(sa)m(y)g Fo(y)2668 4840 y Fv(n)2749 4825 y Fp(=)d Fo(\037)p Fp(\(2)p Fo(\031)t(!)27 b Fp(+)e Fo(\036)p Fp(\),)37 b(for)f(some)-236 4946 y(con)m(tin)m(uous) 27 b Fo(\037)p Fp(,)g(then)g(the)g(sequence)i(of)d(the)h(F)-8 b(ourier)25 b(co)s(e\016cien)m(ts)i(of)f Fo(\037)p Fp(,)h(\()p Fo(\037)2541 4961 y Fv(m)2608 4946 y Fp(\))2646 4961 y Fv(m)p Ft(2)p Fj(Z)2804 4946 y Fp(,)g(w)m(ould)g(b)s(e)f(a)g (solution)f(of)-236 5066 y(\(12\))c(b)s(elonging)f(to)i Fo(l)530 5030 y Fu(2)570 5066 y Fp(\()p Fm(Z)p Fp(\).)37 b(The)23 b(sequences)j(\()p Fo( )1503 5081 y Fv(m)1569 5066 y Fp(\))1607 5081 y Fv(m)p Ft(2)p Fj(Z)1787 5066 y Fp(and)d(\()p Fo(\037)2066 5081 y Fv(m)2132 5066 y Fp(\))2170 5081 y Fv(m)p Ft(2)p Fj(Z)2350 5066 y Fp(w)m(ould)f(b)s(e)g (t)m(w)m(o)h(linearly)d(indep)s(enden)m(t)-236 5186 y(solutions)31 b(of)h(\(12\))g(whic)m(h)h(b)s(elong)f(b)s(oth)g(to)h Fo(l)1460 5150 y Fu(2)1499 5186 y Fp(\()p Fm(Z)p Fp(\).)-90 5307 y(This)28 b(is)f(a)h(con)m(tradiction,)g(b)s(ecause)h(for)e(b)s (ounded)i(p)s(oten)m(tials,)e(lik)m(e)h(the)g(cosine,)h(w)m(e)g(are)f (alw)m(a)m(ys)g(in)f(the)-236 5427 y(limit-p)s(oin)m(t)k(case)37 b(\(see)g([7)o(,)f(9])g(for)f(the)i(con)m(tin)m(uous)f(case\).)54 b(In)37 b(our)e(discrete)i(case,)h(this)d(is)g(ev)m(en)j(simpler,)-236 5547 y(since)h(an)m(y)g(solution)f(in)g Fo(l)726 5511 y Fu(2)765 5547 y Fp(\()p Fm(Z)p Fp(\))e(of)i(the)i(eigen)m(v)-5 b(alue)38 b(equation)g(m)m(ust)h(tend)g(to)g(zero)g(at)f Fn(\0061)p Fp(.)62 b(Hence,)42 b(the)-236 5668 y(existence)24 b(of)d(t)m(w)m(o)i(linearly)e(indep)s(enden)m(t)i(solutions)f(b)s (elonging)e(b)s(oth)i(to)g Fo(l)2514 5632 y Fu(2)2554 5668 y Fp(\()p Fm(Z)p Fp(\))d(w)m(ould)k(b)s(e)f(in)g(con)m(tradiction) -236 5788 y(with)32 b(the)h(preserv)-5 b(ation)33 b(of)f(the)h(W)-8 b(ronskian.)1747 6037 y(7)p eop %%Page: 8 8 8 7 bop -90 219 a Fp(Therefore,)34 b(t)m(w)m(o)f(quasi-p)s(erio)s(dic)e (solutions)h(with)g(frequency)j Fo(!)g Fp(cannot)e(co)s(exist)g(if)f Fo(b)c Fn(6)p Fp(=)g(0.)43 b(A)33 b(similar)-236 339 y(argumen)m(t)f(sho)m(ws)i(that)f(quasi-p)s(erio)s(dic)d(solutions)i (of)g(the)h(form)1358 522 y(\()p Fn(\000)p Fp(1\))1560 481 y Fv(n)1607 522 y Fo( )t Fp(\(2)p Fo(\031)t(!)t(n)21 b Fp(+)h Fo(\036)p Fp(\))p Fo(;)1421 b Fp(\(13\))-236 706 y(for)32 b(a)g(con)m(tin)m(uous)h Fo( )f Fp(:)c Fm(T)j Fn(!)c Fm(R)44 b Fp(cannot)32 b(co)s(exist.)-90 826 y(Finally)-8 b(,)29 b(note)i(that)g(the)g(co)s(existence)i(of)e(t)m(w)m(o)g(quasi-p) s(erio)s(dic)e(solutions)h(with)h(frequency)i Fo(!)i Fp(of)30 b(equa-)-236 947 y(tion)36 b(\(11\))h(is)g(equiv)-5 b(alen)m(t)37 b(to)g(the)g(reducibilit)m(y)f(of)h(the)g(corresp)s (onding)h(t)m(w)m(o-dimensional)d(sk)m(ew-pro)s(duct)-236 1067 y(\015o)m(w)i(\(3\),)g(with)f(the)h(iden)m(tit)m(y)f(as)h(Flo)s (quet)f(matrix.)53 b(Similarly)33 b(the)k(co)s(existence)g(of)f(t)m(w)m (o)i(quasi-p)s(erio)s(dic)-236 1188 y(solutions)d(of)g(the)i(t)m(yp)s (e)g(\(13\))e(is)g(equiv)-5 b(alen)m(t)36 b(to)g(the)g(reducibilit)m(y) f(of)g(the)h(\015o)m(w)h(with)f(min)m(us)f(the)i(iden)m(tit)m(y)-236 1308 y(as)c(Flo)s(quet)e(matrix.)-236 1635 y Fq(3)161 b(The)42 b(Strong)e(T)-13 b(en)42 b(Martini)i(Problem)e(for)g(small)h (\(and)f(large\))6 1818 y Fg(j)p Ff(b)p Fg(j)-236 2037 y Fp(In)35 b(this)g(section)g(w)m(e)h(will)c(sho)m(w)k(that)f(for)f(0)e Fo(<)f Fn(j)p Fo(b)p Fn(j)h Fo(<)f(C)7 b Fp(,)36 b(where)g Fo(C)i(>)32 b Fp(0)j(is)f(a)h(suitable)f(constan)m(t,)i(and)f(for)-236 2157 y Fn(j)p Fo(b)p Fn(j)27 b Fo(>)h Fp(4)p Fo(=C)39 b Fp(all)30 b(sp)s(ectral)j(gaps)f(are)h(op)s(en.)-90 2278 y(The)g(theorem)g(from)e(whic)m(h)i(w)m(e)g(will)d(deriv)m(e)j (Corollary)e(4)h(is)g(due)h(to)f(Eliasson)f(and)i(it)e(w)m(as)i (originally)-236 2398 y(stated)c(for)g(the)g(con)m(tin)m(uous)h(case,)h (based)f(on)e(a)h(KAM)g(sc)m(heme.)44 b(It)29 b(can)g(b)s(e)h(adapted)f (to)g(the)g(discrete)h(case)-236 2518 y(to)i(obtain)f(the)i(follo)m (wing:)-236 2705 y Fl(Theorem)k(7)g(\([16,)h(17]\))48 b Fk(Assume)31 b(that)h Fo(!)j Fk(is)c(Diophantine)f(with)h(c)-5 b(onstants)31 b Fo(c)g Fk(and)g Fo(r)s Fk(.)43 b(Then)30 b(ther)-5 b(e)32 b(is)f(a)-236 2826 y(c)-5 b(onstant)35 b Fo(C)7 b Fp(\()p Fo(c;)17 b(r)s Fp(\))35 b Fk(such)h(that,)g(if)g Fn(j)p Fo(b)p Fn(j)29 b Fo(<)g(C)7 b Fp(\()p Fo(c;)17 b(r)s Fp(\))35 b Fk(and)g Fp(rot\()p Fo(a;)17 b(b)p Fp(\))36 b Fk(is)g(either)f(r)-5 b(ational)36 b(or)f(Diophantine,)g(then)-236 2946 y(the)g(quasi-p)-5 b(erio)g(dic)33 b(skew-pr)-5 b(o)g(duct)34 b(\015ow)328 3042 y Fi(\022)443 3121 y Fo(x)498 3136 y Fv(n)p Fu(+1)488 3241 y Fo(x)543 3256 y Fv(n)677 3042 y Fi(\023)778 3182 y Fp(=)881 3042 y Fi(\022)996 3121 y Fo(a)22 b Fn(\000)h Fo(b)17 b Fp(cos)g Fo(\022)1419 3136 y Fv(n)1578 3121 y Fn(\000)p Fp(1)1207 3241 y(1)399 b(0)1745 3042 y Fi(\023)17 b(\022)1995 3121 y Fo(x)2050 3136 y Fv(n)1950 3241 y Fo(x)2005 3256 y Fv(n)p Ft(\000)p Fu(1)2184 3042 y Fi(\023)2274 3182 y Fo(;)216 b(\022)2562 3197 y Fv(n)p Fu(+1)2727 3182 y Fp(=)28 b Fo(\022)2876 3197 y Fv(n)2945 3182 y Fp(+)22 b(2)p Fo(\031)t(!)394 b Fp(\(14\))-236 3428 y Fk(on)41 b Fm(R)-24 3392 y Fu(2)49 3428 y Fn(\002)27 b Fm(T)45 b Fk(is)d(r)-5 b(e)g(ducible)41 b(to)h(c)-5 b(onstant)41 b(c)-5 b(o)g(e\016cients,)42 b(with)g(Flo)-5 b(quet)41 b(matrix)g Fo(B)5 b Fk(,)44 b(by)d(me)-5 b(ans)41 b(of)h(a)f(quasi-) -236 3548 y(p)-5 b(erio)g(dic)33 b(\(with)h(fr)-5 b(e)g(quency)35 b Fo(!)t(=)p Fp(2)p Fk(\))e(and)h(analytic)g(tr)-5 b(ansformation.)43 b(Mor)-5 b(e)g(over,)35 b(if)f Fo(a)g Fk(is)h(at)f(an)g(endp)-5 b(oint)34 b(of)-236 3669 y(a)g(sp)-5 b(e)g(ctr)g(al)33 b(gap)g(of)h Fo(\033)541 3684 y Fv(b)576 3669 y Fk(,)g(then)g(the)g(tr) -5 b(ac)g(e)34 b(of)f Fo(B)39 b Fk(is)34 b Fn(\006)p Fp(2)p Fk(,)g(b)-5 b(eing)33 b Fo(B)g Fp(=)28 b Fn(\006)p Fo(I)42 b Fk(if,)34 b(and)f(only)h(if,)g(the)g(gap)f(c)-5 b(ol)5 b(lapses.)-236 3789 y(Final)g(ly,)34 b(if)g Fo(B)f Fp(=)28 b Fn(\006)p Fo(I)43 b Fk(then)35 b(the)g(tr)-5 b(ansformation)34 b Fo(Z)41 b Fk(c)-5 b(an)35 b(b)-5 b(e)34 b(chosen)g(to)h(have)f(fr)-5 b(e)g(quency)35 b Fo(!)t Fk(.)-90 3976 y Fp(F)-8 b(or)46 b(other)h(reducibilit)m(y)f (results)h(in)f(the)i(con)m(text)g(of)e(quasi-p)s(erio)s(dic)f(Sc)m (hr\177)-49 b(odinger)48 b(op)s(erators)e(see)-236 4096 y(Dinaburg)27 b(&)i(Sinai)e([13])i(and)g(Moser)g(&)g(P\177)-49 b(osc)m(hel)30 b([29])e(for)h(the)g(con)m(tin)m(uous)h(case)g(and)f (Krik)m(orian)e([25)o(])i(and)-236 4216 y(Avila)i(&)h(Krik)m(orian)f ([2])i(for)f(the)h(discrete)g(case.)-90 4337 y(T)-8 b(aking)42 b(in)m(to)g(accoun)m(t)h(the)g(argumen)m(ts)g(from)f(the)h(previous)g (section,)i(Corollary)c(4)h(is)g(immediate.)-236 4457 y(Indeed,)k(let)c Fn(j)p Fo(b)p Fn(j)j Fo(<)g(C)7 b Fp(,)45 b(where)f Fo(C)49 b Fp(is)42 b(the)h(constan)m(t)h(giv)m(en)f(b)m(y)g (the)g(theorem)g(for)f(a)g(\014xed)i(Diophan)m(tine)-236 4578 y(frequency)h Fo(!)t Fp(.)75 b(Then)45 b(the)e(sk)m(ew-pro)s(duct) j(\015o)m(w)d(\(14\))g(is)g(reducible)g(to)g(constan)m(t)h(co)s (e\016cien)m(ts)h(and)e(the)-236 4698 y(Flo)s(quet)26 b(matrix)f(has)j(trace)f Fn(\006)p Fp(2)g(if)f Fo(a)h Fp(is)f(an)h(endp)s(oin)m(t)g(of)g(a)f(sp)s(ectral)h(gap.)41 b(Moreo)m(v)m(er)29 b(the)e(gap)g(is)f(collapsed)-236 4818 y(if,)38 b(and)h(only)e(if,)i(the)f(Flo)s(quet)f(matrix)g Fo(B)43 b Fp(is)38 b Fn(\006)p Fo(I)8 b Fp(.)60 b(Since)39 b(w)m(e)g(ha)m(v)m(e)g(seen)h(in)d(the)i(previous)f(section)h(that)-236 4939 y(\(14\))32 b(for)g Fo(b)c Fn(6)p Fp(=)f(0)33 b(cannot)f(b)s(e)h (reducible)g(to)f(these)i(Flo)s(quet)d(matrices,)h(Corollary)f(4)h (follo)m(ws.)-236 5266 y Fq(4)161 b(Non-p)t(erturbativ)l(e)83 b(lo)t(calization)88 b(and)d(Can)l(tor)f(sp)t(ectrum)6 5449 y(for)54 b Ff(b)39 b Fg(6)p Fe(=)h(0)-236 5668 y Fp(In)c(this)f(section)h(w)m(e)g(will)e(see)i(ho)m(w)h(Corollary)c(2)j (is)f(a)g(consequence)k(of)c(the)h(follo)m(wing)d(theorem)i(on)h(non-) -236 5788 y(p)s(erturbativ)m(e)d(lo)s(calization,)28 b(due)34 b(to)e(Jitomirsk)-5 b(a)m(y)m(a:)1747 6037 y(8)p eop %%Page: 9 9 9 8 bop -236 219 a Fl(Theorem)37 b(8)g(\([20]\))49 b Fk(L)-5 b(et)39 b Fo(!)k Fk(b)-5 b(e)38 b(Diophantine.)57 b(De\014ne)38 b(the)h(set)g Fp(\010)h Fk(of)60 b Fp(resonan)m(t)38 b(phases)i Fk(as)f(the)h(set)f(of)-236 339 y(those)34 b Fo(\036)28 b Fn(2)g Fm(T)38 b Fk(such)d(that)g(the)g(r)-5 b(elation)1112 562 y Fn(j)p Fp(sin)16 b(\()p Fo(\036)22 b Fp(+)g Fo(\031)t(n!)t Fp(\))o Fn(j)28 b Fo(<)f Fp(exp)2036 451 y Fi(\020)2096 562 y Fn(\000j)p Fo(n)p Fn(j)2312 494 y Fd(1)p 2297 506 61 3 v 2297 547 a(2)p Fc(r)2372 451 y Fi(\021)3606 562 y Fp(\(15\))-236 785 y Fk(holds)39 b(for)i(in\014nitely)f(many)g(values)g(of)g Fo(n)p Fk(,)i(b)-5 b(eing)39 b Fo(r)k Fk(the)e(c)-5 b(onstant)40 b(in)g(the)g (de\014nition)g(of)g(a)g(Diophantine)-236 905 y(numb)-5 b(er.)78 b(Then,)49 b(if)d Fo(\036)i Fn(62)h Fp(\010)e Fk(and)f Fn(j)p Fo(b)p Fn(j)i Fo(>)h Fp(2)d Fk(the)g(op)-5 b(er)g(ator)46 b Fo(H)2143 920 y Fv(b;\036)2285 905 y Fk(has)g(only)g(pur)-5 b(e)47 b(p)-5 b(oint)46 b(sp)-5 b(e)g(ctrum)46 b(with)-236 1026 y(exp)-5 b(onential)5 b(ly)35 b(de)-5 b(c)g(aying)35 b(eigenfunctions.)48 b(Mor)-5 b(e)g(over,)36 b(any)g(of)g(these)g(eigenfunctions)f Fp(\()p Fo( )3189 1041 y Fv(n)3236 1026 y Fp(\))3274 1041 y Fv(n)p Ft(2)p Fj(Z)3448 1026 y Fk(satis\014es)-236 1146 y(that)817 1313 y Fo(\014)6 b Fp(\()p Fo(b)p Fp(\))28 b(=)f Fn(\000)61 b Fp(lim)1220 1379 y Ft(j)p Fv(n)p Ft(j!1)1470 1239 y Fp(log)1613 1159 y Fi(\000)1658 1239 y Fo( )1725 1203 y Fu(2)1721 1264 y Fv(n)1791 1239 y Fp(+)22 b Fo( )1956 1203 y Fu(2)1952 1264 y Fv(n)p Fu(+1)2090 1159 y Fi(\001)p 1470 1290 666 4 v 1722 1381 a Fp(2)p Fn(j)p Fo(n)p Fn(j)2173 1313 y Fp(=)27 b(log)2419 1172 y Fi(\022)2502 1246 y Fn(j)p Fo(b)p Fn(j)p 2502 1290 97 4 v 2526 1381 a Fp(2)2609 1172 y Fi(\023)2699 1313 y Fo(:)880 b Fp(\(16\))-90 1577 y(No)m(w)35 b(w)m(e)g(pro)m(v)m(e)h(Corollary)d(2.)48 b(Let)35 b Fn(j)p Fo(b)p Fn(j)30 b Fo(>)h Fp(2.)48 b(Then,)36 b(according)e(to)g(Theorem)h(8,)g(the)f(op)s(erators)g Fo(H)3690 1592 y Fv(b;)p Fu(0)-236 1697 y Fp(and)43 b Fo(H)45 1713 y Fv(b;\031)r(=)p Fu(2)255 1697 y Fp(ha)m(v)m(e)h(only)f (pure)g(p)s(oin)m(t)g(sp)s(ectrum)g(with)g(exp)s(onen)m(tially)f(deca)m (ying)h(eigenfunctions.)75 b(The)-236 1818 y(eigen)m(v)-5 b(alue)32 b(equation)g(asso)s(ciated)h(to)f(these)i(op)s(erators)e(has) h(the)g(follo)m(wing)d(prop)s(erties:)-236 2025 y Fl(Lemma)37 b(9)49 b Fk(L)-5 b(et)35 b Fp(\()p Fo(x)527 2040 y Fv(n)574 2025 y Fp(\))612 2040 y Fv(n)p Ft(2)p Fj(Z)785 2025 y Fk(b)-5 b(e)35 b(a)g(solution)f(of)h(the)g(di\013er)-5 b(enc)g(e)33 b(e)-5 b(quation)671 2226 y Fo(x)726 2241 y Fv(n)p Fu(+1)885 2226 y Fp(+)22 b Fo(x)1038 2241 y Fv(n)p Ft(\000)p Fu(1)1198 2226 y Fp(+)g Fo(b)17 b Fp(cos)q(\(2)p Fo(\031)t(n!)25 b Fp(+)d Fo(\036)p Fp(\))p Fo(x)2024 2241 y Fv(n)2099 2226 y Fp(=)27 b Fo(ax)2308 2241 y Fv(n)2356 2226 y Fo(;)216 b(n)28 b Fn(2)g Fm(Z)p Fo(;)-236 2427 y Fk(for)35 b(some)f(c)-5 b(onstants)34 b Fo(a;)17 b(b)35 b Fk(and)g Fo(\036)27 b Fn(2)i Fm(T)p Fk(.)48 b(Then,)34 b(if)h Fo(\036)27 b Fp(=)h(0)p Fk(,)34 b Fp(\()p Fo(x)2053 2442 y Ft(\000)p Fv(n)2156 2427 y Fp(\))2194 2442 y Fv(n)p Ft(2)p Fj(Z)2367 2427 y Fk(is)h(also)f(a)h(solution)f(of)h(this)g(e)-5 b(quation)-236 2548 y(and,)34 b(if)h Fo(\036)27 b Fp(=)h Fn(\000)p Fo(\031)t(=)p Fp(2)34 b Fk(then)h Fp(\()p Fn(\000)p Fo(x)922 2563 y Ft(\000)p Fv(n)1025 2548 y Fp(\))1063 2563 y Fv(n)p Ft(2)p Fj(Z)1236 2548 y Fk(is)f(also)h(a)f(solution.)-90 2754 y Fp(Let)39 b(us)h(consider)g(\014rst)f(the)h(case)g(of)e(the)i (op)s(erator)e Fo(H)1977 2769 y Fv(b;)p Fu(0)2066 2754 y Fp(.)63 b(According)39 b(to)g(Theorem)g(8,)i(there)f(exists)-236 2875 y(a)c(sequence)j(of)d(eigen)m(v)-5 b(alues)37 b(\()p Fo(a)970 2839 y Fv(k)1012 2875 y Fp(\()p Fo(b)p Fp(\)\))1167 2890 y Fv(k)r Ft(2)p Fj(Z)1338 2875 y Fp(with)f(eigen)m(v)m(ectors)i (\()p Fo( )2218 2839 y Fv(k)2261 2875 y Fp(\()p Fo(b)p Fp(\)\))2416 2890 y Fv(k)r Ft(2)p Fj(Z)2550 2875 y Fp(,)g(exp)s(onen)m (tially)d(lo)s(calized)f(and)-236 2995 y(whic)m(h)h(form)f(a)h (complete)f(orthonormal)e(basis)j(of)g Fo(l)1723 2959 y Fu(2)1763 2995 y Fp(\()p Fm(Z)p Fp(\).)47 b(Moreo)m(v)m(er)37 b(the)e(set)h(of)f(eigen)m(v)-5 b(alues)34 b(\()p Fo(a)3447 2959 y Fv(k)3490 2995 y Fp(\()p Fo(b)p Fp(\)\))3645 3010 y Fv(k)r Ft(2)p Fj(Z)-236 3116 y Fp(m)m(ust)g(b)s(e)h(dense)g(in)f(the) g(sp)s(ectrum)h Fo(\033)1177 3131 y Fv(b)1212 3116 y Fp(.)48 b(Again,)34 b(w)m(e)h(do)f(not)g(write)g(the)h(dep)s(endence)i (on)d Fo(b)g Fp(for)g(simplicit)m(y)-236 3236 y(in)27 b(what)i(follo)m(ws.)40 b(None)29 b(of)f(these)h(eigen)m(v)-5 b(alues)28 b(can)h(b)s(e)f(rep)s(eated,)i(since)f(w)m(e)g(are)f(in)g (the)g(limit)d(p)s(oin)m(t)i(case.)-236 3356 y(W)-8 b(riting)30 b(eac)m(h)k(of)e(the)h Fo( )687 3320 y Fv(k)762 3356 y Fp(as)1475 3477 y Fo( )1542 3436 y Fv(k)1613 3477 y Fp(=)27 b(\()p Fo( )1821 3436 y Fv(k)1817 3501 y(n)1865 3477 y Fp(\))1903 3492 y Fv(n)p Ft(2)p Fj(Z)2041 3477 y Fo(;)-236 3643 y Fp(w)m(e)33 b(de\014ne)1387 3737 y(~)1367 3763 y Fo( )1434 3722 y Fv(k)1476 3763 y Fp(\()p Fo(\022)s Fp(\))28 b(=)1732 3669 y Fi(X)1739 3881 y Fv(k)r Ft(2)p Fj(Z)1892 3763 y Fo( )1959 3722 y Fv(k)1955 3788 y(n)2003 3763 y Fo(e)2048 3722 y Fv(ik)r(\022)2149 3763 y Fo(;)-236 4024 y Fp(for)34 b Fo(\022)h Fn(2)d Fm(T)p Fp(.)54 b(All)33 b(these)j(functions)f(b)s(elong)f(to)g Fo(C)1586 3988 y Fv(a)1579 4050 y(\014)1628 4024 y Fp(\()p Fm(T)p Fo(;)17 b Fm(R)8 b Fp(\),)41 b(the)36 b(set)f(of)g(real)f(analytic)f(functions) i(of)g Fm(T)j Fp(with)-236 4144 y(analytic)31 b(extension)i(to)f Fn(j=)p Fo(\022)s Fn(j)c Fo(<)f(\014)38 b Fp(and)33 b(they)g(are)g(ev)m (en)h(functions)e(of)g Fo(\022)s Fp(,)h(b)s(ecause)h(of)e(Lemma)f(9)h (\(here)h(w)m(e)-236 4265 y(ha)m(v)m(e)c(applied)e(again)f(that)i(w)m (e)h(are)e(in)h(the)g(limit)c(p)s(oin)m(t)j(case\).)43 b(P)m(assing)28 b(to)g(the)g(dual)f(equation,)i(w)m(e)g(obtain)-236 4385 y(that,)j(for)g(eac)m(h)i Fo(k)d Fn(2)d Fm(Z)p Fp(,)i(the)j (sequence)i(\()1303 4359 y(~)1283 4385 y Fo( )1350 4349 y Fv(k)1393 4385 y Fp(\(2)p Fo(\031)t(!)t(n)p Fp(\)\))1738 4400 y Fv(n)p Ft(2)p Fj(Z)1907 4385 y Fp(is)d(a)h(quasi-p)s(erio)s(dic) d(solution)h(of)366 4631 y Fo(x)421 4646 y Fv(n)p Fu(+1)580 4631 y Fp(+)22 b Fo(x)733 4646 y Fv(n)p Ft(\000)p Fu(1)893 4631 y Fp(+)1001 4563 y(4)p 1001 4608 49 4 v 1005 4699 a Fo(b)1076 4631 y Fp(cos)c Fo(\022)1269 4646 y Fv(n)1316 4631 y Fo(x)1371 4646 y Fv(n)1446 4631 y Fp(=)1560 4563 y(2)p Fo(a)p 1560 4608 100 4 v 1589 4699 a(b)1670 4631 y(x)1725 4646 y Fv(n)1772 4631 y Fo(;)212 b(\022)2056 4646 y Fv(n)p Fu(+1)2221 4631 y Fp(=)28 b Fo(\022)2370 4646 y Fv(n)2439 4631 y Fp(+)22 b(2)p Fo(\031)t(!)198 b(n)28 b Fn(2)g Fm(Z)p Fo(;)426 b Fp(\(17\))-236 4871 y(pro)m(vided)40 b Fo(a)h Fp(is)e(no)m(w)i(replaced)g(b)m(y)g Fo(a)1167 4834 y Fv(k)1210 4871 y Fp(.)66 b(W)-8 b(e)40 b(are)h(no)m(w)f(going)f(to)h(see)h(that)f(2)p Fo(a)2738 4834 y Fv(k)2781 4871 y Fo(=b)g Fp(is)g(at)g(an)g(endp)s(oin)m(t)g(of) -236 4991 y(a)d(sp)s(ectral)f(gap)h(and)g(that)g(this)g(is)f (collapsed.)56 b(T)-8 b(o)38 b(do)e(so)i(w)m(e)g(will)c(use)k (reducibilit)m(y)e(as)h(in)f(the)i(pro)s(of)e(of)-236 5111 y(Theorem)g(4.)54 b(F)-8 b(or)35 b(a)h(direct)g(pro)s(of)f(that)h (2)p Fo(a)1425 5075 y Fv(k)1468 5111 y Fo(=b)g Fp(is)g(at)g(an)g(endp)s (oin)m(t)g(of)f(a)h(gap)g(\(it)f(has)h(rational)e(rotation)-236 5232 y(n)m(um)m(b)s(er\),)f(see)g(again)f(Herman)g([18)o(].)-90 5352 y(The)h(fact)e(that)g(\()569 5326 y(~)549 5352 y Fo( )616 5316 y Fv(k)659 5352 y Fp(\(2)p Fo(\031)t(!)t(n)p Fp(\)\))1004 5367 y Fv(n)p Ft(2)p Fj(Z)1173 5352 y Fp(is)g(a)g(quasi-p) s(erio)s(dic)e(solution)h(of)h(\(17\))g(means)h(that,)f(for)g(all)f Fo(\022)g Fn(2)f Fm(T)3709 5316 y Fv(d)3752 5352 y Fp(,)-236 5472 y(the)k(follo)m(wing)d(equation)i(is)g(satis\014ed)446 5594 y Fi(\022)581 5647 y Fp(~)561 5673 y Fo( )628 5637 y Fv(k)671 5673 y Fp(\(4)p Fo(\031)t(!)25 b Fp(+)d Fo(\022)s Fp(\))581 5776 y(~)561 5802 y Fo( )628 5766 y Fv(k)671 5802 y Fp(\(2)p Fo(\031)t(!)j Fp(+)d Fo(\022)s Fp(\))1128 5594 y Fi(\023)1229 5734 y Fp(=)1333 5594 y Fi(\022)1458 5634 y Fu(2)p Fv(a)p 1458 5650 73 4 v 1479 5708 a(b)1563 5673 y Fn(\000)1672 5634 y Fu(4)p 1672 5650 36 4 v 1675 5708 a Fv(b)1734 5673 y Fp(cos)17 b Fo(\022)114 b Fn(\000)p Fp(1)1664 5794 y(1)405 b(0)2208 5594 y Fi(\023)17 b(\022)2433 5647 y Fp(~)2413 5673 y Fo( )2480 5637 y Fv(k)2522 5673 y Fp(\(2)p Fo(\031)t(!)25 b Fp(+)d Fo(\022)s Fp(\))2579 5776 y(~)2559 5802 y Fo( )2626 5766 y Fv(k)2668 5802 y Fp(\()p Fo(\022)s Fp(\))2980 5594 y Fi(\023)3070 5734 y Fo(:)1747 6037 y Fp(9)p eop %%Page: 10 10 10 9 bop -236 219 a Fp(The)33 b(follo)m(wing)d(lemma)h(sho)m(ws)j (that,)e(if)g(this)g(is)g(the)h(case,)g(then)h(the)f(quasi-p)s(erio)s (dic)d(sk)m(ew-pro)s(duct)35 b(\015o)m(w)283 352 y Fi(\022)398 431 y Fo(x)453 446 y Fv(n)p Fu(+1)443 552 y Fo(x)498 567 y Fv(n)632 352 y Fi(\023)733 492 y Fp(=)837 352 y Fi(\022)962 399 y Fu(2)p Fv(a)1034 375 y Fc(k)p 962 415 111 4 v 1002 472 a Fv(b)1105 438 y Fn(\000)1214 399 y Fu(4)p 1214 415 36 4 v 1217 472 a Fv(b)1276 438 y Fp(cos)17 b Fo(\022)1468 453 y Fv(n)1626 438 y Fn(\000)p Fp(1)1209 558 y(1)446 b(0)1794 352 y Fi(\023)17 b(\022)2044 431 y Fo(x)2099 446 y Fv(n)1999 552 y Fo(x)2054 567 y Fv(n)p Ft(\000)p Fu(1)2233 352 y Fi(\023)2323 492 y Fo(;)212 b(\022)2607 507 y Fv(n)p Fu(+1)2772 492 y Fp(=)27 b Fo(\022)2920 507 y Fv(n)2990 492 y Fp(+)22 b(2)p Fo(\031)t(!)349 b Fp(\(18\))-236 766 y(is)32 b(reducible)g(to)g(constan)m(t)i(co)s (e\016cien)m(ts.)-236 989 y Fl(Lemma)j(10)49 b Fk(L)-5 b(et)36 b Fo(A)29 b Fp(:)h Fm(T)j Fn(!)c Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)t Fp(\))42 b Fk(b)-5 b(e)36 b(a)f(r)-5 b(e)g(al)36 b(analytic)f(map,)g(with)h(analytic)g(extension)e(to)i Fn(j=)p Fo(\022)s Fn(j)29 b Fo(<)h(\016)-236 1109 y Fk(for)38 b(some)g Fo(\016)h(>)c Fp(0)p Fk(.)55 b(Assume)39 b(that)g(ther)-5 b(e)39 b(is)f(a)h(nonzer)-5 b(o)38 b(r)-5 b(e)g(al)38 b(analytic)g(map)g Fo(v)h Fp(:)c Fm(T)j Fn(!)c Fm(R)3115 1073 y Fu(2)3161 1109 y Fk(,)39 b(with)g(analytic)-236 1229 y(extension)34 b(to)h Fn(j=)p Fo(\022)s Fn(j)27 b Fo(<)g(\016)39 b Fk(such)c(that)1287 1350 y Fo(v)t Fp(\()p Fo(\022)25 b Fp(+)d(2)p Fo(\031)t(!)t Fp(\))k(=)i Fo(A)p Fp(\()p Fo(\022)s Fp(\))p Fo(v)t Fp(\()p Fo(\022)s Fp(\))-236 1522 y Fk(holds)34 b(for)h(al)5 b(l)34 b Fo(\022)d Fn(2)d Fm(T)p Fo(:)38 b Fk(Then,)c(the)h(quasi-p)-5 b(erio)g(dic)34 b(skew-pr)-5 b(o)g(duct)34 b(\015ow)g(given)g(by)954 1737 y Fo(u)1010 1752 y Fv(n)p Fu(+1)1174 1737 y Fp(=)27 b Fo(A)p Fp(\()p Fo(\022)1433 1752 y Fv(n)1481 1737 y Fp(\))p Fo(u)1575 1752 y Fv(n)1621 1737 y Fo(;)216 b(\022)1909 1752 y Fv(n)p Fu(+1)2075 1737 y Fp(=)27 b Fo(\022)2223 1752 y Fv(n)2293 1737 y Fp(+)22 b(2)p Fo(\031)t(!)t(;)1015 b Fp(\(19\))-236 1952 y Fk(with)38 b Fp(\()p Fo(u)73 1967 y Fv(n)119 1952 y Fo(;)17 b(\022)208 1967 y Fv(n)255 1952 y Fp(\))33 b Fn(2)h Fm(R)492 1915 y Fu(2)562 1952 y Fn(\002)25 b Fm(T)41 b Fk(for)d(al)5 b(l)38 b Fo(n)33 b Fn(2)h Fm(Z)i Fk(is)h(r)-5 b(e)g(ducible)38 b(to)g(c)-5 b(onstant)37 b(c)-5 b(o)g(e\016cients)37 b(by)i(me)-5 b(ans)37 b(of)g(a)h(quasi-)-236 2072 y(p)-5 b(erio)g(dic)26 b(tr)-5 b(ansformation)25 b(which)h(is)h(analytic)f(in)h Fn(j=)p Fo(\022)s Fn(j)g Fo(<)h(\016)i Fk(and)c(has)h(fr)-5 b(e)g(quency)26 b Fo(!)t Fk(.)42 b(Mor)-5 b(e)g(over)26 b(the)h(Flo)-5 b(quet)-236 2192 y(matrix)34 b(c)-5 b(an)35 b(b)-5 b(e)34 b(chosen)g(to)h(b)-5 b(e)35 b(of)f(the)h(form)1448 2460 y Fo(B)d Fp(=)1658 2319 y Fi(\022)1773 2399 y Fp(1)114 b Fo(c)1773 2519 y Fp(0)c(1)2022 2319 y Fi(\023)3606 2460 y Fp(\(20\))-236 2727 y Fk(for)34 b(some)h Fo(c)27 b Fn(2)h Fm(R)5 b Fk(.)-236 2949 y Fl(Pro)s(of:)48 b Fp(Since)36 b Fo(v)g Fp(=)c(\()p Fo(v)649 2964 y Fu(1)688 2949 y Fo(;)17 b(v)779 2964 y Fu(2)818 2949 y Fp(\))856 2913 y Fv(T)947 2949 y Fp(do)s(es)35 b(not)g(v)-5 b(anish,)36 b Fo(d)c Fp(=)g Fo(v)1919 2913 y Fu(2)1915 2974 y(1)1982 2949 y Fp(+)24 b Fo(v)2133 2913 y Fu(2)2129 2974 y(2)2207 2949 y Fp(is)35 b(alw)m(a)m(ys)h(di\013eren)m(t)f(from)f(zero)i(and)f (the)-236 3070 y(transformation)1054 3223 y Fo(Z)7 b Fp(\()p Fo(\022)s Fp(\))28 b(=)1383 3083 y Fi(\022)1498 3162 y Fo(v)1545 3177 y Fu(1)1585 3162 y Fp(\()p Fo(\022)s Fp(\))110 b Fn(\000)p Fo(v)1943 3177 y Fu(2)1983 3162 y Fp(\()p Fo(\022)s Fp(\))p Fo(=d)p Fp(\()p Fo(\022)s Fp(\))1498 3283 y Fo(v)1545 3298 y Fu(2)1585 3283 y Fp(\()p Fo(\022)s Fp(\))188 b Fo(v)1944 3298 y Fu(1)1983 3283 y Fp(\()p Fo(\022)s Fp(\))p Fo(=d)p Fp(\()p Fo(\022)s Fp(\))2372 3083 y Fi(\023)2462 3223 y Fo(;)-236 3454 y Fp(is)35 b(an)g(analytic)g(map)f Fo(Z)40 b Fp(:)33 b Fm(T)j Fn(!)d Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)t Fp(\).)59 b(The)36 b(transformation)e Fo(Z)42 b Fp(de\014nes)37 b(a)f(conjugation)e(of)h Fo(A)h Fp(with)-236 3574 y Fo(B)-157 3538 y Fu(1)-118 3574 y Fp(,)d(b)s(eing)1129 3694 y Fo(A)p Fp(\()p Fo(\022)s Fp(\))p Fo(Z)7 b Fp(\()p Fo(\022)s Fp(\))27 b(=)h Fo(Z)7 b Fp(\(2)p Fo(\031)t(!)25 b Fp(+)d Fo(\022)s Fp(\))p Fo(B)2224 3653 y Fu(1)2264 3694 y Fp(\()p Fo(\022)s Fp(\))p Fo(;)-236 3867 y Fp(whic)m(h)33 b(means)f(that)h Fo(B)632 3830 y Fu(1)704 3867 y Fp(is)1248 4020 y Fo(B)1327 3979 y Fu(1)1367 4020 y Fp(\()p Fo(\022)s Fp(\))27 b(=)1622 3880 y Fi(\022)1737 3959 y Fp(1)110 b Fo(b)1937 3923 y Fu(1)1937 3984 y(12)2013 3959 y Fp(\()p Fo(\022)s Fp(\))1737 4080 y(0)206 b(1)2178 3880 y Fi(\023)2268 4020 y Fo(;)-236 4255 y Fp(for)27 b(some)g(analytic)g Fo(b)555 4218 y Fu(1)555 4279 y(12)657 4255 y Fp(:)h Fm(T)j Fn(!)d Fm(R)5 b Fp(.)47 b(The)29 b(transformed)e(sk)m(ew-pro)s(duct)j(\015o)m(w,)f (de\014ned)g(b)m(y)g(the)f(matrix)e Fo(B)3647 4218 y Fu(1)3714 4255 y Fp(is)-236 4375 y(reducible)e(to)f(constan)m(t)i(co)s (e\016cien)m(ts)g(b)s(ecause)h(it)d(is)h(in)f(triangular)f(form,)i(the) h(frequency)h Fo(!)h Fp(is)d(Diophan)m(tine)-236 4495 y(and)31 b Fo(b)-7 4459 y Fu(1)-7 4520 y(12)99 4495 y Fp(is)f(analytic.)42 b(Indeed,)32 b(if)e Fo(y)1082 4510 y Fu(12)1184 4495 y Fp(:)e Fm(T)j Fn(!)c Fm(R)42 b Fp(is)30 b(an)h(analytic)f(solution)f(of)i(the)g(small)d(divisors)j(equation)735 4710 y Fo(y)783 4725 y Fu(12)858 4710 y Fp(\(2)p Fo(\031)t(!)25 b Fp(+)d Fo(\022)s Fp(\))g Fn(\000)h Fo(y)1444 4725 y Fu(12)1518 4710 y Fp(\()p Fo(\022)s Fp(\))28 b(=)f Fo(b)1814 4669 y Fu(1)1814 4735 y(12)1889 4710 y Fp(\()p Fo(\022)s Fp(\))c Fn(\000)f Fp([)p Fo(b)2203 4669 y Fu(1)2203 4735 y(12)2279 4710 y Fp(])p Fo(;)211 b(\022)31 b Fn(2)d Fm(T)p Fo(;)-236 4925 y Fp(where)34 b([)p Fo(b)114 4889 y Fu(1)114 4950 y(12)189 4925 y Fp(])e(is)h(the)g(a)m(v)m(erage)g(of)f Fo(b)1016 4889 y Fu(1)1016 4950 y(12)1124 4925 y Fp(\(see)i([1)o(]\),)f (then)g(the)g(transformation)1349 5198 y Fo(Y)21 b Fp(\()p Fo(\022)s Fp(\))28 b(=)1683 5057 y Fi(\022)1798 5137 y Fp(1)110 b Fo(y)2005 5152 y Fu(12)1798 5257 y Fp(0)147 b(1)2121 5057 y Fi(\023)-236 5475 y Fp(conjugates)33 b Fo(B)323 5439 y Fu(1)395 5475 y Fp(with)f(the)h(its)f(a)m(v)m(eraged) i(part:)1235 5743 y Fo(B)f Fp(=)27 b([)p Fo(B)1551 5701 y Fu(1)1591 5743 y Fp(])g(=)1749 5602 y Fi(\022)1864 5682 y Fp(1)110 b([)p Fo(b)2091 5645 y Fu(1)2091 5706 y(12)2167 5682 y Fp(])1864 5802 y(0)171 b(1)2235 5602 y Fi(\023)1723 6037 y Fp(10)p eop %%Page: 11 11 11 10 bop -236 219 a Fp(whic)m(h)33 b(is)f(in)g(the)h(form)e(of)h (\(20\).)p Fb(\003)-90 339 y Fp(Th)m(us,)k(applying)c(this)h(lemma,)f (the)i(\015o)m(w)h(\(18\))e(is)g(reducible)g(to)h(constan)m(t)g(co)s (e\016cien)m(ts)h(with)e(Flo)s(quet)-236 459 y(matrix)i Fo(B)5 b Fp(,)37 b(of)f(the)g(form)f(\(20\).)54 b(That)37 b(is,)g(there)g(exists)g(a)f(real)f(analytic)g(map)g Fo(Z)41 b Fp(:)34 b Fm(T)k Fn(!)33 b Fo(S)6 b(L)p Fp(\(2)p Fo(;)17 b Fm(R)5 b Fp(\))42 b(suc)m(h)-236 580 y(that)1224 700 y Fo(A)p Fp(\()p Fo(\022)s Fp(\))p Fo(Z)7 b Fp(\()p Fo(\022)s Fp(\))28 b(=)f Fo(Z)7 b Fp(\()p Fo(\022)25 b Fp(+)d(2)p Fo(\031)t(!)t Fp(\))p Fo(B)1291 b Fp(\(21\))-236 865 y(for)36 b(all)f Fo(\022)i Fn(2)f Fm(T)p Fp(.)59 b(Moreo)m(v)m(er,)39 b(since)e(the)g(trace)g(of)g Fo(B)k Fp(is)c(2,)g(the)g(rotation)e(n)m(um)m(b)s(er)i(of)g(\(17\))f(is)g (rational,)f(so)-236 985 y(that)d(w)m(e)i(are)e(at)h(the)g(endp)s(oin)m (t)f(of)g(a)g(gap,)h(whic)m(h)g(w)m(e)g(w)m(an)m(t)h(to)e(sho)m(w)i (that)e(is)g(non-collapsed.)-90 1105 y(By)i(the)g(argumen)m(ts)f(of)g (Section)g(2,)g(w)m(e)i(rule)d(out)h(the)h(p)s(ossibilit)m(y)d(of)i Fo(B)38 b Fp(b)s(eing)33 b(the)h(iden)m(tit)m(y)-8 b(.)45 b(Indeed,)-236 1226 y(this)e(w)m(ould)h(imply)e(the)i(co)s(existence)h (of)e(t)m(w)m(o)h(quasi-p)s(erio)s(dic)e(analytic)g(solutions)h(with)g (frequency)j Fo(!)t Fp(,)-236 1346 y(whic)m(h)c(do)s(es)h(happ)s(en)g (in)e(the)h(Almost)f(Mathieu)h(case.)73 b(Therefore)43 b Fo(B)49 b Fn(6)p Fp(=)44 b Fo(I)50 b Fp(and,)45 b(th)m(us,)g Fo(c)f Fn(6)p Fp(=)g(0)e(in)f(the)-236 1467 y(de\014nition)31 b(ab)s(o)m(v)m(e.)-90 1587 y(If)g Fo(B)i Fn(6)p Fp(=)28 b Fo(I)8 b Fp(,)31 b(it)f(is)h(a)g(w)m(ell-kno)m(wn)g(fact)g(of)g(Flo)s (quet)f(theory)i(that)f(2)p Fo(a)2380 1551 y Fv(k)2422 1587 y Fo(=b)h Fp(lies)e(at)h(the)g(endp)s(oin)m(t)g(of)g(a)g(non-)-236 1707 y(collapsed)j(gap)h(\(see,)i(for)e(example,)g(the)h(monograph)e ([35)o(])h(for)g(classical)f(Flo)s(quet)g(theory)i(or)e([6])h(for)g (the)-236 1828 y(con)m(tin)m(uous)j(and)g(quasi-p)s(erio)s(dic)d(Sc)m (hr\177)-49 b(odinger)38 b(case\).)59 b(F)-8 b(or)37 b(the)h(sak)m(e)h(of)e(self-completeness)h(w)m(e)g(sk)m(etc)m(h)-236 1948 y(the)33 b(argumen)m(t.)-90 2068 y(W)-8 b(e)35 b(will)d(see)j (that)f(there)h(exists)g(a)e Fo(\013)1304 2083 y Fu(0)1374 2068 y Fo(>)e Fp(0)i(suc)m(h)j(that)e(if)f(0)d Fo(<)g Fn(j)p Fo(\013)q Fn(j)g Fo(<)g(\013)2591 2083 y Fu(0)2665 2068 y Fp(and)k Fo(\013)h Fp(is)e(either)h(p)s(ositiv)m(e)g(or)-236 2189 y(negativ)m(e)e(\(dep)s(ending)f(on)h(the)g(sign)f(of)g Fo(c)p Fp(\))g(then)h(2)p Fo(a)1697 2153 y Fv(k)1740 2189 y Fo(=b)20 b Fp(+)g Fo(\013)32 b Fp(lies)f(in)g(the)h(resolv)m(en) m(t)g(set)h(of)e Fo(\033)3213 2204 y Fu(4)p Fv(=b)3318 2189 y Fp(.)43 b(T)-8 b(o)32 b(do)f(so,)-236 2309 y(w)m(e)i(will)e(sho) m(w)i(that,)g(for)f(these)i(v)-5 b(alues)32 b(of)g Fo(\013)q Fp(,)h(the)g(sk)m(ew-pro)s(duct)h(\015o)m(w)192 2425 y Fi(\022)307 2504 y Fo(x)362 2519 y Fv(n)p Fu(+1)352 2625 y Fo(x)407 2640 y Fv(n)541 2425 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Fp(are)f(the)h(elemen)m (ts)f(of)g(the)h(matrix)e Fo(Z)44 b Fp(and)37 b(w)m(e)h(ha)m(v)m(e)-236 3732 y(used)j(the)g(relations)e(giv)m(en)i(b)m(y)g(\(21\))f(and)g(the)h (sp)s(ecial)f(form)f(of)h Fo(A)g Fp(and)h Fo(B)5 b Fp(.)67 b(In)41 b(the)g(same)f(calculation,)-236 3852 y(w)m(e)34 b(also)e(see)i(that)f(\()p Fo(z)557 3867 y Fu(11)632 3852 y Fp(\(2)p Fo(\031)t(n!)t Fp(\)\))976 3881 y Fv(n)p Ft(2)p Fj(Z)1147 3852 y Fp(is)g(a)f(quasi-p)s(erio)s(dic)f(solution)h (of)g(equation)h(\(17\))f(and)h(that)g(it)f(is)h(not)-236 3972 y(iden)m(tically)26 b(zero.)42 b(Using)28 b(a)m(v)m(eraging)g (theory)h(\(see,)h(for)e(example,)g([1]\),)h(system)g(\(23\))f(can)g(b) s(e)h(transformed)-236 4093 y(in)m(to)128 4323 y Fo(y)176 4338 y Fv(n)p Fu(+1)396 4323 y Fp(=)555 4183 y Fi(\022\022)743 4262 y Fp(1)22 b(+)g Fo(\013)17 b Fp(\([)p Fo(z)1101 4277 y Fu(11)1176 4262 y Fo(z)1221 4277 y Fu(12)1296 4262 y Fp(])23 b Fn(\000)f Fo(c)p Fp([)p Fo(z)1563 4226 y Fu(2)1559 4287 y(11)1634 4262 y Fp(]\))166 b Fo(c)23 b 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b(done)f(for)h(the)g(op)-5 b(er)g(ator)42 b Fo(H)2171 5544 y Fv(b;\031)r(=)p Fu(2)2381 5528 y Fk(inste)-5 b(ad)41 b(of)h Fo(H)2922 5543 y Fv(b;)p Fu(0)3011 5528 y Fk(.)66 b(In)42 b(this)g(c)-5 b(ase)42 b(the)-236 5648 y(Flo)-5 b(quet)37 b(matrix)g(has)g(tr)-5 b(ac)g(e)38 b Fn(\000)p Fp(2)p Fk(.)53 b(The)37 b(c)-5 b(orr)g(esp)g(onding)36 b(p)-5 b(oint)37 b(eigenvalues)f(c)-5 b(orr)g(esp)g(ond)36 b(to)i(ends)f(of)g(non-)-236 5769 y(c)-5 b(ol)5 b(lapse)-5 b(d)33 b(gaps)h(and)h(ar)-5 b(e)34 b(dense)g(in)h(the)g(sp)-5 b(e)g(ctrum.)1723 6037 y Fp(11)p eop %%Page: 12 12 12 11 bop -236 219 a Fq(Ac)l(kno)l(wledgemen)l(ts)-236 438 y Fp(The)32 b(author)g(is)f(indebted)h(to)g(Hak)-5 b(an)31 b(Eliasson,)h(Raphael)e(Krik)m(orian)h(and)g(Carles)h(Sim\023) -49 b(o)30 b(for)h(stim)m(ulating)-236 558 y(discussions)j(and)f (commen)m(ts)h(on)f(this)g(problem.)45 b(He)33 b(w)m(ould)h(lik)m(e)e (to)h(thank)h(the)g(anon)m(ymous)g(referee)g(for)-236 678 y(his)29 b(useful)g(suggestions.)43 b(He)29 b(also)f(w)m(an)m(ts)j (to)d(thank)i(the)f Fk(Centr)-5 b(e)32 b(de)f(Math)n(\023)-47 b(ematiques)36 b Fp(at)29 b(the)3314 653 y Fk(\023)3299 678 y(Ec)-5 b(ole)32 b(Poly-)-236 799 y(te)-5 b(chnique)45 b Fp(for)37 b(hospitalit)m(y)-8 b(.)58 b(Help)38 b(from)f(the)h (Catalan)f(gran)m(t)h(2000FI71UBPG)f(and)h(gran)m(ts)g(DGICYT)-236 919 y(BFM2000-805)21 b(\(Spain\))h(and)i(CIRIT)g(2000)e(SGR-27,)i(2001) e(SGR-70)g(\(Catalonia\))f(is)i(also)f(ac)m(kno)m(wledged.)-236 1252 y Fq(References)-187 1471 y Fp([1])48 b(V.)29 b(I.)f(Arnol)425 1435 y Ft(0)447 1471 y Fp(d.)36 b Fk(Ge)-5 b(ometric)g(al)31 b(metho)-5 b(ds)30 b(in)g(the)h(the)-5 b(ory)31 b(of)f(or)-5 b(dinary)31 b(di\013er)-5 b(ential)29 b(e)-5 b(quations)p Fp(,)29 b(v)m(olume)-36 1591 y(250)22 b(of)h Fk(Grund)5 b(lehr)-5 b(en)25 b(der)h(Mathematischen)f(Wissenschaften)p Fp(.)i(Springer-V)-8 b(erlag,)23 b(New)g(Y)-8 b(ork,)25 b(1983.)-187 1795 y([2])48 b(A.)41 b(Avila)f(and)h(R.)g(Krik)m(orian.) 66 b(Reducibilit)m(y)39 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