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(\\begin{equation} \\label{eq:Main3relation} \\lim \\limits _{\\varepsilon \\rightarrow 0} \\frac{u_{\\varepsilon }\(t,\\lfloor z/\\varepsilon -lt/\\varepsilon ^{2}\\rfloor \\varepsilon \)}{p\(\\lfloor z/\\varepsilon - lt/\\varepsilon ^{2}\\rfloor \)} \\mathrm{e}^{-E_{0}t/\\varepsilon ^{2}}=u_{0}\(t,z\),\\ z\\in \\mathbb{R}, \\end{equation}) /S /DISPLAYMATH /Pg 158 0 R /ID (767) /P 1082 0 R /A 2082 0 R >> endobj 2080 0 obj << /K 2081 0 R /S /EQNUMBER /Pg 158 0 R /P 2079 0 R /ID (768) >> endobj 2081 0 obj << /K 137 /S /EQNUM /Pg 158 0 R /P 2080 0 R /ID (766) >> endobj 2082 0 obj << /O /Layout /BBox [ 45 81.64 511.26 105.91 ] /Placement /Block >> endobj 2083 0 obj << /K [ 140 2084 0 R 142 2086 0 R 144 2088 0 R 146 ] /S /P /Pg 158 0 R /P 1082 0 R /ID (803) >> endobj 2084 0 obj << /K 141 /Alt ($\\mathbb{R}$) /S /MATH /Pg 158 0 R /ID (769) /P 2083 0 R /A 2085 0 R >> endobj 2085 0 obj << /O /Layout /BBox [ 294 60 301.64 66.87 ] >> endobj 2086 0 obj << /K 143 /Alt ($t\\in \(0,T]$) /S /MATH 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2665 0 R 31 2667 0 R 33 ] /S /P /Pg 235 0 R /P 1082 0 R /ID (1291) >> endobj 2662 0 obj << /K [ << /Obj 238 0 R /Type /OBJR >> 26 ] /S /Link /Pg 235 0 R /P 2661 0 R /ID (1292) >> endobj 2663 0 obj << /K 28 /Alt ($A$) /S /MATH /Pg 235 0 R /ID (1241) /P 2661 0 R /A 2664 0 R >> endobj 2664 0 obj << /O /Layout /BBox [ 354 575 361.5 581.83 ] >> endobj 2665 0 obj << /K 30 /Alt (${\\mathrm{SL}}$) /S /MATH /Pg 235 0 R /ID (1242) /P 2661 0 R /A 2666 0 R >> endobj 2666 0 obj << /O /Layout /BBox [ 423 575 434.81 581.89 ] >> endobj 2667 0 obj << /K [ << /Obj 239 0 R /Type /OBJR >> 32 ] /S /Link /Pg 235 0 R /P 2661 0 R /ID (1293) >> endobj 2668 0 obj << /K [ 34 2669 0 R 37 ] /S /P /Pg 235 0 R /P 1082 0 R /ID (1294) >> endobj 2669 0 obj << /K [ 35 36 ] /Alt ($\\mathcal I:=[2+\\frac{1}{9}\\min \\{g^{2},1\\},2\(\\mathrm{e}^{g}+ \\mathrm{e}^{-g}\)]$) /S /MATH /Pg 235 0 R /ID (1243) /P 2668 0 R /A 2670 0 R >> endobj 2670 0 obj << /O /Layout /BBox [ 211 533.55 363.85 545.45 ] >> endobj 2671 0 obj << /K [ 38 2672 0 R 40 2674 0 R 42 2676 0 R 44 ] /S /P /Pg 235 0 R /P 1082 0 R /ID (1295) >> endobj 2672 0 obj << /K 39 /Alt ($d\\in \\mathbb{N}_{+}\\cup \\{\\infty \\}$) /S /MATH /Pg 235 0 R /ID (1244) /P 2671 0 R /A 2673 0 R >> endobj 2673 0 obj << /O /Layout /BBox [ 140 509.5 203.2 519.5 ] >> endobj 2674 0 obj << /K 41 /Alt ($g>0$) /S /MATH /Pg 235 0 R /ID (1245) /P 2671 0 R /A 2675 0 R >> endobj 2675 0 obj << /O /Layout /BBox [ 210 510.06 233.46 518.3 ] >> endobj 2676 0 obj << /K 43 /Alt ($C_{0}>0$) /S /MATH /Pg 235 0 R /ID (1246) /P 2671 0 R /A 2677 0 R >> endobj 2677 0 obj << /O /Layout /BBox [ 411 510.5 440.96 518.83 ] >> endobj 2678 0 obj << /K 45 /Alt (\\begin{equation*} \\|W_{1}\\|_{\\mathcal B,\\mathcal I}+\\|W_{2}\\|_{\\mathcal B,\\mathcal I}+ \\|V\\|_{\\mathcal B,\\mathcal I}\\leq \\varepsilon \\leq C_{0}\\mathrm{e}^{-12g}g^{6}, \\end{equation*}) /S /DISPLAYMATH /Pg 235 0 R /ID (1247) /P 1082 0 R /A 2679 0 R >> endobj 2679 0 obj << /O /Layout /BBox [ 178 483.14 386.92 494.64 ] /Placement /Block >> endobj 2680 0 obj << /K [ 46 2681 0 R 48 ] /S /P /Pg 235 0 R /P 1082 0 R /ID (1296) >> endobj 2681 0 obj << /K 47 /Alt ($Y, f_{i} \\in \\mathscr B_{\\mathcal I}$) /S /MATH /Pg 235 0 R /ID (1248) /P 2680 0 R /A 2682 0 R >> endobj 2682 0 obj << /O /Layout /BBox [ 120 460.06 165.77 468.97 ] >> endobj 2683 0 obj << /K 49 /Alt (\\begin{equation*} \\mathrm{e}^{-Y\(E,\\theta +\\omega \)}P\(E\)^{-1}S_{W_{1},W_{2},V}^{E,g}\( \\theta \)P\(E\)\\mathrm{e}^{Y\(E,\\theta \)}= \\begin{pmatrix} \\lambda \(E\) e^{ f_{1}\(E,\\theta \)} & 0 \\\\ 0 & \\mu \(E\)e^{ f_{2} \(E,\\theta \)} \\end{pmatrix} , \\end{equation*}) /S /DISPLAYMATH /Pg 235 0 R /ID (1249) /P 1082 0 R /A 2684 0 R >> endobj 2684 0 obj << /O /Layout /BBox [ 112 422.39 453.23 446.61 ] /Placement /Block >> endobj 2685 0 obj << /K [ 50 2686 0 R 52 ] /S /P /Pg 235 0 R /P 1082 0 R /ID (1297) >> endobj 2686 0 obj << /K 51 /Alt ($P\(E\)= \\begin{pmatrix} \\lambda \(E\) & \\mu \(E\) \\\\ 1 & 1 \\end{pmatrix} $) /S /MATH /Pg 235 0 R /ID (1255) /P 2685 0 R /A 2687 0 R >> endobj 2687 0 obj << /O /Layout /BBox [ 78 385.5 185.15 409.5 ] >> endobj 2688 0 obj << /K [ 53 54 55 56 2689 0 R ] /Alt (\\begin{eqnarray} \\label{esf} \\|Y\\|_{\\mathcal B,\\mathcal I}\\leq \\varepsilon ^{\\frac{1}{3}} \\leq\\frac{\\lambda \(E\)}{4\\lambda \(E\)+4\\mu \(E\)}, \\ \\|f_{i}\\|_{\\mathcal B, \\mathcal I}\\leq \\varepsilon ^{\\frac{1}{2}}. \\end{eqnarray}) /S /DISPLAYALIGN /Pg 235 0 R /ID (1257) /P 1082 0 R /A 2690 0 R >> endobj 2689 0 obj << /K 57 /S /EQNUM /Pg 235 0 R /P 2688 0 R /ID (1256) >> endobj 2690 0 obj << /O /Layout /BBox [ 50 348.64 516.26 372.27 ] /Placement /Block >> endobj 2691 0 obj << /K 2692 0 R /S /TRIVLIST /Pg 235 0 R /P 1082 0 R /ID (1258) >> endobj 2692 0 obj << /K [ 2693 0 R 62 2694 0 R 64 2696 0 R 2700 0 R 2705 0 R 2707 0 R 2716 0 R 2718 0 R 2719 0 R 2723 0 R 2726 0 R 2730 0 R 2734 0 R 2737 0 R 2739 0 R 2740 0 R 2744 0 R 2745 0 R 2747 0 R 2750 0 R 2752 0 R 2755 0 R 2757 0 R 2760 0 R 2764 0 R 2767 0 R 2770 0 R 2772 0 R 2778 0 R 2780 0 R 2781 0 R ] /S /Div /Pg 235 0 R /P 2691 0 R /ID (1259) >> endobj 2693 0 obj << /K 61 /S /Span /Pg 235 0 R /P 2692 0 R /ID (1260) >> endobj 2694 0 obj << /K 63 /Alt ($E\\in \\mathcal I$) /S /MATH /Pg 235 0 R /ID (1261) /P 2692 0 R /A 2695 0 R >> endobj 2695 0 obj << /O /Layout /BBox [ 268 328.61 294.36 335.83 ] >> endobj 2696 0 obj << /K [ 65 66 67 68 69 70 71 72 73 74 75 76 77 2697 0 R ] /Alt (\\begin{equation} \\label{eigenvalueseparate} \\begin{aligned} |\\lambda \(E\)/\\mu \(E\)-1|&=\\biggr| \\frac{E+\\sqrt{E^{2}-4}}{E-\\sqrt{E^{2}-4}}-1\\biggr| \\geq\\frac{\\min \\{g^{2},1\\}}{9}, \\\\ |\\mu \(E\)/\\lambda \(E\)-1|&=\\biggr| \\frac{E-\\sqrt{E^{2}-4}}{E+\\sqrt{E^{2}-4}}-1\\biggr|\\geq \\frac{\\min \\{g^{2},1\\}}{27}, \\\\ \\frac{\\lambda \(E\)}{\\lambda \(E\)+\\mu \(E\)}&\\geq\\frac{E-\\sqrt{E^{2}-4}}{2E}\\geq \\frac{\\mathrm{e}^{-2g}}{16}. \\end{aligned} \\end{equation}) /S /DISPLAYMATH /Pg 235 0 R /ID (1263) /P 2692 0 R /A 2699 0 R >> endobj 2697 0 obj << /K 2698 0 R /S /EQNUMBER /Pg 235 0 R /P 2696 0 R /ID (1264) >> endobj 2698 0 obj << /K 79 /S /EQNUM /Pg 235 0 R /P 2697 0 R /ID (1262) >> endobj 2699 0 obj << /O /Layout /BBox [ 50 226.49 516.26 314.51 ] /Placement /Block >> endobj 2700 0 obj << /K [ 83 2701 0 R 85 2703 0 R 87 ] /S /P /Pg 235 0 R /P 2692 0 R /ID (1298) >> endobj 2701 0 obj << /K 84 /Alt ($P\(E\):= \\begin{pmatrix} \\lambda \(E\) & \\mu \(E\) \\\\ 1 & 1 \\end{pmatrix} \\in {\\mathrm{GL}}$) /S /MATH /Pg 235 0 R /ID (1270) /P 2700 0 R /A 2702 0 R >> endobj 2702 0 obj << /O /Layout /BBox [ 96 196.5 187.53 269.05 ] >> endobj 2703 0 obj << /K 86 /Alt ($A_{g}\(E\)$) /S /MATH /Pg 235 0 R /ID (1271) /P 2700 0 R /A 2704 0 R >> endobj 2704 0 obj << /O /Layout /BBox [ 346 196.5 384.62 206.5 ] >> endobj 2705 0 obj << /K 89 /Alt (\\begin{equation*} A'_{g}\(E\):=P\(E\)^{-1}A_{g}\(E\)P\(E\)= diag\(\\lambda \(E\),\\mu \(E\)\). \\end{equation*}) /S /DISPLAYMATH /Pg 235 0 R /ID (1272) /P 2692 0 R /A 2706 0 R >> endobj 2706 0 obj << /O /Layout /BBox [ 173 164.17 393.44 176.64 ] /Placement /Block >> endobj 2707 0 obj << /K [ 91 2708 0 R 93 2710 0 R 95 2712 0 R 97 2714 0 R 99 ] /S /P /Pg 235 0 R /P 2692 0 R /ID (1299) >> endobj 2708 0 obj << /K 92 /Alt ($ \\|\\widetilde{F}_{E,g}\(\\theta \) - \\operatorname{id}\\|_{\\mathcal B, \\mathcal I} \\leq \\mathrm{e}^{g}\\varepsilon $) /S /MATH /Pg 235 0 R /ID (1274) /P 2707 0 R /A 2709 0 R >> endobj 2709 0 obj << /O /Layout /BBox [ 185 139.5 296.91 161.71 ] >> endobj 2710 0 obj << /K 94 /Alt ($F_{g}\\in \\mathscr{B}_{\\mathcal I}$) /S /MATH /Pg 235 0 R /ID (1275) /P 2707 0 R /A 2711 0 R >> endobj 2711 0 obj << /O /Layout /BBox [ 50 126.5 84.3 134.97 ] >> endobj 2712 0 obj << /K 96 /Alt ($ \\widetilde{F}_{E,g}\(\\theta \) =\\mathrm{e}^{F_{g}\(E,\\theta \)} $) /S /MATH /Pg 235 0 R /ID (1277) /P 2707 0 R /A 2713 0 R >> endobj 2713 0 obj << /O /Layout /BBox [ 138 125.5 198.73 160.28 ] >> endobj 2714 0 obj << /K 98 /Alt ($ \\|F_{g}\\|_{\\mathcal B,\\mathcal I} \\leq 2\\mathrm{e}^{g}\\varepsilon $) /S /MATH /Pg 235 0 R /ID (1278) /P 2707 0 R /A 2715 0 R >> endobj 2715 0 obj << /O /Layout /BBox [ 300 125.5 369.91 139.8 ] >> endobj 2716 0 obj << /K 101 /Alt (\\begin{equation*} F'_{g}\(E,\\theta \)=P\(E\)^{-1}F_{g}\(E,\\theta \)P\(E\), \\end{equation*}) /S /DISPLAYMATH /Pg 235 0 R /ID (1279) /P 2692 0 R /A 2717 0 R >> endobj 2717 0 obj << /O /Layout /BBox [ 210 99.17 356.41 111.64 ] /Placement /Block >> endobj 2718 0 obj << /K 103 /S /P /Pg 235 0 R /P 2692 0 R /ID (1300) >> endobj 2719 0 obj << /K [ 105 106 107 108 109 2720 0 R ] /Alt (\\begin{equation} \\label{eq:diagonalreduce} \\begin{aligned} \\|F'_{g}\\|_{\\mathcal B,\\mathcal I}&\\leq \\sup \\limits _{E\\in\\mathcal I} \\frac{2\\|P\(E\)\\|^{2}\\mathrm{e}^{g}\\varepsilon }{|\\det P\(E\)|}\\leq \\sup\\limits _{E\\in \\mathcal I} \\frac{2\(E\\mathrm{e}^{g}+2\)^{2}\\varepsilon }{\\sqrt{E^{2}-4}} \\leq C' \\mathrm{e}^{4g}\\max \\{\\frac{\\varepsilon }{g^{2}},\\varepsilon \\}, \\end{aligned} \\end{equation}) /S /DISPLAYMATH /Pg 235 0 R /ID (1281) /P 2692 0 R /A 2722 0 R >> endobj 2720 0 obj << /K 2721 0 R /S /EQNUMBER /Pg 235 0 R /P 2719 0 R /ID (1282) >> endobj 2721 0 obj << /K 111 /S /EQNUM /Pg 235 0 R /P 2720 0 R /ID (1280) >> endobj 2722 0 obj << /O /Layout /BBox [ 50 40.09 516.26 64.91 ] /Placement /Block >> endobj 2723 0 obj << /K [ 1 2724 0 R 3 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1332) >> endobj 2724 0 obj << /K 2 /Alt ($C'>0$) /S /MATH /Pg 240 0 R /ID (1283) /P 2723 0 R /A 2725 0 R >> endobj 2725 0 obj << /O /Layout /BBox [ 74 678.61 103 686.52 ] >> endobj 2726 0 obj << /K [ 5 2727 0 R 7 2728 0 R 9 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1333) >> endobj 2727 0 obj << /K [ << /Obj 243 0 R /Type /OBJR >> 6 ] /S /Link /Pg 240 0 R /P 2726 0 R /ID (1334) >> endobj 2728 0 obj << /K 8 /Alt ($\(\\omega , A'_{g}\(E\)e^{F'_{g}\(E,\\theta \)}\)$) /S /MATH /Pg 240 0 R /ID (1301) /P 2726 0 R /A 2729 0 R >> endobj 2729 0 obj << /O /Layout /BBox [ 45 648.17 125 661.84 ] >> endobj 2730 0 obj << /K [ 11 2731 0 R ] /Alt (\\begin{equation} \\label{LambdaK} \\Lambda =\\biggr\\{f\\in C\(\\mathcal I\\times \\mathbb{T}^{d},\\mathrm{gl}\(2, \\mathbb R\)\)|f= \\begin{pmatrix} 0 & f_{1} \\\\ f_{2} & 0 \\end{pmatrix} \\in \\mathscr B_{\\mathcal I}\\biggr\\}. \\end{equation}) /S /DISPLAYMATH /Pg 240 0 R /ID (1303) /P 2692 0 R /A 2733 0 R >> endobj 2731 0 obj << /K 2732 0 R /S /EQNUMBER /Pg 240 0 R /P 2730 0 R /ID (1304) >> endobj 2732 0 obj << /K 13 /S /EQNUM /Pg 240 0 R /P 2731 0 R /ID (1302) >> endobj 2733 0 obj << /O /Layout /BBox [ 45 611.5 511.26 635.5 ] /Placement /Block >> endobj 2734 0 obj << /K [ 17 2735 0 R 19 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1335) >> endobj 2735 0 obj << /K 18 /Alt ($Y\\in \\Lambda $) /S /MATH /Pg 240 0 R /ID (1305) /P 2734 0 R /A 2736 0 R >> endobj 2736 0 obj << /O /Layout /BBox [ 169 592.61 196.19 600.16 ] >> endobj 2737 0 obj << /K 21 /Alt (\\begin{equation*} Y\\rightarrow A'_{g}\(E\)^{-1}Y\(E,\\theta +\\omega \)A'_{g}\(E\)-Y\(E,\\theta \) \\end{equation*}) /S /DISPLAYMATH /Pg 240 0 R /ID (1306) /P 2692 0 R /A 2738 0 R >> endobj 2738 0 obj << /O /Layout /BBox [ 185 565.17 372.48 577.64 ] /Placement /Block >> endobj 2739 0 obj << /K 23 /S /P /Pg 240 0 R /P 2692 0 R /ID (1336) >> endobj 2740 0 obj << /K [ 25 2741 0 R ] /Alt (\\begin{equation} \\label{eq:homologicaleq} A'_{g}\(E\)^{-1}Y\(E,\\theta +\\omega \)A'_{g}\(E\)-Y\(E,\\theta \)=G\(\\theta \). \\end{equation}) /S /DISPLAYMATH /Pg 240 0 R /ID (1308) /P 2692 0 R /A 2743 0 R >> endobj 2741 0 obj << /K 2742 0 R /S /EQNUMBER /Pg 240 0 R /P 2740 0 R /ID (1309) >> endobj 2742 0 obj << /K 27 /S /EQNUM /Pg 240 0 R /P 2741 0 R /ID (1307) >> endobj 2743 0 obj << /O /Layout /BBox [ 45 516.17 511.26 528.64 ] /Placement /Block >> endobj 2744 0 obj << /K 31 /S /P /Pg 240 0 R /P 2692 0 R /ID (1337) >> endobj 2745 0 obj << /K 33 /Alt (\\begin{equation*} Y\(E,\\theta \)= \\begin{pmatrix} 0 & Y_{1}\(E,\\theta \) \\\\ Y_{2}\(E,\\theta \) & 0 \\end{pmatrix} ,\\qquad G\(E,\\theta \)= \\begin{pmatrix} 0 & G_{1}\(E,\\theta \) \\\\ G_{2}\(E,\\theta \) & 0 \\end{pmatrix} . \\end{equation*}) /S /DISPLAYMATH /Pg 240 0 R /ID (1310) /P 2692 0 R /A 2746 0 R >> endobj 2746 0 obj << /O /Layout /BBox [ 117 459.5 439.74 483.5 ] /Placement /Block >> endobj 2747 0 obj << /K [ 35 2748 0 R 39 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1338) >> endobj 2748 0 obj << /K [ 36 2749 0 R 38 ] /S /EQNUM /Pg 240 0 R /P 2747 0 R /ID (1311) >> endobj 2749 0 obj << /K [ << /Obj 244 0 R /Type /OBJR >> 37 ] /S /Link /Pg 240 0 R /P 2748 0 R /ID (1339) >> endobj 2750 0 obj << /K [ 41 42 43 44 45 ] /Alt (\\begin{equation*} \\left \\{ \\begin{aligned} \\widehat{Y_{1}\(E\)}\(k\)&= \\frac{\\widehat{G_{1}\(E\)}\(k\)}{\\frac{\\mu}{\\lambda}\(E\)e^{\\mathrm{i}\\langle k,\\omega \\rangle}-1}, \\\\ \\widehat{Y_{2}\(E\)}\(k\)&= \\frac{\\widehat{G_{2}\(E\)}\(k\)}{\\frac{\\lambda}{\\mu}\(E\)\\mathrm{e}^{\\mathrm{i}\\langle k,\\omega \\rangle}-1}. \\end{aligned} \\right . \\end{equation*}) /S /DISPLAYMATH /Pg 240 0 R /ID (1312) /P 2692 0 R /A 2751 0 R >> endobj 2751 0 obj << /O /Layout /BBox [ 211 360.13 345.1 424.87 ] /Placement /Block >> endobj 2752 0 obj << /K [ 47 2753 0 R 51 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1340) >> endobj 2753 0 obj << /K [ 48 2754 0 R 50 ] /S /EQNUM /Pg 240 0 R /P 2752 0 R /ID (1313) >> endobj 2754 0 obj << /K [ << /Obj 245 0 R /Type /OBJR >> 49 ] /S /Link /Pg 240 0 R /P 2753 0 R /ID (1341) >> endobj 2755 0 obj << /K [ 53 54 55 ] /Alt (\\begin{eqnarray*} |\\lambda \(E\)/\\mu \(E\)-\\mathrm{e}^{\\mathrm{i}\\langle k,\\omega \\rangle}| \\geq |1-\\lambda \(E\)/\\mu \(E\)| \\geq \\frac{\\min \\{g^{2},1\\}}{27}, \\\\ |\\mu \(E\)/\\lambda \(E\)-\\mathrm{e}^{\\mathrm{i}\\langle k,\\omega \\rangle}| \\geq |\\mu \(E\)/\\lambda \(E\)-1| \\geq \\frac{\\min \\{g^{2},1\\}}{27}. \\end{eqnarray*}) /S /DISPLAYALIGN /Pg 240 0 R /ID (1314) /P 2692 0 R /A 2756 0 R >> endobj 2756 0 obj << /O /Layout /BBox [ 45 304.14 517.23 325.91 ] /Placement /Block >> endobj 2757 0 obj << /K [ 57 2758 0 R 59 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1342) >> endobj 2758 0 obj << /K 58 /Alt ($Y\\in \\Lambda $) /S /MATH /Pg 240 0 R /ID (1316) /P 2757 0 R /A 2759 0 R >> endobj 2759 0 obj << /O /Layout /BBox [ 105 257.61 132.19 265.16 ] >> endobj 2760 0 obj << /K [ 61 62 2761 0 R ] /Alt (\\begin{equation} \\label{nonresonant} \\|A'_{g}\(E\)^{-1}Y\(E,\\theta +\\omega \)A'_{g}\(E\)-Y\(E,\\theta \)\\|_{ \\mathcal B,\\mathcal I}\\geq \\frac{\\min \\{g^{2},1\\}}{27}\\|Y\\|_{ \\mathcal B,\\mathcal I}, \\end{equation}) /S /DISPLAYMATH /Pg 240 0 R /ID (1318) /P 2692 0 R /A 2763 0 R >> endobj 2761 0 obj << /K 2762 0 R /S /EQNUMBER /Pg 240 0 R /P 2760 0 R /ID (1319) >> endobj 2762 0 obj << /K 64 /S /EQNUM /Pg 240 0 R /P 2761 0 R /ID (1317) >> endobj 2763 0 obj << /O /Layout /BBox [ 45 222.14 511.26 243.91 ] /Placement /Block >> endobj 2764 0 obj << /K [ 68 2765 0 R 71 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1343) >> endobj 2765 0 obj << /K [ 69 70 ] /Alt ($\\Lambda \\subset \\mathscr{B}_{\\mathcal I}^{nre}\( \\frac{\\min \\{g^{2},1\\}}{27}\)$) /S /MATH /Pg 240 0 R /ID (1320) /P 2764 0 R /A 2766 0 R >> endobj 2766 0 obj << /O /Layout /BBox [ 124 195.55 212.92 210.02 ] >> endobj 2767 0 obj << /K [ 73 2768 0 R 75 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1344) >> endobj 2768 0 obj << /K 74 /Alt ($\\det A'_{g}\(E\)=\\lambda \(E\)\\mu \(E\)=e^{2g}$) /S /MATH /Pg 240 0 R /ID (1321) /P 2767 0 R /A 2769 0 R >> endobj 2769 0 obj << /O /Layout /BBox [ 101 182.17 227.74 194.14 ] >> endobj 2770 0 obj << /K [ 77 78 79 ] /Alt (\\begin{equation*} C'\\mathrm{e}^{4g}\\varepsilon \\max \\{\\frac{1}{g^{2}},1\\} \\leq\\frac{e^{4g}\\min \\{g^{4},1\\}}{c_{2} \\sup \\limits _{E\\in \\mathcal I}|\\mu \(E\)|^{6}} ., \\end{equation*}) /S /DISPLAYMATH /Pg 240 0 R /ID (1322) /P 2692 0 R /A 2771 0 R >> endobj 2771 0 obj << /O /Layout /BBox [ 197 137.22 359.33 168.91 ] /Placement /Block >> endobj 2772 0 obj << /K [ 81 2773 0 R 83 2775 0 R 85 2776 0 R 87 ] /S /P /Pg 240 0 R /P 2692 0 R /ID (1345) >> endobj 2773 0 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($\\widetilde B\(E,\\theta \):=\\mathrm{e}^{-Y''\(E,\\theta \)}B'\(E,\\theta \)$) /S /MATH /Pg 266 0 R /ID (1574) /P 3066 0 R /A 3068 0 R >> endobj 3068 0 obj << /O /Layout /BBox [ 75 576.5 201.82 588.75 ] >> endobj 3069 0 obj << /K [ 18 19 ] /Alt ($A^{\\widetilde V\(E,\\cdot \),- \\frac{\\mathrm{e}^{g}+W_{2}}{\\mathrm{e}^{-g}+W_{1}}}=S^{E,g}_{W_{1},W_{2},V'\(E, \\cdot \)}$) /S /MATH /Pg 266 0 R /ID (1576) /P 3066 0 R /A 3070 0 R >> endobj 3070 0 obj << /O /Layout /BBox [ 254 573.84 398.76 592.78 ] >> endobj 3071 0 obj << /K 22 /Alt (\\begin{eqnarray*} \\widetilde B\(E,\\theta +\\omega \)^{-1}S^{E,g}_{W_{1},W_{2},V'\(E,\\cdot \)} \\widetilde B\(E,\\theta \) &=&B'\(E,\\theta +\\omega \)^{-1}A_{K}\(E,\\theta \)B'\(E, \\theta \) \\\\ &=& \\begin{pmatrix} \\lambda \(E\) e^{ \\mathcal{T}_{K} f_{1}\(E,\\theta \)} & 0 \\\\ 0 & \\mu \(E\)e^{ \\mathcal{T}_{K} f_{2} \(E,\\theta \)} \\end{pmatrix} . \\end{eqnarray*}) /S /DISPLAYALIGN /Pg 266 0 R /ID (1577) /P 3005 0 R /A 3072 0 R >> endobj 3072 0 obj << /O /Layout /BBox [ 45 544.84 517.23 559.75 ] /Placement /Block >> endobj 3073 0 obj << /K [ 24 3074 0 R 26 ] /S /P /Pg 266 0 R /P 3005 0 R /ID (1616) >> endobj 3074 0 obj << /K 25 /Alt ($\\omega \\in \\mathbb{R}^{d}\\backslash \\mathbb{Q}^{d}$) /S /MATH /Pg 266 0 R /ID (1579) /P 3073 0 R /A 3075 0 R >> endobj 3075 0 obj << /O /Layout /BBox [ 124 491.5 173.24 502.49 ] >> endobj 3076 0 obj << /K 28 /Alt (\\begin{equation*} \\begin{aligned} y_{i}\(E,\\theta +\\omega \)-y_{i}\(E,\\theta \)=\\mathcal T_{K}f_{i}\(E, \\theta \)-\\widehat {f_{i}\(E\)}\(0\), i=1,2 \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 266 0 R /ID (1580) /P 3005 0 R /A 3077 0 R >> endobj 3077 0 obj << /O /Layout /BBox [ 157 462.92 399.29 478.08 ] /Placement /Block >> endobj 3078 0 obj << /K [ 30 3079 0 R 32 3081 0 R 34 3083 0 R 38 ] /S /P /Pg 266 0 R /P 3005 0 R /ID (1617) >> endobj 3079 0 obj << /K 31 /Alt ($ B\(E,\\theta \) =\\widetilde B\(E,\\theta \) diag\(\\mathrm{e}^{y_{1}\(E\)}, 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(1586) /P 3085 0 R /A 3090 0 R >> endobj 3090 0 obj << /O /Layout /BBox [ 446 412.06 467.09 420.83 ] >> endobj 3091 0 obj << /K 47 /Alt ($\\mathrm{gl}\(2,\\mathbb R\)$) /S /MATH /Pg 266 0 R /ID (1587) /P 3085 0 R /A 3092 0 R >> endobj 3092 0 obj << /O /Layout /BBox [ 45 397.5 77.69 407.5 ] >> endobj 3093 0 obj << /K 49 /Alt ($H\\in \\mathrm{gl}\(2,\\mathbb R\)$) /S /MATH /Pg 266 0 R /ID (1588) /P 3085 0 R /A 3094 0 R >> endobj 3094 0 obj << /O /Layout /BBox [ 200 397.5 254.04 407.5 ] >> endobj 3095 0 obj << /K 52 /Alt (\\begin{equation*} \\mathrm{e}^{B}\\mathrm{e}^{D}=\\mathrm{e}^{B+D+H}, \\end{equation*}) /S /DISPLAYMATH /Pg 266 0 R /ID (1589) /P 3005 0 R /A 3096 0 R >> endobj 3096 0 obj << /O /Layout /BBox [ 240 373.06 316.58 383.91 ] /Placement /Block >> endobj 3097 0 obj << /K [ 54 3098 0 R 56 3100 0 R 58 3102 0 R 60 ] /S /P /Pg 266 0 R /P 3005 0 R /ID (1621) >> endobj 3098 0 obj << /K 55 /Alt ($H$) /S /MATH /Pg 266 0 R /ID (1590) /P 3097 0 R /A 3099 0 R >> endobj 3099 0 obj << /O /Layout /BBox [ 74 350 83.13 356.83 ] >> endobj 3100 0 obj << /K 57 /Alt ($B,D$) /S /MATH /Pg 266 0 R /ID (1591) /P 3097 0 R /A 3101 0 R >> endobj 3101 0 obj << /O /Layout /BBox [ 251 348.06 272.09 356.83 ] >> endobj 3102 0 obj << /K 59 /Alt ($Y\\in \\mathscr B_{0,\\mathcal I}\(\\mathrm{gl}\(2,\\mathbb R\)\)$) /S /MATH /Pg 266 0 R /ID (1592) /P 3097 0 R /A 3103 0 R >> endobj 3103 0 obj << /O /Layout /BBox [ 357 347.14 438.6 357.5 ] >> endobj 3104 0 obj << /K 62 /Alt (\\begin{equation*} \\mathrm{e}^{Y\(E,\\theta \)}=\\mathrm{e}^{-P\(E\)^{-1}Y''\(E,\\theta \)P\(E\)} \\mathrm{e}^{Y'\(E,\\theta \)} \\end{equation*}) /S /DISPLAYMATH /Pg 266 0 R /ID (1593) /P 3005 0 R /A 3105 0 R >> endobj 3105 0 obj << /O /Layout /BBox [ 196 324 360.67 334.48 ] /Placement /Block >> endobj 3106 0 obj << /K 64 /S /P /Pg 266 0 R /P 3005 0 R /ID (1622) >> endobj 3107 0 obj << /K [ 66 67 ] /Alt (\\begin{equation*} \\begin{aligned} \\|Y\(E,\\theta \)\\|_{\\mathcal{I}} &\\leq 2\\|P\(E\)Y''\(E,\\theta \)P\(E\)^{-1}\\|_{ \\mathcal{I}}+2\\|Y'\(E\)\\|_{\\mathcal{I}}\\leq 4\\varepsilon ^{\\frac{1}{3}}. \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 266 0 R /ID (1594) /P 3005 0 R /A 3108 0 R >> endobj 3108 0 obj << /O /Layout /BBox [ 139 269.5 418.89 283.5 ] /Placement /Block >> endobj 3109 0 obj << /K 69 /S /P /Pg 266 0 R /P 3005 0 R /ID (1623) >> endobj 3110 0 obj << /K 71 /Alt (\\begin{eqnarray*} B\(E,\\theta \) &=&\\mathrm{e}^{-Y''\(E,\\theta \)}P\(E\)\\mathrm{e}^{Y'\(E, \\theta \)} diag\(\\mathrm{e}^{y_{1}\(E\)},\\mathrm{e}^{y_{2}\(E\)}\) \\\\ &=&P\(E\)\\mathrm{e}^{Y\(E,\\theta \)} diag\(\\mathrm{e}^{y_{1}\(E\)}, \\mathrm{e}^{y_{2}\(E\)}\), \\end{eqnarray*}) /S /DISPLAYALIGN /Pg 266 0 R /ID (1595) /P 3005 0 R /A 3111 0 R >> endobj 3111 0 obj << /O /Layout /BBox [ 45 220.5 517.31 232.92 ] /Placement /Block >> endobj 3112 0 obj << /K [ 73 3113 0 R ] /S /P /Pg 266 0 R /P 3005 0 R /ID (1624) >> endobj 3113 0 obj << /K 74 /Alt (MATH) /S /MATH /Pg 266 0 R /ID (1597) /P 3112 0 R /A 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N}\({1+|k_{j}|^{\\tau}\\langle j\\rangle ^{\\tau}}\) \\mathrm{e}^{|k_{1}|\(r'-r\)} \\\\ &\\leq \\frac{|f|_{r}}{\\gamma}\\sup \\limits _{k\\in \\mathbb{Z}^{\\infty}_{*} \\backslash \\{0\\}}\\prod \\limits _{j\\in \\mathbb N}\({1+|k_{j}|^{\\tau} \\langle j\\rangle ^{\\tau}}\)\\mathrm{e}^{|k|_{1}\(r'-r\)} \\\\ &\\leq \\frac{|f|_{r}}{\\gamma}\\exp \\bigr\(\\frac{\\mu}{r-r'}\\ln \( \\frac{\\mu}{r-r'}\)\\bigr\).\\qedhere\\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 272 0 R /ID (1632) /P 3131 0 R /A 3148 0 R >> endobj 3148 0 obj << /O /Layout /BBox [ 152 433.23 414.3 569.77 ] /Placement /Block >> endobj 3149 0 obj << /K [ 30 3150 0 R 32 3152 0 R 34 ] /S /P /Pg 272 0 R /P 1082 0 R /ID (1672) >> endobj 3150 0 obj << /K 31 /Alt ($r>0$) /S /MATH /Pg 272 0 R /ID (1633) /P 3149 0 R /A 3151 0 R >> endobj 3151 0 obj << /O /Layout /BBox [ 98 410.61 121.12 417.3 ] >> endobj 3152 0 obj << /K 33 /Alt ($\\mathcal B_{r}:=C^{\\omega}_{r}\(\\mathbb{T}^{\\infty},\\mathrm{gl}\(2, \\mathbb R\)\)$) /S 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($\\widetilde b\(n\)=a\(n-1\)-b\(n\)$) /S /MATH /Pg 302 0 R /ID (1860) /P 3487 0 R /A 3489 0 R >> endobj 3489 0 obj << /O /Layout /BBox [ 80 153.5 177.99 165.86 ] >> endobj 3490 0 obj << /K 102 /Alt ($k\\in \\mathbb{Z}, \\chi _{k}p,\\chi _{k}p_{*}\\in \\ell ^{2}\(\\mathbb{Z}\)$) /S /MATH /Pg 302 0 R /ID (1861) /P 3487 0 R /A 3491 0 R >> endobj 3491 0 obj << /O /Layout /BBox [ 244 153.5 350.89 164.14 ] >> endobj 3492 0 obj << /K 104 /Alt ($\\langle \\chi _{k}\\mathcal L\(\\chi _{k}p\),\\chi _{k}p_{*}\\rangle =$) /S /MATH /Pg 302 0 R /ID (1862) /P 3487 0 R /A 3493 0 R >> endobj 3493 0 obj << /O /Layout /BBox [ 393 153.5 478.64 163.5 ] >> endobj 3494 0 obj << /K [ 107 3495 0 R ] /Alt (\\begin{equation} \\label{eq:E_0=E_0'1} \\sum \\limits _{n=-k}^{k}E_{0}\(pp_{*}\)\(n\)-\(ap_{*}\)\(k\)p\(k+1\)-\( \\widetilde bp_{*}\)\(-k\)p\(-k-1\), \\end{equation}) /S /DISPLAYMATH /Pg 302 0 R /ID (1864) /P 3481 0 R /A 3497 0 R >> endobj 3495 0 obj << /K 3496 0 R /S /EQNUMBER /Pg 302 0 R /P 3494 0 R /ID (1865) >> endobj 3496 0 obj << /K 109 /S /EQNUM /Pg 302 0 R /P 3495 0 R /ID (1863) >> endobj 3497 0 obj << /O /Layout /BBox [ 45 108.14 511.26 140.36 ] /Placement /Block >> endobj 3498 0 obj << /K [ 113 3499 0 R ] /S /P /Pg 302 0 R /P 3481 0 R /ID (1892) >> endobj 3499 0 obj << /K 114 /Alt ($\\langle \\chi _{k} p,\\chi _{k}\\mathcal L^{*}\(\\chi _{k}p_{*}\)\\rangle =$) /S /MATH /Pg 302 0 R /ID (1866) /P 3498 0 R /A 3500 0 R >> endobj 3500 0 obj << /O /Layout /BBox [ 108 85.5 218.44 96.22 ] >> endobj 3501 0 obj << /K [ 117 3502 0 R ] /Alt (\\begin{equation} \\label{eq:E_0=E_0} \\sum \\limits _{n=-k}^{k}E'_{0}\(pp_{*}\)\(n\)-\(ap_{*}\)\(-k-1\)p\(-k\)-\( \\widetilde bp_{*}\)\(k+1\)p\(k\). \\end{equation}) /S /DISPLAYMATH /Pg 302 0 R /ID (1868) /P 3481 0 R /A 3504 0 R >> endobj 3502 0 obj << /K 3503 0 R /S /EQNUMBER /Pg 302 0 R /P 3501 0 R /ID (1869) >> endobj 3503 0 obj << /K 119 /S /EQNUM /Pg 302 0 R /P 3502 0 R /ID (1867) >> endobj 3504 0 obj << /O /Layout /BBox [ 45 40.14 511.26 72.36 ] /Placement /Block >> endobj 3505 0 obj << /K [ 1 3506 0 R 3 3508 0 R 7 3510 0 R 11 ] /S /P /Pg 307 0 R /P 3481 0 R /ID (1933) >> endobj 3506 0 obj << /K 2 /Alt ($\\langle \\chi _{k}\\mathcal L\(\\chi _{k}p\),\\chi _{k}p_{*}\\rangle = \\langle \\chi _{k}p,\\chi _{k}\\mathcal L^{*}\(\\chi _{k}p_{*}\)\\rangle $) /S /MATH /Pg 307 0 R /ID (1870) /P 3505 0 R /A 3507 0 R >> endobj 3507 0 obj << /O /Layout /BBox [ 95 676.5 263.09 686.5 ] >> endobj 3508 0 obj << /K [ 4 3509 0 R 6 ] /S /EQNUM /Pg 307 0 R /P 3505 0 R /ID (1871) >> endobj 3509 0 obj << /K [ << /Obj 310 0 R /Type /OBJR >> 5 ] /S /Link /Pg 307 0 R /P 3508 0 R /ID (1934) >> endobj 3510 0 obj << /K [ 8 3511 0 R 10 ] /S /EQNUM /Pg 307 0 R /P 3505 0 R /ID (1872) >> endobj 3511 0 obj << /K [ << /Obj 311 0 R /Type /OBJR >> 9 ] /S /Link /Pg 307 0 R /P 3510 0 R /ID (1935) >> endobj 3512 0 obj << /K 13 /Alt (\\begin{equation*} \\begin{aligned} \\sum \\limits _{n=-k}^{k}\(E_{0}-E_{0}'\)\(pp_{*}\)\(n\)=&\(ap_{*}\)\(k\)p\(k+1\)+\( \\widetilde bp_{*}\)\(-k\)p\(-k-1\) \\\\ &\\quad \\quad -\(\\widetilde bp_{*}\)\(k+1\)p\(k\) -\(ap_{*}\)\(-k-1\)p\(-k\). \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 307 0 R /ID (1893) /P 3481 0 R /A 3513 0 R >> endobj 3513 0 obj << /O /Layout /BBox [ 124 607.45 441.13 659.55 ] /Placement /Block >> endobj 3514 0 obj << /K [ 15 3515 0 R 17 3517 0 R 19 3519 0 R ] /S /P /Pg 307 0 R /P 3481 0 R /ID (1936) >> endobj 3515 0 obj << /K 16 /Alt ($p,p_{*}$) /S /MATH /Pg 307 0 R /ID (1894) /P 3514 0 R /A 3516 0 R >> endobj 3516 0 obj << /O /Layout /BBox [ 76 584.06 95.1 590.31 ] >> endobj 3517 0 obj << /K 18 /Alt ($E_{0}\\neq E_{0}'$) /S /MATH /Pg 307 0 R /ID (1895) /P 3514 0 R /A 3518 0 R >> endobj 3518 0 obj << /O /Layout /BBox [ 194 570.52 231.07 580.52 ] >> endobj 3519 0 obj << /K 20 /Alt (MATH) /S /MATH /Pg 307 0 R /ID (1896) /P 3514 0 R /A 3520 0 R >> endobj 3520 0 obj << /O /Layout /BBox [ 351 572.9 358.44 578.04 ] >> endobj 3521 0 obj << /K [ 24 3522 0 R 26 ] /S /P /Pg 307 0 R /P 1082 0 R /ID (1937) >> endobj 3522 0 obj << /K 25 /Alt ($E_{0}$) /S /MATH /Pg 307 0 R /ID (1897) /P 3521 0 R /A 3523 0 R >> endobj 3523 0 obj << /O /Layout /BBox [ 184 544.5 195.87 552.83 ] >> endobj 3524 0 obj << /K [ 27 3525 0 R 29 3527 0 R 31 3529 0 R 33 3531 0 R 35 ] /S /P /Pg 307 0 R /P 1082 0 R /ID (1938) >> endobj 3525 0 obj << /K 28 /Alt ($p,p_{*}\\in \\ell ^{\\infty}\(\\mathbb{Z}\)$) /S /MATH /Pg 307 0 R /ID (1898) /P 3524 0 R /A 3526 0 R >> endobj 3526 0 obj << /O /Layout /BBox [ 121 488.5 180.04 498.5 ] >> endobj 3527 0 obj << /K 30 /Alt ($\\mathcal Lp=E_{0}p$) /S /MATH /Pg 307 0 R /ID (1899) /P 3524 0 R /A 3528 0 R >> endobj 3528 0 obj << /O /Layout /BBox [ 305 489.06 347.16 497.83 ] >> endobj 3529 0 obj << /K 32 /Alt ($\\mathcal L^{*}p_{*}=E_{0}p_{*}$) /S /MATH /Pg 307 0 R /ID (1900) /P 3524 0 R /A 3530 0 R >> endobj 3530 0 obj << /O /Layout /BBox [ 353 489.06 408.95 497.89 ] >> endobj 3531 0 obj << /K 34 /Alt ($\\inf p,\\inf p_{*}>0$) /S /MATH /Pg 307 0 R /ID (1901) /P 3524 0 R /A 3532 0 R >> endobj 3532 0 obj << /O /Layout /BBox [ 426 489.06 491.13 497.89 ] >> endobj 3533 0 obj << /K 36 /Alt (\\begin{equation*} p_{*}\(\\mathcal L-E_{0}\)\(pu\)=D^{*}\\bigr\(\\widetilde a\(n\)Du\\bigr\)+\\bar c\(D^{*}u+Du\), \\end{equation*}) /S /DISPLAYMATH /Pg 307 0 R /ID (1902) /P 1082 0 R /A 3534 0 R >> endobj 3534 0 obj << /O /Layout /BBox [ 179 445.5 387.41 457.5 ] /Placement /Block >> endobj 3535 0 obj << /K [ 37 3536 0 R 39 3538 0 R 41 ] /S /P /Pg 307 0 R /P 1082 0 R /ID (1939) >> endobj 3536 0 obj << /K 38 /Alt ($\\bar c\\equiv {\\mathrm{const}}$) /S /MATH /Pg 307 0 R /ID (1904) /P 3535 0 R /A 3537 0 R >> endobj 3537 0 obj << /O /Layout /BBox [ 78 422 118.49 428.3 ] >> endobj 3538 0 obj << /K 40 /Alt ($\\inf \\widetilde a>|\\bar c|$) /S /MATH /Pg 307 0 R /ID (1907) /P 3535 0 R /A 3539 0 R >> endobj 3539 0 obj << /O /Layout /BBox [ 141 419.5 183.35 429.5 ] >> endobj 3540 0 obj << /K 3541 0 R /S /TRIVLIST /Pg 307 0 R /P 1082 0 R /ID (1908) >> endobj 3541 0 obj << /K [ 3542 0 R 45 3543 0 R 47 3545 0 R 49 3547 0 R 51 3549 0 R 3551 0 R 3554 0 R 3557 0 R 3559 0 R 3560 0 R 3562 0 R 3567 0 R 3571 0 R 3576 0 R 3578 0 R ] /S /Div /Pg 307 0 R /P 3540 0 R /ID (1909) >> endobj 3542 0 obj << /K 44 /S /Span /Pg 307 0 R /P 3541 0 R /ID (1910) >> endobj 3543 0 obj << /K 46 /Alt ($c'=c-E_{0}$) /S /MATH /Pg 307 0 R /ID (1911) /P 3541 0 R /A 3544 0 R >> endobj 3544 0 obj << /O /Layout /BBox [ 116 393.5 164.88 402.52 ] >> endobj 3545 0 obj << /K 48 /Alt ($u\\in \\ell ^{\\infty}\(\\mathbb{Z}\)$) /S /MATH /Pg 307 0 R /ID (1912) /P 3541 0 R /A 3546 0 R >> endobj 3546 0 obj << /O /Layout /BBox [ 208 392.5 253.66 402.5 ] >> endobj 3547 0 obj << /K 50 /Alt ($\\mathcal Lp=E_{0}p$) /S /MATH /Pg 307 0 R /ID (1913) /P 3541 0 R /A 3548 0 R >> endobj 3548 0 obj << /O /Layout /BBox [ 284 393.06 326.16 401.83 ] >> endobj 3549 0 obj << /K 52 /Alt (\\begin{equation*} \\begin{aligned} \\widetilde{\\mathcal L}u&:=p_{*}\(\\mathcal L-E_{0}\)\(pu\) \\\\ &=p_{*}\(n\)D^{*}\\biggr\(a\(n\)D\\bigr\(p\(n\)u\\bigr\)\\biggr\)+\(bp_{*}\)\(n\)D^{*} \\bigr\(p\(n\)u\\bigr\)+\(p_{*}c'p\)\(n\)u \\\\ &=p_{*}\(n\)D^{*}\\biggr\(a\(n\)D\\bigr\(p\(n\)u\\bigr\)\\biggr\)+\(bp_{*}\)\(n\)D^{*} \\bigr\(p\(n\)u\\bigr\) \\\\ &\\quad \\quad \\qquad \\qquad \\qquad -p_{*}\(n\)\\biggr[\\biggr\(D^{*}\\bigr\(a\(n\)Dp\(n\) \\bigr\)\\biggr\)+b\(n\)D^{*}p\(n\)\\biggr]u \\\\ &=D^{*}\\bigr\(a\(n\)p_{*}\(n\)p\(n+1\)Du\\bigr\)+\\widetilde c\(n\)D^{*}u=\(I\), \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 307 0 R /ID (1914) /P 3541 0 R /A 3550 0 R >> endobj 3550 0 obj << /O /Layout /BBox [ 136 253.65 430.41 375.35 ] /Placement /Block >> endobj 3551 0 obj << /K [ 54 3552 0 R 56 ] /S /P /Pg 307 0 R /P 3541 0 R /ID (1940) >> endobj 3552 0 obj << /K 55 /Alt ($\\widetilde c\(n\)=\(ap_{*}\)\(n-1\)p\(n\)-\(ap\)\(n-1\)p_{*}\(n\)+\(bp_{*}\)\(n\)p\(n-1\)$) /S /MATH /Pg 307 0 R /ID (1916) /P 3551 0 R /A 3553 0 R >> endobj 3553 0 obj << /O /Layout /BBox [ 79 229.5 351.41 239.5 ] >> endobj 3554 0 obj << /K [ 58 3555 0 R 60 ] /S /P /Pg 307 0 R /P 3541 0 R /ID (1941) >> endobj 3555 0 obj << /K 59 /Alt ($\\widetilde b\(n\)=a\(n-1\)-b\(n\)$) /S /MATH /Pg 307 0 R /ID (1918) /P 3554 0 R /A 3556 0 R >> endobj 3556 0 obj << /O /Layout /BBox [ 129 215.5 226.99 227.86 ] >> endobj 3557 0 obj << /K 62 /Alt (\\begin{equation*} \(\\mathcal L-E_{0}\)u=D^{*}\(a\(n\)Du\)+b\(n\)D^{*}u+c'\(n\)u=D\(\\widetilde b\(n\)D^{*}u\)+b\(n+1\)Du+c'\(n\)u. \\end{equation*}) /S /DISPLAYMATH /Pg 307 0 R /ID (1919) /P 3541 0 R /A 3558 0 R >> endobj 3558 0 obj << /O /Layout /BBox [ 98 186.5 468.19 198.86 ] /Placement /Block >> endobj 3559 0 obj << /K 64 /S /P /Pg 307 0 R /P 3541 0 R /ID (1942) >> endobj 3560 0 obj << /K 66 /Alt (\\begin{equation*} \\begin{aligned} \(\\widetilde{\\mathcal L}u\)=&p_{*}\(n\)D\\biggr\(\\widetilde b\(n\)D^{*}\\bigr\(p\(n\)u \\bigr\)\\biggr\)+b\(n+1\)p_{*}\(n\)D\\bigr\(p\(n\)u\\bigr\)+\(c'pp_{*}\)\(n\)u \\\\ =&p_{*}\(n\)D\\biggr\(\\widetilde b\(n\)D^{*}\\bigr\(p\(n\)u\\bigr\)\\biggr\)+b\(n+1\)p_{*}\(n\)D \\bigr\(p\(n\)u\\bigr\) \\\\ &\\quad \\quad \\qquad \\qquad \\qquad -p_{*}\(n\)\\biggr[D\\bigr\(\\widetilde b\(n\)D^{*}p\(n\) \\bigr\)+b\(n+1\)Dp\(n\)\\biggr]u \\\\ =&D\\bigr\(\\widetilde b\(n\)p_{*}\(n\)p\(n-1\)D^{*}u\\bigr\)+\\widetilde c\(n+1\)Du=\(II\). \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 307 0 R /ID (1920) /P 3541 0 R /A 3561 0 R >> endobj 3561 0 obj << /O /Layout /BBox [ 124 39.85 441.05 143.15 ] /Placement /Block >> endobj 3562 0 obj << /K [ 1 3563 0 R 3 3565 0 R 5 ] /S /P /Pg 312 0 R /P 3541 0 R /ID (2011) >> endobj 3563 0 obj << /K 2 /Alt ($\\mathcal Lpp_{*}=\\mathcal L^{*}p_{*}p=E_{0}pp_{*}$) /S /MATH /Pg 312 0 R /ID (1921) /P 3562 0 R /A 3564 0 R >> endobj 3564 0 obj << /O /Layout /BBox [ 211 677.06 311.9 685.89 ] >> endobj 3565 0 obj << /K 4 /Alt ($\\widetilde c\(n\)=\\widetilde c\(n+1\)$) /S /MATH /Pg 312 0 R /ID (1924) /P 3562 0 R /A 3566 0 R >> endobj 3566 0 obj << /O /Layout /BBox [ 337 676.5 403.77 686.5 ] >> endobj 3567 0 obj << /K [ 7 3568 0 R ] /Alt (\\begin{equation} \\widetilde c\(n\)=a\(n-1\)p_{*}\(n-1\)p\(n\)-\\widetilde b\(n\)p\(n-1\)p_{*}\(n\) \\equiv const:=2\\bar c. \\label{eq4.6} \\end{equation}) /S /DISPLAYMATH /Pg 312 0 R /ID (1926) /P 3541 0 R /A 3570 0 R >> endobj 3568 0 obj << /K 3569 0 R /S /EQNUMBER /Pg 312 0 R /P 3567 0 R /ID (1927) >> endobj 3569 0 obj << /K 9 /S /EQNUM /Pg 312 0 R /P 3568 0 R /ID (1925) >> endobj 3570 0 obj << /O /Layout /BBox [ 45 651.5 511.26 663.86 ] /Placement /Block >> endobj 3571 0 obj << /K [ 13 3572 0 R 15 3574 0 R 17 ] /S /P /Pg 312 0 R /P 3541 0 R /ID (2012) >> endobj 3572 0 obj << /K 14 /Alt ($\(I\)$) /S /MATH /Pg 312 0 R /ID (1943) /P 3571 0 R /A 3573 0 R >> endobj 3573 0 obj << /O /Layout /BBox [ 80 626.5 107.62 636.5 ] >> endobj 3574 0 obj << /K 16 /Alt ($\(II\)$) /S /MATH /Pg 312 0 R /ID (1944) /P 3571 0 R /A 3575 0 R >> endobj 3575 0 obj << /O /Layout /BBox [ 116 626.5 151.24 636.5 ] >> endobj 3576 0 obj << /K 19 /Alt (\\begin{equation*} \\begin{aligned} 2\\widetilde{\\mathcal L}u&=D\\bigr\(\\widetilde b\(n\)p\(n-1\)p_{*}\(n\)D^{*}u \\bigr\)+2\\bar cDu+D^{*}\\bigr\(a\(n\)p_{*}\(n\)p\(n+1\)Du\\bigr\)+2\\bar cD^{*}u, \\\\ &=2D^{*}\\bigr\(\\widetilde a\(n\)Du\\bigr\)+2\\bar c\\bigr\(Du+D^{*}u\\bigr\), \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 312 0 R /ID (1945) /P 3541 0 R /A 3577 0 R >> endobj 3577 0 obj << /O /Layout /BBox [ 103 580.85 454.44 612.15 ] /Placement /Block >> endobj 3578 0 obj << /K [ 21 3579 0 R 24 3581 0 R 26 3583 0 R ] /S /P /Pg 312 0 R /P 3541 0 R /ID (2013) >> endobj 3579 0 obj << /K [ 22 23 ] /Alt ($\\widetilde a\(n\)=\\frac{1}{2}\\bigr[\(ap\)\(n\)p_{*}\(n+1\)-\(bp_{*}\)\(n+1\)p\(n\)+\(ap_{*}\)\(n\)p\(n+1\) \\bigr]$) /S /MATH /Pg 312 0 R /ID (1955) /P 3578 0 R /A 3580 0 R >> endobj 3580 0 obj << /O /Layout /BBox [ 74 556.5 362.08 568.5 ] >> endobj 3581 0 obj << /K 25 /Alt ($\\inf \\widetilde a>0$) /S /MATH /Pg 312 0 R /ID (1957) /P 3578 0 R /A 3582 0 R >> endobj 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p,\\inf p_{*}>0$) /S /MATH /Pg 312 0 R /ID (1962) /P 3585 0 R /A 3593 0 R >> endobj 3593 0 obj << /O /Layout /BBox [ 443 519.06 498.17 533.18 ] >> endobj 3594 0 obj << /K 40 /Alt ($E_{0}\\in E_{\\max}$) /S /MATH /Pg 312 0 R /ID (1963) /P 3585 0 R /A 3595 0 R >> endobj 3595 0 obj << /O /Layout /BBox [ 68 506.5 100.53 521.77 ] >> endobj 3596 0 obj << /K 3597 0 R /S /TRIVLIST /Pg 312 0 R /P 1082 0 R /ID (1964) >> endobj 3597 0 obj << /K [ 3598 0 R 45 3599 0 R 47 3601 0 R 3605 0 R 3611 0 R 3613 0 R 3624 0 R 3628 0 R 3637 0 R 3639 0 R 3644 0 R 3646 0 R 3663 0 R 3670 0 R 3672 0 R 3675 0 R 3677 0 R ] /S /Div /Pg 312 0 R /P 3596 0 R /ID (1965) >> endobj 3598 0 obj << /K 44 /S /Span /Pg 312 0 R /P 3597 0 R /ID (1966) >> endobj 3599 0 obj << /K 46 /Alt ($\\Re \(E-E_{0}\)>0$) /S /MATH /Pg 312 0 R /ID (1967) /P 3597 0 R /A 3600 0 R >> endobj 3600 0 obj << /O /Layout /BBox [ 96 480.5 161.38 490.5 ] >> endobj 3601 0 obj << /K [ 48 3602 0 R ] /Alt (\\begin{equation} \\label{eq:resolventeq} 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438.83 ] >> endobj 3611 0 obj << /K 62 /Alt (\\begin{equation*} p_{*}\(n\)\(E_{0}-\\mathcal L\)\\bigr\(p\(n\)v\\bigr\)=-D^{*}\(\\widetilde a\(n\)Dv\)- \\bar c\(Dv+D^{*}v\), \\end{equation*}) /S /DISPLAYMATH /Pg 312 0 R /ID (1973) /P 3597 0 R /A 3612 0 R >> endobj 3612 0 obj << /O /Layout /BBox [ 158 403.5 399.68 415.5 ] /Placement /Block >> endobj 3613 0 obj << /K [ 64 3614 0 R 66 3616 0 R 68 3618 0 R 70 3620 0 R 72 3622 0 R 76 ] /S /P /Pg 312 0 R /P 3597 0 R /ID (2017) >> endobj 3614 0 obj << /K 65 /Alt ($\\inf \\widetilde a>0$) /S /MATH /Pg 312 0 R /ID (1975) /P 3613 0 R /A 3615 0 R >> endobj 3615 0 obj << /O /Layout /BBox [ 74 380.61 111.47 388.22 ] >> endobj 3616 0 obj << /K 67 /Alt ($\\bar c\\in \\mathbb{R}$) /S /MATH /Pg 312 0 R /ID (1977) /P 3613 0 R /A 3617 0 R >> endobj 3617 0 obj << /O /Layout /BBox [ 118 380.61 142.19 387.87 ] >> endobj 3618 0 obj << /K 69 /Alt ($v=u/p\\in \\ell ^{2}\(\\mathbb{Z}\)$) /S /MATH /Pg 312 0 R /ID (1978) /P 3613 0 R /A 3619 0 R >> endobj 3619 0 obj << /O /Layout /BBox [ 204 378.5 274.25 389.14 ] >> endobj 3620 0 obj << /K 71 /Alt ($\\Re \(E-E_{0}\)>0$) /S /MATH /Pg 312 0 R /ID (1979) /P 3613 0 R /A 3621 0 R >> endobj 3621 0 obj << /O /Layout /BBox [ 298 378.5 363.38 388.5 ] >> endobj 3622 0 obj << /K [ 73 3623 0 R 75 ] /S /EQNUM /Pg 312 0 R /P 3613 0 R /ID (1980) >> endobj 3623 0 obj << /K [ << /Obj 316 0 R /Type /OBJR >> 74 ] /S /Link /Pg 312 0 R /P 3622 0 R /ID (2018) >> endobj 3624 0 obj << /K [ 78 3625 0 R ] /Alt (\\begin{equation} \\label{resolventestimate} -D^{*}\(\\widetilde a\(n\)Dv\)-\\bar c\\bigr\(D^{*}v+Dv\\bigr\)+\(E-E_{0}\)\(pp_{*}\)\(n\)v=\(p_{*}f\)\(n\). \\end{equation}) /S /DISPLAYMATH /Pg 312 0 R /ID (1982) /P 3597 0 R /A 3627 0 R >> endobj 3625 0 obj << /K 3626 0 R /S /EQNUMBER /Pg 312 0 R /P 3624 0 R /ID (1983) >> endobj 3626 0 obj << /K 80 /S /EQNUM /Pg 312 0 R /P 3625 0 R /ID (1981) >> endobj 3627 0 obj << /O /Layout /BBox [ 45 352.5 511.26 364.5 ] /Placement /Block >> endobj 3628 0 obj << /K [ 84 3629 0 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/A 3802 0 R >> endobj 3802 0 obj << /O /Layout /BBox [ 175 44.5 390.98 54.5 ] /Placement /Block >> endobj 3803 0 obj << /K [ 0 3804 0 R 2 3806 0 R 4 3808 0 R 6 ] /S /P /Pg 322 0 R /P 1082 0 R /ID (2149) >> endobj 3804 0 obj << /K 1 /Alt ($\\widehat W\(0\)\\geq 0$) /S /MATH /Pg 322 0 R /ID (2085) /P 3803 0 R /A 3805 0 R >> endobj 3805 0 obj << /O /Layout /BBox [ 56 676.5 97.95 689.03 ] >> endobj 3806 0 obj << /K 3 /Alt ($W\\not \\equiv 0$) /S /MATH /Pg 322 0 R /ID (2086) /P 3803 0 R /A 3807 0 R >> endobj 3807 0 obj << /O /Layout /BBox [ 121 677.06 150.17 685.94 ] >> endobj 3808 0 obj << /K 5 /Alt ($E_{0}>0$) /S /MATH /Pg 322 0 R /ID (2087) /P 3803 0 R /A 3809 0 R >> endobj 3809 0 obj << /O /Layout /BBox [ 179 677.5 209.2 685.83 ] >> endobj 3810 0 obj << /K 3811 0 R /S /TRIVLIST /Pg 322 0 R /P 1082 0 R /ID (2088) >> endobj 3811 0 obj << /K [ 3812 0 R 10 3813 0 R 12 3815 0 R 14 3817 0 R 16 3819 0 R 3823 0 R 3828 0 R 3830 0 R 3833 0 R 3835 0 R 3840 0 R 3842 0 R 3845 0 R 3847 0 R 3854 0 R 3859 0 R 3861 0 R ] /S /Div /Pg 322 0 R /P 3810 0 R /ID (2089) >> endobj 3812 0 obj << /K 9 /S /Span /Pg 322 0 R /P 3811 0 R /ID (2090) >> endobj 3813 0 obj << /K 11 /Alt ($A_{2}\\leq 0$) /S /MATH /Pg 322 0 R /ID (2091) /P 3811 0 R /A 3814 0 R >> endobj 3814 0 obj << /O /Layout /BBox [ 150 649.5 180.32 657.83 ] >> endobj 3815 0 obj << /K 13 /Alt ($p$) /S /MATH /Pg 322 0 R /ID (2092) /P 3811 0 R /A 3816 0 R >> endobj 3816 0 obj << /O /Layout /BBox [ 227 649.06 232.03 655.31 ] >> endobj 3817 0 obj << /K 15 /Alt ($\\mathcal Lp=E_{0}p$) /S /MATH /Pg 322 0 R /ID (2093) /P 3811 0 R /A 3818 0 R >> endobj 3818 0 obj << /O /Layout /BBox [ 318 649.06 360.16 657.83 ] >> endobj 3819 0 obj << /K [ 17 18 19 20 3820 0 R ] /Alt (\\begin{equation} \\label{eq:groundstateenergy} E_{0}=\\lim \\limits _{n\\to \\infty}\\frac{1}{n}\\sum \\limits _{i=1}^{n} \\biggr\(\\frac{D^{*}\\bigr\(A_{1}\(\\omega i\)Dp\(i\)\\bigr\)}{p\(i\)}+A_{2} \\frac{D^{*}p\(i\)}{p\(i\)}+W\(\\omega i\)\\biggr\). \\end{equation}) /S 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/DISPLAYMATH /Pg 322 0 R /ID (2117) /P 3811 0 R /A 3829 0 R >> endobj 3829 0 obj << /O /Layout /BBox [ 216 525.25 340.79 558.51 ] /Placement /Block >> endobj 3830 0 obj << /K [ 35 3831 0 R 39 ] /S /P /Pg 322 0 R /P 3811 0 R /ID (2151) >> endobj 3831 0 obj << /K [ 36 3832 0 R 38 ] /S /EQNUM /Pg 322 0 R /P 3830 0 R /ID (2118) >> endobj 3832 0 obj << /K [ << /Obj 325 0 R /Type /OBJR >> 37 ] /S /Link /Pg 322 0 R /P 3831 0 R /ID (2152) >> endobj 3833 0 obj << /K [ 41 42 43 44 45 46 ] /Alt (\\begin{equation*} \\label{dominatedtermpo} \\begin{aligned} \\lim \\limits _{n\\to \\infty}\\frac{1}{n}\\sum \\limits _{i=1}^{n} \\frac{D^{*}\\bigr\(A_{1}\(\\omega i\)Dp\(i\)\\bigr\)}{p\(i\)}&=\\lim \\limits _{n \\to \\infty}\\sum \\limits _{i=1}^{n}\\frac{A_{1}\(\\omega i\)}{n}\\biggr\( \\frac{\\bigr\(Dp\(i\)\\bigr\)^{2}}{p\(i\)p\(i+1\)}\\biggr\) \\\\ &=\\int _{\\mathbb{T}^{d}} \\frac{A_{1}\(\\theta \)\\bigr\(P\(\\theta +\\omega \)-P\(\\theta \)\\bigr\)^{2}}{P\(\\theta \)P\(\\theta +\\omega \)}d \\theta 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255.75 ] >> endobj 4500 0 obj << /K 106 /Alt ($\\theta =\\omega z$) /S /MATH /Pg 392 0 R /ID (2661) /P 4497 0 R /A 4501 0 R >> endobj 4501 0 obj << /O /Layout /BBox [ 408 246 437.98 252.94 ] >> endobj 4502 0 obj << /K 108 /Alt ($z\\in \\mathbb{R}$) /S /MATH /Pg 392 0 R /ID (2662) /P 4497 0 R /A 4503 0 R >> endobj 4503 0 obj << /O /Layout /BBox [ 444 245.61 468.95 252.87 ] >> endobj 4504 0 obj << /K 110 /Alt (\\begin{equation*} \\begin{aligned} Lu:=D^{*}\(A_{1}\(\\omega z\) Du\)+A_{2}\(\\omega z\) D^{*}u+W\(\\omega z\)u=E_{0}u, \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 392 0 R /ID (2663) /P 1082 0 R /A 4505 0 R >> endobj 4505 0 obj << /O /Layout /BBox [ 163 215.78 402.76 229.22 ] /Placement /Block >> endobj 4506 0 obj << /K [ 111 4507 0 R 113 4509 0 R 115 4511 0 R 117 4513 0 R 119 4515 0 R 122 4517 0 R 125 4519 0 R 127 4521 0 R 129 4523 0 R 131 4524 0 R 133 ] /S /P /Pg 392 0 R /P 1082 0 R /ID (2729) >> endobj 4507 0 obj << /K 112 /Alt ($D=D_{1}$) /S /MATH /Pg 392 0 R /ID (2664) /P 4506 0 R /A 4508 0 R >> endobj 4508 0 obj << /O /Layout /BBox [ 79 191.5 113.65 199.83 ] >> endobj 4509 0 obj << /K 114 /Alt ($D_{\\varepsilon }$) /S /MATH /Pg 392 0 R /ID (2665) /P 4506 0 R /A 4510 0 R >> endobj 4510 0 obj << /O /Layout /BBox [ 141 191.5 153.56 199.83 ] >> endobj 4511 0 obj << /K 116 /Alt ($\\ell ^{2}\(\\mathbb{Z}\)$) /S /MATH /Pg 392 0 R /ID (2666) /P 4506 0 R /A 4512 0 R >> endobj 4512 0 obj << /O /Layout /BBox [ 206 190.5 229.73 201.14 ] >> endobj 4513 0 obj << /K 118 /Alt ($L^{2}\(\\mathbb{R}\)$) /S /MATH /Pg 392 0 R /ID (2667) /P 4506 0 R /A 4514 0 R >> endobj 4514 0 obj << /O /Layout /BBox [ 342 190.5 368.71 201.14 ] >> endobj 4515 0 obj << /K [ 120 121 ] /Alt ($D_{\\varepsilon }^{*}u\(z\)=\\frac{u\(z\)-u\(z-\\varepsilon \)}{\\varepsilon }$) /S /MATH /Pg 392 0 R /ID (2668) /P 4506 0 R /A 4516 0 R >> endobj 4516 0 obj << /O /Layout /BBox [ 385 189.55 478.92 203.1 ] >> endobj 4517 0 obj << /K [ 123 124 ] /Alt ($D_{\\varepsilon }u=\\frac{u\(z+\\varepsilon \)-u\(z\)}{\\varepsilon }$) /S /MATH /Pg 392 0 R /ID (2669) /P 4506 0 R /A 4518 0 R >> endobj 4518 0 obj << /O /Layout /BBox [ 487 189.55 567.34 203.1 ] >> endobj 4519 0 obj << /K 126 /Alt ($u\\in L^{2}\(\\mathbb{R}\)$) /S /MATH /Pg 392 0 R /ID (2670) /P 4506 0 R /A 4520 0 R >> endobj 4520 0 obj << /O /Layout /BBox [ 138 175.5 182.65 186.14 ] >> endobj 4521 0 obj << /K 128 /Alt ($\\varepsilon >0$) /S /MATH /Pg 392 0 R /ID (2671) /P 4506 0 R /A 4522 0 R >> endobj 4522 0 obj << /O /Layout /BBox [ 223 177.61 246 184.3 ] >> endobj 4523 0 obj << /K [ << /Obj 402 0 R /Type /OBJR >> 130 ] /S /Link /Pg 392 0 R /P 4506 0 R /ID (2730) >> endobj 4524 0 obj << /K 132 /Alt ($ \\bar c\(z\)$) /S /MATH /Pg 392 0 R /ID (2673) /P 4506 0 R /A 4525 0 R >> endobj 4525 0 obj << /O /Layout /BBox [ 477 175.5 494.2 185.5 ] >> endobj 4526 0 obj << /K 134 /Alt (\\begin{equation*} P_{*}\(\\omega z\)\(L-E_{0}\)\\bigr\(P\(\\omega z\) u\(z\)\\bigr\)=D^{*}\\bigr\( \\widetilde A_{1}\(\\omega z\) Du\\bigr\)+\\bar c\(\\omega z\)\(D+D^{*}\)u, \\end{equation*}) /S /DISPLAYMATH /Pg 392 0 R /ID (2674) /P 1082 0 R /A 4527 0 R >> endobj 4527 0 obj << /O /Layout /BBox [ 137 135.5 428.75 148.75 ] /Placement /Block >> endobj 4528 0 obj << /K [ 135 4529 0 R 138 4531 0 R 140 4533 0 R 142 4534 0 R 144 4536 0 R 146 4538 0 R 148 4540 0 R 150 4542 0 R 152 4544 0 R 154 ] /S /P /Pg 392 0 R /P 1082 0 R /ID (2731) >> endobj 4529 0 obj << /K [ 136 137 ] /Alt ($\\widetilde A_{1}\(\\theta \)=\\frac{1}{2}\\bigr[\(A_{1}P\)\(\\theta \)P_{*}\( \\theta +\\omega \)-\(A_{2}P_{*}\)\(\\theta +\\omega \)P\(\\theta \)+\(A_{1}P_{*}\)\( \\theta \)P\(\\theta +\\omega \)\\bigr]$) /S /MATH /Pg 392 0 R /ID (2684) /P 4528 0 R /A 4530 0 R >> endobj 4530 0 obj << /O /Layout /BBox [ 78 110.5 543.57 124.98 ] >> endobj 4531 0 obj << /K 139 /Alt ($\\bar c\(\\theta \)=\(A_{1}P_{*}\)\(\\theta -\\omega \)P\(\\theta \)-\(A_{1}P\)\( \\theta -\\omega \)P_{*}\(\\theta \)+\(A_{2}P_{*}\)\(\\theta \)P\(\\theta -\\omega \)$) /S /MATH /Pg 392 0 R /ID (2686) /P 4528 0 R /A 4532 0 R >> endobj 4532 0 obj << /O /Layout /BBox [ 396 110.5 820.74 120.5 ] >> endobj 4533 0 obj << /K [ << /Obj 403 0 R /Type /OBJR >> 141 ] /S /Link /Pg 392 0 R /P 4528 0 R /ID (2732) >> endobj 4534 0 obj << /K 143 /Alt ($\\bar c\(\\theta \)=\\bar c\(\\theta +\\omega \)$) /S /MATH /Pg 392 0 R /ID (2689) /P 4528 0 R /A 4535 0 R >> endobj 4535 0 obj << /O /Layout /BBox [ 426 97.5 518.17 107.5 ] >> endobj 4536 0 obj << /K 145 /Alt ($\\bar c\(\\theta \)$) /S /MATH /Pg 392 0 R /ID (2691) /P 4528 0 R /A 4537 0 R >> endobj 4537 0 obj << /O /Layout /BBox [ 72 83.5 105.52 93.5 ] >> endobj 4538 0 obj << /K 147 /Alt ($\\{k\\omega |k\\in \\mathbb{Z}\\}$) /S /MATH /Pg 392 0 R /ID (2692) /P 4528 0 R /A 4539 0 R >> endobj 4539 0 obj << /O /Layout /BBox [ 214 83.5 281.39 93.5 ] >> endobj 4540 0 obj << /K 149 /Alt ($\\mathbb{T}^{\\infty}$) /S /MATH /Pg 392 0 R /ID (2693) /P 4528 0 R /A 4541 0 R >> endobj 4541 0 obj << /O /Layout /BBox [ 316 86 327.08 98.1 ] >> endobj 4542 0 obj << /K 151 /Alt ($\\widetilde A_{1}>|\\bar c\(\\theta \)|$) /S /MATH /Pg 392 0 R /ID (2696) /P 4528 0 R /A 4543 0 R >> endobj 4543 0 obj << /O /Layout /BBox [ 358 83.5 434.11 97.98 ] >> endobj 4544 0 obj << /K 153 /Alt ($\\bar c\(\\theta \)=\\bar c\\in \\mathbb{R}$) /S /MATH /Pg 392 0 R /ID (2699) /P 4528 0 R /A 4545 0 R >> endobj 4545 0 obj << /O /Layout /BBox [ 82 70.5 149.39 80.5 ] >> endobj 4546 0 obj << /K [ 155 4547 0 R ] /Alt (\\begin{equation} \\label{eq:reducinR} P_{*}\(\\omega z\)\(L-E_{0}\)\\bigr\(P\(\\omega z\) u\(z\)\\bigr\)=D^{*}\\bigr\( \\widetilde A_{1}\(\\omega z\) Du\\bigr\)+\\bar c\(D+D^{*}\)u. \\end{equation}) /S /DISPLAYMATH /Pg 392 0 R /ID (2701) /P 1082 0 R /A 4549 0 R >> endobj 4547 0 obj << /K 4548 0 R /S /EQNUMBER /Pg 392 0 R /P 4546 0 R /ID (2702) >> endobj 4548 0 obj << /K 157 /S /EQNUM /Pg 392 0 R /P 4547 0 R /ID (2700) >> endobj 4549 0 obj << /O /Layout /BBox [ 50 43.5 516.26 56.75 ] /Placement /Block >> endobj 4550 0 obj << /K [ 0 4551 0 R 4 4553 0 R 8 4555 0 R ] /S /P /Pg 404 0 R /P 1082 0 R /ID (2832) >> endobj 4551 0 obj << /K [ 1 4552 0 R 3 ] /S /EQNUM /Pg 404 0 R /P 4550 0 R /ID (2703) >> endobj 4552 0 obj << /K [ << /Obj 407 0 R /Type /OBJR >> 2 ] /S /Link /Pg 404 0 R /P 4551 0 R /ID (2833) >> endobj 4553 0 obj << /K [ 5 4554 0 R 7 ] /S /EQNUM /Pg 404 0 R /P 4550 0 R /ID (2704) >> endobj 4554 0 obj << /K [ << /Obj 408 0 R /Type /OBJR >> 6 ] /S /Link /Pg 404 0 R /P 4553 0 R /ID (2834) >> endobj 4555 0 obj << /K 9 /Alt ($[0,T]\\times \\mathbb{R}$) /S /MATH /Pg 404 0 R /ID (2705) /P 4550 0 R /A 4556 0 R >> endobj 4556 0 obj << /O /Layout /BBox [ 454 676.5 496.15 686.5 ] >> endobj 4557 0 obj << /K [ 11 12 13 14 15 16 4558 0 R ] /Alt (\\begin{equation} \\label{eq:familyopnew} L_{\\varepsilon }u_{\\varepsilon }:=D_{\\varepsilon }^{*}\\biggr\(A_{1} \\bigr\(\\frac{\\omega z}{\\varepsilon }\\bigr\)D_{\\varepsilon }u_{ \\varepsilon }\\biggr\)+\\frac{1}{\\varepsilon }A_{2}\\bigr\( \\frac{\\omega z}{\\varepsilon }\\bigr\)D^{*}_{\\varepsilon }u_{ \\varepsilon }+\\frac{1}{\\varepsilon ^{2}}W\\bigr\( \\frac{\\omega z}{\\varepsilon }\\bigr\)u_{\\varepsilon }, \\ z\\in\\mathbb{R}, \\end{equation}) /S /DISPLAYMATH /Pg 404 0 R /ID (2707) /P 1082 0 R /A 4560 0 R >> endobj 4558 0 obj << /K 4559 0 R /S /EQNUMBER /Pg 404 0 R /P 4557 0 R /ID (2708) >> endobj 4559 0 obj << /K 18 /S /EQNUM /Pg 404 0 R /P 4558 0 R /ID (2706) >> endobj 4560 0 obj << /O /Layout /BBox [ 45 637.5 511.26 661.5 ] /Placement /Block >> endobj 4561 0 obj << /K 21 /S /P /Pg 404 0 R /P 1082 0 R /ID (2835) >> endobj 4562 0 obj << /K [ 22 4563 0 R ] /Alt (\\begin{equation} \\label{eq:extendedeq} \\left \\{ \\begin{aligned} \(u_{\\varepsilon }\)_{t}&=L_{\\varepsilon }u_{\\varepsilon },\\ \(t,z\)\\in [0,T] \\times \\mathbb{R} \\\\ u_{\\varepsilon }\(0,z\)&=\\varphi \(z\), \\ z\\in \\mathbb{R}, \\end{aligned} \\right . \\end{equation}) /S /DISPLAYMATH /Pg 404 0 R /ID (2734) /P 1082 0 R /A 4565 0 R >> endobj 4563 0 obj << /K 4564 0 R /S /EQNUMBER /Pg 404 0 R /P 4562 0 R /ID (2735) >> endobj 4564 0 obj << /K 24 /S /EQNUM /Pg 404 0 R /P 4563 0 R /ID (2733) >> endobj 4565 0 obj << /O /Layout /BBox [ 45 570.81 511.26 602.19 ] /Placement /Block >> endobj 4566 0 obj << /K [ 27 4567 0 R 31 4569 0 R 35 4571 0 R 39 4573 0 R 43 4575 0 R 45 4577 0 R 49 ] /S /P /Pg 404 0 R /P 1082 0 R /ID (2836) >> endobj 4567 0 obj << /K [ 28 4568 0 R 30 ] /S /EQNUM /Pg 404 0 R /P 4566 0 R /ID (2736) >> endobj 4568 0 obj << /K [ << /Obj 409 0 R /Type /OBJR >> 29 ] /S /Link /Pg 404 0 R /P 4567 0 R /ID (2837) >> endobj 4569 0 obj << /K [ 32 4570 0 R 34 ] /S /EQNUM /Pg 404 0 R /P 4566 0 R /ID (2737) >> endobj 4570 0 obj << /K [ << /Obj 410 0 R /Type /OBJR >> 33 ] /S /Link /Pg 404 0 R /P 4569 0 R /ID (2838) >> endobj 4571 0 obj << /K [ 36 4572 0 R 38 ] /S /EQNUM /Pg 404 0 R /P 4566 0 R /ID (2738) >> endobj 4572 0 obj << /K [ << /Obj 411 0 R /Type /OBJR >> 37 ] /S /Link /Pg 404 0 R /P 4571 0 R /ID (2839) >> endobj 4573 0 obj << /K [ 40 4574 0 R 42 ] /S /EQNUM /Pg 404 0 R /P 4566 0 R /ID (2739) >> endobj 4574 0 obj << /K [ << /Obj 412 0 R /Type /OBJR >> 41 ] /S /Link /Pg 404 0 R /P 4573 0 R /ID (2840) >> endobj 4575 0 obj << /K 44 /Alt ($\\mathbb{Z}$) /S /MATH /Pg 404 0 R /ID (2740) /P 4566 0 R /A 4576 0 R >> endobj 4576 0 obj << /O /Layout /BBox [ 502 550 508.87 557.3 ] >> endobj 4577 0 obj << /K [ 46 4578 0 R 48 ] /S /EQNUM /Pg 404 0 R /P 4566 0 R /ID (2741) >> endobj 4578 0 obj << /K [ << /Obj 413 0 R /Type /OBJR >> 47 ] /S /Link /Pg 404 0 R /P 4577 0 R /ID (2841) >> endobj 4579 0 obj << /K [ 50 51 52 53 54 55 4580 0 R ] /Alt (\\begin{equation} \\label{eq:reduclinemulsca} P_{*}\(\\frac{\\omega z}{\\varepsilon }\)\\biggr\(L_{\\varepsilon }- \\frac{E_{0}}{\\varepsilon ^{2}}\\biggr\) \\bigr\(P\( \\frac{\\omega z}{\\varepsilon }\)u_{\\varepsilon }\\bigr\)=D^{*}_{ \\varepsilon }\\biggr\(\\widetilde A_{1}\\bigr\( \\frac{\\omega z}{\\varepsilon }\\bigr\) D_{\\varepsilon }u_{\\varepsilon } \\biggr\)+\\frac{\\bar c}{\\varepsilon }\(D_{\\varepsilon }+D_{\\varepsilon }^{*}\)u_{ \\varepsilon }. \\end{equation}) /S /DISPLAYMATH /Pg 404 0 R /ID (2743) /P 1082 0 R /A 4582 0 R >> endobj 4580 0 obj << /K 4581 0 R /S /EQNUMBER /Pg 404 0 R /P 4579 0 R /ID (2744) >> endobj 4581 0 obj << /K 57 /S /EQNUM /Pg 404 0 R /P 4580 0 R /ID (2742) >> endobj 4582 0 obj << /O /Layout /BBox [ 45 494.5 511.26 518.5 ] /Placement /Block >> endobj 4583 0 obj << /K [ 60 4584 0 R 62 4586 0 R 65 4588 0 R 69 4590 0 R 73 4592 0 R 78 4594 0 R 82 4596 0 R 84 ] /S /P /Pg 404 0 R /P 1082 0 R /ID (2842) >> endobj 4584 0 obj << /K 61 /Alt ($Q=PP_{*}$) /S /MATH /Pg 404 0 R /ID (2745) /P 4583 0 R /A 4585 0 R >> endobj 4585 0 obj << /O /Layout /BBox [ 93 472.06 133.06 480.83 ] >> endobj 4586 0 obj << /K [ 63 64 ] /Alt ($P_{*}\(\\frac{\\omega z}{\\varepsilon }\)$) /S /MATH /Pg 404 0 R /ID (2746) /P 4583 0 R /A 4587 0 R >> endobj 4587 0 obj << /O /Layout /BBox [ 201 470.55 231.58 481.5 ] >> endobj 4588 0 obj << /K [ 66 4589 0 R 68 ] /S /EQNUM /Pg 404 0 R /P 4583 0 R /ID (2747) >> endobj 4589 0 obj << /K [ << /Obj 414 0 R /Type /OBJR >> 67 ] /S /Link /Pg 404 0 R /P 4588 0 R /ID (2843) >> endobj 4590 0 obj << /K [ 70 4591 0 R 72 ] /S /EQNUM /Pg 404 0 R /P 4583 0 R /ID (2748) >> endobj 4591 0 obj << /K [ << /Obj 415 0 R /Type /OBJR >> 71 ] /S /Link /Pg 404 0 R /P 4590 0 R /ID (2844) >> endobj 4592 0 obj << /K [ 74 75 76 77 ] /Alt ($v_{\\varepsilon }\(t,z\)=\\mathrm{e}^{-\\frac{E_{0}t}{\\varepsilon ^{2}}} \\frac{u_{\\varepsilon }\(t,z-\\frac{lt}{\\varepsilon }\)}{P\(z'\)}$) /S /MATH /Pg 404 0 R /ID (2749) /P 4583 0 R /A 4593 0 R >> endobj 4593 0 obj << /O /Layout /BBox [ 45 452.8 154.32 469.67 ] >> endobj 4594 0 obj << /K [ 79 80 81 ] /Alt ($z'=\\omega \\bigr\(\\frac{z}{\\varepsilon }-\\frac{lt}{\\varepsilon ^{2}} \\bigr\)$) /S /MATH /Pg 404 0 R /ID (2758) /P 4583 0 R /A 4595 0 R >> endobj 4595 0 obj << /O /Layout /BBox [ 181 454.5 246.77 466.8 ] >> endobj 4596 0 obj << /K 83 /Alt ($\(t,z\)\\in [0,T]\\times \\mathbb{R}$) /S /MATH /Pg 404 0 R /ID (2759) /P 4583 0 R /A 4597 0 R >> endobj 4597 0 obj << /O /Layout /BBox [ 391 455.5 466.29 465.5 ] >> endobj 4598 0 obj << /K [ 85 86 87 88 89 4599 0 R ] /Alt (\\begin{equation} \\label{eq:transpara} \\left \\{ \\begin{aligned} &\\mathcal P_{\\varepsilon }v_{\\varepsilon }:=Q\(z'\) \(v_{\\varepsilon }\)_{t}-D^{*}_{ \\varepsilon }\\biggr\(\\widetilde A_{1}\(z'\)D_{\\varepsilon }v_{ \\varepsilon }\\biggr\)-\\biggr[\\frac{\\bar c}{\\varepsilon }\(D^{*}_{ \\varepsilon }+D_{\\varepsilon }\)-\\frac{l}{\\varepsilon }Q\(z'\)\\partial _{z} \\biggr]v_{\\varepsilon }=0, \\\\ &v_{\\varepsilon }\(0,z\)= \\frac{\\varphi \(z\)}{P\(\\frac{\\omega z}{\\varepsilon }\)} \\text{ in } \\mathbb{R}, \\end{aligned} \\right . \\end{equation}) /S /DISPLAYMATH /Pg 404 0 R /ID (2761) /P 1082 0 R /A 4601 0 R >> endobj 4599 0 obj << /K 4600 0 R /S /EQNUMBER /Pg 404 0 R /P 4598 0 R /ID (2762) >> endobj 4600 0 obj << /K 91 /S /EQNUM /Pg 404 0 R /P 4599 0 R /ID (2760) >> endobj 4601 0 obj << /O /Layout /BBox [ 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/ID (2847) >> endobj 4612 0 obj << /K [ 106 4613 0 R 108 ] /S /EQNUM /Pg 404 0 R /P 4611 0 R /ID (2767) >> endobj 4613 0 obj << /K [ << /Obj 417 0 R /Type /OBJR >> 107 ] /S /Link /Pg 404 0 R /P 4612 0 R /ID (2848) >> endobj 4614 0 obj << /K 110 /Alt (\\begin{equation*} H^{1}\(\\mathbb{R}\)=\\{u:\\mathbb{R}\\rightarrow \\mathbb{R}:\\|u\\|^{2}_{H^{1}}:= \\| u'\\|^{2}_{L^{2}}+\\|u\\|^{2}_{L^{2}}<\\infty \\}, \\end{equation*}) /S /DISPLAYMATH /Pg 404 0 R /ID (2768) /P 1082 0 R /A 4615 0 R >> endobj 4615 0 obj << /O /Layout /BBox [ 158 305.38 399.27 316.64 ] /Placement /Block >> endobj 4616 0 obj << /K [ 111 4617 0 R 113 4619 0 R 115 4621 0 R 117 4623 0 R 119 4625 0 R 121 4627 0 R 123 4629 0 R 125 ] /S /P /Pg 404 0 R /P 1082 0 R /ID (2849) >> endobj 4617 0 obj << /K 112 /Alt ($H^{-1}\(\\mathbb{R}\)$) /S /MATH /Pg 404 0 R /ID (2769) /P 4616 0 R /A 4618 0 R >> endobj 4618 0 obj << /O /Layout /BBox [ 113 279.5 148.28 290.14 ] >> endobj 4619 0 obj << /K 114 /Alt ($H^{1}\(\\mathbb{R}\)$) /S /MATH /Pg 404 0 R /ID (2770) /P 4616 0 R /A 4620 0 R >> endobj 4620 0 obj << /O /Layout /BBox [ 232 279.5 261.03 290.14 ] >> endobj 4621 0 obj << /K 116 /Alt ($\\mathscr S:=L^{2}\([0, T];H^{1}\(\\mathbb{R}\)\)$) /S /MATH /Pg 404 0 R /ID (2771) /P 4616 0 R /A 4622 0 R >> endobj 4622 0 obj << /O /Layout /BBox [ 286 279.5 387.49 290.14 ] >> endobj 4623 0 obj << /K 118 /Alt ($\\|\\cdot \\|_{\\mathscr S}$) /S /MATH /Pg 404 0 R /ID (2772) /P 4616 0 R /A 4624 0 R >> endobj 4624 0 obj << /O /Layout /BBox [ 71 265.5 96.81 275.5 ] >> endobj 4625 0 obj << /K 120 /Alt ($\\mathscr S'$) /S /MATH /Pg 404 0 R /ID (2773) /P 4616 0 R /A 4626 0 R >> endobj 4626 0 obj << /O /Layout /BBox [ 123 268 136.36 275.52 ] >> endobj 4627 0 obj << /K 122 /Alt ($\\mathscr S$) /S /MATH /Pg 404 0 R /ID (2774) /P 4616 0 R /A 4628 0 R >> endobj 4628 0 obj << /O /Layout /BBox [ 204 268 214.56 274.97 ] >> endobj 4629 0 obj << /K 124 /Alt ($\\|\\cdot \\|_{\\mathscr S'}$) /S /MATH /Pg 404 0 R /ID (2775) /P 4616 0 R /A 4630 0 R >> endobj 4630 0 obj << /O /Layout /BBox [ 363 265.5 391.51 275.5 ] >> endobj 4631 0 obj << /K [ 126 4632 0 R 128 4634 0 R 130 ] /S /P /Pg 404 0 R /P 1082 0 R /ID (2850) >> endobj 4632 0 obj << /K 127 /Alt ($f\\in \\mathscr S'\\cap C^{1}\([0,T]\\times \\mathbb{R}\),\\phi \\in L^{2}\( \\mathbb{R}\)\\cap C^{1}\(\\mathbb{R}\)$) /S /MATH /Pg 404 0 R /ID (2776) /P 4631 0 R /A 4633 0 R >> endobj 4633 0 obj << /O /Layout /BBox [ 138 239.5 331.14 250.14 ] >> endobj 4634 0 obj << /K 129 /Alt ($v_{\\varepsilon }\\in \\mathscr S\\cap C^{1}\([0,T]\\times \\mathbb{R}\)$) /S /MATH /Pg 404 0 R /ID (2777) /P 4631 0 R /A 4635 0 R >> endobj 4635 0 obj << /O /Layout /BBox [ 45 226.5 150.28 237.14 ] >> endobj 4636 0 obj << /K [ 131 4637 0 R ] /Alt (\\begin{equation} \\label{eq:generalizedeq} \\left \\{ \\begin{aligned} \\mathcal P_{\\varepsilon }v_{\\varepsilon }&=f & & \\text{ in } [0,T] \\times \\mathbb{R}, \\\\ v_{\\varepsilon }\(0,z\)&=\\phi \(z\)& & \\text{ in }\\mathbb{R}. \\end{aligned} \\right . \\end{equation}) /S /DISPLAYMATH /Pg 404 0 R /ID (2779) /P 1082 0 R /A 4639 0 R >> endobj 4637 0 obj << /K 4638 0 R /S /EQNUMBER /Pg 404 0 R /P 4636 0 R /ID (2780) >> endobj 4638 0 obj << /K 133 /S /EQNUM /Pg 404 0 R /P 4637 0 R /ID (2778) >> endobj 4639 0 obj << /O /Layout /BBox [ 45 179.81 511.26 211.19 ] /Placement /Block >> endobj 4640 0 obj << /K 136 /S /P /Pg 404 0 R /P 1082 0 R /ID (2851) >> endobj 4641 0 obj << /K [ 137 4642 0 R ] /Alt (\\begin{equation} \\label{eq:energyestimates} \\sup \\limits _{0\\leq t\\leq T}\\|v_{\\varepsilon }\(t,\\cdot \)\\|^{2}_{L^{2}\( \\mathbb{R}\)}\\leq C\\bigr\(\\|f\\|^{2}_{\\mathscr S'}+\\|\\phi \\|^{2}_{L^{2}\( \\mathbb{R}\)}\\bigr\), \\end{equation}) /S /DISPLAYMATH /Pg 404 0 R /ID (2782) /P 1082 0 R /A 4644 0 R >> endobj 4642 0 obj << /K 4643 0 R /S /EQNUMBER /Pg 404 0 R /P 4641 0 R /ID (2783) >> endobj 4643 0 obj << /K 139 /S /EQNUM /Pg 404 0 R /P 4642 0 R /ID (2781) >> endobj 4644 0 obj << /O /Layout /BBox [ 45 122.29 511.26 141.64 ] 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\\label{eq:1:solve_nondivergenceeq_VTEX1} \\begin{aligned} &\\widetilde D^{*}\\bigr\(\(A\(\\theta \)+\\bar c\)\\widetilde DN\\bigr\)+2\\bar c \\widetilde D^{*}N=F. \\end{aligned} \\end{equation}) /S /DISPLAYMATH /Pg 420 0 R /ID (2894) /P 4721 0 R /A 4730 0 R >> endobj 4728 0 obj << /K 4729 0 R /S /EQNUMBER /Pg 420 0 R /P 4727 0 R /ID (2895) >> endobj 4729 0 obj << /K 83 /S /EQNUM /Pg 420 0 R /P 4728 0 R /ID (2893) >> endobj 4730 0 obj << /O /Layout /BBox [ 50 339.61 516.26 353.39 ] /Placement /Block >> endobj 4731 0 obj << /K [ 87 4732 0 R 89 4734 0 R 93 4736 0 R 95 4738 0 R 97 4740 0 R 99 ] /S /P /Pg 420 0 R /P 4721 0 R /ID (2937) >> endobj 4732 0 obj << /K 88 /Alt ($Y=\(A+\\bar c\)\\widetilde D N+2\\bar cN$) /S /MATH /Pg 420 0 R /ID (2899) /P 4731 0 R /A 4733 0 R >> endobj 4733 0 obj << /O /Layout /BBox [ 154 315.5 255.55 327.75 ] >> endobj 4734 0 obj << /K [ 90 4735 0 R 92 ] /S /EQNUM /Pg 420 0 R /P 4731 0 R /ID (2900) >> endobj 4735 0 obj << /K [ << /Obj 428 0 R /Type /OBJR >> 91 ] 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A}\\rangle}/{\\langle\\frac{1}{\\widetilde A}\\rangle}$) /S /MATH /Pg 431 0 R /ID (2921) /P 4766 0 R /A 4774 0 R >> endobj 4774 0 obj << /O /Layout /BBox [ 401 674.07 469.2 687.88 ] >> endobj 4775 0 obj << /K 14 /Alt ($\\langle G\\rangle =0$) /S /MATH /Pg 431 0 R /ID (2922) /P 4766 0 R /A 4776 0 R >> endobj 4776 0 obj << /O /Layout /BBox [ 45 663.5 78.97 673.5 ] >> endobj 4777 0 obj << /K [ 17 18 4778 0 R ] /Alt (\\begin{equation} \\label{eq:convergent} N\(\\theta \)=\\sum \\limits _{k\\in \\mathbb{Z}^{\\infty}_{*}\\backslash \\{0 \\}} \\frac{\\widehat{ G}\(k\)}{\\mathrm{e}^{\\mathrm{i}{\\langle k,\\omega \\rangle}}-1} \\mathrm{e}^{i\\langle k,\\theta \\rangle}. \\end{equation}) /S /DISPLAYMATH /Pg 431 0 R /ID (2945) /P 4721 0 R /A 4780 0 R >> endobj 4778 0 obj << /K 4779 0 R /S /EQNUMBER /Pg 431 0 R /P 4777 0 R /ID (2946) >> endobj 4779 0 obj << /K 20 /S /EQNUM /Pg 431 0 R /P 4778 0 R /ID (2944) >> endobj 4780 0 obj << /O /Layout /BBox [ 45 614.58 511.26 646.52 ] /Placement /Block >> 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511.26 556.6 ] /Placement /Block >> endobj 4799 0 obj << /K [ 51 4800 0 R 53 4801 0 R 55 4803 0 R 57 ] /S /P /Pg 431 0 R /P 4721 0 R /ID (3022) >> endobj 4800 0 obj << /K [ << /Obj 436 0 R /Type /OBJR >> 52 ] /S /Link /Pg 431 0 R /P 4799 0 R /ID (3023) >> endobj 4801 0 obj << /K 54 /Alt ($ M\\in C^{\\omega}_{r_{1}}\(\\mathbb{T}^{\\infty},\\mathbb{R}\) $) /S /MATH /Pg 431 0 R /ID (2962) /P 4799 0 R /A 4802 0 R >> endobj 4802 0 obj << /O /Layout /BBox [ 272 507.53 346.77 518.5 ] >> endobj 4803 0 obj << /K 56 /Alt ($\\widetilde M\(\\theta \):=\\mathrm{e}^{M\(\\theta \)}$) /S /MATH /Pg 431 0 R /ID (2964) /P 4799 0 R /A 4804 0 R >> endobj 4804 0 obj << /O /Layout /BBox [ 395 508.5 458.25 521.03 ] >> endobj 4805 0 obj << /K [ 59 60 61 4806 0 R ] /Alt (\\begin{equation} \\label{eq:changeofvar} \\widetilde M\(\\theta \)=\\frac{ A-\\bar c}{\\bar c+A}\\widetilde M\(\\theta + \\omega \)\\xch{\\mathrm{e}^{-\\langle \\ln \(\\frac{ A-\\bar c}{\\bar c+A}\)\\rangle},}{\\mathrm{e}^{-\\langle \\ln 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112 425.5 154.35 438.03 ] >> endobj 4815 0 obj << /K [ 75 4816 0 R 77 ] /S /EQNUM /Pg 431 0 R /P 4812 0 R /ID (2973) >> endobj 4816 0 obj << /K [ << /Obj 437 0 R /Type /OBJR >> 76 ] /S /Link /Pg 431 0 R /P 4815 0 R /ID (3026) >> endobj 4817 0 obj << /K [ 80 81 ] /Alt (\\begin{equation*} \\widetilde M\(\\theta +\\omega \)N\(\\theta +\\omega \)-\(1- \\frac{2\\bar c}{A+\\bar c}\)\\widetilde M\(\\theta +\\omega \)N\(\\theta \)= \\widetilde M\(\\theta +\\omega \)G\(\\theta \), \\end{equation*}) /S /DISPLAYMATH /Pg 431 0 R /ID (2974) /P 4721 0 R /A 4818 0 R >> endobj 4818 0 obj << /O /Layout /BBox [ 133 388.31 422.99 409.06 ] /Placement /Block >> endobj 4819 0 obj << /K [ 83 4820 0 R 87 ] /S /P /Pg 431 0 R /P 4721 0 R /ID (3027) >> endobj 4820 0 obj << /K [ 84 4821 0 R 86 ] /S /EQNUM /Pg 431 0 R /P 4819 0 R /ID (2975) >> endobj 4821 0 obj << /K [ << /Obj 438 0 R /Type /OBJR >> 85 ] /S /Link /Pg 431 0 R /P 4820 0 R /ID (3028) >> endobj 4822 0 obj << /K [ 89 90 ] /Alt (\\begin{equation*} \\widetilde M\(\\theta +\\omega \)N\(\\theta +\\omega \)-\\mathrm{e}^{\\langle\\ln \(\\frac{ A-\\bar c}{\\bar c+A}\)\\rangle}\\widetilde M\(\\theta \)N\( \\theta \)=\\widetilde M\(\\theta +\\omega \)G\(\\theta \). \\end{equation*}) /S /DISPLAYMATH /Pg 431 0 R /ID (2976) /P 4721 0 R /A 4823 0 R >> endobj 4823 0 obj << /O /Layout /BBox [ 148 334.5 408.32 347.74 ] /Placement /Block >> endobj 4824 0 obj << /K [ 92 4825 0 R 94 4827 0 R 96 4829 0 R 101 4831 0 R 103 4833 0 R 105 4835 0 R ] /S /P /Pg 431 0 R /P 4721 0 R /ID (3029) >> endobj 4825 0 obj << /K 93 /Alt ($c'$) /S /MATH /Pg 431 0 R /ID (2977) /P 4824 0 R /A 4826 0 R >> endobj 4826 0 obj << /O /Layout /BBox [ 79 306 87.19 312.43 ] >> endobj 4827 0 obj << /K 95 /Alt ($G$) /S /MATH /Pg 431 0 R /ID (2978) /P 4824 0 R /A 4828 0 R >> endobj 4828 0 obj << /O /Layout /BBox [ 99 306 105.83 312.83 ] >> endobj 4829 0 obj << /K [ 97 98 99 100 ] /Alt ($- \\frac{\\langle \\widetilde M\(\\theta +\\omega \)\\frac{Y_{0}\(\\theta \)}{A\(\\theta 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128 /Alt ($v_{\\varepsilon }\\rightarrow u_{0}$) /S /MATH /Pg 431 0 R /ID (2996) /P 4849 0 R /A 4855 0 R >> endobj 4855 0 obj << /O /Layout /BBox [ 324 131.5 358.89 137.31 ] >> endobj 4856 0 obj << /K 130 /Alt ($\\mathscr S$) /S /MATH /Pg 431 0 R /ID (2997) /P 4849 0 R /A 4857 0 R >> endobj 4857 0 obj << /O /Layout /BBox [ 422 133 432.56 139.97 ] >> endobj 4858 0 obj << /K 132 /Alt ($L^{2}\([0,T],L^{2}\(\\varepsilon \\mathbb{Z}\)\)$) /S /MATH /Pg 431 0 R /ID (2998) /P 4849 0 R /A 4859 0 R >> endobj 4859 0 obj << /O /Layout /BBox [ 45 117.5 121.83 128.14 ] >> endobj 4860 0 obj << /K 134 /Alt ($u_{0}$) /S /MATH /Pg 431 0 R /ID (2999) /P 4849 0 R /A 4861 0 R >> endobj 4861 0 obj << /O /Layout /BBox [ 157 118.5 167.21 124.31 ] >> endobj 4862 0 obj << /K [ 136 4863 0 R ] /Alt (\\begin{equation} \\label{eq:averagedeq} \\left \\{ \\begin{aligned} \\widehat{\\mathcal P}u_{0}:&=\\langle Q\\rangle \(u_{0}\)_{t}-\\bar a \\partial _{zz}u_{0}=0& & \\text{ in }[0,T]\\times \\mathbb{R}, \\\\ 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439 0 R /ID (3098) /P 4910 0 R /A 4912 0 R >> endobj 4912 0 obj << /O /Layout /BBox [ 367 514.53 410.78 523.94 ] >> endobj 4913 0 obj << /K [ 65 66 ] /Alt ($O\(\\frac{1}{\\varepsilon }\) $) /S /MATH /Pg 439 0 R /ID (3099) /P 4910 0 R /A 4914 0 R >> endobj 4914 0 obj << /O /Layout /BBox [ 420 513.55 442.06 525.45 ] >> endobj 4915 0 obj << /K 68 /Alt ($\\mathscr S'$) /S /MATH /Pg 439 0 R /ID (3100) /P 4910 0 R /A 4916 0 R >> endobj 4916 0 obj << /O /Layout /BBox [ 92 503 105.36 510.52 ] >> endobj 4917 0 obj << /K [ 71 4918 0 R ] /Alt (\\begin{equation} \\label{eq:approximate} D^{*}\\bigr\(\\widetilde A_{1}\(\\omega z\)DN\(\\omega z\)\\bigr\)+\\bar c\(D^{*}+D\) N\(\\omega z\)+D^{*} \\widetilde A_{1}\(\\omega z\)+2\\bar c-lQ\(\\omega z\)=0. \\end{equation}) /S /DISPLAYMATH /Pg 439 0 R /ID (3102) /P 4870 0 R /A 4920 0 R >> endobj 4918 0 obj << /K 4919 0 R /S /EQNUMBER /Pg 439 0 R /P 4917 0 R /ID (3103) >> endobj 4919 0 obj << /K 73 /S /EQNUM /Pg 439 0 R /P 4918 0 R /ID (3101) >> endobj 4920 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0 R /ID (3109) /P 4921 0 R /A 4930 0 R >> endobj 4930 0 obj << /O /Layout /BBox [ 129 438.5 157.58 448.5 ] >> endobj 4931 0 obj << /K 90 /Alt ($N\\in C^{\\omega}_{r'}\(\\mathbb{T}^{\\infty},\\mathbb{R}\)$) /S /MATH /Pg 439 0 R /ID (3110) /P 4921 0 R /A 4932 0 R >> endobj 4932 0 obj << /O /Layout /BBox [ 182 438.25 254.12 448.5 ] >> endobj 4933 0 obj << /K [ 92 93 ] /Alt ($l=\\frac{2\\bar c}{\\langle Q\\rangle}$) /S /MATH /Pg 439 0 R /ID (3112) /P 4921 0 R /A 4934 0 R >> endobj 4934 0 obj << /O /Layout /BBox [ 298 435.8 329.68 449.45 ] >> endobj 4935 0 obj << /K 95 /Alt ($O\(1\)$) /S /MATH /Pg 439 0 R /ID (3113) /P 4921 0 R /A 4936 0 R >> endobj 4936 0 obj << /O /Layout /BBox [ 492 438.5 512.68 448.5 ] >> endobj 4937 0 obj << /K [ 98 99 100 101 102 4938 0 R ] /Alt (\\begin{equation} \\label{eq:averaged} \\mathcal P_{\\varepsilon }v_{\\varepsilon }^{a}=\\langle Q\\rangle \(u_{0}\)_{t}- \\bar a\\partial _{zz}u_{0}+o\(1\)+f_{\\varepsilon }+R_{0} \( \\frac{\\omega z}{\\varepsilon 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/Placement /Block >> endobj 4966 0 obj << /K [ 146 4967 0 R 152 ] /S /P /Pg 439 0 R /P 4870 0 R /ID (3162) >> endobj 4967 0 obj << /K [ 147 148 149 150 151 ] /Alt ($R_{i}\(\\frac{\\omega z}{\\varepsilon }- \\frac{\\omega lt}{\\varepsilon ^{2}}\)=\\varepsilon \\partial _{z}w_{i}\( \\frac{z}{\\varepsilon }-\\frac{lt}{\\varepsilon ^{2}}\)$) /S /MATH /Pg 439 0 R /ID (3132) /P 4966 0 R /A 4968 0 R >> endobj 4968 0 obj << /O /Layout /BBox [ 114 199.55 244.95 211.8 ] >> endobj 4969 0 obj << /K [ 154 155 156 157 158 ] /Alt (\\begin{equation*} \\|R_{0}\(\\frac{\\omega z}{\\varepsilon }- \\frac{\\omega lt}{\\varepsilon ^{2}}\)\(u_{0}\)_{t}\\|_{\\mathscr S'}=\\sup\\limits _{\\|v\\|_{\\mathscr S}\\leq 1}\\int _{0}^{T}\\int _{\\mathbb{R}}|R_{0}\( \\frac{\\omega z}{\\varepsilon }-\\frac{\\omega lt}{\\varepsilon ^{2}}\)\(u_{0}\)_{t}v| \\leq C\\varepsilon , \\end{equation*}) /S /DISPLAYMATH /Pg 439 0 R /ID (3133) /P 4870 0 R /A 4970 0 R >> endobj 4970 0 obj << /O /Layout /BBox [ 135 148.58 430.66 186.39 ] /Placement /Block >> endobj 4971 0 obj << /K 160 /S /P /Pg 439 0 R /P 4870 0 R /ID (3163) >> endobj 4972 0 obj << /K [ 162 163 164 ] /Alt (\\begin{equation*} \\|R_{1}\(\\frac{\\omega z}{\\varepsilon }- \\frac{\\omega lt}{\\varepsilon ^{2}}\)\\partial _{zz}u_{0}\\|_{\\mathscr S'} \\leq C\\varepsilon , \\end{equation*}) /S /DISPLAYMATH /Pg 439 0 R /ID (3134) /P 4870 0 R /A 4973 0 R >> endobj 4973 0 obj << /O /Layout /BBox [ 215 94.14 351.07 114.71 ] /Placement /Block >> endobj 4974 0 obj << /K [ 166 4975 0 R 168 4977 0 R 172 4979 0 R 176 4981 0 R 180 4983 0 R 184 4985 0 R 190 4987 0 R 192 ] /S /P /Pg 439 0 R /P 4870 0 R /ID (3164) >> endobj 4975 0 obj << /K 167 /Alt ($C$) /S /MATH /Pg 439 0 R /ID (3135) /P 4974 0 R /A 4976 0 R >> endobj 4976 0 obj << /O /Layout /BBox [ 147 74 154.55 80.83 ] >> endobj 4977 0 obj << /K [ 169 170 171 ] /Alt ($\\|w_{0}\(\\frac{\\omega z}{\\varepsilon }- \\frac{\\omega lt}{\\varepsilon ^{2}}\)\(u_{0}\)_{t}\\|_{L^{2}\([0,T]\\times\\mathbb{R}\)}$) /S 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(3139) /P 4974 0 R /A 4984 0 R >> endobj 4984 0 obj << /O /Layout /BBox [ 371 57.5 479.35 103.25 ] >> endobj 4985 0 obj << /K [ 185 186 187 188 189 ] /Alt ($o\(1\)+R_{0} \(\\frac{\\omega z}{\\varepsilon }- \\frac{\\omega lt}{\\varepsilon ^{2}}\)\(u_{0}\)_{t}+R_{1}\( \\frac{\\omega z}{\\varepsilon }-\\frac{\\omega lt}{\\varepsilon ^{2}}\) \\partial _{zz}u_{0}\\rightarrow 0$) /S /MATH /Pg 439 0 R /ID (3140) /P 4974 0 R /A 4986 0 R >> endobj 4986 0 obj << /O /Layout /BBox [ 113 44.5 363.44 60.27 ] >> endobj 4987 0 obj << /K 191 /Alt ($\\mathscr S'$) /S /MATH /Pg 439 0 R /ID (3141) /P 4974 0 R /A 4988 0 R >> endobj 4988 0 obj << /O /Layout /BBox [ 389 47 401.77 53.97 ] >> endobj 4989 0 obj << /K 1 /S /P /Pg 453 0 R /P 4870 0 R /ID (3252) >> endobj 4990 0 obj << /K 3 /Alt (\\begin{equation*} \\left \\{ \\begin{aligned} &\\mathcal P_{\\varepsilon }\(v_{\\varepsilon }-v_{\\varepsilon }^{a}\)=o\(1\)+R_{0} \(z'\)\(u_{0}\)_{t}+R_{1}\(z'\)\\partial _{zz}u_{0} & & \\text{ in } [0,T] \\times 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(3175) /P 5014 0 R /A 5016 0 R >> endobj 5016 0 obj << /O /Layout /BBox [ 173 551.14 239.29 567.1 ] >> endobj 5017 0 obj << /K [ 39 5018 0 R 41 ] /S /EQNUM /Pg 453 0 R /P 5014 0 R /ID (3176) >> endobj 5018 0 obj << /K [ << /Obj 457 0 R /Type /OBJR >> 40 ] /S /Link /Pg 453 0 R /P 5017 0 R /ID (3256) >> endobj 5019 0 obj << /K [ 43 5020 0 R 45 ] /S /EQNUM /Pg 453 0 R /P 5014 0 R /ID (3177) >> endobj 5020 0 obj << /K [ << /Obj 458 0 R /Type /OBJR >> 44 ] /S /Link /Pg 453 0 R /P 5019 0 R /ID (3257) >> endobj 5021 0 obj << /K [ 47 5022 0 R 49 ] /S /P /Pg 453 0 R /P 1082 0 R /ID (3258) >> endobj 5022 0 obj << /K 48 /Alt ($v_{\\varepsilon }\(t,z\)$) /S /MATH /Pg 453 0 R /ID (3178) /P 5021 0 R /A 5023 0 R >> endobj 5023 0 obj << /O /Layout /BBox [ 215 515.5 245.05 525.5 ] >> endobj 5024 0 obj << /K [ 50 51 5025 0 R ] /Alt (\\begin{equation} \\label{eq:oscillationinitial} \\left \\{ \\begin{aligned} \\mathcal P_{\\varepsilon }v_{\\varepsilon }&=0 & & \\text{ in }[0,T] \\times \\mathbb{R}, \\\\ 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endobj 5051 0 obj << /K 81 /Alt ($v=0$) /S /MATH /Pg 453 0 R /ID (3194) /P 5040 0 R /A 5052 0 R >> endobj 5052 0 obj << /O /Layout /BBox [ 485 404 508.54 410.3 ] >> endobj 5053 0 obj << /K [ 83 5054 0 R 85 ] /S /P /Pg 453 0 R /P 5040 0 R /ID (3261) >> endobj 5054 0 obj << /K 84 /Alt ($w\\in C_{0}^{\\infty}\([0,T]\\times \\mathbb{R}\)$) /S /MATH /Pg 453 0 R /ID (3195) /P 5053 0 R /A 5055 0 R >> endobj 5055 0 obj << /O /Layout /BBox [ 94 387.5 179.91 397.5 ] >> endobj 5056 0 obj << /K [ 87 5057 0 R ] /Alt (\\begin{equation} \\label{eq:dual_averaging_problem_VTEX1} \\left \\{ \\begin{aligned} &\\mathcal P_{\\varepsilon }^{*} w_{\\varepsilon }=\\widehat{\\mathcal P}^{*}w & & \\text{ in }[0,T]\\times \\mathbb{R}, \\\\ & w_{\\varepsilon }\(T,z\)=w\(T,z\) & & \\text{ in }\\varepsilon \\mathbb{Z}, \\end{aligned} \\right . \\end{equation}) /S /DISPLAYMATH /Pg 453 0 R /ID (3197) /P 5040 0 R /A 5059 0 R >> endobj 5057 0 obj << /K 5058 0 R /S /EQNUMBER /Pg 453 0 R /P 5056 0 R /ID (3198) >> endobj 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P}$) /S /MATH /Pg 453 0 R /ID (3205) /P 5060 0 R /A 5068 0 R >> endobj 5068 0 obj << /O /Layout /BBox [ 376 319 385.75 326.78 ] >> endobj 5069 0 obj << /K [ 102 5070 0 R 104 ] /S /EQNUM /Pg 453 0 R /P 5060 0 R /ID (3206) >> endobj 5070 0 obj << /K [ << /Obj 460 0 R /Type /OBJR >> 103 ] /S /Link /Pg 453 0 R /P 5069 0 R /ID (3263) >> endobj 5071 0 obj << /K [ << /Obj 461 0 R /Type /OBJR >> 106 ] /S /Link /Pg 453 0 R /P 5060 0 R /ID (3264) >> endobj 5072 0 obj << /K [ 108 5073 0 R 110 ] /S /EQNUM /Pg 453 0 R /P 5060 0 R /ID (3207) >> endobj 5073 0 obj << /K [ << /Obj 462 0 R /Type /OBJR >> 109 ] /S /Link /Pg 453 0 R /P 5072 0 R /ID (3265) >> endobj 5074 0 obj << /K 112 /Alt ($w_{\\varepsilon }$) /S /MATH /Pg 453 0 R /ID (3208) /P 5060 0 R /A 5075 0 R >> endobj 5075 0 obj << /O /Layout /BBox [ 431 303.5 438.32 312.16 ] >> endobj 5076 0 obj << /K [ 115 5077 0 R 117 5079 0 R 119 5081 0 R 121 ] /S /P /Pg 453 0 R /P 5040 0 R /ID (3266) >> endobj 5077 0 obj << /K 116 /Alt ($ w_{\\varepsilon }\\rightarrow w_{0}$) /S /MATH /Pg 453 0 R /ID (3209) /P 5076 0 R /A 5078 0 R >> endobj 5078 0 obj << /O /Layout /BBox [ 180 276.5 218.63 282.31 ] >> endobj 5079 0 obj << /K 118 /Alt ($\\mathscr S$) /S /MATH /Pg 453 0 R /ID (3210) /P 5076 0 R /A 5080 0 R >> endobj 5080 0 obj << /O /Layout /BBox [ 266 278 276.56 284.97 ] >> endobj 5081 0 obj << /K 120 /Alt ($ w_{0}$) /S /MATH /Pg 453 0 R /ID (3211) /P 5076 0 R /A 5082 0 R >> endobj 5082 0 obj << /O /Layout /BBox [ 312 276.5 323.64 282.31 ] >> endobj 5083 0 obj << /K [ 123 5084 0 R ] /Alt (\\begin{equation} \\label{eq:averaged_parabolic_VTEX1} \\left \\{ \\begin{aligned} &-\\langle Q\\rangle \(w_{0}\)_{t}-\\bar a_{*}\\partial _{zz}w_{0}= \\widehat{\\mathcal P}^{*}w= -\\langle Q\\rangle w_{t}-\\bar a\\partial _{zz}w& &\\text{ in }[0,T]\\times \\mathbb{R}, \\\\ & w_{0}\(T,z\)=w\(T,z\) & & \\text{ in }\\mathbb{R}, \\end{aligned} \\right . \\end{equation}) /S /DISPLAYMATH /Pg 453 0 R /ID (3213) /P 5040 0 R /A 5086 0 R >> endobj 5084 0 obj << 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192.21 146.5 ] >> endobj 5103 0 obj << /K [ << /Obj 465 0 R /Type /OBJR >> 152 ] /S /Link /Pg 453 0 R /P 5098 0 R /ID (3272) >> endobj 5104 0 obj << /K 154 /Alt ($N_{*}$) /S /MATH /Pg 453 0 R /ID (3223) /P 5098 0 R /A 5105 0 R >> endobj 5105 0 obj << /O /Layout /BBox [ 45 124.5 57.63 132.83 ] >> endobj 5106 0 obj << /K [ 157 5107 0 R ] /Alt (\\begin{equation} \\label{eq:dualapproximate} D^{*}\\bigr\(\\widetilde A_{1} DN_{*}\\bigr\)-\\bar c\(D^{*}+D\)N_{*}+D^{*} \\widetilde A_{1}-\\bigr\(2\\bar c-lQ\\bigr\)=0, \\end{equation}) /S /DISPLAYMATH /Pg 453 0 R /ID (3225) /P 5040 0 R /A 5109 0 R >> endobj 5107 0 obj << /K 5108 0 R /S /EQNUMBER /Pg 453 0 R /P 5106 0 R /ID (3226) >> endobj 5108 0 obj << /K 159 /S /EQNUM /Pg 453 0 R /P 5107 0 R /ID (3224) >> endobj 5109 0 obj << /O /Layout /BBox [ 45 95.5 511.26 108.75 ] /Placement /Block >> endobj 5110 0 obj << /K [ 163 5111 0 R 169 5113 0 R 171 5115 0 R 173 5117 0 R 175 5119 0 R 181 5121 0 R 183 ] /S /P /Pg 453 0 R /P 5040 0 R /ID (3273) >> 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-\\langle \\widetilde A_{1} \\widetilde D N\\widetilde D N_{*}\\rangle + \\bar c\\langle \(\\widetilde D+\\widetilde D^{*}\) N N_{*}\\rangle &= \\langle \\widetilde D^{*}\(\\widetilde A_{1} \\widetilde D N\) N_{*} \\rangle +\\bar c\\langle \(\\widetilde D+\\widetilde D^{*}\) N N_{*} \\rangle\\\\ &=-\\langle \\widetilde D^{*}\\widetilde A_{1} N_{*}\\rangle -\\langle \(2 \\bar c-lQ\) N_{*}\\rangle\\\\ &=\\langle \\widetilde A_{1} \\widetilde D N_{*}\\rangle -\\langle \(2 \\bar c-lQ\) N_{*}\\rangle , \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 466 0 R /ID (3291) /P 5040 0 R /A 5156 0 R >> endobj 5156 0 obj << /O /Layout /BBox [ 120 498.67 446.33 548.33 ] /Placement /Block >> endobj 5157 0 obj << /K 55 /S /P /Pg 466 0 R /P 5040 0 R /ID (3342) >> endobj 5158 0 obj << /K 57 /Alt (\\begin{equation*} \\begin{aligned} -\\langle \\widetilde A_{1}\\widetilde DN\\widetilde DN_{*}\\rangle + \\bar c\\langle \(\\widetilde D+\\widetilde D^{*}\) N N_{*}\\rangle &= \\langle \\widetilde D^{*}\(\\widetilde A_{1}\\widetilde DN_{*}\) N\\rangle - \\bar c\\langle \(\\widetilde D+\\widetilde D^{*}\) N_{*} N\\rangle\\\\ &=-\\langle \\widetilde D^{*}\\widetilde A_{1}N\\rangle +\\langle \(2\\bar c-lQ\) N\\rangle\\\\ &=\\langle \\widetilde A_{1}\\widetilde DN\\rangle +\\langle \(2\\bar c-lQ\) N \\rangle . \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 466 0 R /ID (3292) /P 5040 0 R /A 5159 0 R >> endobj 5159 0 obj << /O /Layout /BBox [ 120 412.67 446.33 462.33 ] /Placement /Block >> endobj 5160 0 obj << /K [ 59 5161 0 R 61 5163 0 R 65 ] /S /P /Pg 466 0 R /P 5040 0 R /ID (3343) >> endobj 5161 0 obj << /K 60 /Alt ($\\bar a=\\bar a_{*}$) /S /MATH /Pg 466 0 R /ID (3295) /P 5160 0 R /A 5162 0 R >> endobj 5162 0 obj << /O /Layout /BBox [ 79 390.5 107.5 398.3 ] >> endobj 5163 0 obj << /K [ 62 5164 0 R 64 ] /S /EQNUM /Pg 466 0 R /P 5160 0 R /ID (3296) >> endobj 5164 0 obj << /K [ << /Obj 473 0 R /Type /OBJR >> 63 ] /S /Link /Pg 466 0 R /P 5163 0 R /ID (3344) >> endobj 5165 0 obj 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0 R /ID (3347) >> endobj 5174 0 obj << /K 78 /Alt ($w_{\\varepsilon }$) /S /MATH /Pg 466 0 R /ID (3300) /P 5173 0 R /A 5175 0 R >> endobj 5175 0 obj << /O /Layout /BBox [ 116 292.5 127.44 298.31 ] >> endobj 5176 0 obj << /K 80 /Alt ($[0,T]\\times \\mathbb{R}$) /S /MATH /Pg 466 0 R /ID (3301) /P 5173 0 R /A 5177 0 R >> endobj 5177 0 obj << /O /Layout /BBox [ 225 291.5 267.15 301.5 ] >> endobj 5178 0 obj << /K 82 /Alt ($\\mathcal P_{\\varepsilon }v_{\\varepsilon }=0$) /S /MATH /Pg 466 0 R /ID (3302) /P 5173 0 R /A 5179 0 R >> endobj 5179 0 obj << /O /Layout /BBox [ 353 292.5 391.69 300.83 ] >> endobj 5180 0 obj << /K [ 85 86 87 88 5181 0 R ] /Alt (\\begin{equation} \\label{eq:parabolicand_dual_VTEX1} \\begin{aligned} 0&=\\int _{0}^{T}\\int _{\\mathbb{R}}\\mathcal P_{\\varepsilon }v_{ \\varepsilon }w_{\\varepsilon }dtdz=\\int _{0}^{T}\\int _{\\mathbb{R}}v_{ \\varepsilon }\\mathcal P_{\\varepsilon }^{*}w_{\\varepsilon }dtdz+ \\\\ &\\qquad \\qquad \\qquad \\int _{\\mathbb{R}}\\bigr[Q\( 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(3315) /P 5184 0 R /A 5206 0 R >> endobj 5206 0 obj << /O /Layout /BBox [ 200 140 219.83 146.3 ] >> endobj 5207 0 obj << /K 124 /Alt ($w\\in C_{0}^{\\infty}\([0,T]\\times \\mathbb{R}\)$) /S /MATH /Pg 466 0 R /ID (3316) /P 5184 0 R /A 5208 0 R >> endobj 5208 0 obj << /O /Layout /BBox [ 341 137.5 450.09 147.5 ] >> endobj 5209 0 obj << /K 127 /Alt (\\begin{equation*} \\int _{0}^{T}\\int _{\\mathbb{R}} v\(t,z\)\\widehat{\\mathcal P}^{*}w\(t,z\)dtdz+ \\langle Q\\rangle \\int _{\\mathbb{R}} v\(T,z\)w\(T,z\)dz=\\xch{0.}{0,} \\end{equation*}) /S /DISPLAYMATH /Pg 466 0 R /ID (3317) /P 5040 0 R /A 5210 0 R >> endobj 5210 0 obj << /O /Layout /BBox [ 162 84.58 404.56 122.39 ] /Placement /Block >> endobj 5211 0 obj << /K [ 129 5212 0 R 131 5214 0 R 133 5216 0 R 135 5218 0 R 137 5220 0 R 139 5222 0 R ] /S /P /Pg 466 0 R /P 5040 0 R /ID (3352) >> endobj 5212 0 obj << /K 130 /Alt ($v$) /S /MATH /Pg 466 0 R /ID (3318) /P 5211 0 R /A 5213 0 R >> endobj 5213 0 obj << /O /Layout /BBox [ 79 61 84.21 65.31 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5226 0 R 5 5228 0 R 7 5229 0 R 9 5231 0 R 11 5233 0 R 5237 0 R 5244 0 R 5249 0 R 5251 0 R 5254 0 R 5256 0 R 5263 0 R 5272 0 R 5276 0 R 5284 0 R 5286 0 R 5290 0 R 5292 0 R 5299 0 R 5303 0 R 5307 0 R 5324 0 R 5327 0 R 5331 0 R 5343 0 R 5347 0 R 5354 0 R 5356 0 R 5388 0 R 5390 0 R ] /S /Div /Pg 477 0 R /P 5224 0 R /ID (3326) >> endobj 5226 0 obj << /K [ 2 5227 0 R 4 ] /S /Span /Pg 477 0 R /P 5225 0 R /ID (3327) >> endobj 5227 0 obj << /K [ << /Obj 480 0 R /Type /OBJR >> 3 ] /S /Link /Pg 477 0 R /P 5226 0 R /ID (3440) >> endobj 5228 0 obj << /K [ << /Obj 481 0 R /Type /OBJR >> 6 ] /S /Link /Pg 477 0 R /P 5225 0 R /ID (3441) >> endobj 5229 0 obj << /K 8 /Alt ($P, P_{*}\\in C\(\\mathbb{T}^{\\infty},\\mathbb{R}\)$) /S /MATH /Pg 477 0 R /ID (3328) /P 5225 0 R /A 5230 0 R >> endobj 5230 0 obj << /O /Layout /BBox [ 385 676.5 463.74 686.5 ] >> endobj 5231 0 obj << /K 10 /Alt ($p\(n\)=P\(n\\omega \), p_{*}\(n\)=P_{*}\(n\\omega \)$) /S /MATH /Pg 477 0 R /ID (3329) /P 5225 0 R /A 5232 0 R >> endobj 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/K 32 /Alt (\\begin{equation*} D^{*}\\bigr\(A_{1}\(n\\omega +k\\omega \) Dp_{k}\\bigr\)+A_{2}\(n\\omega +k \\omega \)D^{*}p_{k}+W\(n\\omega +k\\omega \)p_{k}=E_{0}p_{k}, \\end{equation*}) /S /DISPLAYMATH /Pg 477 0 R /ID (3362) /P 5225 0 R /A 5250 0 R >> endobj 5250 0 obj << /O /Layout /BBox [ 126 546.5 431.12 558.5 ] /Placement /Block >> endobj 5251 0 obj << /K [ 34 5252 0 R 36 ] /S /P /Pg 477 0 R /P 5225 0 R /ID (3444) >> endobj 5252 0 obj << /K 35 /Alt ($q_{k}\(n\)=P_{*}\(n\\omega +k\\omega \)$) /S /MATH /Pg 477 0 R /ID (3363) /P 5251 0 R /A 5253 0 R >> endobj 5253 0 obj << /O /Layout /BBox [ 65 522.5 157.2 532.5 ] >> endobj 5254 0 obj << /K 38 /Alt (\\begin{equation*} D^{*}\\bigr\(A_{1}\(n\\omega +k\\omega \)Dq_{k}\\bigr\)- D\\bigr\(A_{2}\(n \\omega +k\\omega \)q_{k}\\bigr\)+W\(n\\omega +k\\omega \)q_{k}=E_{0}q_{k}. \\end{equation*}) /S /DISPLAYMATH /Pg 477 0 R /ID (3364) /P 5225 0 R /A 5255 0 R >> endobj 5255 0 obj << /O /Layout /BBox [ 125 496.5 432.43 508.5 ] /Placement /Block >> 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/Alt ($u_{\\varepsilon }\\in L^{2}\([0,T], L^{2}\(\\varepsilon \\mathbb{Z}\)\)$) /S /MATH /Pg 477 0 R /ID (3369) /P 5263 0 R /A 5267 0 R >> endobj 5267 0 obj << /O /Layout /BBox [ 306 458.35 442.43 467.87 ] >> endobj 5268 0 obj << /K [ 57 58 59 60 ] /Alt ($v_{\\varepsilon }\(t,x\)=\\mathrm{e}^{-\\frac{E_{0}t}{\\varepsilon ^{2}}} \\frac{u_{\\varepsilon }\(t,x\)}{P\(\\frac{\\omega x}{\\varepsilon }\)}, x\\in\\varepsilon \\mathbb{Z}$) /S /MATH /Pg 477 0 R /ID (3370) /P 5263 0 R /A 5269 0 R >> endobj 5269 0 obj << /O /Layout /BBox [ 45 433.41 158.86 461.81 ] >> endobj 5270 0 obj << /K [ 62 5271 0 R 64 ] /S /EQNUM /Pg 477 0 R /P 5263 0 R /ID (3371) >> endobj 5271 0 obj << /K [ << /Obj 484 0 R /Type /OBJR >> 63 ] /S /Link /Pg 477 0 R /P 5270 0 R /ID (3449) >> endobj 5272 0 obj << /K [ 67 68 69 70 71 72 5273 0 R ] /Alt (\\begin{equation} \\label{eq:originalredu} \\left \\{ \\begin{aligned} Q\(\\frac{\\omega x}{\\varepsilon }\)\(v_{\\varepsilon }\)_{t}&=D_{ \\varepsilon }^{*}\(\\widetilde A_{1}\\bigr\(\\frac{\\omega x}{\\varepsilon }\)D_{ \\varepsilon }v_{\\varepsilon }\\bigr\)-\\frac{\\bar c}{\\varepsilon }\(D_{ \\varepsilon }^{*}+D_{\\varepsilon }\)v_{\\varepsilon }& &\\text{ in }[0,T] \\times \\varepsilon \\mathbb{Z}, \\\\ v_{\\varepsilon }\(0,x\)&= \\frac{\\varphi \(x\)}{P\(\\frac{\\omega x}{\\varepsilon }\)} & & \\text{ in } \\varepsilon \\mathbb{Z}. \\end{aligned} \\right . \\end{equation}) /S /DISPLAYMATH /Pg 477 0 R /ID (3373) /P 5225 0 R /A 5275 0 R >> endobj 5273 0 obj << /K 5274 0 R /S /EQNUMBER /Pg 477 0 R /P 5272 0 R /ID (3374) >> endobj 5274 0 obj << /K 74 /S /EQNUM /Pg 477 0 R /P 5273 0 R /ID (3372) >> endobj 5275 0 obj << /O /Layout /BBox [ 45 377.26 511.26 427.74 ] /Placement /Block >> endobj 5276 0 obj << /K [ 78 5277 0 R 80 5279 0 R 83 5281 0 R 85 5283 0 R 87 ] /S /P /Pg 477 0 R /P 5225 0 R /ID (3450) >> endobj 5277 0 obj << /K 79 /Alt ($\\varphi \\in C^{1}\(\\mathbb{R}\)$) /S /MATH /Pg 477 0 R /ID (3375) /P 5276 0 R /A 5278 0 R >> endobj 5278 0 obj 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\(x_{0},y_{0}\)\)^{-1}D_{y}\\Psi \(x,y\) \\|_{L\(Y,Y\)}\\leq \\frac{1}{2}, \\end{equation}) /S /DISPLAYMATH /Pg 490 0 R /ID (3518) /P 1082 0 R /A 5436 0 R >> endobj 5434 0 obj << /K 5435 0 R /S /EQNUMBER /Pg 490 0 R /P 5433 0 R /ID (3519) >> endobj 5435 0 obj << /K 150 /S /EQNUM /Pg 490 0 R /P 5434 0 R /ID (3517) >> endobj 5436 0 obj << /O /Layout /BBox [ 50 70.46 516.26 96.06 ] /Placement /Block >> endobj 5437 0 obj << /K [ 153 5438 0 R 156 5440 0 R 158 ] /S /P /Pg 490 0 R /P 1082 0 R /ID (3550) >> endobj 5438 0 obj << /K [ 154 155 ] /Alt ($y\\in C^{1}\(B_{s}\(x_{0}\),\\overline{B_{\\delta}\(y_{0}\)}\)$) /S /MATH /Pg 490 0 R /ID (3520) /P 5437 0 R /A 5439 0 R >> endobj 5439 0 obj << /O /Layout /BBox [ 165 44.5 266.03 56.5 ] >> endobj 5440 0 obj << /K 157 /Alt ($\\Psi \(x,y\(x\)\)=0$) /S /MATH /Pg 490 0 R /ID (3521) /P 5437 0 R /A 5441 0 R >> endobj 5441 0 obj << /O /Layout /BBox [ 313 44.5 375.8 54.5 ] >> endobj 5442 0 obj << /K 5443 0 R /S /TRIVLIST /Pg 501 0 R /P 1082 0 R /ID (3522) 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5449 0 obj << /K 10 /Alt (\\begin{equation*} \\Psi \(\\xi ,Y,f\)=\\mathbb P_{nre}\\ln \(\\mathrm{e}^{-A\(\\xi \)^{-1}Y\(\\xi , \\theta +\\omega \)A\(\\xi \)}\\mathrm{e}^{f\(\\xi ,\\theta \)}\\mathrm{e}^{Y\( \\xi ,\\theta \)}\). \\end{equation*}) /S /DISPLAYMATH /Pg 501 0 R /ID (3551) /P 5443 0 R /A 5450 0 R >> endobj 5450 0 obj << /O /Layout /BBox [ 160 597.5 396.22 610.48 ] /Placement /Block >> endobj 5451 0 obj << /K [ 12 5452 0 R 14 5454 0 R 16 5456 0 R 18 5457 0 R 20 ] /S /P /Pg 501 0 R /P 5443 0 R /ID (3588) >> endobj 5452 0 obj << /K 13 /Alt ($X=\\mathcal I\\times \\mathscr B_{\\mathcal I}, Y=Z=\\mathscr B^{nre}_{ \\mathcal I}\(\\eta \)$) /S /MATH /Pg 501 0 R /ID (3552) /P 5451 0 R /A 5453 0 R >> endobj 5453 0 obj << /O /Layout /BBox [ 63 571.25 200.73 581.5 ] >> endobj 5454 0 obj << /K 15 /Alt ($\\Psi \\in C^{1}\(X\\times Y,Z\)$) /S /MATH /Pg 501 0 R /ID (3553) /P 5451 0 R /A 5455 0 R >> endobj 5455 0 obj << /O /Layout /BBox [ 277 571.5 356.76 582.14 ] >> endobj 5456 0 obj << /K 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\\end{equation*}) /S /DISPLAYMATH /Pg 501 0 R /ID (3556) /P 5443 0 R /A 5464 0 R >> endobj 5464 0 obj << /O /Layout /BBox [ 156 477.78 400.69 491.22 ] /Placement /Block >> endobj 5465 0 obj << /K 30 /S /P /Pg 501 0 R /P 5443 0 R /ID (3592) >> endobj 5466 0 obj << /K 32 /Alt (\\begin{equation*} \\|D_{Y}\\Psi \(\\xi ,0,0\)\(Y'\)\\|\\geq |A\(\\xi \)^{-1}Y'\(\\xi ,\\theta +\\omega \)A\( \\xi \)-Y'\(\\xi ,\\theta \)|_{\\mathcal B}\\geq \\eta |Y'|_{\\mathcal B}. \\end{equation*}) /S /DISPLAYMATH /Pg 501 0 R /ID (3557) /P 5443 0 R /A 5467 0 R >> endobj 5467 0 obj << /O /Layout /BBox [ 129 426.5 428.4 437.64 ] /Placement /Block >> endobj 5468 0 obj << /K [ 34 5469 0 R 36 ] /S /P /Pg 501 0 R /P 5443 0 R /ID (3593) >> endobj 5469 0 obj << /K 35 /Alt ($\\|\(D_{Y}\\Psi \(\\xi ,0,0\)\)^{-1}\\|\\leq \\eta ^{-1}$) /S /MATH /Pg 501 0 R /ID (3558) /P 5468 0 R /A 5470 0 R >> endobj 5470 0 obj << /O /Layout /BBox [ 97 400.5 209.39 411.14 ] >> endobj 5471 0 obj << /K [ 38 5472 0 R 40 5474 0 R 42 5476 0 R 44 5478 0 R 46 5480 0 R 48 5482 0 R 51 5484 0 R 55 ] /S /P /Pg 501 0 R /P 5443 0 R /ID (3594) >> endobj 5472 0 obj << /K 39 /Alt ($\\xi _{0}\\in \\mathcal I$) /S /MATH /Pg 501 0 R /ID (3559) /P 5471 0 R /A 5473 0 R >> endobj 5473 0 obj << /O /Layout /BBox [ 75 388.06 107.31 396.94 ] >> endobj 5474 0 obj << /K 41 /Alt ($A\\in C\(\\mathcal I,{\\mathrm{GL}}\)$) /S /MATH /Pg 501 0 R /ID (3560) /P 5471 0 R /A 5475 0 R >> endobj 5475 0 obj << /O /Layout /BBox [ 133 387.5 197.85 404.1 ] >> endobj 5476 0 obj << /K 43 /Alt ($\\rho =\\rho \(\\xi _{0}\)>0$) /S /MATH /Pg 501 0 R /ID (3561) /P 5471 0 R /A 5477 0 R >> endobj 5477 0 obj << /O /Layout /BBox [ 253 387.5 325.9 397.5 ] >> endobj 5478 0 obj << /K 45 /Alt ($|\\xi _{0}-\\xi |\\leq \\rho $) /S /MATH /Pg 501 0 R /ID (3562) /P 5471 0 R /A 5479 0 R >> endobj 5479 0 obj << /O /Layout /BBox [ 337 387.5 410.39 397.5 ] >> endobj 5480 0 obj << /K 47 /Alt ($\\|A\(\\xi _{0}\)-A\(\\xi \)\\|+\\|A\(\\xi _{0}\)^{-1}-A\(\\xi \)^{-1}\\|\\leq\\varepsilon $) /S /MATH /Pg 501 0 R /ID (3563) /P 5471 0 R /A 5481 0 R >> endobj 5481 0 obj << /O /Layout /BBox [ 435 387.5 689.43 404.36 ] >> endobj 5482 0 obj << /K [ 49 50 ] /Alt ($s=\\min \\{\\rho ,\\varepsilon \\},\\delta =\\varepsilon ^{\\frac{1}{2}}$) /S /MATH /Pg 501 0 R /ID (3564) /P 5471 0 R /A 5483 0 R >> endobj 5483 0 obj << /O /Layout /BBox [ 192 372.5 281.1 391.74 ] >> endobj 5484 0 obj << /K [ 52 53 54 ] /Alt ($\\eta \\geq \\frac{60\(\\|A\\|^{3}+1\)}{\\inf |\\det A|}\\varepsilon ^{ \\frac{1}{2}}$) /S /MATH /Pg 501 0 R /ID (3565) /P 5471 0 R /A 5485 0 R >> endobj 5485 0 obj << /O /Layout /BBox [ 307 373.06 355.65 419.85 ] >> endobj 5486 0 obj << /K [ 57 58 59 60 ] /Alt (\\begin{equation*} 2\\sup \\limits _{\\overline {B_{s}\(\(\\xi _{0},0\)\)}}\\|\\Psi \(\\xi ,0,f\) \\|\\times \\|\(D_{Y}\\Psi \(\\xi _{0},0,0\)\)^{-1}\\|\\leq 2\\varepsilon\\frac{\\inf |\\det A|}{60\(\\|A\\|^{3}+1\)}\\varepsilon ^{-\\frac{1}{2}}\\leq\\delta , \\end{equation*}) /S /DISPLAYMATH /Pg 501 0 R /ID (3566) /P 5443 0 R /A 5487 0 R >> endobj 5487 0 obj << /O /Layout /BBox [ 121 327.99 435.08 355.27 ] /Placement /Block >> endobj 5488 0 obj << /K [ 62 5489 0 R 66 ] /S /P /Pg 501 0 R /P 5443 0 R /ID (3595) >> endobj 5489 0 obj << /K [ 63 5490 0 R 65 ] /S /EQNUM /Pg 501 0 R /P 5488 0 R /ID (3567) >> endobj 5490 0 obj << /K [ << /Obj 507 0 R /Type /OBJR >> 64 ] /S /Link /Pg 501 0 R /P 5489 0 R /ID (3596) >> endobj 5491 0 obj << /K [ 68 5492 0 R 70 5494 0 R 72 ] /S /P /Pg 501 0 R /P 5443 0 R /ID (3597) >> endobj 5492 0 obj << /K 69 /Alt ($Y,f$) /S /MATH /Pg 501 0 R /ID (3568) /P 5491 0 R /A 5493 0 R >> endobj 5493 0 obj << /O /Layout /BBox [ 168 292.06 190.02 300.94 ] >> endobj 5494 0 obj << /K 71 /Alt ($\\mathrm{gl}\(2,\\mathbb{R}\)$) /S /MATH /Pg 501 0 R /ID (3569) /P 5491 0 R /A 5495 0 R >> endobj 5495 0 obj << /O /Layout /BBox [ 229 291.5 275.67 301.83 ] >> endobj 5496 0 obj << /K [ 74 75 76 ] /Alt (\\begin{equation*} \\begin{aligned} &\\quad D_{Y}\\Psi \(\\xi ,Y,f\)\(Y'\)-D_{Y}\\Psi \(\\xi _{0},0,0\)\(Y'\) \\\\ &=\\mathbb P_{nre}\(\(A\(\\xi _{0}\)^{-1}-A\(\\xi \)^{-1}\)Y'\(\\xi ,\\theta + \\omega \)A\(\\xi _{0}\)\)-A\(\\xi \)^{-1}Y'\(\\xi ,\\theta +\\omega \)\(A\(\\xi \)-A\( \\xi _{0}\)\) \\\\ &\\quad \\quad -\\mathbb P_{nre}\(O\(A\(\\xi \)^{-1}Y\(\\xi ,\\theta +\\omega \)A\( \\xi \)\)A\(\\xi \)^{-1}Y'\(\\xi ,\\theta +\\omega \)A\(\\xi \)+\\frac{1}{2}[Y''',M+H]+ \\cdots \) \\\\ &\\quad \\quad +\\mathbb P_{nre}\(O\(Y\(\\xi ,\\theta \)\)Y'\(\\xi ,\\theta \)+ \\frac{1}{2}[D+G,-Y'']+\\cdots \\xch{\),}{\).} \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 501 0 R /ID (3570) /P 5443 0 R /A 5497 0 R >> endobj 5497 0 obj << /O /Layout /BBox [ 88 193.91 468.36 277.09 ] /Placement /Block >> endobj 5498 0 obj << /K [ 78 5499 0 R 80 5501 0 R 82 5503 0 R 84 5505 0 R 86 5507 0 R 88 ] /S /P /Pg 501 0 R /P 5443 0 R /ID (3598) >> endobj 5499 0 obj << /K 79 /Alt ($Y''\(\\xi ,\\theta \)=Y'\(\\xi ,\\theta \)+O\(Y\(\\xi ,\\theta \)\)Y'\(\\xi ,\\theta \)$) /S /MATH /Pg 501 0 R /ID (3571) /P 5498 0 R /A 5500 0 R >> endobj 5500 0 obj << /O /Layout /BBox [ 75 169.5 247.19 179.52 ] >> endobj 5501 0 obj << /K 81 /Alt ($Y'''\(\\xi ,\\theta \)=-A\(\\xi \)^{-1}Y'\(\\xi ,\\theta +\\omega \)A\(\\xi \)-O\(A\( \\xi \)^{-1}Y\(\\xi ,\\theta +\\omega \)A\(\\xi \)\)A\(\\xi \)^{-1}Y'\(\\xi ,\\theta + \\omega \)A\(\\xi \)$) /S /MATH /Pg 501 0 R /ID (3572) /P 5498 0 R /A 5502 0 R >> endobj 5502 0 obj << /O /Layout /BBox [ 256 169.5 647.57 180.14 ] >> endobj 5503 0 obj << /K 83 /Alt ($D\(\\xi ,\\theta \)=-A\(\\xi \)^{-1}\(Y\(\\xi ,\\theta +\\omega \)+Y'\(\\xi , \\theta +\\omega \)\)A\(\\xi \)+f\(\\xi ,\\theta \)+Y\(\\xi ,\\theta \)$) /S /MATH /Pg 501 0 R /ID (3573) /P 5498 0 R /A 5504 0 R >> endobj 5504 0 obj << /O /Layout /BBox [ 191 156.5 496.7 167.14 ] >> endobj 5505 0 obj << /K 85 /Alt ($M\(\\xi ,\\theta \)=-A\(\\xi \)^{-1}Y\(\\xi ,\\theta +\\omega \)A\(\\xi \)+f\(\\xi , \\theta \) +Y\(\\xi ,\\theta \)$) /S /MATH /Pg 501 0 R /ID (3574) /P 5498 0 R /A 5506 0 R >> endobj 5506 0 obj << /O /Layout /BBox [ 45 142.5 281.26 153.14 ] >> endobj 5507 0 obj << /K 87 /Alt ($G, H$) /S /MATH /Pg 501 0 R /ID (3575) /P 5498 0 R /A 5508 0 R >> endobj 5508 0 obj << /O /Layout /BBox [ 304 143.06 325.43 151.83 ] >> endobj 5509 0 obj << /K [ 90 5510 0 R 92 5512 0 R 94 5514 0 R 96 ] /S /P /Pg 501 0 R /P 5443 0 R /ID (3599) >> endobj 5510 0 obj << /K 91 /Alt ($\\mathcal B$) /S /MATH /Pg 501 0 R /ID (3576) /P 5509 0 R /A 5511 0 R >> endobj 5511 0 obj << /O /Layout /BBox [ 130 132 136.87 138.83 ] >> endobj 5512 0 obj << /K 93 /Alt ($\(H\)$) /S /MATH /Pg 501 0 R /ID (3577) /P 5509 0 R /A 5513 0 R >> endobj 5513 0 obj << /O /Layout /BBox [ 214 129.5 230.9 139.5 ] >> endobj 5514 0 obj << /K 95 /Alt ($\\|Y\\|_{\\mathcal B,\\mathcal I}\\leq \\delta , \\|f\\|_{\\mathcal B, \\mathcal I}\\leq s$) /S /MATH /Pg 501 0 R /ID (3578) /P 5509 0 R /A 5515 0 R >> endobj 5515 0 obj << /O /Layout /BBox [ 252 129.14 352.66 139.5 ] >> endobj 5516 0 obj << /K [ 98 99 100 101 ] /Alt (\\begin{equation*} \\begin{aligned} &\\quad \\|D_{Y}\\Psi \(\\xi ,Y,f\)\(Y'\)-D_{Y}\\Psi \(\\xi _{0},0,0\)\(Y'\)\\| \\\\ &\\leq \\frac{6|Y'|_{\\mathcal B}\(\\|A\\|^{3}+1\)}{|\\det A\(\\xi _{0}\)|} \\bigr\(2\\|A\(\\xi \)-A\(\\xi _{0}\)\\|+2\\|A\(\\xi _{0}\)^{-1}-A\(\\xi \)^{-1}\\|+|Y|_{ \\mathcal B}+|f|_{\\mathcal B}\\bigr\) \\\\ &\\leq \\frac{6\(\\|A\\|^{3}+1\)|Y'|_{\\mathcal B}}{\\inf |\\det A\(\\xi \)|} \\bigr\(3\\varepsilon +\\varepsilon ^{\\frac{1}{2}}\\xch{\\bigr\),}{\\bigr\).} \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 501 0 R /ID (3579) /P 5443 0 R /A 5517 0 R >> endobj 5517 0 obj << /O /Layout /BBox [ 110 40.53 447.29 114.47 ] /Placement /Block >> endobj 5518 0 obj << /K 1 /S /P /Pg 508 0 R /P 5443 0 R /ID (3728) >> endobj 5519 0 obj << /K [ 3 4 5 6 7 ] /Alt (\\begin{equation*} \\sup \\limits _{\\overline{B_{s}\(\(\\xi _{0},0\)\)}\\times\\overline{B_{\\delta}\(0\)}}\\|D_{Y}\\Psi \(\\xi _{0},0,0\)-D_{Y}\\Psi \(\\xi ,Y,f\) \\|\\leq\\frac{6\(\\|A\\|^{3}+1\)\(\\varepsilon ^{\\frac{1} {2}}+3\\varepsilon \)}{\\inf |\\det A\(\\xi \)|}. \\end{equation*}) /S /DISPLAYMATH /Pg 508 0 R /ID (3580) /P 5443 0 R /A 5520 0 R >> endobj 5520 0 obj << /O /Layout /BBox [ 126 633.99 439.07 663.23 ] /Placement /Block >> endobj 5521 0 obj << /K 9 /S /P /Pg 508 0 R /P 5443 0 R /ID (3729) >> endobj 5522 0 obj << /K [ 11 12 13 14 15 16 17 18 19 20 ] /Alt (\\begin{equation*} \\begin{aligned} &\\sup \\limits _{\\overline{B_{s}\(\(\\xi _{0},0\)\)}\\times\\overline{B_{\\delta}\(0\)}}\\|\\operatorname{id}_{\\mathcal B_{\\xi _{0}}^{nre}\( \\eta \)}-\(D_{Y}\\Psi \(\\xi _{0},0,0\)\)^{-1}\\times D_{Y}\\Psi \(\\xi ,Y,f\)\\| \\\\ &\\quad \\leq \\sup \\limits _{\\overline{B_{s}\(\(\\xi _{0},0\)\)}\\times\\overline{B_{\\delta}\(0\)}}\\|D_{Y}\\Psi \(\\xi _{0},0,0\)-D_{Y}\\Psi \(\\xi ,Y,f\) \\|\\|\(D_{Y}\\Psi \(\\xi _{0},0,0\)\)^{-1}\\| \\\\ &\\quad \\leq\\frac{6\(\\|A\\|^{3}+1\)\(\\varepsilon ^{\\frac{1} {2}}+3\\varepsilon \)}{\\inf |\\det A\(\\xi \)|} \\frac{\\inf |\\det A\(\\xi \)|}{60\(\\|A\\|^{3}+1\)}\\varepsilon ^{-\\frac{1}{2}} \\leq \\frac{1}{2} \\end{aligned} \\end{equation*}) /S /DISPLAYMATH /Pg 508 0 R /ID (3600) /P 5443 0 R /A 5523 0 R >> endobj 5523 0 obj << /O /Layout /BBox [ 127 518.3 439.31 600.7 ] /Placement /Block >> endobj 5524 0 obj << /K [ 22 5525 0 R 26 5527 0 R 28 5528 0 R 30 5530 0 R 34 5532 0 R 36 5534 0 R 39 ] /S /P /Pg 508 0 R /P 5443 0 R /ID (3730) >> endobj 5525 0 obj << /K [ 23 5526 0 R 25 ] /S /EQNUM /Pg 508 0 R /P 5524 0 R /ID (3601) >> endobj 5526 0 obj << /K [ << /Obj 511 0 R /Type /OBJR >> 24 ] /S /Link /Pg 508 0 R /P 5525 0 R /ID (3731) >> endobj 5527 0 obj << /K [ << /Obj 512 0 R /Type /OBJR >> 27 ] /S /Link /Pg 508 0 R /P 5524 0 R /ID (3732) >> endobj 5528 0 obj << /K 29 /Alt ($\\|f\\|_{\\mathcal B,\\mathcal I}\\leq \\varepsilon $) /S /MATH /Pg 508 0 R /ID (3602) /P 5524 0 R /A 5529 0 R >> endobj 5529 0 obj << /O /Layout /BBox [ 313 490.14 360.27 500.5 ] >> endobj 5530 0 obj << /K [ 31 32 33 ] /Alt ($\\eta \\geq\\frac{60\(1+\\|A\\|^{3}\)\\varepsilon ^{\\frac{1}{2}}}{\\inf |\\det A|}$) /S /MATH /Pg 508 0 R /ID (3603) /P 5524 0 R /A 5531 0 R >> endobj 5531 0 obj << /O /Layout /BBox [ 387 487.8 460.58 506.35 ] >> endobj 5532 0 obj << /K 35 /Alt ($Y=Y\(\\xi ,f\),\\ F^{re}=F^{re}\(\\xi ,f\)\\in C\(\\mathcal I\\times \\mathscr B_{ \\mathcal I},\\mathscr B^{re}_{\\mathcal I}\)$) /S /MATH /Pg 508 0 R /ID (3604) /P 5524 0 R /A 5533 0 R >> endobj 5533 0 obj << /O /Layout /BBox [ 50 474.25 261.54 484.5 ] >> endobj 5534 0 obj << /K [ 37 38 ] /Alt ($\\|Y\(\\xi ,f\)\\|_{\\mathcal B,\\mathcal I}\\leq \\varepsilon ^{\\frac{1}{2}}$) /S /MATH /Pg 508 0 R /ID (3605) /P 5524 0 R /A 5535 0 R >> endobj 5535 0 obj << /O /Layout /BBox [ 349 474.14 427.65 486.46 ] >> endobj 5536 0 obj << /K 41 /Alt (\\begin{equation*} \\Psi \(A\(\\xi \),Y\(\\xi ,f\),f\)=0, \\end{equation*}) /S /DISPLAYMATH /Pg 508 0 R /ID (3606) /P 5443 0 R /A 5537 0 R >> endobj 5537 0 obj << /O /Layout /BBox [ 231 448.5 333.7 458.5 ] /Placement /Block >> endobj 5538 0 obj << /K 43 /S /P /Pg 508 0 R /P 5443 0 R /ID (3733) >> endobj 5539 0 obj << /K 45 /Alt (\\begin{equation*} \\mathrm{e}^{-A\(\\xi \)^{-1}Y\(\\xi ,f\)\(\\theta +\\omega \)A\(\\xi \)}\\mathrm{e}^{f\( \\theta \)}\\mathrm{e}^{Y\(\\xi ,f\)\(\\theta \)}=\\mathrm{e}^{F^{re}\(\\xi ,f\)\( \\theta \)}. \\end{equation*}) /S /DISPLAYMATH /Pg 508 0 R /ID (3607) /P 5443 0 R /A 5540 0 R >> endobj 5540 0 obj << /O /Layout /BBox [ 173 401 391.91 411.48 ] /Placement /Block >> endobj 5541 0 obj << /K 47 /S /P /Pg 508 0 R /P 5443 0 R /ID (3734) >> endobj 5542 0 obj << /K 49 /Alt (\\begin{equation*} \\mathrm{e}^{-Y\(\\xi ,\\theta +\\omega \)}A\(\\xi \)\\mathrm{e}^{F\(\\xi , \\theta \)}\\mathrm{e}^{Y\(\\xi ,\\theta \)}=A\(\\xi \)\\mathrm{e}^{F^{re}\(\\xi , \\theta \)}. \\end{equation*}) /S /DISPLAYMATH /Pg 508 0 R /ID (3608) /P 5443 0 R /A 5543 0 R >> endobj 5543 0 obj << /O /Layout /BBox [ 187 347.5 379.45 359.38 ] /Placement /Block >> endobj 5544 0 obj << /K [ 51 5545 0 R 53 5547 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0 R 1976 0 R 1981 0 R 1982 0 R 1983 0 R 1982 0 R 1981 0 R 1984 0 R 1981 0 R 1986 0 R 1981 0 R 1988 0 R 1981 0 R 1990 0 R 1981 0 R 1992 0 R 1994 0 R 1995 0 R 1994 0 R 1997 0 R 1994 0 R 1999 0 R 1994 0 R 2001 0 R 1994 0 R 2003 0 R 1994 0 R 2005 0 R 2005 0 R 2005 0 R 2005 0 R 2005 0 R 2005 0 R null 2007 0 R null null 2009 0 R 2010 0 R 2009 0 R 2012 0 R 2009 0 R 2014 0 R 2009 0 R 2016 0 R 2009 0 R 2018 0 R 2009 0 R 2020 0 R 2021 0 R 2020 0 R 2009 0 R 2022 0 R 2009 0 R 2024 0 R 2009 0 R 2026 0 R 2027 0 R 2026 0 R 2028 0 R null 2030 0 R null null 2032 0 R 2033 0 R 2032 0 R 2034 0 R 2035 0 R 2034 0 R 2032 0 R 2036 0 R 2037 0 R 2036 0 R 2039 0 R 2036 0 R 2041 0 R 2036 0 R 2043 0 R 2036 0 R 2045 0 R 2036 0 R 2047 0 R 2036 0 R 2049 0 R 2051 0 R 2052 0 R 2053 0 R 2052 0 R 2051 0 R 2054 0 R 2051 0 R 2056 0 R 2051 0 R 2058 0 R 2051 0 R 2060 0 R 2051 0 R 2062 0 R 2051 0 R 2064 0 R 2065 0 R 2064 0 R 2067 0 R 2064 0 R 2069 0 R 2070 0 R 2069 0 R 2071 0 R 2069 0 R 2073 0 R 2074 0 R 2073 0 R 2069 0 R 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253 3218 0 R 254 3229 0 R 255 3242 0 R 256 3249 0 R ] >> endobj 5986 0 obj [ 2263 0 R 2264 0 R 2263 0 R 2265 0 R 2263 0 R 2267 0 R 2263 0 R 2269 0 R 2270 0 R 2269 0 R 2263 0 R 2271 0 R 2263 0 R 2273 0 R 2263 0 R 2275 0 R 2277 0 R 2278 0 R 2277 0 R 2279 0 R 2277 0 R 2281 0 R 2282 0 R 2281 0 R 2277 0 R 2283 0 R 2284 0 R 2285 0 R 2284 0 R 2286 0 R 2284 0 R 2287 0 R 2284 0 R 2289 0 R 2284 0 R 2291 0 R 2284 0 R 2293 0 R 2294 0 R 2295 0 R 2295 0 R 2294 0 R 2297 0 R 2294 0 R 2299 0 R null 2301 0 R null null 2303 0 R 2304 0 R 2303 0 R 2306 0 R 2303 0 R 2308 0 R 2303 0 R 2310 0 R 2312 0 R 2313 0 R null 2312 0 R 2315 0 R 2312 0 R 2317 0 R null 2312 0 R 2319 0 R 2312 0 R 2321 0 R 2321 0 R 2321 0 R 2321 0 R 2321 0 R 2321 0 R 2323 0 R 2324 0 R null 2325 0 R 2324 0 R 2327 0 R null 2324 0 R 2329 0 R 2330 0 R 2329 0 R 2324 0 R 2331 0 R 2331 0 R 2331 0 R 2324 0 R 2333 0 R 2324 0 R 2335 0 R 2335 0 R 2335 0 R 2324 0 R 2337 0 R 2337 0 R 2337 0 R 2324 0 R 2339 0 R 2324 0 R 2341 0 R 2342 0 R 2343 0 R 2342 0 R 2344 0 R 2342 0 R 2346 0 R 2342 0 R 2348 0 R 2342 0 R 2350 0 R 2342 0 R 2352 0 R 2354 0 R 2355 0 R 2354 0 R 2356 0 R 2354 0 R 2358 0 R 2354 0 R 2360 0 R 2354 0 R 2362 0 R 2354 0 R 2364 0 R ] endobj 5987 0 obj [ 2366 0 R 2367 0 R 2368 0 R 2369 0 R 2368 0 R 2367 0 R 2370 0 R 2367 0 R 2372 0 R 2372 0 R 2372 0 R 2372 0 R 2372 0 R 2367 0 R 2374 0 R null 2376 0 R null null 2378 0 R 2379 0 R 2379 0 R 2379 0 R 2378 0 R 2381 0 R 2381 0 R 2381 0 R 2381 0 R 2381 0 R 2378 0 R 2383 0 R 2383 0 R 2383 0 R 2378 0 R 2385 0 R 2386 0 R 2385 0 R 2378 0 R 2387 0 R 2378 0 R 2389 0 R 2378 0 R 2391 0 R 2391 0 R 2391 0 R 2393 0 R 2394 0 R 2393 0 R 2396 0 R 2397 0 R 2398 0 R 2397 0 R 2400 0 R 2397 0 R 2402 0 R 2397 0 R 2404 0 R 2397 0 R 2406 0 R 2397 0 R 2408 0 R 2410 0 R 2411 0 R 2412 0 R 2411 0 R 2414 0 R 2411 0 R 2416 0 R null 2418 0 R null null 2420 0 R 2421 0 R 2420 0 R 2423 0 R 2420 0 R 2425 0 R 2427 0 R 2428 0 R 2427 0 R 2430 0 R 2430 0 R null 2432 0 R null null 2434 0 R 2435 0 R 2434 0 R 2437 0 R null 2439 0 R null null 2441 0 R 2442 0 R 2441 0 R ] endobj 5988 0 obj [ 2444 0 R 2445 0 R 2444 0 R 2447 0 R 2444 0 R 2448 0 R 2444 0 R 2449 0 R 2450 0 R 2449 0 R 2451 0 R 2449 0 R 2452 0 R 2449 0 R null null 2456 0 R 2455 0 R 2457 0 R 2455 0 R 2459 0 R 2460 0 R 2459 0 R 2455 0 R 2461 0 R 2461 0 R 2461 0 R null 2463 0 R null 2464 0 R null 2466 0 R 2467 0 R 2466 0 R 2469 0 R 2470 0 R 2469 0 R 2466 0 R null 2471 0 R 2472 0 R 2471 0 R null 2474 0 R null 2476 0 R 2477 0 R 2476 0 R 2479 0 R 2476 0 R 2481 0 R 2482 0 R 2481 0 R 2476 0 R 2483 0 R 2476 0 R 2485 0 R 2476 0 R 2487 0 R 2476 0 R 2489 0 R 2476 0 R null 2491 0 R null 2493 0 R 2494 0 R 2495 0 R 2494 0 R 2493 0 R null 2496 0 R 2496 0 R null 2498 0 R 2499 0 R 2498 0 R 2501 0 R null null null 2503 0 R 2504 0 R 2505 0 R 2505 0 R 2505 0 R 2507 0 R 2508 0 R 2507 0 R 2510 0 R 2507 0 R 2512 0 R 2507 0 R 2514 0 R 2507 0 R 2516 0 R 2507 0 R ] endobj 5989 0 obj [ 2518 0 R 2519 0 R 2520 0 R 2519 0 R 2522 0 R 2519 0 R 2524 0 R 2519 0 R 2526 0 R 2519 0 R 2528 0 R 2529 0 R 2528 0 R 2519 0 R 2530 0 R null 2532 0 R null null 2534 0 R 2535 0 R 2534 0 R 2537 0 R 2534 0 R 2539 0 R null 2541 0 R null null 2543 0 R 2544 0 R 2543 0 R 2546 0 R 2543 0 R 2548 0 R 2548 0 R 2548 0 R 2548 0 R 2548 0 R 2550 0 R 2551 0 R 2552 0 R 2551 0 R 2550 0 R 2553 0 R 2550 0 R 2555 0 R 2550 0 R 2557 0 R 2559 0 R 2560 0 R 2559 0 R 2562 0 R 2559 0 R 2564 0 R 2559 0 R 2566 0 R 2559 0 R 2568 0 R 2569 0 R 2568 0 R 2571 0 R 2568 0 R 2573 0 R 2568 0 R 2575 0 R 2568 0 R 2577 0 R 2568 0 R 2579 0 R 2568 0 R 2581 0 R 2568 0 R 2583 0 R 2568 0 R 2585 0 R 2586 0 R 2585 0 R 2588 0 R 2585 0 R 2590 0 R 2585 0 R 2592 0 R 2585 0 R 2594 0 R 2595 0 R 2594 0 R 2597 0 R 2594 0 R 2599 0 R 2594 0 R 2601 0 R 2603 0 R 2604 0 R 2603 0 R 2606 0 R 2603 0 R 2608 0 R 2603 0 R 2610 0 R 2603 0 R 2612 0 R 2603 0 R 2614 0 R 2615 0 R 2614 0 R 2617 0 R 2614 0 R 2619 0 R 2614 0 R 2621 0 R 2614 0 R 2623 0 R 2614 0 R 2625 0 R 2627 0 R 2628 0 R 2627 0 R 2630 0 R 2627 0 R 2632 0 R 2627 0 R 2634 0 R 2627 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null ] endobj 5991 0 obj [ null 2723 0 R 2724 0 R 2723 0 R null 2726 0 R 2727 0 R 2726 0 R 2728 0 R 2726 0 R null 2730 0 R null 2732 0 R null null null 2734 0 R 2735 0 R 2734 0 R null 2737 0 R null 2739 0 R null 2740 0 R null 2742 0 R null null null 2744 0 R null 2745 0 R null 2747 0 R 2748 0 R 2749 0 R 2748 0 R 2747 0 R null 2750 0 R 2750 0 R 2750 0 R 2750 0 R 2750 0 R null 2752 0 R 2753 0 R 2754 0 R 2753 0 R 2752 0 R null 2755 0 R 2755 0 R 2755 0 R null 2757 0 R 2758 0 R 2757 0 R null 2760 0 R 2760 0 R null 2762 0 R null null null 2764 0 R 2765 0 R 2765 0 R 2764 0 R null 2767 0 R 2768 0 R 2767 0 R null 2770 0 R 2770 0 R 2770 0 R null 2772 0 R 2773 0 R 2772 0 R 2775 0 R 2772 0 R 2776 0 R 2772 0 R null 2778 0 R null ] endobj 5992 0 obj [ null 2780 0 R null 2781 0 R 2781 0 R 2781 0 R 2781 0 R 2781 0 R 2781 0 R 2781 0 R null null 2783 0 R 2784 0 R 2785 0 R 2784 0 R 2786 0 R null 2788 0 R null null 2790 0 R 2791 0 R 2790 0 R 2793 0 R 2790 0 R 2795 0 R 2796 0 R 2797 0 R 2796 0 R 2799 0 R 2796 0 R 2801 0 R 2796 0 R 2803 0 R 2805 0 R 2806 0 R 2805 0 R 2808 0 R 2805 0 R 2810 0 R 2805 0 R 2812 0 R 2813 0 R 2812 0 R 2814 0 R 2815 0 R 2814 0 R 2817 0 R 2814 0 R 2819 0 R 2814 0 R 2821 0 R 2814 0 R 2823 0 R 2814 0 R 2825 0 R 2827 0 R 2828 0 R 2827 0 R 2830 0 R 2832 0 R 2833 0 R 2832 0 R 2835 0 R 2837 0 R 2838 0 R 2838 0 R 2838 0 R null null 2842 0 R 2841 0 R 2843 0 R 2841 0 R 2845 0 R 2841 0 R 2846 0 R 2841 0 R 2848 0 R null ] endobj 5993 0 obj [ null 2850 0 R 2851 0 R 2850 0 R 2853 0 R 2850 0 R 2855 0 R 2850 0 R null 2857 0 R null 2859 0 R 2860 0 R 2859 0 R null 2862 0 R null 2864 0 R 2865 0 R null null null 2867 0 R 2868 0 R 2869 0 R 2868 0 R 2871 0 R 2868 0 R 2873 0 R 2874 0 R 2873 0 R 2868 0 R 2875 0 R 2868 0 R 2877 0 R 2868 0 R 2878 0 R 2868 0 R 2880 0 R 2881 0 R 2880 0 R 2868 0 R 2882 0 R 2868 0 R 2884 0 R 2868 0 R 2886 0 R 2868 0 R 2888 0 R 2868 0 R 2889 0 R 2890 0 R 2889 0 R 2892 0 R 2894 0 R 2895 0 R 2894 0 R 2897 0 R 2894 0 R 2899 0 R 2894 0 R 2901 0 R 2902 0 R 2901 0 R 2904 0 R 2904 0 R 2904 0 R 2901 0 R 2906 0 R 2901 0 R 2908 0 R 2901 0 R 2910 0 R 2901 0 R 2912 0 R 2901 0 R 2914 0 R 2901 0 R 2916 0 R 2918 0 R 2919 0 R 2918 0 R 2921 0 R 2918 0 R 2923 0 R 2918 0 R 2925 0 R 2918 0 R 2927 0 R 2928 0 R 2927 0 R 2930 0 R 2927 0 R 2932 0 R 2927 0 R null null 2935 0 R 2934 0 R 2936 0 R 2934 0 R 2938 0 R 2934 0 R 2940 0 R 2934 0 R 2942 0 R 2934 0 R 2944 0 R 2934 0 R 2946 0 R 2946 0 R 2946 0 R 2934 0 R 2948 0 R 2934 0 R 2950 0 R 2950 0 R 2950 0 R null 2952 0 R 2953 0 R 2952 0 R 2955 0 R 2952 0 R 2957 0 R 2952 0 R 2959 0 R 2952 0 R 2961 0 R 2952 0 R 2963 0 R 2952 0 R 2965 0 R 2965 0 R 2965 0 R 2952 0 R 2967 0 R null null null ] endobj 5994 0 obj [ 2969 0 R 2970 0 R 2971 0 R 2970 0 R 2973 0 R 2970 0 R 2975 0 R 2970 0 R 2977 0 R 2970 0 R 2979 0 R 2970 0 R 2981 0 R 2983 0 R 2984 0 R 2983 0 R 2986 0 R 2983 0 R 2988 0 R 2983 0 R 2990 0 R 2983 0 R 2992 0 R null 2994 0 R null null 2996 0 R 2997 0 R 2996 0 R 2999 0 R 3001 0 R 3002 0 R 3002 0 R 3002 0 R null null 3006 0 R 3005 0 R 3007 0 R 3005 0 R 3009 0 R 3005 0 R 3011 0 R 3005 0 R 3012 0 R 3005 0 R 3014 0 R 3014 0 R 3014 0 R 3005 0 R 3016 0 R 3005 0 R 3018 0 R null 3020 0 R 3021 0 R 3020 0 R 3023 0 R 3020 0 R null 3025 0 R null 3027 0 R null 3028 0 R null 3030 0 R 3031 0 R 3030 0 R 3033 0 R 3030 0 R null 3035 0 R null 3037 0 R 3038 0 R 3037 0 R 3040 0 R 3037 0 R null 3042 0 R 3043 0 R 3043 0 R 3043 0 R 3042 0 R 3045 0 R 3045 0 R 3045 0 R 3045 0 R 3045 0 R 3042 0 R 3047 0 R 3042 0 R 3048 0 R 3048 0 R 3042 0 R 3050 0 R 3042 0 R null 3052 0 R 3052 0 R null 3054 0 R null ] endobj 5995 0 obj [ null 3055 0 R 3055 0 R null 3057 0 R 3058 0 R 3057 0 R 3060 0 R 3057 0 R 3062 0 R 3057 0 R null 3064 0 R 3064 0 R null 3066 0 R 3067 0 R 3066 0 R 3069 0 R 3069 0 R 3066 0 R null 3071 0 R null 3073 0 R 3074 0 R 3073 0 R null 3076 0 R null 3078 0 R 3079 0 R 3078 0 R 3081 0 R 3078 0 R 3083 0 R 3084 0 R 3083 0 R 3078 0 R null 3085 0 R 3086 0 R 3085 0 R 3088 0 R 3085 0 R 3089 0 R 3085 0 R 3091 0 R 3085 0 R 3093 0 R 3085 0 R null 3095 0 R null 3097 0 R 3098 0 R 3097 0 R 3100 0 R 3097 0 R 3102 0 R 3097 0 R null 3104 0 R null 3106 0 R null 3107 0 R 3107 0 R null 3109 0 R null 3110 0 R null 3112 0 R 3113 0 R null null null 3115 0 R 3116 0 R 3117 0 R 3118 0 R 3117 0 R 3120 0 R 3117 0 R 3122 0 R 3117 0 R 3124 0 R 3117 0 R 3126 0 R 3127 0 R 3126 0 R 3117 0 R 3128 0 R 3117 0 R null null 3132 0 R 3131 0 R ] endobj 5996 0 obj [ null 3133 0 R 3134 0 R 3133 0 R 3135 0 R 3133 0 R null 3137 0 R 3137 0 R 3137 0 R null 3139 0 R 3140 0 R 3139 0 R null 3142 0 R 3143 0 R 3142 0 R 3145 0 R 3142 0 R null 3147 0 R 3147 0 R 3147 0 R 3147 0 R 3147 0 R 3147 0 R 3147 0 R null null 3149 0 R 3150 0 R 3149 0 R 3152 0 R 3149 0 R 3154 0 R 3156 0 R 3157 0 R 3156 0 R 3159 0 R 3156 0 R 3161 0 R 3156 0 R 3163 0 R 3164 0 R 3163 0 R 3166 0 R 3163 0 R 3168 0 R 3163 0 R 3170 0 R 3163 0 R 3172 0 R 3163 0 R 3174 0 R 3176 0 R 3177 0 R 3176 0 R 3179 0 R 3176 0 R 3181 0 R 3183 0 R 3184 0 R 3183 0 R 3186 0 R 3188 0 R 3189 0 R 3189 0 R 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2737 0 R (1307) 2742 0 R (1308) 2740 0 R (1309) 2741 0 R (131) 1259 0 R (1310) 2745 0 R (1311) 2748 0 R (1312) 2750 0 R (1313) 2753 0 R (1314) 2755 0 R (1316) 2758 0 R (1317) 2762 0 R (1318) 2760 0 R (1319) 2761 0 R (132) 1261 0 R (1320) 2765 0 R (1321) 2768 0 R (1322) 2770 0 R (1323) 2773 0 R (1324) 2776 0 R (1325) 2778 0 R (133) 1263 0 R (1332) 2723 0 R (1333) 2726 0 R (1334) 2727 0 R (1335) 2734 0 R (1336) 2739 0 R (1337) 2744 0 R (1338) 2747 0 R (1339) 2749 0 R (134) 1265 0 R (1340) 2752 0 R (1341) 2754 0 R (1342) 2757 0 R (1343) 2764 0 R (1344) 2767 0 R (1345) 2772 0 R (1346) 2775 0 R (1347) 2781 0 R (1349) 2788 0 R (135) 1268 0 R (1350) 2786 0 R (1351) 2787 0 R (1352) 2791 0 R (1353) 2793 0 R ] /Limits [ (1289) (1353) ] >> endobj 6032 0 obj << /Names [ (1354) 2797 0 R (1355) 2799 0 R (1356) 2801 0 R (1357) 2803 0 R (1358) 2806 0 R (1359) 2808 0 R (136) 1270 0 R (1360) 2810 0 R (1361) 2815 0 R (1362) 2817 0 R (1363) 2819 0 R (1364) 2821 0 R (1365) 2823 0 R (1366) 2825 0 R (1367) 2828 0 R (1368) 2830 0 R (1369) 2833 0 R (137) 1272 0 R (1370) 2835 0 R (1371) 2838 0 R (1372) 2840 0 R (1373) 2841 0 R (1374) 2842 0 R (138) 1274 0 R (1380) 2843 0 R (1381) 2846 0 R (1382) 2848 0 R (1383) 2851 0 R (1384) 2853 0 R (1385) 2855 0 R (139) 1276 0 R (1391) 2780 0 R (1392) 2783 0 R (1393) 2784 0 R (1394) 2785 0 R (1395) 2790 0 R (1396) 2795 0 R (1397) 2796 0 R (1398) 2805 0 R (1399) 2812 0 R (14) 1113 0 R (140) 1278 0 R (1400) 2813 0 R (1401) 2814 0 R (1402) 2827 0 R (1403) 2832 0 R (1404) 2837 0 R (1405) 2845 0 R (1406) 2857 0 R (1407) 2860 0 R (1408) 2862 0 R (1409) 2865 0 R (141) 1280 0 R (1410) 2869 0 R (1411) 2871 0 R (1412) 2873 0 R (1413) 2875 0 R (1414) 2878 0 R (1415) 2880 0 R (1416) 2882 0 R (1417) 2884 0 R (1418) 2886 0 R (142) 1282 0 R (1424) 2890 0 R ] /Limits [ (1354) (1424) ] >> endobj 6033 0 obj << /Names [ (1425) 2892 0 R (1426) 2895 0 R (1427) 2897 0 R (1428) 2899 0 R (1429) 2902 0 R (143) 1284 0 R (1432) 2904 0 R (1433) 2906 0 R (1434) 2908 0 R (1436) 2910 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0 R (1649) 3181 0 R ] /Limits [ (1584) (1649) ] >> endobj 6036 0 obj << /Names [ (165) 1165 0 R (1650) 3184 0 R (1651) 3186 0 R (1652) 3189 0 R (1653) 3191 0 R (1654) 3192 0 R (1655) 3193 0 R (1656) 3195 0 R (1657) 3198 0 R (1658) 3204 0 R (166) 1167 0 R (1660) 3206 0 R (1661) 3210 0 R (1662) 3212 0 R (1668) 3133 0 R (1669) 3134 0 R (167) 1171 0 R (1670) 3139 0 R (1671) 3142 0 R (1672) 3149 0 R (1673) 3156 0 R (1674) 3163 0 R (1675) 3176 0 R (1676) 3183 0 R (1677) 3188 0 R (1678) 3194 0 R (1679) 3196 0 R (168) 1173 0 R (1680) 3197 0 R (1681) 3214 0 R (1682) 3215 0 R (1683) 3216 0 R (1684) 3217 0 R (1685) 3219 0 R (1686) 3223 0 R (1687) 3221 0 R (1688) 3222 0 R (1689) 3226 0 R (169) 1179 0 R (1690) 3228 0 R (1691) 3230 0 R (1692) 3232 0 R (1693) 3234 0 R (1694) 3236 0 R (1695) 3238 0 R (1696) 3241 0 R (1697) 3243 0 R (1698) 3246 0 R (1699) 3250 0 R (170) 1183 0 R (1700) 3252 0 R (1702) 3254 0 R (1703) 3256 0 R (1704) 3258 0 R (1705) 3262 0 R (1706) 3265 0 R (1707) 3267 0 R (1708) 3269 0 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(2019) 3628 0 R (202) 1378 0 R (2020) 3632 0 R (2021) 3639 0 R (2022) 3646 0 R (2023) 3652 0 R (2024) 3663 0 R (2025) 3672 0 R (2026) 3678 0 R (2027) 3680 0 R (2028) 3683 0 R (2029) 3685 0 R (203) 1380 0 R (2030) 3688 0 R (2031) 3690 0 R (2032) 3692 0 R (2033) 3694 0 R (2034) 3696 0 R (2035) 3698 0 R (2036) 3699 0 R (2037) 3700 0 R (2038) 3701 0 R (2039) 3703 0 R (204) 1382 0 R (2040) 3705 0 R (2041) 3707 0 R (2042) 3709 0 R (2043) 3711 0 R (2044) 3713 0 R (2045) 3715 0 R (2046) 3717 0 R ] /Limits [ (1984) (2046) ] >> endobj 6042 0 obj << /Names [ (2047) 3719 0 R (2048) 3721 0 R (2049) 3723 0 R (205) 1322 0 R (2050) 3725 0 R (2051) 3727 0 R (2052) 3729 0 R (2053) 3731 0 R (2054) 3733 0 R (2055) 3736 0 R (2056) 3738 0 R (2057) 3740 0 R (2058) 3742 0 R (2059) 3744 0 R (206) 1326 0 R (2060) 3746 0 R (2061) 3749 0 R (2062) 3752 0 R (2063) 3754 0 R (2064) 3756 0 R (2065) 3759 0 R (2066) 3761 0 R (2067) 3763 0 R (2068) 3765 0 R (2069) 3767 0 R (207) 1324 0 R (2070) 3769 0 R (2071) 3771 0 R 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endobj 6048 0 obj << /Names [ (2494) 4218 0 R (2495) 4221 0 R (2496) 4225 0 R (2497) 4235 0 R (2498) 4240 0 R (2499) 4245 0 R (25) 1132 0 R (250) 1349 0 R (2500) 4256 0 R (2501) 4261 0 R (2502) 4276 0 R (2503) 4278 0 R (2506) 4280 0 R (2508) 4282 0 R (2509) 4284 0 R (251) 1350 0 R (2510) 4287 0 R (2511) 4289 0 R (2513) 4291 0 R (2514) 4293 0 R (2515) 4295 0 R (2519) 4297 0 R (252) 1351 0 R (2520) 4299 0 R (2522) 4302 0 R (2523) 4304 0 R (2524) 4307 0 R (2525) 4308 0 R (2526) 4309 0 R (2527) 4313 0 R (2528) 4320 0 R (2529) 4321 0 R (253) 1352 0 R (2530) 4322 0 R (2531) 4326 0 R (2532) 4324 0 R (2533) 4325 0 R (2536) 4329 0 R (2537) 4331 0 R (2538) 4333 0 R (2539) 4336 0 R (254) 1353 0 R (2540) 4338 0 R (2541) 4340 0 R (2542) 4344 0 R (2543) 4346 0 R (2544) 4348 0 R (2545) 4350 0 R (2546) 4352 0 R (2547) 4354 0 R (2548) 4356 0 R (2549) 4358 0 R (255) 1354 0 R (2550) 4360 0 R (2551) 4362 0 R (2552) 4364 0 R (2553) 4366 0 R (2554) 4368 0 R (2555) 4370 0 R (2556) 4372 0 R (2557) 4374 0 R 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(Jacobi operator,Almost periodicity,Ground state,KAM theory) /Subject (Journal de Mathmatiques Pures et Appliques, 207 \(2026\) 103845. 10.1016/j.matpur.2025.103845) /CrossMarkDomains#5B2#5D (sciencedirect.com) /Author (Xing Liang) >> endobj 6086 0 obj << /Dests 713 0 R >> endobj 6087 0 obj << /Length 4910 /Subtype /XML /Type /Metadata >> stream 1 https://www.elsevier.com/tdm/tdmrep-policy.json application/pdf doi:10.1016/j.matpur.2025.103845 Elsevier Masson SAS Journal de Mathématiques Pures et Appliquées, 207 (2026) 103845. 10.1016/j.matpur.2025.103845 Jacobi operator Almost periodicity Ground state KAM theory Almost-periodic ground state of the non-self-adjoint Jacobi operator and its applications Xing Liang Hongze Wang Qi Zhou journal Journal de Mathématiques Pures et Appliquées © 2025 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies. 0021-7824 207 103845 103845 10.1016/j.matpur.2025.103845 https://doi.org/10.1016/j.matpur.2025.103845 March 2026 2010-04-23 true 10.1016/j.matpur.2025.103845 elsevier.com sciencedirect.com VoR 10.1016/j.matpur.2025.103845 noindex 2010-04-23 true elsevier.com sciencedirect.com 1 Elsevier 2026-02-03T14:28:37+02:00 2026-02-03T14:26:51+02:00 2026-02-03T14:28:37+02:00 True Acrobat Distiller 25.0 (Windows) Jacobi operator,Almost periodicity,Ground state,KAM theory uuid:16f4fdee-e02f-4b4a-9d29-8d1f6acf5903 uuid:26ef081e-7bbb-479b-a5e5-46a9605a1dfe endstream endobj xref 0 6088 0000000000 65535 f 0000204899 00000 n 0000205328 00000 n 0000219064 00000 n 0000219282 00000 n 0000219489 00000 n 0000219699 00000 n 0000219906 00000 n 0000220136 00000 n 0000220366 00000 n 0000220597 00000 n 0000220829 00000 n 0000221060 00000 n 0000221292 00000 n 0000221522 00000 n 0000221753 00000 n 0000221985 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