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",[536,883,884],{},"\\sum_{i=1}^n",[380,886,888,916],{"className":887},[383],[380,889,891],{"className":890},[387],[389,892,893],{"xmlns":391},[393,894,895,914],{},[396,896,897],{},[789,898,899,902,911],{},[403,900,901],{},"∑",[396,903,904,906,908],{},[399,905,796],{},[403,907,410],{},[412,909,910],{},"1",[399,912,913],{},"n",[418,915,884],{"encoding":420},[380,917,919],{"className":918,"ariaHidden":426},[425],[380,920,922,926],{"className":921},[430],[380,923],{"className":924,"style":925},[434],"height:1.104em;vertical-align:-0.2997em;",[380,927,930,936],{"className":928},[929],"mop",[380,931,901],{"className":932,"style":935},[929,933,934],"op-symbol","small-op","position:relative;top:0em;",[380,937,939],{"className":938},[820],[380,940,942,985],{"className":941},[682,683],[380,943,945,982],{"className":944},[687],[380,946,949,970],{"className":947,"style":948},[691],"height:0.8043em;",[380,950,952,955],{"style":951},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[380,953],{"className":954,"style":837},[699],[380,956,958],{"className":957},[704,705,706,707],[380,959,961,964,967],{"className":960},[439,707],[380,962,796],{"className":963},[439,440,707],[380,965,410],{"className":966},[470,707],[380,968,910],{"className":969},[439,707],[380,971,973,976],{"style":972},"top:-3.2029em;margin-right:0.05em;",[380,974],{"className":975,"style":837},[699],[380,977,979],{"className":978},[704,705,706,707],[380,980,913],{"className":981},[439,440,707],[380,983,744],{"className":984},[743],[380,986,988],{"className":987},[687],[380,989,992],{"className":990,"style":991},[691],"height:0.2997em;",[380,993],{},[495,995,997],{"id":996},"greek-letters","Greek Letters",[357,999,1000],{},"Greek letters are typed by prefixing the letter name with a backslash:",[369,1002,1003,1176],{},[372,1004,1005,1006,1009,1010,1040,1041,1009,1044,1040,1075,1009,1078,1040,1109,1009,1112,1040,1143,1009,1146,1175],{},"Lowercase: ",[536,1007,1008],{},"\\alpha"," (",[380,1011,1013,1027],{"className":1012},[383],[380,1014,1016],{"className":1015},[387],[389,1017,1018],{"xmlns":391},[393,1019,1020,1025],{},[396,1021,1022],{},[399,1023,1024],{},"α",[418,1026,1008],{"encoding":420},[380,1028,1030],{"className":1029,"ariaHidden":426},[425],[380,1031,1033,1036],{"className":1032},[430],[380,1034],{"className":1035,"style":459},[434],[380,1037,1024],{"className":1038,"style":1039},[439,440],"margin-right:0.0037em;","), ",[536,1042,1043],{},"\\beta",[380,1045,1047,1061],{"className":1046},[383],[380,1048,1050],{"className":1049},[387],[389,1051,1052],{"xmlns":391},[393,1053,1054,1059],{},[396,1055,1056],{},[399,1057,1058],{},"β",[418,1060,1043],{"encoding":420},[380,1062,1064],{"className":1063,"ariaHidden":426},[425],[380,1065,1067,1071],{"className":1066},[430],[380,1068],{"className":1069,"style":1070},[434],"height:0.8889em;vertical-align:-0.1944em;",[380,1072,1058],{"className":1073,"style":1074},[439,440],"margin-right:0.0528em;",[536,1076,1077],{},"\\gamma",[380,1079,1081,1095],{"className":1080},[383],[380,1082,1084],{"className":1083},[387],[389,1085,1086],{"xmlns":391},[393,1087,1088,1093],{},[396,1089,1090],{},[399,1091,1092],{},"γ",[418,1094,1077],{"encoding":420},[380,1096,1098],{"className":1097,"ariaHidden":426},[425],[380,1099,1101,1105],{"className":1100},[430],[380,1102],{"className":1103,"style":1104},[434],"height:0.625em;vertical-align:-0.1944em;",[380,1106,1092],{"className":1107,"style":1108},[439,440],"margin-right:0.0556em;",[536,1110,1111],{},"\\theta",[380,1113,1115,1129],{"className":1114},[383],[380,1116,1118],{"className":1117},[387],[389,1119,1120],{"xmlns":391},[393,1121,1122,1127],{},[396,1123,1124],{},[399,1125,1126],{},"θ",[418,1128,1111],{"encoding":420},[380,1130,1132],{"className":1131,"ariaHidden":426},[425],[380,1133,1135,1139],{"className":1134},[430],[380,1136],{"className":1137,"style":1138},[434],"height:0.6944em;",[380,1140,1126],{"className":1141,"style":1142},[439,440],"margin-right:0.0278em;",[536,1144,1145],{},"\\lambda",[380,1147,1149,1163],{"className":1148},[383],[380,1150,1152],{"className":1151},[387],[389,1153,1154],{"xmlns":391},[393,1155,1156,1161],{},[396,1157,1158],{},[399,1159,1160],{},"λ",[418,1162,1145],{"encoding":420},[380,1164,1166],{"className":1165,"ariaHidden":426},[425],[380,1167,1169,1172],{"className":1168},[430],[380,1170],{"className":1171,"style":1138},[434],[380,1173,1160],{"className":1174},[439,440],")",[372,1177,1178,1179,1009,1182,1040,1213,1009,1216,1040,1245,1009,1248,1040,1277,1009,1280,1175],{},"Uppercase: ",[536,1180,1181],{},"\\Gamma",[380,1183,1185,1200],{"className":1184},[383],[380,1186,1188],{"className":1187},[387],[389,1189,1190],{"xmlns":391},[393,1191,1192,1198],{},[396,1193,1194],{},[399,1195,1197],{"mathvariant":1196},"normal","Γ",[418,1199,1181],{"encoding":420},[380,1201,1203],{"className":1202,"ariaHidden":426},[425],[380,1204,1206,1210],{"className":1205},[430],[380,1207],{"className":1208,"style":1209},[434],"height:0.6833em;",[380,1211,1197],{"className":1212},[439],[536,1214,1215],{},"\\Delta",[380,1217,1219,1233],{"className":1218},[383],[380,1220,1222],{"className":1221},[387],[389,1223,1224],{"xmlns":391},[393,1225,1226,1231],{},[396,1227,1228],{},[399,1229,1230],{"mathvariant":1196},"Δ",[418,1232,1215],{"encoding":420},[380,1234,1236],{"className":1235,"ariaHidden":426},[425],[380,1237,1239,1242],{"className":1238},[430],[380,1240],{"className":1241,"style":1209},[434],[380,1243,1230],{"className":1244},[439],[536,1246,1247],{},"\\Theta",[380,1249,1251,1265],{"className":1250},[383],[380,1252,1254],{"className":1253},[387],[389,1255,1256],{"xmlns":391},[393,1257,1258,1263],{},[396,1259,1260],{},[399,1261,1262],{"mathvariant":1196},"Θ",[418,1264,1247],{"encoding":420},[380,1266,1268],{"className":1267,"ariaHidden":426},[425],[380,1269,1271,1274],{"className":1270},[430],[380,1272],{"className":1273,"style":1209},[434],[380,1275,1262],{"className":1276},[439],[536,1278,1279],{},"\\Lambda",[380,1281,1283,1297],{"className":1282},[383],[380,1284,1286],{"className":1285},[387],[389,1287,1288],{"xmlns":391},[393,1289,1290,1295],{},[396,1291,1292],{},[399,1293,1294],{"mathvariant":1196},"Λ",[418,1296,1279],{"encoding":420},[380,1298,1300],{"className":1299,"ariaHidden":426},[425],[380,1301,1303,1306],{"className":1302},[430],[380,1304],{"className":1305,"style":1209},[434],[380,1307,1294],{"className":1308},[439],[495,1310,1312],{"id":1311},"delimiters-and-brackets","Delimiters and Brackets",[357,1314,1315,1316,877,1319,1322],{},"Use ",[536,1317,1318],{},"\\left",[536,1320,1321],{},"\\right"," to auto-size delimiters:",[1324,1325,1330],"pre",{"className":1326,"code":1327,"language":1328,"meta":1329,"style":1329},"language-latex shiki shiki-themes material-theme-lighter material-theme material-theme-palenight","\\left( \\frac{a}{b} \\right)\n\\left[ \\sum_{i=1}^n x_i \\right]\n\\left\\{ x \\in \\mathbb{R} : x > 0 \\right\\}\n","latex","",[536,1331,1332,1339,1345],{"__ignoreMap":1329},[380,1333,1336],{"class":1334,"line":1335},"line",1,[380,1337,1338],{},"\\left( \\frac{a}{b} \\right)\n",[380,1340,1342],{"class":1334,"line":1341},2,[380,1343,1344],{},"\\left[ \\sum_{i=1}^n x_i \\right]\n",[380,1346,1348],{"class":1334,"line":1347},3,[380,1349,1350],{},"\\left\\{ x \\in \\mathbb{R} : x > 0 \\right\\}\n",[357,1352,1353,1354,1357,1358,1357,1361,1357,1364,1357,1367,758],{},"Common delimiter pairs: ",[536,1355,1356],{},"\\left( ... \\right)",", ",[536,1359,1360],{},"\\left[ ... \\right]",[536,1362,1363],{},"\\left\\{ ... \\right\\}",[536,1365,1366],{},"\\left| ... \\right|",[536,1368,1369],{},"\\left\\langle ... \\right\\rangle",[357,1371,1372,1373,1357,1376,1357,1379,1357,1382,1385,1386,1357,1389,1357,1391,1357,1394,1357,1397,1357,1400,1357,1403,1357,1406,1409],{},"For manual sizing: ",[536,1374,1375],{},"\\big",[536,1377,1378],{},"\\Big",[536,1380,1381],{},"\\bigg",[536,1383,1384],{},"\\Bigg"," (each with ",[536,1387,1388],{},"(",[536,1390,1175],{},[536,1392,1393],{},"[",[536,1395,1396],{},"]",[536,1398,1399],{},"\\{",[536,1401,1402],{},"\\}",[536,1404,1405],{},"|",[536,1407,1408],{},"\u002F"," variants).",[361,1411,1413],{"id":1412},"math-spacing","Math Spacing",[357,1415,1416],{},"LaTeX provides fine control over spacing in math mode:",[500,1418,1419,1432],{},[503,1420,1421],{},[506,1422,1423,1426,1429],{},[509,1424,1425],{},"Command",[509,1427,1428],{},"Description",[509,1430,1431],{},"Example",[523,1433,1434,1523,1575,1627,1713,1766],{},[506,1435,1436,1441,1444],{},[528,1437,1438],{},[536,1439,1440],{},"\\,",[528,1442,1443],{},"Thin space (3\u002F18 em)",[528,1445,1446],{},[380,1447,1449,1483],{"className":1448},[383],[380,1450,1452],{"className":1451},[387],[389,1453,1454],{"xmlns":391},[393,1455,1456,1480],{},[396,1457,1458,1461,1464,1467,1469,1471,1475,1478],{},[403,1459,1460],{},"∫",[399,1462,1463],{},"f",[403,1465,1388],{"stretchy":1466},"false",[399,1468,793],{},[403,1470,1175],{"stretchy":1466},[1472,1473,1474],"mtext",{}," ",[399,1476,1477],{},"d",[399,1479,793],{},[418,1481,1482],{"encoding":420},"\\int f(x)\\,dx",[380,1484,1486],{"className":1485,"ariaHidden":426},[425],[380,1487,1489,1493,1497,1501,1505,1508,1511,1514,1517,1520],{"className":1488},[430],[380,1490],{"className":1491,"style":1492},[434],"height:1.1111em;vertical-align:-0.3061em;",[380,1494,1460],{"className":1495,"style":1496},[929,933,934],"margin-right:0.1945em;position:relative;top:-0.0006em;",[380,1498],{"className":1499,"style":1500},[444],"margin-right:0.1667em;",[380,1502,1463],{"className":1503,"style":1504},[439,440],"margin-right:0.1076em;",[380,1506,1388],{"className":1507},[674],[380,1509,793],{"className":1510},[439,440],[380,1512,1175],{"className":1513},[757],[380,1515],{"className":1516,"style":1500},[444],[380,1518,1477],{"className":1519},[439,440],[380,1521,793],{"className":1522},[439,440],[506,1524,1525,1530,1533],{},[528,1526,1527],{},[536,1528,1529],{},"\\:",[528,1531,1532],{},"Medium space (4\u002F18 em)",[528,1534,1535],{},[380,1536,1538,1557],{"className":1537},[383],[380,1539,1541],{"className":1540},[387],[389,1542,1543],{"xmlns":391},[393,1544,1545,1554],{},[396,1546,1547,1549,1552],{},[399,1548,401],{},[1472,1550,1551],{}," ",[399,1553,654],{},[418,1555,1556],{"encoding":420},"a \\: b",[380,1558,1560],{"className":1559,"ariaHidden":426},[425],[380,1561,1563,1566,1569,1572],{"className":1562},[430],[380,1564],{"className":1565,"style":1138},[434],[380,1567,401],{"className":1568},[439,440],[380,1570],{"className":1571,"style":445},[444],[380,1573,654],{"className":1574},[439,440],[506,1576,1577,1582,1585],{},[528,1578,1579],{},[536,1580,1581],{},"\\;",[528,1583,1584],{},"Thick space (5\u002F18 em)",[528,1586,1587],{},[380,1588,1590,1609],{"className":1589},[383],[380,1591,1593],{"className":1592},[387],[389,1594,1595],{"xmlns":391},[393,1596,1597,1606],{},[396,1598,1599,1601,1604],{},[399,1600,401],{},[1472,1602,1603],{},"  ",[399,1605,654],{},[418,1607,1608],{"encoding":420},"a \\; b",[380,1610,1612],{"className":1611,"ariaHidden":426},[425],[380,1613,1615,1618,1621,1624],{"className":1614},[430],[380,1616],{"className":1617,"style":1138},[434],[380,1619,401],{"className":1620},[439,440],[380,1622],{"className":1623,"style":466},[444],[380,1625,654],{"className":1626},[439,440],[506,1628,1629,1634,1637],{},[528,1630,1631],{},[536,1632,1633],{},"\\!",[528,1635,1636],{},"Negative thin space (-3\u002F18 em)",[528,1638,1639],{},[380,1640,1642,1673],{"className":1641},[383],[380,1643,1645],{"className":1644},[387],[389,1646,1647],{"xmlns":391},[393,1648,1649,1670],{},[396,1650,1651,1653,1656,1658,1660,1662,1664,1666,1668],{},[403,1652,1460],{},[1472,1654,1655],{}," ⁣",[399,1657,1463],{},[403,1659,1388],{"stretchy":1466},[399,1661,793],{},[403,1663,1175],{"stretchy":1466},[1472,1665,1474],{},[399,1667,1477],{},[399,1669,793],{},[418,1671,1672],{"encoding":420},"\\int\\! f(x)\\,dx",[380,1674,1676],{"className":1675,"ariaHidden":426},[425],[380,1677,1679,1682,1685,1689,1692,1695,1698,1701,1704,1707,1710],{"className":1678},[430],[380,1680],{"className":1681,"style":1492},[434],[380,1683,1460],{"className":1684,"style":1496},[929,933,934],[380,1686],{"className":1687,"style":1688},[444],"margin-right:-0.1667em;",[380,1690],{"className":1691,"style":1500},[444],[380,1693,1463],{"className":1694,"style":1504},[439,440],[380,1696,1388],{"className":1697},[674],[380,1699,793],{"className":1700},[439,440],[380,1702,1175],{"className":1703},[757],[380,1705],{"className":1706,"style":1500},[444],[380,1708,1477],{"className":1709},[439,440],[380,1711,793],{"className":1712},[439,440],[506,1714,1715,1720,1723],{},[528,1716,1717],{},[536,1718,1719],{},"\\quad",[528,1721,1722],{},"1 em space",[528,1724,1725],{},[380,1726,1728,1747],{"className":1727},[383],[380,1729,1731],{"className":1730},[387],[389,1732,1733],{"xmlns":391},[393,1734,1735,1744],{},[396,1736,1737,1739,1742],{},[399,1738,401],{},[444,1740],{"width":1741},"1em",[399,1743,654],{},[418,1745,1746],{"encoding":420},"a \\quad b",[380,1748,1750],{"className":1749,"ariaHidden":426},[425],[380,1751,1753,1756,1759,1763],{"className":1752},[430],[380,1754],{"className":1755,"style":1138},[434],[380,1757,401],{"className":1758},[439,440],[380,1760],{"className":1761,"style":1762},[444],"margin-right:1em;",[380,1764,654],{"className":1765},[439,440],[506,1767,1768,1773,1776],{},[528,1769,1770],{},[536,1771,1772],{},"\\qquad",[528,1774,1775],{},"2 em space",[528,1777,1778],{},[380,1779,1781,1800],{"className":1780},[383],[380,1782,1784],{"className":1783},[387],[389,1785,1786],{"xmlns":391},[393,1787,1788,1797],{},[396,1789,1790,1792,1795],{},[399,1791,401],{},[444,1793],{"width":1794},"2em",[399,1796,654],{},[418,1798,1799],{"encoding":420},"a \\qquad b",[380,1801,1803],{"className":1802,"ariaHidden":426},[425],[380,1804,1806,1809,1812,1816],{"className":1805},[430],[380,1807],{"className":1808,"style":1138},[434],[380,1810,401],{"className":1811},[439,440],[380,1813],{"className":1814,"style":1815},[444],"margin-right:2em;",[380,1817,654],{"className":1818},[439,440],[361,1820,1822],{"id":1821},"matrices","Matrices",[357,1824,1825,1826,1829],{},"The ",[536,1827,1828],{},"amsmath"," package provides several matrix environments:",[1324,1831,1833],{"className":1326,"code":1832,"language":1328,"meta":1329,"style":1329},"\\begin{pmatrix}  % parenthesized\n  a & b \\\\ c & d\n\\end{pmatrix}\n\n\\begin{bmatrix}  % bracketed\n  a & b \\\\ c & d\n\\end{bmatrix}\n\n\\begin{vmatrix}  % vertical bars (determinant)\n  a & b \\\\ c & d\n\\end{vmatrix}\n\n\\begin{Bmatrix}  % curly braces\n  a & b \\\\ c & d\n\\end{Bmatrix}\n\n\\begin{Vmatrix}  % double vertical bars\n  a & b \\\\ c & d\n\\end{Vmatrix}\n",[536,1834,1835,1840,1845,1850,1857,1863,1868,1874,1879,1885,1890,1896,1901,1907,1912,1918,1923,1929,1934],{"__ignoreMap":1329},[380,1836,1837],{"class":1334,"line":1335},[380,1838,1839],{},"\\begin{pmatrix}  % parenthesized\n",[380,1841,1842],{"class":1334,"line":1341},[380,1843,1844],{},"  a & b \\\\ c & d\n",[380,1846,1847],{"class":1334,"line":1347},[380,1848,1849],{},"\\end{pmatrix}\n",[380,1851,1853],{"class":1334,"line":1852},4,[380,1854,1856],{"emptyLinePlaceholder":1855},true,"\n",[380,1858,1860],{"class":1334,"line":1859},5,[380,1861,1862],{},"\\begin{bmatrix}  % bracketed\n",[380,1864,1866],{"class":1334,"line":1865},6,[380,1867,1844],{},[380,1869,1871],{"class":1334,"line":1870},7,[380,1872,1873],{},"\\end{bmatrix}\n",[380,1875,1877],{"class":1334,"line":1876},8,[380,1878,1856],{"emptyLinePlaceholder":1855},[380,1880,1882],{"class":1334,"line":1881},9,[380,1883,1884],{},"\\begin{vmatrix}  % vertical bars (determinant)\n",[380,1886,1888],{"class":1334,"line":1887},10,[380,1889,1844],{},[380,1891,1893],{"class":1334,"line":1892},11,[380,1894,1895],{},"\\end{vmatrix}\n",[380,1897,1899],{"class":1334,"line":1898},12,[380,1900,1856],{"emptyLinePlaceholder":1855},[380,1902,1904],{"class":1334,"line":1903},13,[380,1905,1906],{},"\\begin{Bmatrix}  % curly braces\n",[380,1908,1910],{"class":1334,"line":1909},14,[380,1911,1844],{},[380,1913,1915],{"class":1334,"line":1914},15,[380,1916,1917],{},"\\end{Bmatrix}\n",[380,1919,1921],{"class":1334,"line":1920},16,[380,1922,1856],{"emptyLinePlaceholder":1855},[380,1924,1926],{"class":1334,"line":1925},17,[380,1927,1928],{},"\\begin{Vmatrix}  % double vertical bars\n",[380,1930,1932],{"class":1334,"line":1931},18,[380,1933,1844],{},[380,1935,1937],{"class":1334,"line":1936},19,[380,1938,1939],{},"\\end{Vmatrix}\n",[361,1941,1943],{"id":1942},"cases-and-piecewise-functions","Cases and Piecewise Functions",[357,1945,1825,1946,1949],{},[536,1947,1948],{},"cases"," environment is used for piecewise-defined functions:",[1324,1951,1953],{"className":1326,"code":1952,"language":1328,"meta":1329,"style":1329},"f(x) = \\begin{cases}\n  x^2 & \\text{if } x \\geq 0 \\\\\n  -x  & \\text{if } x \u003C 0\n\\end{cases}\n",[536,1954,1955,1960,1965,1970],{"__ignoreMap":1329},[380,1956,1957],{"class":1334,"line":1335},[380,1958,1959],{},"f(x) = \\begin{cases}\n",[380,1961,1962],{"class":1334,"line":1341},[380,1963,1964],{},"  x^2 & \\text{if } x \\geq 0 \\\\\n",[380,1966,1967],{"class":1334,"line":1347},[380,1968,1969],{},"  -x  & \\text{if } x \u003C 0\n",[380,1971,1972],{"class":1334,"line":1852},[380,1973,1974],{},"\\end{cases}\n",[361,1976,1978],{"id":1977},"advanced-math-amsmath","Advanced Math (amsmath)",[357,1980,1981,1982,1984,1985,1988],{},"For document structures containing numerous complex formulas, load ",[536,1983,1828],{}," or ",[536,1986,1987],{},"mathtools"," in your preamble:",[1324,1990,1992],{"className":1326,"code":1991,"language":1328,"meta":1329,"style":1329},"\\usepackage{amsmath}\n",[536,1993,1994],{"__ignoreMap":1329},[380,1995,1996],{"class":1334,"line":1335},[380,1997,1991],{},[357,1999,2000],{},"This unlocks advanced alignment environments:",[1324,2002,2004],{"className":1326,"code":2003,"language":1328,"meta":1329,"style":1329},"\\begin{align}\n  f(x) &= (x + a)^2 \\\\\n       &= x^2 + 2ax + a^2\n\\end{align}\n",[536,2005,2006,2011,2016,2021],{"__ignoreMap":1329},[380,2007,2008],{"class":1334,"line":1335},[380,2009,2010],{},"\\begin{align}\n",[380,2012,2013],{"class":1334,"line":1341},[380,2014,2015],{},"  f(x) &= (x + a)^2 \\\\\n",[380,2017,2018],{"class":1334,"line":1347},[380,2019,2020],{},"       &= x^2 + 2ax + a^2\n",[380,2022,2023],{"class":1334,"line":1852},[380,2024,2025],{},"\\end{align}\n",[495,2027,2029],{"id":2028},"other-amsmath-environments","Other amsmath Environments",[500,2031,2032,2040],{},[503,2033,2034],{},[506,2035,2036,2038],{},[509,2037,515],{},[509,2039,1428],{},[523,2041,2042,2052,2062,2072],{},[506,2043,2044,2049],{},[528,2045,2046],{},[536,2047,2048],{},"gather",[528,2050,2051],{},"Multiple centered equations, auto-numbered",[506,2053,2054,2059],{},[528,2055,2056],{},[536,2057,2058],{},"multline",[528,2060,2061],{},"Long equations split across lines, first left-aligned, last right-aligned",[506,2063,2064,2069],{},[528,2065,2066],{},[536,2067,2068],{},"flalign",[528,2070,2071],{},"Full-length alignment (spans entire line width)",[506,2073,2074,2079],{},[528,2075,2076],{},[536,2077,2078],{},"alignat{N}",[528,2080,2081],{},"Alignment with N alignment columns",[495,2083,2085],{"id":2084},"custom-operators","Custom Operators",[357,2087,2088,2089,2092],{},"Define your own operators with ",[536,2090,2091],{},"\\DeclareMathOperator",":",[1324,2094,2096],{"className":1326,"code":2095,"language":1328,"meta":1329,"style":1329},"\\DeclareMathOperator{\\Tr}{Tr}      % Trace\n\\DeclareMathOperator{\\diag}{diag}  % Diagonal\n\\DeclareMathOperator*{\\esssup}{ess\\,sup}  % ess sup with limits above\u002Fbelow\n",[536,2097,2098,2103,2108],{"__ignoreMap":1329},[380,2099,2100],{"class":1334,"line":1335},[380,2101,2102],{},"\\DeclareMathOperator{\\Tr}{Tr}      % Trace\n",[380,2104,2105],{"class":1334,"line":1341},[380,2106,2107],{},"\\DeclareMathOperator{\\diag}{diag}  % Diagonal\n",[380,2109,2110],{"class":1334,"line":1347},[380,2111,2112],{},"\\DeclareMathOperator*{\\esssup}{ess\\,sup}  % ess sup with limits above\u002Fbelow\n",[361,2114,2116],{"id":2115},"common-math-symbols","Common Math Symbols",[495,2118,2120],{"id":2119},"relations","Relations",[500,2122,2123,2134],{},[503,2124,2125],{},[506,2126,2127,2130,2132],{},[509,2128,2129],{},"Symbol",[509,2131,1425],{},[509,2133,1428],{},[523,2135,2136,2178,2219,2300,2342,2384,2426],{},[506,2137,2138,2171,2175],{},[528,2139,2140],{},[380,2141,2143,2158],{"className":2142},[383],[380,2144,2146],{"className":2145},[387],[389,2147,2148],{"xmlns":391},[393,2149,2150,2155],{},[396,2151,2152],{},[403,2153,2154],{},"≤",[418,2156,2157],{"encoding":420},"\\leq",[380,2159,2161],{"className":2160,"ariaHidden":426},[425],[380,2162,2164,2168],{"className":2163},[430],[380,2165],{"className":2166,"style":2167},[434],"height:0.7719em;vertical-align:-0.136em;",[380,2169,2154],{"className":2170},[470],[528,2172,2173],{},[536,2174,2157],{},[528,2176,2177],{},"Less than or equal",[506,2179,2180,2212,2216],{},[528,2181,2182],{},[380,2183,2185,2200],{"className":2184},[383],[380,2186,2188],{"className":2187},[387],[389,2189,2190],{"xmlns":391},[393,2191,2192,2197],{},[396,2193,2194],{},[403,2195,2196],{},"≥",[418,2198,2199],{"encoding":420},"\\geq",[380,2201,2203],{"className":2202,"ariaHidden":426},[425],[380,2204,2206,2209],{"className":2205},[430],[380,2207],{"className":2208,"style":2167},[434],[380,2210,2196],{"className":2211},[470],[528,2213,2214],{},[536,2215,2199],{},[528,2217,2218],{},"Greater than or equal",[506,2220,2221,2293,2297],{},[528,2222,2223],{},[380,2224,2226,2241],{"className":2225},[383],[380,2227,2229],{"className":2228},[387],[389,2230,2231],{"xmlns":391},[393,2232,2233,2238],{},[396,2234,2235],{},[403,2236,2237],{"mathvariant":1196},"≠",[418,2239,2240],{"encoding":420},"\\neq",[380,2242,2244],{"className":2243,"ariaHidden":426},[425],[380,2245,2247,2250],{"className":2246},[430],[380,2248],{"className":2249,"style":1070},[434],[380,2251,2253,2286,2290],{"className":2252},[470],[380,2254,2256],{"className":2255},[470],[380,2257,2260],{"className":2258},[439,2259],"vbox",[380,2261,2264],{"className":2262},[2263],"thinbox",[380,2265,2268,2271,2282],{"className":2266},[2267],"rlap",[380,2269],{"className":2270,"style":1070},[434],[380,2272,2275],{"className":2273},[2274],"inner",[380,2276,2278],{"className":2277},[439],[380,2279,2281],{"className":2280},[470],"",[380,2283],{"className":2284},[2285],"fix",[380,2287],{"className":2288},[444,2289],"nobreak",[380,2291,410],{"className":2292},[470],[528,2294,2295],{},[536,2296,2240],{},[528,2298,2299],{},"Not equal",[506,2301,2302,2335,2339],{},[528,2303,2304],{},[380,2305,2307,2322],{"className":2306},[383],[380,2308,2310],{"className":2309},[387],[389,2311,2312],{"xmlns":391},[393,2313,2314,2319],{},[396,2315,2316],{},[403,2317,2318],{},"≈",[418,2320,2321],{"encoding":420},"\\approx",[380,2323,2325],{"className":2324,"ariaHidden":426},[425],[380,2326,2328,2332],{"className":2327},[430],[380,2329],{"className":2330,"style":2331},[434],"height:0.4831em;",[380,2333,2318],{"className":2334},[470],[528,2336,2337],{},[536,2338,2321],{},[528,2340,2341],{},"Approximately equal",[506,2343,2344,2377,2381],{},[528,2345,2346],{},[380,2347,2349,2364],{"className":2348},[383],[380,2350,2352],{"className":2351},[387],[389,2353,2354],{"xmlns":391},[393,2355,2356,2361],{},[396,2357,2358],{},[403,2359,2360],{},"≡",[418,2362,2363],{"encoding":420},"\\equiv",[380,2365,2367],{"className":2366,"ariaHidden":426},[425],[380,2368,2370,2374],{"className":2369},[430],[380,2371],{"className":2372,"style":2373},[434],"height:0.4637em;",[380,2375,2360],{"className":2376},[470],[528,2378,2379],{},[536,2380,2363],{},[528,2382,2383],{},"Equivalent",[506,2385,2386,2419,2423],{},[528,2387,2388],{},[380,2389,2391,2406],{"className":2390},[383],[380,2392,2394],{"className":2393},[387],[389,2395,2396],{"xmlns":391},[393,2397,2398,2403],{},[396,2399,2400],{},[403,2401,2402],{},"∼",[418,2404,2405],{"encoding":420},"\\sim",[380,2407,2409],{"className":2408,"ariaHidden":426},[425],[380,2410,2412,2416],{"className":2411},[430],[380,2413],{"className":2414,"style":2415},[434],"height:0.3669em;",[380,2417,2402],{"className":2418},[470],[528,2420,2421],{},[536,2422,2405],{},[528,2424,2425],{},"Similar to",[506,2427,2428,2460,2464],{},[528,2429,2430],{},[380,2431,2433,2448],{"className":2432},[383],[380,2434,2436],{"className":2435},[387],[389,2437,2438],{"xmlns":391},[393,2439,2440,2445],{},[396,2441,2442],{},[403,2443,2444],{},"∝",[418,2446,2447],{"encoding":420},"\\propto",[380,2449,2451],{"className":2450,"ariaHidden":426},[425],[380,2452,2454,2457],{"className":2453},[430],[380,2455],{"className":2456,"style":459},[434],[380,2458,2444],{"className":2459},[470],[528,2461,2462],{},[536,2463,2447],{},[528,2465,2466],{},"Proportional to",[495,2468,2470],{"id":2469},"operators","Operators",[500,2472,2473,2483],{},[503,2474,2475],{},[506,2476,2477,2479,2481],{},[509,2478,2129],{},[509,2480,1425],{},[509,2482,1428],{},[523,2484,2485,2527,2568,2610,2651,2692,2733],{},[506,2486,2487,2520,2524],{},[528,2488,2489],{},[380,2490,2492,2507],{"className":2491},[383],[380,2493,2495],{"className":2494},[387],[389,2496,2497],{"xmlns":391},[393,2498,2499,2504],{},[396,2500,2501],{},[403,2502,2503],{},"∪",[418,2505,2506],{"encoding":420},"\\cup",[380,2508,2510],{"className":2509,"ariaHidden":426},[425],[380,2511,2513,2517],{"className":2512},[430],[380,2514],{"className":2515,"style":2516},[434],"height:0.5556em;",[380,2518,2503],{"className":2519},[439],[528,2521,2522],{},[536,2523,2506],{},[528,2525,2526],{},"Union",[506,2528,2529,2561,2565],{},[528,2530,2531],{},[380,2532,2534,2549],{"className":2533},[383],[380,2535,2537],{"className":2536},[387],[389,2538,2539],{"xmlns":391},[393,2540,2541,2546],{},[396,2542,2543],{},[403,2544,2545],{},"∩",[418,2547,2548],{"encoding":420},"\\cap",[380,2550,2552],{"className":2551,"ariaHidden":426},[425],[380,2553,2555,2558],{"className":2554},[430],[380,2556],{"className":2557,"style":2516},[434],[380,2559,2545],{"className":2560},[439],[528,2562,2563],{},[536,2564,2548],{},[528,2566,2567],{},"Intersection",[506,2569,2570,2603,2607],{},[528,2571,2572],{},[380,2573,2575,2590],{"className":2574},[383],[380,2576,2578],{"className":2577},[387],[389,2579,2580],{"xmlns":391},[393,2581,2582,2587],{},[396,2583,2584],{},[403,2585,2586],{},"∈",[418,2588,2589],{"encoding":420},"\\in",[380,2591,2593],{"className":2592,"ariaHidden":426},[425],[380,2594,2596,2600],{"className":2595},[430],[380,2597],{"className":2598,"style":2599},[434],"height:0.5782em;vertical-align:-0.0391em;",[380,2601,2586],{"className":2602},[470],[528,2604,2605],{},[536,2606,2589],{},[528,2608,2609],{},"Element of",[506,2611,2612,2644,2648],{},[528,2613,2614],{},[380,2615,2617,2632],{"className":2616},[383],[380,2618,2620],{"className":2619},[387],[389,2621,2622],{"xmlns":391},[393,2623,2624,2629],{},[396,2625,2626],{},[403,2627,2628],{},"⊂",[418,2630,2631],{"encoding":420},"\\subset",[380,2633,2635],{"className":2634,"ariaHidden":426},[425],[380,2636,2638,2641],{"className":2637},[430],[380,2639],{"className":2640,"style":2599},[434],[380,2642,2628],{"className":2643},[470],[528,2645,2646],{},[536,2647,2631],{},[528,2649,2650],{},"Subset",[506,2652,2653,2685,2689],{},[528,2654,2655],{},[380,2656,2658,2673],{"className":2657},[383],[380,2659,2661],{"className":2660},[387],[389,2662,2663],{"xmlns":391},[393,2664,2665,2670],{},[396,2666,2667],{},[399,2668,2669],{"mathvariant":1196},"∀",[418,2671,2672],{"encoding":420},"\\forall",[380,2674,2676],{"className":2675,"ariaHidden":426},[425],[380,2677,2679,2682],{"className":2678},[430],[380,2680],{"className":2681,"style":1138},[434],[380,2683,2669],{"className":2684},[439],[528,2686,2687],{},[536,2688,2672],{},[528,2690,2691],{},"For all",[506,2693,2694,2726,2730],{},[528,2695,2696],{},[380,2697,2699,2714],{"className":2698},[383],[380,2700,2702],{"className":2701},[387],[389,2703,2704],{"xmlns":391},[393,2705,2706,2711],{},[396,2707,2708],{},[399,2709,2710],{"mathvariant":1196},"∃",[418,2712,2713],{"encoding":420},"\\exists",[380,2715,2717],{"className":2716,"ariaHidden":426},[425],[380,2718,2720,2723],{"className":2719},[430],[380,2721],{"className":2722,"style":1138},[434],[380,2724,2710],{"className":2725},[439],[528,2727,2728],{},[536,2729,2713],{},[528,2731,2732],{},"There exists",[506,2734,2735,2767,2771],{},[528,2736,2737],{},[380,2738,2740,2755],{"className":2739},[383],[380,2741,2743],{"className":2742},[387],[389,2744,2745],{"xmlns":391},[393,2746,2747,2752],{},[396,2748,2749],{},[399,2750,2751],{"mathvariant":1196},"∇",[418,2753,2754],{"encoding":420},"\\nabla",[380,2756,2758],{"className":2757,"ariaHidden":426},[425],[380,2759,2761,2764],{"className":2760},[430],[380,2762],{"className":2763,"style":1209},[434],[380,2765,2751],{"className":2766},[439],[528,2768,2769],{},[536,2770,2754],{},[528,2772,2773],{},"Nabla\u002Fgradient",[495,2775,2777],{"id":2776},"arrows","Arrows",[500,2779,2780,2790],{},[503,2781,2782],{},[506,2783,2784,2786,2788],{},[509,2785,2129],{},[509,2787,1425],{},[509,2789,1428],{},[523,2791,2792,2833,2874,2915,2956,2997],{},[506,2793,2794,2826,2830],{},[528,2795,2796],{},[380,2797,2799,2814],{"className":2798},[383],[380,2800,2802],{"className":2801},[387],[389,2803,2804],{"xmlns":391},[393,2805,2806,2811],{},[396,2807,2808],{},[403,2809,2810],{},"→",[418,2812,2813],{"encoding":420},"\\rightarrow",[380,2815,2817],{"className":2816,"ariaHidden":426},[425],[380,2818,2820,2823],{"className":2819},[430],[380,2821],{"className":2822,"style":2415},[434],[380,2824,2810],{"className":2825},[470],[528,2827,2828],{},[536,2829,2813],{},[528,2831,2832],{},"Right arrow",[506,2834,2835,2867,2871],{},[528,2836,2837],{},[380,2838,2840,2855],{"className":2839},[383],[380,2841,2843],{"className":2842},[387],[389,2844,2845],{"xmlns":391},[393,2846,2847,2852],{},[396,2848,2849],{},[403,2850,2851],{},"←",[418,2853,2854],{"encoding":420},"\\leftarrow",[380,2856,2858],{"className":2857,"ariaHidden":426},[425],[380,2859,2861,2864],{"className":2860},[430],[380,2862],{"className":2863,"style":2415},[434],[380,2865,2851],{"className":2866},[470],[528,2868,2869],{},[536,2870,2854],{},[528,2872,2873],{},"Left arrow",[506,2875,2876,2908,2912],{},[528,2877,2878],{},[380,2879,2881,2896],{"className":2880},[383],[380,2882,2884],{"className":2883},[387],[389,2885,2886],{"xmlns":391},[393,2887,2888,2893],{},[396,2889,2890],{},[403,2891,2892],{},"↔",[418,2894,2895],{"encoding":420},"\\leftrightarrow",[380,2897,2899],{"className":2898,"ariaHidden":426},[425],[380,2900,2902,2905],{"className":2901},[430],[380,2903],{"className":2904,"style":2415},[434],[380,2906,2892],{"className":2907},[470],[528,2909,2910],{},[536,2911,2895],{},[528,2913,2914],{},"Left-right arrow",[506,2916,2917,2949,2953],{},[528,2918,2919],{},[380,2920,2922,2937],{"className":2921},[383],[380,2923,2925],{"className":2924},[387],[389,2926,2927],{"xmlns":391},[393,2928,2929,2934],{},[396,2930,2931],{},[403,2932,2933],{},"⇒",[418,2935,2936],{"encoding":420},"\\Rightarrow",[380,2938,2940],{"className":2939,"ariaHidden":426},[425],[380,2941,2943,2946],{"className":2942},[430],[380,2944],{"className":2945,"style":2415},[434],[380,2947,2933],{"className":2948},[470],[528,2950,2951],{},[536,2952,2936],{},[528,2954,2955],{},"Double right arrow",[506,2957,2958,2990,2994],{},[528,2959,2960],{},[380,2961,2963,2978],{"className":2962},[383],[380,2964,2966],{"className":2965},[387],[389,2967,2968],{"xmlns":391},[393,2969,2970,2975],{},[396,2971,2972],{},[403,2973,2974],{},"⇐",[418,2976,2977],{"encoding":420},"\\Leftarrow",[380,2979,2981],{"className":2980,"ariaHidden":426},[425],[380,2982,2984,2987],{"className":2983},[430],[380,2985],{"className":2986,"style":2415},[434],[380,2988,2974],{"className":2989},[470],[528,2991,2992],{},[536,2993,2977],{},[528,2995,2996],{},"Double left arrow",[506,2998,2999,3032,3036],{},[528,3000,3001],{},[380,3002,3004,3019],{"className":3003},[383],[380,3005,3007],{"className":3006},[387],[389,3008,3009],{"xmlns":391},[393,3010,3011,3016],{},[396,3012,3013],{},[403,3014,3015],{},"↦",[418,3017,3018],{"encoding":420},"\\mapsto",[380,3020,3022],{"className":3021,"ariaHidden":426},[425],[380,3023,3025,3029],{"className":3024},[430],[380,3026],{"className":3027,"style":3028},[434],"height:0.522em;vertical-align:-0.011em;",[380,3030,3015],{"className":3031},[470],[528,3033,3034],{},[536,3035,3018],{},[528,3037,3038],{},"Maps to",[361,3040,1825,3042,3044],{"id":3041},"the-mathtools-package",[536,3043,1987],{}," Package",[357,3046,1825,3047,3049,3050,3052],{},[536,3048,1987],{}," package extends ",[536,3051,1828],{}," with additional 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