[{"data":1,"prerenderedAt":21192},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-05-30-about-the-tensor":3},{"post":4,"nextPost":6090,"prevPost":21141},{"id":5,"title":6,"body":7,"description":6064,"draft":6076,"enableComment":6077,"extension":6078,"image":6064,"important":6076,"location":6079,"meta":6080,"navigation":6077,"ogImage":6079,"onday":6081,"path":6082,"seo":6083,"stem":6084,"summary":6085,"tags":6086,"__hash__":6089},"blog\u002Fblog\u002F2026\u002F2026-05-30-about-the-tensor.md","About the Tensor",{"type":8,"value":9,"toc":6063},"minimark",[10,17,20,32,39,42,47,50,66,77,137,141,144,155,444,447,803,806,1111,1114,1154,1178,1339,1346,1351,1355,1358,1511,1514,1999,2002,2016,2019,2104,2107,2111,2114,2213,2220,2226,2343,2351,2507,2510,2582,2588,2649,2757,2868,2976,2980,2986,3084,3087,3269,3272,3439,3494,3497,3501,3652,5299,5302,5617,5620,5624,5627,6014,6018,6025,6043,6050,6054,6057,6060],[11,12,13],"blockquote",{},[14,15,16],"p",{},"A tensor is a higher-dimensional generalization of a scalar, a vector, and a matrix",[14,18,19],{},"You have almost certainly encountered this sentence. It appears on the first page of virtually every textbook, and it is also the point of departure for virtually all confusion.",[14,21,22,23,27,28,31],{},"The reason is that this sentence describes only the ",[24,25,26],"strong",{},"storage format"," of a tensor, while saying nothing whatsoever about its ",[24,29,30],{},"essence",". It is akin to stating that \"a human being is a collection of cells\"—not strictly false, yet it does not answer the genuine question of what a human being is.",[14,33,34,35,38],{},"This article pursues a single objective: ",[24,36,37],{},"why are physicists and mathematicians so insistent on distinguishing vectors from \"dual vectors,\" and on distinguishing \"upper indices\" from \"lower indices\" in tensors?"," If you can grasp this one point, all the mystique surrounding tensors evaporates.",[14,40,41],{},"We shall proceed in the following order: first, we reexamine the vector itself; next, we introduce its twin sibling, the dual vector; then we furnish a rigorous definition of a tensor; we dismantle a common point of confusion with a concrete example; and finally, we elucidate the critical bridge connecting vectors and dual vectors—the metric.",[43,44,46],"h2",{"id":45},"arrows","Arrows",[14,48,49],{},"Suppose there is a physical arrow lying on a plane, pointing 30° east of north, with a length of 5 meters. This arrow has an objective existence—no matter how you erect a coordinate system, it remains exactly where it is; its length and direction do not alter merely because you have changed your frame of reference.",[14,51,52,53,57,58,61,62,65],{},"Yet when you write down coordinates such as ",[54,55,56],"code",{},"(3, 4)",", you have performed a concealed operation: ",[24,59,60],{},"you have selected a coordinate basis"," and subsequently \"translated\" the arrow into a tuple of numbers with respect to that basis. Switch to a different basis (for instance, by rotating the coordinate axes by 45°), and the very same arrow is translated into an entirely different set of numbers, say ",[54,63,64],{},"(4.95, 0.71)",".",[14,67,68,69,72,73,76],{},"The numbers changed; the arrow did not. ",[24,70,71],{},"This is precisely the most fundamental distinction between a vector as a geometric object and an array as its coordinate representation."," A tensor in deep learning (e.g., ",[54,74,75],{},"torch.tensor([3,4])",") is merely this \"translation result\"; it never concerns itself with whether an invariant geometric object stands behind it. This is perfectly acceptable when one is engaged in purely numerical computation, but the moment one wishes to entertain propositions such as \"physical laws do not depend on the choice of observer,\" one must return to the arrow itself.",[14,78,79,80,128,129,132,133,136],{},"Setting coordinates aside, the definition of a vector space ",[81,82,86,108],"span",{"className":83,"translate":85},[84],"katex","no",[81,87,90],{"className":88},[89],"katex-mathml",[91,92,94],"math",{"xmlns":93},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[95,96,97,104],"semantics",{},[98,99,100],"mrow",{},[101,102,103],"mi",{},"V",[105,106,103],"annotation",{"encoding":107},"application\u002Fx-tex",[81,109,113],{"className":110,"ariaHidden":112},[111],"katex-html","true",[81,114,117,122],{"className":115},[116],"base",[81,118],{"className":119,"style":121},[120],"strut","height:0.6833em;",[81,123,103],{"className":124,"style":127},[125,126],"mord","mathnormal","margin-right:0.2222em;"," is in fact remarkably parsimonious: so long as a collection of objects supports ",[24,130,131],{},"addition"," (tip-to-tail composition) and ",[24,134,135],{},"scalar multiplication"," (stretching or shrinking), and satisfies a handful of natural algebraic laws, it constitutes a vector space, and its elements are vectors. Velocity, force, and displacement are all objects of this kind.",[43,138,140],{"id":139},"dual-space","Dual Space",[14,142,143],{},"A vector is an arrow; what, then, is the instrument that \"measures\" that arrow?",[14,145,146,147,150,151,154],{},"It is an object of an altogether different type, called a ",[24,148,149],{},"dual vector"," (also termed a covector or a 1-form). It is not an arrow but rather a ",[24,152,153],{},"ruler","—a family of equally spaced parallel lines. When you lay the arrow upon it, the number of lines the arrow crosses is the number it outputs.",[14,156,157,158,265,266,298,299,357,358,392,393,443],{},"A rigorous definition: ",[24,159,160,161,236,237],{},"the dual space ",[81,162,164,185],{"className":163,"translate":85},[84],[81,165,167],{"className":166},[89],[91,168,169],{"xmlns":93},[95,170,171,182],{},[98,172,173],{},[174,175,176,178],"msup",{},[101,177,103],{},[179,180,181],"mo",{},"∗",[105,183,184],{"encoding":107},"V^*",[81,186,188],{"className":187,"ariaHidden":112},[111],[81,189,191,195],{"className":190},[116],[81,192],{"className":193,"style":194},[120],"height:0.6887em;",[81,196,198,201],{"className":197},[125],[81,199,103],{"className":200,"style":127},[125,126],[81,202,205],{"className":203},[204],"msupsub",[81,206,209],{"className":207},[208],"vlist-t",[81,210,213],{"className":211},[212],"vlist-r",[81,214,217],{"className":215,"style":194},[216],"vlist",[81,218,220,225],{"style":219},"top:-3.063em;margin-right:0.05em;",[81,221],{"className":222,"style":224},[223],"pstrut","height:2.7em;",[81,226,232],{"className":227},[228,229,230,231],"sizing","reset-size6","size3","mtight",[81,233,181],{"className":234},[235,231],"mbin"," is the set of all linear functionals on ",[81,238,240,253],{"className":239,"translate":85},[84],[81,241,243],{"className":242},[89],[91,244,245],{"xmlns":93},[95,246,247,251],{},[98,248,249],{},[101,250,103],{},[105,252,103],{"encoding":107},[81,254,256],{"className":255,"ariaHidden":112},[111],[81,257,259,262],{"className":258},[116],[81,260],{"className":261,"style":121},[120],[81,263,103],{"className":264,"style":127},[125,126],". That is, each element ",[81,267,269,284],{"className":268,"translate":85},[84],[81,270,272],{"className":271},[89],[91,273,274],{"xmlns":93},[95,275,276,281],{},[98,277,278],{},[101,279,280],{},"ω",[105,282,283],{"encoding":107},"\\omega",[81,285,287],{"className":286,"ariaHidden":112},[111],[81,288,290,294],{"className":289},[116],[81,291],{"className":292,"style":293},[120],"height:0.4306em;",[81,295,280],{"className":296,"style":297},[125,126],"margin-right:0.0359em;"," in ",[81,300,302,319],{"className":301,"translate":85},[84],[81,303,305],{"className":304},[89],[91,306,307],{"xmlns":93},[95,308,309,317],{},[98,310,311],{},[174,312,313,315],{},[101,314,103],{},[179,316,181],{},[105,318,184],{"encoding":107},[81,320,322],{"className":321,"ariaHidden":112},[111],[81,323,325,328],{"className":324},[116],[81,326],{"className":327,"style":194},[120],[81,329,331,334],{"className":330},[125],[81,332,103],{"className":333,"style":127},[125,126],[81,335,337],{"className":336},[204],[81,338,340],{"className":339},[208],[81,341,343],{"className":342},[212],[81,344,346],{"className":345,"style":194},[216],[81,347,348,351],{"style":219},[81,349],{"className":350,"style":224},[223],[81,352,354],{"className":353},[228,229,230,231],[81,355,181],{"className":356},[235,231]," is a rule: feed it a vector ",[81,359,361,377],{"className":360,"translate":85},[84],[81,362,364],{"className":363},[89],[91,365,366],{"xmlns":93},[95,367,368,374],{},[98,369,370],{},[101,371,373],{"mathvariant":372},"bold","v",[105,375,376],{"encoding":107},"\\mathbf{v}",[81,378,380],{"className":379,"ariaHidden":112},[111],[81,381,383,387],{"className":382},[116],[81,384],{"className":385,"style":386},[120],"height:0.4444em;",[81,388,373],{"className":389,"style":391},[125,390],"mathbf","margin-right:0.016em;",", and it emits a real number ",[81,394,396,419],{"className":395,"translate":85},[84],[81,397,399],{"className":398},[89],[91,400,401],{"xmlns":93},[95,402,403,416],{},[98,404,405,407,411,413],{},[101,406,280],{},[179,408,410],{"stretchy":409},"false","(",[101,412,373],{"mathvariant":372},[179,414,415],{"stretchy":409},")",[105,417,418],{"encoding":107},"\\omega(\\mathbf{v})",[81,420,422],{"className":421,"ariaHidden":112},[111],[81,423,425,429,432,436,439],{"className":424},[116],[81,426],{"className":427,"style":428},[120],"height:1em;vertical-align:-0.25em;",[81,430,280],{"className":431,"style":297},[125,126],[81,433,410],{"className":434},[435],"mopen",[81,437,373],{"className":438,"style":391},[125,390],[81,440,415],{"className":441},[442],"mclose",", and this rule is linear with respect to vector addition and scalar multiplication.",[14,445,446],{},"In finite-dimensional linear algebra, this is so straightforward as to be almost banal:",[448,449,450,622],"ul",{},[451,452,453,454,482,483],"li",{},"A vector ",[81,455,457,470],{"className":456,"translate":85},[84],[81,458,460],{"className":459},[89],[91,461,462],{"xmlns":93},[95,463,464,468],{},[98,465,466],{},[101,467,373],{"mathvariant":372},[105,469,376],{"encoding":107},[81,471,473],{"className":472,"ariaHidden":112},[111],[81,474,476,479],{"className":475},[116],[81,477],{"className":478,"style":386},[120],[81,480,373],{"className":481,"style":391},[125,390]," is a column vector: ",[81,484,486,533],{"className":485,"translate":85},[84],[81,487,489],{"className":488},[89],[91,490,491],{"xmlns":93},[95,492,493,530],{},[98,494,495,498,527],{},[179,496,497],{"fence":112},"[",[499,500,504,518],"mtable",{"rowspacing":501,"columnalign":502,"columnspacing":503},"0.16em","center","1em",[505,506,507],"mtr",{},[508,509,510],"mtd",{},[511,512,514],"mstyle",{"scriptlevel":513,"displaystyle":409},"0",[515,516,517],"mn",{},"2",[505,519,520],{},[508,521,522],{},[511,523,524],{"scriptlevel":513,"displaystyle":409},[515,525,526],{},"3",[179,528,529],{"fence":112},"]",[105,531,532],{"encoding":107},"\\begin{bmatrix} 2 \\\\ 3 \\end{bmatrix}",[81,534,536],{"className":535,"ariaHidden":112},[111],[81,537,539,543],{"className":538},[116],[81,540],{"className":541,"style":542},[120],"height:2.4em;vertical-align:-0.95em;",[81,544,547,556,616],{"className":545},[546],"minner",[81,548,552],{"className":549,"style":551},[435,550],"delimcenter","top:0em;",[81,553,497],{"className":554},[555,230],"delimsizing",[81,557,559],{"className":558},[125],[81,560,562],{"className":561},[499],[81,563,566],{"className":564},[565],"col-align-c",[81,567,570,607],{"className":568},[208,569],"vlist-t2",[81,571,573,602],{"className":572},[212],[81,574,577,590],{"className":575,"style":576},[216],"height:1.45em;",[81,578,580,584],{"style":579},"top:-3.61em;",[81,581],{"className":582,"style":583},[223],"height:3em;",[81,585,587],{"className":586},[125],[81,588,517],{"className":589},[125],[81,591,593,596],{"style":592},"top:-2.41em;",[81,594],{"className":595,"style":583},[223],[81,597,599],{"className":598},[125],[81,600,526],{"className":601},[125],[81,603,606],{"className":604},[605],"vlist-s","​",[81,608,610],{"className":609},[212],[81,611,614],{"className":612,"style":613},[216],"height:0.95em;",[81,615],{},[81,617,619],{"className":618,"style":551},[442,550],[81,620,529],{"className":621},[555,230],[451,623,624,625,653,654],{},"A dual vector ",[81,626,628,641],{"className":627,"translate":85},[84],[81,629,631],{"className":630},[89],[91,632,633],{"xmlns":93},[95,634,635,639],{},[98,636,637],{},[101,638,280],{},[105,640,283],{"encoding":107},[81,642,644],{"className":643,"ariaHidden":112},[111],[81,645,647,650],{"className":646},[116],[81,648],{"className":649,"style":293},[120],[81,651,280],{"className":652,"style":297},[125,126]," is a row vector: ",[81,655,657,692],{"className":656,"translate":85},[84],[81,658,660],{"className":659},[89],[91,661,662],{"xmlns":93},[95,663,664,689],{},[98,665,666,668,687],{},[179,667,497],{"fence":112},[499,669,671],{"rowspacing":501,"columnalign":670,"columnspacing":503},"center center",[505,672,673,680],{},[508,674,675],{},[511,676,677],{"scriptlevel":513,"displaystyle":409},[515,678,679],{},"1",[508,681,682],{},[511,683,684],{"scriptlevel":513,"displaystyle":409},[515,685,686],{},"4",[179,688,529],{"fence":112},[105,690,691],{"encoding":107},"\\begin{bmatrix} 1 & 4 \\end{bmatrix}",[81,693,695],{"className":694,"ariaHidden":112},[111],[81,696,698,702],{"className":697},[116],[81,699],{"className":700,"style":701},[120],"height:1.2em;vertical-align:-0.35em;",[81,703,705,712,797],{"className":704},[546],[81,706,708],{"className":707,"style":551},[435,550],[81,709,497],{"className":710},[555,711],"size1",[81,713,715],{"className":714},[125],[81,716,718,755,760,763],{"className":717},[499],[81,719,721],{"className":720},[565],[81,722,724,746],{"className":723},[208,569],[81,725,727,743],{"className":726},[212],[81,728,731],{"className":729,"style":730},[216],"height:0.85em;",[81,732,734,737],{"style":733},"top:-3.01em;",[81,735],{"className":736,"style":583},[223],[81,738,740],{"className":739},[125],[81,741,679],{"className":742},[125],[81,744,606],{"className":745},[605],[81,747,749],{"className":748},[212],[81,750,753],{"className":751,"style":752},[216],"height:0.35em;",[81,754],{},[81,756],{"className":757,"style":759},[758],"arraycolsep","width:0.5em;",[81,761],{"className":762,"style":759},[758],[81,764,766],{"className":765},[565],[81,767,769,789],{"className":768},[208,569],[81,770,772,786],{"className":771},[212],[81,773,775],{"className":774,"style":730},[216],[81,776,777,780],{"style":733},[81,778],{"className":779,"style":583},[223],[81,781,783],{"className":782},[125],[81,784,686],{"className":785},[125],[81,787,606],{"className":788},[605],[81,790,792],{"className":791},[212],[81,793,795],{"className":794,"style":752},[216],[81,796],{},[81,798,800],{"className":799,"style":551},[442,550],[81,801,529],{"className":802},[555,711],[14,804,805],{},"Their \"pairing\" is simply matrix multiplication:",[81,807,810],{"className":808,"translate":85},[809],"katex-display",[81,811,813,888],{"className":812,"translate":85},[84],[81,814,816],{"className":815},[89],[91,817,819],{"xmlns":93,"display":818},"block",[95,820,821,885],{},[98,822,823,825,827,829,831,834,856,880,882],{},[101,824,280],{},[179,826,410],{"stretchy":409},[101,828,373],{"mathvariant":372},[179,830,415],{"stretchy":409},[179,832,833],{},"=",[98,835,836,838,854],{},[179,837,497],{"fence":112},[499,839,840],{"rowspacing":501,"columnalign":670,"columnspacing":503},[505,841,842,848],{},[508,843,844],{},[511,845,846],{"scriptlevel":513,"displaystyle":409},[515,847,679],{},[508,849,850],{},[511,851,852],{"scriptlevel":513,"displaystyle":409},[515,853,686],{},[179,855,529],{"fence":112},[98,857,858,860,878],{},[179,859,497],{"fence":112},[499,861,862,870],{"rowspacing":501,"columnalign":502,"columnspacing":503},[505,863,864],{},[508,865,866],{},[511,867,868],{"scriptlevel":513,"displaystyle":409},[515,869,517],{},[505,871,872],{},[508,873,874],{},[511,875,876],{"scriptlevel":513,"displaystyle":409},[515,877,526],{},[179,879,529],{"fence":112},[179,881,833],{},[515,883,884],{},"14",[105,886,887],{"encoding":107},"\\omega(\\mathbf{v}) = \\begin{bmatrix} 1 & 4 \\end{bmatrix} \\begin{bmatrix} 2 \\\\ 3 \\end{bmatrix} = 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space formed by all row vectors is precisely the dual space of the column-vector space. 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20-1h17z",[81,1722,1723,1726],{"style":1655},[81,1724],{"className":1725,"style":583},[223],[81,1727,1729,1759,1762,1765,1768,1771,1774,1777,1780],{"className":1728},[125],[81,1730,1732,1735],{"className":1731},[125],[81,1733,103],{"className":1734,"style":127},[125,126],[81,1736,1738],{"className":1737},[204],[81,1739,1741],{"className":1740},[208],[81,1742,1744],{"className":1743},[212],[81,1745,1747],{"className":1746,"style":1637},[216],[81,1748,1750,1753],{"style":1749},"top:-3.113em;margin-right:0.05em;",[81,1751],{"className":1752,"style":224},[223],[81,1754,1756],{"className":1755},[228,229,230,231],[81,1757,181],{"className":1758},[235,231],[81,1760],{"className":1761,"style":127},[912],[81,1763,1197],{"className":1764},[235],[81,1766],{"className":1767,"style":127},[912],[81,1769,1551],{"className":1770},[546],[81,1772],{"className":1773,"style":127},[912],[81,1775,1197],{"className":1776},[235],[81,1778],{"className":1779,"style":127},[912],[81,1781,1783,1786],{"className":1782},[125],[81,1784,103],{"className":1785,"style":127},[125,126],[81,1787,1789],{"className":1788},[204],[81,1790,1792],{"className":1791},[208],[81,1793,1795],{"className":1794},[212],[81,1796,1798],{"className":1797,"style":1637},[216],[81,1799,1800,1803],{"style":1749},[81,1801],{"className":1802,"style":224},[223],[81,1804,1806],{"className":1805},[228,229,230,231],[81,1807,181],{"className":1808},[235,231],[81,1810,606],{"className":1811},[605],[81,1813,1815],{"className":1814},[212],[81,1816,1819],{"className":1817,"style":1818},[216],"height:0.7313em;",[81,1820],{},[81,1822,606],{"className":1823},[605],[81,1825,1827],{"className":1826},[212],[81,1828,1831],{"className":1829,"style":1830},[216],"height:1.3688em;",[81,1832],{},[81,1834],{"className":1835,"style":127},[912],[81,1837,1197],{"className":1838},[235],[81,1840],{"className":1841,"style":127},[912],[81,1843,1845,1849,1979,1982,1985],{"className":1844},[116],[81,1846],{"className":1847,"style":1848},[120],"height:2.0522em;vertical-align:-1.3688em;",[81,1850,1852],{"className":1851},[546,1535],[81,1853,1855,1971],{"className":1854},[208,569],[81,1856,1858,1968],{"className":1857},[212],[81,1859,1861,1875],{"className":1860,"style":121},[216],[81,1862,1863,1866],{"style":1640},[81,1864],{"className":1865,"style":583},[223],[81,1867,1869],{"className":1868},[228,229,230,231],[81,1870,1872],{"className":1871},[125,231],[81,1873,1417],{"className":1874,"style":297},[125,126,231],[81,1876,1877,1880],{"style":1655},[81,1878],{"className":1879,"style":583},[223],[81,1881,1883],{"className":1882},[546,1535],[81,1884,1886,1960],{"className":1885},[208,569],[81,1887,1889,1957],{"className":1888},[212],[81,1890,1892,1922],{"className":1891,"style":121},[216],[81,1893,1895,1898],{"className":1894,"style":1675},[1674],[81,1896],{"className":1897,"style":583},[223],[81,1899,1901,1908,1915],{"className":1900,"style":1683},[1682],[81,1902,1904],{"className":1903,"style":1688},[1687],[1690,1905,1906],{"xmlns":1692,"width":1693,"height":1694,"viewBox":1695,"preserveAspectRatio":1696},[1698,1907],{"d":1700},[81,1909,1911],{"className":1910,"style":1688},[1704],[1690,1912,1913],{"xmlns":1692,"width":1693,"height":1694,"viewBox":1695,"preserveAspectRatio":1707},[1698,1914],{"d":1710},[81,1916,1918],{"className":1917,"style":1688},[1714],[1690,1919,1920],{"xmlns":1692,"width":1693,"height":1694,"viewBox":1695,"preserveAspectRatio":1717},[1698,1921],{"d":1720},[81,1923,1924,1927],{"style":1655},[81,1925],{"className":1926,"style":583},[223],[81,1928,1930,1933,1936,1939,1942,1945,1948,1951,1954],{"className":1929},[125],[81,1931,103],{"className":1932,"style":127},[125,126],[81,1934],{"className":1935,"style":127},[912],[81,1937,1197],{"className":1938},[235],[81,1940],{"className":1941,"style":127},[912],[81,1943,1551],{"className":1944},[546],[81,1946],{"className":1947,"style":127},[912],[81,1949,1197],{"className":1950},[235],[81,1952],{"className":1953,"style":127},[912],[81,1955,103],{"className":1956,"style":127},[125,126],[81,1958,606],{"className":1959},[605],[81,1961,1963],{"className":1962},[212],[81,1964,1966],{"className":1965,"style":1818},[216],[81,1967],{},[81,1969,606],{"className":1970},[605],[81,1972,1974],{"className":1973},[212],[81,1975,1977],{"className":1976,"style":1830},[216],[81,1978],{},[81,1980],{"className":1981,"style":913},[912],[81,1983,1589],{"className":1984},[917],[81,1986],{"className":1987,"style":913},[912],[81,1989,1991,1995],{"className":1990},[116],[81,1992],{"className":1993,"style":1994},[120],"height:0.6889em;",[81,1996,1593],{"className":1997},[125,1998],"mathbb",[14,2000,2001],{},"Two keywords, neither dispensable:",[448,2003,2004,2010],{},[451,2005,2006,2009],{},[24,2007,2008],{},"Multilinear",": linear in each slot individually (additive and homogeneous with respect to scalar multiplication). This guarantees that tensor operations can be fully expanded as weighted sums of components, just as matrix operations can.",[451,2011,2012,2015],{},[24,2013,2014],{},"Scalar output",": irrespective of the number of slots or the variety of input types, what ultimately emerges is an absolute number that does not vary with changes of coordinates.",[14,2017,2018],{},"From this we directly obtain the formula for the order of a tensor:",[81,2020,2022],{"className":2021,"translate":85},[809],[81,2023,2025,2050],{"className":2024,"translate":85},[84],[81,2026,2028],{"className":2027},[89],[91,2029,2030],{"xmlns":93,"display":818},[95,2031,2032,2047],{},[98,2033,2034,2038,2040,2042,2045],{},[2035,2036,2037],"mtext",{},"Order",[179,2039,833],{},[101,2041,14],{},[179,2043,2044],{},"+",[101,2046,1417],{},[105,2048,2049],{"encoding":107},"\\text{Order} = p + q",[81,2051,2053,2076,2095],{"className":2052,"ariaHidden":112},[111],[81,2054,2056,2060,2067,2070,2073],{"className":2055},[116],[81,2057],{"className":2058,"style":2059},[120],"height:0.6944em;",[81,2061,2064],{"className":2062},[125,2063],"text",[81,2065,2037],{"className":2066},[125],[81,2068],{"className":2069,"style":913},[912],[81,2071,833],{"className":2072},[917],[81,2074],{"className":2075,"style":913},[912],[81,2077,2079,2083,2086,2089,2092],{"className":2078},[116],[81,2080],{"className":2081,"style":2082},[120],"height:0.7778em;vertical-align:-0.1944em;",[81,2084,14],{"className":2085},[125,126],[81,2087],{"className":2088,"style":127},[912],[81,2090,2044],{"className":2091},[235],[81,2093],{"className":2094,"style":127},[912],[81,2096,2098,2101],{"className":2097},[116],[81,2099],{"className":2100,"style":1477},[120],[81,2102,1417],{"className":2103,"style":297},[125,126],[14,2105,2106],{},"This is also why the term \"order\" alone carries insufficient information—it tells you how many slots a tensor possesses, but not which slots consume vectors and which consume dual vectors. This distinction is by no means a verbal quibble; the following chapter will illustrate, with a concrete example, why it is a matter of life and death.",[43,2108,2110],{"id":2109},"the-slot-philosophy","The Slot Philosophy",[14,2112,2113],{},"Physics textbooks frequently describe second-order tensors in the following manner:",[11,2115,2116],{},[14,2117,2118,2119,2149,2150,2180,2181,2212],{},"\"The stress tensor ",[81,2120,2122,2137],{"className":2121,"translate":85},[84],[81,2123,2125],{"className":2124},[89],[91,2126,2127],{"xmlns":93},[95,2128,2129,2134],{},[98,2130,2131],{},[101,2132,2133],{},"σ",[105,2135,2136],{"encoding":107},"\\sigma",[81,2138,2140],{"className":2139,"ariaHidden":112},[111],[81,2141,2143,2146],{"className":2142},[116],[81,2144],{"className":2145,"style":293},[120],[81,2147,2133],{"className":2148,"style":297},[125,126]," acts on a normal vector ",[81,2151,2153,2168],{"className":2152,"translate":85},[84],[81,2154,2156],{"className":2155},[89],[91,2157,2158],{"xmlns":93},[95,2159,2160,2165],{},[98,2161,2162],{},[101,2163,2164],{"mathvariant":372},"n",[105,2166,2167],{"encoding":107},"\\mathbf{n}",[81,2169,2171],{"className":2170,"ariaHidden":112},[111],[81,2172,2174,2177],{"className":2173},[116],[81,2175],{"className":2176,"style":386},[120],[81,2178,2164],{"className":2179},[125,390]," to produce a stress vector ",[81,2182,2184,2199],{"className":2183,"translate":85},[84],[81,2185,2187],{"className":2186},[89],[91,2188,2189],{"xmlns":93},[95,2190,2191,2196],{},[98,2192,2193],{},[101,2194,2195],{"mathvariant":372},"t",[105,2197,2198],{"encoding":107},"\\mathbf{t}",[81,2200,2202],{"className":2201,"ariaHidden":112},[111],[81,2203,2205,2209],{"className":2204},[116],[81,2206],{"className":2207,"style":2208},[120],"height:0.6349em;",[81,2210,2195],{"className":2211},[125,390],".\"",[14,2214,2215,2216,2219],{},"This sentence sounds like \"",[24,2217,2218],{},"input a vector, output a vector",".\" Yet the definition in Chapter 3 explicitly states that a tensor is something that \"inputs several vectors and dual vectors and outputs a scalar.\" The two statements appear utterly irreconcilable—and this is precisely where the vast majority of readers become stuck.",[14,2221,2222,2223],{},"The key that resolves this contradiction is: ",[24,2224,2225],{},"a multilinear map does not require you to fill all of its slots at once.",[14,2227,2228,2229,2257,2258,2310,2311,2339,2340,1533],{},"The stress tensor ",[81,2230,2232,2245],{"className":2231,"translate":85},[84],[81,2233,2235],{"className":2234},[89],[91,2236,2237],{"xmlns":93},[95,2238,2239,2243],{},[98,2240,2241],{},[101,2242,2133],{},[105,2244,2136],{"encoding":107},[81,2246,2248],{"className":2247,"ariaHidden":112},[111],[81,2249,2251,2254],{"className":2250},[116],[81,2252],{"className":2253,"style":293},[120],[81,2255,2133],{"className":2256,"style":297},[125,126]," is a tensor of type ",[81,2259,2261,2283],{"className":2260,"translate":85},[84],[81,2262,2264],{"className":2263},[89],[91,2265,2266],{"xmlns":93},[95,2267,2268,2280],{},[98,2269,2270,2272,2274,2276,2278],{},[179,2271,410],{"stretchy":409},[515,2273,679],{},[179,2275,1414],{"separator":112},[515,2277,679],{},[179,2279,415],{"stretchy":409},[105,2281,2282],{"encoding":107},"(1,1)",[81,2284,2286],{"className":2285,"ariaHidden":112},[111],[81,2287,2289,2292,2295,2298,2301,2304,2307],{"className":2288},[116],[81,2290],{"className":2291,"style":428},[120],[81,2293,410],{"className":2294},[435],[81,2296,679],{"className":2297},[125],[81,2299,1414],{"className":2300},[1441],[81,2302],{"className":2303,"style":1025},[912],[81,2305,679],{"className":2306},[125],[81,2308,415],{"className":2309},[442],", possessing two slots: one dual-vector slot and one vector slot. When you insert only the normal vector ",[81,2312,2314,2327],{"className":2313,"translate":85},[84],[81,2315,2317],{"className":2316},[89],[91,2318,2319],{"xmlns":93},[95,2320,2321,2325],{},[98,2322,2323],{},[101,2324,2164],{"mathvariant":372},[105,2326,2167],{"encoding":107},[81,2328,2330],{"className":2329,"ariaHidden":112},[111],[81,2331,2333,2336],{"className":2332},[116],[81,2334],{"className":2335,"style":386},[120],[81,2337,2164],{"className":2338},[125,390]," into its ",[24,2341,2342],{},"vector slot",[448,2344,2345,2348],{},[451,2346,2347],{},"The vector slot is now filled;",[451,2349,2350],{},"The dual-vector slot remains vacant.",[14,2352,2353,2354,2412,2413,2416,2417,1153],{},"What has ",[81,2355,2357,2382],{"className":2356,"translate":85},[84],[81,2358,2360],{"className":2359},[89],[91,2361,2362],{"xmlns":93},[95,2363,2364,2379],{},[98,2365,2366,2368,2370,2373,2375,2377],{},[101,2367,2133],{},[179,2369,410],{"stretchy":409},[179,2371,2372],{},"⋅",[179,2374,1414],{"separator":112},[101,2376,2164],{"mathvariant":372},[179,2378,415],{"stretchy":409},[105,2380,2381],{"encoding":107},"\\sigma(\\cdot, \\mathbf{n})",[81,2383,2385],{"className":2384,"ariaHidden":112},[111],[81,2386,2388,2391,2394,2397,2400,2403,2406,2409],{"className":2387},[116],[81,2389],{"className":2390,"style":428},[120],[81,2392,2133],{"className":2393,"style":297},[125,126],[81,2395,410],{"className":2396},[435],[81,2398,2372],{"className":2399},[125],[81,2401,1414],{"className":2402},[1441],[81,2404],{"className":2405,"style":1025},[912],[81,2407,2164],{"className":2408},[125,390],[81,2410,415],{"className":2411},[442]," become? It has become an ",[24,2414,2415],{},"intermediate product still awaiting a dual-vector input","—and \"a machine that awaits a dual vector and then emits a scalar\" happens to be another rigorous way of defining a vector (via the natural isomorphism ",[81,2418,2420,2447],{"className":2419,"translate":85},[84],[81,2421,2423],{"className":2422},[89],[91,2424,2425],{"xmlns":93},[95,2426,2427,2444],{},[98,2428,2429,2431,2434],{},[101,2430,103],{},[179,2432,2433],{},"≅",[174,2435,2436,2438],{},[101,2437,103],{},[98,2439,2440,2442],{},[179,2441,181],{},[179,2443,181],{},[105,2445,2446],{"encoding":107},"V \\cong V^{**}",[81,2448,2450,2468],{"className":2449,"ariaHidden":112},[111],[81,2451,2453,2456,2459,2462,2465],{"className":2452},[116],[81,2454],{"className":2455,"style":121},[120],[81,2457,103],{"className":2458,"style":127},[125,126],[81,2460],{"className":2461,"style":913},[912],[81,2463,2433],{"className":2464},[917],[81,2466],{"className":2467,"style":913},[912],[81,2469,2471,2474],{"className":2470},[116],[81,2472],{"className":2473,"style":194},[120],[81,2475,2477,2480],{"className":2476},[125],[81,2478,103],{"className":2479,"style":127},[125,126],[81,2481,2483],{"className":2482},[204],[81,2484,2486],{"className":2485},[208],[81,2487,2489],{"className":2488},[212],[81,2490,2492],{"className":2491,"style":194},[216],[81,2493,2494,2497],{"style":219},[81,2495],{"className":2496,"style":224},[223],[81,2498,2500],{"className":2499},[228,229,230,231],[81,2501,2503],{"className":2502},[125,231],[81,2504,2506],{"className":2505},[125,231],"∗∗",[14,2508,2509],{},"Thus:",[448,2511,2512,2576],{},[451,2513,2514,2517,2518,2546,2547,2575],{},[24,2515,2516],{},"Only one slot filled",": ",[81,2519,2521,2534],{"className":2520,"translate":85},[84],[81,2522,2524],{"className":2523},[89],[91,2525,2526],{"xmlns":93},[95,2527,2528,2532],{},[98,2529,2530],{},[101,2531,2133],{},[105,2533,2136],{"encoding":107},[81,2535,2537],{"className":2536,"ariaHidden":112},[111],[81,2538,2540,2543],{"className":2539},[116],[81,2541],{"className":2542,"style":293},[120],[81,2544,2133],{"className":2545,"style":297},[125,126]," degenerates into a \"pending\" vector, which is precisely the \"output stress vector ",[81,2548,2550,2563],{"className":2549,"translate":85},[84],[81,2551,2553],{"className":2552},[89],[91,2554,2555],{"xmlns":93},[95,2556,2557,2561],{},[98,2558,2559],{},[101,2560,2195],{"mathvariant":372},[105,2562,2198],{"encoding":107},[81,2564,2566],{"className":2565,"ariaHidden":112},[111],[81,2567,2569,2572],{"className":2568},[116],[81,2570],{"className":2571,"style":2208},[120],[81,2573,2195],{"className":2574},[125,390],"\" referred to in textbooks;",[451,2577,2578,2581],{},[24,2579,2580],{},"Both slots filled"," (the normal vector + a dual vector \u002F ruler along some direction): the tensor finally emits a scalar, namely the magnitude of the stress component in that direction.",[14,2583,2584,2587],{},[24,2585,2586],{},"Hence \"input a vector, output a vector\" and \"input two arguments, output a scalar\" are not at all contradictory; the former is merely an intermediate state of the latter."," Physics textbooks, privileging intuition, typically exhibit only the \"half-fed\" result; the mathematical definition rigorously describes the ultimate behavior once all slots are filled. Once this layer is understood, virtually all disputes over whether a tensor \"is a mapping or an operation\" dissolve of their own accord.",[14,2589,2590,2591,2619,2620,2648],{},"Why, then, must ",[81,2592,2594,2607],{"className":2593,"translate":85},[84],[81,2595,2597],{"className":2596},[89],[91,2598,2599],{"xmlns":93},[95,2600,2601,2605],{},[98,2602,2603],{},[101,2604,14],{},[105,2606,14],{"encoding":107},[81,2608,2610],{"className":2609,"ariaHidden":112},[111],[81,2611,2613,2616],{"className":2612},[116],[81,2614],{"className":2615,"style":1477},[120],[81,2617,14],{"className":2618},[125,126]," and ",[81,2621,2623,2636],{"className":2622,"translate":85},[84],[81,2624,2626],{"className":2625},[89],[91,2627,2628],{"xmlns":93},[95,2629,2630,2634],{},[98,2631,2632],{},[101,2633,1417],{},[105,2635,1417],{"encoding":107},[81,2637,2639],{"className":2638,"ariaHidden":112},[111],[81,2640,2642,2645],{"className":2641},[116],[81,2643],{"className":2644,"style":1477},[120],[81,2646,1417],{"className":2647,"style":297},[125,126]," be distinguished, rather than merely speaking of the order?",[14,2650,2651,2652,2703,2704,2756],{},"A tensor of type ",[81,2653,2655,2676],{"className":2654,"translate":85},[84],[81,2656,2658],{"className":2657},[89],[91,2659,2660],{"xmlns":93},[95,2661,2662,2674],{},[98,2663,2664,2666,2668,2670,2672],{},[179,2665,410],{"stretchy":409},[515,2667,679],{},[179,2669,1414],{"separator":112},[515,2671,679],{},[179,2673,415],{"stretchy":409},[105,2675,2282],{"encoding":107},[81,2677,2679],{"className":2678,"ariaHidden":112},[111],[81,2680,2682,2685,2688,2691,2694,2697,2700],{"className":2681},[116],[81,2683],{"className":2684,"style":428},[120],[81,2686,410],{"className":2687},[435],[81,2689,679],{"className":2690},[125],[81,2692,1414],{"className":2693},[1441],[81,2695],{"className":2696,"style":1025},[912],[81,2698,679],{"className":2699},[125],[81,2701,415],{"className":2702},[442]," and a tensor of type ",[81,2705,2707,2729],{"className":2706,"translate":85},[84],[81,2708,2710],{"className":2709},[89],[91,2711,2712],{"xmlns":93},[95,2713,2714,2726],{},[98,2715,2716,2718,2720,2722,2724],{},[179,2717,410],{"stretchy":409},[515,2719,513],{},[179,2721,1414],{"separator":112},[515,2723,517],{},[179,2725,415],{"stretchy":409},[105,2727,2728],{"encoding":107},"(0,2)",[81,2730,2732],{"className":2731,"ariaHidden":112},[111],[81,2733,2735,2738,2741,2744,2747,2750,2753],{"className":2734},[116],[81,2736],{"className":2737,"style":428},[120],[81,2739,410],{"className":2740},[435],[81,2742,513],{"className":2743},[125],[81,2745,1414],{"className":2746},[1441],[81,2748],{"className":2749,"style":1025},[912],[81,2751,517],{"className":2752},[125],[81,2754,415],{"className":2755},[442],", when written out as arrays, are both two-dimensional matrices—visually indistinguishable. Yet their behavior under coordinate transformations is altogether different:",[448,2758,2759,2814],{},[451,2760,2761,2762,2813],{},"Type ",[81,2763,2765,2786],{"className":2764,"translate":85},[84],[81,2766,2768],{"className":2767},[89],[91,2769,2770],{"xmlns":93},[95,2771,2772,2784],{},[98,2773,2774,2776,2778,2780,2782],{},[179,2775,410],{"stretchy":409},[515,2777,679],{},[179,2779,1414],{"separator":112},[515,2781,679],{},[179,2783,415],{"stretchy":409},[105,2785,2282],{"encoding":107},[81,2787,2789],{"className":2788,"ariaHidden":112},[111],[81,2790,2792,2795,2798,2801,2804,2807,2810],{"className":2791},[116],[81,2793],{"className":2794,"style":428},[120],[81,2796,410],{"className":2797},[435],[81,2799,679],{"className":2800},[125],[81,2802,1414],{"className":2803},[1441],[81,2805],{"className":2806,"style":1025},[912],[81,2808,679],{"className":2809},[125],[81,2811,415],{"className":2812},[442]," (e.g., linear transformations, the inertia tensor): one upper index and one lower index; under transformation one component is \"contravariant\" and the other \"covariant,\" and the two cancel each other out.",[451,2815,2761,2816,2867],{},[81,2817,2819,2840],{"className":2818,"translate":85},[84],[81,2820,2822],{"className":2821},[89],[91,2823,2824],{"xmlns":93},[95,2825,2826,2838],{},[98,2827,2828,2830,2832,2834,2836],{},[179,2829,410],{"stretchy":409},[515,2831,513],{},[179,2833,1414],{"separator":112},[515,2835,517],{},[179,2837,415],{"stretchy":409},[105,2839,2728],{"encoding":107},[81,2841,2843],{"className":2842,"ariaHidden":112},[111],[81,2844,2846,2849,2852,2855,2858,2861,2864],{"className":2845},[116],[81,2847],{"className":2848,"style":428},[120],[81,2850,410],{"className":2851},[435],[81,2853,513],{"className":2854},[125],[81,2856,1414],{"className":2857},[1441],[81,2859],{"className":2860,"style":1025},[912],[81,2862,517],{"className":2863},[125],[81,2865,415],{"className":2866},[442]," (e.g., the metric tensor): two lower indices; under transformation both components are \"covariant\" and accumulate in the same direction.",[14,2869,2870,2871,2922,2923,2975],{},"If one speaks only of \"order\" while neglecting the ",[81,2872,2874,2895],{"className":2873,"translate":85},[84],[81,2875,2877],{"className":2876},[89],[91,2878,2879],{"xmlns":93},[95,2880,2881,2893],{},[98,2882,2883,2885,2887,2889,2891],{},[179,2884,410],{"stretchy":409},[101,2886,14],{},[179,2888,1414],{"separator":112},[101,2890,1417],{},[179,2892,415],{"stretchy":409},[105,2894,1422],{"encoding":107},[81,2896,2898],{"className":2897,"ariaHidden":112},[111],[81,2899,2901,2904,2907,2910,2913,2916,2919],{"className":2900},[116],[81,2902],{"className":2903,"style":428},[120],[81,2905,410],{"className":2906},[435],[81,2908,14],{"className":2909},[125,126],[81,2911,1414],{"className":2912},[1441],[81,2914],{"className":2915,"style":1025},[912],[81,2917,1417],{"className":2918,"style":297},[125,126],[81,2920,415],{"className":2921},[442]," distinction, one will inevitably misapply formulas when computing operations such as divergence or contraction—and in general relativity, an error of this kind is sufficient to cause a physical law such as \"conservation of energy\" to collapse computationally. This is why mathematicians insist on labeling tensors by their ",[81,2924,2926,2948],{"className":2925,"translate":85},[84],[81,2927,2929],{"className":2928},[89],[91,2930,2931],{"xmlns":93},[95,2932,2933,2945],{},[98,2934,2935,2937,2939,2941,2943],{},[179,2936,410],{"stretchy":409},[101,2938,14],{},[179,2940,1414],{"separator":112},[101,2942,1417],{},[179,2944,415],{"stretchy":409},[105,2946,2947],{"encoding":107},"(p,q)",[81,2949,2951],{"className":2950,"ariaHidden":112},[111],[81,2952,2954,2957,2960,2963,2966,2969,2972],{"className":2953},[116],[81,2955],{"className":2956,"style":428},[120],[81,2958,410],{"className":2959},[435],[81,2961,14],{"className":2962},[125,126],[81,2964,1414],{"className":2965},[1441],[81,2967],{"className":2968,"style":1025},[912],[81,2970,1417],{"className":2971,"style":297},[125,126],[81,2973,415],{"className":2974},[442]," type rather than by the catch-all term \"order.\"",[43,2977,2979],{"id":2978},"the-metric-tensor","The Metric Tensor",[14,2981,2982,2983],{},"One question remains unanswered at this stage: ",[24,2984,2985],{},"vectors and dual vectors reside in two manifestly distinct spaces, yet physicists routinely treat them interchangeably (for instance, speaking of \"lowering an index on a vector\"). What is going on?",[14,2987,2988,2989,2992,2993,3080,3081,1533],{},"In a space equipped with a ",[24,2990,2991],{},"metric tensor"," ",[81,2994,2996,3022],{"className":2995,"translate":85},[84],[81,2997,2999],{"className":2998},[89],[91,3000,3001],{"xmlns":93},[95,3002,3003,3019],{},[98,3004,3005],{},[3006,3007,3008,3011],"msub",{},[101,3009,3010],{},"g",[98,3012,3013,3016],{},[101,3014,3015],{},"i",[101,3017,3018],{},"j",[105,3020,3021],{"encoding":107},"g_{ij}",[81,3023,3025],{"className":3024,"ariaHidden":112},[111],[81,3026,3028,3032],{"className":3027},[116],[81,3029],{"className":3030,"style":3031},[120],"height:0.7167em;vertical-align:-0.2861em;",[81,3033,3035,3038],{"className":3034},[125],[81,3036,3010],{"className":3037,"style":297},[125,126],[81,3039,3041],{"className":3040},[204],[81,3042,3044,3071],{"className":3043},[208,569],[81,3045,3047,3068],{"className":3046},[212],[81,3048,3051],{"className":3049,"style":3050},[216],"height:0.3117em;",[81,3052,3054,3057],{"style":3053},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[81,3055],{"className":3056,"style":224},[223],[81,3058,3060],{"className":3059},[228,229,230,231],[81,3061,3063],{"className":3062},[125,231],[81,3064,3067],{"className":3065,"style":3066},[125,126,231],"margin-right:0.0572em;","ij",[81,3069,606],{"className":3070},[605],[81,3072,3074],{"className":3073},[212],[81,3075,3078],{"className":3076,"style":3077},[216],"height:0.2861em;",[81,3079],{}," (as nearly all physical spaces are), a natural \"translator\" exists, capable of translating a vector into a dual vector, and vice versa. This operation is called ",[24,3082,3083],{},"index raising and lowering",[14,3085,3086],{},"Lowering an index: vector to dual vector",[81,3088,3090],{"className":3089,"translate":85},[809],[81,3091,3093,3129],{"className":3092,"translate":85},[84],[81,3094,3096],{"className":3095},[89],[91,3097,3098],{"xmlns":93,"display":818},[95,3099,3100,3126],{},[98,3101,3102,3108,3110,3120],{},[3006,3103,3104,3106],{},[101,3105,373],{},[101,3107,3015],{},[179,3109,833],{},[3006,3111,3112,3114],{},[101,3113,3010],{},[98,3115,3116,3118],{},[101,3117,3015],{},[101,3119,3018],{},[174,3121,3122,3124],{},[101,3123,373],{},[101,3125,3018],{},[105,3127,3128],{"encoding":107},"v_i = g_{ij} v^j",[81,3130,3132,3189],{"className":3131,"ariaHidden":112},[111],[81,3133,3135,3139,3180,3183,3186],{"className":3134},[116],[81,3136],{"className":3137,"style":3138},[120],"height:0.5806em;vertical-align:-0.15em;",[81,3140,3142,3145],{"className":3141},[125],[81,3143,373],{"className":3144,"style":297},[125,126],[81,3146,3148],{"className":3147},[204],[81,3149,3151,3171],{"className":3150},[208,569],[81,3152,3154,3168],{"className":3153},[212],[81,3155,3157],{"className":3156,"style":3050},[216],[81,3158,3159,3162],{"style":3053},[81,3160],{"className":3161,"style":224},[223],[81,3163,3165],{"className":3164},[228,229,230,231],[81,3166,3015],{"className":3167},[125,126,231],[81,3169,606],{"className":3170},[605],[81,3172,3174],{"className":3173},[212],[81,3175,3178],{"className":3176,"style":3177},[216],"height:0.15em;",[81,3179],{},[81,3181],{"className":3182,"style":913},[912],[81,3184,833],{"className":3185},[917],[81,3187],{"className":3188,"style":913},[912],[81,3190,3192,3196,3239],{"className":3191},[116],[81,3193],{"className":3194,"style":3195},[120],"height:1.1608em;vertical-align:-0.2861em;",[81,3197,3199,3202],{"className":3198},[125],[81,3200,3010],{"className":3201,"style":297},[125,126],[81,3203,3205],{"className":3204},[204],[81,3206,3208,3231],{"className":3207},[208,569],[81,3209,3211,3228],{"className":3210},[212],[81,3212,3214],{"className":3213,"style":3050},[216],[81,3215,3216,3219],{"style":3053},[81,3217],{"className":3218,"style":224},[223],[81,3220,3222],{"className":3221},[228,229,230,231],[81,3223,3225],{"className":3224},[125,231],[81,3226,3067],{"className":3227,"style":3066},[125,126,231],[81,3229,606],{"className":3230},[605],[81,3232,3234],{"className":3233},[212],[81,3235,3237],{"className":3236,"style":3077},[216],[81,3238],{},[81,3240,3242,3245],{"className":3241},[125],[81,3243,373],{"className":3244,"style":297},[125,126],[81,3246,3248],{"className":3247},[204],[81,3249,3251],{"className":3250},[208],[81,3252,3254],{"className":3253},[212],[81,3255,3258],{"className":3256,"style":3257},[216],"height:0.8747em;",[81,3259,3260,3263],{"style":1749},[81,3261],{"className":3262,"style":224},[223],[81,3264,3266],{"className":3265},[228,229,230,231],[81,3267,3018],{"className":3268,"style":3066},[125,126,231],[14,3270,3271],{},"Raising an index: dual vector to vector",[81,3273,3275],{"className":3274,"translate":85},[809],[81,3276,3278,3314],{"className":3277,"translate":85},[84],[81,3279,3281],{"className":3280},[89],[91,3282,3283],{"xmlns":93,"display":818},[95,3284,3285,3311],{},[98,3286,3287,3293,3295,3305],{},[174,3288,3289,3291],{},[101,3290,373],{},[101,3292,3015],{},[179,3294,833],{},[174,3296,3297,3299],{},[101,3298,3010],{},[98,3300,3301,3303],{},[101,3302,3015],{},[101,3304,3018],{},[3006,3306,3307,3309],{},[101,3308,373],{},[101,3310,3018],{},[105,3312,3313],{"encoding":107},"v^i = g^{ij} v_j",[81,3315,3317,3361],{"className":3316,"ariaHidden":112},[111],[81,3318,3320,3323,3352,3355,3358],{"className":3319},[116],[81,3321],{"className":3322,"style":3257},[120],[81,3324,3326,3329],{"className":3325},[125],[81,3327,373],{"className":3328,"style":297},[125,126],[81,3330,3332],{"className":3331},[204],[81,3333,3335],{"className":3334},[208],[81,3336,3338],{"className":3337},[212],[81,3339,3341],{"className":3340,"style":3257},[216],[81,3342,3343,3346],{"style":1749},[81,3344],{"className":3345,"style":224},[223],[81,3347,3349],{"className":3348},[228,229,230,231],[81,3350,3015],{"className":3351},[125,126,231],[81,3353],{"className":3354,"style":913},[912],[81,3356,833],{"className":3357},[917],[81,3359],{"className":3360,"style":913},[912],[81,3362,3364,3367,3399],{"className":3363},[116],[81,3365],{"className":3366,"style":3195},[120],[81,3368,3370,3373],{"className":3369},[125],[81,3371,3010],{"className":3372,"style":297},[125,126],[81,3374,3376],{"className":3375},[204],[81,3377,3379],{"className":3378},[208],[81,3380,3382],{"className":3381},[212],[81,3383,3385],{"className":3384,"style":3257},[216],[81,3386,3387,3390],{"style":1749},[81,3388],{"className":3389,"style":224},[223],[81,3391,3393],{"className":3392},[228,229,230,231],[81,3394,3396],{"className":3395},[125,231],[81,3397,3067],{"className":3398,"style":3066},[125,126,231],[81,3400,3402,3405],{"className":3401},[125],[81,3403,373],{"className":3404,"style":297},[125,126],[81,3406,3408],{"className":3407},[204],[81,3409,3411,3431],{"className":3410},[208,569],[81,3412,3414,3428],{"className":3413},[212],[81,3415,3417],{"className":3416,"style":3050},[216],[81,3418,3419,3422],{"style":3053},[81,3420],{"className":3421,"style":224},[223],[81,3423,3425],{"className":3424},[228,229,230,231],[81,3426,3018],{"className":3427,"style":3066},[125,126,231],[81,3429,606],{"className":3430},[605],[81,3432,3434],{"className":3433},[212],[81,3435,3437],{"className":3436,"style":3077},[216],[81,3438],{},[14,3440,3441,3442,3493],{},"The metric tensor itself is a tensor of type ",[81,3443,3445,3466],{"className":3444,"translate":85},[84],[81,3446,3448],{"className":3447},[89],[91,3449,3450],{"xmlns":93},[95,3451,3452,3464],{},[98,3453,3454,3456,3458,3460,3462],{},[179,3455,410],{"stretchy":409},[515,3457,513],{},[179,3459,1414],{"separator":112},[515,3461,517],{},[179,3463,415],{"stretchy":409},[105,3465,2728],{"encoding":107},[81,3467,3469],{"className":3468,"ariaHidden":112},[111],[81,3470,3472,3475,3478,3481,3484,3487,3490],{"className":3471},[116],[81,3473],{"className":3474,"style":428},[120],[81,3476,410],{"className":3477},[435],[81,3479,513],{"className":3480},[125],[81,3482,1414],{"className":3483},[1441],[81,3485],{"className":3486,"style":1025},[912],[81,3488,517],{"className":3489},[125],[81,3491,415],{"className":3492},[442],"; its role is precisely to \"ingest two vectors and emit their inner product\"—and the inner product itself is what defines length and angle. It is only by virtue of the metric that we can speak of \"the length of a vector,\" a quantity that sounds as though it were an intrinsic property of the vector but in fact cannot be defined without recourse to a dual pairing.",[14,3495,3496],{},"In Euclidean space (such as the familiar Cartesian coordinate system), the metric happens to be the identity matrix, and index raising and lowering \"appear\" to do nothing at all. This is also why, in secondary-school physics, the distinction between vectors and dual vectors is entirely concealed. The moment the coordinate system becomes more involved (polar coordinates, spherical coordinates), or the space itself is curved (the spacetime of general relativity), the metric ceases to be the identity matrix, and the distinction between vectors and dual vectors reasserts itself with decisive importance.",[43,3498,3500],{"id":3499},"the-coordinate-transformation-rule","The Coordinate Transformation Rule",[14,3502,3503,3504,3533,3534,3599,3600,3651],{},"The preceding chapters have all appealed to intuition; here we provide the concrete component transformation rule as an \"anchor.\" Let coordinates transform from ",[81,3505,3507,3521],{"className":3506,"translate":85},[84],[81,3508,3510],{"className":3509},[89],[91,3511,3512],{"xmlns":93},[95,3513,3514,3519],{},[98,3515,3516],{},[101,3517,3518],{},"x",[105,3520,3518],{"encoding":107},[81,3522,3524],{"className":3523,"ariaHidden":112},[111],[81,3525,3527,3530],{"className":3526},[116],[81,3528],{"className":3529,"style":293},[120],[81,3531,3518],{"className":3532},[125,126]," to ",[81,3535,3537,3557],{"className":3536,"translate":85},[84],[81,3538,3540],{"className":3539},[89],[91,3541,3542],{"xmlns":93},[95,3543,3544,3554],{},[98,3545,3546],{},[174,3547,3548,3550],{},[101,3549,3518],{},[179,3551,3553],{"mathvariant":1133,"lspace":3552,"rspace":3552},"0em","′",[105,3555,3556],{"encoding":107},"x'",[81,3558,3560],{"className":3559,"ariaHidden":112},[111],[81,3561,3563,3567],{"className":3562},[116],[81,3564],{"className":3565,"style":3566},[120],"height:0.7519em;",[81,3568,3570,3573],{"className":3569},[125],[81,3571,3518],{"className":3572},[125,126],[81,3574,3576],{"className":3575},[204],[81,3577,3579],{"className":3578},[208],[81,3580,3582],{"className":3581},[212],[81,3583,3585],{"className":3584,"style":3566},[216],[81,3586,3587,3590],{"style":219},[81,3588],{"className":3589,"style":224},[223],[81,3591,3593],{"className":3592},[228,229,230,231],[81,3594,3596],{"className":3595},[125,231],[81,3597,3553],{"className":3598},[125,231],". The component transformation rule for a tensor of type ",[81,3601,3603,3624],{"className":3602,"translate":85},[84],[81,3604,3606],{"className":3605},[89],[91,3607,3608],{"xmlns":93},[95,3609,3610,3622],{},[98,3611,3612,3614,3616,3618,3620],{},[179,3613,410],{"stretchy":409},[101,3615,14],{},[179,3617,1414],{"separator":112},[101,3619,1417],{},[179,3621,415],{"stretchy":409},[105,3623,2947],{"encoding":107},[81,3625,3627],{"className":3626,"ariaHidden":112},[111],[81,3628,3630,3633,3636,3639,3642,3645,3648],{"className":3629},[116],[81,3631],{"className":3632,"style":428},[120],[81,3634,410],{"className":3635},[435],[81,3637,14],{"className":3638},[125,126],[81,3640,1414],{"className":3641},[1441],[81,3643],{"className":3644,"style":1025},[912],[81,3646,1417],{"className":3647,"style":297},[125,126],[81,3649,415],{"className":3650},[442]," is:",[81,3653,3655],{"className":3654,"translate":85},[809],[81,3656,3658,3901],{"className":3657,"translate":85},[84],[81,3659,3661],{"className":3660},[89],[91,3662,3663],{"xmlns":93,"display":818},[95,3664,3665,3898],{},[98,3666,3667,3711,3713,3749,3751,3785,3787,3822,3824,3858,3860],{},[3668,3669,3670,3672,3691],"msubsup",{},[101,3671,1379],{},[98,3673,3674,3677,3683,3685],{},[2035,3675,3676],{},"  ",[3006,3678,3679,3681],{},[101,3680,3018],{},[515,3682,679],{},[179,3684,1551],{},[3006,3686,3687,3689],{},[101,3688,3018],{},[101,3690,1417],{},[98,3692,3693,3695],{},[179,3694,3553],{"mathvariant":1133},[98,3696,3697,3703,3705],{},[3006,3698,3699,3701],{},[101,3700,3015],{},[515,3702,679],{},[179,3704,1551],{},[3006,3706,3707,3709],{},[101,3708,3015],{},[101,3710,14],{},[179,3712,833],{},[1199,3714,3715,3734],{},[98,3716,3717,3720],{},[101,3718,3719],{"mathvariant":1133},"∂",[174,3721,3722,3724],{},[101,3723,3518],{},[98,3725,3726,3728],{},[179,3727,3553],{"mathvariant":1133},[3006,3729,3730,3732],{},[101,3731,3015],{},[515,3733,679],{},[98,3735,3736,3738],{},[101,3737,3719],{"mathvariant":1133},[174,3739,3740,3742],{},[101,3741,3518],{},[3006,3743,3744,3747],{},[101,3745,3746],{},"k",[515,3748,679],{},[179,3750,1551],{},[1199,3752,3753,3771],{},[98,3754,3755,3757],{},[101,3756,3719],{"mathvariant":1133},[174,3758,3759,3761],{},[101,3760,3518],{},[98,3762,3763,3765],{},[179,3764,3553],{"mathvariant":1133},[3006,3766,3767,3769],{},[101,3768,3015],{},[101,3770,14],{},[98,3772,3773,3775],{},[101,3774,3719],{"mathvariant":1133},[174,3776,3777,3779],{},[101,3778,3518],{},[3006,3780,3781,3783],{},[101,3782,3746],{},[101,3784,14],{},[179,3786,2372],{},[1199,3788,3789,3804],{},[98,3790,3791,3793],{},[101,3792,3719],{"mathvariant":1133},[174,3794,3795,3797],{},[101,3796,3518],{},[3006,3798,3799,3802],{},[101,3800,3801],{},"l",[515,3803,679],{},[98,3805,3806,3808],{},[101,3807,3719],{"mathvariant":1133},[174,3809,3810,3812],{},[101,3811,3518],{},[98,3813,3814,3816],{},[179,3815,3553],{"mathvariant":1133},[3006,3817,3818,3820],{},[101,3819,3018],{},[515,3821,679],{},[179,3823,1551],{},[1199,3825,3826,3840],{},[98,3827,3828,3830],{},[101,3829,3719],{"mathvariant":1133},[174,3831,3832,3834],{},[101,3833,3518],{},[3006,3835,3836,3838],{},[101,3837,3801],{},[101,3839,1417],{},[98,3841,3842,3844],{},[101,3843,3719],{"mathvariant":1133},[174,3845,3846,3848],{},[101,3847,3518],{},[98,3849,3850,3852],{},[179,3851,3553],{"mathvariant":1133},[3006,3853,3854,3856],{},[101,3855,3018],{},[101,3857,1417],{},[179,3859,2372],{},[3668,3861,3862,3864,3882],{},[101,3863,1379],{},[98,3865,3866,3868,3874,3876],{},[2035,3867,3676],{},[3006,3869,3870,3872],{},[101,3871,3801],{},[515,3873,679],{},[179,3875,1551],{},[3006,3877,3878,3880],{},[101,3879,3801],{},[101,3881,1417],{},[98,3883,3884,3890,3892],{},[3006,3885,3886,3888],{},[101,3887,3746],{},[515,3889,679],{},[179,3891,1551],{},[3006,3893,3894,3896],{},[101,3895,3746],{},[101,3897,14],{},[105,3899,3900],{"encoding":107},"T'^{i_1 \\cdots i_p}_{\\ \\ j_1 \\cdots j_q} = \\frac{\\partial x'^{i_1}}{\\partial x^{k_1}} \\cdots \\frac{\\partial x'^{i_p}}{\\partial x^{k_p}} \\cdot \\frac{\\partial x^{l_1}}{\\partial x'^{j_1}} \\cdots \\frac{\\partial x^{l_q}}{\\partial x'^{j_q}} \\cdot T^{k_1 \\cdots k_p}_{\\ \\ l_1 \\cdots 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56,5159],{"className":5155},[125,231],[81,5157,3801],{"className":5158,"style":4785},[125,126,231],[81,5160,5162],{"className":5161},[204],[81,5163,5165,5185],{"className":5164},[208,569],[81,5166,5168,5182],{"className":5167},[212],[81,5169,5171],{"className":5170,"style":4021},[216],[81,5172,5173,5176],{"style":4800},[81,5174],{"className":5175,"style":3980},[223],[81,5177,5179],{"className":5178},[228,3984,711,231],[81,5180,1417],{"className":5181,"style":297},[125,126,231],[81,5183,606],{"className":5184},[605],[81,5186,5188],{"className":5187},[212],[81,5189,5191],{"className":5190,"style":4042},[216],[81,5192],{},[81,5194,5195,5198],{"style":4047},[81,5196],{"className":5197,"style":224},[223],[81,5199,5201],{"className":5200},[228,229,230,231],[81,5202,5204,5244,5247],{"className":5203},[125,231],[81,5205,5207,5210],{"className":5206},[125,231],[81,5208,3746],{"className":5209,"style":4244},[125,126,231],[81,5211,5213],{"className":5212},[204],[81,5214,5216,5236],{"className":5215},[208,569],[81,5217,5219,5233],{"className":5218},[212],[81,5220,5222],{"className":5221,"style":3973},[216],[81,5223,5224,5227],{"style":4259},[81,5225],{"className":5226,"style":3980},[223],[81,5228,5230],{"className":5229},[228,3984,711,231],[81,5231,679],{"className":5232},[125,231],[81,5234,606],{"className":5235},[605],[81,5237,5239],{"className":5238},[212],[81,5240,5242],{"className":5241,"style":3997},[216],[81,5243],{},[81,5245,1551],{"className":5246},[546,231],[81,5248,5250,5253],{"className":5249},[125,231],[81,5251,3746],{"className":5252,"style":4244},[125,126,231],[81,5254,5256],{"className":5255},[204],[81,5257,5259,5279],{"className":5258},[208,569],[81,5260,5262,5276],{"className":5261},[212],[81,5263,5265],{"className":5264,"style":4021},[216],[81,5266,5267,5270],{"style":4259},[81,5268],{"className":5269,"style":3980},[223],[81,5271,5273],{"className":5272},[228,3984,711,231],[81,5274,14],{"className":5275},[125,126,231],[81,5277,606],{"className":5278},[605],[81,5280,5282],{"className":5281},[212],[81,5283,5285],{"className":5284,"style":4042},[216],[81,5286],{},[81,5288,606],{"className":5289},[605],[81,5291,5293],{"className":5292},[212],[81,5294,5297],{"className":5295,"style":5296},[216],"height:0.4986em;",[81,5298],{},[14,5300,5301],{},"There is no need to be intimidated by this formula; it does nothing more than reiterate a single point:",[448,5303,5304,5461],{},[451,5305,5306,5307,5310,5311,5460],{},"Every ",[24,5308,5309],{},"upper index"," (corresponding to a \"contravariant\" component of a vector slot) is multiplied by a Jacobian factor of the form ",[81,5312,5314,5344],{"className":5313,"translate":85},[84],[81,5315,5317],{"className":5316},[89],[91,5318,5319],{"xmlns":93},[95,5320,5321,5341],{},[98,5322,5323],{},[1199,5324,5325,5335],{},[98,5326,5327,5329],{},[101,5328,3719],{"mathvariant":1133},[174,5330,5331,5333],{},[101,5332,3518],{},[179,5334,3553],{"mathvariant":1133,"lspace":3552,"rspace":3552},[98,5336,5337,5339],{},[101,5338,3719],{"mathvariant":1133},[101,5340,3518],{},[105,5342,5343],{"encoding":107},"\\frac{\\partial x'}{\\partial x}",[81,5345,5347],{"className":5346,"ariaHidden":112},[111],[81,5348,5350,5354],{"className":5349},[116],[81,5351],{"className":5352,"style":5353},[120],"height:1.3185em;vertical-align:-0.345em;",[81,5355,5357,5360,5457],{"className":5356},[125],[81,5358],{"className":5359},[435,1247],[81,5361,5363],{"className":5362},[1199],[81,5364,5366,5449],{"className":5365},[208,569],[81,5367,5369,5446],{"className":5368},[212],[81,5370,5373,5390,5398],{"className":5371,"style":5372},[216],"height:0.9735em;",[81,5374,5375,5378],{"style":1263},[81,5376],{"className":5377,"style":583},[223],[81,5379,5381],{"className":5380},[228,229,230,231],[81,5382,5384,5387],{"className":5383},[125,231],[81,5385,3719],{"className":5386,"style":4206},[125,231],[81,5388,3518],{"className":5389},[125,126,231],[81,5391,5392,5395],{"style":1278},[81,5393],{"className":5394,"style":583},[223],[81,5396],{"className":5397,"style":1286},[1285],[81,5399,5400,5403],{"style":1289},[81,5401],{"className":5402,"style":583},[223],[81,5404,5406],{"className":5405},[228,229,230,231],[81,5407,5409,5412],{"className":5408},[125,231],[81,5410,3719],{"className":5411,"style":4206},[125,231],[81,5413,5415,5418],{"className":5414},[125,231],[81,5416,3518],{"className":5417},[125,126,231],[81,5419,5421],{"className":5420},[204],[81,5422,5424],{"className":5423},[208],[81,5425,5427],{"className":5426},[212],[81,5428,5431],{"className":5429,"style":5430},[216],"height:0.8278em;",[81,5432,5434,5437],{"style":5433},"top:-2.931em;margin-right:0.0714em;",[81,5435],{"className":5436,"style":3980},[223],[81,5438,5440],{"className":5439},[228,3984,711,231],[81,5441,5443],{"className":5442},[125,231],[81,5444,3553],{"className":5445},[125,231],[81,5447,606],{"className":5448},[605],[81,5450,5452],{"className":5451},[212],[81,5453,5455],{"className":5454,"style":1311},[216],[81,5456],{},[81,5458],{"className":5459},[442,1247],";",[451,5462,5306,5463,5466,5467,5616],{},[24,5464,5465],{},"lower index"," (corresponding to a \"covariant\" component of a dual-vector slot) is multiplied by a Jacobian factor of the form ",[81,5468,5470,5500],{"className":5469,"translate":85},[84],[81,5471,5473],{"className":5472},[89],[91,5474,5475],{"xmlns":93},[95,5476,5477,5497],{},[98,5478,5479],{},[1199,5480,5481,5487],{},[98,5482,5483,5485],{},[101,5484,3719],{"mathvariant":1133},[101,5486,3518],{},[98,5488,5489,5491],{},[101,5490,3719],{"mathvariant":1133},[174,5492,5493,5495],{},[101,5494,3518],{},[179,5496,3553],{"mathvariant":1133,"lspace":3552,"rspace":3552},[105,5498,5499],{"encoding":107},"\\frac{\\partial x}{\\partial x'}",[81,5501,5503],{"className":5502,"ariaHidden":112},[111],[81,5504,5506,5510],{"className":5505},[116],[81,5507],{"className":5508,"style":5509},[120],"height:1.2251em;vertical-align:-0.345em;",[81,5511,5513,5516,5613],{"className":5512},[125],[81,5514],{"className":5515},[435,1247],[81,5517,5519],{"className":5518},[1199],[81,5520,5522,5605],{"className":5521},[208,569],[81,5523,5525,5602],{"className":5524},[212],[81,5526,5529,5577,5585],{"className":5527,"style":5528},[216],"height:0.8801em;",[81,5530,5531,5534],{"style":1263},[81,5532],{"className":5533,"style":583},[223],[81,5535,5537],{"className":5536},[228,229,230,231],[81,5538,5540,5543],{"className":5539},[125,231],[81,5541,3719],{"className":5542,"style":4206},[125,231],[81,5544,5546,5549],{"className":5545},[125,231],[81,5547,3518],{"className":5548},[125,126,231],[81,5550,5552],{"className":5551},[204],[81,5553,5555],{"className":5554},[208],[81,5556,5558],{"className":5557},[212],[81,5559,5562],{"className":5560,"style":5561},[216],"height:0.6828em;",[81,5563,5565,5568],{"style":5564},"top:-2.786em;margin-right:0.0714em;",[81,5566],{"className":5567,"style":3980},[223],[81,5569,5571],{"className":5570},[228,3984,711,231],[81,5572,5574],{"className":5573},[125,231],[81,5575,3553],{"className":5576},[125,231],[81,5578,5579,5582],{"style":1278},[81,5580],{"className":5581,"style":583},[223],[81,5583],{"className":5584,"style":1286},[1285],[81,5586,5587,5590],{"style":1289},[81,5588],{"className":5589,"style":583},[223],[81,5591,5593],{"className":5592},[228,229,230,231],[81,5594,5596,5599],{"className":5595},[125,231],[81,5597,3719],{"className":5598,"style":4206},[125,231],[81,5600,3518],{"className":5601},[125,126,231],[81,5603,606],{"className":5604},[605],[81,5606,5608],{"className":5607},[212],[81,5609,5611],{"className":5610,"style":1311},[216],[81,5612],{},[81,5614],{"className":5615},[442,1247]," (which is precisely the inverse of the former).",[14,5618,5619],{},"This is the rigorous mathematical version of the intuitive example from Chapter 2—\"centimeters to meters\": the Jacobian factors for upper and lower indices are mutual inverses; when paired and contracted they cancel one another exactly, thereby ensuring that the tensor as a whole (qua geometric object) remains invariant, with only its components relative to a particular coordinate system undergoing change.",[43,5621,5623],{"id":5622},"common-tensors","Common Tensors",[14,5625,5626],{},"Below are examples of tensors encountered across a variety of domains.",[5628,5629,5630,5708],"table",{},[5631,5632,5633],"thead",{},[5634,5635,5636,5640,5671,5702,5705],"tr",{},[5637,5638,5639],"th",{},"Tensor type",[5637,5641,5642,5670],{},[81,5643,5645,5658],{"className":5644,"translate":85},[84],[81,5646,5648],{"className":5647},[89],[91,5649,5650],{"xmlns":93},[95,5651,5652,5656],{},[98,5653,5654],{},[101,5655,14],{},[105,5657,14],{"encoding":107},[81,5659,5661],{"className":5660,"ariaHidden":112},[111],[81,5662,5664,5667],{"className":5663},[116],[81,5665],{"className":5666,"style":1477},[120],[81,5668,14],{"className":5669},[125,126]," (dual-vector slots)",[5637,5672,5673,5701],{},[81,5674,5676,5689],{"className":5675,"translate":85},[84],[81,5677,5679],{"className":5678},[89],[91,5680,5681],{"xmlns":93},[95,5682,5683,5687],{},[98,5684,5685],{},[101,5686,1417],{},[105,5688,1417],{"encoding":107},[81,5690,5692],{"className":5691,"ariaHidden":112},[111],[81,5693,5695,5698],{"className":5694},[116],[81,5696],{"className":5697,"style":1477},[120],[81,5699,1417],{"className":5700,"style":297},[125,126]," (vector slots)",[5637,5703,5704],{},"Total order",[5637,5706,5707],{},"Canonical example",[5709,5710,5711,5726,5740,5754,5768,6000],"tbody",{},[5634,5712,5713,5717,5719,5721,5723],{},[5714,5715,5716],"td",{},"Scalar",[5714,5718,513],{},[5714,5720,513],{},[5714,5722,513],{},[5714,5724,5725],{},"Temperature, mass",[5634,5727,5728,5731,5733,5735,5737],{},[5714,5729,5730],{},"Vector",[5714,5732,513],{},[5714,5734,679],{},[5714,5736,679],{},[5714,5738,5739],{},"Velocity, force",[5634,5741,5742,5745,5747,5749,5751],{},[5714,5743,5744],{},"Dual vector (1-form)",[5714,5746,679],{},[5714,5748,513],{},[5714,5750,679],{},[5714,5752,5753],{},"Gradient, row vector",[5634,5755,5756,5759,5761,5763,5765],{},[5714,5757,5758],{},"Linear transformation \u002F inertia tensor",[5714,5760,679],{},[5714,5762,679],{},[5714,5764,517],{},[5714,5766,5767],{},"Moment of inertia, stress tensor",[5634,5769,5770,5773,5775,5777,5779],{},[5714,5771,5772],{},"Metric tensor",[5714,5774,513],{},[5714,5776,517],{},[5714,5778,517],{},[5714,5780,5781,5782],{},"Length definition ",[81,5783,5785,5835],{"className":5784,"translate":85},[84],[81,5786,5788],{"className":5787},[89],[91,5789,5790],{"xmlns":93},[95,5791,5792,5832],{},[98,5793,5794,5797,5804,5806,5816,5818,5824,5826],{},[101,5795,5796],{},"d",[174,5798,5799,5802],{},[101,5800,5801],{},"s",[515,5803,517],{},[179,5805,833],{},[3006,5807,5808,5810],{},[101,5809,3010],{},[98,5811,5812,5814],{},[101,5813,3015],{},[101,5815,3018],{},[101,5817,5796],{},[174,5819,5820,5822],{},[101,5821,3518],{},[101,5823,3015],{},[101,5825,5796],{},[174,5827,5828,5830],{},[101,5829,3518],{},[101,5831,3018],{},[105,5833,5834],{"encoding":107},"ds^2 = g_{ij}dx^i dx^j",[81,5836,5838,5886],{"className":5837,"ariaHidden":112},[111],[81,5839,5841,5845,5848,5877,5880,5883],{"className":5840},[116],[81,5842],{"className":5843,"style":5844},[120],"height:0.8141em;",[81,5846,5796],{"className":5847},[125,126],[81,5849,5851,5854],{"className":5850},[125],[81,5852,5801],{"className":5853},[125,126],[81,5855,5857],{"className":5856},[204],[81,5858,5860],{"className":5859},[208],[81,5861,5863],{"className":5862},[212],[81,5864,5866],{"className":5865,"style":5844},[216],[81,5867,5868,5871],{"style":219},[81,5869],{"className":5870,"style":224},[223],[81,5872,5874],{"className":5873},[228,229,230,231],[81,5875,517],{"className":5876},[125,231],[81,5878],{"className":5879,"style":913},[912],[81,5881,833],{"className":5882},[917],[81,5884],{"className":5885,"style":913},[912],[81,5887,5889,5893,5936,5939,5968,5971],{"className":5888},[116],[81,5890],{"className":5891,"style":5892},[120],"height:1.1108em;vertical-align:-0.2861em;",[81,5894,5896,5899],{"className":5895},[125],[81,5897,3010],{"className":5898,"style":297},[125,126],[81,5900,5902],{"className":5901},[204],[81,5903,5905,5928],{"className":5904},[208,569],[81,5906,5908,5925],{"className":5907},[212],[81,5909,5911],{"className":5910,"style":3050},[216],[81,5912,5913,5916],{"style":3053},[81,5914],{"className":5915,"style":224},[223],[81,5917,5919],{"className":5918},[228,229,230,231],[81,5920,5922],{"className":5921},[125,231],[81,5923,3067],{"className":5924,"style":3066},[125,126,231],[81,5926,606],{"className":5927},[605],[81,5929,5931],{"className":5930},[212],[81,5932,5934],{"className":5933,"style":3077},[216],[81,5935],{},[81,5937,5796],{"className":5938},[125,126],[81,5940,5942,5945],{"className":5941},[125],[81,5943,3518],{"className":5944},[125,126],[81,5946,5948],{"className":5947},[204],[81,5949,5951],{"className":5950},[208],[81,5952,5954],{"className":5953},[212],[81,5955,5957],{"className":5956,"style":4318},[216],[81,5958,5959,5962],{"style":219},[81,5960],{"className":5961,"style":224},[223],[81,5963,5965],{"className":5964},[228,229,230,231],[81,5966,3015],{"className":5967},[125,126,231],[81,5969,5796],{"className":5970},[125,126],[81,5972,5974,5977],{"className":5973},[125],[81,5975,3518],{"className":5976},[125,126],[81,5978,5980],{"className":5979},[204],[81,5981,5983],{"className":5982},[208],[81,5984,5986],{"className":5985},[212],[81,5987,5989],{"className":5988,"style":4318},[216],[81,5990,5991,5994],{"style":219},[81,5992],{"className":5993,"style":224},[223],[81,5995,5997],{"className":5996},[228,229,230,231],[81,5998,3018],{"className":5999,"style":3066},[125,126,231],[5634,6001,6002,6005,6007,6009,6011],{},[5714,6003,6004],{},"Riemann curvature tensor",[5714,6006,679],{},[5714,6008,526],{},[5714,6010,686],{},[5714,6012,6013],{},"Spacetime curvature in general relativity",[43,6015,6017],{"id":6016},"two-registers-and-their-rules","Two Registers and Their Rules",[14,6019,6020,6021,6024],{},"A final clarification is in order: ",[24,6022,6023],{},"the word \"tensor\" as used in deep learning and as used in physics \u002F differential geometry refers to concepts at two distinct levels of abstraction",", and this is a fertile source of confusion.",[448,6026,6027,6037],{},[451,6028,6029,6032,6033,6036],{},[24,6030,6031],{},"In deep learning",": a ",[54,6034,6035],{},"tensor"," is simply a multidimensional array; the sole concern is the efficiency of parallel computation. There is no such thing as \"invariance under coordinate transformation,\" nor any distinction between covariance and contravariance. Here, understanding a tensor as a \"generalized array\" is entirely sufficient; there is no need to overburden the term with philosophical weight.",[451,6038,6039,6042],{},[24,6040,6041],{},"In physics and differential geometry",": a tensor is first and foremost a geometric or physical object that does not depend on any choice of coordinates; an array is merely its \"provisional snapshot\" with respect to a particular coordinate basis. Covariance, contravariance, the metric, contraction—these concepts exist for one purpose and one purpose only: to guarantee that the physical quantities computed from these provisional snapshots (length, energy, curvature) are objective, absolute, and invariant under changes of the observer.",[14,6044,6045,6046,6049],{},"Neither register is \"right\" or \"wrong\"; they serve radically different ends. Genuine confusion tends to arise from ",[24,6047,6048],{},"using deep-learning intuition to understand the tensors of general relativity",", or vice versa. Once one recognizes that these are two distinct enterprises, much of the frustration that takes the form of \"the textbook explains this poorly\" dissolves of its own accord.",[43,6051,6053],{"id":6052},"conclusion","Conclusion",[14,6055,6056],{},"Let us return to the original question: what, after all, is a tensor?",[14,6058,6059],{},"It is a machine furnished with a certain number of \"slots\": some slots await \"arrows\" (vectors), others await \"rulers\" (dual vectors). When every slot has been filled, the number it emits is forever the same—regardless of the coordinate system from which you observe it, regardless of the units in which you measure. The metric tensor, for its part, is the bridge connecting the world of vectors to the world of dual vectors, making it possible to define rigorously a notion as seemingly innocent as \"length.\"",[14,6061,6062],{},"The next time you write down a lower index or an upper index, you might pause to consider: is this position waiting for an arrow, or is it waiting for a ruler?",{"title":6064,"searchDepth":6065,"depth":6065,"links":6066},"",2,[6067,6068,6069,6070,6071,6072,6073,6074,6075],{"id":45,"depth":6065,"text":46},{"id":139,"depth":6065,"text":140},{"id":1353,"depth":6065,"text":1354},{"id":2109,"depth":6065,"text":2110},{"id":2978,"depth":6065,"text":2979},{"id":3499,"depth":6065,"text":3500},{"id":5622,"depth":6065,"text":5623},{"id":6016,"depth":6065,"text":6017},{"id":6052,"depth":6065,"text":6053},false,true,"md",null,{},"2026-05-30","\u002Fblog\u002F2026\u002F2026-05-30-about-the-tensor",{"title":6,"description":6064},"blog\u002F2026\u002F2026-05-30-about-the-tensor","Vectors are arrows, dual vectors are rulers—their pairing yields invariant scalars. Tensors are multi-slot linear machines with fillable slots, resolving the vector-in\u002Fvector-out vs. scalar-out paradox. The metric tensor bridges the two, keeping lengths and energies coordinate-invariant.",[6087,6088],"mathematics","machine-learning","G5WPwX-MIBzvMW5I_TliZgVzNDr3X4aovgq1B1upyOI",{"id":6091,"title":6092,"body":6093,"description":6064,"draft":6076,"enableComment":6077,"extension":6078,"image":6064,"important":6076,"location":6079,"meta":21133,"navigation":6077,"ogImage":6079,"onday":21134,"path":21135,"seo":21136,"stem":21137,"summary":21138,"tags":21139,"__hash__":21140},"blog\u002Fblog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory.md","Proof of the UAT, by Weierstrass Theory",{"type":8,"value":6094,"toc":21124},[6095,6099,6104,6107,6201,6285,6288,6345,6394,6589,6592,6597,6600,7127,7130,7133,7349,7380,7383,7486,7489,7495,7498,7937,7940,8042,8045,8246,8249,8566,8569,8572,8575,8732,8735,9056,9059,9367,9370,9543,9590,9593,9790,10096,10098,10235,10266,10322,10485,10488,10701,10703,11042,11045,11049,11216,11303,11306,11491,11494,11887,11890,12077,12080,12233,12236,12425,12565,12800,12903,13057,13234,13404,13407,13848,13850,14152,14184,14411,14413,14590,14621,14624,14893,14895,15005,15008,15011,15015,15018,15175,15510,15513,15698,15730,15735,15782,15981,16247,16312,16417,16420,16743,16791,16985,16988,17355,17358,17540,17797,17859,17867,17872,18248,18518,18521,18773,18776,19429,19434,19437,19686,19689,20031,20034,20037,20041,20044,20047,20591,20594,20897,20900,21121],[43,6096,6098],{"id":6097},"weierstrass-theorem","Weierstrass Theorem",[6100,6101,6103],"h3",{"id":6102},"intuitive-explanation","Intuitive Explanation",[14,6105,6106],{},"Suppose",[81,6108,6110],{"className":6109,"translate":85},[809],[81,6111,6113,6148],{"className":6112,"translate":85},[84],[81,6114,6116],{"className":6115},[89],[91,6117,6118],{"xmlns":93,"display":818},[95,6119,6120,6145],{},[98,6121,6122,6125,6128,6131,6133,6135,6137,6139,6141,6143],{},[101,6123,6124],{},"f",[179,6126,6127],{},"∈",[101,6129,6130],{},"C",[179,6132,410],{"stretchy":409},[179,6134,497],{"stretchy":409},[515,6136,513],{},[179,6138,1414],{"separator":112},[515,6140,679],{},[179,6142,529],{"stretchy":409},[179,6144,415],{"stretchy":409},[105,6146,6147],{"encoding":107},"f\\in C([0,1])",[81,6149,6151,6171],{"className":6150,"ariaHidden":112},[111],[81,6152,6154,6158,6162,6165,6168],{"className":6153},[116],[81,6155],{"className":6156,"style":6157},[120],"height:0.8889em;vertical-align:-0.1944em;",[81,6159,6124],{"className":6160,"style":6161},[125,126],"margin-right:0.1076em;",[81,6163],{"className":6164,"style":913},[912],[81,6166,6127],{"className":6167},[917],[81,6169],{"className":6170,"style":913},[912],[81,6172,6174,6177,6181,6185,6188,6191,6194,6197],{"className":6173},[116],[81,6175],{"className":6176,"style":428},[120],[81,6178,6130],{"className":6179,"style":6180},[125,126],"margin-right:0.0715em;",[81,6182,6184],{"className":6183},[435],"([",[81,6186,513],{"className":6187},[125],[81,6189,1414],{"className":6190},[1441],[81,6192],{"className":6193,"style":1025},[912],[81,6195,679],{"className":6196},[125],[81,6198,6200],{"className":6199},[442],"])",[14,6202,6203,6204,6232,6233,65],{},"i.e., ",[81,6205,6207,6220],{"className":6206,"translate":85},[84],[81,6208,6210],{"className":6209},[89],[91,6211,6212],{"xmlns":93},[95,6213,6214,6218],{},[98,6215,6216],{},[101,6217,6124],{},[105,6219,6124],{"encoding":107},[81,6221,6223],{"className":6222,"ariaHidden":112},[111],[81,6224,6226,6229],{"className":6225},[116],[81,6227],{"className":6228,"style":6157},[120],[81,6230,6124],{"className":6231,"style":6161},[125,126]," is continuous on ",[81,6234,6236,6258],{"className":6235,"translate":85},[84],[81,6237,6239],{"className":6238},[89],[91,6240,6241],{"xmlns":93},[95,6242,6243,6255],{},[98,6244,6245,6247,6249,6251,6253],{},[179,6246,497],{"stretchy":409},[515,6248,513],{},[179,6250,1414],{"separator":112},[515,6252,679],{},[179,6254,529],{"stretchy":409},[105,6256,6257],{"encoding":107},"[0,1]",[81,6259,6261],{"className":6260,"ariaHidden":112},[111],[81,6262,6264,6267,6270,6273,6276,6279,6282],{"className":6263},[116],[81,6265],{"className":6266,"style":428},[120],[81,6268,497],{"className":6269},[435],[81,6271,513],{"className":6272},[125],[81,6274,1414],{"className":6275},[1441],[81,6277],{"className":6278,"style":1025},[912],[81,6280,679],{"className":6281},[125],[81,6283,529],{"className":6284},[442],[14,6286,6287],{},"Then for any",[81,6289,6291],{"className":6290,"translate":85},[809],[81,6292,6294,6314],{"className":6293,"translate":85},[84],[81,6295,6297],{"className":6296},[89],[91,6298,6299],{"xmlns":93,"display":818},[95,6300,6301,6311],{},[98,6302,6303,6306,6309],{},[101,6304,6305],{},"ε",[179,6307,6308],{},">",[515,6310,513],{},[105,6312,6313],{"encoding":107},"\\varepsilon>0",[81,6315,6317,6336],{"className":6316,"ariaHidden":112},[111],[81,6318,6320,6324,6327,6330,6333],{"className":6319},[116],[81,6321],{"className":6322,"style":6323},[120],"height:0.5782em;vertical-align:-0.0391em;",[81,6325,6305],{"className":6326},[125,126],[81,6328],{"className":6329,"style":913},[912],[81,6331,6308],{"className":6332},[917],[81,6334],{"className":6335,"style":913},[912],[81,6337,6339,6342],{"className":6338},[116],[81,6340],{"className":6341,"style":1107},[120],[81,6343,513],{"className":6344},[125],[14,6346,6347,6348,6393],{},"there exists a polynomial ",[81,6349,6351,6372],{"className":6350,"translate":85},[84],[81,6352,6354],{"className":6353},[89],[91,6355,6356],{"xmlns":93},[95,6357,6358,6369],{},[98,6359,6360,6363,6365,6367],{},[101,6361,6362],{},"P",[179,6364,410],{"stretchy":409},[101,6366,3518],{},[179,6368,415],{"stretchy":409},[105,6370,6371],{"encoding":107},"P(x)",[81,6373,6375],{"className":6374,"ariaHidden":112},[111],[81,6376,6378,6381,6384,6387,6390],{"className":6377},[116],[81,6379],{"className":6380,"style":428},[120],[81,6382,6362],{"className":6383,"style":1394},[125,126],[81,6385,410],{"className":6386},[435],[81,6388,3518],{"className":6389},[125,126],[81,6391,415],{"className":6392},[442]," such that",[81,6395,6397],{"className":6396,"translate":85},[809],[81,6398,6400,6465],{"className":6399,"translate":85},[84],[81,6401,6403],{"className":6402},[89],[91,6404,6405],{"xmlns":93,"display":818},[95,6406,6407,6462],{},[98,6408,6409,6412,6414,6416,6418,6420,6423,6425,6427,6429,6431,6433,6436,6438,6440,6443,6446,6448,6450,6452,6454,6456,6458,6460],{},[101,6410,6411],{"mathvariant":1133},"∣",[101,6413,6124],{},[179,6415,410],{"stretchy":409},[101,6417,3518],{},[179,6419,415],{"stretchy":409},[179,6421,6422],{},"−",[101,6424,6362],{},[179,6426,410],{"stretchy":409},[101,6428,3518],{},[179,6430,415],{"stretchy":409},[101,6432,6411],{"mathvariant":1133},[179,6434,6435],{},"\u003C",[101,6437,6305],{},[179,6439,1414],{"separator":112},[912,6441],{"width":6442},"2em",[101,6444,6445],{"mathvariant":1133},"∀",[101,6447,3518],{},[179,6449,6127],{},[179,6451,497],{"stretchy":409},[515,6453,513],{},[179,6455,1414],{"separator":112},[515,6457,679],{},[179,6459,529],{"stretchy":409},[101,6461,65],{"mathvariant":1133},[105,6463,6464],{"encoding":107},"|f(x)-P(x)|\u003C\\varepsilon,\n\\qquad\n\\forall x\\in [0,1].",[81,6466,6468,6498,6528,6562],{"className":6467,"ariaHidden":112},[111],[81,6469,6471,6474,6477,6480,6483,6486,6489,6492,6495],{"className":6470},[116],[81,6472],{"className":6473,"style":428},[120],[81,6475,6411],{"className":6476},[125],[81,6478,6124],{"className":6479,"style":6161},[125,126],[81,6481,410],{"className":6482},[435],[81,6484,3518],{"className":6485},[125,126],[81,6487,415],{"className":6488},[442],[81,6490],{"className":6491,"style":127},[912],[81,6493,6422],{"className":6494},[235],[81,6496],{"className":6497,"style":127},[912],[81,6499,6501,6504,6507,6510,6513,6516,6519,6522,6525],{"className":6500},[116],[81,6502],{"className":6503,"style":428},[120],[81,6505,6362],{"className":6506,"style":1394},[125,126],[81,6508,410],{"className":6509},[435],[81,6511,3518],{"className":6512},[125,126],[81,6514,415],{"className":6515},[442],[81,6517,6411],{"className":6518},[125],[81,6520],{"className":6521,"style":913},[912],[81,6523,6435],{"className":6524},[917],[81,6526],{"className":6527,"style":913},[912],[81,6529,6531,6534,6537,6540,6544,6547,6550,6553,6556,6559],{"className":6530},[116],[81,6532],{"className":6533,"style":6157},[120],[81,6535,6305],{"className":6536},[125,126],[81,6538,1414],{"className":6539},[1441],[81,6541],{"className":6542,"style":6543},[912],"margin-right:2em;",[81,6545],{"className":6546,"style":1025},[912],[81,6548,6445],{"className":6549},[125],[81,6551,3518],{"className":6552},[125,126],[81,6554],{"className":6555,"style":913},[912],[81,6557,6127],{"className":6558},[917],[81,6560],{"className":6561,"style":913},[912],[81,6563,6565,6568,6571,6574,6577,6580,6583,6586],{"className":6564},[116],[81,6566],{"className":6567,"style":428},[120],[81,6569,497],{"className":6570},[435],[81,6572,513],{"className":6573},[125],[81,6575,1414],{"className":6576},[1441],[81,6578],{"className":6579,"style":1025},[912],[81,6581,679],{"className":6582},[125],[81,6584,529],{"className":6585},[442],[81,6587,65],{"className":6588},[125],[14,6590,6591],{},"Next, let's try to prove this theorem.",[14,6593,6594],{},[24,6595,6596],{},"Step 1: Construct the Bernstein polynomial",[14,6598,6599],{},"Define",[81,6601,6603],{"className":6602,"translate":85},[809],[81,6604,6606,6713],{"className":6605,"translate":85},[84],[81,6607,6609],{"className":6608},[89],[91,6610,6611],{"xmlns":93,"display":818},[95,6612,6613,6710],{},[98,6614,6615,6622,6624,6626,6628,6630,6632,6634,6636,6652,6654,6657,6669,6682,6688,6690,6692,6694,6696,6708],{},[3006,6616,6617,6620],{},[101,6618,6619],{},"B",[101,6621,2164],{},[179,6623,410],{"stretchy":409},[101,6625,6124],{},[179,6627,415],{"stretchy":409},[179,6629,410],{"stretchy":409},[101,6631,3518],{},[179,6633,415],{"stretchy":409},[179,6635,833],{},[6637,6638,6639,6642,6650],"munderover",{},[179,6640,6641],{},"∑",[98,6643,6644,6646,6648],{},[101,6645,3746],{},[179,6647,833],{},[515,6649,513],{},[101,6651,2164],{},[101,6653,6124],{},[2035,6655,6656],{}," ⁣",[98,6658,6659,6661,6667],{},[179,6660,410],{"fence":112},[1199,6662,6663,6665],{},[101,6664,3746],{},[101,6666,2164],{},[179,6668,415],{"fence":112},[98,6670,6671,6673,6680],{},[179,6672,410],{"fence":112},[1199,6674,6676,6678],{"linethickness":6675},"0px",[101,6677,2164],{},[101,6679,3746],{},[179,6681,415],{"fence":112},[174,6683,6684,6686],{},[101,6685,3518],{},[101,6687,3746],{},[179,6689,410],{"stretchy":409},[515,6691,679],{},[179,6693,6422],{},[101,6695,3518],{},[174,6697,6698,6700],{},[179,6699,415],{"stretchy":409},[98,6701,6702,6704,6706],{},[101,6703,2164],{},[179,6705,6422],{},[101,6707,3746],{},[101,6709,65],{"mathvariant":1133},[105,6711,6712],{"encoding":107},"B_n(f)(x)\n=\n\n\\sum_{k=0}^{n}\nf\\!\\left(\\frac{k}{n}\\right)\n\\binom{n}{k}\nx^k(1-x)^{n-k}.",[81,6714,6716,6792,7076],{"className":6715,"ariaHidden":112},[111],[81,6717,6719,6722,6765,6768,6771,6774,6777,6780,6783,6786,6789],{"className":6718},[116],[81,6720],{"className":6721,"style":428},[120],[81,6723,6725,6729],{"className":6724},[125],[81,6726,6619],{"className":6727,"style":6728},[125,126],"margin-right:0.0502em;",[81,6730,6732],{"className":6731},[204],[81,6733,6735,6757],{"className":6734},[208,569],[81,6736,6738,6754],{"className":6737},[212],[81,6739,6742],{"className":6740,"style":6741},[216],"height:0.1514em;",[81,6743,6745,6748],{"style":6744},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[81,6746],{"className":6747,"style":224},[223],[81,6749,6751],{"className":6750},[228,229,230,231],[81,6752,2164],{"className":6753},[125,126,231],[81,6755,606],{"className":6756},[605],[81,6758,6760],{"className":6759},[212],[81,6761,6763],{"className":6762,"style":3177},[216],[81,6764],{},[81,6766,410],{"className":6767},[435],[81,6769,6124],{"className":6770,"style":6161},[125,126],[81,6772,415],{"className":6773},[442],[81,6775,410],{"className":6776},[435],[81,6778,3518],{"className":6779},[125,126],[81,6781,415],{"className":6782},[442],[81,6784],{"className":6785,"style":913},[912],[81,6787,833],{"className":6788},[917],[81,6790],{"className":6791,"style":913},[912],[81,6793,6795,6799,6876,6879,6882,6886,6889,6967,6970,7031,7061,7064,7067,7070,7073],{"className":6794},[116],[81,6796],{"className":6797,"style":6798},[120],"height:2.9535em;vertical-align:-1.3021em;",[81,6800,6804],{"className":6801},[6802,6803],"mop","op-limits",[81,6805,6807,6867],{"className":6806},[208,569],[81,6808,6810,6864],{"className":6809},[212],[81,6811,6814,6836,6849],{"className":6812,"style":6813},[216],"height:1.6514em;",[81,6815,6817,6821],{"style":6816},"top:-1.8479em;margin-left:0em;",[81,6818],{"className":6819,"style":6820},[223],"height:3.05em;",[81,6822,6824],{"className":6823},[228,229,230,231],[81,6825,6827,6830,6833],{"className":6826},[125,231],[81,6828,3746],{"className":6829,"style":4244},[125,126,231],[81,6831,833],{"className":6832},[917,231],[81,6834,513],{"className":6835},[125,231],[81,6837,6839,6842],{"style":6838},"top:-3.05em;",[81,6840],{"className":6841,"style":6820},[223],[81,6843,6844],{},[81,6845,6641],{"className":6846},[6802,6847,6848],"op-symbol","large-op",[81,6850,6852,6855],{"style":6851},"top:-4.3em;margin-left:0em;",[81,6853],{"className":6854,"style":6820},[223],[81,6856,6858],{"className":6857},[228,229,230,231],[81,6859,6861],{"className":6860},[125,231],[81,6862,2164],{"className":6863},[125,126,231],[81,6865,606],{"className":6866},[605],[81,6868,6870],{"className":6869},[212],[81,6871,6874],{"className":6872,"style":6873},[216],"height:1.3021em;",[81,6875],{},[81,6877],{"className":6878,"style":1025},[912],[81,6880,6124],{"className":6881,"style":6161},[125,126],[81,6883],{"className":6884,"style":6885},[912],"margin-right:-0.1667em;",[81,6887],{"className":6888,"style":1025},[912],[81,6890,6892,6898,6961],{"className":6891},[546],[81,6893,6895],{"className":6894,"style":551},[435,550],[81,6896,410],{"className":6897},[555,230],[81,6899,6901,6904,6958],{"className":6900},[125],[81,6902],{"className":6903},[435,1247],[81,6905,6907],{"className":6906},[1199],[81,6908,6910,6950],{"className":6909},[208,569],[81,6911,6913,6947],{"className":6912},[212],[81,6914,6917,6928,6936],{"className":6915,"style":6916},[216],"height:1.3714em;",[81,6918,6919,6922],{"style":4196},[81,6920],{"className":6921,"style":583},[223],[81,6923,6925],{"className":6924},[125],[81,6926,2164],{"className":6927},[125,126],[81,6929,6930,6933],{"style":1278},[81,6931],{"className":6932,"style":583},[223],[81,6934],{"className":6935,"style":1286},[1285],[81,6937,6938,6941],{"style":4290},[81,6939],{"className":6940,"style":583},[223],[81,6942,6944],{"className":6943},[125],[81,6945,3746],{"className":6946,"style":4244},[125,126],[81,6948,606],{"className":6949},[605],[81,6951,6953],{"className":6952},[212],[81,6954,6956],{"className":6955,"style":4385},[216],[81,6957],{},[81,6959],{"className":6960},[442,1247],[81,6962,6964],{"className":6963,"style":551},[442,550],[81,6965,415],{"className":6966},[555,230],[81,6968],{"className":6969,"style":1025},[912],[81,6971,6973,6979,7025],{"className":6972},[125],[81,6974,6976],{"className":6975,"style":551},[435,550],[81,6977,410],{"className":6978},[555,230],[81,6980,6982],{"className":6981},[1199],[81,6983,6985,7017],{"className":6984},[208,569],[81,6986,6988,7014],{"className":6987},[212],[81,6989,6992,7003],{"className":6990,"style":6991},[216],"height:1.1076em;",[81,6993,6994,6997],{"style":4196},[81,6995],{"className":6996,"style":583},[223],[81,6998,7000],{"className":6999},[125],[81,7001,3746],{"className":7002,"style":4244},[125,126],[81,7004,7005,7008],{"style":4290},[81,7006],{"className":7007,"style":583},[223],[81,7009,7011],{"className":7010},[125],[81,7012,2164],{"className":7013},[125,126],[81,7015,606],{"className":7016},[605],[81,7018,7020],{"className":7019},[212],[81,7021,7023],{"className":7022,"style":4385},[216],[81,7024],{},[81,7026,7028],{"className":7027,"style":551},[442,550],[81,7029,415],{"className":7030},[555,230],[81,7032,7034,7037],{"className":7033},[125],[81,7035,3518],{"className":7036},[125,126],[81,7038,7040],{"className":7039},[204],[81,7041,7043],{"className":7042},[208],[81,7044,7046],{"className":7045},[212],[81,7047,7050],{"className":7048,"style":7049},[216],"height:0.8991em;",[81,7051,7052,7055],{"style":1749},[81,7053],{"className":7054,"style":224},[223],[81,7056,7058],{"className":7057},[228,229,230,231],[81,7059,3746],{"className":7060,"style":4244},[125,126,231],[81,7062,410],{"className":7063},[435],[81,7065,679],{"className":7066},[125],[81,7068],{"className":7069,"style":127},[912],[81,7071,6422],{"className":7072},[235],[81,7074],{"className":7075,"style":127},[912],[81,7077,7079,7083,7086,7124],{"className":7078},[116],[81,7080],{"className":7081,"style":7082},[120],"height:1.1491em;vertical-align:-0.25em;",[81,7084,3518],{"className":7085},[125,126],[81,7087,7089,7092],{"className":7088},[442],[81,7090,415],{"className":7091},[442],[81,7093,7095],{"className":7094},[204],[81,7096,7098],{"className":7097},[208],[81,7099,7101],{"className":7100},[212],[81,7102,7104],{"className":7103,"style":7049},[216],[81,7105,7106,7109],{"style":1749},[81,7107],{"className":7108,"style":224},[223],[81,7110,7112],{"className":7111},[228,229,230,231],[81,7113,7115,7118,7121],{"className":7114},[125,231],[81,7116,2164],{"className":7117},[125,126,231],[81,7119,6422],{"className":7120},[235,231],[81,7122,3746],{"className":7123,"style":4244},[125,126,231],[81,7125,65],{"className":7126},[125],[14,7128,7129],{},"This is the Bernstein polynomial.",[14,7131,7132],{},"Note that",[81,7134,7136],{"className":7135,"translate":85},[809],[81,7137,7139,7189],{"className":7138,"translate":85},[84],[81,7140,7142],{"className":7141},[89],[91,7143,7144],{"xmlns":93,"display":818},[95,7145,7146,7186],{},[98,7147,7148,7160,7166,7168,7170,7172,7174],{},[98,7149,7150,7152,7158],{},[179,7151,410],{"fence":112},[1199,7153,7154,7156],{"linethickness":6675},[101,7155,2164],{},[101,7157,3746],{},[179,7159,415],{"fence":112},[174,7161,7162,7164],{},[101,7163,3518],{},[101,7165,3746],{},[179,7167,410],{"stretchy":409},[515,7169,679],{},[179,7171,6422],{},[101,7173,3518],{},[174,7175,7176,7178],{},[179,7177,415],{"stretchy":409},[98,7179,7180,7182,7184],{},[101,7181,2164],{},[179,7183,6422],{},[101,7185,3746],{},[105,7187,7188],{"encoding":107},"\\binom{n}{k}\nx^k(1-x)^{n-k}",[81,7190,7192,7302],{"className":7191,"ariaHidden":112},[111],[81,7193,7195,7198,7258,7287,7290,7293,7296,7299],{"className":7194},[116],[81,7196],{"className":7197,"style":542},[120],[81,7199,7201,7207,7252],{"className":7200},[125],[81,7202,7204],{"className":7203,"style":551},[435,550],[81,7205,410],{"className":7206},[555,230],[81,7208,7210],{"className":7209},[1199],[81,7211,7213,7244],{"className":7212},[208,569],[81,7214,7216,7241],{"className":7215},[212],[81,7217,7219,7230],{"className":7218,"style":6991},[216],[81,7220,7221,7224],{"style":4196},[81,7222],{"className":7223,"style":583},[223],[81,7225,7227],{"className":7226},[125],[81,7228,3746],{"className":7229,"style":4244},[125,126],[81,7231,7232,7235],{"style":4290},[81,7233],{"className":7234,"style":583},[223],[81,7236,7238],{"className":7237},[125],[81,7239,2164],{"className":7240},[125,126],[81,7242,606],{"className":7243},[605],[81,7245,7247],{"className":7246},[212],[81,7248,7250],{"className":7249,"style":4385},[216],[81,7251],{},[81,7253,7255],{"className":7254,"style":551},[442,550],[81,7256,415],{"className":7257},[555,230],[81,7259,7261,7264],{"className":7260},[125],[81,7262,3518],{"className":7263},[125,126],[81,7265,7267],{"className":7266},[204],[81,7268,7270],{"className":7269},[208],[81,7271,7273],{"className":7272},[212],[81,7274,7276],{"className":7275,"style":7049},[216],[81,7277,7278,7281],{"style":1749},[81,7279],{"className":7280,"style":224},[223],[81,7282,7284],{"className":7283},[228,229,230,231],[81,7285,3746],{"className":7286,"style":4244},[125,126,231],[81,7288,410],{"className":7289},[435],[81,7291,679],{"className":7292},[125],[81,7294],{"className":7295,"style":127},[912],[81,7297,6422],{"className":7298},[235],[81,7300],{"className":7301,"style":127},[912],[81,7303,7305,7308,7311],{"className":7304},[116],[81,7306],{"className":7307,"style":7082},[120],[81,7309,3518],{"className":7310},[125,126],[81,7312,7314,7317],{"className":7313},[442],[81,7315,415],{"className":7316},[442],[81,7318,7320],{"className":7319},[204],[81,7321,7323],{"className":7322},[208],[81,7324,7326],{"className":7325},[212],[81,7327,7329],{"className":7328,"style":7049},[216],[81,7330,7331,7334],{"style":1749},[81,7332],{"className":7333,"style":224},[223],[81,7335,7337],{"className":7336},[228,229,230,231],[81,7338,7340,7343,7346],{"className":7339},[125,231],[81,7341,2164],{"className":7342},[125,126,231],[81,7344,6422],{"className":7345},[235,231],[81,7347,3746],{"className":7348,"style":4244},[125,126,231],[14,7350,7351,7352,65],{},"is itself a polynomial in ",[81,7353,7355,7368],{"className":7354,"translate":85},[84],[81,7356,7358],{"className":7357},[89],[91,7359,7360],{"xmlns":93},[95,7361,7362,7366],{},[98,7363,7364],{},[101,7365,3518],{},[105,7367,3518],{"encoding":107},[81,7369,7371],{"className":7370,"ariaHidden":112},[111],[81,7372,7374,7377],{"className":7373},[116],[81,7375],{"className":7376,"style":293},[120],[81,7378,3518],{"className":7379},[125,126],[14,7381,7382],{},"Therefore",[81,7384,7386],{"className":7385,"translate":85},[809],[81,7387,7389,7419],{"className":7388,"translate":85},[84],[81,7390,7392],{"className":7391},[89],[91,7393,7394],{"xmlns":93,"display":818},[95,7395,7396,7416],{},[98,7397,7398,7404,7406,7408,7410,7412,7414],{},[3006,7399,7400,7402],{},[101,7401,6619],{},[101,7403,2164],{},[179,7405,410],{"stretchy":409},[101,7407,6124],{},[179,7409,415],{"stretchy":409},[179,7411,410],{"stretchy":409},[101,7413,3518],{},[179,7415,415],{"stretchy":409},[105,7417,7418],{"encoding":107},"B_n(f)(x)",[81,7420,7422],{"className":7421,"ariaHidden":112},[111],[81,7423,7425,7428,7468,7471,7474,7477,7480,7483],{"className":7424},[116],[81,7426],{"className":7427,"style":428},[120],[81,7429,7431,7434],{"className":7430},[125],[81,7432,6619],{"className":7433,"style":6728},[125,126],[81,7435,7437],{"className":7436},[204],[81,7438,7440,7460],{"className":7439},[208,569],[81,7441,7443,7457],{"className":7442},[212],[81,7444,7446],{"className":7445,"style":6741},[216],[81,7447,7448,7451],{"style":6744},[81,7449],{"className":7450,"style":224},[223],[81,7452,7454],{"className":7453},[228,229,230,231],[81,7455,2164],{"className":7456},[125,126,231],[81,7458,606],{"className":7459},[605],[81,7461,7463],{"className":7462},[212],[81,7464,7466],{"className":7465,"style":3177},[216],[81,7467],{},[81,7469,410],{"className":7470},[435],[81,7472,6124],{"className":7473,"style":6161},[125,126],[81,7475,415],{"className":7476},[442],[81,7478,410],{"className":7479},[435],[81,7481,3518],{"className":7482},[125,126],[81,7484,415],{"className":7485},[442],[14,7487,7488],{},"is also a polynomial.",[14,7490,7491,7494],{},[24,7492,7493],{},"Step 2: Understand what it does."," This polynomial is a bit abstract, but we can interpret it in another way.",[14,7496,7497],{},"First note the identity",[81,7499,7501],{"className":7500,"translate":85},[809],[81,7502,7504,7596],{"className":7503,"translate":85},[84],[81,7505,7507],{"className":7506},[89],[91,7508,7509],{"xmlns":93,"display":818},[95,7510,7511,7593],{},[98,7512,7513,7527,7539,7545,7547,7549,7551,7553,7565,7567,7569,7571,7573,7575,7577,7579,7581,7583,7589,7591],{},[6637,7514,7515,7517,7525],{},[179,7516,6641],{},[98,7518,7519,7521,7523],{},[101,7520,3746],{},[179,7522,833],{},[515,7524,513],{},[101,7526,2164],{},[98,7528,7529,7531,7537],{},[179,7530,410],{"fence":112},[1199,7532,7533,7535],{"linethickness":6675},[101,7534,2164],{},[101,7536,3746],{},[179,7538,415],{"fence":112},[174,7540,7541,7543],{},[101,7542,3518],{},[101,7544,3746],{},[179,7546,410],{"stretchy":409},[515,7548,679],{},[179,7550,6422],{},[101,7552,3518],{},[174,7554,7555,7557],{},[179,7556,415],{"stretchy":409},[98,7558,7559,7561,7563],{},[101,7560,2164],{},[179,7562,6422],{},[101,7564,3746],{},[179,7566,833],{},[179,7568,410],{"stretchy":409},[101,7570,3518],{},[179,7572,2044],{},[179,7574,410],{"stretchy":409},[515,7576,679],{},[179,7578,6422],{},[101,7580,3518],{},[179,7582,415],{"stretchy":409},[174,7584,7585,7587],{},[179,7586,415],{"stretchy":409},[101,7588,2164],{},[179,7590,833],{},[515,7592,679],{},[105,7594,7595],{"encoding":107},"\\sum_{k=0}^{n}\n\\binom{n}{k}\nx^k(1-x)^{n-k}\n=(x+(1-x))^n\n=1",[81,7597,7599,7779,7835,7856,7877,7928],{"className":7598,"ariaHidden":112},[111],[81,7600,7602,7605,7672,7675,7735,7764,7767,7770,7773,7776],{"className":7601},[116],[81,7603],{"className":7604,"style":6798},[120],[81,7606,7608],{"className":7607},[6802,6803],[81,7609,7611,7664],{"className":7610},[208,569],[81,7612,7614,7661],{"className":7613},[212],[81,7615,7617,7637,7647],{"className":7616,"style":6813},[216],[81,7618,7619,7622],{"style":6816},[81,7620],{"className":7621,"style":6820},[223],[81,7623,7625],{"className":7624},[228,229,230,231],[81,7626,7628,7631,7634],{"className":7627},[125,231],[81,7629,3746],{"className":7630,"style":4244},[125,126,231],[81,7632,833],{"className":7633},[917,231],[81,7635,513],{"className":7636},[125,231],[81,7638,7639,7642],{"style":6838},[81,7640],{"className":7641,"style":6820},[223],[81,7643,7644],{},[81,7645,6641],{"className":7646},[6802,6847,6848],[81,7648,7649,7652],{"style":6851},[81,7650],{"className":7651,"style":6820},[223],[81,7653,7655],{"className":7654},[228,229,230,231],[81,7656,7658],{"className":7657},[125,231],[81,7659,2164],{"className":7660},[125,126,231],[81,7662,606],{"className":7663},[605],[81,7665,7667],{"className":7666},[212],[81,7668,7670],{"className":7669,"style":6873},[216],[81,7671],{},[81,7673],{"className":7674,"style":1025},[912],[81,7676,7678,7684,7729],{"className":7677},[125],[81,7679,7681],{"className":7680,"style":551},[435,550],[81,7682,410],{"className":7683},[555,230],[81,7685,7687],{"className":7686},[1199],[81,7688,7690,7721],{"className":7689},[208,569],[81,7691,7693,7718],{"className":7692},[212],[81,7694,7696,7707],{"className":7695,"style":6991},[216],[81,7697,7698,7701],{"style":4196},[81,7699],{"className":7700,"style":583},[223],[81,7702,7704],{"className":7703},[125],[81,7705,3746],{"className":7706,"style":4244},[125,126],[81,7708,7709,7712],{"style":4290},[81,7710],{"className":7711,"style":583},[223],[81,7713,7715],{"className":7714},[125],[81,7716,2164],{"className":7717},[125,126],[81,7719,606],{"className":7720},[605],[81,7722,7724],{"className":7723},[212],[81,7725,7727],{"className":7726,"style":4385},[216],[81,7728],{},[81,7730,7732],{"className":7731,"style":551},[442,550],[81,7733,415],{"className":7734},[555,230],[81,7736,7738,7741],{"className":7737},[125],[81,7739,3518],{"className":7740},[125,126],[81,7742,7744],{"className":7743},[204],[81,7745,7747],{"className":7746},[208],[81,7748,7750],{"className":7749},[212],[81,7751,7753],{"className":7752,"style":7049},[216],[81,7754,7755,7758],{"style":1749},[81,7756],{"className":7757,"style":224},[223],[81,7759,7761],{"className":7760},[228,229,230,231],[81,7762,3746],{"className":7763,"style":4244},[125,126,231],[81,7765,410],{"className":7766},[435],[81,7768,679],{"className":7769},[125],[81,7771],{"className":7772,"style":127},[912],[81,7774,6422],{"className":7775},[235],[81,7777],{"className":7778,"style":127},[912],[81,7780,7782,7785,7788,7826,7829,7832],{"className":7781},[116],[81,7783],{"className":7784,"style":7082},[120],[81,7786,3518],{"className":7787},[125,126],[81,7789,7791,7794],{"className":7790},[442],[81,7792,415],{"className":7793},[442],[81,7795,7797],{"className":7796},[204],[81,7798,7800],{"className":7799},[208],[81,7801,7803],{"className":7802},[212],[81,7804,7806],{"className":7805,"style":7049},[216],[81,7807,7808,7811],{"style":1749},[81,7809],{"className":7810,"style":224},[223],[81,7812,7814],{"className":7813},[228,229,230,231],[81,7815,7817,7820,7823],{"className":7816},[125,231],[81,7818,2164],{"className":7819},[125,126,231],[81,7821,6422],{"className":7822},[235,231],[81,7824,3746],{"className":7825,"style":4244},[125,126,231],[81,7827],{"className":7828,"style":913},[912],[81,7830,833],{"className":7831},[917],[81,7833],{"className":7834,"style":913},[912],[81,7836,7838,7841,7844,7847,7850,7853],{"className":7837},[116],[81,7839],{"className":7840,"style":428},[120],[81,7842,410],{"className":7843},[435],[81,7845,3518],{"className":7846},[125,126],[81,7848],{"className":7849,"style":127},[912],[81,7851,2044],{"className":7852},[235],[81,7854],{"className":7855,"style":127},[912],[81,7857,7859,7862,7865,7868,7871,7874],{"className":7858},[116],[81,7860],{"className":7861,"style":428},[120],[81,7863,410],{"className":7864},[435],[81,7866,679],{"className":7867},[125],[81,7869],{"className":7870,"style":127},[912],[81,7872,6422],{"className":7873},[235],[81,7875],{"className":7876,"style":127},[912],[81,7878,7880,7883,7886,7889,7919,7922,7925],{"className":7879},[116],[81,7881],{"className":7882,"style":428},[120],[81,7884,3518],{"className":7885},[125,126],[81,7887,415],{"className":7888},[442],[81,7890,7892,7895],{"className":7891},[442],[81,7893,415],{"className":7894},[442],[81,7896,7898],{"className":7897},[204],[81,7899,7901],{"className":7900},[208],[81,7902,7904],{"className":7903},[212],[81,7905,7908],{"className":7906,"style":7907},[216],"height:0.7144em;",[81,7909,7910,7913],{"style":1749},[81,7911],{"className":7912,"style":224},[223],[81,7914,7916],{"className":7915},[228,229,230,231],[81,7917,2164],{"className":7918},[125,126,231],[81,7920],{"className":7921,"style":913},[912],[81,7923,833],{"className":7924},[917],[81,7926],{"className":7927,"style":913},[912],[81,7929,7931,7934],{"className":7930},[116],[81,7932],{"className":7933,"style":1107},[120],[81,7935,679],{"className":7936},[125],[14,7938,7939],{},"Thus",[81,7941,7943],{"className":7942,"translate":85},[809],[81,7944,7946,7975],{"className":7945,"translate":85},[84],[81,7947,7949],{"className":7948},[89],[91,7950,7951],{"xmlns":93,"display":818},[95,7952,7953,7973],{},[98,7954,7955,7961,7963,7965,7967,7969,7971],{},[3006,7956,7957,7959],{},[101,7958,6619],{},[101,7960,2164],{},[179,7962,410],{"stretchy":409},[101,7964,6124],{},[179,7966,415],{"stretchy":409},[179,7968,410],{"stretchy":409},[101,7970,3518],{},[179,7972,415],{"stretchy":409},[105,7974,7418],{"encoding":107},[81,7976,7978],{"className":7977,"ariaHidden":112},[111],[81,7979,7981,7984,8024,8027,8030,8033,8036,8039],{"className":7980},[116],[81,7982],{"className":7983,"style":428},[120],[81,7985,7987,7990],{"className":7986},[125],[81,7988,6619],{"className":7989,"style":6728},[125,126],[81,7991,7993],{"className":7992},[204],[81,7994,7996,8016],{"className":7995},[208,569],[81,7997,7999,8013],{"className":7998},[212],[81,8000,8002],{"className":8001,"style":6741},[216],[81,8003,8004,8007],{"style":6744},[81,8005],{"className":8006,"style":224},[223],[81,8008,8010],{"className":8009},[228,229,230,231],[81,8011,2164],{"className":8012},[125,126,231],[81,8014,606],{"className":8015},[605],[81,8017,8019],{"className":8018},[212],[81,8020,8022],{"className":8021,"style":3177},[216],[81,8023],{},[81,8025,410],{"className":8026},[435],[81,8028,6124],{"className":8029,"style":6161},[125,126],[81,8031,415],{"className":8032},[442],[81,8034,410],{"className":8035},[435],[81,8037,3518],{"className":8038},[125,126],[81,8040,415],{"className":8041},[442],[14,8043,8044],{},"is actually a weighted average of the sequence",[81,8046,8048],{"className":8047,"translate":85},[809],[81,8049,8051,8102],{"className":8050,"translate":85},[84],[81,8052,8054],{"className":8053},[89],[91,8055,8056],{"xmlns":93,"display":818},[95,8057,8058,8099],{},[98,8059,8060,8062,8064,8066,8068,8070,8072,8084,8086,8089,8091,8093,8095,8097],{},[101,8061,6124],{},[179,8063,410],{"stretchy":409},[515,8065,513],{},[179,8067,415],{"stretchy":409},[179,8069,1414],{"separator":112},[101,8071,6124],{},[98,8073,8074,8076,8082],{},[179,8075,410],{"fence":112},[1199,8077,8078,8080],{},[515,8079,679],{},[101,8081,2164],{},[179,8083,415],{"fence":112},[179,8085,1414],{"separator":112},[179,8087,8088],{},"…",[179,8090,1414],{"separator":112},[101,8092,6124],{},[179,8094,410],{"stretchy":409},[515,8096,679],{},[179,8098,415],{"stretchy":409},[105,8100,8101],{"encoding":107},"f(0),\nf\\left(\\frac1n\\right),\n\\dots,\nf(1)",[81,8103,8105],{"className":8104,"ariaHidden":112},[111],[81,8106,8108,8111,8114,8117,8120,8123,8126,8129,8132,8135,8213,8216,8219,8222,8225,8228,8231,8234,8237,8240,8243],{"className":8107},[116],[81,8109],{"className":8110,"style":542},[120],[81,8112,6124],{"className":8113,"style":6161},[125,126],[81,8115,410],{"className":8116},[435],[81,8118,513],{"className":8119},[125],[81,8121,415],{"className":8122},[442],[81,8124,1414],{"className":8125},[1441],[81,8127],{"className":8128,"style":1025},[912],[81,8130,6124],{"className":8131,"style":6161},[125,126],[81,8133],{"className":8134,"style":1025},[912],[81,8136,8138,8144,8207],{"className":8137},[546],[81,8139,8141],{"className":8140,"style":551},[435,550],[81,8142,410],{"className":8143},[555,230],[81,8145,8147,8150,8204],{"className":8146},[125],[81,8148],{"className":8149},[435,1247],[81,8151,8153],{"className":8152},[1199],[81,8154,8156,8196],{"className":8155},[208,569],[81,8157,8159,8193],{"className":8158},[212],[81,8160,8163,8174,8182],{"className":8161,"style":8162},[216],"height:1.3214em;",[81,8164,8165,8168],{"style":4196},[81,8166],{"className":8167,"style":583},[223],[81,8169,8171],{"className":8170},[125],[81,8172,2164],{"className":8173},[125,126],[81,8175,8176,8179],{"style":1278},[81,8177],{"className":8178,"style":583},[223],[81,8180],{"className":8181,"style":1286},[1285],[81,8183,8184,8187],{"style":4290},[81,8185],{"className":8186,"style":583},[223],[81,8188,8190],{"className":8189},[125],[81,8191,679],{"className":8192},[125],[81,8194,606],{"className":8195},[605],[81,8197,8199],{"className":8198},[212],[81,8200,8202],{"className":8201,"style":4385},[216],[81,8203],{},[81,8205],{"className":8206},[442,1247],[81,8208,8210],{"className":8209,"style":551},[442,550],[81,8211,415],{"className":8212},[555,230],[81,8214],{"className":8215,"style":1025},[912],[81,8217,1414],{"className":8218},[1441],[81,8220],{"className":8221,"style":1025},[912],[81,8223,8088],{"className":8224},[546],[81,8226],{"className":8227,"style":1025},[912],[81,8229,1414],{"className":8230},[1441],[81,8232],{"className":8233,"style":1025},[912],[81,8235,6124],{"className":8236,"style":6161},[125,126],[81,8238,410],{"className":8239},[435],[81,8241,679],{"className":8242},[125],[81,8244,415],{"className":8245},[442],[14,8247,8248],{},"where the weights are",[81,8250,8252],{"className":8251,"translate":85},[809],[81,8253,8255,8327],{"className":8254,"translate":85},[84],[81,8256,8258],{"className":8257},[89],[91,8259,8260],{"xmlns":93,"display":818},[95,8261,8262,8324],{},[98,8263,8264,8276,8278,8280,8282,8284,8296,8302,8304,8306,8308,8310,8322],{},[3006,8265,8266,8268],{},[101,8267,14],{},[98,8269,8270,8272,8274],{},[101,8271,2164],{},[179,8273,1414],{"separator":112},[101,8275,3746],{},[179,8277,410],{"stretchy":409},[101,8279,3518],{},[179,8281,415],{"stretchy":409},[179,8283,833],{},[98,8285,8286,8288,8294],{},[179,8287,410],{"fence":112},[1199,8289,8290,8292],{"linethickness":6675},[101,8291,2164],{},[101,8293,3746],{},[179,8295,415],{"fence":112},[174,8297,8298,8300],{},[101,8299,3518],{},[101,8301,3746],{},[179,8303,410],{"stretchy":409},[515,8305,679],{},[179,8307,6422],{},[101,8309,3518],{},[174,8311,8312,8314],{},[179,8313,415],{"stretchy":409},[98,8315,8316,8318,8320],{},[101,8317,2164],{},[179,8319,6422],{},[101,8321,3746],{},[101,8323,65],{"mathvariant":1133},[105,8325,8326],{"encoding":107},"p_{n,k}(x)\n=\n\\binom{n}{k}\nx^k(1-x)^{n-k}.",[81,8328,8330,8406,8516],{"className":8329,"ariaHidden":112},[111],[81,8331,8333,8337,8388,8391,8394,8397,8400,8403],{"className":8332},[116],[81,8334],{"className":8335,"style":8336},[120],"height:1.0361em;vertical-align:-0.2861em;",[81,8338,8340,8343],{"className":8339},[125],[81,8341,14],{"className":8342},[125,126],[81,8344,8346],{"className":8345},[204],[81,8347,8349,8380],{"className":8348},[208,569],[81,8350,8352,8377],{"className":8351},[212],[81,8353,8356],{"className":8354,"style":8355},[216],"height:0.3361em;",[81,8357,8359,8362],{"style":8358},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[81,8360],{"className":8361,"style":224},[223],[81,8363,8365],{"className":8364},[228,229,230,231],[81,8366,8368,8371,8374],{"className":8367},[125,231],[81,8369,2164],{"className":8370},[125,126,231],[81,8372,1414],{"className":8373},[1441,231],[81,8375,3746],{"className":8376,"style":4244},[125,126,231],[81,8378,606],{"className":8379},[605],[81,8381,8383],{"className":8382},[212],[81,8384,8386],{"className":8385,"style":3077},[216],[81,8387],{},[81,8389,410],{"className":8390},[435],[81,8392,3518],{"className":8393},[125,126],[81,8395,415],{"className":8396},[442],[81,8398],{"className":8399,"style":913},[912],[81,8401,833],{"className":8402},[917],[81,8404],{"className":8405,"style":913},[912],[81,8407,8409,8412,8472,8501,8504,8507,8510,8513],{"className":8408},[116],[81,8410],{"className":8411,"style":542},[120],[81,8413,8415,8421,8466],{"className":8414},[125],[81,8416,8418],{"className":8417,"style":551},[435,550],[81,8419,410],{"className":8420},[555,230],[81,8422,8424],{"className":8423},[1199],[81,8425,8427,8458],{"className":8426},[208,569],[81,8428,8430,8455],{"className":8429},[212],[81,8431,8433,8444],{"className":8432,"style":6991},[216],[81,8434,8435,8438],{"style":4196},[81,8436],{"className":8437,"style":583},[223],[81,8439,8441],{"className":8440},[125],[81,8442,3746],{"className":8443,"style":4244},[125,126],[81,8445,8446,8449],{"style":4290},[81,8447],{"className":8448,"style":583},[223],[81,8450,8452],{"className":8451},[125],[81,8453,2164],{"className":8454},[125,126],[81,8456,606],{"className":8457},[605],[81,8459,8461],{"className":8460},[212],[81,8462,8464],{"className":8463,"style":4385},[216],[81,8465],{},[81,8467,8469],{"className":8468,"style":551},[442,550],[81,8470,415],{"className":8471},[555,230],[81,8473,8475,8478],{"className":8474},[125],[81,8476,3518],{"className":8477},[125,126],[81,8479,8481],{"className":8480},[204],[81,8482,8484],{"className":8483},[208],[81,8485,8487],{"className":8486},[212],[81,8488,8490],{"className":8489,"style":7049},[216],[81,8491,8492,8495],{"style":1749},[81,8493],{"className":8494,"style":224},[223],[81,8496,8498],{"className":8497},[228,229,230,231],[81,8499,3746],{"className":8500,"style":4244},[125,126,231],[81,8502,410],{"className":8503},[435],[81,8505,679],{"className":8506},[125],[81,8508],{"className":8509,"style":127},[912],[81,8511,6422],{"className":8512},[235],[81,8514],{"className":8515,"style":127},[912],[81,8517,8519,8522,8525,8563],{"className":8518},[116],[81,8520],{"className":8521,"style":7082},[120],[81,8523,3518],{"className":8524},[125,126],[81,8526,8528,8531],{"className":8527},[442],[81,8529,415],{"className":8530},[442],[81,8532,8534],{"className":8533},[204],[81,8535,8537],{"className":8536},[208],[81,8538,8540],{"className":8539},[212],[81,8541,8543],{"className":8542,"style":7049},[216],[81,8544,8545,8548],{"style":1749},[81,8546],{"className":8547,"style":224},[223],[81,8549,8551],{"className":8550},[228,229,230,231],[81,8552,8554,8557,8560],{"className":8553},[125,231],[81,8555,2164],{"className":8556},[125,126,231],[81,8558,6422],{"className":8559},[235,231],[81,8561,3746],{"className":8562,"style":4244},[125,126,231],[81,8564,65],{"className":8565},[125],[14,8567,8568],{},"And the sum of these weights is 1.",[14,8570,8571],{},"But we can also interpret it from a probabilistic perspective.",[14,8573,8574],{},"Let the random variable",[81,8576,8578],{"className":8577,"translate":85},[809],[81,8579,8581,8636],{"className":8580,"translate":85},[84],[81,8582,8584],{"className":8583},[89],[91,8585,8586],{"xmlns":93,"display":818},[95,8587,8588,8633],{},[98,8589,8590,8597,8600,8621,8623,8625,8627,8629,8631],{},[3006,8591,8592,8595],{},[101,8593,8594],{},"X",[101,8596,2164],{},[179,8598,8599],{},"∼",[98,8601,8602,8604,8606,8608,8611,8614,8616,8619],{},[101,8603,6619],{"mathvariant":1133},[101,8605,3015],{"mathvariant":1133},[101,8607,2164],{"mathvariant":1133},[101,8609,8610],{"mathvariant":1133},"o",[101,8612,8613],{"mathvariant":1133},"m",[101,8615,3015],{"mathvariant":1133},[101,8617,8618],{"mathvariant":1133},"a",[101,8620,3801],{"mathvariant":1133},[179,8622,410],{"stretchy":409},[101,8624,2164],{},[179,8626,1414],{"separator":112},[101,8628,3518],{},[179,8630,415],{"stretchy":409},[101,8632,65],{"mathvariant":1133},[105,8634,8635],{"encoding":107},"X_n\\sim \\mathrm{Binomial}(n,x).",[81,8637,8639,8697],{"className":8638,"ariaHidden":112},[111],[81,8640,8642,8646,8688,8691,8694],{"className":8641},[116],[81,8643],{"className":8644,"style":8645},[120],"height:0.8333em;vertical-align:-0.15em;",[81,8647,8649,8653],{"className":8648},[125],[81,8650,8594],{"className":8651,"style":8652},[125,126],"margin-right:0.0785em;",[81,8654,8656],{"className":8655},[204],[81,8657,8659,8680],{"className":8658},[208,569],[81,8660,8662,8677],{"className":8661},[212],[81,8663,8665],{"className":8664,"style":6741},[216],[81,8666,8668,8671],{"style":8667},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[81,8669],{"className":8670,"style":224},[223],[81,8672,8674],{"className":8673},[228,229,230,231],[81,8675,2164],{"className":8676},[125,126,231],[81,8678,606],{"className":8679},[605],[81,8681,8683],{"className":8682},[212],[81,8684,8686],{"className":8685,"style":3177},[216],[81,8687],{},[81,8689],{"className":8690,"style":913},[912],[81,8692,8599],{"className":8693},[917],[81,8695],{"className":8696,"style":913},[912],[81,8698,8700,8703,8711,8714,8717,8720,8723,8726,8729],{"className":8699},[116],[81,8701],{"className":8702,"style":428},[120],[81,8704,8706],{"className":8705},[125],[81,8707,8710],{"className":8708},[125,8709],"mathrm","Binomial",[81,8712,410],{"className":8713},[435],[81,8715,2164],{"className":8716},[125,126],[81,8718,1414],{"className":8719},[1441],[81,8721],{"className":8722,"style":1025},[912],[81,8724,3518],{"className":8725},[125,126],[81,8727,415],{"className":8728},[442],[81,8730,65],{"className":8731},[125],[14,8733,8734],{},"That is,",[81,8736,8738],{"className":8737,"translate":85},[809],[81,8739,8741,8811],{"className":8740,"translate":85},[84],[81,8742,8744],{"className":8743},[89],[91,8745,8746],{"xmlns":93,"display":818},[95,8747,8748,8808],{},[98,8749,8750,8752,8754,8760,8762,8764,8766,8768,8780,8786,8788,8790,8792,8794,8806],{},[101,8751,6362],{},[179,8753,410],{"stretchy":409},[3006,8755,8756,8758],{},[101,8757,8594],{},[101,8759,2164],{},[179,8761,833],{},[101,8763,3746],{},[179,8765,415],{"stretchy":409},[179,8767,833],{},[98,8769,8770,8772,8778],{},[179,8771,410],{"fence":112},[1199,8773,8774,8776],{"linethickness":6675},[101,8775,2164],{},[101,8777,3746],{},[179,8779,415],{"fence":112},[174,8781,8782,8784],{},[101,8783,3518],{},[101,8785,3746],{},[179,8787,410],{"stretchy":409},[515,8789,679],{},[179,8791,6422],{},[101,8793,3518],{},[174,8795,8796,8798],{},[179,8797,415],{"stretchy":409},[98,8799,8800,8802,8804],{},[101,8801,2164],{},[179,8803,6422],{},[101,8805,3746],{},[101,8807,65],{"mathvariant":1133},[105,8809,8810],{"encoding":107},"P(X_n=k)\n=\n\\binom{n}{k}\nx^k(1-x)^{n-k}.",[81,8812,8814,8875,8896,9006],{"className":8813,"ariaHidden":112},[111],[81,8815,8817,8820,8823,8826,8866,8869,8872],{"className":8816},[116],[81,8818],{"className":8819,"style":428},[120],[81,8821,6362],{"className":8822,"style":1394},[125,126],[81,8824,410],{"className":8825},[435],[81,8827,8829,8832],{"className":8828},[125],[81,8830,8594],{"className":8831,"style":8652},[125,126],[81,8833,8835],{"className":8834},[204],[81,8836,8838,8858],{"className":8837},[208,569],[81,8839,8841,8855],{"className":8840},[212],[81,8842,8844],{"className":8843,"style":6741},[216],[81,8845,8846,8849],{"style":8667},[81,8847],{"className":8848,"style":224},[223],[81,8850,8852],{"className":8851},[228,229,230,231],[81,8853,2164],{"className":8854},[125,126,231],[81,8856,606],{"className":8857},[605],[81,8859,8861],{"className":8860},[212],[81,8862,8864],{"className":8863,"style":3177},[216],[81,8865],{},[81,8867],{"className":8868,"style":913},[912],[81,8870,833],{"className":8871},[917],[81,8873],{"className":8874,"style":913},[912],[81,8876,8878,8881,8884,8887,8890,8893],{"className":8877},[116],[81,8879],{"className":8880,"style":428},[120],[81,8882,3746],{"className":8883,"style":4244},[125,126],[81,8885,415],{"className":8886},[442],[81,8888],{"className":8889,"style":913},[912],[81,8891,833],{"className":8892},[917],[81,8894],{"className":8895,"style":913},[912],[81,8897,8899,8902,8962,8991,8994,8997,9000,9003],{"className":8898},[116],[81,8900],{"className":8901,"style":542},[120],[81,8903,8905,8911,8956],{"className":8904},[125],[81,8906,8908],{"className":8907,"style":551},[435,550],[81,8909,410],{"className":8910},[555,230],[81,8912,8914],{"className":8913},[1199],[81,8915,8917,8948],{"className":8916},[208,569],[81,8918,8920,8945],{"className":8919},[212],[81,8921,8923,8934],{"className":8922,"style":6991},[216],[81,8924,8925,8928],{"style":4196},[81,8926],{"className":8927,"style":583},[223],[81,8929,8931],{"className":8930},[125],[81,8932,3746],{"className":8933,"style":4244},[125,126],[81,8935,8936,8939],{"style":4290},[81,8937],{"className":8938,"style":583},[223],[81,8940,8942],{"className":8941},[125],[81,8943,2164],{"className":8944},[125,126],[81,8946,606],{"className":8947},[605],[81,8949,8951],{"className":8950},[212],[81,8952,8954],{"className":8953,"style":4385},[216],[81,8955],{},[81,8957,8959],{"className":8958,"style":551},[442,550],[81,8960,415],{"className":8961},[555,230],[81,8963,8965,8968],{"className":8964},[125],[81,8966,3518],{"className":8967},[125,126],[81,8969,8971],{"className":8970},[204],[81,8972,8974],{"className":8973},[208],[81,8975,8977],{"className":8976},[212],[81,8978,8980],{"className":8979,"style":7049},[216],[81,8981,8982,8985],{"style":1749},[81,8983],{"className":8984,"style":224},[223],[81,8986,8988],{"className":8987},[228,229,230,231],[81,8989,3746],{"className":8990,"style":4244},[125,126,231],[81,8992,410],{"className":8993},[435],[81,8995,679],{"className":8996},[125],[81,8998],{"className":8999,"style":127},[912],[81,9001,6422],{"className":9002},[235],[81,9004],{"className":9005,"style":127},[912],[81,9007,9009,9012,9015,9053],{"className":9008},[116],[81,9010],{"className":9011,"style":7082},[120],[81,9013,3518],{"className":9014},[125,126],[81,9016,9018,9021],{"className":9017},[442],[81,9019,415],{"className":9020},[442],[81,9022,9024],{"className":9023},[204],[81,9025,9027],{"className":9026},[208],[81,9028,9030],{"className":9029},[212],[81,9031,9033],{"className":9032,"style":7049},[216],[81,9034,9035,9038],{"style":1749},[81,9036],{"className":9037,"style":224},[223],[81,9039,9041],{"className":9040},[228,229,230,231],[81,9042,9044,9047,9050],{"className":9043},[125,231],[81,9045,2164],{"className":9046},[125,126,231],[81,9048,6422],{"className":9049},[235,231],[81,9051,3746],{"className":9052,"style":4244},[125,126,231],[81,9054,65],{"className":9055},[125],[14,9057,9058],{},"Then",[81,9060,9062],{"className":9061,"translate":85},[809],[81,9063,9065,9130],{"className":9064,"translate":85},[84],[81,9066,9068],{"className":9067},[89],[91,9069,9070],{"xmlns":93,"display":818},[95,9071,9072,9127],{},[98,9073,9074,9080,9082,9084,9086,9088,9090,9092,9094,9097,9099,9125],{},[3006,9075,9076,9078],{},[101,9077,6619],{},[101,9079,2164],{},[179,9081,410],{"stretchy":409},[101,9083,6124],{},[179,9085,415],{"stretchy":409},[179,9087,410],{"stretchy":409},[101,9089,3518],{},[179,9091,415],{"stretchy":409},[179,9093,833],{},[101,9095,9096],{},"E",[2035,9098,6656],{},[98,9100,9101,9103,9105,9107,9123],{},[179,9102,497],{"fence":112},[101,9104,6124],{},[2035,9106,6656],{},[98,9108,9109,9111,9121],{},[179,9110,410],{"fence":112},[1199,9112,9113,9119],{},[3006,9114,9115,9117],{},[101,9116,8594],{},[101,9118,2164],{},[101,9120,2164],{},[179,9122,415],{"fence":112},[179,9124,529],{"fence":112},[101,9126,65],{"mathvariant":1133},[105,9128,9129],{"encoding":107},"B_n(f)(x)\n=\n\nE\\!\\left[\nf\\!\\left(\\frac{X_n}{n}\\right)\n\\right].",[81,9131,9133,9206],{"className":9132,"ariaHidden":112},[111],[81,9134,9136,9139,9179,9182,9185,9188,9191,9194,9197,9200,9203],{"className":9135},[116],[81,9137],{"className":9138,"style":428},[120],[81,9140,9142,9145],{"className":9141},[125],[81,9143,6619],{"className":9144,"style":6728},[125,126],[81,9146,9148],{"className":9147},[204],[81,9149,9151,9171],{"className":9150},[208,569],[81,9152,9154,9168],{"className":9153},[212],[81,9155,9157],{"className":9156,"style":6741},[216],[81,9158,9159,9162],{"style":6744},[81,9160],{"className":9161,"style":224},[223],[81,9163,9165],{"className":9164},[228,229,230,231],[81,9166,2164],{"className":9167},[125,126,231],[81,9169,606],{"className":9170},[605],[81,9172,9174],{"className":9173},[212],[81,9175,9177],{"className":9176,"style":3177},[216],[81,9178],{},[81,9180,410],{"className":9181},[435],[81,9183,6124],{"className":9184,"style":6161},[125,126],[81,9186,415],{"className":9187},[442],[81,9189,410],{"className":9190},[435],[81,9192,3518],{"className":9193},[125,126],[81,9195,415],{"className":9196},[442],[81,9198],{"className":9199,"style":913},[912],[81,9201,833],{"className":9202},[917],[81,9204],{"className":9205,"style":913},[912],[81,9207,9209,9212,9216,9219,9222,9361,9364],{"className":9208},[116],[81,9210],{"className":9211,"style":542},[120],[81,9213,9096],{"className":9214,"style":9215},[125,126],"margin-right:0.0576em;",[81,9217],{"className":9218,"style":6885},[912],[81,9220],{"className":9221,"style":1025},[912],[81,9223,9225,9231,9234,9237,9240,9355],{"className":9224},[546],[81,9226,9228],{"className":9227,"style":551},[435,550],[81,9229,497],{"className":9230},[555,230],[81,9232,6124],{"className":9233,"style":6161},[125,126],[81,9235],{"className":9236,"style":6885},[912],[81,9238],{"className":9239,"style":1025},[912],[81,9241,9243,9249,9349],{"className":9242},[546],[81,9244,9246],{"className":9245,"style":551},[435,550],[81,9247,410],{"className":9248},[555,230],[81,9250,9252,9255,9346],{"className":9251},[125],[81,9253],{"className":9254},[435,1247],[81,9256,9258],{"className":9257},[1199],[81,9259,9261,9338],{"className":9260},[208,569],[81,9262,9264,9335],{"className":9263},[212],[81,9265,9268,9279,9287],{"className":9266,"style":9267},[216],"height:1.3603em;",[81,9269,9270,9273],{"style":4196},[81,9271],{"className":9272,"style":583},[223],[81,9274,9276],{"className":9275},[125],[81,9277,2164],{"className":9278},[125,126],[81,9280,9281,9284],{"style":1278},[81,9282],{"className":9283,"style":583},[223],[81,9285],{"className":9286,"style":1286},[1285],[81,9288,9289,9292],{"style":4290},[81,9290],{"className":9291,"style":583},[223],[81,9293,9295],{"className":9294},[125],[81,9296,9298,9301],{"className":9297},[125],[81,9299,8594],{"className":9300,"style":8652},[125,126],[81,9302,9304],{"className":9303},[204],[81,9305,9307,9327],{"className":9306},[208,569],[81,9308,9310,9324],{"className":9309},[212],[81,9311,9313],{"className":9312,"style":6741},[216],[81,9314,9315,9318],{"style":8667},[81,9316],{"className":9317,"style":224},[223],[81,9319,9321],{"className":9320},[228,229,230,231],[81,9322,2164],{"className":9323},[125,126,231],[81,9325,606],{"className":9326},[605],[81,9328,9330],{"className":9329},[212],[81,9331,9333],{"className":9332,"style":3177},[216],[81,9334],{},[81,9336,606],{"className":9337},[605],[81,9339,9341],{"className":9340},[212],[81,9342,9344],{"className":9343,"style":4385},[216],[81,9345],{},[81,9347],{"className":9348},[442,1247],[81,9350,9352],{"className":9351,"style":551},[442,550],[81,9353,415],{"className":9354},[555,230],[81,9356,9358],{"className":9357,"style":551},[442,550],[81,9359,529],{"className":9360},[555,230],[81,9362],{"className":9363,"style":1025},[912],[81,9365,65],{"className":9366},[125],[14,9368,9369],{},"This step is very elegant. The Bernstein polynomial is actually the expectation of",[81,9371,9373],{"className":9372,"translate":85},[809],[81,9374,9376,9408],{"className":9375,"translate":85},[84],[81,9377,9379],{"className":9378},[89],[91,9380,9381],{"xmlns":93,"display":818},[95,9382,9383,9405],{},[98,9384,9385,9387,9403],{},[101,9386,6124],{},[98,9388,9389,9391,9401],{},[179,9390,410],{"fence":112},[1199,9392,9393,9399],{},[3006,9394,9395,9397],{},[101,9396,8594],{},[101,9398,2164],{},[101,9400,2164],{},[179,9402,415],{"fence":112},[101,9404,65],{"mathvariant":1133},[105,9406,9407],{"encoding":107},"f\\left(\\frac{X_n}{n}\\right).",[81,9409,9411],{"className":9410,"ariaHidden":112},[111],[81,9412,9414,9417,9420,9423,9537,9540],{"className":9413},[116],[81,9415],{"className":9416,"style":542},[120],[81,9418,6124],{"className":9419,"style":6161},[125,126],[81,9421],{"className":9422,"style":1025},[912],[81,9424,9426,9432,9531],{"className":9425},[546],[81,9427,9429],{"className":9428,"style":551},[435,550],[81,9430,410],{"className":9431},[555,230],[81,9433,9435,9438,9528],{"className":9434},[125],[81,9436],{"className":9437},[435,1247],[81,9439,9441],{"className":9440},[1199],[81,9442,9444,9520],{"className":9443},[208,569],[81,9445,9447,9517],{"className":9446},[212],[81,9448,9450,9461,9469],{"className":9449,"style":9267},[216],[81,9451,9452,9455],{"style":4196},[81,9453],{"className":9454,"style":583},[223],[81,9456,9458],{"className":9457},[125],[81,9459,2164],{"className":9460},[125,126],[81,9462,9463,9466],{"style":1278},[81,9464],{"className":9465,"style":583},[223],[81,9467],{"className":9468,"style":1286},[1285],[81,9470,9471,9474],{"style":4290},[81,9472],{"className":9473,"style":583},[223],[81,9475,9477],{"className":9476},[125],[81,9478,9480,9483],{"className":9479},[125],[81,9481,8594],{"className":9482,"style":8652},[125,126],[81,9484,9486],{"className":9485},[204],[81,9487,9489,9509],{"className":9488},[208,569],[81,9490,9492,9506],{"className":9491},[212],[81,9493,9495],{"className":9494,"style":6741},[216],[81,9496,9497,9500],{"style":8667},[81,9498],{"className":9499,"style":224},[223],[81,9501,9503],{"className":9502},[228,229,230,231],[81,9504,2164],{"className":9505},[125,126,231],[81,9507,606],{"className":9508},[605],[81,9510,9512],{"className":9511},[212],[81,9513,9515],{"className":9514,"style":3177},[216],[81,9516],{},[81,9518,606],{"className":9519},[605],[81,9521,9523],{"className":9522},[212],[81,9524,9526],{"className":9525,"style":4385},[216],[81,9527],{},[81,9529],{"className":9530},[442,1247],[81,9532,9534],{"className":9533,"style":551},[442,550],[81,9535,415],{"className":9536},[555,230],[81,9538],{"className":9539,"style":1025},[912],[81,9541,65],{"className":9542},[125],[14,9544,9545,9546,65],{},"Now let's analyze why it approximates ",[81,9547,9549,9569],{"className":9548,"translate":85},[84],[81,9550,9552],{"className":9551},[89],[91,9553,9554],{"xmlns":93},[95,9555,9556,9566],{},[98,9557,9558,9560,9562,9564],{},[101,9559,6124],{},[179,9561,410],{"stretchy":409},[101,9563,3518],{},[179,9565,415],{"stretchy":409},[105,9567,9568],{"encoding":107},"f(x)",[81,9570,9572],{"className":9571,"ariaHidden":112},[111],[81,9573,9575,9578,9581,9584,9587],{"className":9574},[116],[81,9576],{"className":9577,"style":428},[120],[81,9579,6124],{"className":9580,"style":6161},[125,126],[81,9582,410],{"className":9583},[435],[81,9585,3518],{"className":9586},[125,126],[81,9588,415],{"className":9589},[442],[14,9591,9592],{},"For the binomial distribution we have",[81,9594,9596],{"className":9595,"translate":85},[809],[81,9597,9599,9637],{"className":9598,"translate":85},[84],[81,9600,9602],{"className":9601},[89],[91,9603,9604],{"xmlns":93,"display":818},[95,9605,9606,9634],{},[98,9607,9608,9610,9612,9628,9630,9632],{},[101,9609,9096],{},[2035,9611,6656],{},[98,9613,9614,9616,9626],{},[179,9615,410],{"fence":112},[1199,9617,9618,9624],{},[3006,9619,9620,9622],{},[101,9621,8594],{},[101,9623,2164],{},[101,9625,2164],{},[179,9627,415],{"fence":112},[179,9629,833],{},[101,9631,3518],{},[179,9633,1414],{"separator":112},[105,9635,9636],{"encoding":107},"E\\!\\left(\\frac{X_n}{n}\\right)\n=\nx,",[81,9638,9640,9778],{"className":9639,"ariaHidden":112},[111],[81,9641,9643,9646,9649,9652,9655,9769,9772,9775],{"className":9642},[116],[81,9644],{"className":9645,"style":542},[120],[81,9647,9096],{"className":9648,"style":9215},[125,126],[81,9650],{"className":9651,"style":6885},[912],[81,9653],{"className":9654,"style":1025},[912],[81,9656,9658,9664,9763],{"className":9657},[546],[81,9659,9661],{"className":9660,"style":551},[435,550],[81,9662,410],{"className":9663},[555,230],[81,9665,9667,9670,9760],{"className":9666},[125],[81,9668],{"className":9669},[435,1247],[81,9671,9673],{"className":9672},[1199],[81,9674,9676,9752],{"className":9675},[208,569],[81,9677,9679,9749],{"className":9678},[212],[81,9680,9682,9693,9701],{"className":9681,"style":9267},[216],[81,9683,9684,9687],{"style":4196},[81,9685],{"className":9686,"style":583},[223],[81,9688,9690],{"className":9689},[125],[81,9691,2164],{"className":9692},[125,126],[81,9694,9695,9698],{"style":1278},[81,9696],{"className":9697,"style":583},[223],[81,9699],{"className":9700,"style":1286},[1285],[81,9702,9703,9706],{"style":4290},[81,9704],{"className":9705,"style":583},[223],[81,9707,9709],{"className":9708},[125],[81,9710,9712,9715],{"className":9711},[125],[81,9713,8594],{"className":9714,"style":8652},[125,126],[81,9716,9718],{"className":9717},[204],[81,9719,9721,9741],{"className":9720},[208,569],[81,9722,9724,9738],{"className":9723},[212],[81,9725,9727],{"className":9726,"style":6741},[216],[81,9728,9729,9732],{"style":8667},[81,9730],{"className":9731,"style":224},[223],[81,9733,9735],{"className":9734},[228,229,230,231],[81,9736,2164],{"className":9737},[125,126,231],[81,9739,606],{"className":9740},[605],[81,9742,9744],{"className":9743},[212],[81,9745,9747],{"className":9746,"style":3177},[216],[81,9748],{},[81,9750,606],{"className":9751},[605],[81,9753,9755],{"className":9754},[212],[81,9756,9758],{"className":9757,"style":4385},[216],[81,9759],{},[81,9761],{"className":9762},[442,1247],[81,9764,9766],{"className":9765,"style":551},[442,550],[81,9767,415],{"className":9768},[555,230],[81,9770],{"className":9771,"style":913},[912],[81,9773,833],{"className":9774},[917],[81,9776],{"className":9777,"style":913},[912],[81,9779,9781,9784,9787],{"className":9780},[116],[81,9782],{"className":9783,"style":1477},[120],[81,9785,3518],{"className":9786},[125,126],[81,9788,1414],{"className":9789},[1441],[81,9791,9793],{"className":9792,"translate":85},[809],[81,9794,9796,9857],{"className":9795,"translate":85},[84],[81,9797,9799],{"className":9798},[89],[91,9800,9801],{"xmlns":93,"display":818},[95,9802,9803,9854],{},[98,9804,9805,9814,9816,9832,9834,9852],{},[98,9806,9807,9809,9811],{},[101,9808,103],{"mathvariant":1133},[101,9810,8618],{"mathvariant":1133},[101,9812,9813],{"mathvariant":1133},"r",[2035,9815,6656],{},[98,9817,9818,9820,9830],{},[179,9819,410],{"fence":112},[1199,9821,9822,9828],{},[3006,9823,9824,9826],{},[101,9825,8594],{},[101,9827,2164],{},[101,9829,2164],{},[179,9831,415],{"fence":112},[179,9833,833],{},[1199,9835,9836,9850],{},[98,9837,9838,9840,9842,9844,9846,9848],{},[101,9839,3518],{},[179,9841,410],{"stretchy":409},[515,9843,679],{},[179,9845,6422],{},[101,9847,3518],{},[179,9849,415],{"stretchy":409},[101,9851,2164],{},[101,9853,65],{"mathvariant":1133},[105,9855,9856],{"encoding":107},"\\mathrm{Var}\n\\!\\left(\n\\frac{X_n}{n}\n\\right)\n=\n\\frac{x(1-x)}{n}.",[81,9858,9860,10002],{"className":9859,"ariaHidden":112},[111],[81,9861,9863,9866,9873,9876,9879,9993,9996,9999],{"className":9862},[116],[81,9864],{"className":9865,"style":542},[120],[81,9867,9869],{"className":9868},[125],[81,9870,9872],{"className":9871},[125,8709],"Var",[81,9874],{"className":9875,"style":6885},[912],[81,9877],{"className":9878,"style":1025},[912],[81,9880,9882,9888,9987],{"className":9881},[546],[81,9883,9885],{"className":9884,"style":551},[435,550],[81,9886,410],{"className":9887},[555,230],[81,9889,9891,9894,9984],{"className":9890},[125],[81,9892],{"className":9893},[435,1247],[81,9895,9897],{"className":9896},[1199],[81,9898,9900,9976],{"className":9899},[208,569],[81,9901,9903,9973],{"className":9902},[212],[81,9904,9906,9917,9925],{"className":9905,"style":9267},[216],[81,9907,9908,9911],{"style":4196},[81,9909],{"className":9910,"style":583},[223],[81,9912,9914],{"className":9913},[125],[81,9915,2164],{"className":9916},[125,126],[81,9918,9919,9922],{"style":1278},[81,9920],{"className":9921,"style":583},[223],[81,9923],{"className":9924,"style":1286},[1285],[81,9926,9927,9930],{"style":4290},[81,9928],{"className":9929,"style":583},[223],[81,9931,9933],{"className":9932},[125],[81,9934,9936,9939],{"className":9935},[125],[81,9937,8594],{"className":9938,"style":8652},[125,126],[81,9940,9942],{"className":9941},[204],[81,9943,9945,9965],{"className":9944},[208,569],[81,9946,9948,9962],{"className":9947},[212],[81,9949,9951],{"className":9950,"style":6741},[216],[81,9952,9953,9956],{"style":8667},[81,9954],{"className":9955,"style":224},[223],[81,9957,9959],{"className":9958},[228,229,230,231],[81,9960,2164],{"className":9961},[125,126,231],[81,9963,606],{"className":9964},[605],[81,9966,9968],{"className":9967},[212],[81,9969,9971],{"className":9970,"style":3177},[216],[81,9972],{},[81,9974,606],{"className":9975},[605],[81,9977,9979],{"className":9978},[212],[81,9980,9982],{"className":9981,"style":4385},[216],[81,9983],{},[81,9985],{"className":9986},[442,1247],[81,9988,9990],{"className":9989,"style":551},[442,550],[81,9991,415],{"className":9992},[555,230],[81,9994],{"className":9995,"style":913},[912],[81,9997,833],{"className":9998},[917],[81,10000],{"className":10001,"style":913},[912],[81,10003,10005,10009,10093],{"className":10004},[116],[81,10006],{"className":10007,"style":10008},[120],"height:2.113em;vertical-align:-0.686em;",[81,10010,10012,10015,10090],{"className":10011},[125],[81,10013],{"className":10014},[435,1247],[81,10016,10018],{"className":10017},[1199],[81,10019,10021,10082],{"className":10020},[208,569],[81,10022,10024,10079],{"className":10023},[212],[81,10025,10028,10039,10047],{"className":10026,"style":10027},[216],"height:1.427em;",[81,10029,10030,10033],{"style":4196},[81,10031],{"className":10032,"style":583},[223],[81,10034,10036],{"className":10035},[125],[81,10037,2164],{"className":10038},[125,126],[81,10040,10041,10044],{"style":1278},[81,10042],{"className":10043,"style":583},[223],[81,10045],{"className":10046,"style":1286},[1285],[81,10048,10049,10052],{"style":4290},[81,10050],{"className":10051,"style":583},[223],[81,10053,10055,10058,10061,10064,10067,10070,10073,10076],{"className":10054},[125],[81,10056,3518],{"className":10057},[125,126],[81,10059,410],{"className":10060},[435],[81,10062,679],{"className":10063},[125],[81,10065],{"className":10066,"style":127},[912],[81,10068,6422],{"className":10069},[235],[81,10071],{"className":10072,"style":127},[912],[81,10074,3518],{"className":10075},[125,126],[81,10077,415],{"className":10078},[442],[81,10080,606],{"className":10081},[605],[81,10083,10085],{"className":10084},[212],[81,10086,10088],{"className":10087,"style":4385},[216],[81,10089],{},[81,10091],{"className":10092},[442,1247],[81,10094,65],{"className":10095},[125],[14,10097,7382],{},[81,10099,10101],{"className":10100,"translate":85},[809],[81,10102,10104,10126],{"className":10103,"translate":85},[84],[81,10105,10107],{"className":10106},[89],[91,10108,10109],{"xmlns":93,"display":818},[95,10110,10111,10123],{},[98,10112,10113],{},[1199,10114,10115,10121],{},[3006,10116,10117,10119],{},[101,10118,8594],{},[101,10120,2164],{},[101,10122,2164],{},[105,10124,10125],{"encoding":107},"\\frac{X_n}{n}",[81,10127,10129],{"className":10128,"ariaHidden":112},[111],[81,10130,10132,10136],{"className":10131},[116],[81,10133],{"className":10134,"style":10135},[120],"height:2.0463em;vertical-align:-0.686em;",[81,10137,10139,10142,10232],{"className":10138},[125],[81,10140],{"className":10141},[435,1247],[81,10143,10145],{"className":10144},[1199],[81,10146,10148,10224],{"className":10147},[208,569],[81,10149,10151,10221],{"className":10150},[212],[81,10152,10154,10165,10173],{"className":10153,"style":9267},[216],[81,10155,10156,10159],{"style":4196},[81,10157],{"className":10158,"style":583},[223],[81,10160,10162],{"className":10161},[125],[81,10163,2164],{"className":10164},[125,126],[81,10166,10167,10170],{"style":1278},[81,10168],{"className":10169,"style":583},[223],[81,10171],{"className":10172,"style":1286},[1285],[81,10174,10175,10178],{"style":4290},[81,10176],{"className":10177,"style":583},[223],[81,10179,10181],{"className":10180},[125],[81,10182,10184,10187],{"className":10183},[125],[81,10185,8594],{"className":10186,"style":8652},[125,126],[81,10188,10190],{"className":10189},[204],[81,10191,10193,10213],{"className":10192},[208,569],[81,10194,10196,10210],{"className":10195},[212],[81,10197,10199],{"className":10198,"style":6741},[216],[81,10200,10201,10204],{"style":8667},[81,10202],{"className":10203,"style":224},[223],[81,10205,10207],{"className":10206},[228,229,230,231],[81,10208,2164],{"className":10209},[125,126,231],[81,10211,606],{"className":10212},[605],[81,10214,10216],{"className":10215},[212],[81,10217,10219],{"className":10218,"style":3177},[216],[81,10220],{},[81,10222,606],{"className":10223},[605],[81,10225,10227],{"className":10226},[212],[81,10228,10230],{"className":10229,"style":4385},[216],[81,10231],{},[81,10233],{"className":10234},[442,1247],[14,10236,10237,10238,65],{},"becomes more and more concentrated around ",[81,10239,10241,10254],{"className":10240,"translate":85},[84],[81,10242,10244],{"className":10243},[89],[91,10245,10246],{"xmlns":93},[95,10247,10248,10252],{},[98,10249,10250],{},[101,10251,3518],{},[105,10253,3518],{"encoding":107},[81,10255,10257],{"className":10256,"ariaHidden":112},[111],[81,10258,10260,10263],{"className":10259},[116],[81,10261],{"className":10262,"style":293},[120],[81,10264,3518],{"className":10265},[125,126],[14,10267,10268,10269,1414],{},"As ",[81,10270,10272,10292],{"className":10271,"translate":85},[84],[81,10273,10275],{"className":10274},[89],[91,10276,10277],{"xmlns":93},[95,10278,10279,10289],{},[98,10280,10281,10283,10286],{},[101,10282,2164],{},[179,10284,10285],{},"→",[101,10287,10288],{"mathvariant":1133},"∞",[105,10290,10291],{"encoding":107},"n\\to\\infty",[81,10293,10295,10313],{"className":10294,"ariaHidden":112},[111],[81,10296,10298,10301,10304,10307,10310],{"className":10297},[116],[81,10299],{"className":10300,"style":293},[120],[81,10302,2164],{"className":10303},[125,126],[81,10305],{"className":10306,"style":913},[912],[81,10308,10285],{"className":10309},[917],[81,10311],{"className":10312,"style":913},[912],[81,10314,10316,10319],{"className":10315},[116],[81,10317],{"className":10318,"style":293},[120],[81,10320,10288],{"className":10321},[125],[81,10323,10325],{"className":10324,"translate":85},[809],[81,10326,10328,10356],{"className":10327,"translate":85},[84],[81,10329,10331],{"className":10330},[89],[91,10332,10333],{"xmlns":93,"display":818},[95,10334,10335,10353],{},[98,10336,10337,10347,10349,10351],{},[1199,10338,10339,10345],{},[3006,10340,10341,10343],{},[101,10342,8594],{},[101,10344,2164],{},[101,10346,2164],{},[179,10348,10285],{},[101,10350,3518],{},[101,10352,65],{"mathvariant":1133},[105,10354,10355],{"encoding":107},"\\frac{X_n}{n}\n\\to x.",[81,10357,10359,10473],{"className":10358,"ariaHidden":112},[111],[81,10360,10362,10365,10464,10467,10470],{"className":10361},[116],[81,10363],{"className":10364,"style":10135},[120],[81,10366,10368,10371,10461],{"className":10367},[125],[81,10369],{"className":10370},[435,1247],[81,10372,10374],{"className":10373},[1199],[81,10375,10377,10453],{"className":10376},[208,569],[81,10378,10380,10450],{"className":10379},[212],[81,10381,10383,10394,10402],{"className":10382,"style":9267},[216],[81,10384,10385,10388],{"style":4196},[81,10386],{"className":10387,"style":583},[223],[81,10389,10391],{"className":10390},[125],[81,10392,2164],{"className":10393},[125,126],[81,10395,10396,10399],{"style":1278},[81,10397],{"className":10398,"style":583},[223],[81,10400],{"className":10401,"style":1286},[1285],[81,10403,10404,10407],{"style":4290},[81,10405],{"className":10406,"style":583},[223],[81,10408,10410],{"className":10409},[125],[81,10411,10413,10416],{"className":10412},[125],[81,10414,8594],{"className":10415,"style":8652},[125,126],[81,10417,10419],{"className":10418},[204],[81,10420,10422,10442],{"className":10421},[208,569],[81,10423,10425,10439],{"className":10424},[212],[81,10426,10428],{"className":10427,"style":6741},[216],[81,10429,10430,10433],{"style":8667},[81,10431],{"className":10432,"style":224},[223],[81,10434,10436],{"className":10435},[228,229,230,231],[81,10437,2164],{"className":10438},[125,126,231],[81,10440,606],{"className":10441},[605],[81,10443,10445],{"className":10444},[212],[81,10446,10448],{"className":10447,"style":3177},[216],[81,10449],{},[81,10451,606],{"className":10452},[605],[81,10454,10456],{"className":10455},[212],[81,10457,10459],{"className":10458,"style":4385},[216],[81,10460],{},[81,10462],{"className":10463},[442,1247],[81,10465],{"className":10466,"style":913},[912],[81,10468,10285],{"className":10469},[917],[81,10471],{"className":10472,"style":913},[912],[81,10474,10476,10479,10482],{"className":10475},[116],[81,10477],{"className":10478,"style":293},[120],[81,10480,3518],{"className":10481},[125,126],[81,10483,65],{"className":10484},[125],[14,10486,10487],{},"Hence",[81,10489,10491],{"className":10490,"translate":85},[809],[81,10492,10494,10539],{"className":10493,"translate":85},[84],[81,10495,10497],{"className":10496},[89],[91,10498,10499],{"xmlns":93,"display":818},[95,10500,10501,10536],{},[98,10502,10503,10505,10507,10523,10526,10528,10530,10532,10534],{},[101,10504,6124],{},[2035,10506,6656],{},[98,10508,10509,10511,10521],{},[179,10510,410],{"fence":112},[1199,10512,10513,10519],{},[3006,10514,10515,10517],{},[101,10516,8594],{},[101,10518,2164],{},[101,10520,2164],{},[179,10522,415],{"fence":112},[179,10524,10525],{},"≈",[101,10527,6124],{},[179,10529,410],{"stretchy":409},[101,10531,3518],{},[179,10533,415],{"stretchy":409},[101,10535,65],{"mathvariant":1133},[105,10537,10538],{"encoding":107},"f\\!\\left(\\frac{X_n}{n}\\right)\n\\approx\nf(x).",[81,10540,10542,10680],{"className":10541,"ariaHidden":112},[111],[81,10543,10545,10548,10551,10554,10557,10671,10674,10677],{"className":10544},[116],[81,10546],{"className":10547,"style":542},[120],[81,10549,6124],{"className":10550,"style":6161},[125,126],[81,10552],{"className":10553,"style":6885},[912],[81,10555],{"className":10556,"style":1025},[912],[81,10558,10560,10566,10665],{"className":10559},[546],[81,10561,10563],{"className":10562,"style":551},[435,550],[81,10564,410],{"className":10565},[555,230],[81,10567,10569,10572,10662],{"className":10568},[125],[81,10570],{"className":10571},[435,1247],[81,10573,10575],{"className":10574},[1199],[81,10576,10578,10654],{"className":10577},[208,569],[81,10579,10581,10651],{"className":10580},[212],[81,10582,10584,10595,10603],{"className":10583,"style":9267},[216],[81,10585,10586,10589],{"style":4196},[81,10587],{"className":10588,"style":583},[223],[81,10590,10592],{"className":10591},[125],[81,10593,2164],{"className":10594},[125,126],[81,10596,10597,10600],{"style":1278},[81,10598],{"className":10599,"style":583},[223],[81,10601],{"className":10602,"style":1286},[1285],[81,10604,10605,10608],{"style":4290},[81,10606],{"className":10607,"style":583},[223],[81,10609,10611],{"className":10610},[125],[81,10612,10614,10617],{"className":10613},[125],[81,10615,8594],{"className":10616,"style":8652},[125,126],[81,10618,10620],{"className":10619},[204],[81,10621,10623,10643],{"className":10622},[208,569],[81,10624,10626,10640],{"className":10625},[212],[81,10627,10629],{"className":10628,"style":6741},[216],[81,10630,10631,10634],{"style":8667},[81,10632],{"className":10633,"style":224},[223],[81,10635,10637],{"className":10636},[228,229,230,231],[81,10638,2164],{"className":10639},[125,126,231],[81,10641,606],{"className":10642},[605],[81,10644,10646],{"className":10645},[212],[81,10647,10649],{"className":10648,"style":3177},[216],[81,10650],{},[81,10652,606],{"className":10653},[605],[81,10655,10657],{"className":10656},[212],[81,10658,10660],{"className":10659,"style":4385},[216],[81,10661],{},[81,10663],{"className":10664},[442,1247],[81,10666,10668],{"className":10667,"style":551},[442,550],[81,10669,415],{"className":10670},[555,230],[81,10672],{"className":10673,"style":913},[912],[81,10675,10525],{"className":10676},[917],[81,10678],{"className":10679,"style":913},[912],[81,10681,10683,10686,10689,10692,10695,10698],{"className":10682},[116],[81,10684],{"className":10685,"style":428},[120],[81,10687,6124],{"className":10688,"style":6161},[125,126],[81,10690,410],{"className":10691},[435],[81,10693,3518],{"className":10694},[125,126],[81,10696,415],{"className":10697},[442],[81,10699,65],{"className":10700},[125],[14,10702,7939],{},[81,10704,10706],{"className":10705,"translate":85},[809],[81,10707,10709,10783],{"className":10708,"translate":85},[84],[81,10710,10712],{"className":10711},[89],[91,10713,10714],{"xmlns":93,"display":818},[95,10715,10716,10780],{},[98,10717,10718,10724,10726,10728,10730,10732,10734,10736,10738,10740,10742,10768,10770,10772,10774,10776,10778],{},[3006,10719,10720,10722],{},[101,10721,6619],{},[101,10723,2164],{},[179,10725,410],{"stretchy":409},[101,10727,6124],{},[179,10729,415],{"stretchy":409},[179,10731,410],{"stretchy":409},[101,10733,3518],{},[179,10735,415],{"stretchy":409},[179,10737,833],{},[101,10739,9096],{},[2035,10741,6656],{},[98,10743,10744,10746,10748,10750,10766],{},[179,10745,497],{"fence":112},[101,10747,6124],{},[2035,10749,6656],{},[98,10751,10752,10754,10764],{},[179,10753,410],{"fence":112},[1199,10755,10756,10762],{},[3006,10757,10758,10760],{},[101,10759,8594],{},[101,10761,2164],{},[101,10763,2164],{},[179,10765,415],{"fence":112},[179,10767,529],{"fence":112},[179,10769,10525],{},[101,10771,6124],{},[179,10773,410],{"stretchy":409},[101,10775,3518],{},[179,10777,415],{"stretchy":409},[101,10779,65],{"mathvariant":1133},[105,10781,10782],{"encoding":107},"B_n(f)(x)\n=\nE\\!\\left[\nf\\!\\left(\\frac{X_n}{n}\\right)\n\\right]\n\\approx\nf(x).",[81,10784,10786,10859,11021],{"className":10785,"ariaHidden":112},[111],[81,10787,10789,10792,10832,10835,10838,10841,10844,10847,10850,10853,10856],{"className":10788},[116],[81,10790],{"className":10791,"style":428},[120],[81,10793,10795,10798],{"className":10794},[125],[81,10796,6619],{"className":10797,"style":6728},[125,126],[81,10799,10801],{"className":10800},[204],[81,10802,10804,10824],{"className":10803},[208,569],[81,10805,10807,10821],{"className":10806},[212],[81,10808,10810],{"className":10809,"style":6741},[216],[81,10811,10812,10815],{"style":6744},[81,10813],{"className":10814,"style":224},[223],[81,10816,10818],{"className":10817},[228,229,230,231],[81,10819,2164],{"className":10820},[125,126,231],[81,10822,606],{"className":10823},[605],[81,10825,10827],{"className":10826},[212],[81,10828,10830],{"className":10829,"style":3177},[216],[81,10831],{},[81,10833,410],{"className":10834},[435],[81,10836,6124],{"className":10837,"style":6161},[125,126],[81,10839,415],{"className":10840},[442],[81,10842,410],{"className":10843},[435],[81,10845,3518],{"className":10846},[125,126],[81,10848,415],{"className":10849},[442],[81,10851],{"className":10852,"style":913},[912],[81,10854,833],{"className":10855},[917],[81,10857],{"className":10858,"style":913},[912],[81,10860,10862,10865,10868,10871,10874,11012,11015,11018],{"className":10861},[116],[81,10863],{"className":10864,"style":542},[120],[81,10866,9096],{"className":10867,"style":9215},[125,126],[81,10869],{"className":10870,"style":6885},[912],[81,10872],{"className":10873,"style":1025},[912],[81,10875,10877,10883,10886,10889,10892,11006],{"className":10876},[546],[81,10878,10880],{"className":10879,"style":551},[435,550],[81,10881,497],{"className":10882},[555,230],[81,10884,6124],{"className":10885,"style":6161},[125,126],[81,10887],{"className":10888,"style":6885},[912],[81,10890],{"className":10891,"style":1025},[912],[81,10893,10895,10901,11000],{"className":10894},[546],[81,10896,10898],{"className":10897,"style":551},[435,550],[81,10899,410],{"className":10900},[555,230],[81,10902,10904,10907,10997],{"className":10903},[125],[81,10905],{"className":10906},[435,1247],[81,10908,10910],{"className":10909},[1199],[81,10911,10913,10989],{"className":10912},[208,569],[81,10914,10916,10986],{"className":10915},[212],[81,10917,10919,10930,10938],{"className":10918,"style":9267},[216],[81,10920,10921,10924],{"style":4196},[81,10922],{"className":10923,"style":583},[223],[81,10925,10927],{"className":10926},[125],[81,10928,2164],{"className":10929},[125,126],[81,10931,10932,10935],{"style":1278},[81,10933],{"className":10934,"style":583},[223],[81,10936],{"className":10937,"style":1286},[1285],[81,10939,10940,10943],{"style":4290},[81,10941],{"className":10942,"style":583},[223],[81,10944,10946],{"className":10945},[125],[81,10947,10949,10952],{"className":10948},[125],[81,10950,8594],{"className":10951,"style":8652},[125,126],[81,10953,10955],{"className":10954},[204],[81,10956,10958,10978],{"className":10957},[208,569],[81,10959,10961,10975],{"className":10960},[212],[81,10962,10964],{"className":10963,"style":6741},[216],[81,10965,10966,10969],{"style":8667},[81,10967],{"className":10968,"style":224},[223],[81,10970,10972],{"className":10971},[228,229,230,231],[81,10973,2164],{"className":10974},[125,126,231],[81,10976,606],{"className":10977},[605],[81,10979,10981],{"className":10980},[212],[81,10982,10984],{"className":10983,"style":3177},[216],[81,10985],{},[81,10987,606],{"className":10988},[605],[81,10990,10992],{"className":10991},[212],[81,10993,10995],{"className":10994,"style":4385},[216],[81,10996],{},[81,10998],{"className":10999},[442,1247],[81,11001,11003],{"className":11002,"style":551},[442,550],[81,11004,415],{"className":11005},[555,230],[81,11007,11009],{"className":11008,"style":551},[442,550],[81,11010,529],{"className":11011},[555,230],[81,11013],{"className":11014,"style":913},[912],[81,11016,10525],{"className":11017},[917],[81,11019],{"className":11020,"style":913},[912],[81,11022,11024,11027,11030,11033,11036,11039],{"className":11023},[116],[81,11025],{"className":11026,"style":428},[120],[81,11028,6124],{"className":11029,"style":6161},[125,126],[81,11031,410],{"className":11032},[435],[81,11034,3518],{"className":11035},[125,126],[81,11037,415],{"className":11038},[442],[81,11040,65],{"className":11041},[125],[14,11043,11044],{},"Intuitively, this holds.",[6100,11046,11048],{"id":11047},"rigorous-proof","Rigorous Proof",[14,11050,11051,11052,11080,11081,11109,11110,11160,11161,11215],{},"Since ",[81,11053,11055,11068],{"className":11054,"translate":85},[84],[81,11056,11058],{"className":11057},[89],[91,11059,11060],{"xmlns":93},[95,11061,11062,11066],{},[98,11063,11064],{},[101,11065,6124],{},[105,11067,6124],{"encoding":107},[81,11069,11071],{"className":11070,"ariaHidden":112},[111],[81,11072,11074,11077],{"className":11073},[116],[81,11075],{"className":11076,"style":6157},[120],[81,11078,6124],{"className":11079,"style":6161},[125,126]," is continuous on a closed interval, by properties of continuous functions on closed intervals: ",[81,11082,11084,11097],{"className":11083,"translate":85},[84],[81,11085,11087],{"className":11086},[89],[91,11088,11089],{"xmlns":93},[95,11090,11091,11095],{},[98,11092,11093],{},[101,11094,6124],{},[105,11096,6124],{"encoding":107},[81,11098,11100],{"className":11099,"ariaHidden":112},[111],[81,11101,11103,11106],{"className":11102},[116],[81,11104],{"className":11105,"style":6157},[120],[81,11107,6124],{"className":11108,"style":6161},[125,126]," must be uniformly continuous. Therefore, for any ",[81,11111,11113,11130],{"className":11112,"translate":85},[84],[81,11114,11116],{"className":11115},[89],[91,11117,11118],{"xmlns":93},[95,11119,11120,11128],{},[98,11121,11122,11124,11126],{},[101,11123,6305],{},[179,11125,6308],{},[515,11127,513],{},[105,11129,6313],{"encoding":107},[81,11131,11133,11151],{"className":11132,"ariaHidden":112},[111],[81,11134,11136,11139,11142,11145,11148],{"className":11135},[116],[81,11137],{"className":11138,"style":6323},[120],[81,11140,6305],{"className":11141},[125,126],[81,11143],{"className":11144,"style":913},[912],[81,11146,6308],{"className":11147},[917],[81,11149],{"className":11150,"style":913},[912],[81,11152,11154,11157],{"className":11153},[116],[81,11155],{"className":11156,"style":1107},[120],[81,11158,513],{"className":11159},[125]," there exists 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whenever",[81,11217,11219],{"className":11218,"translate":85},[809],[81,11220,11222,11249],{"className":11221,"translate":85},[84],[81,11223,11225],{"className":11224},[89],[91,11226,11227],{"xmlns":93,"display":818},[95,11228,11229,11246],{},[98,11230,11231,11233,11236,11238,11240,11242,11244],{},[101,11232,6411],{"mathvariant":1133},[101,11234,11235],{},"u",[179,11237,6422],{},[101,11239,373],{},[101,11241,6411],{"mathvariant":1133},[179,11243,6435],{},[101,11245,11175],{},[105,11247,11248],{"encoding":107},"|u-v|\u003C\\delta",[81,11250,11252,11273,11294],{"className":11251,"ariaHidden":112},[111],[81,11253,11255,11258,11261,11264,11267,11270],{"className":11254},[116],[81,11256],{"className":11257,"style":428},[120],[81,11259,6411],{"className":11260},[125],[81,11262,11235],{"className":11263},[125,126],[81,11265],{"className":11266,"style":127},[912],[81,11268,6422],{"className":11269},[235],[81,11271],{"className":11272,"style":127},[912],[81,11274,11276,11279,11282,11285,11288,11291],{"className":11275},[116],[81,11277],{"className":11278,"style":428},[120],[81,11280,373],{"className":11281,"style":297},[125,126],[81,11283,6411],{"className":11284},[125],[81,11286],{"className":11287,"style":913},[912],[81,11289,6435],{"className":11290},[917],[81,11292],{"className":11293,"style":913},[912],[81,11295,11297,11300],{"className":11296},[116],[81,11298],{"className":11299,"style":2059},[120],[81,11301,11175],{"className":11302,"style":11196},[125,126],[14,11304,11305],{},"we 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\\delta}.",[81,12291,12293,12337],{"className":12292,"ariaHidden":112},[111],[81,12294,12296,12299,12328,12331,12334],{"className":12295},[116],[81,12297],{"className":12298,"style":7907},[120],[81,12300,12302,12305],{"className":12301},[125],[81,12303,12097],{"className":12304},[125,126],[81,12306,12308],{"className":12307},[204],[81,12309,12311],{"className":12310},[208],[81,12312,12314],{"className":12313},[212],[81,12315,12317],{"className":12316,"style":7907},[216],[81,12318,12319,12322],{"style":1749},[81,12320],{"className":12321,"style":224},[223],[81,12323,12325],{"className":12324},[228,229,230,231],[81,12326,12257],{"className":12327},[125,126,231],[81,12329],{"className":12330,"style":913},[912],[81,12332,833],{"className":12333},[917],[81,12335],{"className":12336,"style":913},[912],[81,12338,12340,12343,12422],{"className":12339},[116],[81,12341],{"className":12342,"style":428},[120],[81,12344,12346,12349,12389,12392,12395,12398,12401,12404,12407,12410,12413,12416,12419],{"className":12345},[125],[81,12347,6411],{"className":12348},[125],[81,12350,12352,12355],{"className":12351},[125],[81,12353,8594],{"className":12354,"style":8652},[125,126],[81,12356,12358],{"className":12357},[204],[81,12359,12361,12381],{"className":12360},[208,569],[81,12362,12364,12378],{"className":12363},[212],[81,12365,12367],{"className":12366,"style":6741},[216],[81,12368,12369,12372],{"style":8667},[81,12370],{"className":12371,"style":224},[223],[81,12373,12375],{"className":12374},[228,229,230,231],[81,12376,2164],{"className":12377},[125,126,231],[81,12379,606],{"className":12380},[605],[81,12382,12384],{"className":12383},[212],[81,12385,12387],{"className":12386,"style":3177},[216],[81,12388],{},[81,12390,1134],{"className":12391},[125],[81,12393,2164],{"className":12394},[125,126],[81,12396],{"className":12397,"style":127},[912],[81,12399,6422],{"className":12400},[235],[81,12402],{"className":12403,"style":127},[912],[81,12405,3518],{"className":12406},[125,126],[81,12408,6411],{"className":12409},[125],[81,12411],{"className":12412,"style":913},[912],[81,12414,12282],{"className":12415},[917],[81,12417],{"className":12418,"style":913},[912],[81,12420,11175],{"className":12421,"style":11196},[125,126],[81,12423,65],{"className":12424},[125],[14,12426,12427,12430,12431,1414],{},[24,12428,12429],{},"First part:"," When ",[81,12432,12434,12468],{"className":12433,"translate":85},[84],[81,12435,12437],{"className":12436},[89],[91,12438,12439],{"xmlns":93},[95,12440,12441,12465],{},[98,12442,12443,12445,12451,12453,12455,12457,12459,12461,12463],{},[101,12444,6411],{"mathvariant":1133},[3006,12446,12447,12449],{},[101,12448,8594],{},[101,12450,2164],{},[101,12452,1134],{"mathvariant":1133},[101,12454,2164],{},[179,12456,6422],{},[101,12458,3518],{},[101,12460,6411],{"mathvariant":1133},[179,12462,6435],{},[101,12464,11175],{},[105,12466,12467],{"encoding":107},"|X_n\u002Fn-x|\u003C\\delta",[81,12469,12471,12535,12556],{"className":12470,"ariaHidden":112},[111],[81,12472,12474,12477,12480,12520,12523,12526,12529,12532],{"className":12473},[116],[81,12475],{"className":12476,"style":428},[120],[81,12478,6411],{"className":12479},[125],[81,12481,12483,12486],{"className":12482},[125],[81,12484,8594],{"className":12485,"style":8652},[125,126],[81,12487,12489],{"className":12488},[204],[81,12490,12492,12512],{"className":12491},[208,569],[81,12493,12495,12509],{"className":12494},[212],[81,12496,12498],{"className":12497,"style":6741},[216],[81,12499,12500,12503],{"style":8667},[81,12501],{"className":12502,"style":224},[223],[81,12504,12506],{"className":12505},[228,229,230,231],[81,12507,2164],{"className":12508},[125,126,231],[81,12510,606],{"className":12511},[605],[81,12513,12515],{"className":12514},[212],[81,12516,12518],{"className":12517,"style":3177},[216],[81,12519],{},[81,12521,1134],{"className":12522},[125],[81,12524,2164],{"className":12525},[125,126],[81,12527],{"className":12528,"style":127},[912],[81,12530,6422],{"className":12531},[235],[81,12533],{"className":12534,"style":127},[912],[81,12536,12538,12541,12544,12547,12550,12553],{"className":12537},[116],[81,12539],{"className":12540,"style":428},[120],[81,12542,3518],{"className":12543},[125,126],[81,12545,6411],{"className":12546},[125],[81,12548],{"className":12549,"style":913},[912],[81,12551,6435],{"className":12552},[917],[81,12554],{"className":12555,"style":913},[912],[81,12557,12559,12562],{"className":12558},[116],[81,12560],{"className":12561,"style":2059},[120],[81,12563,11175],{"className":12564,"style":11196},[125,126],[81,12566,12568],{"className":12567,"translate":85},[809],[81,12569,12571,12623],{"className":12570,"translate":85},[84],[81,12572,12574],{"className":12573},[89],[91,12575,12576],{"xmlns":93,"display":818},[95,12577,12578,12620],{},[98,12579,12580,12582,12584,12586,12592,12594,12596,12598,12600,12602,12604,12606,12608,12610,12612,12618],{},[101,12581,6411],{"mathvariant":1133},[101,12583,6124],{},[179,12585,410],{"stretchy":409},[3006,12587,12588,12590],{},[101,12589,8594],{},[101,12591,2164],{},[101,12593,1134],{"mathvariant":1133},[101,12595,2164],{},[179,12597,415],{"stretchy":409},[179,12599,6422],{},[101,12601,6124],{},[179,12603,410],{"stretchy":409},[101,12605,3518],{},[179,12607,415],{"stretchy":409},[101,12609,6411],{"mathvariant":1133},[179,12611,6435],{},[1199,12613,12614,12616],{},[101,12615,6305],{},[515,12617,517],{},[101,12619,65],{"mathvariant":1133},[105,12621,12622],{"encoding":107},"|f(X_n\u002Fn)-f(x)|\n\u003C\n\\frac{\\varepsilon}{2}.",[81,12624,12626,12699,12729],{"className":12625,"ariaHidden":112},[111],[81,12627,12629,12632,12635,12638,12641,12681,12684,12687,12690,12693,12696],{"className":12628},[116],[81,12630],{"className":12631,"style":428},[120],[81,12633,6411],{"className":12634},[125],[81,12636,6124],{"className":12637,"style":6161},[125,126],[81,12639,410],{"className":12640},[435],[81,12642,12644,12647],{"className":12643},[125],[81,12645,8594],{"className":12646,"style":8652},[125,126],[81,12648,12650],{"className":12649},[204],[81,12651,12653,12673],{"className":12652},[208,569],[81,12654,12656,12670],{"className":12655},[212],[81,12657,12659],{"className":12658,"style":6741},[216],[81,12660,12661,12664],{"style":8667},[81,12662],{"className":12663,"style":224},[223],[81,12665,12667],{"className":12666},[228,229,230,231],[81,12668,2164],{"className":12669},[125,126,231],[81,12671,606],{"className":12672},[605],[81,12674,12676],{"className":12675},[212],[81,12677,12679],{"className":12678,"style":3177},[216],[81,12680],{},[81,12682,1134],{"className":12683},[125],[81,12685,2164],{"className":12686},[125,126],[81,12688,415],{"className":12689},[442],[81,12691],{"className":12692,"style":127},[912],[81,12694,6422],{"className":12695},[235],[81,12697],{"className":12698,"style":127},[912],[81,12700,12702,12705,12708,12711,12714,12717,12720,12723,12726],{"className":12701},[116],[81,12703],{"className":12704,"style":428},[120],[81,12706,6124],{"className":12707,"style":6161},[125,126],[81,12709,410],{"className":12710},[435],[81,12712,3518],{"className":12713},[125,126],[81,12715,415],{"className":12716},[442],[81,12718,6411],{"className":12719},[125],[81,12721],{"className":12722,"style":913},[912],[81,12724,6435],{"className":12725},[917],[81,12727],{"className":12728,"style":913},[912],[81,12730,12732,12735,12797],{"className":12731},[116],[81,12733],{"className":12734,"style":11425},[120],[81,12736,12738,12741,12794],{"className":12737},[125],[81,12739],{"className":12740},[435,1247],[81,12742,12744],{"className":12743},[1199],[81,12745,12747,12786],{"className":12746},[208,569],[81,12748,12750,12783],{"className":12749},[212],[81,12751,12753,12764,12772],{"className":12752,"style":6991},[216],[81,12754,12755,12758],{"style":4196},[81,12756],{"className":12757,"style":583},[223],[81,12759,12761],{"className":12760},[125],[81,12762,517],{"className":12763},[125],[81,12765,12766,12769],{"style":1278},[81,12767],{"className":12768,"style":583},[223],[81,12770],{"className":12771,"style":1286},[1285],[81,12773,12774,12777],{"style":4290},[81,12775],{"className":12776,"style":583},[223],[81,12778,12780],{"className":12779},[125],[81,12781,6305],{"className":12782},[125,126],[81,12784,606],{"className":12785},[605],[81,12787,12789],{"className":12788},[212],[81,12790,12792],{"className":12791,"style":4385},[216],[81,12793],{},[81,12795],{"className":12796},[442,1247],[81,12798,65],{"className":12799},[125],[14,12801,12802,12803,65],{},"The contribution does not exceed ",[81,12804,12806,12824],{"className":12805,"translate":85},[84],[81,12807,12809],{"className":12808},[89],[91,12810,12811],{"xmlns":93},[95,12812,12813,12821],{},[98,12814,12815],{},[1199,12816,12817,12819],{},[101,12818,6305],{},[515,12820,517],{},[105,12822,12823],{"encoding":107},"\\frac{\\varepsilon}{2}",[81,12825,12827],{"className":12826,"ariaHidden":112},[111],[81,12828,12830,12834],{"className":12829},[116],[81,12831],{"className":12832,"style":12833},[120],"height:1.0404em;vertical-align:-0.345em;",[81,12835,12837,12840,12900],{"className":12836},[125],[81,12838],{"className":12839},[435,1247],[81,12841,12843],{"className":12842},[1199],[81,12844,12846,12892],{"className":12845},[208,569],[81,12847,12849,12889],{"className":12848},[212],[81,12850,12853,12867,12875],{"className":12851,"style":12852},[216],"height:0.6954em;",[81,12854,12855,12858],{"style":1263},[81,12856],{"className":12857,"style":583},[223],[81,12859,12861],{"className":12860},[228,229,230,231],[81,12862,12864],{"className":12863},[125,231],[81,12865,517],{"className":12866},[125,231],[81,12868,12869,12872],{"style":1278},[81,12870],{"className":12871,"style":583},[223],[81,12873],{"className":12874,"style":1286},[1285],[81,12876,12877,12880],{"style":1289},[81,12878],{"className":12879,"style":583},[223],[81,12881,12883],{"className":12882},[228,229,230,231],[81,12884,12886],{"className":12885},[125,231],[81,12887,6305],{"className":12888},[125,126,231],[81,12890,606],{"className":12891},[605],[81,12893,12895],{"className":12894},[212],[81,12896,12898],{"className":12897,"style":1311},[216],[81,12899],{},[81,12901],{"className":12902},[442,1247],[14,12904,12905,12908,12909,13056],{},[24,12906,12907],{},"Second part:"," Let ",[81,12910,12912,12957],{"className":12911,"translate":85},[84],[81,12913,12915],{"className":12914},[89],[91,12916,12917],{"xmlns":93},[95,12918,12919,12954],{},[98,12920,12921,12924,12926,12948,12950,12952],{},[101,12922,12923],{},"M",[179,12925,833],{},[3006,12927,12928,12936],{},[98,12929,12930,12933],{},[101,12931,12932],{},"max",[179,12934,12935],{},"⁡",[98,12937,12938,12940,12942,12944,12946],{},[179,12939,497],{"stretchy":409},[515,12941,513],{},[179,12943,1414],{"separator":112},[515,12945,679],{},[179,12947,529],{"stretchy":409},[101,12949,6411],{"mathvariant":1133},[101,12951,6124],{},[101,12953,6411],{"mathvariant":1133},[105,12955,12956],{"encoding":107},"M=\\max_{[0,1]}|f|",[81,12958,12960,12979],{"className":12959,"ariaHidden":112},[111],[81,12961,12963,12966,12970,12973,12976],{"className":12962},[116],[81,12964],{"className":12965,"style":121},[120],[81,12967,12923],{"className":12968,"style":12969},[125,126],"margin-right:0.109em;",[81,12971],{"className":12972,"style":913},[912],[81,12974,833],{"className":12975},[917],[81,12977],{"className":12978,"style":913},[912],[81,12980,12982,12986,13044,13047,13050,13053],{"className":12981},[116],[81,12983],{"className":12984,"style":12985},[120],"height:1.1052em;vertical-align:-0.3552em;",[81,12987,12989,12992],{"className":12988},[6802],[81,12990,12932],{"className":12991},[6802],[81,12993,12995],{"className":12994},[204],[81,12996,12998,13035],{"className":12997},[208,569],[81,12999,13001,13032],{"className":13000},[212],[81,13002,13005],{"className":13003,"style":13004},[216],"height:0.3448em;",[81,13006,13008,13011],{"style":13007},"top:-2.5198em;margin-right:0.05em;",[81,13009],{"className":13010,"style":224},[223],[81,13012,13014],{"className":13013},[228,229,230,231],[81,13015,13017,13020,13023,13026,13029],{"className":13016},[125,231],[81,13018,497],{"className":13019},[435,231],[81,13021,513],{"className":13022},[125,231],[81,13024,1414],{"className":13025},[1441,231],[81,13027,679],{"className":13028},[125,231],[81,13030,529],{"className":13031},[442,231],[81,13033,606],{"className":13034},[605],[81,13036,13038],{"className":13037},[212],[81,13039,13042],{"className":13040,"style":13041},[216],"height:0.3552em;",[81,13043],{},[81,13045],{"className":13046,"style":1025},[912],[81,13048,6411],{"className":13049},[125],[81,13051,6124],{"className":13052,"style":6161},[125,126],[81,13054,6411],{"className":13055},[125],". Then",[81,13058,13060],{"className":13059,"translate":85},[809],[81,13061,13063,13113],{"className":13062,"translate":85},[84],[81,13064,13066],{"className":13065},[89],[91,13067,13068],{"xmlns":93,"display":818},[95,13069,13070,13110],{},[98,13071,13072,13074,13076,13078,13084,13086,13088,13090,13092,13094,13096,13098,13100,13102,13104,13106,13108],{},[101,13073,6411],{"mathvariant":1133},[101,13075,6124],{},[179,13077,410],{"stretchy":409},[3006,13079,13080,13082],{},[101,13081,8594],{},[101,13083,2164],{},[101,13085,1134],{"mathvariant":1133},[101,13087,2164],{},[179,13089,415],{"stretchy":409},[179,13091,6422],{},[101,13093,6124],{},[179,13095,410],{"stretchy":409},[101,13097,3518],{},[179,13099,415],{"stretchy":409},[101,13101,6411],{"mathvariant":1133},[179,13103,11907],{},[515,13105,517],{},[101,13107,12923],{},[101,13109,65],{"mathvariant":1133},[105,13111,13112],{"encoding":107},"|f(X_n\u002Fn)-f(x)|\n\\le\n2M.",[81,13114,13116,13189,13219],{"className":13115,"ariaHidden":112},[111],[81,13117,13119,13122,13125,13128,13131,13171,13174,13177,13180,13183,13186],{"className":13118},[116],[81,13120],{"className":13121,"style":428},[120],[81,13123,6411],{"className":13124},[125],[81,13126,6124],{"className":13127,"style":6161},[125,126],[81,13129,410],{"className":13130},[435],[81,13132,13134,13137],{"className":13133},[125],[81,13135,8594],{"className":13136,"style":8652},[125,126],[81,13138,13140],{"className":13139},[204],[81,13141,13143,13163],{"className":13142},[208,569],[81,13144,13146,13160],{"className":13145},[212],[81,13147,13149],{"className":13148,"style":6741},[216],[81,13150,13151,13154],{"style":8667},[81,13152],{"className":13153,"style":224},[223],[81,13155,13157],{"className":13156},[228,229,230,231],[81,13158,2164],{"className":13159},[125,126,231],[81,13161,606],{"className":13162},[605],[81,13164,13166],{"className":13165},[212],[81,13167,13169],{"className":13168,"style":3177},[216],[81,13170],{},[81,13172,1134],{"className":13173},[125],[81,13175,2164],{"className":13176},[125,126],[81,13178,415],{"className":13179},[442],[81,13181],{"className":13182,"style":127},[912],[81,13184,6422],{"className":13185},[235],[81,13187],{"className":13188,"style":127},[912],[81,13190,13192,13195,13198,13201,13204,13207,13210,13213,13216],{"className":13191},[116],[81,13193],{"className":13194,"style":428},[120],[81,13196,6124],{"className":13197,"style":6161},[125,126],[81,13199,410],{"className":13200},[435],[81,13202,3518],{"className":13203},[125,126],[81,13205,415],{"className":13206},[442],[81,13208,6411],{"className":13209},[125],[81,13211],{"className":13212,"style":913},[912],[81,13214,11907],{"className":13215},[917],[81,13217],{"className":13218,"style":913},[912],[81,13220,13222,13225,13228,13231],{"className":13221},[116],[81,13223],{"className":13224,"style":121},[120],[81,13226,517],{"className":13227},[125],[81,13229,12923],{"className":13230,"style":12969},[125,126],[81,13232,65],{"className":13233},[125],[14,13235,13236,13237,65],{},"Thus the contribution does not exceed ",[81,13238,13240,13286],{"className":13239,"translate":85},[84],[81,13241,13243],{"className":13242},[89],[91,13244,13245],{"xmlns":93},[95,13246,13247,13283],{},[98,13248,13249,13251,13253,13255,13257,13259,13261,13267,13269,13271,13273,13275,13277,13279,13281],{},[515,13250,517],{},[101,13252,12923],{},[179,13254,2372],{"separator":112},[101,13256,6362],{},[179,13258,410],{"stretchy":409},[101,13260,6411],{"mathvariant":1133},[3006,13262,13263,13265],{},[101,13264,8594],{},[101,13266,2164],{},[101,13268,1134],{"mathvariant":1133},[101,13270,2164],{},[179,13272,6422],{},[101,13274,3518],{},[101,13276,6411],{"mathvariant":1133},[179,13278,12282],{},[101,13280,11175],{},[179,13282,415],{"stretchy":409},[105,13284,13285],{"encoding":107},"2M·P(|X_n\u002Fn-x|\\ge\\delta)",[81,13287,13289,13371,13392],{"className":13288,"ariaHidden":112},[111],[81,13290,13292,13295,13298,13301,13304,13307,13310,13313,13316,13356,13359,13362,13365,13368],{"className":13291},[116],[81,13293],{"className":13294,"style":428},[120],[81,13296,517],{"className":13297},[125],[81,13299,12923],{"className":13300,"style":12969},[125,126],[81,13302,2372],{"className":13303},[1441],[81,13305],{"className":13306,"style":1025},[912],[81,13308,6362],{"className":13309,"style":1394},[125,126],[81,13311,410],{"className":13312},[435],[81,13314,6411],{"className":13315},[125],[81,13317,13319,13322],{"className":13318},[125],[81,13320,8594],{"className":13321,"style":8652},[125,126],[81,13323,13325],{"className":13324},[204],[81,13326,13328,13348],{"className":13327},[208,569],[81,13329,13331,13345],{"className":13330},[212],[81,13332,13334],{"className":13333,"style":6741},[216],[81,13335,13336,13339],{"style":8667},[81,13337],{"className":13338,"style":224},[223],[81,13340,13342],{"className":13341},[228,229,230,231],[81,13343,2164],{"className":13344},[125,126,231],[81,13346,606],{"className":13347},[605],[81,13349,13351],{"className":13350},[212],[81,13352,13354],{"className":13353,"style":3177},[216],[81,13355],{},[81,13357,1134],{"className":13358},[125],[81,13360,2164],{"className":13361},[125,126],[81,13363],{"className":13364,"style":127},[912],[81,13366,6422],{"className":13367},[235],[81,13369],{"className":13370,"style":127},[912],[81,13372,13374,13377,13380,13383,13386,13389],{"className":13373},[116],[81,13375],{"className":13376,"style":428},[120],[81,13378,3518],{"className":13379},[125,126],[81,13381,6411],{"className":13382},[125],[81,13384],{"className":13385,"style":913},[912],[81,13387,12282],{"className":13388},[917],[81,13390],{"className":13391,"style":913},[912],[81,13393,13395,13398,13401],{"className":13394},[116],[81,13396],{"className":13397,"style":428},[120],[81,13399,11175],{"className":13400,"style":11196},[125,126],[81,13402,415],{"className":13403},[442],[14,13405,13406],{},"Using Chebyshev's inequality:",[81,13408,13410],{"className":13409,"translate":85},[809],[81,13411,13413,13501],{"className":13412,"translate":85},[84],[81,13414,13416],{"className":13415},[89],[91,13417,13418],{"xmlns":93,"display":818},[95,13419,13420,13498],{},[98,13421,13422,13424,13426,13428,13434,13436,13438,13440,13442,13444,13446,13448,13450,13452,13478,13480,13496],{},[101,13423,6362],{},[179,13425,410],{"stretchy":409},[101,13427,6411],{"mathvariant":1133},[3006,13429,13430,13432],{},[101,13431,8594],{},[101,13433,2164],{},[101,13435,1134],{"mathvariant":1133},[101,13437,2164],{},[179,13439,6422],{},[101,13441,3518],{},[101,13443,6411],{"mathvariant":1133},[179,13445,12282],{},[101,13447,11175],{},[179,13449,415],{"stretchy":409},[179,13451,11907],{},[1199,13453,13454,13468],{},[98,13455,13456,13458,13460,13462,13464,13466],{},[101,13457,3518],{},[179,13459,410],{"stretchy":409},[515,13461,679],{},[179,13463,6422],{},[101,13465,3518],{},[179,13467,415],{"stretchy":409},[98,13469,13470,13472],{},[101,13471,2164],{},[174,13473,13474,13476],{},[101,13475,11175],{},[515,13477,517],{},[179,13479,11907],{},[1199,13481,13482,13484],{},[515,13483,679],{},[98,13485,13486,13488,13490],{},[515,13487,686],{},[101,13489,2164],{},[174,13491,13492,13494],{},[101,13493,11175],{},[515,13495,517],{},[101,13497,65],{"mathvariant":1133},[105,13499,13500],{"encoding":107},"P(|X_n\u002Fn-x|\\ge\\delta)\n\\le\n\\frac{x(1-x)}\n{n\\delta^2}\n\\le\n\\frac1{4n\\delta^2}.",[81,13502,13504,13574,13595,13616,13744],{"className":13503,"ariaHidden":112},[111],[81,13505,13507,13510,13513,13516,13519,13559,13562,13565,13568,13571],{"className":13506},[116],[81,13508],{"className":13509,"style":428},[120],[81,13511,6362],{"className":13512,"style":1394},[125,126],[81,13514,410],{"className":13515},[435],[81,13517,6411],{"className":13518},[125],[81,13520,13522,13525],{"className":13521},[125],[81,13523,8594],{"className":13524,"style":8652},[125,126],[81,13526,13528],{"className":13527},[204],[81,13529,13531,13551],{"className":13530},[208,569],[81,13532,13534,13548],{"className":13533},[212],[81,13535,13537],{"className":13536,"style":6741},[216],[81,13538,13539,13542],{"style":8667},[81,13540],{"className":13541,"style":224},[223],[81,13543,13545],{"className":13544},[228,229,230,231],[81,13546,2164],{"className":13547},[125,126,231],[81,13549,606],{"className":13550},[605],[81,13552,13554],{"className":13553},[212],[81,13555,13557],{"className":13556,"style":3177},[216],[81,13558],{},[81,13560,1134],{"className":13561},[125],[81,13563,2164],{"className":13564},[125,126],[81,13566],{"className":13567,"style":127},[912],[81,13569,6422],{"className":13570},[235],[81,13572],{"className":13573,"style":127},[912],[81,13575,13577,13580,13583,13586,13589,13592],{"className":13576},[116],[81,13578],{"className":13579,"style":428},[120],[81,13581,3518],{"className":13582},[125,126],[81,13584,6411],{"className":13585},[125],[81,13587],{"className":13588,"style":913},[912],[81,13590,12282],{"className":13591},[917],[81,13593],{"className":13594,"style":913},[912],[81,13596,13598,13601,13604,13607,13610,13613],{"className":13597},[116],[81,13599],{"className":13600,"style":428},[120],[81,13602,11175],{"className":13603,"style":11196},[125,126],[81,13605,415],{"className":13606},[442],[81,13608],{"className":13609,"style":913},[912],[81,13611,11907],{"className":13612},[917],[81,13614],{"className":13615,"style":913},[912],[81,13617,13619,13622,13735,13738,13741],{"className":13618},[116],[81,13620],{"className":13621,"style":10008},[120],[81,13623,13625,13628,13732],{"className":13624},[125],[81,13626],{"className":13627},[435,1247],[81,13629,13631],{"className":13630},[1199],[81,13632,13634,13724],{"className":13633},[208,569],[81,13635,13637,13721],{"className":13636},[212],[81,13638,13640,13681,13689],{"className":13639,"style":10027},[216],[81,13641,13642,13645],{"style":4196},[81,13643],{"className":13644,"style":583},[223],[81,13646,13648,13651],{"className":13647},[125],[81,13649,2164],{"className":13650},[125,126],[81,13652,13654,13657],{"className":13653},[125],[81,13655,11175],{"className":13656,"style":11196},[125,126],[81,13658,13660],{"className":13659},[204],[81,13661,13663],{"className":13662},[208],[81,13664,13666],{"className":13665},[212],[81,13667,13670],{"className":13668,"style":13669},[216],"height:0.7401em;",[81,13671,13672,13675],{"style":4228},[81,13673],{"className":13674,"style":224},[223],[81,13676,13678],{"className":13677},[228,229,230,231],[81,13679,517],{"className":13680},[125,231],[81,13682,13683,13686],{"style":1278},[81,13684],{"className":13685,"style":583},[223],[81,13687],{"className":13688,"style":1286},[1285],[81,13690,13691,13694],{"style":4290},[81,13692],{"className":13693,"style":583},[223],[81,13695,13697,13700,13703,13706,13709,13712,13715,13718],{"className":13696},[125],[81,13698,3518],{"className":13699},[125,126],[81,13701,410],{"className":13702},[435],[81,13704,679],{"className":13705},[125],[81,13707],{"className":13708,"style":127},[912],[81,13710,6422],{"className":13711},[235],[81,13713],{"className":13714,"style":127},[912],[81,13716,3518],{"className":13717},[125,126],[81,13719,415],{"className":13720},[442],[81,13722,606],{"className":13723},[605],[81,13725,13727],{"className":13726},[212],[81,13728,13730],{"className":13729,"style":4385},[216],[81,13731],{},[81,13733],{"className":13734},[442,1247],[81,13736],{"className":13737,"style":913},[912],[81,13739,11907],{"className":13740},[917],[81,13742],{"className":13743,"style":913},[912],[81,13745,13747,13751,13845],{"className":13746},[116],[81,13748],{"className":13749,"style":13750},[120],"height:2.0074em;vertical-align:-0.686em;",[81,13752,13754,13757,13842],{"className":13753},[125],[81,13755],{"className":13756},[435,1247],[81,13758,13760],{"className":13759},[1199],[81,13761,13763,13834],{"className":13762},[208,569],[81,13764,13766,13831],{"className":13765},[212],[81,13767,13769,13812,13820],{"className":13768,"style":8162},[216],[81,13770,13771,13774],{"style":4196},[81,13772],{"className":13773,"style":583},[223],[81,13775,13777,13780,13783],{"className":13776},[125],[81,13778,686],{"className":13779},[125],[81,13781,2164],{"className":13782},[125,126],[81,13784,13786,13789],{"className":13785},[125],[81,13787,11175],{"className":13788,"style":11196},[125,126],[81,13790,13792],{"className":13791},[204],[81,13793,13795],{"className":13794},[208],[81,13796,13798],{"className":13797},[212],[81,13799,13801],{"className":13800,"style":13669},[216],[81,13802,13803,13806],{"style":4228},[81,13804],{"className":13805,"style":224},[223],[81,13807,13809],{"className":13808},[228,229,230,231],[81,13810,517],{"className":13811},[125,231],[81,13813,13814,13817],{"style":1278},[81,13815],{"className":13816,"style":583},[223],[81,13818],{"className":13819,"style":1286},[1285],[81,13821,13822,13825],{"style":4290},[81,13823],{"className":13824,"style":583},[223],[81,13826,13828],{"className":13827},[125],[81,13829,679],{"className":13830},[125],[81,13832,606],{"className":13833},[605],[81,13835,13837],{"className":13836},[212],[81,13838,13840],{"className":13839,"style":4385},[216],[81,13841],{},[81,13843],{"className":13844},[442,1247],[81,13846,65],{"className":13847},[125],[14,13849,7382],{},[81,13851,13853],{"className":13852,"translate":85},[809],[81,13854,13856,13922],{"className":13855,"translate":85},[84],[81,13857,13859],{"className":13858},[89],[91,13860,13861],{"xmlns":93,"display":818},[95,13862,13863,13919],{},[98,13864,13865,13867,13869,13871,13873,13875,13877,13883,13885,13887,13889,13891,13893,13895,13897,13899,13901,13917],{},[515,13866,517],{},[101,13868,12923],{},[179,13870,2372],{"separator":112},[101,13872,6362],{},[179,13874,410],{"stretchy":409},[101,13876,6411],{"mathvariant":1133},[3006,13878,13879,13881],{},[101,13880,8594],{},[101,13882,2164],{},[101,13884,1134],{"mathvariant":1133},[101,13886,2164],{},[179,13888,6422],{},[101,13890,3518],{},[101,13892,6411],{"mathvariant":1133},[179,13894,12282],{},[101,13896,11175],{},[179,13898,415],{"stretchy":409},[179,13900,11907],{},[1199,13902,13903,13905],{},[101,13904,12923],{},[98,13906,13907,13909,13911],{},[515,13908,517],{},[101,13910,2164],{},[174,13912,13913,13915],{},[101,13914,11175],{},[515,13916,517],{},[101,13918,65],{"mathvariant":1133},[105,13920,13921],{"encoding":107},"2M·\nP(|X_n\u002Fn-x|\\ge\\delta)\n\\le\n\\frac{M}{2n\\delta^2}.",[81,13923,13925,14007,14028,14049],{"className":13924,"ariaHidden":112},[111],[81,13926,13928,13931,13934,13937,13940,13943,13946,13949,13952,13992,13995,13998,14001,14004],{"className":13927},[116],[81,13929],{"className":13930,"style":428},[120],[81,13932,517],{"className":13933},[125],[81,13935,12923],{"className":13936,"style":12969},[125,126],[81,13938,2372],{"className":13939},[1441],[81,13941],{"className":13942,"style":1025},[912],[81,13944,6362],{"className":13945,"style":1394},[125,126],[81,13947,410],{"className":13948},[435],[81,13950,6411],{"className":13951},[125],[81,13953,13955,13958],{"className":13954},[125],[81,13956,8594],{"className":13957,"style":8652},[125,126],[81,13959,13961],{"className":13960},[204],[81,13962,13964,13984],{"className":13963},[208,569],[81,13965,13967,13981],{"className":13966},[212],[81,13968,13970],{"className":13969,"style":6741},[216],[81,13971,13972,13975],{"style":8667},[81,13973],{"className":13974,"style":224},[223],[81,13976,13978],{"className":13977},[228,229,230,231],[81,13979,2164],{"className":13980},[125,126,231],[81,13982,606],{"className":13983},[605],[81,13985,13987],{"className":13986},[212],[81,13988,13990],{"className":13989,"style":3177},[216],[81,13991],{},[81,13993,1134],{"className":13994},[125],[81,13996,2164],{"className":13997},[125,126],[81,13999],{"className":14000,"style":127},[912],[81,14002,6422],{"className":14003},[235],[81,14005],{"className":14006,"style":127},[912],[81,14008,14010,14013,14016,14019,14022,14025],{"className":14009},[116],[81,14011],{"className":14012,"style":428},[120],[81,14014,3518],{"className":14015},[125,126],[81,14017,6411],{"className":14018},[125],[81,14020],{"className":14021,"style":913},[912],[81,14023,12282],{"className":14024},[917],[81,14026],{"className":14027,"style":913},[912],[81,14029,14031,14034,14037,14040,14043,14046],{"className":14030},[116],[81,14032],{"className":14033,"style":428},[120],[81,14035,11175],{"className":14036,"style":11196},[125,126],[81,14038,415],{"className":14039},[442],[81,14041],{"className":14042,"style":913},[912],[81,14044,11907],{"className":14045},[917],[81,14047],{"className":14048,"style":913},[912],[81,14050,14052,14055,14149],{"className":14051},[116],[81,14053],{"className":14054,"style":10135},[120],[81,14056,14058,14061,14146],{"className":14057},[125],[81,14059],{"className":14060},[435,1247],[81,14062,14064],{"className":14063},[1199],[81,14065,14067,14138],{"className":14066},[208,569],[81,14068,14070,14135],{"className":14069},[212],[81,14071,14073,14116,14124],{"className":14072,"style":9267},[216],[81,14074,14075,14078],{"style":4196},[81,14076],{"className":14077,"style":583},[223],[81,14079,14081,14084,14087],{"className":14080},[125],[81,14082,517],{"className":14083},[125],[81,14085,2164],{"className":14086},[125,126],[81,14088,14090,14093],{"className":14089},[125],[81,14091,11175],{"className":14092,"style":11196},[125,126],[81,14094,14096],{"className":14095},[204],[81,14097,14099],{"className":14098},[208],[81,14100,14102],{"className":14101},[212],[81,14103,14105],{"className":14104,"style":13669},[216],[81,14106,14107,14110],{"style":4228},[81,14108],{"className":14109,"style":224},[223],[81,14111,14113],{"className":14112},[228,229,230,231],[81,14114,517],{"className":14115},[125,231],[81,14117,14118,14121],{"style":1278},[81,14119],{"className":14120,"style":583},[223],[81,14122],{"className":14123,"style":1286},[1285],[81,14125,14126,14129],{"style":4290},[81,14127],{"className":14128,"style":583},[223],[81,14130,14132],{"className":14131},[125],[81,14133,12923],{"className":14134,"style":12969},[125,126],[81,14136,606],{"className":14137},[605],[81,14139,14141],{"className":14140},[212],[81,14142,14144],{"className":14143,"style":4385},[216],[81,14145],{},[81,14147],{"className":14148},[442,1247],[81,14150,65],{"className":14151},[125],[14,14153,14154,14155,14183],{},"When ",[81,14156,14158,14171],{"className":14157,"translate":85},[84],[81,14159,14161],{"className":14160},[89],[91,14162,14163],{"xmlns":93},[95,14164,14165,14169],{},[98,14166,14167],{},[101,14168,2164],{},[105,14170,2164],{"encoding":107},[81,14172,14174],{"className":14173,"ariaHidden":112},[111],[81,14175,14177,14180],{"className":14176},[116],[81,14178],{"className":14179,"style":293},[120],[81,14181,2164],{"className":14182},[125,126]," is sufficiently large,",[81,14185,14187],{"className":14186,"translate":85},[809],[81,14188,14190,14228],{"className":14189,"translate":85},[84],[81,14191,14193],{"className":14192},[89],[91,14194,14195],{"xmlns":93,"display":818},[95,14196,14197,14225],{},[98,14198,14199,14215,14217,14223],{},[1199,14200,14201,14203],{},[101,14202,12923],{},[98,14204,14205,14207,14209],{},[515,14206,517],{},[101,14208,2164],{},[174,14210,14211,14213],{},[101,14212,11175],{},[515,14214,517],{},[179,14216,6435],{},[1199,14218,14219,14221],{},[101,14220,6305],{},[515,14222,517],{},[101,14224,65],{"mathvariant":1133},[105,14226,14227],{"encoding":107},"\\frac{M}{2n\\delta^2}\n\u003C\n\\frac{\\varepsilon}{2}.",[81,14229,14231,14340],{"className":14230,"ariaHidden":112},[111],[81,14232,14234,14237,14331,14334,14337],{"className":14233},[116],[81,14235],{"className":14236,"style":10135},[120],[81,14238,14240,14243,14328],{"className":14239},[125],[81,14241],{"className":14242},[435,1247],[81,14244,14246],{"className":14245},[1199],[81,14247,14249,14320],{"className":14248},[208,569],[81,14250,14252,14317],{"className":14251},[212],[81,14253,14255,14298,14306],{"className":14254,"style":9267},[216],[81,14256,14257,14260],{"style":4196},[81,14258],{"className":14259,"style":583},[223],[81,14261,14263,14266,14269],{"className":14262},[125],[81,14264,517],{"className":14265},[125],[81,14267,2164],{"className":14268},[125,126],[81,14270,14272,14275],{"className":14271},[125],[81,14273,11175],{"className":14274,"style":11196},[125,126],[81,14276,14278],{"className":14277},[204],[81,14279,14281],{"className":14280},[208],[81,14282,14284],{"className":14283},[212],[81,14285,14287],{"className":14286,"style":13669},[216],[81,14288,14289,14292],{"style":4228},[81,14290],{"className":14291,"style":224},[223],[81,14293,14295],{"className":14294},[228,229,230,231],[81,14296,517],{"className":14297},[125,231],[81,14299,14300,14303],{"style":1278},[81,14301],{"className":14302,"style":583},[223],[81,14304],{"className":14305,"style":1286},[1285],[81,14307,14308,14311],{"style":4290},[81,14309],{"className":14310,"style":583},[223],[81,14312,14314],{"className":14313},[125],[81,14315,12923],{"className":14316,"style":12969},[125,126],[81,14318,606],{"className":14319},[605],[81,14321,14323],{"className":14322},[212],[81,14324,14326],{"className":14325,"style":4385},[216],[81,14327],{},[81,14329],{"className":14330},[442,1247],[81,14332],{"className":14333,"style":913},[912],[81,14335,6435],{"className":14336},[917],[81,14338],{"className":14339,"style":913},[912],[81,14341,14343,14346,14408],{"className":14342},[116],[81,14344],{"className":14345,"style":11425},[120],[81,14347,14349,14352,14405],{"className":14348},[125],[81,14350],{"className":14351},[435,1247],[81,14353,14355],{"className":14354},[1199],[81,14356,14358,14397],{"className":14357},[208,569],[81,14359,14361,14394],{"className":14360},[212],[81,14362,14364,14375,14383],{"className":14363,"style":6991},[216],[81,14365,14366,14369],{"style":4196},[81,14367],{"className":14368,"style":583},[223],[81,14370,14372],{"className":14371},[125],[81,14373,517],{"className":14374},[125],[81,14376,14377,14380],{"style":1278},[81,14378],{"className":14379,"style":583},[223],[81,14381],{"className":14382,"style":1286},[1285],[81,14384,14385,14388],{"style":4290},[81,14386],{"className":14387,"style":583},[223],[81,14389,14391],{"className":14390},[125],[81,14392,6305],{"className":14393},[125,126],[81,14395,606],{"className":14396},[605],[81,14398,14400],{"className":14399},[212],[81,14401,14403],{"className":14402,"style":4385},[216],[81,14404],{},[81,14406],{"className":14407},[442,1247],[81,14409,65],{"className":14410},[125],[14,14412,7939],{},[81,14414,14416],{"className":14415,"translate":85},[809],[81,14417,14419,14469],{"className":14418,"translate":85},[84],[81,14420,14422],{"className":14421},[89],[91,14423,14424],{"xmlns":93,"display":818},[95,14425,14426,14466],{},[98,14427,14428,14430,14436,14438,14440,14442,14444,14446,14448,14450,14452,14454,14456,14458,14460,14462,14464],{},[101,14429,6411],{"mathvariant":1133},[3006,14431,14432,14434],{},[101,14433,6619],{},[101,14435,2164],{},[179,14437,410],{"stretchy":409},[101,14439,6124],{},[179,14441,415],{"stretchy":409},[179,14443,410],{"stretchy":409},[101,14445,3518],{},[179,14447,415],{"stretchy":409},[179,14449,6422],{},[101,14451,6124],{},[179,14453,410],{"stretchy":409},[101,14455,3518],{},[179,14457,415],{"stretchy":409},[101,14459,6411],{"mathvariant":1133},[179,14461,6435],{},[101,14463,6305],{},[101,14465,65],{"mathvariant":1133},[105,14467,14468],{"encoding":107},"|B_n(f)(x)-f(x)|\n\u003C\n\\varepsilon.",[81,14470,14472,14548,14578],{"className":14471,"ariaHidden":112},[111],[81,14473,14475,14478,14481,14521,14524,14527,14530,14533,14536,14539,14542,14545],{"className":14474},[116],[81,14476],{"className":14477,"style":428},[120],[81,14479,6411],{"className":14480},[125],[81,14482,14484,14487],{"className":14483},[125],[81,14485,6619],{"className":14486,"style":6728},[125,126],[81,14488,14490],{"className":14489},[204],[81,14491,14493,14513],{"className":14492},[208,569],[81,14494,14496,14510],{"className":14495},[212],[81,14497,14499],{"className":14498,"style":6741},[216],[81,14500,14501,14504],{"style":6744},[81,14502],{"className":14503,"style":224},[223],[81,14505,14507],{"className":14506},[228,229,230,231],[81,14508,2164],{"className":14509},[125,126,231],[81,14511,606],{"className":14512},[605],[81,14514,14516],{"className":14515},[212],[81,14517,14519],{"className":14518,"style":3177},[216],[81,14520],{},[81,14522,410],{"className":14523},[435],[81,14525,6124],{"className":14526,"style":6161},[125,126],[81,14528,415],{"className":14529},[442],[81,14531,410],{"className":14532},[435],[81,14534,3518],{"className":14535},[125,126],[81,14537,415],{"className":14538},[442],[81,14540],{"className":14541,"style":127},[912],[81,14543,6422],{"className":14544},[235],[81,14546],{"className":14547,"style":127},[912],[81,14549,14551,14554,14557,14560,14563,14566,14569,14572,14575],{"className":14550},[116],[81,14552],{"className":14553,"style":428},[120],[81,14555,6124],{"className":14556,"style":6161},[125,126],[81,14558,410],{"className":14559},[435],[81,14561,3518],{"className":14562},[125,126],[81,14564,415],{"className":14565},[442],[81,14567,6411],{"className":14568},[125],[81,14570],{"className":14571,"style":913},[912],[81,14573,6435],{"className":14574},[917],[81,14576],{"className":14577,"style":913},[912],[81,14579,14581,14584,14587],{"className":14580},[116],[81,14582],{"className":14583,"style":293},[120],[81,14585,6305],{"className":14586},[125,126],[81,14588,65],{"className":14589},[125],[14,14591,14592,14593,65],{},"Moreover, this estimate is independent of ",[81,14594,14596,14609],{"className":14595,"translate":85},[84],[81,14597,14599],{"className":14598},[89],[91,14600,14601],{"xmlns":93},[95,14602,14603,14607],{},[98,14604,14605],{},[101,14606,3518],{},[105,14608,3518],{"encoding":107},[81,14610,14612],{"className":14611,"ariaHidden":112},[111],[81,14613,14615,14618],{"className":14614},[116],[81,14616],{"className":14617,"style":293},[120],[81,14619,3518],{"className":14620},[125,126],[14,14622,14623],{},"Consequently,",[81,14625,14627],{"className":14626,"translate":85},[809],[81,14628,14630,14704],{"className":14629,"translate":85},[84],[81,14631,14633],{"className":14632},[89],[91,14634,14635],{"xmlns":93,"display":818},[95,14636,14637,14701],{},[98,14638,14639,14664,14666,14672,14674,14676,14678,14680,14682,14684,14686,14688,14690,14692,14694,14696,14698],{},[1535,14640,14641,14648],{},[98,14642,14643,14646],{},[101,14644,14645],{},"sup",[179,14647,12935],{},[98,14649,14650,14652,14654,14656,14658,14660,14662],{},[101,14651,3518],{},[179,14653,6127],{},[179,14655,497],{"stretchy":409},[515,14657,513],{},[179,14659,1414],{"separator":112},[515,14661,679],{},[179,14663,529],{"stretchy":409},[101,14665,6411],{"mathvariant":1133},[3006,14667,14668,14670],{},[101,14669,6619],{},[101,14671,2164],{},[179,14673,410],{"stretchy":409},[101,14675,6124],{},[179,14677,415],{"stretchy":409},[179,14679,410],{"stretchy":409},[101,14681,3518],{},[179,14683,415],{"stretchy":409},[179,14685,6422],{},[101,14687,6124],{},[179,14689,410],{"stretchy":409},[101,14691,3518],{},[179,14693,415],{"stretchy":409},[101,14695,6411],{"mathvariant":1133},[179,14697,10285],{},[515,14699,14700],{},"0.",[105,14702,14703],{"encoding":107},"\\sup_{x\\in[0,1]}\n|B_n(f)(x)-f(x)|\n\\to 0.",[81,14705,14707,14854,14884],{"className":14706,"ariaHidden":112},[111],[81,14708,14710,14714,14781,14784,14787,14827,14830,14833,14836,14839,14842,14845,14848,14851],{"className":14709},[116],[81,14711],{"className":14712,"style":14713},[120],"height:1.9104em;vertical-align:-1.1604em;",[81,14715,14717],{"className":14716},[6802,6803],[81,14718,14720,14772],{"className":14719},[208,569],[81,14721,14723,14769],{"className":14722},[212],[81,14724,14726,14759],{"className":14725,"style":293},[216],[81,14727,14729,14732],{"style":14728},"top:-2.1146em;margin-left:0em;",[81,14730],{"className":14731,"style":583},[223],[81,14733,14735],{"className":14734},[228,229,230,231],[81,14736,14738,14741,14744,14747,14750,14753,14756],{"className":14737},[125,231],[81,14739,3518],{"className":14740},[125,126,231],[81,14742,6127],{"className":14743},[917,231],[81,14745,497],{"className":14746},[435,231],[81,14748,513],{"className":14749},[125,231],[81,14751,1414],{"className":14752},[1441,231],[81,14754,679],{"className":14755},[125,231],[81,14757,529],{"className":14758},[442,231],[81,14760,14761,14764],{"style":1655},[81,14762],{"className":14763,"style":583},[223],[81,14765,14766],{},[81,14767,14645],{"className":14768},[6802],[81,14770,606],{"className":14771},[605],[81,14773,14775],{"className":14774},[212],[81,14776,14779],{"className":14777,"style":14778},[216],"height:1.1604em;",[81,14780],{},[81,14782],{"className":14783,"style":1025},[912],[81,14785,6411],{"className":14786},[125],[81,14788,14790,14793],{"className":14789},[125],[81,14791,6619],{"className":14792,"style":6728},[125,126],[81,14794,14796],{"className":14795},[204],[81,14797,14799,14819],{"className":14798},[208,569],[81,14800,14802,14816],{"className":14801},[212],[81,14803,14805],{"className":14804,"style":6741},[216],[81,14806,14807,14810],{"style":6744},[81,14808],{"className":14809,"style":224},[223],[81,14811,14813],{"className":14812},[228,229,230,231],[81,14814,2164],{"className":14815},[125,126,231],[81,14817,606],{"className":14818},[605],[81,14820,14822],{"className":14821},[212],[81,14823,14825],{"className":14824,"style":3177},[216],[81,14826],{},[81,14828,410],{"className":14829},[435],[81,14831,6124],{"className":14832,"style":6161},[125,126],[81,14834,415],{"className":14835},[442],[81,14837,410],{"className":14838},[435],[81,14840,3518],{"className":14841},[125,126],[81,14843,415],{"className":14844},[442],[81,14846],{"className":14847,"style":127},[912],[81,14849,6422],{"className":14850},[235],[81,14852],{"className":14853,"style":127},[912],[81,14855,14857,14860,14863,14866,14869,14872,14875,14878,14881],{"className":14856},[116],[81,14858],{"className":14859,"style":428},[120],[81,14861,6124],{"className":14862,"style":6161},[125,126],[81,14864,410],{"className":14865},[435],[81,14867,3518],{"className":14868},[125,126],[81,14870,415],{"className":14871},[442],[81,14873,6411],{"className":14874},[125],[81,14876],{"className":14877,"style":913},[912],[81,14879,10285],{"className":14880},[917],[81,14882],{"className":14883,"style":913},[912],[81,14885,14887,14890],{"className":14886},[116],[81,14888],{"className":14889,"style":1107},[120],[81,14891,14700],{"className":14892},[125],[14,14894,8734],{},[81,14896,14898],{"className":14897,"translate":85},[809],[81,14899,14901,14929],{"className":14900,"translate":85},[84],[81,14902,14904],{"className":14903},[89],[91,14905,14906],{"xmlns":93,"display":818},[95,14907,14908,14926],{},[98,14909,14910,14916,14918,14920,14922,14924],{},[3006,14911,14912,14914],{},[101,14913,6619],{},[101,14915,2164],{},[179,14917,410],{"stretchy":409},[101,14919,6124],{},[179,14921,415],{"stretchy":409},[179,14923,10285],{},[101,14925,6124],{},[105,14927,14928],{"encoding":107},"B_n(f)\n\\to f",[81,14930,14932,14996],{"className":14931,"ariaHidden":112},[111],[81,14933,14935,14938,14978,14981,14984,14987,14990,14993],{"className":14934},[116],[81,14936],{"className":14937,"style":428},[120],[81,14939,14941,14944],{"className":14940},[125],[81,14942,6619],{"className":14943,"style":6728},[125,126],[81,14945,14947],{"className":14946},[204],[81,14948,14950,14970],{"className":14949},[208,569],[81,14951,14953,14967],{"className":14952},[212],[81,14954,14956],{"className":14955,"style":6741},[216],[81,14957,14958,14961],{"style":6744},[81,14959],{"className":14960,"style":224},[223],[81,14962,14964],{"className":14963},[228,229,230,231],[81,14965,2164],{"className":14966},[125,126,231],[81,14968,606],{"className":14969},[605],[81,14971,14973],{"className":14972},[212],[81,14974,14976],{"className":14975,"style":3177},[216],[81,14977],{},[81,14979,410],{"className":14980},[435],[81,14982,6124],{"className":14983,"style":6161},[125,126],[81,14985,415],{"className":14986},[442],[81,14988],{"className":14989,"style":913},[912],[81,14991,10285],{"className":14992},[917],[81,14994],{"className":14995,"style":913},[912],[81,14997,14999,15002],{"className":14998},[116],[81,15000],{"className":15001,"style":6157},[120],[81,15003,6124],{"className":15004,"style":6161},[125,126],[14,15006,15007],{},"uniformly.",[14,15009,15010],{},"Weierstrass theorem is proved.",[43,15012,15014],{"id":15013},"proof-of-the-uat","Proof of the UAT",[14,15016,15017],{},"Next, we will go through the process of proving the Universal Approximation Theorem (UAT) using polynomial approximation, step by step, in language understandable with basic calculus. We take the univariate continuous function as an example for clarity; the vector case can be generalized similarly.",[14,15019,15020,15023,15024,15121,15122,15174],{},[24,15021,15022],{},"Goal:"," Given a continuous function ",[81,15025,15027,15058],{"className":15026,"translate":85},[84],[81,15028,15030],{"className":15029},[89],[91,15031,15032],{"xmlns":93},[95,15033,15034,15055],{},[98,15035,15036,15038,15040,15042,15044,15046,15049,15051,15053],{},[101,15037,6124],{},[179,15039,1533],{},[179,15041,497],{"stretchy":409},[101,15043,8618],{},[179,15045,1414],{"separator":112},[101,15047,15048],{},"b",[179,15050,529],{"stretchy":409},[179,15052,10285],{},[101,15054,1593],{"mathvariant":1592},[105,15056,15057],{"encoding":107},"f:[a,b]\\to \\mathbb{R}",[81,15059,15061,15079,15112],{"className":15060,"ariaHidden":112},[111],[81,15062,15064,15067,15070,15073,15076],{"className":15063},[116],[81,15065],{"className":15066,"style":6157},[120],[81,15068,6124],{"className":15069,"style":6161},[125,126],[81,15071],{"className":15072,"style":913},[912],[81,15074,1533],{"className":15075},[917],[81,15077],{"className":15078,"style":913},[912],[81,15080,15082,15085,15088,15091,15094,15097,15100,15103,15106,15109],{"className":15081},[116],[81,15083],{"className":15084,"style":428},[120],[81,15086,497],{"className":15087},[435],[81,15089,8618],{"className":15090},[125,126],[81,15092,1414],{"className":15093},[1441],[81,15095],{"className":15096,"style":1025},[912],[81,15098,15048],{"className":15099},[125,126],[81,15101,529],{"className":15102},[442],[81,15104],{"className":15105,"style":913},[912],[81,15107,10285],{"className":15108},[917],[81,15110],{"className":15111,"style":913},[912],[81,15113,15115,15118],{"className":15114},[116],[81,15116],{"className":15117,"style":1994},[120],[81,15119,1593],{"className":15120},[125,1998]," and any ",[81,15123,15125,15144],{"className":15124,"translate":85},[84],[81,15126,15128],{"className":15127},[89],[91,15129,15130],{"xmlns":93},[95,15131,15132,15141],{},[98,15133,15134,15137,15139],{},[101,15135,15136],{},"ϵ",[179,15138,6308],{},[515,15140,513],{},[105,15142,15143],{"encoding":107},"\\epsilon>0",[81,15145,15147,15165],{"className":15146,"ariaHidden":112},[111],[81,15148,15150,15153,15156,15159,15162],{"className":15149},[116],[81,15151],{"className":15152,"style":6323},[120],[81,15154,15136],{"className":15155},[125,126],[81,15157],{"className":15158,"style":913},[912],[81,15160,6308],{"className":15161},[917],[81,15163],{"className":15164,"style":913},[912],[81,15166,15168,15171],{"className":15167},[116],[81,15169],{"className":15170,"style":1107},[120],[81,15172,513],{"className":15173},[125],", there exists a single-hidden-layer neural network",[81,15176,15178],{"className":15177,"translate":85},[809],[81,15179,15181,15249],{"className":15180,"translate":85},[84],[81,15182,15184],{"className":15183},[89],[91,15185,15186],{"xmlns":93,"display":818},[95,15187,15188,15246],{},[98,15189,15190,15193,15195,15197,15199,15201,15215,15222,15224,15226,15233,15235,15237,15244],{},[101,15191,15192],{},"N",[179,15194,410],{"stretchy":409},[101,15196,3518],{},[179,15198,415],{"stretchy":409},[179,15200,833],{},[6637,15202,15203,15205,15213],{},[179,15204,6641],{},[98,15206,15207,15209,15211],{},[101,15208,3015],{},[179,15210,833],{},[515,15212,679],{},[101,15214,2164],{},[3006,15216,15217,15220],{},[101,15218,15219],{},"α",[101,15221,3015],{},[101,15223,2133],{},[179,15225,410],{"stretchy":409},[3006,15227,15228,15231],{},[101,15229,15230],{},"β",[101,15232,3015],{},[101,15234,3518],{},[179,15236,2044],{},[3006,15238,15239,15242],{},[101,15240,15241],{},"γ",[101,15243,3015],{},[179,15245,415],{"stretchy":409},[105,15247,15248],{"encoding":107},"N(x) = \\sum_{i=1}^{n} \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[81,15250,15252,15279,15460],{"className":15251,"ariaHidden":112},[111],[81,15253,15255,15258,15261,15264,15267,15270,15273,15276],{"className":15254},[116],[81,15256],{"className":15257,"style":428},[120],[81,15259,15192],{"className":15260,"style":12969},[125,126],[81,15262,410],{"className":15263},[435],[81,15265,3518],{"className":15266},[125,126],[81,15268,415],{"className":15269},[442],[81,15271],{"className":15272,"style":913},[912],[81,15274,833],{"className":15275},[917],[81,15277],{"className":15278,"style":913},[912],[81,15280,15282,15286,15355,15358,15400,15403,15406,15448,15451,15454,15457],{"className":15281},[116],[81,15283],{"className":15284,"style":15285},[120],"height:2.9291em;vertical-align:-1.2777em;",[81,15287,15289],{"className":15288},[6802,6803],[81,15290,15292,15346],{"className":15291},[208,569],[81,15293,15295,15343],{"className":15294},[212],[81,15296,15298,15319,15329],{"className":15297,"style":6813},[216],[81,15299,15301,15304],{"style":15300},"top:-1.8723em;margin-left:0em;",[81,15302],{"className":15303,"style":6820},[223],[81,15305,15307],{"className":15306},[228,229,230,231],[81,15308,15310,15313,15316],{"className":15309},[125,231],[81,15311,3015],{"className":15312},[125,126,231],[81,15314,833],{"className":15315},[917,231],[81,15317,679],{"className":15318},[125,231],[81,15320,15321,15324],{"style":6838},[81,15322],{"className":15323,"style":6820},[223],[81,15325,15326],{},[81,15327,6641],{"className":15328},[6802,6847,6848],[81,15330,15331,15334],{"style":6851},[81,15332],{"className":15333,"style":6820},[223],[81,15335,15337],{"className":15336},[228,229,230,231],[81,15338,15340],{"className":15339},[125,231],[81,15341,2164],{"className":15342},[125,126,231],[81,15344,606],{"className":15345},[605],[81,15347,15349],{"className":15348},[212],[81,15350,15353],{"className":15351,"style":15352},[216],"height:1.2777em;",[81,15354],{},[81,15356],{"className":15357,"style":1025},[912],[81,15359,15361,15365],{"className":15360},[125],[81,15362,15219],{"className":15363,"style":15364},[125,126],"margin-right:0.0037em;",[81,15366,15368],{"className":15367},[204],[81,15369,15371,15392],{"className":15370},[208,569],[81,15372,15374,15389],{"className":15373},[212],[81,15375,15377],{"className":15376,"style":3050},[216],[81,15378,15380,15383],{"style":15379},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[81,15381],{"className":15382,"style":224},[223],[81,15384,15386],{"className":15385},[228,229,230,231],[81,15387,3015],{"className":15388},[125,126,231],[81,15390,606],{"className":15391},[605],[81,15393,15395],{"className":15394},[212],[81,15396,15398],{"className":15397,"style":3177},[216],[81,15399],{},[81,15401,2133],{"className":15402,"style":297},[125,126],[81,15404,410],{"className":15405},[435],[81,15407,15409,15413],{"className":15408},[125],[81,15410,15230],{"className":15411,"style":15412},[125,126],"margin-right:0.0528em;",[81,15414,15416],{"className":15415},[204],[81,15417,15419,15440],{"className":15418},[208,569],[81,15420,15422,15437],{"className":15421},[212],[81,15423,15425],{"className":15424,"style":3050},[216],[81,15426,15428,15431],{"style":15427},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[81,15429],{"className":15430,"style":224},[223],[81,15432,15434],{"className":15433},[228,229,230,231],[81,15435,3015],{"className":15436},[125,126,231],[81,15438,606],{"className":15439},[605],[81,15441,15443],{"className":15442},[212],[81,15444,15446],{"className":15445,"style":3177},[216],[81,15447],{},[81,15449,3518],{"className":15450},[125,126],[81,15452],{"className":15453,"style":127},[912],[81,15455,2044],{"className":15456},[235],[81,15458],{"className":15459,"style":127},[912],[81,15461,15463,15466,15507],{"className":15462},[116],[81,15464],{"className":15465,"style":428},[120],[81,15467,15469,15472],{"className":15468},[125],[81,15470,15241],{"className":15471,"style":4206},[125,126],[81,15473,15475],{"className":15474},[204],[81,15476,15478,15499],{"className":15477},[208,569],[81,15479,15481,15496],{"className":15480},[212],[81,15482,15484],{"className":15483,"style":3050},[216],[81,15485,15487,15490],{"style":15486},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[81,15488],{"className":15489,"style":224},[223],[81,15491,15493],{"className":15492},[228,229,230,231],[81,15494,3015],{"className":15495},[125,126,231],[81,15497,606],{"className":15498},[605],[81,15500,15502],{"className":15501},[212],[81,15503,15505],{"className":15504,"style":3177},[216],[81,15506],{},[81,15508,415],{"className":15509},[442],[14,15511,15512],{},"such that",[81,15514,15516],{"className":15515,"translate":85},[809],[81,15517,15519,15577],{"className":15518,"translate":85},[84],[81,15520,15522],{"className":15521},[89],[91,15523,15524],{"xmlns":93,"display":818},[95,15525,15526,15574],{},[98,15527,15528,15530,15532,15534,15536,15538,15540,15542,15544,15546,15548,15550,15552,15554,15556,15558,15560,15562,15564,15566,15568,15570,15572],{},[101,15529,6411],{"mathvariant":1133},[101,15531,15192],{},[179,15533,410],{"stretchy":409},[101,15535,3518],{},[179,15537,415],{"stretchy":409},[179,15539,6422],{},[101,15541,6124],{},[179,15543,410],{"stretchy":409},[101,15545,3518],{},[179,15547,415],{"stretchy":409},[101,15549,6411],{"mathvariant":1133},[179,15551,6435],{},[101,15553,15136],{},[179,15555,1414],{"separator":112},[912,15557],{"width":503},[101,15559,6445],{"mathvariant":1133},[101,15561,3518],{},[179,15563,6127],{},[179,15565,497],{"stretchy":409},[101,15567,8618],{},[179,15569,1414],{"separator":112},[101,15571,15048],{},[179,15573,529],{"stretchy":409},[105,15575,15576],{"encoding":107},"|N(x) - f(x)| \u003C \\epsilon, \\quad \\forall x\\in [a,b]",[81,15578,15580,15610,15640,15674],{"className":15579,"ariaHidden":112},[111],[81,15581,15583,15586,15589,15592,15595,15598,15601,15604,15607],{"className":15582},[116],[81,15584],{"className":15585,"style":428},[120],[81,15587,6411],{"className":15588},[125],[81,15590,15192],{"className":15591,"style":12969},[125,126],[81,15593,410],{"className":15594},[435],[81,15596,3518],{"className":15597},[125,126],[81,15599,415],{"className":15600},[442],[81,15602],{"className":15603,"style":127},[912],[81,15605,6422],{"className":15606},[235],[81,15608],{"className":15609,"style":127},[912],[81,15611,15613,15616,15619,15622,15625,15628,15631,15634,15637],{"className":15612},[116],[81,15614],{"className":15615,"style":428},[120],[81,15617,6124],{"className":15618,"style":6161},[125,126],[81,15620,410],{"className":15621},[435],[81,15623,3518],{"className":15624},[125,126],[81,15626,415],{"className":15627},[442],[81,15629,6411],{"className":15630},[125],[81,15632],{"className":15633,"style":913},[912],[81,15635,6435],{"className":15636},[917],[81,15638],{"className":15639,"style":913},[912],[81,15641,15643,15646,15649,15652,15656,15659,15662,15665,15668,15671],{"className":15642},[116],[81,15644],{"className":15645,"style":6157},[120],[81,15647,15136],{"className":15648},[125,126],[81,15650,1414],{"className":15651},[1441],[81,15653],{"className":15654,"style":15655},[912],"margin-right:1em;",[81,15657],{"className":15658,"style":1025},[912],[81,15660,6445],{"className":15661},[125],[81,15663,3518],{"className":15664},[125,126],[81,15666],{"className":15667,"style":913},[912],[81,15669,6127],{"className":15670},[917],[81,15672],{"className":15673,"style":913},[912],[81,15675,15677,15680,15683,15686,15689,15692,15695],{"className":15676},[116],[81,15678],{"className":15679,"style":428},[120],[81,15681,497],{"className":15682},[435],[81,15684,8618],{"className":15685},[125,126],[81,15687,1414],{"className":15688},[1441],[81,15690],{"className":15691,"style":1025},[912],[81,15693,15048],{"className":15694},[125,126],[81,15696,529],{"className":15697},[442],[14,15699,15700,15701,15729],{},"where ",[81,15702,15704,15717],{"className":15703,"translate":85},[84],[81,15705,15707],{"className":15706},[89],[91,15708,15709],{"xmlns":93},[95,15710,15711,15715],{},[98,15712,15713],{},[101,15714,2133],{},[105,15716,2136],{"encoding":107},[81,15718,15720],{"className":15719,"ariaHidden":112},[111],[81,15721,15723,15726],{"className":15722},[116],[81,15724],{"className":15725,"style":293},[120],[81,15727,2133],{"className":15728,"style":297},[125,126]," is a \"non-polynomial, continuous\" activation function (e.g., sigmoid).",[14,15731,15732],{},[24,15733,15734],{},"Step 1: Use the Weierstrass Theorem",[14,15736,15737,15738,15781],{},"We have already proved this theorem earlier: any continuous function can be approximated by a polynomial ",[81,15739,15741,15760],{"className":15740,"translate":85},[84],[81,15742,15744],{"className":15743},[89],[91,15745,15746],{"xmlns":93},[95,15747,15748,15758],{},[98,15749,15750,15752,15754,15756],{},[101,15751,6362],{},[179,15753,410],{"stretchy":409},[101,15755,3518],{},[179,15757,415],{"stretchy":409},[105,15759,6371],{"encoding":107},[81,15761,15763],{"className":15762,"ariaHidden":112},[111],[81,15764,15766,15769,15772,15775,15778],{"className":15765},[116],[81,15767],{"className":15768,"style":428},[120],[81,15770,6362],{"className":15771,"style":1394},[125,126],[81,15773,410],{"className":15774},[435],[81,15776,3518],{"className":15777},[125,126],[81,15779,415],{"className":15780},[442],". That is,",[81,15783,15785],{"className":15784,"translate":85},[809],[81,15786,15788,15854],{"className":15787,"translate":85},[84],[81,15789,15791],{"className":15790},[89],[91,15792,15793],{"xmlns":93,"display":818},[95,15794,15795,15851],{},[98,15796,15797,15799,15801,15803,15805,15807,15810,15812,15814,15816,15818,15821,15823,15825,15827,15829,15831,15833,15835,15837,15839,15841,15843,15845,15847,15849],{},[101,15798,6445],{"mathvariant":1133},[101,15800,15136],{},[179,15802,6308],{},[515,15804,513],{},[179,15806,1414],{"separator":112},[101,15808,15809],{"mathvariant":1133},"∃",[101,15811,6362],{},[179,15813,410],{"stretchy":409},[101,15815,3518],{},[179,15817,415],{"stretchy":409},[2035,15819,15820],{}," s.t. ",[101,15822,6411],{"mathvariant":1133},[101,15824,6124],{},[179,15826,410],{"stretchy":409},[101,15828,3518],{},[179,15830,415],{"stretchy":409},[179,15832,6422],{},[101,15834,6362],{},[179,15836,410],{"stretchy":409},[101,15838,3518],{},[179,15840,415],{"stretchy":409},[101,15842,6411],{"mathvariant":1133},[179,15844,6435],{},[101,15846,15136],{},[101,15848,1134],{"mathvariant":1133},[515,15850,517],{},[105,15852,15853],{"encoding":107},"\\forall \\epsilon>0, \\exists P(x) \\text{ s.t. } |f(x)-P(x)|\u003C\\epsilon\u002F2",[81,15855,15857,15878,15938,15968],{"className":15856,"ariaHidden":112},[111],[81,15858,15860,15863,15866,15869,15872,15875],{"className":15859},[116],[81,15861],{"className":15862,"style":11192},[120],[81,15864,6445],{"className":15865},[125],[81,15867,15136],{"className":15868},[125,126],[81,15870],{"className":15871,"style":913},[912],[81,15873,6308],{"className":15874},[917],[81,15876],{"className":15877,"style":913},[912],[81,15879,15881,15884,15887,15890,15893,15896,15899,15902,15905,15908,15914,15917,15920,15923,15926,15929,15932,15935],{"className":15880},[116],[81,15882],{"className":15883,"style":428},[120],[81,15885,513],{"className":15886},[125],[81,15888,1414],{"className":15889},[1441],[81,15891],{"className":15892,"style":1025},[912],[81,15894,15809],{"className":15895},[125],[81,15897,6362],{"className":15898,"style":1394},[125,126],[81,15900,410],{"className":15901},[435],[81,15903,3518],{"className":15904},[125,126],[81,15906,415],{"className":15907},[442],[81,15909,15911],{"className":15910},[125,2063],[81,15912,15820],{"className":15913},[125],[81,15915,6411],{"className":15916},[125],[81,15918,6124],{"className":15919,"style":6161},[125,126],[81,15921,410],{"className":15922},[435],[81,15924,3518],{"className":15925},[125,126],[81,15927,415],{"className":15928},[442],[81,15930],{"className":15931,"style":127},[912],[81,15933,6422],{"className":15934},[235],[81,15936],{"className":15937,"style":127},[912],[81,15939,15941,15944,15947,15950,15953,15956,15959,15962,15965],{"className":15940},[116],[81,15942],{"className":15943,"style":428},[120],[81,15945,6362],{"className":15946,"style":1394},[125,126],[81,15948,410],{"className":15949},[435],[81,15951,3518],{"className":15952},[125,126],[81,15954,415],{"className":15955},[442],[81,15957,6411],{"className":15958},[125],[81,15960],{"className":15961,"style":913},[912],[81,15963,6435],{"className":15964},[917],[81,15966],{"className":15967,"style":913},[912],[81,15969,15971,15974,15977],{"className":15970},[116],[81,15972],{"className":15973,"style":428},[120],[81,15975,15136],{"className":15976},[125,126],[81,15978,15980],{"className":15979},[125],"\u002F2",[14,15982,15983,15984,65],{},"Here we do not consider neural networks yet; we transform the problem into \"approximating a polynomial\". As we know, a polynomial can be written as ",[81,15985,15987,16033],{"className":15986,"translate":85},[84],[81,15988,15990],{"className":15989},[89],[91,15991,15992],{"xmlns":93},[95,15993,15994,16030],{},[98,15995,15996,16002,16004,16010,16012,16014,16016,16018,16024],{},[3006,15997,15998,16000],{},[101,15999,8618],{},[515,16001,513],{},[179,16003,2044],{},[3006,16005,16006,16008],{},[101,16007,8618],{},[515,16009,679],{},[101,16011,3518],{},[179,16013,2044],{},[179,16015,1551],{},[179,16017,2044],{},[3006,16019,16020,16022],{},[101,16021,8618],{},[101,16023,8613],{},[174,16025,16026,16028],{},[101,16027,3518],{},[101,16029,8613],{},[105,16031,16032],{"encoding":107},"a_0 + a_1 x + \\dots + a_m x^m",[81,16034,16036,16093,16151,16170],{"className":16035,"ariaHidden":112},[111],[81,16037,16039,16043,16084,16087,16090],{"className":16038},[116],[81,16040],{"className":16041,"style":16042},[120],"height:0.7333em;vertical-align:-0.15em;",[81,16044,16046,16049],{"className":16045},[125],[81,16047,8618],{"className":16048},[125,126],[81,16050,16052],{"className":16051},[204],[81,16053,16055,16076],{"className":16054},[208,569],[81,16056,16058,16073],{"className":16057},[212],[81,16059,16062],{"className":16060,"style":16061},[216],"height:0.3011em;",[81,16063,16064,16067],{"style":8358},[81,16065],{"className":16066,"style":224},[223],[81,16068,16070],{"className":16069},[228,229,230,231],[81,16071,513],{"className":16072},[125,231],[81,16074,606],{"className":16075},[605],[81,16077,16079],{"className":16078},[212],[81,16080,16082],{"className":16081,"style":3177},[216],[81,16083],{},[81,16085],{"className":16086,"style":127},[912],[81,16088,2044],{"className":16089},[235],[81,16091],{"className":16092,"style":127},[912],[81,16094,16096,16099,16139,16142,16145,16148],{"className":16095},[116],[81,16097],{"className":16098,"style":16042},[120],[81,16100,16102,16105],{"className":16101},[125],[81,16103,8618],{"className":16104},[125,126],[81,16106,16108],{"className":16107},[204],[81,16109,16111,16131],{"className":16110},[208,569],[81,16112,16114,16128],{"className":16113},[212],[81,16115,16117],{"className":16116,"style":16061},[216],[81,16118,16119,16122],{"style":8358},[81,16120],{"className":16121,"style":224},[223],[81,16123,16125],{"className":16124},[228,229,230,231],[81,16126,679],{"className":16127},[125,231],[81,16129,606],{"className":16130},[605],[81,16132,16134],{"className":16133},[212],[81,16135,16137],{"className":16136,"style":3177},[216],[81,16138],{},[81,16140,3518],{"className":16141},[125,126],[81,16143],{"className":16144,"style":127},[912],[81,16146,2044],{"className":16147},[235],[81,16149],{"className":16150,"style":127},[912],[81,16152,16154,16158,16161,16164,16167],{"className":16153},[116],[81,16155],{"className":16156,"style":16157},[120],"height:0.6667em;vertical-align:-0.0833em;",[81,16159,1551],{"className":16160},[546],[81,16162],{"className":16163,"style":127},[912],[81,16165,2044],{"className":16166},[235],[81,16168],{"className":16169,"style":127},[912],[81,16171,16173,16177,16217],{"className":16172},[116],[81,16174],{"className":16175,"style":16176},[120],"height:0.8144em;vertical-align:-0.15em;",[81,16178,16180,16183],{"className":16179},[125],[81,16181,8618],{"className":16182},[125,126],[81,16184,16186],{"className":16185},[204],[81,16187,16189,16209],{"className":16188},[208,569],[81,16190,16192,16206],{"className":16191},[212],[81,16193,16195],{"className":16194,"style":6741},[216],[81,16196,16197,16200],{"style":8358},[81,16198],{"className":16199,"style":224},[223],[81,16201,16203],{"className":16202},[228,229,230,231],[81,16204,8613],{"className":16205},[125,126,231],[81,16207,606],{"className":16208},[605],[81,16210,16212],{"className":16211},[212],[81,16213,16215],{"className":16214,"style":3177},[216],[81,16216],{},[81,16218,16220,16223],{"className":16219},[125],[81,16221,3518],{"className":16222},[125,126],[81,16224,16226],{"className":16225},[204],[81,16227,16229],{"className":16228},[208],[81,16230,16232],{"className":16231},[212],[81,16233,16236],{"className":16234,"style":16235},[216],"height:0.6644em;",[81,16237,16238,16241],{"style":219},[81,16239],{"className":16240,"style":224},[223],[81,16242,16244],{"className":16243},[228,229,230,231],[81,16245,8613],{"className":16246},[125,126,231],[14,16248,16249],{},[24,16250,16251,16252,16311],{},"Step 2: Approximate monomials ",[81,16253,16255,16273],{"className":16254,"translate":85},[84],[81,16256,16258],{"className":16257},[89],[91,16259,16260],{"xmlns":93},[95,16261,16262,16270],{},[98,16263,16264],{},[174,16265,16266,16268],{},[101,16267,3518],{},[101,16269,3746],{},[105,16271,16272],{"encoding":107},"x^k",[81,16274,16276],{"className":16275,"ariaHidden":112},[111],[81,16277,16279,16282],{"className":16278},[116],[81,16280],{"className":16281,"style":4767},[120],[81,16283,16285,16288],{"className":16284},[125],[81,16286,3518],{"className":16287},[125,126],[81,16289,16291],{"className":16290},[204],[81,16292,16294],{"className":16293},[208],[81,16295,16297],{"className":16296},[212],[81,16298,16300],{"className":16299,"style":4767},[216],[81,16301,16302,16305],{"style":219},[81,16303],{"className":16304,"style":224},[223],[81,16306,16308],{"className":16307},[228,229,230,231],[81,16309,3746],{"className":16310,"style":4244},[125,126,231]," using an activation function",[14,16313,16314,16315,16373,16374,65],{},"If we can approximate ",[81,16316,16318,16335],{"className":16317,"translate":85},[84],[81,16319,16321],{"className":16320},[89],[91,16322,16323],{"xmlns":93},[95,16324,16325,16333],{},[98,16326,16327],{},[174,16328,16329,16331],{},[101,16330,3518],{},[101,16332,3746],{},[105,16334,16272],{"encoding":107},[81,16336,16338],{"className":16337,"ariaHidden":112},[111],[81,16339,16341,16344],{"className":16340},[116],[81,16342],{"className":16343,"style":4767},[120],[81,16345,16347,16350],{"className":16346},[125],[81,16348,3518],{"className":16349},[125,126],[81,16351,16353],{"className":16352},[204],[81,16354,16356],{"className":16355},[208],[81,16357,16359],{"className":16358},[212],[81,16360,16362],{"className":16361,"style":4767},[216],[81,16363,16364,16367],{"style":219},[81,16365],{"className":16366,"style":224},[223],[81,16368,16370],{"className":16369},[228,229,230,231],[81,16371,3746],{"className":16372,"style":4244},[125,126,231]," by a linear combination of neurons in one hidden layer, then we can approximate the polynomial ",[81,16375,16377,16396],{"className":16376,"translate":85},[84],[81,16378,16380],{"className":16379},[89],[91,16381,16382],{"xmlns":93},[95,16383,16384,16394],{},[98,16385,16386,16388,16390,16392],{},[101,16387,6362],{},[179,16389,410],{"stretchy":409},[101,16391,3518],{},[179,16393,415],{"stretchy":409},[105,16395,6371],{"encoding":107},[81,16397,16399],{"className":16398,"ariaHidden":112},[111],[81,16400,16402,16405,16408,16411,16414],{"className":16401},[116],[81,16403],{"className":16404,"style":428},[120],[81,16406,6362],{"className":16407,"style":1394},[125,126],[81,16409,410],{"className":16410},[435],[81,16412,3518],{"className":16413},[125,126],[81,16415,415],{"className":16416},[442],[14,16418,16419],{},"The single-hidden-layer network has the form:",[81,16421,16423],{"className":16422,"translate":85},[809],[81,16424,16426,16490],{"className":16425,"translate":85},[84],[81,16427,16429],{"className":16428},[89],[91,16430,16431],{"xmlns":93,"display":818},[95,16432,16433,16487],{},[98,16434,16435,16437,16439,16441,16443,16445,16459,16465,16467,16469,16475,16477,16479,16485],{},[101,16436,15192],{},[179,16438,410],{"stretchy":409},[101,16440,3518],{},[179,16442,415],{"stretchy":409},[179,16444,833],{},[6637,16446,16447,16449,16457],{},[179,16448,6641],{},[98,16450,16451,16453,16455],{},[101,16452,3015],{},[179,16454,833],{},[515,16456,679],{},[101,16458,2164],{},[3006,16460,16461,16463],{},[101,16462,15219],{},[101,16464,3015],{},[101,16466,2133],{},[179,16468,410],{"stretchy":409},[3006,16470,16471,16473],{},[101,16472,15230],{},[101,16474,3015],{},[101,16476,3518],{},[179,16478,2044],{},[3006,16480,16481,16483],{},[101,16482,15241],{},[101,16484,3015],{},[179,16486,415],{"stretchy":409},[105,16488,16489],{"encoding":107},"  N(x) = \\sum_{i=1}^{n} \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[81,16491,16493,16520,16694],{"className":16492,"ariaHidden":112},[111],[81,16494,16496,16499,16502,16505,16508,16511,16514,16517],{"className":16495},[116],[81,16497],{"className":16498,"style":428},[120],[81,16500,15192],{"className":16501,"style":12969},[125,126],[81,16503,410],{"className":16504},[435],[81,16506,3518],{"className":16507},[125,126],[81,16509,415],{"className":16510},[442],[81,16512],{"className":16513,"style":913},[912],[81,16515,833],{"className":16516},[917],[81,16518],{"className":16519,"style":913},[912],[81,16521,16523,16526,16593,16596,16636,16639,16642,16682,16685,16688,16691],{"className":16522},[116],[81,16524],{"className":16525,"style":15285},[120],[81,16527,16529],{"className":16528},[6802,6803],[81,16530,16532,16585],{"className":16531},[208,569],[81,16533,16535,16582],{"className":16534},[212],[81,16536,16538,16558,16568],{"className":16537,"style":6813},[216],[81,16539,16540,16543],{"style":15300},[81,16541],{"className":16542,"style":6820},[223],[81,16544,16546],{"className":16545},[228,229,230,231],[81,16547,16549,16552,16555],{"className":16548},[125,231],[81,16550,3015],{"className":16551},[125,126,231],[81,16553,833],{"className":16554},[917,231],[81,16556,679],{"className":16557},[125,231],[81,16559,16560,16563],{"style":6838},[81,16561],{"className":16562,"style":6820},[223],[81,16564,16565],{},[81,16566,6641],{"className":16567},[6802,6847,6848],[81,16569,16570,16573],{"style":6851},[81,16571],{"className":16572,"style":6820},[223],[81,16574,16576],{"className":16575},[228,229,230,231],[81,16577,16579],{"className":16578},[125,231],[81,16580,2164],{"className":16581},[125,126,231],[81,16583,606],{"className":16584},[605],[81,16586,16588],{"className":16587},[212],[81,16589,16591],{"className":16590,"style":15352},[216],[81,16592],{},[81,16594],{"className":16595,"style":1025},[912],[81,16597,16599,16602],{"className":16598},[125],[81,16600,15219],{"className":16601,"style":15364},[125,126],[81,16603,16605],{"className":16604},[204],[81,16606,16608,16628],{"className":16607},[208,569],[81,16609,16611,16625],{"className":16610},[212],[81,16612,16614],{"className":16613,"style":3050},[216],[81,16615,16616,16619],{"style":15379},[81,16617],{"className":16618,"style":224},[223],[81,16620,16622],{"className":16621},[228,229,230,231],[81,16623,3015],{"className":16624},[125,126,231],[81,16626,606],{"className":16627},[605],[81,16629,16631],{"className":16630},[212],[81,16632,16634],{"className":16633,"style":3177},[216],[81,16635],{},[81,16637,2133],{"className":16638,"style":297},[125,126],[81,16640,410],{"className":16641},[435],[81,16643,16645,16648],{"className":16644},[125],[81,16646,15230],{"className":16647,"style":15412},[125,126],[81,16649,16651],{"className":16650},[204],[81,16652,16654,16674],{"className":16653},[208,569],[81,16655,16657,16671],{"className":16656},[212],[81,16658,16660],{"className":16659,"style":3050},[216],[81,16661,16662,16665],{"style":15427},[81,16663],{"className":16664,"style":224},[223],[81,16666,16668],{"className":16667},[228,229,230,231],[81,16669,3015],{"className":16670},[125,126,231],[81,16672,606],{"className":16673},[605],[81,16675,16677],{"className":16676},[212],[81,16678,16680],{"className":16679,"style":3177},[216],[81,16681],{},[81,16683,3518],{"className":16684},[125,126],[81,16686],{"className":16687,"style":127},[912],[81,16689,2044],{"className":16690},[235],[81,16692],{"className":16693,"style":127},[912],[81,16695,16697,16700,16740],{"className":16696},[116],[81,16698],{"className":16699,"style":428},[120],[81,16701,16703,16706],{"className":16702},[125],[81,16704,15241],{"className":16705,"style":4206},[125,126],[81,16707,16709],{"className":16708},[204],[81,16710,16712,16732],{"className":16711},[208,569],[81,16713,16715,16729],{"className":16714},[212],[81,16716,16718],{"className":16717,"style":3050},[216],[81,16719,16720,16723],{"style":15486},[81,16721],{"className":16722,"style":224},[223],[81,16724,16726],{"className":16725},[228,229,230,231],[81,16727,3015],{"className":16728},[125,126,231],[81,16730,606],{"className":16731},[605],[81,16733,16735],{"className":16734},[212],[81,16736,16738],{"className":16737,"style":3177},[216],[81,16739],{},[81,16741,415],{"className":16742},[442],[14,16744,16745,16746,16790],{},"Choose a suitable ",[81,16747,16749,16769],{"className":16748,"translate":85},[84],[81,16750,16752],{"className":16751},[89],[91,16753,16754],{"xmlns":93},[95,16755,16756,16766],{},[98,16757,16758,16760,16762,16764],{},[101,16759,2133],{},[179,16761,410],{"stretchy":409},[101,16763,3518],{},[179,16765,415],{"stretchy":409},[105,16767,16768],{"encoding":107},"\\sigma(x)",[81,16770,16772],{"className":16771,"ariaHidden":112},[111],[81,16773,16775,16778,16781,16784,16787],{"className":16774},[116],[81,16776],{"className":16777,"style":428},[120],[81,16779,2133],{"className":16780,"style":297},[125,126],[81,16782,410],{"className":16783},[435],[81,16785,3518],{"className":16786},[125,126],[81,16788,415],{"className":16789},[442],", such as the sigmoid:",[81,16792,16794],{"className":16793,"translate":85},[809],[81,16795,16797,16840],{"className":16796,"translate":85},[84],[81,16798,16800],{"className":16799},[89],[91,16801,16802],{"xmlns":93,"display":818},[95,16803,16804,16837],{},[98,16805,16806,16808,16810,16812,16814,16816],{},[101,16807,2133],{},[179,16809,410],{"stretchy":409},[101,16811,3518],{},[179,16813,415],{"stretchy":409},[179,16815,833],{},[1199,16817,16818,16820],{},[515,16819,679],{},[98,16821,16822,16824,16826],{},[515,16823,679],{},[179,16825,2044],{},[174,16827,16828,16831],{},[101,16829,16830],{},"e",[98,16832,16833,16835],{},[179,16834,6422],{},[101,16836,3518],{},[105,16838,16839],{"encoding":107},"   \\sigma(x) = \\frac{1}{1+e^{-x}}",[81,16841,16843,16870],{"className":16842,"ariaHidden":112},[111],[81,16844,16846,16849,16852,16855,16858,16861,16864,16867],{"className":16845},[116],[81,16847],{"className":16848,"style":428},[120],[81,16850,2133],{"className":16851,"style":297},[125,126],[81,16853,410],{"className":16854},[435],[81,16856,3518],{"className":16857},[125,126],[81,16859,415],{"className":16860},[442],[81,16862],{"className":16863,"style":913},[912],[81,16865,833],{"className":16866},[917],[81,16868],{"className":16869,"style":913},[912],[81,16871,16873,16877],{"className":16872},[116],[81,16874],{"className":16875,"style":16876},[120],"height:2.0908em;vertical-align:-0.7693em;",[81,16878,16880,16883,16982],{"className":16879},[125],[81,16881],{"className":16882},[435,1247],[81,16884,16886],{"className":16885},[1199],[81,16887,16889,16973],{"className":16888},[208,569],[81,16890,16892,16970],{"className":16891},[212],[81,16893,16895,16951,16959],{"className":16894,"style":8162},[216],[81,16896,16897,16900],{"style":4196},[81,16898],{"className":16899,"style":583},[223],[81,16901,16903,16906,16909,16912,16915],{"className":16902},[125],[81,16904,679],{"className":16905},[125],[81,16907],{"className":16908,"style":127},[912],[81,16910,2044],{"className":16911},[235],[81,16913],{"className":16914,"style":127},[912],[81,16916,16918,16921],{"className":16917},[125],[81,16919,16830],{"className":16920},[125,126],[81,16922,16924],{"className":16923},[204],[81,16925,16927],{"className":16926},[208],[81,16928,16930],{"className":16929},[212],[81,16931,16934],{"className":16932,"style":16933},[216],"height:0.6973em;",[81,16935,16936,16939],{"style":4228},[81,16937],{"className":16938,"style":224},[223],[81,16940,16942],{"className":16941},[228,229,230,231],[81,16943,16945,16948],{"className":16944},[125,231],[81,16946,6422],{"className":16947},[125,231],[81,16949,3518],{"className":16950},[125,126,231],[81,16952,16953,16956],{"style":1278},[81,16954],{"className":16955,"style":583},[223],[81,16957],{"className":16958,"style":1286},[1285],[81,16960,16961,16964],{"style":4290},[81,16962],{"className":16963,"style":583},[223],[81,16965,16967],{"className":16966},[125],[81,16968,679],{"className":16969},[125],[81,16971,606],{"className":16972},[605],[81,16974,16976],{"className":16975},[212],[81,16977,16980],{"className":16978,"style":16979},[216],"height:0.7693em;",[81,16981],{},[81,16983],{"className":16984},[442,1247],[14,16986,16987],{},"Its Taylor expansion is non-zero locally:",[81,16989,16991],{"className":16990,"translate":85},[809],[81,16992,16994,17051],{"className":16993,"translate":85},[84],[81,16995,16997],{"className":16996},[89],[91,16998,16999],{"xmlns":93,"display":818},[95,17000,17001,17048],{},[98,17002,17003,17005,17007,17009,17011,17013,17019,17021,17027,17029,17031,17038,17044,17046],{},[101,17004,2133],{},[179,17006,410],{"stretchy":409},[101,17008,3518],{},[179,17010,415],{"stretchy":409},[179,17012,833],{},[1199,17014,17015,17017],{},[515,17016,679],{},[515,17018,517],{},[179,17020,2044],{},[1199,17022,17023,17025],{},[515,17024,679],{},[515,17026,686],{},[101,17028,3518],{},[179,17030,6422],{},[1199,17032,17033,17035],{},[515,17034,679],{},[515,17036,17037],{},"48",[174,17039,17040,17042],{},[101,17041,3518],{},[515,17043,526],{},[179,17045,2044],{},[179,17047,8088],{},[105,17049,17050],{"encoding":107},"  \\sigma(x) = \\frac12 + \\frac14 x - \\frac1{48} x^3 + \\dots",[81,17052,17054,17081,17158,17238,17345],{"className":17053,"ariaHidden":112},[111],[81,17055,17057,17060,17063,17066,17069,17072,17075,17078],{"className":17056},[116],[81,17058],{"className":17059,"style":428},[120],[81,17061,2133],{"className":17062,"style":297},[125,126],[81,17064,410],{"className":17065},[435],[81,17067,3518],{"className":17068},[125,126],[81,17070,415],{"className":17071},[442],[81,17073],{"className":17074,"style":913},[912],[81,17076,833],{"className":17077},[917],[81,17079],{"className":17080,"style":913},[912],[81,17082,17084,17087,17149,17152,17155],{"className":17083},[116],[81,17085],{"className":17086,"style":13750},[120],[81,17088,17090,17093,17146],{"className":17089},[125],[81,17091],{"className":17092},[435,1247],[81,17094,17096],{"className":17095},[1199],[81,17097,17099,17138],{"className":17098},[208,569],[81,17100,17102,17135],{"className":17101},[212],[81,17103,17105,17116,17124],{"className":17104,"style":8162},[216],[81,17106,17107,17110],{"style":4196},[81,17108],{"className":17109,"style":583},[223],[81,17111,17113],{"className":17112},[125],[81,17114,517],{"className":17115},[125],[81,17117,17118,17121],{"style":1278},[81,17119],{"className":17120,"style":583},[223],[81,17122],{"className":17123,"style":1286},[1285],[81,17125,17126,17129],{"style":4290},[81,17127],{"className":17128,"style":583},[223],[81,17130,17132],{"className":17131},[125],[81,17133,679],{"className":17134},[125],[81,17136,606],{"className":17137},[605],[81,17139,17141],{"className":17140},[212],[81,17142,17144],{"className":17143,"style":4385},[216],[81,17145],{},[81,17147],{"className":17148},[442,1247],[81,17150],{"className":17151,"style":127},[912],[81,17153,2044],{"className":17154},[235],[81,17156],{"className":17157,"style":127},[912],[81,17159,17161,17164,17226,17229,17232,17235],{"className":17160},[116],[81,17162],{"className":17163,"style":13750},[120],[81,17165,17167,17170,17223],{"className":17166},[125],[81,17168],{"className":17169},[435,1247],[81,17171,17173],{"className":17172},[1199],[81,17174,17176,17215],{"className":17175},[208,569],[81,17177,17179,17212],{"className":17178},[212],[81,17180,17182,17193,17201],{"className":17181,"style":8162},[216],[81,17183,17184,17187],{"style":4196},[81,17185],{"className":17186,"style":583},[223],[81,17188,17190],{"className":17189},[125],[81,17191,686],{"className":17192},[125],[81,17194,17195,17198],{"style":1278},[81,17196],{"className":17197,"style":583},[223],[81,17199],{"className":17200,"style":1286},[1285],[81,17202,17203,17206],{"style":4290},[81,17204],{"className":17205,"style":583},[223],[81,17207,17209],{"className":17208},[125],[81,17210,679],{"className":17211},[125],[81,17213,606],{"className":17214},[605],[81,17216,17218],{"className":17217},[212],[81,17219,17221],{"className":17220,"style":4385},[216],[81,17222],{},[81,17224],{"className":17225},[442,1247],[81,17227,3518],{"className":17228},[125,126],[81,17230],{"className":17231,"style":127},[912],[81,17233,6422],{"className":17234},[235],[81,17236],{"className":17237,"style":127},[912],[81,17239,17241,17244,17306,17336,17339,17342],{"className":17240},[116],[81,17242],{"className":17243,"style":13750},[120],[81,17245,17247,17250,17303],{"className":17246},[125],[81,17248],{"className":17249},[435,1247],[81,17251,17253],{"className":17252},[1199],[81,17254,17256,17295],{"className":17255},[208,569],[81,17257,17259,17292],{"className":17258},[212],[81,17260,17262,17273,17281],{"className":17261,"style":8162},[216],[81,17263,17264,17267],{"style":4196},[81,17265],{"className":17266,"style":583},[223],[81,17268,17270],{"className":17269},[125],[81,17271,17037],{"className":17272},[125],[81,17274,17275,17278],{"style":1278},[81,17276],{"className":17277,"style":583},[223],[81,17279],{"className":17280,"style":1286},[1285],[81,17282,17283,17286],{"style":4290},[81,17284],{"className":17285,"style":583},[223],[81,17287,17289],{"className":17288},[125],[81,17290,679],{"className":17291},[125],[81,17293,606],{"className":17294},[605],[81,17296,17298],{"className":17297},[212],[81,17299,17301],{"className":17300,"style":4385},[216],[81,17302],{},[81,17304],{"className":17305},[442,1247],[81,17307,17309,17312],{"className":17308},[125],[81,17310,3518],{"className":17311},[125,126],[81,17313,17315],{"className":17314},[204],[81,17316,17318],{"className":17317},[208],[81,17319,17321],{"className":17320},[212],[81,17322,17325],{"className":17323,"style":17324},[216],"height:0.8641em;",[81,17326,17327,17330],{"style":1749},[81,17328],{"className":17329,"style":224},[223],[81,17331,17333],{"className":17332},[228,229,230,231],[81,17334,526],{"className":17335},[125,231],[81,17337],{"className":17338,"style":127},[912],[81,17340,2044],{"className":17341},[235],[81,17343],{"className":17344,"style":127},[912],[81,17346,17348,17352],{"className":17347},[116],[81,17349],{"className":17350,"style":17351},[120],"height:0.123em;",[81,17353,8088],{"className":17354},[546],[14,17356,17357],{},"One can generate arbitrary polynomial terms using linear combinations of its shifted versions.",[14,17359,17360,17361,17539],{},"By adjusting ",[81,17362,17364,17398],{"className":17363,"translate":85},[84],[81,17365,17367],{"className":17366},[89],[91,17368,17369],{"xmlns":93},[95,17370,17371,17395],{},[98,17372,17373,17379,17381,17387,17389],{},[3006,17374,17375,17377],{},[101,17376,15219],{},[101,17378,3015],{},[179,17380,1414],{"separator":112},[3006,17382,17383,17385],{},[101,17384,15230],{},[101,17386,3015],{},[179,17388,1414],{"separator":112},[3006,17390,17391,17393],{},[101,17392,15241],{},[101,17394,3015],{},[105,17396,17397],{"encoding":107},"\\alpha_i, \\beta_i, \\gamma_i",[81,17399,17401],{"className":17400,"ariaHidden":112},[111],[81,17402,17404,17407,17447,17450,17453,17493,17496,17499],{"className":17403},[116],[81,17405],{"className":17406,"style":6157},[120],[81,17408,17410,17413],{"className":17409},[125],[81,17411,15219],{"className":17412,"style":15364},[125,126],[81,17414,17416],{"className":17415},[204],[81,17417,17419,17439],{"className":17418},[208,569],[81,17420,17422,17436],{"className":17421},[212],[81,17423,17425],{"className":17424,"style":3050},[216],[81,17426,17427,17430],{"style":15379},[81,17428],{"className":17429,"style":224},[223],[81,17431,17433],{"className":17432},[228,229,230,231],[81,17434,3015],{"className":17435},[125,126,231],[81,17437,606],{"className":17438},[605],[81,17440,17442],{"className":17441},[212],[81,17443,17445],{"className":17444,"style":3177},[216],[81,17446],{},[81,17448,1414],{"className":17449},[1441],[81,17451],{"className":17452,"style":1025},[912],[81,17454,17456,17459],{"className":17455},[125],[81,17457,15230],{"className":17458,"style":15412},[125,126],[81,17460,17462],{"className":17461},[204],[81,17463,17465,17485],{"className":17464},[208,569],[81,17466,17468,17482],{"className":17467},[212],[81,17469,17471],{"className":17470,"style":3050},[216],[81,17472,17473,17476],{"style":15427},[81,17474],{"className":17475,"style":224},[223],[81,17477,17479],{"className":17478},[228,229,230,231],[81,17480,3015],{"className":17481},[125,126,231],[81,17483,606],{"className":17484},[605],[81,17486,17488],{"className":17487},[212],[81,17489,17491],{"className":17490,"style":3177},[216],[81,17492],{},[81,17494,1414],{"className":17495},[1441],[81,17497],{"className":17498,"style":1025},[912],[81,17500,17502,17505],{"className":17501},[125],[81,17503,15241],{"className":17504,"style":4206},[125,126],[81,17506,17508],{"className":17507},[204],[81,17509,17511,17531],{"className":17510},[208,569],[81,17512,17514,17528],{"className":17513},[212],[81,17515,17517],{"className":17516,"style":3050},[216],[81,17518,17519,17522],{"style":15486},[81,17520],{"className":17521,"style":224},[223],[81,17523,17525],{"className":17524},[228,229,230,231],[81,17526,3015],{"className":17527},[125,126,231],[81,17529,606],{"className":17530},[605],[81,17532,17534],{"className":17533},[212],[81,17535,17537],{"className":17536,"style":3177},[216],[81,17538],{},", the linear combination",[81,17541,17543],{"className":17542,"translate":85},[809],[81,17544,17546,17592],{"className":17545,"translate":85},[84],[81,17547,17549],{"className":17548},[89],[91,17550,17551],{"xmlns":93,"display":818},[95,17552,17553,17589],{},[98,17554,17555,17561,17567,17569,17571,17577,17579,17581,17587],{},[1535,17556,17557,17559],{},[179,17558,6641],{},[101,17560,3015],{},[3006,17562,17563,17565],{},[101,17564,15219],{},[101,17566,3015],{},[101,17568,2133],{},[179,17570,410],{"stretchy":409},[3006,17572,17573,17575],{},[101,17574,15230],{},[101,17576,3015],{},[101,17578,3518],{},[179,17580,2044],{},[3006,17582,17583,17585],{},[101,17584,15241],{},[101,17586,3015],{},[179,17588,415],{"stretchy":409},[105,17590,17591],{"encoding":107},"   \\sum_i \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[81,17593,17595,17748],{"className":17594,"ariaHidden":112},[111],[81,17596,17598,17602,17647,17650,17690,17693,17696,17736,17739,17742,17745],{"className":17597},[116],[81,17599],{"className":17600,"style":17601},[120],"height:2.3277em;vertical-align:-1.2777em;",[81,17603,17605],{"className":17604},[6802,6803],[81,17606,17608,17639],{"className":17607},[208,569],[81,17609,17611,17636],{"className":17610},[212],[81,17612,17615,17626],{"className":17613,"style":17614},[216],"height:1.05em;",[81,17616,17617,17620],{"style":15300},[81,17618],{"className":17619,"style":6820},[223],[81,17621,17623],{"className":17622},[228,229,230,231],[81,17624,3015],{"className":17625},[125,126,231],[81,17627,17628,17631],{"style":6838},[81,17629],{"className":17630,"style":6820},[223],[81,17632,17633],{},[81,17634,6641],{"className":17635},[6802,6847,6848],[81,17637,606],{"className":17638},[605],[81,17640,17642],{"className":17641},[212],[81,17643,17645],{"className":17644,"style":15352},[216],[81,17646],{},[81,17648],{"className":17649,"style":1025},[912],[81,17651,17653,17656],{"className":17652},[125],[81,17654,15219],{"className":17655,"style":15364},[125,126],[81,17657,17659],{"className":17658},[204],[81,17660,17662,17682],{"className":17661},[208,569],[81,17663,17665,17679],{"className":17664},[212],[81,17666,17668],{"className":17667,"style":3050},[216],[81,17669,17670,17673],{"style":15379},[81,17671],{"className":17672,"style":224},[223],[81,17674,17676],{"className":17675},[228,229,230,231],[81,17677,3015],{"className":17678},[125,126,231],[81,17680,606],{"className":17681},[605],[81,17683,17685],{"className":17684},[212],[81,17686,17688],{"className":17687,"style":3177},[216],[81,17689],{},[81,17691,2133],{"className":17692,"style":297},[125,126],[81,17694,410],{"className":17695},[435],[81,17697,17699,17702],{"className":17698},[125],[81,17700,15230],{"className":17701,"style":15412},[125,126],[81,17703,17705],{"className":17704},[204],[81,17706,17708,17728],{"className":17707},[208,569],[81,17709,17711,17725],{"className":17710},[212],[81,17712,17714],{"className":17713,"style":3050},[216],[81,17715,17716,17719],{"style":15427},[81,17717],{"className":17718,"style":224},[223],[81,17720,17722],{"className":17721},[228,229,230,231],[81,17723,3015],{"className":17724},[125,126,231],[81,17726,606],{"className":17727},[605],[81,17729,17731],{"className":17730},[212],[81,17732,17734],{"className":17733,"style":3177},[216],[81,17735],{},[81,17737,3518],{"className":17738},[125,126],[81,17740],{"className":17741,"style":127},[912],[81,17743,2044],{"className":17744},[235],[81,17746],{"className":17747,"style":127},[912],[81,17749,17751,17754,17794],{"className":17750},[116],[81,17752],{"className":17753,"style":428},[120],[81,17755,17757,17760],{"className":17756},[125],[81,17758,15241],{"className":17759,"style":4206},[125,126],[81,17761,17763],{"className":17762},[204],[81,17764,17766,17786],{"className":17765},[208,569],[81,17767,17769,17783],{"className":17768},[212],[81,17770,17772],{"className":17771,"style":3050},[216],[81,17773,17774,17777],{"style":15486},[81,17775],{"className":17776,"style":224},[223],[81,17778,17780],{"className":17779},[228,229,230,231],[81,17781,3015],{"className":17782},[125,126,231],[81,17784,606],{"className":17785},[605],[81,17787,17789],{"className":17788},[212],[81,17790,17792],{"className":17791,"style":3177},[216],[81,17793],{},[81,17795,415],{"className":17796},[442],[14,17798,17799,17800,17858],{},"can approximate ",[81,17801,17803,17820],{"className":17802,"translate":85},[84],[81,17804,17806],{"className":17805},[89],[91,17807,17808],{"xmlns":93},[95,17809,17810,17818],{},[98,17811,17812],{},[174,17813,17814,17816],{},[101,17815,3518],{},[101,17817,3746],{},[105,17819,16272],{"encoding":107},[81,17821,17823],{"className":17822,"ariaHidden":112},[111],[81,17824,17826,17829],{"className":17825},[116],[81,17827],{"className":17828,"style":4767},[120],[81,17830,17832,17835],{"className":17831},[125],[81,17833,3518],{"className":17834},[125,126],[81,17836,17838],{"className":17837},[204],[81,17839,17841],{"className":17840},[208],[81,17842,17844],{"className":17843},[212],[81,17845,17847],{"className":17846,"style":4767},[216],[81,17848,17849,17852],{"style":219},[81,17850],{"className":17851,"style":224},[223],[81,17853,17855],{"className":17854},[228,229,230,231],[81,17856,3746],{"className":17857,"style":4244},[125,126,231]," to arbitrary precision (similar to Taylor polynomial approximation in calculus).",[11,17860,17861],{},[14,17862,17863,17866],{},[24,17864,17865],{},"Important intuition here:"," Each neuron is a \"nonlinear function block\", and summing many such blocks can combine into a polynomial form.",[14,17868,17869],{},[24,17870,17871],{},"Step 3: Combine to approximate the polynomial",[14,17873,17874,17875,18101,18102,18160,18161,18247],{},"For a polynomial ",[81,17876,17878,17926],{"className":17877,"translate":85},[84],[81,17879,17881],{"className":17880},[89],[91,17882,17883],{"xmlns":93},[95,17884,17885,17923],{},[98,17886,17887,17889,17891,17893,17895,17897,17911,17917],{},[101,17888,6362],{},[179,17890,410],{"stretchy":409},[101,17892,3518],{},[179,17894,415],{"stretchy":409},[179,17896,833],{},[3668,17898,17899,17901,17909],{},[179,17900,6641],{},[98,17902,17903,17905,17907],{},[101,17904,3746],{},[179,17906,833],{},[515,17908,513],{},[101,17910,8613],{},[3006,17912,17913,17915],{},[101,17914,8618],{},[101,17916,3746],{},[174,17918,17919,17921],{},[101,17920,3518],{},[101,17922,3746],{},[105,17924,17925],{"encoding":107},"P(x) = \\sum_{k=0}^m a_k x^k",[81,17927,17929,17956],{"className":17928,"ariaHidden":112},[111],[81,17930,17932,17935,17938,17941,17944,17947,17950,17953],{"className":17931},[116],[81,17933],{"className":17934,"style":428},[120],[81,17936,6362],{"className":17937,"style":1394},[125,126],[81,17939,410],{"className":17940},[435],[81,17942,3518],{"className":17943},[125,126],[81,17945,415],{"className":17946},[442],[81,17948],{"className":17949,"style":913},[912],[81,17951,833],{"className":17952},[917],[81,17954],{"className":17955,"style":913},[912],[81,17957,17959,17963,18029,18032,18072],{"className":17958},[116],[81,17960],{"className":17961,"style":17962},[120],"height:1.1488em;vertical-align:-0.2997em;",[81,17964,17966,17971],{"className":17965},[6802],[81,17967,6641],{"className":17968,"style":17970},[6802,6847,17969],"small-op","position:relative;top:0em;",[81,17972,17974],{"className":17973},[204],[81,17975,17977,18020],{"className":17976},[208,569],[81,17978,17980,18017],{"className":17979},[212],[81,17981,17984,18005],{"className":17982,"style":17983},[216],"height:0.8043em;",[81,17985,17987,17990],{"style":17986},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[81,17988],{"className":17989,"style":224},[223],[81,17991,17993],{"className":17992},[228,229,230,231],[81,17994,17996,17999,18002],{"className":17995},[125,231],[81,17997,3746],{"className":17998,"style":4244},[125,126,231],[81,18000,833],{"className":18001},[917,231],[81,18003,513],{"className":18004},[125,231],[81,18006,18008,18011],{"style":18007},"top:-3.2029em;margin-right:0.05em;",[81,18009],{"className":18010,"style":224},[223],[81,18012,18014],{"className":18013},[228,229,230,231],[81,18015,8613],{"className":18016},[125,126,231],[81,18018,606],{"className":18019},[605],[81,18021,18023],{"className":18022},[212],[81,18024,18027],{"className":18025,"style":18026},[216],"height:0.2997em;",[81,18028],{},[81,18030],{"className":18031,"style":1025},[912],[81,18033,18035,18038],{"className":18034},[125],[81,18036,8618],{"className":18037},[125,126],[81,18039,18041],{"className":18040},[204],[81,18042,18044,18064],{"className":18043},[208,569],[81,18045,18047,18061],{"className":18046},[212],[81,18048,18050],{"className":18049,"style":8355},[216],[81,18051,18052,18055],{"style":8358},[81,18053],{"className":18054,"style":224},[223],[81,18056,18058],{"className":18057},[228,229,230,231],[81,18059,3746],{"className":18060,"style":4244},[125,126,231],[81,18062,606],{"className":18063},[605],[81,18065,18067],{"className":18066},[212],[81,18068,18070],{"className":18069,"style":3177},[216],[81,18071],{},[81,18073,18075,18078],{"className":18074},[125],[81,18076,3518],{"className":18077},[125,126],[81,18079,18081],{"className":18080},[204],[81,18082,18084],{"className":18083},[208],[81,18085,18087],{"className":18086},[212],[81,18088,18090],{"className":18089,"style":4767},[216],[81,18091,18092,18095],{"style":219},[81,18093],{"className":18094,"style":224},[223],[81,18096,18098],{"className":18097},[228,229,230,231],[81,18099,3746],{"className":18100,"style":4244},[125,126,231],", for each ",[81,18103,18105,18122],{"className":18104,"translate":85},[84],[81,18106,18108],{"className":18107},[89],[91,18109,18110],{"xmlns":93},[95,18111,18112,18120],{},[98,18113,18114],{},[174,18115,18116,18118],{},[101,18117,3518],{},[101,18119,3746],{},[105,18121,16272],{"encoding":107},[81,18123,18125],{"className":18124,"ariaHidden":112},[111],[81,18126,18128,18131],{"className":18127},[116],[81,18129],{"className":18130,"style":4767},[120],[81,18132,18134,18137],{"className":18133},[125],[81,18135,3518],{"className":18136},[125,126],[81,18138,18140],{"className":18139},[204],[81,18141,18143],{"className":18142},[208],[81,18144,18146],{"className":18145},[212],[81,18147,18149],{"className":18148,"style":4767},[216],[81,18150,18151,18154],{"style":219},[81,18152],{"className":18153,"style":224},[223],[81,18155,18157],{"className":18156},[228,229,230,231],[81,18158,3746],{"className":18159,"style":4244},[125,126,231]," find the corresponding neuron combination ",[81,18162,18164,18188],{"className":18163,"translate":85},[84],[81,18165,18167],{"className":18166},[89],[91,18168,18169],{"xmlns":93},[95,18170,18171,18185],{},[98,18172,18173,18179,18181,18183],{},[3006,18174,18175,18177],{},[101,18176,15192],{},[101,18178,3746],{},[179,18180,410],{"stretchy":409},[101,18182,3518],{},[179,18184,415],{"stretchy":409},[105,18186,18187],{"encoding":107},"N_k(x)",[81,18189,18191],{"className":18190,"ariaHidden":112},[111],[81,18192,18194,18197,18238,18241,18244],{"className":18193},[116],[81,18195],{"className":18196,"style":428},[120],[81,18198,18200,18203],{"className":18199},[125],[81,18201,15192],{"className":18202,"style":12969},[125,126],[81,18204,18206],{"className":18205},[204],[81,18207,18209,18230],{"className":18208},[208,569],[81,18210,18212,18227],{"className":18211},[212],[81,18213,18215],{"className":18214,"style":8355},[216],[81,18216,18218,18221],{"style":18217},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[81,18219],{"className":18220,"style":224},[223],[81,18222,18224],{"className":18223},[228,229,230,231],[81,18225,3746],{"className":18226,"style":4244},[125,126,231],[81,18228,606],{"className":18229},[605],[81,18231,18233],{"className":18232},[212],[81,18234,18236],{"className":18235,"style":3177},[216],[81,18237],{},[81,18239,410],{"className":18240},[435],[81,18242,3518],{"className":18243},[125,126],[81,18245,415],{"className":18246},[442]," that approximates it:",[81,18249,18251],{"className":18250,"translate":85},[809],[81,18252,18254,18310],{"className":18253,"translate":85},[84],[81,18255,18257],{"className":18256},[89],[91,18258,18259],{"xmlns":93,"display":818},[95,18260,18261,18307],{},[98,18262,18263,18265,18271,18273,18275,18277,18279,18285,18287,18289],{},[101,18264,6411],{"mathvariant":1133},[3006,18266,18267,18269],{},[101,18268,15192],{},[101,18270,3746],{},[179,18272,410],{"stretchy":409},[101,18274,3518],{},[179,18276,415],{"stretchy":409},[179,18278,6422],{},[174,18280,18281,18283],{},[101,18282,3518],{},[101,18284,3746],{},[101,18286,6411],{"mathvariant":1133},[179,18288,6435],{},[1199,18290,18291,18293],{},[101,18292,15136],{},[98,18294,18295,18297,18299,18301,18303,18305],{},[515,18296,517],{},[179,18298,410],{"stretchy":409},[101,18300,8613],{},[179,18302,2044],{},[515,18304,679],{},[179,18306,415],{"stretchy":409},[105,18308,18309],{"encoding":107},"  |N_k(x) - x^k| \u003C \\frac{\\epsilon}{2 (m+1)}",[81,18311,18313,18380,18427],{"className":18312,"ariaHidden":112},[111],[81,18314,18316,18319,18322,18362,18365,18368,18371,18374,18377],{"className":18315},[116],[81,18317],{"className":18318,"style":428},[120],[81,18320,6411],{"className":18321},[125],[81,18323,18325,18328],{"className":18324},[125],[81,18326,15192],{"className":18327,"style":12969},[125,126],[81,18329,18331],{"className":18330},[204],[81,18332,18334,18354],{"className":18333},[208,569],[81,18335,18337,18351],{"className":18336},[212],[81,18338,18340],{"className":18339,"style":8355},[216],[81,18341,18342,18345],{"style":18217},[81,18343],{"className":18344,"style":224},[223],[81,18346,18348],{"className":18347},[228,229,230,231],[81,18349,3746],{"className":18350,"style":4244},[125,126,231],[81,18352,606],{"className":18353},[605],[81,18355,18357],{"className":18356},[212],[81,18358,18360],{"className":18359,"style":3177},[216],[81,18361],{},[81,18363,410],{"className":18364},[435],[81,18366,3518],{"className":18367},[125,126],[81,18369,415],{"className":18370},[442],[81,18372],{"className":18373,"style":127},[912],[81,18375,6422],{"className":18376},[235],[81,18378],{"className":18379,"style":127},[912],[81,18381,18383,18386,18415,18418,18421,18424],{"className":18382},[116],[81,18384],{"className":18385,"style":7082},[120],[81,18387,18389,18392],{"className":18388},[125],[81,18390,3518],{"className":18391},[125,126],[81,18393,18395],{"className":18394},[204],[81,18396,18398],{"className":18397},[208],[81,18399,18401],{"className":18400},[212],[81,18402,18404],{"className":18403,"style":7049},[216],[81,18405,18406,18409],{"style":1749},[81,18407],{"className":18408,"style":224},[223],[81,18410,18412],{"className":18411},[228,229,230,231],[81,18413,3746],{"className":18414,"style":4244},[125,126,231],[81,18416,6411],{"className":18417},[125],[81,18419],{"className":18420,"style":913},[912],[81,18422,6435],{"className":18423},[917],[81,18425],{"className":18426,"style":913},[912],[81,18428,18430,18434],{"className":18429},[116],[81,18431],{"className":18432,"style":18433},[120],"height:2.0436em;vertical-align:-0.936em;",[81,18435,18437,18440,18515],{"className":18436},[125],[81,18438],{"className":18439},[435,1247],[81,18441,18443],{"className":18442},[1199],[81,18444,18446,18506],{"className":18445},[208,569],[81,18447,18449,18503],{"className":18448},[212],[81,18450,18452,18484,18492],{"className":18451,"style":6991},[216],[81,18453,18454,18457],{"style":4196},[81,18455],{"className":18456,"style":583},[223],[81,18458,18460,18463,18466,18469,18472,18475,18478,18481],{"className":18459},[125],[81,18461,517],{"className":18462},[125],[81,18464,410],{"className":18465},[435],[81,18467,8613],{"className":18468},[125,126],[81,18470],{"className":18471,"style":127},[912],[81,18473,2044],{"className":18474},[235],[81,18476],{"className":18477,"style":127},[912],[81,18479,679],{"className":18480},[125],[81,18482,415],{"className":18483},[442],[81,18485,18486,18489],{"style":1278},[81,18487],{"className":18488,"style":583},[223],[81,18490],{"className":18491,"style":1286},[1285],[81,18493,18494,18497],{"style":4290},[81,18495],{"className":18496,"style":583},[223],[81,18498,18500],{"className":18499},[125],[81,18501,15136],{"className":18502},[125,126],[81,18504,606],{"className":18505},[605],[81,18507,18509],{"className":18508},[212],[81,18510,18513],{"className":18511,"style":18512},[216],"height:0.936em;",[81,18514],{},[81,18516],{"className":18517},[442,1247],[14,18519,18520],{},"Then combine:",[81,18522,18524],{"className":18523,"translate":85},[809],[81,18525,18527,18581],{"className":18526,"translate":85},[84],[81,18528,18530],{"className":18529},[89],[91,18531,18532],{"xmlns":93,"display":818},[95,18533,18534,18578],{},[98,18535,18536,18538,18540,18542,18544,18546,18560,18566,18572,18574,18576],{},[101,18537,15192],{},[179,18539,410],{"stretchy":409},[101,18541,3518],{},[179,18543,415],{"stretchy":409},[179,18545,833],{},[6637,18547,18548,18550,18558],{},[179,18549,6641],{},[98,18551,18552,18554,18556],{},[101,18553,3746],{},[179,18555,833],{},[515,18557,513],{},[101,18559,8613],{},[3006,18561,18562,18564],{},[101,18563,8618],{},[101,18565,3746],{},[3006,18567,18568,18570],{},[101,18569,15192],{},[101,18571,3746],{},[179,18573,410],{"stretchy":409},[101,18575,3518],{},[179,18577,415],{"stretchy":409},[105,18579,18580],{"encoding":107},"  N(x) = \\sum_{k=0}^m a_k N_k(x)",[81,18582,18584,18611],{"className":18583,"ariaHidden":112},[111],[81,18585,18587,18590,18593,18596,18599,18602,18605,18608],{"className":18586},[116],[81,18588],{"className":18589,"style":428},[120],[81,18591,15192],{"className":18592,"style":12969},[125,126],[81,18594,410],{"className":18595},[435],[81,18597,3518],{"className":18598},[125,126],[81,18600,415],{"className":18601},[442],[81,18603],{"className":18604,"style":913},[912],[81,18606,833],{"className":18607},[917],[81,18609],{"className":18610,"style":913},[912],[81,18612,18614,18617,18681,18684,18724,18764,18767,18770],{"className":18613},[116],[81,18615],{"className":18616,"style":6798},[120],[81,18618,18620],{"className":18619},[6802,6803],[81,18621,18623,18673],{"className":18622},[208,569],[81,18624,18626,18670],{"className":18625},[212],[81,18627,18629,18649,18659],{"className":18628,"style":6813},[216],[81,18630,18631,18634],{"style":6816},[81,18632],{"className":18633,"style":6820},[223],[81,18635,18637],{"className":18636},[228,229,230,231],[81,18638,18640,18643,18646],{"className":18639},[125,231],[81,18641,3746],{"className":18642,"style":4244},[125,126,231],[81,18644,833],{"className":18645},[917,231],[81,18647,513],{"className":18648},[125,231],[81,18650,18651,18654],{"style":6838},[81,18652],{"className":18653,"style":6820},[223],[81,18655,18656],{},[81,18657,6641],{"className":18658},[6802,6847,6848],[81,18660,18661,18664],{"style":6851},[81,18662],{"className":18663,"style":6820},[223],[81,18665,18667],{"className":18666},[228,229,230,231],[81,18668,8613],{"className":18669},[125,126,231],[81,18671,606],{"className":18672},[605],[81,18674,18676],{"className":18675},[212],[81,18677,18679],{"className":18678,"style":6873},[216],[81,18680],{},[81,18682],{"className":18683,"style":1025},[912],[81,18685,18687,18690],{"className":18686},[125],[81,18688,8618],{"className":18689},[125,126],[81,18691,18693],{"className":18692},[204],[81,18694,18696,18716],{"className":18695},[208,569],[81,18697,18699,18713],{"className":18698},[212],[81,18700,18702],{"className":18701,"style":8355},[216],[81,18703,18704,18707],{"style":8358},[81,18705],{"className":18706,"style":224},[223],[81,18708,18710],{"className":18709},[228,229,230,231],[81,18711,3746],{"className":18712,"style":4244},[125,126,231],[81,18714,606],{"className":18715},[605],[81,18717,18719],{"className":18718},[212],[81,18720,18722],{"className":18721,"style":3177},[216],[81,18723],{},[81,18725,18727,18730],{"className":18726},[125],[81,18728,15192],{"className":18729,"style":12969},[125,126],[81,18731,18733],{"className":18732},[204],[81,18734,18736,18756],{"className":18735},[208,569],[81,18737,18739,18753],{"className":18738},[212],[81,18740,18742],{"className":18741,"style":8355},[216],[81,18743,18744,18747],{"style":18217},[81,18745],{"className":18746,"style":224},[223],[81,18748,18750],{"className":18749},[228,229,230,231],[81,18751,3746],{"className":18752,"style":4244},[125,126,231],[81,18754,606],{"className":18755},[605],[81,18757,18759],{"className":18758},[212],[81,18760,18762],{"className":18761,"style":3177},[216],[81,18763],{},[81,18765,410],{"className":18766},[435],[81,18768,3518],{"className":18769},[125,126],[81,18771,415],{"className":18772},[442],[14,18774,18775],{},"Error bound:",[81,18777,18779],{"className":18778,"translate":85},[809],[81,18780,18782,18918],{"className":18781,"translate":85},[84],[81,18783,18785],{"className":18784},[89],[91,18786,18787],{"xmlns":93,"display":818},[95,18788,18789,18915],{},[98,18790,18791,18793,18795,18797,18799,18801,18803,18805,18807,18809,18811,18813,18815,18829,18831,18837,18839,18841,18847,18849,18851,18853,18855,18861,18863,18865,18879,18881,18887,18889,18907,18909,18911,18913],{},[101,18792,6411],{"mathvariant":1133},[101,18794,15192],{},[179,18796,410],{"stretchy":409},[101,18798,3518],{},[179,18800,415],{"stretchy":409},[179,18802,6422],{},[101,18804,6362],{},[179,18806,410],{"stretchy":409},[101,18808,3518],{},[179,18810,415],{"stretchy":409},[101,18812,6411],{"mathvariant":1133},[179,18814,11907],{},[6637,18816,18817,18819,18827],{},[179,18818,6641],{},[98,18820,18821,18823,18825],{},[101,18822,3746],{},[179,18824,833],{},[515,18826,513],{},[101,18828,8613],{},[101,18830,6411],{"mathvariant":1133},[3006,18832,18833,18835],{},[101,18834,8618],{},[101,18836,3746],{},[101,18838,6411],{"mathvariant":1133},[101,18840,6411],{"mathvariant":1133},[3006,18842,18843,18845],{},[101,18844,15192],{},[101,18846,3746],{},[179,18848,410],{"stretchy":409},[101,18850,3518],{},[179,18852,415],{"stretchy":409},[179,18854,6422],{},[174,18856,18857,18859],{},[101,18858,3518],{},[101,18860,3746],{},[101,18862,6411],{"mathvariant":1133},[179,18864,6435],{},[6637,18866,18867,18869,18877],{},[179,18868,6641],{},[98,18870,18871,18873,18875],{},[101,18872,3746],{},[179,18874,833],{},[515,18876,513],{},[101,18878,8613],{},[101,18880,6411],{"mathvariant":1133},[3006,18882,18883,18885],{},[101,18884,8618],{},[101,18886,3746],{},[101,18888,6411],{"mathvariant":1133},[1199,18890,18891,18893],{},[101,18892,15136],{},[98,18894,18895,18897,18899,18901,18903,18905],{},[515,18896,517],{},[179,18898,410],{"stretchy":409},[101,18900,8613],{},[179,18902,2044],{},[515,18904,679],{},[179,18906,415],{"stretchy":409},[179,18908,6435],{},[101,18910,15136],{},[101,18912,1134],{"mathvariant":1133},[515,18914,517],{},[105,18916,18917],{"encoding":107},"  |N(x) - P(x)| \\le \\sum_{k=0}^m |a_k||N_k(x)-x^k| \u003C \\sum_{k=0}^m |a_k| \\frac{\\epsilon}{2(m+1)} \u003C \\epsilon\u002F2",[81,18919,18921,18951,18981,19159,19206,19417],{"className":18920,"ariaHidden":112},[111],[81,18922,18924,18927,18930,18933,18936,18939,18942,18945,18948],{"className":18923},[116],[81,18925],{"className":18926,"style":428},[120],[81,18928,6411],{"className":18929},[125],[81,18931,15192],{"className":18932,"style":12969},[125,126],[81,18934,410],{"className":18935},[435],[81,18937,3518],{"className":18938},[125,126],[81,18940,415],{"className":18941},[442],[81,18943],{"className":18944,"style":127},[912],[81,18946,6422],{"className":18947},[235],[81,18949],{"className":18950,"style":127},[912],[81,18952,18954,18957,18960,18963,18966,18969,18972,18975,18978],{"className":18953},[116],[81,18955],{"className":18956,"style":428},[120],[81,18958,6362],{"className":18959,"style":1394},[125,126],[81,18961,410],{"className":18962},[435],[81,18964,3518],{"className":18965},[125,126],[81,18967,415],{"className":18968},[442],[81,18970,6411],{"className":18971},[125],[81,18973],{"className":18974,"style":913},[912],[81,18976,11907],{"className":18977},[917],[81,18979],{"className":18980,"style":913},[912],[81,18982,18984,18987,19051,19054,19057,19097,19101,19141,19144,19147,19150,19153,19156],{"className":18983},[116],[81,18985],{"className":18986,"style":6798},[120],[81,18988,18990],{"className":18989},[6802,6803],[81,18991,18993,19043],{"className":18992},[208,569],[81,18994,18996,19040],{"className":18995},[212],[81,18997,18999,19019,19029],{"className":18998,"style":6813},[216],[81,19000,19001,19004],{"style":6816},[81,19002],{"className":19003,"style":6820},[223],[81,19005,19007],{"className":19006},[228,229,230,231],[81,19008,19010,19013,19016],{"className":19009},[125,231],[81,19011,3746],{"className":19012,"style":4244},[125,126,231],[81,19014,833],{"className":19015},[917,231],[81,19017,513],{"className":19018},[125,231],[81,19020,19021,19024],{"style":6838},[81,19022],{"className":19023,"style":6820},[223],[81,19025,19026],{},[81,19027,6641],{"className":19028},[6802,6847,6848],[81,19030,19031,19034],{"style":6851},[81,19032],{"className":19033,"style":6820},[223],[81,19035,19037],{"className":19036},[228,229,230,231],[81,19038,8613],{"className":19039},[125,126,231],[81,19041,606],{"className":19042},[605],[81,19044,19046],{"className":19045},[212],[81,19047,19049],{"className":19048,"style":6873},[216],[81,19050],{},[81,19052],{"className":19053,"style":1025},[912],[81,19055,6411],{"className":19056},[125],[81,19058,19060,19063],{"className":19059},[125],[81,19061,8618],{"className":19062},[125,126],[81,19064,19066],{"className":19065},[204],[81,19067,19069,19089],{"className":19068},[208,569],[81,19070,19072,19086],{"className":19071},[212],[81,19073,19075],{"className":19074,"style":8355},[216],[81,19076,19077,19080],{"style":8358},[81,19078],{"className":19079,"style":224},[223],[81,19081,19083],{"className":19082},[228,229,230,231],[81,19084,3746],{"className":19085,"style":4244},[125,126,231],[81,19087,606],{"className":19088},[605],[81,19090,19092],{"className":19091},[212],[81,19093,19095],{"className":19094,"style":3177},[216],[81,19096],{},[81,19098,19100],{"className":19099},[125],"∣∣",[81,19102,19104,19107],{"className":19103},[125],[81,19105,15192],{"className":19106,"style":12969},[125,126],[81,19108,19110],{"className":19109},[204],[81,19111,19113,19133],{"className":19112},[208,569],[81,19114,19116,19130],{"className":19115},[212],[81,19117,19119],{"className":19118,"style":8355},[216],[81,19120,19121,19124],{"style":18217},[81,19122],{"className":19123,"style":224},[223],[81,19125,19127],{"className":19126},[228,229,230,231],[81,19128,3746],{"className":19129,"style":4244},[125,126,231],[81,19131,606],{"className":19132},[605],[81,19134,19136],{"className":19135},[212],[81,19137,19139],{"className":19138,"style":3177},[216],[81,19140],{},[81,19142,410],{"className":19143},[435],[81,19145,3518],{"className":19146},[125,126],[81,19148,415],{"className":19149},[442],[81,19151],{"className":19152,"style":127},[912],[81,19154,6422],{"className":19155},[235],[81,19157],{"className":19158,"style":127},[912],[81,19160,19162,19165,19194,19197,19200,19203],{"className":19161},[116],[81,19163],{"className":19164,"style":7082},[120],[81,19166,19168,19171],{"className":19167},[125],[81,19169,3518],{"className":19170},[125,126],[81,19172,19174],{"className":19173},[204],[81,19175,19177],{"className":19176},[208],[81,19178,19180],{"className":19179},[212],[81,19181,19183],{"className":19182,"style":7049},[216],[81,19184,19185,19188],{"style":1749},[81,19186],{"className":19187,"style":224},[223],[81,19189,19191],{"className":19190},[228,229,230,231],[81,19192,3746],{"className":19193,"style":4244},[125,126,231],[81,19195,6411],{"className":19196},[125],[81,19198],{"className":19199,"style":913},[912],[81,19201,6435],{"className":19202},[917],[81,19204],{"className":19205,"style":913},[912],[81,19207,19209,19212,19276,19279,19282,19322,19325,19408,19411,19414],{"className":19208},[116],[81,19210],{"className":19211,"style":6798},[120],[81,19213,19215],{"className":19214},[6802,6803],[81,19216,19218,19268],{"className":19217},[208,569],[81,19219,19221,19265],{"className":19220},[212],[81,19222,19224,19244,19254],{"className":19223,"style":6813},[216],[81,19225,19226,19229],{"style":6816},[81,19227],{"className":19228,"style":6820},[223],[81,19230,19232],{"className":19231},[228,229,230,231],[81,19233,19235,19238,19241],{"className":19234},[125,231],[81,19236,3746],{"className":19237,"style":4244},[125,126,231],[81,19239,833],{"className":19240},[917,231],[81,19242,513],{"className":19243},[125,231],[81,19245,19246,19249],{"style":6838},[81,19247],{"className":19248,"style":6820},[223],[81,19250,19251],{},[81,19252,6641],{"className":19253},[6802,6847,6848],[81,19255,19256,19259],{"style":6851},[81,19257],{"className":19258,"style":6820},[223],[81,19260,19262],{"className":19261},[228,229,230,231],[81,19263,8613],{"className":19264},[125,126,231],[81,19266,606],{"className":19267},[605],[81,19269,19271],{"className":19270},[212],[81,19272,19274],{"className":19273,"style":6873},[216],[81,19275],{},[81,19277],{"className":19278,"style":1025},[912],[81,19280,6411],{"className":19281},[125],[81,19283,19285,19288],{"className":19284},[125],[81,19286,8618],{"className":19287},[125,126],[81,19289,19291],{"className":19290},[204],[81,19292,19294,19314],{"className":19293},[208,569],[81,19295,19297,19311],{"className":19296},[212],[81,19298,19300],{"className":19299,"style":8355},[216],[81,19301,19302,19305],{"style":8358},[81,19303],{"className":19304,"style":224},[223],[81,19306,19308],{"className":19307},[228,229,230,231],[81,19309,3746],{"className":19310,"style":4244},[125,126,231],[81,19312,606],{"className":19313},[605],[81,19315,19317],{"className":19316},[212],[81,19318,19320],{"className":19319,"style":3177},[216],[81,19321],{},[81,19323,6411],{"className":19324},[125],[81,19326,19328,19331,19405],{"className":19327},[125],[81,19329],{"className":19330},[435,1247],[81,19332,19334],{"className":19333},[1199],[81,19335,19337,19397],{"className":19336},[208,569],[81,19338,19340,19394],{"className":19339},[212],[81,19341,19343,19375,19383],{"className":19342,"style":6991},[216],[81,19344,19345,19348],{"style":4196},[81,19346],{"className":19347,"style":583},[223],[81,19349,19351,19354,19357,19360,19363,19366,19369,19372],{"className":19350},[125],[81,19352,517],{"className":19353},[125],[81,19355,410],{"className":19356},[435],[81,19358,8613],{"className":19359},[125,126],[81,19361],{"className":19362,"style":127},[912],[81,19364,2044],{"className":19365},[235],[81,19367],{"className":19368,"style":127},[912],[81,19370,679],{"className":19371},[125],[81,19373,415],{"className":19374},[442],[81,19376,19377,19380],{"style":1278},[81,19378],{"className":19379,"style":583},[223],[81,19381],{"className":19382,"style":1286},[1285],[81,19384,19385,19388],{"style":4290},[81,19386],{"className":19387,"style":583},[223],[81,19389,19391],{"className":19390},[125],[81,19392,15136],{"className":19393},[125,126],[81,19395,606],{"className":19396},[605],[81,19398,19400],{"className":19399},[212],[81,19401,19403],{"className":19402,"style":18512},[216],[81,19404],{},[81,19406],{"className":19407},[442,1247],[81,19409],{"className":19410,"style":913},[912],[81,19412,6435],{"className":19413},[917],[81,19415],{"className":19416,"style":913},[912],[81,19418,19420,19423,19426],{"className":19419},[116],[81,19421],{"className":19422,"style":428},[120],[81,19424,15136],{"className":19425},[125,126],[81,19427,15980],{"className":19428},[125],[14,19430,19431],{},[24,19432,19433],{},"Step 4: Triangle inequality completes the approximation",[14,19435,19436],{},"We have two sources of error:",[19438,19439,19440,19563],"ol",{},[451,19441,19442,19562],{},[81,19443,19445,19487],{"className":19444,"translate":85},[84],[81,19446,19448],{"className":19447},[89],[91,19449,19450],{"xmlns":93},[95,19451,19452,19484],{},[98,19453,19454,19456,19458,19460,19462,19464,19466,19468,19470,19472,19474,19476,19478,19480,19482],{},[101,19455,6411],{"mathvariant":1133},[101,19457,6124],{},[179,19459,410],{"stretchy":409},[101,19461,3518],{},[179,19463,415],{"stretchy":409},[179,19465,6422],{},[101,19467,6362],{},[179,19469,410],{"stretchy":409},[101,19471,3518],{},[179,19473,415],{"stretchy":409},[101,19475,6411],{"mathvariant":1133},[179,19477,6435],{},[101,19479,15136],{},[101,19481,1134],{"mathvariant":1133},[515,19483,517],{},[105,19485,19486],{"encoding":107},"|f(x) - P(x)| \u003C \\epsilon\u002F2",[81,19488,19490,19520,19550],{"className":19489,"ariaHidden":112},[111],[81,19491,19493,19496,19499,19502,19505,19508,19511,19514,19517],{"className":19492},[116],[81,19494],{"className":19495,"style":428},[120],[81,19497,6411],{"className":19498},[125],[81,19500,6124],{"className":19501,"style":6161},[125,126],[81,19503,410],{"className":19504},[435],[81,19506,3518],{"className":19507},[125,126],[81,19509,415],{"className":19510},[442],[81,19512],{"className":19513,"style":127},[912],[81,19515,6422],{"className":19516},[235],[81,19518],{"className":19519,"style":127},[912],[81,19521,19523,19526,19529,19532,19535,19538,19541,19544,19547],{"className":19522},[116],[81,19524],{"className":19525,"style":428},[120],[81,19527,6362],{"className":19528,"style":1394},[125,126],[81,19530,410],{"className":19531},[435],[81,19533,3518],{"className":19534},[125,126],[81,19536,415],{"className":19537},[442],[81,19539,6411],{"className":19540},[125],[81,19542],{"className":19543,"style":913},[912],[81,19545,6435],{"className":19546},[917],[81,19548],{"className":19549,"style":913},[912],[81,19551,19553,19556,19559],{"className":19552},[116],[81,19554],{"className":19555,"style":428},[120],[81,19557,15136],{"className":19558},[125,126],[81,19560,15980],{"className":19561},[125]," (Weierstrass theorem)",[451,19564,19565,19685],{},[81,19566,19568,19610],{"className":19567,"translate":85},[84],[81,19569,19571],{"className":19570},[89],[91,19572,19573],{"xmlns":93},[95,19574,19575,19607],{},[98,19576,19577,19579,19581,19583,19585,19587,19589,19591,19593,19595,19597,19599,19601,19603,19605],{},[101,19578,6411],{"mathvariant":1133},[101,19580,6362],{},[179,19582,410],{"stretchy":409},[101,19584,3518],{},[179,19586,415],{"stretchy":409},[179,19588,6422],{},[101,19590,15192],{},[179,19592,410],{"stretchy":409},[101,19594,3518],{},[179,19596,415],{"stretchy":409},[101,19598,6411],{"mathvariant":1133},[179,19600,6435],{},[101,19602,15136],{},[101,19604,1134],{"mathvariant":1133},[515,19606,517],{},[105,19608,19609],{"encoding":107},"|P(x) - N(x)| \u003C \\epsilon\u002F2",[81,19611,19613,19643,19673],{"className":19612,"ariaHidden":112},[111],[81,19614,19616,19619,19622,19625,19628,19631,19634,19637,19640],{"className":19615},[116],[81,19617],{"className":19618,"style":428},[120],[81,19620,6411],{"className":19621},[125],[81,19623,6362],{"className":19624,"style":1394},[125,126],[81,19626,410],{"className":19627},[435],[81,19629,3518],{"className":19630},[125,126],[81,19632,415],{"className":19633},[442],[81,19635],{"className":19636,"style":127},[912],[81,19638,6422],{"className":19639},[235],[81,19641],{"className":19642,"style":127},[912],[81,19644,19646,19649,19652,19655,19658,19661,19664,19667,19670],{"className":19645},[116],[81,19647],{"className":19648,"style":428},[120],[81,19650,15192],{"className":19651,"style":12969},[125,126],[81,19653,410],{"className":19654},[435],[81,19656,3518],{"className":19657},[125,126],[81,19659,415],{"className":19660},[442],[81,19662,6411],{"className":19663},[125],[81,19665],{"className":19666,"style":913},[912],[81,19668,6435],{"className":19669},[917],[81,19671],{"className":19672,"style":913},[912],[81,19674,19676,19679,19682],{"className":19675},[116],[81,19677],{"className":19678,"style":428},[120],[81,19680,15136],{"className":19681},[125,126],[81,19683,15980],{"className":19684},[125]," (neuron approximation of polynomials)",[14,19687,19688],{},"By the triangle inequality:",[81,19690,19692],{"className":19691,"translate":85},[809],[81,19693,19695,19797],{"className":19694,"translate":85},[84],[81,19696,19698],{"className":19697},[89],[91,19699,19700],{"xmlns":93,"display":818},[95,19701,19702,19794],{},[98,19703,19704,19706,19708,19710,19712,19714,19716,19718,19720,19722,19724,19726,19728,19730,19732,19734,19736,19738,19740,19742,19744,19746,19748,19750,19752,19754,19756,19758,19760,19762,19764,19766,19768,19770,19772,19774,19776,19778,19780,19782,19784,19786,19788,19790,19792],{},[101,19705,6411],{"mathvariant":1133},[101,19707,6124],{},[179,19709,410],{"stretchy":409},[101,19711,3518],{},[179,19713,415],{"stretchy":409},[179,19715,6422],{},[101,19717,15192],{},[179,19719,410],{"stretchy":409},[101,19721,3518],{},[179,19723,415],{"stretchy":409},[101,19725,6411],{"mathvariant":1133},[179,19727,11907],{},[101,19729,6411],{"mathvariant":1133},[101,19731,6124],{},[179,19733,410],{"stretchy":409},[101,19735,3518],{},[179,19737,415],{"stretchy":409},[179,19739,6422],{},[101,19741,6362],{},[179,19743,410],{"stretchy":409},[101,19745,3518],{},[179,19747,415],{"stretchy":409},[101,19749,6411],{"mathvariant":1133},[179,19751,2044],{},[101,19753,6411],{"mathvariant":1133},[101,19755,6362],{},[179,19757,410],{"stretchy":409},[101,19759,3518],{},[179,19761,415],{"stretchy":409},[179,19763,6422],{},[101,19765,15192],{},[179,19767,410],{"stretchy":409},[101,19769,3518],{},[179,19771,415],{"stretchy":409},[101,19773,6411],{"mathvariant":1133},[179,19775,6435],{},[101,19777,15136],{},[101,19779,1134],{"mathvariant":1133},[515,19781,517],{},[179,19783,2044],{},[101,19785,15136],{},[101,19787,1134],{"mathvariant":1133},[515,19789,517],{},[179,19791,833],{},[101,19793,15136],{},[105,19795,19796],{"encoding":107},"  |f(x) - N(x)| \\le |f(x)-P(x)| + |P(x)-N(x)| \u003C \\epsilon\u002F2 + \\epsilon\u002F2 = \\epsilon",[81,19798,19800,19830,19860,19890,19920,19950,19980,20001,20022],{"className":19799,"ariaHidden":112},[111],[81,19801,19803,19806,19809,19812,19815,19818,19821,19824,19827],{"className":19802},[116],[81,19804],{"className":19805,"style":428},[120],[81,19807,6411],{"className":19808},[125],[81,19810,6124],{"className":19811,"style":6161},[125,126],[81,19813,410],{"className":19814},[435],[81,19816,3518],{"className":19817},[125,126],[81,19819,415],{"className":19820},[442],[81,19822],{"className":19823,"style":127},[912],[81,19825,6422],{"className":19826},[235],[81,19828],{"className":19829,"style":127},[912],[81,19831,19833,19836,19839,19842,19845,19848,19851,19854,19857],{"className":19832},[116],[81,19834],{"className":19835,"style":428},[120],[81,19837,15192],{"className":19838,"style":12969},[125,126],[81,19840,410],{"className":19841},[435],[81,19843,3518],{"className":19844},[125,126],[81,19846,415],{"className":19847},[442],[81,19849,6411],{"className":19850},[125],[81,19852],{"className":19853,"style":913},[912],[81,19855,11907],{"className":19856},[917],[81,19858],{"className":19859,"style":913},[912],[81,19861,19863,19866,19869,19872,19875,19878,19881,19884,19887],{"className":19862},[116],[81,19864],{"className":19865,"style":428},[120],[81,19867,6411],{"className":19868},[125],[81,19870,6124],{"className":19871,"style":6161},[125,126],[81,19873,410],{"className":19874},[435],[81,19876,3518],{"className":19877},[125,126],[81,19879,415],{"className":19880},[442],[81,19882],{"className":19883,"style":127},[912],[81,19885,6422],{"className":19886},[235],[81,19888],{"className":19889,"style":127},[912],[81,19891,19893,19896,19899,19902,19905,19908,19911,19914,19917],{"className":19892},[116],[81,19894],{"className":19895,"style":428},[120],[81,19897,6362],{"className":19898,"style":1394},[125,126],[81,19900,410],{"className":19901},[435],[81,19903,3518],{"className":19904},[125,126],[81,19906,415],{"className":19907},[442],[81,19909,6411],{"className":19910},[125],[81,19912],{"className":19913,"style":127},[912],[81,19915,2044],{"className":19916},[235],[81,19918],{"className":19919,"style":127},[912],[81,19921,19923,19926,19929,19932,19935,19938,19941,19944,19947],{"className":19922},[116],[81,19924],{"className":19925,"style":428},[120],[81,19927,6411],{"className":19928},[125],[81,19930,6362],{"className":19931,"style":1394},[125,126],[81,19933,410],{"className":19934},[435],[81,19936,3518],{"className":19937},[125,126],[81,19939,415],{"className":19940},[442],[81,19942],{"className":19943,"style":127},[912],[81,19945,6422],{"className":19946},[235],[81,19948],{"className":19949,"style":127},[912],[81,19951,19953,19956,19959,19962,19965,19968,19971,19974,19977],{"className":19952},[116],[81,19954],{"className":19955,"style":428},[120],[81,19957,15192],{"className":19958,"style":12969},[125,126],[81,19960,410],{"className":19961},[435],[81,19963,3518],{"className":19964},[125,126],[81,19966,415],{"className":19967},[442],[81,19969,6411],{"className":19970},[125],[81,19972],{"className":19973,"style":913},[912],[81,19975,6435],{"className":19976},[917],[81,19978],{"className":19979,"style":913},[912],[81,19981,19983,19986,19989,19992,19995,19998],{"className":19982},[116],[81,19984],{"className":19985,"style":428},[120],[81,19987,15136],{"className":19988},[125,126],[81,19990,15980],{"className":19991},[125],[81,19993],{"className":19994,"style":127},[912],[81,19996,2044],{"className":19997},[235],[81,19999],{"className":20000,"style":127},[912],[81,20002,20004,20007,20010,20013,20016,20019],{"className":20003},[116],[81,20005],{"className":20006,"style":428},[120],[81,20008,15136],{"className":20009},[125,126],[81,20011,15980],{"className":20012},[125],[81,20014],{"className":20015,"style":913},[912],[81,20017,833],{"className":20018},[917],[81,20020],{"className":20021,"style":913},[912],[81,20023,20025,20028],{"className":20024},[116],[81,20026],{"className":20027,"style":293},[120],[81,20029,15136],{"className":20030},[125,126],[14,20032,20033],{},"QED.",[14,20035,20036],{},"In other words: the nonlinear units of a neural network act as an \"adjustable polynomial basis\", and a single layer can combine them into any polynomial, thereby approximating any continuous function.",[43,20038,20040],{"id":20039},"uat-and-the-machine-learning-perspective","UAT and the Machine Learning Perspective",[14,20042,20043],{},"This proof is actually quite interesting:",[14,20045,20046],{},"The Bernstein polynomial essentially expresses",[81,20048,20050],{"className":20049,"translate":85},[809],[81,20051,20053,20148],{"className":20052,"translate":85},[84],[81,20054,20056],{"className":20055},[89],[91,20057,20058],{"xmlns":93,"display":818},[95,20059,20060,20145],{},[98,20061,20062,20064,20066,20068,20070,20072,20078,20080,20082,20094,20096],{},[101,20063,6124],{},[179,20065,410],{"stretchy":409},[101,20067,3518],{},[179,20069,415],{"stretchy":409},[179,20071,10525],{},[1535,20073,20074,20076],{},[179,20075,6641],{},[101,20077,3746],{},[101,20079,6124],{},[2035,20081,6656],{},[98,20083,20084,20086,20092],{},[179,20085,410],{"fence":112},[1199,20087,20088,20090],{},[101,20089,3746],{},[101,20091,2164],{},[179,20093,415],{"fence":112},[179,20095,2372],{},[1535,20097,20098,20142],{},[1535,20099,20100,20140],{},[98,20101,20102,20114,20120,20122,20124,20126,20128],{},[98,20103,20104,20106,20112],{},[179,20105,410],{"fence":112},[1199,20107,20108,20110],{"linethickness":6675},[101,20109,2164],{},[101,20111,3746],{},[179,20113,415],{"fence":112},[174,20115,20116,20118],{},[101,20117,3518],{},[101,20119,3746],{},[179,20121,410],{"stretchy":409},[515,20123,679],{},[179,20125,6422],{},[101,20127,3518],{},[174,20129,20130,20132],{},[179,20131,415],{"stretchy":409},[98,20133,20134,20136,20138],{},[101,20135,2164],{},[179,20137,6422],{},[101,20139,3746],{},[179,20141,1562],{"stretchy":112},[2035,20143,20144],{},"base function",[105,20146,20147],{"encoding":107},"f(x)\n\\approx\n\\sum_k\nf\\!\\left(\\frac{k}{n}\\right)\n\\cdot\n\\underbrace{\n\\binom{n}{k}\nx^k(1-x)^{n-k}\n}_{\\text{base function}}",[81,20149,20151,20178,20327],{"className":20150,"ariaHidden":112},[111],[81,20152,20154,20157,20160,20163,20166,20169,20172,20175],{"className":20153},[116],[81,20155],{"className":20156,"style":428},[120],[81,20158,6124],{"className":20159,"style":6161},[125,126],[81,20161,410],{"className":20162},[435],[81,20164,3518],{"className":20165},[125,126],[81,20167,415],{"className":20168},[442],[81,20170],{"className":20171,"style":913},[912],[81,20173,10525],{"className":20174},[917],[81,20176],{"className":20177,"style":913},[912],[81,20179,20181,20185,20229,20232,20235,20238,20241,20318,20321,20324],{"className":20180},[116],[81,20182],{"className":20183,"style":20184},[120],"height:2.7521em;vertical-align:-1.3021em;",[81,20186,20188],{"className":20187},[6802,6803],[81,20189,20191,20221],{"className":20190},[208,569],[81,20192,20194,20218],{"className":20193},[212],[81,20195,20197,20208],{"className":20196,"style":17614},[216],[81,20198,20199,20202],{"style":6816},[81,20200],{"className":20201,"style":6820},[223],[81,20203,20205],{"className":20204},[228,229,230,231],[81,20206,3746],{"className":20207,"style":4244},[125,126,231],[81,20209,20210,20213],{"style":6838},[81,20211],{"className":20212,"style":6820},[223],[81,20214,20215],{},[81,20216,6641],{"className":20217},[6802,6847,6848],[81,20219,606],{"className":20220},[605],[81,20222,20224],{"className":20223},[212],[81,20225,20227],{"className":20226,"style":6873},[216],[81,20228],{},[81,20230],{"className":20231,"style":1025},[912],[81,20233,6124],{"className":20234,"style":6161},[125,126],[81,20236],{"className":20237,"style":6885},[912],[81,20239],{"className":20240,"style":1025},[912],[81,20242,20244,20250,20312],{"className":20243},[546],[81,20245,20247],{"className":20246,"style":551},[435,550],[81,20248,410],{"className":20249},[555,230],[81,20251,20253,20256,20309],{"className":20252},[125],[81,20254],{"className":20255},[435,1247],[81,20257,20259],{"className":20258},[1199],[81,20260,20262,20301],{"className":20261},[208,569],[81,20263,20265,20298],{"className":20264},[212],[81,20266,20268,20279,20287],{"className":20267,"style":6916},[216],[81,20269,20270,20273],{"style":4196},[81,20271],{"className":20272,"style":583},[223],[81,20274,20276],{"className":20275},[125],[81,20277,2164],{"className":20278},[125,126],[81,20280,20281,20284],{"style":1278},[81,20282],{"className":20283,"style":583},[223],[81,20285],{"className":20286,"style":1286},[1285],[81,20288,20289,20292],{"style":4290},[81,20290],{"className":20291,"style":583},[223],[81,20293,20295],{"className":20294},[125],[81,20296,3746],{"className":20297,"style":4244},[125,126],[81,20299,606],{"className":20300},[605],[81,20302,20304],{"className":20303},[212],[81,20305,20307],{"className":20306,"style":4385},[216],[81,20308],{},[81,20310],{"className":20311},[442,1247],[81,20313,20315],{"className":20314,"style":551},[442,550],[81,20316,415],{"className":20317},[555,230],[81,20319],{"className":20320,"style":127},[912],[81,20322,2372],{"className":20323},[235],[81,20325],{"className":20326,"style":127},[912],[81,20328,20330,20334],{"className":20329},[116],[81,20331],{"className":20332,"style":20333},[120],"height:3.7341em;vertical-align:-2.2841em;",[81,20335,20337],{"className":20336},[546,1535],[81,20338,20340,20582],{"className":20339},[208,569],[81,20341,20343,20579],{"className":20342},[212],[81,20344,20346,20365],{"className":20345,"style":576},[216],[81,20347,20349,20353],{"style":20348},"top:-1.1659em;",[81,20350],{"className":20351,"style":20352},[223],"height:3.45em;",[81,20354,20356],{"className":20355},[228,229,230,231],[81,20357,20359],{"className":20358},[125,231],[81,20360,20362],{"className":20361},[125,2063,231],[81,20363,20144],{"className":20364},[125,231],[81,20366,20368,20371],{"style":20367},"top:-3.45em;",[81,20369],{"className":20370,"style":20352},[223],[81,20372,20374],{"className":20373},[546,1535],[81,20375,20377,20570],{"className":20376},[208,569],[81,20378,20380,20567],{"className":20379},[212],[81,20381,20383,20414],{"className":20382,"style":576},[216],[81,20384,20387,20390],{"className":20385,"style":20386},[1674],"top:-1.852em;",[81,20388],{"className":20389,"style":20352},[223],[81,20391,20393,20400,20407],{"className":20392,"style":1683},[1682],[81,20394,20396],{"className":20395,"style":1688},[1687],[1690,20397,20398],{"xmlns":1692,"width":1693,"height":1694,"viewBox":1695,"preserveAspectRatio":1696},[1698,20399],{"d":1700},[81,20401,20403],{"className":20402,"style":1688},[1704],[1690,20404,20405],{"xmlns":1692,"width":1693,"height":1694,"viewBox":1695,"preserveAspectRatio":1707},[1698,20406],{"d":1710},[81,20408,20410],{"className":20409,"style":1688},[1714],[1690,20411,20412],{"xmlns":1692,"width":1693,"height":1694,"viewBox":1695,"preserveAspectRatio":1717},[1698,20413],{"d":1720},[81,20415,20416,20419],{"style":20367},[81,20417],{"className":20418,"style":20352},[223],[81,20420,20422,20482,20511,20514,20517,20520,20523,20526,20529],{"className":20421},[125],[81,20423,20425,20431,20476],{"className":20424},[125],[81,20426,20428],{"className":20427,"style":551},[435,550],[81,20429,410],{"className":20430},[555,230],[81,20432,20434],{"className":20433},[1199],[81,20435,20437,20468],{"className":20436},[208,569],[81,20438,20440,20465],{"className":20439},[212],[81,20441,20443,20454],{"className":20442,"style":6991},[216],[81,20444,20445,20448],{"style":4196},[81,20446],{"className":20447,"style":583},[223],[81,20449,20451],{"className":20450},[125],[81,20452,3746],{"className":20453,"style":4244},[125,126],[81,20455,20456,20459],{"style":4290},[81,20457],{"className":20458,"style":583},[223],[81,20460,20462],{"className":20461},[125],[81,20463,2164],{"className":20464},[125,126],[81,20466,606],{"className":20467},[605],[81,20469,20471],{"className":20470},[212],[81,20472,20474],{"className":20473,"style":4385},[216],[81,20475],{},[81,20477,20479],{"className":20478,"style":551},[442,550],[81,20480,415],{"className":20481},[555,230],[81,20483,20485,20488],{"className":20484},[125],[81,20486,3518],{"className":20487},[125,126],[81,20489,20491],{"className":20490},[204],[81,20492,20494],{"className":20493},[208],[81,20495,20497],{"className":20496},[212],[81,20498,20500],{"className":20499,"style":7049},[216],[81,20501,20502,20505],{"style":1749},[81,20503],{"className":20504,"style":224},[223],[81,20506,20508],{"className":20507},[228,229,230,231],[81,20509,3746],{"className":20510,"style":4244},[125,126,231],[81,20512,410],{"className":20513},[435],[81,20515,679],{"className":20516},[125],[81,20518],{"className":20519,"style":127},[912],[81,20521,6422],{"className":20522},[235],[81,20524],{"className":20525,"style":127},[912],[81,20527,3518],{"className":20528},[125,126],[81,20530,20532,20535],{"className":20531},[442],[81,20533,415],{"className":20534},[442],[81,20536,20538],{"className":20537},[204],[81,20539,20541],{"className":20540},[208],[81,20542,20544],{"className":20543},[212],[81,20545,20547],{"className":20546,"style":7049},[216],[81,20548,20549,20552],{"style":1749},[81,20550],{"className":20551,"style":224},[223],[81,20553,20555],{"className":20554},[228,229,230,231],[81,20556,20558,20561,20564],{"className":20557},[125,231],[81,20559,2164],{"className":20560},[125,126,231],[81,20562,6422],{"className":20563},[235,231],[81,20565,3746],{"className":20566,"style":4244},[125,126,231],[81,20568,606],{"className":20569},[605],[81,20571,20573],{"className":20572},[212],[81,20574,20577],{"className":20575,"style":20576},[216],"height:1.598em;",[81,20578],{},[81,20580,606],{"className":20581},[605],[81,20583,20585],{"className":20584},[212],[81,20586,20589],{"className":20587,"style":20588},[216],"height:2.2841em;",[81,20590],{},[14,20592,20593],{},"This closely resembles a neural network",[81,20595,20597],{"className":20596,"translate":85},[809],[81,20598,20600,20659],{"className":20599,"translate":85},[84],[81,20601,20603],{"className":20602},[89],[91,20604,20605],{"xmlns":93,"display":818},[95,20606,20607,20656],{},[98,20608,20609,20611,20613,20615,20617,20619,20625,20631,20633,20635,20637,20644,20646,20648,20654],{},[101,20610,6124],{},[179,20612,410],{"stretchy":409},[101,20614,3518],{},[179,20616,415],{"stretchy":409},[179,20618,10525],{},[1535,20620,20621,20623],{},[179,20622,6641],{},[101,20624,3015],{},[3006,20626,20627,20629],{},[101,20628,8618],{},[101,20630,3015],{},[179,20632,2372],{"separator":112},[101,20634,2133],{},[179,20636,410],{"stretchy":409},[3006,20638,20639,20642],{},[101,20640,20641],{},"w",[101,20643,3015],{},[101,20645,3518],{},[179,20647,2044],{},[3006,20649,20650,20652],{},[101,20651,15048],{},[101,20653,3015],{},[179,20655,415],{"stretchy":409},[105,20657,20658],{"encoding":107},"f(x)\n\\approx\n\\sum_i\na_i·\\sigma(w_i x+b_i)",[81,20660,20662,20689,20848],{"className":20661,"ariaHidden":112},[111],[81,20663,20665,20668,20671,20674,20677,20680,20683,20686],{"className":20664},[116],[81,20666],{"className":20667,"style":428},[120],[81,20669,6124],{"className":20670,"style":6161},[125,126],[81,20672,410],{"className":20673},[435],[81,20675,3518],{"className":20676},[125,126],[81,20678,415],{"className":20679},[442],[81,20681],{"className":20682,"style":913},[912],[81,20684,10525],{"className":20685},[917],[81,20687],{"className":20688,"style":913},[912],[81,20690,20692,20695,20739,20742,20782,20785,20788,20791,20794,20836,20839,20842,20845],{"className":20691},[116],[81,20693],{"className":20694,"style":17601},[120],[81,20696,20698],{"className":20697},[6802,6803],[81,20699,20701,20731],{"className":20700},[208,569],[81,20702,20704,20728],{"className":20703},[212],[81,20705,20707,20718],{"className":20706,"style":17614},[216],[81,20708,20709,20712],{"style":15300},[81,20710],{"className":20711,"style":6820},[223],[81,20713,20715],{"className":20714},[228,229,230,231],[81,20716,3015],{"className":20717},[125,126,231],[81,20719,20720,20723],{"style":6838},[81,20721],{"className":20722,"style":6820},[223],[81,20724,20725],{},[81,20726,6641],{"className":20727},[6802,6847,6848],[81,20729,606],{"className":20730},[605],[81,20732,20734],{"className":20733},[212],[81,20735,20737],{"className":20736,"style":15352},[216],[81,20738],{},[81,20740],{"className":20741,"style":1025},[912],[81,20743,20745,20748],{"className":20744},[125],[81,20746,8618],{"className":20747},[125,126],[81,20749,20751],{"className":20750},[204],[81,20752,20754,20774],{"className":20753},[208,569],[81,20755,20757,20771],{"className":20756},[212],[81,20758,20760],{"className":20759,"style":3050},[216],[81,20761,20762,20765],{"style":8358},[81,20763],{"className":20764,"style":224},[223],[81,20766,20768],{"className":20767},[228,229,230,231],[81,20769,3015],{"className":20770},[125,126,231],[81,20772,606],{"className":20773},[605],[81,20775,20777],{"className":20776},[212],[81,20778,20780],{"className":20779,"style":3177},[216],[81,20781],{},[81,20783,2372],{"className":20784},[1441],[81,20786],{"className":20787,"style":1025},[912],[81,20789,2133],{"className":20790,"style":297},[125,126],[81,20792,410],{"className":20793},[435],[81,20795,20797,20801],{"className":20796},[125],[81,20798,20641],{"className":20799,"style":20800},[125,126],"margin-right:0.0269em;",[81,20802,20804],{"className":20803},[204],[81,20805,20807,20828],{"className":20806},[208,569],[81,20808,20810,20825],{"className":20809},[212],[81,20811,20813],{"className":20812,"style":3050},[216],[81,20814,20816,20819],{"style":20815},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[81,20817],{"className":20818,"style":224},[223],[81,20820,20822],{"className":20821},[228,229,230,231],[81,20823,3015],{"className":20824},[125,126,231],[81,20826,606],{"className":20827},[605],[81,20829,20831],{"className":20830},[212],[81,20832,20834],{"className":20833,"style":3177},[216],[81,20835],{},[81,20837,3518],{"className":20838},[125,126],[81,20840],{"className":20841,"style":127},[912],[81,20843,2044],{"className":20844},[235],[81,20846],{"className":20847,"style":127},[912],[81,20849,20851,20854,20894],{"className":20850},[116],[81,20852],{"className":20853,"style":428},[120],[81,20855,20857,20860],{"className":20856},[125],[81,20858,15048],{"className":20859},[125,126],[81,20861,20863],{"className":20862},[204],[81,20864,20866,20886],{"className":20865},[208,569],[81,20867,20869,20883],{"className":20868},[212],[81,20870,20872],{"className":20871,"style":3050},[216],[81,20873,20874,20877],{"style":8358},[81,20875],{"className":20876,"style":224},[223],[81,20878,20880],{"className":20879},[228,229,230,231],[81,20881,3015],{"className":20882},[125,126,231],[81,20884,606],{"className":20885},[605],[81,20887,20889],{"className":20888},[212],[81,20890,20892],{"className":20891,"style":3177},[216],[81,20893],{},[81,20895,415],{"className":20896},[442],[14,20898,20899],{},"The difference is only:",[448,20901,20902,21047],{},[451,20903,20904,20905],{},"Weierstrass uses polynomial bases ",[81,20906,20908,20946],{"className":20907,"translate":85},[84],[81,20909,20911],{"className":20910},[89],[91,20912,20913],{"xmlns":93},[95,20914,20915,20943],{},[98,20916,20917,20923,20925,20927,20929,20931],{},[174,20918,20919,20921],{},[101,20920,3518],{},[101,20922,3746],{},[179,20924,410],{"stretchy":409},[515,20926,679],{},[179,20928,6422],{},[101,20930,3518],{},[174,20932,20933,20935],{},[179,20934,415],{"stretchy":409},[98,20936,20937,20939,20941],{},[101,20938,2164],{},[179,20940,6422],{},[101,20942,3746],{},[105,20944,20945],{"encoding":107},"x^k(1-x)^{n-k}",[81,20947,20949,21000],{"className":20948,"ariaHidden":112},[111],[81,20950,20952,20956,20985,20988,20991,20994,20997],{"className":20951},[116],[81,20953],{"className":20954,"style":20955},[120],"height:1.0991em;vertical-align:-0.25em;",[81,20957,20959,20962],{"className":20958},[125],[81,20960,3518],{"className":20961},[125,126],[81,20963,20965],{"className":20964},[204],[81,20966,20968],{"className":20967},[208],[81,20969,20971],{"className":20970},[212],[81,20972,20974],{"className":20973,"style":4767},[216],[81,20975,20976,20979],{"style":219},[81,20977],{"className":20978,"style":224},[223],[81,20980,20982],{"className":20981},[228,229,230,231],[81,20983,3746],{"className":20984,"style":4244},[125,126,231],[81,20986,410],{"className":20987},[435],[81,20989,679],{"className":20990},[125],[81,20992],{"className":20993,"style":127},[912],[81,20995,6422],{"className":20996},[235],[81,20998],{"className":20999,"style":127},[912],[81,21001,21003,21006,21009],{"className":21002},[116],[81,21004],{"className":21005,"style":20955},[120],[81,21007,3518],{"className":21008},[125,126],[81,21010,21012,21015],{"className":21011},[442],[81,21013,415],{"className":21014},[442],[81,21016,21018],{"className":21017},[204],[81,21019,21021],{"className":21020},[208],[81,21022,21024],{"className":21023},[212],[81,21025,21027],{"className":21026,"style":4767},[216],[81,21028,21029,21032],{"style":219},[81,21030],{"className":21031,"style":224},[223],[81,21033,21035],{"className":21034},[228,229,230,231],[81,21036,21038,21041,21044],{"className":21037},[125,231],[81,21039,2164],{"className":21040},[125,126,231],[81,21042,6422],{"className":21043},[235,231],[81,21045,3746],{"className":21046,"style":4244},[125,126,231],[451,21048,21049,21050],{},"Neural networks use activation function bases ",[81,21051,21053,21079],{"className":21052,"translate":85},[84],[81,21054,21056],{"className":21055},[89],[91,21057,21058],{"xmlns":93},[95,21059,21060,21076],{},[98,21061,21062,21064,21066,21068,21070,21072,21074],{},[101,21063,2133],{},[179,21065,410],{"stretchy":409},[101,21067,20641],{},[101,21069,3518],{},[179,21071,2044],{},[101,21073,15048],{},[179,21075,415],{"stretchy":409},[105,21077,21078],{"encoding":107},"\\sigma(wx+b)",[81,21080,21082,21109],{"className":21081,"ariaHidden":112},[111],[81,21083,21085,21088,21091,21094,21097,21100,21103,21106],{"className":21084},[116],[81,21086],{"className":21087,"style":428},[120],[81,21089,2133],{"className":21090,"style":297},[125,126],[81,21092,410],{"className":21093},[435],[81,21095,20641],{"className":21096,"style":20800},[125,126],[81,21098,3518],{"className":21099},[125,126],[81,21101],{"className":21102,"style":127},[912],[81,21104,2044],{"className":21105},[235],[81,21107],{"className":21108,"style":127},[912],[81,21110,21112,21115,21118],{"className":21111},[116],[81,21113],{"className":21114,"style":428},[120],[81,21116,15048],{"className":21117},[125,126],[81,21119,415],{"className":21120},[442],[14,21122,21123],{},"Thus from a modern perspective, the UAT theorem can be understood as a generalization of the Weierstrass theorem to \"neural network basis functions\". Weierstrass says \"the polynomial basis is dense\", while UAT says \"the neuron basis is also dense\".",{"title":6064,"searchDepth":6065,"depth":6065,"links":21125},[21126,21131,21132],{"id":6097,"depth":6065,"text":6098,"children":21127},[21128,21130],{"id":6102,"depth":21129,"text":6103},3,{"id":11047,"depth":21129,"text":11048},{"id":15013,"depth":6065,"text":15014},{"id":20039,"depth":6065,"text":20040},{},"2026-06-06","\u002Fblog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory",{"title":6092,"description":6064},"blog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory","The core purpose of UAT (Universal Approximation Theorem) is that neural networks can approximate any continuous function, and continuous functions can be approximated by polynomials (Weierstrass theorem).",[6088,6087],"FXBw_kctBq_P9VgwI_OR4cZjAgM-1RBjox6VpeUkrds",{"id":21142,"title":21143,"body":21144,"description":6064,"draft":6076,"enableComment":6077,"extension":6078,"image":6064,"important":6076,"location":6079,"meta":21183,"navigation":6077,"ogImage":6079,"onday":21184,"path":21185,"seo":21186,"stem":21187,"summary":21188,"tags":21189,"__hash__":21191},"blog\u002Fblog\u002F2026\u002F2026-05-13-the-pale-blue-dot.md","暗淡蓝点",{"type":8,"value":21145,"toc":21181},[21146,21152,21155,21158,21161,21164,21167,21170,21173,21176],[14,21147,21148],{},[21149,21150],"img",{"alt":6064,"src":21151},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202605132235273.png",[14,21153,21154],{},"再看看那个小点。就在这里。就是家。就是我们。",[14,21156,21157],{},"所有你爱的人，所有你认识的人，所有你听说过的人，所有曾经存在过的人类，都在上面度过了他们的一生。",[14,21159,21160],{},"我们欢乐和痛苦的总和，成千上万自信的宗教、意识形态和经济学说，每一个猎人和觅食者，每一个英雄和懦夫，每一个文明的创造者和毁灭者，每一个国王和农民，每一对相爱的情侣，每一位母亲和父亲，充满希望的孩子，发明家和探险家，每一个道德教师，每一个腐败的政客，每一个“超级巨星”，每一个“最高领导人”，每一个圣人和罪人，都在那里生活过——在一个悬浮在阳光中的尘埃微粒上。",[14,21162,21163],{},"地球在浩瀚的宇宙舞台上只是一个小小的舞台。想想所有那些将军和皇帝流淌的血河，为了荣耀和胜利，他们可以成为一个小点的一小部分的瞬间主人。",[14,21165,21166],{},"想想这个像素的一个角落的居民对另一个几乎无法区分的角落的居民所施加的无休止的残忍，他们之间的误解是多么频繁，他们是多么渴望互相残杀，他们的仇恨是多么强烈。",[14,21168,21169],{},"我们的姿态，我们想象的自命不凡，我们认为我们在宇宙中拥有某种特权地位的错觉，都受到了这个苍白的光点的挑战。我们的星球是浩瀚的宇宙黑暗中孤独的一点。",[14,21171,21172],{},"在我们的默默无闻中，在所有这些广阔的空间中，没有任何迹象表明会有来自其他地方的帮助来拯救我们自己。到目前为止，地球是唯一已知孕育生命的星球。至少在不久的将来，我们的物种无处可去。参观，可以。定居，还不行。不管你喜欢与否，目前地球就是我们站立的地方。",[14,21174,21175],{},"有人说，天文学是一种令人谦卑和塑造性格的体验。对于人类自负的愚蠢，也许没有比我们这个小世界的遥远图像更好的证明了。对我来说，它强调了我们彼此更友善地相处，并保护和珍惜这个苍白蓝点的责任，这是我们唯一知道的家园。",[11,21177,21178],{},[14,21179,21180],{},"所有的战争与和平\n爱恨与悲欢\n都在这颗名为「地球」的微尘上\n日复一日地上演。",{"title":6064,"searchDepth":6065,"depth":6065,"links":21182},[],{},"2026-05-13","\u002Fblog\u002F2026\u002F2026-05-13-the-pale-blue-dot",{"title":21143,"description":6064},"blog\u002F2026\u002F2026-05-13-the-pale-blue-dot","所有的战争与和平，爱恨与悲欢，都在这颗名为「地球」的微尘上，日复一日地上演。",[21190],"thoughts","fVZwpQuTB11Vpr_Nh2S5mrdM3Z98pnu5QfBsLvO025o",1786880521000]