wikipedia_math_equations.json

{"Linear algebra": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f0f2986d54c01f3bccf464d266dfac923c80f3", "tex": "{\\displaystyle a_{1}x_{1}+\\cdots +a_{n}x_{n}=b,}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4931f234eb04e66a7a146d8bfd02b18bc3f595fe", "tex": "{\\displaystyle (x_{1},\\ldots ,x_{n})\\mapsto a_{1}x_{1}+\\ldots +a_{n}x_{n},}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d861c6f0917de60abeaaa77df3ffc664eeb10a18", "tex": "{\\displaystyle {\\overline {wz}}\\ \\ {\\text{and}}\\ \\ {\\overline {0(w-z)}}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d3da97120143aed537cf303f6d098fce0df7a6c4", "tex": "{\\displaystyle {\\overline {pq}}.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4c59f606d4e06de82ae3016ab89884480356b3d6", "tex": "T:V\\to W", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/250391d618ab20ba1a2f62449f33cd0d59a35fd7", "tex": "T(u+v)=T(u)+T(v),\\quad T(av)=aT(v)", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ea5b00aaacea3327dfdb3d9a9933e10e7aa8058a", "tex": "{\\displaystyle T(au+bv)=T(au)+T(bv)=aT(u)+bT(v)}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e15d3619d42b28662c5952c9ece60cd928e31774", "tex": "a_{1}v_{1}+a_{2}v_{2}+\\cdots +a_{k}v_{k},", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d0639dddd9856616169190754bcdf160c0a5c379", "tex": "a_i.", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1698332588b03324a24cd0b66b722d1def37a132", "tex": "{\\displaystyle S\\subseteq T,}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/671fb2c70e269885f7154c3c30f3edfe2f2d2163", "tex": "{\\displaystyle S\\subseteq B\\subseteq T.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f0694e66688bc88f285b1ae33a68d046cb3df0", "tex": "{\\displaystyle \\dim(U_{1}+U_{2})=\\dim U_{1}+\\dim U_{2}-\\dim(U_{1}\\cap U_{2}),}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/40a8145e6ceb15b6301392306db9613c6f8b5911", "tex": "{\\displaystyle U_{1}+U_{2}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/32da1e39d8ec566edb2a5c499ee8f170240d85f9", "tex": "{\\displaystyle U_{1}\\cup U_{2}.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6b1a11cc0ee40895945fc2332a011791669a0f", "tex": "{\\displaystyle {\\begin{aligned}(a_{1},\\ldots ,a_{m})&\\mapsto a_{1}v_{1}+\\cdots a_{m}v_{m}\\\\F^{m}&\\to V\\end{aligned}}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfa13d061be0189351c58a7131361db66d4320b", "tex": "{\\displaystyle F^{m},}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3f36605ff92be21e23d2053fce4b0372f2be2e68", "tex": "{\\displaystyle F^{m}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2102f815e7f1039f18216d6a3d1a6be3a9130293", "tex": "{\\displaystyle (a_{1},\\ldots ,a_{m})}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/70a031c92741d94ef93fa6613a3f993940c41943", "tex": "{\\displaystyle {\\begin{bmatrix}a_{1}\\\\\\vdots \\\\a_{m}\\end{bmatrix}}.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1af1b20f156f231f2d96e24ea114a873ae877a3e", "tex": "{\\displaystyle (w_{1},\\ldots ,w_{n}),}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e49f988b3d64efad8962ab17221b6f29561e6219", "tex": "{\\displaystyle (f(w_{1}),\\ldots ,f(w_{n})).}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/866d92a12b845daa665a5d1ea59a0052137b1a37", "tex": "{\\displaystyle f(w_{j})=a_{1,j}v_{1}+\\cdots +a_{m,j}v_{m},}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f14f5e0e845d14f74676250abe249ab1f27000", "tex": "{\\displaystyle {\\begin{bmatrix}a_{1,1}&\\ldots &a_{1,n}\\\\\\vdots &\\ldots &\\vdots \\\\a_{m,1}&\\ldots &a_{m,n}\\end{bmatrix}},}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d85855e5827184dd78769ede30040585097a9eba", "tex": "{\\displaystyle {\\begin{alignedat}{7}2x&&\\;+\\;&&y&&\\;-\\;&&z&&\\;=\\;&&8\\\\-3x&&\\;-\\;&&y&&\\;+\\;&&2z&&\\;=\\;&&-11\\\\-2x&&\\;+\\;&&y&&\\;+\\;&&2z&&\\;=\\;&&-3\\end{alignedat}}\\qquad {\\text{(S)}}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c05624c39dd4221664021e29a30add5bd090a0a2", "tex": "{\\displaystyle M\\left[{\\begin{array}{rrr}2&1&-1\\\\-3&-1&2\\\\-2&1&2\\end{array}}\\right]{\\text{.}}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6e9d17caf05d75997dae01b9aeeba5091a75fc", "tex": "{\\displaystyle v={\\begin{bmatrix}8\\\\-11\\\\-3\\end{bmatrix}}.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0ce759a36464073f2f5ae2bf664c9186b90a0b", "tex": "{\\displaystyle X={\\begin{bmatrix}x\\\\y\\\\z\\end{bmatrix}}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/40a9387fd9d653a81588c8bc45f1c156da76a628", "tex": "{\\displaystyle T(X)=v,}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/67d8077b50a0f5dd2f0e54e1efb629a0d53231f1", "tex": "{\\displaystyle M\\left[{\\begin{array}{rrr|r}2&1&-1&8\\\\-3&-1&2&-11\\\\-2&1&2&-3\\end{array}}\\right]}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/70f22fd6c23b6497862c719a5a3a9deda47bf58d", "tex": "{\\displaystyle M\\left[{\\begin{array}{rrr|r}1&0&0&2\\\\0&1&0&3\\\\0&0&1&-1\\end{array}}\\right],}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9995c77c5af33f793e99b15e577480243041c6c5", "tex": "{\\displaystyle {\\begin{aligned}x&=2\\\\y&=3\\\\z&=-1.\\end{aligned}}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9256a6bf21c144488d5efa785da63950abd46958", "tex": "{\\displaystyle Mz=az.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/22d42e0d9e0b62bc4e78cfd102b21e4a7217ac6c", "tex": "{\\displaystyle (M-aI)z=0.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/515c11bb7e78026a7517c44bf90976661d094e8d", "tex": "{\\displaystyle \\det(M-aI)}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/29c19c2b35265004add12e16a97d849ba94764d0", "tex": "{\\displaystyle \\det(xI-M).}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/eb69343e096235ec162a5f08f9a59c684687c041", "tex": "{\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2951349c258f7b3cbd282c6578aea933aa76f9fd", "tex": "{\\displaystyle V^{*}.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fb40a91abab8b7bfb0e84b074732b2f044fd56ef", "tex": "{\\displaystyle v_{1},\\ldots ,v_{n}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/97661cc042bf758736628eec83f2755015d70941", "tex": "{\\displaystyle v_{i}^{*}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/97a7c3865585aff9d1cd1e58dc51635ac950a235", "tex": "{\\displaystyle v_{i}^{*}(e_{i})=1}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/12d4f9c0c66ea58e9d6a0a3b10bc31815652edf2", "tex": "{\\displaystyle v_{i}^{*}(e_{j})=0}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/88213f842ca974a6f411b4965e5a14e51a3cdb24", "tex": "V^{*},", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/86a24bd73cf7a53e79739e6bc21c58f680e42f29", "tex": "{\\displaystyle v_{1},\\ldots ,v_{n}.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/97661cc042bf758736628eec83f2755015d70941", "tex": "v_{i}^{*}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3a4e146963d0c437898229b1444151d32ba33e5e", "tex": "{\\displaystyle f\\to f(v)}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2951349c258f7b3cbd282c6578aea933aa76f9fd", "tex": "{\\displaystyle V^{*}.}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5013d54f25c7766a6596a47a5cfcca3982bcd685", "tex": "{\\displaystyle V^{**},}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/88213f842ca974a6f411b4965e5a14e51a3cdb24", "tex": "V^{*},", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0127917745cbf68c20c60069f7b0f38470e2ca20", "tex": "{\\displaystyle \\langle f,x\\rangle }", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/574dffa1c85efaef6b6ef553ebd8ad9cf7f87fd6", "tex": "f:V\\to W", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d73927ebc5f7b531d76d7fa92c13722386219041", "tex": "{\\displaystyle f^{*}:W^{*}\\to V^{*}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/190a73fde235865b8d2a783334f90194331c7f19", "tex": "f^{*}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/afffcc243abefa84734c19b0ca26e261efc8703d", "tex": "{\\displaystyle M^{\\mathsf {T}}}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/26a8ca13a55394f2329ef562f47a79a29dae9e94", "tex": "{\\displaystyle \\langle h^{\\mathsf {T}},Mv\\rangle =\\langle h^{\\mathsf {T}}M,v\\rangle .}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd58674bdacd48bc582179f217b77cfcfe76b45", "tex": "{\\displaystyle \\langle h^{\\mathsf {T}}\\mid M\\mid v\\rangle .}", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4cdbcb35a0d89f3c292633a21c274cf0133bbc56", "tex": "\\langle \\cdot ,\\cdot \\rangle :V\\times V\\rightarrow F", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/eb7089af54e8541a5001df6372522cad5123b874", "tex": "\\langle u,v\\rangle ={\\overline {\\langle v,u\\rangle }}.", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/93969eff808bf65e827f36dae63a6af5de671793", "tex": "\\langle au,v\\rangle =a\\langle u,v\\rangle .", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b57e52a26d6303480a65514c10b7c3aa0755a817", "tex": "\\langle u+v,w\\rangle =\\langle u,w\\rangle +\\langle v,w\\rangle .", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/018740c46d48b28da21dbe16e5dea2f9b795416e", "tex": "\\langle v,v\\rangle \\geq 0", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2462c96226b54a4f16a8aadf2142d68a93b894", "tex": "\\|v\\|^{2}=\\langle v,v\\rangle ,", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c8a0d368b00efdcb5bfb0ac0b39e98250738a324", "tex": "|\\langle u,v\\rangle |\\leq \\|u\\|\\cdot \\|v\\|.", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b957bdc84ba692b9ab6936add01cac528b35d00b", "tex": "{\\frac {|\\langle u,v\\rangle |}{\\|u\\|\\cdot \\|v\\|}}\\leq 1,", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/387ba1508966878a7437d870071b2057ae4a27fb", "tex": "\\langle u,v\\rangle =0", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c28af6afe0a2c25c8df2dae4f238bbb75a9d6e1c", "tex": "a_{i}=\\langle v,v_{i}\\rangle ", "domain": "Algrebra", "topic": "Linear algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a79d34cec7daccbc00f8d5dcdbbcb113670eab", "tex": "\\langle Tu,v\\rangle =\\langle u,T^{*}v\\rangle .", "domain": "Algrebra", "topic": "Linear algebra"}], "Elementary_algebra": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e", "tex": "ax^2+bx+c=0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f", "tex": "{\\displaystyle a\\neq 0}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d", "tex": "a,b,c", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/61efa8d731eb6affd257bb15138b7bef53e5ed05", "tex": "y = x^2 - x - 2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e64f206e30ab6fe182a8aa38cbd30353bbfbb202", "tex": "3x^{2}-2xy+c", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": 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"3x^2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e83c3f914bae95e8d54fe8ab8b1c9886ff287dea", "tex": "2 \\times x \\times y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf28a82088fc2021c794fee1de241f90622d5ae", "tex": "2xy", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93", "tex": "x^{2}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/11216e2eac7b4b023de3753b8ba582dd7324014a", "tex": "1x^{2}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": 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"https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab", "tex": "d", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a", "tex": "\\pi ", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d5687cadd1fdb6197673e771331876c11b1013ca", "tex": "{\\displaystyle C=P+20}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/939c7ab17c10efa4dfaed57bcddc1b6e24660a74", "tex": "60 \\times 5 = 300", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b6634f8738ffb1a017147fa9ffbef3134dcb1f68", "tex": "s = 60 \\times m", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": 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"https://wikimedia.org/api/rest_v1/media/math/render/svg/4e2d4389a3b6f20cb8a118506601a68c2263143a", "tex": "x^{3}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5328816e9887a6f0081c3a0483e6869494cb17f0", "tex": "2x^2 + 3ab - x^2 + ab", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/61e65e40a5d5471dae0c0d5a7588d614b904cf01", "tex": "x^2 + 4ab", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93", "tex": "x^{2}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d", "tex": "ab", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/80fffddd92727a5c5d71c7583af8ca32afb30d4e", "tex": "x (2x + 3)", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e63c90413397ab755981d0f2e37c34f05f9076", "tex": "(x \\times 2x) + (x \\times 3)", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b504e73b6e1812fe65d1f827d0bd08c3fcfe5", "tex": "2x^2 + 3x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/81b597d7c2a7ccb6943812c240ffd627217a4e6b", "tex": "6x^5 + 3x^2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f94e37e2833e124c2339e013806d641bba54f4ce", "tex": "3x^2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7db4c92ee8fcdd3b2e806fb9372f7b500d5c93", "tex": "3x^2 (2x^3 + 1)", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743", "tex": "=", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5bace667e8cfea3ef573af86a1f3e72984b10755", "tex": "c^2 = a^2 + b^2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f3386a00382ce857fb0b3b04b9fa2bbe5cfae9", "tex": "c^{2}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc", "tex": "a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": 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"tex": " a > b ", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1b27b77ab4e3293ea9ce65cef60fea655c398423", "tex": " > ", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642", "tex": " a < b ", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1", "tex": "<", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7e808eb0d800bf37c7707d0ded5d243f58459a1d", "tex": "b = b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4", "tex": "a=b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2e4888d97a754d4bfa4da297b226788a73c6b8", "tex": "b=a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4", "tex": "a=b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b133a00dc90e54130a96482c99750f845feb955e", "tex": "b=c", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1beb3f1b1ad87e99791ba713839204a88b27239a", "tex": "a=c", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4", "tex": "a=b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d8498d2878942ddeef7d6a8ec870959f0d38d32e", "tex": "c = d", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/81ef6bd66c184b2f27a91e0bcac1d41fb2073063", "tex": "a + c = b + d", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7585cd9fe78dd1fb4e21d6528fb92f260664e9cc", "tex": "ac = bd", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4", "tex": "a=b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8d7c402d7bd2bba2f1538877a9f92b458c4f68", "tex": "a + c = b + c", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61", "tex": "f", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4", "tex": "a=b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f70209bf143b8417feef2aed98b2e86bc8f447e2", "tex": "f(a) = f(b)", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1", "tex": "<", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1b27b77ab4e3293ea9ce65cef60fea655c398423", "tex": " > ", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642", "tex": "a<b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f3667e3266421420148b93988e6f7db6b8aada82", "tex": "b < c", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1f61bdc0f32d5cbd3c62c118ef526397ab4c1e1b", "tex": "a < c", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642", "tex": "a<b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/826f129a304df73d8c5c71cc0ea8787fad9d40ce", "tex": "c < d", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffff35f8aa2508180de411cb546968bc63bd6c27", "tex": "a + c < b + d", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642", "tex": "a<b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba126f626d61752f62eaacaf11761a54de4dc84", "tex": "c>0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d0265433404d1a45f8a1cdda0343945d9dbed5f5", "tex": "ac < bc", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642", "tex": "a<b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/26a48dc798956117afd8c429c39886678c0e7204", "tex": "c < 0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f92b4137cd9315a7d23ed92f33ea900d97aeb95c", "tex": "bc < ac", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/33737c89a17785dacc8638b4d66db3d5c8670de1", "tex": "<", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1b27b77ab4e3293ea9ce65cef60fea655c398423", "tex": " > ", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/91a7698e4c7401bb321f97888b872b583a9e4642", "tex": "a<b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5b1cfc86ca957eea4f09d683db2412a173f6f404", "tex": "b>a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/144c47acca25cef5368659a7106d0483a08f9803", "tex": "a^2:=a*a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc", "tex": "a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/84cba38173d6b69364f2016245721c333282e0d7", "tex": "3^2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/346c990e543c1b807d25efad677f17a910c44f22", "tex": "2x + 4 = 12", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/346c990e543c1b807d25efad677f17a910c44f22", "tex": "2x + 4 = 12", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1254b15209bd6094b209d4615078d94051bc7ea6", "tex": "2x + 4 - 4 = 12 - 4", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fc58ebcf67c4f2be2a0ad26b646f576dd1617674", "tex": "2x = 8", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbec859078a96fde4ef8382044974dd75c20ace", "tex": "\\frac{2x}{2} = \\frac{8}{2}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/73bfdd8100b6a4ce07d011900560f102e3965064", "tex": "x = 4", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/08bf93625b100c9a4838fb52ddb9e65acfdb1234", "tex": "{\\displaystyle ax+b=c}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3", "tex": "b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc", "tex": "a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8f1e0d24f24022000a106a0088ca9e9e6db538", "tex": "x=\\frac{c-b}{a}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3bafcee05572b20028e3e7c8a97309e21ae5d065", "tex": "y = x + 22", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/81c702d220faee32fcbfc38802e680a19b18a534", "tex": "y + 10 = 2 \\times (x + 10)", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c3dc92c96940338211ac8989322463dfc52a2d60", "tex": "y = 2 \\times (x + 10) - 10", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fbe34799ae095755b4463e66abc9998446954ac5", "tex": "y = 2x + 20 - 10", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9e89c65a498801caff3359888848151b86607f", "tex": "y = 2x + 10", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9e89c65a498801caff3359888848151b86607f", "tex": "y = 2x + 10", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3bafcee05572b20028e3e7c8a97309e21ae5d065", "tex": "y = x + 22", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5ce7c570cc65c28ea60c84db990f3db5eca1aadc", "tex": "(y - y) = (2x - x) +10 - 22", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/eaecdb1418ccd1486ab8369d1401815378accd53", "tex": "0 = x - 12", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/81fc2b5512ee91e395e41b7154383088b17efdc5", "tex": "12 = x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/dc88c3d66d535ded040dcb909b1c57c8c47f1bd4", "tex": "x=12", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/542a39aa589dcbb42f429ee7fd21563e68fd2011", "tex": "y = x^2 + 3x - 10", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0190f8b44849ea92222977907c26e938566b98a2", "tex": "x = -5", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7", "tex": "x = 2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/98e3cbe17adefaec352d99657fbd6bbe6f02ea3d", "tex": "y = (x + 5)(x - 2)", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93", "tex": "x^{2}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e", "tex": "ax^{2}+bx+c=0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc", "tex": "a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a63fb574ed624044abbe6aeebcb600d0cb9802", "tex": "ax^2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f", "tex": "a\\neq 0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc", "tex": "a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b1447b7f80b86966dc9f431ba415b2de2608e50e", "tex": "{\\displaystyle x^{2}+px+q=0}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4863d307e928bab8d79085004f614d545a010b64", "tex": "{\\displaystyle p={\\frac {b}{a}}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/35385a31b58ac8828c8bb9c9f92e6e9ec5372ef8", "tex": "{\\displaystyle q={\\frac {c}{a}}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6dca3ce631f83b33ad881aed5f6e12e2f6c3afbd", "tex": "x=\\frac{-b \\pm \\sqrt {b^2-4ac}}{2a},", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d03b5b5af5ed622791d827b7e3661efb8a3db1f8", "tex": "x={\\frac {-b+{\\sqrt {b^{2}-4ac}}}{2a}}\\quad {\\text{and}}\\quad x={\\frac {-b-{\\sqrt {b^{2}-4ac}}}{2a}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/21d195781d3bba5b1c3a35b2790f73fe42f9f637", "tex": "{\\displaystyle x^{2}+3x-10=0,}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/01091b269645322d27a8f5b12cc54060862707cc", "tex": "{\\displaystyle (x+5)(x-2)=0.}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7", "tex": "x = 2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0190f8b44849ea92222977907c26e938566b98a2", "tex": "x = -5", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e01c67127b28bb80e2102c934d0d01daa5c20a61", "tex": "{\\displaystyle x^{2}+1=0}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b839672eac67f8ac41cabb303c08b87f802869a9", "tex": "{\\displaystyle (x+1)^{2}=0.}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea674f8b3cef9f4350a8ce21f48fc09d99cb539", "tex": "[x-(-1)][x-(-1)]=0.", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5e55ad884c42814df1c0e098670f34d685d0eb", "tex": "x^{2}+x+1=0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bd41f4fbb95aaf7dcc832f376d228077f3874ba3", "tex": "x={\\frac  {-1+{\\sqrt  {-3}}}{2}}\\quad \\quad {\\text{and}}\\quad \\quad x={\\frac  {-1-{\\sqrt  {-3}}}{2}}.", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/49e01b99dd42d59d290fdef7722b870efe61eb2e", "tex": "{\\sqrt  {-3}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/009ac5470a4ecd0afab8866426ad920c494316c9", "tex": "a^x = b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9", "tex": "a>0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/073a89019885af05c44f82023ede62981c2b4fd8", "tex": "X = \\log_a b = \\frac{\\ln b}{\\ln a}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/94436473a90bd55191a79c59474cb5456dcbec00", "tex": "b > 0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d77071814152240876f6c13181b8738ecb611e3e", "tex": "3 \\cdot 2^{x - 1} + 1 = 10", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3ac896aac2850d21edf38fbda9988efd7cb16469", "tex": "{\\displaystyle 2^{x-1}=3}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/96ccb53695d9611a82433817e2405e6ed7df35c9", "tex": "{\\displaystyle x-1=\\log _{2}3}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b9720b577b6c98343f0032293a3dcb5190174e7b", "tex": "{\\displaystyle x=\\log _{2}3+1.}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0b748099342737433ad3523aae1afdf87ac82e", "tex": "log_a(x) = b", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9", "tex": "a>0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ac8838d9bf17eab356d441c0757234645c95d0f4", "tex": "{\\displaystyle X=a^{b}.}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/48756f6a38589bbfc6e5e3ccc614db69447da988", "tex": "{\\displaystyle 4\\log _{5}(x-3)-2=6}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a77c02b98ff14bed821b26f955e1cd545ceb4f", "tex": "{\\displaystyle \\log _{5}(x-3)=2}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/57a64460e334e40242f98165c068c36d2eab5dd1", "tex": "{\\displaystyle x-3=5^{2}=25}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c829eab2e5b164e590a5bfeb4f83b0ea3a7fcbce", "tex": "{\\displaystyle x=28.}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b9763d9ff8638b882608213f6dc8e9fc8ddc79d3", "tex": "{\\displaystyle {\\sqrt {x}},}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9a55f866116e7a86823816615dd98fcccde75473", "tex": "{\\sqrt[{3}]{x}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3ba2638d05cd9ed8dafae7e34986399e48ea99", "tex": "{\\sqrt[{n}]{x}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7b3ba2638d05cd9ed8dafae7e34986399e48ea99", "tex": "{\\sqrt[{n}]{x}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/55c86c68cf18b5f2f38c440bee62562ff4f304f1", "tex": "x^{\\frac{1}{n}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6689fad02ba04851cff57ef80164ad8b1049f847", "tex": "\\sqrt[2]{x^3}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1a19ab17bfacbb7976f7f90799c9fd79241d5433", "tex": "x^{\\frac{3}{2}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7c088e9f58767bab398f26af86cca6ec2ff0fde1", "tex": "{\\sqrt[ {n}]{x^{m}}}=a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2b63addf2cbcdd590d1c725337bffa31f8edc439", "tex": "x^{{\\frac  {m}{n}}}=a", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc", "tex": "m", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b", "tex": "n", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc", "tex": "m", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc", "tex": "m", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c33f2f7ac8b81e1296d581427bdb2d30a50a2b12", "tex": "a \\ge 0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc", "tex": "m", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b", "tex": "n", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7ca60f6ed64b99649dcee21847295fedf206c", "tex": "a<0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc", "tex": "m", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b", "tex": "n", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7ca60f6ed64b99649dcee21847295fedf206c", "tex": "a<0", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cd09966c4c1505671df512298eda6fc36dd1a456", "tex": "x = \\sqrt[m]{a^n}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e2527ddffa8d54db65cc7c01b9d438378c796abb", "tex": "x = \\left(\\sqrt[m]a\\right)^n", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/26a5a6e030a43a7f81c76d76ef0c2c05d645110e", "tex": "x = \\pm \\sqrt[m]{a^n}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/124b0e0acfea5e41274b8d6593ced4b49834021e", "tex": "x = \\pm \\left(\\sqrt[m]a\\right)^n", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/26a5a6e030a43a7f81c76d76ef0c2c05d645110e", "tex": "x=\\pm {\\sqrt[ {m}]{a^{n}}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/28f0811034e880ed24e6af617729f1742458f421", "tex": "{\\displaystyle (x+5)^{2/3}=4,}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4534653d5fffb0254eceef586281290f8d653475", "tex": "\\begin{align}\nx + 5 & = \\pm (\\sqrt{4})^3\\\\\nx + 5 & = \\pm 8\\\\\nx &  = -5 \\pm 8\\\\\nx & = 3,-13\n\\end{align}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/107763e42385143d3d69090af16c1f45c066d49a", "tex": "x - y = -1", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/047b72c6cc8d183ab9fedb331ff24d76d6703c52", "tex": "3x + y = 9", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2d87cc6ac7e78a26fdc02b68941dc01dc41b55", "tex": "{\\displaystyle {\\begin{cases}4x+2y&=14\\\\2x-y&=1.\\end{cases}}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f808b7f9403c7ed113e148457240a3fea14726b8", "tex": "{\\displaystyle 4x+2y=14}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5e83a51011ff3cf00285e3892383560f609410", "tex": "{\\displaystyle 4x-2y=2.}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d2643249fdfb0c7f1c4d5d4125335be274cdc073", "tex": "{\\displaystyle 8x=16}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba0d302dee657b740f239df7d781071f6c247b5", "tex": "{\\displaystyle x=2.}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7", "tex": "x = 2", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6284ea4bb2d82b7a988082dd286adbb9dd095356", "tex": "y = 3", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f4bd4bc969d06560b090bcd79ec60c912d368d6d", "tex": "{\\displaystyle {\\begin{cases}x=2\\\\y=3.\\end{cases}}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2d87cc6ac7e78a26fdc02b68941dc01dc41b55", "tex": "{\\displaystyle {\\begin{cases}4x+2y&=14\\\\2x-y&=1.\\end{cases}}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5e08b49c74c2109c877dfd8893ee2e12ef4227b2", "tex": "{\\displaystyle 2x-y=1}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e50b849d3a7cd902f0ae3fa6ad6d1cad49987c39", "tex": "2x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/188d10b5e0e191e0d83eefa48e3c030960c3f864", "tex": "\\begin{align}2x - 2x - y & = 1 - 2x  \\\\\n- y & = 1 - 2x\n\\end{align}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/65064963b28ae18d96fa2e4218a97cc176fff7e1", "tex": "{\\displaystyle y=2x-1.}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b204003cd2f3139da95931466a361534cdf5026a", "tex": "\\begin{align}4x + 2(2x - 1) &= 14\\\\\n4x + 4x - 2 &= 14 \\\\\n8x - 2 &= 14 \\end{align}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/47382f34735f0fc0d4923ec1496ca70c8c930311", "tex": "\\begin{align}8x - 2 + 2 &= 14 + 2 \\\\\n8x &= 16 \\end{align}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7", "tex": "{\\displaystyle x=2}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f4bd4bc969d06560b090bcd79ec60c912d368d6d", "tex": "{\\displaystyle {\\begin{cases}x=2\\\\y=3.\\end{cases}}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c069a4331c966648337f5ea1db220afce23514d8", "tex": "3x + 2y = 6", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/58c127ab08640639903e2d5e1412709d6a7d35c5", "tex": "3x + 2y = 12", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d16b7301ea325d3bed4a98ce404bf5c12f00b964", "tex": "\\begin{cases}\\begin{align} x + y &= 1 \\\\\n0x + 0y &= 2\\,. \\end{align} \\end{cases}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb2eaef955f8943956aa58acdca1719c6d618e0", "tex": "\\begin{cases}\\begin{align}4x + 2y &= 12 \\\\\n-2x - y &= -4\\,. \\end{align}\\end{cases}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/19223a6e22bc6207f62f1794a23e336228eccb21", "tex": "0x+0y = 4 \\,,", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a7d5f209da787f90421c56fb21b9561b1db753", "tex": "{\\displaystyle {\\begin{cases}{\\begin{aligned}4x+2y&=12\\\\-2x-y&=-6\\end{aligned}}\\end{cases}}}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b7a0f213eacc838e16f1b74ab4644665cef145", "tex": "{\\displaystyle y=-2x+6}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/62ec343dafcb09463d30f4183660379740d8a19a", "tex": "\\begin{align}4x + 2(-2x + 6) = 12 \\\\\n4x - 4x + 12 = 12 \\\\\n12 = 12 \\end{align}", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b7a0f213eacc838e16f1b74ab4644665cef145", "tex": "y = -2x + 6", "domain": "Algrebra", "topic": "Elementary_algebra"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4ae0ac2282dbd3ee48d3333da4ba22535ca78e", "tex": "{\\begin{cases}{\\begin{aligned}x+2y&=10\\\\y-z&=2.\\end{aligned}}\\end{cases}}", "domain": "Algrebra", "topic": "Elementary_algebra"}], "Number theory": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ae973a762a92b9cd3eafe7f283890ccfa9b887e8", "tex": "(a,b,c)", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2", "tex": "a^{2}+b^{2}=c^{2}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e9be9111eb813cf2c8b31c740ddff47308c1e1a3", "tex": "{\\displaystyle \\left({\\frac {1}{2}}\\left(x-{\\frac {1}{x}}\\right)\\right)^{2}+1=\\left({\\frac {1}{2}}\\left(x+{\\frac {1}{x}}\\right)\\right)^{2},}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/11e52b3c6d3960cfc6efa96a811ec2feb5efa2b4", "tex": "c/a", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff", "tex": "{\\sqrt {2}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff", "tex": "{\\sqrt {2}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4c30069c6e8ec13aa767b2a0fe071bba15bedcc2", "tex": "{\\sqrt  {3}},{\\sqrt  {5}},\\dots ,{\\sqrt  {17}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3d49febe8c5fb88fb25d4376433469c13bdbc4", "tex": "f(x,y)=z^{2}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1004a1f588de9d793e07ee911e4378555096b606", "tex": "f(x,y,z)=w^{2}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7baab52cb78d79450c244b52a4275daf66feef29", "tex": "f(x_{1},x_{2},x_{3})=0", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cec6d1991738e8fd73ba3aacfb23be87fb0f3dc8", "tex": "g_{1},g_{2},g_{3}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538", "tex": "r", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632", "tex": "s", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f633174796bcccd1b7358e73094c42947b476b56", "tex": "x_{i}=g_{i}(r,s)", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4a87b7e5d81c62b040a35078417fb9294b1ec7c7", "tex": "i=1,2,3", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e631519582bc45b9a08f9ab887544a1d67800a33", "tex": "f(x_{1},x_{2},x_{3})=0.", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c663611bcfb301bc7826f1b93ccd106dec042744", "tex": "{\\displaystyle n\\equiv a_{1}{\\bmod {m}}_{1}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/70c47219ea418a20f473f14a09b48959cbe71125", "tex": "{\\displaystyle n\\equiv a_{2}{\\bmod {m}}_{2}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/24ec1de006b31585777b0bd54505db183dc0281e", "tex": "{\\displaystyle a^{p-1}\\equiv 1{\\bmod {p}}.}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/14fc28103c5d2aa9276728469f82c9f415f4b257", "tex": "a^2 + b^2", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/14fc28103c5d2aa9276728469f82c9f415f4b257", "tex": "a^2 + b^2", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0fa2a18557e1553bfe01d2ed1dfeafed52b42c02", "tex": "x^{2}-Ny^{2}=1", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bfc85e410302961db48dfb90dfb4f27372961922", "tex": "x^{{4}}+y^{{4}}=z^{{4}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2dff0aa9dde87ce8348763c196bd299ad434b822", "tex": "x^{3}+y^{3}=z^{3}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c34dcc46ebfb58e1b91a6c0caa1470e76139543a", "tex": "x^{n}+y^{n}=z^{n}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/73136e4a27fe39c123d16a7808e76d3162ce42bb", "tex": "n\\geq 3", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fe433e9ba9c42efe1bc9fd51bb439c698b8f6843", "tex": "p=x^{2}+y^{2}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d504d3c492a1b5a4fcf14acb15f857fb0ea705ab", "tex": "{\\displaystyle p\\equiv 1{\\bmod {4}}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f35aa613c26db550f2733d251ae8cbcb906831ec", "tex": "x^{4}+y^{4}=z^{2}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/de0d4c34b673da9926f76d0642c4ba92e09e882d", "tex": "x^{2}+Ny^{2}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/af5c33baa55ad20007ec7c02f6351cdb98980ec3", "tex": "mX^{2}+nY^{2}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7c75ed5d8d24057cd1160d2dfc659972e903af", "tex": "ax^{2}+by^{2}+cz^{2}=0", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb41e9a10a8fd7179b9170149a8d70949ba5d03", "tex": "n=5", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85883d74b75fe35ca8d3f2b44802df078e4fa1", "tex": "f(x)=0", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9503fed2affe4705838d28dd802e295c5df1d2b8", "tex": "x^{5}+(11/2)x^{3}-7x^{2}+9=0", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6c46254eee2978b88135fbd1d75e89d836dbcc", "tex": "a+b{\\sqrt  {d}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc", "tex": "a", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3", "tex": "b", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab", "tex": "d", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6cc38ada9b0a58b1d3248d62f616e40341e8e5", "tex": "{\\sqrt  {-5}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253", "tex": "6", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/36bd35c05d4411ba15b1890e917fc1370381a8db", "tex": "6=2\\cdot 3", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6ec35824d17526a7edf178c52be24823bb3091f7", "tex": "6=(1+{\\sqrt  {-5}})(1-{\\sqrt  {-5}})", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f", "tex": "2", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f", "tex": "3", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f4625b87f83984dd8727b066cc0f19f801185b79", "tex": "1+{\\sqrt  {-5}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a1831393106d87542d5282bf65d7ec460390ee09", "tex": "1-{\\sqrt  {-5}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1", "tex": "x^{2}+y^{2}=1", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386", "tex": "(x,y)", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef0a5a4b8ab98870ae5d6d7c7b4dfe3fb6612e2", "tex": "a^{2}+b^{2}=c^{2}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d355166b22e17a160f930cee7316594fcb8f80ae", "tex": "x = a/c", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fc1ee1bcb03c82c95a3e8587b13db23cb274c434", "tex": "y = b/c", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1", "tex": "x^2 + y^2 = 1", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543", "tex": "f(x,y)=0", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61", "tex": "f", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543", "tex": "f(x,y)=0", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543", "tex": "f(x,y)=0", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/01ede27471a9cc9a0c5eb6e1ebdc7afc8a086543", "tex": "f(x,y)=0", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/8286e068898b57215dbd608e8c97520a4ce3db0d", "tex": "a/q", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9caf09a62819517d7404f4839223e5944d455f22", "tex": "\\gcd(a,q)=1", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e57fea4b868d34d02adcc805fd4669fec9caa3c3", "tex": "|x-a/q|<{\\frac  {1}{q^{c}}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455", "tex": "c", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950", "tex": "{\\displaystyle 0}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc", "tex": "a", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9fae7f0c412e074931fb8f7c39e3ff32642bd577", "tex": "{\\displaystyle a+b,a+2b,a+3b,\\ldots ,a+10b}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/18c6ebc8e6d1bbf9f61da8a7e870949abb404c65", "tex": "A+A", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2a39c048e6c827a005e417be0e1bd5ac314e6fcc", "tex": "ax+by=c", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6e71bfeb1edd0b7451eff65e6c7f3c8c3eaa944b", "tex": "{\\displaystyle a\\equiv b{\\bmod {m}}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6b88601064cc54c2ab369f4a06bd6a7a9f0fa74d", "tex": "{\\displaystyle xa\\equiv 1{\\bmod {p}}}", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5b7a09643d5da81c32abf12cb0a4816442299b", "tex": "y^2 = x^3 + 7", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2b98401d1ddd1f356020afe4f6b817c2351be8c6", "tex": "(a + b i)^2 = (c + d i)^3 + 7", "domain": "Arithmetic", "topic": "Number theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/61e679a80d8e3df3cfd591b045627eecb10262f6", "tex": "(p^{2}-q^{2},2pq,p^{2}+q^{2})", "domain": "Arithmetic", "topic": "Number theory"}], "Mathematical analysis": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d78e6f2ddf5baee227ee2a9f164726ba0c23c263", "tex": "(M,d)", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd", "tex": "M", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab", "tex": "d", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd", "tex": "M", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf5a1da6629cc651072dab329ed2e4874d19e34", "tex": "d\\colon M\\times M\\rightarrow \\mathbb {R} ", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ba283a127121ad64c98d3f69ced0ac4a86ec6414", "tex": "x,y,z\\in M", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3329b396b395dedb4a470dea0aacea4b485070a8", "tex": "d(x,y) = 0", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366", "tex": "x=y", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fea33d0e60116abd16287351eb6bf142a61fdd", "tex": "d(x,y)=d(y,x)", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/43ae751284c2944886e1effbfe4e0c1293f98419", "tex": "d(x,z)\\leq d(x,y)+d(y,z)", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a61c98d99e4f3fea95d3ace9687c25dcab83ac44", "tex": "z=x", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cec878e6d9effccd8a2d2ed065a8e82bfa82", "tex": "d(x,y)\\geq 0", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/39ceac2fba6ab5c1246eb71dac752723cef1635d", "tex": "\\lim_{n\\to\\infty} a_n = x.", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b", "tex": "n", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d", "tex": "\\mathbb {R} ^{n}", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c57121c2b6c63c0b2f38eb96b1f7a543b5d1c522", "tex": "\\left[0, 1\\right]", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab", "tex": "X", "domain": "Calculus", "topic": "Mathematical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36", "tex": "\\sigma ", "domain": "Calculus", "topic": "Mathematical analysis"}], "Differential equations": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1cd29ae411a2d09205b3b39aec038af563d5e2", "tex": "{\\displaystyle {\\begin{aligned}&{\\frac {dy}{dx}}=f(x)\\\\[5pt]&{\\frac {dy}{dx}}=f(x,y)\\\\[5pt]&x_{1}{\\frac {\\partial y}{\\partial x_{1}}}+x_{2}{\\frac {\\partial y}{\\partial x_{2}}}=y\\end{aligned}}}", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0df4fcdfdb40fe6b609da7321a8c1d1a63c90eb2", "tex": "y'+P(x)y=Q(x)y^{n}\\,", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/945804f346250140666c0a6523fd20eedf499eb6", "tex": "{\\frac {du}{dx}}=cu+x^{2}.", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/173b1430314cc300bf90b4a967eae6380711b47b", "tex": "{\\frac {d^{2}u}{dx^{2}}}-x{\\frac {du}{dx}}+u=0.", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7cdb026da11e5a24ca46f43bb8af325e5f3289", "tex": "{\\frac {d^{2}u}{dx^{2}}}+\\omega ^{2}u=0.", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d03496c5764c42adfecd1c9660d876a4d0670e1d", "tex": "{\\frac {du}{dx}}=u^{2}+4.", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3272632b022f8ed937acc24597fb8dc386faf388", "tex": "L{\\frac {d^{2}u}{dx^{2}}}+g\\sin u=0.", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/edf368fadab981484268ec32723105c65a1e7028", "tex": "{\\frac {\\partial u}{\\partial t}}+t{\\frac {\\partial u}{\\partial x}}=0.", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/de38a82a39e63e77a24a423a9e430924b3539b27", "tex": "{\\frac {\\partial ^{2}u}{\\partial x^{2}}}+{\\frac {\\partial ^{2}u}{\\partial y^{2}}}=0.", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5fe49befc15be8cc358653332fde7f46398699ef", "tex": "{\\frac {\\partial u}{\\partial t}}=6u{\\frac {\\partial u}{\\partial x}}-{\\frac {\\partial ^{3}u}{\\partial x^{3}}}.", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c", "tex": "(a,b)", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd", "tex": "Z", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/515b71ad3b5fc01ff532df8cb71018baca811973", "tex": "Z=[l,m]\\times [n,p]", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c", "tex": "(a,b)", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd", "tex": "Z", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/71f88cadc10af603ac9f11db657ece34d52b9f94", "tex": "{\\frac {\\mathrm {d} y}{\\mathrm {d} x}}=g(x,y)", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/15a2b7edf933b5c4ff37ac2bca32cb5edc0c596c", "tex": "y=b", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae23950e96a955ab5b07015a168fd931d4d82b", "tex": "x=a", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bf358a54b0375e22ae5f3ab2c3e1a22c0c87e11c", "tex": "g(x,y)", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2f437ef83d659d7fd3c1230047001b7f3f4f1f45", "tex": "{\\frac {\\partial g}{\\partial x}}", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd", "tex": "Z", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc", "tex": "a", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/dabb3e9327f84cafa3277ac6a1effbe5df4e6492", "tex": "f_{n}(x){\\frac {\\mathrm {d} ^{n}y}{\\mathrm {d} x^{n}}}+\\cdots +f_{1}(x){\\frac {\\mathrm {d} y}{\\mathrm {d} x}}+f_{0}(x)y=g(x)", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a79f3811d3e8e76210a5f4038d723ed1ecb53f20", "tex": "y(x_{0})=y_{0},y'(x_{0})=y'_{0},y''(x_{0})=y''_{0},\\cdots ", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b86f950ad1f8135017742e12a8da5e2367bfdc1a", "tex": "f_{{n}}(x)", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/55698552cace4f20a7a6ebe30f27817255a8b3f9", "tex": "\\{f_{0},f_{1},\\cdots \\}", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77", "tex": "g", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf", "tex": "x_{0}", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d", "tex": "y", "domain": "Calculus", "topic": "Differential equations"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5784be3732078094538b9f4b94468d3f242dc23b", "tex": "{\\displaystyle {\\frac {\\partial k(t)}{\\partial t}}=s[k(t)]^{\\alpha }-\\delta k(t)}", "domain": "Calculus", "topic": "Differential equations"}], "Numerical analysis": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/eb27471bb2fc45e4efb1852e3563fe0871d8c882", "tex": "3x^{3}+4=28", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/697d815a1f48e45ef5b0b569d2ef502ebb8876fa", "tex": "{\\displaystyle a+b+c+d+e}", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff", "tex": "{\\sqrt {2}}", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bd01a68a0d14cee9d71920b6a63169d72c1301f1", "tex": " \nf(x)=x\\left(\\sqrt{x+1}-\\sqrt{x}\\right)\n\\text{ and } g(x)=\\frac{x}{\\sqrt{x+1}+\\sqrt{x}}.\n", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1d65cd6786b3580abd1ea0be9f17df79b4182063", "tex": " f(500)=500 \\left(\\sqrt{501}-\\sqrt{500} \\right)=500 \\left(22.38-22.36 \\right)=500(0.02)=10", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/238361d9898342c90597f79db114ba2eeadd6f9b", "tex": "\n\\begin{alignat}{3}g(500)&=\\frac{500}{\\sqrt{501}+\\sqrt{500}}\\\\\n      &=\\frac{500}{22.38+22.36}\\\\\n      &=\\frac{500}{44.74}=11.17\n\\end{alignat}\n", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fa4c3b409027082f75cffe26932bf76806bfaea5", "tex": " \\begin{alignat}{4}\nf(x)&=x \\left(\\sqrt{x+1}-\\sqrt{x} \\right)\\\\\n    &=x \\left(\\sqrt{x+1}-\\sqrt{x} \\right)\\frac{\\sqrt{x+1}+\\sqrt{x}}{\\sqrt{x+1}+\\sqrt{x}}\\\\\n    &=x\\frac{(\\sqrt{x+1})^2-(\\sqrt{x})^2}{\\sqrt{x+1}+\\sqrt{x}}\\\\\n    &=x\\frac{x+1-x}{\\sqrt{x+1}+\\sqrt{x}}  \\\\\n    &=x\\frac{1}{\\sqrt{x+1}+\\sqrt{x}}  \\\\\n    &=\\frac {x}{\\sqrt{x+1}+\\sqrt{x}}  \\\\\n    &=g(x)\n\\end{alignat}", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/01c0adb58ed2acc4d9b540e5f2df6dd913d77fce", "tex": "2x+5=3", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/93f96893bb2543d1c02eb664f26e9c6cfa7b5bc6", "tex": "2x^2+5=3", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/20ede42f65265feef76c0662d694c6c3a0ce2444", "tex": "x=(x^2-2)^2+x=f(x)", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff", "tex": "{\\sqrt {2}}", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7b27daf9286032e04f9c26d8021405066d4ff5", "tex": "f(x)\\geq x", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf28323c0b7ae68daa0e3d30e951711e39aa9a6", "tex": "x_1=1.4<\\sqrt{2}", "domain": "Calculus", "topic": "Numerical analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/368ce7debfd89ebf6c043f9af67bc603543ffbf6", "tex": "x_1=1.42>\\sqrt{2}", "domain": "Calculus", "topic": "Numerical analysis"}], "Mathematical optimization": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e825429becf7730acee7e5a9da089c6b98ca759d", "tex": "{\\displaystyle f\\colon A\\to \\mathbb {R} }", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a1140d98da6ff57f2b4d3ec152c2ee32769c3eba", "tex": "{\\displaystyle \\mathbf {x} _{0}\\in A}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd844a63f0693c5d581f183013ae80b92285659", "tex": "{\\displaystyle f\\left(\\mathbf {x} _{0}\\right)\\leq f\\left(\\mathbf {x} \\right)}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/652191b134e9c23ce0dfe2defd4a9a5c4cadc9be", "tex": "{\\displaystyle \\mathbf {x} \\in A}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d197184c5c8d397e176575e92feb8312b793c7b2", "tex": "{\\displaystyle f\\left(\\mathbf {x} _{0}\\right)\\geq f\\left(\\mathbf {x} \\right)}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/652191b134e9c23ce0dfe2defd4a9a5c4cadc9be", "tex": "{\\displaystyle \\mathbf {x} \\in A}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61", "tex": "f", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d", "tex": "\\mathbb {R} ^{n}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61", "tex": "f", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61", "tex": "f", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a902678bff3be2ae29f828d78594ce61eb5031", "tex": "{\\displaystyle \\mathbf {x} ^{\\ast }}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/595d5cea06fdcaf2642caf549eda2cfc537958a9", "tex": "\\delta >0", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/652191b134e9c23ce0dfe2defd4a9a5c4cadc9be", "tex": "{\\displaystyle \\mathbf {x} \\in A}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e5c1b2175adc969f4a49b5d41cd57ba59a1ee4b2", "tex": "{\\displaystyle \\left\\Vert \\mathbf {x} -\\mathbf {x} ^{\\ast }\\right\\Vert \\leq \\delta ,\\,}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ea411eec240cdc4cd48718ae9965f10a8e86af96", "tex": "{\\displaystyle f\\left(\\mathbf {x} ^{\\ast }\\right)\\leq f\\left(\\mathbf {x} \\right)}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a902678bff3be2ae29f828d78594ce61eb5031", "tex": "{\\displaystyle \\mathbf {x} ^{\\ast }}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7dab584903c901e13c4cf7efa1226c84b25f2f94", "tex": "\\min _{x\\in \\mathbb {R} }\\;(x^{2}+1)", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/92a3a8d23f9f8123651e496dcf8490990c65cf9a", "tex": "x^{2}+1", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc", "tex": "\\mathbb {R} ", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf", "tex": "1", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc", "tex": "x=0", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a128d2c279975694f4c98533554a03f500780804", "tex": "\\max _{x\\in \\mathbb {R} }\\;2x", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/39c071258fcecb43aaf25920b9833589bc35036c", "tex": "{\\underset {x\\in (-\\infty ,-1]}{\\operatorname {arg\\,min} }}\\;x^{2}+1,", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ecc7504e271936684860c5d738740d0841edac", "tex": "{\\underset {x}{\\operatorname {arg\\,min} }}\\;x^{2}+1,\\;{\\text{subject to:}}\\;x\\in (-\\infty ,-1].", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3159526ce3102ba58727dfc08cb732b19fdd0b96", "tex": "(-\\infty ,-1]", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/92a3a8d23f9f8123651e496dcf8490990c65cf9a", "tex": "x^{2}+1", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4fefa55268918f98da2e0dcc19ea86d78f84ac56", "tex": "{\\displaystyle x=-1}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc", "tex": "x=0", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5a821b5f32ba0a33ed4f712423e44edd613e3696", "tex": "{\\underset {x\\in [-5,5],\\;y\\in \\mathbb {R} }{\\operatorname {arg\\,max} }}\\;x\\cos(y),", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a588eccb13c01b752b4fae06f4b90fff204f06ea", "tex": "{\\underset {x,\\;y}{\\operatorname {arg\\,max} }}\\;x\\cos(y),\\;{\\text{subject to:}}\\;x\\in [-5,5],\\;y\\in \\mathbb {R} ,", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/41cf50e4a314ca8e2c30964baa8d26e5be7a9386", "tex": "(x,y)", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6380b1ab1185be3f64791b973b3bdb6745670f7c", "tex": "x\\cos(y)", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e33603930f175fbb5b1bb25c0a106f4564cb47", "tex": "[-5,5]", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/10ef1c8b158c74011787ba7c9ad4b12cb44e4bd7", "tex": "{\\displaystyle (5,\\,2k\\pi )}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3626d43a4fbb01ca70a66653b8f89745abdf5c40", "tex": "{\\displaystyle (-5,\\,(2k+1)\\pi )}", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40", "tex": "k", "domain": "Calculus", "topic": "Mathematical optimization"}, {"png": 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"https://wikimedia.org/api/rest_v1/media/math/render/svg/94935b097f13ccb916e178c3d3c1f423aa733690", "tex": "\\,\\ell^{\\,p\\ }", "domain": "Calculus", "topic": "Functional analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab", "tex": "X", "domain": "Calculus", "topic": "Functional analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3289fce0da007ab48e5369f327e9972963b051", "tex": "\\sup\\nolimits_{T \\in F} \\|T(x)\\|_Y  < \\infty, ", "domain": "Calculus", "topic": "Functional analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/77594b0fbe8b147b2c2a2be2716ffcb4311891dc", "tex": "\\sup\\nolimits_{T \\in F} \\|T\\|_{B(X,Y)}  < \\infty.", "domain": "Calculus", "topic": "Functional analysis"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2a1001cebb78470dcc715499bf8002579825d024", "tex": "[A\\varphi ](t)=t\\varphi (t).\\;", "domain": "Calculus", "topic": 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"\\sin A=\\frac{\\textrm{opposite}}{\\textrm{hypotenuse}}=\\frac{a}{\\,c\\,}\\,.", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1e53b5926e121689a8feb472c0bdd6242c237f25", "tex": "\\cos A=\\frac{\\textrm{adjacent}}{\\textrm{hypotenuse}}=\\frac{b}{\\,c\\,}\\,.", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5282f1c0166fe30866b541c4ea7417af913ea576", "tex": "\\tan A=\\frac{\\textrm{opposite}}{\\textrm{adjacent}}=\\frac{a}{\\,b\\,}=\\frac{a}{\\,c\\,}*\\frac{c}{\\,b\\,}=\\frac{a}{\\,c\\,} / \\frac{b}{\\,c\\,}=\\frac{\\sin A}{\\cos A}\\,.", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5002e2b8611901edd865b97573ef959826292241", "tex": "\\csc A={\\frac  {1}{\\sin A}}={\\frac  {{\\textrm  {hypotenuse}}}{{\\textrm  {opposite}}}}={\\frac  {c}{a}},", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b0fad5e45d6027da193f6ae9392d9a0d95277d", "tex": "\\sec A={\\frac  {1}{\\cos A}}={\\frac  {{\\textrm  {hypotenuse}}}{{\\textrm  {adjacent}}}}={\\frac  {c}{b}},", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/087f418e780f9d7adac7c4c9f90e241f60838b86", "tex": "\\cot A={\\frac  {1}{\\tan A}}={\\frac  {{\\textrm  {adjacent}}}{{\\textrm  {opposite}}}}={\\frac  {\\cos A}{\\sin A}}={\\frac  {b}{a}}.", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/47377e0ec401e4565bc198dedecdb5870d5a64e8", "tex": "{\\displaystyle e^{x+iy}=e^{x}(\\cos y+i\\sin y).}", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/90e1e648d37897b7cf601fc95a51d3ae90615737", "tex": "\\sin^2 A + \\cos^2 A = 1 \\ ", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b0484ed6255ac72bda9bc78d14a12c622ecccd94", "tex": "\\tan ^{2}A+1=\\sec ^{2}A\\ ", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/676d3de0aa5f6fa889e8513727b41966a57e944f", "tex": "{\\displaystyle \\cot ^{2}A+1=\\csc ^{2}A\\ }", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3a27194e335db5cae63aea7a50d73e64a9433cfd", "tex": "\\sin (A \\pm B) = \\sin A \\ \\cos B \\pm \\cos A \\ \\sin B", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5712c1a21f0b18f57d9b608f20e0e9fadfb329ab", "tex": "\\cos (A \\pm B) = \\cos A \\ \\cos B \\mp \\sin A \\ \\sin B", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/e06113364d8af3bb9a6e06eeda3e196f1d6f071e", "tex": "{\\displaystyle \\tan(A\\pm B)={\\frac {\\tan A\\pm \\tan B}{1\\mp \\tan A\\ \\tan B}}}", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/8aca4af47eed4b14486eb40831d2e41d65706ce1", "tex": "\\cot (A \\pm B) = \\frac{ \\cot A \\ \\cot B \\mp 1}{ \\cot B \\pm \\cot A } ", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/46b51315d543d24a8fae0af70a6278a71c8cee79", "tex": "{\\frac  {a}{\\sin A}}={\\frac  {b}{\\sin B}}={\\frac  {c}{\\sin C}}=2R={\\frac  {abc}{2\\Delta }},", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2", "tex": "\\Delta ", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/94ad4e6664b2f3240fa73558cc9506009e46ab88", "tex": "R = \\frac{abc}{\\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/98c3e83d4e3db5978463dd5f0469e9b31061f5f6", "tex": "{\\mbox{Area}}=\\Delta ={\\frac  {1}{2}}ab\\sin C.", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7dfdcae91b8940c352d127428000a9270b544928", "tex": "c^2=a^2+b^2-2ab\\cos C ,\\,", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f74fccac0abd11430ccf804c1c1da0faf369e248", "tex": "\\cos C=\\frac{a^2+b^2-c^2}{2ab}.\\,", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6700e4d84c028818eab0c624626fd723c18d24", "tex": "s=\\frac{1}{2}(a+b+c),", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f87978af4f82a52da49cfd23dcd0f628081dfae0", "tex": "{\\mbox{Area}}=\\Delta ={\\sqrt  {s(s-a)(s-b)(s-c)}}={\\frac  {abc}{4R}}", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a1da4e06eb6f25cd7f7fc1a7784a11a82ae53f9f", "tex": "\\frac{a-b}{a+b}=\\frac{\\tan\\left[\\tfrac{1}{2}(A-B)\\right]}{\\tan\\left[\\tfrac{1}{2}(A+B)\\right]}", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4907c0489ab08ce550c7700a1587d4634801dff8", "tex": "e^{ix} = \\cos x + i \\sin x", "domain": "Geometry", "topic": "Trigonometry"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5b30e164c54b1788e201501d9e510a16e1a4921b", "tex": "\\sin x = \\frac{e^{ix} - e^{-ix}}{2i}, \\qquad \\cos x = \\frac{e^{ix} + e^{-ix}}{2}, \\qquad \\tan x = \\frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}.", "domain": "Geometry", "topic": "Trigonometry"}], "Mathematical logic": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/037b84dcf7eb85313f964632c939bce6464a657c", "tex": "L_{\\omega_1,\\omega}", "domain": "Foundations of Mathematics", "topic": "Mathematical logic"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/037b84dcf7eb85313f964632c939bce6464a657c", "tex": "L_{\\omega_1,\\omega}", "domain": "Foundations of Mathematics", "topic": "Mathematical logic"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f1ab1d9af54fdcb661a5ef228fcef266605adf56", "tex": "(x = 0) \\lor (x = 1) \\lor (x = 2) \\lor \\cdots.", "domain": "Foundations of Mathematics", "topic": "Mathematical logic"}], "Probability theory": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/59dfbc94007edbe75da5a93799873a3ea6cc8d07", "tex": "{\\displaystyle X(heads)=0}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/beb323b43601737ec3c4b6155aa592ba14b96391", "tex": "{\\displaystyle X(tails)=1}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5c61b1e501afd8b9b7f7956e67688bf6e62f63e8", "tex": "{\\tfrac {3}{6}}={\\tfrac {1}{2}}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f", "tex": "\\Omega ", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3da19a78c4c9c44c208a3966b4559350364d6b30", "tex": "x\\in \\Omega \\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/78b2b66021c2cac2b5654495678c63ff142952e5", "tex": "f(x)\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d9fb6901c7d0c1686386da77b3a2af38bc0d7952", "tex": "f(x)\\in [0,1]{\\mbox{ for all }}x\\in \\Omega \\,;", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa8747238c133bea47b792e62debb7895f8cacd", "tex": "\\sum _{x\\in \\Omega }f(x)=1\\,.", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9123abddc2ec35f72035ec59f443c79ee052c9ef", "tex": "E\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/771ad27abf999ca4ae152aeb523d93eca2447b7e", "tex": "\\Omega \\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9123abddc2ec35f72035ec59f443c79ee052c9ef", "tex": "E\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/dbdfbd709b02d46695aceed6dc515f22bbf7ec3f", "tex": "P(E)=\\sum _{x\\in E}f(x)\\,.", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/78b2b66021c2cac2b5654495678c63ff142952e5", "tex": "f(x)\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc", "tex": "\\mathbb {R} ", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/af686cd9f8a742bde6d8ee773ddebc793960d0ee", "tex": "F\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1402b024ca9849c9f9cab1906d1ee3696c8eb553", "tex": "F(x)=P(X\\leq x)\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/af686cd9f8a742bde6d8ee773ddebc793960d0ee", "tex": "F\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/fab48bd8aa8e8a3b68c5cc9cf5692c9dcafc10dc", "tex": "\\lim _{x\\rightarrow -\\infty }F(x)=0\\,;", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5b257faa636faaa653d6b5b119f589379e5638f9", "tex": "\\lim _{x\\rightarrow \\infty }F(x)=1\\,.", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/af686cd9f8a742bde6d8ee773ddebc793960d0ee", "tex": "F\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b26f5319ac182790459b3cbb220799fc38b31c6c", "tex": "f(x)={\\frac {dF(x)}{dx}}\\,.", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/651e39fa02a0a98bc7f719fb35a883abe09bd8c5", "tex": "E\\subseteq \\mathbb {R} ", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9123abddc2ec35f72035ec59f443c79ee052c9ef", "tex": "E\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cae717be66873a6576dc7aff66ccb3d246b26379", "tex": "P(X\\in E)=\\int _{x\\in E}dF(x)\\,.", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b53ca69694950757b27f0456ea0bf1cf765c4540", "tex": "P(X\\in E)=\\int _{x\\in E}f(x)\\,dx\\,.", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/795f63bc49a9e48a6167df6ef5d50f494d996d65", "tex": "\\mathbb {R} \\,.", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d", "tex": "\\mathbb {R} ^{n}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b59e7c343760eb33e805760c20aff9d9c3831260", "tex": "(\\delta [x]+\\varphi (x))/2", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3953eacd8865ec4c182566de1728419117164d2c", "tex": "\\delta [x]", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/771ad27abf999ca4ae152aeb523d93eca2447b7e", "tex": "\\Omega \\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/749d43d5c2bdf63bc56b4be305f22c6fd9ad8d90", "tex": "{\\mathcal {F}}\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f60e07ebc2aadee94e4cbbccef71fd9bcf44f5f9", "tex": "P\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/749d43d5c2bdf63bc56b4be305f22c6fd9ad8d90", "tex": "{\\mathcal {F}}\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/af0688f8cbf26ef29787d7170301e75de67a06ff", "tex": "P(\\Omega )=1.\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/749d43d5c2bdf63bc56b4be305f22c6fd9ad8d90", "tex": "{\\mathcal {F}}\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/749d43d5c2bdf63bc56b4be305f22c6fd9ad8d90", "tex": "{\\mathcal {F}}\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9123abddc2ec35f72035ec59f443c79ee052c9ef", "tex": "E\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/749d43d5c2bdf63bc56b4be305f22c6fd9ad8d90", "tex": "{\\mathcal {F}}\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3408adc76f4b674b0c7f9bce20413b2be7c9b571", "tex": "P(E)=\\int _{\\omega \\in E}\\mu _{F}(d\\omega )\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a1bc0e9d7428ce837085047acdfa5ca8297baded", "tex": "\\mu _{F}\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/35805e894b77f9a5eca1324b665a4448cfed57b8", "tex": "F\\,.", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d", "tex": "\\mathbb {R} ^{n}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc6ac013652da6fd682f84ae20ea3f1b5a77bcb", "tex": "X_{1},X_{2},\\dots ,\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7028e89b7722d12ec0ea8780f26a9912456b63f5", "tex": "X\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4c5a2d183491e4e48bd711f6414346cf97b59c", "tex": "F_{1},F_{2},\\dots \\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/af686cd9f8a742bde6d8ee773ddebc793960d0ee", "tex": "F\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7028e89b7722d12ec0ea8780f26a9912456b63f5", "tex": "X\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/af686cd9f8a742bde6d8ee773ddebc793960d0ee", "tex": "F\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ebafc8acdec0751c11fef107e1a05717167ad22e", "tex": "{\\displaystyle \\displaystyle X_{n}\\,{\\xrightarrow {\\mathcal {D}}}\\,X}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a1cb9701e43a8b90bb111c42bab27149d89a9f6c", "tex": "X_{1},X_{2},\\dots \\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7028e89b7722d12ec0ea8780f26a9912456b63f5", "tex": "X\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0862214947bd75c5e550621cdaa47bd962a5d2d2", "tex": "\\lim _{n\\rightarrow \\infty }P\\left(\\left|X_{n}-X\\right|\\geq \\varepsilon \\right)=0", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a9eebab26b9d2fbdba9b696ad0bec6c1c27f1fd8", "tex": "{\\displaystyle \\displaystyle X_{n}\\,{\\xrightarrow {P}}\\,X}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a1cb9701e43a8b90bb111c42bab27149d89a9f6c", "tex": "X_{1},X_{2},\\dots \\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7028e89b7722d12ec0ea8780f26a9912456b63f5", "tex": "X\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7a5800715bfaed48fea5702d8153ba1868a683", "tex": "P(\\lim _{n\\rightarrow \\infty }X_{n}=X)=1", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ce4719ed0b899257c74aad20217bc3aa3564b98d", "tex": "{\\displaystyle \\displaystyle X_{n}\\,{\\xrightarrow {\\mathrm {a.s.} }}\\,X}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bfdf015b3311f4d3fab8f2a67d0e53a8dc50a3f9", "tex": "{\\overline {X}}_{n}={\\frac {1}{n}}{\\sum _{k=1}^{n}X_{k}}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/33c25229c6989c235f9cbb7908331f6d01d0abfe", "tex": "X_{k}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161", "tex": "\\mu ", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d00970763c764e0beb2d9ae1e934f0d07de648af", "tex": "|X_{k}|", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/dc0b668266398d0df9fa42138487013271173078", "tex": "{\\displaystyle \\displaystyle {\\overline {X}}_{n}\\,{\\xrightarrow {P}}\\,\\mu }", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1", "tex": "n\\to \\infty ", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/8e1c4d954c7b4504b3a15d686168612c9e0dc4f1", "tex": "{\\displaystyle \\displaystyle {\\overline {X}}_{n}\\,{\\xrightarrow {\\mathrm {a.\\,s.} }}\\,\\mu }", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a34a9f62668de90200a6cbde865c27af2cdbb7", "tex": "{\\displaystyle n\\to \\infty .}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac52a1d59dcfd2e9464d555f0e27a44f7726128", "tex": "Y_{1},Y_{2},...\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5becbac176657bfd973aa97fe5285c9af68760", "tex": "{\\textrm {E}}(Y_{i})=p", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b90b164b2586db6dc7d985579a33e1fa27620650", "tex": "{\\bar {Y}}_{n}", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a1cb9701e43a8b90bb111c42bab27149d89a9f6c", "tex": "X_{1},X_{2},\\dots \\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161", "tex": "\\mu ", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/eb252c392475049156215fc45d3ceede93ecade2", "tex": "\\sigma ^{2}>0.\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c5891ac594b4619bb3214875c00549aad8660f41", "tex": "Z_{n}={\\frac {\\sum _{i=1}^{n}(X_{i}-\\mu )}{\\sigma {\\sqrt {n}}}}\\,", "domain": "Applied Mathematics", "topic": "Probability theory"}], "Game theory": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8", "tex": "\\mathbf {C} ", "domain": "Applied Mathematics", "topic": "Game theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bd297b1df8f7204c60c6c8f2d18795a156d73b26", "tex": "\\left({\\frac {\\mathbf {N} }{\\mathbf {C} }}\\right)", "domain": "Applied Mathematics", "topic": "Game theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa66d7f588aeeff5ccfecb8408c32802c1cb231", "tex": "v:2^{N}\\to \\mathbf {R} ", "domain": "Applied Mathematics", "topic": "Game theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b", "tex": "n", "domain": "Applied Mathematics", "topic": "Game theory"}], "Information theory": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/182a05385b07bc194d5aa35023abee98dbc5256a", "tex": "\\lim _{p\\rightarrow 0+}p\\log p=0", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a7673eba0df7fb924d538fb6b1c94174f812af", "tex": "{\\displaystyle H=-\\sum _{i}p_{i}\\log _{2}(p_{i})}", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/93b167a7fd856749b6844408ead71c3667da3358", "tex": "x\\in \\mathbb {X} ", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/971ffd75b32f284123036d4ae8fc3dd6e377e030", "tex": "H(X)=\\mathbb {E} _{X}[I(x)]=-\\sum _{x\\in \\mathbb {X} }p(x)\\log p(x).", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/9430bcaab47b309a20d10f8b9a299b35a90d3bd9", "tex": "H_{\\mathrm {b} }(p)=-p\\log _{2}p-(1-p)\\log _{2}(1-p).", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a10572b1a21187833168e50a998f66856658674d", "tex": "H(X,Y)=\\mathbb {E} _{X,Y}[-\\log p(x,y)]=-\\sum _{x,y}p(x,y)\\log p(x,y)\\,", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdf0f6f2421309afb521010957e1e556669e479", "tex": "{\\displaystyle H(X|Y)=\\mathbb {E} _{Y}[H(X|y)]=-\\sum _{y\\in Y}p(y)\\sum _{x\\in X}p(x|y)\\log p(x|y)=-\\sum _{x,y}p(x,y)\\log p(x|y).}", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/55a800109119215c303f549c60dd667d707ff58c", "tex": "H(X|Y)=H(X,Y)-H(Y).\\,", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7d86fba391a52b75e1cf36dae6bfaa1ccbccf461", "tex": "I(X;Y)=\\mathbb {E} _{X,Y}[SI(x,y)]=\\sum _{x,y}p(x,y)\\log {\\frac {p(x,y)}{p(x)\\,p(y)}}", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ff968f71052c9abe1da3e3e414327077de6f2721", "tex": "I(X;Y)=H(X)-H(X|Y).\\,", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c8ca328d60dbd8eaab8833e9ec1e882daaa3b6", "tex": "I(X;Y)=I(Y;X)=H(X)+H(Y)-H(X,Y).\\,", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d90fc72e685981072e77a3933c7ad6c4c83d13", "tex": "I(X;Y)=\\mathbb {E} _{p(y)}[D_{\\mathrm {KL} }(p(X|Y=y)\\|p(X))].", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/03b108331b20bad2137992fa16f9fdd48906072d", "tex": "I(X;Y)=D_{\\mathrm {KL} }(p(X,Y)\\|p(X)p(Y)).", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6882199003743a8f6d6d0d4723b2bd2e496f98", "tex": "D_{\\mathrm {KL} }(p(X)\\|q(X))=\\sum _{x\\in X}-p(x)\\log {q(x)}\\,-\\,\\sum _{x\\in X}-p(x)\\log {p(x)}=\\sum _{x\\in X}p(x)\\log {\\frac {p(x)}{q(x)}}.", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2344c3096dbc21ce050a9b70a4ddf2062ff41e", "tex": "r=\\lim _{n\\to \\infty }H(X_{n}|X_{n-1},X_{n-2},X_{n-3},\\ldots );", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/324db7538097abf899fe20ce454b645c2e7ee5ba", "tex": "r=\\lim _{n\\to \\infty }{\\frac {1}{n}}H(X_{1},X_{2},\\dots X_{n});", "domain": "Applied Mathematics", "topic": "Information theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/6d6698ae5326003e7530add14e24f875f834fb68", "tex": "C=\\max _{f}I(X;Y).\\!", "domain": "Applied Mathematics", "topic": "Information theory"}], "Computer science": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4c96ef28327313f53f6deb407cd795c885e4be57", "tex": "\\Gamma \\vdash x:{\\text{Int}}", "domain": "Applied Mathematics", "topic": "Computer science"}], "Theory of computation": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2b0511d452842ec71e40223d223472b7527ca7", "tex": "\\alpha \\rightarrow \\beta ", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1173552bcbf68bb06baf9b0a2f543dbc845caefd", "tex": "\\alpha A \\beta \\rightarrow \\alpha \\gamma \\beta", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/91998fd76701871c7df169d4e9114b5970c61877", "tex": "A \\rightarrow \\gamma", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f2c4fbdf7d3a3af41e9c6b2f8e9586d9ada6317d", "tex": "A \\rightarrow a", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/01b2bfd09f27542fbcf3811786b84f6a43fd17d3", "tex": "A \\rightarrow aB", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/34109fe397fdcff370079185bfdb65826cb5565a", "tex": "O(n)", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a", "tex": "\\lambda ", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a", "tex": "\\lambda ", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074", "tex": "f(x)", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59", "tex": "g(x)", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7235c6799c7d4112231c9941bd428fe6a4111fe4", "tex": "h(x,y)", "domain": "Applied Mathematics", "topic": "Theory of computation"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6cce0a24b4df2a31c2c214940795920f1ea777", "tex": "f(x)=h(x,g(x))", "domain": "Applied Mathematics", "topic": "Theory of computation"}], "Control theory": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a017597b49ed2396ca6b2dd6685c5b67091bee0c", "tex": "{\\displaystyle Y(s)=P(s)U(s)}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6f7688193f812a5fa3c3609c3d250529044993", "tex": "{\\displaystyle U(s)=C(s)E(s)}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a787e2eecc1f81323f15d20384ddf2ad2c419032", "tex": "{\\displaystyle E(s)=R(s)-F(s)Y(s).}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2ded87a715d1d1b77ddcf2cb4bd88997f540f9d1", "tex": "{\\displaystyle Y(s)=\\left({\\frac {P(s)C(s)}{1+P(s)C(s)F(s)}}\\right)R(s)=H(s)R(s).}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f2490516044765db8e04dcd227c81a27716c5d00", "tex": "H(s)={\\frac {P(s)C(s)}{1+F(s)P(s)C(s)}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2731f366bed2c595293672f798a869f770fbff2a", "tex": "|P(s)C(s)|\\gg 1", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/37beeed5677b9485b73900c76cb56f10a68bc3af", "tex": "|F(s)|\\approx 1", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/98cb518a61ada87bcd636f531d4d9fd2e67876c3", "tex": "e(t)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/801274e6424890bbd73a54c0350a75f66bda967d", "tex": "e(t)=r(t)-y(t)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/14b8569ef4dd4ea1c5a5611765ea74cc273e3abd", "tex": "{\\displaystyle u(t)=K_{P}e(t)+K_{I}\\int e(\\tau ){\\text{d}}\\tau +K_{D}{\\frac {\\text{de(t)}}{{\\text{d}}t}}.}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/3efe20e22e60f8defdec7c8b246a76934356a618", "tex": "K_{P}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c52214955bed2bd43b066ffb7daebc3bddaaa33c", "tex": "K_{I}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bb828bb2eeba1fd11782462f29e274346406279d", "tex": "K_{D}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1f1b371a8b9b147b9b61a2a9e07f6e44b4809f", "tex": "u(s)=K_{P}e(s)+K_{I}{\\frac {1}{s}}e(s)+K_{D}se(s)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/842e237127cf16ee7e961c84116df3946c887b99", "tex": "u(s)=\\left(K_{P}+K_{I}{\\frac {1}{s}}+K_{D}s\\right)e(s)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/05adc98b2b79fcad6a23a3d57eb604b8423c51b4", "tex": "C(s)=\\left(K_{P}+K_{I}{\\frac {1}{s}}+K_{D}s\\right).", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b91390324fc9c33ec00fe57e3924ad7118cc1e", "tex": "H(s)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/971eedbf43e9b4ae951eecf88a2d0363d427a80e", "tex": "{\\displaystyle P(s)={\\frac {A}{1+sT_{P}}}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3", "tex": "A", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a26b906b5e41d3e67f53482df5203583ca779f66", "tex": "{\\displaystyle T_{P}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7ee165b5becac24c5ba7e6b79cafe41b42d1a3", "tex": "{\\displaystyle F(s)={\\frac {1}{1+sT_{F}}}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/cc73116f4f9552211edfb92a1f08757caa6b24aa", "tex": "{\\displaystyle T_{F}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a9a4507c23017bc3b4a418604e0532deef298632", "tex": "{\\displaystyle K_{P}=K\\left(1+{\\frac {T_{D}}{T_{I}}}\\right)}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a3838be72bac0d7059dbf87b84d2c4d540cade", "tex": "{\\displaystyle K_{D}=KT_{D}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/ae35c621895197d1b4326d9a78cc15614a349317", "tex": "{\\displaystyle K_{I}={\\frac {K}{T_{I}}}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa1f945339571ff1e8fbb4ae781e00906b361b1", "tex": "{\\displaystyle C(s)=K\\left(1+{\\frac {1}{sT_{I}}}\\right)(1+sT_{D})}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/7927ac2ad8a94b105f536045494e75f3193930ec", "tex": "P(s)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d633e75b0c0edb9b5cf174df6f79f4b90634718b", "tex": "F(s)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/045b88e2ab20cae3cd42b634a823b24461993ae4", "tex": "C(s)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b91390324fc9c33ec00fe57e3924ad7118cc1e", "tex": "H(s)", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/601cadb75337fbd250b242c078d4d9a106751e4c", "tex": "{\\displaystyle K={\\frac {1}{A}},T_{I}=T_{F},T_{D}=T_{P}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/938b5468fc5a0821b3d2fc6e1afa6788dd346e28", "tex": "{\\displaystyle H(s)=1}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4", "tex": "x", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64", "tex": "\\rho ", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/c3fe9bf89c5cffaf461081935fd41745dc768063", "tex": "\\ x[n]=0.5^{n}u[n]", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/4f65c87b92a3611ffc58d3df40792ff4e5e69b65", "tex": "{\\displaystyle \\ X(z)={\\frac {1}{1-0.5z^{-1}}}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/d9267ea9749ae11b3cde9c735c21f2507de379f3", "tex": "z=0.5", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/b769b726a2a55b9fc5e5c8d800187d7715cf84cd", "tex": "\\ x[n]=1.5^{n}u[n]", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/81dd2f0c11aecabe8b5bf08505c7d52974f4ec35", "tex": "{\\displaystyle \\ X(z)={\\frac {1}{1-1.5z^{-1}}}}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5f374dfe3fe7ce1355695d53ea69482094aa8935", "tex": "z=1.5", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/acd3c480f7bd6fa14fd42e56521994a3b4ad8e2d", "tex": "Re[\\lambda ]<-{\\overline {\\lambda }}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/de9d2ee37eb0cd550de8d8f34b9d13700f79384e", "tex": "{\\overline {\\lambda }}", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/57bd3912e4d0e7aafac442e28a10f4748da7b90d", "tex": "Re[\\lambda ]<0", "domain": "Applied Mathematics", "topic": "Control theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/1eb22dba60937019ecc7cd8dd87f67d6977c2209", "tex": "m{\\ddot {x}}(t)=-Kx(t)-\\mathrm {B} {\\dot {x}}(t)", "domain": "Applied Mathematics", "topic": "Control theory"}], "Order theory": [{"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/70669bcaf5f66ffa92993979daf1a0a3f412f3ea", "tex": "\\bigvee S", "domain": "Others", "topic": "Order theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/811212099611ddf57acae7ddeb8ee1e42b109ca9", "tex": "\\bigwedge S", "domain": "Others", "topic": "Order theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/304b00d1f1cf4a707c7863e8fae02a2dff7d5a8a", "tex": "x\\vee y", "domain": "Others", "topic": "Order theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/5e39c22fab294b953b40e439378be357dea68150", "tex": "x\\wedge y", "domain": "Others", "topic": "Order theory"}, {"png": "https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6615a01565e8491dcf08b698ae6d5a96cf3721", "tex": "a\\leq a^{*}\\Rightarrow a=0", "domain": "Others", "topic": "Order theory"}]}